Singularities in Geometry and Topology: Proceedings of the Trieste Singularity Summer School and Workshop Ictp, Trieste, Italy, 15 August - 3 September 2005 ( World Scientific )

Singularities in Geometry and Topology -

Proceedings of the Trieste Singular@ Summer School and Workshop

This page intentionally left blank

Singularities in Geometry and TOpOlOgy Proceedings of the Trieste Singularity Summer School and Workshop 15 August - 3 September 2005

ICTI? Trieste, Italy

Editors

Jean-Pau I Brasselet lnstitut de Mathematiques de Luminy-CNRS, France

James Damon University of North Carolina, USA

L5 DcngTrang ICTR Trieste, Italy

Mutsuo Oka Tokyo University of Science, Japan

N E W JERSEY

- LONDON

*

v

World Scientific

SINGAPORE

*

BElJlNG

-

SHANGHAI

- HONG KONG

*

TAIPEI

- CHENNAI

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224

USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

SINGULAIUTIES IN GEOMETRY AND TOPOLOGY Proceedings of the Trieste Singularity Summer School Copyright Q 2007 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-I3 978-981-270-022-3 ISBN-I0 981-270-022-6

Printed in Singapore by World Scientific Printers (S)Pte Ltd

V

INTRODUCTION In August-September 1991, the first meeting on singularities at the International Centre of Theoretical Physics (ICTP) was held. It was organized by V.I. Arnold, L6 Diing T r h g , K. Saito and B. Teissier. That meeting can be viewed historically as a continuation of earlier major conferences in singularities held in Liverpool (1970-71), Cargese (Corsica) in 1972, Arcata (California) in 1981 and Lille (June 1991). Since then, there have been a number of international meetings on singularities, in Luminy (France), Cracow (Poland), Sapporo (Japan) and Liverpool (UK) among others. In 2000 an entire semester at the Isaac Newton Institute was devoted to singularities, making it one of the important events of the past decade. With the exception of the 1991 Trieste meeting (pub1ishe.d by World Scientific Publishers in 1993), the proceedings for these conferences concentrated on the most current research advances. What has been lacking is a collection of surveys and introductions on diverse areas of singularities which a non-expert reader could use to learn the basic ideas in these areas. With this in mind, the editors decided to gather in a new volume the contents of the lectures given at the ICTP School on Singularities and Applications in August-September 2005. For this volume, we have included surveys and basic introductions by leading experts to various aspects of singularity theory, exactly to provide students and non-experts with the desired access. At the same time, where appropriate certain advanced and noteworthy articles have been included. To help guide the reader, we have organized the book so that papers which survey results and provide an introduction to a subfield are gathered in the first part of the book. The second part contains original research papers principally describing results from invited lectures. All the papers that appear here have been refereed. This volume complements the volume of proceedings of the Luminy School on Singularity Theory and Applications held in January-February 2005, which is also being published by World Scientific Publishers. The meetings had different emphasis, with that in Trieste placing greater emphasis on the underlying theory which is a basis for more advanced work. We hope that this presentation of recent trends in singularity theory will become a basic tool for scientists who desire to understand the role of singularities in mathematics. We wish to acknowledge the generous support of ICTP, with its emphasis on financial assistance for participating students and researchers from

vi

developing countries. Our thanks are also extended to the staff of ICTP, especially Mabilo Koutou for their considerable help in the organization of the conference. Finally, we thank all of the speakers whose contributions appear in the proceedings and all of the participants who contributed to the success of the event.

Jean-Paul Brasselet James Damon Le Diing TrAng Mutsuo Oka

This page intentionally left blank

ix

CONTENTS Iiitroduction

V

Part I Elementary School on Singularity Theory Introduction to Basic Toric Geometry G. Barthel, L. Kaup and K. -H. Fieseler

3

PoincarBHopf Theorems on Singular Varieties J. -P. Brasselet

57

Notes on Real and Complex Analytic and Semianalytic Singularities D.B. Massey and D. T. L6 On Milnor’s Fibration Theorem for Real and Complex Singularities J. Seade

127

Introduction to Complex Analytic Geometry T. Suwa

161

Part I1 Advanced School on Singularity Theory Metric Theory of Singularities. Lipschitz Geometry of Singular Spaces L . Birbrair

223

Lectures on Monodromy W. Ebeling and S. M. Gusein-Zade

234

81

Computational Aspects of Singularities A . Friihbis-Kriiger Lagrangian and Legendrian Varieties and Stability of Their Projections V. V. Goryunov and V. M. Zakalyukin

328

A Resolution of Singularities of a Toric Variety and a Non-degenerate Hypersurface S. Ishii

354

Problems in Topology of the Complements to Plane Singular Curves A . Libgober

370

253

X

Topology of Degeneration of Riemann Surfaces Y. Matsumoto

388

Graded Roots and Singularities A . Ndmethi

394

Chern Classes and Thom Polynomials T. Ohmoto

464

McKay Correspondence for Quotient Surface Singularities 0. Riemenschneider

483

Lectures on the Topology of Polynomial Functions and Singularities at Infinity D. Siersma and M. Taba'r

520

Hodge Theory: The Search for Purity C. A . M. Peters and J. H. M. Steenbrink

556

Part I11 Workshop on Singularities in Geometry and Topology On Sufficiency of Jets H. Brodersen Singularity Theory Approach to Time Averaged Optimization A . Davydov and H. Mena-Matos Indices of Collections of l-Forms W. Ebeling and S. M. Gusein-Zade

585 598 629

A Lefschetz Theorem on the Picard Group of Complex Projective Varieties H. A . Hamm and D. T. Le^ Braid Monodromy and M. Lonne

640

of Discriminant Complements 661

Tangential Alexander Polynomials and Non-reduced Degeneration M. Oka

669

On Rigidity of Germs of Holomorphic Dicritic Foliations and Formal Normal Forms E. Rosales- Gonza'lez

705

Polar Multiplicities and Euler Obstruction of the Stable Types in Weighted Homogeneous Map Germs from @" to C3, n 2 3 E. C. Rizziolli and M. J. Saia

723

xi

Logarithmic Vector Fields and Multiplication Table S. Tanabe'

749

Some Surface Singularities Obtained via Lie Algebras K. Nakamoto and M. Tosun

779

Maxwell Strata and Caustics M. wan Manen Complex Curve Singularities: A Biased Introduction B. Teissier

825

Programme of the Conference

889

List of Participants

895

787

This page intentionally left blank

PART I

Elementary School on Singularity Theory

This page intentionally left blank

3

INTRODUCTION TO BASIC TORIC GEOMETRY

GOTTFRIED BARTHEL AND LUDGER KAUP

Fachbereich Mathematik und Statistik Universitat Konstanz, Fach D 203 0 - 7 8 4 5 7 Konstanz, Deutschland E-moil: Gottfried. BarthelBuni-konstanz.de E-mail: Ludger. KaupBuni-konstanz. de KARL-HEINZ FIESELER

Maternatiska Institutionen Uppsala Universitet SE-751 06 Uppsala, Sverige E-mail: Karl-Heinz. [email protected]

Introduction

The aim of these notes is to give a concise introduction to some fundamental notions of toric geometry, with applications t o singularity theory in mind. Toric varieties and their singularities provide a lot of particularly interesting examples: Though belonging t o a restricted class, they illustrate many central concepts for the general study of algebraic varieties and singularities. Quoting from the introduction of [5], one may say that “toric varieties have provided a remarkably fertile testing ground for general theories”. Whereas a singular variety may not be “globally” toric, singularities often are “toroidal”, i.e., locally analytically equivalent to toric ones, so toric geometry can help for a better understanding even of non-toric singular varieties. In addition to that, for studying certain classes of non-toroidal singularities, methods of toric geometry turn out to be most useful, e.g., for the resolution of “non-degenerate complete intersection singularities” . As a key feature, toric varieties admit a surprisingly simple, yet elegant and powerful description that prominently uses objects from elementary convex and combinatorial geometry. These objects are “rational” convex polyhedral cones and compatible collections thereof, called “fans”, in a real vector space of dimension equal to the complex dimension of the variety. The attribute “toric” refers to the algebraic torus of algebraic group theory. In the complex setting we are dealing with exclusively, the complex algebraic ntorus is an n-fold product T n := (C*)n, endowed both with its group structure

4 and its structure as an affine algebraic variety. (It is the complexification of the familiar real n-torus (S1)n and includes the latter as an equivariant deformation retract.) A toric variety is an algebraic variety including T, as an open dense subset such that the group structure extends t o an action on the variety. It turns out that many familiar algebraic varieties actually are toric; basic singular ones are the quadric cones V(C3;zy-z2)and V(C4;zy-zw). In these notes, we focus on fundamental parts of the theory that are indispensable if one wants t o apply toric methods as a tool for singularity theory. The picture presented here is by no means complete since important applications to singularity theory, let alone to other parts of mathematics, had to be left out. As examples, we just mention the role of toric geometry in studying non-degenerate complete intersection singularities or in the general resolution of singularities. We assume that the reader is familiar with elementary concepts of algebraic geometry. Affine complex algebraic varieties and their morphisms are in one-toone contravariant correspondence t o finitely generated reduced C-algebras and their homomorphisms: The elements of the algebra yield the regular functions on the variety, and the points of the variety correspond t o the maximal ideals of the algebra. Ideals determine closed subvarieties; conversely, to any closed subvariety corresponds its vanishing ideal. General varieties are obtained from affine ones by a natural gluing procedure that respects the separation condition. All varieties to be considered here are of finite type, i.e., they admit a finite covering by open affine subspaces. Moreover, we exclusively deal with (connected) normal varieties, i.e., the “coordinate algebras” corresponding to their f i n e open subsets are integral domains and integrally closed in their field of fractions. Besides these fundamental notions of algebraic geometry, we use some basic concepts of group actions like orbits, invariant subsets, isotropy subgroups, and fixed points.

1. Fundamental Notions 1.1. Group embeddings

Let G be a complex algebraic group, which means that G is both, a group and a complex algebraic variety, and these structures are compatible: The group multiplication GxG-G, and the inversion

G

(g,h)-gh

- G, g

g-’

are morphisms of algebraic varieties. A (homo-)morphism between algebraic groups is a homomorphism of groups which at the same time is a

morphism of varieties.

5

Standard examples are the general linear group GL,(C) and its closed subgroups like SL,(C), regular upper and lower triangular matrices, and regular diagonal matrices. The latter family of commutative connected complex algebraic groups plays the key role in these notes:

Definition 1.1. A (complex) algebraic (n-)torus, usually denoted by T or T, , is an algebraic group isomorphic to the n-fold Cartesian product of the multiplicative group @* of nonzero complex numbers:

T = T, = (@*)”. With such a torus, we usually associate a fixed isomorphism T, =” (@*)n. The toric varieties, to be considered in the sequel, are embeddings of algebraic tori. We first define that notion for an arbitrary algebraic group G:

-

Definition 1.2. A G-embedding is an algebraic variety X together with (1) an algebraic action GxX

-

X , (g,x)

~f

g*z=ga:

of the group G on X , i.e., that mapping is both, a G-action and a morphism of algebraic varieties; (2) an open embedding j : G --$ X with dense image such that j ( g h ) = g . j ( h ) holds for arbitrary elements g , h E G, i.e., the action of G on X extends the G-action on G j(G) by left translation. An immediate example for the group G = GL,(C) is its embedding into of square matrices. the vector space PXn

Remark 1.1. Condition (2) may be equivalently restated as follows: There is a “big” (i.e., open and dense) G-orbit 0 = G-xo in X such that the isotropy subgroup G,, is trivial. The embedding j then is just the orbit map g H g-xo, where zo is the j-image of the unit element of G. - The point xo is often called the base point. We usually identify G with its image j(G) in X . - Next, we consider morphisms of group embeddings:

Definition 1.3. Given a homomorphism q : G -+ H of algebraic groups, a morphism cp: X -+ Y from a G-embedding X to an H-embedding Y is called a) a q-extension if cp o jx = jy o q ; b) q-equivariant if cp(gx) = q(g)cp(x) for arbitrary g E G and x E X.

6 Every q-extension cp is q-equivariant, since the equality cp(g-z) = q(g).cp(z)holds on the dense open subset G of X and thus on all of X . Conversely, for an abelian group G, if cp: X -+Y is q-equivariant and the image cp(z0) of the base point z o of X lies in the big orbit H 2 H.yo of Y - say cp(z0) = hoeyo -, then := h o l y is a q-extension. So a q-equivariant morphism 9: X -+ Y is a q-extension if (and only if) it maps the base point of X t o the base point of Y . $J

Referring to the subsequent remarks for the notion of a “normal” variety, we introduce the main object of the present course:

Definition 1.4. For a torus T, a T-embedding into a normala algebraic variety X is called a (T-) toric variety. Sometimes it is useful t o use a more precise notation for a toric variety, writing a pair ( X ,T) or even a triplet ( X ,T, 5,) instead of X . Normal varieties We briefly recall that an algebraic variety X is called normal if all its local rings OX,^ are normal integral domains, i.e., they are integrally closed in their respective field of fractions Q(Ox,,). A connected normal variety is irreducible. If X is affine, then normality is equivalent to the fact that the restriction of functions O ( X ) +O(Xreg) from all of X to the regular locus Xreg := X \ S ( X ) is an isomorphism of rings. (This is a strong “Riemann removable singularity” property.) If furthermore X is irreducible, then X is normal if (and only if) the ring O ( X ) of globally regular functions is integrally closed in its field of fractions Q ( O ( X ) ) = C ( X ) , the function field of X . Smooth varieties are normal, since their local rings are factorial, and normal varieties are “not too singular”: If an n-dimensional irreducible variety X is normal, then its singular locus satisfies dim S ( X ) 5 n-2. For hypersurfaces, the converse holds. A standard example of an irreducible variety that is not normal is provided by “Neil’s parabola” X = V ( C 2 ;y2-z3): The rational function h = y/z E C ( X ) satisfies the integral equation h3 = y, but it is not regular. (The TI-action t . (z, y) := (t2z,t3y) with base point (1,l) actually would make this singular curve a non-normal TI-embedding.) ~

aFor some problems in algebraic geometry, the normality condition is unnecessarily restrictive. Since those problems lie outside the scope of these notes, we stick here to the

“classical” definition.

7

1.2. Toric varieties: Basic examples After these preparations, we proceed to discuss our main object of interest, namely, the toric varieties. According to Remark 1.1,their definition sums up to the following: A normal algebraic T-variety X is toric if and only if X has a base point xo with trivial isotropy and dense orbit. The embedding then is provided by the orbit map j = jx:T

5 T.so Q x ,t H t xo, *

where “a”indicates the inclusion of an open subspace. We present a few fundamental examples, most of which will be considered repeatedly in these notes.

Example 1.1. The following varieties, endowed with the torus action and base point as indicated, are toric: (0) The torus T, acting on itself by translation, with the natural base point (1,.. . ,l). (1) The linear space C”, with Tn = (C*)nQ C” acting by componentwise multiplication, and the natural base point (1,. . . ,1). (2) The two-dimensional affine quadric cone Y := V(C3 ;xz-y2) with the Tz-action (s,t).(x,y, z ) := (SIC, sty, st’z) and base point (1,1,1). - The Tz-action (s,t).(z,y, z ) := (sx,ty, s-’t2z) on that variety yields another toric structure, denoted Y‘ for distinction, which is q-isomorphic to the first one (for which q ? ) in the sense of Definition 1.5, but not isomorphic. - Normality is assured by the fact that Y is a hypersurface with an isolated singularity.

Figure 1. The set of real points of Y

With regard to Figure 1, the reader should keep in mind that it does not faithfully reflect the situation in the complex case:

8

Whereas the punctured real part YR\ (0) is disconnected, its complex counterpart Y \ (0) is connected. (3) The three-dimensional “determinantal variety” 2 ~t C 2 x 2 consisting of all singular 2 x 2 -matrices

with the ”3-action (S,t,U).

(”

y ) :=

Z W

and base point

(suzsx tuwt y )

(; :>. Normality is seen as above. - This variety 2

will also be interpreted as the three-dimensional L‘Segrecone” : The obvious identification CZx2S C4 yields 2 V(C4; xw-yz), the affine cone over the smooth projective quadric surface in P3 that is the image of the Segre embedding of PIx P I . (4) The projective space P, with T,-action t.[x]:= [ X O ,t1x1,. . . ,tnx,] and base point [I,.. . ,1].This is the most basic example of a compact toric variety. We remark that compact toric surfaces are always projective, whereas higher-dimensional compact toric varieties in general are not. The two-dimensional quadric cone Y actually is the lowest-degree member in an infinite family of singular two-dimensional hypersurfaces in C3 that are toric varieties, playing an important role in the theory of surface singularities:

Remark 1.2. For every integer k >= 2 , the variety Y k := V(C3; z z - y k ) , with the Tz-action ( s , t ) . ( z y, , z ) := (sz,s t y , s k - ’ t k z ) and the natural base point (1,1,l), is toric. In the literature, the singularity at the origin of Y k is called a “rational double point of type & - I ” .

With respect to the projection onto the (z, z ) plane, the surface y k is a finite covering branched along the coordinate axes - such coverings occur during the resolution of arbitrary singular surfaces. w e note that Yk is a “cyclic quotient singularity”: The cyclic group c k of k-th roots of unity acts on the plane C z as a subgroup of sL2(C) via C.(u,v):= (Cu,Ck-’v). The quotient variety ( c 2 / c k is a normal surface. The map C2 -+ C3, (u,v ) H (uk,uv,v k ) given by invariant polynomials induces an isomorphism ( c 2 / c k --% Y k (see also Example 3.1). The restriction of this map to R2 yields a parametrization of ( Y k ) R = Y k n R3 if k is odd.

9

Figure 2.

Three views of the set

( y 3 )with ~

the real

A2

surface singularity

Remark 1.3. There are some natural ways of constructing new toric varieties from given ones: (1) Every nonempty open T-invariant subset of a toric variety is itself toric. (2) A finite product of toric varieties is again toric (with respect to the direct product of the involved tori). (3) Let X be a T-toric variety and G, a closed subgroup of the torus T. The residue class group T/G is again a torus of dimension n-dim G, see Remark 1.6 (4). This quotient torus acts on the topological orbit space X/G of the induced G-action on X. The embedding T a X induces an open T/G-equivariant inclusion T/G X/G. If G is finite, then X/G has a natural T/G-toric structure that makes the projection X --H. X/G a toric morphism. This is discussed at the end of Subsection 2.3 when X is affine; using Sumihiro’stheorem 1.1, the general case then follows by a natural guing procedure. If G is not finite, then the G-orbit G . t of a point t E T is closed in T,but it may fail to be closed in X ; see Example 2.8 for three typical subgroups G ”= C* of Tz acting on X = C2. In that case, the topological orbit space is not separated; in particular, it is not an algebraic variety! To obtain a “categorical quotient”, a more involved approach is needed, since such a quotient morphism identifies different orbits if their closures intersect. For the affine case, we discuss some aspects of the “algebraic quotient” in the paragraph on quotients at the end of Subsection 2.3. The general situation lies outside the scope of these notes.

Morphisms of toric varieties are as in Definition 1.3:

Definition 1.5. Let q : T -+ T’be a homomorphism of algebraic tori, and (X,T, zo) and (X’, T’, zb) be toric varieties. Then a base point preserving q-equivariant morphism (i.e., a q-extension) is called a q-(toric) morphism. - In the case T = T’and q = idT, we simply speak of a toric morphism.

10

The theory of toric varieties heavily relies on the following result:

Theorem 1.1. (Sumihiro's Theorem) Every point in a (normal) toric variety admits a n affine open T-invariant neighbourhood. Thus, in order to analyse arbitrary T-toric varieties, it suffices to consider affine T-toric varieties, what we shall do in Section 2, and then to study how they can be patched together, see Section 3.2. Without assuming normality, the conclusion of Sumihoro's theorem is no longer valid:

Example 1.2. The binary cubic forms (-4tu(t+u), -4tu(t-u), ( t + ~ )define ~) a morphism P1 * C ~ -P2 t onto C = V(P2; y2z - z2(z+z)), the projective nodal cubic curve. The map is injective except for identifying the points 0 := [0,1] and 00 := [1,0], and it induces an isomorphism Pl/(O m) C. The C*action s . [t,u] := [ s t , u] on P1 thus defines a almost transitive algebraic action of the 1-torus on that (non-normal) projective curve with one big orbit and the single fixed point [O,O, 11, so the fixed point does not have an invariant affme neighbourhood. N

1.3. Characters and one-parameter subgroups of tori

In the study of toric varieties, algebraic group homomorphisms between the acting torus and C* play a key role. This starts with the following easy but crucial fact: Every algebraic group endomorphism of the algebraic 1-torus T1 = C* is of the form s H sk with a unique integer k E Z. The resulting canonical group isomorphism Hom(C*,C*) E Z sending id@. to 1 can be generalized in two ways:

Definition 1.6. Let T, E (C*), be an algebraic n-torus. A homomorphism of algebraic groups x: T, -+ C* is called a character of T,, and a homomorphism A : C* -+ T, is called a one-parameter subgroup. With respect to the argumentwise multiplication, the sets

X(T,)

:= Hom(T,,

C*) and U(T,)

:= Hom(C*, T ,)

(1)

are abelian groups. The canonical group isomorphism X(C*)= U(C*) 2 Z is generalized as follows:

Remark 1.4. The set X(T,) is a lattice (i.e., a free abelian group) of rank n : In fact, the mapping

M

:= (Z", +)

-

(X(T,), .) , p

- (f

:t

H

P

:=

n" t'i >

i=l

11

(with coordinates p = (PI,. . . , p n ) E Z"and t = ( t l , .. . , t n ) E (cC*), ) is an isomorphism of abelian groups. Hence, every character of T is a Laurent monomial in the coordinate functions (i.e., the basis characters) t l , . . . , t , on T,. The Laurent algebra generated by these monomials is the coordinate ring of the torus as an affine algebraic variety, i.e.,

(3(T,)

=

c[t,,t;l, ..., t,,t,l]

=

@

C.X.

(2)

XEWTTL)

-

Dually, the set Y(T,) is a lattice of rank n, too: There is an isomorphism

N := Zn

-+

Y(Tn), v = ( ~ 1 , .. . , vn)

(A,:

s

. . , sVn)) .

(#I,.

By a slight abuse of terminology, we occasionally call M and N the lattice of characters and of one-parameter subgroups, respectively. - Each of these lattices M and N determines the torus T: Using the canonical Zmodule structure on the abelian group C*, there are functorial isomorphisms

T

S

Homz(M,cC*) and

T

N

2

cC*

I&

=:

TN .

(3)

Remark 1.5. Via M = Z"-% X(T) and N = Z"5 Y(T) as provided by the fixed identification T, (C*)", the composition pairing X(T) x Y(T)

-

Hom(cC*,C*) ,

(x",A,)

corresponds to the usual inner product

(-,-) : M x N

-

-

Z,( p , ~ )

-

(x",A,)

:=

x" o A,

(4)

n

( p l y ):= C p i v i

,

(5)

i=l

i.e.,

(x" o A,)(s)

= s(">,)

holds for every s E C* .

(6)

We use the same symbol for the extended dual pairing

(-,-): Mw x Nw ---+ R of real vector spaces, where for a lattice L E Z", we set

Lw

:= L

@zR

2 R".

(7)

Moreover, for the dual pair of standard lattice bases in M and in N and thus, dual vector space bases of Mw and Nw, we shall use this notation: ( e l , . . . ,en)

in M ,

and

(f1,

... ,fn)

in N .

(8)

12 To better understand the mutual relations between the torus T and the lattice N expressed in formulae (1) and (3) as well as the structure of closed subgroups of the torus, it is helpful t o use an intermediate “analytic” object.b

Remark 1.6. We use the exact “exponential sequence” 0 + Z -+ C 2 C* -+ 1 (where exp(z) := e 2 T i z ) . “Tensoring” with the lattice N yields a new exact sequence exp O-+ N --t Nc TN - - t l ~ , (9) -+

’YZn

:=N&@%C”

”((c*)n

where the function exp is applied componentwise, and lg is the unit element of T := TN. Occasionally, we interpret Nc as tangent space Tl(T)of the torus at the unit element. The exact sequence immediately provides identifications

N = ker(exp: Nc

-+

T)

and

T

E

NcIN.

In this setting, the isomorphism N S Y(T) can be seen as follows: There is a canonical identification N E Hom(Z, N ) , with v E N corresponding to the map Z -+ N , lz H v and conversely. Scalar extension of lattice homomorphisms then yields a natural identification of N with the group Hom((C,Z), (Nc,N ) ) of vector space homomorphisms respecting the given lattices. Passing to the quotient modulo these sublattices, such a linear homomorphim then uniquely “descends” to a homomorphism of tori C’ 4 T. The inverse homomorphism Y(T) --+ N is obtained by “lifting” a one-parameter subgroup X E Y(T) to such a vector space homomorphism 1 = dX: (C, Z)4 ( N c ,N ) . Any homomorphism q : T‘ + T of tori lifts t o a vector space homomorphism dq: N& -+ Nc that respects the lattices and hence induces a lattice homomorphism S:N’ + N . We thus have a commutative “lad-

F=:

(10) exp

In the “differential” interpretation, the map dq actually is the derivative

of q at la,. Conversely, to any lattice homomorphism ‘p: N’ -+ N corresponds a homomorphism T(’p) := ‘p 8 id@. : T N ~-+ TN of tori. After a choice bFrom the “categorical” point of view, this is quite natural since a lattice of positive rank, being an infinite discrete group, is not an algebraic group, but rather an analytic group.

13

of bases, the map cp is explicitly represented by a matrix A = ( a i j ) in Znxm . In the corresponding coordinates for the tori, the homomorphism T(p) := cp @ id@*is given by

-

-

(3) The pair of covariant functors N TN and T Y(T) E N actually establishes an equivalence of categories. One should note, however, that the behaviour of morphisms is not quite straightforward since the functor N TN is right-exact only: For the inclusion L : N’ L) N of a sublattice, the resulting homomorphism T ( L ) TN, : + TN is injective if and only if N‘ is a saturated sublattice (i.e., if N/N‘ is torsion-free and thus, free: In that case, there is a one-sided inverse to L that, upon tensoring with @*, provides a one-sided inverse on the level of tori). In particular, if rank(N’) = rank(N), then the homomorphism T ( L )is surjective with the finite abelian group N/N‘ as kernel. Analogously, the inverse T ---+ N is left-exact only. ---+

Readers with an interest in “categorical” aspects might wonder why this behaviour does not contradict the equivalence property. Looking closer, one notes a subtle difference between the two categories: both are additive, neither of them is abelian, but they fail “on different sides”. In fact, in the category of finitely generated lattices, morphisms do not always have cokernels, whereas in the category of tori, morphisms do not always have kernels.

(4) Any closed subgroup G of T can be diagonalised: There is an isomorphism of tori transforming (T, G) into nZ1(@*,Gi) := ((C*)”, Gi), where each Gi, a closed subgroup of the one-torus C*, is either @* or

nZ1

~~

a finite (cyclic) group of roots of unity. To obtain such a diagonalization, consider the inverse image exp-l(G) in Nc. It splits (non-canonically) into a direct sum V @ L of the vector subspace V := exp-’(G0) S T l G and a “transversal” lattice L, where Godenotes the connected component of the identity in G. The lattice L spans a vector subspace Lc of Nc that is complementary t o V, and L‘ := N fl Lc is included in L as a sublattice of finite index. Splitting N n V into sublattices of rank one, applying the structure theorem for subgroups of finitely generated free abelian groups to the pair of lattices ( L ,L’), and then passing to the image in T then yields a diagonalization. This immediately implies that the residue class group T/G is a torus, as stated in Remark 1.3 (3).

If G is finite, then its inverse image exp-’(G) is discrete, so it consists only of the lattice L. The latter includes N, and the exponential mapping induces an isomorphism L/N G. Conversely, given the inclusion L : N’ N of a sublattice of finite index, the corresponding surjective

-

=

14

homomorphism T ( L :) TNI -+ TN of tori identifies TN with the quotient TN/G by a finite subgroup G N/N' (see also (2) and (3) above). For the study of quotient structures, it is useful to have an alternative approach to the closed subgroups of a torus T: Associating to G the sublattice K := { p E A4 ; x p I ~= l}, and to such a sublattice K C M the subgroup G := ker(Xp), establishes a one-to-one correspondence.

npEK

-

TN on the level of morphisms with We illustrate the correspondence N an example. The idea will be used again several times (see Examples 2.3, 2.5, and 2.6) since it is essential for one of our standard examples: Example 1.3. We consider the standard lattice N = Z2 with the standard basis f i , f 2 , and the sublattice fi spanned by 01 = 2f1- f2 and 02 = f2. With respect to this basis, the inclusion L : fi -+ N is given by the matrix ( !). Under the basis isomorphism fi 2 Z2, the torus Tfi = fi @zC* is identified with (C*)2by sending 0 1 8 s to (s, l),and v z @ t to (1,t). Thus, with the standard identification TN (C*)2,the associated morphism of tori takes the form T ( L :) (C*)2 -+ (C*)2, (s,t ) H ( s 2 ,t / s ) , with kernel &(1,1). 2. Affine Toric Varieties

2.1. Algebraic description: The coordinate ring An affine variety X is completely determined by its ring O ( X ) of regular functions. If X is toric, then the restriction of global regular functions to the (open dense) embedded torus T provides an injective algebra homomorphism from O ( X )into O(T), the Laurent monomial algebraof Equation (2). We may thus identify O ( X ) with a subalgebra of the latter:

O(X)

2

O ( X ) ) T c O(T) =

@ C.X.

(11)

XWT)

The characters of T that extend to a regular function on X - and thus, are elements of O ( X )- play a key role in the study of the coordinate ring. Evidently, the set

S = Sx := O ( X )n X(T)

(12)

of these characters is a (multiplicative) submonoid of the character group. We first study its role for the vector space structure of O ( X ) :

Lemma 2.1. The set S = S X provides a vector space basis of the coordinate ring:

15

-

-

Proof. The torus action on X induces an action on the coordinate ring:

T x O(X)

O ( X ) , (t,f )

ft

where f'(z) := f(t.z).

By Formula (ll),each non-zero function f E O ( X )can uniquely be written as f = Xixi with distinct characters x i c X(T) and non-zero complex coefficients Xi. Applying the torus action yields

'&

r ft =

C~i.xi(t).xi E O ( X ) for every t E T . i=l

Since the characters X I , . . . ,x, E O(T) are linearly independent, we find points u 1 , .. . ,u, in T such that the matrix ( x i ( u j ) )is nonsingular. Hence, each Xixi and thus, each xi lies in the span of the functions f " j ; conse0 quently, all xi belong to O(X). We now study an additive description of O ( X ) using the corresponding subset of "exponents"

E = Ex := { p E M ; X'

E

Sx}

(14)

in M = Zn. Since SX is a monoid, this set EX is a sub-semigroup of M , so we can describe O(X)as the semigroup algebra:

We list some essential properties:

Remark (2) E (3) E (4)E

2.1. (1) 0 E E; is finitely generated; generates M as a group; is a saturated sub-semigroup of M : If lcp E E holds for some k E N,1 and p E M , then p E E.

Proof. (1) holds, since 1 = xo E O ( X ) . (2) This assertion is true since O ( X ) is a finitely generated C-algebra that is spanned by characters; so we find elements p l , . . . ,pT E M with' E = C$, N . p i . ( 3 ) We have to verify that E ( - E ) = M . To that end, let x := X P with p := Ci=lpLias in the proof of (2). Since none of the generators x p 1 , . . . , xbr of SX = X(T) n O ( X ) has a zero on the principal open subset U := ( x # 0) of X , every character in S X even

+

=We adopt the convention that

N :=N 20 =Zto.

16

yields an (invertible) function on U . In particular, this implies that the character x-' = (xp' . . .xpLT)-lbelongs to O(U)x , and since O(U) = O(X)[x-'], this yields SU := O(U) nX(T) c O ( U ) x .It now suffices to show that U is just the embedded torus T of X, since then O ( U ) = @ [E N.(-p)] agrees with O(T) = CC [MI,so

+

M = E+N-(-p) G E + ( - E ) S M . The (open) inclusion T C U being obvious, we have to verify that the complementary subset 2 := U \ T of U is empty. This complement being a proper closed T-invariant subset, its vanishing ideal I ( 2 ) is non-zero and spanned by characters in Su. We have seen that such characters are invertible functions on U . Hence, the ideal I ( 2 ) is the unit ideal in O ( U ) ,and thus Z is empty. (4) If some power ( x " ) ~ = xkp of a character xp E X(T) c Q(O(X)) lies in O ( X ) ,then xp is integral over this normal ring and thus lies in it. 0 For any affine variety X and any finite system (fl, . . . ,fr) of regular functions, the corresponding morphism ( f l , . . . , fr): X 4 Cr is a closed embedding if and only if this system generates the coordinate ring O(X). In the toric case, we may thus characterize the generators of the semigroup of exponents:

Remark 2.2. Let X be an affine toric variety, and fix p l , . . . ,pr E EX with corresponding character functions xi := x p i E S x . Then i=l i.e., p', . . . ,p r generate the semigroup Ex, if and only if the morphism

is a closed embedding (see also Lemma 2.2). In that case, the "-action extends to the ambient space @' in the form t'(Zl,...,%-= ) (xl(t).zl,...,Xr(t).zr)

*

(17)

On the other hand, if the equivariant morphism of Formula (16) given by the vectors p l , . . . , p r is a closed embedding, then these vectors generate Ex as a semigroup.

17

We apply this remark to our standard examples (see also Remark 2.4):

Example 2.1. Using the numbering of Example 1.1, we obtain: (1) The semigroup Ex for X = C" is generated by e l , . ... e n . (2) The semigroup E y for the quadric cone Y is generated by e l , el+e2, el +2e2. More generally, for the two-dimensional toric hypersurfaces Yk (with k 2 2) discussed in Remark 1.2, the semigroup Eyk is generated by e l , e l + e 2 , (k-l)ei+ke2.

Figure 3.

Points and generators of Ecz and of E y

(3) The semigroup E z for the Segre cone 2 is generated by e l , e2, el+e3, e2+e3. (See Figure 7 in Example 2.2: these generators are the four points on the first cross-section.) 2.2. Geometric description: Polyhedral lattice cones

We now prepare for the geometric description of toric varieties in terms of cones and fans as announced in the introduction.

(I) Recollection: Polyhedral cones The semigroup of exponents E is the set of lattice points E = y n M in a suitable n-dimensional "lattice cone" y in Mw. We briefly recall the general notion of a (lattice) cone:

Definition 2.1. For a lattice L E Z", let y be a subset of V := Lw. (1) The set y is called a polyhedral cone if there are finitely many vectors 211,. ... II, in V such that T

y=

C 1w2O.q i=l

=: cone(v1,. .. , v T ) .

18

These vectors are called spanning vectors or generators of the cone. A cone spanned by a single non-zero vector v is called a ray, denoted by ray(v) := cone(v) . (2) A polyhedral cone y is called an L-cone (or lattice cone) if its spanning vectors 211,. . . ,v, can be chosen in the lattice L. (3) A polyhedral cone y is called strongly convex (or pointed) if it does not include a line through the origin. (4) A subset 6 of a polyhedral cone y is called a face, denoted 6 5 y, if it is of the form

6 = y n {v* = 0)

for some linear form v* E V* with v*17 2 0 .

Figure 4. A strongly convex polyhedral cone and its faces

Since the linear form v* = 0 E V* is admissible, each cone is a face of itself. We use the notation 6 $ y if we want to emphasize that a face 6 is proper. - For future use, we introduce some conventions:

Conventions 2.1. All cones to be considered in the sequel will be polyhedral, and usually, they are assumed to be lattice cones. A system of spanning vectors 211,. . . ,v, for a cone y is usually assumed to be irredundant. Moreover, if y is a lattice cone, then each such vector vi is usually assumed to be a primitive lattice vector, i.e., not a non-trivial positive integer multiple of another lattice vector. We add a few remarks and introduce some additional notions and not ations:

19

Remark 2.3. A polyhedral cone y as in Definition 2.1 (1) can be equivalently described as finite intersection of closed half spaces

n{w; 9

y=

2 0) with suitable linear forms w; E V*.

j=1

The boundary hyperplanes {v: = 0) with w; # 0 are called supporting hyperplanes of y. - In the case of a lattice cone, the linear forms wj* can be chosen as vectors of the dual lattice Hom(L, Z).Hence, each face of a lattice cone is again a lattice cone. The intersection y n ( - 7 ) is the largest linear subspace included in y. In particular, y n (-7) is the zero cone 0

:= (0)

if and only if y is strongly convex. A proper face 6 of y is cut out by a supporting hyperplane. The relative interior yo is the set of all points in y not included in a proper face. - The dimension of y is defined as dim y := dim lin(y)

where lin(y) = y

+ (-7)

is the linear subspace of V spanned by y. A cone of dimension d is usually called a d-cone; a (d-1) -face of y is called a facet, and and a 1-face, an edge. As a notational convention, we mostly use symbols like u, T,,Q for N-cones, and y 1 6 etc. for M-cones.

(11) “Contravariant” description For the next results. we need some more notation:

Notation 2.1. For a ring R, we denote by Sp(R) its maximal spectrum. Furthermore, we denote by

IuZ’x3.1rthe category of affine T-toric varieties with (idT-)toric morphisms, for a lattice L , the category of L-cones (in Lw) and their inclusions, C L , ~the full subcategory of d-cones, GCL and G & L the , ~ respective full subcategories of strongly convex cones. We may now start the construction of the bridge between the algebraic geometry of toric varieties and the elementary real convex geometry of cones that will be provided by the Equivalence Theorem 2.2.

20

Theorem 2.1. (Anti-Equivalence Theorem) For a n n-torus T with corresponding group M of exponents for X(T), the assignment C M , -+ ~ UZ’&

, y H Xy := Sp(C[y n MI)

is a n anti-equivalence of categories. Proof. On the one hand, for an n-cone y = cone(p’, . . . ,p‘) with primitive generators p i , one verifies that E := y n M shares the properties of Remark 2.1 and that this fact guarantees that X7 actually is a toric variety: The semigroup E is generated by the finite set P n M , where P := {Ci=,tipi ; 0 5 ti 5 1 for i = 1,.. . ,r } is the “fundamental polytope” of the cone y (see the left-hand side of Figure 5 in Remark 2.4). Furthermore, E (-19)= M (this guarantees that T -+ X , is an open embedding), since for every sufficiently “long” vector p E M n yo we have p+ei E E for i = 1,.. . ,n. Finally, we may write y = Hj, a n intersection of closed half spaces Hj = { v j 2 0) in Mw with suitable vj E N M*. To this corresponds a description

+

n,”=,

S

~ [ y n =~ ~ ] c [ H ~ ~ M ] . j=l

Since the subrings C [Hj n MI 2 O(C x (C*)n-’) of the Laurent algebra C [MI are normal, so is their intersection. On the other hand, t o an affine toric variety X corresponds a semigroup of exponents E x as defined in Formula (14) with finitely many generators, say p’, . . . ,p r , in M . Then, for the cone y := cone(p’,. . . , p ‘ ) , the variety X is isomorphic t o X,. For more details, see Lemma 2.2. 0 Whereas the above elements p l , . . . ,p‘ of E also generate the corresponding cone y, the converse need not be true, even if the generating vectors of the cone are primitive; cf. Example 2.2 (2). Remark 2.4. For the semigroup E, := y n M cut out by a strongly convex cone y, there is a canonical minimal system of generators, sometimes called a Hilbert basis: It is the set E \ (E + 8 ) of indecomposable elements in E , with E := E \ (0). In the two-dimensional case, it consists of those primitive lattice vectors p a which lie on the boundary of the (unbounded) “polyhedron” K := conv(8). This can be seen as follows: According to the regularity criterion in Corollary 3.1, two neighbouring points pi,pi+‘ generate a regular cone, so each lattice point in this cone is a linear combination of p i and pi+’.

21

.. ....... ...... ......... . . . . . . . . . . . . . . . . . . . . . . .

-

Figure 5 . Fundamental cell P, and polyhedron K , for y = cone(-5el+3ez,3el+2ez)

We illustrate the correspondence X = XY y between affine toric varieties and M-cones with our basic examples from 2.1:

Example 2.2. (1) The linear space X = C” corresponds to y = cone(e1,. ... en). (2) The affine quadric cone Y corresponds to y = cone(e1, el +2ez) . More generally, the toric surface Y k (with k 2 2) discussed in Remark 1.2 corresponds t o y = cone(e1, (k-l)el+kez).

.. .. .. .. .. .. ....... .. ..

.. .. .. .. .. ......... .. ..

Figure 6. The M-cones cutting out

EC2

and E y (cf. Figure 3)

In Figure 6, the encircled dots represent a Hilbert basis of the pertinent semigroups EQ and E y . (3) The Segre cone 2 corresponds to y = cone(e1, e2, el e3, e2 e3) with generators forming a Hilbert basis (see Figure 7).

+

+

(III) Dual cones and the “covariant” description For a truly elegant and powerful geometric description of toric varieties, it is essential to complement the correspondence of the Anti-Equivalence Theorem 2.1 with a LLcovariant” version, which is obtained by dualization of cones.

22

Figure 7. The M-cone 7 = cone(el,e;?,el+ea,e;?+e3)cutting out Ez

For a lattice L, its dual lattice is defined as L* := Homz(L,Z):

Definition 2.2. For an L-cone y, its dual is the cone YV := {u E

(L*)n ; (%Y) 2 0)

7

where (u, y) 2 0 means that (u, w) 2 0 holds for every w E y.

:.

.

Figure 8. A pair of dual cones y, y" for dual lattices L , L '

Remark 2.5. The dualization of cones enjoys the following properties: (1) The dual of an L-cone is an L*-cone, so dualization defines a map CL

4

CL*.

(2) Dualization is inclusion-reversing, i.e., b C y implies bV 2 yv . (3) Dualization is involutive, i.e., ( Y " ) ~ = y. (4) If y is a linear subspace, then yv = yl holds. ) ' =' 6 yv . (5) (6+y)" = bV n yv and ( b n y ( 6 ) dirnc7'=n--dim(c7n(-c7)).

+

23

In particular, according to properties (3) and ( 6 ) , there is a one-to-one correspondence between the objects of the category ~ C ofLstrongly convex L-cones, and of the category C p ,12 of full-dimensional L*-cones. Applying this observation to the dual lattices M and N, we obtain:

Corollary 2.1. The dualization of cones

is a n anti-equivalence of categories. Combining this result with the Anti-Equivalence Theorem 2.1, we achieve the construction of the bridge, which is fundamental for the toric geometry:

-

-

Corollary 2.2. (Equivalence Theorem) The functor 6~~

~ T B0~ ,

xu := xu"= s ~ ( c [ ~ ~ ~ M I )

is a (covariant) equivalence of categories,

Conventions 2.2. In view of the above correspondence, all N-cones considered in the sequel will usually be assumed to be strongly convex. The symbol (T will denote such a cone.

-

We illustrate the correspondence X = X u u between afine toric varieties and N-cones again with our basic examples of 2.1.

Example 2.3. (1) The linear space X = C" corresponds to u = cone(f1, . . . ,fn). (2) The affine quadric cone Y corresponds to (T = cone(2fl- f2, f 2 ) (see Figure 9). More generally, the toric surface Y b (with k 2 2) discussed in Remark 1.2 corresponds to 0 = cone (kfl-(k--l)f2, f 2 ) . (3) The Segre cone 2 corresponds to u = cone(f1, f 2 , f3, f1 + f 2 - f 3 ) . Let us also give an example for the correspondence of morphisms in Corollary 2.2 that at the same time exhibits the important role played by faces:

Example 2.4. To the face relation T 5 (T of N-cones corresponds an open embedding X , + X , of affine toric varieties. Proof. According to Remark 2.5, the "face equality" some p E uv n M ) translates into rv = (T' Rep = u"

+

T

=

(T

n p'

+ ray(-p).

(with This in

24

*

*

.*..*

.......... .@ .

..*.

Figure 9. The N-cones for C2 and Y

turn implies that O(X,) = O(X,)[x-'], where x := x p . As a consequence, the morphism X , -+ X u decomposes into an isomorphism X , g and the open inclusion of the principal open subset ( X u ) xinto its ambient variety X u . 0

So far, we mainly have been interested in full-dimensional N-cones. Fortunately, their investigation is essentially sufficient for the general theory. This fact, to be applied time and again, is a consequence of the following result, where we use this notation: Nu := N n l i n a

-

N

and

T, := Nu @z@*

T = N @ z C * . (18)

~ - f

Proposition 2.1. For a d-dimensional N-cone a , there exists a complementary subtorus Tn-d of T, in T n and a T,-toric variety Z, such that

endowed with the product action, where Tn-d acts on itself by translation. Proof. Since o is an N-cone, the lattice Nu is a saturated sublattice of rank d in N. Hence, the residue class group N/N, is free of rank n - d , and hence, the sublattice Nu admits a complement N' in N. We then choose Tn-d := N' @z @*. Eventually, we define 2, as the T,-toric variety associated to the cone o,considered as a (full-dimensional) cone in the real vector space (N,)R. 0 The factors 2, and Tn-d actually are naturally embedded in X u as closed subvarieties; see Remark 2.9 for details.

25

A word of warning may be in order concerning the role of the lattice in the correspondence between cones and affine toric varieties, a role that is not apparent from the usual notation: Remark 2.6. Given a lattice N "= Zn,then every finite collection w 1 , . . . , wr in the rational vector space NQ := N @zQ that includes a vector space basis generates a new Z.wi (in general, this sum is not direct). The two lattices lattice fi := intersect in a sublattice of finite index (such lattices are called commensurable in the associated real vector space Nw). An N-cone u then also is an fi-cone, here denoted by Z for distinction. Disregarding the primitivity condition, one may even choose the same set of generators for u and 5. Obviously, the resulting affine toric varieties X and 2 are not isomorphic if N # fi,since then T # ff. Isomorphy may even fail when disregarding the torus action and only considering the underlying structures of affine algebraic varieties :

c;=,

Example 2.5. Let N = Z2 be the standard lattice in W2 and u , the cone spanned by w1 = 2 f l - f ~ and w2 = f 2 as in Example 2.3 ( 2 ) . Hence, X ( N , ~is)the singular two-dimensional quadric cone Y . With respect to the sublattice fi := [email protected], however, the cone is spanned by a lattice basis, and the resulting variety X

(Za

is the affine plane C2. These two surfaces are even topologically distinct since the local fundamental group of Y at the origin is the cyclic group C2. This example fits into an important generalization of the equivalence of categories described in Remark 1.6: Remark 2.7. Given two lattices N1, N2 and Ni-cones ui , a morphism of lattice N2 such that cones ( N 1 ,q )-+ (N2,02) is a lattice homomorphism cp : N1 u2 holds. Then the homomorphism of tori "(9) : T N ~ + extends cpw(u1) to a toric morphism X(p): X ( N ~ + , ~X~( N) 2 , u z )The . correspondence -+

is an equivalence beween the categories of (strictly convex) lattice cones and of &ne toric varieties. We come back to the last example in order t o illustrate this correspondence: Example 2.6. (Example 2.5 continued) The inclusion of lattices L : fi ct N defines a morphism of lattice cones ( K ,Z) -+ (N, a).To describe the corresponding morphism of toric varieties, we identify TN and TE with (C*)2 by the choice ) the of the above bases. According to Example 1.3, the morphism of tori T ( L takes form ( s , t ) H ( s 2 , t / s ) . Composed with the embedding of TN into X(N,,) = Y

26 given by the orbit map ( u , v ). (1,1,1)= (u,uv,uv2) of Example 1.1 (2), we obtain the map (C*)' -+ Y , ( s ,t ) H (s2, st, t 2 )that clearly extends to a map from C2 = - to X ( N , u ) .

x( N > U )

2.3. Cones and orbit structure

We now analyse how the orbit structure of an affine toric variety Xu can be read off from the (strongly convex) N-cone o. The following facts turn out to be easy, but crucial:

Lemma 2.2. Let o be an N-cone, moreover, let Y E N and p E M . Then: (1) p E o" (2) u E o

-

* xI1 extends to a regular function on Xu. * A:, C* T Q Xu extends to a morphism C

--t

Xu.

Proof. (1) is true by definition. (2) Using (l), a system of semigroup generators pi E o" n M , i = 1,.. . ,r, with corresponding characters xi := xPi defines a closed embedding

X q C r , x++ ( ~ ~ ( x ) , . * * , x r ( x ) ) ?

(21)

equivariant with respect to the torus action t.z = ( x l ( t ) z l ,. . . , xr(t)zr)of Equation (17) on C r . The proof now follows from the equivalence of these five properties: (a) The one-parameter subgroup A, extends to a morphism C --f Xu; (b) The group homomorphisms

y i 0 A,

: C*

--t

C* Q C , s H S(P1>4

extend to regular functions C C , (c) ( p i , v ) > O f o r i = l , ..., r ; (d) ( o v , v )2 0 ; (e) v ~ a = ( a " ) " . --f

By the above equivalence (2), given an affine toric variety, we may recover the set of lattice points in the defining cone and thus, recover the cone: Remark 2.8. For an affine toric variety X, the set of lattice points in its defining N-cone ox is explicitly given as follows: ux

nN

=

{v E N ; A,(O)

exists in X}.

In the preceding formula, we have adopted the following convenient not ation:

27

Conventions 2.3. Let X be a (not necessarily affine) toric variety and X E Y(T), a one-parameter subgroup. If X extends to a morphism from C to X , then the limit X(0) := s+o lim X(s)

(22)

exists in X . We only use the symbol X(0) in this situation. In Example 2.4, we have seen that the inclusion r 5 u of a face induces an open embedding X , Q X u of affine toric varieties. We are now ready to prove the converse, a result that plays an essential role for the gluing of affine toric varieties to general ones:

c

Proposition 2.2. Let d u be an inclusion of N-cones. Then the induced morphism Xu,--+ X u is an open embedding (if and) only if u‘ 5 u. Proof. We may interpret U := X u , as an open subset of X := X u . Since X \ U is a closed T-invariant subvariety, its vanishing ideal in O ( X ) is generated by finitely many characters xi = x p ’ E Sx. As a consequence, U is the union of the (affine) principal open toric subvarieties (xi# 0) = X,, corresponding to the faces ~i = CJ n ( p i ) L (cf. Example 2.4). According to Remark 2.8, this description U = UX,” yields an analogous description u’ = ri. Hence, there is a face ri 5 u satisfying dimTi = dimu’. Together with ~i u’ u,this implies that a’ is included in lin(ri) n u = ri, thus 0 proving u’= ri.

u

c

The one-to-one correspondence between faces and affine open toric subvarieties thus established actually is only one aspect of a larger picture: The combinatorial face structure of a cone also corresponds to the orbit structure and the structure of invariant irreducible closed subvarieties. Theorem 2.2. Let u be an N-cone. (1) There is a one-to-one correspondence between the following sets:

(a) The set A(u) := {T c Np. ; T 5 a} of faces of CJ, (b) the sat X u / T := {T.z; z E X u } of T-orbits in X u , (c) the set of non-empty closed irreducible T-invariant subvarieties of x,, (d) the set of open a f i n e toric subvarieties of X u . (2) This correspondence is explicitly given as follows: To a face r 5 CJ, we first associate a “base point” z, := XV(0) E

xu

28

where v is an arbitrary lattice vector in the relative interior 7’. Then the orbit, the closed irreducible “-invariant subvariety, and the a f i n e open toric subvariety corresponding to T are

-

V, :=Or, and X , ,

0,:= T.x,,

-

respectively. Here the closure of 0, is taken with respect to X u , and X, is identified with the image of the open embedding into Xu. (5’) The correspondence r V, is inclusion-reversing. Each orbit 0, is a locally closed subvariety in X u , and it is isomorphic to a torus of the complementary dimension dim0, = dimV, = dima - dim7.

Moreover, this orbit is the unique closed orbit in X,; o n the other hand, it is open in its closure

v,

u

=

0,) 7

T
(4)

the union of 0, andfinitely many orbits of strictly smaller dimension. Dually, the correspondence r X , is inclusion-preseruing, and the afine open toric subvariety

-

x, =

u

7)

0,)

5r

is the union of 0 , and finitely many orbits of strictly larger dimension. The reader might wish to keep in mind that the notation V, has no “absolute” meaning, since the closure depends on the ambient toric variety X u . - Before proving the theorem, we first add a few comments: The zero cone o corresponds to the “big” orbit 0,= T (the embedded torus), whereas the orbit 0, corresponding to a full-dimensional cone u consists of precisely one point xu, which is then the unique fixed point of the affine toric variety X,. Next, we look at a basic example (see Figure 10 for n = 2):

Example 2.7. Assume that u = cone(vl, . . . ,v”) is spanned by a lattice basis of N . Denote by p l , , . . ,p” the corresponding dual basis of M . Then X, is isomorphic to the linear space C”, endowed with the action of Equation (17). The base point is (1,.. . , 1). The faces of u are of the

29

form aJ := cone(d ; j E J ) given by the subsets J of (1,.. . ,n}. To aJ corresponds the orbit

...,

oUJ={(Z1,z n ) ; z i = o

++ i E J } ,

the closure of which is a coordinate subspace. The pertinent orbit base point xuJ is the unique point of 0, with zi = 1 for i @ J.

Figure 10. Faces of cone(f1, f2) and corresponding orbit base points in C2

If a is a general d-cone, then the orbit 0,is isomorphic to a torus of the complementary dimension n - d : Corollary 2.3. In the product decomposition (19), the variety 2, has a (unique) fixed point z,, and the orbit 0,corresponds to (2,) x Tn-d (with isotropy group T, at each point).

In order to reduce the proof of Theorem 2.2 to the case of fulldimensional cones, we add the following complement to the product decomposition of Proposition 2.1: Remark 2.9. In the affine toric variety Xu defined by a d-cone a, let x , and xu denote the base points of the orbits 0,Z T and O,, respectively. In the product decomposition (Xu,") (Z,,T,) x Tn-d of (19), the two factors admit a natural closed embedding into Xu as follows:

--

2, E TUT.xoand

Tn-d

T.2, = 0,.

Proof. To prove Theorem 2.2, we proceed by induction on d := dima. For d = 0 , there is nothing to prove since then a = 0,so X , = 0,= V, = T because of o" = MR. For d > 0 , we may assume that d = n: By the product decomposition Xu 2 2, x Tn-d of Proposition 2.1 together with

30

Remark 2.9, T-objects like orbits, invariant irreducible closed subvarieties, and affine open invariant subvarieties in X u correspond to the respective T,-objects in 2, via Y ++ Y n 2, and vice versa. For convenience, we fix a closed equivariant embedding X ct C‘ given by the character functions xi = xp’ E Sx corresponding to a system of non-zero generators p l , . . . , p r for Ex as in Remark 2.2. These characters also describe the torus action on the ambient space C T . Since they are non-trivial, the origin is the only fixed point on C‘. We have to show that 0E By induction hypothesis, the theorem holds if we replace 0 with any proper face T . From Example 2.4, we know that each natural morphism X , 4 X u is an open T-equivariant embedding. We now consider the open “quasi-affine” toric subvariety Xa, := UTs, X , of X u and its complement, the closed invariant subvariety F := X,\Xa,. The proof of the theorem will essentially be deduced from the following three properties:

xu.

(i) F = {0}, which thus is the unique fixed point on X u , (ii) F is included in each orbit closure V, (taken with respect to X u ) , (iii) the equality X,(O) = 0 holds for any Y E ( T O . In fact, properties (iii) and (i) imply that the base point z, = 0 is well defined and that its orbit 0,= T.0 = F is closed, so 0,equals V,, and it has the asserted dimension n - dimu = 0. This proves part (2) for T = u. To establish the correspondence between faces and orbits, we fix an arbitrary non-zero orbit 0 = T.z through some point x E X \ {0} = Xa,. We thus have x E X, for a suitable proper face T of (T, so by induction hypothesis, there exists a face TO 5 T with 0 = 0, = T.zTo.We may thus replace x with the orbit base point x,,,. By hypothesis, the unicity of 7-0 is valid in every affine open subvariety X,, that includes 0, so it is valid in X u . For the asserted correspondence between faces and invariant irreducible closed subvarieties, we consider such an A L) X , with A := A \ F # 8. Then A = UTEa,(A n X,) implies the equality dim(A n X,) = dimA for some face T E 8u. Again by induction hypothesis, there is a face TO 5 T with A n X, = V(TO) = TO), the closure being taken in X,; moreover, TO) is open in AnX, and hence, has the same dimension. Since a T-orbit is irreducible, this readily implies that A is the closure of O(7-0)in X u . The unicity of TO is seen as above. We still have to prove the properties (i)-(iii): The coordinate functions xi E SX generate O(X,) and thus have at most one common zero. On the

31

other hand, for any lattice vector v E N n uo and for each pi, the inner product satisfies ( p i , v) > 0, and this implies that the limit Xv(0) = 0 exists in X u . 0 F'rom the proof of Theorem 2.2 and Proposition 2.1, we deduce the following consequence:

Corollary 2.4. I n an afjrine toric variety X u , the unique closed orbit 0, is a T-equivariant deformation retract of X u . Proof. It clearly suffices to consider the case of a full-dimensional cone u, or, in other words, the case where X u has a (unique) fixed point x u . We consider an equivariant embedding X u L) C r , 2 H ( x l ( x ) ,. . . ,~ ~ ( 2as ) in embedding (21) with xi := x P i . To each fixed lattice vector Y E N corresponds a set of exponents ki := ( p i ,v) with xi( X, (s)) = ski and thus, an induced C*-action C* x C'

+

Cr 7 (s,2)

-

(xi (Xv(s)) . z i ) , p l , . . . ,= .

(s%1,.

. . ,S k ' Z r )

on Cr that respects X u . For a lattice vector v in the relative interior u o (which here coincides with the topological interior), these exponents satisfy the strict inequality Ici > 0. It follows that the corresponding C*-action extends to s = 0. Restricting to scalars s E [0,1] yields a T-equivariant (why?) homotopy, providing an equivariant retraction by deformation to 0 the fixed point. To study the geometry of orbit closures in an affine toric variety, let r be a face of a cone u. We recall from Corollary 2.3 that the subtorus T, of T is the isotropy group at each point of 0,. Hence, restricting the T-action to 0, provides an action of the quotient torus T/T,on 0,.

Proposition 2.3. Every orbit closure variety:

v,

V, in X u is an afjrine T/T,-toric

xu,,,

where u / r := n(u) with the quotient map

T:

Nw

-, (N/N,)ltp.

Proof. We may regroup the "weight subspaces" C . x P occuring in Equation (15) according to the decomposition E = F U J of the exponent semigroup E := M n u" into the sub-semigroups

F:=Enrl

and J : = E \ F

)

32

+ 5 J).

We claim that this

o(xU) = o(xO)"T @ I(vT)

(23)

(where J actually is an "ideal", i.e., J E regrouping yields a direct sum decomposition

of the coordinate ring into the subalgebra of all ",-invariant

0(Xu)'T =

functions

@@.xP !J€F

and the ideal of

V, in Xu.To that

end, it suffices t o verify that

x

We choose an arbitrary lattice vector u E T O n N . For a character = X P in O(Xu) n X(T) (i.e., with exponent p E E ) , we have the following chain of equivalences (since 2, = x,(o)):

x E I(vT) * X ( 2 T ) = 0 * (/d,u) > 0

*p@

T1

.

As a consequence of formulae (23) and (24), factoring out the ideal yields an isomorphism

0(VT)2 0(Xu)'T

= @@.xP. PEF

Under the natural dual pairing of A4 and N , an arbitrary exponent vector p E F vanishes on the sublattice N,. It thus corresponds to a unique linear form ii on the quotient lattice N/N,. Such a linear form is the exponent of a character for the quotient torus T/T,,and the condition p E ov then is equivalent to p belonging to the dual cone of the image o/7 = n(cJ). 0 Quotients of afine toric varieties

To finish the discussion of affine toric varieties, we briefly come back to quotients, continuing Remark 1.3 (3). We use this notation: Let G be a closed subgroup of a torus T, and let q : T ++ T' := T/G be the quotient map. We let N' Hom(C*,TI) denote the lattice of one-parameter subgroups of TI,and i?i= dq: N --t N ' , the lattice homomorphism corresponding to q. Its kernel NO := ker(dq) is a saturated sublattice of N . Finally, we consider the sublattice K = N t c M corresponding to G introduced in Remark 1.6 (4). We note that for each p E K , the character x!J passes to the quotient torus T';in fact, the lattice K is naturally identified with the lattice M' = (N')* of characters of T'. Remark 2.10. Let o be a full-dimensional (strongly convex) N-cone with dual M-cone ov. The image o' := &(a) is an "-cone, but it may fail to be strongly

33 convex (this is illustrated by the “elliptic” case of the following example). The strong convexity of u’ is equivalent to the condition that NO does not intersect the relative interior u o . If u’ is strongly convex, then dq induces a map of cones (N, u ) -+ (N’, u ’ ) , so the surjective homomorphism q : T --k T’extends t o a q-toric map

This map is surjective, since each orbit base point z,t of X’ lies in the image: For a face T’ 5 u’, there is a face T 3 u and a vector Y E T O such that d q ( v ) lies in the relative interior of 7’.Then cp maps zT = Xv(0) to X d q ( v ) ( 0 ) = x,!. The comorphism cp* : O ( X ’ ) -+ O ( X )induces an isomorphism between O(X’) and the ring

O ( X ) G:= {f E O ( X ) ;V t E G : f t = f} of G-invariant functions on X . This provides an identification

X’

%

X / / G := SP(O(X)~)

with the algebraic quotient of X by G. The quotient morphism X a factorisation

X

++

X/G

++

(25) ++

X / / G admits

X//G

through the topological orbit space. As stated in Remark 1.3 (3), this orbit space need not be separated if G is not finite: Example 2.8. We consider three closed subgroups G = G* := L * ( @ * ) C TZ with embeddings L * : C’ L* T2 given by Lh: s H (s,s-’), L ~ s: H ( s , l ) , and L ~ s: H (s,s), where the index * = e , h , p indicates “elliptic, hyperbolic, parabolic”. The quotient map q : T2 ++ Tz/G E @* is then given by a character x*’, namely X h ( t , u ) = tu, X p ( t , u ) = u, and x e ( t , u )= tu-’. The map of lattices dq: 2’ -+ Z is represented by the (2x1)-matrices (1,l) for Gh, (0,l) for G,, and (1,-1) for G,. The respective sublattices ker(dq) of oneparameter subgroups are Nh = Z.(1, -1), Np := Z.(l,O), and Ne = Z.(1, 1). Let u = cone(f1, fz) be the standard lattice cone defining X := C2.The respective image cone d q ( u ) in NL E R is u’ = ray(1) for Gh and G,. For G,, we obtain u’ = cone(1, -1) = NL, so it is not strongly convex; accordingly, the lattice point (1,l)E Ne lies in the relative interior of u. The @*-actionson X given by these groups are as follows: (1) For Gh, we have the hyperbolic action s.(z, y) = (sz,s-’y). Each general orbit is a fibre of the quotient map ‘p: @’ + C2//Gh G2 C, (z,y) I-+ zy, so it is closed. The fibre (zy = 0) consists of three orbits: the fixed point (0,O) and the punctured coordinate axes @* x0 and OxC*. In the topological orbit space, these punctured coordinate axes are non-closed points having the fixed point in their closure. The map C2/Gh + @’//Gh identifies these three points. (2) For G,, we get the parabolic action s.(z,y) = (sz,y). The pertinent quotient map (o: C2 + C2//Gp E @ is given by (z,y) H y. Each fibre ‘p-l(y) = C x y consists of two orbits, namely the (non-closed) punctured horizontal line @*x(y}, and the (closed) fixed point (0, y). The topological orbit space consists of two

34 copies of @, say O X @ and 1xC. The points in the first copy are closed, whereas the closure of any point (1,y) in the second copy consists of the two points (1,y), (0,y). The map C 2 / G p-W C 2 / / G pidentifies these two points.

(3) For G,, we obtain the elliptic action s.(z,y) = (sz,sy). The fixed point (0,O) is the only closed orbit; any other orbit is a punctured line. The topological orbit space consists of the (closed) unique fixed point and a projective line, with the closure of any point consisting of that point and the fixed point. The only invariant functions are the constants, so the algebraic quotient C2//G, consists of a single point, with the obvious quotient map.

To end this discussion of quotients, we add a few further remarks that might help for a better understanding of the situation. Using the notation introduced above, we again assume that u’ is strongly convex.

Remark 2.11. (Remark 2.10 continued) For each T‘-orbit O(T’) in X’, given by a face r15 u1, there is a unique face T 5 u such that the T-orbit O(T) in X is relatively closed in the preimage cp-’ (O(T’)):We write T’ = u’n (p’)’ with a suitably chosen lattice vector p’ E M’ n (u’)”.The map xp’ o q : T C* is a character of T;so it is of the form xp, and its exponent p = q*(p’) clearly lies in u”. It is not difficult t o see that in fact, the T’-orbit O(T’) is the G-orbit space O(T)/G. If G is finite, then q: N + N’ is the inclusion of a sublattice of finite index IGI (see item (4)in Remark 1.6). Any (strongly convex) N-cone also is an “-cone; the face structure is of course independent of the lattice, so there is a one-toone correspondence between T-orbits and TI-orbits. In that case, the induced map X/G --H X//G is a homeomorphism, so the algebraic quotient is just the topological orbit space. In such a situation, one calls X//G a geometric quotient. -+

We finally note that the algebraic quotient X//G := S P ( U ( X ) ~ )is defined for any closed subgroup G of T, and it always admits a natural T/G-action. The strong convexity condition for the image cone u’ = q(u)implies that the canonical morphism T/G -+ X//G is an open embedding, and conversely.

3. Toric Singularities

3.1. Local structure at a fixed point The common theme of the notes collected in the present volume is t h e geometry of singularities. I n our context, we have to study the singularities occuring o n afine toric varieties. As a consequence of t h e product decomposition obtained in Proposition 2.1, it suffices to consider the variety defined by a full-dimensional N-cone. Here, the fixed point is the natural candidate for a singularity. So first of all we have to find o u t under which conditions a fixed point is or is not singular:

35 Proposition 3.1. Let u be a full-dimensional (strongly convex) N-cone.

Then the following statements are equivalent: (1) The afine variety X u is smooth. (2) The unique fixed point xu of X u is a regular point. (3) The M-cone uv is spanned by a lattice basis p l , . . . , p n , that is, uv = cone(pl,. . . ,p n ) . (4) The N-cone u is spanned by a lattice basis u l , ... , v n , that is, u = cone(vl,. . . , v n ) . (5) X u Z CC” with the base point ( 1 , . . . , 1) and the T-action

t . z = (XI (t)zl . . ., X n (t)zn) for a lattice basis p l , . . . , punof M 1

where

xi = x P i

A cone satisfying these properties is called regular (see Definition 3.1 for the case of not full-dimensional cones). Proof. Since the implication “(4) =+ (5)” has already been discussed in Example 2.3, the only non-trivial implication is “(2) =+ (3)”: The maximal ideal of all regular functions vanishing at xu is of the form

m

=

@c.~P P€E

with E := E \ (0) for E = 0” n M , so m2 = @,,-fi+~C.xP. Hence, the Zariski tangent space T z c X , = (m/m2)* has the dual

m/m2 z @ C . X ~ , PEB

where 23 := fi \ (,!+A) denotes the Hilbert basis of the semigroup E (cf. Remark 2.4). Now

123

I = dim@(m/m2) = dim X u = n

since xu is a regular point. As a semigroup, E is generated by 93 (why?); furthermore, we know that M = E ( - E ) by Remark 2.1 (2). Hence, the elements in 23 generate M as a lattice and thus form a lattice basis, since there are only n of them. 0

+

Somewhat more general are the cones spanned by linearly independent lattice vectors: Definition 3.1. A d-dimensional L-cone is called simplicial if it has exactly d edges. It is called regular if it is simplicial and if the primitive spanning vectors of the edges are part of a lattice basis of L.

36

The name “simplicial” is to indicate that transversal sections of such a cone are (d - 1)-simplices. (In [3], they are called “simplex cones”, whereas “simplicial cone” there means a cone that has simplicial polytopes as cross-sections.) Two-cones are always simplicial, whereas cones of dimension d 2 3 in general are not. Our standard example is provided by u = cone(f1, f 2 , f3, fi+f2-f3) defining the Segre cone 2 of Example 2.3 (3) (see also Example 2.2 (3) and Figure 7 for its dual). To a simplicial d-cone u = cone(v1, .... Wd), we associate the sublattice d

Li

:=

5 L,

with

Li := Lnlin(vi)

i=l

(recalling L, := L n lin(a), cf. Proposition 2.1), generated by the lattice points on the edges (see Figure 11 for an example with d = 2). Then

Figure 11.

Sublattice r mfor o = cone(-5fl+3f2,3fi+2f2)

the regularity of u can be characterized by the equality l?, = L,. More precisely, the deviation of a simplicial L-cone from regularity is measured by the index of this sublattice:

Definition 3.2. (Multiplicity of a simplicial cone) For a d-dimensional simplicial L-cone u,the positive integer mult(a) := mu := [L, : I?,]

= JLU/ruJ

is called its multiplicity.

It follows from the structure theorem for finitely generated abelian groups that the multiplicity of a full-dimensionalsimplicial L-cone is readily computed in terms of a primitive generating system:

37

Remark 3.1. If the generators v1, . . . ,v, E L of a simplicial n-cone are primitive and n = rank(L), then, after fixing an isomorphism L Z”, mult(cone(v1,. . . , vn)) = Idet(v1,. . . , v,)l.

A topological interpretation of the multiplicity for full-dimensional cones is given in Corollary 3.2. The next observation follows immediately from the definition of the multiplicity. Remark 3.2. (Geometric interpretation of the multiplicity) For a simplicial d-cone a spanned by primitive lattice vectors v1, . . . ,v d E L , let

be the “half open” fundamental parallelotope of the sublattice the multiplicity is given by the number of N-lattice points in P :

Then

mult(a) = # ( P n L ) .

As a useful consequence, one may check for regularity by counting lattice points: Corollary 3.1. (Geometric regularity criterion) A simplicial dcone a as above is regular i f its fundamental parallelotope P intersects the lattice L, only at the origin. I n dimension d = 2, the same conclusion holds if we replace P with the d-simplex conv(0, v1,. . . ,Vd). Toric varieties given by a simplicial N-cone have a remarkably simple structure that we are now going to describe. In the two-dimensional case (where cones are always simplicial), we have already seen that the affine toric surface Yk (with k 2 2) introduced in Remark 1.2 is a quotient (c2/ck of the atfine plane by a suitable action of a finite group, namely the cyclic group of order k. We want to show that an analoguous result, except for the cyclic structure of the group, holds for arbitrary affine toric varieties in the “simplicia1 class”. Using the product decomposition of Proposition 2.1, we again may assume that dima = n. We then consider on Cn the action of T 2 (C*), by componentwise multiplication, thus identifying the elements in T with diagonal matrices in GL,(C):

38

Proposition 3.2. For a full-dimensional simplicia1 N-cone (T, the toric variety X u is the algebraic quotient Cn/G := S P ( O ( C " ) ~ ) of C" by the induced action of a finite subgroup G E N / r , of the torus T , so Xu is just the usual orbit space of the action of G on C". In the literature, varieties that locally are orbit spaces of a smooth manifold with respect to an action of a finite group often are called "orbifolds".

Proof. With respect to the sublattice N' := rc of the lattice N , the cone (T =: (T'is regular. By Proposition 3.1, the associated toric variety X ( N I , ~is, )the affine space C". The assertion now follows from our discussion of quotients in the remarks 1.6 and 2.11. 0 The group G 2 N / r , actually can be recovered from the toric variety by topological means, namely as fundamental group of the open invariant subvariety of regular points:

Remark 3.3. There is an isomorphism

G

2 7r1 ( ( X u ) r e g ) .

Proof. We let v l ,. . . ,v n E N denote the primitive generators of (T,and L : fi H vz, the inclusion of the sublattice Z" 2 N' := ru into N . On the side of tori, we have the finite morphism

T ( L )=: q : T'-s T S T'/G,

(26)

with T' := T N and ~ T := TN for ease of notation. Then q extends to the quotient morphism cp = X ( L ) :X ' = C"

+ Cn/G 2

X

,

with X' := X ( N , , ~ ,and ) X = X ( N , ~ )Furthermore, . let 2 ~t C" be the set of points where the G-action has non-trivial isotropy. It suffices to show t h a t 9 induces an unramified covering C" \ Z -+ (Xu)regwhich is the universal covering. First of all, we verify that C" \ 2 is simply connected: We use the fact that a lattice vector v E N is primitive if and only if the corresponding one-parameter subgroup A, : @* -+ T is an injective homomorphism. Since L( = vi,this implies for each coordinate subtorus T i := Xf; (C*) of that the group G n T i is trivial. Restated in other words: If a matrix in G c T c GL,(C) has the eigenvalue 1with multiplicity a t least n-1, then it is the identity; such a group G is called a small subgroup of GL,(C).

fi)

39

So the subset 2 c-$ @" is a union of coordinate subspaces of dimension at most n-2. Using the fact that spheres of dimension at least 2 are simply connected, the complement @" \ 2 can be seen to be simply connected as well. Eventually, we have to show that the preimage of the singular locus S(X,) coincides with 2. Since 2 = cp-'(cp(Z)), this follows from the equality S(X,) = ~ ( 2 )The . inclusion is clear, while "2" follows from the fact that the set of points where a dominant morphism between equidimensional smooth varieties is not a local analytic isomorphism has codimension one.

","

Corollary 3.2. The multiplicity of a full-dimensional simplicia1 N-cone u satisfies

mult(0) =

lm((xo)reg)l .

Example 3.1. (1) Let n = 2. For any two coprime integers k,C with C > 0, we consider (T = cone(Cf1- k f 2 , f 2 ) in R2. Then the morphism q : T' + T of (26) satisfies q(t1,tz) = ( t f , t T k t 2 ) ; thus G is the cyclic subgroup of order C in GLz(C) generated by (C,Ck) with ,f := e2.rri'e. The singularity C2/G thus obtained is a cyclic quotient singula~ty, and mult(a) = (GI = C. The group G lies in SL2(C) if and only if k = C - 1. In that case, for 2 2, the semigroup aVnM is generated by e l , el+e2, (t--l)el+Ce2. The closed equivariant embedding (21) shows that Xu is a toric hypersurface in C3, namely the surface 5 of Remark 1.2. In particular, for C = 2, we thus obtain one of our standard examples, the quadric cone Y of Example 1.1 (2).

C

(2) Under the primitivity hypothesis of Remark 3.3, for any dimension n, the group G has a system of at most n-1 generators. Non-cyclic groups actually occur already in dimension n = 3. An example is furnished by (T = cone(wl,v2,~3) with w1 = f l , w2 = f l + 2 f 2 , and w3 = f l +2f3. To close this subsection, we have to add at least some remarks on the nonsimplicial case. As usual, we may restrict to full-dimensional cones.

Remark 3.4. The structure of general toric singularities is considerably more complicated. There is a close relation with polytopes spanned by lattice vectors (called lattice polytopes for short): Intersecting a given n-cone o with the affine hyperplanes H p := {u E N R ; (w,p) = 1 ) for any p E ((T')" n M (on a suitable integral "level" 1 > 0) associates to (T a family of (n-1)-dimensional lattice polytopes with fixed combinatorial type. Conversely, any lattice polytope in Rn-',

40

placed in the affine hyperplanes (zn = 1 ) on different levels 1 E N>o, spans a family of n-cones. For n = 3, the associated polytopes are plane polygons. Their combinatorial type is just given by the number of vertices (or edges). For any fixed number k, however, there are countably many non-equivalent realizations of a k-gon as a lattice polygon. For n = 4, we have to look at three-dimensional polytopes (classically also called “polyhedra”). Here the situation gets much worse already on the side of combinatorial types: The enumeration results known so far show that the number of such types, considered as a function of the number fi of i-faces (vertices, edges, or facets for i = 0, 1, or 2), grows rather rapidly. (An empirical formula gives (k - 6 ) ( k - - 8 ) / 3as approximate number of types for f1 = k.) For special types of toric singularities, there are satisfactory classification results: Toric complete intersection singularities, for example, correspond to the so-called Nakajima polytopes placed on the level 1 = 1. Starting from points and line segments, these polytopes are inductively constructed as follows: One takes a sufficiently high prism over a Nakajima base polytope and then makes a “skew” truncation by a linear height function that is strictly positive on the relative interior of the base and integer-valued at lattice points. - For a survey of results about toric singularities, we refer to section 2 in [2]. A more detailed exposition is outside the scope of the present notes.

3 . 2 . General toric varieties and fans

Except for the general definition, the case of the projective space in Example 1.1, and Sumihiro’s Theorem 1.1,we so far only have studied affine toric varieties. This is sufficient for the local investigation of toric singularities, but it does not allow to deal with problems like resolution of such singularities. Since we intend t o address this topic in the final subsection, we have to provide the necessary tools. As a consequence of Sumihiro’s Theorem, every toric variety, say X, can be covered by open affine toric subvarieties. From Remark 2.8, we know that t o each a6ne toric variety, say U , corresponds a unique N-cone u = uu such that U = X u . To the general toric variety X, we may thus associate the following collection of N-cones:

A

:= A(X) := {u = m~ E

Ob(6CN) ; U QI X} ,

where U runs through the affine open toric subvarieties of X. For any two cones u,u’ E A, the intersection X u n Xu, is a T-invariant affine open subspace of X, and thus X u n Xu) = X, with a cone r E A. Since X is separated, a one-parameter subgroup X E Y(T) has at most one limit

41

X(0) E X . Hence, according to Remark 2.8 (2), the following holds:

I - n N = {V E N ; x,(o)

E

xunXul}

Xu}n {V =(anN)n(dnN). = {V E N ; X,(o)

E

E N ; X,(O)

E Xul}

This readily implies that I- = a n a’. Furthermore, it follows from Proposition 2.2 that a n a‘ is a common face of both, a and a’.That leads to the following notion:

Definition 3.3. An (N-lattice) fan in Nw is a finite non-empty set A of (strongly convex) N-cones satisfying (1) r 5 u E A j T E A ; (2) a, a’ E A ----7- a n a’ 5 0,a’; in particular, u n a’ E A.

A fan is called simplicia1 or regular, respectively, if each of its cones has that property. There are two fans that naturally correspond to a cone:

Remark 3.5. A single (strongly convex) N-cone a generates its full face fan A(a) := ( 7 ; 7 5 a}, also called an afine fan. If a

# 0,then

da := ( 7 ;

I-

ja}

is a subfan, called the boundary f a n of a.

We have just seen that a toric variety determines a fan. On the other hand, given a fan, there is a unique corresponding toric variety:

Proposition 3.3. To every N-fan A, one associates the toric variety X A :=

u xu

(TEA

by gluing the family of afine toric varieties ( X v ) u E ~where , X u and X z are glued along the common open afine invariant subvariety X u n z .

The fact that the prevariety XA is separated and thus, a variety, can be seen as follows: For cones a and 5 in A, the cone I- := a n 5 lies in A, too. Hence, the intersection X u n Xz; X , again is affine. We have to verify that the corresponding comorphism

d*: O(Xu ~ X , ) = c C [ M n a ~ ] ~ ~ c C [ M n 5 ~ ] - O ( x , ) = c C [ M n 7 ~

42

of affine algebras, given by xfi 8 x p xfi+z, is surjective. This is an immediate consequence of the equality T" = a" Z", see Remark 2.5 (5). Obviously, Proposition 3.1 implies that the variety X A is smooth if and only if the fan A is regular.

+

Example 3.2. In

Nw

fo := - Cyzl fi and

2 Rn, we set

A := {aJ := cone(fi ; i

E

J); J

5 (0,. . . ,n}} .

(27)

Then X A is isomorphic to the projective space P, with the toric structure of Example 1.1 (4).- In Figure 12, we depict the fan A in the two-dimensional case.

Figure 12. The fan for the projective plane P2

Proof. We write N , in order to indicate the rank of the lattice under consideration; thus A is a fan in (N,)w. In N,+l, we exceptionally denote the standard lattice basis by (go,. . . ,gn), and consider the regular cone a := cone(g0,. . . ,g,). The homomorphism p : T,+l

-

T, , u := (uo,. . . ,un)

-

-

,

(u1uO1,. . . unu;l)

induces the linear map

d p : (N,+l)W

n

-Cfi,gi

(Nn)w, go H

H

fz for i = 1 , . . . , n ,

i=l

which maps the cones in da onto the cones in A. Hence, p extends to a morphism

cnfl\ { 0)

=

X&

--$

XA .

The kernel of p is the diagonal D := { (5,.. . , s) E (C*),+'}. As a consequence, the map p is D-invariant and factors through P, = ((C"+' \{O})/D.

43

In order to establish the isomorphism P, E' X A , we consider a cone ffJ E A and denote by I - ~the unique cone in da with dp(TJ) = aJ.Then

p-l(XgJ

= x , E'

ZTJXD

-

XOJ2

z.,

is the projection onto the first factor. That proves the claim.

Many of the general remarks already made in the affine case remain valid in this more general setting. Firstly, the dependence on the lattice N has to be kept in mind (cf. Remark 2.6): If N and @ are commensurable lattices, then any N-fan A can be considered as an %-fan i.The resulting toric varieties need not be isomorphic as abstract varieties: Example 3.3. (Weighted projective spaces) For any given (n+ 1)-vector a := (ao,. . . ,a,) of integers ai 2 1 with gcd(ao,.. . ,an) = 1, we consider the fan A of Equation (27), but replace the standard lattice N z Zn with the finer lattice fi generated by the rational vectors ( l / u i ) f i for i = 0,. . . ,n. Then A of course remains simplicial, but in general, it is no longer regular. The resulting toric variety is called the weighted projective space P(a). The open affine "charts" given by the n-cones U, (for J (0,. . . ,n} as above) are cyclic quotients Cn/Gj. There is an isomorphism P(a) z Pn/G(a) with G(a) := Cai acting coordinatewise on P,, so in particular, P(1,. . . ,1) equals P,. Moreover, the description P, E (C"+')/D given above generalizes t o the weighted projective space if one replaces the diagonal 1-subtorus D c (C*)n+l with D(a) := {t" = ( P o , . . , t a n ); t E C*}.

5

ny=o

Secondly, there is a similar equivalence of suitable categories (cf. Remark 2.7); see (3) below. Thirdly, the results pertaining to orbits and orbit closures carry over to the general case:

Remark 3.6. (1) For each cone c E A, there exists an associated orbit

0, := T*A,(O), where v E a ' . Again, Theorem 2.2 and Formula (2.3) hold. In fact, the orbit 0,is the unique closed orbit in the open subvariety X , of X A , and the T/T,-toric variety V,,cf X A satisfies vo

xA/o

with the quotient fan A / a := { T / o ; T E A , a 2: T } in (N/N,)R. In Figure 13, we indicate schematically the correspondence between cones and orbits.

44

e3

e4

OU,

Figure 13. Correspondence between fan cones and orbits

(2) A toric variety X A is complete if and only if the fan A is complete, that is, its support lAl := U a E Ais~ the entire linear space Nw. In the strong topology of complex varieties, “complete” means compact. (3) Let A, A be fans in NR. Then there is a toric morphism X A --f X A if and only if each cone of A is included in some cone of A. - More generally, the equivalence of categories stated for the affine case as in Expression (20) easily carries over to lattice fans and general toric varieties. (For a Iattice homomorphism ‘p: N1 -+ N2 and Ni-lattice fans Ai, the condition for a morphism of lattice fans is that the cpw-image of each cone of A1 be included in some cone of A2.) (4) For fans A and A in Nw, we assume that each cone of A is included in some cone of A. The resulting morphism X A + X A is proper if and only if lA( = lAl; and in that case A is called a subdivision or refinement of A. - In the strong topology of complex varieties, “proper” means that the inverse image of a compact subset is again compact.

For a proof of (2) and (4),we refer to the standard literature. Exercise 1. In a two-dimensional fan A, let a ray e be the common edge of two 2-cones. Describe Ale and show that V, is a projective line. Exercise 2. Given the lattice vectors v1 = f 2 , v2 = 2f1 - f 2 , and 213 = -f1, the 2-cones a i j := cone(vi,vj) determine a fan A, indicated in Figure 14. Show that two of the cones are regular and one is singular, defining the affine quadric cone of Example 1.1 (2).

45

Figure 14. The fan for the projective quadric cone The fan A is complete, and X a is the projective quadric cone. - Figure 15 shows the set ( X A ) of~ real points, also called the “pinched torus” (tore pince‘). The caveat about interpreting “real” pictures of complex varieties stated in Example 1.1 (2) also applies here.

Figure 15. The set of real points of the projective quadric cone The projective quadric cone is the closure in B3 of the affine quadric cone Y in C3. In general, the projective closure of an affine toric variety needs not be normal:

Example 3.4. Among the singular twedimensional toric hypersurfaces Yk = V ( C 3 ; x z - y k )considered in Remark 1.2, only the quadric cone Y = Y2 and the cubic Y3 with the A2 singularity have projective closures in B3 with i s e lated (and thus, normal) singularities. Since the torus action on the affine part clearly extends, the closure then is a (normal) toric variety. We add a few remarks on the closure of Y3 in P3: In homogeneous coordinates [x,y, z , w],it is given by the homogenized equation wxz = y3. Hence,

46 there are two additional A2 singularities at infinity, namely at the origins of the affine charts (x = 1) and ( z = l),respectively. The affine quotient representation Y3 C2/C3 explained in Remark 1.2 also carries over to the closure: In homogeneous coordinates [T, s,t]for P2,the action takes the form 6 . [T, s,t] := [r,cs,C2t], and the quotient P2/C3 is embedded in P3 via [r,s,t] H [ r 3,s3,rst, t 3 ] . - The defining fan is spanned by v2 = f2, 01 = 3f1-2f2, and vo = -3f1+ f2, so it consists of three 2-cones oi with mult(ai) = 3. Moreover, the projective quadric cone is isomorphic to the weighted projective plane P ( l , 1,2). This is a particular case of the following exercise (whereas the cubic is not of this form): Exercise 3. Let A(v0, vl, vz) be the complete two-dimensional lattice fan given by primitive lattice vectors vi, and let mi be the multiplicity of cone(vj,vk). Then the toric surface X A is a weighted projective plane - and then isomorphic to P(m0, ml, m2) - if and only if gcd(m0, m l , m2) = 1. That condition in turn is equivalent to the fact that N coincides with its sublattice C:=o Z.vi.

Toric divisors For applications in Shihoko Ishii’s course, we introduce the concept of toric divisors. First, we briefly recall the general notion: In order to study the zeros and poles of a rational function f on a normal variety 2 , one first notes that the locus of zeros and poles of f has only finitely many irreducible components, all of codimension one. Hence, one associates to every a “multiplicity” vA( f ) E Z, 1-codimensional irreducible subvariety A the vanishing order of f along V. Now the pertinent information may be encoded in the “divisor” ( f ) of f: By definition, a divisor on 2 is a formal sum

-

A-Z

over all one-codimensional irreducible subvarieties, where only finitely many of the integer coefficients VA are allowed to be non-zero. Then the divisor of the rational function f is A-Z

Returning to the toric situation, the T-invariant irreducible subvarieties of codimension one in a toric variety X A are just the orbit closures DQi:= V,,, where the ei denote the rays of A. The characters of the torus are regular functions on T without zeros. Considering them as rational

47

functions on X A , it follows that non-trivial multiplicities only can occur on the “boundary” X A \ T = D e i . - Let vi E N be the primitive generator of ei = ray(v2).

ufzl

Remark 3.7. The divisor in X A of the character x is k i=l

Proof. For a fixed index i, we set e := ei and X = ,A, . Then X maps C* isomorphically onto the subtorus T, of T. As in Proposition 2.1 and Corollary 2.3, we write X , = 2, x T,-1 = CC x T,-1 with the orbit 0,= ( 0 ) x T,-1 and closure Y := V,. For f E CC(X,), the multiplicity vy(f) in the divisor ( f ) is just the multiplicity of the function s ++ f ( s , t ) at s = 0, for Applying this to the special case f = x, the equation generic t E “,-I. x ( s , 1) = x(X(s)) = s(X>’) implies that x(s,t ) = x(1,t).s(Xy’) has multi0 plicity (x,A) at s = 0 for all t E T,-1.

A divisor on X A of the form k

D = EniDei i=l

is called a toric divisor. The special case where all coefficients ni = 1 is of particular interest. Its negative, k

Kx

Dei ,

:= i=l

is the famous canonical divisor. 3.3. Resolution of toric singularities

In general, smooth (i.e., non-singular) varieties are much better understood and usually enjoy much nicer formal properties than singular ones. In studying singular varieties, it is thus a natural attempt to %esolve” the singularities. This means to find a non-singular “model” 2 of the given singular variety X , i.e., a smooth variety 2 together with a proper morphism 2 -+ X that is an isomorphism over the regular locus Xreg. The general resolution of singularities is rather involved. For complex varieties, it has been achieved by a celebrated result of HIRONAKA.

48

In the toric case, resolution of singularities is much more accessible: We recall that such a variety X A is smooth if and only if the defining fan A is regular. According to Remark 3.6, a subdivision A’ of a (general) fan A corresponds to a proper toric morphism X A 4 ~ X A that induces an isomorphism on the common open invariant subvariety X A ~ AHence, ~ . an equivariant resolution of singularities is given by a regular subdivision A’ of A, such that the subdividing fan A’ contains Areg as a subfan. In that case we call A’ a resolution of A (or of the cone 0 if A = A(0)).

Theorem 3.1. (Equivariant resolution of toric singularities) For every toric variety, there exists a resolution of the defining f a n and thus, an equivariant resolution of singularities. Since a one-dimensional normal variety is smooth, the case of (normal!) toric curves is without relevance for singularities: In fact, the only such curves are C*, C,and PI. Thus, we first discuss the resolution of singular toric surfaces.

(I) The surface case This situation can be dealt with most explicitly: A two-dimensional fan necessarily is simplicial, so the singularities are of the nice “quotient” type discussed in Proposition 3.2, and they are necessarily isolated.

Theorem 3.2. For every toric surface, there exists a unique minimal resolution of the defining fan and thus, a canonical equivariant resolution of singularities.

It obviously suffices to prove this statement in the affine case: Lemma 3.1. A two-dimensional N-cone admits a unique minimal resolution; in particular, every resolution is a refinement of the minimal one. Proof. We may assume that N = No. To construct a minimal resolution, we consider the convex hull K of the set N n \ (0) as in Remark 2.4. The boundary of this polyhedron consists of two unbounded half-lines, each one included in an edge of 0,and finitely many bounded line segments. It thus contains only finitely many primitive lattice points, say YO,. . . ,Y‘+’ E N in clockwise order. We set pi := ray(vi), ( ~ := i ei+ei+l, and

A := {T ; T 5 ( ~ for i some i , 0 5 i 5 r )

49

..

. .

Figure 16. Polyhedron K , and minimal resolution for c = cone(-5fl+3fz, 3 f l - t Z . f ~ )

(see Figure 16). According to Remark 3.1, each cone ui is regular, since the triangle with vertices 0, vi,vi+' contains no further lattice vector. Now let A' be an arbitrary regular subdivision of 0.To verify that A' is a refinement of A - thus also proving minimality and unicity of A -, it suffices to show that each ray ei of A is a ray of A': The ray ei is included in a 2-cone r := cone(b1,b2) of A'. Since r is regular, we may assume that ( b l , b2) is a basis of N . Hence, by the regularity criterion of Remark 3.1, there is no further lattice point in the triangle spanned by 0, b l , b2. Moreover, the lattice points b l , b2, vz lie in K , so the line segments [bl,b2] and 10, vi]intersect in a point of K . Since vi is an element of d K , it lies on the line segment [bl, b z ] , so it is one of the endpoints. We indicate how to construct the first subdividing vector v'; iterating that step then enables a recursive computation of all vectors v2: For the primitive spanning vectors y o , v E N of u,there is a lattice basis b l , b2 of N 2 Z 2 such that

vo = b2

and

v = m,bl

-

kb2 with an integer 1 5 k

(see the following exercise). Then v1 is the vector bl.

< mu

(29) 0

Exercise 4. (1) Given two primitive lattice vectors q , w 2 E Z 2 with det(v1,q) = m > 1, prove that there exists a (positively oriented) lattice basis ( b l , b 2 ) and an integer k with 1 5 k < m, gcd(m, k ) = 1 such that v1 = m b l - kb2 and v2 = b2.

(2) In the proof of Lemma 3.1, show that the first subdivision yields a resolution if and only if k = 1. (3) Show that the maximal number of necessary subdivisions equals mu - 1, and characterize the case when this occurs.

The theory of toric surface singularities and their resolution is particularly rich and fascinating. Within the scope of these notes, we have to content ourselves to indicating some key results:

50

Remark 3.8. In the situation of Lemma 3.1, let X A --f X u be the minimal resolution of the singular affine surface X , , and T : X A --t ~ X u , an arbitrary resolution. We denote by eo, . . . , p r + l the rays of A’ in clockwise order. (1) The “exceptional fiber” E := T - ’ ( Z ~ )over the singular point of X u consists of the “new” orbit closures Ei := VQi ES PI (cf. Exercise 1) corresponding to the subdividing rays el,.. . , e r . They form a “chain” E l , . . . ,E, of curves as depicted in Figure 17: Each Ei

Figure 17. System of exceptional curves

only intersects its neighbours; the intersection is transverse and consists of one point (transversality means that the curves meet like coordinate axes). (2) To curves Ei and E j , one attaches their “intersection number” Ei . Ej. For i # j , this is just the number of intersection points since a non-empty intersection is transverse. For i = j , the “self intersection number” Ei.Ei = E: expresses the “twisting” of the ambient smooth surface along the curve. In the present case, this number equals -ai, where ai 2 1 is the multiplicity of the cone ei-l+ei+l spanned by the two neighbouring rays of ei. For a minimal resolution, we determine the ai in (3). The resulting “intersection matrix” (Ei.E j ) is an integral symmetric ( r x r)-matrix satisfying E i . Ej = 0 for li - j l 2 2. It is negative definite with det (Ei.E j ) = mult (CT). Instead of schematically indicating the chain of curves as in Figure 17, it is customary to depict its dual graph (see Figure 18): The dual graph has one node for each of the curves; two nodes are

1

1

Figure 18. Weighted dual graph of exceptional curves

joined by a simple edge if the corresponding curves meet. Moreover, the nodes are weighted by their self-intersection numbers.

51

As A’ is a refinement of A, there is a factorization XA/ -+ XA -+ X , of 7 ~ .The first morphism X a , -+ X A is a finite composition of “blow ups” in (regular) fixed points: Such a blow up of X A in the fixed point x, corresponding to a 2-cone T = cone(b1,bz) E A spanned by a lattice basis, corresponds to subdividing r with the diagonal

e := ray(bl+bz). In the exceptional fibre, the blow up introduces a new chainlink E, P1 of self-intersection -1. (Such a curve that may be “contracted” to a smooth point is called an “exceptional curve of the first kind”). The self-intersection number of each “old” curve Ei passing through z, drops by 1. In particular, if all exceptional curves Ei of the resolution have self-intersection’number -ai 5 -2, then the resolution is minimal. For the minimal resolution A of cr, each multiplicity ai is at least two. Furthermore, in the situation of Equation (29), the numbers ai are the integers occurring in the following “Hirzebruch-Jung continued fraction”

=

- --

k

1

al -

(30)

1 a2 -

1

.a,-1-

1 ap

Proof.

(3) It suffices to verify the description of blow ups for the “affine” fan A generated by the regular 2-cone r := cone(f l , fz). We have to show that the diagonal ray e spanned by f := fl+f2 belongs to every nontrivial regular subdivision A’ of T , or, equivalently, that a regular subdivision A’ of T not containing e as an edge coincides with A, In fact, assuming p 9 A’, then the primitive lattice vector f lies in the interior of a regular cone T’ := cone(b1,bz) & T spanned by a lattice basis. We thus may write f = nlbl +nzbz with integers 721,722 > 0. Since both, bl and b2, are non-zero, non-negative integral linear combinations of f l , fz and the decomposition f = f1+ fz is unique, we necessarily have T’ = cone(b1,bz) = cone(f1, fz) = T . (4) We verify formula (30) by induction on T . For T = 1, we find k = 1 and a1 = mu.In the induction step, we apply the induction hypothesis to the cone CT’ := cone(b1,m,bl-kbz). It has multiplicity m’ = k . We write mu = qm’ - k’ = qk - k’ with 0 I k’ < k and

52

consider the lattice basis ba := bl, bi := qbl - bz. Then a1 = q and m’ k’

1

= a2a3

1

-

1

.ar-l-

1 -

a,

by induction hypothesis. Formula (30) now is an immediate consequence. 0 We describe the blowing up of the origin in local coordinates:

Example 3.5. Let A be the regular fan obtained by subdividing u = cone(f1, f2) with e = ray(w) for w := f l + f 2 . The basic characters e1,eZ yield coordinates (z,y) on X , = Cz. For 01 = cone(f1,v) and a 2 = cone(w,f2), the dual cones are uy = cone(el-ez,e2) and a; = cone(ez-el, e l ) , thus providing coordinates (ui,wi) for Xui E C2. The inclusions ui ~ - - u t then correspond t o the assignments ( ~ i , wH ) ( z , ~ )= ( ~ 1 ~ 1 and ~ ~ (u2,v2) 1 ) ++ (z,Y) = ( 7 ~ 2 , ~ 2 ~ 2 Hence, ). the ui-axes wi = 0 get collapsed t o the origin. The two coordinate charts for X A are glued along C* x C by the transition functions uz = l / u l - thus gluing together the two ui-axes t o the exceptional fibre of the blow-up map X A -+ C2 over the origin, a projective line - and v2 = u1w1 = q / u 2 . Hence, the constant function w1 = 1 is transformed into the rational function 212 = 1/74 with a simple pole at the origin. This vanishing order -1 is the self-intersection number of the exceptional curve.

(11) The general case The basic idea for the toric resolution in higher dimensions is to proceed in two steps: Firstly, the fan is made simplicial, secondly, it is regularized. Both steps rely on the process of “stellar subdivision”. In the following definition, we do not assume that the ray e is an edge of the cone (T; we even allow e not to be included as a subset.

Definition 3.4. (Stellar subdivision) Let trary ray. Then the union of face fans

is a fan subdividing

D,

(T

be a cone and

e, an arbi-

called the stellar subdivision of u with center

Q.

53

If A is a fan and Q an arbitrary ray included in the support of A, then the stellar subdivision of A with center Q is the fan

E,(A)

:=

U E,(a). aEA

Remark 3.9. (1) If the ray Q even is an element of A, then no simplicial cone of A gets subdivided. (2) The resolution of toric surface singularities described in the proof of Lemma 3.1 is obtained by an iterated stellar subdivision. Similarly, the blowing up of regular fixed points in a toric surface discussed in Remark 3.8 (4)is nothing but the stellar subdivision of cone(b1,bz) with respect to ray(bl+ bz). This procedure generalizes from n = 2 to regular cones of arbitrary dimension n 2 3, thus describing the higher-dimensional “blowing up” of regular fixed points. (3) For a reader familiar with the notion of “blowing up an ideal”, we add the following: On the level of toric varieties, a stellar subdivision CQ(o) of a cone u with respect to e := ray(u) corresponds to the blow up of a T-invariant ideal I in c3(Xo): There is a positive integer k such that the affine hyperplane (. . . , u ) = k in Mw intersects each edge of the dual cone in a lattice point. Then I may be chosen as the a” not contained in ideal generated by the characters corresponding to those lattice points.

We now apply the stellar subdivision to achieve the first step of a resolution.

Lemma 3.2. (“Simplicialization”) Every fan admits a simplicial subdivision. Proof. We introduce the following terminology: An edge e of a cone u is said to split u if there is a ‘‘complementary” facet r of u, i.e., such that (T = T e ; a cone is called stout if it has no splitting edges. A cone is simplicial if (and only if) it does not include any stout face. If a ray e is included in a cone of a fan A, then E,(A) \ A obviously does not contain any stout cone. As a consequence, a stellar subdivision with center included in a stout cone lowers the number of such cones. Hence, after finitely many subdivisions with centers in stout cones, one arrives at a simplicial fan.

+

Remark 3.10. For such an iterated stellar subdivision A’ of A , there exist two extreme possibilities for the choice of the centers:

54 (1) We call the subdivision A‘ thin if each center e is an edge in A; in other words, there are no “new” rays. We denote by E the “exceptional locus” of the associated toric morphism X A -, ~ X A , that is, the union of all infinite fibres. Then each irreducible component Ei of E is of the form V ( T ~where ) T; is a minimal new cone of A’, so Ei has codimension at least 2. (2) The subdivision A’ is called fat if each center e is generated by a lattice vector in the relative interior of a stout cone. Then each Ei is of codimension 1.

This follows from the fact that the irreducible components of E are the orbit closures corresponding to cones a E A’ with boundary da included in A.

As an example, we discuss the Segre cone: Example 3.6. For the three-dimensional variety X,, = N(C4; 2124-,2223) of Example 3.1 (2), one can show that there are exactly two different thin simplicial subdivisions A , , A2 of the cone a. The fans Ai actually are regular, and in both cases, the corresponding exceptional set E := 7r-l (z,) is a projective line. - We indicate such a dividing cone ri in the next figure. A fat simplicial subdivision for the pertinent cone a is provided by the stellar subdivision A’ := .E,(a) with center e := cone(f1 + f 2 ) ; the exceptional fiber E is isomorphic to the surface PI x PI. We remark that X a , 4 X,, factors through the two thin simplicial resolutions Ai of the preceding exercise: The fan A‘ consists of all cones which are the intersection of a cone in A1 with a cone in A,. In fact, A’ also is a resolution. Finally we show

Lemma 3.3. (“Regularization”) Every simplicial fan admits a regular subdivision.

Proof. We first remark that a stellar subdivision .E,(A) of a simplicial fan again is simplicial, and that a ray e not included in a regular cone of the fan A provides an inclusion

Areg c c,(A>reg. Obviously a fan A is regular if all its maximal cones are regular. One now successively lowers the multiplicities of maximal cones: We fix such a cone (T of maximal multiplicity m, = m > 1 and a minimal

55

Figure 19. A thin simplicia1 subdivision

face T 5 0 of multiplicity m, > 1. It is of the form T = COne(b1,. . . ,bd) with linearly independent primitive lattice vectors b l , . . . , bd. According to the regularity criterion in Remark 3.1, there is a (w.1.o.g. primitive) lattice vector u E N in the parallelotope spanned by b l , . . ,bd which is not a vertex. Then necessarily v = d a& with rational coefficients ai strictly between 0 and 1. For e := ray(v), we now consider the stellar subdivision Z,(A) and show that the multiplicity of each new maximal cone 8 in E,(A) is strictly less than mu. Such a cone 8 includes a face of the form ? = y Q with a facet y of T ,say y = cone(b1,. . . ,bd-1). Now an easy computation, using the fact that b l , . . . , bd-1 are part of a basis of N , (a7 being a regular fan) shows that m? = admT < m,. Finally there are exact sequences

.

+

o and

- - -

o -+

x/r,

-.+

&/rs -+

Nu/(ru+N,)

N5/(ra+&)

-

-

o

o

of finite abelian groups. Since the third terms are isomorphic, counting elements yields that ma = ad.mu < mu.

56 Since each maximal cone of A that includes r may take the role of C , the fan E,(A) has less maximal cones of multiplicity m than the original 0 fan A.

Remark 3.11. If the simplicialization step only consists of fat stellar subdivisions, the subsequent regularization yields a resolution where the exceptional set

-

E = ..-l(s(x)) X’ is a divisor, as it only has irreducible components of codimension 1. Being a smooth toric variety, X’can be covered by invariant coordinate patches of the type Cd x (C*)n-d. Hence, the invariant subvariety E intersects them in unions of coordinate hyperplanes. Such a divisor E is called a “divisor with normal crossings”. Exercise 5. Consider the singular simplicia1 cone c = cone(f1, f 2 , dimension three.

cj=l of 3

j.fj)

(1) Prove that its boundary subfan ac is regular. (2) Prove that ray(fl+2(fi+f3)) passes through the relative interior of c and that the stellar subdivision with respect to it yields cones of smaller multiplicity.

References

Eventails et varie’te‘s toriques, in: Se‘minaire sur les Sin1. J.-L. BRYLINSKI, gularitks des Surfaces, Lecture Notes in Mathematics 777,Springer-Verlag, 1980. 2. D. Cox, Update on toric geometry, pp. 1-41 in Geometry of Toric Varieties, L. Bonavero, M. Brion (ed.), SBminaires et Congrks 6,SociBtB Math. de France, 2002. 3. G. EWALD,Combinatorial Convexity and Algebraic Geometry, Grad. Texts in Math. 168,Springer-Verlag, 1996. 4. K.-H. FIESELER, L. KAUP, Algebraische Geornetrie - Grundlagen, Berliner Studienreihe zur Mathematik 13,Heldermann-Verlag, 2005. 5. W. FULTON, Introduction to Toric Varieties, Annals of Math. Studies 131, Princeton Univ. Press, 1993. 6. L. KAUP,Vorlesungen iiber torische Van’etaten, Konstanzer Schriften in Mathematik und Informatik 130, 2001. web address: h t t p : / / m . inf .mi-konstanz. de/Schriften. 7. T. ODA,Convex Bodies and Algebraic Geometry, Ergebnisse Math. Grenzgeb. (n.S.) 15,Springer-Verlag,1988. 8. M. OKA, Non-Degenerate Complete Intersection Singularity, ActualitBs Mathematiques, Hermann, 1997.

57

Poincare-Hopf Theorems on Singular Varieties Jean-Paul BRASSELET Directeur de recherche CNRS Institut de Mathkmatiques de Luminy - Marseille - France E-mail: j p b @iml.univ-mrs.fr These notes are based on the course given during the Trieste School on Singularities. Aim of the course was to present in an elementary way the Poincar.6-Hopf Theorems for manifolds and for singular varieties. Keywords: Euler-Poincar.6 characteristic, Betti numbers, index of vector field, obstruction theory, Gauss map, PoincarbHopf theorem, Singular Varieties

1. Introduction

The Euler-Poincar6 characteristic has been the first characteristic class to be introduced. For a triangulated (possibly singular) compact variety X without boundary, it has been defined as

where ni is the number of i-dimensional simplices of the triangulation of X.It is also equal to C(-l)ifipiwhere pi is the i-th Betti number, rank of H i ( X ) . The PoincarBHopf theorem says that if M is a (compact) manifold and v a continuous vector field with a finite number of isolated singularities ak with indices I(v,a k ) , then

This means that the Euler-Poincar6 characteristic is a measure of the obstruction to the construction of a non-zero vector field tangent to M . A similar result has been proved for manifolds with boundary and one has the same formula if the vector field is pointing outward along the boundary (Theorem 4.2). In the case of a singular complex variety X,equipped with a Whitney stratification and embedded in a smooth analytic manifold M, one can consider the union of tangent bundles t o the strata, that is a subspace E

58

of the tangent bundle to M . The space E is not a bundle but it generalizes the notion of tangent bundle in the following sense: A section of E over X is a section v of T M l x such that at each point x E X, then v(x) belongs to the tangent space of the stratum containing x. Such a section is called a stratified vector field over X . To consider E as a substitute for the tangent bundle of X and to use obstruction theory is the M.-H. Schwartz point of view (1965, [Scl]), in the case of analytic varieties. In fact, if one considers a stratified vector field v on X with finitely many singularities ak,one can compute the index I(v,ak) inside the stratum containing ak, that is a manifold. But, in general, the sum I(v,ak) for all singularities of v is not equal to the Euler-PoincarB characteristic of X . We give counterexamples in 55.1. M.-H. Schwartz’s very nice construction of radial extension of vector fields provides a class of vector fields for which one recovers the PoincarB Hopf Theorem in the case of stratified singular varieties. We give the main ideas and data for the construction and the proof of the Theorem (Theorem 5.2). References [Brl], [Br2], [Br3], [BSS] provide generalization of the M.-H. Schwartz’s construction to Chern classes of complex analytic varieties. The author wants to send special thanks to the referee for correction of all english mistakes and patience.

Ck

2. Euler-PoincarB characteristic In this section, the varieties we consider are possibly singular varieties.

2.1. Combinatorial definition History of characteristic classes begins with the discovery of the so-called Euler formula, by Leonhard Euler around 1750 : Let P be a 2-dimensional polyhedron in R3,homeomorphic to the sphere S2, one has

ko - kl + kz

=2

where ko is the number of vertices in P , kl is the number of segments and k2 the number of faces. That is the case for the tetraedron: 4 - 6 4, for the cube (with diagonals on the faces): 8 - 18 12. According to different authors, that formula was first proven by Euler himself, by Legendre in 1794 or by Cauchy. In fact, it seems that this formula was already known by R. Descartes (around 1620) and even by Archimedes.

+

+

59

Simon Antoine-Jean Lhuilier, a Swiss mathematician, gave (in 1812) a slight generalization of Euler 's formula taking into account orientable 2dimensional polyhedra with holes. The number g of holes is called genus. Lhuilier's formula is

ko

-

kl

+ k2 = 2 - 29,

where g is the genus. Thus one obtains 0 for a torus-like polyhedron. For a general 2-dimensional polyhedron P in R3,the alternative sum

x ( P ) = ko - kl

+

k2

is called Euler characteristic of P. H. Poincark [Po21 generalized the result in 1893 for finite polyhedra P of higher dimensions and proved the so-called Poincark-Hopf Theorem, which is the bridge to differential geometry. One defines

Definition 2.1. Let us denote by ki the number of i-dimensional simplices of a finite n-dimensional polyhedron P in R", the Euler-Poincar6 characteristic of the polyhedron P is defined by n

x ( P ) = E(-1)Zki. i=O

An important result due t o Poincark is that for a triangulable compact topological space this number does not depend on the triangulation. More precisely, one defines the following: Definition 2.2. Let P be a finite polyhedron in R". The union of simplices of P is a compact subspace of R", denoted by IPI and called geometric realization of P. A topological space X is said to be triangulable if there exists a polyhedron P and a homeomorphism h : IPI -, X. Such a pair (P,h ) , or simply the polyhedron P , is called a triangulation of X .

Poincar6's result is the following: Theorem 2.1. (Poincare', PO^]) Let (P,h) and (P',h') be two triangulations of the same topological space X . Then one has x ( P ) = x(P').

By this result the following definition makes sense: Definition 2.3. The Euler-Poincare' characteristic of a triangulable space X , denoted by x ( X ) , is defined as x ( P ) for any triangulation P of X .

60

Theorem 2.1 also proves that the Euler-Poincar6 characteristic is a topological invariant of triangulable spaces. The Euler-PoincarB characteristics of the sphere Sn, the 2-dimensional real torus T and the complex projective space CPn are respectively x(Sn) = 1 (-l)n, x ( T ) = 0 and x(cCPn)= n 1.

+

+

2.1.1. Betti numbers In 1871, Betti [Be] defined numbers related to 3-dimensional compact manifolds without boundary and announced a duality property. A form of PoincarB duality was first stated, without proof, by Henri Poincar6 in 1893. It was stated in terms of Betti numbers: The k-th and (n- k)-th Betti numbers of a closed (i.e. compact and without boundary) orientable n-manifold are equal. In his 1895 paper Analysis Situs, Poincar6 tried to prove the theorem using topological intersection theory, which he had invented. In his dissertation thesis of 1898 Poul Heegaard gives a counter-example to the above first version of Poincar6 duality. Poincar6 had overlooked the possibility of the appearance of torsion in the homology groups of a space. This counter-example led PoincarB to realize that his proof had to be modified. In the first two complements to Analysis Situs, Poincar6 gave a new proof in terms of triangulations and dual cell decompositions. Let us denote by P an n-dimensional finite polyedra and by Ci(P) the finitely generated free abelian group whose generators are (oriented) i-simplices of the triangulation P. The boundary operator is classically defined as a complex map ai : Ci(P) -+ Ci-l(P). The subgroups of cycles and boundaries

& ( P ) = Ker[& : Ci(P)-+ Ci-l(P)] & ( P ) = Im[&+l : Ci+l(P)

4

Ci(P)]

are finitely generated, as subgroups of a finitely generated group. The homology groups H i ( P , Z ) = Z i ( P ) / B i ( P )are also finitely generated, as quotient group of a finitely generated one. One can write

Hi(P, Z)= Fi(P) Ti(P) where Fi(P) is the free subgroup and Ti(P) the torsion subgroup of

Hi(P,Z). The Betti numbers of P are defined as

/3i(P)= rk (Hi(P, Z)) = rk (Fi(P)).

61

Equivalently, one can define it as the dimension of the vector space Hi(X;Q). Noting that Pi(P)= 0 if i > d i m P = n, one has the Poincar6 Theorem:

Theorem 2.2. (Poincare‘ Theorem) [Poi?]Let P be a finite polyhedron in Rm, with Betti numbers Pi(P), one has

x(P)= I(-l)iPi(P). i=O

-

Proof. From the long exact sequence

. . . d ci+l(P)

Ci(P) 3Ci-l(P)

...

one deduces the following equalities (where ni = rk (Ci(P)) and n = dim( P)) n

n

x ( - l ) i n i = x(-l)i(dimZi i=O

i=O n

n

= x ( - l ) i ( d i m Z i -dim&)

= x(-l)iPi(P), i=O

i=O

and the Theorem.

J. W. Alexander proved in 1915 that two triangulations (P,h) and (P‘, h’) of the same topological space X have the same Betti numbers Pi(P)= Pi(P’)for every i. One can define Pi(X)as being Pi(P)for any triangulation (P,h) of X and one has n

i=O

This result proves that each Betti number is a topological invariant. In this sense, it is more precise than PoincarQTheorem 2.1, which proves the topological invariance of Euler-PoincarQ characteristic only.

Theorem 2.3. The Betti numbers of a compact orientable manifold satisfy

Pi = Pn-i

for i = 0,1,. . . ,n.

Corollary 2.1. If M is a compact orientable n-manifold with odd n, then X(M)= 0.

62

Example 2.1. If n is odd, the Euler-Poincar6 characteristic of the sphere S", the real projective space RP", a compact hypersurface in IW"+l, are zero.

Proposition 2.1. If X is a n orientable (connected) surface of genus g , then PO = 1, g = and /3z = 1. One has x ( X ) = 2 - 29. If X is a non-orientable surface of genus g, then PO = 1, g = 1 /31 and Pz = 0 . One has x ( X ) = 2 - g .

+

We will later use the following proposition: Proposition 2.2. If X is a deformation retract of Y , then

X ( X >= X W . Proof. It is a direct consequence of the fact that X and Y have the same homology groups. One concludes the result by Theorem 2.2. 0

3. The index of a vector field In this section, one gives different ways to define the index of a vector field at an isolated singular point. The obstruction-theoretical definition will be useful for the following and provides a geometrical meaning to characteristic classes.

3.1. T h e index: index as a degree Firstly, let us recall the definition of the (Brower) degree of a map. Let M be an oriented n-manifold without boundary. The tangent bundle to M denoted by T M is a real vector bundle of rank n whose fiber at a point x of M is the tangent space to M at x denoted by T,(M). It is isomorphic to IWn. Let M and N be two oriented n-manifolds without boundary, M compact and N connected, and let

f:M-+N be a smooth map. Let x E M be a regular point of f, so that df, : T,M -+ Tf(,,N is a linear isomorphism between oriented vector spaces. Define the sign of df, to be +1 or -1 according as df, preserves or reverses orientation. Definition 3.1. Let y defined as

E

N be any regular value of f. The degree of f is

deg(f) =

signdf,. X € f -1b)

(1)

63

This integer does not depend on the choice of the regular value y (see [Mi], $5, Theorem A). 3.1.1. Vector fields in Euclidean space

In a first step, let us consider vector fields in Euclidean space, then we will define the index for vector fields tangent to a manifold. Let R be an open subset in Rn with coordinates ( X I , . . . ,x,). Let n i=l

be a vector field on R. The vector field is said to be continuous, smooth, analytic, according as its components { f1, . . . ,f,} are continuous, smooth, analytic, respectively. A singularity a of w is a point where all of its components vanish, i.e., f i ( a ) = 0 for all i = 1,.. . ,n. The singularity is isolated if at every point x near a there is at least one component of w which is not zero. Let w be a continuous vector field on R with an isolated singularity at a, and let B ( a ) be a small ball in R around a so that there is no other singularity of w within B ( a ) . Let us define the Gauss map

-

y : a B ( a ) = ~ ( aE) s"-~ by

Y(X)

sn-'

= ~(X>/II.(X>II.

Definition 3.2. The (local) index of w at a , denoted by I ( w , a ) , is the degree of the Gauss map y : Sn-l -+ Sn-l. The degree of the Gauss map is geometrically the number of times the cycle y(S(a)) covers P-', i.e. the degree of the map "I* : Hn_1(S,-1)

=z

-

Hn-l(P-l) 2

z.

This index does not depend on either the choice of the small ball B ( a ) or the choice of orientations. If w and w' are two such vector fields with an isolated singularity at a , then their indices at a coincide if and only if they are homotopic through a family of vector fields, defined on a neighbourhood of a. 3.1.2. Vector fields o n a manifold

Let M be a smooth n-manifold. Then one has the following Property:

64

Property 3.1. For every point x on an n-dimensional smooth manifold M , there is a neighbourhood U, of x in M homeomorphic to a ball B" c R" by a homeomorphism $ : B" + U, such that $(O) = x and the boundary of U,, called the link of x, is homeomorphic to the sphere S P 1 . A continuous vector field on the n-dimensional smooth manifold M is a section of its tangent bundle T M . Giving a local chart (Ua,$) on M , a vector field on M is locally expressed as follows:

Definition 3.3. Let us denote by x = (XI,.. . ,x,) the coordinates of Bn in R". For x E B", the vector field v($(x)) can be written in terms of the basis g ( x ) of the tangent vector space T+(,)M

The functions

(f1,.

. . ,fn) are called coordinates of the vector 21.

A singularity a of the vector field v is a point where the vector field is zero or is not defined. In fact, given a vector field on the boundary S (a ) of the image $(En), a sphere centered at a , there are many ways to extend the vector field inside B(a) = q5(Bn). Here are two natural ones: we can consider on each Se(a)= b'$(E;), 0 < E 5 1, either EX) = EV(X) or ~ ( € 2=)v(x). In the first case, the vector field admits an extension by 0 at a, and in the second case the extension is not defined at a (see [SC~]). Whatever the type of singularity is, at such a point Definition 3.2 of the index at the isolated singularity extends in the obvious way. The index does not depend on the local chart. 3.2. The index

-

Definition by obstruction theory

The index can be also defined in the following way: Let M be a differentiable manifold of dimension n. The vector bundle T M is locally trivial, i.e. there is a covering of M by open subsets { U } such that the restriction of T M to each U is homeomorphic to U x R". Let us denote by SO the zero section of T M , we will consider the bundle (not any more a vector bundle) T M o = T M \ so(M). Its fiber in a point x E M is TM: = Rn \ (0). Let us consider a ball B ( a ) centered in a, contained in an open chart U, over which T M is trivial and sufficiently small so that a is the only singular point of v in B ( a ) . One has to think of B ( a ) as an n-cell, in view of the generalization we will make later. The vector field v defines a section

65

of T M without zero over S(a) = aB(a),hence a map S(a) "= sn-l

5T M * I &

"=

v,

x (Rn \ (0))

2Rn \ (0)

(3)

where p7-2 is the second projection. One obtains a map

s d B ( a ) przqvR" \ {O}, hence an element X ( w , a) in 7rT,-1(Rn\ (0)). One knows that this homotopy group is Z.The generator +1 corresponds to the radial unitary vector field S"-l

whose image in R" \{O) is S"-l and for which the map p7-2 o 21 is the identity of sn-l. The previous construction shows that X ( w , a) corresponds to the index I(w,a) by the isomorphism 7rn-1(Rn \ (0)) E Z. By classical homotopy theory, the map gn-1

s aB(a) przov R" \ (0)

extends to a map

B(a)

-

R"

\ (01

if and only if the element X(w,u) is zero in 7rn-1(R" \ {0}), i.e. the vector field w extends inside the ball B ( a ) if and only if the index I(v,a) is zero. This construction is the most elementary example of obstruction theory, but is the basis of it. 3.3. The index

- Nondegenemte singularities

In this section we use (and abuse) the very nice book by Milnor [Mi]. 3.3.1. In Euclidean space

The vector field w on an open subset R c R" is a mapping 0 -+ R", so that the linear transformation, differential map dw, : R" + Rn is well defined. In terms of local coordinates, it is expressed as the matrix

One defines particular singularities of the vector field v,called nondegenerate ones:

Definition 3.4. One says that the vector field is nondegenerate at its isolated singularity a if the differential map dw, is nonsingular.

66

Lemma 3.1. The index of v at a nondegenerate singularity a is either +1 or -1 according as the determinant of dv, is positive or negative. Proof. We may assume that a = 0 and we consider u as a diffeomorphism from a convex neighbourhood U of a into R". If v preserves orientation, the restriction vlu can be deformed smoothly into the identity without introducing new zeros (see [Mi], $6, Lemma 1,2). Hence the index is equal to +l. If w reverses orientation, then u can be deformed into a reflection, hence the index is -1. 0 3.3.2. In a manifold More generally, a vector field v on a manifold M c W" can be considered as a map from M to W", so that the differential map dv, : T,M -+ W" is defined. Let us consider a singularity a of v. In the same way as in the proof of Lemma 3.1, one can show ( [Mi], $6, Lemma 5) that:

Lemma 3.2. The differential dv, is a map from T,M into the subspace T,M c R". If this linear transformation has determinant det(dv,) # 0 then a is an isolated zero of v with index equal to +1 or -1 according as det(dv,) is positive or negative. Proposition 3.1. A n y continuous vectorfield o n a manifold can be approximated (in C1-topology) by a differentiable vector field whose all singularities are nondegenerate. Moreover, the sum of indices of the two vector fields are the same. Proof. Let us consider a vector field on an open set R in Euclidean space Rm with an isolated zero at a. Let us define a (differentiable) function q5 : R -+ [0,1] taking the value 1 inside a small neighbourhood U of a and the value 0 outside a larger neighbourhood V. Let us consider a sufficiently small regular value y of v. The vector field

v ' ( 4 = 4.) - 4 b ) Y is nondegenerate inside U and does not vanish inside V - U . Then the vector field u' is nondegenerate inside V. The sum of the indices at the zeros within V is the degree of the map

v' : dV

-+

sm-l.

It is the same as the index of v at a. On an rn-manifold M , one uses the same proof locally. 0

67

3.4. Relation with the Gauss map In this section, as well, we use (and abuse) Milnor's book [Mi]. Let N be a compact m-manifold with boundary in R". The Gauss map g : d N --+ Sm-'

assigns to each x E d N the outward unit normal vector at x.

Lemma 3.3. (Hopf) If u is a smooth vector field o n N with isolated singularities ai in N \ d N , and u points outward of N along the boundary, then the sum of indices C I(u,ai) equals the degree of the Gauss map f r o m d N to S"-1. Proof. Removing an €-ball around each zero, we obtain a new manifold with boundary. The function V ( x ) = u(x)/IIu(z)IImaps this manifold to gm-1 . The sum of the degrees of V restricted to the various boundary components is zero, because on the complement of the small balls, u has no zero. But V l a ~is homotopic to the Gauss map and the degrees on the other boundary components add up to - C I(u,ai). The minus sign occurs since each small sphere gets the opposite orientation. Therefore

as required.

0

Let us consider a compact, n-manifold M c Rm without boundary. Let N, denote the closed E-neighbourhood of M (i.e. the set of all x E Rm with 1 1 2 - yII < E for some y E M ) . For E sufficiently small, N, is a smooth manifold without boundary.

Theorem 3.1. For any vector field u on M with non degenerate zeros ai, the index sum C I ( u , a i ) is equal to the degree of the Gauss m a p g : dN,

--+

Sm-l.

We reproduce the proof due to Milnor [Mi]. His proof was delivered in his lectures at University of Virginia in December 1963. The procedure used by him is exactly the method simultaneously developed by M.-H. Schwartz [ScO] in her definition of radial extension. Namely, the idea is to extend a vector field u defined on the manifold M with index I(u,a; M ) at the isolated singularity a, to a vector field 20 in the ambient space Rm that has also an isolated singularity at a with the same index I ( w ,a; Rm) = I ( u ,a; M ) . The principle is to add a transversal vector field to the parallel

68

extension of v in a neighbourhood. M.-H. Schwartz, as we will see, developed this idea in the context of stratified singular varieties. Proof. For x E N E ,let ~ ( xbe ) the closest point of M . The vector x - ~ ( 2 ) is orthogonal to the tangent space of M at ~ ( z )for , otherwise ~ ( xwould ) not be the closest point of M . If E is sufficiently small, then the restriction T ( Z ) is smooth and well defined. We also consider the squared distance function =

Ib - +>112

whose gradient vector is grad4 = 2(x - r(x)). On the one hand, the gradient vector field is a vector field that is zero along M , that is transverse to M going outward and that increases with the distance to M . For each point 2 at the level surface aN, = $-'(c2), the outward unit normal vector is given by g ( x ) = grad$/llgrad4ll = .( - + ) ) / E .

On the other hand, at each point x E N , the vector v(r(x)) is a parallel extension of v. Extend v to a vector field w on the neighbourhood N, by setting

w(x)= (x - +))

+

V(T(Z)).

Then w points outward N, along the boundary, since the inner product ~ ( x.)g(x) is equal to E > 0. In fact w can vanish only at the zeros of v in M . That is clear because the two summands (x - ~ ( z ) and ) v ( T ( x ) )are mutually orthogonal. Computing the derivative of w at a zero a E M , we see that

dwa(h)=

{

:(h)

for all h E T,M for h E (7'aM)l.

Thus the determinant of dw, is equal to the determinant of dv,. Hence the index of w at the zero a, computed in Rk,is equal t o the index of v at a , computed in M . Now, according to Lemma 3.3, the index sum C I(v,a) is equal to the degree of g, which proves the theorem. 0 Theorem 3.1 is still true for a vector field with degenerate zeros. That is a consequence of Proposition 3.1.

69

4. Poincar&Hopf Theorem The PoincarB-Hopf Theorem was proved by PoincarB [Pol] in the 2-dimensional case in 1885, and by Hopf [Ho] for higher dimensions in 1927. In between, partial results had been proved by Brouwer and Hadamard. The PoincarB-Hopf Theorem is the first appearance of Euler-PoincarB characteristic in differential topology, out of combinatorial topology. A meaning of Poincard-Hopf Theorem is that obstruction to the construction of a vector field without singularity is evaluated by the EulerPoincarB characteristic. The origin of the PoincarB-Hopf Theorem is the study of differential equations in terms of integral curves of an appropriate vector field. The singular points of the vector field are points of equilibrium in dynamical systems. PoincarB-Hopf Theorem has many applications: mathematical economics, electrical engineering, applied probability, statistical complexity, particle physics, material structure, crystallography, graphics applications, astrophysics, etc... There are many ways to prove PoincarB-Hopf Theorem. They correspond gross0 mod0 to the different definitions of the index.

4.1. The smooth case without boundary Theorem 4.1. [PoincarbHopf Theorem] [Pol] Let M be a compact differentiable manifold, and let v be a continuous vector field on M with finitely many isolated singularities ai. One has

x ( M )=

c

q v , 4.

i

Proof. Let us firstly give a proof in the orientable case: In general, the idea of the proof is the following: In a first step, one shows that the sum of indices of a continuous tangent vector field with isolated singularities does not depend of the choice of the vector field. The second step of the proof consists in describing a particular vector field for which the sum of indices is equal to x ( M ) . For the first step, one can use the Milnor way to prove it: Extend the vector field in a tubular neighbourhood NE of M in R" and one obtains the result noting that the sum of indices is the degree of the Gauss normal map, defined on the boundary of the tubular neighbourhood (see Theorem 3.1 and [Mi], p 38).

70

For the second step, such a vector field is given by the Hopf vector field H of which we recall the construction (see [Ste], p. 202). Let us consider a triangulation K of M and a barycentric subdivision K' of K . The Hopf vector field is tangent to simplices of K', with a singularity at every vertex of K', i.e. at every barycenter of K . On every 1-simplex [6, ?] of K', where 6 and ? are barycenters of IJ and r , and IJ < r , the vector field H is going in the direction from 6 to ?. For example it is pointing outward all vertices of K . The Hopf vector field H has a singularity of index (-l)i at the barycenter of every i-simplex of K . The sum of indices of H at all singularities is C ~ = o ( - l ) i k iwhere ki is the number of i-dimensional simplices of K , so it is equal to x ( M ) . In the non-orientable case, one proceeds in the following way: Let us consider the (connected) oriented double covering IT : M + M . On the one side, if v is a continuous vector field on M with isolated singular points ai of index I(v;ai),then one can define a lifting ij of which is a continuous vector field on M with isolated singular points a!, j = 1 , 2 such that IT(a:) = ai. One has I ( v ; a ! ) = I ( v ; a i ) for j = 1 , 2 . One obtains I(v;a!) = 2 I(v;ai). On the other side, a triangulation of M provides a triangulation of M with double number of simplices, then x ( M ) = 2 x ( M ) . One concludes the Poincarb-Hopf Theorem :

xi

x ( M ) = 1/2 . x ( M ) = 1/2

C I(v;a!) = C I(v;ai).

0

4.1.1. Consequences of Poincard-Hopf Theorem

As an important consequence of the Poincarb-Hopf Theorem, one has the following Corollary 4.1. If x ( M ) # 0, then any continuous vector field tangent to a compact manifold M admits at least a singular point. Conversely, every compact manifold such that x ( M ) = 0 admits a continuous tangent vector field without singularities.

The unitary sphere 9" with odd n satisfies ~ ( 9 "=) 0 and admits continuous tangent vector fields without singularities. Corollary 4.2. Every compact odd dimensional manifold admits a continuous tangent vector field without singularities.

The torus and the Klein bottle are the only compact 2-dimensional surfaces admitting a non-zero continuous tangent vector field.

71 4.2. The smooth case with boundary

If M is now an oriented manifold with boundary, one has a similar theorem: Theorem 4.2. (Poincar6-Hopf Theorem with boundary] Let M a compact submanifold with boundary d M in a n oriented diflerentiable manifold W (possibly Euclidean space). Let v be a non-singular continuous vector field tangent to W , strictly pointing outward (resp. inward) M along the bounda r y d M . Then:

(1) v can be extended to the interior of M as a vector field tangent to M with finitely many isolated singularities ai. (2) The total index ojv in M is independent of the way we extend it to the interior of M . (3) If v is everywhere transverse to the boundary and pointing outward from M, then one has a

If v is everywhere transverse t o d M and pointing inward M , then X(intM) = x(M) - x(8M) = C l ( ~ , a i ) .

(5)

i

Proof. The proof of (1) is by ordinary obstruction theory. Following the “stepwise” process of [Ste] one easily shows that the vector field can be extended without singularities on the (n - 1)-skeleton of a triangulation of M . Then we extend it to the n-simplices. If necessary, we introduce a singular point ai for each n-simplex. Statement (2) is a consequence of the following proof of Statement (3). Statement (3) can be proved in the following way: Again, let us denote by N , the closed E-neighbourhood of M . If the vector field is pointing outward along d M , then it can be extended over the neighbourhood N, so that the extended one points outward along dN,. The extension w is defined as before by w(z) = (z - r ( x ) ) v ( r ( z ) )and is a continuous vector field near d M . In this case, N , is not necessarily of class C”, but only a C1-manifold. Nevertheless, the same argument as in the case “without boundary” can be carried out (see [Mi] 56). If the vector field is pointing inward along d M , one can extend v inside M with finitely many isolated singularities ai of index I(v,ai). One proceeds to the following construction: the boundary d M admits a neighbourhood d M x [0,1] in M and one can extend this neighbourhood

+

72

as d M x [0,2]. Let us call M’ the new manifold MU ( d M x [0,2]).One has d M . Let us call C the collar dA4 x [1,2]. One x ( M ‘ ) = x ( M ) and dM’

has x ( C ) = x ( d M ) . At the level C1 = d M x {l}, one has the vector field v pointing inward M and outward C. At the level Cz = d M x {2}, one considers any vector field v‘ pointing outward M’ along dM’. Let us call w the vector field defined on dC which is equal to v and v’ on C1 and Cz respectively. The vector field w is defined on the boundary of C and pointing outward C along the boundary. By (4)on C , one can extend w inside C with finitely many isolated singularities b j and one has

x ( C ) = x ( a M )=

c

I ( w , bj).

j

On M‘ one consider the vector field v‘, which is v on M and w on C. It has isolated singularities ai and bj and it is pointing outward M‘. Again one can apply (4) (on M’) and one has

and the result.

0

5. PoincarB-Hopf Theorem: The singular case 5.1. Poincark-Hopf Theorem fails in general

A singular variety is a variety which contains points (the singular points) for which Property 3.1 is not satisfied. Examples of singular varieties are the following: The pinched torus : the pinched point a does not admit any neighbourhood satisfying Property 3.1. In this case, the link of an “elementary neighbourhood” of a is the disjoint union of two not connected circles. Another example is provided by the suspension of the torus. The two pinched points a and b of the suspension of the torus are singular points, in which case, the link of a (or b) is a torus, not a sphere. If X is a singular variety, the Poincark-Hopf Theorem fails to be true. The main reason is that there is no longer tangent space at each singularity. The definition of the index of a vector field in one of its singular points takes sense on a smooth m-manifold only. In particular the singular point must have a neighbourhood isomorphic to the ball Bm and whose boundary is isomorphic to the sphere Sm-l. Let us consider the example of the pinched torus X in R3.The pinched point a is a singular point of X , in fact it

73

constitutes the singular part of the pinched torus. The only ‘natural’ way to define an index of a vector field at the point a is to consider a vector field defined in a ball B 3 ( a )centered in a, in R3, with an isolated singularity at a , such that if z E X \ { a } , then v(z) is tangent t o the smooth manifold X \ { a } and such that v does not have other singularities in B 3 ( a ) . Let us consider two examples of such vector fields: a) The vector field tangent to the parallels of the torus T determines, on the pinched torus X , a vector field v pointing inward the ball B 3 ( a )along one of the two unlinked circles, which are the intersection of aB3(a)and X , and pointing outward the ball along the other circle. On the one hand, this vector field, defined on aB3(a)nX,is the restriction of a vector field w defined on 6’B3(a)with index 0 at a. On the other hand, there is no more singularity of v on X \ { a } . In this case, the PoincarBHopf Theorem is not satisfied : one has

x ( X ) = 1 # 0 = I(w,a). b) Let us consider a vector field v pointing outward the ball B 3 ( a )along aB3(a) and tangent to X along the restriction aB3(a)n X . This vector field has index +1 at a. It is orthogonal to the two meridians aB3(u)n X and it can be extended on the pinched torus as a continuous vector field without other singularity. In fact, one can define an extension v such that, on each meridian, the angle of v(x) with the tangent space to the meridian is constant and this angle goes down continuously as the distance to a grows till being 0 for the meridian opposed to a. In this case, the PoincarBHopf Theorem is valid:

x ( X )= 1

= I(v,a).

The vector field is the first example of M.-H. Schwartz’s radial vector field, of which we will make a systematic study. 5.2. Whitney stratifications

As we have seen, on a singular variety, there is no longer tangent space in the singular points. One way to find a substitute for the tangent bundle is to stratify the singular variety in submanifolds. One can proceed the following construction: If X is a singular analytic variety, equipped with a stratification and embedded in a smooth analytic manifold M , one can consider the union of tangent bundles to the strata, that is a subspace E of the tangent bundle to M . The space E is not a bundle but it generalizes the notion of tangent bundle in the following sense: A section of E over

74

X is a section w of T M ( x such that at each point IC E X the vector ~ ( I c ) belongs to the tangent space of the stratum containing x. Such a section is called a stratified vector field over X. To consider E as the substitute for the tangent bundle of X and to use obstruction theory is the M.-H. Schwartz point of view (1965, [Scl]) in the case of analytic varieties. When one consider stratification of singular varieties, it is natural to ask for conditions with which the strata glue together. The sc+called Whitney conditions are those which allow one to proceed to the construction of radial extension of vector fields. According to a result of Whitney, every analytic variety can be equipped with a Whitney stratification. Definition 5.1. One says that the Whitney conditions (a) and (b) are satisfied for the stratification {V,} of X if, for any pair of strata (Va,Vp) such that V, is in the closure of Vp, one has: (a) if (2,) is a sequence of points in Vp with limit y E V, and if the sequence of tangent spaces Txn(Vp) admits a limit T (in the suitable Grassmanian space) when n goes to +m, then Ty(Va) is included in T . (b) if (x,) is a sequence of points in Vp with limit y E V, and if (9,) is a sequence of points in V, with limit y, such that the sequence of tangent spaces Txn(Vp) admits a limit T for n going to +m and such that the sequence of directions admits a limit X when n goes to +m, then X lies in T. 5.3. Radial extension process

- the

local case

One gives a description of the local radial extension process. This will be used for the global process in the next section. The local radial extension, defined by M.-H. Schwartz, is similar to the one defined by Milnor in order to prove Theorem 3.1. Let X be a singular variety embedded in an m-dimensional manifold M . Let V, c X be a stratum, B(a) c V, a neighbourhood of the point a in V, and w a vector field defined on B ( a ) with an isolated singularity at a. One can construct two vector fields: 1. Parallel Extension. We will denote by N ( a ) a tubular neighbourhood homeomorphic to B ( a ) x IDk, where IDk is a disk transverse to V, and k = m - dimV,. Let us consider the parallel extension 6 of w in the tube N ( a ) . Let Vp be a stratum such that a E At a point x E N ( a ) the parallel extension $(.) is not necessarily tangent to V,. However, the Whitney condition (a) guarantees that if N ( a ) is sufficiently small, then the angle between T,(V,) and T,(Vp) is small. That implies that the orthogonal pro-

G.

75

jection of +(z) onto T,(Vp) does not vanish. Of course, considering for each stratum the projection of the parallel extension on the tangent space to the stratum at the given point does not provide a continuous vector field. In order to obtain a continuous vector field, one has to consider a slight modification of the construction in the neighbourhood of the strata, which is easy to understand, but complicated to describe in details. A good extension will be +(x) away from Vp and continuously going to the projection of 6(z) onto T,(Vp) when approaching Vp, using a suitable partition of unity. The construction of such an extension is correctly and entirely described in M.-H. Schwartz’s book [SC~]. In fact, one has to work simultaneously for all strata Vo such that a E G,that complicates a detailed construction. In conclusion, the Whitney condition (a) implies that one can proceed to the construction of a stratified vector field, still denoted by S(z), which is a “parallel extension” of the given vector field on V,, in a suitably small tubular neighbourhood around B ( a ) in M . One observes that the singular locus of 6 corresponds to an (m-dim V,)dimensional disk which is transversal to B ( a ) in M . 2. Transversal vector field. Let us consider the transversal vector field g(x), as in the proof of Theorem 3.1. This vector field is essentially the gradient of the function “square of the distance to V,”, for an appropriate Riemannian metric. The vector field g(z) is not necessarily tangent to the strata Vp such that a E However, the Whitney condition (b) guarantees that in a sufficiently small “tube” around B ( a ) ,the angle between g(z) and T,(Vp) is small. That means that the orthogonal projection of g(z) onto T,(Vp) does not vanish. In the same way as for the parallel extension, for each stratum one could consider the projection of g(z) onto the tangent space to the stratum at the given point. However, this does not provide a continuous vector field. In order to obtain a continuous vector field, one has to consider a similar modification of the construction. The correct vector field is g(z) away from Vp and continuously going to the projection of g(z) onto T,(Vp) when approaching V,. That construction is also completely described in M.-H. Schwartz’s book [ S C ~ and ] , one has t o work simultaneously for all strata Vp such that a E In the boundary of the tube N ( a ) 2 B ( a ) x D k the part B ( a ) x dDk = B ( a ) x Sk-l shall be called the horizontal part. In conclusion, one obtains a stratified “transversal” vector field, still denoted by g(x), which is zero along V, and growing with the distance to V, and which is pointing outward the horizontal part of the boundary of the tube N ( a ) provided that the tube is sufficiently small.

F.

G.

76

Definition 5.2. The radial extension of the vector field v defined on B ( a ) c V, is the vector field 6 defined on the tube N ( a ) as the sum G(2)=

q . ) +g(2).

Proposition 5.1. The radial extension 6 of the vector field v is transversal to the boundaries of the tube N ( a ) around B ( a ) , pointing outward the horizontal part of a N ( a ) . Its unique singularity inside N ( a ) is a, i.e. the same singularity as the initial vector field v. Moreover, the index of ij in a, computed in the tube N ( a ) , is the same as the index of v at a, computed in the manifold V,, namely we have I ( 6 , a; M ) = I(., a; V,).

This property, i.e., the above equality, is the main property of the radial extension, that is precisely the property which allows one to prove the Poincark-Hopf Theorem for singular varieties. 5.4. Poincard-Hopf Theorem for singular varieties

In this section one proceeds to the construction of a “global” radial extension of a vector field, which M.-H. Schwartz called radial vector field and one shows the following:

Theorem 5.1. Let X c M be a (compact) singular variety embedded in a manafold M . One can construct o n X a (stratified) radial vector field, in the sense of M.-H. Schwartz. That is a vector field v defined in a tubular neighbourhood N , ( X ) of X in M , pointing outward N E ( X )along the boundary. I t has finitely many isolated singularities ai in N E ( X ) ,all situated in X , and one has I(v7 ai; M)= I ( V I V ~1 ai; ( ~ Va(i)), )

where V,(i)is the stratum of X containing ai. Proof. The “global” construction of the radial vector field is as follows: One consider a Whitney stratification on X as before and one adds the stratum M \ X in order to obtain a Whitney stratification of M . The aim of the process is to construct, by increasing induction on the dimension of the strata of X , a stratified vector field v in a neighbourhood of X in M , with finitely many isolated singularities ai in the strata V,, such that if ai E Val the index of v at ai is the same, computed in V, or in M.

77

By the induction process on the dimension of the strata, we will show the following:

(P) For each stratum V,, there is a neighbourhood N," around 7, and a stratified vector field w defined on N,", pointing outward N," along the boundary, with isolated singularities ai in and such that if Vp is the stratum containing ai, then the index of ai computed in Vp is the same

v,

as the index ofai computed in N,", i.e. in M . Namely, we will denote

I(w,ai) = I(v,a i ; Vp) = I(v,ai; M ) . The neighbourhood N," is the set of points in M with distance less than E from points in That is not a fibre bundle over but the following construction shows that for each stratum Vp c there is a neighbourhood Ap of v p \ Vp in Vp such that the restriction of N," to Ap is a fibre bundle with fibre a disk whose dimension is the codimension of Vp in M . Let us show that induction property (P) is true for the lowest dimensional stratum. If the lowest dimensional strata in X are 0-dimensional ones, i.e. Vo is a set of finitely many points ak, then one considers a radial vector field w in a ball B E ( a k )centered at each of these points. According to the Bertini-Sard Lemma, the boundary aB,(ak) is transverse to the strata V, containing a k in their closure (one takes for E the smallest of E for all a k ) . For each point z E V, n aB,(ak), the radial vector field w(z)is not orthogonal to T,(V,). One can deform the radial vector field to a stratified vector field w pointing outward B,(ak) along aB,(ak). In this case, the index I ( w , a k ) is +1 and obviously one has

v,.

v,, v,,

k

In this case, N," is the union of B E ( a k ) . If the lowest dimensional stratum is a stratum V, of dimension s > 0, then one constructs, by classical obstruction theory, a vector field w on V, with finitely many isolated singularities ai. We notice that V, is a manifold without boundary and it has to be compact if X is compact. According to the classical PoincarbHopf Theorem 4.1, one has i

The extension process in a neighbourhood N," of V,, described in the proof of Theorem 3.1, can be slightly deformed according to the local case (cf

78

Proposition 5.1) in order to obtain a stratified vector field defined on N,", pointing outward N," along its boundary and such that the index of v at each singularity ai is the same, whether it is computed in V, or in N,", i.e. in M . Let us now suppose that induction property (P) holds for all strata up to V, (i.e. for all strata whose dimension is less than or equal to dimV,) and let us call V, the following one. One has to show that (P) holds for V,. The vector field v is defined on U, = V, n N,* and is pointing inward V, along XJ,= \ U,. By classical obstruction theory, one can extend v inside V,, as a vector field still denoted by v , with finitely many isolated singularities aj in V, \ U,. For t ~ ] 0 , 1 ]let , us denote by Nt", the (open) neighbourhood of which is the set of points whose distance to is less than t E . The vector field v defined on V, \ N Z 2 can be extended, using the local extension process, as a vector field v' defined in a tube N,!,around V, \ N Z 2 without other singularities than the points aj and such that

u,

v,

I(v,a j ; V,)

v,,

= I ( v , a j ; NA).

Let us notice that N,!,is a disk bundle with the basis V, \ N."/2and the fiber a disk whose dimension is the codimension of V, in Ad. On the intersection N," n N,!,, one has two vector fields: v defined on N," and v' defined on N,!,.They coincide on V,. The vector field defined by w = ( 2 - 2t)v (at - 1)v' on each N,",, for t E [1/2,1], coincides with Y on aN$, and with v' on aN,!,. The vector field defined as v on N Z 2 , v' on N,!,\ N," and w on N," n N,!, satisfies the property (P) for V, with the neighbourhood N,' defined as

+

N,"12U N,!,. At the last step, one denotes by N E ( X )the neighbourhood of X constructed by the induction process. By construction, the stratified radial 0 vector field v satisfies the statement of the theorem.

Theorem 5.2. (PoincarbHopf Theorem f o r singular varieties) Let X c M be a (compact) singular variety embedded in a manifold M , and v be a stratified radial vector field defined in the neighbourhood N E ( X )of X (Theorem 5.1). Then one has

79

Proof. Using the stratified radial vector field constructed in the proof of Theorem 5.1 and Theorem 4.2, one has i

Noticing that X is a deformation retract of its neighbourhood N E ( X ) one , concludes the theorem by Proposition 2.2. Here we remark that at each stage of the proof of Theorem 5.1 one has

the first summand being for all strata Vp in

va,including V,.

0

References Be. Betti, Sopra gli spazi di un numero qualcunque di dimensioni, Ann. Math. Pura Appl. 2 (4), (1871) 140-158. Brl. J.P. Brasselet, T h e Schwartz Classes of complex analytic singular Varieties, to appear in the Proceedings of the Singularity Conference, Luminy, February 2005, World Scientific. Br2. J.P. Brasselet, Characteristic classes, book in preparation. Br3. J.P. Brasselet, Characteristic classes and Singular Varieties, Vietnam Journal of Mathematics, 33 (2005) 1-16. BS. J.P. Brasselet, M.-H. Schwartz, Sur les classes de Chern d’un ensemble analytique complexe, Asterisque 82-83 (1981), 93-147. BSS. J.P. Brasselet, J. Seade, T. Suwa, Indices of Vector fields and characteristic Classes of singular Varieties, book in preparation. Ch. S.S. Chern, Characteristic classes of hermitian manifold, Ann. Math. 47 (1946), 85-121. Ho. H. Hopf, Vektorfelden in n-dimensionalen Mannigfaltikeiten, Math. Annalen 96 (1927), 225-250. Mi. J. Milnor, Topology f r o m the Differentiable Viewpoint , Univ. Press of Virginia, Charlot tesville, 1965. Pol. H. Poincarb, Mimoire sur les courbes difinies par une iquation diffirentielle, Jour. Math. Pures et Appl. (4)1, (1885), 167-244. P02. H. PoincarB, Sur la gkniralisation d’un thiordme d’Euler relatif aux poly&dres, CRAS vol. 117 (1893) 144. ScO. M.-HSchwartz, Classes obstructrices d’un sous-ensemble analytique complexe d’une v a r i i t i lisse. Lille 1964, second version in Publ. de l’U.F.R. de Mathematiques de Lille, 11, 1986. Scl. M.-H.Schwartz, Classes caractkristiques difinies par une stratification d’une v a r i i t i analytique complexe, CRAS 260, (1965), 3262-3264 et 3535-3537. Sc2. M.-H. Schwartz, Champs radiaux sur une stratification analytique, Tkavaux en cours, 39 (1991), Hermann, Paris. Sc3. M.-H. Schwartz, Classes obstructrices des ensembles analytiques 2001.

80

Ste. N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press (1951). Wh. H. Whitney, Tangents to a n analytic variety, Ann of Math 81, 496 - 549 (1965).

81

Notes on Real and Complex Analytic and Semianalytic Singularities David B. Massey and Lb DBng Tr8ng

Department of Mathematics, Northeastern University, Boston, M A 02115, USA *E-mail: [email protected] www.massey.math.neu. edu Mathematics Section, Abdus Salam ICTP, IPrieste, I-34014, Italy 'E-mail: [email protected]

1. Manifolds and Local, Ambient, Topological-type

We assume that the reader is familiar with the notion of a manifold, but we want to clarify the different types of manifolds that one can discuss. Throughout these notes, a topological manifold is a non-empty, Hausdorff, second-countable, topological space M such that, for all x E M , there exists an open neighborhood U, of x in M , a natural number n,, an open subset V , of EX"., and a homeomorphism h, : U, V,. Such an h, (or, sometimes, U, itself) is called a coordinate chart (or patch) around x. The number n, is uniquely determined and is called the dimension of M at x. If n, is independent of the point x E M , we say that M is pure-dimensional;if we call the common value n, then we say that M is (purely) n-dimensional or that M is an n-manifold, and write d i m M = n. If M is a connected manifold, then M is pure-dimensional. Suppose that M is a topological manifold. If we have two coordinate charts h, : U, -+ U, and h, : U, -+ V , such that U, n U, # 0, then we have the transition function t,, : h,(U, nu,) -+ h,(U, nu,) given by t x y ( v ):= h,(h;'(v)). Note that t,, = t;;. A collection of coordinate charts such that the domains of the coordinate charts cover M and such that all of the transition functions are continuously differentiable (or, smooth) is called a smooth atlas on M . Two smooth atlases on M are equivalent if and only if -+

82

their union is again a smooth atlas on M . A smooth manifold is a topological manifold together with an equivalence class of smooth atlases on M ; such an equivalence class is referred to as a differentiablestructure on M . Given a smooth atlas A on M , there is a maximal smooth atlas on M which is equivalent t o A; one takes the union of all of the smooth atlases equivalent to A. Many authors define a smooth manifold to be a topological manifold M together with a maximal smooth atlas on M , and refer to the maximal smooth atlas as the differentiable structure. In practice, one usually writes something along the lines of “let M be a smooth manifold with smooth atlas d” to mean that A is a representative of the differentiable structure on M . One has various notions of smooth manifolds, which correspond to exactly what one means by “continuously differentiable”. If the transition functions are all C‘ (continuously r times differentiable), then we have a C’ manifold. A C1manifold would be the weakest notion that one could use for a “smooth manifold”. The case of topological manifolds is the Co case. We also have C“ (infinitely differentiable) manifolds, and C” (real analytic) manifolds. (Recall, that analytic, over the real or complex numbers, means a function which can be represented by a convergent power series in a neighborhood of each point.) We refer to all of these cases by simply writing C‘, and allowing r to have the values 0 , 1 , 2 , 3 ,. . . ,00, w . Note that in all of these cases, since t,, = t;;, all of the transition functions are invertible and the inverses are required to be equally as “smooth”, i.e., the transition functions are homeomorphisms (in the Co case), C‘ diffeomorphisms, C” diffeomorphisms, or real analytic isomorphisms; we refer to all of these as C’ isomorphisms. One obtains the notion of a complex (analytic) manifold by replacing the open sets V , 2 Rnx in the discussion above by open subsets V , C Px, and using an atlas for which the transition functions are complex analytic, i.e., holomorphic. The number n, here is usually referred to as simply the dimension of M at 2,since it is usually obvious that we mean the dimension over the complex numbers. However, occasionally, it is necessary to explicitly use the terminology complex dimension for this n, to distinguish it from the real dimension of the complex manifold, which would be 2n,. Once one has the notions of C ‘ and complex analytic manifolds, it is not difficult to define the analogous morphisms between two manifolds of the same type; that is, to define C‘ functions between two C ‘ manifolds and complex analytic functions between two complex manifolds. Suppose that M and N are C‘ (rap., complex analytic) manifolds, with smooth (resp.,

83

complex analytic) atlases d M and d N , and that W is an open subset of N . We say that a function g : W + R (resp., -+ @) is a C' (resp., complex analytic) function o n W if and only if, for all h : U -+ V in dN, the function g o h-l : V n h(U n W )-+ R (resp., -+ @) is C' (resp., complex analytic). The reader should verify that this definition is independent of the representative of the differentiable structure on N . A continuous function f : M -+ N is a C', (resp., complex analytic) function if and only if, for all open subsets W C N, for all C' (resp., complex analytic) g : W -+ R (resp., -+ C), the function go f : f-l(W) -+ R (resp., -+ C ) is C' (resp., complex analytic). It is an easy exercise to verify that compositions of C' (resp., complex analytic) functions between C' (resp., complex analytic) manifolds are again C' (resp., complex analytic) functions. Equivalently, a C' (resp., complex analytic) function f : M -+ N is a continuous function such that, for all h : U V in dN and k : U' -+ V' in d M , the function h o f o k-' : V' n k(U' n f-l(U)) 4 V is C ' (resp., complex analytic). The reader is referred to for a complete treatment. When we say that we have a C' or complex analytic function between two manifolds, we mean that the two manifolds are also of the corresponding type (i.e., in the appropriate category). --f

Suppose that N is a smooth n-dimensional manifold. A subset M C N is a C' submanifold of N if and only if, for all x E M, there exists an open neighborhood U of x in N , an open neighborhood V of 0 in Rn, a natural number m,, and a C' isomorphism f : U -+ V such that f ( x ) = 0,and f (Un M ) = V n (Rmz x (0)).A C' submanifold inherits a C' atlas from the manifold in which it sits, and so C' submanifolds are C' manifolds with a prescribed atlas. When discussing submanifolds, the space in which the submanifold sits is frequently referred to as the ambient space. Note that the m, here is the dimension of the manifold M at x . If M is a pure-dimensional submanifold of the pure-dimensional manifold N , then the codimension of M i n N isdimN-dimM. We shall also be interested in complex submanifolds of a complex ndimensional manifold N . To obtain the definition of a complex submanifold, one simply takes the definition above, replaces R with @, and replaces "C' isomorphism" with complex analytic isomorphism. Note that when one is dealing with purely topological questions, there is no need to distinguish between the real and complex cases, since @" is homeomorphic to R2". In general, we are interested in the topology of how various subsets X

84

are embedded in R". If x E X G R", then the local, ambient, topological type of X in R" at x is determined by the homeomorphism-type of triples of the form ( W ,W n X ,x ) , where W is an open neighborhood of x in B".This means that if x E X G Rn and y E Y C R", then we would say that the local, ambient, topological-type of X at x is the same as the local, ambient, topological-type of Y at y if and only if there exist open neighborhoods W , and W , in Bn around x and y , respectively, and a homeomorphism g : W , 4 W , such that g(Wxn X) = W , n Y and g ( x ) = y . For a topological submanifold of R", there is no interesting local, ambient topology - the local, ambient topological-type at each point x is prescribed to be the same as that of Rm= x ( 0 ) inside Rn at the origin; we refer to such topological-types as Euclidean or trivial. 2. The Geometry of the Implicit Function Theorem

We all encountered l-manifolds in high school. We studied lines, parabolas, circles, ellipses, and hyperbolas. We encountered 2-manifolds in multivariable Calculus, where we saw spheres, ellipsoids, paraboloids, and hyperboloids. Actually, all of these examples are examples of smooth submanifolds of Euclidean space. One may wonder why equations with such forms as y = x2,x 2 y2 = 4, x y = 1, and 3x2 2y2 5z2 = 1 should define smooth (actually, real analytic) submanifolds of R2 or R3.In fact this follows from the Implicit Function Theorem (see, for instance, 2 , Theorem 9.28 and 3 , p. 19), which we state in a geometric form.

+

+

+

Theorem 2.1. (Implicit Function Theorem: geometric form) Let r 2 1 and let f be a C' function f r o m a n ( m c)-dimensional C' manifold N to a c-dimensional C' manifold P . Suppose that the rank of the derivative, d, f , of f at a point x E N is equal to c, i.e., d, f is surjective. Then, there exist open subsets U 5 N , V 5 P , V' Rc,and W' C Rm such that x E U, f(U)= V , and such that there exist C' isomorphisms 4 : W' x V' 4 U and 1c, : V + V' such that 1c, o flu o 4 : W' x V' + V' is equal to the projection onto V'. I n addition, the complex analytic version of the above statement is also true.

+

The following corollary is immediate.

Corollary 2.2. Let r 2 1 and let f be a C' function from a n ( m + c)dimensional C' manifold N to a c-dimensional C ' manifold P . Suppose

85

that d, f is surjective. Then, there exists a n open neighborhood U of x in N such that: (1) f l u : U -+ P is a n open map; (2) f o r all y E U , d, f is surjective; and (3) f o r every C' submanifold Q of codimension c' in (the open set) f ( U ) , U n f - ' ( Q ) is a C' submanifold of U of codimension c'; in particular, U n f - l ( f (x)) is a C ' submanifold of U of codimension c.

In addition, the complex analytic versions of the above statemen.ts are also true. Example 2.3. Let us see how the Implicit Function Theorem or, rather, its corollary implies that the set, M , of points in W2 which satisfy z2 y2 = 4 form a 1-dimensional C" submanifold of W2. Consider the function f : W2 -+ W1 given by f(z,y ) = x 2 y2 - 4. Then, f is a real analytic function, and the rank of d(a,b)f will be equal t o 1 precisely if one of the partial derivatives of f at ( a , b) is non-zero. As the partial derivatives o f f are 2 x and 2y, we see that d(a,b)f has rank equal to 1 everywhere except at (0,O). However, (0,O) is certainly not in M . Thus, Corollary 2.2 tells us that, for every point ( a ,b) E M , there exists an open neighborhood U' of ( a ,b) in EX2 such that U n f - l ( f ( a ,b ) ) = U n M is a C" 1-submanifold of R2. Therefore, A4 is a C" 1-submanifold of EX2.

+ +

For the remainder of these notes, we let f : N + P be a map between pure-dimensional manifolds, where f is a C' or complex analytic function, r 2 1, and dim N 2 dim P. Corollary 2.2 tells us that if z E N is such that d, f is surjective, then the local, ambient, topological-type of f -'( f (x))at x is Euclidean. Therefore, we can obtain non-Euclidean - that is, interesting - local topological-types in f -l( f (z)) only at points where the derivative is not surjective. Hence, we make some definitions.

Definition 2.4. A point x E N such that d,f is surjective is called a regular point of f . A point z E N at which d, f is not surjective is called a critical point of f . The set of critical points of f is denoted C f . The set of critical values of f is f ( C f ) . The set of regular values of f is P - f (Cf ) . If Cf = 0, then f is called a submersion.

86

Note that Item 2 of Corollary 2.2 implies that Cf is a closed subset of

N. Using the terminology of Definition 2.4, and arguing as in Example 2.3, we immediately obtain the following corollary to the Implicit Function Theorem.

Corollary 2.5. Suppose v E f ( N ) is a regular value o f f . Then, f-'(v) is a C' (or complex analytic) submanafold of N of codimension equal to dim P . Example 2.6. If x E Cf,then the local topology of f-l(f(x)) can certainly be non-trivial. Consider the example from high school: f : R2 4 R given by f ( x , y ) = z y , where we are interested in the local topology of f-l(f(0)) = f-l(O) at 0 . The set f - l ( O ) consists of the union of the zand y- axes and, hence, is not even a topological manifold - much less, a topological submanifold of IR2 - in a neighborhood of the origin. As we are interested in points where spaces fail to be manifolds or submanifolds, as was the case with the origin above, we make a definition. Definition 2.7. Suppose that X is a topological space, and x E X. Then, we say that x is a singular point of X or is a singularity of X if and only if there is no neighborhood of x which is a topological manifold. Suppose that X is a subspace of a C', or complex, manifold N , and x E X. Then, we say that x is a C' singular point of X in N or is a C' singularity of X in N if and only if there is no neighborhood, U , of x in N such that U n X is a C', or complex, submanifold of U . Example 2.8. Consider the set of points X in R2 which satisfy y2 = x3. This is called a cusp.

I

/

87

The origin is a C1singular point of X in R2, for there is no smooth way to flatten out the sharp point at the origin. On the other hand, the origin is not a topological (i.e., Co)singular point of X in R2. We leave the verification of this as an exercise for the reader. Note that “singular point” is a type of point associated to a topological space, while a “critical point” is a type of point associated to a function. Of course, the Implicit Function Theorem and its corollaries tell us that these two notions are closely related. We shall return to examples of critical points and singular points throughout these notes. Henceforth, for simplicity, we will restrict our attention to three cases: the C“ case, which we will refer to as the smooth case, and the real and complex analytic cases. Thus, we will always assume, from now on, that f : N + P is at least smooth. 3. The Theorem of Ehresmann and Integrating Vector

Fields The theorem of Ehresmann is a theorem which describes a nice topological property of proper submersions. Recall that a continuous function g : X -+ Y between topological spaces is called proper provided that for every compact subset C of Y , g-’(C) is compact. Also recall that a topological space 2 is compactly generated if it satisfies the following condition: a set A is open in 2 if and only if, for all compact subsets C of 2, AnC is open in C. See 5 , section 7-4. If 2 is locally compact or first countable, then 2 is compactly generated; in particular, manifolds are compactly generated. It is an easy exercise to show that if g :X Y is proper, and Y is Hausdorff and compactly-generated, then g is a closed map. On the other hand, Item 1 of the Corollary 2.2 tells us that submersions are open maps. Therefore, we have: --$

Proposition 3.1. I f f : N -+ P is a proper submersion, and P is connected, then f is a surjection. Ehresmann’s Theorem refers to smooth, locally trivial, fibrations. We need to define this concept.

Definition 3.2. A smooth function g : M -+ Q between two manifolds is a smooth, trivial fibration if and only if there exists a smooth manifold

88

F such that g : M + Q is diffeomorphic to the projection T : Q x F + Q, i.e., there exists a diffeomorphism $ : M --t Q x F such that g = T o $. Note that, if g is a smooth trivial fibration, then each fiber g-’(q), for q E Q, is diffeomorphic t o F . The manifold F (or, actually, its diffeomorphismtype) is referred to as the fiber of the trivial fibration. A smooth function g : M -+ Q between two manifolds is a smooth, locally trivial fibration if and only if, for all q E Q, there exists an open neighborhood U of q in Q such that g l g - l ( v ) : g-l(V) + U is a trivial fibration. It is easy to show that if g : M Q is a smooth, locally trivial fibration, and Q is connected, then the diffeomorphism-type of each of the fibers g - l ( q ) is independent of q; this common diffeomorphism-type is referred to as the fiber of the locally trivial fibration. --f

It is important to note that the “locally” in the term “locally trivial fibration” refers to local in the base space Q, not local in the space M . Later, in Theorem 7.17, we shall need the notion of a (topological) locally trivial fibration; one obtains this notion exactly as above, replacing “smooth by %ontinuous” and “diffeomorphism” by LLhomeomorphism’l. Before we state the theorem below, we should point out to the reader that two smooth manifolds are smoothly homotopy-equivalent if and only if they are (topologically) homotopy-equivalent. See 6, p. 36. In particular, there is no difference between a contractible smooth manifold and a smoothly contractible smooth manifold. The following theorem is well-known; see, for instance,

7 , 11.6.

Theorem 3.3. Iff : N -+ P is a smooth, locally trivial fibration and P is contractible, then f is a smooth trivial fibration. Now we state the theorem of Ehresmann,

Theorem 3.4. Iff : N locally trivial fibration.

t

‘.

P is a proper submersion, then it is a smooth,

One might hope that if f is assumed to be real or complex analytic, then a generalization of Ehresmann’s Theorem would imply that the local trivializations could be made real or complex analytic; this is not the case, even though the Implicit Function Theorem tells us that, locally in N , we obtain such analytic trivializations. The problem is that one cannot analytically “patch together” the local trivializations in N to obtain an analytic

89 trivialization over open subsets of P. We wish to say more about this, and discuss some important aspects of the proof of Ehresmann's Theorem. The idea in the proof of Theorem 3.4 is fairly simple. Suppose p E P. As we are interested in the local situation at p , we may assume, without loss of generality, that P = Rd. By the Implicit Function Theorem, at every z E N , there is a local trivialization of f. Each of these local trivializations yields a collection of d vector fields v1,. . . , Vd on an open subset of N , which project by df to the standard basis vector fields .. ., on an open subset of Rd. One then takes a C" partition of unity (see ', p. 69-70), subordinate to the collection of local trivializations, and use this partition of unity to smoothly patch together the local vector fields to obtain a collection of smooth vector fields on N . Then, using the properness o f f , one shows that this collection of vector fields can be simultaneously integrated, i.e., the d integral curves, with velocities given by the vector fields (see ', p. 203-204 or 8 , p. 9-11) can be composed to yield a "flow" of f - ' ( p ) , which gives a local trivialization. For details of this proof, see 8.12 of '. Integrating along vector fields is a fundamental differential technique for obtaining diffeomorphisms and trivializations. We shall return to this topic later. Note that the existence of C" partitions of unity is used in a strong way above. The non-existence of analytic partitions of unity is what prevents us from proving a real or complex analytic version of Ehresmann's Theorem.

&, &

4. Basic Morse Theory

Ehresmann's Theorem is a theorem about smooth functions with no critical points. Morse Theory is the study of what happens at the most basic type of critical point of a smooth map. The classic, beautiful references for Morse Theory are and lo. We also recommend the excellent, new introductory treatment in ll. From this point, through Section 4,f : N R will be a smooth function from a smooth manifold of dimension n into R. For all a E R, let N<, - := f - l ( ( - c o , a ] ) . Note that if a is a regular value o f f , then N
Theorem 4.1. Suppose that a, b E R and a < b. Suppose that f - l ( [ a , b ] ) is compact and contains no critical points o f f . Then, N l a is a deformation

90

retract of N
Definition 4.2. The point p is a non-degenerate critical point of f provided that p is a critical point of f , and that the Hessian matrix (&(PI)

id is non-singular.

The index of f at a non-degenerate critical point p is the number of negative eigenvalues of ( & ( p ) ) , ,,counted with multiplicity. J

It is left to the reader to check that p being a non-degenerate critical point of f is independent of the choice of local coordinates on N . The reader may also try, as an exercise, to prove that the index of f at a nondegenerate critical is independent of the coordinate choice; this is also true, but not quite so easy (see p. 4-5). Note that since the Hessian matrix is a real symmetric matrix, it is diagonalizable and, hence, the algebraic and geometric multiplicities of eigenvalues are the same. The following is Lemma 2.2 of which tells us the basic structure of f near a non-degenerate critical point.

Lemma 4.3. (The Morse Lemma) Let p be a non-degenerate critical point o f f . Then, there is a local coordinate system ( y l , . . . ,y,) in an open neighborhood U of p , with y i ( p ) = 0, f o r all i , and such that, f o r all x E U ,

f >.(

(.N2

= f (P)- (Vl

- (Y2(x))2

-. .

*-

(Y/X(x))2+(YX+1(xN2+.. * + ( Y , ( 4 ) 2 ,

where X is the index o f f at p . I n particular, the point p is a n isolated critical point o f f , i.e., there is a n open neighborhood of p (namely, U ) in which p is the only critical point

off.

91

The fundamental result of Morse Theory is a description of how Nib is obtained from N<,, - where a < b, and where f - l ( [ a , b ] ) is compact and contains a single critical point of f , and that critical point is contained in f - l ( ( a , b ) ) and is non-degenerate. In 8 , Theorem 3.2, , Milnor gives this result up to homotopy. However, we wish to give the stronger “handle” result, as given in lo, Theorems 3.13 and 3.14, and in 11, Theorem 3.2. First, we need a definition. Let Bk denote a closed ball of dimension k.

Definition 4.4. A smooth n-dimensional manifold M‘ with boundary is obtained from a smooth n-dimensional manifold M with boundary by smoothly attaching a A-handle provided that there is an embedding i : dBX x Bn-X -+ d M such that M‘ is diffeomorphic to the space obtained by attaching the space B X x B“-’ to M via i and then “smoothing the corners” (see 11, p. 78). Theorem 4.5. Suppose that a < b, f - l ( [ a ,b ] ) is compact and contains exactly one critical point of f , and that this critical point is contained in f - l ( ( a , b ) ) and is non-degenerate of index A. Then, N
Definition 4.6. The smooth function f : N -+ IR is a Morse function if and only if all of the critical points of f are non-degenerate. Remark 4.7. The reader should be careful when encountering the term LLMorse function” in various references. We have given the weakest possible definition. Other authors sometimes mean that f is proper, or that N must be compact, or that the critical values at distinct critical points must be distinct. If a and b are not critical values of f and M := f - l ( [ a ,b ] ) is compact, then the change in diffeomorphism-type between N<, and N
Definition 4.8. The triple ( M ;VO,V1) is a smooth manifold triad if and only if M is a compact, smooth, pure-dimensional manifold, and the

92

boundary a M is the disjoint union of two open and closed submanifolds VO and V1. A Morse function on the smooth manifold triad ( M ;VO, V1) is a smooth function g : M + [a,b] R such that g - l ( a ) = VO,g-'(b) = 6 , and all of the critical points of g lie in M - a M and are non-degenerate. Note that a Morse function on a smooth manifold triad has at most a finite number of critical points, since the manifold must be compact and Morse critical points are isolated. Also, note that a smooth manifold triad includes as a special case a compact manifold without boundary, i.e., VO= V1 = 0;in this case, a Morse function g : M 4 [a,b] would have a below the minimum value of g and b above the maximum. Definitions 4.6 and 4.8 would not be terribly useful if there were very few Morse functions. However, there are a number of theorems which tell us that Morse functions are very plentiful. We remind the reader that "almost all" means except for a set of measure zero. Theorem 4.9. to, p. 11) If g is a C2 function from a n open subset U of R" to R, then, f o r almost all linear functions L : R" -+ R, the function g L : U + R has no degenerate critical points.

+

Theorem 4.10. to, p. 14-17} Let ( M ;V,,V1) be a smooth manifold triad, and suppose a < b. Then, in the C2topology, the Morse functions form an V1) + open, dense subset of the space of all smooth functions g : ( M ,VO, ( [ a ,b],a , b) . I n particular, there exists a Morse function o n the triad. Theorem 4.11. (", Theorem 6.6) Let M be a smooth submanifold of I%", which is a closed subset of R". For all p E R", let L, : M + R be given by L p ( z ):= 112 - p1I2. Then, for almost all p E R", L, is a proper Morse function such that M<, - is compact for all a. Corollary 4.12. (", p . 36) Every smooth manifold M possesses a Morse function g : M -+IR such that M5a is compact for all a E R. Given such a function g, M has the homotopy-type of a CW-complex with one cell of dimension X for each critical point of g of index A.

While we stated the above as a corollary to Theorem 4.11, it also strongly uses two other results: Theorem 3.5 of * and Whitney's Embedding Theorem, which tells us that any smooth manifold can be smoothly embedded as a closed subset of some Euclidean space.

93

To apply Theorem 4.5 at each critical point of a Morse function g on the smooth manifold triad ( M ;h,VI), we need to know that the critical values of g are distinct.

Theorem 4.13. to, Lemma 2.8) Let g be a Morse function o n the VI) with critical points P I , . . . ,Pk of insmooth manifold triad ( M ;VO, dices X I , . . . ,X k , respectively. Then, there exists a Morse function h on ( M ;Vo,VI) with critical points pl ,. . . ,P k of indices XI, . . . , X k , respectively, such that all of the critical values h(pi) are distinct for distinct pi. Moreover, h can be chosen arbitrarily close to g in the C2 topology. By combining Theorems 4.5, 4.10, and 4.13, we immediately obtain:

Corollary 4.14. Every smooth compact n-manifold has a finite handlebody structure, i.e., is formed by successively attaching a finite number of handles of various indices. In the above corollary, one begins at the global minimum of a Morse function with distinct critical values, and "attaches" a O-handle (a closed n-ball) to the empty set. The final attaching occurs at the global maximum, where one attaches an n-handle, i.e., attaches an n-ball along its bounding ( n- 1)-sphere. We now wish to mention a few complex analytic results which are of importance.

Theorem 4.15. (", p. 39-41) Suppose that M is an m-dimensional complex analytic submanifold of C". For all p E C", let L, : M --t I% be given by L p ( x ):= IJx- p1I2. If x E M is a non-degenerate critical point of L,, then the index of L, at x is less than or equal to m. Corollary 4.12 immediately implies:

Corollary 4.16. (", Theorem 7.2) I f M is an m-dimensional complex analytic submanifold of en,which is a closed subset of C", then M has the homotopy-type of an m-dimensional CW-complex. I n particular, H i ( M ; Z)= O for i > m. Note that this result should not be considered obvious; m is the complex dimension of M . Over the real numbers, M is 2m-dimensional, and so m is frequently referred to as the middle dimension. Thus, the above corollary says that the homology of a complex analytic submanifold of C", which is closed in C", has trivial homology above the middle dimension.

94

The reader might hope that the corollary above would allow one to obtain nice results about compact complex manifolds; this is not the case. The maximum modulus principle, applied to the coordinate functions on @", implies that the only compact, connected, complex submanifold of @" is a point. Suppose now that M is a complex m-manifold, and that c : M 4 @ is a complex analytic function. Let p E M , and let (ZI,..., z,) be a complex analytic coordinate system for M in an open neighborhood of p . Analogous to our definition in the smooth case, we have:

Definition 4.17. The point p is a complex non-degenerate critical point of c provided that p is a critical point of c, and that the Hessian matrix ( p ) ) is non-singular.

(&

,

,

273

There is a complex analytic version of the Morse Lemma, Lemma 4.3:

Lemma 4.18. Let p be a complex non-degenerate critical point of c . Then, there is a local complex analytic coordinate system ( ~ 1 , .. . ,y m ) in an open neighborhood U of p , with yi(p) = 0, for all i, and such that, for all x E U ,

4%) = C ( P ) + ( Y l ( x ) ) 2+ (Y2(x))2+ * . + ( Y n ( X N 2 . *

I n particular, the point p is an isolated critical point of c. The first statement of the following theorem is proved in exactly the same manner as Theorem 4.9; one uses the open mapping principle for complex analytic functions to obtain the second statement.

Theorem 4.19. If c is a complex analytic function f r o m a n open subset U of Cm to @, then, for almost all complex linear functions L : CW + @, the function c L : U + CC has no complex degenerate critical points. In addition, for all x E U , there exists an open, dense subset W in Horn@(@",@)E Cm such that, for all L E W , there exists an open neighborhood U' C U of x such that c + L has no complex degenerate critical points in U'.

+

Finally, we leave the following theorem as an exercise for the reader. We denote the real and imaginary parts of c by Rec and Imc, respectively.

Theorem 4.20. Let p be a complex non-degenerate critical point of c. Then, the real functions Rec : M + R and Imc : M + R each have a

95

(real, smooth) non-degenerate critical point at p of index precisely equal to m, the complex dimension of M . In addition, if c ( p ) # 0, then the real function lcI2 : M -+ R also has a non-degenerate critical point of index m at p .

5. Real and Complex Analytic Sets

In Definition 2.7, we defined a singular point as a point where a topological space is not locally homeomorphic to an open subset of Euclidean space. While this is, in fact, a reasonable definition of a singular point, it is unreasonable to expect to be able to obtain nice results about the local topology of arbitrary topological spaces at such singular points. We need to restrict our attention to topological spaces which are more manageable, and occur “naturally” .

Thus, we shall restrict our attention to algebraic and analytic sets (varieties) over a field ff, which we assume to be R or C;we will define these notions below. When we write analytic below, we mean real analytic if ff = R, and complex analytic if ff = C.There are two topologies on An which we will consider; the classical topology on Iw” or C”, which is what we used up to this point, and the Zariski topology, which the reader may be familiar with, but which we will define below. When we use the terms “open” and “closed” without qualification, we mean open or closed in the classical topology. When we want to refer to the Zariski topology, we shall do so explicitly. In the algebraic setting, we shall consider global algebraic subsets of f f n , i.e., f i n e algebraic sets. In the analytic setting, instead of restricting ourselves to connected open subsets of f f n , it is useful to allow the ambient space t o be more general. Thus, throughout these notes, unless we specifically state otherwise, M denotes a connected n-dimensional analytic manifold. There are many, many basic references for algebraic geometry, and we shall not need very much of the theory. The references for analytic geometry, even in the complex case, are not so widely known. While we shall discuss most of the results that we need, we recommend: 12, 52, for the real and complex algebraic cases; 13, Chapters 2 and 3, for the real algebraic case; 14, 1.1, and 15, Chapter 1, $1, for the complex algebraic case; Chapter 1 of l6 for the real analytic case; for the complex analytic case, 15, Chapter 4A and ”, Chapters I1 and IV; for a combined treatment of the complex

96

algebraic and analytic cases, 18, Chapter 0, $1and $2; and, finally, for both the real and complex analytic cases, l9 and 20. We let O p := ff[zl, . . . ,z], denote the polynomial ring in n variables over ff. We let 0;’ denote the ring of analytic functions from M to -R; these are functions which are locally, on coordinate patches, given by convergent power series in n variables over the field ff; over C , these are simply the holomorphic functions on M . It is important to note that an analytic function is not required to have one fixed power series representation that can be used at all points. We write (z1, . . . ,z), for local analytic coordinates on M . Recall the notions of the germ of a topological space, X , and g e m of a f u n c t i o n o n a topological space, h : X + Y at a point p E X ; intuitively, the germ at p means the space X or the function h in an arbitrarily small neighborhood of p . Rigorously, the germ of X (resp., h) at p is the equivalence class under the equivalence relation: two spaces (resp., functions) X and X’ (resp., h and h‘) have the same germ at p E X n X’ if and only if there is an open neighborhood U of p in X and U‘ of p in X‘ such that the topological spaces (resp., functions) U and 24’ (resp., hlu and hi,,) are equal. We denote the germ of X (resp., h) at p by X , (resp., [h],). Now, if p = ( p i , . . . ,p,) E M , we let 0;; denote the ring of germs of functions which are analytic on some open neighborhood of p in M ; these are the power series that converge in some neighborhood of p . Note that 0;; is a local ring, whose maximal ideal, in local coordinates (xi,. .. ,z,), is

mM,p:=< 21 - p i , . . . ,2,

-p,

>C @{xi - p i , . . . ,Z,

anal

- p,} = OM,,.

The following result is fundamental.

Theorem 5.1. ( T h e Principle of Analytic Continuation) Suppose that p E anal anal M , f E O r ’ , and [f], = 0 in OM,,. Then, f = 0 in OM . Proof. Let 2 := {z E M I [ f I z = 0). Then, 2 is non-empty (since p E 2 ) and is clearly open in M . However, as power series representations of functions are unique on the domain of convergence, 2 is also closed in M . 0 As M is connected and non-empty, Z = M , i.e., f = 0 in O r ‘ .

We have the following basic algebraic result (see 20).

97

Theorem 5.2. The rings Ofin and 0;; are Noetherian, unique factorization domains (UFDs). The ring 0;' is a n integral domain, but need not be Noetherian or a UFD.

c

c

Definition 5.3. Let A ORn (resp., A 0;'). Then, we define the vanishing locus (or, zero locus) of A, V ( A ) to , be the set of points in An (resp., the set of points in M ) V ( A ):= {x E An (resp., M ) I f(x) = 0 for all f E A , If A C O K p ,then one makes an analogous definition of Vp(A)as a germ of a set of points in M at p . (We leave this formulation as an exercise for the reader.) Let E C An (resp., E C M ) . Then, we define the ideal of polynomials (resp., analytic functions) which vanish on E to be Z ( E ) := {f E OBn(resp., 0;') I f ( e ) = o for all e E E } . If Ep is the germ of a set of points in M at p , then one makes an analogous definition of Z ( E p )as an ideal in 0;;. (We leave this formulation as an exercise for the reader.) If A = { f i , . . . , fj}, we write V ( f i , . .. , f j ) in place of V ( A ) . The following proposition is a straightforward exercise. Recall that the radical, of an ideal J in a ring R is equal to { f E R I there exists k E N such that f k E J } . Also, recall that for A R, ( A )is the ideal generated by A, i.e., the intersection of all ideals in R which contain A.

a,

c

98

uES ‘ LyES a! E S , V (

(11) for all

PET

a )= V ( J a ) ;

PET

Moreover, the analogous statements for 0;;

and 0;’

are also true.

Definition 5.5. An algebraic subset of f f n or an affine algebraic set is a set of the form V ( A ) ,where A C O R n . As V ( A )= V ( ( A ) )and O R n is Noetherian, this is equivalent to saying that an algebraic subset of f f n is defined by the vanishing of a finite number of polynomials, i.e., is of the form V (f l , . . . ,fj). A subset X & M is an analytic subset of M if and only if X is closed in M and, for all x E X , there exists an open neighborhood W of 3: in M and a finite collection f l , . . . , f j E 0;’ such that V ( f 1 , . . ,f j ) = W n X. A subset E of M is locally analytic if and only if, for all p E E , there exists an open neighborhood W of p in M such that W n E is an analytic subset of W .

Remark 5.6. Item 6 of Proposition 5.4 tells us that if X is an algebraic set (resp., X p is the germ of an analytic subset), then V ( Z ( X ) )= X (resp., However, an analytic subset X of M need not be defined by the vanishing of a collection of analytic functions which are defined everywhere on M . In this case, while it is still true that X V ( Z ( X ) )the , containment may well be proper. A particularly striking example is given in 19, $11, where a real analytic subset S C R3 is defined by the vanishing of a C“ function, and it is shown that Z(S) = (0). We should also remark that some authors do not require an analytic subset to be closed. Their notion of an analytic subset of M is what we have called a locally analytic subset. One must be careful when reading the definition of “analytic subset” in a given source. Another source of possible confusion is that it is common to require an analytic subset X to be closed without explicitly stating that X is closed; some authors take our definition, omit the phrase “X is closed in M ” , but change “for all x E X” to “for all x E M”.

99

The following theorem follows from Item 10 of Proposition 5.4 in the algebraic case. In the analytic case, it is substantially more difficult; see 21, Theorem 9C and 20, Cor. 2, p. 100.

Theorem 5.7. Intersections of a f i n e algebraic sets are afine algebraic sets. Intersections of analytic subsets of M are analytic subsets of M ; in particular, if A C O r ’ , then V ( A ) is an analytic subset of M . In light of this theorem, and the fact that V(1) = 8 and V ( 0 ) is the entire ambient space, An or M , we may make the following definition.

Definition 5.8. The algebraic Zariski topology on An is the topology in which the closed sets are precisely the algebraic subsets. The analytic Zariski topology on M is the topology in which the closed sets are precisely the analytic subsets. We denote the topological closure of E M by F and the analytic -anal closure, i.e., the closure in the analytic Zariski topology, by E . Note that the Zariski topologies are very coarse; the nonempty open subsets are very big and special. In particular, a Zariski-open (resp., closed) set is open (resp., closed) in the classical topology. We remind the reader that when we use the terms ‘Lopen”and ”closed”, or other topological notions, without qualification, we mean in the classical topology.

Definition 5.9. A non-empty algebraic subset in An (resp., analytic subset of M ) is irreducible if and only if it cannot be written as the union of two proper algebraic (resp., analytic) subsets. An analytic subset X M is irreducible at a point p E X (or the germ X , is irreducible) if and only if the X , cannot be written as the union of two proper germs of analytic spaces at p. Note that it is immediate from the definition that an irreducible analytic subset of M must be connected. Moreover, the following proposition is an easy exercise:

Proposition 5.10. (1) I f X is a n irreducible algebraic set in An, then Z ( X ) is a prime ideal in OR%.Moreover, the analogous statements for irreducible analytic

sets in M and irreducible germs are also true. (2) If A G O R n , X := V ( A ) ,a n d Z ( X ) is a prime ideal in ORn, then X is irreducible. Moreover, the analogous statement in 0;; is also true.

100

In particular, a n algebraic set (resp., a n analytic germ) i s irreducible i f and only if the ideal of polynomials vanishing o n the set (resp., the ideal of germs of analytic functions vanishing o n the germ) i s prime. Note that we are not claiming that Item 2 of Proposition 5.10 holds for analytic subsets X C M .

Example 5.11. Consider X := V ( y 2- z3 - z2)

R2.

This algebraic/analytic set is both algebraically and analytically irreducible in R2.However, X is not analytically irreducible at 0 . In the graph, one sees that in a small neighborhood of the origin, X consists of two smooth curves which transversely intersect each other. One detects this analytically by factoring y 2 - x 3 - z2 as ( y z G ) ( y - z-). Here, G denotes one of the two convergent power series at 0 whose square is 1 z, and the equality y 2 - z3 - z2 = ( y z G ) ( y - x G ) holds only on neighborhoods of the origin where the power series for G converges, 0 i.e, inside an open ball B of radius 1. Thus, by Item 9 of Proposition 5.4,

+

+

+

0

inside B ,

x = V ( y+ z

r n )

u V ( y- x

r n ) ,

and so the germ of X at 0 has two irreducible “components” (see below). Note that the Inverse Function Theorem ( 2 , 9.24; 16, 1.8.1; 21, Appendix 11, Lemma 2.A) implies that the map (z, y) H (z-, y ) is a local, analytic change of coordinates at the origin. This means that, up to an analytic isomorphism in a neighborhood of the origin, the analytic set V ( y 2 - z 3 - z 2 ) is the same as V(y x) U V ( y - x), i.e., two intersecting lines. Our final comment on this example is that we could have written all of the above with x and y as complex variables and X being a complex analytic subset of C2. Of course, C2 is real 4-dimensional, and so we have no hope of

+

101 drawing an accurate picture over C. It is common in low-dimensional complex analytic geometry to draw the picture of the corresponding situation over R and hope that the picture over the real numbers provides one with some intuition for what happens over C. There are, unfortunately, quite a few definitions of smooth, regular, and singular points in the literature. The issues involved in the various definitions are whether one uses globally-defined polynomials or local analytic functions, and whether the space must have maximal dimension at a regular point. Throughout these notes, we will deal with the local analytic situation, and below we adopt notation and terminology for this situation. In the remark following the definition, we discuss the algebraic situation.

Definition 5.12. Let X be an analytic subset of M . A point p E X is called smooth, of dimension d, if and only if there exists an open neighborhood W of p in M such that W n X is an analytic submanifold of W of dimension d (over the field ff). Thus, p E X is smooth of dimension d if and only if there exists an open neighborhood W of p in M and f d + l , . . . f,, E 0;’ such that W fl X = V ( f d + l , . . . ,fn) and, for all 2 E W , d z f d + l , . . . , d, fn are linearly independent. 0 We denote the set of smooth points of X by x,and the set of smooth A smooth point of X of highest dimension points of dimension d by is a regular point; we denote the set of regular points of X by Xreg.

i(d),

A point p E X which is not smooth is called a singular point (or, a singularity).We denote the set of singular points, the singular locus, of X by EX. The set of exceptional points of X is X -Xreg, and is denoted by X e x . Remark 5.13. In the above definition, we have followed the standard analytic definition, as given in 19, 20, 17, and 21 - though, we have used the term smooth where some authors write regular. Our uses of the terms regular and exceptional are not standard in many references. The reader should be careful to look up the definitions of any such terms in any given source. The reader should also note that the definitions of “smooth”, “regular”, “simple”, “non-singular”, and “singular” points given in l2 are different; Milnor is dealing with the algebraic case and, in addition, gives a definition which is satisfactory only when the dimension of the space is the same at all of the regular points; see 1 2 , p.10 and be sure to read the footnote. In

102

the real or complex algebraic setting, it is standard to use globally-defined polynomials in the definitions of regular and singular points, as Milnor does in 1 2 ; see, in addition, 13, $3.3 and 1 5 , $1A and 51.B. Using global polynomial functions, one can still define smooth points of dimension d (as in 1 3 ) , and avoid the “problem” mentioned in the footnote of 1 2 , p. 10. The set of algebraic smooth points i a l g is the union of the smooth points of all dimensions, and the complement of this set is one notion of the algebraic singular locus of X , which we will denote by CalgX.The algebraic regular points are the algebraic smooth points of highest dimension; we denote the set of algebraic regular points by X$:.

If X is algebraically irreducible, then X,”: = i a l g . The set of algebraic exceptional points of X is X - X,”:, and is denoted by XzA:. Clearly, C X G Cal,X C Xt$. An important question, of course, is whether or not E X = &,X. In fact, in the complex algebraic case, Milnor’s remark and proof in 1 2 , p. 13-14, shows that C X = Cal,X. However, in the real algebraic case, it is false, in general, that EX = CalgX;see Example 3.3.12a of 13. However, the algebraic situation over the real numbers is not so badly behaved; over the reals or complexes, X,$t is an algebraic set; see 1 3 , Proposition 3.3.14, which is essentially the easy argument given by Milnor in 1 2 , p. 11. The real analytic situation is much more problematic; see Example 5.31. Theorem 5.14. Let X be a n analytic subset of M . Suppose that 0 5 d 5 n.

i(d) is a d-dimensional analytic submanifold subset of X ; the i(d) are disjoint f o r different d , i =

of M , and is a n open U

. . .icn), i is a n

0

analytic submanifold of M , and x is open in X ; i f p is a smooth point of X , then the germ X , is irreducible; 0

x

is dense in X ; and EX is a closed, nowhere dense subset of X .

Proof. Items 1 and 2 follow at once from the definitions. At a smooth point p of dimension d, the Implicit Function Theorem tells us that there is an analytic isomorphism in a neighborhood of p which takes the ideal ([f d + l ] p , . . . , [ f & ) in 0:; to the ideal ( [ z d + l ] , , . . . , [z,],) in where U is an open subset of f t n and ( X I , . . . ,z,) are coordianal nates on U. As ( [ z d + l ] , , . . . , [z,],) is obviously a prime ideal in Ou,p,

a,.,;.

103 anal

([fd+l],,. . . , [fn],) is prime in OM,p.Item 2 of Proposition 5.10 now implies that X, is irreducible. Item 5 follows immediately from Items 2 and 4. Item 4 is difficult; see 20, p. 41, Theorem 1. 0 Definition 5.15. The dimension (over R), dimX, of an analytic set X C M is the largest d such that ;(dl is non-empty. The dimension of X at a point p E X , dim, X, is the largest d such that p is in the closure of We say that X is pure-dimensional if and only if the dimension of X at each point p E X is independent of p. We make the analogous definitions for local analytic sets.

i(d).

We have defined dimension geometrically above. However, there are algebraic characterizations. See 20, p. 32-40. Below, we use the Krull dimension; this follows quickly from 20, since the Krull dimension of an integral extension of a ring R is the same as that of R (for Noetherian rings). For the global dimension statement for an algebraic set, there are many references; see, for instance, 15, Chapter 1, in the complex case, and 13, 53.3 in the real case.

Theorem 5.16. The dimension of a n analytic subset X C M at a point p E X is equal to the (Krull) dimension of the quotient ring O K J T ( X , ) and, thus, dimX is the maximum of these dimensions as p varies over all of the points of X . If X is a n algebraic subset of An, then dimX is equal to the dimension of ORn/Z(X). We have the following corollary, over R or C , which is a very easy exercise, but is still extremely useful for proving local irreducibility:

Corollary 5.17. Suppose that a is a n ideal in 0;; prime ideal p, and such that

such that

fi

is a

- = dim V,(a). (Of9 Then, Z(V,(a)) = p and, hence, V,(a) is irreducible. I n particular, i f f is a n irreducible element (in particular, not a unit or zero) o f O z ; s u c h t h a t d i m V , ( f ) = n - 1 , t h e n ( f ) = Z ( V , ( f ) ) a n d V , ( f ) is irreducible. dim

104

Remark 5.18. It is tempting to read more into Theorem 5.16 and Corollary 5.17 than they actually say. Suppose that X, is irreducible of dimension d , and f E 0;; is such that f ( p ) = 0 and X, g V,(f). Then, one might think that Krull’s Hauptidealsatz would easily imply that dim(X,nV,(f)) = d - 1 . While this is true if A = @, it is not true if A = R. What the Hauptidealsatz actually does tell us is that dim

W,) + (f)

=d-l.As

dim

Oz;

0;;

=dim

+ (f)

J V P ) + (f). and, by Item 1 of Proposition 5.4,Z(X, n V,(f)) is equal to its own radical, (f)? The answer is the question is: does Z(X, n V,(f)) equal &(X,) “yes” if A = C , by Hilbert’s Nullstellensatz (see Theorem 5.19 below), and “no”, in general, if A = R. It is easy to see/explain this failure over R; if fl , . . . , fk are real analytic functions and f := ff + . .. then V ( f 1 , .. . , fk) = V(f). Thus, being locally defined by the vanishing of a single real analytic function should not have any nice dimensionality properties. For instance, if p = ( P I , . . . , p n ) , dim X, 2: 2, and we choose f = (z1 - ~ 1 ). . ~ (zn - P,)~ E O;;, then

Z(X,)

+

+ fz,

+. +

/nanal

dim(X, n V,(f)) = 0, while dim

v M , ~

JZ(X,>

+ (f)

= (dimX,) - 1

2 1.

In the complex algebraic setting, there are a large number of references for Hilbert’s Nullstellensatz; see, for instance, 15, Theorem 1.5. In the local complex analytic setting, see 17, 111.4.1.

Theorem 5.19. (Halbert’s Nullstellensatz) Let R = C. Suppose that a i s a n ideal in (resp., in 0;;). Then, fi = Z(V(a)) (resp., = Z(V,(a))). Our discussion in Remark 5.18 immediately yields:

Proposition 5.20. Let A = C, f E O r ’ , and p E V(f). Suppose that [f],# 0. Then, dimV,(f) = n - 1. I n particular, if f $ 0, then V(f) is purely ( n - 1)-dimensional (this vacuously allows f o r V( f ) = 0). In analogy with the manifold terminology, we say, in the setting above, that V(f) has codimension 1 everywhere.

105

It should come as no surprise that Proposition 5.20 fails over the real numbers, even if one assumes that V (f) is irreducible of dimension n - 1. Example 5.21. A famous example of a real algebraic set which is "troublesome" is the Whitney umbrella, X := V(y2 - zx2) C R3.

Note the 2-dimensional portion above the xy-plane, and that when z < 0, the only points of X are on the z-axis. One might suspect that X is not irreducible, as X is certainly the union of the z-axis (the handle of the umbrella) and the 2-dimensional "umbrella" portion. However, X n {(x,y, z ) I z 2 0) is not an analytic set (use Corollary 5.17 at the origin), and X is, in fact, irreducible. Thus, the Whitney umbrella is an irreducible analytic set which is not pure-dimensional.

Proposition 5.22. Suppose that f E 0:; has an irreducible decomposition uf,"'f;' . . . f:', where u is a unit, and the f i are non-associated, irreducible elements of 0:;. Then, V p ( f )is an (n - 1)-dimensional analytic submanifold inside the germ of M at p if and only i f there exists i o such that dpfio # 0 and such that, for all i # 20, V ( f i ) V ( f i , ) and dim V,( f i ) < dim V,( fi,). Therefore, if ff = C and f E O;;, then V,(f) is a n (n-1)-dimensional analytic submanifold inside the germ of M at p i f and only i f there exists a n irreducible g E 0;; and a E N such that f = ga and dpg # 0. Proof. Suppose that there exists io such that dpfio # 0 and such that, for all i # io, V (f i ) 2 V (f i , ) and dim V,( f i ) < dim Vp(fi,). Then, Item 9 of Proposition 5.4 tells us that V,( f) = V,( fi,), and the Implicit Function Theorem (Corollary 2.2) tells us that V,(fi,) is an ( n - 1)-dimensional analytic submanifold of the germ of M at p.

106

Now suppose that V,(f) is an ( n- 1)-dimensional analytic submanifold inside the germ of M at p. Then, Z(V,( f)) equals a prime ideal p and, since V,(f) = UiV,(fi), we also have that Z(V,(f)) = niz(V,(fi)). Therefore, there exists io such that p = T(Vp(fio));it follows that V,(f) = V,(fio) and dimVp(fio)= n - 1. Thus, for all i, V(fi) C V(fi,). Now, note that we are in the situation of the last statement of Corollary 5.17, and so we conclude that T(V,(f)) = Z(V,(fi,)) = (fi,).As V,(f) is an ( n - 1)-dimensional analytic submanifold, there exists some g E 0;; such that V,(g) = V,(f) and d,g # 0. Since g E Z(V,(f)) = (ti,), there exists q E 0;; such that g = q fi,. The product rule yields dpg = d p ) d p f i , .fi,(p)dpq = q ( p ) d p f i o ,and so d p f i o must be unequal to zero. Suppose that for some i # io, dimV,(fi) = n - 1. Then, applying Corollary 5.17 again, we conclude that Z(V,(fi)) = (fi). Thus, we obtain a containment of prime ideals (fi,) = Z(V,( f ) ) Z(V,(fi)) = (fi), where anal dim OM,,/(fi0)= dim 0K;/(fi). Hence, (fi,) = (fi), which contradicts that fi is not associated to fi,.

+

-

This finishes the proof, except for the final statement, which now follows at once from Proposition 5.20 (and the fact that one can extract arbitrary roots of units over C). 0 Example 5.23. Over R, one does not have the nice result that one has over C in Proposition 5.22. Consider the germ of the real function f = -(x2 + y 2 ) 2 x 2 at the origin in R2. Then, V(f) = V(x) is certainly a smooth 1-manifold at the origin, and yet one cannot eliminate the x2 y 2 factors from f nor “absorb” the unit -1 into the powers of irreducible elements.

+

For lack of a convenient reference, we will now prove: Theorem 5.24. Suppose that X i s a n analytic subset of M , and E is a

connected, pure-dimensional, locally analytic subset of M such that, for all x E E , Ex is irreducible. Finally, suppose that p E E is such that Ep C X,. Then, E C X . Proof. Let F := {x E E 1 Ex C Xx}.We will show that F is both open and closed in E. As E is connected and p E F , it will follow that F = E , and so E C_ X. Let d denote the dimension of E at each of its points. By definition of the containment of germs, the set F is open in E. It remains for us to show that the complement of F in E is open in E.

107

Suppose that q E E and E, 9 X,. As E, $L X,, Z(X,) $L Z(E,). Therefore, there exists an open neighborhood W C M of q and f E 0;‘ such that f i w n x = 0 and f # Z(E,). It follows that E, n V(f) is a proper analytic germ inside the irreducible germ E,. By Proposition 7, p. 41, of 20, dim,(E n V( f)) < d. Let W’ C W be an open neighborhood of q in M such that dim(W’ n E n V(f)) < d. We claim that, for all IC E W’ n E, Ex g X,, i.e., that the complement of F in E is open. Suppose, to the contrary, that IC E W’ n E and E, C X,. Then, as flwnx = 0, IC E V ( f ) and , so dim E,nV( f ) < d. However, Ex C X , C V(f) and, hence, dim E, n V(f) = dim Ex = d. This contradiction concludes the proof. 0 The following corollary is an easy exercise. Corollary 5.25. Suppose that E is a connected, pure-dimensional, locally -anal analytic subset of M such that, f o r all x E E, E, i s irreducible. Then, E is irreducible in M . Example 5.26. Theorem 5.24 is false without the assumption that E is pure-dimensional. Consider the set E := V(z(z2 y2) - z3)C &I3. This is an example of H. Cartan from p. 93 of 19; see also 20, Example 1, p. 106. We refer to this example as the Cartan top.

+

The Implicit Function Theorem tells us that outside Z := ((0, 0, z ) I z E R} (i.e., the z-axis), E is an an analytic 2-manifold. Therefore, E - 2 is irreducible at each point. In addition, at any point p on Z - {0}, Ep is certainly irreducible since it agrees with 2,.Finally, use Corollary 5.17, or refer to l9 and ‘O, for the irreducibility of E at 0 . Thus, E is connected and locally irreducible. Let p := (O,O, -1). Then, Ep C Z,,but certainly E g 2.

108

The following result is very useful.

Theorem 5.27. Suppose that X i s a connected, pure-dimensional, locally irreducible, analytic subset of M . Suppose that W is a non-empty, open subset of X . Then, the analytic closure of W in M is equal t o X . -anal

Proof. Let R be the interior of W in X . Then, R is a non-empty open subset of X . We shall show that R is closed in X . As X is connected, that will conclude the proof. Let d denote the common dimension of X at each of its points. Let x E C we will show that z E R. For every open neigh-anal borhood V of x in X, R n V # 0 and, thus, dim(W ), = d. As C X,, and X, is irreducible is dimension d , we may use Proposi-

a manal;

(rnanal),

tion 7, p. 41, of

2o

-anal

to conclude that (W

),

= X,. Therefore, there is an --anal

open neighborhood V of z in X such that V n W Thus, by definition of the interior, x E R. V

mana1.

=

V , i.e., such that 0

Corollary 5.28. Suppose that Y is a n a f i n e linear subspace of Rn, and that W i s a non-empty open subset of Y . Then, the analytic closure of W in ffn i s Y . We shall now give a number of fundamental results which hold over the complex numbers, but not over the real numbers. In Item 1 below, we use the convention that dim 0 = -cm.

Theorem 5.29. Let X be a non-empty analytic subset of M . Let V be a n irreducible analytic subset of M . Then, the following statements hold for R = C and are false, in general, for = R. (1) E X is a n analytic subset of M of dimension strictly less than dimX. (2) V is pure-dimensional.

(3) If p E V and V, C_ X,, then V G X . (4) X is irreducible i f and only i f it i s the topological closure of a nonempty, connected analytic submanifold of M . 0 (5) If V X , then V - X and -X are connected and dense in V .

v

i,

c

(6) For every connected component C of the closure i s a n irreducible analytic subset of M , and distinct connected components C yield distinct The collection K := I C a connected component of }; is

c.

{c

109

countable and locally finite. None of the elements of K: is contained in the union of the others, and the union of all of the elements of K: is X . If V C X , then there exists Y E K: such that V C Y . (7) Suppose that X = U i X i , where the X i are irreducible analytic subsets of M , { X i } i is locally finite (or countable), and X i , X i , if il # i p . Then, { X i } i equals the set K: from Item 6. (8) If X is connected and irreducible at each point, then X is irreducible. (9) If X C V, and dimX = dimV, then X = V. Proof. We give references for each of these results over C, except we will prove Item 8, and leave Item 9 as an exercise. We give examples below which show that the general statements are false if fi = R. While these results can be found in many places, all of our references are to 17. Item 1 is Theorem 1, p. 211. Item 2 is immediate from Item 4. Item 3 is Proposition 4, p. 217. Item 4 is Corollary 3 on p. 216. Item 5 is Corollary 1 to Proposition 3 on p. 216. Items 6 and 7 are the corollary and Theorem 4 on p. 217. Finally, we use Items 3 and 6 to prove Item 8. Suppose that X is connected and irreducible at each point. Write X = &??, where the C’s 0

are the connected components of X . We need to show that there is only one such C. Suppose-not. Fix a C,. As X is connected, cannot be disjoint from Ucpco C (locally finite unions of closed sets are closed). Let p E ~ n U , , , C. As X = Uc ??,which is a locally finite union, and we are assuming that X , is irreducible, there must be a Ci such that X , = However, this implies that for every other distinct Cj such that p E C X, = By Item 3, this implies C a contradiction of Item 6. 0

a

(c

(q)p

(c),.

q c;

q,

Definition 5.30. When ff = C, we refer to the ?? of Item 6, or the Xi of Item 7, in Theorem 5.29 as the irreducible components of X .

We will now give examples, or references to examples, which show that all of the items of Theorem 5.29 fail over R. Example 5.31. 0 Looking at Definition 5.12, it is tempting to believe that an easy argument -such as that mentioned in 1 2 , p. 11-in terms of determinants of the minors of the matrix of partial derivatives would allow one to conclude in the real or complex analytic case that E X is analytic. However, the result is simply

110

false if R = R; see Example 3 on pages 106-107 of 20. In this example, an -anal analytic subset A & R3 is described such that dim(CA ) = dim A . The Whitney umbrella of Example 5.21 or the Cartan top of Example 5.26 are examples of irreducible real analytic sets which are not puredimensional. 0

0 Let V equal the Whitney umbrella or the Cartan top, and let X denote the z-axis in R3. Let p = (O,O, -1). Then V is irreducible, V, X,, but V X. Thus, Item 3 of Theorem 5.29 fails over R. Also, even though V is irreducible, it is not the closure of a connected analytic submanifold; therefore, Item 4 of Theorem 5.29 fails over R. Note, however, that the other implication in Item 4 remains valid even when R = R. That is, if an analytic subset X of M is the topological closure of a non-empty, connected analytic submanifold, then X is irreducible. This follows immediately from Corollary 5.25. 0 As W - (0) is not connected, certainly the connectedness conclusion of Item 5 is terribly false over R. In addition, removing the z-axis from V , where V is the Whitney umbrella or Cartan top, leaves one with a nondense subset of V , which means that the other conclusion of Item 5 is not generally true over R.

0

No reasonable concept of an analytic irreducible component of a space

X, as appears in Items 6 and 7, exists over the real numbers. Certainly the topological closure of the 2-dimensional connected component of the smooth points of the Whitney umbrella or the Cartan top are not analytic sets. However, the situation is significantly worse than this. Example 4 on p. 107 of 2o is an example of an analytic subset S of R3 such that if S = B U C, where B and C are analytic subsets and B is irreducible, then C = S.

+

0 Suppose that Y is the Cartan top, V ( z ( x 2 y2) - z3) C R3. Let 2 be a shifted copy of the Cartan top, 2 := V ( ( z l)(x2 y2) - x3). Then, Y n 2 is the z-axis. Let X := Y U 2. Then, X is connected and locally irreducible, but not irreducible. Therefore, Item 8 does not hold over W.

+

+

Example 5 on p. 108 of 2o is an example of an irreducible 2-dimensional analytic subset S of R3 containing proper 2-dimensional analytic subsets. Thus, Item 9 fails over R. 0

111

6. Real and Complex Semianalytic Sets

When dealing with analytic subsets of an analytic manifold M , we frequently use constructions which require intersecting the analytic set with a closed ball, or use some other non-analytic intersection. Therefore, it is useful to expand our point-of-view somewhat and discuss semianalytic subsets. See 22 and 23.

Definition 6.1. Suppose that ff = R. Let W be an open subset of M . A basic semianalytic subset of W is a subset of the form V(f1, * . . f d n {. E where

fl,

. . . ,fk, g1 ,. . . ,g1

w I gx(xc>> 0 ,

=17..

. 7

Q,

E 0;'.

A subset X in M is semianalytic if and only if, for all p E M , there exists a open neighborhood W of p such that W i l X is a finite union of basic semianalytic subsets of W . Note that, by negating functions, the definition of a basic semianalytic subset can also contain gx(x) < 0 and so, after taking unions, we can also obtain g x ( x ) # 0 , gx(x) 2 0 , and g x ( x ) 5 0 . Now, suppose that ff = C. Let W be an open subset of M . A basic semianalytic subset of W is a subset of the form v(fi,...fk)n{xE

where

fl,

. . . ,fk, 9 1 , . . . ,g1

w I gx(x) #O,x=17..*71},

E 0;'.

A subset X in M is semianalytic if and only if, for all p E M , there exists a open neighborhood W of p such that W n X is a finite union of basic semianalytic subsets of W . If X is a semianalytic subset of M , and Y is a semianalytic subset of an analytic manifold N , then a function f : X Y is semianalytic if and only if the graph of f is a semianalytic subset of A4 x N . --f

Remark 6.2. What we have referred t o above as a semianalytic subset in the complex case has classically been called an analytically constructible subset. By calling such a set semianalytic, we are able to state simultaneously results in both the real and complex settings. We summarize a number of properties of semianalytic sets below. The first item is an easy exercise and the others can be found in 23, 52.

Theorem 6.3. Let ff be R or @. Let X be a semianalytic subset of M

112

( 1 ) The collection of semianalytic subsets of M is closed under finite

unions, finite intersections, and taking complements. (2) Every connected component of X is semianalytic. (3) The family of connected components of X is locally finite. (4) X is locally connected. (5) The closure and interior of X is semianalytic. I n fact, if fi = C, an analytic subset of M .

is

Remark 6.4. Later, we shall see that Item 4 above can be substantially strengthened; semianalytic subsets are, in fact, locally contractible.

The following is Corollary 2.9 of

23.

Proposition 6.5. Let fi = R. Let X be a semianalytic subset of M . X , and V is open in X . Then, f o r all x E X , there exists an open neighborhood W of x in M such that W n V is a finite union of sets of the form

(1) Let V be a semianalytic subset of M such that V

w n {x x I fl(x) > 0 , . .

*

> fk(x)

> o},

where f 1 , . . . ,fk E OK1, (2) If X is closed in M , then, for all x E X , there exists an open neighborhood W of x in M such that W n X is a finite union of sets of the form {x E where

fl,.

. . ,f k

w I f l ( x ) 2 0,. .. lfk(x) 2 o},

E 0;’.

The following lemma is an extremely useful tool; see 12, $3 and 24, 52.1. The complex analytic statement uses Lemma 3.3 of 12. Below, Ib6 denotes an open disk of radius 6 > 0 centered at the origin in C.

Lemma 6.6. (Curve Selection Lemma) Let R = R, and let p E M . Let Z be a semianalytic subset of M such that p E Then, there exists a real analytic curve y : [0,6) -+ M with y(0) = p and y ( t ) E 2 for t E (0,b). Let fi = C, and let p E M . Let Z be a semianalytic subset of M such Then, there exists a complex analytic curve y : M with that p E

z.

z.

y(0) = p and y ( t ) E 2 f o r t E

bs

.--)

Ibs- ( 0 ) .

113

Theorem 6.7. Let f E 0;'. Let p E M . Then, there exists an open neighborhood W of p in M such that the critical locus WnCf C f - ' ( f ( p ) ) , i.e., locally in M , the critical values o f f are isolated. Proof. Suppose to the contrary that, for every open neighborhood W of n (Cf- f - ' ( f ( p ) ) ) # 0, i.e., that p E Cf - f - l ( f ( p ) ) . Let y denote either a real or complex.curve as guaranteed by the Curve Selection Lemma; so, y(0) = p and y ( t ) E Cf - f - ' ( f ( p ) ) for small t # 0. Then, the derivative (f o y ) ' ( t ) is an analytic function which is zero for all small t # 0, and so must also be zero at t = 0. Therefore, (f 0 y ) ( t ) must be constant. As f (y(0)) = f(p), it follows that f ( y ( t ) ) = f ( p ) for all small t , i.e., y ( t ) E f - l ( f ( p ) ) for all small t. This is a contradiction.

p in M , W

Remark 6.8. In the algebraic setting, there is a much stronger result. For an algebraic set X C ff", recall the definition of i a l g from Remark 5.13. The following is Corollary 2.8 of 12: Let X be an algebraic subset of ff", and let f E Ofin. Then, the restriction of the polynomial map f to the set i a l g has a finite number of critical values. Definition 6.9. Let X be a semianalytic subset of M . Then, we define smooth points, regular points, singular points, and exceptional points exactly as we did for analytic subsets in Definition 5.12, and we use the same notations. As we shall see below (Corollary 7.7), the smooth points of a semianalytic set are dense, and so we define the dimension of X , and the dimension of X at a point exactly as we did for analytic subsets in Definition 5.15. Remark 6.10. As with analytic sets, it follows at once from the definitions and Xregare that, for 0 5 d 5 n, ,$(d) is open in X and, hence, that open in X . Thus, C X and X,,, are closed in X . The following result is from

22,

and

23,

Theorem 7.2 and Remark 7.3..

Theorem 6.11. Let X be a non-empty semianalytic subset of M . Suppose is semianalytic and, thus, so are E X , Xreg, that 0 5 d <_ n. Then, and X,,, . I n addition, dim X,,, < dim X .

i(d)

2,

114

7. Partitions and Stratifications For the remainder of these notes, it will be necessary to consider both the smooth and semianalytic case. Thus, below, when we are in the smooth case, X will denote a subset of a smooth, connected ambient manifold M and, when we are in the analytic/semianalytic case, X will denote a semianalytic subset of the connected analytic manifold M. If N is a smooth (resp., analytic) manifold, then we define f : X + N to be smooth (resp., analytic) if and only if it extends to a smooth (resp., analytic) function on an open subset of M. One approach to describing a singular subset X C M is to “subdivide” or “partition” the set into “smooth pieces”, and then try to describe how these pieces are put together to form the analytic subset. As a first attempt 0 at such a partitioning, one would proceed inductively: write X = X U EX, 0

and then write EX =(EX) U C ( C X ) ,and so on. If X is complex analytic, then Item 1 of Theorem 5.29 tells us that EX is complex analytic, and so this “inductive” process works well. However, Example 5.31 tells us that, for a real analytic X , C X need not be real analytic. On the other hand, Theorem 6.11 says that if X is semianalytic, then EX is also semianalytic. Thus, we make the following definition (see 23). Definition 7.1. A collection S := {S,}, of non-empty subsets of M is a smooth (resp., semianalytic) partition of X if and only if (1) X is the disjoint union of the Sa; (2) Each S, is a smooth (resp., analytic) submanifold of M and is a connected (semianalytic) subset of M ; (3) S is locally finite.

Given a smooth (resp., semianalytic) partition S of X , we refer to the elements of S as strata of X. A smooth (resp., semianalytic) stratification S of X is a smooth (resp., semianalytic) partition of X which satisfies (the Condition of the and Frontier): if S,, Sp E S, S, # So, and S, n # 0, then S, G dim S, < dim So. A smooth (resp., semianalytic) partition/stratification S’ of X is a refinement of a smooth (resp., semianalytic) partition/stratification S of X if and only if the strata in S are unions of the strata in S’. If X is a complex analytic subset of M , and S is a semianalyic stratification of X , then, for all S E S, in addition to S being a complex analytic

115

submanifold of M , 3 is an irreducible complex analytic subset of M , and a complex analytic subset of M . Therefore, we refer to semianalytic stratifications of complex analytic subsets X of M simply as (complex) analytic stratifications of X .

S - S is

Note that, in the Condition of the Frontier, it is irrelevant whether the closures are in X or in M ; however, using closures in X , the Condition of the Frontier says precisely that the closure of a stratum S, is equal to the union of S, and smaller-dimensional strata. Also, note that, in our definition, we have required the strata to be connected; not all authors require this. Example 7.2. Consider the analytic set X := V ( x y ) the x- and y-axes. Then,

R2 consisting of

s:= ( V ( 4 , { ( x , O ) I 2 > 01, ( ( 2 7 0 ) I x < O}} is a semianalytic partition, but is not a stratification. One must refine the partition by including the origin, in order to make the Condition of the Frontier hold. Thus, S’ :=

{{(O,Y) I Y > 01, {(O,Y) I Y < 01, {(GO)

I 2 > 01,

{(x,O) I z < O), (01)

is the “best”, i.e., most coarse, stratification of X . Definition 7.3. Let S be a partition of X , let N be a smooth (resp., analytic) manifold, and let f : X -+ N be a smooth (resp., analytic) function. Then, the stratified critical locus of f (with respect to S) is C s f := USES C(f1,). Naturally, we define a stratified critical value of f (with respect to S) to be be an element of f (CSf ) . We leave the proof of the following generalization of Theorem 6.7 as an exercise. Theorem 7.4. Let S be a semianalytic partition of X , and let f : X -+ A be a n analytic function. Let p E X . Then, there exists a n open neighborhood W of p in X such that WnCsf C f -’( f ( p ) ) , i.e., locally in X , the stratified critical values off are isolated. If, in addition, f i s a proper function, then the set of stratified critical values off is discrete.

116

The following theorem is in

22

and is Corollary 2.11 of

23.

Theorem 7.5. Let { X i } i be a locally finite collection of semianalytic subsets of M . Then, there exists a semianalytic stratification S := {S,}, of M which is compatible with { X i } i , i.e., each X i is a union of elements of S. Remark 7.6. Let X be a semianalytic subset of M . Then, one easily concludes from Theorem 7.5 that there is a semianalytic stratification S of 0 M such that M - X and the connected components of x are strata, and X is a union of strata. Therefore, S’ := { S E S I S GX} is a semianalytic stratification of X .

As an example of how one can use Theorem 7.5, we leave it as an exercise for the reader to prove: Corollary 7.7. Let X be a semianalytic subset of M . Let S be a semianalytic stratification of X . Let p E X , and let S be a stratum in S such that S has maximal dimension among all strata contained p in their closures. Then, S 5 0 I n particular, x is dense in X .

i.

A stratification is useful since it decomposes a space into pieces which are smooth manifolds. However, to understand the original space better, one would like to know how a given stratum behaves as it approaches points in another stratum. The classic conditions to desirelrequire are the Whitney conditions: Whitney’s condition a) and condition b). See 2 5 , 26, 27, and 28. Definition 7.8. Let S be a smooth (resp., semianalytic) partition of X . Let S,, Sp E S, and let p E S,. Fix a local coordinate system for M at p . A Whitney a) sequence in So at p is a sequence of points pi E So such that p i 4 p and the tangent spaces TpiSp converge to some ‘ I in the appropriate Grassmanian. A Whitney b) pair of sequences in (Sp,S,) at p is a pair of sequences of points pi E So and qi E S, such that {pi} is a Whitney a) sequence in Sp at p , qi + p and the lines (using our fixed coordinate system) PiQi converge to some line I. The pair (So, S,) satisfies Whitney’s condition a) at p if and only if, for all Whitney a) sequences { p i } in So at p , TpS, C_ lim TpiSp. 2-00

117

The pair (So, S,) satisfies Whitney’s condition b) at p if and only if, for all pairs of Whitney b) sequences {pi}, { q i } in (So, S), at p , lim PiQi C z-+m lim TpiSp. i+oo

The Whitney conditions are independent of the choice of local coordinates on M , and are vacuously satisfied if p @ G. The smooth (resp., semianalytic) partition S is a smooth (resp., semianalytic) W h i t n e y a ) partition of X if and only if, for all So, € S, the pair (Sp,S,) satisfies Whitney’s condition a) at each point of S,. If S is, in fact, a stratification, we naturally refer to S as a W h i t n e y a) stratification. The smooth (resp., semianalytic) partition S is a smooth (resp., semianalytic) W h i t n e y b) partition of X if and only if, for all So, € S, the pair (So, S a ) satisfies Whitney’s condition b) at each point of S,. By 27, Proposition 2.4 and Corollary 10.5, a Whitney b) partition also satisfies Whitney’s condition a) and the condition of the frontier; hence, we generally refer to a Whitney b) partition as a W h i t n e y stratification.

s,

s,

Example 7.9. Consider a family of nodes degenerating to a cusp, given by V(f)C A3, wh-ere A is JR or C and f = y2 - z3- tz2.

Then, CV(f)= V(z,y), and {V(f)- V(z,y), V(z,y)} is a semianalytic stratification of V(f). However, this Stratification is not a Whitney a) stratification; the tangent planes to V(f)- V(z,y) along the line where y = 0 and x = -t approach a plane which does not contain ToV(x,y). Now consider another family of nodes degenerating to a cusp, given by

V(g)C_ A3, where g = y2 - z3- t2x2.

118

Then, C V ( g ) = V ( x ,y ) , and { V ( g ) - V ( z ,y ) , V(z, y)} is a semianalytic Whitney a) stratification of V ( g ) . However, this stratification is not a Whitney stratification. We leave it as an exercise for the reader to show that Whitney’s condition b) fails at the origin. The following proposition is an easy exercise. Proposition 7.10. Let N be a smooth manifold and let f : X -+ N be a smooth function o n X . Suppose that S is a Whitney a) partition of X . Then, Cs f i s closed in X . The following strengthened version of Theorem 7.5 follows from

22.

Theorem 7.11. Let { X i } i be a locally finite collection of semianalytic subsets of M . Then, there exists a semianalytic Whitney stratification S := {Sff}, of M which is compatible with { X i } i , i.e., each X i i s a union of elements of S .

Remark 7.12. In the general smooth case, one is not guaranteed that Whitney stratifications exist. We could look at subanalytic subsets of M , for which subanalytic Whitney stratifications always exist (see 29 and 30), but we do not wish to discuss this generalization here. Henceforth, we fix a smooth (resp., semianalytic) Whitney stratification S of X . In the smooth case, the existence of a Whitney stratification is a new condition which we place on the space X . The following proposition is a good exercise.

119

Proposition 7.13. Let N be a smooth (resp., a n analytic) submanifold of M which transversely intersects a stratum SOE S at a point p E SO. Then, there exists an open neighborhood U of p in M such that, f o r all S E S , U n N transversely intersects U n S in U ;in addition, in evey such neighborhood U ,{U n S n N 1 S E S } is a smooth (resp., semianalytic) Whitney stratification of Z4 n X n N .

We naturally refer to the stratification {U n S n N 1 S E S} above as the stratafication induced by the transverse intersection. We now wish to consider definitions that involve “closed balls in M”. Locally, we could simply use closed balls with respect to any coordinate choice. However, we wish t o compare the structure of Whitney stratified spaces at different points in a given stratum, and thus will need to take closed balls at various points. Hence, we now assume that M is endowed with a Riemannian metric. By taking infima of lengths of piecewise smooth paths between pairs of points, this Riemannian metric induces a global (topological) metric r : M x M -+ R. The closed ball of radius E > 0 centered at a point p E M is, of course, B,(p) := {x E M 1 r ( x , p ) 5 E } . We let S,(p) denote the sphere

a(PI.

We shall also need to use the notion of the abstract cone on a set. If Y is a topological space, then, by cone(Y), we shall mean the pair of spaces (Y x [O, 1]/Y x {O}, Y x {0}), where Y x (0) is the equivalence class which is normally referred to as the cone point. The following result follows from

22, 27, and

$1.4 of

28.

Theorem 7.14. Let p E X . Then, for all suficiently small E > 0, S,(p) transversely intersects all of the strata of S , and there is a homeomorphism h, : cone(S,(p) n X) 4 ( B , ( p ) n X , { p } ) which ”presemes the strata”, i.e., for all S E S , h,((S,(p) n S ) x (0,1]) G S , and which is a diffeomorphism when restricted to each (S,(p) n S ) x (0,l) f o r S E S . I n particular, X is locally contractible. The stratified homeomorphism-type of S,(p) n X is independent of the choice of the Riemannian metric o n M and of the choice of small E , i.e., i f S,(p) and S:,(p) are spheres centered at p with respect to possibly different Riemannian metrics and of different radii, then, for both E and E‘ sufficiently small, there is a homeomorphism k : S,(p) n X -+ S:, ( p ) n X such that, f o r all S E S , k(S,(p) n S ) = S:,(p) n S and kls,(p)ns is a diffeomorphism.

120

Definition 7.15. The stratified homeomorphism-type of S,(p) n orem 7.14 is called the real link of X at p .

x in The-

Note that Theorem 7.14 implies that B,(p) n X , itself, has a canonical Whitney stratification for small E > 0. We now wish to give a substantial generalization of the Theorem of Ehresmann (Theorem 3.4). First, we need another definition. Let N be a smooth manifold.

Definition 7.16. Let f : X -+ N be a smooth function on X . Then, f is a stratified submersion (with respect to our fixed stratification S) if and only if, for all S E S, fl, is a submersion. The complete proof of the following theorem can be found in

27

Theorem 7.17. (Thorn’s first isotopy lemma) Suppose that f : X 4 N is a smooth, proper, stratified submersion. Then, f is a (topological) locally trivial fibration, and the local trivializations can be chosen to respect the strata and to be diffeomorphisms when restricted to strata. We wish to be clear about the conclusion of Thorn’s first isotopy lemma. Suppose that f : X + N is a smooth, proper, stratified submersion, and let q E im f . By Proposition 7.13, {S n f - l ( q ) I S E S} is a smooth Whitney stratification of f - ’ ( q ) . The conclusion of Theorem 7.17 is that there exists an open neighborhood U of q in N and a homeomorphism h : f-’(U) -+ U x f - l ( q ) such that flf-l(u) = T o h, where T is the projection from U x f - l ( q ) onto U ; moreover, for all S E S, the restriction of h is a diffeomorphism from S n f-’(U) to U x ( S n f - l ( q ) ) . Below, we will describe this type of situation by saying that f-’(U) and U x f - l ( q ) have the same stratified topological-type, and the same smoothness structure along the strata. As we discussed briefly earlier, the Theorem of Ehresmann is proved by integrating smooth vector fields. Thorn’s first isotopy lemma is also proved by integrating vector fields; however, these “controlled” vector fields need not even be continuous. See 27. The following corollary tells us that X is topologically trivial along each Whitney stratum, and that the topological-type is determined by the

121

normal slice to the stratum; see 28, 51.4. Recall that our strata are assumed to be connected.

Corollary 7.18. Let p E SOE S . Let U be an open neighborhood of p in M , and let N be a smooth submanifold of U ,which transversely intersects all of the strata of S in U ,such that N n SO= { p } (and, hence, dim N = dim M - dim S). Then, for all suficiently small E > 0 , the stratified topological-type and l B , ( p ) ,N n X n smoothness structure along the strata of the pair ( N n X r S e ( p ) ) is independent of the point p in So, the choice of the Riemannian metric o n M , the transverse submanifold N , and 6. I n addition, for a fixed choice of all of the above data, there exists a (nonopen) neighborhood T of p in X , a n open neighborhood W of p in SO,and a stratum-preserving homeomorphism h : W x ( N n X nB , ( p ) ) 4 T which is a diffeomorphism when restricted to each stratum and is the “identity map” on { p } x ( N n X n B,(p)). I n particular, the stratified topological-type of X is trivial along a Whitney stratum. Definition 7.19. The N n x n B , ( p ) above (or, sometimes, simply N itself) is referred to as a normal slice to SO at p , and N n X n S,(p) is called the real link of So at p (with respect to N at radius E ) . The stratified topological-type, and smoothness structure along the strata, of the pair ( N n X n B , ( p ) ,N n X n S,(p)) is called the normal data of SO(in X ) . Remark 7.20. Suppose that p E SO E S. Then we may simply declare { p } to be a “point-stratum” by refining the Whitney stratification S; one defines a new Whitney stratification S’ := 1s E s I

# So} u {So - { P ) , {P}).

In this sense, the real-link defined in Definition 7.15 can be considered the real link of a stratum.

8. Basic Stratified Morse Theory

Thom’s first isotopy lemma is a generalization of the Theorem of Ehresmann to the Whitney stratified case. For the remainder of these notes, we will discuss Goresky and MacPherson’s generalization of Morse Theory to the Whitney stratified case, as presented in 31 and 28.

122

It is easy to state an intuitive form of the main theorem of stratified Morse Theory: at a well-behaved isolated critical point p of a smooth function f on a Whitney stratified space, the local Morse data, which measures how the topology changes at p with respect f , is a product of tangential Morse data and Morse data which is normal to the stratum containing p . Making this statement precise is the goal of the remainder of these notes. We now assume that X is a semianalytic subset of a real analytic Riemannian manifold M , that S is a semianalytic Whitney stratification of X,and that f : X + R is a proper function which extends to a real analytic function f’ on an open neighborhood of X in M . We fix an SOE S and a point p E SO. Remark 8.1. We could, in fact, deal with a more general situation; the condition that we need is that p is a nondepraved critical point of f ; see 28, $2.3. However, we do not wish to go into the details of this definition. By s2.4 and $2.6 of 28, isolated critical points of real analytic functions on real analytic manifolds are nondepraved. Definition 8.2. The point p is nondegenerate (with respect to f and S) if and only if, for all S E S such that S # So and SO 3,for all sequences pi E S such that pi -+ p and TpiS converges to some 2,it follows that d p f ” ( 2 )f 0. The point of saying that p is nondegenerate is that it implies that small perturbations of f will not have critical points near p , unless those critical points are on SOitself.

return to our earlier notation: for b E R, we let xg, := f-’((-oo, b]). In addition, if a , b E R, we let X[a,b]:= f - ’ ( [ a , b ] ) , and

we

X, := f - l ( a ) . A technical issue arises in what follows: if D is an arbitrary subset of X , we still wish to keep track of the intersections of strata with D ,even when this does not produce a stratification of D. Definition 8.3. Let A , B X . Then, an S-function g : A -+ B is a function such that, for all S E S, g ( A n S ) & B n S. Naturally, an S-homeomorphism h : A -+ B is a homeomorphism such that h and h-’ are S-functions, i.e., for all S E S, h(A n S ) = B n S.

123

We say that A and B have the same S-topological-type if and only if there exists an S-homeomorphism h : A + B.

We now assume that p is a nondegenerate, isolated critical and that p is the only stratified critical point with point of fi, critical value v := f ( p ) . Let 6 > 0 be small enough so that the only stratified critical value off in [v-6, v+6] is v (which we may assume by Theorem 7.4); thus, we are assuming that C s f nf -l( [v-6, v+6]) = { p } . Note that this implies that Xv-6, Xv+6, X l v - 6 , and Xl,,+a all have induced Whitney stratifications. The following is a combination of Propositions 3.5.3 and 3.6.2 of

28.

Theorem 8.4. For all sufficiently small E > 0, f o r all sufficiently small 6‘ > 0 (small compared t o E ) , the S-topological-type of the pair ( B E ( p )n X[v-6t,u+6t~, B , ( p ) nXv-6t)i s independent of the choice of the Riemannian metric and the choices of E and 6’. If N i s a smooth (not necessarily analytic) normal slice t o SO at p , then, for all suficiently small E > 0, f o r all suficiently small 6’ > 0, the S-topological-type of the pair ( N n B€(P)n X[v--6’,v+6’], N n BdP) n Xv-SJ) is also independent of the choice of the Riemannian metric and the choices of E and 6‘.

Definition 8.5. The S-topological-type of the pair

(B€(P) n X[v-6’,v+6’],

n xv-6’)

above is the local Morse data for f at p . The S-topological-type of the pair

( N n B ~ Pn)X[v--6t,v+6t~, N n &(PI

nX , - S ~

above is the normal Morse data for f at p. The local Morse data for fl, at p is the tangential Morse data for f at P. Note that, in Definition 8.5, one may assume that 6’ 5 6; moreover, by re-choosing 6 after E , one may assume that 6‘ = 6. However, below, it is important that 6 does not depend on the choice of E .

124

The theorem below is Theorem 3.5.4 of 28. The point of this theorem is that the change in the topological-type as one passes from x 5 v - 6 to X5wf6 is a result of activity in a small neighborhood of the unique stratified critical point in x[w-6,w+6]. Theorem 8.6. The stratified space Xsw+6is obtained from x<+6 by a stratified attaching of the local Morse data f o r f at p; more precisely, with E and IS as in Theorem 8.4 and Definition 8.5, there is a n S-embedding j : B,(p) nX,,-p + Xv-6 and an S-homeomorphism ( i n the obvious sense) to the identification space from

Therefore, to understand the change in the topology as one passes from x<w-,5 to Xsw+6,it is enough to understand the local Morse data of f at P-

We are now ready to state the main theorem of stratified Morse Theory. We use our notation from Theorem 8.4 and Definition 8.5. Recall that the product, (P,Q) x ( J ,K ) , of two pairs of spaces is defined to be

( P x J , ( P x K ) U (Q x J)). Theorem 8.7. (The Fundamental Theorem of Stratified Morse Theory) Let (A,B ) equal the local Morse data (B,(p) flX[v-6’,~+6’], &(I)) n xw-6t) f o r f at p . Let (P,Q ) equal the tangential Morse data

(B,(P)n (SO)[w-6’,~+6‘], &(P) n (SO)w-6’) for f at p . Let ( J ,K ) equal the normal Morse data

( N n BE@)n X [ W - ~ ~ , WN + Pn]Be@) , n xw-6’)

f atp. Then, there is a homeomorphism of pairs h : (A,B ) -+ (P,Q ) x ( J ,K ) such that, f o r all S E S , h ( An S ) P x S, i.e., local Morse data is, u p to homeomorphism, the product of the tangential Morse data and the normal Morse data.

for

The reader needs to understand the point of this theorem. The tangential Morse data is Morse data on an analytic manifold; this is the classical situation that we discussed earlier. The tangential Morse data at an isolated critical point is analyzed by perturbing f slightly (as in Theorem 4.9, for

125 instance) and thereby “splitting” the isolated critical point into a collection of non-degenerate critical points, and then applying Theorem 4.5. The normal Morse data is the really new piece of d a t a that needs t o be understood in order for one t o apply the Fundamental Theorem of Stratified Morse Theory. This data can, in fact, be very complicated. However, a stunning thing happens in the case where X is a complex semianalytic subset of a complex analytic manifold, and is endowed with a complex analytic Whitney stratification: the normal Morse data is independent of the function f. This fact underlies much of the beauty of applications of stratified Morse Theory t o complex stratified spaces.

References 1. Spivak, M., A Comprehensive Introduction to Differential Geometry (Publish or Perish, Inc., 1970). 2. Rudin, W., Principles of Mathematical Analysis (McGraw-Hill, 1953). 3. Griffiths, P. and Harris, J., Principles of Algebraic Geometry (Wiley, 1978). 4. Ehresmann, C., Colloque de Topologie, Brmxelles , 29 (1950). 5. Munkres, J., Topology: a first course (Prentice-Hall, 1975). 6. Bott, R. and Tu, L., Differential Forms in Algebraic Topology, Grad. Texts in Math., Vol. 82 (Springer-Verlag, 1982). 7. Steenrod, N., The Topology of Fibre Bundles (Princeton Univ. Press, 1951). 8. Milnor, J., Morse Theory, Annals of Math. Studies, Vol. 51 (Princeton Univ. Press, 1963). Based on lecture notes by Spivak, M. and Wells, R. 9. Brocker, Th. and Janich, K., Introduction to differential topology (Cambridge Univ. Press, 1973). Translated by Thomas, C. B. and Thomas, M. J. 10. Milnor, J., Lectures o n the h-Cobordism Theorem, Mathematical Notes, Vol. 1 (Princeton Univ. Press, 1965). Notes by Siebenmann, L. and Sondow, J. 11. Matsumoto, Y., A n Introduction to Morse Theory, Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs, Vol. 208 (AMS, 1997). Translated by Hudson, K. and Saito, M. 12. Milnor, J., Singular Points of Complex Hypersurfaces, Annals of Math. Studies, Vol. 77 (Princeton Univ. Press, 1968). 13. Bochnak, J., Coste, M., and Roy, M.-F., Real Algebraic Geometry, Ergeb. der Math., Vol. 36 (Springer-Verlag, 1998). 14. Hartshorne, R., Algebraic Geometry, Grad. Texts in Math., Vol. 52 (SpringerVerlag, 1977). 15. Mumford, D., Algebraic Geometry I, Complex Projective Varieties, Grund. der math. Wissen., Vol. 221 (Springer-Verlag, 1976). 16. Krantz, S. and Parks, H., A Primer of Real Analytic Functions, Basler Lehrbucher, Vol. 4 (Birkhauser, 1992). 17. Lojasiewicz, S., Introduction t o Complex Analytic Geometry (Birkhauser, 1991). Translated by Klimek, M. 18. Griffiths, P. and Adams, J., Topics in Algebraic and Analytic Geometry, Mathematical Notes, Vol. 13 (Princeton Univ. Press, 1974).

126 19. Cartan, H., Bulletin de la SOC.Math. de France 85, 77 (1957). 20. Narasimhan, R., Introduction to the Theory of Analytic Spaces, Lecture Notes in Math., Vol. 25 (Springer-Verlag, 1966). 21. Whitney, H., Complex Analytic Varieties (Addison-Wesley, 1972). 22. Lojasiewicz, S., Ensemble semi-analytiques (IHES Lecture notes, 1965). 23. Bierstone, E. and Milman, P., Publ. math. de I.H.E.S. 67,5 (1988). 24. Looijenga, E., Isolated Singular Points on Complete Intersections (Cambridge Univ. Press, 1984). 25. Whitney, H . , Ann. of Math. 81, 496 (1965). 26. Whitney, H., Local properties of analytic varieties, in Differential and Combinatorial Topology, (Princeton Univ. Press, 1965), pp. 205-244. S. Cairns, ed . 27. Mather, J., Notes on Topological Stability, Notes from Harvard Univ., (1970). 28. Goresky, M. and MacPherson, R., Stratified Morse Theory, Ergeb. der Math., Vol. 14 (Springer-Verlag, 1988). 29. Hardt, R., Trans. A M S 2 1 1 , 57 (1975). 30. Hironaka, H., Subanalytic Sets, in Number Theory, Algebraic Geometry and Commutative Algebra, (Kinokunya Tokyo, 1973), pp. 453-493. volume in honor of A. Akizuki. 31. Goresky, M. and MacPherson, R., Stratified Morse Theory, Proc. Symp. Pure Math. Vol. 40 (AMS, 1983), pp. 517-533.

127

On Milnor’s fibration Theorem for real and complex singularities Jose Seade * Imtituto de Matemdticas, UNAM, Unidad Cuernavaca, A . P. 873-3,Cuernavaca, Morelos, Mkxico. *E-mail: [email protected]

Milnor’s fibration theorem for complex singularities is a key-stone in singularity theory. This is a result about the topology of the fibres of analytic functions near their critical points. This result, proved in the late 1960’s, has given rise to a vast literature and it has gripped the attention of many people for a long time. In this expository article we give an introduction to Milnor’s theorem as well as to some of its generalisations and refinements that have appeared in the literature. We focus on explaining the main ideas in each of the topics we envisage and we refer to the literature for details and complete proofs. This article does not pretend to be comprehensive of the subject. In fact this is threaded towards the end of presenting certain aspects in which we have been particularly interested in the last few years, and in which we have made some contributions, both by myself and in collaboration with several colleagues. Yet, we do cover in this work a rather wide spectrum that allows us to give an overall picture of some of the developments in this branch of singularity theory. We first motivate the fibration theorem by proving it in a concrete example, the case of the celebrated Pham-Brieskorn singularities, where things are much simpler and yet one has the whole complexity and richness of the situation in general. Understanding this example helps us to grasp better the general case. We then present Milnor’s theorem for holomorphic functions in its two usual ways: as a fibration on a small sphere around a critical point, and also as a fibration of the so-called Milnor tube. As

*Supported by I C T P (Italy), C O N A C Y T and DGAPA-UNAM (Mexico)

128

mentioned before, we only sketch here some of the main ideas and refer to the literature for complete proofs. In section 3 we mention briefly generalisations of Milnor's theorem that have been given by various authors, such as Helmut Hamm, L2 Diing Trdng and others. We then look, in $4, at the case of real analytic functions with a Milnor fibration. We explain briefly Milnor's theorem for real singularities and two of its main differences with the complex case, namely: i) that the theorem holds only for analytic map-germs f : (IRn+k,Q)+ ( I R k , O ) which are submersions on a punctured neighbourhood of Q E IRn+', a rather stringent hypothesis that we call the Milnor condition for the map-germ; and ii) that even if f satisfies the Milnor condition, it is not always true that the projection map of the associated fibration can be taken to be the obvious map f / l l f 11, as in the complex case. When this is also satisfied we say that f satisfies the strong Milnor condition. In $5 we study a specially interesting family of real analytic singularities with the strong Milnor condition, called the twisted Pham-Brieskorn singularities in [39]. These have a rich geometry behind, coming from the theory of holomorphic vector fields, and I believe that the understanding of their associated Milnor fibrations can provide interesting new insights on some aspects of the geometry of singularities. The difference between satisfying the Milnor condition and the strong Milnor condition was first addressed by Jacquemard in [16], though this terminology was introduced later in [35]. The technique used in [37], and explained in $5, to prove that the twisted Pham-Brieskorn singularities satisfy the strong Milnor condition was used later in [34] by Ruas and Dos Santos to improve the results of [16] using also Bekka's (c)-regularity. This actually gives an interesting method for studying the geometry of analytic maps into IR2 that we describe in $6. This method is used in $7 for f holomorphic maps (C:",Q)--+ (C,O), where it yields a slight refinement of the classical Milnor fibration theorem for complex singularities. We obtain a fibre bundle IB, \ f - ' ( O ) 4 S1,for every sufficiently small ball BEaround 0 , whose restriction to dIB, is the classical Milnor fibration on the sphere, and its restriction to a Milnor tube IB, n f-l(dlD,) is the other classical fibration (up to multiplication by the scalar q ) . So this has the nice feature of unifying the two classical fibrations. A particularly interesting class of real analytic maps is provided by maps Cn 4 C of the form f g with f , g holomorphic functions. This type of maps have already appeared in articles by N. A'Campo [l],L. Rudolph [14,36] and A. Pichon [31], as well as in [32]. In $8 we present a fibration theorem

129

from [33]for such germs. A key point, first observed in [14], is noting that away from the zero loci V(fg) o f f and g the map f j j / l f i j l equals the map (f /g)/lf/gl. Then one can follow the proof of Milnor’s fibration theorem in [26], essentially step by step, and arrive to a fibration theorem for maps fij. Of course this can be also regarded as a result for meromorphic mapgerms, and a similar theorem has been also proved in [4] by the same method. In [33] we ask that f i j has an isolated critical value, while in [4] this condition is replaced by demanding the germ f /g to be semi-tame. We remark that both of these theorems yield fibrations 9, \ V(fg) -+ S1. Local fibrations for meromorphic germs have been studied previously by GuseinZade, Luengo and Melle in a series of articles (see for instance [9,10]); these are local fibrations of “Milnor tubes”, not on the sphere. In [5] we study the relation between all these local fibrations. In particular, for semitame germs in two complex variables we show that the fibre of the Milnor fibration on the sphere is essentially the connected sum of the fibres of the local fibrations of [9,10]at 0 and 00. Finally, in 39 we explain briefly one of the main results in [33]. So far we have discussed fibrations of the complement of the link of certain singularities, but we have not payed special attention to what happens near the link. There is actually a richer structure one can look at, the multilink structure. Beware that a link here has two distinct meanings. On one hand it means the intersection V n 8, of some analytic set V c (EN with a small sphere around a singular point. But a link also means a finite, disjoint union of knots (codimension 2 connected submanifolds) in some manifold M . For simplicity we restrict to the case M = S3,so a link L is a disjoint union of circles embedded in the 3-sphere, L = K1 U. . .UK,. A multilink means that we have fixed an orientation on each component Ki and we have assign a multiplicity ni E Z to each, with the convention that -ni Ki = ni (-Ki), where -Ki is Ki with the opposite orientation. A fibration of a multilink means a fibration of its complement whose behaviour near each Ki “takes into account” the multilink structure (see 39). For instance, if f : (C2,0) (C,O) is holomorphic and f = fTa is its decomposition into irreducible factors, then

-

nf=,

1

L f = IJ niKfi

1

i=l

is the multilink associated with f , where K f i = S2nft71(0), and the Milnor fibration f(x)/lf(x)l : Sz \ L f + S’ is a fibration of the multilink L f . In

$9 we briefly explain the corresponding theorem of [33]for germs fij.

130

I am most grateful to David Massey, Anne Pichon and LB Ddng T r h g for very enjoyable and helpful conversations and comments. I am also indebted to the ICTP at Trieste, Italy, and the ENS of Lyon, France, for hosting me and supporting me while I wrote this work. This article grew out from the notes for the lectures I gave at the passed Advanced School and Workshop on Singularities held at the ICTP of Trieste in Summer 2005, and I thank the organizers for inviting me to such an interesting meeting. Finally, I thank the referee for her/his kind, accurate and helpful comments.

CONTENTS $1:An example: Pham-Brieskorn singularities. $2: The classical fibration theorem of Milnor. $3: A glance at generalisations. $4:Real analytic germs with a Milnor fibration. $5: Twisted Pham-Brieskorn singularities. $6: A method for studying maps into R2. $7: A refinement of Milnor's classical theorem. $8: Singularities f and Milnor fibrations for meromorphic germs. $9: Fibrations of multilinks. 1. An example: Pham-Brieskorn singularities

Let us begin with an example which is the paradigm for Milnor's Fibration Theorem and was a motivation for it. Consider the Pham-Brieskorn polynomial f(g) = z?

+ . . . + zZn

, ai > 1,

where z= (ZO,..., z,). It is clear that the origin Q E is the only critical point of f, so the fibres & = f-'(t) are all complex n-manifolds for t # 0 and V = f - l ( O ) is a complex hypersurface with an isolated singularity at 0. We want to study the topology of V and of the K's.For this let d be the least common multiple of the ai and define an action I? of the non-zero complex numbers C* on (En+' by:

x . (zo, . . ,2,) f

Notice this action satisfies:

H

(Xd'%g,

. . . ,Ad'"" z,) .

131

f ( X . (ZO,.. . ,z,)) = Ad * f ( z o , * * * ,z,)

,

Hence V is an invariant set of the action and one has the following properties: Property 1.1. Restricting the action to t E IR+ we get a real analytic flow (or a vectorfield) o n whose orbits are real lines (arcs) which converge to 0 when t tends to 0 , they escape to 00 when t --+ 00 being transversal to all spheres around 0, and they leave V invariant (i.e., V is union of orbits).

en+'

Fig. 1. The conical structure

Property 1.2. Restricting the action to the unit circle {eie} we get an S1-

132

action o n (En+'

such that each sphere around Q is invariant and: f ( e i e . (zo, ..., z,)) = e i d e f ( z o , .. ,z,) ,

< = f ( z o , . . . ,z,)~ then multiplication by eie

that is, if we set transports the fibre

f-l(c)

into the fibre over eide.

c.

in

an+'

Property 1.3. The real analytic flow defined by restricting the C*-action t o t E IR+ has the additional property that for points in Cn+' \V, the argument of the complex number f(2) is constant o n each orbit, i.e., f(g)/lf(g)l= f(tz)/lf(tz)lfor t E IR+,and the norm of f(g) is an strictly increasing function o f t . Each of these three properties has important implications. The first property (1.1)implies:

+

Property 1.4. The variety V intersects transversally every (2n 1)-sphere 9, around the origin; hence the intersection K , = V n 9, is a smooth manifold of real dimension 2n- 1 embedded as a codimension 2 submanifold of the sphere 9;, Property 1.5. The flow determines a 1-parameter group of diffeomorphisms that preserve V , thus the diffeomorphism type of K , is independent of the choice of the sphere 9,; and Property 1.6. The embedded topological type of V in Cn+' is determined by the pair ( S , , K,); more precisely, the pair ((En+', V ) is homeomorphic to the (global) cone over the pair ( S T ,K,), with the origin Q being carried into the vertex of the cone. In the following sections we explain how the essential features of these properties extend to holomorphic functions in general. But in order to grasp better some important consequences of these properties, in this section we envisage only the Pham-Brieskorn singularities. The manifold K = K , (for some r ) is called the link of the singularity. The manifolds that arise in this way have been studied by several authors obtaining remarkable results. Notice that the link is a codimension 2 submanifold of the sphere, so it is a knot if it is connected (which is always the case when n > 1 or if V is irreducible, by [26]).This type of knots (S,, K ) are called algebraic knots, and the above properties 1.3 to 1.5 imply that the embedded topological type of V in (En+' is determined by the knot ( S , , K ) . We refer to Chapter 1 in [39]for an overview of the topic, including a large bibliography.

133

Fig. 2.

The trefoil knot determines the curve {x2 = v3}

On the other hand, for t # 0 the fibres f-'(t) are all complex manifolds, because 0 E C is the only critical value of f . Furthermore, the second property (1.2) above implies that for constant It1 = 6 > 0 the manifolds f - ' ( t ) are all diffeomorphic and we have a flow on the "tube" f-'(Ca) given by the $-action, where Ca = { t E C It1 = 6). This flow is transversal to all the fibres and carries fibres into fibres. Thus we have a (locally trivial) fibre bundle over the circle Ca:

I

f : f-l(Cg)

-

ca.

Let us focus our attention near the origin, say restricted to the unit ball B := B ~ c Cn+l. ~ + Since ~ v = f-'(o) meets Pn+l= dlB transversally, it follows that for 7 > 0 sufficiently small, all fibres f - ' ( t ) with It1 = 7 meet Szn+' transversally. Let Ca be the circle in (E around 0 of radius 6 with 77 > 6 > 0, and look at the set f-'(Ca). Since the action of 8' on is by isometries, it preserves the ball IB and determines a fibre bundle:

an+'

-

f : B n f-l(ca)

ca.

(1)

Notice that one can also restrict the map f to the unit sphere Szn+' and define

It is not hard to see that the map q5 has no critical points at all, so its fibres are smooth submanifolds of the sphere, and we see from the previous

134

Fig. 3.

The Milnor fibration

discussion that the S1 action r on (En+' preserves the unit sphere and leaves the link K invariant; thus it also carries its complement SZn+l \ K into itself. Furthermore, (1.2) tells us that the orbits of this action are also transversal to the fibres of 4 and carry fibres of &I into fibres of 4, showing that (2) is a locally trivial fibre bundle too. In other words, given the polynomial map f,we have already associated to it two different fibre bundles. One of them is given by (1) and the other by (2). We claim that these two fibre bundles are actually equivalent. To prove this, notice that we lose nothing if in (2), instead of removing K , we remove from SZn+l a compact tubular neighborhood N ( K ) = SZn+l n f-'(lD~) of the link, where D6 is the disc in (E with boundary CJ.Let us now define a diffeomorphism h : IB n f-l(C~) + SZn+l \ N ( K ) 7 0

as follows: for each point g in f-l(t)nIB, follow the orbit of the IR+-action that passes through till this orbit meets SZn+l at some point, say 2. Then define h(g) = 2. That this map is a well-defined diffeomorphism follows from Property (1.3); and h obviously extends as the identity to the points in ~ N ( Kc) SZn+l. Summarizing: Property (1.1) implies one has a conical structure on V.

135

Property (1.2) implies one has the fibrations (1) and (2), and Property (1.3) implies that these two fibrations are equivalent. Both of these are known as the Milnor fibration of f ; we indicate in the following section how they generalize to other situations. 2. The classical fibration theorem of Milnor

Consider now, more generally, a holomorphic function

(U

c C="+1,0).L(C,O),

defined on an open neighbourhood U of the origin in Cn+' with a critical value at 0 E C. Assume for simplicity that 0 E a="+'is the only critical point of f in U . Let V be the singular variety defined by f , i.e., V := { f - l ( O ) } ; thus V* = (V \ (0)) is a smooth complex manifold of dimension n. We know (by work of Milnor and others) that one has in this general setting similar properties to those of the Pham-Brieskorn singularities explained in the previous section, the main difference being that in general one must restrict the discussion to a "sufficiently small" neighbourhood of the singular point. Let us explain this briefly. First, one has that V is locally a cone [26]: there exists E > 0 sufficiently small so that given the ball IB, in centered at 0 of radius E , one can construct a vector field (flow) similar to the one in the previous section given by the It+-action: its orbits are transversal to all the spheres in B,centered at 0, and it leaves V n IB, invariant. Hence each sphere SEi in an+'centered at 0 of radius E' 5 E meets V transversally. The intersection K , = V n 9,is a smooth manifold of real dimension 2n - 1embedded as a submanifold of the (272 1)-sphere 8,. The diffeomorphism type of the manifold K , and the isotopy class of the pair (S,, K E )do not depend on the choice of the sphere S,; the pair ( B EK,) , is homeomorphic to the cone over the pair (S,, K,), where (IB, is the ball bounded by S, so that the topology of V near 0, and its embedding in (En+', are determined by the pair (S,, K,). As mentioned in Section 1, the manifold K = K, is called the link of the singularity and the pair ( S E ,K,) is called an algebraic knot, a name introduced by L6 Dung T r h g in 1971, [18]. One may thus consider the obvious map:

an+'

+

f

$=-:(S

If I

E

\ K, ) +S1.

Theorem 2.1. (Milnor [26]) This is a (locally trivial) Coofiber bundle.

136

Milnor gave two proofs of this theorem; we already had glimpses of both of them in the previous section; each proof brings out different insights and lends itself to different generalisations. Let us sketch the key-points in each of them. lStProof: This is along the lines of the above proof of (2). The idea is simple: first show that the map 4 has no critical points at all, so the fibres of 4 are all smooth, codimension-1 submanifolds of (9, \ K,); then construct a tangent vector field on ( S , \ K E )which is transversal to the fibres of 4 and the corresponding flow moves at constant speed with respect to the argument of the complex number &), so it carries fibres of 4 into fibres of 4. This proves one has a product structure around each fibre of 4. For this, 4 Is1 , if there to begin, Milnor shows that the critical points of (9, \ K E )--+ were such points, are exactly the points 2 = (20,.. . , 2,) where the vector (i grad log f ( 2 ) )is a real multiple of 2. To prove this, set

I ff 1

4(4) = -(2)

:= e"(4)

,

-

so one has: O(g) = Re

(-i log f ( 2 ) ).

An easy computation shows that given any curve 2 = p ( t ) in en+'\.f-'(O), the chain rule implies:

dO(p(t))/dt

=

dP Re ( -d(t t ) , i g r a d l o g f (4) ,

(3)

where (., .) denotes the usual hermitian product in en+'. Hence, given a based at g, the derivative of O ( 2 ) in the direction of vector v(g) in v(z) is:

en+'

Re ( ~ ( z ) , i g r a d l o g f ( z ) ) . Since the real part of the hermitian product is the usual inner product in ELzn, it follows that if ~(z) is tangent to the sphere S:,-', then the corresponding derivative vanishes whenever (i grad log f (z))is orthogonal to the sphere, i.e., when it is a real multiple of 2; conversely, if this inner product vanishes for all vectors tangent to the sphere then 2 is a critical point of 4, and the claim follows. Once we know how to characterize the critical points of 4 and how the argument of the complex number 4(2) varies as 2 moves along paths in 9, \ K , Milnor uses his Curve Selection L e m m a (see [26]) to conclude that 4 has no critical points at all. This part is a little technical and we refer

137

to Milnor's book (Chapter 4) for details. It follows that all fibres of q5 are smooth submanifolds of the sphere 9, of real codimension 1. In order to show that 4 is actually the projection map of a Coofibre bundle one must prove that one has a local product structure around each fibre. This is achieved in [26] by constructing a vector field w on 9, \ K satisfying: i) the real part of the hermitian product (w(z), i grad log f(z))is identically equal to 1; recall that this is the derivative of the argument of q5 in the direction of ~ ( 2 ) . ii) the absolute value of the corresponding imaginary part is less than 1:

IRe (W, g r a d k 7 f ( z ) ) I < 1. Consider now the integral curves of this vector field, i.e., the solutions p-( t ) of the differential equation dp/dt = w ( p ( t ) ) . Set eio(z) = $(z)as before. Since the derivative of e(z) ;It p-( t ) in the direction w@(t)) is identically equal to 1 we have:

e ( p ( t ) ) = t + constant. Therefore the path p ( t ) projects to a path which winds around the unit circle in the positive direction with unit velocity. In other words, these paths are transversal to the fibres of q5 and for each t they carry a point z E q5-l(ezt0) into a point in #-l(ei(to+t)). If there is a real number to > 0 so that all these paths are defined for at least a time to then, being solutions of the above differential equation, they will carry each fibre of 4 diffeomorphically into all the nearby fibres, proving that one has a local product structure and q5 is the projection of a locally trivial fibre bundle. Milnor proves this by showing that condition (ii) above implies that all these paths are actually defined for all t E R, so we arrive to Theorem 2.1. 0

2nd Proof: This has two big steps, the first is showing that given a map-

germ (C:"+',Q) (C,O) one has a fibre bundle as in (2): Theorem 2.2. Let 9, be a suficiently small sphere in Cn+' centered at and choose 6 > 0 small enough with respect to E . Let Ca = dlDg be the circle in C of radius 6 and centered at 0 , and set N ( E 6) , = f-l(Ca) nB E . Then:

0

is a C" fibre bundle.

138

The manifold N(E,6) is usually called a Milnor tube for f. Notice that if f has an isolated critical point at 0, as we are assuming in (2.2), then Thorn’s transversality implies that all the fibres f-l(t) with E >> It1 > 0 are transversal to S,, and this is what we mean by “6 > 0 small enough with respect to E” in the statement of the theorem. In fact this same statement and the theorem above, also hold for map-germs

(Cn+l,Q) (C!,O) with a non-isolated critical point at 0 E but in this case one must use Hironaka’s theorem in [15], that f has the Thom approperty, to conclude that the tube N(E,6) is transversal to 9,. Once we know this, the proof of (2.2) for non-isolated critical points is essentially the same as in the isolated singularity case. This was done by Li. in [17], where he actually extends (2.2) to all map-germs defined on complex varieties. Theorem 2.2 is essentially an extension of Ehresmann’s fibration lemma. The idea of the proof is the following. Notice first that 0 E C! is an isolated critical value o f f , and therefore given E >> It1 > 0 as above, one has that f is a submersion at each point of the Milnor tube N(E,6). Hence we can lift a vector field on the circle CSto a vector field on the tube N ( E ,S), transversal to the fibres of f. We may further choose this lifting to be integrable and tangent to N ( E6) , n S,, by transversality. Since the fibres are compact, we can assume that for each fibre F, there is a time to > 0 so that all solutions passing by F,, are defined for at least time t o .Then the corresponding local flow identifies F, with all nearby fibres, which are thus diffeomorphic to Fa and one has a local product structure, hence a fibre bundle. The second step for proving (2.1) in this way is to show [26, Ch. 51 that there exists a vector field on BE\ f-’(O) whose solutions move away from the origin being transversal to all the spheres around 0, transversal to all the tubes f-’(Ci) and the argument of the complex number f(g) is constant along the path of each solution. This allows us to “inflate” the Milnor tube f-’(aIDh) n B, in (2.2) to become the complement of a neighbourhood T(E,~ of )the link K in the sphere S E , taking the fibres of (2.2) into the fibres of the map 4 in Theorem 2.1. Then one must show that this fibration extends to T ( E6) , \ K,. Notice that (2.2) implies that the fibres of Milnor’s fibration are diffeomorphic to complex Stein manifolds of dimension n in (En+’,hence a theorem of Andreotti-Frankel implies they have the homotopy type of a CW-complex of middle-dimension n. This is also proved by Milnor in his book, where he proves too that the link K is always ( n- 2)-connected. When f has an isolated critical point at 0, Milnor proves more: the

139

fibres of 4 have the homotopy type of a bouquet ASn of n-spheres. The number of spheres in this wedge is by definition the Milnor number of f, an important invariant of f . For instance the Milnor fibre of the Morse singularity z," z; +. z i is diffeomorphic to the total space of the tangent bundle of the unit sphere S", so it has Milnor number 1.

+

3. A glance at generalisations

Several natural generalisations of Milnor's fibration theorem have been considered by various authors. We briefly mention here some of them.

i) As noticed by Hamm in [ll],Milnor's fibration theorem extends, with essentially the same proof, to the case of holomorphic functions defined on complex analytic spaces with an isolated singularity. In this more general setting the fibre still has the homotopy type of a CW-complex of middle dimension, but it may not be homotopically equivalent to a bouquet of spheres (see for instance the articles [12,13] of Hamm-Li5 about rectified homotopical, or homological, depth). Hamm's interest in [ll]was mainly to study complete intersection germs

f : (C"+"Q) 4 (C=",Q). The critical points off are the points where the rank of the jacobian matrix drops down, and their image A in C k is called the discriminant of f . It is known that if k > 1 and Q E C k is a critical value, then the sub-analytic set A c C kmay have dimension > 0 at 0. Hence, in general, one can not possibly have a fibration theorem over a small punctured disc ID6 around 0 E Ckl as one does when k = 1. However, if Q E C n f k is an isolated singularity of V = f-' (Q), then Hamm proved that one has a fibration over ID6 \ A analogous to that in (2.2). A key for understanding this fibration (see for instance [23] for details) is to notice that in this case one can choose the functions (f1, ..., fk) that define the germ (V,Q)so that

F = fCl(0)n ... n f&(O)

+

is a complete intersection germ of dimension n 1 with an isolated singularity, and the restriction of fk to F has an isolated critical point at Q; so one has the corresponding Milnor fibration. It is remarkable (Hamm's theorem) that for complete intersection germs the fibres have the homotopy type of a bouquet of spheres of middle dimension, just as for hypersurfaces.

140

We remark that this fibration theorem for complete intersection germs is false in general if the critical points of f are non-isolated in V, unlike the hypersurface case were the theorem holds for every map-germ. This fact was first noticed by LC Dting Tr6ng (see Le's example in [39]), and the reason behind (noticed also by LC) is that Hironaka's result in [15],that every holomorphic map into C has the Thom af-property, is false in general for maps into C kwith k > 1.

ii) Given .f : (aN, 0) .+ (C, 0) holomorphic and an analytic singular variety X c C N ,one may consider the restriction f of f to X . The concept of critical points of f on X makes perfect sense once we equip X with a Whitney stratification (this goes back to the work of Thom, Lazzeri, Goresky-MacPherson, LC and others). It is proved in [17] that one has in this case a fibration theorem as in (2.2);these are called Milnor-Le^fibrations and they have given rise to a vast literature. We refer particularly to LC's paper [19] and to [24,40,41], which contain remarkable generalisations of Milnor's theorem about the topology of the Milnor fibre. Somehow these spring out from the handle-body decomposition of the Milnor fibre given in [20].

(en,

iii) Consider polynomial maps f , g : 0) --f (C, 0) and the meromorphic map f/g defined away from the zero loci of f,g. In [9,10] Gusein-Zade, Luengo and Melle proved that one has a local fibration around each point in CP' , and they studied carefully some aspects of the corresponding local fibrations at 0 and m. These fibrations appear also in [4,33] and we discuss them briefly in section 8 below. iv) Consider real analytic germs f : (U c IR"+',Q) -+ (Rk,Q). This situation was first considered by Milnor himself in his book [26] and several authors have worked on this topic afterwards. This is explained in sections 4, 5 and 6 below (we refer also to [39] for more on the subject). A specially interesting case is when the real analytic map f is of the form h3 with h and g holomorphic, or a sum of functions of this type. These singularities are also discussed in section 8. 4. Real analytic germs with a Milnor fibration

For real analytic germs, Milnor's fibration theorem in [25,26] states:

Theorem 4.1. Let (R"+',Q) f,

(R',,),n 2 0 , be the germ of a real

141

analytic map with an isolated critical point at the origin. Then f o r every sufficiently small sphere 9, = dB, around Q E one has that the complement 9, \ K of the link K = f -l(Q)ns, fibres over the sphere Sk-’. The proof of this result is by noticing first that for 6 > 0 sufficiently small the tube f - ’ ( S : - ’ ) n IB, fibres over the sphere Sf’ c C of radius 6 (just as in (2.2) above), and then constructing a vector field that “inflates” this tube taking it to the complement of (a regular neighbourhood of) the link in the sphere; see [26] or [33] for details. When the map f is from C“+’ into C and is holomorphic, Milnor shows that one actually has a much richer structure: i) first, one does not actually need to have an isolated critical point of f to have such a fibration: here the critical value is automatically isolated and that is enough in this case to have a fibration. For real analytic germs, isolated critical value is not enough in general and we need to ask for the Thom upproperty (see [33]).

ii) for holomorphic germs the projection map Cp : S, \ K -+ S1 can be taken to be the obvious map $J = f / l f I; as Milnor shows in his book, this statement is false in general when f is not holomorphic, even if one does have the fibration in (4.1). In [16,34] the authors give sufficient conditions to insure that a map f as in (4.1) defines a fibration where the projection map can be taken to be f / l f l . We shall return to this point in 56. The geometry of the Milnor fibrations associated with holomorphic singularity germs has given rise to a rich literature, both within singularity theory itself, as well as in nearby areas like fibered (or Neuwirth-Stallings) knots and links, open-book decompositions, the results of Lawson and others about codimension 1 foliations, etc. That is not the case for real analytic germs. There are various reasons for this, in particular because it is difficult to find examples of real analytic singularities with an isolated critical point, and it is even harder to study their underlying geometry and topology. Several natural -related- problems arise, as for instance: i) find interesting families of real analytic germs with a rich geometry and a Milnor fibration; ii) relax the conditions in Milnor’s fibration theorems in order to include larger families. These questions have been addressed by several authors in various ways, and in the previous section we mentioned some of these. Let us discuss briefly some general facts about real analytic germs with a Milnor fibration as above. For simplicity we restrict the discussion to real analytic functions f : (IRn,Q) (IR2,0), n > 2. -+

142

Definition 4.1. We say that the map f satisfies the Milnor condition at 0 if the derivative D f (x)has rank 2 at every point x E U - 0, where U is an open neighbourhood of 0 E IR", i.e., if f is a local submersion at every point in a punctured neighbourhood of

0 E IR".

One has the above theorem (4.1) of Milnor for functions satisfying (4.1). For instance, every complex valued holomorphic function with an isolated critical point in its domain satisfies these conditions, and so does the composition of such functions with a real analytic local diffeomorphism in the source C"+', or in the target C , but these are somehow "fake" examples of real analytic functions. The interesting point is to find examples which are honestly real. As Milnor pointed out in his book, the hypothesis of D f having maximal rank everywhere near 0 is too stringent; the generic case is to have real curves in IR2 converging to (0,O) whose inverse image contains points where the jacobian matrix has rank less than 2. And even if we have that (0,O) E IR2 is an isolated critical value, for (4.1) we need more: 0 E IR" must be an isolated critical point. Milnor actually asked whether there exist %on-trivial" examples satisfying the condition of (4.1). This question was answered positively by Looijenga [22] for n even and by Church and Lamotke [6] for n odd, using Looijenga's technique. The key point in [22]is to show that if (S2,-l, K ) is a fibred knot with non-empty K , which is odd (in the sense that the antipodal map of S2m-1preserves K and the fibration of S2,-l \ K over S1 is equivariant with respect to the antipodal maps of both spheres), then one has that (Szrn-l, K ) arises from the isolated singularity of some real polynomial map. Furthermore, given such a fibred knot (S2m-1,K ) , the connected sum (S2,-l, K)#(S2"-l, K ) is again fibred and odd. Hence there exist many non-complex examples of isolated singularities satisfying the Milnor condition. However , essentially no explicit examples of such singularities were given. An explicit non-trivial example of a real analytic singularity satisfying the Milnor condition at 0, other than those of Milnor, was given by -+ C defined by A'Campo in [I].This is the map (21,21,

z1,

...,z,)

-

uv(e+ a) + 2 ;

+ ... + z;

,

(4)

which is not holomorphic due to the presence of complex conjugation. Later Perron in [28] proved that the figure-eight knot, which was known to be fibered, can also be realised by real algebraic equations. Lee Rudolph, in his review of Perron's article, gives a very nice set of equations that

143

+

+

define this knot: if z = x i y and w = u iv are complex coordinates for R4,then the figure-eight knot is the link of the 2-dimensional singularity defined by the polynomials: g(z, w ) = w3 - 3(x2

+ y2)(1+ i y ) w - 22,

f(2, w )

= g(z2, w ) .

In [35,37] there are given infinite families of singularities satisfying Milnor's condition, which are somehow in the same vein ils the example in equation (4). Before explaining these examples, let us look at a more subtle question for which we introduce the following notation from [35]:

Definition 4.2. Let f = ( f 1 , f z ) : (R",Q) -+ (R2,Q), n 2 2 , be an analytic function which satisfies the Milnor condition at 0. Let K = f;l(o)n fF'(0) n 9, be its link; f satisfies the strong Milnor condition at Q if for every sufficiently small sphere S , around Q the map

-f: S , - K

If I

--f

S1,

is the projection of a fibre bundle. As shown by Milnor himself in [25, p. 991, there exist maps that satisfy condition (4.1) but not the stronger condition in (4.2). So the question is: given a real analytic map-germ f satisfying the Milnor condition (4.1), when does it satisfy the strong Milnor condition? This question was first studied by Jacquemard in [16] and we return to this point in section 6 , but before doing so we describe below, as an example, a family of real singularities with a rich geometry and satisfying (4.2). 5. Twisted Pham-Brieskorn singularities

In the previous section we mentioned Looijenga's construction of real singularities having the Milnor condition. We also gave the corresponding examples of A'Campo and Perron. The first explicit family of non-trivial examples of real singularities satisfying the (strong) Milnor condition (4.2) were given in [37] (see also [35,39]).These are defined in R2" E Cn by: f(g) =

+ . + zznZuT,

, ai > 1, (5) is any permutation of (1,. . . ,n). Notice these are Z;LIEul

* *

where a = (al,.. . ,a,) reminiscent of the Pham-Brieskorn singularities f(2) = ':2 +. . .+ z:n and they are called in [39] twisted Pham-Brieskorn singularities; the permutation a being the twisting. A reason for this name is that when there is no

144

twisting, i. e., when c is the identity, these singularities are topologically equivalent to Pham-Brieskorn singularities, by [35]. The variety defined by such a polynomial map consists of the points in Cn where the holomorphic vector field:

is tangent to the spheres around the origin 0 E (En.In general, given a holomorphic vector field <(z)= Cy=lai(z) with an isolated singularity at 0, one can look at the holomorphic foliation F by complex curves that it defines, and consider the points of contact of this foliation with the spheres around the origin, i.e., the points were 3 is tangent to some sphere centered at 0. This variety is defined by the equation:

&

V, = {zE C nI (t(z),z)= 0) , where (<(z),g)= Cy'l ai(z)Ziis the usual hermitian product. Notice that Vt is therefore defined by two real analytic equations, which are the real and imaginary parts of the analytic map z-(<(g),zJ E C. One can prove in general that Vc is smooth of codimension 2 at a point z if and only if the contact of F and the sphere through z and center 0 is generic, and this is equivalent to saying that z is either a local minimal point or a saddle in its corresponding leaf in F (see [39, Chap. VI]). Besides this, it is rather hard (and interesting) to say anything about the varieties that one gets in this way (see [39]). For instance consider the linear vector field in C 3 ,

+

where the eigenvalues (X1,Xz,X3) are either ( l , i , l i), (l,i,-1 - i) or (1,i, 4).In the first case we have a vector field in the so-called Poincare' domain, which means that 0 E C is not contained in the convex hull of the eigenvalues. It is an easy exercise to show that in this case the only solution to the equation ( < ( t ) , g )= 0 is z= 0, hence V, = (0). In the second case we have a vector field in the Siege1 domain with generic eigenvalues. The variety Vc is a cone with vertex at 0 and base its intersection with the unit sphere, which is the link K t . Notice that the natural diagonal action of the 3-torus 'IT3 = S1 x S1 x S1 on C3 preserves V, and is by isometries, so it preserves the link K t ; it is easy to show that this action of T3 on Kc is free and transitive, hence Kt is T3.

145 In the third case the variety Vc is defined by the equations 1zll = 0 and 1221 = 1231, so it is a real 3-dimensional quadric in the plane { z z , z 3 } . Surprisingly, for the vector fields in (6) one has an amazing regularity. The following more general theorem is proved in [35,37] (see also [39]):

Theorem 5.1. Let f : Cn -+ C be a polynomial map of the form: f(z1,. .

I

,

2,)

= A1 zal 1

-

ZOl

+ . . . + A,

22 Q, ,

where the X i are arbitrary non-zero complex numbers, all ai are 2 2 and a = (01,. . . ,a,) is a permutation of (1,.. . ,n). Then: (i) The variety V = f-'(O) is a (codimension 2) real algebraic complete intersection in C n with an isolated singularity at 0. (ii) f satisfies the strong Milnor condition at Q. We notice that this result is extended in [35]by allowing some exponents to be 1 in certain cases, depending on the permutation 0. The proof of the first statement in the theorem is by a straightforward computation, showing that Q is actually the only point where the rank of the jacobian matrix is less than 2. This implies (i) and also implies that f satisfies the Milnor condition at 0. The proof that these singularities satisfy the strong Milnor condition uses a construction that actually proves more. Given f we consider a 1parameter family of real analytic maps he, 8 E [0,T ) , where each he is the composition of f followed by the projection into the line in C that has angle 8 with the positive real axis. This family fills out the entire C n by real analytic hypersurfaces Xe,which meet at V = f - l ( O ) and they have each a unique singularity at Q E ( E n . One can define a diagonal action of IR+ on a=", whose orbits are transversal to all the spheres around Q and leave each Xe invariant. Thus one has a global conical structure: each Xe meets transversally all the spheres around Q and its link is a smooth codimension 1 submanifold of the unit sphere. Furthermore, one has also an S1 action on C" that permutes the varieties X e , leaving V invariant, and endows the whole space ( E n \ V with the structure of a fibre bundle over S1;each fibre is transversal to all the spheres around Q and each pair of antipodal fibres is naturally glued together along V forming the analytic space Xe. The specific R+ and S1 actions can be described explicitly as follows: look at the permutation (T of the indices (1, ..., n} in the definition of f and split it into cycles. Then split C n into linear subspaces according to the cycles of (T.A 1-cycle corresponds to a monomial of the form Z;~.Z~; a

146

+

two-cycle corresponds to a monomial z:iZj z p Z i , and so on. Given an r-cycle, 1 5 r 5 n,re-label the components so that this cycle is: ' : 2

ZZ

+z;'z~+ - +zFr~1, *

then the R+-action is of the form

t . (21,.. . ,z,)

= (PZ1,.

. . .t - z , ) ,

where the m i are rational numbers determined by the equation A . ( m l , . -. ,m,) = (1,. . ' , l),with A being the non-singular matrix: 1 o o * * . * *0* 0 a 3 1 0 . . . * * .0

a2

.. .. .. .. . . . .

A =

0 0 o o . . . a, 1 , 1 0 0 O . S . 0 a]

Of course we can multiply them by the least common multiple of the denominators in order to get integral weights. Similarly, the $-action is defined by: eie

. (zl,.. . . z,)

. . . eiss.2 )

= (,5i'3s1z1,.

T

7

1

where the si are the unique solution to B . ( ~ 1 ,.. . . s,) = (1,... ,I) where B is the matrix: -1 a1 0 . . . . . .

B=

(I

. ;l:. . 0.

*-.

0 0 0 0 -1 a,-1 a, 0 0 ...... 0 -1

The obvious problem now is to study the topology of these singularities. Notice that the monodromy map of the corresponding fibre bundle over S1 is just the first return map of this S1-action, so it is periodic and its period can be computed explicitly from the weights of the action. The simplest of these fibrations is when the permutation ( ~ 1 , .. . . gn) is the identity, so the corresponding singularities are of the form:

f(z) = z?'Z1+ ... + zFZn

.

ai

> 1.

As mentioned before, this case was studied in [35]where it is proved that these singularities are topologically (not analytically) equivalent to the Pham-Brieskorn singularities: z;l-l

+ ... + z2-1,

147

whose topology is well understood. This implies, for instance, that the famous Poincar6 homology sphere C(2,3,5) can be regarded as the set of points in the unit sphere S5 in C3 where the holomorphic vector field (zf, 224,z,") is tangent to the sphere. For n = 2 the remaining case is when f is of the form: f ( z 1 , z z ) = zpk?

+ z;z1

, p , q > 1.

This case was studied thoroughly in [32].The results in that article, together with [30], imply that the link K = f -'(O) n S 3 of this singularity is isotopic to the link of the complex singularity defined by h

h(z1, z 2 ) = z1 z2 (zf+'

+ z;+l),

(7)

but their corresponding Milnor fibrations are not equivalent. In fact one has that the real analytic singularity

also satisfies condition (4.2) and its Milnor fibration is equivalent to that of f , by [30,32].The components of K corresponding to the axes zlz2 = 0 get different orientations in (7) and (8), in a sense that can be made precise, and this implies (by [30]) that the corresponding Milnor fibrations: and are not equivalent: the fibres have different Euler characteristic and the monodromy maps have different period (see [32]). 6. A method for studying maps into

R2

As mentioned before, the problem of studying real singularities satisfying the strong Milnor condition (4.2) was first studied in [16] and turns out to be quite subtle. Such singularities define open-book decompositions on the spheres with the link as binding. Given an analytic map-germ f : (IR", 0) + (IR2, 0) that satisfies the Milnor condition (4.1), Jacquemard gave two conditions that were sufficient to guarantee that f actually satisfies the strong Milnor condition. He proved:

Theorem 6.1. Let f : (R",Q + (IR2,Q) be an analytic map-germ. If the components ( f 1 , f 2 ) = f satisfy the following two conditions, then f satisfies the strong Milnor condition:

148

A ) there exists a neighbourhood U of the origin in IR” and a real number 0 < p < 1 such that for all x 6 U - Q one has:

where (., .) is the usual inner product in R”; and, B ) if sn denotes the local ring of analytic map-germs at the origin in R”, then the integral closures in sn of the ideals generated b y the partial derivatives

3fl 3fl ... 8x1 ’ ax2 ’ ’ ax,

3f2

3x1

afz

’ 3x2’

’

coincide. It was noted in [35] that the above condition (B) can be relaxed and (6.1) still holds. It is enough to demand only condition (BR):that the real integral closures of the jacobian ideals of fi and fz coincide. This slight improvement of (6.1) was used in [35] to prove a stability theorem for real singularities with the strong Milnor condition. This was used also by Ruas and Dos Santos in [34] where they refine (6.1) using “regularity conditions” instead of Jacquemard’s conditions. Their construction is inspired in the proof of (5.1) that we sketched in the previous section (see also [38]), but beautifully taken into a general setting using Bekka’s (c)-regularity. This provides a new method for studying the geometry of maps into R2 and motivates the refinement of Milnor’s theorem that we give in the following section. This same method can be adapted to study, more generally, the geometry of maps Rn+k Rk,though we do not know whether or not one can actually get something interesting in this way for Ic # 2. Let us explain briefly the main ideas in [34]. Consider a map-germ f : (R”,Q) + (R2,Q) satisfying the Milnor condition at 0. As before, let : C + Le be the projection into the line Le forming an angle 0 with the horizontal axis in C = R2, and consider the family of maps he = r e o f. The projections are all submersions, hence the {he} are all real valued analytic maps with an isolated critical point at Q. Setting V = f -‘(Q) and X s = h,’(Q), is easy to see that one has (see for instance [39, VI.41 for details): ---f

B = UXe and V = nXe = Xel n Xe, , for each pair The main result of [34] says:

81

# 02.

149

Theorem 6.2. Let f : (R",Q)-+ (R2,Q)be a real analytic map-germ with an isolated critical point at the origin that satisfies the Milnor condition at 0. If the family he : (Rn,Q) .+ (R,O) satisfies the (c)-regularity condition of Bekka with respect to the function p(x, 6') = x:, then f satisfies the strong Milnor condition.

xi

They also prove that if a function f satisfies Jacquemard's conditions A and B (or even BR) as in (6.1), then the corresponding family {he} is (c)-regular at 0, but not conversely. Hence 6.2 improves (6.1). As a corollary they obtain that for quasi-homogeneous germs (as for instance the twisted Pham-Brieskorn singularities) the Milnor condition (4.1) and the strong Milnor condition (4.2) are actually equivalent. Bekka's (c)-regularity (cf. [2,3]) is reminiscent of Thom af-condition (Thom regularity), and in our setting it is actually a special case. To explain this, define F : (R" x 10,T ) ) 4 R by F ( x , 13) = he(x) and set Y f = Q x [0,T ) and X f = F-'(O) \ Y f .Notice that if we set X ; = X e - {Q} c R" for each 6' E [O,T), then one has X f = for all 19 E [ O , T ) . Now consider the x:. The pair (Xf, Yf) controlfinction on R"x [0,T ) defined by p(x, 6') = is (c)-regular at a point yo E Y f with respect to p if for each sequence of points {xi} -+ yo, xi E X f , such that the sequence of planes {ker D p ( x i ) n T Z i X f converges } to a plane T in the grassmannian of (dim X f - 1)-planes in R",one has TyoYfc T . The family {he} is (c)-regular at Q with respect to the control function p if the pair ( X f ,Y f )is (c)-regular at every point yo E Y f . That is, (c)-regularity is, in this setting, Thom regularity with respect to the control function p. We remark that in the proof of (5.1) described above we used an action of R+that leaves all X s invariant in order to prove that one has a unzfomn conical structure for all of them; in particular they are all transverse to every small sphere around 0. Then we used an S1-action that leaves invariant all spheres around 0 and permutes the X e to exhibit the fibre bundle structure. Two key points in [34] are to notice first that the (c)-regularity condition implies that the manifolds X ; are all transversal to every sufficiently small sphere centered at 0, and then use [2,43] to get a family of germs of homeomorphisms he : (R",Q) 4 (R", Q), that leave invariant each small sphere around the origin and permute the X e . Hence the hypersurfaces Xe are homeomorphic and one has a local product structure. This yields to 6.2.

UX;

xi

150

7. A refinement of Milnor's classical theorem Let us now adapt the previous method for studying the classical Milnor's fibration theorem for complex singularitites. We only sketch the main points here, the proofs are given in [7]. Consider a holomorphic function f : (IB,Q) 4 (C, 0) , defined on an open ball B in Cm,m 2 2, centered at 0, with an isolated critical value at 0 E C ,We are going to study the geometry of f by looking at the 1-parameter family of real valued functions associated to it as above. Recall there are two equivalent ways of defining the Milnor fibration, either as:

$ = - :fS , \ K + S

1

If I where K = f-'(O)

(9)

n S, is the link, or equivalently by:

f :N(E,r]) -a%,

(10)

where E >> r] > 0 are sufficiently small, ID, C (E is the disc of radius r] around 0 E C , BEis the ball of radius E around 0 E Cn and N ( E ,r ] ) is the Milnor tube IB, n f-l(dID,). For each 8 E [ O , T ) consider the real line .Ce c C passing through the origin with an angle 8 with respect to the real axis, measured in the usual of , way. Let rie: C -+ Ce be the orthogonal projection and set he = so that ho and hq are, respectively, the real and imaginary parts of f . Since each r i g is a submersion everywhere, one has that the critical points of f are exactly the critical points of each he. Hence {h,} is a 1-parameter family of real analytic functions with a critical value at 0, and if we set X e = h i l ( 0 ) and V = f - l ( O ) , then each X e is a real hypersurface with Sing Xe = Sing V . As before (see for instance [39, VI.41 for details), it is an exercise to show that one has a decomposition:

B = UXe and V = nXe = Xel n Xez ,for each pair dl # 82 ,

(11)

that we denote by XB.That is, XB denotes the ball IB regarded as union of the hypersurfaces X e , with 8 E [O,n).Then, using [26] one can easily prove:

Lemma 7.1. For every suficiently small ball BE c IB around 0, one has that each manifold Xt\V meets the boundary sphere 9, = dlB, transversally. where K = V n 6, is the link, Thus one has a decomposition (Xs,iK),

151

induced from the above decomposition XB, restricted to B, is difleomorphic to the cylinder X~,\K) x [0,1).

\ V , and XIB<\V

Now observe that each X t is naturally the union of three sets: the points Ce,and the points that map under this projection into the two half lines of Lo \ (0). Write this as: z E IB that map to 0 under the projection IT@: C

X0 = Ee

--f

U VU E O + ~ .

Similarly one has: ( X 0 n S E )=

(Ben S,) u (V S,) u (Ee+?,n S,) .

The previous lemma, together with Milnor's proof of his Theorem 4.8 in [26]imply that there exists a differentiable flow { + t } on BB,\V,defined for all t E R,which leaves invariant each sphere in IB, around 0, whose orbits are transversal to the manifolds X0 \ V and permutes these manifolds: for each fixed time t , the flow carries each Xe \ V into Xe+t \V, where the angle 8+ t must be taken modulo IT.In particular, for t = IT the flow interchanges the two halves of X e \ V . Thus we arrive to the following slight refinement of Milnor's fibration theorem:

Theorem 7.2. Let f : (IBE,Q) + (C,O) be holomorphic with a critical value at 0 E C.For each 0 E [O,n),let he be the composition o f f followed by the projection into the real line in C with angle 8. Then: i) The he are all real analytic maps, whose critical points are those of f . Thus each X e = h s ' ( 0 ) is a real analytic hypersurface in B that contains V = f - l ( O ) and its singular set is Sing(V). ii) The union of all X e is the whole ball B, and the intersection of any two of them is V .

iii) The manifolds X0 \ V are the fibres of the fibre bundle:

whose restriction to the sphere gives Milnor's fibration (9), and whose restriction to f-'(aD,,) n IB,, E >> v > 0 , gives the equivalent Milnor-L2 fibration (lo), up to multiplication by 7.. In [7] we actually prove this theorem (as in [25]) for holomorphic m a p germs defined on analytic spaces with arbitrary singularities.

152

8. Singularities fg and Milnor fibrations for meromorphic germs

We notice that the singularities in equation (8) as well as that in A’Campo’s example (4.3) for m = 0, are all of the form f g with f , g being holomorphic functions in two complex variables. These types of singularities also appear in Lee Rudolph’s work [36] and in [14]. This motivated the study in [31,33] of singularities in of the form f i j . Notice that the zero-set of f i j is {f = 0) U { g = 0 ) , so for n > 2 the link K f g = { f i j = 0) n SZnP1is necessarily a singular variety and f ij must have non-isolated critical points. Also notice that, since the link of f g is also the link of f g which is a holomorphic function, Milnor’s theorem tells us that its complement SZn-’ \ K f gfibres over the circle S1. But the same manifold can fibre over the circle in very many different ways, and Milnor’s fibration (2.1) is one of them. The following theorem is proved in [33].

an

Theorem 8.1. Let f , g : (Cn,O)-+ (C,O) be holomorphic map-germs such that the function f g has an isolated critical value at 0. Then,

is a C” fibre bundle. (In general not equivalent to the corresponding fibration for the holomorphic germ f g . ) Notice that in Milnor’s theorem (2.1) the origin 0 E C is automatically an isolated critical value (unless it is a regular value) because the function in question is holomorphic. However maps of the form f i j can have nonisolated critical values. The starting point for the proof in [33] of this result is to notice that, as pointed out in [14], on SP-’ \ K f g the map q5 coincides with so Theorem (8.1)can be regarded as a statement for meromorphic maps. Then the proof follows step by step the method used by Milnor in [2G] to prove his fibration theorem for complex singularities: first show that q5 has no critical points, so that all the fibres q5-l (eie) are smooth manifolds of codimension 1 in S 2 - l ; then construct a smooth, complete vector field on the complement SZn-’ \ K f , whose integral lines are transversal to the fibres of q5 and the corresponding flow moves the points at constant speed with respect to the argument of q5(z),so it carries each fibre diffeomorphically into the nearby fibres, giving a local product structure. We notice that a similar theorem is proved in [4] replacing the condition that f has an isolated critical value by the condition that the meromorphic

&,

153

germ f/g is semi-tame at the origin (cf. [27]). It is of course interesting to compare this fibration with the local fibrations for meromorphic germs studied in [9,10] (see also [42]); this is done in [ 5 ] . More precisely, given f,g as above, one has a well-defined map f/g : C n \ V ( f g ) -+ C \ {0}, where V ( f g ) = f - l ( O ) U g-l(O). One can consider local "Milnor tubes" for this map around 0, 00, and around each critical point of f,as in Theorem (2.2); it is proved in [9,10] that each of these tubes has a natural fibre bundle structure. This is achieved in those articles by "going up" to a resolution of the singularity and looking at the induced map from there into C . I am grateful to David Massey for explaining me how to obtain these fibrations from the Milnor-L6 fibration theorem for holomorphic germs. The idea is simple and beautiful, so we sketch it now. For simplicity we do it for the local fibration at 0, but this can be easily adapted to all critical points of f/g. Consider the hypersurface X in C"+l defined by

x = ((2,t ) E C" x C I f ( z )= tg(z)} , and let 7r : X -+ C be the obvious map ( z , t ) I+ t. As mentioned before, Theorem (2.2) was proved by L6 in [17] in a more general setting, where the ambient space can be taken to be an arbitrary complex analytic space. In our setting, L6's theorem says that for every E , Q > 0 sufficiently small, E >> Q , one has a fibre bundle

nx n~ - l ( d ~ 5 , ) d ~ ,, where ID, is a small disc around 0 in C. The fibre of this bundle over a point to E dID, is: ~ 2 n + 2 E

n (({f

= 9 = 0)

u {f/g = t o with 9 # 0))

x

Ito})

By the lStThom-Mather Isotopy Theorem (see [21]) one can lift the usual vector field on dID, to a stratified, integrable vector field on Bz"+2n X n 7r-'(dID,), which is transversal to all the fibres of 7r and preserves the strata {f = g = 0) in the fibres. This vector field can be also chosen to be tangent to the link K of X. Thus one gets the usual Milnor-L6 fibration. Now remove from each fibre the stratum {f = g = 0) to get a fibre bundle over dID, with fibre {f/g = t o }n B,2"+2.This is the local fibration for f/g at 0. Of course we can do the same for the germ g/f and get the local fibration for f / g at 03. In [5] we show that, with appropriate conditions, these are both subfibrations of the fibration given in 8.1.

154 9. Fibrations of multilinks

The material in this section is taken from [33] and we refer to that article and to [8] for more on the subject. See also the seminal article [30] of Anne Pichon, as well as [4,31,32]. So far we have envisaged fibre bundle decompositions of the complement of the link of some (real or complex) singularity, but we have ignored significant additional information that one has: what is the behaviour of the fibration near the link? Let us illustrate this with a trivial example. Take your favourite holomorphic map f : Cz+ C,for instance (z1,zz)H z;+zz, whose link K f is the trefoil knot in S 3 . Milnor’s theorem tells us that - : S3 \

f If1

Kf

+

S1,

is a fibre bundle. Now take the same map f to some power k > 1. The vanishing set is that of f , so the link is the same K f , and Milnor’s theorem says that

-f:kS 3 \ K f + S 1 ,

Ifkl is also a fibre bundle. In the first case one actually has an open-book decomposition of the 3-sphere, i. e., we can find a tubular neighbourhood N of K f in S3 diffeomorphic to S1 x ID2 where the projection map takes the form

&

7.

( t ,z ) H 4 In the second case the corresponding map is ( t ,z ) H

& and

this can be considered as a “generalized open-book” (or rather a multilink fibration, as we explain below). Similarly, consider the singularities given by equations (7) and (8). In both cases the link is the union of the Hopf link

L~ = s3n ( { z l = o} u { z 2 = o}) with the link L1 of { Z : + ~ + Z : + ~ - 0 } , which is a torus link of type (p+ 1, q+ 1). Choosing appropriate orientations one has that in a neighbourhood of each component of LOthe projection map corresponding to the holomorphic map in (7) is (t,z ) H while for the map in (8) it is (t,z ) H Around the components in L1 both projection maps coincide. What we see in these examples is the multilink structure. Let us make this concept precise. For simplicity we restrict to the case of links in S3. A link in S3 is a finite union of disjoint circles S1 embedded in the 3-sphere; its connected components are called knots. We let L = K1 U . . .UKl be the connected components of a link L in S3. We say that L is oriented if an orientation is fixed on each Ki.Notice that

G,

6.

155 the normal bundle of each Ki in S3 is trivial. Hence, if L is oriented then one has orientation preserving diffeomerphisms N ( K i ) S 0 ' x S1defined on a neighbourhood of each Ki, that carry the oriented circles Ki into (0) x S1 with its usual orientation. A multilink is the data of an oriented link L = K1 U . . . U K1 together with a multiplicity ni E Z associated with each component Ki. We denote such a multilink by

L = nlK1 U . . . U n l K l , and we fix the convention that ni Ki = (-ni) (- K i ) , where -Ki means Ki with the opposite orientation.

-

Definition 9.1. A multilink L = nlK1U.. .UnlKl in S3 is fiberable if there exists a map @ : S3 \ L 9 which satisfies: i) the map

is a Coolocally trivial fibration; and

ii) for each i = 1,. . . ,1, there exists a tubular neighbourhood N ( K i ) of Ki in S3 \ ( L \ K i ) , an orientation-preserving diffeomorphism T : 8' x ID2 -+ N ( K i ) such that T(S' x ( 0 ) ) = Ki, and an integer ki E Z such that for all ( t , ~E)8' x (ID2 \ (0)) one has:

-

In this case we say that @ is a fibrution of the multilink L.

nf=,fyi

For instance, if f : (C2,0 ) (C,0) is holomorphic and f = is its decomposition into irreducible factors, then the multilink associated with f is 1

i=l

where K f i = Sz n f x y l ( 0 ) ,and the Milnor fibration f (x)// f )I(. : Sz \ L f + 8' is a fibration of the multilink L f . A plumbing multilink in S3 is a multilink L = nlK1U.. .UnlKl such that there is a plumbing decomposition of S3 for which each Ki is an $-leaf. These links are classified in [8], where there is also an easy combinatorial algorithm (11.2) to determine from the plumbing graph of the corresponding plumbing decomposition, whether or not a given plumbing multilink is fiberable. This is purely topological. Notice that all links defined by a holomorphic equation are automatically plumbing links.

156

One has the following theorem of [33], which improves an earlier version given in [31] (see also [4]).

Theorem 9.1. Let f,g : (C2,0 ) --t (C,0 ) be two holomorphic germs with n o common branches. T h e n the real analytic germ fij : ( S 3 , p ) + (R2,0) has 0 as a n isolated critical value if and only if the multilink L f - L, is fibred. Moreover, if these conditions hold, then the Milnor fibration (of 8.1):

fs : @ \ ( L f If4

UL,)

-s

I

is a fibration of the multilink L f - L,. We refer to [33] for the proof and details about this result, which is actually proved in the more general setting of plumbing multilinks in any 3manifold which is the link of an isolated complex surface singularity ( X ,p ) . There is also given in [33]one more equivalence in the theorem: the multilink Lf - L, is fibred if and only if f and g have different multiplicities a t each rupture vertex of the graph of a resolution of the holomorphic germ

f 9 : @,PI

+

(CO).

References 1. N. A’Campo, Le Nombre de Lefschetz d’une Monodromie, Indag. Math. 35, 1973, 113-118. 2. K. Bekka, Regular quasi-homogeneous stratifications. Travaux en Cours, Hermann, 55, p 1-14 (1997). Ed. D. Trotman and L. C. Wilson. 3. K. Bekka and S. Koike, The Kuo condition, an inequality of Thom’s type and (c)-regularity. Topology 37 (1998), 45-62. 4. A. Bodin and A. Pichon, Meromorphic junctions, real singularities and fibered links, preprint 2005. 5. A. Bodin, A. Pichon, J. Seade Local fibrations for meromorphic germs (provisional title), in preparation. 6. P. T. Church and K. Lamotke, Non-trivial polynomial isolated singularities, Indag. Math. 37 (1975), 149-154. 7. J. L. Cisneros, J. Seade and J. Snoussi, A refinement of Milnor’s fibration theorem; provisional title, work in progress. 8. D. Eisenbud, W. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, 110. Princeton University Press, Princeton, NJ, 1985. 9. S. Gusein-Zade, I. Luengo, A. Melle-HBrnandez, Zeta functions of germs of meromorphic functions, and the Newton diagram, Funct. Anal. Appl. 32 (1998), no. 2, 93-99. 10. S. Gusein-Zade, I. Luengo, A. Melle-HBrnandez, On the topology of germs of meromorphic functions and its applications, St. Petersburg Math. J. 11 (ZOOO), no. 5, 775-780.

157 11. H. Hamm, Lokale topologische Eigenschaften komplexer Raume, Math. Ann. 191 (1971), 235-252. 12. H. Hamm, A f i n e varieties and Lefschetz theorems. In “Singularity theory” (Trieste, 1991), 248-262, edited by D. T. L6 et al. World Sci. Publishing, River Edge, NJ, 1995. 13. H. Hamm, D. T. L6, Rectified homotopical depth and Grothendieck conjectures. The Grothendieck Festschrift, Vol. 11, 311-351, edited by P. Cartier et al. Progr. Math., 87, Birkhuser Boston, Boston, MA, 1990. 14. M. Hirasawa, L. Rudolph, Constructions of Morse maps for knots and links, and upper bounds on the Morse-Nowikov number, Preprint 2003, math.GT/0311134. 15. H. Hironaka, Stratification and Flatness, in “Real and Complex singularities”, Proceedings of the Nordic Summer School, Oslo 1976; edited by P. Holm; Publ. Sijthoff & Noordhoff Int. Publishers (1977), 199-265. 16. A. Jacquemard, Fibrations de Milnor pour des applications re’elles, Boll. Un. Mat. Ital. 3-B (1989), 591-600. 17. D. T. L6, Some remarks on relative monodromy, in “Real and complex singularities”, Proc. Nordic Summer School, edit. P. Holm, p. 397-403, Sijthoff and Noordhoff 1977. 18. D. T. L6, D e w noeuds alge‘briques FM-e‘quivalents sont e‘gaux, C. R. Acad. Sci. Paris 272 (1971), 214-216. 19. D. T. L6, Complex analytic functions with isolated singularities, J. Algebraic Geom. 1 (1992), no. 1, 83-99. 20. D. T. L6 et B. Perron Sur la fibre de Milnor d’une singularite‘ isole‘e e n dimension complexe trois, C. R. Acad. Sci. Paris Sbr. A-B 289 (1979), no. 2, A115-A118. 21. D. T. L6, B. Teissier, Cycles Evanescents et conditions de Whitney, in Proc. Symp. Pure Math 40, Part 2 (1983), 65-103. 22. E. Looijenga, A note on polynomial isolated singularities. Indag. Math. 33 (1971), 418-421. 23. E. Looijenga, Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series, 77. Cambridge University Press, Cambridge, 1984. 24. D. Massey, Le^ cycles and hypersurface singularities, Lecture Notes in Mathematics 1615 (1995). 25. J. Milnor, O n isolated singularities of hypersurfaces, unpublished, 1966. 26. J. Milnor, Singular points of complex hypersurfaces, Ann. of Maths. Studies 61, 1968, Princeton Univ. Press. 27. A. Nemethi, A. Zaharia, Milnor fibration at infinity. Indag. Math. 3 (1992), 323-335. 28. B. Perron, Le nmud ”huit” est algdbrique re‘el Invent. Math. 65 (1981/82), no. 3, 441-451. 29. F. Pham, Formules de Picard-Lefschetz ge‘n6ralise‘es et ramification des inte‘grales, Bull. SOC.Math. France 93 (1965), 333-367. 30. A. Pichon, Fibrations sur le cercle et surfaces complexes. Ann. Inst. Fourier (Grenoble), 51, 337-374, 2001.

158 31. A. Pichon, Real analytic germs f i j and open-book decompositions of the 3sphere, Int. J. Math. 16 (2005), 1-12. 32. A. Pichon, J. Seade, Real singularities and open-book decompositions of the 3-sphere, Ann. Fac. Sci. Toulouse 12, 2003, 245-265. 33. A. Pichon, J. Seade, Fibered Multilinks and singularities f i j , Preprint 2005, math.AG/0505312. 34. M.A.S. Ruas and R.N.A. dos Santos, Real Milnor Fibration and CRegularity, Manus, Math. 117 (ZOOS), 207-218. 35. M.A.S. Ruas, J. Seade and A. Verjovsky, O n real singularities with a Milnor fibrationjn “Trends in Singularities”, ed. A. Libgober and M. TibZir”, p. 191-213, Birkhauser 2002. 36. L. Rudolph, Isolated critical points of mappings from R4 to R2 and a natural splitting of the Milnor number of a classical fibered link. I. Basic theory; examples. Comment. Math. Helv. 62 (1987), no. 4, 630-645. 37. J. Seade, Open book decompositions associated to holomorphic vector fields. Bol. SOC.Mat. Mexicana (3) 3 (1997), no. 2, 323-335. 38. J. Seade, O n the topology of hypersurface singularities. Real and complex singularities, 201-205, Lecture Notes in Pure and Appl. Math., 232, Dekker, New York, 2003. 39. J. Seade, O n the topology of isolated singularities in analytic spaces, to appear as v. 241, Progress in Mathematics, Birkhauser. 40. D. Siersma, A bouquet theorem for the Milnor fibre, J. Algebraic Geom. 4 (1995), no. 1, 51-66. 41. M. T i b k , Bouquet decomposition of the Milnor fibre, Topology 35 (1996), no. 1, 227-241. 42. M. T i b k , Singularities and topology of meromorphic functions. In “Trends in singularities”, 223-246, ed. A. Libgober and M. Tibzr. Trends in Math., Birkhuser, Basel, 2002. 43. A. Verona, Stratified Mappings: Structure and Rangulability, SpringerVerlag Lecture Notes in Maths. 1102, 1984.

on the previous page, Miramare, by Tatsuo Suwa, 31 July 2005 with authorization of Tatsuo Suwa

161

INTRODUCTION TO COMPLEX ANALYTIC GEOMETRY Tatsuo Suwa’ Department of Information Engineering, Niigata University 2-8050 Ikarashi, Niigata 950-2181, Japan * [email protected]

These are the notes I prepared for my course of the title same as above. It was given in “Advanced School and Workshop on Singularities in Geometry and Topology”, held at ICTP, Miramare-Trieste, Italy, August 15September 3, 2005. I tried to present fundamental material in Complex Analytic Geometry, but at the same time, I set up the goal to give a unified description of various important invariants in the singularity theory as “residues of Chern classes” of certain vector bundles or virtual bundles on singular varieties. The idea behind is the localization theory of characteristic classes in the framework of Chern-Weil theory adapted to the Cech-de Rham cohomology. I would like to thank the referees of this article for reading the manuscripts very carefully and giving many valuable comments.

Contents Chapter I. Analytic functions of several complex variables and analytic varieties 162 1. Analytic functions of one complex variable 163 2. Analytic functions of several complex variables 167 3. Germs of holomorphic functions 4. Complex manifolds and analytic varieties 169 5. Germs of varieties 175 Chapter 11. Differential forms and Cech-de Rham cohomology 1. Vector bundles 178 2. Vector fields and differential forms 184 3. Integration and the Stokes theorem 187 Partially supported by a grant from the Japan Society for the Promotion of Science.

162

4, de Rham cohomology 187 5, Cech-de Rham cohomology 189 Chapter 111. Chern-Weil theory of characteristic classes and some more complex analytic geometry 1, Chern classes via connections 194 2. Virtual bundles 197 3. Characteristic classes in the Cech-de Rham cohomology and a vanishing theorem 198 4. Divisors 199 5. Complete intersections and local complete intersections 200 6. Grothendieck residues 203 Chapter IV. Localization of Chern classes and associated residues 1. Localization of the top Chern class 204 2. Residues at an isolated zero 206 3. Examples I 208 4. Residues of Chern classes on singular varieties 209 5. Residues at an isolated singularity 211 6. Examples I1 214 References 218

Chapter I. Analytic functions of several complex variables and analytic varieties 1. Analytic functions of one complex variable

Let D be an open set in the complex plane C and f a complex valued function on D.

Definition 1.1. We say that f is analytic at a point a in D if there is ~ ( - a)n, z which converges at each point z in a neigha power series borhood of a, suckthat

in a neighborhood of a. We say that f is analytic in D if it is analytic at every point of D.

Definition 1.2. We say that f is holomorphic at a point a in D if the limit

exists. We say that f is holomorphic in D if it is holomorphic at every point of D.

163

The above limit, if it exists, is denoted by % ( a ) and is called the derivative off at a. Iff is holomorphic in D, then we may think of as a function on D. Let z * x f l y with x and y the real and imaginary parts, respectively. We may think of f as a function of (x,y). We write f = u G v with u and Y the ,real and imaginary parts. In general, we’say that a function of real variables is (of class) C‘, if the partial derivatives exist up to order T and are continuous. If all the partial derivatives exist we say it is C”.

%

+

+

Theorem 1.3. The following are equivalent: (1) f is analytic in D, (2) f is holornorphic in D , (3) f is C1 in (x,y ) and satisfies the “Cauchy-Riemann equations” in D; du dv dv - -- --au ax_ ay ax’

ay7

Note that, if we introduce the operators --I(---E) d d and --I(-+--!?), d a az 2 dY dz 2 ax we may write the Cauchy-Riemann equations as

ax

af

- = 0.

dz If this is the case, we have = We finish this section by recalling the Cauchy integral formula. Let f be an analytic function in a neighborhood of a and y the boundary of a small disk about a , oriented counterclockwise. Then we have

2 g.

2. Analytic functions of several complex variables

Let C” = { z = ( 2 1 , . . . ,zn) I zi E C } be the product of n copies of C. For an n-tuple v = (v1,. , . ,vn) of non-negative integers, we set zV = ZT . . ez?, IvI = v1 . . . v, and Y! = y !. . .vn!. Let D be an open set in C” and f a complex valued function on D.

+ +

Definition 2.1. We say that f is analytic at a point a in D if there is a power series

c

1420

c v ( z - a)V = Vl

c

,...,V&O

c”l...”n(zl - a1)”l

* * *

(2” - a J V n ,

164

which converges absolutely at each point z in a neighborhood of a , such that

in a neighborhood of a. We say that f is analytic in D if it is analytic at every point of D. The following can be proved by a repeated use of the Cauchy integral formula:

Theorem 2.2. The following conditions are equivalent: (1) f is analytic in D . (2) f is continuous and is analytic in each variable zi in D , f o r i = 1,.. . ,n. It is known that we may remove the continuity condition in (2) above (Hartogs’ theorem). From Theorems 1.3 and 2.2, we have:

Theorem 2.3. The following are equivalent: (1) f is analytic in D .

af = 0 in D , for (2) f is C1and satisfies the Cauchy-Riemann equations 2 =

1,.. . ,n.

aza

In the sequel, we call analytic function also a holomorphic function and use the words “analytic” and “holomorphic” interchangeably. Note that, if f is holomorphic, for arbitrary v, the partial derivative

exists and is holomorphic in D. If f ( z ) = c,(z - a ) , is a power series expansion of f, then each coefficient c, is given by 1 d”f c, = --(a). v! dz” This series is called the Taylor series of f at a. Let D be an open set in Cn and f : D -+ Cm a map. We say that f is holomorphic if, when we write f componentwise as f = (fl, . . . ,f m ) , each fi is holomorphic. Let D and D’ be two open sets in C” and f : D -+ D’ a map. We say that f is biholomorphic, if f is bijective and if both f and f are holomorphic. It is not difficult to see that the composition of holomorphic maps is holomorphic.

165

For a holomorphic map f into C", we set

= ( f l ,.

I

. . ,fm) from

afl

an open set D in C"

"I

... az, -. .. . . .

az, -

- ... \

821

az,

and call it the Jacobian matrix of f with respect to z .

Definition 2.4. We say that a point a in D is a regular point of f , if the rank of the Jacobian matrix (a(f1.. . . ,f,)/a(zl,. . . ,z n ) ) ( a ) , evaluated at a, is maximal possible, i.e., min(n, m). Otherwise we say that a is a critical (or singular) point of f. When n = m, the determinant of the Jacobian matrix is called the Jacobian of f with respect to z. Thus, in this case, a is a regular point of f if and only if det(a(f l , . . . ,f,)/a(z1,... ,z,)) ( a ) # 0. If we denote by ui and vi the real and the imaginary parts of f i , we compute:

The following two theorems show how a holomorphic map looks in a neighborhood of a regular point. Without loss of generality, we may only consider maps f from a neighborhood of the origin 0 in C" into C" with f (0) = 0.

Theorem 2.6 (Inverse mapping theorem). Let f be a holomorphic map from a neighborhood of 0 in en into C" with f (0) = 0. If 0 is a regular point off, then there are open neighborhoods U and V of 0 such that f is a biholomorphic map f r o m U onto V . This theorem follows from (2.5), the inverse mapping theorem in the real case and the Cauchy-Ftiemann equations. From this theorem, we get the following theorem as in the real variable case.

Theorem 2.7 (Implicit function theorem). Let f be a holomorphic map from a neighborhood of the origin 0 in C" into C" with f ( 0 ) = 0. W e assume that 0 is a regular point off. (I) Suppose n 2 m. Thus the rank of the Jacobian matrix is m and, by renumbering the functions and the variables, if necessary, we m a y

166

assume that

In this case, there exist neighborhoods U and V of 0 in @" and a biholomorphic map h from U onto V with h(0) = 0 such that

(f 0 h ) ( Z 1 , . . ., zm,. . ., zn) = (21,.. ., zm) for ( ~ 1 , .. . ,2") in a neighborhood of 0 .

(11) Suppose n 5 m. Thus the rank of the Jacobian matrix is n and we may assume that

I n this case, there exist neighborhoods U and V of 0 in Cm and a biholomorphic map h from U onto V with h(0) = 0 such that ( h o f ) ( z i ..., , 2") =(zi ,..., zn,O ,...,0) for ( ~ 1 , ... ,2,) in a neighborhood of 0. (111) Suppose n > m. Thus the rank of the Jacobian matrix is m and we may assume as in (I) that det

a(fl,* > fm) (0) # 0. f

q21,..

*

.,

In this case, there is a holomorphic m a p g from a neighborhood of 0 in en-" znto Cm with g(0) = 0 such that '

=o

f(91(Zm+l,...,~n),...,gm(Zm+l ,...,Z n),Zm+l,...,Zn)

for (zm+l,. . . ,2") in a neighborhood of 0.

Remark 2.8. In the case (I) above, f is a submersion in a neighborhood of 0, in the case (11), f is an embedding in a neighborhood of 0 and in the case (111),we may solve the equation

f ( 2 1 , . . . ,Zm, . . . ,2,) = 0 for 21,.. . , 2 , as functions 91,. . . ,gm of (zm+1,. . . , z , ) in a neighborhood of 0 and the set f - l ( o ) is the graph of the map g = (91,. . . ,gm). The following theorem can be proved as in the one variable case.

Theorem 2.9 (Uniqueness of analytic continuation). Let D be an open connected subset of @" and let f and g be holomorphic functions in D . If there i s a non-empty open set U in D such that f = g on U , then f=gonD.

167

The following can be proved using the corresponding result in the case of one variable.

Theorem 2.10 (Maximum principle). Let D be a connected open set in Cn and let f be a holomorphic function in D . If there is a point a in D such that If (.)I 2 If (z)l for all z in a neighborhood of a, then f is a constant function on D . 3. Germs of holomorphic functions

We list, for example, 119,351 as references for this section. Let H be the set of functions holomorphic in some neighborhood of 0 in C". We define a relation in H as follows. For two elements f and g in H , f g if there is a neighborhood U of 0 such that the restrictions of f and g to U are identical. Then it is easily checked that is an equivalence relation in H . The equivalence class of a function f is called the germ of f at 0, which we also denote by f , if there is no fear of confusion. We let 0, be the quotient set of H by this equivalence relation. The set 0, has the structure of a commutative ring with respect to the operations induced from the addition and the multiplication of functions. It has the unity which is the equivalence class of the function constantly equal to 1. If we denote by C { z l , . . . ,z,} the set of power series which converge absolutely in some neighborhood of 0 , this set also has the structure of a ring. Since, as in the one variable case, f g if and only if f and g have the same power series expansion, we may identify 0, with C(z1,. . . ,z,}. In what follows we denote by R a commutative ring with unity 1. A zero divisor in R is an element a in R such that there is an element b # 0 in R with ab = 0. A ring R # 0 is an integral domain if there are no non-zero zero divisors, i.e., if ab = 0 , for a, b E R, then a = 0 or b = 0. As a consequence of Theorem 2.9, the ring 0, is an integral domain. Thus we may form the quotient field of On,which we denote by M,. Each element in M , can be expressed as f / g and two expressions f / g and f '/g' stand for the same element if and only if f g ' = f ' g . We call an element of M , a germ of meromorphic function at 0 in Cn. We say that an element u in a ring R is a unit if there is an element v in R such that uv = 1. It is not difficult to see that a germ u in 0, is a unit if and only if it is the germ of a function u with u(0) # 0. We say that an ideal I in a ring R is maximal if I # R and if there are no ideals J with I J fc R. This is equivalent to saying that the quotient R/I is a field. Let m denote the set of non-units in 0,. Then it is an ideal N

N

N

N

5

168

in On. Moreover, we have the following proposition.

Proposition 3.1. The ideal m is the unique maximal ideal in On.

A ring with a unique maximal ideal is called a local ring. We analyze the structure of the ring On by induction on n. First, for a germ f in On, we write f = ~ , , l l o a , z u . We say that the order o f f is k, if a, = 0 for all v with IvI < k and a, # 0 for some vo with lvol = k. We define the order of the germ 0 to be +CQ. We say that the order of f in z, is k, if the order of f (0,. . . ,0, zn), as a power series in z,, is k. Then we have; Lemma 3.2. If the order off is k, then we may find a suitable coordinate of @" such that the order off in is k. system (
cn)

<,

We consider the ring On-l[zn]of polynomials in z, with coefficients in On-l: On-l[znI = { f(z) = a0

+ alzn + . + *

1

U ~ Z ;

I ai E On-1 1.

Definition 3.3. A Weierstrasspolynomial in z, of degree k is an element h of On-l[zn] of the form

h = uo

+ U~Z, + . + ak-1~k-l + z:, * *

where k is a positive integer and ao, al,. . . ,arc-1 are non-units in

On-1.

Note that in the above, h(0,.. . , 0 , zn) = .2: Hence the order of h in z, is k. In general, any germ f in On is written as f (2) = a0

+ alzn + . - + a@; + . . . *

with ai E On-l.The order of f in z, is k if and only if ao, al,. . . ,ak-1 are non-units in On-l and ak is a unit in On-l.In this case, a;'(ao alz, . . akzh) is a Weierstrass polynomial in zn of degree k. The Weierstrass preparation theorem stated below shows that such an f is essentially equal to a Weierstrass polynomial of degree k.

+

+. +

Theorem 3.4 (Weierstrass division theorem). If h is a Weierstrass polynomial in zn of degree k, then for any germ f in On,there exist uniquely determined elements q in 0, and r in O,-l[zn] with degr < k such that f =qh+r.

Moreover, if f is in O,-l[zn], so is q. Thus we also have a division theorem in the ring On-,[~n].

169

Theorem 3.5 (Weierstrass preparation theorem). Let f be a germ in On whose order in z, i s k (< +m). Then there is a unique Weierstrass polynomial h in z, of degree k such that f = uh with u a unit in On. Next we discuss some important properties of the ring On which follow from the above theorems. First we recall some more terms from algebra. Let R be an integral domain. An element a in R is irreducible if a is not a unit and if the identity a = bc for elements b and c in R implies that either b or c is a unit. Note that 0 is not irreducible. We say that R is a unique factorization domain, or simply a UFD, if every element a in R which is not 0 or a unit can be expressed as a product of irreducible elements in R and the expression is unique up to the order and multiplications by units. It is known that if R is a UFD, so is the polynomial ring R[X]in the variable X (Gauss' Theorem). Theorem 3.6. The ring On is a unique factorization domain. Let R be a UFD. For elements a and b in R,there is always the greatest common divisor gcd(a, b), which is unique up to multiplication by units. We say that a and b are relatively prime if gcd(a, b ) is a unit. For a point z in C", let On,* be the ring of germs of holomorphic functions at z , which is naturally isomorphic with On.Using Theorem 3.5, we can also prove that i f f and g are relatively prime in On,then they are relatively prime in On,z for all z sufficiently close to 0. We say that a ring R is a Noetherian ring if every ideal in R has a finite number of generators, namely, if I is an ideal in R,there exist a finite number of elements a l , . . . ,a, in I such that every element a in I is written as a = xiai with xi E R. It is known that if R is Noetherian, so is R[X](Hilbert basis theorem).

EL,

Theorem 3.7. The ring 0, is a Noetherian ring.

The following is a consequence of the %iemann extension theorem", which is proved using the Weierstrass preparation theorem. Theorem 3.8. Let D be an open set in @" and f a function holomorphic and not identically 0 in D. W e set V = { z E D I f(z) = 0 ) . If D is connected, then D \ V is also connected. 4. Complex manifolds and analytic varieties

References for this section will be [17,27]. The notion of complex manifold is obtained by replacing C" maps by holomorphic maps in the definition

170

of a C" manifold;

Definition 4.1. Let M be a Hausdorff topological space with countable basis. We say that M is a complex manifold if it admits an open covering U = (U,),,g with the following properties: (1) for each a,there is a homeomorphism cp, from U, onto an open set D, in C", for some n, (2) for each pair ( a , P ) , the map cp, 0 'p0-l is biholomorphic from cpdUa n UP) onto cpa(Va n Up). The natural number n, which is uniquely determined on each connected component of M , is called the (complex) dimension of the component. If all the components have dimension n, we say the dimension of M is n. Let U = {U,)aE~ be an open covering as above. We call (U,,p,) a (holomorphic) local coordinate system on M . For a point p in U,, we call U, a coordinate neighborhood of p and cpa(P>= b ? ( P ) ,

f

.

1 ,

.,"(P))

the local coordinates of p (with respect to (pa). Sometimes we identify U, with D, by the homeomorphism pa and identify p with the point (z,O(p),. . . ,z,"(p)) in D, c C". In this case we call (z?, . ,. ,z,") a coordinate system on U,. The collection {(U,,(P,)),EI of pairs (U,,cp,) as above is called a system of (holomorphic) coordinate neighborhoods on M .

Examples 4.2. 1. A (non-empty) open subset in C" is an n-dimensional complex manifold. 2. The complex projective space (CP". We introduce a relation in en+'\ {0 ) by setting, for C = (Q, . . . ,5") and C' = (<&,. . . ,
-

N

N

m).

+

ui = { [ ( O , . . ., t]E M I Ci # 0 ) . Then the map

'pi:

cpi([bo,* *

Ui --+ C" defined by * 7

Cn])=

(Co/Cii * . * > Ci-l/
* * 7

Cn/Ci)

is a homeomorphism. Moreover, it is not difficult to check that for each pair (i,j), the map cpi o 9j-l is a biholomorphic map from cpj(Ui n U j ) onto cpi(Ui n U j ) . Thus M becomes a (connected) complex manifold of

171 dimension n, which we denote by CP" and call the n-dimensional complex projective space. We call [coo,.. ., homogeneous coordinates on CP".Note that CP' is the Riemann sphere. From the construction, the projective space CP"-' is interpreted as the set of complex lines through 0 (one-dimensional subspaces) in Cn. Likewise the Grassmannian Gp(n) is defined to be the set of pdimensional subspaces of C". It admits also naturally the structure of a compact complex manifold of dimension p(n - p) (cf. [17, Ch. 1, 51). 3. If M and M' are complex manifolds of dimensions n and n', respectively, the product M x M' has naturally the structure of a complex manifold of dimension n n'.

cn]

+

cn)

Exercise 4.3. Let S a n f l = { (co,. . . , E C"+' I IQI2 + - .. + = 1} be the (2n+l)-dimensional unit sphere and T the restriction of the canonical surjection Cn+l \ ( 0 ) -+ CP" to S2"+l.Show that T is surjective (thus CP" is compact) and find the inverse image ~ - ' ( p )for each point p in CP". A complex valued function f on an open set U in a complex manifold M is said to be holomorphic if, for each local coordinate system (Ua,(pa), the function f o p('; is holomorphic on (p,(U n U,). Also, a map f : M -+ M' from a complex manifold M into another M' is said to be holomorphic if, for local coordinate systems (U,, ( p o l ) on M and (Vx,$A) on M', the map o f o (p;l is holomorphic on cpa(Ua n f -'(Vx)). A biholomorphic map is a bijective holomorphic map f such that f - l is also holomorphic. If M is a complex manifold of dimension n, since we may identify C" with W2" and a holomorphic map is of class C", M has the structure of a C" manifold of real dimension 2n. If ( 2 1 , . . . ,2,) is a coordinate system on a neighborhood U of a point p in M , then writing zi = x i f l yi with xi and yi the real and the imaginary parts of zi, we see that ( X I , y1,. . . ,zn,yn) is a C" coordinate system on U . Let TR,,M denote the tangent space of M at p as a C" manifold. We may think of the vectors

+

d

-2) dyi

and

-='(-+--) d d azi

2

ax,

d ayi

as being in the complexification Tg,pM = Tw,,M 8~CC of T R , ~ MIt. is not difficult to see that, if we denote by T p M and F p M the subspaces of the C-vector space Tg,pM spanned, respectively, by d / d z l , . . . ,d/dz, and d/dzl,. . . , d/dZn, then they do not depend on the choice of the coordinates ( 2 1 , . . . ,2"). Thus we have:

172

Proposition 4.4. For a complex manifold M we have a decomposition

Ti,pM = TpM @ T p M . We call TpM and T,M, respectively, the holomorphic and antiholomorphic parts of T i , p M . There is an important class of subsets in a complex manifold, namely, analytic varieties.

Definition 4.5. Let D be an open subset of a complex manifold M and V a subset of D. We say that V is an (analytic) variety in D if, for any point p in D,there exist a neighborhood U of p and a finite number of holomorphic functions f1,. . . ,fr on U such that

V n U ={q

E

U 1 fl(q) = . . . = fr(Q) =O}.

We call (f1, . . . ,f,.) a system of local defining functions of V and f1 = . . = fr = 0 local equations for V near p.

A variety in D is sometimes called a subvariety of D. Note that a variety in D is a closed subset of D. If V is a closed subset of D , it is a variety in D if (and only if) each point po in V admits a neighborhood U with the properties in Definition 4.5. A non-empty variety which is locally defined by a single (not identically zero) holomorphic function is called a hypersurface (cf. Theorem 5.11 below). The first part of the following is obvious from the uniqueness of analytic continuation and the second part follows from Theorem 3.8. Theorem 4.6. Let V be a variety in a connected open set D . If it is a proper subset of D , it does not have interior points. Moreover, D \ V is connected. Definition 4.7. Let V be a variety. A point p in V is called a regular point of V if there is a system of local defining functions (fi, . . . ,f,.) of V in a neighborhood of p such that p is a regular point of the map f = (fi, . . . ,f,.). We say p is a singular point of V if it is not a regular point. Note that if p is a regular point of V , by the inverse mapping theorem, we may assume without loss of generality that r 5 n in the above. Exercises 4.8. In what follows, let p be a regular point of a variety V. (1) Show that, if (fl, . . . ,f r ) is a system as in 4.7, then there is a neighborhood U of p such that V n U has the structure of a complex manifold of dimension n - r so that the inclusion map L : V n U + U is holomorphic.

173

(2) Show that, in this situation, the differential 1,: TpV -+ TpU = TpM is injective. Thus we may identify TpV with a subspace of TpM. We call the quotient space TpM/TpV the (holomorphic) normal space of V in M at p . (3) Let ( 2 1 , .. . , 2 % ) be a coordinate system in a neighborhood of p in M. We identify TpM with C” = { ( ( I , . . . ,G)) by taking (a/az1, . . . ,a/az,) as its basis. Show that, in C”, TpV is given by

CG(P).Cj afi =o,

2

=1, ...,?-.

j=1

For a variety V, we denote by Reg(V) and Sing(V), respectively, the sets of regular and singular points of V. By 4.8,Reg(V) is a complex manifold. It is shown that Sing(V) is again an analytic variety (cf. Ch. 111, Proposition 5.3 and Remark 5.4). Hence Sing(V) is a closed set in V and Reg(V) is an open set in V. An analytic set V in D is said to be a (closed) submanifold of D if V = Reg(V). In this case, it is a locally closed submanifold of M .

Examples 4.9. 1. Let M be C2 with coordinates (z1,22). We set f(z1,zz) = 2122 and V = { (z1,22) I f ( z l , 2 2 ) = 0). Thus V consists of two “complex lines” (zl and 22 “axes”) intersecting in C2 at one point (the origin 0). By definition we see that V\{O) c Reg(V), while by looking at the neighborhood structure of 0, we see that 0 is a singular point of V (cf. Exercise 4.10, (1) below). This can be also checked by studying the behavior of the tangent spaces of the regular part. See also Ch. 111, Proposition 5.3. Thus Reg(V) = V \ {0), which has two connected components each being a one-dimensional complex manifold biholomorphic to C* = C \ (0). 2. Again let M be C2.We set f ( z 1 , z2) = 2: - 2; and let V be the variety defined by {f = 0). By definition we see that V \ (0) c Reg(V), while 0 is a singular point of V (cf. Exercise 4.10, (2)). Thus Reg(V) = V\{O), which has one component biholomorphic to C*. Note that V is homeomorphic to C. 3. Let M be C3 with coordinates (21, z2,23). We set f(zl,z2,23) = 212: -2: and let V be the variety defined by {f = 0). Then Reg(V) is a twodimensional complex manifold and Sing(V) is the 21-axis. The set V is called the Whitney umbrella. 4. Let M be C3. We set ~ ( z ~ , z z , = z ~z;) - 2fz; - 2: and let V be the variety defined by {f = 0). Then Reg(V) is a two-dimensional complex manifold and Sing(V) is the zs-axis.

174

+

Exercises 4.10. (1) Let S3 = { ( ~ 1 ~ 2 2I )1z1I2 1z2I2 = 1 ) be the threedimensional unit sphere in C2 = R4. Show that, in Example 4.9, 1, the intersection L = V n S3 consists of two circles which are unknotted but link with each other. (2) Show that, in Example 4.9, 2, L = V n S3 is the "torus knot of type (2,3)". (3) In Example 4.9, 2, find an explicit (holomorphic) homeomorphism from C onto V. (4) Let V be the variety in C4 = {(z1,z2,z3,z4)) defined by the three equations:

z1zq - ~

2 =~0, 3 z$ - 2123 = 0

and

zi

-~

2

=~0. 4

Find Reg(V) and Sing(V). What is the dimension of each connected component of Reg(V)? Consider the n-dimensional complex projective space CP" with homogeneous coordinates [coo,. . . ,<"] and let, for each j = 1 , . . . , r , Pj(c0,. . . , be a homogeneous polynomial in ( ( 0 , . . . , of degree d j . Then the set

v = { [CO, . . . ,<"I

E CP"

I Pj(C0,.

en) . . ,en) = 0, j

en)

= 1,.

1 .

,r}

is a well-defined subset of CP" and is, moreover, a variety in CP". In fact, in each open set Ui = { c i # 0}, V is defined by the holomorphic functions d. f j = Pj(c0,. .., 3 , j = 1,.. . , r . Such a variety is called a (projective) algebraic variety. It is known that every variety in CP" is algebraic (Chow's theorem). In particular, the "hyperplane" defined by (0 = 0 is an ( n- 1)dimensional submanifold of CP" which may be identified with CP"-'. Thus we may express CP" as a disjoint union CP" = C" U CEOn-', which leads to a cellular decomposition of CP"; CP" = C" U C"-l U. . UCo. Using this we may compute the homology of CP"; for p H,(CP",Z) = Z, 0,

= 0,2,.

otherwise.

..,2n,

(4.11)

Exercise 4.12. For complex numbers a , p and y, let V a , ~ , be y the variety in CiP2 defined by

Va,p,r = { [coo,c1, c21 E @P2I eoe; - (el - aco)(cl - PCO)(Cl - 7 6 ) = 0 ) (1) Show that, if a , /3 and y are mutually distinct, Va,p,? has no singular points. (2) Show that, if y # 0, VO,O,~ has only one singular point a t p = [1,0,0], which is equivalent t o the one in Example 4.9, 1.

175

(3) Show that VO,O,O has only one singular point at p = [l,O,O], which is equivalent to the one in Example 4.9, 2. 5. Germs of varieties

In this section, we consider the germs of varieties and the relation between these germs and the ideals in 0,. See, e.g., [22] for the corresponding theory in Algebraic Geometry. We first introduce a relation in the set of subsets of C”. Let A and B be two subsets of C”. We define A B if there is a neighborhood U of 0 such that A n U = B n U . It is easily checked that this is an equivalence relation. We call the equivalence class of A the germ of A at 0 and we denote it also by A unless it is necessary to distinguish the two. Usual operations of sets induce those of germs. Thus for two germs A and B at 0, A n B , A U B and A \ B are well-defined. The relation A c B is also well-defined. Let fl, . . . ,f r be germs in 0,. We choose a neighborhood U of 0 such that these germs are represented by holomorphic functions on U ,which we also denote by f 1 , . . . ,f r . We set N

N

V(f1, . . . , f r ) = thegermat O o f { z E U I fl(z) = . . . = f r ( z ) = O } and call it the germ of the variety defined by f l , . . . , f r . More generally, let I be an ideal in 0,. By the Noetherian property of 0, (Theorem 3.7), there exist a finite number of germs f l , . . . ,f r such that I = (f1, . . . ,fr) (the ideal generated by f1,. . . , f r ) . We set V ( I ) = V ( f 1 , .. . ,fr) and call it the germ of the variety defined by I . It is easily checked that it does not depend on the choice of generators of I . Thus each ideal in 0, defines a germ of variety at 0. Conversely suppose we are given a germ V of variety at 0. We choose a neighborhood U of 0 such that the germ is represented by a variety in U ,which we also denote by V. We set

I ( V ) = { f E 0, I f ( z ) = 0 for all z in V and near 0). It is easily checked that this is an ideal in 0,. For an ideal I in On, we set

Ji =

f E O,I

fk E

I for some positive integer IC 1

and call it the radical of I . This is again an ideal in 0, and it contains I . Exercise 5.1. For k = 1,.. . , n, we consider the “coordinate functions” z1, . . . ,Zk as germs in 0,. Show that

I ( v ( z l , . * .7zk)) = (z17...,zk).

176

There are various relations between germs of varieties and ideals, most of which follow rather straightforward from definition. The most important and deep fact will be the following theorem. We refer to [19], for example, for the proof.

Theorem 5.2 (Hilbert Nullstellensatz). For any ideal I in On,

I ( V ( I ) )= Ji. Exercise 5.3. Show that, for an ideal I (# 0,) in On, the complex vector space 0,/I is finite dimensional if and only if V ( I )= (0). In general, let R be a commutative ring with identity. For an ideal I in R, its radical J? is defined similarly as for the ones in 0,. An ideal p in R is said to be prime if Rlp is an integral domain, i.e., p # R and ab E p implies a E p or b E p. If p is prime, then f i = p.

Definition 5.4. Let V be a germ of variety at 0. We say that V is irreducible if V # 0 and if V = V1 U V2 implies V1 = V or V2 = V . Theorem 5.5. A germ of variety V is irreducible af and only i f the ideal I ( V ) is prime. Corollary 5.6. For a germ f , which is not 0 or a unit, in On, the following are equivalent: (i) V ( f )is irreducible. (ii) I ( V ( f ) )(= is a prime ideal. (iii) There is an irreducible element p in 0, such that f = p” f o r some positive integer m.

m)

The following is a consequence of the “primary decomposition theorem” :

Theorem 5.7. Every (non-empty) germ V of variety can be written as

V = V1 U ... U Vr, where Vl,. . . , V , are germs of varieties such that each V , is irreducible and that V, $ V,, i f i # j . Moreover, Vl,. . . , Vr are uniquely determined by V up to order. The proof of the following is not difficult.

Theorem 5.8. Let f be a germ, which is not 0 or a unit, in 0,. I f f = p;”’ . . .p r r is the irreducible decomposition of f , then

V(f)

= V(P1)U . . . U V ( P r )

177

is the irreducible decomposition of V(f).

Let f be a germ in On, which is not 0 or a unit. We say that f is reduced if the irreducible decomposition of f has no multiple factors, z.e., in the irreducible decomposition f = p y l . . . p F , we have mi = 1 for all i. We represent f by a holomorphic function f in a neighborhood of 0. Then, it can be proved that, i f f is reduced at 0 , the germ fz in On,z is reduced for all z sufficiently close to 0. Note that, on the other hand, even iff is irreducible at 0 , f z may not be irreducible. For example, consider the “Whitney umbrella” (Example 4.9, 3). Exercise 5.9. Show that f is reduced if and only if I(V(f)) = (f).

We define the dimension of a variety on the basis of the following theorem. For the proof we refer to [19].

Theorem 5.10. Let V be a n irreducible germ of variety. W e m a y f i n d a representative V of V such that Reg(V) is connected and dense in V. For a germ of variety V at 0 , we define its dimension (at 0 ) , denoted by dimV, as follows. If V is irreducible, then we define dimV to be the dimension of the complex manifold Reg(V) . In general, if V = V1 U . . . U V, is the irreducible decomposition of V, we set dim V = maxlsi<, dim V,. We also define the codimension (denoted by codim V) by codim V = n - dim V. Note that in this case we have the corresponding decomposition Reg(V) = C1 U ... U C, of Reg(V) into its connected components Ci.Each Ci is a complex manifold whose closure coincides with V,. However, in general, Ci does not coincide with Reg(V,). We say that V is pure dimensional if all the components Vi have the same dimension. The “if” part of the following theorem follows from Theorem 5.8. For the “only if” part, we refer to [19]. Theorem 5.11. A germ V of variety is pure ( n - 1)-dimensional i f and only if there is a germ f in On, not 0 or a unit, such that I ( V ) = (f).

Let D be an open set in a complex manifold M . A variety V in D is said to be (globally) irreducible if it cannot be expressed as the union of two varieties Vl and V2 in D with Vl, V2 # V. This notion should be distinguished from the “local” irreducibility (Definition 5.4). For example the variety VO,O,? of Exercise 4.12, ( 2 ) is globally irreducible, but locally not irreducible at p . Note that every variety is written as a union of irreducible varieties. Note also that V is irreducible if and only if the regular part

178

Reg(V) is connected. Hence, for an irreducible variety V and a point p in V, the dimension of V at p remains constant. We call it the dimension of V. In general, we say that V is pure dimensional, if all the irreducible components of V have the same dimension.

Chapter 11. Differential forms and Cech-de Rham cohomology 1. Vector bundles

In what follows, we denote by K either R or C and M T ( K )the sets of r x r matrices with entries in K. It is naturally identified with K T2 .Also we set

GL(r,K) = { A E M r ( K ) 1 det A # 0). It has the structure of a real or complex Lie group. Namely, it is a group with respect to the multiplication of matrices and, moreover, it is a C" or a complex manifold according as K is R or C, since it is an open set of M T ( K ) ,and the group operation is C" or holomorphic.

Definition 1.1. Let M be a C" manifold. A (C") vector bundle of rank r over M is a topological space E together with a continuous map IT: E + M such that there exists an open covering U = (U,},,, of M with the following properties: (1) for each a,there is a homeomorphism $,:

7r-'(u,) 1u,

x KT

with a o $, = 7r, where a denotes the projection U, x K' (2) for each pair ( a , @ ,there is a C" map

+ U,,

h a p : U, n Up -+ GL(r,K) with $a 0 $jl(P,

c>= (P,hap(P>c>

for

(P, C) E

u, n up x KT.

We say that E is a real or complex vector bundle according as K is R or C. Thus if IT: E + M is a vector bundle of rank r over a C" manifold M of dimension m, then E has the structure of a C" manifold of dimension m r or m 2r, according as K = R or C, so that 7r is a C" surjective submersion (surmersion) and each fiber Ep = n - l ( p ) , p E M , has the structure of a vector space of dimension r over K. We call a trivialization of E on U,. We also call hap the transition matrix of E on U, n Ua and the

+

+

$ J ,

179

collection {hap} the system of transition matrices of E . For each point p in U, n Up n U,, we have the identity h*P(p)hPr(p) = h",(p).

(1.2)

Thus, in particular, hQQ(p) = I (the identity matrix) and hp"(p) = (h"D(p))-l. We may think of the system {(U,,$,,hap)} as defining a vector bundle structure on E. If we are given an open covering {U,} of M and a collection {hap} of C" maps hap: U, n Up -+ GL(r, K)

satisfying (1.2) for p in U, n Up n U,, we may construct a vector bundle as follows. For ( p a , C,) in U, x K' and (pp, Cp) in Up x K', we define (Pa,Ca) (PO,Cp) if and only if

-

i

Pa = P o (=PI

I" = h"P(p)CP.

Then it is easy to see that this is an equivalence relation in the disjoint union u,(U, x K'). Let E be the quotient space. Then, since

(U, x R')/- = u, x K', E has a vector bundle structure with {hap} as a system of transition matrices. Let E and F be two vector bundles on M . A C" map 'p: E --+ F is said to be a vector bundle homomorphism if it commutes with the projections and if the induced map ' p p: Ep -+ Fp on each fiber is K-linear. We say that cp is an isomorphism if it is a C" diffeomorphism. In this case 'p induces a K-isomorphism on each fiber. We also say that E and F are isomorphic (or E is isomorphic to F ) , and write E N F , if there is an isomorphism of E onto F . A vector bundle is called trivial if it is isomorphic to the product M x K'. Exercise 1.3. Let E and F be two vector bundles on M with systems of transition matrices {hap} and { g * p } , respectively, on an open covering {U,}. Show that E and F are isomorphic if and only if there exists a C" map h a : U, 4 GL(r,K), for each a,such that

h""d for p in

uan Up.

= h"(P)-'gap(P)hp(P)7

180

We say that a sequence of vector bundle homomorphisms

E X F ~ G is exact if, for each p in M , the induced sequence Ep -% Fp 4 1c, Gp is exact, ie., Ker$, = Impp. Let T : E -+ M be a vector bundle of rank r. A subset E' of E is said to be a subbundle of E , if there is a system { (U,, $, hap)} as in Definition 1.1 such that each $, maps T'-~(U,) onto U, x K", where T' denotes the restriction of IT to E' and Kr' is identified with the subspace (the transposed of of Kr consisting of (column) vectors c = t ( c l , .. . , ([I,. . . ,&)) with [,.)+I = . . = [,.= 0. In this case, each hap is of the form

cr)

where hlaP and h'laP are C" maps from U, n Up into GL(r',K) and GL(r",K),r" = r - r', respectively. Note that each of the systems {h'@} and {h'lLYp} satisfies (1.2). Thus E' has the structure of a vector bundle of rank r' with {h'"'} as a system of transition matrices. The vector bundle of rank r" defined by the system {h"ap}is called the quotient bundle of E by E' and is denoted by EIE'. Note that there is a surjective vector bundle homomorphism cp: E -+ E/E' so that the sequence 0 -+ E' -k, E 2 E/E'

-+ 0

is exact, where L denotes the inclusion. In general, if we may choose a system {hap} of transition matrices of a vector bundle E so that each hap is of the form (1.4), then E admits a subbundle with {h'"'} as a system of transition matrices. Exercise 1.5. Let cp: E -+ F be a homomorphism of vector bundles. Show that, if the rank of the restriction cpp of cp t o each fiber Ep, p E M , is constant, then the kernel Kercp = UpEMKercppand the image I m p = UpEMImp, of cp are subbundles of E and F , respectively. Show also that the quotient bundle E/Ker cp is isomorphic to Im cp. The quotient F/Im cp is called the cokernel of cp and is denoted by Coker cp. If f : M' t M is a C" map of C" manifolds and if T : E -+ M is a vector bundle over M , we define the pull-back f * E of E by f by

f * E = { (P, e) E M'

x E

I f (PI = 4.) 1.

181

It is a vector bundle over M' with projection the restriction of the projection onto the first factor. Note that ( f * E ) ,= Ef(,). In particular, if V is a submanifold of M with inclusion map i and if E is a vector bundle on M , the pull-back i*E is called the restriction of E to V and is denoted by Elv. A complex vector bundle over a complex manifold M is said to be holomorphic if E admits a system of transition matrices {h"o} such that each hap is holomorphic. Note that in this case, E has the structure of a complex manifold so that the projection E -, M is a holomorphic submersion. Let T :E -, M be a vector bundle of rank r and U an open set in M . A (C") section of E on U is a C" map s: U + E such that T O S = l u , the identity map of U . A vector bundle E always admits the "zero section", i.e., the map M --+ E which assigns to each point p in M the zero of the vector space E,, The set of C" sections of E on U is denoted by C"(U, E ) . This has a natural structure of vector space by the operations defined by (s1+ sz)(p) = s l ( p ) s2(p) and (cs)(p)= cs(p) for s1, s2 and s in C"(U, E ) , c in K and p in U . If E is a holomorphic vector bundle over a complex manifold M , a section over U is said to be holomorphic if it is a holomorphic map from U into E. The set of holomorphic sections of E over U is denoted by r ( U ,E ) . This has the structure of a complex vector space. A section s on U can be described as follows. We fix a system of transition matrices {hap} of E on an open covering {U,}. Using the C" diffeomorphism $Ia: T - ~ ( U , ) 2 U, x Kr, we may write

+

$ I M P ) ) = (?A S " ( P ) ) for P

E

u n ua,

where sa is a C" map from U nU, into Kr. For each point p in U nU, nUp, we have

Conversely suppose we have a system {sa} of C" maps satisfying (1.6). Then by setting s ( p ) = $I;'(p, s * ( p ) ) for p in U n U,, we have a section s over U . For k = 1, . . . ,r, a k-frame of E on an open set U in M is a collection s = ( ~ 1 , .. . , sk) of k sections si of E on U linearly independent at each point in U . An r-frame is simply called a frame. Note that a frame of E on U determines a trivialization of E over U . Example 1.7. Let M be a C" manifold of dimension m. We may give naturally a vector bundle structure on the (disjoint) union TwM = UpEMT~,,M of the tangent spaces of M . First, define T : TwM -, M by

182

assigning to each tangent vector its base point. Then let {U,} be a covering of A4 by coordinate neighborhoods U, with coordinates (x?,.. . ,x;). By taking (a/axy,.. . ,a/ax;) as a basis of Tw+,M for each p in U,, we have a bijection $, : 7r-'(Ua) + U, x R". Since we have the relation

for p in U,

n UP, we see that $,

o

$il(plc) = (p,t@(p)()

for ( p , o E

(U, n U P )x W", where t f f p= a(xy,.. . ,X K ) P

d(X, 1 . .

P ' . ,Xm)

Hence we see that TwM admits the structure of a real vector bundle of rank m with { P o } as a system of transition matrices. We call it the (real) tangent bundle of M . A (C"") vector field w on an open set U is a (C") section of T R M . Thus it is expressed as, on each U n U,,

where the fr's are (C"") functions on U, n U . In U n U, n U P ,we have f" = t*PfP, f a = . , f:). Note that (a/ax?,. . . , a/ax&) is a frame of TwM on U,. If V is a submanifold of dimension l? of M , then we may cover V with coordinate neighborhoods U, on M with coordinates (xy,. . . , x;) such that

t(fr,..

V n U,

={p E

U, I x;+l (p)= . . . = x;(p) = 0 }.

Then the restriction t"PIv of t a p to V n U,

n UP is of the form

where tIapand t1IQPdenote the Jacobian matrices ~ ( X ~ , . . . ~ X ~ ) / ~P( X ~ ~ . . and a(x;+,, . . . ,X%)/~(X:+~, . . . , xk), respectively, both restricted to V . Since the restriction of (xy,. . . , x;) to V form a coordinate system on VnU,, we see that TwMlv admits TwV as a subbundle. We call the quotient bundle the normal bundle of V in M and denote it by Nw,v. It is known that there exist a neighborhood U of V in M , a neighborhood W of (the image of) the zero section 2 in Nw,v and a diffeomorphism $ of U onto W such that $ ( V ) = 2 ([18, p. 761). Such a neighborhood U is called a tubular neighborhood. Usually we take an open disk

183

bundle (or NR,V itself) as W so that V is deformation retract of U with a C" retraction p : U -+ V .

Example 1.8. Let M be a complex manifold of dimension n and {U,} a covering of M by coordinate neighborhoods U, with complex coordinates ( z r , . . . ,z,*). Then, as in Example 1.7, the union T M = U p E M T p M of the holomorphic parts of the complexified tangent spaces of M admits the structure of a complex vector bundle of rank n with {Po},

as a system of transition matrices. Since, for each pair ( a ,p), r"o is a holomorphic map from U,nUp into GL(n,C ) ,T M is a holomorphic bundle. We call it the holomorphic tangent bundle of M . Note that, as a real bundle, T M is isomorphic to TRM (see Proposition 2.2 of the following section). A holomorphic section v of T M is called a holomorphic vector field. On U,, we may write as

where the fr's are holomorphic functions on U,. If V is a complex submanifold of M , then as in 1.7, T M l v admits T V as a subbundle. We call the quotient the holomorphic normal bundle of V in M and denote it by N v so that we have the exact sequence 0 + TV -+ T M l v

+Nv

--+

0..

For each point p in V , we have the situation considered in Ch. I, Exercise 4.8 (2). Note that, again by Proposition 2.2, N v is isomorphic to NR,V as a real bundle.

Example 1.9. Let M be a complex manifold of dimension n and V a hypersurface (possibly with singularities) in M . We cover M by open sets U, so that in each U,, V is defined by a "reduced equation" f a = 0, i.e., the germ of f a at each point in V n U , is reduced (see Ch. I, Section 5). Note that if V n U , = 8, then we may take a non-zero constant as fa. Then, for each pair (a,p), f ap = f*/f p is a non-vanishing holomorphic function on U, n Up and the system { f a o } defines a complex vector bundle of rank one (a line bundle) on M . We call this bundle the line bundle defined by V and denote it by L ( V ) .Note that L ( V ) is a holomorphic bundle and admits a natural holomorphic section whose zero set is exactly V , i.e., the section determined by the collection { f *}.

184

In particular, the line bundle on the projective space CP“defined by the “hyperplane” is called the hyperplane bundle and denoted by H,. If we use the notation of Ch. I, Example 4.2, 2, the bundle Hn is defined by the system transition functions {hij}with haj = C j / & on the covering {Vi}. Exercise 1.10. Show that, if V is a non-singular hypersurface of M , then there is a natural isomorphism L(V)Iv N Nv.

If we are given some vector bundles, we may construct new ones by algebraic operations. Thus we let E and F be vector bundles on M . We may construct the direct sum E @ F , the homomorphism Hom(E, F ) and the tensor product E @ F . Note that there is a natural isomorphism Hom(E, F ) N E*@ F.

Ak

We also have the k-th exterior power E. For a complex vector bundle E, we have the complex conjugate F and for a real vector bundle E , the complexification E“ = E & C. The complex vector space C‘ is naturally considered as a real vector space of dimension 2r and this defines a natural homomorphism

GL(r,C ) -+ GL(2r,R). Thus if E is a complex vector bundle of rank r , it has the structure of a real vector bundle of rank 2r. 2. Vector fields and differential forms

We denote by T M the holomorphic tangent bundle of a complex manifold M as in Section 1. The following two propositions are consequences of the Cauchy-Riemann equations. Proposition 2.1. If M is a complex manifold, there is a natural isomorphism

TiM Proposition 2.2. We have T M

2~ pv

T M @TM. TwM as real bundles.

The following shows how a complex vector field (a section of T M ) and a real vector field (a section of TwM) correspond in the above isomorphism, when they are expressed in terms of local coordinates:

185

where f i = U i

+a

v i with ui and vi real valued functions.

Example 2.3. In C = { z } the complex vector field .z& corresponds to the real vector field x& y a and z2& to (x2- y2)& 2 2 a ~ ~ .

+

+

aY

Let M be a Coomanifold of dimension m. We call a C" section w of the bundle A"(T;M)* on an open set U in M a (complex valued) differential p-form of class C" (simply, a C" pform) on U . We denote by Ap(U) the set of C" pforms on U , which has naturally the structure of a C-vector space. The set A o ( U ) is thought of as the ring of C" functions on U . We have the exterior product

A P ( U )x A 9 ( U ) + Ap+q(U),

(w,8)

Hw

A 8.

It is bilinear in w and 0 and satisfies w A 0 = (-1)PqO A w. We also have the exterior derivative d = d P : A P ( U )-+ Ap+'(U),

which is a C-linear map satisfying dP+' o d p = 0 and

d(w A e) = dw A e + (-i)pw A de

(2.4)

for w E AP(U) and 0 E Aq(U). For a complex vector bundle E on M and an open set U in M , we set A p ( U , E ) = C"(U, Ap(T;M)* @ E ) . An element 0 in AP(U, E ) , called a differential pform with coefficients in E , is expressed locally as a finite sum C wi 63 si with wi pforms and si sections of E. The exterior product induces a bilinear map

AP(U) x Aq(U,E ) + Ap+q(U,E ) . Now let M be a complex manifold. Recall that the holomorphic cotangent bundle is the vector bundle T * M dual to the holomorphic tangent bundle T M . By Proposition 2.1, we have a natural isomorphism

( T i M ) * 2: T*M

@

T*M .

(2.5)

Hence we have an isomorphism P

r

9

A(T;M)*@ AT*M@ A T * M . N

p+q=r We call a section of A" T * M @ AqT*M a differential form of type ( p , q) (simply, a ( p , q)-form). Thus a differential r-form is expressed as a sum of ( p , q)-forms with p q = r. Suppose that a point z in M is in a coordinate

+

186

neighborhood U, with coordinates ( z y ,. . . ,z,"). We write z: = x $ + & T y , " and identify T,*M and T f M with subspaces of (Trip,; M ) * by the isomorphism (2.5). Then, if we set

d z r = dxq

+a

d y p

and dZr = dxq - &id?/?,

a straightforward computation shows that dz?, ...,dz," are in T,*M and form a basis dual to the basis (a/&?, ...,a/&:) of T z M and that dz?,. . . ,dZ," are in T f M and form a basis dual to the basis (a/a.Z?,. . . ,a/%:) of T z M . Since d z z A A dz& where (il,. . . ,i,) runs through ptuples of integers with 1 5 i l < ... < i, 5 n, form a basis of ApT,*M and d"; A . . A ax,where ( j l , . . . ,j,) runs through q-tuples of integers with 1 5 j 1 < . . . < j , 5 n, form that of AqT : M , a ( p ,q)-form w is written as, on U,,

fz

C

W =

,,,,,i p , j l ,...,j q ( z a ) d r z A . . . A d d Z Z Q p A d Z ~ A . . . A d Z j D l q ,

l
(2.6) where fi":,...,i,,,jl, ...,j, are C" functions on U,. By setting I = and J = (jl,.. . ,j,) we may write (2.6) simply as

(21,.

. . , ip)

In particular, a (p,O)-form w can be written as, on U,,

f ~ , , , , , i , ( z a ) d z A.--AdzzpP. z

w= l
If each fg,,,,,i, is holomorphic, we say that w is a holomorphic pform. It is nothing but a holomorphic section of A" T*M . We denote by Ap,q(U) the set of (p,q)-forms on an open set U in M . For each ( p , q ) , it is an Ao(U)-module and we have the decomposition Ar(U) =

@ Aptq(U). p+q=r

Thus we may express the exterior derivative d as a sum d = a

a:

Ap?'J(U)--t AP+'?Q( U )

and

a:

A p 7 q ( U ) - - + A p ,(W. q+1

From d o d = 0, we have doa=O,

8 0 8 = 0 and

+ 8 with

aoa+aoa=O.

187

3. Integration and the Stokes theorem

Let M be an oriented C" manifold of dimension m. Recall that, for a Co m-form w with compact support, we may define the integral w. More generally, let o = C nioi be a (C"") singular pchain in M . Thus each oi is a C" map from (a neighborhood of) the standard psimplex AP into M . For a pform w on M , we define

,s

Let D be an open set in M and assume that the boundary d D of D is C", i.e., for any point p of d D there is a coordinate neighborhood U with coordinates ( X I , . . . ,x,) such that

R n U = { q E U I zi(q) I ~ i ( p},) (the closure of D in M ) . In this case, dR = d D is an (m - 1)-

where R = dimensional C" submanifold of M . In fact if (XI,... ,zm) is a coordinate system as above, then ( z ~.,. . ,)z, is a coordinate system on d R n U . Moreover, if M is orientable, so is dR. If M is oriented so that a coordinate system ( 2 1 , .. . ,z,) as above is positive, we orient d R so that (XZ,. . . z ,), is positive. Suppose M is oriented and R as above is compact. Then we may define, for a Co m-form w on a neighborhood of R, the integral w.

sR

Theorem 3.1 (Stokes theorem). Let D be a relatively compact open set in M with C" boundary (which m a y be empty). For a Co (m - l ) - f o m w in a neighborhood of R = 0,

where

L:

dR

L)

M denotes the inclusion.

Note that the above formula makes sense even if the boundary dR is only piecewise C" . If A4 is a complex manifold, M is orientable, since by Ch. I, (2.5), we may find a C" local coordinate system on M such that the Jacobian of every coordinate change is positive where it is defined. We orient M so that, if ( ~ 1 ,... , z,) is a complex coordinate system on M , ( X I ,y1,. . . , z, y,) is a positive coordinate system, where zi = xi G y i , i = 1,.. . ,n.

+

4. de Rham cohomology

We list [4] as a basic reference for this section. Let M be a Coo manifold of dimension m. For an open set U in M , we denote by A p ( U ) the space of

188

complex valued CO" pforms on U , i.e. Coosections of the bundle A p ( T i M ) * on U. The exterior derivative d defines the de Rham complex of M :

o -+A O ( M )2 A ~ ( M % ) ...

dm-'

A"(M)

+ 0.

The p t h de Rham cohomology H I ( M ) (with complex coefficients) is the p t h cohomology of this complex; H Z ( M ) = KerdP/ImdP-'. For a closed pform w , we denote its class in Hdp(M) by [u]. If M is connected, we easily see that @ ( M ) 21 C.

Lemma 4.1 (Poincard lemma). T h e de R h a m complex of R" i s acyclic, i.e.,

Hz(R") = O

for p > O .

This is a special case of the following de Rham theorem, in fact it is a key ingredient in the proof of the theorem. Let H p ( M ,@) and Hp(M,C ) denote the singular (or simplicial) homology and cohomology of M . The integration of a pform on a (piecewise Cm) singular pchain of M induces a homomorphism

H,p(M)

+

HP(M,C ) ,

which is shown to be an isomorphism;

Theorem 4.2 (de Rham theorem). For a Coo manifold M ,

H,p(M) 21 HP(M,C). Now, by (2.4), the pairing

H:(M) x H j ( M ) + H I + q ( M ) given by ( [ w ] ,[el) H [w A e] is well-defined and it corresponds to the cup product in the isomorphism of Theorem 4.2. We write [w A 01 = [w] v [el. This product makes the direct sum H : ( M ) = H z ( M ) a graded ring. If M is compact, connected and oriented then, by the Stokes theorem, the integration on M induces a linear map

ep

Then it is proved that the bilinear form Hd

(M ) x H Y - P ( M )

is non-degenerate (e.g., [4, Ch. I, § 5 ] ) :

H r ( M ) =k C

189

Theorem 4.3 (Poincar6 duality). For a compact, connected and oriented C” manifold M of dimension m, the above pairing induces an isomorphism P : H Z ( M ) 1Hm-P(M,C)* = H,-p(M,C).

In the isomorphism of Theorem 4.3, a class [w] in H p ( M ,C) corresponds to the class of a (piecewise C’) singular (rn - p)-cycle C in M satisfying J M w ~ e =J‘B

(4-4)

for all closed (m - p)-form 0 on M. In particular,

H”(M,C)

N

H o ( M , C )N C

and, for the class [w] of a closed m-form w , the corresponding homology w. Also class may be thought of as a complex number, which is given by

sM

H,(M,C)

II

H o ( M , C )I I C

and the homology class in H,(M,C) corresponging to the class [l]of the function constantly equal to 1 is represented by an m-cycle C such that JM 8 = 8 for all closed rn-form 8. Thus this homology class coincides with the class [MI of M considered as an m-cycle, the fundamental class of the compact oriented manifold M . Note that it is the canonical generator of the integral homology H,(M,Z) N Z.

,s

Remark 4.5. Let M be a complex manifold and V a compact analytic variety of dimension -t in M . Then we may think of V as a 2Lcycle, for example, by triangulation, in which case we have the class [V] in H2e(V, Z), or by integration of 2Gforms on M ([17, Ch. O ] ) , in which case we have the class [V] in H2e(M,C). Moreover, if V is (globally) irreducible, then H2e(V,Z) N Z and [V] is the fundamental class (e.g., [5]). Recall that the Poincare isomorphism P in Theorem 4.3 is given by the “cap product” with the fundamental class;

mJ1)

= [WI

A

[MI.

5. Cech-de Rham cohomology The Cech-de %am cohomology is defined for arbitrary coverings of a manifold M , however for simplicity here we only consider coverings of M consisting of only two open sets. For details, we refer to [4], see also [47].

190

Let M be a C" manifold of dimension m and U = {UO,U l } an open covering of M . We set Uol = Uo n U1. Define a vector space AP(U) as follows:

A p ( U )= AP(Uo)CBAp(U1)CBAp-l(Uoi). Therefore an element a E A p ( U ) is given by a triple a = ( a o , a 1 , ( ~ 0 1 ) with a0 a pf or m on Uo, a1 a p fo rm on U1 and a01 a ( p - 1)-form on U O ~ . We define the operator D : Ap(U) -+ AP+l(U) by

Da

= (dao,da~,al - a0

-

dao1).

Then it is not difficult to see that D o D = 0. This allows us to define a cohomological complex, the Cech-de Rham complex:

Set ZP(U) = Ker DP, BP(Z4)= Im DP-' and

H g ( u ) = ZP(U)/BP(U), which is called the p t h Cech-de Rham cohomology of U.We denote the image of u by the canonical surjection ZP(ZA) 4 H g ( U ) by [a]. Theorem 5.1. The map A p ( M ) + AP(U) given by w an isomorphism a : H,p(M)

H

(w,w,O) induces

= H;(u).

Proof. It is not difficult to show that a is well-defined. To prove that a is surjective, let a = (ao,a1,a01)be such that D o = 0. Let { p o , p l } be a partition of unity subordinated to the covering U.Define w = pogo plal - dpo A 001. Then it is easy to see that dw = 0 and [(w,w , O)] = [a]. The injectivity of a is not difficult to show. 0

+

We define the k u p product"

A p ( U )x Aq(U) 4 AP+q(U) by assigning t o a in Ap(U) and given by (av

~ ) =i ai

A ~ i , = 0, 1,

T

in Aq(U) the element a (av

T ) O ~=

v

(-l)po~ A TO^

T

in AP+q(U)

+ 001 A 7 1 . (5.2)

191

Then we have D ( g w cup product

T)

=Da w

+ (-1)Pa

w

Or.Thus it induces the

H g ( U ) x H;(U) -+Hg+’((U) compatible, via the isomorphism of Theorem 5.1, with the cup product in the de Rham cohomology. Now we recall the integration on the Cech-de Rham cohomology (cf. [30]). Suppose that the m-dimensional manifold M is oriented and compact and let U = {UO, U l } be a covering of M . Let &,R1 c M be two compact manifolds of dimension m with C” boundary with the following properties: (1) Rj c U j for j = 0,1, (2) Int R,, n Int R1 = 8 and (3) Ro U R1 = M . Let &I = & n R1 and give &I the orientation as the boundary of &; Rol = ~ R oequivalently , give Rol the orientation opposite to that of the boundary of R1; &I = 4 R 1 . We define the integration

Then by the Stokes theorem, we see that if D a = 0 then Su , is independent of {Ro,R1} and that if a = DT for some r € AP-’(U) then a = 0. Thus we may define the integration

sM

which is compatible with the integration on the de Rham cohomology via the isomorphism of Theorem 5.1. The cup product followed by the integration gives a bilinear paring

AP(U)x Am-P(U)4 C, which induces the Poincare duality

P : H p ( M , C )I IH&(U) 2 H,”-’(U)*

N

Hm-p(M,(C).

Next we define the relative Cech-de Rham cohomology and describe the Alexander duality. Let M is an m-dimensional oriented manifold (not necessarily compact) and S a compact subset of M . Let UO= M \ S and let U1 be an open neighborhood of S. We consider the covering U = {UO, Ul} of M . We set

AP(M,Uo) = { u = (go, ai,~

0 1E )

AP(U)I 00= 0 }.

192

Then we see that if (T is in AP(LI,Uo), D(T is in AP+'(U, Uo). This gives rise to another complex, called the relative Cech-de Rham complex and we may define the p t h relative Cech-de Rham cohomology of the pair ( U J o ) as

H g (U, Uo) = Ker DP/Im DP-'. By the five lemma, we see that there is a natural isomorphism

Hk(U,Uo) 1: H P ( M , M \ S ; C ) . Let R1 be a compact manifold of dimension m with C" boundary such that S c Int R1 c R1 c U1. Let & = M \ Int R1. Note that & C UO. The integral operator (which is not defined in general for Am(LI) unless M is compact) is well defined on Am(U,UO):

sM

sM

and induces an operator : H E ( U , UO)t C. In the cup product Ap(U) x A"-p(U) --+ A"(U) given as (5.2), we see that if (TO = 0, the right hand side depends only on (~1,001and 71. Thus , followed by we have a pairing AP(24,UO)x A"-P(Ul) t A"(U, U O ) which, the integration, gives a bilinear pairing

AP(U,Uo) x A"-'(Ui)

-+

C.

If we further assume that U1 is a regular neighborhood of S , this induces the Alexander duality

A : H P ( M , M \ S ; C ) 2 ~ H g ( L f , U o5) H H " - " ( U 1 , C ) *-Hm--p(S,C). (5.3)

-

Proposition 5.4. If M is compact, we have the commutative diagram

HP(M,M

\ s;C )

1lA

Hm-p(S, C )

j '

.i

-

HP(M,C ) 11.

H"-P(M,

-

@I,

where A , P, i and j denote, respectively, the Alexander and Poincare' isomorphisms and the inclusions S M and ( M ,0) ( M ,M \ S ). Example 5.5. Let M = R" and S = (0) with m 2 2. Then UO= R"\{O}, which retracts to S"-l. Let U1 = R". In this situation, we compute Hg(U,Uo). For p = 0, each element (T in A0(U,Uo) can be written as

193

a = (0, f , O ) for some C" function f on U1. If D a = 0, we have f = 0 and therefore Hi(U, UO)= (0). Next, an element a in A1(U,UO)can be written as a = (0, g1, f) with 01 a 1-form on U1 and f a C" function on UOn U1. If a is a cocycle then d q = 0 on U1 and df = a1 on UOn U1. By the Poincar6 lemma the first condition implies that a1 = dg for some C" function g on U1 and the second condition implies that f 3 g c for some c E C. Therefore f has a Cooextension, still denoted by f , over (0) and IJ = (O,df, f ) = D(0,f , O ) . Hence HA(U,Uo) = (0). For p 2 2 the map

+

Hdp-l(Uo)

+

H;(U,UO)

given by

[w]H [(O,O, -w)]

can be shown to be an isomorphism (we leave the details to the reader) and we have

H;(U,uo)

2L

Hdp-l(Uo) cv Hp-l(Sm-l,C) =

C for p = m 0 for p = 2 , . . . ,m - 1.

An explicit closed ( m - 1)-form representing a basis of the de Rham pv C is given as follows ([17, p. 3701). For z = cohomology Hr-l(Sm-l) ( ~ 1 ,... ,)z , in Rm, we set @(z)= dzl A . A dz, and h

@i(z) = ( - l ) i - l ~ i d z 1 A

* *

A dzi A . .

. A dxm.

Also, let C, be the constant given by

{ A,

cm=

(e-iy 2.rr',

for m = 2C for m = 2 e + 1.

Then the form

is a closed ( m- 1)-form on Rm\(0) whose integral on the unit sphere S m P 1 (in fact a sphere of arbitrary radius) is 1. Now we identify C" with R2", where then 7 h n = ( A+z)/2,

Then

pn

on

~sz,-l

is a closed (n,n - 1)-form on C" \ {0}, real on S2n-' and . . = 1. We call pn the Bochner-Martinelli kernel on C". Note that 1 dz p1 = ___-

2 7 4 3 z '

the Cauchy kernel on C.

194

Chapter 111. Chern-Weil theory of characteristic classes and some more complex analytic geometry 1. Chern classes via connections

In this section, we review the Chern-Weil theory of characteristic classes of complex vector bundles. For details, we refer to [2,3,17,37]. Let M be a C" manifold of dimension m. For an open set U in M , we denote by A o ( U )the C-algebra of C" functions on U . Also, for a Coocomplex vector bundle E of rank r on M , we let AP(U, E ) be the vector space of C" sections of A p ( T . M ) *@ E on U , which are called differential forms on U with values in E . Thus Ao(U,E ) is the Ao(U)-module of Coosections of E.

Definition 1.1. A connection for E is a C-linear map

v:

AO(M,E)

+A

~(M,E)

satisfying

V(fs) = d f @ s + f V ( s ) for f E A o ( M ) and s E A o ( M , E ) . Example 1.2. The exterior derivative d : A o ( M )+ A 1 ( M )

is a connection for the trivial line bundle M x C. From the definition we have the following:

Lemma 1.3. A connection V is a local operator, i.e., i f a section s is identically 0 o n an open set U , so is V(s). Thus the restriction of V to an open set U makes sense and it is a connection for E ( u .From the definition we also have the following lemma.

Lemma 1.4. Let V1,. . . , Vk be connectionsfor E and f l , . . . , f k C" funcfi = 1. Then fiVi is a connection for E . tions o n M with

xt=l

zt=,

Exercises 1.5. (1) Prove Lemmas 1.3 and 1.4. (2) Show that every vector bundle admits a connection. If V is a connection for E , it induces a @-linear map

v:

A ~ ( M , E+ ) A ~ ( ME, )

satisfying

V ( w @ s) = dw @ s - w A V(s) for w E A 1 ( M )and s E A o ( M ,E ) .

195

The composition

K =v V :

E ) 3 A ~ ( ME,)

AO(M,

is called the curvature of V . It is not difficult to see that

K ( f s ) = f K ( s ) for f E A o ( M )and s E A o ( M ,E ) . The fact that a connection is a local operator allows us to get local representations of it and its curvature by matrices whose entries are differential forms. Thus suppose that V is a connection for a vector bundle E of rank r and that E is trivial on U ; Elv 11 U x C'. If s = ( ~ 1 , ... ,S,) is a frame of E on U , then we may write, for i = 1,.. . ,r ,

j=1

We call 8 = ( O i j ) the connection matrix with respect to s. For an arbitrary section s on U , we may write s = CL, fisi with fi Coofunctions on U and we compute

The connection V is trivial with respect to s, if and only if 8 = 0. Thus in this case we have V(s) = C:='=, dfi 8 si. Also, from the definition we compute to get r

T

K ( ~ i ) = x ~ i j @ ~~23 j. ,.-j=1

d 823 . . -x

e i k Aekj. k=l

We call K = ( ~ i j the ) curvature matrix with respect to s. If s' = (si,. .. ,sk) is another frame of E on U', we have s: = aijsj for some C" functions aij on U n U'. The matrix A = ( a i j ) is non-singular at each point of U n U'. If we denote by 8' and K' the connection and curvature matrices of V with respect to s',

8'

= dA

. Ad'

+ A8A-l

and

K'

= A K A - ~ in

U n U'.

[TI,

(1.6)

Let n = the largest integer not exceeding T, and, for each i = 1,. . , n, let oi denote the i-th elementary symmetric function in n variables X I , .. . X n , i.e., o i ( X 1 , .. . ,X n ) is a polynomial of degree i defined by

.

n(~+ n

Xi) = 1

i= 1

+ (TI( X I ,

* * . l

Xn)

+ . . . + on(X1

* *

,X n ) .

196

Since differential forms of even degrees commute with one another with respect to the exterior product, we may treat IC as an ordinary matrix whose entries are numbers. Thus we define a 22-form O ~ ( I Con ) U by det(l,+K) = l + c l ( r c ) + ~ ~ ~ + c , ( r c ) , where I, denotes the identity matrix of rank T . Note that . ~ ( I c ) = 0 for i = T 1 , . . . ,n and, in particular, O ~ ( I C= ) tr(K) and O,(IC) = det(rc). Although O ~ ( I Cdepends ) on the connection V, by (1.6), it does not depend on the choice of the frame of E and it defines a global 2i-form on M , which we denote by ci(0).It is shown that the form is closed ([17,Ch. 3, 3 Lemma], [37, Appendix C, Fundamental Lemma]). We set

+

and call it the i-th Chern form. If we have two connections V and 8’for E , there is a (2i - 1)-form ci(V,0’) with q ( V , 0‘) = -ci(V‘, V) and satisfying

d c i ( V , V ’ ) = ~ i ( 0-’~ ) i(0).

(1.7)

The form ci(V,V’) is known as the Bott difference form and is constructed as follows ([3, p. 651).We consider the vector bundle E x B + M x R and let V be the connection for it given by

v = ( 1 -t)V+tV’, where t denotes a coordinate on R. Denoting by [0,1]the unit interval and by T : M x [0,1]-+ M the projection, we have the integration along the fiber 7r,,:

A2i(M x [0,1])+ A2i-1( M I .

Then we set ~ ( 0’) 0, =~*(ci(V)). One of the consequences of (1.7) is the following Proposition 1.8. The class [ci(V)] of the closed 22-form ci(V) in the de R h a m cohomology H F ( M ) 21 H 2 i( M , @ )depends only o n E but not o n the choice of the connection V.

We denote this class by ci(E)and call it the i-th Chern class c i (E ) of E. We call

197

the total Chern class of E , which is considered as an element in the cohomology ring H * ( M , C ) .Note that the class c ( E ) is invertible in H * ( M , C ) .

Remarks 1.9. 1. It can be shown that the top Chern class c T ( E )is equal to the Euler class e ( E ) of the underlying real bundle. 2. It is known that Q ( E )is in the image of the canonical homomorphism

P i ( M , Z)

4

H y M , C).

In fact it is possible to define c i ( E ) in H 2 i ( M ,Z) using the obstruction theory; it is the primary obstruction to constructing r-Z+l sections linearly independent everywhere [45]. 3. For the hyperplane bundle H, on CP" (Ch. 11, Example 1.9),

+

c(Hn) = 1 h,, where h, denotes the canonical generator of H2(CPn,C) (the Poincark dual of the homology class [CP"-']). More generally, if we have a symmetric polynomial cp, we may write cp = P(c1,02,. . . ) for some polynomial P . We define, for a connection V for E , the characteristic form cp(V)for cp by cp(0)= P ( c l ( V ) ,c z ( V ) ,. . . ), which is a closed form and defines the characteristic class cp(E)of E for cp in the de Rham cohomology. We may also define the difference form cp(V,0') by a similar construction. 2. Virtual bundles For simplicity, we consider only virtual bundles given as a difference of two vector bundles. If we have two complex vector bundles E and F , the total Chern class of the "virtual bundle" F - E is defined by

c(F - E ) = c ( F ) / c ( E ) .

(2.1) Let O E and V Fbe connections for E and F , respectively. We write the degree i term in the right hand side of (2.1) as

ci(F - E ) =

C cpy'(E) $ f ) ( F ) j

with cpii)(E)and $ i i ) ( F ) polynomials in the Chern classes of E and F , respectively. Then the i-th Chern class of F - E is represented by the differential form

cp:)(VE)A $ji)(VF),

ci(V*)= j

198

where V' denotes the pair ( V E VF). , Also, for a polynomial cp in the Chern classes of E, we may define a closed form cp(V') which represents the class cp(F - E ) . If we have two pairs of connections V:, Vi for E and F , there is a form cp(V:, 0:) satisfying an identity similar to (1.7),see [47,p. 721. Now let Q + E ~ F - loG + O (2.2) be a sequence of vector bundles on M , and VE, VF and VG connections for E , F and G , respectively. We say that the family (VE,VF,VG) is compatible with the sequence if the following diagram is commutative: Ao(M,E)

Ao(M,F)

lvE

lvF

Ao(M,G)

lVG

m1 L A ~ ( MF, ) --+ A ~ ( MGI. ,

A ~ ( ME) ,

If the above sequence is exact, there is always a family (VE,VF,VG) of connections compatible with the sequence and for such a family we have ( [ 2 , (4.22) Lemma]) C(V') = C(VE). 3. Characteristic classes in the Cech-de Rham cohomology

and a vanishing theorem Let M be a C" manifold and U = {UO,U I } an open covering of M . For a complex vector bundle E -+ M , we take a connection V j on U j , j = O , l . Then let ci(V,) be the element of A2i(U)given by

do*)= ( ~ i ( ~ O ) , ~ i ~ ~ l ~ , ~ i ~ ~ O , ~ l(3.1) ~ ) . Then we see that Dci(V,) = 0 and this defines a class [ ~ ( v ,E) ]H g ( U ) . Comparing with the class defined by a global connection, we have the following

Theorem 3.2. The class [cj(V,)] E H Z ( U ) corresponds to the Chern class ci(E) E H F ( M ) under the isomorphism of Ch. II, Theorem 5.1. By a similar construction, we may define the characteristic class cp(E)for a polynomial cp in the Chern polynomials in the Cech-de Rham cohomology. It can be done also for virtual bundles.

199

Let E be a complex vector bundle of rank r on a C" manifold M . Let s = (s1,. . . ,se) be an C-frame of E on an open set U , i.e. , C sections linearly independent everywhere on U . We say that a connection V for E on U is s-trivial, if V ( s i )= 0 for i = 1,.. . ,C.

Proposition 3.3. If V is s-trivial, then

ci(V)-O

for i > r - C + l .

Proof. For simplicity, we prove the proposition when C = 1. Let U C M be an open set such that Elu N U x C'. Since s1 # 0 everywhere on M , we may take a frame e = (el,. . . ,e,) on U so that el = s1. Then all the entries of the first row of the curvature matrix IC of V with respect to e are 0 zero. Since G ( V )= det K , up to a constant, we have c,(V) = 0. 4. Divisors

Let M be a complex manifold of dimension n. A meromorphic function cp on M is defined by a data { (U,, f ", g a ) } , where { Ua} is a covering of M , and f" and g" are holomorphic functions on U, such that the germ g,* at z is non-zero for all z in U,, the germs f," and g,* are relatively prime for all z in U, (cf. the phrase after Ch. I, Theorem 3.6) and that f O'gp = f pg" in U, n Up.We write cp = fa/g" on U,. A divisor D on M is a finite formal sum D = CE1niK, where the K's are irreducible hypersurfaces in M and the ni's are integers. Thus, if we cover M by open sets {U,} so that K is defined by f,P on U, (cf. Ch. 11, Example 1.9), we have the meromorphic function cp" = on U,, which is determined by the expression of D. For each pair (a,P), f @ = cp"/cpp is a non-vanishing holomorphic function on U, n Up and the system { f @} defines a line bundle on M . We call this bundle the line bundle defined by D and denote it by L ( D ) .We may write L ( D ) = where L(V,)nadenotes the tensor product of ni copies of L ( K ) ,for ni > 0, and the tensor product of -ni copies of L ( E ) * ,for ni < 0. A divisor D = niK is called effective if ni L 0 for all i. Thus an effective divisor is defined locally by a holomorphic function. If cp is a meromorphic function on M given by cp = f " / g a on U,, taking the irreducible decompositions of f" and g", we may consider a divisor, which we call the divisor of cp and denote by (cp). We may write (cp) = DO- D,, where DOand D , are defined, respectively, by f a and g" on U,. Clearly the bundle L(cp) is trivial. Conversely, it is shown that, if L ( D ) is trivial for a divisor D, then D = (cp) for some meromorphic function cp

nI='=,(fip)ni

EL='=,

200

([17, Ch. 1, 11, [23, $151). We say that two divisors D1 and 0 2 are linearly equivalent if D1 - 0 2 = (cp) for some meromorphic function cp. Thus this is equivalent to saying L(D1) = L(D2). For a divisor D = ni&, we set ID1 = & and call it the support of D. If each Vi is compact, the divisor D defines a homology class [D]= ni[&] in N2,-2(M,Z) (or in Hz,-z(lDI,Z)). It is known that, if M is compact, [D]is the Poincare dual of c l ( L ( D ) )([17, Ch. 1, 1 Proposition], see [48]for a more “precise” duality). Thus, if D1 and 0 2 are linearly equivalent, then [Dl] = [ D z ] .Also, for n divisors D1,. . . ,D, the “global” intersection number ( 0 1 . . . D,) is given by JM cl(L(D1))* * Cl(L(Dn)), where the product is the cup product. If the intersection lDil consists of isolated points, then this number is the sum of the intersection numbers at the points of intersection. See Example 6.2 below for this “local” intersection number when the divisors are effective.

xi=,

u;=,

xi=,

ny=,

Example 4.1. Let V be the algebraic variety in CP” = {[CO,. . . ,[,I} defined by a homogeneous polynomial P of degree d (Ch. I, 4). The function cp = P/c,d is a well-defined meromorphic function on CPn, which is given as the quotient of P/cf by C,d/cf on each &ne open set Ui = {Ci # 0). Thus, if we denote by D, the hyperplane defined by Co = 0, then V is linearly equivalent to d D , and [V]= d[D,]. Recall that [Dm] = [CPn-’] is the generator of H2,-2(CPn,Z) 21 Z. Also, the intersection of k copies of [D,] generates H2,-2k(CPn,Z) 21 z, for k = 1 , . . . , n (cf. Ch. I, (4.11)). 5. Complete intersections and local complete intersections We start with the local situation. Let On+k be the ring of convergent power series in (21,. . . ,z , + k ) .

Definition 5.1. Let V be a germ of variety at 0 of pure dimension n in Cn+’. We call V a complete intersection if the ideal I ( V ) is generated by k germs of holomorphic functions. In particular, if k = 1, V is a (germ of) hypersurface.

. germs f1, . . . ,f,. In general, let V be a germ of variety at 0 in C n f k Take in O,+k and set gi = aij fj with aij E On+k, for i = 1 , . . . ,s. Then

c,‘=,

rank Thus we have

a(gl, ’ ‘ * ,gs) (0) 5 rank a(zl, * * * 7 &+k)

a(f1,.. .,f,.) (0). . .,%+k)

a(Z1,.

201

Lemma 5.2. If the g e m s f1,. . . ,f r generate I(V), then the rank of the Jacobian matrix a(f1,.. . ,f r ) / a ( z l , .. . ,z,+k) at 0 does not depend o n fl,...,.fr.

Let V be a complete intersection of dimension n and f1,. . . ,f k generators of I(V). We take a neighborhood U of 0 in Cnfk such that the germs V and f1,. . . , fk have representatives on U . We may assume that the germs f l , z , .. . , fk,z generate I(Vz) for all z in U (“coherence of the ideal sheaf”, e.g., [19]) and hence we may write

v = { z E u I f1 (x)=

* *

. = fk(2)

= 0 }.

We call f l , . . . ,f r as above “reduced defining functions” for V. With these functions, we may describe the singular set of V as follows: Proposition 5.3. If V is a k-codimensional complete intersection,

Remark 5.4. If V is a pure k-codimensional subvariety, which may not be a complete intersection, the set Sing(V) may be expressed similarly as above, replacing f l , . . . ,fk by (arbitrary number of) generators f1,. . . ,fr of I(V) (e.g., [39, Ch. I, 511). Thus, for an analytic variety V, Sing(V) is also an analytic variety.

which In general, let V be a variety in a neighborhood U of 0 in hasOasitsonlysingularpoint.LetB, = ( ( ~ 1 ..., , z,+k) 1z1I2+...+ Iz,+kI2 L E~ } be the closed disk of radius E and S, the (2(n k) - 1)-sphere of radius E , which is the boundary of B E .It is known that, for sufficiently , nV) is homeomorphic to the cone over (SE,S, nV) small E , the pair ( B EBE (136, Theorem 2.101, see also 139, Ch. I, 511). In this case, S, and V are transverse and L = S, n V is a (2n - 1)-dimensional C” manifold, which is called the link of the singularity of V at 0. Now let V be a (germ of) complete intersection of dimension n and f1,. . . , f k generators of I(V). We suppose that these germs have representatives in U and we think of f = (fi,. . . ,fk) as a holomorphic map from U onto a neighborhood W of 0 in Ck.Let C ( f ) be the set of critical points of f . Then, by Proposition 5.3, Sing(V) = V n C ( f ) .Note that, if k = 1, Sing(V) = C ( f )(cf. [33, Proof of (1.2) Proposition]). We have the following “fibration theorem”, which is due to [36] when k = 1 and to [20] for general k, see also [21,29,33,391.

+

I

202

Theorem 5.5. Let V be a complete intersection of dimension n with isolated singularity at 0 in U F k . Then there exist small disks B, about 0 in U and B; about 0 in W such that D ( f )= Binf(C(f)) is a hypersurface in Bi and that f induces afiber bundle structure B,nf-l(Bi\D(f)) + Bi\D(f). Moreover, the (typical) fiber F of this bundle has the homotopy type of a bouquet of n-spheres.

The fiber F is called the Milnor fiber and the number of spheres appearing in the above is called the Milnor number of V at 0 and is denoted by p ( V , 0). The number p ( V , 0) does not depend on the choice of generators of I ( V ) . There is an algebraic formula for this number ([16,28], see also [33]). We set, for i = 1,.. . ,k, ai = dim@on+k/(J(.fl,. .. 7 fi),

fly-.., fi-l),

where the denominator in the right hand side is the ideal generated by the Jacobians det(d(f1,. . . , f i ) / d ( z v l , ,. . , z v i ) ) , 1 5 v1 < ... < vi 5 n + k , and f1,. . . , fi-1. Then k

p(V, 0) = C(-l)k-"i. i= 1

In particular, if k = 1, p ( V , 0) = dim@on+l/J(f),

(5.6)

where J ( f ) = ( d f / d z l , . . . ,df/dzn+l), f = f l (cf. [40]). Now we consider the global situation. Let W be a complex manifold of dimension n k and V an analytic variety in W . Suppose V is pure n-dimensional. We call V a local complete intersection (LCI) if the germ of V at each point of V is a complete intersection. Thus each point of V has a neighborhood, where V admits k reduced defining functions. In this case, there is a vector bundle N v over V of rank k,which extends the normal bundle Nv, of V' in W . We have a commutative diagram with an exact row (e.g., [32, Proposition 11)

+

TWlv

T

0

TV'

incl.

Ny

T

(5.7)

incl.

TWIv, KNvl

A

0.

If f1,. . . , fk are local reduced defining functions of V , then there is a frame of NV which extends the frame ( ~ ( d / d f i ).,. . , n(d/dfk)) of Nvr. (Note that near a regular point off = (f1, . . . , f k ) , i.e., a regular point of

v,

203

we may take (fl, . . . ,fk) as a part of a local coordinate system on W . ) w e call it the frame of Nv associated to f = ( f i , . . . , f k ) . For an LCI V in W ,we call TV = TWlv -Nv the virtual tangent bundle of V . The image of the total Chern class of r v by the Poincark homomorphism (see Ch. IV, Section 4 below) is the so-called Fblton-Johnson class of V , which is one of the important characteristic classes of singular varieties. Now let N be a holomorphic vector bundle over W of rank k and s a holomorphic section of N . We call the zero set V of s in W an LCI defined by s if it is an LCI with local components of s (with respect t o some local holomorphic frame of N ) as its reduced defining functions. In this case, we have Nv = Nlv. Examples 5.8. As examples of LCIs defined by a section of a holomorphic vector bundle, we have the following: 1. V a hypersurface in W (k = 1).In this case, we may take as N the line bundle L(V) defined by V and as s the natural section described in Ch. 11, Example 1.9. 2. V a complete intersection. In this case we may take as N the trivial bundle and as s a system of generators of the ideal of holomorphic functions vanishing on V . 3. V a (projective algebraic) complete intersection in the projective space @Pn+k. This means that the ideal of homogeneous polynomials vanishing on V is generated by k homogeneous polynomials PI,.. . ,P k . Let di denote the degree of Pi for i = 1,.. . , k and U j the affine coordinate <j # 0 for j = 0 , . . . , n+k. Then, in U j , V is defined by fi = . - . = fk = 0, fi = Pi/c,"i Note that it is only locally a complete intersection. In this case, we may take as N the bundle H d l @ . . . @I H d k , where H denotes the hyperplane bundle (Ch. 11, Example 1.9).

6. Grothendieck residues

For details on this subject, we refer to [17].Let Ondenote the ring of germs of holomorphic functions at the origin 0 in C" and fi, . . . ,fn germs in On such that V ( f 1 , . . ,fn) = (0) (Ch. I, 5 ) . For a germ w at 0 of holomorphic n-form we choose a neighborhood U of 0 in @" where f 1 , . . . ,fn and w have representatives and let r be the n-cycle in U defined by

204

where, E is a small positive number. We orient J? so that the form d01 A . d0, is positive, Bi = arg fi. Then we set

-

A

Note that this residue is alternating in (fl,. . . ,fn).

Example 6.1. When n = 1, the above residue is the usual Cauchy residue at 0 of the meromorphic 1-form w / f l (cf. Ch. I, Section 1).

-

Example 6.2. If w = dfl A * . A dfn, then

is a positive integer which is simultaneously equal to (i) the intersection number (Dl . . . Dn)0 at 0 of the divisors Di defined by fi (see Section 4 and [17, Ch. 5 , 21, [48]), (ii) dim@O n / ( f l , . . ,fn) and (iii) the (Poincark-Hopf) index at 0 in C" of the vector field v = Cy=lfi d / a Z i (the mapping degree of f = ( f i ,. . . ,fn)), see Ch. IV, Sections 2 and 3.

Example 6.3. In particular, if fi = af/az, for some f in On,then it is the Milnor number p(V,O) of the hypersurface V defined by f at 0 (see Section 5 ) :

We also call this number the multiplicity of f at 0 and denote it by m(f,O) (cf. Ch. IV, 3 below).

Chapter IV. Localization of Chern classes and associated residues 1. Localization of the top Chern class

E -+ M be a C" complex vector bundle of rank T over an oriented C" manifold M of dimension m. Let s be a non-vanishing section of E on some open set U . Recall that a connection V for E on U is s-trivial, if V(s) = 0. If V is an s-trivial connection, we have the vanishing (Ch. 111, Proposition 3.3) Let

7r:

Cr(V)= 0.

(1.1)

205

Let S be a closed set in M and suppose we have a C" non-vanishing section s of E on M \ S . Then, from the above fact, we will see that there is a natural lifting cT(E,s) in H2'(M, M \ S ;C ) of the top Chern class c T ( E ) in H2'(M, C). Letting Uo = M \ S and Ul a neighborhood of S , we consider the covering U = {Uo,Ul} of M . Recall the Chern class c T ( E )is represented by the cocycle c,.(V*) in A2'(U) given by

where V Oand V1 denote connections for E on UOand Ul,respectively. If we take as V Oan s-trivial connection, then ~ ( 0 0 =) 0 by (1.1) and thus the cocycle is in A2T(M,C70)and it defines a class in the relative cohomology H2'(M,M \ S ; C ) , which we denote by & ( E , s ) . It is sent to the class c,.(,(E) by the canonical homomorphism j * : H2'(M, M\S; C ) H2'(M, C). It does not depend on the choice of the connection V1 or on the choice of the s-trivial connection Vo. This fact is a localized version of Ch. 111, Proposition 1.8 and a precise proof of it is in [47, Ch. 111, Lemma 3.11. We call cT(E,s) the localization of cT(E)with respect to the section s at S. In the above situation, suppose that S is a compact set, with a finite , a regular neighborhood. number of connected components (S X ) ~admitting Then we have the Alexander duality Ch. I1 (5.3):

A : H 2 ' ( M , M \ S ; C ) 7Hm-2T(S,C) =@Hm-aT(Sx,C).

x Thus the class c,.(E, s) defines a class in Hm--ar(Sx,C ) , which we call the residue of c T ( E )at S, with respect to s and denote by Res,,.(s, E ; Sx). This residue corresponds to what is called the "localized top Chern class" of E with respect to s in [12, g14.11. For each A, we choose a neighborhood Ux of Sx in U1, so that the Ux's are mutually disjoint. Let Rx be an m-dimensional manifold with C" boundary in Ux containing Sx in its interior. We set Rox = 4 R x . Then the residue Res,,.(s, E ; Sx) is represented by an (m - 2r)-cycle C in Sx such that

for any closed (m - 2r)-form T on Ux.In particular, if 2r = m, the residue is a complex number given by Resc,(s, E ; Sx)=

s,, c'(vd Lo, +

C'(V0,Vl).

(1.3)

By Ch. 11, Proposition 5.4, we have the following "residue formula".

206

Proposition 1.4. In the above situation, if M is compact,

-

x(ix)* Res,,.(s, E ; Sx) = cr(E) , , [MI x

where ax denotes the inclusion Sx

in Hm-zT(M,C),

M.

2. Residues at an isolated zero

Let T : E + M be a holomorphic vector bundle of rank n over a complex manifold M of dimension n. Suppose we have a section s with an isolated zero at p in M . In this situation, we have Res,, (s,E;p ) in H o ( { p } ,C ) = C. In the following, we give explicit expressions of this residue. Let U be an open neighborhood of p where the bundle E is trivial with holomorphic frame (el,. . . , en). We write s = x7=l f i ei with fi holomorphic functions on U .

(I) Analytic expression Theorem 2.1. In the above situation, we have

Res,, (s,E ;p)

= Res,

[df;: :::,ynl .

Proof. We indicate the proof for the case n = 1 (for n > 1,we use the Cechde Rham cohomology theory for n open sets, see [48,50]). Thus s = f el for some holomorphic function f on U . Let R be a closed disk about p in U . In the expression (1.3) of the residue, we may take as V1 an el-trivial connection on U , thus cl(V1) = 0 and

with VO an s-trivial connection on U' = U \ {p}. Now we recall how the Bott difference form cl(V0, V1) is defined (Ch. 111, 1).Consider the bundle E = E x R over U x R, and let t be a coordinate on R. Define a connection for E on U' x Iw by V = (1 - t)Vo tV1. Let T : U' x [0,1] -+ U' be the canonical projection and let 7r* be the integration along the fibers of 7r. Then we define cl(V0, V I ) = T*cI(V). Let Bi be the connection matrix of Vi, i = 0 , l with respect to the frame el. Therefore 81 = 0. To find 6 0 , we use Ch. 111, (1.6). Since the connection matrix with respect to s is zero, we get

+

eo = --.df f

207

Hence

e = (1- t)& = (t - I)?

and the curvature matrix

I(,

is given by

- - -

I(,

=dB

- 0 A 0 = dt A -.df f

Thus

which proves the theorem (for the case n = 1). Remark 2.2. For general n, if we take suitable connections we see that the difference form is given by G o o , 01)= -f*Pn,

where f = (fl, . . . ,fn) and Pn denotes the Bochner-Martinelli kernel on Cn (Ch. II,5). This gives a direct proof of Theorem 2.4 below. Thus we reprove the fact that the Grothendieck residue in the above theorem is equal to the mapping degree o f f (cf. [17, Ch. 5, 1. Lemma]). (11) Algebraic expression

Theorem 2.3. I n the above situation, we have Res,,(s,E;p) = d i m W ( f i , . . . , f n ) . This can be proved, for example, by perturbing the sections and using the theory of Cohen-Macaulay rings (e.g., [50]). (111) Topological expression

Let S:"-' denote a small ( 2 n - 1)-sphere in U with center p . Then we have the mapping

where S2"-l denotes the unit shere in C". Theorem 2.4. I n the above situation, we have Res,, (s,E ;p ) = deg p. This can also be proved by perturbing the sections, see [17,50].

208

3. Examples I (a) PoincarB-Hopf index theorem

Let M be a complex manifold of dimension n. We take as E the holomorphic tangent bundle T M . Then a section of TM is a (complex) vector field w. We define the PoincarB-Hopf index PH(w, Sx) of v at a connected component Sx of its zero set S by PH(w, Sx) = Rescn(w,TM;Sx). Then, if M is compact, by Proposition 1.4, we have

where & ( M ) = % ( T M ) and it is known that the right hand side coincides with the Euler-PoincarB characteristic x ( M ) of M (“Gauss-Bonnet formula”). Thus, by Theorem 2.4, we recover the PoincarB-Hopf theorem in case w is holomorphic and the zeros are isolated.

Exercise 3.1. Find all the holomorphic vector fields on the Ftiemann sphere CP1 and verify the PoincarB-Hopf formula for each of them.

(b) Multiplicity formula Let M be a complex manifold of dimension n. We take as E the holomorphic cotangent bundle T*M. For a holomorphic function f on M , its differential df is a section of T * M . The zero set S of df coincides with the critical set C ( f ) of f. We define the multiplicity m ( f , S x ) of f at a connected component Sx of C ( f )by

m ( f , S x )= Resc,(df,T*M;Sx). Note that, if Sx consists of a point p , it coincides with the multiplicity m ( f , p ) o f f at p described in Example 6.3 of Ch. 111. Now we consider the global situation. Let f : M --f C be a holomorphic map of M onto a complex curve (Riemann surface) C. The differential df: T M

-+

f*TC

of f determines a section of the bundle T*M @ f *TC,which is also denoted by d f . The set of zeros of df is the critical set C (f) of f. Suppose C(f) is ~. a compact set with a finite number of connected components ( S A ) Then

209

we have the residue Res,, ( d f ,T*M@f *TC;Sx) for each A. If M is compact, by Proposition 1.4, we have

Al4

E R e s c n ( d f , T * M @f*TC;Sx)= x

k ( T * M @f*TC).

We look at the both sides of the above more closely. In the sequel, we set D(f) = f ( C ( f ) )the , set of critical values. Then, if M is compact, f defines a C" fiber bundle structure on M \ C(f ) + C \ D(f ) . We refer to [25] for a precise proof of the following

Lemma 3.2. If M is compact, Cn(T*M8 f*TC) = (-l)"(X(M) - X ( F ) X(C)),

where F denotes a general fiber o f f . Suppose that f ( S x )is a point. Taking a coordinate on C around f ( S x ) , we think of f as a holomorphic function near Sx. Then we may write

Res,,(df,T*M@ f * T C ; S x )= Res,,,(df,T*M;Sx) = m ( f , S x ) , the multiplicity of f at SX.Thus we have

Theorem 3.3. Let f : M 4 C be a holomorphic map of a compact complex manifold M of dimension n onto a complex curve C . Then we have

c

m u , Sx) = (-1)"(X(M) - X ( F ) X ( C > > ,

x

where the sum is taken over the connected components Sx of C(f ) . In particular, we have ([24], see also [12, Example 14.1.51):

Corollary 3.4. I n the above situation, i f the critical set C(f ) o f f consists of only isolated points,

c

m(f7 PI = (-1)"(X(M)

- X(F)

X(C>).

PEC(f)

4. Residues of Chern classes on singular varieties

In this section, we deal with the situation more general than the one we discussed in Section 1, in two ways. Namely, we consider Chern classes other than the top ones for vector bundles on possibly singular varieties. We refer to [47,50] for details of the material in this section, see also [7].

210

Let V be an analytic variety of pure dimension n in a complex manifold W of dimension n+ k. We denote by Sing(V) the singular set of V and set V’ = V \ Sing(V). First, suppose V is compact and let 0 be a regular neighborhood of V in W . Then, as in Ch. 11, 4,the cup product in H * ( 0 ) N H*(V) and the integration

induces the “PoincarB homomorphism”

P : HP(V,C) --t H2n-p(v,@), which is not an isomorphism in general. Note that in [5] the above homomorphism P , as well as the Alexander homomorphism defined below, are described in a combinatorial way for (co)homology with integral coefficients. The homomorphism P is given by the cap product with the fundamental class [ V] . Now suppose V may not be compact. Let S be a compact set in V. We assume that S has a finite number of connected components, S 3 Sing(V) and that S admits a regular neighborhood in W . Let 0 1 be a regular neighborhood of S in W and 00a tubular neighborhood of UO= V \ S in W . We consider the covering U = {OO,01} of the union 0 = 00U01,which may be assumed to have the same homotopy type as V. We define the subcomplex A*(U,00)of A*(U) as in Ch. 11, 5. Then we see that

H g ( u ,00) N HP(V, v \ s C). Again, as in Ch. 11, 5 , the cup product and the integration induces the “Alexander homomorphism”

A : Hp(V,V \ S;C ) 4 H2n-p(S1 C ) , which is not an isomorphism in general. Suppose V is compact. Then the following diagram is commutative: HP(V, v \ s;@)

lA

H2n-p(S7 C )

j’ ___+

HP(V, @)

lp

- .i

HZn-p(V,

where i and j denote, respectively, the inclusions S (V, v \ S).

(4.1)

el, V and (V,8)

~t

211

For a complex vector bundle E over 0 of rank T , the i-th Chern class ( E )is in H22(0) N H2i ( V ). The corresponding class in ( V )is denoted by ci(Elv). The class q ( E ) is represented by a Cech-de Rham cocycle ~ ( 0 ,on) U given as (3.1)in Ch. I11 with Vo and V1 connections for E on 0 0 and 01,respectively. Note that it is sufficient if VO is defined only on UO,since there is a C“ retraction of 00onto UO.Suppose we have an Gtuple s = (sl,.. . , se) of C” sections linearly independent everywhere on UOand let Vo be s-trivial. Then we have the vanishing ~ ( 0 0 =) 0, for i 2 r-l+l (Ch. 111, Proposition 3.3), and the above cocycle ci(0,) defines aclass c i ( E ( v , s )in H$(U,00) N_ H2i(V,V\S;C). It is sent t o q ( E ( v )by the canonical homomorphism j * : H2i(V,V \ S; C ) --+ H2i(V,C ) . Let (Sx)x be the connected components of S. Then, for each A, ci(Elv,s) defines the residue Resci(s,E l v ; Sx) via the Alexander homomorphism Q

A : H2i(V,V\S;C) -+ H2,-2i(S,C) =@HZn-zi(Sx,C). x

For each A, we choose a neighborhood 0~of SA in 01,so that the Ox’s are mutually disjoint. Let f i x be a real 2(n k)-dimensional manifold with C” boundary in containing Sx in its interior such that the boundary aRx is transverse to V . We set = -aRx n V . Then the residue Resci(s,E l v ; Sx) is represented by a 2(n - 2)-cycle C in Sx satisfying the identity as (1.2) for every closed 2(n - i)-form T on Ox. In particular, if i = n, the residue is a number given by a formula as (1.3). From the commutativity of (4.1),we have the “residue formula” (cf. [47, Ch. VI, Theorem 4.81):

+

Proposition 4.2. In the above situation, if V is compact, we have, for i 2 r - f2+ 1,

C(ix)* ~es,~(s, x where i x : Sx

-

E I V ;S A ) = G ( E )n [VI

in

H~,-~~(v,c),

V denotes the inclusion.

Note that the Resc,(s, E l v ; Sx)’s are in fact in the integral homology and the above formula holds in the integral homology. 5. Residues at an isolated singularity

Let V be a subvariety of dimension n in a complex manifold W of dimension n + Ic, as before. Suppose now that V has at most an isolated singularity at p and let E be a holomorphic vector bundle of rank r (2 n) on a small

212

coordinate neighborhood 0of p in W . We may assume that E is trivial and let e = ( e l ,. . . ,en) be a holomorphic frame of E on 0. Let l = r - n 1 and suppose we have an &tuple of holomorphic sections i of E on 0.Suppose that S(i)n V = {p}. Then we have Res,,(s,Elv;p) with s = Slv. In the following, we give various expressions of this number. fij ej, i = 1,.. . , t?, with f i j holomorphic functions We write 3i = Let F be the l x r matrix whose (i,j)-entry is fij. We set on 0.

+

xiz1

Z = { (il, ..., it) 11 5 il

<

e

.

0


5 r}.

For an element I = ( i l l . . . ,it) in Z, let FI denote the l x l matrix consisting of the columns of F corresponding to I and set ( P I = det FI. If we write el = ei, A . A ei!, we have IEI

Note that S(S) is the set of common zeros of the

(PI'S.

(I) Analytic expression First we recall:

Grothendieck residues relative to a subvariety Let 0 be a neighborhood of 0 in Cn+k and V a subvariety of dimension n in 0 which contains 0 as at most an isolated singular point. Also, let f1,. . . ,fn be holomorphic functions on 0 and V ( f 1 , .. . ,fn) the variety defined by them. We assume that V ( f 1 , .. . fn) n V = (0). For a holomorthe Grothendieck residue relative to V is defined by phic n-form w on 0, (e.g., [47, Ch. IV, 81)

where I? is the n-cycle in V given by

r = { q E 0n v I lfi(q)I = ci,

i = 1,. . . ,n }

for small positive numbers Q. It is oriented so that darg(f1) A ... A d x g ( f n ) 2 0. If k = 0, it reduces to the usual Grothendieck residue (Ch. 111, S), in which case we omit the suffix V . If V is a complete intersection defined by hl = = hk = 0 in U ,we have

213

To get the analytic expression, we first note that, from the assumption S(S) n V = { p } , we have ([49,Lemma 5.61):

Lemma 5.1. We may choose a holomorphic frame e = ( e l , . . . ,e,) of E so that there exist n elements I(1), . . . ,I(,) in Z with V(pI(l,,. . . ,p l ( n ) )nV = {P). Theorem 5.2. W e have

where pI(l),. . . ,v I ( n ) are chosen so that they satisfy the conditions in Lemma 5.1 and a,(@) is a holomorphic n-from given in terms of the matrix F (see (491 for the precise expression). Here are some special cases: 1. The case C = 1 and r = n. Let e = ( e l , . . . ,en) be an arbitrary frame of E and write s = C;=,fiei. Then we may set pI(i) = f i , i = 1,.. . ,n, and we have an(@)= df1 A . A df,. 2. The case n = 1 and C = r. Let e = ( e l , . . . ,e,) be an arbitrary frame of E and write si = C>=l fij e j , i = 1, . . . ,r. Let F = ( f i j ) and set p = det F. Then we may set pl(l) = cp and we have on(@) = d p . See 1491 for more cases where the form a,(@) is computed explicitly.

(11) Algebraic expression Let O C ,denote ~ the ring of germs of holomorphic functions on 0at p , which is isomorphic to the ring On+k of convergent power series in n+ k variables. We assume that V is a complete intersection defined by h l , . . . ,hk near p and let .F(V)p denote the ideal in 0 0 ,generated ~ by (the germs of) the ( P I ' S and hl, . . . ,h k . A detailed proof, involving the theory of Cohen-Macaulay rings, of the following is given in [50].

Theorem 5.3. W e have

(111) Topological expression We again assume that V is a complete intersection in 0. Let We(C') denote the Stiefel manifold of C-frames in C'. It is known that the space We(Cr) is 2(r - C)-connected and ~2,-1(Wg(P>) II Z (recall 2r - 2C+ 1 = 2 n - 1).

214

Note also that We(@')has a natural generator for the (2n - 1)-st homology (cf. [45]).Let L denote the link of ( V , p ) (cf. Ch. 111,Section 5) and ( L i ) l l i l s its connected components. Note that, if k = 1 and n 2 2, L is connected (cf. [36]). We have the degree deg pi of the map pi = SILi : Li

---f

We(C').

The following can be proved by noting that both the residue and the mapping degree satisfy the conservation law under perturbation of sections.

Theorem 5.4. We have

6. Examples I1

(a) Index of a holomorphic 1-form of Ebeling and Gusein-Zade Let V be a complete intersection in 0 with an isolated singularity at p and defined by ( h l ,. . . ,h k ) , as before. Also, let L be the link of ( V , p ) and ( L i ) l l i l sits connected components. For a holomorphic 1-form 6' on 0, we consider the (k 1)-tuple S = (0, dhl, . . . , d h k ) of sections of T * 0 ,which is of rank n k. Thus T - l 1 = n k - (k 1) 1 = n. We assume that S(S) n V = { p } , which means that the pull-back of 0 to V \ { p } by the inclusion V \ { p } c-) 0 does not vanish. Let s = slv, which defines a map of V \ { p } to W[(C').It should be emphasized that here we take the restrictions of components of 5 as sections and not as differential forms. Following [9,10], with different naming and notation, we define the V-index Indv(O,p) of 0 at p by

+

+

+

+

+ +

S

i=l

Then by Theorem 5.4, it coincides with Resc,(s,T*fi;Jv;p)and by Theorems 5.2 and 5.3, it has analytic and algebraic expressions. In fact the algebraic one is already given in [9,10].

Remark 6.1. The above index is an analogy of the one earlier introduced for vector fields, i e . , the so-called GSV-index ([15,43]).Namely, in the above situation let v be a holomorphic vector field on 0. Assume that v is tangent

215

to V \ { p } and non-vanishing there. Set S = (v, grad hl,. . . ,grad h k ) and s = Slv. Then the GSV-index of v at p is defined by S

GSV(v,p) = x d e g s l ~ ~ . i=l

Note that, for this index, since s involves anti-holomorphic objects, we cannot directly apply our previous results. It coincides with the “virtual index” of v ([31,44]) and an explicit formula for the index in terms of a Grothendieck residue is given in [31]. There is also an algebraic formula for it as a homological index, in the case k = 1, in [14].

(b) Multiplicity of a function on a local complete intersection We refer to [25] for details of this subsection. Let V be a subvariety of dimension n in a complex manifold W of dimension n k. We assume that V is a local complete intersection defined by a section s of a holomorphic vector bundle N of rank k over W (see Ch. 111, 5). Recall that the restriction of N to the non-singular part V’ coincides with the normal bundle of V’ in W . We denote the virtual bundle (T*W - N * ) l v by T; and call it the virtual cotangent bundle of V. Let g be a holomorphic function on W and let f and f’ be its restrictions to V and V‘, respectively. We define the singular set S ( f ) of f by S(f) = Sing(V) U C(f’). As in the case of vector bundles, we may define the localization of the n-th Chern class of 7; by df , which in turn defines the residue Res,,(df, 7;; S ) at each compact connected component S of S(f). We define the virtual multiplicity &( f , S ) of f at S by

+

& ( f , S )= Res,,(df,~;;S).

(6.2)

The multiplicity of f at S is then defined by

m(f

7

S ) = &(f1 S ) - p(V,S)l

(6.3)

where, p(V,S ) denotes the (generalized) Milnor number of V at S as defined in [7] (cf. [1,41,42] in the case k = 1). Note that if S consists of a point p , it is the usual Milnor number p(V,p)of the isolated complete intersection singularity (V,p)(see Ch. 111, 5). Note that, if S is in V’, we have Res,,(df,~;; S ) = Res,,(df,T*V’; S ) . On the other hand, in this case we have p(V, S ) = 0 so that m ( f , S )coincides with the one in 3 (b).

216

Let g : W 4 C be a holomorphic map onto a complex curve C and set f = glv, f' = g(vl and S ( f ) = Sing(V) U C ( f ' ) .We assume that S ( f ) is compact. We further set VO= V \ S ( f ) and fo = glvo.Thus dfo is a nonvanishing section of the bundle T*Vo8 fGTC, which is of rank n. If we look at c,(E), E = 7; 8 f * T C and we see that there is a canonical localization % ( E , df) in H2,(V, V \ S; C ) of %(,(E). Let ( S X ) be ~ the connected components of S and let ( R A )be~ as in 4. Then % ( E , df) defines, for each A, the residue Res,, (df,T; 8 f *TC;Sx). If V is compact, by Proposition 4.2, we have x

Res,, (df,7 ; €3 f *TC;Sx) =

s,

c,(T; 8 f * T C ) .

The both sides in the above are reduced as follows. If f(S(f))consists of isolated points, we may write Resc,(df,

T;

€3 f * T C ;sx) = f i ( f ,sx) = m(f, sx) -

~ ( sx) v,

and, if moreover, V is compact, we have

where F is a general fiber of f ([25, Lemma 5.21). Thus, in the above situation, we have ([25, Theorem 5.51):

c c

m(f7 SX) = (-l)n(X(V)- X ( F )X(C>>. x In particular, if S(f)consists only of isolated points, m(f,p) = ( - l ) Y X ( V )- X ( F )x ( C ) > ,

(6.4)

PES(f)

which generalizes Corollary 3.4 for a singular variety V . If Sx consists of a single point p , the residue Res,, (df,7;; p ) is given as follows. Let 0 be a small neighborhood of p in W so that the bundle N k admits a frame ( ~ 1 , .. . ,uk) on 0.We write s = Ci=lhi ui with hi holomorphic functions on 0. Then V is defined by ( h l ,. . . ,hk) in 0. Consider the ( k 1)-tuple of sections

+

ii = (dg,dhl, . . . , d h k ) of T*0.By the assumption, we have S(G) n V = { p } . Since the rank of T * 0 is n k , we have the residue Resc,(s,T*Olv;p),s = Slv. Then we have ([25, Theorem 4.61)

+

f i (f,p ) = Res,, (s, T*01v ;p ).

(6.5)

217

The virtual multiplicity %(f,p) was defined as the residue of df on the virtual bundle and this definition led us to a global formula as (6.4). The identity (6.5) shows that it coincides with the residue of s = (dgJV,dhllV,.. . dhklv) on the vector bundle T*filv. Thus we have various expressions for %(f,p) as given in the previous sections; by Theorem 5.2 we have a way to compute %(f,p) explicitly, by Theorem 5.3 we may express

I-;

%(f,P) = dim@On+k/(J(g, h 1 7 .. . hk)7 hl7 - * . hk), (6.6) where J(g, h17. . . ,h k ) denotes the Jacobian ideal of the map (gl hl, . . . h k ) , I

7

+

ie., the ideal generated by the (k 1) x (k matrix a(Zl,...,Zn+k) %7hl,. ..,hk) and by Theorem 5.4,

+ 1) minors of the Jacobian

f 4 f 7 P ) = Indv(dg,p).

(6.7)

From (6.3), (6.6) and the identity (cf. [16,28]) p(v7p ) + p(vg,p) = dim@on+k/(J(g,

h17

...

7

hk)7

. ..

h17

7

hk)

7

where V, denotes the complete intersection defined by (9, hl, . . . h k ) , assuming g(p) = 0, we get m(f7P) = 0

- g 7 P).

(6.8)

(c) Some others

Let V be a complete intersection defined by ( h l ,. . . h k ) in fi and p an isolated singularity of V, as before. The n-the polar multiplicity m,(V,p) of Gaffney ([13]) is defined by mn(v7 p ) = dim@O,+k/(J(C7 hl, . . . 7 hk)7 hl7 ' ' ' hk) 7

7

where C is a general linear function. By (6.6) and (6.7), we may write mn(V1p)

= Indv(dt1p)= fi(ClvlP).

Also, in the expression Eu(V7p)= 1

+ (-1)"+'p(Ve,p>

for the Euler obstruction Eu(Vlp ) of V at p (cf. [8,26], see also [S]), we have by (6.81, P(ve1P) = m(Clv1p). Note that these local invariants appear in the comparison of the Schwartz-MacPherson, Mather and F'ulton-Johnson classes of a local complete intersection with isolated singularities (cf. [38,46]).

218

References 1. P. Aluffi, Chern classes for singular hypersurfaces, Trans. Amer. Math. SOC. 351 (1999), 3989-4026. 2. P. Baum and R. Bott, Singularities of holomorphic foliations, J. Differential Geom. 7 (1972), 279-342. 3. R. Bott, Lectures on characteristic classes and foliations, Lectures on Algebraic and Differential Topology, Lecture Notes in Mathematics 279, SpringerVerlag, New York, Heidelberg, Berlin, 1972, pp. 1-94. 4. R. Bott and L. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer-Verlag, New York, Heidelberg, Berlin, 1982. 5. J.-P. Brasselet, De'finition combinatoire des homomorphismes de Poincare', Alexander et Thom, pour une pseudo-varie'tk, CaractBristique, d'EulerPoincarB, AstBrisque, 82-83, SociBtB MathBmatiques de France, 1981, pp. 7191. 6. J.-P. Brasselet, D.-T. Le and J. Seade, Euler obstruction and indices of vector fields, Topology 39 (2000), 1193-1208. 7. J.-P. Brasselet, D. Lehmann, J . Seade and T. Suwa, Milnor classes of local complete intersections, Trans. Amer. Math. SOC.354 (2002), 1351-1371. 8. A. Dubson, Classes caractkristiques des varikte's singuliires, C . R. Acad. Sci. Paris 287 (1978), 237-240. 9. W. Ebeling and S. M. Gusein-Zade, O n the index of a holomorphic 1-form on an isolated complete intersection singularity, Doklady Math. 64 (2001), 221-224. 10. W. Ebeling and S. M. Gusein-Zade, Indices of 1-forms on an isolated complete intersection singularity, Moscow Math. J. 3 (2003), 439-445. 11. G. Fischer, Complex Analytic Geometry, Lecture Notes in Mathematics 538, Springer-Verlag, New York, Heidelberg, Berlin, 1976. 12. W. Fulton, Intersection Theory, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984. 13. T. Gaffney, Multiplicities and equisingularity of ICIS germs, Invent. Math. 123 (1996), 209-220. 14. X. G6mez-Mont, An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, J. Algebraic Geometry 7 (1998), 731-752. 15. X. G6mez-Mont, J. Seade and A. Verjovsky, The index of a holomorphic flow with an isolated singularity, Math. Ann. 291 (1991), 737-751. 16. G.-M. Greuel, Der GauJ-Manin Zusammenhang isolierter Singularitaten von vollstandigen Durchschnitten, Math. Ann. 214 (1975), 235-266. 17. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, 1978. 18. V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, 1974. 19. R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. 20. H. Hamm, Lokale topologische Eigenschaften komplexer Raume, Math. Ann. 191 (1971), 235-252.

219 21. H. Hamm and D.-T. L6, Un thkorkme de Zariski du type de Lefschetz, Ann. scient. Ec. Norm. Sup. 6 (1973), 317-366. 22. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, Heidelberg, Berlin, 1977. 23. F. Hirzebruch, Topological Methods i n Algebraic Geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1966. 24. B. Iversen, Critical points of an algebraic function, Invent. Math. 12 (1971), 2 10-224. 25. T . Izawa and T. Suwa, Multiplicity of functions on singular varieties, Intern. J . Math. 14 (2003), 541-558. 26. M. Kashiwara, Index theorem f o r a maximally overdetermined system of linear digerential equations, Proc. Japan Acad. 49 (1973), 803-804. 27. K . Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer-Verlag, New York, Heidelberg, Berlin, 1986. 28. D.-T. LB, Calculation of Milnor number of isolated singularity of complete intersection, Funct. Anal. Appl. 8 (1974), 127-131. 29. D.-T. LB, Some remarks on relative monodromy, Real and Complex Singularities, Oslo 1976, ed. P. Holm, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1977, pp. 397-403. 30. D. Lehmann, Systbmes d’alve‘oles et intkgration sur le complexe de Cechde Rham, Publications de l’IRMA, 23, No VI, Universitk de Lille I, 1991. 31. D. Lehmann, M. Soares and T. Suwa, On the index of a holomorphic vector field tangent to a singular variety, Bol. SOC.Bras. Mat. 26 (1995), 183-199. 32. D. Lehmann and T . Suwa, Residues of holomorphic vector fields relative to singular invariant subvarieties, J. Differential Geom. 42 (1995), 165-192. 33. E. Looijenga, Isolated Singular Points on Complete Intersections, London Mathematical Society Lecture Note Series 77, Cambridge Univ. Press, Cambridge, London, New York, New Rochelle, Melbourne, Sydney, 1984. 34. R. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974), 423-432. 35. H. Matsumura, Commutative Algebra, Benjamin/Cummings, 1980. 36. J. Milnor, Singular Points of Complex Hypersurfaces, Annales of Mathematics Studies 61, Princeton University Press, Princeton, 1968. 37. J. Milnor and J. Stasheff, Characteristic Classes, Annales of Mathematics Studies 76, Princeton University Press, Princeton, 1974. 38. T. Ohmoto, T. Suwa and S. Yokura, A remark on the Chern classes of local complete intersections, Proc. Japan Acad. 73 (1997), 93-95. 39. M. O h , Non-Degenerate Complete Intersection Singularity, Actualitks Mathkmatiques, Hermann, Paris, 1997. 40. P. Orlik, The multiplicity of a holomorphic map at an isolated critical point, Real and Complex Singularities, Oslo 1976, ed. P. Holm, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1977, pp. 405-474. 41. A. Parusiriski, A generalization of the Milnor number, Math. Ann. 281 (1988), 247-254. 42. A. Parusiriski and P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Algebraic Geom. 10 (2001), 63-79.

220 43. J. Seade and T. Suwa, A residue formula for the index of a holomorphic flow, Math. Ann. 304 (1996), 621-634. 44. J. Seade and T. Suwa, An adjunction formula for local complete intersections, Intern. J. Math. 9 (1998), 759-768. 45. N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, 1951. 46. T. Suwa, Classes de Chern des intersections complites locales, C. R. Acad. Sci. Paris 324 (1996), 67-70. 47. T. Suwa, Indices of Vector Fields and Residues of Singular Holomorphic Foliations, Actualit& MathBmatiques, Hermann, Paris, 1998. 48. T. Suwa, Dual class of a subvariety, Tokyo J. Math. 23 (2000), 51-68. 49. T. Suwa, Residues of Chern classes, J. Math. SOC.Japan 55 (2003), 269-287. 50. T. Suwa, Residues of Chern classes o n singular varieties, Singularit& fiancoJaponaises, Marseille 2002, SBminaires et Congrhs 10, SOC.Math. fiance, 2005, pp. 265-285.

PART I1

Advanced School on Singularity Theory

This page intentionally left blank

223

METRIC THEORY OF SINGULARITIES. LIPSCHITZ GEOMETRY OF SINGULAR SPACES. L. BIRBRAIR Departamento de Matemdtica, Universidade Federal do Ceard Fortaleza, Ceard, Brazil E-mail: birbaujc. b r www.ujc. b r We present a basic overview of Metric Theory of Singularities: Finiteness Theorems, Normal Embeddings, Classification of Germs of Subanalytic Surfaces, Metric Homology and Characteristic Exponents. Keywords: Subanalytic Sets; Singularity; Lipschitz Geometry.

1. Metric viewpoint. Comparison of metrics. Normal

embedding. Modern Metric Geometry nowday can be considered as a generalization of the classical Differential Geometry or, sometime, as a non-smooth Differential Geometry. We consider metric spaces as objects of a category. In fact, there are two ways to define morphisms. One can consider isometric embeddings as morphisms. This viewpoint works well if the objects are smooth. Another natural morphisms are Lipschitz maps. The set of Lipschitz maps is sufficiently big and these morphisms are better adopted to non-smooth objects. Recall that a map F : X --+ Y is called Lipschitz if there exists a positive constant K E R such that, for every ~ 1 ~ Ex X2 , we have: d y ( F ( z l ) , F ( z Z ) )5 K d x ( z 1 , z ~ )A. map F is called bi-Lipschitz if F is bijective, Lipschitz and F-l is Lipschitz. Clearly, bi-Lipschitz maps are isomorphisms in this category. We study a question of classification of "singular" metric spaces up to isomorphisms in this category, i.e. up to biLipschitz isomorphisms ( bi-Lipschitz equivalence ). By "singular spaces" we mean semialgebraic or globally subanalytic subsets of Rn.(For an information on subanalytic or semialgebraic geometry we refer to the exelent course of M.Coste presented in ICTP (see also [12]). This family of sets is rather wide and have many good topological properties. For example,

224

any compact subanalytic set admits a finite subanalytic triangulation (see [20]) and also admits a finite topologically trivial Coo-stratification (see, for example, [18, 161). From the other side, very many objects of Classical Singularity Theory belong to this category (real algebraic sets, complex algebraic sets, constructible sets, semialgebraic sets ). We apply the question of bi-Lipschitz classification to "singular spaces". Let X C IR" be a compact connected subanalytic set. We consider two natural metrics on X. The first one is the euclidean metric d e ( z 1 , z 2 ) = 11x1 - 2211. The second one is a so-called inner (or intrinsic ) metric di(51,22) = inf{l(y) : y E I'(21,22)} where I'(zl,z2) is a set of piecewise smooth arcs connecting 2 1 and 5 2 and l(y) is the length of y. In fact, di(51,22) = min{l(y) : y E I ' ( z l , z 2 ) } because Hopf-Rnov theorem is true for compact singular spaces (see [6]). These two metrics define the same topology on X but they are not necessary bi-Lipschitz isomorphic. To see it, consider the following semialgebraic subset of R2:

x = {(51,52)1

5:

- 5 ; = 0,

lzll 5 1,

1x21

5 1)

Clearly, d i ( z 1 , 52)/de(51,5 2 ) is unbounded, for 2 1 , 5 2 sufficiently close to 0. This example motivates the following definition. A set A c IRn is called normally embedded if ( X ,d,) and ( X ,d i ) are bi-Lipschitz isomorphic. All compact smooth submanifolds of R" are normally embedded. A so-called ,&horn Hp gives an example of a normally embedded singular set

where p = p / q 2 1 is a rational number. These notes are devoted to the first steps of bi-Lipschitz classification of subanalytic sets with respect to the inner metric. The following theorem shows that this question is equivalent to a question of bi-Lipschitz classification of normally embedded singular sets.

Theorem 1.1 (Normal Embedding Theorem). [lo] Let X c IRn be a compact subanalytic set. T h e n there exists a normally embedded subanalytic set 2 c IR" such that 2 i s bi-Lipschitz equivalent t o X with respect to the inner metric. One of the most important steps in the proof of Normal Embedding Theorem is a so-called "pancake decomposition".

225

Theorem 1.2 (Pancake Decomposition Theorem). /19, 101 Let X c Rn be a globally subanalytic set. T h e n there exists a collection of globally subanalytic subsets { X i } of X satisfying the following conditions: (1)

x =UiXi.

(2) All subsets X i are normally embedded.

(3) dim X i n X j < min{dim X i , dim X j } . The partition { X i } is called "a pancake decomposition of X " .

Question. Let X c Rn be a compact real algebraic set. Is it possible to construct a real algebraic set 2 c R" with the same properties as in the theorem above? Namely, 2 must be normally embedded and bi-Lipschitz equivalent to X with respect to the inner metric? 2. Finiteness results.

There are many classification problems in Singularity Theory. If one restricts a classification problem to the algebraic setting the following question is important. Consider the set of singular spaces (singular maps, singular germs) defined by polynomials of degree bounded from above by some number K . Is the set of equivalence classes (by the given equivalence relation) finite? Positive answers for this question are called Finiteness results. Lipschitz Geometry of Singularities admits a finiteness result.

Definition. Upper complexity of a semialgebraic set X c R" is a number defined as follows: U ( X )= min(N+D+E), where N is a number of variables, E is a number of equations and inequalities from a given formula defining X and D is the maximal degree of all the polynomials appear in this formula. The minimum is taken on the set of all the presentations X . Theorem 2.1 (Mostowski). [22] Let K be a positive integer number. Consider the set of all semialgebraic sets with upper complexity bounded f r o m above by K . T h e n the number of equivalence classes f o r the bi-Lipschitz equivalence with respect t o the euclidean metric is finite. If two semialgebraic (or subanalytic) sets are bi-Lipschitz isomorphic with respect to the euclidean metric they are bi-Lipschitz isomorphic with respect to the inner metric. Hence, the bi-Lipschitz equivalence with respect to the inner metric also admits a finiteness result. Recently G.Valette proved the following generalization of finiteness theorem of Mostowski.

226

Theorem 2.2 (24). Consider a finite-dimensional globally subanalytic family of globally subanalytic sets. Then the number of equivalence classes with respect t o a bi-Lipschitz isomorphism (with respect to the euclidean metric) is finite. Moreover, the Lipschitz types define a globally subanalytic stratification of the space of parameters of the family. If two sets belong t o the same stratum the corresponding bi-Lipschitz homeomorphism can be chosen globally subanalytic. In fact, Valette proved this theorem for definable sets in polynomially bounded o-minimal structures. Note, that the last statement of the theorem is new in comparison to the original theorem of Mostowski. For definable sets in o-minimal structures (especially, if a structure is not polynomially bounded), the finiteness result cited above does not take place. To see it one can consider the following family Tx:

Tx = ( ( ~ 1 ~ ~ E 2 )R21 0 5 21 5 1, 0 I zz 5 xi}, for X E [l,00). For different XI, X2, the sets Txl , Txz are not bi-Lipschitz equivalent. This family is definable in log - exp o-minimal structure. Some results related to finiteness theorems and bi-Lipschitz equivalence appears recently in Classical Singularity Theory. In [17], J-P.Henry and A.Parusinski showed that there is no finiteness result for the problem of biLipschitz classification of germs of polynomial functions f : R2 -+ R.From the other hand, the problem of K-bi-Lipschitz classification of polynomial function-germs does not have moduli. It is a recent joint work of the author, J.Costa, A.Fernandes and M.Ruas (see [7]). 3. Germs of subanalytic surfaces.

A subanalytic surface X c R" is a subanalytic set of dimension 2. This section is devoted t o a local bi-Lipschitz classification of subanalytic surfaces. Recall that two germs of subanalytic sets are called bi-Lipschitz equivalent if there exists a germ of a bi-Lipschitz isomorphism h : ( X , X O+ ) (Y,yo). We start our classification from the following result.

Theorem 3.1 (2). Let X c R" be a subanalytic surface and let 50 E X be a singular point such that, for suficiently small E > 0, the intersection X n Szo,r is connected. T h e n there exists a rational p 2 1 such that the germ of X at xo i s bi-Lipschitz isomorphic with respect to the inner metric to the germ of Ho at 0 E R3.Moreover, for # pz, the germs of Hpl and Hp, are not bi-Lipschitz isomorphic.

227

Here we say that ZO is a singular point of X if X is not a smooth submanifold of R" without boundary near ZO. It means that 0 E Tp is not an isolated singular point. (See the definition below). In fact, this theorem is a corollary of a more general result so-called Holder Complex Theorem. Holder Complex Theorem is a Lipschitz version of the Triangulation Theorem [20].In 2-dimensional case a triangulation can be chosen canonically and this canonical triangulation presents a complete Lipschitz invariant for germs of semialgebraic surfaces.

Definition. An Abstract Holder Complex is a pair ( I ' D ) where I' is a finite graph without loops and p : Er + Q (where Er is the set of edges of I') is a rational-valued function such that, for all g E Er, we have: p(g) 2 1. A Holder triangle Tp is a semialgebraic set in R2 defined as follows: Tp = { ( z I , z ~E)R2\ 0 5 21 51,

P

0 5x2 5 ~ 1 ) .

Let (I?, p) be an Abstract Holder Complex. A germ of subanalytic surface X at a point xo E X is called a Geometric Holder Complex associated to (r,p) if there exists a subanalytic triangulation such that xo is a vertex of this triangulation, l? is isomorphic to the star of xo in this triangulation and, for any p E E r , the corresponding 2-dimensional simplex is bi-Lipschitz isomorphic with respect t o the intrinsic metric to Tp(,). Moreover, the triangulation map maps 0 E To(,) to ZO. The following result was proved by several authors independently (using different notations and terminology). Theorem 3.2 (19, 13, 2, 3). Let X be a subanalytic surface and let xo E X . Then there exists an Abstract Holder Complex (I',p) such that the g e r m of X at xo is a Geometric Holder Complex associated to (I',p).

Note that this Abstract Holder Complex is not unique. Now we are going to make a modification the notion of an Abstract Holder Complex in order to obtain a complete bi-Lipschitz invariant. Let (r,p)be a Holder complex. A vertex a E Vr (here Vr is the set of vertices of I?) is called smooth if a is connected with two other vertices by only two corresponding edges. Suppose that a is connected by edges g1 and 92 with vertices a1 and a2. Let us construct a new graph such that the vertex a and the edges g1,g2 are removed, the vertices a1 and a2 are connected by a new edge g. Set p(g) = min{P(gl),P(g2)}. This operation is called "elimination of a smooth vertex ".

r

228

A vertex a is called a loop vertex if it is connected with only two edges g1 and g2 and these edges connect a with the same vertex a l . A loop vertex is called simple if p(g1) = p(g2). Let (r,p)be a Holder complex with a non-simple loop vertex a. Let us construct a new Holder complex (F, in the following way. We make this loop vertex simple just redefining p(g1) = p(g2) = min(p(gl), p(g2)). This operation is called "correction near a loop vertex". An Abstract Holder Complex is called simplified if it has no smooth vertices and all the loop vertices are simple. Abstract Holder Complex (F,6) is called a simplification of (r,p) if it is simplified and can be obtained from (I?, p) using the operations described above.

p)

Theorem 3.3. Let a germ of a subanalytic surface X at xo be a Geometric Holder Complex associated to two Abstract Holder Complexes (rl,PI) and (I'2,,92). Then the simplifications of them are isomorphic. This motivates the following definition. A Canonical Holder Complex is the simplification associated to a germ of a subanalytic surface (X,xo) of any Abstract Holder Complex ( r , p ) such that (X,XO) is the Geometric Holder Complex associated to (r,p). By Theorem 3.3, the Canonical Holder Complex is well defined.

Theorem 3.4 (Classification Theorem of Subanalytic Surfaces). 121 Two g e r m s of subanalytic surfaces ( X ,20) and (Y,yo) are bi-Lipschitz isomorphic if and only i f the corresponding Canonical Holder Complexes are isomorphic. We finish this section by the following Realization Theorem.

Theorem 3.5 (14). Let (r,p) be an Abstract Holder Complex. Then there exists a Geometric Holder Complex ( X ,XO) associated to (I?, p). Moreover, (X,xo) can chosen as a germ of a semialgebraic set. Question. When does an Abstract Holder Complex admit a real algebraic realization? Remark. R.Benedetti with M.Dedo (see [9]) and independently S.Akbulut with H.King (see [l])gave conditions for a topological realization of a 2-dimensional simplicia1 complex. Our problem is a geometric version of their results.

229 4. Metric homology.

A theory presented in Section 3 is a 2-dimensional theory. Here we are going to discuss some invariants of similar nature for higher dimensions. Let Y and 2 be two bounded subanalytic subsets of Bn.Let U E ( Z )be an E-neighbourhood of 2. We define a function f ( ~as) follows:

f(&)= VoldirnY(YnU,(Z)). Let

a ( r ) = lim &+O

-.f ( E l &T

By results of Lion-Rolin (see [21]) on volume-functions for subanalytic sets, there exists a rational number p such that

a(.) =

{

0, if 0 < r < p 00, if p < r < 00.

This number p is called a volume growth number of Y with respect to 2. We use the notation p ( Y , 2).

Remark. If 2 is a point and Y is a smooth manifold then p ( Y , Z ) = dimY. If Y and 2 are submanifolds of R" and if they intersect transversally such that Y n 2 # 0 then p ( Y , 2 ) = n - dimZ. Let X be a subanalytic set. A partition { X i } of X is called Lipschitz trivial stratification if (1) All X i are Lipschitz submanifolds of R" (2) { X i } is a topological stratification of X . (3) For any i and for any two points 51,5 2 E X i , there exist a pair of neighbourhoods U,, and U,, and a bi-Lipschitz homeomorphism h : U,, + U,, such that, for all other stratum X j , we have: h(UxlnXj) = U x 2 n X j . Now we are going define Metric Homology [4,51 in "Intersection Homology" [18] style. Let 8: {0,1,. . . ,n - 1) -+ Q n [l,00[ be a function called volume-perversity . Now we fix a field of coefficients L and consider, for each k, the set of subanalytic singular k-chains, i.e. the set of expressions P

i=l where Ak is a k-dimensional simplex, Fi: Ak + X are subanalytic maps and ai are elements of L. A subanalytic singular k-chain 7 is called admissible with respect to a stratification { X j } and a perversity function 8 if, for

230

each stratum X j , one has: ~ ( S U P PXj) ~ ,2 p(codimXj),

p(suppdq, Xj) 2 p(c0dimXj).

Admissible chains form a chain complex and the homology of this chain complex is called Metric Homology with respect to the stratification { X j } and the perversity function p. We use a notation M H @ ( X {, X j } ) .

Theorem 4.1 (Basic properties of Metric Homology). Let p be a volume-perversity function satisfying the following condition:

+

p ( i ) 5 p ( i + 1) I p ( i ) 1. This condition is called G-M condition. (1) M H p ( X , { X j } ) does not depend o n a Lipschitz trivial stratification

{ X j } . Thus, one can use the notation M H p ( X ) . (2) Let F : X 4 Y be a subanalytic bi-Lipschitz map. Then M H @ ( X )= MHP(Y).

5. Characteristic exponents of germs of subanalytic sets. This section is devoted to some generalizations of results of section 3 for higher dimension. In order to do it we are going to give another interpretation of Theorem 3.1. Let X C R" be a subanalytic surface and let xo E X be a singular point with a connected link. We can define a volume growth number p ( X , xo).

Theorem 5.1. Let ( X ,XO) and (Y,yo) be two germs of subanalytic surfaces such that the links of the singular points xo E X and yo E Y are connected. Then the germs are bi-Lipschitz equivalent with respect to the inner metric if, and only if, p ( X 1X O ) = p ( Y ,YO). Theorem 5.2. The number p ( X , x o ) can be interpreted as a vanishing order of the fundamental cycle o n the link of X at XO. For any subanalytic set X c R" of dimension m and for each k < m, we are going to define characteristic exponents as vanishing orders of k-dimensional cycles. We remind the following result of Federer and Fleming [15].

Theorem 5.3. Let M Ic < dim M , there exists q with a k-dimensional cycle, i.e. there exists a

be a compact riemannian manifold. Then, for all a constant c(k) such that each k-dimensional cycle volume of suppq less or equal to c ( k ) is a trivial (k + 1)-dimensional chain 1c, such that = q.

231

The following result is an analog of this theorem for singular spaces.

Theorem 5.4 (8). Let X c Rn be a closed subanalytic set and let xo E X be a point. Then, for every k 5 dim(X), there exists a constant p k satisfying the following property: "Let U,, be a a suficiently small neighbourhood of 20. Consider a subanalytic k-cycle c such that supp(c) c U,, - {XO} and dq = C, for a chain q with supp(v) c U,,,. Assume that p (su p p ( q ) ,x ~ )> pk. Then [c]= 0 in Hk(U,, - {xo}). (In other words, c = dql for a chain ql with SUPP(rl1) c ux, - {XO))l'. Remark. The usual interpretation of Theorem 5.1 is the following: small cycles are trivial. The interpretation of Theorem 5.2 can be presented as follows: the cycles collapsing sufficiently fast at the singular point are trivial on the link of this singular point. This theorem permits to define a characteristic exponent of any element of H ( L ( X , x o ) )where L(X,xo) means the link of X a t ZO. Let P: Hk( x, xo) 4 €3 be a function defined in the following way. Set P(h) be the supremum of the value p(supp(q),XO) over all cycles q E h and over all chains $ c X such that a$ = q. By Theorem 5.2, the value P(h) is well defined. Moreover, the function ji has the following "valuation" property: P(h1 hz) 2 max{p(hl),P(hz)}, for any pair h l , hz E H k ( L ( x , x o ) ) ,and p(ah) = p(h).Now we can suppose that the homology of link is considered with real coefficients. Let f i k , A be a subspace of Hk(L(X,xo)) such that, for all h E E ~ J we, have ji(h) 2 A.

+

Theorem 5.5. There exists aflag of subspaces V, C V,-l C . . . C Vz C V1 of Hk(L(x,ZO)) (here V1 = H k ( L ( X , x o ) ) , V , = 0 ) such that, for any A E R, one has H k , x = &, for some i = 1,.. . , s .

-

The flag {&} is called a characteristic flag of the point 20.The characteristic flag is also a bi-Lipschitz invariant. Namely, if the germs ( X ,xo) and (Y,yo) of subanalytic sets are bi-Lipschitz equivalent then their corresponding flags are linearly equivalent, i.e. there exists a linear isomorphism between H k( L ( x, 2 0 ) )and Hk(L(Y, yo)) preserving the characteristic flags.

Acknowledgments The author was supported by CAPES grant N AEX0591/05-0.

232

References 1. S.Akbulut, H.King Topology of real algebraic sets. Mathematical Sciences Research Institute publications, 25. Springer-Verlag, New York, 1992 2. L.Birbrair. Local bi-Lipschitz classification of 2-dimensional semialgebraic sets. - Houston Journal of Mathematics, N 3, vol. 25, pp.453-472 (1999). 3. L.Birbrair. Lipschitz geometry of curves and surfaces, definable in o-minimal structures. - Preprint RAAG, 2005 4. L.Birbrair, J.-P.Brasselet. Metric homology theory . - Communications on Pure and Applied Mathematics, N 11, vo1.53, pp.1434-1447 (2000). 5. L.Birbrair, J.-P.Brasselet. Metric homology f o r isolated conical singularities. Bulletin des Sciences Mathmatiques, ~01.126,pp 87-95 (2002). 6. D.Burago, Yu.Burago, S.Ivanov A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. 415 pp. 7. L.Birbrair, J.Costa, A.Fernamdes, M.Ruas. K-bi-Lipschitz equivalence of Real Function-Germs . - To appear in Proceedings of AMS. 8. L.Birbrair, F.Cano. Characteristic exponents of semialgebraic singularities . Math. Nachr., ~01.276,pp.23-30 (2004). 9. R.Benedetti, M.Dedo The topology of two-dimensional real algebraic varieties. Ann. Mat. Pura Appl. (4) 127 (1981), 141-171. 10. L.Birbrair, T.Mostowski. Normal embedding of semialgebraic sets . - Michigan Math. Journal, vo1.47, pp.125-132 (2000). 11. Birbrair,L.; Fernandes,A.G. Metric theory of semialgebraic curves. - Revista Matematica Complutence, vol.XII1, N 2 (2000), 369-382. 12. Benedetti, J.J.Risler Real algebraic and semi-algebraic sets. Actualits Mathmatiques. Hermann, Paris, 1990. 340 pp. 13. J.Bochnak, J.J.Risler. Sur les exposants de Lojasiewicz. - Comment Math. Helvetici, 50, pp.493-507, 1975 14. L.Birbrair, M.Sobolevsky. Realization of Holder complexes. - Annales de la Faculte des Sciences de Toulouse, v.VIII, N1 (1999) pp.35-44. 15. H.Federer, W.H.Fleming. Normal and integral currents. - Ann. of Math., 72, pp.458-520 (1960). 16. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, 11. Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 17. J.-P. Henry, A. Parusinski, Existence of moduli for bi-Lipschitz equivalence of analytic functions, Compositio Math. 136 (2003), no. 2, 217-235. 18. M.Goresky, R.MacPherson. Intersection Homology Theory. - Topology, 19 (1980), pp.135-162. 19. K.Kurdyka. On a subanalytic stratification satisfying a Whitney property with exponent 1 . - Real algebraic geometry (Rennes, 1991), pp.316-322. Lecture Notes in Math., 1524, Springer, Berlin 1992. 20. S.Lojasiewicz. Triangulation of semi-analytic sets .- Ann. Scuola Norm. Sup. di Pisa, 18, no 4, pp.449-474 (1989). 21. J.-M.Lion, J.-P.Rolin. Integration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques. - Ann. Inst. Fourier (Grenoble), v01.48, N3, (1998), pp.755-767.

233 22. T.Mostowski. Lipschitz equisingularity. - Dissertationes Math. CCXLIII (1985). 23. A.Parusinski. Lipschitz stratifications of subanalytic sets. - Ann. scient. Ec. Norm. Sup., 4 serie, t.27 (1994), pp. 661-696. 24. G.Valette, Lipschitz triangulations. Illinois J. Math. 49 (2005), no. 3, 953-979 (electronic).

234

LECTURES ON MONODROMY W. EBELING' Institut fur Algebraische Geometrie, Universztat Hannover, Postfach 6009, 0-30060Hannover, Germany 'E-mail: [email protected]. de

S. M. GUSEIN-ZADE Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 11 9992, Russia E-mail: [email protected]

Introduction In the course of J. Seade l1 the Milnor fibration of a hypersurface singularity was introduced. In our course we want to continue the study of this fibration. The basic object to study is the monodromy of this fibration. The word 'monodromy' comes from the greek words povo - Gpopor and means something like 'uniformly running' or 'uniquely running'. It goes back to B. Ftiemann. It arose first in keeping track of the solutions of the hypergeometric differential equation going once around a singular point along a closed path. In $1 we introduce the notion of the geometric monodromy of a germ f : (Cn+l,O) -+ (C'O) of a holomorphic function at the origin. In $2 we show how one can use a deformation of an isolated singularity to study the monodromy. We review the Picard-Lefschetz theory. In $3 we show how one can compute the zeta function of the monodromy from a resolution of the singularity. This leads to a proof of the famous monodromy theorem. In $4 we show how the zeta function of the monodromy of a hypersurface singularity with a good C*-action can be computed. In particular we derive a relation between the zeta function of the monodromy and the Poincar6 series of the coordinate algebra of the singularity which was recently discovered by the authors. Basic references for the first two sections are and 6. For a survey on

235

further results on the monodromy of isolated singularities see 1. Geometric Monodromy

Let f : (@"+l,O) -+ (@,O) be the germ of a holomorphic function with a critical point at the origin. Let us recall some of the results of Seade's course: Let E > 0 be small enough and let 0 < Q << E . Let B, be the closed ball of radius E around the origin in Cn+' and let A be the closed disc of radius Q around the origin in @. Let

Xt := f-'(t) n BE for t E A,

x := f-l(a) n B,, x*:= x \ x,, a* := a \ (01. By a result of J. Milnor, the mapping fix. : X* + a* is the projection of a (locally trivial) C"-differentiable fibre bundle. This fibre bundle ( X * ,fix., A*, X,) is called the Milnorfibration. The fibre X , over the point Q E A* is an n-dimensional complex analytic manifold (or a 2n-dimensional differentiable manifold) with boundary. The manifold X , is called the Milnor fibre of the germ f . It is homotopy equivalent to a CW-complex of dimension n and therefore H,(X,) = 0 for q > n. (Unless otherwise stated, we consider homology with integer coefficients.) As a complex analytic manifold, the Milnor fibre X , has a natural orientation. If the germ f has an isolated critical point at the origin, the Milnor fibre X , has the homotopy type of a bouquet of several n-dimensional spheres. The number p of these spheres is called the Milnor number of the germ f. In this case H,(X,) E Zp, H,(X,) = 0 for q # 0 , n (here we suppose n > 0; otherwise Ho(X,) 2 Zp+'). In our course we want to study the Milnor fibration. For this purpose we start with a general discussion of fibre bundles. Let (E,T, B , F ) be a differentiable fibre bundle with total space E , base space B , projection T : E -+ B and fibre F . Assume that the fibre F is compact. Let y : I 4 B be a piecewise differentiable path in B. There exists a (piecewise smooth) family rt of diffeomorphisms of the initial fibre E,(o) = r-l(y(0)) to the fibre E,(t) over the point y(t). These diffeomorphisms are well defined up to isotopy. Let h, = I'(1) : E,(o) E,(l). The diffeomorphism h, (well defined up to isotopy) is called the translation of the fibre E,(o) along the path y.Moreover, if two (piecewise smooth) paths y1 and 72 are homotopic in the class of paths with fixed starting point a. = yl(0) = yz(0) and end point b = n(1) = y2(l), the translations h,, -+

236

-

and h,, are isotopic. In this way one gets an (anti-)homomorphism PO : nl(B, b)

DiffO(Eb)

of the fundamental group of the base space B to the group of diffeomorphisms of the fibre Eb modulo isotopy. The image of PO is called the geometric monodromy gmup of the differentiable fibre bundle ( E , n ,B , F ) . One way to construct the translation h, is the following. Consider a Riemannian metric on the manifold E and consider the corresponding splitting of the tangent space TxE of E at a point x E E. The subspace

T,E = {V E TxE I dnX(v)= 0). of TxE is called the vertical tangent space of E at the point x. It coincides with the tangent space TXE,(,) to the fibre E+) = T - ~ ( T ( X > ) above the point n(x).The orthogonal complement

of TE : in TxE is called the horizontal tangent space to E at the point x. One has

T,E = T,"E @ T f E . The linear mapping d7rx maps T,hE isomorphically to the tangent space T,(,)B of the base space B at the point n ( x ) .This defines a splitting of the tangent bundle T E as the Whitney sum

T E = T*E @ T h E of two subbundles of TE. This is called an Ehresmann connection on the fibre bundle ( E ,n,B , F ) . As above, let y : I + B be a piecewise differentiable path in the base space B and let x E E,(o) be a point in the fibre over y(0). Then there is a unique lifting y : I + E of the path y with y(0) = x, which is horizontal, i.e. y(t)E T$(t,E for all t E I. This defines a diffeomorphism

h, : E,(O)

-

E,(l)7

which in this case is called the parallel translation of E,(o) along the path y (cf. Fig. 1). We return to the Milnor fibration (X*, flp,A*, Xv). Let y be the loop (closed path)

y : [0,1] -+ A,

t H ve2Tit.

237

E

Fig. 1. Parallel translation along the path y

Definition 1.1. The diffeomorphism h f = h, : X , + X , is called the geometric monodromy of the singularity f. The induced homomorphism h:! : H,(X,) + H,(X,) on the q-th homology group of the Milnor fibre X , (for any q E Z) is called the (classical) monodromy (or the (classical) monodromy operator) of the singularity f. Our objective is to study this operator. 2. Deformation and monodromy

The first method is to use a deformation of a singularity. Suppose that the germ f has an isolated critical point at the origin. We consider a morsification fx of f. This is a perturbation of (a representative of) f (i.e., fo = f) defined in a neighbourhood U of the origin in (Cn+', depending on a parameter X E (C,O) and such that, for X # 0 small enough, in a neighbourhood of the origin the function fx has only non-degenerate critical points with distinct critical values. One can show

238

that the number of these critical points is equal to the Milnor number p of the germ f . Let Y := f y l ( A ) n B , and yt := fyl(t)nB, fort E A. Assume that X # 0 is chosen so small that all the critical points of the function fx are contained in the interior of Y and the fibre fyl(t) for t E A intersects the ball B, transversely. Denote the critical points of the function fx by p l , . . . , p , and the corresponding critical values by sl,. . . ,s., The number r] E dA is a non-critical value of the function fx. Let A' := A - ( ~ 1 , . . . ,s}, and Y':= Ynfy'(A/). Then the mapping fxly) : Y' + A' is the projection of a differentiable fibre bundle. The fibre yt of this bundle is diffeomorphic to the fibre X t of the Milnor fibration. In particular, Y, is diffeomorphic to X,. We therefore identify these fibres. Let y : [0,1] + A' be a piecewise differentiable loop in A' with starting and end point r]. The diffeomorphism h, : X , -+X, is called the geometric monodromy with respect to the loop y. The induced homomorphism h t i : H4(X,) -+ H4(X,) on the q-th homology group of the fibre X , (for any q E Z)is called the monodromy (or the monodromy operator) with respect to the loop y. Let y : [0,1] -+ A be a piecewise differentiable path which connects the critical value si with 71 and does not pass through any other critical value, i.e. y(0) = si, y(1) = r] and y((O,l]) c A'. By the complex Morse lemma there exists a neighbourhood of the (non-degenerate) critical point pi and local coordinates ( ~ 1 ,... ,z,+1) on it (centred at the point p i ) such that fx can be written in the form

fx(z1,. . . ,Z,+l) For sufficiently small t

= si

+ + . . . + z,,,. 2 2.1

2

> 0 the fibre X,(t) contains an n-sphere S ( t ) := J-.

S"

where S" is the n-dimensional unit sphere

S"

=

{z = (z1,. . . ,z,+1) E

@"+l

I Im zi = 0, c z :

= I}.

By translation along y one obtains an n-sphere S ( t ) c X,(t) for each t E (0,1]. For t = 0 the sphere S ( t ) shrinks to the critical point pi (cf. Fig. 2). We choose an arbitrary orientation of S(1). Then S(1) is an n-cycle in X , and represents a homology class b E H,(X,). This homology class is called a vanishing cycle of fx (along the path y). It is well defined up to orient ation. It is important to know the self-intersection number (6,b) of the vanishing cycle 6. For that one can compute the self-intersection number of the

239

r

O

1

t Fig. 2.

Vanishing cycle

{cz:

unit sphere S" in the complex manifold F := = 1). It is easy to see that the latter ma.nifold is diffeomorphic to the total space TS" of the tangent bundle of the sphere S". Namely, the space TS" can be considered as the submanifold of the space Rn+' x Rnfl defined by the equations C us = 1, C ui'ui = 0 where u1, . . . , un+l and q ,, . . , wn+l are coordinates in the first and in the second factor Rn+' respectively. Now a diffeomorphism from the manifold F to TS" can be defined by ui = xi///x//, 'ui = yi where xi and yi are the real and the imaginary parts of the coordinate zi respectively, llzll = This diffeomorphism sends the unit sphere S" c F to the zero section of the tangent bundle TS". It is well known that the self-intersection number of the zero section in the space of the tangent bundle of a manifold is equal to its Euler characteristic. In our case it is equal to x(S")= 1 (-l)n (2 for n even and 0 for n odd). However, the described diffeomorphism does not respect the natural orientations of the manifolds F (as a complex analytic manifold) and TS" (as the total space of a tangent bundle). One can easily see that the orientations differ by the sign (-l)n(n-1)/2 (the sign of the permutation (1,3,. . . , (2n - l ) , 2 , 4 , . . . ,271)). Therefore the self-intersection number of the unit sphere S" in the manifold F (and thus the self intersection number of the vanishing cycle b in the Milnor fibre X,) is equal to

m.

+

(-q"("-W

0 for n odd, 2 for n = 0 (mod 4), -2 for n = 2 (mod 4).

240

A path y : I + A with y(0) = si, y(1) = q, y((O,l]) c A’ defines a loop around the critical value si in the following way: Let Ai be a disc of sufficiently small radius around si such that y ( I ) intersects the boundary dAi of Ai exactly once, namely at time t = 8 at the point si ui.Let r : I -t hi, t H s i + u i e z f l n t ,be the path starting at si+ui which goes once around si along the boundary of the disc Ai in counterclockwise direction. Moreover, set 7 := y l [ ~ , l ]The . loop w = with base point 7 is called the simple loop associated to the path y (cf. Fig. 3). The monodromy

+

Fig. 3.

h, := h:)

Simple loop associated to y

: Hn(Xv) ---+Hn(Xv)

corresponding to the simple loop w associated to y is called the PicardLefschetz transformation corresponding to the path y (or to the vanishing cycle 6; see below). One has the following basic theorem.

Theorem 2.1 (Picard-Lefschetz formula). For

Q

E

Hn(Xv) one has

.n n-1

h*(a)= Q - (4)*(Q,6)6, where (., .) denotes the intersection f o r m o n the homology group Hn(Xv). Let us consider an ordered system (71,. . . ,y p ) of paths yi : I --f A with yi(0) = si, yi(1) = 7 and yi((0, 11) c A’. The system (71,. . . , y p ) of paths is called distinguished if the following conditions are satisfied: (i) the paths yi are non-self-intersecting; (ii) the only common point of yi and yj for i # j is q; (iii) the paths axe numbered in the order in which they arrive at q where one has to count clockwise from the boundary of the disc A (cf. Fig. 4).

241

Fig. 4. Distinguished system of paths

A system (61,. . . ,S,) of cycles 6i E H n ( X q ) is called distinguished, if there exists a distinguished system ( 7 1 , . . . ,7), of paths such that 6i is a cycle vanishing along the path yi. One has the following theorem. Theorem 2.2. A distinguished system (61, . . . ,S,) of vanishing cycles is a basis of the homology group H,(X,,). Let (71,. . . ,7,) be a distinguished system of paths and let hi be the Picard-Lefschetz transformation corresponding to the path yi. Then the classical monodromy operator of f can be expressed as follows:

hf, = h l . * . h,. For the loop w corresponding to h f * is homotopic to the combination wpw,-l . . .w1 of the simple loops associated to h,, hp-l,. . . ,hl (cf. Fig. 5). Let us recall that the Milnor fibre X,,has a natural orientation and therefore there is a well defined intersection form on the homology group H n ( X q ) . Let A = ((&,Sj)) be the intersection matrix with respect to a distinguished basis (61, . . . , 6 , ) of vanishing cycles. The diagonal entries of the matrix A (the self-intersection numbers of the vanishing cycles) are equal to (-1)*(1 (see above). This matrix is of the form A = -V-(-l)nVt for some upper triangular matrix V with all the diagonal elements equal to -(-1)n(n-1)/2. In fact, V is the matrix of the so called

+

242

Fig. 5 . w is homotopic t o wILwIL--l . . . w1

Seifert form (or variation operator) associated to the singularity f. Let H be the matrix of the monodromy operator h f . with respect to the basis (61,. . . ,dp). Using the Picard-Lefschetz formula, it is an algebraic exercise to derive the following formula for H (cf. ):

Proposition 2.1. One has

H = (-l)nflV-lVt. Example 2.1. Let us consider the map germ f : ((C2,0) -+ (C,O) given by f ( z ,y ) = z3- y2. A morsification of f can be taken in the form fx = z3- 3Xz - y2. It has critical values s 1 , 2 = f 2 X 3 I 2 . Zero is not a critical value. The level sets of the function fx corresponding to the critical values s1 and s2 and the value 0 are shown in Fig. 6 (for X real and positive). From this figure one can derive that the intersection matrix A of this singularity in a distinguished basis has the form

(2) *

Therefore

243

s2

fx

= SI

Fig. 6. The level sets of the function

fx

(one can check that H6 is the identity matrix).

Example 2.2. Let us consider the map germ f : (C2,0) + (C, 0) given by f(z, y) = z5- y3. The curve

c = {(z, y) E C2 1x5 - y3 = 0) is parametrized by x = t 3 ,y = t 5 .We consider the following perturbation of this parametrization:

+ 1.7Xt2 + 0.66X2t - 0.04X3, y = t 5+ 2.3Xt4 + 1.68X2t3+ 0.356X3t2

z = t3

-

0.0245X4t.

The real part of the curve for real X > 0 is depicted in Fig. 7. Let f x ( x ,y) = 0 be a real equation of this curve. The real function f x has 8 non-degenerate critical points: the 4 double points of the curve and 4 maximum points which are situated in the closed regions of the curve. To each of these critical points (after a suitable choice of a distinguished system of paths: one should

244

demand that all the paths lie in the upper half-plane { z 2 0)) there is associated a vanishing cycle in the Milnor fibre X,, of the complex function f . According to results of N. A’Campo and S. M. Gusein-Zade 9, with a suitable orientation these vanishing cycles intersect as follows: A vanishing cycle 6i corresponding to a maximum point in a closed region intersects a vanishing cycle 6, corresponding to a double point on the boundary of that region with intersection number ( S i , S j ) = 1 (and therefore ( S j , S i ) = -l), cf. Example 1. All other intersection numbers are equal to zero. The corresponding incidence graph is the Coxeter-Dynkin diagram of type ES (see Fig. 8).

Fig. 7.

Real morsification of Es

Fig. 8. Coxeter-Dynkin diagram of type E8

Problem 2.1. Let p ( t ) = det(hE) -t-id,) be the characteristic polynomial of the monodromy operator hE’ of a germ f : ( ~ n + l0) ,

+

(c,0)

with an

245

isolated critical point at the origin. Show that p(1) = (-l)p'

n+1 n+2 ;)

' detA.

3. Resolution and monodromy

We now review work of H. Clemens and N. A'Campo which uses a resolution to study the classical monodromy operator. Let Y be a finite CW-complex. Let h : Y -+ Y be a homeomorphism and let W C Y be a subcomplex preserved by h. The zeta function of the mapping hlw is a rational function of the variable t defined by

Cw(Q :=

(_1)4+1

{det (id - t(h(w)*;Hq(W,C))} ,220

(Note that there are different sign conventions, some authors use the inverse of this function.) The zeta function has the following properties: (1) (Mayer-Vietoris) If Y is the union Y = Y1 UY2 of two subcomplexes Y1 and Y2, h : Y t Y preserves Y1, Y2, and Y1 n Y2, then

(2) If ( E ,T , B , F ) is a differentiable fibre bundle with a connected base space B and h : E -+E is a fibre preserving diffeomorphism then

CE(t) = CF(t)X ( B ) , Let f be a germ of a function (Cn+',O) at the origin (not necessarily isolated). Let the germ f . This means the following:

+ T

:

(5,O)with a critical point U

-+

U be a resolution of

(1) U is a neighbourhood of the origin in Cn+' where the function f

is defined, B, c U ; 6 is a non-singular complex analytic manifold of dimension n 1, T is a proper analytic map which is an isomorphism outside of the zero level set (f = 0) c U of the function f ; (3) in a neighbourhood of each point x E ~ - ' ( { f = 0}), there exist local coordinates z1, . . . , zn+l on 6 centred at the point x such that the lifting = f o T of the function f has the form f(z) = mn+1 z y l . . . . . zn+l with mi E 2510.

+

(2)

7

Such a resolution always exists due to Hironaka's theorem.

246

-

Since the map 7r : U + U is an isomorphism outside of the zero level set {f = O}, the map Trestricted to ?-'(A*) nr-l(B,) (as a map to A*) coincides with the Milnor fibration f : f-'(A*) n B, + A*. Therefore, instead for the function f, one can construct the geometric monodromy for the function y (i. e. in the manifold 6).Let us do it in a neighbourhood of a point x E T--'(O)n ..-'(BE). In some local coordinates 2 1 , . . . , z,+l on U (centred at the point x) one has

-

-

f(z) = ZI"' . . . . . z?' with T 2 1, mi # 0 for i = 1,.. . ,T. Consider a closed ball B, of (small) radius p around the origin (in the coordinates z1, . . . , z,+l). Let d = gcd(m1 . . . m,) and let d = klml+. . .+k,m, for some ki E Z. A geometric monodromy h of the function finside the ball B, can be defined by the family

given by the equation

.. In this case h is defined by h(z17..

.

7

=

The zeta function of the mapping h is different for T = 1 and for T > 1. In both cases the geometric monodromy has finite order (equal to d) and determines a free action of the cyclic group Z d of order d o n the Milnor fibre X,. In the first case the set q } consists of ml copies of a closed 2ndimensional ball and the map h is a cyclic permutation of these copies. This implies that the corresponding zeta function is equal to (1-tml)-'. Let S, be the set of points x E f=l(O) n 7r-' ( B E such ) that, in a neighbourhood of x, in some local coordinates the function ?is written as z y . In the second case there is the following free S1-action (S' = {A E C : llXll = 1)) on the level set {y = q}. Let kimi = 0, k: # 0, gcd(ki,. . . ,ki) = 1. Then the action is defined by

{fl=

xi=,

* ( a , ... zn+l) = exp (27r-t) (exp ( 2 n G k i t ) z l , .. . exp ( 2 7 r G k i t ) z , , z,+i,. . . ,z,+i)

.

247

This implies that the base space of the covering X, -3 X,/& (the latter space is the quotient space by the group generated by the mapping h ) inherits an S1-action and therefore (each connected component of) the base space of this covering has Euler characteristic equal to zero. Applying the property (2) above one gets that the corresponding zeta function is trivial (equal to 1). These local data can be glued together using the construction of A'Campo l . Applying the Mayer-Vietoris formula (property (1)) one gets the following result: Theorem 3.1 (A'Campo). One has

m2l This implies that the eigenvalues of the classical monodromy operator of an isolated singularity are roots of unity. One can even get a more precise result. Let M be the least common multiple of the numbers m such that Sm # 0. Let 2,.be the part of r-l(O)which is the union of all r-fold intersections of its non-singular components:

-( 0 )

21=p

3

22 3 . . . 3 Zn+l3 Zn+2= 0.

The described construction implies that in a tubular neighbourhood of 2,. minus a tubular neighbourhood of Zr+1 one can construct a geometric monodromy the M-th power of which is homotopic to the identity. Fix q arbitrary. A q-cycle in the Milnor fibre is homologous to a q-cycle a which does not intersect a tubular neighbourhood of the set 2,+2.Since, in a tubular neighbourhood of 21 minus a tubular neighbourhood of 22,the M th power hM of the geometric monodromy is homotopic t o the identity, the cycle hMa-a is homologous to a cycle which lies in a tubular neighbourhood of 22 minus a tubular neighbourhood of 2,+2.Continuing in this way, one gets the following statement: Theorem 3.2 (Monodromy theorem). One has

Corollary 3.1. One has

( ( h g ) ) M- l)n+l= 0. Example 3.1. Consider again the plane curve C = { (2,y) E C2I x5 -y3 = 0). The (minimal) embedded resolution of the curve C (a resolution of the

248

germ f(z,y) = x5 -y3) is indicated in Figure 9. Here the segments show the

5

9

Fig. 9.

Resolution of C = {(I,y) E C2I z5 - y3 = 0)

components of the exceptional divisor n-'(O) (each of them is isomorphic to the complex projective line (CP1),the numbers indicate their multiplicities in the zero divisor of the function the arrow shows the strict transform of the curve C. From this we compute

7,

. us conThe curve C can be parametrized as X ( T ) = T ~y (, ~ )= T ~ Let sider the initial terms of the Taylor series decompositions of the functions g(z(T), y ( ~ ) for ) all germs g of functions of two variables (at the origin). One can see that the exponents of these terms (in increasing order) are v1 = 0, v2 = 3, v3 = 5, 214 = 6, 715 = 8, v6 = 9, . . .(starting from 8 each integer occurs as the exponent of such a term). Let us consider the corresponding generating series 02

P(t)

=Ct"t= 1 + t 3 + t 5 + t 6 + t 8 + t 9 +

... .

i= I

One can see that it is the power series decomposition of the rational function ( 1 4 5 ) (l-t3)(l-t5), i. e. (as a rational function) it coincides with the zeta function of the germ f. This (mysterious) coincidence holds for all irreducible plane curve singularities. It was observed in lo. The proof consists of a direct computation of the zeta function c(t) and of the generating series P ( t ) in the same terms (say, in terms of so called Puiseux pairs) and of a comparison of the results. Up to now no real explanation of this coincidence exists.

249

Problem 3.1. Let f have an isolated critical point at the origin and let cp(t) be the characteristic polynomial of the monodromy operator hf.. Show that if - x(Sm)= 1 then

4. @*-actionand monodromy

Now we assume that f is weighted homogeneous of degree d with respect to weights 41,. . . ,qn+l. Here 41,. . . ,qn+l are positive integers with gcd(q1,. . . ,q n + l ) = 1. This means that for X E C*

. . ,XQn+lzn+l)= Ad f (21,. . . , & & + I ) .

f(XQlZ1,.

There is a natural C*-action on the space Cn+ldefined by

x * (XI,.. . ,zn+l)= (XQlz1,.. . ,XQn+lzn+l), x E c*. Let F := f-l(l). The geometric monodromy h : F as the transformation z

*z

H eanild

(z E F

c Cn+',

+F

can be defined

eanildE C*).

We again want to compute the zeta function of h. Let V = f - l ( O ) be the zero set of f and let Y := (V \ {O})/C* be the space of orbits of the C*-action on V \ (0). We consider a decomposition of V, Y , and F . For I c 10 = {I,.. . ,n l}, I # 0, define

+

TI := {z E Cn+' I z i = 0 for i @ I , z i # 0 for i E I } ;

VI := V n T I ;

YI := (V n T I ) / @ * ; FI := F T I . Let mI := gcd(qi, i E I ) . The integer mI is the order of the isotropy group of the C*-action on the torus T I . Note that if mI does not divide d then YI is empty. The geometric monodromy maps FI to itself. Let hI := h(pI and let
250

transformation of it and acts as a cyclic permutation of the d / m I points of a fibre. Therefore t ( t ) = (1- td/mr)-x((Tr/@*)\fi),

This formula even makes sense if mI does not divide d since in this case (TI/C*)\YI is empty and therefore the exponent is equal to zero. Therefore =

n

(1 - t

d/mr )-x((Tr/c*)\Yr).

I:I I12 1

We want to relate this function to the Poincari! series of the singularity f . This is defined as follows: Let A = C[z]/(f)be the coordinate ring of V. There is a natural grading on the ring A: Ak is the set of functions g E A such that g(X * z) = X k g ( z ) . Let P ( t ) = cp=o dim Ak . tk be the Poincarh series of the graded algebra A = Ak. One has

@Eo

We see that the zeta function is a function of the form

$(t)=

n(l t y , -

a[ E

z.

[Id

Following K. Saito, we define the Saito dual to $(t) to be the rational function

(b*(t)= n(1-tm)-a-. mld

Let

i3)= c ( W - t ) be the reduced zeta function, i.e. the zeta function defined by using the action of the monodromy on the reduced homology groups. Let Ym C Y be the set of orbits for which the isotropy group is the cyclic group of order m. Let

Or(t) :=

n

(1 - tm)x(Ym).

m>l

We derive the following theorem due to the authors '. Theorem 4.1.

251

Example 4.1. We consider again the function f ( z , y ) = z5 - y3. This function is weighted homogeneous of degree 15 with respect t o weights 3,5. In this case, Y = Y1 = pt. Therefore we have

From this it follows t h a t

as we have already seen above.

Problem 4.1. Let

References 1. A'Campo, N.: La fonction z6ta d'une monodromie. Comment. Math. Helv. 50, 233-248 (1975). 2. A'Campo, N.: Le groupe de monodromie du dkploiement des singularit& isolees de courbes planes I. Math. Ann. 213, 1-32 (1975). 3. Arnold, V. I., Gusein-Zade, S. M., Varchenko, A. N.: Singularities of Differentiable Maps, Volume 11. Birkhauser, Boston-Basel-Berlin, 1988.

252 4. Bourbaki, N.: Groupes et algbbres de Lie, Chapitres 4,5 et 6. Hermann, Paris, 1968. 5. Clemens, C. H.: Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities. Trans. Amer. Math. SOC.136,93108 (1969). 6. Ebeling, W.: Funktionentheorie, Differentialtopologie und Singularitaten. Vieweg, Braunschweig, Wiesbaden, 2001. 7. Ebeling, W.: Monodromy. math.AG/0507171. To appear in Proceedings of a Conference on occasion of G.-M. Greuel’s 60th birthday, LMS Lecture Notes, Cambridge University Press. 8. Ebeling, W., Gusein-Zade, S. M.: Monodromies and Poincark series of quasihomogeneous complete intersections. Abh. Math. Sem. Univ. Hamburg 74, 175-179 (2004). 9. Gusein-Zade, S. M.: Intersection matrices for certain singularities of functions of two variables. @ y H K q a o H a n . h a n . M npMnoxeH. 8 , l l - 1 5 (1974) (Engl. translation in Funct. Anal. Appl. 8 , 10-13 (1974)). 10. Gusein-Zade, S. M., Delgado, F., Campillo, A.: On the monodromy of a plane curve singularity and the Poincark series of its ring of functions. @YHKUHoHan. AHan. H n p H n o x c e H . 33,n o . l , 6 6 4 8 (1999) (Engl. translation in Funct. Anal. Appl. 33,no.1, 56-57 (1999)). 11. Seade, J.: On Milnor’s fibration theorem for real and complex singularities. These proceedings.

253

COMPUTATIONAL ASPECTS OF SINGULARITIES

ANNE FRUHBIS-KRUGER

Fachbereich Mathematik University of Kaiserslautern 67653 Kois erslautern Germany

These notes of a mini-course are written as an introduction to methods from computer algebra which have turned out to be useful in singularity theory. Starting with rather basic examples (and explicit solutions using the computer algebra system SINGULAR) we proceed to more complex applications and eventually tasks which are not purely computational such as the use of computational methods in the construction of hypersurfaces with assigned singularities. The final chapter is then devoted to the computational problems arising in algorithmic resolution of singularities.

Preface This text is a set of notes of a mini-course entitled Computational Aspects of Singularities given at the School o n Singularities in Geometry and Topology in August 2005. The aim of these talks was to introduce the participants to the use of computational methods for checking and studying properties of explicit examples of singularities; to this end, the first part of the course was devoted to an overview of computational tools for tasks from singularity theory. We then proceed step by step from the simple application of predefined computational tools to more complex applications, like studying families of singularities and constructing hypersurfaces with prescribed singularities. Using the algorithmic resolution of singularities as an example, we also show, how a rather complex computational task can be tackled by decomposing it into several smaller tasks. As this set of talks was embedded into a school on singularities, some familiarity of the readers with the (singularity theory) background of the treated computational tasks is assumed, but references providing a starting point for reading on the theoretical background are also specified for

254

each topic. Hence, this text only puts each task into context by recalling basic definitions and some properties of the discussed objects from the algebraic point of view before outlining the computational approach to it and discussing a practical example. For the convenience of the readers, all practical examples have been treated using the same Computer Algebra System SINGULAR (see 22); but, of course, many of the discussed algorithms and applications are also available in other Computer Algebra Systems. In the first section of this article, the calculation of the singular locus is used as an example on how to describe a given singularity in a computer algebra system and on how to determine basic properties of it using standard algorithms from computer algebra. The subsequent section is then devoted to the computational study of germs of singularities and related invariants like e.g. dimension, multiplicity and Milnor number, whereas the last two chapters lead to more complex applications including deformations, hypersurfaces with assigned singularities and algorithmic resolution of singularities in characteristic zero. I would like to thank the organizers of this conference, the professors J.-P. Brasselet, J. Damon, M. Lejeune-Jalabert and L& D.-T. for the opportunity to take part in and contribute to this interesting meeting. In addition to that I would like to thank H. SchBnemann, G. Pfister and several students at the University of Kaiserslautern for many useful comments on earlier versiohs of this text, and F.-0. Schreyer and the referee for many helpful comments on the exposition of the material.

1. Studying the Singular Locus

For novice users of CA-systems, a first obstacle to using software for studying explicit examples is often the question how to encode the geometric object in the language of the computer algebra system. Using SINGULAR as an example of a CA-system, we shall illustrate these first steps with the calculation of the singular locus of a given variety. Along the way, we see applications of standard methods of computational commutative algebra like elimination of variables, primary decomposition and normalization. For a more detailed discussion of these techniques and for a description of the underlying algorithms see e.g. 21.

255

1.1. The Jacobian Criterion

Computational Task Given an affine variety V ( I ) c K" over a (perfect) field K , corresponding to an ideal I = (ti,.. .,f m ) c K[x1,..., x , ] , our first task is to to determine its singular locus by means of the Jacobian criterion.

Background For a detailed discussion of the following notions and related topics, see, for instance, 21 ,section 5.7. Definition 1. (Singular Locus) Given a ring A = K [ x l , ... ,z n ] / I the set

Sing(A) := {P E Spec(A) I A, is not regular} is called the singular locus of A. The singular locus can be determined by means of the Jacobian criterion:

Lemma 2 . (Jacobian Criterion) Let K be a perfect field, I = ( f i , . . .,fm) and let A = K[xl,. .. , x n ] / I be equidimensional. Let J c A be the ideal generated b y the (n - dim(A))-minorsof the Jacobian matrix Then

(g).

Sing(A) = V ( J ) . Computational Solution (Irreducible Case) Example 1 (irreducible curve specified by parametrization) In this first example, we consider a curve in @. which is specified by the parametrization

G-4 t

c--)

(t3 , t4 , t5 ).

Our computational subtasks here will be 0

0

computation of the ideal of the curve application of the Jacobian criterion to compute the singular locus simplifying the description of the resulting set of points.

At this point, it is important to mention that, in general, all calculations in a computer algebra system are performed over the rationals or over suitable

256

field extensions thereof, but not over the real or complex numbers. This does not change the calculations, but we need to be aware of this when discussing computational results. We now determine the ideal of the curve by using an elimination technique for the parameter t. More precisely, we already know that the ideal of the graph of the parametrization map in x A$ is

A&

(2 - t3,y - t4,z

-t5)

c q2,y ,z , t ],

where the variable t corresponds to A&. We can now determine the image x +A$ via elimination of of the graph under the projection map the variable t. Elimination is available as a standard command in many CA-systems; details on its implementation using Grobner bases w.r.t. elimination orderings can, for example, be found in 21, sections 1.2 and 1.8.

A& 4

In the language of SINGULAR the computations which we just described can be performed as followsa: > ring r=O,(t,x,y,z),dp;

// ring of char 0 containing // variables x,y,z and t

> ideal Ip=x-t-3,y-t-4,z-t-5; // input of the parametrization > ideal Ie=eliminate(I,t); > Ie; - [I] =ya-xz - [a] =x2y-z2 - [31=x3-YZ

// compute ideal of the curve: / / elimination provides equations / / of the projection to A-3

> ring r2=0,(x,y,z),dp;

/ / we do not need t any more: / / move the ideal of the / / curve to this ring

> ideal I=imap (r ,Ie) ;

As the variety was specified by means of its parametrization, we already know that we are dealing with an irreducible curve. Hence we do not need aThe sequence of characters ’11’ in SINGULARQUtpUt marks a comment and should not be considered a8 input. The sequence of characters ’dp’ in the declaration of the ring specifies the ordering on the monomials. The effects of the choice of ordering on the computations are discussed in detail in 21, section 1.5

257

to worry about the equidimensionality condition in the Jacobian criterion. We can simply proceed by computing the Jacobian matrix and the ideal of minors of the appropriate size. For didactic reasons, however, we also include the computation of the dimension of K [ z ,y,.]/I (of which we know that it is 1):

> int dimA=dim(std(I)); > did;

// // // // //

compute dimension of K[x,y,zl/I; ’std’ (Groebner basis) required for using command ’dim’ we already knew that it is I

1

> matrix Jac=jacob (I) ; > print (Jac) ;

// determine Jacobian matrix // show the matrix

-z, 2y,-x, 2xy,x2,-2z, 3x2,- Z ,-y

> ideal J=rninor(Jac,S-dimA); > ideal sL=J+I; > sL;

// determine the minors // ideal of singular locus / / show content of variable sL

SL [I] =-x2y-222 SL [2] =-2xy2+6~2~ SL [3] =-2y2-xz sL [41=3x3+yz sL [5] =3x4+2xyz SL [6]=6~2y-z2 SL [7]=x3-4yz sL [8] =2x2y+2z2 sL [9]=4xy2+x2z SL [I01 =y2-xz SL [I 11=x2y-z2 SL [I23 =x3-yz 12 generators for the ideal of the singular locus seem to be quite a lot. Indeed, ideals generated by minors of matrices tend to have a high number of redundant generators and we can try to find a smaller set of generators by applying appropriate commands such as the SINGULAR-command mstd.

258

But in our particular case, we are only interested in the set of singular points, which is by the Hilbert Nullstellensatzb also the vanishing locus of the radicalCof the ideal which we computed:

> LIB “primdec .lib” ; > radical (sL) ; - c11=z

// // // //

the command radical is contained in a Singular library which needs to be loaded before using the command

- C2l =y - C3l =x Thus we see that the only singular point of this curve is the origin.

Background (continued)

In the following examples, we will use primary decomposition. To fix notation we briefly sketch some definitions and statements of this field. For a detailed discussion of the theoretical aspects see e.g. 12, section 3.3, for a treatment from the algorithmic point of view see e.g. 21, section 4.3. Definition 3. (Associated Primes) Let R be a noetherian ring and let

I

C

R be an ideal. The set of associated primes of I is defined as Ass(I) := { P c RIP prime, P = AnnR/I(b) for some b E A}.

Let P,Q E Ass(I) and Q 2 P,then P is called an embedded prime ideal of I ; the elements of Ass(I) which are not embedded are referred to as minimal prime ideals of I . Theorem 4. (Primary Decomposition for Ideals) Let R be an noetherian

ring. Every ideal I

cR

is the intersection of finitely m a n y primary ideals.

bOver an algebraically closed field K, Hilbert’s Nullstellensata states that, given an ideal I C K [ z l ,...,z n ] , the ideal of V ( I ) is precisely the radical f i = {f E K[z1,... , z n ] l f m E I for some m}. This is one of the situations, where we need to remind ourselves that even though we are computing over Q, our reasoning takes place over C CThere are different algorithms for the computation of the radical of a given ideal, e.g. the one of Krick and Logar 27 or the one of Eisenbud, Huneke and Vasconcelos 13. As different approaches have different advantages and drawbacks, a CA-system often provides several different commands for it. It is often worthwhile to try another algorithm, if the first one did not show good performance on a particular class of examples.

259

Furthermore, if I = & Q i is an i r r e d u n d a d p r i m a y decomposition of I (with Qi being Pi-primary), then the Pi are precisely the associated prime ideals of RII. Computational Solution (equidimensional case) Example 2 (reducible, equidimensional variety) In the second example, we consider a very simple reducible variety consisting of 3 smooth hyperplanes. Our computational subtasks will be: 0

0

0

defining each of the hyperplanes separately and forming their union by intersecting the corresponding ideals applying the predefined procedure slocuswhich performs precisely the same steps as we did in example 1 studying the singular locus using primary decomposition.

> ring r=O,(x,y,z,w) ,dp; > ideal Il=x,w; > ideal 12=y,z; > ideal 13=z-x^2-y*2,w; > ideal Itemp=intersect(Il,I2); > Itemp; Itemp [11=yw Itemp[Z]=zw Itemp[3l =xy Itemp[4]=xz > ideal I=intersect(Itemp,I3); > I; I Cll =zw IC2l=yw I [3] =-X~ZW-Y~ZW+Z~W I [4]=-x2yw-y3w+yzw I[5] =-X~Z-XY~Z+XZ~ I[S] =-X~Y-XY~+XYZ I [7]=-Y~W+ZW

// // // // // //

polynomial ring: char. 0, 4 var. the y-z plane the x-w plane another smooth surface union of y-z and x-w planes

/ / union of all three surfaces

dA primary decomposition is called irredundant, if no Qi can be omitted in the intersection.

260

In this case, we expect the singular locus to be the locus where the surfaces meet, since each of the surfaces is smooth. Using the ideals of the respective surfaces, an easy calculation by hand shows that the surfaces V(I1) and V(I2) respectively V(I2) and V(I3) meet in the point V((z,y,z,w)), whereas the intersection locus of V(I1) and V(I3) is the curve V((z, w, z - y2}). We now determine the singular locus using the procedure slocus and compare this result to the result of our computation by hand.

> LIB "sing.lib" ; > ideal sL=slocus(I); > size(sL) ;

/ / slocus is in the library // 'sing.lib' / / compute singular locus / / size of set of generators

91

To see both components of the singular locus, we cannot restrict our considerations to the radical or the minimal associated primes in this case, because one component, the point, is contained in the other component. We need to consider a full primary decompositione of the ideal sL, to find both components:

> LIB "primdec .lib" ;

// library for primary decomp.

> minAssGTZ (sL) ;

/ / minimal associated primes / / just one minimal prime

Cll:

- [ll =y2-z - r21 =w - C3l =x > primdecGTZ(sL) ; Cil: 111 :

/ / complete primary decomp. / / first primary component / / * primary ideal

- c11 =w eSome aspects of the theoretical background of primary decomposition are briefly sketched at the beginning of section 1.3; an introduction to the computation of a primary decomposition, of the radical and of several related tasks can be found in 21, chapter 4. As for the radical there are also different approaches to the calculation of the primary decomposition, e.g. the one of Gianni, Trager and Zacharias and the one of Shimoyama and Yokoyama 36. In the primary decomposition commands implemented in SINGULAR, the list containing the result is not ordered. Therefore permutations of the list entries occur quite often.

26 1

//

*

corresponding prime

/ / 2nd primary component // * primary ideal

//

*

corresponding prime

1.2. The Non-equidimensional Case The two previous examples were constructed in a suitable way to make sure that they are equidimensional. But in general this is a priori unknown. Hence the variety needs to be decomposed first. Using primary decomposition at this point is expensive and often leads to a rather high number of components whose singular loci and intersections need to be computed. Moreover, it is unnecessary, because we only need to satisfy the condition that each of the parts is equidimensional; a task which is performed by the algorithm of equidimensional decomposition.

262

Background A detailed discussion on equidimensional decomposition and its computation can be found e.g. in 21, section 4.4. Here we only recall the definitions: Definition 5. (Equidimensional Part) Let R be a noetherian ring, I c R an ideal and let I = Qi be an irredundant primary decomposition. The equidimensional part of I is the intersection of all primary ideals Qi for which dirn(R/Qi) = dirn(R/I);I is called equidimensionalif it coincides with its equidimensional part. Remark 6. (Equidimensional Decomposition) Iterating the process of determining the equidimensional part and removing it for ideals without embedded primes, we obtain a decomposition of I into equidimensional ideals. In the presence of embedded primes, however, we only obtain a decomposition into equidimensional ideals such that the intersection of their radicals is the radical of 1.

Computational Solution (general case, no embedded primes) Example 3 We now consider the union of the space curve of example 1 and the surface V ( x 3- y2) which possesses a non-isolated singularity along the z-axis. Here the computational subtasks which we consider for this example will be 0

0 0

0

defining the ideals of the components separately and intersecting them to obtain the ideal describing our variety computing an equidimensional decomposition of the ideal computing the singular locus by determining the singular loci and intersections of the equidimensional parts computing a primary decomposition of the singular locus

// polynomial ring: > ring r=O,(x,y,z),dp; // char 0, 3 var. // the previously computed ideal > ideal I1=y2-xz,x2y-z2,x3-yz; / / of the curve in example I / / the singular surface > ideal 12=x-3-yA2;

> ideal I=intersect(Il,I2); > I; I [ 13 =x3y2-x4z-y4+xy2z I [23=x5y-x2y3-x3z2+y2z2

/ / the union of the two varieties

263

I[3] = x ~ - x ~ Y ~ - x ~ Y z + Y ~ z

> LIB "primdec.lib"; > list li=equidim(l) ; > li; li [I] :

/ / equidim. decomp. is in // library 'primdec.lib' / / compute list of // equidim. parts of I

/ / part of dim. I

- [I1 =y2-xz - C2l=x2y-z2

- [31=X~-YZ li [2] : - [1]=~3-~2

/ / part of dim. 2

Using this equidirnensional decomposition, we can then compute the singular locus of each of the equidimensional parts by the Jacobian criterion. The union of these singular loci and of the intersection locus of the various parts is precisely the singular locus of the whole variety.

Example 4 (example 3 continued)

> LIB "sing.libtl; > ideal sLl=slocus(li [l] ) ; > ideal sL2=slocus(li [2] ) ; > sL2;

/ / 'sing.lib' contains slocus / / sing. locus of I-dim. part // i.e. point from example 1 / / sing. locus of 2-dim. part

sL2[11=~3-y2 sL2[21=-2y sL2[3]=3x2

> ideal interl2=li[I] +li [a] ; > interl2;

// intersection of I- and // 2-dim. parts

inter12[I] =y2-xz inter12[2]=x2y-z2 inter12[3] =x3-yz inter12[4]=x3-y2

/ / union of contributions

264

// to sing. locus > ideal sL=intersect(sLlYsL2,interl2); Now let us check whether we can identify the various contributions in the primary decomposition of the singular locus:

> primdecGTZ(sL); C11:

Cll : -Cll=y - [21 =x2 C2l : - C11 =y C2l =x

-

C2l : Cll :

- C 11 =z2

- C2l =ya-yz - C3l =xyz - [S]=x2z - [51=~3-y~

/ / first primary component: // singular locus of surface

// // // //

second primary component: singular locus of curve but also one of intersection points of the two parts

C2l :

- Cll =z

- [21=2y-z - C3l =x

C31:

Cll :

- [ll =z-1 - C2l =y-1

/ / third primary component: / / other intersection point // of the two parts

-[3]=~-1 C21 : [ 11 =z-1 - c21 =y-1 [31=x- 1

-

-

1.3. Finding the C o m c t Number of Components

In the previous examples, we have already used primary decomposition to determine the components of a given variety. This is, however, one of the

265

situations where the fact that we are calculating over the rationals, but arguing over the complex numbers can easily lead to wrong conclusions.

Computational Task Given a curve V(I) c A$, determine the number of branches.

Background There are several possible approaches to this task. We shall only consider two, the first one based on primary decomposition (see section l.l), the other one on normalization (see e.g. 21, sections 3.2 and 3.6 and second half of section 5.7 for algorithmic aspects, or ’, section 4.4 for a treatment from the point of view of singularities). For the second one, we briefly recall notation and some important statements here:

Definition 7. (Normalization) Let R be a reduced ring. The normalization of R is the integral closure of R in its total ring of fractions. Theorem 8. (Serre’s Normality Criterion) Let R be a reduced noetherian ring. Then R is normal i f and only if the following two conditions are satisfied:

( R l ) Rp is a regular local ring for every prime ideal P of height one. (S2) Let f E R be a non-zerodivisor, then rninAss((f ) ) = Ass( ( f ) ) . Remark 9. A regular local ring is normal. Remark 10. A normal one-dimensional variety is non-singular.

Computational Pitfalls and Solutions Example 5 Consider the variety V((x4 - yz2,xy - z3,y 2 - x3.z>)C @. It is a curve which has only one singular point at the origin. The task is to compute the number of branches of this space curve.

> ring r=O, (x,y,z) ,dp; > ideal I=x4-yz2,xy-z3,y2-~3z; > primdecGTZ (I) ; c11 : c11 :

- [11 =z8+yz6+y2~4+y3~2+y4

/ / the ideal of the curve / / primary decomposition

266

- [21=xz5+~6+yz4+y2~2+y3 - [3l=-z3+xy - c41 =x2z2+xz3+xyz+yz2+y2

- [5l = x ~ + x ~ z + x z ~ + x ~ + ~ z c21 :

- [I1 =z8+yz6+y2~4+y3~2+y4 - [2] =xz5+~6+yz4+y2~2+y3 - [31 =-Z3+Xy

- c41 =x2z2+xz3+xyz+yz2+y2 - C5I =~3+~2z+xz2+xy+yz 121 : [I1 :

- [I] =-z2+y - [21=x-z

C2l :

- [ 11 =-z2+y - [a] =x-2 This result seems to imply that the number of branches could be 2. To check the plausibility of this conclusion, we consider the Milnor numberf of the singularity at the origin, which turns out to be 12. But by the formula p = 26 - T 1 (see *), an even number of branches T would imply an odd Milnor number. Therefore, the conclusion that there are two branches is not plausible. The reason for this strange 'result' is that we are calculating over the rationals, but all arguments and considerations are performed over the complex numbers. In particular, the first primary component actually consists of 4 components, as we see by considering the norma1ization.g

+

// normalization is in

> LIB "normal.lib" ; // the library 'normal .lib' > list li=normal(I); // compute normalization // 'normal' created a list of 1 ring(s). Obviously, this singularity is neither a hypersurface nor an ICIS. Therefore its Milnor number cannot be computed directly by means of the tools discussed below in section 2.3. But the singularity is a quasihomogeneous curve singularity of Cohen-Macaulay type 2 and the tools of section 2.3 allow us t o determine the Tjurina number which is 13 here. Hence we can compute the Milnor number by the formula ,u = T - t 1 for quasihomogeneous curve singularities, where t is the Cohen-Macaulay type (cf. 2o for details on this formula). gRecall that for curves the normalization coincides with a parametrization.

+

267

// To see the rings, type (if the name of your list is nor): show( nor);

/ / To access the I-st ring and map (similar for the others), type: def R = nor111 ; setring R; norid; normap;

/ / R/norid is the 1-st ring of the normalization and // normap the map from the original basering to R/norid

// how many branches (over Q)

size(li) ;

def norring=li[ 11 ; setring norring; basering ; // characteristic : 0 / / number of vars : 2 // block 1 : ordering // : names // : weights // block 2 : ordering // : names // block 3 : ordering > norid; norid [11=T (2) -5-1

/ / consider branch more closely

a T(1) T ( 2 ) 1 0 dp T(1) T(2)

C

At first glance, it might seem strange that the ring describing the normalization has two variables, although it describes a 1-dimensional object. But looking at norid,we see that the second variable is only used to specify an appropriate field extension over the rationals such that all branches are separated. In particular, we see that we have 5 branches. Other possibilities to determine the correct number of branches, are the use of a generic projection to the plane followed by a Puiseux expansion (see 2.4 below) or the use of absolute primary decomposition. A detailed example of the first of these alternative approaches, however, is beyond the scope of this article; an application of absolute primary decomposition (in a different context) can be found in section 4.2.5.

268

2. Computing Invariants of Isolated Singularities

After briefly discussing some aspects of algorithmic calculations in local rings, this section contains a few examples of invariants which can be computed in practice.h To each example, we also mention where further information on the algorithmic aspects can be found. This list of examples is by no means exhaustive, it is intended as a kind of appetizer for the reader to start discovering what is available as algorithmic tools for their field of research; for simplicity of the presentation, we only discuss examples whose implementation is also available in SINGULAR and not even half of in this area is mentioned. the functionality of SINGULAR 2.1. Local and Global Considemtions

Up to this point, we have only studied varieties, but not germs. As a consequence all computations have been performed in polynomial rings, not in power series rings.' Actually, a full implementation of power series rings on a computer is not feasible, but nevertheless many practical tasks can be tackled by using the localization of the respective polynomial ring at the origin instead. (For obvious reasons, the input and output still need to be specified in terms of polynomial data.) To understand the basic idea behind the implementation of this type of localizations of polynomial rings, we first consider the representation of polynomials on the computer. The need to represent polynomials on the computer in a unique way forces us to use a total ordering on the set of all monomials; this ordering has to be compatible with multiplication of monomials. If the monomial 1 is the smallest monomial, the monomial ordering is called global and the ring is a polynomial ring; if 1 is the largest monomial, all elements whose largest term is a constant are considered as units; the ring is therefore a localization of the polynomial ring at the origin and the monomial ordering is called local. Orderings, in which some, hReaders who are not familiar with the theoretical background of some of the examples in subsections 2.3 t o 2.6 may safely skip the respective subsection. 'In SINGULAR,the functionality of primary decomposition, radical and normalization is only available in polynomial rings, not in localizations thereof. For a primary decomposition in a localization of a polynomial ring it is, however, possible to compute the primary decomposition in the polynomial ring and t o subsequently drop those c o m p e nents which are irrelevant in the localization. But this approach might be misleading as the following example shows: Consider the plane curve defined by y2 - z2 z3= 0; a primary decomposition in the polynomial ring provides just one component, although the germ of this curve at the origin consists of two branches.

+

269

but not all monomials are smaller than 1, are also possible and are usually referred to as mixed orderings. A detailed discussion of the influence of the choice of ordering on the ring is beyond the scope of this set of two talks and we refer the participants to a suitable textbook, e.g. 21, section 1.5, or 8 , sections 3.2 and 9.1.

Computational Task Given an ideal I c K[xl,. .. ,x,], compare results of Grobner basis computations w.r.t local and global orderings, i.e. in the rings K[z1,* * * '%](el, ...,z,) and in K [ z l , .. .,zn].

Computational Example

Example 6 In this example, we show some very simple calculations to illustrate the contrast between the local and global monomial orderings. The first of these small tasks is the calculation of a Grobner basis resp. standard basis for the ideal of the variety consisting of the plane V ( z 1) and the two lines V(z,g) and V(z - 1,y - 1) in @:

+

> ring rl=O,(x,y,z) ,ds;

/ / polynomial ring in 3 var.: // global ordering dp // localization at origin: // local ordering ds

> > > >

// // // //

> ring rg=O,(x,y,z) ,dp;

setring rg; ideal Il=z+l; ideal 12=x,y; ideal 13=x-I,y-l;

> ideal Itemp=intersect(I1,12); > ideal I=intersect(Itemp,I3); > I; I[13=-xz+yz-x+y Ic23=xyz+xy-yz-y

go back to polynomial ring the plane V(z+l) first line second line

/ / union of the first two / / union of all three

/ / remark: I is radical by // construction > ideal J=groebner(I) ;

/ / compute Groebner basis

270

> J;

- c11 =xz-yz+x-y - [21=yaz+ya-yz-y > setring rl; > def I=imap(rg,I); > I; I[ 11 =-x+y-xz+yz I [2]=-y+xy-yz+xyz

/ / now go to localization / / map ideal via identity map

/ / observe the different way / / of writing I[Il

> ideal J=groebner(I) ; > J;

/ / compute standard basis

-[lI=x - c21 =y

// we only see the components // meeting the origin

Continuing with the same example, we now compute the dimensions and check whether the variety/germ is contained in the plane V(z):

> setring rg;

> dim(J); 2

> setring rl; > dim(J); 1

/ / back to polynomial ring / / and compute dimension / / (remark: 'dim' needs // Groebner/standard basis) / / dimension of the plane / / back to localization / / applying 'dim' at 0 / / dimension of components / / meeting 0

> setring rg;

X

// ideal membership test: // xinJ? / / (remark: 'reduce' needs // Groebner/standard basis // of J) // answer: no

>setring rl; > reduce(x, J) ; 0

// same question locally // answer: yes

> reduce(x,J) ;

271

2.2. Dimension and Multiplicity

As we already used the dimension of a variety or a germ in previous examples, this seems to be a good moment to look at its calculation and at related data. The notion of dimension itself can be phrased in several ways (e.g. for a local ring (R, m): maximal length of chains of prime ideals, minimal number of generators of an m-primary ideal in a local ring (R, m),etc.), but most accessible to the use in practical calculations is the definition by means of the degree of the Hilbert-Samuel polynomial.

Computational Task Given an ideal I c K [ z l , .. . ,x,], compute the dimension and multiplicity of V ( I )at the origin.

Background

A detailed discussion on the Hilbert-Samuel polynomial and its calculation can be found in 21, sections 5.4 and 5.5. Here we only recall its definition to fix notation.

Definition 11. (Hilbert-Samuel Polynomial) Let (R, m) be a noetherian local ring where m is generated by T elements, and assume for simplicity that K = R/m. The Hilbert-Samuel function of R (w.r.t. the filtration {mk}kGN) is defined as X ( k ) := dirnK(R/m'). There exists a polynomial f ( t ) E Q t ] ,the Hilbert-Samuel polynomial of R, of degree at most r such that X(k) = f ( k ) for all sufficiently large k E N. Writing f ( t ) as E!=oaiti, the degree d of f coincides with the dimension of (R, m) and the multiplicity of (R, m) is defined as d! a d .

-

Remark 12. It is possible to explicitly compute a Hilbert-Samuel polynomial of a given ideal with polynomial generators in the localization of a polynomial ring at the origin. The general idea of this calculation is to find a suitable system of generators (i.e. a standard basis of the ideal w.r.t. a local degree ordering), then pass to the ideal generated by the largest terms of the generators (the so-called leading ideal) and compute the desired C vector space dimensions for this new (monomial) ideal in a combinatorial way.

272

Computational Example In SINGULAR, the dimension and multiplicity are directly accessible as kernel commands dim and mult which require a standard basis w.r.t. a local degree ordering as an input> Example 7 To illustrate the use of these commands, we now consider a space curve singularity at the origin consisting of an Ee singularity in the x-y plane and the z-axis.

> ring r=O,(x,y,z),ds; > ideal I=xz,yz,x3-y4; > I=groebner(I) ; > I; I[ll =xz Ic21=yz IC3]=~3-y4

> lead(1) ; - [11 =xz

// a local degree ordering // a space curve singularity / / compute standard basis

// ideal generated by largest / / monomials of the generators // of 1

- c21=yz - [3] =x3 > dim(1) ;

/ / compute dimension

1

> mult (I) ; 4 > dim(lead(1)) ,mult(lead(I)) ; / / ** - is no standardbasis / / ** - is no standardbasis

// multiplicity // should give the same values // ** automatic warnings can be // ** ignored if ideal monomial

1 4

jNote that the command h i l b does not compute the Hilbert-Samuel polynomial. It computes the Hilbert-Poincarb series of the homogeneous ideal generated by the initial terms of the given generators of the ideal. From these it is, of course, possible to compute both the Hilbert polynomial and a Hilbert-Samuel polynomial of the given ideal, the generators formed a standard basis w.r.t. t o an appropriate monomial ordering as is explained e.g. in 21, sections 5.2 and 5.5 respectively.

273

2.3. Milnor and Tjurina Number

Computational Task Given a polynomial f E C[x1,.. . ,x,] such that all singularities of V (f) are isolated, determine the Milnor and Tjurina numbers at all critical (resp. singular) points. In particular, determine these invariants at the origin.

Background For a discussion of the Milnor and Tjurina numbers of hypersurface singularities see any book on singularity theory, e.g. 7 , section 3.4.

Definition 13. Let f E @{a}define a germ of an isolated hypersurface singularity at the origin. The Milnor number of the germ ( V ( f ) 0) , is defined as the C-vector space dimension

Its Tjurina number is defined as the C-vector space dimension

C{z}/(f, -, af . ..,3)). ax, ax, Computational Solution Example 8 In this example, we compute the Milnor and Tjurina number at the origin for the plane curve consisting of two cusps V ( x 2 - g 3 ) and

V(x3 - 9 2 ) ) .

> r i n g r=O, (x,y), d s ; > i d e a l I=(x^~-Y‘~)*(x-~-Y-~) ; > i d e a l Jac=j acob (I) ; > groebner (Jac) ; - [13=2x2y-5y4 - [2] =2xy2-5x4

// // // //

l o c a l r i n g i n 2 variables t h e curve jacobian i d e a l of I compute standard b a s i s

- [31=~5-y5 - C41 =y6

> vdim(groebner(Jac)); 11

/ / t h e command ’vdim’ needs // a standard b a s i s a s input // t h e Milnor number

274

> vdim(groebner(Jac+I))

;

// the Tjurina number

10

Alternatively, we can also use the corresponding predefined commands in the library 'sing.lib':

> LIB "sing.lib"; > milnor(1) ; 11 > tjurina(1); 10

/ / the Milnor number // the Tjurina number

But what would have happened, if we had specified a global ordering instead of the local ordering?

> ring r2=0,(x,y) ,dp; > ideal I=(x-2-y-3) * (x-3-y-2) ; > ideal Jac=jacob (I) ; > groebner (Jac) ; - c13=3x2y3-5x4+2xy2

// // // //

global ordering the curve jacobian ideal of I compute Groebner basis

- C21=3x3y2-5y4+2x2y - r31=x5-y5 - c41=9y7-19x4y+lOxy3 > vdim(groebner(Jac)); 21 > vdim(groebner (Jac+I)) ; 15

The numbers, which we computed here, are precisely the sums over the Milnor resp. Tjurina numbers of all singularities of the affine curve. Therefore we expect to find further critical points outside the origin whose multiplicities add up to 10 and further singular points whose Tjurina numbers add up to 5. To check this, we determine the singular locus, move to each of the other singular points and compute Milnor and Tjurina numbers there. Subsequently, we also study the critical locus, which, of course, contains the singular locus.

> LIB "primdec.lib"; > minAssGTZ(slocus(I)) ; c13 :

- Cll =y - c23=x

/ / components of sing. locus / / the origin -- we already // considered this one

275

/ / the point (1,l)

C2l :

- C11 =y-I - c21=x-I 131 :

- [I1 =y4+y3+y2+y+i

/ / a set of 4 points // <--- keep this in mind (*)

- 121=y3+y2+x+y+l

> setring r; > map ml=r2,x+l,y+l; > def 12=ml(I); > 12;

/ / go back to local ring / / translation of (1,i) // to origin / / apply translation to curve

12[1]~6~2-13~~+6~2+9~3-11~2~-11~~2+9~3+5~4~3~3~+x5-3x3y2-3x2y3+y5-x3y3 / / Milnor number > milnor (12) ;

I > tjurina(I2)

;

/ / Tjurina number

I

> ring r2a=(O,a), (x,y>,ds; > minpoly=a4+a3+a2+a+i;

> map m2=r2,x-a3-a2-a-i,y+a; > def I3=m2(I) ; > milnor (12) ;

// // // // //

extend basefield to look at the 4 points adjoining parameter a minimal polynomial, see (*) above

/ / translation of one of the // 4 points to origin / / apply translation to curve // Milnor number

1

> tjurina(I2)

;

/ / Tjurina number

1

> setring r2; > minAssGTZ(jacob(1)); C11 :

- Cil =y

- C2l=x

/ / go back to r2 (global) / / decompose set of // crit. points / / origin // already considered

276

C2l:

- C 11 =y- 1

- C2l =x- I

C3l:

- C11 =y4+y3+y2+y+i - c21 =y3+y2+x+y+l C41 :

- [I] =81~4+54y3+36~2+24~+16

// (1,l) // already considered / / 4 points // already considered / / 4 c r i t i c a l points

- [2] =27y3+18y2+12~+12y+8 C5l :

/ / 1 c r i t i c a l point

- [11 =3y-2

- [2]=3~-2 Actually, it would not have been necessary to move to each of the points and check the Milnor and Tjurina numbers explicitly, because we only had a difference of 10 for the Milnor and of 5 for the Tjurina number and this equals the number of additional points in the critical resp. singular locus. The Milnor and Tjurina numbers for isolated complete intersection singularities are available by the same command (in the case of the Milnor number by use of the LBGreuel formula, see 30). The Tjurina number for Cohen-Macaulay codimension 2 singularities, which are not ICIS, is provided in the library 'spcurve.lib'; in the general case it can be obtained via the command T1, see 3.1 below. For further details on these invariants see any textbook on singularities, e.g. 40 in the curve case, for hypersurfaces or 31 in the case of complete intersections. 2.4. Pzliseua: Ezpansion

Computational Task Given an element 0 # f E C{x,y} (which is assumed to be a polynomial for practical reasons) defining an isolated singularity at 0, determine the number of branches of the corresponding germ of the plane curve at the origin. Additionally compute the &invariants, the degree of the conductors for the branches and the intersection multiplicities of the branches.

Background Here we do not state the rather technical definition of the HamburgerNoether expansion of which a detailed discussion can be found in 5 , chapter

1. Instead, we focus on recalling the definitions of the invariants which we want to determine. A compact and accessible discussion of these topics can be found in ', chapter 5 .

Definition 14. (Puiseux Expansion) Let 0 # f E cC{z,y} be irreducible. Then there exist power series z ( t ) , y ( t ) E @{t}such that

f(z(t),y(t)) = 0 dimc(@{t}/@{j:(t),Y(t)})

< 00

Definition 15. (&Invariant, Conductor) Let (R,m) be the local ring of an irreducible plane curve singularity; R is a subring of its normalization R = @{t}. 0

0

0

The &invariant is defined as 6(R):= dimc(fi/R). The semigroup of values of R is defined as r ( R ) := {ordt(a)l a E R , a # 0). The conductor is defined as c(R) := min{b E r(R)I b + N C F(R)}.

Lemma 16. AnnR(ii/R) = tc(R)fi Lemma 17. S(R) = #(N

\ I?(R))

Definition 18. (Intersection Multiplicity) Let 0 # f,g E cC{z,y} define two germs of plane curve singularities (at 0) not having a common component. The intersection multiplicity of the two germs is defined as dimc(@{z,y } l ( f , 9 ) ) . Computational Example

Example 9 (example 8 continued) Continuing where we stopped in our calculations in the previous example, we now apply Hamburger-Noether expansion and extract information about the given plane curve from it. Recall that this curve consisted of two branches V(z2 - y3) and V(y2 -z3).

> LIB "hnoether.lib" ;

/ / load corresponding library / / hnexpansion needs argument

> poly f=I c11;

//

> hnexpansion (f) ;

/ / call Hamburger-Noether // expansion // result lives in a new ring

111: //

characteristic : 0

of type POlY

278

// // // //

number of vars : 2 block 1 : ordering 1s : names x y block 2 : ordering C > def S=-ClI ; / / give that ring the name S > setring S; / / and change to it

> hne;

/ / result can be found in hne / / technical data, not // really readable

Cll : c11 :

- C1,lI =o - C 1 21 =x - Cl 31=0 - C2 ,11 =o - C2 21=1 Y Y

Y

- C2,31 =x

C21 : 192 C31 : 0

C41 : 0 C21 : c11:

- c1,ll=O

- Cl

Y

21 =x

- C1,31 =O - C2 ,11 =o - C2 ,21 =l - C2 ,31=x C21 : 1Y2 C31 : 1 C41 : 0

> displayHNE(hne) ;

/ / a more readable way to // look at it ;-)

/ / Hamburger-Noether development of branch nr.1:

279

/ / Hamburger-Noether development of branch nr.2: HNE [ I]=-x+z (0)*z( I) HNE[2]=-y+z(l)^2 // Caution! / / numbering of branches may / / change when calling // hnexpansion a 2nd time on / / the same input Usually, we are not interested in the Hamburger-Noether or Puiseux expansion itself, but rather in invariants which can easily be extracted from it. Therefore these invariants are provided by a separate post-processing command:

> displayInvariants(hne);

---

invariants of branch number 1 : characteristic exponents : 2,3 generators of semigroup : 2,3 Puiseux pairs : (3,2) degree of the conductor : 2 delta invariant : I sequence of multiplicities: 2,1,1

---

--- invariants of branch number 2

---

:

characteristic exponents : 2,3 generators of semigroup : 2,3 Puiseux pairs : (3,2) degree of the conductor : 2 delta invariant : 1 sequence of multiplicities: 2,1,1

-------------- contact numbers

:

--------------

280

-------------branch I

intersection multiplicities :

--------------

2

-------+-----

1 1

4

_______-----__ delta invariant of the curve

:

6

2.5. Classification of Hgpersurface Singularities

Sometimes, we also want to check whether a given singularity is in Arnold's list of hypersurface singularities This test is implemented in SINGULAR as well:

'.

Example 10 Still continuing with the singularity which we have been considering in the previous examples, we now use the Arnold-classifier to determine its type:

> LIB "classify.lib"; > setring r ;

// // // // // //

classifier library needs local ring input needs to be > poly f=I[11; of type 'poly' first guess via > quickclass(f) ; invariants Z [k, 12k+6r-l] =Z [l ,111 or Singularity R-equivalent to : Y [k,r ,s]=Y [l ,1 ,11 Hilbert-Code of Jf-2 We have 2 cases to test null form

Cll : Z[k,12k+6r-l]=Z [l, 113 Y [k ,I ,s]=Y [l,1 ,13 121 : 2 / / following Arnold's > classify(f); // algorithm About the singularity : Milnor number(f) Corank(f) Determinacy

= 11

= 2 <= 8

281

Guessing type via Milnorcode:

Z[k, 12k+6r-l]=Z[lyll] or Y [k,r s] =Y [l,1,1]

Computing normal form ... Arnold step number 16 The singularity -x2y2+x5+y5-x3y3 is R-equivalent to Y[l,p,q] = T[2,4+p,4+q] Milnor number = 11 modality = I

2.6. Monodmmy and Spectral Numbers

Computational Task Given an element 0 # f E C(z1,. . . ,zn} (which is assumed to be a polynomial for practical reasons) defining an isolated singularity at the origin, determine the Jordan normal form of the monodromy of f and the spectral numbers of f .

Background

A very brief introduction to these topics an be found in lo, which also provides many references for further reading on various aspects of the topick, a more detailed discussion of the definitions and basic statements can be found in the textbook 9 , chapter 5. A description of algorithmic and computational methods in this area can be found in 34. Remark 19. (General Situation) Let f : (Cn+l,0) + (GO) be a germ of an analytic function defining an isolated hypersurface singularity at the origin. Passing to representatives, let BE c (C"+l be a ball of sufficiently small radius E > 0 around the origin in (C"+l such that it intersects f-l(O) transversally. Let 0 6 c C be a closed disc of sufficiently small radius 6 > 0 around the origin in C such that f-l(s) intersects BEtransversally for all s E Da. Now set

X,

:= f-l(s)

nBc for all

this volume there are articles by Ebeling from these areas.

l1

sE~ a .

and Steenbrink 39 discussing material

282

Milnor (see 33) showed that in this situation

f -l(Da \ ( 0 ) ) -+ Da \ ( 0 ) is a locally trivial differentiable fiber bundle and that the fibers x8 have the homotopy type of a bouquet of p ( f ) n-spheres. Hence for s E 0 6 \ { 0 } , Ho(X,,Z) = Z and H n ( X 8 ,74)= Zn are the only non-zero homology groups. Parallel translation along the path

y : [o, 11 + DJ t yields a diffeomorphism h : Xa

-+

I-+

6 - e2?rit

Xa

Definition 20. (Monodromy) The induced morphism h, : Hn(Xa,C) -+ Hn(Xa,C) is called the complex monodromy of the singularity. Theorem 21. (Monodromy Theorem) The eigenvalues of h, are roots of unity. The size of the blocks in the Jordan normal form of h, is at most (n 1) x (n 1).

+

+

Definition 22. (Hodge Filtration and Spectrum) Steenbrink proved in 37 that there is a mixed Hodge structure on the Milnor fiber consisting of an increasing weight filtration

o=w-1 cwo C - . * C w 2 n = H n ( X 6 , z ) @ Q on H n ( X a ,Z)€3Q and a decreasing Hodge filtration

H " ( X g , Z ) @ C = FO 3 F 1 3 * - . 3 F" 3 F"+l

=o.

0 5 p 5 n. A rational number (Y E Q To define the spectrum of f , let p E Z, with n - p - 1< Q! 5 n - p is in the spectrum of f if and only if e2?riais an eigenvalue of the semisimple part of h* on FPH"(Xa,Z ) / F H ' H " ( X a , Z). Example 11 Let us consider the example of the isolated hypersurface singularity defined by the polynomial f = x5 -t y5 x2y2. We first want to compute a matrix M such that e-2?riMis the monodromy matrix of the given f :

+

> LIB "gmssing.lib"; > ring r=O, (x,y>,ds; > poly f=x5+y5+x2y2; > monodromy (f) ;

// // // // //

monodromy, spectrum etc. as usual: first a 'ring', then the polynomial compute data of the monodromy:

283

/ / eigenvalues of M

// sizes of blocks

/ / multiplicities Therefore, the Jordan normal form of M has the following structure:

($10 0 0 0 0 0 0 4 0 0 0 0 0 0 oo&o 0 000 ooo&o 000 0 0 0 o & o o o 0 0 0 0 o & o o 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 ~ 00

oogo ogo

0 0 %

The same library also provides support for calculation of the spectral numbers of f using standard basis methods for the microlocal structure of the Brieskorn lattice. For details on this algorithmic approach and on more sophisticated data which can also be acquired along these lines see 34.

> spectrum(f) ;

111 :

// compute the spectrum / / spectral numbers

- Ci1=-1/2 - [21=-3/10

- C31=-1/10 - C41=O - C51=1/10 - C61=3/10

- C71=1/2 c21 :

/ / their multiplicities

284

3. Deformations of Singularities

After using computational methods for studying the singular locus of a variety and for determining invariants of isolated singularities, we now turn our interest to families of singularities. More precisely, we first consider the computation of T1 and T2of an isolated singularity and the construction of versal families and then proceed to a more detailed study of certain special families of singularities. 3.1. T1 a n d T 2

Computational Task Let (X,O) be a germ of an isolated singularity. Compute the vector space of first order deformations and the Ti,o. Then proceed to determining a versal deformation of the given singularity up to a given degree.

Background

A very accessible introduction to the definitions and properties of T1, T 2 and versal deformations can e.g. be found in the textbook 10.3.

',

sections 10.2-

(First Order Embedded Deformations) Let (X,O) be a germ space, defined by a n ideal I = ( f l , . . . ,fr) C C { q , . . . , z n } , be further elements of C{zl,. . .,xn} and denote by (T,0 ) the f a t point corresponding t o the ring C { E } / ( E ~ )T.h e n fl €91,. . . ,fr Egr defines a flat deformation (X,O)+ (T,0 ) i f and only i f

Lemma 23. of a complez let g l , . . . ,gr the germ of

+

+

fi * 9 1 , - . - , f r * ~ r

yields a well-defined element of Homq,, ,...,z,,}(I,C{xl,. - ., z n } / I ) . Definition 24. (Normal Module) The normal module of the singularity ( X ,0) is the module Homq,, ,...,.,,}(I, C{z1,. . . ,x n } / I ) .

285

Definition 25. (T1) Let 6 denote the free module of C { x 1 , ...,xn}derivations. Then the cokernel of the map :e

6 is denoted by

-

+N (f I+

~

,

~

W))

Lemma 26. (First Order Deformations) The isomorphism classes of deformations of (X,O) over (T,O) are in one-to-one correspondence to the elements of Lemma 27. (First Order Deformations of an Isolated Singularity) For a of an isolated singularity, the Ti,o is a finite dimensional C germ (X,O) vector space.

As shorthand notation, we now use 0 for @ { X I , . . .,x,} and we denote the module of syzygies of I by R and the submodule of Koszul relations by '&; that is we have an exact sequence O+R+

0 k

+0+0/I-+O

and the submodule '& is generated by the relations i ,j 5 r.

fj

- ei - fi

ej, 1

5

Definition 28. (Obstructions) The module T$,o is defined as

Ti,O := Hmo(R/Ro,O x , o ) / H m o ( O k0,x , o ). Lemma 29. (Obstructions of an Isolated Singularity) Let ( X ,0 ) be a germ of an isolated singularity, then T$,, is a finite dimensional @ vector space. Theorem 30. (Obstructions and Lifting of Deformations) Consider an exact sequence

O-+J+B+A+O where (A,mA) and (B,mB) are local Artinian @-algebras and m B J = 0. Let ( T A , ~and ) ( T B , ~denote ) the complex space g e m s corresponding to A and B respectively and let ( X A , ~ + ) ( T A , ~be) a frat deformation of (X,O) (denoted by for short). There exists a well-defined obstruction element

T$,O 8 J. This obstruction element is zero if and only i f there exists a frat deformation ( X B ,0 ) + ( T B ,extending ~) 6. o ~ ( ~ ) B + AE

286

This general approach to computing T 1 and T 2 can be rather time consuming; therefore additional information on special cases which allows a more direct calculation should be used whenever available.

Lemma 31. (TI and T 2 of an ICI,!?) Let ( X , O ) be an ICIS. Then all relations are generated b y the Koszul relations, i.e. R = % and hence T$,o = 0. The normal module is ( O / I ) r and thus = (U/I)r/Im(a). Computational Solutions: Example 12 As a complete intersection example, let us consider the isolated space curve singularity defined by the ideal I = (x2 y2 z 3 ,yz).

+ +

> ring r=O,(x,y,z) ,ds; > ideal I=x2+y2+z3,yz; > def N=I*freemodule(2); > def T=jacob(I)+N; > vdim(std(T)) 6

;

> kbase(std (T)) ; - [I] =z2*gen( I)

/ / the singularity / / presentation of the // normal module / / presentation of T-I // the Tjurina number // base of vector space // of 1st order deform.

- [2]=z*gen (I) - C3l=gen(l) - 141=z*gen(2)

- C5l =x*gen(2) - C6l=gen (2)

This is, of course also available as a SINGULAR command:

> LIB "sing.lib" ; > Tjurina(1) ;

/ / command is in 'sing.lib' / / compute TI, ICIS case

// Tjurina number = 6 -[I1 =x*gen(l) 'In the case of Cohen-Macaulay codimension 2 singularities, there is a direct method for computing these data, too. Whenever there is such a direct approach, it tends t o be much more efficient than the general one and hence should be preferred.

287

- [2] =y*gen (2)+3z2*gen ( 1) - [3] =2y*gen( 1) +z*gen(2)

- [4]=~2*gen(2)+~2*gen(2)+~3*gen(2) - [5]=xz*gen(2) - [S]=z2*gen(2) - C71 =z3*gen(l) > kbase(std(-)); - [11 =z2*gen( 1)

// base of 1st order // miniversal deform.

-[2]=z*gen(i)

- C3l =genf 1) - [4]=z*gen (2) - [51 =x*gen(2) - [Sl =gen(2) In the general case, however, we cannot avoid computing T 1and T 2 as described at the the beginning of this section. To illustrate this, we consider the isolated singularity at the origin of the cone over the rational normal curve of degree 4:

Example 13 As an example in the general case, let us consider the isolated singularity defined by the 2-minors of the matrix

and compute its T 1and T 2 using the appropriate built-in commands of SINGULAR.

> > > > >

/ / Tl, T2 are in 'sing.lib' ring rl=O,(x,y,z,u,v),ds; / / local ring in 5 var. matrix M[2l C41 = x,y,z,u,y,z,u,v; / / the matrix, see above ideal I=minor (M, 2) ; // the ideal LIB "sing.lib'I;

I; I [11 =-u2+zv I [2] =-zu+yv I [S]=-YU+XV I [4]=Z~-YU I [5]=YZ-XU I [S] =-Y~+xz

288

> T-i2(1); / / dim T-i = 4 / / dim T-2 = 3

// compute Ti and T2

Ci1:

// standard basis for T-I

- [I] =gen(8)+2*gen(4) -C21=gen(7)

- C3l =gen (6)+gen (2)

- C41 =gen(5)+gen(i) - C5l =gen(3) - C6l =x*gen(9)

,[7] =2x*gen(4)+z*gen(2)

- [8]=x*gen(2)

- 191=x*gen( 1) +y*gen (2) - [lo] =y*gen (9)+z*gen( 2)

- Ciil=2y*gen(4)-z*gen(i)+u*gen(2) - C12l =y*gen(2) - CIS] =y*gen( i) +z*gen( 2) - [14]=z*gen(S)+u*gen(2)

-[15]=2z*gen(4)-u*gen(9)-u*gen(i)

- [16]=z*gen(2) - [17]=3z*gen( i)-u*gen(2) - [18]=u*gen(9)+3u*gen(i) - Cl91=2u*gen(4)-v*gen(9)-v*gen(i) - C201=u*gen(2) - C2 11 =2u*gen (i) -v*gen ( 2

- [22]=v*gen(9)+v*gen(i)

)

- 1231=v*gen(4) - 1241=v*gen(2) - C251 =v*gen (1)

C21 : - Cll=gen(9) - C21 =gen(7) +gen(5) - C31 =gen (6) C4l =gen(3) C5l =gen(2) CSl =gen(1 1 - C71=x*gen(8) C81 =x*gen(5)

-

// standard basis for T2

289

- [9]=x*gen(4)

- [lO]=y*gen(8)-z*gen(5)-u*gen(4) - I:113=y*gen(5)+z*gen(4) - C12l=y*gen(4) - C13l =z*gen(8)

- [141=z*gen(5)+u*gen( 4) - [is] =z*gen(4) -[is] =u*gen(8) - [171=u*gen(5)+v*gen(4) - 1183=u*gen(4) - C191 =v*gen(8) - [20]=v*gen(5) - [21] =v*gen(4) > list li=-;

/ / basis of finite dim. // vector space T-1

// basis of finite dim. / / vector space T-2

These results for the bases of the vector spaces seem rather difficult to interpret at first glance. But as soon as we know the system of generators of the normal module (respectively of H o m o (R/%,OX,^)) with respect to which the results have been expressed, we have all the data we need. These additional pieces of information can be extracted from the same command by supplying an optional second parameter of arbitrary type. As this creates rather lengthy output (approx. 4 pages in our example), we only state the respective systems of generators: For the normal module the system of generators can be expressed in terms of the perturbations of I

290

defined by I

0

-U

0

-U

0

0

0

z

>

-Y

0

,

z

V

V

-2

Y

0

0

-Y 0

X

z

0 0 ,0.

z

U

,

0 0

0

-X

.

-U

0

," 2 where the lst, 2nd, 4th and 9th form a vector space basis of the T1according 0

2

-5

to the output of our previous computation, namely of kbase(li[l]).For the module Homo(R/%, Ox,o) the corresponding generators are

/O' 0

0'

X

0'

2

0

Y

X

0

0

0 0

0

U

V

0

V

Y

, Y0

Y

0

0

0 0 z 0

0

z

U

U

z.

>

-u.

-V

0' 0

'0'

z

U

0

0

0

U

0

,v,

z

,

-U

U

L

,

-V

0 0

0

,

Background (continued)m

Definition 32. (Versa1 Deformation) Let (Xs, 0) + (S,0) be a deformation of ( X ,0) over a base space (S,0). This deformation is called versal if for any deformation (Xs,, 0) + (S',0) of ( X ,0) over a base space (S',0 ) , there exists a map (S',O) + ($0) such that (Xst,O) is isomorphic to the pull-back of (XS, 0) over this map. Theorem 33. (Grauert) Let (X,O) be a germ of a complex space and suppose that d i m T i , o < 00. Then there exists an analytic semi-universal deformation of ( X ,0 ) . Being able to compute T1 and T 2 explicitly, the natural subsequent step is to ask whether we can also determine versal deformations up to a given degree in practice. The answer is affirmative and the corresponding mAs in the first part of this section, a good reference for a discussion of these topics is e.g. 7,sections 10.2-10.3

291

algorithm is implemented in the library deform .l i b ; a detailed description of the algorithm can be found in 32.

Computational Solutions (continued) Example 14 (example 13 continued)

> L I B "deform.lib"; > list L=versal ( I ,5) ;

/ / compute v e r s a l deformation // up t o degree 5

// ready: T-1 and T-2 // start computation i n degree 2. / / ' v e r s a l ' returned a l i s t , say L , of f o u r r i n g s . I n Lei] a r e s t o r e d : as matrix F s : Equations of t o t a l space of miniversal deform., // as matrix Js: Equations of miniversal base space, // / / as matrix R s : syzygies of F s mod J s . // To access t h e s e d a t a , type def Px=L[l] ; s e t r i n g Px; p r i n t (Fs) ; p r i n t (Js) ; p r i n t (Rs) ;

/ / r e s u l t i s l i s t of r i n g s

> L;

[I1 : // // // // // // //

characteristic : 0 number of v a r s : 9 block 1 : ordering : names block 2 : ordering : names block 3 : ordering

ds

A B C D ds x y z u v C

C2l :

// characteristic : 0 // number of v a r s : 9 // block I : ordering d s // : names A B C D // block 2 : ordering d s // : names x y z u v // block 3 : ordering C / / q u o t i e n t r i n g from i d e a l .. .

292

C3l: // characteristic : 0 / / number of v a r s : 4 // block 1 : ordering d s // : names A B C D // block 2 : ordering C C41:

//

// // // // // // //

characteristic : 0 number of v a r s : 9 block I : ordering d s : names A B C D block 2 : ordering d s : names x y z u v block 3 : ordering C q u o t i e n t r i n g from i d e a l ...

> def R l = L [ l ] ; > s e t r i n g R1;

// go t o 1st of r e t u r n e d r i n g s

> Js;

// equations of m i n i v e r s a l // base space

Js[l,l]=BD JS [ 1 21 =-AD+D2 JS [ 1,3] =-CD

> Fs;

/ / e q u a t i o n s of miniveral // t o t a l space

Fs [ 1 l] =-u~+zv+Bu+Dv F s [I 23 =-zu+yv-Au+Du FS [1,3]=-yu+xv+Cu+Dz FS [ I , 41=zZ-yu+A~+By FS [ 1,~]=Yz-xu+Bx-CZ F s [I 61=-y2+xz+Ax+Cy

3.2. Studying Families of Singularities

Having constructed versa1 families in the previous example, we now proceed to study the question of stratifying the base space of a certain classes of

293

families of singularities with respect to the Tjurina number. This question can be dealt with algorithmically for versal families of simple hypersurface and Cohen-Macaulay codimension 2 singularities and for families of semiquasihomogeneoussingularities (again hypersurfaces or CM codimension 2) with fixed initial part. In the f i s t case, it can be used as one ingredient to determining an adjacency to another singularity explicitly"; in the second case, it is one step in the construction of moduli spaces for semiquasihomogeneous singularities with fixed initial part (for more details on this topic see e.g. 14). We only consider the first situation here, as the latter one involves the use of a rather technical modification of the standard basis algorithm .

Computational Task Given a simple hypersurface singularity (X, 0), determine the stratification of the base space of its versal deformation by the Tjurina number.

Background The computational tools we are using here axe Fitting Ideals and Flattening Stratification. We only state the definitions, a detailed discussion can e.g. be found in 21, sections 7.2-7.3. From the point of view of singularity theory a treatment of the background of the topics discussed in this section can be found in 29.

Definition 34. (Fitting Ideal) Let R be a ring and M a R-module with presentation

R"

-%R" -+

M

+o

and denote by A a matrix of the map q5 w.r.t. some chosen bases of R" and R". For k E Z, define Fk ( M ) to be the ideal generated by the ( n - k)minors of the matrix A , the k-th Fitting idealo L e m m a 3 5 . (Fitting Ideals) Fitting ideals are independent of the choice of the presentation and of the choice of bases for Rm and R". firthermore,

they are compatible with base change. "Along these lines it was possible to complete the list of adjacencies for Giusti's list of simple ICIS, see 15. OFor the remaining values of k the following definitions are used: if k 2 n , then Fk ( M) := R; if n - k > min{n,m), then Fk(M) := 0.

294

Definition 36. (Flattening Stratification) For subset

T

2 0, the locally closed

FZatTM := { P c R prime ideal IP 3 FT-l(M) and P

3 FT(M)}

of Spec(R) is called the flattening stratum of rank T of M . The collection of these strata is called the flattening stratification.

Definition 37. (The Relative T 1 of the Versa1 Family) Let the power series f E C { x l , . .. ,x,} define an isolated hypersurface singularity ( X ,0), let 91,... ,gTO be generators for and let 9 denote a versal deformation of (X,O) specified by 70

C

F = f -t

sigi

E ~ ( $ 1 ,..., E n , 3 1 , . . ,sTo}.

i=l

Passing to sufficiently small representatives X of the total space and S of the base space of the deformation, we can consider a family of deformations with section, i.e. a morphism of complex spaces 9 : X + S and a section o : S + X such that : (X, o(s)) -+(S,s) is flat and defines a deformation for all s E S and such that (9-,(s),o(s)) is an isolated Singularity. The relative T 1of this family of deformations can be computed its

T i = C { X ~. .,. ,x,,

$1,.

dF dF . . ,S,}/(F, -, .. . -).

ax,

axn

Remark 38. Note that fixing a value for the parameters s1,. .. ,sTo,the C vector space dimension of the stalk of T,'at this points of S provides the Tjurina number of the respective isolatedhypersurface singularity. Computational Solution Example 15 To keep the calculations as simple as possible, we only consider a very small but well-known example, an A3-singu1arity.P We first compute a versal family by means of calculation of a vector space basis for the T' (Tjurina algebra) and the relative T1of this family: PAS these calculations involve ideals generated by minors of a matrix and as this matrix is, in general, not of a simple structure, the computations tend to become very lengthy and this approach should hence be considered as a brute force approach which should only be used in combination with the use of all additional information that could possibly lower the complexity.

295

> > > >

r i n g r=O,(x,y),ds; poly f=x-4+y-2; i d e a l kb=kbase(Tjurina(f)) ; kb; kb [I] =x2 kb [21 =x kb [3]=1

// the s i n g u l a r i t y // v e c t o r space b a s i s f o r T I

> r i n g r t = O , (a,b,c,x,y) ,ds; > poly F=x-4+y-2+a+b*x+c*x-2;

// move t o s u i t a b l e r i n g // for t o t a l space // versa1 family

> i d e a l j F = d i f f (F,x) , d i f f (F,y) , F ;

/ / p r e s e n t a t i o n of r e l . T I // // //

b u t as module over r t , w e need it as C [a ,b ,c] -module

> jF; j F [I] =b+2cx+4x3 j F [2] =2y j F [3] =a+bx+y2+cx2+x4

We know that j F is a finitely presentable K[a,b,c]module. As f is a hypersurface singularity, we can determine the corresponding presentation matrix by looking at the Euler relation and suitable products of it with monomials in x and y. (In this example only the products with x and x2 are relevant .) // s u i t a b l e r i n g f o r // finding Euler rel. ( Q [ a , b , c l > Cx,yl > r i n g rg=O, ( x , y , a , b , c ) ,(dp(2) , d p ) ; / / > def j F = i m a p ( r t , jF) ; > jF=mstd(jF) [21;

> jF;

/ / f e t c h j F from rt / / f i n d m i n i a l system of // g e n e r a t o r s f o r jF / / look a t j F

j F 111=y j F 121=4x3+2xc+b j F [31=2~2c+3xb+4a

// <-- E u l e r r e l a t i o n

> m a t r i x Tmat [3] [3] ; > def tempmat=coef (jF[31 ,xy);

// g i v e temporary name,

296

> > > > > >

// // //

because lists can only be formed from named objects

Tmat[l,l. .3]=tempmat[2,1. -31; tempmat=coef(reduce(jF[S]*x, jF[2]) ,xy); Tmat[2,1. .3]=tempmat[2,1. .31; tempmat=coef(reduce( jF [31 *x-2,jF [21) ,XY) ; Tmat[3,1. .3]=tempmat[2,1. .31; / / presentation matrix of Ti print (Tmat) ; // as Q [a ,b ,c] -module 2c, 3b, 4a, 3b , -c2+4a,-1/2bcY -c2+4a,-2bc, -3/4b2

The strata of constant Tjurina number can now be obtained by means of the flattening stratification of the relative TI.This implies that we need to determine the Fitting ideals of the module - that is we need to determine the minors of size one, two and three:

> ideal minl=mstd(minor(Tmat,l))[2]; > mini;

/ / minimal set of gen. / / for 1-minors

mini [13=c minl[2] =b mini [3] =a > ideal min2=mstd(minor(Tmat,2))[2]; / / dito for 2-minors > min2; min2 [11=2c3+9b2-8ac min2 [2] =bc2+12ab min2 [3] =3b2~-8ac2+32a2 > ideal min3=mstd(minor (Tmat,3)) [2] ; / / and f o r 3-minors > min3;

From this computation, we can see that the maximal value of the Tjurina number is attained exactly for the fiber over the point V(a,b, c) of the base. For fibers over points outside of V(4b2c3 - 16ac4 + 27b4 - 144ab2c 128a2c2- 256a3),which is the swallowtail singularity (6.figure l),on the other hand there are no singularities. The Tjurina number is 2 for points in V(min2) \ V(a, b, c) (cf. figure 2). It is 1for points in V(min3) \ V(min2), i.e. points on the swallowtail which do not lie on the curve V(min2).

+

297

Figure 1. The swallowtail singularity: V(min3).

Figure 2. The singular locus of the swallowtail singularity: V(min2).

4. Varieties with Singularities

In this last section, we consider two areas of more complex applications of computational methods in singularity theory: the task of finding hypersurfaces with prescribed singularities and the task of resolution of singularities. In the first case, the goal is more of theoretical nature and explicit calculations are basically used to check whether certain conditions are satisfied or for finding good examples which show certain properties. In the second case, the set-up is different: The task itself is computational, but it consists of many different computational aspects each of which needs to be treated carefully in order to obtain a usable implementation.

298

4.1. Hypersurfaces with Pwscribed Singularities

Here, we briefly sketch two applications of computer algebra tools in this area: first we treat the question of finding an upper bound for how many singularities of a given type can fit on a hypersurface of a given degree, then we outline how computational tools aided in the search for examples of surfaces of fixed degree with a high number of double points.

Computational Task Given a hypersurface of fixed degree and a singularity type, give an upper bound for the number of singularities of this type which can appear on such a hypersurface.

Background In this application we use another property of the singularity spectrum. Recall that the spectrum was already considered in section 2.6; the property which we are going to use now was proved by Steenbrink in 3a. Lemma 39. (Properties of the Spectrum) T h e spectrum of a n isolated hypersurface singularity is constant under p-constant deformations. The number of spectral numbers in a half open internal (a,a + 11 is uppersemicontinuous under small deformations of isolated hypersurface singularities; f o r semi-quasihomogeneous isolated hypersurface singularities the same property also holds f o r intervals (a,a + 1).

Computational Solution Example 16 The question, which we are treating in this example, is the following: What is the maximal number of singularities of type T3,3,3 that can occur on a surface of degree 7 in B3? Let us fist recall that the singularities of type T3,3,3 form a p-constant 1-parameter family given by equations of the kind

x3 + y3 + z3 + t .xyz = 0, where t3 # -27. To obtain the desired bound, we now use the semicontinuity property of the spectrum. More precisely, the number of spectral numbers of the singularities of a deformation of a given hypersurface in an interval (a,a 11 cannot exceed the number of spectral numbers of the original singularity

+

299

in this interval; for semiquasihomogeneoussingularities the same statement also holds for the intervals (a,a 1).

+

> L I B "gmssing.lib"; > ring R=O,(x,y,z),ds; > poly f=x^3+y^3+z"3; > list sl=spectrum(f); > sl; C11 : - Cil =o - C21=1/3 - C31=2/3 - 141=1

// // // //

C21 : 1,3,3,1

// multiplicities

> poly g = x-7+y-7+z-7; > list s2 = spectrum(g);

spectrum related commands local ring in 3 variables a singularity of type T-333 compute its spectrum

/ / spectral numbers

// any surface of degree 7 is // deformation of this surface // compute its spectrum // * takes some time!!

> s2;

Cil : - c11=-4/7 - [21=-3/7 - [31=-2/7

// spectral numbers

- [4l=-1/7 - C5l=O - C61=1/7

- C71=2/7 - C81=3/7 - C91=4/7

- CiOI=5/7 - [ll] =6/7 - C121=l - C131=8/7

- 1141=9/7 - C151=10/7 - 1161=11/7 C2l : // multiplicities 1,3,6,10,15,21,25,27,27,25,2i,15yi0y6,3,1

300

> spsemicont(s2,list(sl));

// checking semicont.condition

c11 : 18

> spsemicont(s2,~ist(si) , I ) ; c11:

// checking sqh.semicont.cond.

17

Thus a septic in @ can at most contain 17 singularities of type

T3,3,3.

A Non-Computational Task Tackled b y Computational Means On the other hand, computer algebra methods have recently been successfully used by 0. Labs and D. van Straten to construct a septic with 99 nodes (see figure 3 for a picture of the singularity, 28 for details on the approach).q The basic idea behind the approach of Labs and van Straten is the following: They start with a 7-parameter family of septics and develop conditions to easily determine the number of nodes on a given septic from a 5-parameter subfamily of this family. Then they pass to small prime fields (with primes 11 5 p 5 53) and explicitly check the actual number of nodes on the septic for all possible parameter combinations to obtain those which provide exactly 99 nodes. Further geometric considerations in characteristic zero lead to a condition for the parameters which can be described as the zero locus of a single univariate polynomial of degree 150, which is, of course, still too large to be of any direct use. Therefore they factorize the polynomial and plug into each of the factors the solutions which were previously obtained over the small prime fields. This leads to only one factor of degree 3 whose vanishing locus contains one real solution; it can then be checked by explicit calculation that the surface corresponding to this parameter value has precisely 99 nodes and no other singularities. 4.2. Resolution of Singularities

The last computational aspect which we want to consider is how to tackle more complex algorithmic tasks, in this case the task of resolution of sinQUpto degree 6 the maximal number of nodes on a surface is known, that is there are known examples possessing exactly the number of nodes specified by an upper bound. In degree 7, however, Varchenko’s spectrum bound and Giventhal’s bound both lead to an upper bound of 104 for the number of nodes on a septic, the septic with the highest number of nodes that had been known prior to the example of Labs and van Straten had 93 nodes.

30 1

Figure 3. The septic surface with 99 real nodes found by 0. Labs and D. van Straten.

gularities. As the series of talks of H. Hauser at this school was devoted to the theoretical background of this topic, we only recall the most important definitions and statements in section 4.2.1 before considering the practical side of it. 4.2.1. Theoretical Background The existence of an embedded resolution of singularities in characteristic zero has been proved by H. Hironaka in his article 25 in 1964. But it took another 25 years, until algorithmic approaches to this task were found independently by E. Bierstone and P. Milman and by the group of 0. Villamayor (see e.g. and for recent articles on these approaches). For recent and accessible introductions to the topic, the article 24 of H. Hauser, which contains many illustrations, and the notes of the Seattle lecture of J. Kollar 26 are good references. Theorem 40. (Embedded Resolution of Singularities) Let W be a smooth

302

algebraic scheme (over a field K of characteristic zero) and let X be a subscheme (with ideal sheaf ZX C Ow). There exists a sequence of

w = wo of blowups 7ri : Wi -+

*

WI

P- . . .

* w,

Wi-Iat smooth centers Ci-1

C Wi-1

such that

(a) The exceptional divisor of the induced morphism Wi + W has only normal crossings and Ci has normal crossings with it. (b) Let X i c Wi be the strict transform of X . All centers Ci are disjoint f r o m R e g ( X ) C X i , the set of points where X is smooth.' (c) X , is smooth and has normal crossings with the exceptional divisor of the morphism Wr + W . (d) T h e morphism (WT,X,) + (W,X ) is equivariant under group actions. Considering the above theorem from a computational point of view, we immediately see two central tasks: the calculation of the blowing up which will be discussed in section 4.2.2 and the choice of the centers which is by far the more difficult task and will be dealt with in sections 4.2.34.2.4. Further computational tasks are added due to the fact that in this context we are not dealing with affine varieties, but with algebraic varieties or schemes. This implies, in particular, that we cannot assume that the whole situation can be encoded (for computational purposes) by means of an ideal/ideals in one polynomial ring; instead we may only assume that W is specified by means of affine charts in each of which we can again work with ideals in a polynomial ring. This poses the problem of gluing and identification of objects that appear in more than one chart; section 4.2.5 discusses these practical aspects. The central point of desingularization by a sequence of blow ups is the appropriate choice of the centers. Therefore algorithms for resolution of singularities are often just stated as algorithms for the choice of centers. This choice is controlled by assigning a rather elaborate invariant value (from a totally ordered set) to each point of X ; the locus of maximal value 'This is not a typographical error, it is redly R e g ( X ) , not Re g ( X i ) . This condition simply ensures that the sequence of blow-ups is an isomorphism on R e g ( X ) . 8A priori, we can make sure that each of the charts of W simply looks like an d . But it will turn out in section 4.2.2 that, after blowing up, it might be necessary to consider charts in which we do no longer have this special structure. Nevertheless, each of these charts can still be expressed as an f i n e variety in some larger A N , a fact which will be used in the subsequent sections.

303

is then used as the subsequent center. The rather complicated structure of such invariants is a consequence of the facts that the center has to satisfy the properties which were stated in the above theorem (e.g. it has to be a closed set, normal crossing with the exceptional divisors) and that the improvement of the situation has to be measured by the invariant, eventually ensuring termination of the algorithm. The definition of the invariant and the computation of its maximal locus is based on an inductive construction which first assigns a part of the invariant to the given point and then constructs an auxiliary object embedded in some ambient space of smaller dimension such that a value has already been assigned to the corresponding point there by induction hypothesis. As already mentioned at the beginning of this section, there are currently two main kinds of algorithms, the ones based on the work of E. Bierstone and P. Milman, which are very close to the original proof of Hironaka, and the ones based on Villamayor’s approach, which is more accessible to practical calculations. The latter point of view is the one we shall use here. Its basic idea is to achieve a reduction of the order of a given ideal by means of order reductions of auxiliary ideals. Before we discuss the underlying notions and the invariant in detail, we now give an example of the difficulties: Example 17 Consider Zx = (z2- x2y2)c C[z, y, z] = 0%.The singular locus of X can easily be determined to be V ( z ,zy), i.e. the union of the zand the y-axis, which is clearly singular itself. As there is nothing special about either of the lines, any choice between the two would be completely at random. Hence the only natural choice for a center here is the coordinate origin V(z,y,z). After blowing up, however, we are facing one chart in which there are no singularities and two other charts (which look identical) in which we encounter the original situation with just one small change: the presence of an exceptional divisor which contains one of the lines (allowing a choice between the two lines).

As the previous example shows, the data we need t o consider does not only consist of ZX and the ambient space W , we also need to take into account the set E of exceptional divisors, which we assume to be ordered chronologically according to the moment of birth of the respective divisors in the resolution process. The last piece of data which is added is an integer

304

b, which by default states the maximal ordert of the ideal 2,. In the induction step, the construction of auxiliary objects, however, it is usually a different value is assigned.

Definition 41. (Basic Objects) A collection of data (W,Zx,b,E) as described above is called a basic objectu. Definition 42. (Resolution Algorithm) Let (3, <) be a totally ordered set. A family of functions" f ( w , z x , b , E ):

x +3,

which is equivariant under isomorphism of basic objects, is said to be governing a blow upw T : (W1, Zx,,b, El) + (WO, Zx0,b, Eo),if the following conditions hold: (8) The setofpointsMaxf(wo,zx,,b,Eo) c XO,where f(Wo,Zxo,b,Eo) takes its maximal value m a x f ( ~ o , ~ x o , bis, ~aoclosed ), subset of WO. (b) Maxf(wo,zxo,b,Eo) is a permissible center, that is, it is regular, has normal crossings with Eo and is disjoint from the set of pointsx {. E Xolz $2Singb(Xo),s# EOi v1 5 i 5 #Eo}. (c) Maxf(wo,zxo, b , ~ is ~ )the center of the blow-up T. (dl maxf(WoJxo ,b,Eo) > maxf(Wi Jxl $,El)(e) (Compatibility with open restrictions) f(wo,zxo,b,Eo)(4 = f(w1,zx1,b,E1)(4 for each point 2 E X O\Maxf(woIzx0, b , E o ) , where 2 E X I is identified with the corresponding point in XO by means of the fact that the blow-up is an isomorphism outside the center.

An algorithm of resolution of basic objects consists of such a family of functions dictating the subsequent blow-ups for any given basic object (W,ZX,b, E ) subject to the additional conditions that tAn exact definition of the order of an ideal will be given below. For now it is sufficient to see the order of an ideal as the generalization of the order of a power series. UOther notions for these types of collections of data include Marked Ideal and Present ation. "By the notation f(w,lx , b , E ) we want t o emphasize that the function itself depends on the whole basic object, not just on X , although it assigns values to the points of X . WTheprecise definition of the transform of a basic object under a blow up can be found at the end of the next section. xSee the subsequent definition of the order of an ideal and of the b-singular locus for an explanation of the notion Singb(X0).

305

(a) (Termination of the algorithm) There is an index N depending on the basic object such that the object is resolved after N steps. (b) If X O is a regular pure-dimensional subscheme of dimension r , b = 1 and EO = 0, then there is a value s ( r ) E 3 such that f(Wc,,Xo,b,Eo)(x) = 5 ( T ) for all 2 E XODefinition 43. (Order of an Ideal and Mingular Locus) The order of an ideal Z c Ow,, at a point x E X is defined as

ord,(Z) := max{m E N I Z c mg,,}. Singb(X),the &singular locus of a basic objecty (W,ZX,b, E ) is defined as the closed set of all points at which the order of the ideal of X is at least b. L e m m a 44. (Semicontinuity of the Order) The order of a n ideal at a point is infinitesimally upper-semicontinuous and Zariski-upper-semicontinuous, that is it does not increase under blow ups and in a suficiently small Zariski open neighborhood of a point there are n o points at which the ideal has a higher order. Using this definition it is now possible to state the first two entries of Villamayor’s invariant for a given basic object (W,ZX,b, E ) :

where N E ( ~is) an integer counting the exceptional divisors containing x which have been born before the order at z attained its current value. To continue the definition of the invariant, we need to pass to an auxiliary basic object in an ambient space of lower dimension to apply the induction hypothesis there. This new lower dimensional ambient space is a so-called hypersurface of maximal contact. A detailed discussion of its construction and properties may be found in 19, a slightly different point of view is outlined in 2 6 . Definition 45. (Hypersurface of Maximal Contact) Given a basic object (W,ZX,b, E ) , where b is the maximal order of ZX, a smooth hypersurface 2 c W is called a hypersurface of maximal contact, if (a) for every open set U c W , the locus of maximal value of the truncated governing function is contained in ZIu, YThis is, of course, the concept of the idealistic exponent of Hironaka.

306

(b) for every open set U c W and every sequence of blow ups at centers of maximal value of the truncated governing function starting at the basic object (U,ZXIU,~,EIU) the center of every blow up is again contained in the respective strict transform of Z I U , (c) 2 has transversal intersections with each exceptional divisor which arose after the maximal order dropped to the current value, (d) the set {Eifl ZlEi E E born after maximal order dropped to b } is normal crossing. These conditions seem to be rather strange at first glance. But conditions (a) and (b) simply ensure that we do not loose any points of maximal value of the truncated governing function when passing to the hypersurface of maximal contact and that we do not need to choose a new hypersurface as long as our maximal value does not drop. Conditions (c), on the other hand, ensures that the centers determined by means of passing to the hypersurface 2 are also permissible as centers for the given basic object, whereas condition (d) is ensuring the normal crossing condition after passing to 2. Note that although (c) and (d) are very similar, neither one implies the other.z

Remark 46. (Existence of Hypersurface of Maximal Contact) A hypersurface of maximal contact does not always exist globally; Iocally, however, it does. One of the central points in the proof of algorithmic desingularization is the independence of the construction of the (local) choice of the hypersurfaces of maximal contact. By the construction of the coefficient ideal, which will be discussed in section 4.2.4, it is now possible to obtain an auxiliary basic object, using the hypersurface of maximal contact as the new ambient space, and determine the values of the governing function for this object. As the construction of the auxiliary object is rather technical and involves notions that will be considered in detail in the following sections, we do not state the definition of the coefficient ideal yet. Marking the descent in dimension of the ambient space by ’;’, we can now (at least) state the general structure of =To see that conditions (c) and (d) are truly different, consider the following two ex-

amples: For the set of exceptional divisors {V(z),V(y)} in 4 and the hypersurface 2 = V(z y) (c) is satisfied, but (d) obviously fails, since the two Ei fl 2 coincide; for the set of exceptional divisors {V(z-l), V(y+l)} and the hypersurface 2 = V(z2+y2-1) condition (d) is satisfied in a trivial way as the Ei f l 2 do not meet, but of course each of the two exceptional divisors is tangent to 2 contradicting to condition (c).

+

307

Villamayor’s invariantA for a given basic object B = (W,ZX, b, E) and its auxiliary objects 2 3 d i m ~ - l , . ..,B2 :

f ~ ( x=) (fB,trunc(x)i fi3aimw-l,trunc(x);

*. *

;f ~ z ( 2 ) ) -

From the computational point of view, these considerations show that the choice of the center can be split up into several subtasks: 0 0

0

Computation of the locus of maximal order of a given basic object Computation of the locus of maximal NE inside the locus of maximal order Descent in dimension and construction of the auxiliary basic object

While the second task is straight forward, the other two require several non-trivial considerations before an efficient implementation is possible. Therefore section 4.2.3 is devoted to the first subtask and the subsequent one 4.2.4 to the third. Based on the implementation of the resolution process, it is also possible to determine resolution related invariants explicitly. As an example, we discuss how to determine the intersection matrix of the exceptional divisors in a (non-embedded) resolution of a surface. For a detailed discussion of practical aspects of other applications see 16. 4.2.2. Blowing Up

Computational Task Given a basic object (W,Zx, b, E) and a permissible center C of a blow up, determine the total/weak/strict transform of it under this blow up.

Background Since we need to blow up at general smooth centers, not just at points, we would like to recall the definitions and basic properties of blow ups (for a detailed discussion see e.g. 23 section 11.7).

Definition 47. (Blowing up) Let W be a scheme, Y C W a subscheme corresponding to the coherent ideal sheaf J . The blowing up of W with *In fact, we omit one special case here, the so-called monomial case which is solved by a separate algorithm of combinatorial nature and does not pose any special computational problems. It can, for instance, be found in 3.

308

center Y is R :

W

J ~+ ) W.

:= Proj($ d20

The notations W , W ,Y and J stay the same as in the previous definition for the rest of this section on the mathematical background.

Lemma 48. (Basic Properties of Blowing Up) Let R : W + W be the blow up of W at a center Y . Then R-'JO* is an invertible sheaf on W ; the corresponding subscheme of W is called the exceptional divisor of the blow up. R : x-l(W \ Y ) + W \ Y is an isomorphism. Lemma 49. (Universal Property of Blowing Up) Iff : Z + W is any morphism such that f - l J O z is an invertible sheaf on Z , then there exists a unique morphism g : Z + W factoring f .

4

Lemma W be a closed subscheme. Let - 50. (Strict Bansform) Let Z1 R I : Z1 + Z1 be the blow up of Z1 along i - l J O z l . Then the following diagram commutes

8, c) W m-1 -1. z1 c)

w.

is called the strict transform of Z1 under the blow up x : W + W , whereas R * ( Z ~is) called the total transform of Z1 under this blow up.

21

On the other hand, we shall need some standard techniques from computer algebra which are discussed in detail in 2 1 , section 1.8. Lemma 51. (Kernel of a Ring Homomorphism, Simplest Case) Let Q : K [ x l , ... ,x,] + K [ y l , .. . ,ym] be a morphism of rings over a field K specified by $(xi) = fi E K[yl,.. .,ym] for 1 5 i 5 n. Then the ideal Ker($) c K [ x l ,. . . ,x,] can be determined by elimination of the variables y 1 , . . . ,ym from the ideal ( 2 1 - f 1 , . . . , x , - f,). Definition 52. (Saturation) Let I1,Iz E K [ x l , ... ,x,] be ideals. The ideal quotient of I1 by I2 is defined as (I1 : 12) := { g E K [ x l , ... ,xn]1g12 C I I } . Iterating the operation of taking the ideal quotient, we obtain an ascending chain of ideals 11

c (11: 1 2 ) c (I1 : 1;) c . ..

309

which eventually stabilizes, since the polynomial ring is noetherian.

(I1 : I F ) :=

u1

O(I1 : I;)

i

is called the saturation of Il w.r.t. the ideal with (I1 : I,) for all sufficiently large s.

12.

Obviously, it coincides

Lemma 53. (Computation of Ideal Quotients) Let I1 C K [ x l , . . .,xn] and let 0 # h E K [ x l , ... ,x,]. Denoting a set of generators for 11n ( h ) by 91 h,. . .,9r h), (11 : ( h ) )= (g t,. . . ,9r>. For a non-principal ideal 12,this approach can be generalized by choosing a set of generators hl,. .. ,h, for I2:

-

m

:I

~)= n ( I l : (hi)). i= 1

Saturation can then be computed by iterating the operation of taking ideal quotients and checking f o r stabilization. Computational Approach When dealing with explicit examples, it is usually more convenient to pass to a covering by affine charts. In particular, the calculations of the blow ups can be formulated from this point of view allowing a direct implementation. Let U c W be an affine open subset and and denote F(U,Ow) by A and r ( U ,3)= (fi,. . . ,f m ) A by J . Then the blowing up of U at the center Y n U is

..-'(v)

= Proj(@

P).

d10

To compute this blow up explicitly, we consider the canonical graded Aalgebra homomorphism @ : A[yl,. . .,ym]

+@ Jntn E A[t] n20

defined by @(pi) = tfi. Then @,>oJn is obviously isomorphic to the ring A[yl,. .. ,ymJ/Ker(@) and we can hence describe the situation by means of the embedding x - l ( U ) E V ( K e r ( @ ) )g S p c ( A ) x Pm-l. In particular, .-'(V) can again be covered by &ne charts D(yi) (each of which is, of course, precisely the complement of the vanishing locus of the corresponding variable yi).

310

For simplicity of presentation, we assume from now on that we are only dealing with the situation W c AN for some N E M, by passing to an affine covering as described above and considering each of these charts separately. Let I c Ow be an ideal and X the corresponding subscheme of W . Then the exceptional divisor and the different transforms of X under the blow up of W at center Y can be computed in the following way: exceptional divisor I(H)= JOe total transform A*(I) = IOW strict transform I3 = ( I O e :J O W ~ ) weak transform (IOW : J O p k ) where k = max(1 E M/(IU@ : J 0 ~ l - l ) = ( I O e : JOW') * J O W ) controlled transform (w.r.t. a control c)

( I O e : JO+') At this point, we are now ready to state how the transform (W1,ZX, ,b, E l ) of a basic object (Wo,ZX,, b, Eo) under a blow up A : (Wl ,ZX, ,b, E l ) + (WO, ZX, ,b, Eo)is defined: W1 = I@ ZX,,,is the ideal of the weakB transform of Zx,, b remains unchanged and El is the union of the strict transforms of the exceptional divisors from EO and the new exceptional divisor. Obviously, the difficulty of the computation of the blow up, which is a preimage computation, depends very much on the generators gi of the center and on the total number of variables involved, because in the very heart of the computation there is an elimination, that is a Griibner basis computation in n + s 1 variables w.r.t. an elimination ordering for t . In particular, this causes successive blowing-ups in smooth irreducible centers to be by far less expensive than blowing-up at several smooth (disjoint) irreducible centers simultaneously. Therefore it is usually a good idea to apply primary decomposition of the center and then blow up at each of the components separately. Clearly, this is possible because, a blow-up is an isomorphism outside of the center and because the components of the non-singular center are obviously disjoint. The draw-back of this improvement is the fact that more charts are produced and hence more duplicate calculations can occur in future steps of the resolution process; but this

+

BThis is the case in the algorithms following Villamayors approach, in the approach of Bierstone and Milman strict transforms are used at this point. Although we are not discussing the latter algorithm here, this principal difference needs to be mentioned.

311

trade-off still pays off in a very large number of practical applications. Another enhancement to the resolution process follows from the fact that not all s charts arising from a single blow-up contain new information. It may very well happen that in one or more charts we do not see any new information that is not already provided by the other charts. In this case, such charts may be dropped. We have been careful not to state what the relevant information is in the previous phrase, because that can depend very much on the data that is to be computed from the resolution: If the goal is, e.g., to compute the intersection matrix of the exceptional divisors of a resolved surface, no relevant information can be provided by data at points outside of the strict transform of the surface. If on the other hand, the goal is the computation of a C-function, the information on the intersections of the exceptional divisors (even outside the weak transform of the variety) is vital and no charts may be dropped. 4.2.3. Computing the Locus of Maximal Order Computational Task

Given a basic object (W,ZX ,b, E ) , determine the locus of maximal order of ZX

.

Background

In the exposition of this part, we follow closely the definitions in

3.

Definition 54. (A(Zx)) For a basic object (W,Zx,b,E), we define A(Zx) C Ow as the sheaf of ideals locally generated by

where z1,. ..,X d is a regular system of parameters for ow,,and 91,. . .,gB are a set of generators for I,. Ai(Zx) is then inductively defined as a(ai-l(zx)). Lemma 5 5 . (Locus of Order at least c) The locus of order at least c of Zx coincides with V(Ac-'(J)).

Computational Approach

The definition of A(J) and hence the calculation of the locus of maximal order heavily rely on using generators for the ideal I, C Ow,,and a regular

312

system of parameters for Ow,, at the given closed point w E W . Theoretically this is fine, but in practice it is, of course, not feasible to compute at each point of W . Here, the use of a set of generators of I , c OW,, does not cause any problems, since we are working on affine charts and on each chart we are specifying ZX by a set of generators anyway. For simplicity, we now assume again that our ambient space W is contained in some AN as in the previous section. The difficulties arise from the computational need to have a global system 91,.. .,Y d C Ow inducing a local regular parameters for OW,, on the whole W , which, in general, does not exist. To avoid this problem, it is necessary to pass to a suitable open covering { U j } of AN such that for each Uj we can find a global system giving rise to a regular system of parameters at each point of U j . This, of course, increases the number of charts, a drawback which can, in turn, be eliminated by recombining the results on the Uj to one on AN in a suitable way. More precisely, A( J) is determined by the following algorithm:

Algorithm Delta (91,... ,gT) generating ZW c @[XI,. . . ,x,] = OU, (f1,. . . ,fs) generating ZX c 4x1,.. . ,x,] such that V(Zw)is equidimensional and regular and Output A(Zx) c @[xi,. . .,2,] = Oui (1) if Zw = (0) then return((f1,. . ., fs, Q.fL azl > . ., a)) az, (2) Initialization

Input

ZW c ZX

c ={ f l , . . . , f s } D = (1) L1 = { n-dim(W) square submatrices of the Jacobian matrix of ZW whose determinant is non-zero} (3) while (L1!=0) 0

0 0

choose M E L1 L1= L 1 \ { M } q = det(M) determine an n - dim(W) square matrix A such that D A ' M = q &-dim(W)

cFor simplicity, the row and column indices used inside the submatrices will be the ones of the corresponding rows resp. columns in the Jxobian matrix. DEj denotes the j x j unit matrix. As before, we use row and column indices come sponding to those of A4 for simplicity.

313 0

determine components of A(J) not lying inside V ( q ) :

I row of M

0

C M = sat(CM, 9 ) Add these components to the previously found ones: D=DnCM

(4) return(D) The basic idea behind this algorithm is that W is regular and hence at each point there is at least one ( n - dim(W))-minor of the Jacobian matrix of Zw which does not vanish. This allows us to pass to the open covering defined by the complements of the vanishing loci of these (n-dim(W))-minors. Fixing one such (n - dim(W))-submatrix M of the Jacobian matrix and working on the complement of the minor detM , the desired global system of local parameters can be determined based on the following observation: As detM is now invertible, the generators of the ideal of W corresponding to the respective rows in M can be used as a local system of parameters at each point of our open set. Direct calculation then yields expressions for the derivatives of the generators of ZX. To obtain elements of the polynomial ring, these expressions need to be multiplied by the unit detM which provides precisely

used in the algorithm. The set containing these expressions and the original generators of ZX generates an ideal Cn in the polynomial ring which contains ZW as we assumed at the beginning of the algorithm that Zw c Zx. The subscheme corre sponding to this ideal coincides with the desired A(Zx) on the complement of detM; on V ( d e t M ) ,however, we are likely to see components which do not have any relation to ZX. So we remove all components contained in V ( d e t M ) by saturation. As the missing components of Zx inside V ( d e t M ) appear automatically when considering the other Uj, we make sure that we have determined all components by forming the intersection of all ideals Cn in the last step of the algorithm.

314

Figure 4. As an example for the problem of computing A(J), let us consider the situation illustrated in the above picture: There are three minors whose determinant does not vanish (each one illustrated by one of the curves in the above picture) and V ( A ( J ) ) consists of the three points. Then computing on the complement of just one of the mi, each of the curves meets at least nors will not provide all points of V ( A ( J ) ) because one point.

From the practical point of view the above algorithm still needs to be improved to avoid redundant calculations. In particular, one should first check whether there is a minor of the appropriate size which is itself an element of C In this case, the complement of the minor is the whole AN and the other minors obviously do not provide any new contributions.

4.2.4. Descent in Dimension Computational Task Given a basic object (W,Zx, b, E ) , where b is the maximal order of Zx, find (at least locally) a hypersurface of maximal contact and determine an auxiliary basic object permitting the induction step of the resolution process.

B ackground Among the notions and facts necessary for this task, the definition of a hypersurface of maximal contact has already been defined in section 4.2.1; for the other statements a good reference is or 24. A different point of view for the construction is taken in 26 discussing conditions on the possible constructions of an auxiliary object rather than just stating one construction.

Lemma 56. (Choice of a Hypersurface of M w i m d Contact) Given a basic object (W,Zx, b, E ) , where b is the maximal order of ZX, and a point

315

w E X c W , any order I element of Ab-l(Tx), C OW,,which satisfies conditions (c) and (d) can be chosen as a hypersurface of maximal contact in a suficiently small neighborhood of the point w . Definition 57. (Auxilliary Basic Object (Villamayor's Construction)) Given a basic object (W,ZX,b, E ) , where b is the maximal order of Zx, and an open set U c W on which the hypersurface 2 can be chosen as hypersurface of maximal contact for the given basic object, the auxiliary basic object (2,Zne,, c, En,,) (on U ) is defined as b- 1 i=O

c := b!

Enew := {Ei n ZlEi E E born after maximal order dropped to b} Lemma 58. (Coeficient Ideal and Blow Up 'commute') Let B = (W,Zx, b, E ) be a basic object and let A = (2, ,,Z , ,c, Enew) be a n auxiliary basic object as defined above. T h e n the controlled transform of A w.r.t. the control c under a blow u p at a center determined by the governing invariant coincides with the auxiliary basic object constructed f r o m the weak transform of B under the same blow u p using the strict transform of 2 as the hypersurface of maximal contact.

Computational Approach For the descent in dimension, that is the computation of the coefficient ideal, the crucial point is hence the choice of the smooth hypersurface 2 which is subject to two normal crossing conditions regarding the exceptional divisors. As soon as such a hypersurface is found, the computation of the coefficient ideal only involves determining the Ai of the ideal which has previously been discussed and basic operations on ideals such as taking powers and sums. As already mentioned, such a hypersurface 2 usually does not exist globally. In an implementation, the choice of the hypersurface involves passing to a suitable open covering such that on each open set Uj there is a hypersurface which can be used as Z for each point w E U j . The basic idea for finding such a covering is to consider AC-l(ZX) . As the intersection of the singular loci of the generators of A"-'(Tx) is empty (c is the maximal order), it is possible to express 1 as a combination of the generators of the

316

ideals of these singular loci and use the complements of those generators appearing with non-zero coefficients as the open coveringE The need to pass to an open covering can enlarge the number of charts significantly which slows down the subsequent steps of the resolution process due to duplicate calculations for points/subvarieties/centers appearing in more than one chart, as we already mentioned before. The first idea to keep the number of open sets as low as possible is to recombine in the end in the same way as in the algorithm for determining A (that is passing to the closure, dropping components which do not meet the respective open set and taking the intersection of the resulting ideals). Unfortunately, the auxiliary objects really depend on the chosen hypersurface, although the resulting value of the governing function at each point is independent of this choice. Therefore, we cannot recombine directly as before; instead, we continue with the algorithm for finding the maximal locus of the governing function in each of the open sets and then (carefully) recombine those maximal loci in the following way: After passing to the closure and dropping components not meeting the open set Uj,each open set Uj provides an ideal 2% describing a candidate for the next center. To this ideal 2yj we can associate the value v j of the governing function corresponding to one (and hence any) point in Yj.The next center then corresponds to the ideal j

v,

s u c h that

maximal

4.2.5. Identification of Exceptional Divisors

The last subtask, which we want to discuss, is the identification of points resp. subvarieties which occur in more than one chart; in particular, we need to decide whether two given exceptional divisors living in two different charts actually belong to the same exceptional divisor (after gluing the charts). To this end, we move through the tree of charts arising during the resolution process, first blowing-down from the first chart to the one in which the history of the two charts in question branched, and then blowingup again to the other chart with which we want to compare (cf. figure 5 ) . As blow-ups are isomorphisms away from the center, this process of successively blowing-down and then blowing-up again does not cause any EOf course, it is necessary to check that the two normal crossing conditions hold and, if necessary, pass to a different way of expressing 1 in terms of the generators of the singular loci.

317

chart 2:V(x2!42-z2)

EI:V(x)

EI:V(x):

E2:V(u)

chart 5:V(!~2-22) E l : V ( y ) t E3:VCx)

I

-V(u.z) chart 7:Vd-z2)

E3:V(x):

E4:V(y)

Figure 5. The tree of blow-up in the rescAion of the singularity V , ~ - ~z 2 ) C y A3.~ For simplicity of notation the variable names in all charts have been chosen t o be z,y, z. To determine whether two exceptional divisors in two different charts actually belong to the same exceptional divisor, we need to move through the tree by first blowing down and then blowing up again; for instance, the question, whether the divisors V(z) in chart 6 and V(y) in chart 7 belong to the same divisor, can only be answered by comparing the centers in charts 4 and 5. To this end, we have to move from chart 4 to chart one by blowing down twice and then proceed to chart 5 by blowing up twice.

problems for points which do not lie on an exceptional divisor at all or only lie on exceptional divisors, which already exist in the chart at which the history of the considered charts branched. If, however, the point lies on an exceptional divisor which arises later, then blowing-down beyond the moment of birth of this divisor will inevitably lead to incorrect results. To see this let us consider an exceptional divisor Y x Pk originating from a blow up at Y and a point y E Y : two distinct points p and q in y x IPk are always blown down to the same point y. To avoid this problem, we need to represent the points on the exceptional divisor as the locus of intersection of the exceptional divisor with an auxiliary variety which is not contained in the exceptional divisor. More formally speaking, we use the following simple observation from commutative alge bra:

Remark 59. Let I c K [ z l , .. .,z,] be a prime ideal, J c K[zl,...,$,I another ideal such that I J is equidimensional and ht(I) = ht(1 J) - T

+

+

318

for some integer 0 < T < n. Then there exist polynomials p l , . .. , p r E I+J and a polynomial f E K [ q , .. . ,xn] such that

dG-7 = J ( I+ ( P I , . .. , p r ) ) : f. In our situation, the ideal I is, of course, the ideal of the intersection of the exceptional divisors in which the point or subvariety V ( J )is contained. Any sufficiently general set of polynomials PI,.. .,p,. E J \ (In J ) leading to the correct height of I (PI,. . . ,p r ) may be chosen and the only truly restricting condition on f is that it has to exclude all extra components (PI,. . . , p r ) . Thus we also have enough freedom of choice of the of I pl ,. . . ,pr ,f to achieve that none of these is contained in any further exceptional divisor whose moment of birth has to be crossed when blowing-down. Having solved the problem of identification of points existing in more than one chart, we are now prepared to determine which exceptional divisor in one chart coincides with which one in another chart by simply comparing the centers giving rise to these exceptional divisors. To this end, we start at the root of the tree of charts of the resolution and work our way up to the final charts. The criteria for identification of the centers are quite simple:

+

+

0

0

0

centers do not coincide, if the corresponding values of the governing function do not coincide, centers do not coincide, if the exceptional divisors in which they are contained do not coincide explicit comparison of centers (which are not excluded by the previous criteria) is possible by means of successive blowing down and blowing up as described above.

At this point, we need to recall that computations in a computer algebra system are performed over Q not over the complex numbers although the reasoning often takes place over C. This is particularly important during interpretation of the results here, because using this computational approach we have not yet been able to determine, for instance, the correct number (over C) of exceptional divisors arising during the resolution process. This will be a crucial issue in the subsequent section.

4.2.6. Intersection Matrix of Exceptional Curves Computational Task Given an embedded resolution of a isolated surface singularity (where the singular locus consists of precisely one point), pass to a non-embedded

319

resolution and compute the intersection matrix of the exceptional divisors. This task may be split into three subtasks:

(1) determine a non-embedded resolution from a given embedded one (2) determine the intersections Ei.Ej for exceptional divisors Ei # Ej (3) determine the self-intersection numbers Background (1) As the theorem of Embedded Resolution of Singularities has already been stated in theorem 40,we only state the more general task of non-embedded resolution of singularities here and illustrate the difference in an example. For a detailed comparison of various resolution type tasks and theorems including embedded and non-embedded resolution, see e.g. 2 6 , sections 2-3, and 6 , sections 13 and 17.8. Theorem 60. (Resolution of Singularities) Let X be a n algebraic variety over a field of characteristic zero. T h e n there is a resolution of singularities,

i.e. a proper birational morphism T :

3X

and a non-singular variety

2 such that T is o n isomorphism away from the singular locus of X . Example 18 (Comparison of Embedded and Non-Embedded Task) Given an &ne variety X = V ( z 2- y3) c A$, consider the blow up at the center V(z, y). The total transform of X under the blow up is given by the variety V ( z 2- y3, a y - b z ) c A$ x P& where (a : b) are the variables corresponding to the P&.For the corresponding non-embedded blow up, we may use the same calculation (due to lemma 50) keeping in mind that the exceptional divisor needs to be considered as a subscheme of X in this case. In charts, we thus have:

a # 0 3 n Ul = V(z2 - yie,z3) is the total transform where ynew := :, the exceptional divisor is E = V(z) and the strict transform is V ( l - y:,,z) which is non-singular and does not meet E . In this chart, we have hence achieved both a resolution and an embedded resolution of singularities (the sense that all further blow ups in both resolution processes will be isomorphisms on this chart). b # 0 -% n U2 = V(zie,y2 - y3) is the total transform where zne, := %, the exceptional divisor is E = V(y) and the strict transform is V(z;,, - y) which is non-singular, but tangent to E. Therefore, we have achieved non-embedded resolution of singularities, but not embedded resolution of singularities for this chart.

320

These considerations show that the given blow up is a resolution of singulaxities, but not an embedded resolution of singularities. Note that a comparison of the two tasks as in the above example was possible because X was embedded in W . This is also the condition under which all further computational remarks need to be understood, because we are actually using Villamayor’s algorithm for embedded resolution of singularities to obtain a non-embedded one.F

Remark 61. (F’unctoriality/Canonicity) An algorithm for resolution of singularities is called Eunctorial (or canonical) if it commutes with smooth morphisms and closed embeddings and is compatible with change of fields (see e.g. 26, section 3, or 41, section 2.4). Villamayor’s algorithm, however, is not functorial because it depends on the embedding as the following example shows: Let (W,Zx,b,0) = (A$,(z2-y3,y4z3 -w8),2,0) be a basic object where and consider a closed embedding i : W L) A$ = W1 inducing a basic object (W1,Zl := Z x O , , b, 0) = (4,(z2-y3,y4z3 -w8), 2,s). Choosing the center V(z, y,w) c i$ as required by Villamayor’s algorithm, we obtain order reduction in all three charts of the corresponding blow up. In A$ the corresponding choice of center V(z, y,w,v) also yields order reduction in three of the four charts; in the chart with exceptional divisor V ( v ) ,however, the maximal order stays two which forces us to blow up once again in contradiction to the situation before embedding into @. . This is an important fact to keep in mind when using Villamayor’s algorithm in practical examples, but it is not as much of a problem as it might seem at first, because we choose the embedding at the very beginning of our calculations and do not change it afterwards. Computational Approach (1) Given an embedded resolution of an isolated surface singularity, stored as a tree of charts, we would like to pass to a non-embedded resolution by dropping unnecessary blow-ups at the end of the branches of the tree of charts. To this end, we compute the list of exceptional divisors by identifying them in the different charts as described in the previous section. Starting at the is in general not true that any given scheme X can be embedded into a smooth scheme. An example can e.g. be found in 26, remark 33 in section 3.

321

final charts, we then move backwards through the resolution tree and cancel those blowing ups which are not necessary for the non-embedded resolution (see illustration 6 for an example).

chart l : V ( x Z + u 2 + ~ 3 )

/c=v(x-u-2) chart 2:V(x2+uZ+z)

\ chart 3:V(xz3+$2+1)

Figure 6. Tree of the embedded resolution process of an A2 surface singularity. All charts which are marked by gray background arise from blow-ups which are only necess a r y in the embedded case, but not for a non-embedded resolution.

Then we consider the intersection of the remaining exceptional divisors of the embedded resolution with the strict transform to obtain the exceptional locus of the non-embedded resolution. We can easily decompose these intersections into irreducible components over Q, but these components may still be reducible over C.

Background (2) For computing a decomposition over C of the exceptional divisors which is necessary for determining their intersection matrix, the following theorem (6.17) is used: Theorem 62. (Gao/Ruppert) Let f E Q [ x , y ] be irreducible of bidegree a(glf) = (m,n). Let G = ( 9 E Q@,Y]I(m- 1,n) 2 & d 9 ) , 3 h E Q [ X , Y l , ay

w az }The . vector space G has the following properties

(i) f is irreducible in C [ x , y ] if and only if dimq(G) = 1.

322

(ii) g G c 2 G mod f f o r all g E G. (iii) Let gi, ...,ga E G be a basis and g E G \ Q g , g g i =Caijgjg mod f. Let X ( t ) = d e t ( t E - ( a i j ) ) be the characteristic polynomial. T h e n x is irreducible in Q[t]. (iv) f = gcd( f, g - c g ) is the decomposition off into irreducible factors tn C[x,y].

nc,--.x(c)=q

We use this theorem for the decomposition of the exceptional curves of our surface, which are irreducible over Q, over C! by means of the following corollary:

Corollary 63. Let I c Q x 1 , .. .,x,], ht(I) = 1, be a prime ideal. T h e n there exists a n irreducible polynomial X ( t ) E Qt]such that the complex zeros of X ( t ) = 0 correspond t o the associated prime ideals of IC![x,,.. .,xn].

Background (2 and 3) Definitions and properties on intersection numbers of divisors on surfaces can be found in many books on algebraic geometry, among others in 23, section V.l, and in 35, section IV.l.

Definition 64. (Intersection Numbers) If D1, D2 are divisors on a nonsingular surface X in general positionH, then

c

X E D 1 nDa

is the intersection number of D1 and D2, where ( 0 l . D ~denotes )~ the intersection multiplicity at 2.

Lemma 65. (Invariance under Linear Equivalence) For any divisors D1 and D2 o n the non-singular surface X , there exist divisors D:, D i such that Di Di and D i , 0: are in general position. If D1, D2 and D i , D i are two tuples of divisors in general position, then

-

GFor n = 2 this statement is obvious; in the case n > 2, a (suitable) generic linear coordinate change leads to I n a x n - l ,zn] = (f) where the associated primes of I correspond to the associated primes of (f). Geometrically this is a generic projection of the curve defined by I to the plane. H D 1 and D2 are called in general position, if the intersection Supp(D1) n Supp(D2) is either empty or a finite set of points.

323

Note that this lemma allows the definition of intersection numbers for any two divisors on X by means of passing to linearly equivalent divisors in general position.

Computational Approach (2) Example 19 As a simple example of the situation, let us consider the plane curve V ( 2 3 - 293) which is Qirreducible, but consists of three components over @: ring R=O ,(x y) dp ; poly p=x3-2y3 ; getMinpoly(p) ;

/ / the ring / / the polynomial

Cll : poly p=t-3-2; 121 :

/ / the polynomial \chi(t)

/ / its 3 complex zeros

C11 : (-0.6299605249474365823836+i*1.0911236359717214035601) C2l : (-0.6299605249474365823836-i*1.0911236359717214035601) C31 : 1.25992104989487316476721061

C3l : 3

Using the field extension proposed by the output of GetMinpoly, we now pass to the field extension Q t ] / t 3- 2 and factorize: ring T=(O,t), (x,y),dp; minpoly=t3-2; factorize (x3-2y3);

Cll :

- ClI=i

- [2l=x2+(t)*x*y+(t2)*y2 - C3l =x+ (-t *y 121 : 1,131

/ / new ring, with parameter t / / minimal polynomial / / factorization

324

This is obviously not a complete factorization. But a complete factorization can only be achieved using a Galois extension which is of higher degree (in this case degree 6). Therefore the factorization is more expensive from a computational point of view, i.e. it takes more time and memory, and so are all further calculations over this field. (Just look at the number of summands in the coefficients of y in our example below!). ring T=(O,t) (x,y> ,dp; minpoly=t6+3t5+6t4+llt3+12t2-3t+l; factorize(x3-2y3) ;

To identify the (C-components of an exceptional divisor E (irreducible over Q) in a chart, we, therefore, store E , X ( t ) and the respective numerical root of X ( t ) . Given these data, we can then proceed in the same way as for the identification of the Qcomponents in the previous section. As soon as the exceptional divisors in the different charts are identified, we can directly compute the intersection numbers Ei.Ej for all i # j.

Background (3) To compute the self intersection numbers, we need some more properties of intersection numbers under birational morphisms. For a more detailed discussion of the topic of intersection numbers of divisors and their behavior under blowing up see e.g. 35, sections IV,3-4' Theorem 66. (Intersection Numbers and Blowing Up) Let x : -% + X be a resolution of singularities of the surface X . Let D1 be a divisor o n X all of whose components are exceptional curves of T and let DZ be any 'The statements there are all for non-singular surfaces. For passing from the case of a non-singular surface X to a singular one see the remark in 35 right after section III,1.3, theorem 1.

325

divisor o n X , t h e n W*(DZ).Dl = 0 Computational Approach (3) Let w : X + X be a resolution of the surface X whose singular locus consists of just one point as specified in the statement of the task. Let E l , . . .,E8 the exceptional divisors arising during the respective blow ups and let D be a non-trivial divisor on X (defined by meam of a linear form h : X +C passing through the only singular point of X ) . Then we know 8

r*(D).Ei = 0

and

w*(D) =c.iEi +HI i=l

where the ci are integers and H denotes the strict transform of D . The self intersection numbers can then be computed using the formula 8

0 = r*(h)Ei=

C c.E..Ei + H.Ei

V1 5 i

5 S,

i=l

as all other intersection numbers can be/have already been computed directly. References 1. Arnold,V., Gusein-Zade,S., Varchenko,A.: Singularities of Differentiable Maps I, Birkhauser (1985) 2. Bierstone,E., Milman,P.: Desingularization Algorithms I: The Role of Exceptional Divisors, Mosc. Math. J. 3 (2003), 751-805 3. Bravo,A., Encinas,S., Villamayor,O.:A Simplified Proof of Desingularisation and Applications, Rev. Math. Iberoamericana 21 (2005), 349458 4. Buchweitz,R.-O., Greue1,G.-M.: The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), 241-281 5. Campillo,A.: Algebroid Curves in Positive Characteristic, Springer (1980) 6. Cutkosky,S.D.: Resolution of singulan'taes, Graduate Studies in Mathematics, 63, AMS (2004) 7. de Jong,T., Pf%ter,G.: Local Analytic Geometry, Vieweg, (2000) 8. Decker,W., Lossen$.: Computing in Algebraic Geometry - A quick start using SINGULAR, Algorithms and Computation in Mathematics 16, Springer Verlag (2006). 9. Ebeling,W.: finktionentheorie, Dafferentialtopologie und SingularatEten, Vieweg (2001) 10. Ebeling,W.: Monodromy, to appear in Singulaxities and Computer Algebra, Cambridge University Press (2006)

326 11. Ebeling,W.: Notes to the series of Talks entitled Monodromy of Isolated Singularities at this summer school 12. Eisenbud,D.: Commutative Algebra with a View toward Algebraic Geometry, Springer (1995) 13. Eisenbud,D., Huneke,C., Vasconcelos,W.: Direct Methods for Primary Decomposition, Invent. Math.110 (1992), 207-235 14. F’riihbisKriiger,A.: Construction of Moduli Spaces for Space Curve Singularities JPAA 164 (2001), 165-178 15. FriihbisKriiger,A.,Neumer,A.: Simple Cohen-Macaulay Codimension 2 Singularities, preprint (2004) 16. F’riihbis-Kriiger,A., Pf%ter,G.: Some Applications of Resolution of Singularities from a Practical Point of View, in Proceedings of Computational Commutative and Non-commutative Algebraic Geometry, Chisinau 2004 (ZOOS), 104-117 17. Gao,S.: Factoring Multivariate Polynomials via Partial Differential Equations, Math.Comp 72 (2003), 801-822 18. Gianni,P., Trager,B., Zacharias,G.: Grobner Bases and Primary Decomposition of Polynomial Ideals, JSC 6 (1988), 149-167 19. Giraud,J.: Sur la theorie de contact maximal, Math.Z. 137 (1974), 285-310 20. Greue1,G.-M.: On deformation of curves and a formula of Deligne, in: Algebraic Geometry (Proceedings, La Rbida 1981) Springer (1983), 141-168. 21. Greue1,G.-M., Pfister,G.: A SINGULAR Introduction to Commutative Algebra, Springer (2002) SINGULAR 3.0, 22. Greue1,G.-M., Pfster,G., Schonemann,H.: http://vvw.singular.uni-kl.de/ 23. Hartshorne,R.: Algebraic Geometry, Springer (1977) 24. Hauser,H.: The Hironaka Theorem o n resolution of singularities, Bull. Amer. Math. SOC.40 (2003), 323-403 25. Hironaka,H.: Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero, Annals of Math. 79 (1964), 109-326 26. Kollar,J.: Resolution of Singularities - Seattle Lecture, math.AGl0508332 27. Krick,T., Logar,A. A n Algorithm for the Computation of the Radical of an Ideal in the Ring of Polynomials, AAECC9, Springer LNCS 539 (1991), 195205 28. Labs,O.: A Septic with 99 Real Nodes, math.AG/0409348 29. Laudal,O.A., P&ter,G.: Local Moduli and Singularities, Springer-Verlag, (1988) 30. L&D.-T.: Calculation of the Milnor number of an isolated sangularity of a complete intersection, Funkt. Anal. Ego Prilozheniya 8(2) (1974), 45-49 31. Looijenga,E.: Isolated Singular Points of Complete Intersections, Cambridge University Press, LNS 77 (1984) 32. Martin,B.: Computing versa1 deformations with Singular, in: Algorithmic Algebra ans Number Theory, Springer (1998), 283-294 33. Milnor,J.: Singular Points of Complex Hypersurfaces, Princeton University Press (1968) 34. Schulze,M.: A Normal Form Algorithm for the Brieskorn Lattice,

327 J.Symb.Comp.38 (2004), 1207-1225 35. Shafarevich,I.: Basic Algebraic Geometry, Springer (1977) 36. Shimoyama,T., Yokoyama,K.: Localization and Primary Decomposition of Polynomial Ideals, JSC 22 (1996), 247-277 37. Steenbrink,J.: Mixed Hodge Structure on the Vanishing Cohomology, in Real and Complex Singularities (Proceedings Oslo 1976) SijthoENoordhoff, Alphen a/d Rijn (1977), 525-563 38. Steenbrink,J.: Semiwntinuity of the singularity spectrum, Invent, Math. 79(3) (1985), 557-565 39. Steenbrink,J.: Mixed Hodge Theory: The search for purity, in this volume 40. Wal1,C.T.C.:Singular Points of Plane Curves, London Mathematical Society Student Texts 63 (2004) 41. Wlodarczyk,J.: Simple Hironaka Resolution in Characteristic Zero, J. Amer. Math. SOC.18 (2005), 779-822

328

LAGRANGIAN AND LEGENDRIAN VARIETIES AND STABILITY OF THEIR PROJECTIONS

V. V. GORYUNOV Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK E-mail: [email protected]. uk V. M. ZAKALYUKIN Department of Mechanics and Mathematics, Moscow State University, Leninskie gory, 1 119992 Moscow, Russia, E-mail: vzakalQliv.ac.uk These are notes of a mini-course given at ICTP in August 2005. The first half of the notes contains basic notions and theorems of the local theory of Lagrangian and Legendrian mappings. The second half concentrates on recent stability results, in particular on the stability of the Lagrangian map-germs defined by composite functions.

Almost all applications of singularity theory are related to discriminants and bifurcation diagrams of singularities of functions: they can be visualised and recognised in many physical models. Suppose, for example, that a disturbance (such as a shock wave, light, an epidemic or a flame) is propagating in a medium from a given submanifold (called initial wave front). To determine where the disturbance will be at time t (according to the Huygens principle) we must lay a segment of length t along every normal to the initial front. The resulting variety is called an equidistant or a wave front. Along with wave fronts, ray systems may also be used to describe propagation of disturbances. For example, we can consider the family of all normals to the initial front. This family has the envelope, which is called caustic - “burning” in Greek - since the light concentrates at it. A caustic is clearly visible on the inner surface of a cup put in the sunshine. A rain-

329

bow in the sky is the caustic of a system of rays which have passed through drops of water with total internal reflection. Generic caustics in three-dimensional space have only standard singularities. Besides regular surfaces, cuspidal edges and their generic (transversal) intersections, these are: the swallowtail, the ‘pyramid’ (or ‘elliptic umbilic’) and the ‘purse’ (or ‘hyperbolic umbilic’). They are a part of R.Thom’s famous list of simple catastrophes. It is not so difficult to see that the singularities of a propagating wave front slide along the caustic and trace it out. The study of singularities of wave fronts and caustics was the starting point of the theory of Lagrangian and Legendrian mappings developed by V.I.Arnold and his school some thirty years ago. Since then the significance of Lagrangian and Legendrian submanifolds of symplectic and respectively contact spaces has been recognised throughout all mathematics, from algebraic geometry to differential equations, optimisation problems and physics. In these lecture notes, we do not touch the fascinating results in symplectic and contact topology, a young branch of mathematics which answers questions on global behaviour of Lagrangian and Legendrian submanifolds. An interested reader may be addressed to the book [15] and paper [18] forming a good introduction to that area. Our lectures were designed as an introduction to the original local theory, with an accent on recent results on the stability issues. We hope that they will inspire the reader to do more extensive reading. Items on our bibliography list may be rather useful for this. Bearing the introductory approach in mind, we have devoted the first two sections of the notes to generalities on local Lagrangian and Legendrian singularities. There we just outline basic notions, following the books [2, 3, 5, 41 which we recommend for a detailed description of the theory. The last, third section deals with the study of singular Lagrangian and Legendrian varieties which was initiated about twenty-five years ago by Arnold when he was investigating singularities in the variational problem of obstacle bypassing [l]. The first examples of such varieties, open swallowtails, were related to the discriminants of non-crystallographic Coxeter groups 19, 171. Incorporating these examples into a general context, Givental [9] introduced the notion of stability of Lagrangian and Legendrian varieties with respect to perturbations of the symplectic structure and Lagrangian or, respectively, Legendrian projection only, keeping the diffeomorphic type of the variety fixed. Later, in [16], it was shown that this stability notion has an explicit

330

geometrical meaning in terms of generating families, versal deformations of function singularities and inducing mappings. The interest in the theory of singular Lagrangian and Legendrian varieties has been growing recently due to its possible applications to F'robenius structures, D-modules and in other areas [14]. Due to that, the main object of the last section is a stable Lagrangian projection playing the central r61e in the geometry of Hamiltonian systems and, in particular, in the theory of F-manifolds. We extend the results of [16] to a natural modification of Givental's stability notion and show that a wide class of Lagrangian and Legendrian varieties associated to matrix singularities (see [6, 7, 121) and singularities of composed mappings [ll]are stable [13]. 1. Symplectic and contact geometry 1.1. Symplectic geometry

A symplectic form w on a manifold M is a closed 2-form, non-degenerate as a skew-symmetric bilinear form on the tangent space at each point. So dw = 0 and wn is a volume form, dim M = 2n.

A manifold M equipped with a symplectic form is called symplectic. It is necessarily even-dimensional. If the form is exact, w = dX, the manifold M is called exact symplectic.

Examples. 1. Let K = M = R2" = { q l , . . . ,q n , p l , .. . , p n } be a vector space, and

In these co-ordinates the form w is constant. The corresponding bilinear form on the tangent space at a point is given by the matrix 0 -In 0

J=(In

).

NOTICE: for any non-degenerate skew-symmetric bilinear form on a linear space, there exists a basis (called Darboux basis) in which the form has this matrix.

331 2. M = T * N , w = dX, where X is the Liouville f o r m defined in an invariant (co-ordinate-free) way as X(a) = n(a)(p*(a))1

where

a E T(T*N), and T

: T ( T * N )-+

T*N,

p : T * N -+ N

are the natural projections. This is an exact symplectic manifold. If q1,. . . ,qn are local co-ordinates on the base N , the dual co-ordinates P I , . . . , p , are the coefficients of the decomposition of a covector into a linear combination of the differentials dqi: n

A diffeomorphism cp : M1 -+ M2 which sends the symplectic structure w2 on M2 to the symplectic structure w1 on M I , v*w2 = w1,

is called a symplectomorphism between ( M I ,w l ) and ( M 2 ,w2). When the ( M i ,wi)are the same, a symplectomorphism preserves the symplectic structure. In particular, it preserves the volume form wn. Symplectic group. For K = (R2", d p A d q ) of our first example, the group Sp(2n) of linear symplectomorphisms is isomorphic to the group of matrices S such that

s-l = - J S ~ J . Here t is for transpose. The dimension k of a linear subspace Lk c K and the rank r of the restriction of the bilinear form w on it are the complete set of Sp(2n)-invariants of L. Any two linear subspaces of the same dimensions and same restricted rank are mapped one to the other by a linear symplectomorphism: it is easy to construct a Darboux basis whose first k vectors belong to L thus defining the first 1-12 and k - 1-12of the co-ordinates p and q respectively.

332

Define the skew-orthogonal complement LL of L as

LL = {W E K ~ w ( w , u =)0 Vu E L } . So dimLL = 2 n - Ic. The kernel subspace of the restriction of w to L is L LL . Its dimension is Ic - r.

n

A subspace is called isotropic if L c LL (hence dim L 5 n). Any line is isotropic. A subspace is called co-isotropic if L L c L (hence dim L 2 n). Any hyperplane H is co-isotropic. The line H L is called the characteristic direction on H .

A subspace is called Lagrangian if L L = L (hence dim L = n). The following lemma determines a system of charts on the Grassmannian manifold of all Lagrangian subspaces of K.

Lemma 1. Each Lagrangian subspace L c K has a regular projection to at least one of the 2n co-ordinate Lagrangian planes ( p ~ , q j )along , the complementary Lagrangian plane ( p ~QI). , Here I J = (1,. . . ,n } and

u

1n~=0. A Lagrangian subspace L which projects regularly onto the q-plane is the graph of a self-adjoint operator S from the q-space to the pspace with its matrix symmetric in the Darboux basis. The classical theorem below shows that a symplectic structure, unlike its Riemannian counterpart, has no local invariants even in a non-linear setting.

Darboux Theorem. Any two symplectic manifolds of the same dimension are locally symplectomorphic. In a similar local setting, the inner geometry of a submanifold defines its outer geometry:

Givental’s Theorem. Two germs of submanifolds in a symplectic manifold are symplectomorphic, i f and only if the restrictions of the symplectic

333

structure to the tangent bundles of the submanifolds are diffeomorphic. Proof of Givental’s Theorem. It is sufficient to prove that if the restrictions of two symplectic forms, wo and w1, to the tangent bundle of a submanifold G c M near a point m E G coincide, then there exits a local diffeomorphism of M fixing G point-wise and sending one form to the other. We may assume that the forms coincide on T,M. We use the homotopy method, so common in singularity theory, aiming to find a family of diffeomorphism-germs gt, t E [0,1], such that gtlG

gF(wt) = wo,

= idG,

(*)

90 = idkt

where wt = wo

+ (w1 - wo)t.

Differentiating (*) by t , and using the Cartan formula for the Lie derivative of a form along a vector field Lie,,(wt) = d (g;(wt)) = d(i,,wt) iwtdwt,

+

we get Lie,, (wt) = d(i,,wt)

= wo

- w1 ,

(**)

where vt is the vector field of the flow g t , and i,w is the result of the substitution of the vector field v into the form w as its first argument. Using the “relative Poincar6 lemma”, it is possible to find a 1-form a so that d a = wo - w1 and a vanishes on G. Then a vector field vt which vanishes on G and satisfies (**) exists since wt is non-degenerate. Hence the phase flow of the field satisfies (*) and determines the required equivalence. 0 The Darboux theorem is a particular case of Givental’s theorem: take a point as a submanifold. Clearly, a global embedding of a submanifold yields more symplectic invariants. For instance, the following theorem is very useful in global symplectic geometry and topology.

Weinstein’s Theorem. A submanifold of a symplectic manifold is defined, up to a symplectomorphism of its neighbourhood, by the restriction of the symplectic form to the tangent vectors to the ambient manifold at the points of the submanifold. If at each point x of a submanifold L of a symplectic manifold M the subspace T,L is Lagrangian in the symplectic space T,M, then L is called Lagrangian.

334

Examples. 1. In T * N , the following are Lagrangian submanifolds: the zero section of the bundle, fibres of the bundle, the graph of the differential of a function on N . 2. The graph of a symplectomorphism is a Lagrangian submanifold of the product space, endowed with the symplectic structure which is the difference of the pull-backs of the symplectic structures on the factors. The graph has regular projections onto the factors. An arbitrary Lagrangian submanifold of the product space defines a so-called Lagrangian relation.

3. Weinstein’s theorem implies that a tubular neighbourhood of a Lagrangian submanifold L in any symplectic space is symplectomorphic t o a tubular neighbourhood of the zero section in T * L . The splitting of the ambient tangent space at n E L into the sum of the Lagrangian subspaces of vectors tangent and conormal to N yields the equivalence.

A locally trivial fibration with Lagrangian fibres is called Lagrangian. Locally all Lagrangian fibrations are symplectomorphic (the proof is similar to that of the Darboux theorem). A cotangent bundle is a Lagrangian fibration. Let 1c, : L + T * N be an embedding with a Lagrangian image and p : T * N 4N the fibration. The composition p o 1c, : L + N is called a Lagrangian mapping. Its critical values

EL = { q E N13p : ( p , q ) E L , rankd(p o 1c,) < n } form the caustic of the Lagrangian mapping. The equivalence of Lagrangian mappings is that up to fibre-preserving symplectomorphisms of the ambient symplectic space T*N . Caustics of equivalent Lagrangian mappings are diffeomorphic.

Hamiltonian vector fields. Given a real function h : M 4 R on a symplectic manifold, define a Hamiltonian vector field v h on M by the formula w(., W h ) = dh

335

This field is tangent to the level hypersurfaces H , = h-l(c): Y a € H,

*

dh(TaHc) = O

TaH,=v,L,

but

vh

E v ~ .

The directions of vh on the level hypersurfaces H, of h are the characteristic directions of the tangent spaces of the hypersurfaces. Associating functions:

Vh

to h, we obtain a Lie algebra structure on the space of [vh,vf] = v { h , f }

where { h , f} = vh(f)

7

the latter being the Poisson bracket of the Hamiltonians h and f .

A Hamiltonian flow (even if h depends on time) consists of symplectomorphisms. This is due to the Cartan formula Lie,,w = d(i,,w) = -ddh = 0. Locally (or in R2"), any time-dependent family of symplectomorphisms that starts from the identity is a phase flow of a time-dependent Hamiltonian. However, for example, on a torus R2/Z2 (which is the quotient of the plane by an integer lattice) the family of constant velocity displacements are symplectomorphisms but they cannot be Hamiltonian since a Hamiltonian function on a torus must have critical points.

x

Given a time-dependent Hamiltonian = x(t,p , q ) , consider the extended space M x T * R with auxiliary co-ordinates (s,t ) and the form pdq - sdt. An auxiliary (extended) Hamiltonian = -s determines a flow in the extended space generated by the vector field

+x

.

ax

p = -84

ax

q. = -aP

The restrictions of this flow to the sections t = const are essentially the flow mappings of Ti. The integral of the extended form over a closed chain in M x { t o } is pre= wnst are inserved by the k-Hamiltonian flow. Hypersurfaces -s variant. When is autonomous, the form pdq is also a relative integral invariant.

x

+x

336

A (transversal) intersection of a Lagrangian submanifold L c M with a Hamiltonian level set H, = h-l(c) is an isotropic submanifold L,. All Hamiltonian trajectories emanating from L , form a Lagrangian submani~ the Hamiltonian trajectories on H , fold ezpH(L,) c M . The space Z H of inherits, at least locally, an induced symplectic structure. The image of the projection of ezpH(L,) to E H ~ is a Lagrangian submanifold there. This is a particular case of a symplectic reduction which will be discussed later. Example. The set of all oriented straight lines in R; is T*S"-l as a space of characteristics of the Hamiltonian h = p 2 on its level p 2 = 1 in K = RZn.

1.2. Contact geometry

An odd-dimensional manifold MZn+' equipped with a maximally nonintegrable distribution of hyperplanes (contact elements) in the tangent spaces at its points is called a contact manifold. The maximal non-integrability means that if locally the distribution is determined by zeros of a 1-form cy on M then cy A (da)" # 0 (cf. the Robenius condition a A d a = 0 of complete integrability.) Examples. 1. A projectivised cotangent bundle PT*N"+' with the projectivisation of the Liouville form a = pdq. This is also called a space of contact elements on N . The spherisation of PT*Nn+' is a 2-fold covering of PT*Nn+' and its points are co-oriented contact elements. 2. The space J I N of 1-jets of functions on N". (Two functions have the same rn-jet at a point z if their Taylor polynomials of degree Ic at z coincide). The space of all 1-jets at all points of N has local co-ordinates q E N , p = df ( q ) which are the partial derivatives of a function at q, and z = f ( 4 ) . The contact form is pdq - d z .

Contactomorphisms are diffeomorphisms preserving the distribution of contact elements. Contact Darboux theorem. All equidimensional contact manifolds are locally contactomorphic.

337

An analog of Givental’s theorem also holds.

Symplectisation. Let %?2n+2 be the space of all linear forms vanishing on contact elements of M . The space G2n+2 is a “line” bundle over M (fibres do not contain the zero forms). Let

;ii:G-+M

E,

be the projection. On the symplectic structure (which is homogeneous of degree 1with respect to fibres) is the differential of the canonical 1-form Z on G defined as

Z(E) = P(?*E)

7

E E TPG

A contactomorphism F of M lifts to a symplectomorphism of &)

2:

:= ( G ( x ) ) - l P ,

where x = % ( p ) . This commutes with the multiplication by constants in the fibres and preserves Z. The symplectisation of contact vector fields (= infinitesimal contactomorphisms) yields Hamiltonian vector fields with homogeneous (of degree 1) Hamiltonian functions h ( n ) = r h ( s ) . Assume the contact structure on M is defined by zeros of a fixed 1-form ,B. Then M has a natural embedding x H ,Bxinto G. Using the local model JIRn, ,B = pdq - dz, of a contact space we get the following formulas for components of the contact vector field with a homogeneous Hamiltonian function K ( s ) = h(Px)(notice that K = ,B(X) where X is the corresponding contact vector field): i. = p K p - K , p = -Kq -pK,,

Q = Kp.

where the subscripts mean the partial derivations. Various homogeneous analogs of symplectic properties hold in contact geometry (the analogy is similar t o that between affine and projective geometries). In particular, a hypersurface (transversal to the contact distribution) in a contact space inherits a field of characteristics.

338

Contactisat ion. To an exact symplectic space MZn associate M^ = R x M with an extra co-ordinate z and take the l-form a = X - dz. This gives a contact space.

-&

is such that i,a = 1 and i,da = 0. Such Here the vector field x = a field is called a vector field. Its direction is uniquely defined by a contact structure. It is transversal to the contact distribution. Locally, projection along x produces a symplectic manifold.

A Legendrian submanifold A of MZn+' is an n-dimensional integral submanifold of the contact distribution. This dimension is maximal possible for integral submanifolds. Examples. 1. To a Lagrangian L

c T * M associate A c J I M :

A = {(Z,P,!?)

Iz=

s

pdq, h q ) E L l .

Here the integral is taken along a path on L joining a distinguished point on L with the point ( p ,q). Such a A is Legendrian. 2. The set of all covectors annihilating tangent spaces to a given sub-

manifold (or variety) WOC N form a Legendrian submanifold (variety) in PT*N.

3. If the intersection I of a Legendrian submanifold A with a hypersurface r in a contact space is transversal, then I is transversal to the characteristic vector field on r. The set of characteristics emanating from I form a Legendrian submanifold.

A Legendrian fibration of a contact space is a fibration with Legendrian fibres. For example, PT*N 4 N and J I N 4 JON are Legendrian. Any two Legendrian fibrations of the same dimension are locally contactomorphic. The projection of an embedded Legendrian submanifold A to the base of a Legendrian fibration is called a Legendrian mapping. Its image is called the wave front of A.

339

Examples. 1. Embed a Legendrian submanifold A into J I N . Its projection to JON, wave front W(A), is a graph of a multivalued action function J p d q c (again we integrate along paths on the Lagrangian submanifold L = 7rl(A), where 7r1 : J I N -+ T*N is the projection dropping the z co-ordinate). If q E N is not in the caustic C L of L , then over q the wave front W(A) is a collection of smooth sheets. If at two distinct points (p', q ) , (p", q ) E L with a non-caustical value q , the values z of the action function are equal, then at ( z , q ) the wave front is a transversal intersection of graphs of two regular functions on N . The images under the projection ( z ,q ) H q of the singular and transverand sosal self-intersection loci of W(A) are respectively the caustic called Maxwell (conflict) &.

+

2. To a function f = f ( q ) , q E R", associate its Legendrian lifting A = j l ( f )(also called the 1-jet extension of f) to JIR". Project A along the fibres parallel to the q-space of another Legendrian fibration

.rr:(w,q)

(2

-P%P)

of the same contact structure pdq-dz = -qdp-d(z-pq). The image 7rt(A) is called the Legendre transform of the function f. It has singularities if f is not convex. This is an affine version of the projective duality (which is also related to Legendrian mappings). The space PT*Pn (P" is the projective space) is isomorphic to the projectivised cotangent bundle PT*P"" of the dual space PnA. Elements of both are pairs consisting of a point and a hyperplane, containing the point. The natural contact structures coincide. The set of all hyperplanes in Pn tangent to a submanifold S c Pn is the front of the dual projection of the Legendrian lifting of S . 2. Generating families 2.1. Lagmngian case

Consider a co-isotropic submanifold Cn+' c M2". The skew-orthogonal complements T$C, c E C , of tangent spaces to C define an integrable distribution on Indeed, take two regular functions whose common zero level set contains C. At each point c E C , the vectors of their Hamiltonian fields belong to T t C . So the corresponding flows commute. Trajectories of all such fields emanating from c E C form a smooth submanifold I, integral

c.

340

for the distribution. By Givental's theorem, any co-isotropic submanifold is locally symplectomorphic to a cc-ordinate subspace p~ = 0, I = (1,. . . , n - k } , in K = R2". The fibres are the sets qJ = const.

Proposition 2. Let Ln and be respectively Lagrangian and coisotropic submanifolds of a symplectic manifold M2". Assume L meets C transversally at a point a. Then the intersection X O = L C is transversal to the isotropic fibres I , near a.

n

Proof. If TaX0 contains a vector v E Talc,then w is skew-orthogonal to TaL and also to T a C , that is to any vector in T a M . Hence v = 0. Isotropic fibres define the fibration E : C -+ B over a certain manifold B of dimension 2k (defined at least locally). We can say that B is the manifold of isotropic fibres. It has a well-defined induced symplectic structure W B . Given any two vectors u,w tangent to B at a point b take their liftings, that is vectors Z,i? tangent to C at some point of C-'(b) such that their projections to B are u and v. The value w(Z,5) depends only on the vectors u,v. For any other choice of liftings the result will be the same. This value is taken for the value of the two-form W B on B . Thus, the base B gets a symplectic structure which is called a symplectic reduction of the co-isotropic submanifold C.

Example. Consider a Lagrangian section L of the (trivial) Lagrangian fibration T*(Rkx R"). The submanifold L is the graph of the differential of a function f = f ( x , q ) , x E Rk,q E R". The dual co-ordinates y , p are given on L by y = p = Therefore, the intersection of L with the co-isotropic subspace y = 0 is given by the equations = 0. The intersection is transversal iff the rank of the matrix of the derivatives of these equations, with respect to x and q, is k . If so, the symplectic reduction of is a Lagrangian submanifold L, in T*R" (it may be not a section of T*R" + R"). This example leads to the following definition of a generating function (the idea is due to Hormander).

2,

g.

2

z

z

Definition. A generating family of the Lagrangian mapping of a sub-

341

manifold L c T*N is a function F : E over N such that

+R

defined on a vector bundle E

Here q E N , and x is in the fibre over q. We also assume that the following Morse condition is satisfied: dF 0 is a regular value of the mapping ( 2 ,q ) H - . dX The latter guarantees smoothness of L.

Remark. The points of the intersection of L with the zero section of

T*N are in one-to-one correspondence with the critical points of the function F .

Existence. Any germ L of a Lagrangian submanifold in T*R" has a regular projection to some ( p J ,41) co-ordinate space. In this case there exists a function f = f ( p ~ , q r(defined ) up to a constant) such that

+

Then the family FJ = xqJ f ( x ,q I ) , x E RIJI,is generating for L. If IJI is minimal possible, then Hess,, FJ = HesspJ p f vanishes at the distinguished point.

Uniqueness. Two family-germs Fi(z,q ) , x E Rk, q E R", i = 1,2, at the origin are called %-equivalent if there exists a diffeomorphism 7 : (x,q) H ( X ( x ,q), q ) (i.e. preserving the fibration Rk x R" -+ R") such that F2 = Fl o 7. The family @(x,y, q ) = F ( x ,q ) f y: f . . . ,fy$ is called a stabilisation of F . Two family-germs are called stably 720-equivalent if they are %-equivalent to appropriate stabilisations of the same family (in a lower number of variables).

Lemma 3. Up to addition of a constant, any two generating families of the same germ L of a Lagmngian submanifold are stably %-equivalent.

342

2.2. Legendrian case

Definition. A generating family of the Legendrian mapping T \ L of a Legendrian submanifold L c P T * ( N )is a function F : E + R defined on a vector bundle E over N such that

where q E N and x is in the fibre over q, provided that the following Morse condition is satisfied: aF 0 is a regular value of the mapping ( x ,q ) H { F , -} .

ax

Definition. Two function family-germs F i ( z , q ) , i = 1,2, are called V-equivalent if there exists a fibre-preserving diffeomorphism 0 : (z,q ) H ( X ( x ,q ) , q ) and a function Q ( x ,q ) not vanishing at the distinguished point such that Fz o 0 = QFl. Two function families are called stably V-equivalent if they are stabilisations of a pair of V-equivalent functions (may be in a lower number of variables z).

Theorem 4. Any g e r m T ~ Lof a Legendrian mapping has a generating family. All generating families of a fixed germ are stably V-equivalent.

3. Stability of projections of Lagrangian varieties 3.1. 0-stabilitg

We shall slightly modify the standard notions introduced earlier. 3.1.1. The Lagrangian setup

A singular Lagrangian (sub)variety L of a symplectic space M2" is an n-dimensional analytic subset of M which is Lagrangian in the ordinary sense at all its regular points. A Lagrangian projection T is a projection T : M -+ B" defining a fibre bundle whose fibres are Lagrangian. Fibres of any Lagrangian fibration posses a well-defined aEne structure.

343

Indeed, local co-ordinates on the base rise to regular functions on the total space, which are pairwise in involution. Hence their Hamiltonian vector fields do not vanish, commute and are tangent to the fibres. As before, the restriction n l ~of the Lagrangian projection n to a Lagrangian subvariety L c M is called a Lagrangian mapping. Two Lagrangian mappings, of Lagrangian subvarieties L' and L", are called equivalent if there exists a symplectomorphism of the ambient symplectic spaces sending L' to L" and fibres of one Legendrian projection to fibres of the other. In particular, L' and L" are symplectomorphic. The germ of a Lagrangian map 7 r I ~of a variety L at its point m is called stable if the germ of any Lagrangian map ZIL close to 7 r l ~at any point 6 close to m is equivalent to the germ of T I T J L at a point near m. Notice that only the fibration 7r is allowed to vary in this context while the subvariety L is fixed. According to Givental [9], the stability introduced is essentially equivalent to the following versality of the map-germ nl~. Let OL be the algebra of regular functions on L and m B , m the maximal ideal in the algebra O B , of~ function-germs on the base B at the point 7r(m).We define the local algebra of the germ of 7 r l ~at m as Qm = OL/(TlL)*(mB,m)OL*

The algebra Qm is the algebra of restrictions of functions on L to the intersection of L with the fibre F?r(m)= 7r-'(n(m)). Denote by Am the subspace of aEne (with respect to the corresponding f i n e structure) functions on the fibre F.cm) and by r : Am -+ Qm the restriction homomorphism sending a function on the fibre to its restriction to L n F?r(m). The germ of X ( L at m E L is called versa1if r is surjective, and miniversal if r is an isomorphism. Let p , q be local Darboux co-ordinates on M about m: p(m)= po, q(m)= 0 and n(p,q) = q. The Weierstrass preparation theorem implies that the versality of n l ~at m is equivalent to the existence of a representation of any analytic function-germ cp on M at m = (PO,0) in the form

c&?bj+ n

cpb,Q ) = $(P,4 ) +

j=1

ao(4) >

(1)

344

where the a j , j 2 0 are analytic function-germs on the base B , and the function-germ $ vanishes on L. The decomposition means that any function-germ on M at m is a sum of a function vanishing on L and a function affine on the individual fibres. Therefore, any Hamiltonian vector field near m is a sum of a Hamiltonian vector field tangent to L and a Hamiltonian vector field preserving the fibration n. Hence the homotopy method implies that any symplectomorphismgerm of M at m close to the identity is a composition of a symplectomorphism preserving L and a symplectomorphism preserving the standard projection n. Since any perturbation of the germ of n in the class of Lagrangian projections is a composition of n with an appropriate symplectomorphism, the versality implies stability. See [9] for more details. We now turn to a restricted version of the above setup. Namely, we take M = T*B to be the cotangent bundle of a manifold Bn,and distinguish the zero section TOof T*B. Let Symo(M) be the subgroup of symplectomorphisms of M preserving To. Two Lagrangian mappings of Lagrangian subvarieties of a cotangent bundle are called 0-equivalent if they are equivalent via a symplectomorphism from Symo(M). Replacement of the equivalence by the 0-equivalence in the stability definition yields a definition of the 0-stability of Lagrangian map-germs. The zero section To determines a linear structure on fibres of a cotangent bundle. Replacing the space A, of affine functions on F, by its welldefined subspace A; of linear functions, we obtain the definition of the 0-versality which is equivalent to the existence of the representation of any function-germ cp on M at m such that PIT^ = 0 in the form

where the germs a j and $ are similar to those in (1) and we assume that the Darboux co-ordinates p vanish on the zero section TO. Like before, the 0-versality implies the 0-stability. For the benefit of the exposition, we continue now with the complex case only. Everything below transfers absolutely straightforwardly to the real situation.

345

Lemma 5. The projection 7r : T*Cn -t C", ( p , q ) H q, of a Lagrangian germ L at the origin is 0-stable if and only i f the germs of the products pipj, i ,j = 1,.. . ,n, have decompositions n

PiPj = qij(P,q) -k c P k c t j ( q )

(3)

k=l in which the function-germs cpij and cfj are holomorphic, and the ish o n L.

cpij

van-

Proof. The "only if' part is obvious. To prove the "if" part, we notice that the ideal I generated by all the quadratic polynomials Pij(p) = pipj - C cfj(0)pk, i,j = 1 , . . . ,n, in the space of all holomorphic functiongerms on the fibre Fo is of finite codimension. Modulo I , any function-germ on Fo is an affine function in p. After the projection to the local algebra Qo, that is after a further reduction modulo the functions vanishing on L (more precisely on L f~ Fo), such a function is still d n e in p . Hence, the 0 0-versality condition holds.

For the stability (rather than 0-stability) version of the lemma see [9]. The suspension of a Lagrangian fibration uct

?=

7r :

M

--f

B is its direct prod-

h

(X,TO)

:M

= M x T * C - iB x C

with the canonical projection T O : T*C -i C . A suspension of a Lagrangian variety L c M2" is an (n+l)-dimensional Lagrangian variety L^ c M x T*C which is the product of L with the line C = {p,+l = const # 0 ) in T*C endowed with the standard Darboux co-ordinates p , + l , qn+l. The propositions below follow immediately from the definitions. Proposition 6. A map-germ 7 r l ~at m E M is (mini)versal if and only i f its suspension $12 is 0-(mini)versal at a point of the line m x I (hence at all the points of this lane) in M^. Example. A germ of the standard projection 7r of a Lagrangian submanifold L c T*C" = { ( p , q ) ) determined by a generating family f = f ( z , q ) with parameters q E C" and variables z E C k ,

L = {(P,q)Pz : af/ax = 0 , P = af/aq) 7

346

is stable if and only if the family-germ f is an R+-versal deformation of the function-germ f (-,0). The projection is @stable if f (., .) is an 77.-versa1 deformation of the function germ f (., 0). (a,

a)

Proposition 7. Consider a germ of a Lagrangian subvariety L in M^ = T*Cnx T*C at a point not in the zero section. Assume $ 1 ~ is 0-versa1 and L belongs to a regular hypersurface in M transversal to the l$,n+l -direction. Then $ 1 ~ is 0-equivalent to a suspension of a versa1 map-germ TILI of a Lagrangian subvariety L' c M = T*C". h

3.1.2. The Legendrian case

A singular n-dimensional subvariety of a contact space is called Legendrian if at all its regular points it is Legendrian in the ordinary sense. For standard (and equivalent) local models of contact (2n+l)-spaces we use the projectivised cotangent bundle PT*C"+l and the space J1(C", C ) = {p, q, z } of one-jets of functions on Cn endowed with the contact form a = d z -pdq. The definitions of Legendrian mappings, stability and others are analogous to the Lagrangian case (see also [9]). Symplectisation and contactisation functors relate Lagrangian and Legendrian germs as follows.

A. The projection p : ( p ,q, 2) H ( p ,q ) maps a Legendrian variety A c J1(C",C ) to the Lagrangian variety p ( A ) c T*C".

B. Local Lagrangian fibration and its zero section determine uniquely the Liouville primitive form a = pdq of the symplectic form w = d a . Given a Lagrangian germ L c T*Cn at a point m, denote by LO,, the subset of points s E L such that the integral of a along some path y on L joining m and s vanishes. For simplicity we assume the values of the integral do not depend on the local path y, that is the cohomology class of a vanishes on L (see 191 for examples of the opposite). If LO,, does not meet the zero section of T*Cn,then its projectivisation is a Legendrian (or isotropic) variety in PT*Cn. Its projection Wo(L,m) = LO,,) c C" is called the 0-wave front of L. C. For a Lagrangian germ L c T*C" at a point m, the set A L , C ~ J1(Cn,C ) of points ( p ,q, z ) such that s = ( p ,q ) E L and the integral of

347

a along a path in L joining m and s equals z is a Legendrian variety in J1(C",C ) .

A germ of a symplectomorphism 8 E S y m o ( T * C n ) preserving 7r preserves a. Hence if 8(L') = L" then 8(Lb,,) = L&,(,). In Darboux co-ordinates, 8 has the form:

8 : (P,4 ) l-4 (P,&))

1

where d is the underlying diffeomorphism of the base and P = ( e - l ) * p is the value at p of the linear operator on the fibres dual to the inverse of the derivative of 8. In particular, e(W,(L',m ) ) = Wo(L", 8 ( m ) ) . The proof of the following statement is straightforward.

Proposition 8. Consider a Legendrian germ A c J 1 ( C n , C ) . Assume the variety p ( h ) does not meet the zero section and that its standard Lagrangian projection is 0-stable. Then the projection of A to J o ( C " , is Legendrian stable. Conversely, if the projection of A to J o ( C n ,C ) is Legendrian stable and A is quasihomogeneous with positive weights then p ( A ) is 0-stable. h

c)

h

3.2. Stability of induced mappings

3.2.1. The critical-value theorem The images under a Lagrangian mapping 7 r I ~of singular points of the Lagrangian variety L along with the images of critical points of the restriction of 7 r l ~to the regular part of L form the caustic C L of the Lagrangian mapping. The caustic of a Lagrangian germ L at a point m of finite multiplicity p is a proper analytic subset of the base B of codimension at least 1. For q $! EL close to the distinguished point 7r(m),the inverse image 7r-I(q) L consists of p distinct points mi close to m. We can assume that locally IT is the standard fibration T * B -+ B. This allows us to introduce the Maxwell set M L C B as the closure of the set of the points q 4 C L for which the p values of z on AL,, n ( p o x ) - l ( q ) are not all distinct. If p is finite, the Maxwell set is a germ of a proper analytic subset of the base. The union of the caustic and Maxwell set is called the bifurcation diagram Bif(7r, L ) of the Lagrangian projection.

n

348

Consider the Lagrangian projection n : T*Cn + Cn of a Lagrangian variety-germ L. Let g : C k C" be a germ of a smooth mapping. If the choice of the base points of the germs is consistent, we define the induced Lagrangian mapping g*(nlL) as the projection of g*(L) c T*Ck to Ck. --f

Theorem 9. Assume the germs n l ~at m, m $! TO,and g*(rIL) are 0miniversal and 0-stable respectively. Then the critical value set Eg of the mapping g belongs to the union Wo(L,m )UBif(n, L ) .

Here we consider a source point of a mapping as critical if the derivative at the point is not surjective. In particular, all the source is critical if its dimension is less than that of the target, in which case the theorem implies that g maps C k into Wo(L,m) UBif(n, L ) . The stability analog of the theorem was proved in [16]. Proof. Take a point qo E C" \ CL close to the base point. Its nlL-inverse image consists of n distinct points ml, . . . ,m, E F,,, all different from the origin. The multi-germ of TI T ( L at the finite set { m l ,. .. ,m,} is 0-versa1 (the decomposition (2) holds for multi-germs). This is equivalent to the mi being linearly independent in the fibre F,,: the restriction of any function from the fibre to this set coincides with the restriction of a linear function. Consider now A0 E 9-l(qo). Let I c TqoC" be the image of the derivative g* : Tx,Ck -+ TqoCn.The pullback mapping g* : FZ --+ FX, between the fibres of the cotangent bundles is a composition of the factorisation pr of F,, by the subspace I' of covectors annihilating I and an embedding. Assume the dimension r of I v is positive, that is A0 is a critical point of g. The 0-stability of g*(nlL) implies that the pr-images of the linearly independent points ml, . . . ,m, E F,, form a linearly independent set in the ( n - r)-dimensional space F,,/Iv (the image points counted without the multiplicities). As a result, the vertex set {mo = O,ml,.. . , m,} of the n-simplex a c Fqo is mapped to the vertex set {mh = 0, mi,. . . ,mb-,,} of an ( n - r)-simplex in F,,/Iv. In particular, the rank r subspace I v is spanned by all the differences mi - mj such that pr(mi)= pr(mj),that is by the vectors in all the faces of c contracted by pr to points (the sum of the dimensions of such faces is r ) . Near each of the mi, i = 1,.. . , n, the Lagrangian variety L is locally the graph of the differential of a function z = $ i ( q ) , $(qo) =' 0. The linearly independent points mi E F,, are the differentials of the $i at qo.

349

For any pair i # j, denote by A , c Tg,C" the hyperplane tangent to the hypersurface $i(q) - $ j ( q ) = 0. For any l , let Ae c TgoC"be the hyperplane tangent to the hypersurface $e(q) = 0. The hyperplanes Ae and A , are dual to the directional lines of the 1-dimensional faces of the simplex r~ c Fg0. The condition for the multi-germ g*(T11;) to be 0-stable at the points of Fxo is equivalent to the subspace I being the intersection of all the Ae such that pr(me) = 0 and all the Aij such that pr(mi) = pr(mj) # 0. Hence I is the intersection of the subspaces in TgoC"dual to certain faces of the simplex r ~ .Since I belongs to the tangent cone at qo of the critical value set Z,, the regular strata of E, near qo coincide with the integral manifolds of the distributions defined similarly to I in the spaces TpCnby subsets of the faces of the relevant n-simplices in the fibres Fg. According to [4] (items 7.1 and 7.2), among such integral manifolds, those having the highest dimension and containing ~ ( min) their closures are the regular strata of the caustic, Maxwell set and, as it is easy to see, 0 wavefront Wo(L,m).Hence E, c Wo(L,m)UBif(T, L). Theorem 10. If g is a germ of a proper mapping between spaces of the same dimension, then the 0-stability of g * ( T I L ) is equivalent to g being a ramified covering with the ramification locus contained in WO( L ,m) U Bif ( T , L ). Proof. In this case the regular strata of E, are ( n - 1)-dimensional. By Theorem 9, the 0-stability implies the ramification property. To prove the converse it is sufficient to notice that outside the ramification locus the induced map g*(Tl1;) is 0-miniversal. Also it is versal at points of the regular strata of the ramification set, as it can be seen from the action of the pullback mapping g* on the corresponding simplex in the fibre. Hence any holomorphic function-germ p ( p , q ) possesses a decomposition (2) with the coefficients aj ( q ) uniquely determined on the complement of the analytic subset of codimension at least 2. Now Hartog's theorem extends the decomposition to an entire neighbourhood of the base point. n Remark. Assume the Lagrangian variety-germ L at m E T*Cn is a suspension of a Lagrangian germ L' at a point not contained in the zero section of T*C"-l. The base C" of the suspended Lagrangian fibration contains a distinguished co-ordinate function, let it be qn, corresponding to the second factor of the decomposition L N L' x C . The caustic and Maxwell set for

350

L are also isomorphic to the products of the caustic and Maxwell set for L' with a line, the q,-axis. On the contrary, the hyperplane tangent to the wavefront Wo(L,m) at m is dq, = 0. If, under the conditions of Theorem 10, the ramification locus zgcontains an ( n - 1)-dimensional component of the caustic or of the Maxwell stratum then the direction ,a, belongs to the image I of the differential of g at points arbitrary close to m. Hence the composition qn o g is not singular at the base point. On the other hand, if the ramification locus contains an (n- 1)-dimensional component of the wavefront Wo(L,m ) ,then the composition qn o g must be singular at the base point. Otherwise, the hyperplanes tangent t o Sg near the base point are not close to the hyperplane dq, = 0. 3.2.2. Composite functions

An interesting class of 0-stable Lagrangian projections is provided by versa1 deformations of composite mappings [6]. Given a function-germ f : (Cn,O) --+ (C,O) consider the group Icf (see [ 6 ] ) which consists of diffeomorphism-germs 0 of the product space (C" x Cn, (0,O)) fibred over the projection to the first factor 0 : (2, y ) +-+ ( X ( x ) , Y ( x ,y ) ) , z E C", y E C", and such that f ( Y ( z y, ) ) = f ( y ) for any ( 2 ,Y>. The group I C f acts naturally on the space of map-germs 'p : (C", 0 ) (C",0) sending the graph of one map to the graph of the other. --f

Assume a map-germ 'p at the origin has a finite Tjurina number r with = 'p(z) ~X,'p,(z), X E C', be a respect to the group I c f . Let @(.,A) ICf-miniversal deformation of 'p. Introduce the composition F = f 0 CP.

+

Theorem 11. The Lagrangian projection defined by the generating familygerm F ( x ,A) is 0-stable.

Proof. Let t E (C,O) be an additional parameter. Consider the deformation

of the composite function f o 'p. Since Fijlt=o = F and CP is Kf-versal, there exists a family of Kf-equivalencies depending on t and inducing Fij

351

from F :

Fij(x,X,t) = f o ( v ( x ( x , ~ , t + ) )~ ~ s ( x , t ) ~ ~ ( x ( x , .x , t ) ) ) s=1

Moreover, we choose the family so that for t = 0 the mapping (.,A) ( X ,A) is the identity mapping. Differentiating this equality with respect to t at t = 0 we obtain dF dF dF dX, dF dhk axi dXj

--=CGx+C%dt*

Since dFI8Xi = pi and d F / d x , = 0 on the Lagrangian variety defined by the generating family F , this means that the 0-stability criterium of Lemma 5 holds for it. Assume the germ at the origin of a composed function h = f o p has a finite multiplicity p. The deformation F = f o @ of h is induced from an R-miniversal deformation H of h by a map-germ g : (C',O) + ( C p , O ) between the deformation bases.

Corollary 12. If I- = p and the inducing mapping g is proper, then g is a covering ramified over the 0-wavefront of the 0-stable Lagrangian manifold defined by the generating family H . Proof. The claim is trivial when the function-germ f is regular (if so, the mapping g is a diffeomorphism). So we may assume that f has critical point at the origin. In this case the composition of g with the projection Cp -+ C along the hyperplane tangent to the discriminant of h at the origin is singular at 0 E C'. Now Theorems 1 1 , l O and the Remark after Theorem 10 imply the result. 0

Remark. Under the conditions of Corollary 12, the Kf-discriminant of cp is a free divisor. Example. The covering mapping inducing the determinantal function of a versal matrix deformation of a simple matrix singularity from a versal deformation of the determinantal function of the unperturbed matrix (see [12])is a particular case of Corollary 12. At this point, one should consider either symmetric matrices in 2 variables or arbitrary square matrices in 3 variables. Skew-symmetric matrices in 5 variables will also do.

352

The matrix setting of the Example has been generalised in [ll]to compositions f o cp with functions f not necessarily determinantal. One of the main results of [ll]states that p = T provided the critical locus of f is Cohen-Macaulay and has codimension m+ 1 in C". In this case the critical locus C of the inducing map g turns out to be the set of all those points in C' which correspond to perturbations of cp whose images meet the critical locus of f [8]. Clearly, g maps this set to the discriminant of the function f o cp which agrees with the theorems of section 3.2.1. Now, the space of linear functions on a fibre Fpof the cotangent bundle T*B + B is the tangent space T,B. So the functions c$ defined in (3) for a 0-versa1 Lagrangian map-germ determine a point-wise associative multiplication on the germs of vector fields on the base. When B is the base of a lcf-miniversal deformation this is exactly the multiplication considered in [8]. The only difference is that in [8] certain hypersurfaces were removed from B t o guarantee the multiplication has a unity. However, degeneracy of the multiplication is an interesting question on its own. For example, experiments suggest the following

Conjecture 13. Let 7 be the space of vector fields o n the base of a lcfminiversal deformation of a map-germ cp : (C", 0) + ( C n ,0 ) . Assume the critical locus of f is Cohen-Macaulay and has codimension m 1 in Cn. Assume also that the transversal type off is A l . Then

+

T~= Der(-ZogC) where C is the critical locus of the inducing map g. The inclusion 'T2 c Der(-loge) follows immediately from the results of [lo]. Perhaps this inclusion should not depend on the transversality type of f at all. The results of [lo] also indicate that the Conjecture can be generalised t o higher Ak transversality types if we increase to k the order of tangency of the fields in 'T2 to the relevant components of C.

References 1. V. I. Arnold, Critical points of functions on a manifold with boundary, the simple Lie groups Bk, Ck,F4 and singularities of evolutes, Russian Math. Surveys 33 (1978), no. 5, 99-116. 2. V. I. Arnold, Singularities of caustics and wave fronts, Kluwer Academic Publ., Dordrecht-Boston-London, 1990. 3. V. I. Arnold, Mathematical methods of classical mechanics, Springer Verlag,New-York, 1989.

353 4. V. I. Arnold, V. V. Goryunov, 0. V. Lyashko and V. A. Vassiliev, Singu-

5.

6. 7.

8.

9. 10. 11. 12.

13. 14. 15. 16. 17. 18.

larities II. Classification and Applications, Encyclopaedia of Mathematical Sciences, vo1.39, Dynamical Systems VIII, Springer Verlag, Berlin a.o., 1993. V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable maps. Vol. I , Monographs in Mathematics 82, Birkhauser, Boston, 1985. J. W. Bruce, O n families of symmetric matrices, Moscow Math. J. 3 (2003), no. 2, 335-360. J. W. Bruce, V. V. Goryunov and V. M.Zakalyukin, Sectional singularities and geometry of families of planar quadratic forms, in: Trends in singularities, 83-97, Trends Math., Birkhauser, Basel, 2002. I. de Gregorio and D. Mond, F-manifolds f r o m composed maps, to appear in: Sa6 Carlos-Sur-Mer, Proceedings of VIIIth Workshop on Real and Complex Singularities, Luminy 2004. A. B. Givental, Singular Lagrangian manifolds and their Lagrangian mappings, J. Soviet Math. 52 (1990), no. 4, 3246-3278. V. V. Goryunov, Logarithmic vector fields for the discriminants of composite functions, to appear in Moscow Math. Journal. V. V. Goryunov and D. Mond, Tjurina and Milnor numbers of matrix singularities, J. London Math. SOC.7 2 (2005), 205-224. V. V. Goryunov and V. M. Zakalyukin, Simple symmetric matrix singularities and the subgroups of Weyl groups A,, D,, E,, Moscow Math. Journal 3 (2003), no. 2, 507-530. V. V. Goryunov and V. M. Zakalyukin, O n stability of projections of Lagrangian varieties, Funct. Anal. Appl. 38 (2004), no. 4, 249-255. C. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics 151, Cambridge University Press, 2002. D. McDuff and D. Salomon, Introduction t o symplectic topology, Springer Verlag, 1995. R. M. Roberts and V. M. Zakalyukin, Stability of Lagrangian manifolds with singularities, Funct. Anal. Appl. 26 (1992), no. 3, 174-178. 0. P. Shcherbak, Wave fronts and reflection groups, Russian Math. Surveys 43 (1988), no. 3, 149-194. C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292(1992), no.4, 685-710.

354

A RESOLUTION OF SINGULARITIES OF A TORIC VAFUETY AND A NON-DEGENERATE HYPERSURFACE

SHIHOKO ISHII* Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo, Japan e-mail : shihokoOQmath. titech.ac.jp

This is an expository paper on resolutions of the singularities of toric varieties and non-degenerate hypersurfaces.

1. Introduction

This note is the sequel of “Introduction to basic toric geometry’’ [l] by Gottfried Barthel, Karl-Heinz Fieseler and Ludger Kaup. Based on their expository paper we focus here on a resolution of singularities and weak resolution of the singularities. A weak resolution of singularities of X is, roughly speaking, a non-singular variety which is proper birational over X and a weak resolution of the singularities of X requires a more condition that it is isomorphic away from the singular locus of X. When a singular variety X is given, the existence of a resolution of the singularities of X is an important and basic problem in algebraic geometry. This problem was solved by Heisuke Hironaka [4]for a variety over the base field of characteristic zero. After that many people are working in order to find a simple proof or a canonical way to resolve the singularities of a variety with the base field of characteristic zero. (see for example, [2], [3], [6],[8], [9]) On the other hand, if the base field has the positive characteristic, the existence problem of a resolution is still open. Here, we will construct a resolution of the singularities of a toric variety and a weak resolution of the singularities of a non-degenerate hypersurface by toric methods. We use the same notation and the assumption as in [l]. So we formulate all statement over the base field C. But we should note that every argument here goes well also for arbitrary algebraically closed base *partially supported by the grant-in-aid for scientific research

355

field. So we emphasize that “over the base field of arbitrary characteristic a toric variety and a non-degenerate hypersurface have resolutions or weak resolutions of the singularities”. The earliest reference the author knows on a weak resolution of a toric variety is [5]. The author was informed that a weak resolution of non-degenerate hypersurface had been first introduced by Khovansky. It is sketched in [lo]. This paper is organized as follows: in section two we construct a resolution of the singularities of a toric variety. In section three, we construct a weak resolution of the singularities of a non-degenerate hypersurface. In section four we introduce a canonical divisor on a normal variety which is used for the definition of canonical singularities and log-canonical singularities. In section five we make use of a weak resolution of the singularities of non-degenerate hypersurface to characterize canonical singularities and log-canonical singularities of a non-degenerate hypersurface in terms of the Newton polygon. 2. A resolution of the singularities of a toric variety

First we begin with the definition of a resolution of the singularities of a variety.

Definition 2.1. A morphism f : Y singularities of X if

-

X is called a resolution of the

(1) f is proper, (2) the restricted morphism f I Y \ ~ - I ( s ~ ~ ~ x ) SingX is an isomorphism and (3) Y is non-singular.

A morphism f : Y of

-

:Y

\ f-l(SingX)

-

X\

X is called a weak resolution of the singularities

X if (1) (3) above and the following (2’) are satisfied (2’) f is birational.

-

Theorem 2.2. (Hironaka 141) If X is a variety over a n algebraically closed field of characteristic zero, then there is a resolution f : Y X of the singularities of X .

As Hironaka’s original proof does not give a canonical way for resolving the singularities and also the proof is very difficult, many people later on tried t o construct a canonical resolution and also tried to give a simpler

356

proof ( [2], [3], [ 6 ] ,[8], [9]). On the other hand, for a variety over the base field of positive characteristic, the existence of a resolution is still open. Now we are going to resolve the singularities of a toric variety. Let N , M , Nw, Mw be as in [l].The natural pairing

( , ):MxN-Z can be extended to a pairing

(

, ):MwxNw+IW.

Let X A be the toric variety corresponding to a fan A. Recall that a subdivision A’ of A gives the torus-equivariant proper birational morphism f :XA, X A . Note that the condition (2) in the definition of a resolution of the singularities is a little bit stronger than just “birational”. So, for our purpose, it is sufficient to construct a subdivision A’ of A such that every regular cone in A remains in A’.

-

-

Theorem 2.3. For a toric variety X A , there is a torus-equivariant resolution f : X A , X a of the singularities of XA. This theorem is reduced to the following theorem:

Theorem 2.4. For a fan A in Nw, there is a regular subdivision A’ such that every regular cone in A remains in A’. Proof. First Step. First we construct a simplicial fan A1 by subdividing A. If A is simplicial, then let A1 = A. If A is not simplicial, then let the minimal dimension of non-simplicia1 cones be d and the number of such cones be s. As every one- or two-dimensional cone is simplicial, we have d > 2. Take an d-dimensional non-simplicia1 cone 6. By definition, every proper face of c is simplicial. Take a ray p generated by a lattice point in the relative interior of u, and make a Steller subdivision A’ with respect to p (see [l]).Here, we note that if a cone in A is simplicial, then it remains in A’ and u is replaced by finite number of simplicial cones. We should also note that the number of minimal dimensional non-simplicia1 cones in A’ is s - 1. By continuing Steller subdivisions on d-dimensional non-simplicia1 cones, we obtain a fan whose minimal dimension of nonsimplicial cones is greater than d. Then, perform the same procedures for minimal dimensional non-simplicia1 cones, then we obtain finally a fan A, which has no non-simplicia1 cones.

357

Second Step. For a simplicia1cone cr generated by primitive elements el,. . . ,e, E N , we define the multiplicity mult(a) as follows:

= #P,

nN ,

where P, is a parallelotope {Caiei I 0 5 ai < 1). Here, the number #Po n N coincides with the r-dimensional volume of Po.By definition, cr is regular if and only if mult(cr) = 1. Now, if there is a cone in A1 with the multiplicity > 1, take such D of maximal multiplicity. Then there is a non-zero point p in P,. Take the Stellar subdivision with respect to the ray R>op and obtain a subdivision of A1 such that the multiplicities of the new cones are less than that of cr. By continuing this procedures, all non-regular cones are subdivided into regular cones. Now we obtain a regular fan A'. We should note that in each procedure in the first and second steps we did not change any regular cones. 0 3. A weak resolution of a non-degenerate hypersurface

In order to introduce the concept of non-degenerate hypersurface we define first the Newton polygon.

Definition 3.1. Let M = Zn+l.Let f be a polynomial in (n+l)-variables . . ,xn. By using multi-variables xm = (mo,. . . ,m,) E Mnp, we represent f as follows:

X O ,XI,.

+

where a , E C. Let r+(f) be the convex hull of Uamz0(m RY):' and call it the Newton polygon of f. For a face y of r+(f), a polynomial fy is defined as follows: mEy

Definition 3.2. A polynomial f is called non-degenerate if for every comaf . . . ,$ have no common zero on pact face y, the partial differentials &, (C \ {0}),+'. Note that this is equivalent to that X O ~ . .,. ,znaz, af, have no common zero on (C \ {O}),+l.

358

+ + +

Example 3.3. A polynomial f ( 2 0 , . . . ,x,) = ?x . . . xpn is nondegenerate for every (mo,. . . ,m,) E Z,O. On the other hand, a polynomial 9(20,21,52) = (20 ~ 1 )xi ~is degenerate. Indeed, a segment y spanned by (2,0,0) and (0,2,0) is a face . can check that of the Newton polygon T'+(g) and gr = (20 2 1 ) ~One (i = 0,1,2) have common zero (1, -1,l) E (C \ { O } ) 3 .

+

2

+

Let A be the fan consisting of all faces of the positive octant' :R : Then, X A = en+'.

c Mw.

Theorem 3.4. Let H c Cn+l be a hypersurface of Cn+l passing through the origin 0 defined by a non-degenerate irreducible polynomial f E C [ X O.,. , x,]. Then, there exists a subdivision A' of A such that the restriction

-

(P(H' : H'

H

of the morphism 'p

:Xa,

XA = Cn+'

4

on the proper transform H' of H is a weak resolution of the singularities of H in a neighborhood of the origin. Moreover every orbit 0 , in ' p - ' ( O ) intersects H' transversally. Proof. Construction of A'. For a face y of I?+( f ) we define a cone y* in Nw as follows:

y* = {v

E ':R :

-

c NW

I (u,v) =

min (u', v) for every u E y}.

111

E r+( f )

It is easy to show that y* becomes an N-cone. Let C be the set of such y*'s. Then, C is a fan obtained by a subdivision of A. This fan C is called the dual fan with respect to the Newton polygon I-+(f). Next, subdivide C into a regular fan A' in the same way as in the proof of Theorem 2.4. Verification. Now we prove that A' is our required subdivision. For the first statement of the theorem, it is sufficient to prove that H' is nonsingular along cp-'(O). Take a cone 7 E A' such that 0, c q-'(O). For simplicity we assume that dim7 = 1 and 7 = R ~ o ~ Then o . a0 is in the interior of positive octant "5'; c Nw. Define r* c Mw as follows: r* = {u E r+(f) I ( u , v )= min (u',v) for every v u'Er+(f)

E 7).

359

+

Then, r* is a compact face of I?+( f ) . Take an ( n 1)-dimensional cone a such that T < a. Let a0 = (aoo, sol, ..,son), a1 = (alo,all, .., aln), . . ., a, = (a,~,a,l, ..,ann) be the generators of a in N . Let yo, y1,. . . ,yn be the coordinates of U,, = Cn+' corresponding to the dual basis a& a;, . . . ,a:. Recall that the coordinates system XO, . . . ,x, of X A = Cn+' is corresponding to the dual basis e,*,. . .,et of the canonical generators eo = (1,0,..,O),el = (0,1,0,.., 0) ,..,en = (0,.., 0 , l ) of c NR. By this we obtain the relation:

xi =Yoaoi Y1a l i . . .yEni On U,, the closure 0,is defined by yo = 0. Representing yo, .., yn on U,,, we obtain

..,

f(z0, 2,)

by

f (zo, ..,4 = YO"(9O(Yl,.',Yn) + Yo91(Yo, Y1, ..,Yn)).

+

Here, let g = go(y1, ..,yn) yo91(yo, y1, ..,yn). Then, HI is defined by g = 0 on U, and f,* = yrgo. Assume that HI is singular at a point P = (O,pl, . . , p , ) E 0,. Here we note that pi # 0 for i # 0. Then, for i # 0, we have (1) &O,Pl,

..,Pn)=

%(Ply

..,Pn) = 0

and

(2) S(O,Pl, ..,Pn)= So(P1, ..,Pn)= 0

We claim that (3)

for every i = 0, ,. .,n. This is proved as follows: For i # 0, by (l),we have

where the left hand side equals to

Now we obtain (3) for i # 0. For i = 0, by (2), we have

360

where the left hand side equals to

which yields (3) for i = 0. Here, the matrix (aij) is regular, we have xj*(l,pl, ..,pn)= 0 , which is a contradiction to the assumption that f is non-degenerate. To prove that H’ intersects 0, transversally, it is sufficient to prove that the locus 910, = 0 in 0, is non-singular. Note that 910, = go. If (PI, ..,p,) E 0, is a singular point of the locus of go = 0. Then it follows the equality ( 1 ) and (2). So, by the same argument as above we obtain the 0 contradiction.

Example 3.5. Consider the hypersurface H c C 3 defined by f = xoxlxz+ xt+xf + x i = 0. Then H has an isolated singularity at the origin. The dual f) is as Figure 1. Here, the big triangle shows fan of the Newton polygon r+( the three dimensional cone corresponding to C 3 and the three points inside show the new one-dimensional cones in the dual fan. These three points is the “dual” of the 2-dimensional faces of the Newton polygon r+( f). Now, subdivide the dual fan into a regular fan as in Figure 2. This gives the resolution of the singularities of H . Here, we note that the single blow-up at the origin also gives a resolution of the singularity of H and it is much simpler than by using the above toric method. Our toric method requires the normal crossings of H and the exceptional divisors and therefore we need more steps. 4. Canonical divisors

In this section we introduce canonical divisors on a normal variety X . First, we start with a non-singular variety.

Definition 4.1. Let X be a non-singular variety of dimension n 2 1 and P E X a point. Take a rational function g on X and 2 1 , . . . , x , regular parameters at P. Let w = gdxl A dxz A . . .A dx, be a rational form. Cover X by open covering {Ui} with regular parameters y i l , . . . , gin on Ui. For every i, there is a rational function f i on X such that w = fidyil A

. . . A dyin.

A divisor K on X is defined by Klui := div( fi)lui

36 1

for every i. We should note that K is well defined. Indeed, on Ui n U j , f i / f j and f j / f i are both regular functions, which yields div(fi) = div(fj).

Remark 4.2. K depends on the choice of X I , . . .,x, and g. But the linearly equivalence class does not depend on the choice of X I , . . . ,x, and g. This equivalence class or a divisor belonging to this class is called a canonical divisor and denoted by K x . Definition 4.3. Let X be a normal variety. (Then the singular locus SingX is of codimension greater than 1). Let Kx\singx = aiDi be a canonical divisor on X \ Sing X, where Di's are irreducible Weil divisors on X \ Sing X. Define a canonical divisor K X of X by:

xi

i

where

is the closure of Di in X.

Prop 4.1. Let X = Xa be a toric variety and Di (i = 1,.. . , r ) the irreducible invariant divisors, i.e., Di = 0,for a one-dimensional cone T E A. Then T

i=l

-

Proof. The non-singular part X\Sing X is covered by {X,}'S, where T E A

is regular. It is sufficient to prove that K x - C&,Di on X I , .. . , x, be the coordinate system of T c X. Put W =

1

UX,.

Let

dXl dxn dxl A . . . Adzn = - A . . . A -.

51 "OX,

Xn

21

For a regular cone T , let y1,. . . ,ys, ys+l, ..,yn be the coordinate system of xT -- CS x C*"--" . Note that the union of the invariant divisors is defined by y1. * - y s= O on X,. As ys+l,. . . ,yn are units, we can say that the union of invariant divisors is defined by y1. yn = 0 on X,. We have y'a -- x?li . . . xEni for a unimodular matrix ( a j i ) . By this we obtain dyi - dxali 1 . . . XEni _ . .. Yi

Therefore,

xali

1

x$i

=

C j

dxj aji-.

xcj

362

Hence, we have

Prop 4.2. (Adjunction formula) Let X be a non-singular variety and H a normal subvariety of codimension one. Then

KH

-

( K x + WIH,

in particular K H is a Cartier divisor.

-

Proof. By the definition of a canonical divisor on a normal variety, it is sufficient to prove that K H ~( K~x ~H ) ~ H , . , where ~, HTegis the nonsingular part of H . So we may assume that H is also non-singular. For a point P E H c X , take regular parameters 51,. . . ,xn on a neighborhood V of P such that H = (21 = 0). Let

+

dXl

w = - Adz2 A

.. . A dxn.

51

Let {Ui}i be open subsets of X such that H regular parameters on Ui. Then W =

x1

c

Ui Ui.

Let y1,. . . , yn be

dyl A . . . A dyn

be the restriction of xi and yj onto H . Denote For i,j 2 2 let and x1 = ylu, then u is a unit on Ui n V . Then, for a point Q E H n Ui n V ,

Therefore, it follows that

WIH

(2)$ (2); -

= u-'udet

= det

-

-

Adyz A . . . A d y n

A dy2 A

.. . A dy,.

363

Here, we note that K H

N

div (det

( g ).)By this we obtain that KXIH

KH - HIH.

= 0

Now we are going to introduce certain singularities in terms of a canonical divisor.

Definition 4.4. A normal variety X is called a Q-Gorenstein variety if m K x is a Cartier divisor for some m E N. The minimal such m is called the index of X .

-

Definition 4.5. Let X be a Q-Gorenstein variety of index r. For a surjective morphism cp : Y X we define the pull back of K x by

-

Definition 4.6. A normal variety X is said to have canonical singularities (resp. terminal singularities) if for every weak resolution cp : Y X of the singularities of X m

K y = cp*Kx

+X a i E i

with ai

L 0 (resp. ai > 0),

i=l

where Ei’s are irreducible exceptional divisors for cp.

Definition 4.7. A normal variety X is said to have log-canonical singularities (resp. log-terminal singularities) if for every weak resolution cp : Y X of the singularities of X such that the exceptional set is a normal crossing divisors

-

m

K y = cp*Kx

+X a i E i

with ai 2 -1 (resp. ai

> -l),

i=l

where Ei’s are irreducible exceptional divisors for cp.

Remark 4.8. (1) In the above four definitions on singularities the conditions are for all weak resolutions or all weak resolutions such that the exceptional set is a normal crossing divisors. But, these are equivalent to the conditions for “one” weak resolutions. (2) These singularities are important in birational geometry in particular the minimal model problem. For further study on birational geometry, the reader can refer [7].

364

Example 4.9. For a normal two dimensional variety X , we have the following characterizations: (1) X has at most terminal singularities if and only if X is non-singular. (2) X has non-terminal canonical singularities if and only if X has A , D ,E singularities. (3) X has log-terminal singularities if and only if X has a quotient singularities, i.e., it is locally isomorphic to the quotient C 2 / G by a finite group.

5. Singularities on a non-degenerate hypersurface In this section we study the singularities on a non-degenerate hypersurface. The problem is what kinds of properties on the Newton polygon characterize the singularities. Here, we note that for a hypersurface H c Cn+l a canonical divisor K H is a Cartier divisor by the Adjunction formula (Proposition 4.2). Therefore, if we write K H = ~ cp*(K~) aiEi for a weak resolution cp : H' H of the singularities of H , then ai E Z for every i. By this, H has canonical singularities if and only if H has log-terminal singularities. Let N be Zn+l and A the fan consisting of all faces of the positive octant c NR. Then X A = Cn+l and by Theorem 3.4 there is a subdivision At of A such that the restriction

-

+ czl

of the morphism cp : X ' = xa,+ Xa =

cn+l

on the proper transform H' of H is a weak resolution of the singularities of H in a neighborhood of the origin. Moreover every orbit 0, in cp-'(O) intersects H' transversally. Therefore $ = (PIH, is a weak resolution such that the exceptional set is normal crossing divisor. Here, we state the final theorem:

Theorem 5.1. Let H c Cn+' be the hypersurface defined by a nondegenerate irreducible polynomial f E @[SO,. . . ,z,]. Then,

(i) H has canonical singularities around 0 if and only if 1 = (1,1,.. . ,1) E I?+( f ) " c MR, where r+(f)" is the interior of the Newton polygon I?+ ( f ) (Figure 3);

365

(ii) H has log-canonical singularities around 0 if and only if 1 = (1,1,. . . ,1) E r+(f) c MR (Figure 4). In order to prove the theorem we need some lemmas.

Lemma 5.2. Let A(1) and A’(1) be the set of one dimensional cones in A and A’, respectively. I t follows that

K x , = cp*Kx

c

+

((v,1) - 1) D,,

R~O~EA’(~)\A(~)

where D, = OR,^^ is the irreducible invariant divisor corresponding to the one dimensional cone I w ~ o v . Proof. As cp is isomorphic away from the exceptional divisors, K x , = cp*Kx outside of the divisor

u

D,.

RtovEA’ ( l ) \ A ( l )

Hence, we have

Kxt=(p*Kx+

m,D, R>ov€A’(l)\A(l)

for some m, E Z. The problem is to determine mu for every R~owE A’(1) \ A(1). By Proposition 4.1,

Kx =-

c

D, = div(z0 q . . .x n ) ,

W>ouEA(l)

therefore we have

cp*Kx = -div(so . s1 ex,) = -

(v,l)&. WZovEA’(1)

Hence, it follows that

K x ~=cp*Kx+

mvDu R>ow€A’(l)\A(l)

On the other hand, by Proposition 4.1

366

Therefore we obtain m,

=

(v,1) - 1.

0

Definition 5.3. For a primitive element v ( f ) = min{(v,u) I 21 E F+(f)).

E N we define

v(f) as follows:

Lemma 5.4.

K H= ~ $*KH

+

((v,1) - 1 - v(f)) D,IH~. W>OVEA'(l)

Proof. First we note that p*(H) = H' junction formula we obtain

KH= ~ (Kxt

+ CWtovEA,(l) v(f)D,.

By Ad-

+ H')JH~,

Then, by using the lemmas it follows that

= @*((KX

+H)IH) +C W ~ ~ , E A l) /-(1~-)v(f))DvIH'. ((~,

0

Proof. [Theorem 5.11 Assume that 1 E F+(f)", then for every v E A'(1) \ A ( l ) , it follows that (v,1) > v(f) by the definition of v(f). Then

(v,1) - 1 - v ( f ) 2 0, which shows that the coefficient of every exceptional divisor is non-negative in K H / by the previous lemma. This shows that H has canonical singularities around 0. Conversely, assume that 1 $ I'+(f)". Then there exists a primitive point v E N n R:il such that (v , 1 ) 5 v(f). We can construct a subdivision A' of the dual fan of r+(f) such that 1W>~,v E A'. Then,

(v,1) - 1 - 4.f) < 0, which shows that the coefficient of the exceptional divisor D, is negative in K H , by the previous lemma. This implies that H does not have canonical singularities around 0. The second statement follows in a similar way. 0

367

Figure 1.

Figure 2.

368

Figure 3.

Figure 4.

References 1. G. Barthel, K-H. Fieseler and L. Kaup, Introduction t o basic toric geometry. In this volume. 2. E. Bierstone & P. Milman, A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry, Progress in Math. 94, Birkhauser (1991) 11-30. 3. E. Bierstone & P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128, (1997) , 207-302 4. H. Hironaka, Resolution of singularities of a n algebraic variety over a field of characteristic zero, I, 11, Ann. of Math. (2) 79 (1964) 109-203, 205-326. 5. G. Kempf, F. Knudsen, D. Mumford & B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Math. 339, Springer Verlag (1973). 6. J. Kollir, Resolution of singularities-Seattle lecture,

369 arXiv:math.AG/0508332. 7. J. KollAr & S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134 (1998) 8. 0. Villamayor, Pathching local unifomizations, Ann. Sci. l’Ecole Norm. Sup. (4)25, (1992) 629-677. 9. 0. Villamayor, An introduction to constructive disingularization, arXiv:math.AG/0507537. 10. A. N. Varchenco, Zeta-function of monodromy and Newton’s diagram, Inv. Math. 37 (1976) 253-262.

370

PROBLEMS IN TOPOLOGY OF THE COMPLEMENTS TO PLANE SINGULAR CURVES

A.LIBGOBER* Department of Mathematics University of Illinois at Chicago, 851 S.Morgan, Chicago, IL, 60607, USA E-mail: libgoberOmath.uic.edu

The paper discusses the statements and the background of several problems in the topology of plane singular algebraic curves.

1. Introduction

The purpose of this note is to discuss what it seems to me presently are the main issues which have to be resolved in the topology of plane singular curves. This area was quite active during the last 25 years after 40 years of hibernation which followed its birth in the works of Zariski and Enriques in 20’s and 30’s. The state of affairs in the end of 70’s is beautifully presented by M.Artin and B.Mazur in introduction to Zariski’s Collected papers (cf. 48, section 7. The Fundamental Group p.6-1) and also by D.Mumford in the appendix to 47. Artin and Masur’s survey contains numerous problems. The central issue, as stated by the authors then, still remains the same now: “What can be said about the Poincare group G, that is, the fundamental group of the complement of C (i.e. an algebraic curve) in Pa” However the status of many concrete problems completely changed and others can be stated in a much more specific way. For example the problem of “irreducibility of the family of plane curves with nodes and commutativity of Poincare group” was solved by J.Harris (6.20) which followed the proofs of commutativity by Fulton l8 and Deligne lo. Subsequent works by Z.Ran (6.41) and M.Nori (cf 35) provided alternative views on these issues. Also the problems described in the section on “Cyclic multiple planes and knot ‘Work partially supported by NSF

371

theory” in 48 completely addressed by the theory of Alexander invariants with its topological (”3) and algebro-geometric aspects (24, 31,34). My list of problems is very much incomplete and has many important omissions but I hope that it addresses at least one of the goals which I had in mind in selecting particular problems which was to show the relations between the topology of plane singular curves and other areas such as topology of arrangements, symplectic geometry etc. Before stating each problem I often presented background and in principle much of this can be read without extensive preparation. Some additional details of the background are presented in my lectures in Lumini (cf. 30). In fact these notes can be viewed as a supplement to the latter. The book l1 also contains much of the needed background. It is important to note that a large part of the issues raised in Zariski’s work in the 30s and discussed by Artin and Masur were generalized to higher dimensions with the fundamental group being replaced by the higher homotopy groups (an aspect totally absent in the late 70s). I refer to 30 for discussion of these developments and related questions. Many of the problems from our list were discussed during the workshop in Trieste and I want to thank its participants for lively discussions. I also am grateful to referee for careful reading of the manuscript and the comments. Finally, I want to thank M.Oka for his encouragement to write a contribution to these proceedings. Without him, these notes would not have been written. 2. Alexander Invariants

We shall start by describing problems related to Alexander invariants of plane algebraic curves. While a characterization of the fundamental groups is very difficult, if not impossible, (cf. section 3), the Alexander polynomial of the fundamental group is much better understood a. Recall a definition of this invariant. Let C C C2be an algebraic curve. The fundamental group 7r1 (C2 - C) has a canonical surjection onto Z sending a loop into its linking number with C. Let C2 - C + C2 - C be the corresponding cover and let K = Ker(7r1(C2 - C) + Z). The group Z acts on the abelianization of K/K’ of K which makes K/K’ 8 Q into a Q[Z] = Q[t,t-’] module. It has a cyclic decomposition K/K’ 8 Q = @iQ[t,t-’]/(Ai(t)) for some

-

aThough it does not give complete information on the type of equisingular deformation of a curve cf. 45 , 38

372

Laurent polynomials &(t)defined up to a unit in Q[t,t-l]. The Alexander polynomial of C is defined as

Ac(t)= IIiAi(t). The polynomial Ac(t) is affected by the local type of the singularities of C as follows. For each singular point P of the curve C let us consider the Alexander polynomial Ap(t) of the link in a small 3-sphere in C2 centered at P. If the closure of C in P2is transversal to the line at infinity then one has the following divisibility property:

&(t)

I nPESingCAp(t).

(1)

In particular if C is irreducible and all its singularities are either nodes or cusps (i.e. locally given by x2 y2 = and x2 y3 = 0 resp.) then A,(t) = (t2- t 1)"for some non negative integer s. One can also define the Alexander polynomial at infinity Ac,,(t) as the Alexander polynomial of the link C n S c S where S is a sphere of sufficiently large radius in If C is C2. The global Alexander polynomial Ac(t) divides A , , ( t ) . transversal to the line at infinity and has degree d this is equivalent to

+

+

Ac(t)

+

I (td- l)d-2.

2.1. Realization PTOblems

Problem 2.1. For which polynomials P(t) E Z [ t ,t-'1 does a n algebraic curve C exist such that P(t) = Ac(t). In particular (i) what is the m a x i m u m s(d) of integers s such that there exists a curve C of degree d transversal to the line at infinity, having nodes and cusps as the only singularities and such that Ac(t)= (t2- t 1)8? (ii) Does there exist a bound o n s(d) which is independent of d ? (iii) W h a t are inequalities f o r the degrees of the Alexander polynomials f o r other classes of curves with fixed type of singularities (inculding more general ones than the nodes and cusps)?

+

The largest value for s in (i) which I know is 3: it is achieved for the sextic with 9 cusps (i.e. the dual curve to a non-singular cubic). For nodal curves, since the fundamental group q ( C 2 - C) is abelian, the answer to (iii) is yes ( A ( C )= 1). In 38 it was shown that 4 6 ) = 3 (cf. also ', 36, 37) There is a reformulation of the problem 2.1 which does not involve the Alexander polynomial but is based on the relationship between the latter

373

and the position of singulaxities. With each singular point P E C C P2 in 24 we associated the collection of rational numbers I C ~(called the local constant of quasiadjunction) and the ideals Jn,p,c in the local ring Oppa depending on the local type of the singularity of C at P. In the case of a node this collection is empty and in the case of a cusp it contains single number The corresponding ideal J+,p,c is the maximal ideal of the local ring at P E P2. These ideals define the ideal sheaf Jn,con P2 with the stalk at any P E SingC being the smallest ideal Jn,,p,cwith K’ 5 K and the stalk in any other point P $! SingC coinciding with the whole local ring O p , p ~Then . the Alexander polynomial has the following expression (here E is the set of all local constants of quasiadjunction of all singularities of

i.

C) : A c ( t )=

n

[(t - e x p ( 2 n i n ) ) ( t- e z p ( - 2 n i ~ ) )dimH’ ] (P2,J, ( d - 3 - 4 )

. (3)

KEE

If C has nodes and cusps as the only singularities then dimH1(P2, J ; ( d - 3 - g ) ) = dimHo(P2,J + ( d - 3 - t ) ) - x ( J + ( d - 3 - $ ) ) . The first term in this difference is the dimension of the space of curves of degree d - 3 - $ passing through the cusps of C while the second is the “expected” dimension of this space. Such difference is called the superabundance s of the family of curves. Hence the problem 2.1 is equivalent to the question on possible values of the superabundances of linear systems of curves of degree d - 3 - passing through the cusps of a curve of degree d. A similar reformulation of the problem 2.1 can be made for curves with arbitrary singularities using results of 24 which we leave to the reader. An interesting version of the realization problem deals with twisted Alexander polynomials. In the authors extended the study of twisted Alexander polynomials in knot theory and considered Alexander polynomials associated with a plane curves and a linear representation of the fundamental group. Problem 2.2. (i) Find an expression for the twisted Alexander polynomial generalizing (3). (ii) What is the class of polynomials which can appear as twisted Alexander polynomials of plane curves. 3. Fundamental Groups

One of the oldest problem in the study of the topology of complements is the problem of characterization, in some sense, of the fundamental groups of the

374

complements to plane curves. The lack of examples which was acutely felt in the past (6.2 5 ) has been addressed by now in many respects though there is still a real possibility that we so far have only been probing very special curves. The cyclotomic property of Alexander polynomials rules out many groups as possible candidates for the fundamental groups of the comple ments (27). Below is one very specific question (mentioned by D.Mumford in the footnote to 47) on general properties of fundamental groups of the complements. 3.1. Residually finiteness

Problem 3.1. Are the fundamental groups of plane curve residually finite? In other words: does the intersection of subgroup of finite index consist of the identity only. An equivalent formulation: is the map x1(P2 - C) + n;"(P2 - C) of the topological fundamental group into the algebraic one injective. It is known that non-residually finite fundamental groups can appear as the fundamental groups of algebraic surfaces (46). 3.2. Finite groups

Problem 3.2. Which finite groups can appear as the fundamental groups of the complements? What are finite fundamental groups of the complements to the curves with nodes and cusps only? Some restrictions on the finite groups in the second class come from the divisibility theorem for the Alexander polynomials over finite field i.e. the relation (1) in which one uses the Alexander polynomial defend using the the homology groups with the coefficients in a finite field. A quartic with 3 cusps in P2 has finite fundamental group of the complement (the 3 strings braid group of sphere). Interesting examples of finite groups of the complements are due to Oka (6.39). 3.3. Braid monodromy

Braid monodromy is a more subtle invariant of singular curves than the fundamental group of the complement. Recall its definition. Let us consider the complement to C in an f i n e (rather than projective) plane C2. In the case when the line at infinity is generic, the fundamental group of the complement to projective curve is a quotient of nl(C2 - C n C2) by

375

a subgroup generated by a central element (cf. 27). Also, the classes of equisingular isotopy of curves in P2correspond to the classes of equisingular isotopy of curves transversal to the line at infinity. Let 1 : C 2+ C be a generic linear projection and Zc : C + C be its restriction on C. One has in the target C of I , a subset R consisting of points for which lc has fewer than d (d = degC) preimages. Genericity of 1 also implies that for any T E R, Z-l(r) is not a flex and all but exactly one point in ZF1(r) n C are the transversal intersections at a non singular point of C. Given a path y : [0,1] + C - R we shall restrict the map Z to Z-l(y) and pick a trivialisation of the locally trivial fibration of pairs 1 : (Z-'(y),Z;'(y)) + y. If y is a loop, i.e. y(0) = y(1) then we obtain a diffeomorphism of C = Z-l(y(0)) fixing the finite subset lZl(y(0)) = Z-l(y(0)) n C. We shall pick this diffeomorphism to be constant outside of a disk of a large radius in Z-l(y(0)). The group of isotopy classes of such diffeomorphisms is isomorphic to the Artin's braid group B d on d strings (6.30). This assignment of a braid to a loop in C - R is independent of the choice of y up to homotopy. In fact we obtain the homomorphism:

/3 : xl(C - R)+ B d

(4) called the braid monodromy (its composition with the map of the braid group onto the symmetric group gives the classical Ed valued monodromy of the multivalued function corresponding to C with the domain being the target of 1). It is useful to describe the homomorphism (4) by its values on special generators of (C - R).

Definition 3.1. An ordered good system of generators of nl(C - F ) where F is a finite subset of C is defined as follows. For i = 1, ...,Card F , let Si be the counterclockwise oriented boundary of a sufficiently small disk Ai about a point fi C F (small implies that Ai r l A j = O,i # j ) . Let pi be a collection of non-intersecting paths in C - UAi each having a base point b as the initial point and a point on Si as the end point. Then the ordered good system consists of loops pT1 oSi opi ordered counterclockwiseby their intersections with a small disk about b. It turns out that the product of braids ,b(yi) in the above ordering does not depend on the choice of generators yi of xl(C - R):

376

(A2 is the standard generator of the center of Artin's braid group). The set of ordered good systems of generators of 7rl (C - F, b ) is acted upon by the braid group BCardF via (oiE BCardF are the standard generators):

....,

oi('Y1, * * * ' Y i , " f i + l , 'YCardF)

= (71,

....,'Yi-l,'Yi'Yi+l'Y~1,'Yi,'Yi+2,

**.,'YCardF)

(6) (Hurwitz action). The orbit of this action on a vector

(P('Yl),

P('YCardR))

(where R, as above, is the ramification locus of Zc) is independent of the choice of 1, the ordered good system of generators of x 1 ( C - R), the base point of the latter or a curve within its class of equisingular deformation. So is the case for a simultaneous conjugation:

(P('Yl), ...,P('YCardR)

(ob('Yl)o-',

...,oP(YCardR)o-l)

(0

E

Bd)

(7)

The fundamental problem is the following: Problem 3.3. Characterize factorizations (5) of A2 obtained as in the above construction for algebraic curves. Describe the orbits of Hurwitz action and conjugation on the CardF-fold product B d X ... x B d . Some restrictions were proposed in 32. Let, as above, oi,i = 1, ...,d - 1 be the standard generators of Bd. For the curves with cusps and nodes as the only singularities each /I(? is conjugate ;) to 01 (respectively of or o!) provided that Si is aloop about the point T E R such that CnZ,'(r) contains the tangency point (respectively the node or the cusp). Some restrictions on the number of factors @ ( ~ i ) of each type come from the restrictions on the number of nodes and cusps a given curve can have (cf. 33) and also from Plucker formulas. Closely related to the problems 3.3 is the problem of constructing invariants of the braid monodromy which are invariants of Hurwitz action and conjugation and hence yield invariants of the curve. One method was proposed in 26 (cf. for other invariants of braid monodromy). Let

p :B d + GLk(h) be a linear representation of the Artin's braid group over a commutative ring A. The free module Ak can be considered as a n-1 (C - R)module using the action of the latter given by composition of braid monodromy and p . Then, the A-module Ho(7r1(C - R), p ( P ) ) is invariant under Hurwitz moves

377

and hence is an invariant of the curve. The canonical presentation of this A-module is given by:

Using this presentation one can calculate such invariant of braid monodromy as the A-torsion of Ho(nl(C- R),p(P)). In the case, when p is the reduced Burau representation of the braid group (over Q ) , i.e. A is the ring of Laurent polynomials Q[t,t-'1, the order of the Q[t,t-']-torsion of Ho(nl(C - R ) , p ( P ) ) (which is a Laurent polynomial) is related to the Alexander polynomial of the curve as follows (cf. 26): Ordq[t,t-l$&,(nl(C - R),p(P)) = ( 1

+ t + ... + tdegc-l)Ac(t)

(9)

Problem 3.4. Calculate the module Ho(nl(C-R), p@)) f o r non Burau representations of the braid group e.g for Lawrence representations and find their geometric interpretation (a counterpart to the above relation with superabundances of linear systems).

e2)

3.4.

of the complements to generic projections

An important class of plane curves with cusps and nodes consists of the branching curves of generic projections. They come up as follows. Let S c PN be an algebraic surface and ns be the restriction of a projection on P2from a generic subspace PN-' c PN. The branching curve Brs of ns consists of points having fewer than d = degS preimages. The equisingular isotopy type of the curve Brs depends on (the deformation type of) S only. If the embedding of S C PN is canonically associated with S (i.e. corresponds to the embedding with a fixed multiple of the canonical class) then nl(P2 - CS)is an invariant of the deformation type of S. In the case when the surface is a non-singular degree n hypersurface V, c P3the fundamental group n1 (P2-Brv,,) is isomorphic to the quotient of the Artin's braid group by its center (cf. 32). However for surfaces having higher codimension the corresponding groups can be much simpler. We shall state several results due to A.Robb and M.Teicher on the structure of the group nl(C2 - BTSn C2) where C2 is the complement to a generic line in P2. One has a surjection 0s :nl(C2 - Brs r l C2) + Ed@ onto the symmetric group &gS and also the surjection abs onto its abelianization Z. Let ITS = Keras nKerabs be the intersection of the kernels of these two n n+3 surjections. In the case when S is the image of P2 in P w embedded

378

using Op2(n) (n 2 3) (Veronese surface) the group IIs is solvable (cf. 43). So is IIs if S is a non-singular complete intersection in P” which is not a hypersurface (cf. 42). In fact in the latter case IIs has Z2 as its center and II/Z2 is free abelian). These and other calculations lead to the following: Problem 3.5. (M.Teicher, cf. 44) For which simply connected surfaces of S c PN is the fundamental group of the complement to the branching curve of generic projection is almost solvable i.e. contains a solvable subgroup of finite index.

It is expected that this is the cme for all simply connected surfaces with exception of small number of classes (e.g. hypersurfaces in P3). Another interesting problem concerning the branching curves of generic projections was proposed recently in the work of Auroux, Donaldson, Katzarkov and Yotov. In the authors study the fundamental groups n1 ( C 2- Brsb fl C 2 )where s k is the surface S embedded via the complete is linear system Ho(S,O s ( k ) ) .The stabilized group St(nl(C2-BrSb nc2)) defined as the quotient of n1 (C2- Brsb fl C 2 )by the normal subgroup generated by the commutators (y1,7 2 ) where y1 and 7 2 are good generators (cf. section 3.3) of the fundamental group having disjoint transpositions as their images o ( n ) ,4 7 2 ) in the symmetric group &egSb. If yl,7 2 are the loops about two branches of Brs intersecting at a node then the corresponding y1,72 commute. Therefore, one can view adding such commutation relation to the fundamental group as an algebraic analog of “adding nodes”. In general, a degeneration of a curve adding a node can change the fundamental group (see examples of curves with the same number of cusps but different number of nodes in 1 3 ) . Nevertheless suggested the following: Problem 3.6. For k suficiently large one has the isomorphism: St(nl(C2B T ~n, c2)) = nl(C2- B T n ~ c2) ~

In fact in the authors propose a conjectural structure theorem of the “stabilized” fundamental group St(nl(C2- Brsb n C2)). Problem 3.7. Let A k be the image of the map: H2(S, Z) + Z2 corresponding to the class (cl(OS(k)), KS 2cl(Os(k))) E H 2 ( S ,Z) @ H 2 ( S ,Z) via the identification of the latter with Hom(H2(S,Z), Z2). Let St(nl(C2- Brsk n C2))0be the intersection of the kernel of the homomorphisms of St(nl(C2Brsb fl C2)) onto and the kernel of the abelianization St(nl(C2Brsb r l C 2 ) )+ Z (“the reduced stabilized fundamental group”). Then the commutator of St(n1(C2- Brsk n C2))0is a quotient of Z2 @ Z2 and the

+

379

quotient of S t ( r l ( C 2- Brsk n C2))0 by its commutator is isomorphic to Z 2 / h @ Z[xdegSk]* The analogs of the groups xl(C2- Brsk r l C 2 )and St(r1(C2- Brsk n C2))can be defined for symplectic 4-manifolds and they play an important role in classification. In fact, problems 3.6 and 3.7 were formulated in in the symplectic setting. 3.5. Question of Nori

This problem does not deal with the fundamental groups of plane curve directly but rather comes up naturally in Nori’s theory which gives a clear picture of the reasons for the commutativity of the fundamental groups of plane curves with “mild” singularities. In 35 it is shown that if is a non-singular model of an irreducible nodal curve C with r ( C ) nodes on a non-singular projective surface X and if C2 > 2r(C) then the index of the subgroup of r l ( X ) which is the image of rl(c) in r l ( X ) is finite (in fact does not exceed C2/C2 - 2r(C)). One should compare this with Lefschetz theorem where one makes a weaker assumption that C2 > 0 but the conclusion is that r l ( C ) -+ r l ( X ) is surjective. Note that the group rl(C) is an amalgamated product of and a free group and hence is much bigger than r1(c). This prompted the following:

c

rl(c)

Problem 3.8. (cf. 35) Let D be an effective divisor on a complete surface X with D2 > 0. Is it true that the normal subgroup of r l ( X ) generated by the images of the fundamental groups of normalizations of the components of D has a finite index in 7c1(X)?

3.6. Question of Artin and Mazur In 48 the authors raised a question (cf. p. 8) which bears on the relation between the strata of topological equisingularity of the space of plane curves of fixed degree and the topology of the complement. Geometry and topology of these strata is very interesting and little is understood (cf. discussion in section 5.2). Let U be the connected component of the equisingular stratum of the space of plane curves of degree n(n - 1) containg the branching curves of generic projections of a non singular surface of degree n in P3. Consider an open subset U p of U consisting of curves not passing through a point P E P2. Then the fundamental group r l ( U p ) acts on the group r 1 ( P 2-

380

C0,P) = Bn/(A2) via the monodromy (COE Up,B n is the Artin's braid group and A2 is the generator of the center 6. 32 and section 3.3). Problem 3.9. Determine the homomorphism

+ AUt(Bn/A2).

~1(Up)

The automorphism group of B n is calculated in

15.

4. Multivariable Alexander Invariants

In the case of reducible plane curves the Alexander invariant discussed in section 2 has a multivariable refinement. From now on let C be a curve in C2 having T irreducible components all transversal to the line at infinity. The abelianization of the commutator A(C) = ~ I ( C ~ - C ) ~ / ~ ~ ( C ~ can be viewed as a module over the group ring C[H1(C2 - C, Z)] of the homology group. Since H1(C2 - C, Z) = Z' (6.30) this group ring is isomorphic to the ring of Laurent polynomials A, = C[tl,t,', ...,t,, t;']. A construction of commutative algebra (6. ") associates with a A,-module M its support which is a subscheme of SpecC[tl, t;', ...,t,, t;'] = C*' consisting of the prime ideals p E SpecC[tl,t,', ...,t,,t;'] whose localization M p at p is a non-zero module. This leads to the following Definition 4.1. The i-th characteristic variety of a curve C is a subvariety of C*' which is the reduced subscheme of the support of i-th exterior power of A( C):

K(C) = SuppAi(r1(C2 - C)'/rl(C2

- C)" @ C).

In the irreducible case (T = 1) the characteristic variety Vl(C) is the subset of C* which is the collection of roots of the Alexander polynomial Ac(t) (cf. section 2). If in definition 4.1 one replaces r1(C2 - C) by the fundamental group of a link L in a 3-sphere S3 then the corresponding characteristic variety is the zero set of the multivariable Alexander polynomial of r1(S3 - L ) . However for a fundamental group of a curve the characteristic variety Vl(C) typically has codimension greater than 1 in (C*), and hence cannot be the zero set of a multivariable polynomial. The realization problems mentioned in section 2 have the following multivariable counterparts. Problem 4.1. Find a bound on the number of irreducible components of Vl (C). Are the irreducible components of V1 (C) containing the identity of

381

C*' combinatorially defined i.e. depend only on the isomorphism class of the data consisting of the set of components and local type of singular points each component contains? Let us make some comments on the second part of this problem by providing first some more details on the structure of Vl(C). The irreducible components of & (C) have a remarkably simple geometric structure: each is a coset of a subgroup of C*' = H1(C2 - C,C*) having a finite order in the quotient by this subgroup (cf. 28, ';this is in sharp contrast with the fundamental group of links in S 3 ) . Moreover for each irreducible component of K(C) there exists a holomorphic map f : C2 - C + C - D where CardD 2 2 such that this component coincides with the coset pf*(H1(C - D,C*)) c H1(C2 - C,C*) where p is a point of finite order in C*'. In particular in the case when Vl(C) has components having a positive dimension, K (C) also has the subgroups of H1(C2- C,C*) of the same dimension. In the case when all irreducible components of C c C2 are lines. i.e. C is an arrangement of lines, the components of (C) can be described as follows. Let H*(C2- C,C ) be the cohomology algebra of the complement. It has a combinatorial description known as the Orlik-Solomon algebra of the arrangement (in a more general case of hyperplanes in Cn) (cf. 40). Each w E H1(C2 - C,C) defines the complex

K: : HO(C2- c,C) -3, H'(C2 -

c,C) uw,H2(C2- c,C)

(10)

in which differential is given by the cup product with w. Let us consider the following set { w E H1(C2 - C,C)ldimH1(K:)

2 1).

(11)

This set is combinatorially defined and in fact is a union of linear subspaces in H1(C2 - C,C) (cf. 2 8 , 29). Moreover the exponential map exp : H1(C2 - C,C) + H1(C2 - C,C*) (induced by the map C + C* having as the source the tangent space to C*) provides the one to one correspondence between the components of Vl(C) containing the identity of H1(C2 - C,C*) (i.e. the subgroups) and the irreducible components of the set (11). In particular the set (11) is combinatorially defined i.e. one has a positive answer to the second part of problem 4.1 in the case when C is an arrangement of lines. A generalization of this to arrangements of rational curves in discussed in '. In the case of arrangements one has a more precise conjecture than what is suggested by 4.1 and will be discussed in section 5.

382

Essential components of characteristic varieties having a positive dimension and non coordinate torsion points can be calculated in terms of position of singularities by a formula generalizing the expression (3) (cf. 2 8 ) . We refer to 30 for discussion of related interesting invariants (polytopes of quasiadjunction) and corresponding problems and to l2 for a study of translated components. 5. Complements to Arrangements

Study of the complements to arrangements is a vast subject. Here we shall focus only on few problems, which it seems represent a gateway to more general case of arbitrary reducible curves.

5.1. Dimensions of components of characteristic varieties Let us consider the problem of an estimate on the dimension of the characteristic variety. In order to isolate the central issue we shall review a definition of an essential component of Vl(C) (6. 28, '). Let C be an arrangement of lines in C2 and let C U L be obtained by adding a line L to the arrangement C. The space SpecC[H1(C2 - C,Z)] containing the characteristic varieties can be identified with the space of characters of the fundamental group: Hom(nl(C - C, Z), C') = H1(C2 - C, C') and similar identification can be made for C U L. The (injective) map H1(C2- C, C') + H1(C - C U L, C') induced by inclusion takes a component of V1(C) into a component of VI (C U L ). The corresponding inclusion of components may or may not be strict (cf 3). We call a component HCULC Vl(C U L ) non essential in the latter case, i.e. if it coincides with the image of a component in V1 (C) and essential is it is not a non essential one. Clearly the key issue is to decide what the dimensions of the essential components are and how big these dimensions can be relative to the number of lines in the arrangement (or the degree of the curve in case of an arbitrary reducible curve). For the arrangement of d lines passing through a point in C2 the variety Vi is given by the equation tl ....t d = 1 (cf. 2 8 ) i.e. we have a d - 1dimensional component of (C) which is large relative to the degree of C. It is surprisingly hard to find other arrangements with large dimension of essential components. Problem 5.1. Find a bound 4 ( N ) on the dimension of an essential component of the characteristic variety for an arrangement of N lines. Can the

383

dimension of an essential component of a characteristic variety be greater than 4 for an arrangement different from a pencil of lines? There are quite a few arrangements with dimension of essential components equal to 2 but even in these cases the arrangements are related to beautiful and subtle non-linear geometry. The arrangement of 12 lines formed by all the lines in P2which contain 3 of the 9 inflection points of a non-singular cubic yields an arrangement with the essential component having dimension 3. A class of arrangements for which one has $ ( N ) 5 4 is given in 29 and represents the main evidence for the existence of an interesting answer to the problem 5.1. See l7 and for recent results regarding this problem. 5 . 2 . Arrangement Strata

It was mentioned already in section 4 that the existence of a component of characteristic variety having dimension k is equivalent to the existence of a holomorphic map of C2- C onto complement in C to k distinct points. In other words, for an arrangement C of lines in P2 the existence of a component of Vl (C) having a positive dimension k yields a pencil of curves in P2of fixed degree d with k+ 1 members being the (possibly non reduced) curves having only the lines as their irreducible components. Such a pencil corresponds to a line in the space of plane curves of degree d i.e. the projectivization P(Ho(P2,O(d))) of the space of sections of Op2(d). This projectivization contains subvarieties All ,...,1. corresponding to partitions of d = 11 ... 2, Zi 2 1 and consisting of curves of degree d given by the equations Ly . ... . L’,. = 0 where Li are linear forms. If Zi = 1,i = 1,...,d then the stratum All,...,ldrepresents the d-fold symmetric product of P2 d(d+31 embedded in P 2 . Subvarieties corresponding to other partitions of d are the strata of canonical stratification of the symmetric product and all All,...,la are the strata of equisingular stratification of the space of plane curves (6. 30). A k-secant of All,....,lais a line in P(Ho(P2,O(d 1))) intersecting All,...,lsat k points. A reformulation of the problem 5.1 is the problem of determining for which k the “arrangement stratum’’ All ,...,la admits a k-secant. One way to get a bound on possible degrees of multisecants is to use the degrees of defining equations:

+ +

+

Lemma 5.1. a) If All,...,lais a set theoretical intersection of hgpersurfaces of degrees not exceeding k - 1 then any k-secant of A I , , . . . , must ~ ~ belong

384

to All ,...,!#. In particular the dimension of the characteristic variety of an arrangement composed of curves of degree d satisfies

k 5 rnaxKM(Z1, ...,ZS) where the maximum is taken over all systems of equations K for All,..ls and M(11, ..ls) is the maximum of degrees of equations in K . b)Lines in the space P(Ho(P2, O(d)) of plane curves of degree d which ~ represented b y pencils members of which are unions of belong to A I , . . . , are k e d components and a pencil all members of which are lines containing a fiedpoint (i.e. the pencils have the f o r m XF.Ll. ....L k + ~ F . L : .....L~ where F is a product of linear forms and Li, Li are linear forms all vanishing at the same point). The first part follows from Bezout theorem. To see the second, let us consider the movable part of the pencil i.e. the pencil formed by the components of all curves which do not belong to all elements of the pencil. Let P2 be the blow up of P2 at the base points of the movable part. The map P2 + P1 given by the movable part of the pencil yields the composition C % Pl where has irreducible fibers and II,is finite (Stein factorP2 ization). C must be rational since otherwise the pullback of a holomorphic l-form from C will yields a holmorphic l-form on P2 and P2 does not have non-zero holomorphic l-forms. Hence 4 is a pencil of lines and therefore it consists of lines passing through a point. This leads to the following Problem 5.2. Find the degrees of defining equations of the arrangement strata AI,,...,1.. H.Bril1 showed that if Zi = 1 then Al,...,I is the zero set of a system of equations of degree d 1 (cf. 19). In the case d = 3 this can be seen as follows. A cubic curve C is a union of lines if and only if any point is an inflection point. Therefore the Hessian is vanishing on C and if F is the equation of C then there is a constant y such that Hess(F) = y . F . This is equivalent to saying that the rank of the 2 x 10 matrix formed by coefficients of Hess(F) and F being equal to 1. Since the degree of Hess(F) in = 45 equations for the stratum A ~ J J coefficients of F is 3 we obtain all having degree 4. In fact, the 12 lines containing 9 inflection points of a cubic form a pencil of cubic curves yielding a 4-secant of the arrangement stratum of A ~ J J(and hence the arrangement with 3-dimensional characteristic variety).

+

(120)

385

Example 5.1. (Next two examples were pointed out by 1.Dolgachev) Plane section of the pencil of desmic surfaces (Kummer surfaces corresponding to squares of elliptid‘curves). Such pencil contains 4 tetrahedra yielding in a plane section a pencil of quartics which is tri-secant of the top stratum of Ai,i,i,i (cf. 21). Example 5.2. d) Modular configurations (6. 14) define arrangements of hyperplanes for which plane sections yield the tri-secants of Al,...,l. The hyperplanes can be constructed either using Schrodinger representation of G = (Z/NZ)2 or embedding of modular surfaces parametrising elliptic curves with level N structure. We refer to 1.Dolgachev’spaper for detailed discussion of this configuration. For N = 3 one obtains the Hesse arrangement. Problem 5.3. Find new examples of arrangements with characteristic varieties having essential components of positive dimension and classify secants of the arrangement stratum in the space of plane curves having fixed (at least small) degree.

References 1. D.Arapura, Geometry of cohomology of support loci for local systems I, Alg. Geom., vol. 6, p. 563, (1997). 2. E.Artal Bartolo, R.Carmona, J.I.Cogolludo, Braid monodromy and topology of plane curves. Duke Math. J. 118 (2003), no. 2, 261-278. 3. E.Artal Bartolo, R.Carmona, J.1 Cogolludo, Essential coordinate components of characteristic varieties. Math. Proc. Cambridge Philos. SOC.136 (2004), no. 2, 287-299. 4. E.Artal Bartolo, R. Carmona, Zariski pairs, fundamental groups and Alexander polynomials. J. Math. SOC.Japan 50 (1998), no. 3, 521-543. 5. D.Auroux, S.Donaldson, L. Katzarkov, M.Yotov, Fundamental groups of complements of plane curves and symplectic invariants. Topology 43 (2004), no. 6, 1285-1318. 6. M.A.Marco-Buzunariz, Resonance varieties, admissible line combinatorics, and combinatorial pencils, math.C0/0505435, 2005. 7. J.I.Cogolludo, Topological invariants of the complement to arrangements of rational plane curves. Mem. Amer. Math. SOC.159 (2002), no. 756, 8. J.Cogolludo, V.Florens, Twisted Alexander polynomials of Plane Algebraic Curves math.GT/0504356. 9. A.Degtyarev, Alexander polynomial of a curve of degree six, Journal of knot theory and reamifications. 3 (1994) p.439-454. 10. P.Deligne, Le groupe fondamental du complement d’une courbe plane n’ayant que des points doubles ordinaires est abelien (d’apres W. Fulton). Bourbaki

386 Seminar, Vol. 1979/80, pp. 1-10, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981. 11. A.Dimca, Singularities and topology of hypersurfaces, Universitext, Springer, NY, 1992. 12. A.Dimca, Pencils of plane curves and characteristic varieties, math.AG /0606442. 2006 13. I.Dolgachev, A.Libgober, On the fundamental group of the complement to a discriminant variety. Algebraic geometry (Chicago, Ill., 1980), pp. 1-25, Lecture Notes in Math., 862, Springer, Berlin-New York, 1981. 14. I.Dolgachev, Abstract configurations in algebraic geometry. The Fano Conference, 423-462, Univ. Torino, Turin, 2004. 15. J.Dyer, E.Grossman, The automorphism groups of the braid groups, Amer. J. Math. 103 (1981), p.1151-1169. 16. D.Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. 17. M.Falk, S.Yuzvinski. Multinets, resonance varieties, and pencils of plane curves. math.AG/0603166. 18. W.Fulton, On the fundamental group of the complement of a node curve. Ann. of Math. (2) 111 (1980), no. 2, 407-409. 19. I.Gelfand, M.Kapranov, A.Zelevinsky, Discriminants, resultants, and multidimensional determinants. Mathematics: Theory and Applications. Birkhauser Boston, Inc., Boston, MA, 1994. 20. J.Harris, On the Severi problem. Invent. Math. 84 (1986), no. 3, 445-461. 21. B.Hunt. The geometry of some special arithmetic quotients. Lecture Notes in Mathematics, 1637. Springer-Verlag, Berlin, 1996. 22. R.Lawrence, Braid group representations associated with slm. J. Knot T h e ory Ramifications 5 (1996), no. 5, 637-660. 23. A.Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes. Duke Math. J. 49 (1982), no. 4, 833-851. 24. A.Libgober, Alexander invariants of plane algebraic curves, Proc. Symp. Pure Math. 1983. Amer. Math. SOC.vol. 40. 1983. 25. A.Libgober, Fundamental groups of the complements to plane singular curves. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 29-45, Proc. Sympos. Pure Math., 46, Part 2. 26. A.Libgober, Invariants of plane algebraic curves via representations of the braid groups. Invent. Math. 95 (1989), no. 1, 25-30. 27. A.Libgober, Groups which cannot be realized as fundamental groups of the complements to hypersurfaces in C N .Algebraic geometry and its applications (West Lafayette, IN, 1990), 203-207, Springer, New York, 1994. 28. A.Libgober, Characteristic varieties of algebraic curves. Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), 215-254, NATO Sci. Ser. I1 Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht, 2001. 29. A.Libgober, S.Yuzvinski, Cohomology of the Orlik-Solomon algebras and local systems. Compositio Math. 121 (ZOOO), no. 3, 337-361. 30. A.Libgober, Lectures on topology of complements and fundamental groups.

387 math.AG/0510049. 31. F.Loeser, M.Vaquie, Le polynome d’blexander d’une courbe plane projective. Topology 29 (1990), no. 2, 163-173. 32. B. Moishezon, Stable branch curves and braid monodromies. Algebraic geometry (Chicago, Ill., 1980), pp. 107-192, Lecture Notes in Math., 862, Springer, Berlin-New York, 1981. 33. Y.Myaoka, The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann. 268 (1984), no. 2, 159-171. 34. D.Naie, The irregularity of cyclic multiple planes after Zariski. math.AG/0603427. 35. M.Nori, Zariski’s conjecture and related problems. Ann. Sci. Ecole Norm. Sup. (4) 16 (1983), no. 2, 305-344. 36. M. Oka, Alexander polynomial of sextics. J. Knot Theory Ramifications 12 (2003), no. 5, 619436. 37. M.Oka, A survey on Alexander polynomials of plane curves. Singularites Franco-Japonaises, 209-232, SOC.Math. France, Paris, 2005. 38. M.Oka, A new Alexander-equivalent Zariski pair. Acta Math. Vietnam. 27 (2002), no. 3, 349-357. 39. M.Oka, Two transforms of plane curves and their fundamental groups. Topology of holomorphic dynamical systems and related topics (Japanese) (Kyoto, 1995). 40. P.Orlik, H. Terao, Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften 300. Springer-Verlag, Berlin, 1992. 41. Z.Ran, On nodal plane curve, 1nvent.Math (86) 1986. p.529-534. 42. A.Robb, On branch curves of algebraic surfaces. Singularities and complex geometry (Beijing, 1994), 193-221, AMS/IP Stud. Adv. Math., 5, Amer. Math. SOC.,Providence, RI, 1997. 43. M.Teicher, The fundamental group of a CP2 complement of a branch curve as an extension of a solvable group by a symmetric group. Math. Ann. 314 (1999), no. 1, 19-38. 44. M.Teicher, New invariants for surfaces. Tel Aviv Topology Conference: Rothenberg Festschrift (1998), 271-281, Contemp. Math., 231, Amer. Math. SOC.,Providence, RI, 1999. 45. H.Tokunaga, Some examples of Zariksi pairs arising from certain elliptic K3 surfaces. I1 Degtyrev’s conjecture, Math.Z. 230 (1999) 110.2 389-400. 46. D.Toledo, Projective varieties with non-residually finite fundamental group. Inst. Hautes Etudes Sci. Publ. Math. No. 77 (1993), 103-119. 47. O.Zariski, Algebraic surfaces. Second Edition, Springer Verlag, 1971. 48. O.Zariski, Collected papers Vol. 111. Topology of curves and surfaces, and special topics in the theory of algebraic varieties. Edited and with an introduction by M. Artin and B. Mazur. Mathematicians of Our Time. The MIT Press, Cambridge, Mass.-London, 1978.

388

TOPOLOGY OF DEGENERATION OF RIEMANN SURFACES

YUKIO MATSUMOTO* Graduate School of Mathematical Sciences, T h e University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo, 153-8914, Japan E-mail: [email protected]

This is a summary of the author's joint work with J . M. MontesinosAmilibia [Pseudo-periodicmaps and degeneration of Riemann surfaces, I, II,preprint, Univ. Tokyo and Univ. Complutense de Madrid, 1991/1992] and [Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces, Bull. Amer. Math. SOC.SO (1994), 70-751. We study topological types of degenerations of Riemann surfaces from the viewpoint of their topological monodromy.

1. Degeneration of Riemann surfaces and topological

monodromy By degeneration of Riemann surfaces we mean the following phenomenon: Suppose we have a family of closed Riemann surfaces {Fc} parametrized by complex numbers E E C. As approaches to the origin 0, the shape of the surfaces Fc is distorted, and at the moment when E = 0 the surface gets singularities becoming a degenerate fiber Fo. To be more formal, we regard the family as a comlex surface M (a complex manifold of complex dimension 2 ) and the parameter E as complex numbers in the unit disk: A = {E E C I < 1). Then the degeneration phenomenon is embodied by a holomorphic map

c

f:M-+A

(1)

which is surjective and proper. Furthermore we will assume that the fiber Fc over E can be singular only if = 0. The map f : M --t A is sometimes called a degenerating family of Riemann surfaces of genus g , where g is the genus of the general fiber. *Supported by Grant-in-Aid for Scientific Research (A) 16204003 of the JSPS.

389

Definition 1.1. Degenerating families of Riemann surfaces of genus g , TgP f : M -+ A, and f' : M' -+ A' are topologically equivalent ( ), iff there exist orientation preserving homeomorphisms H : M + M' and h : A -+ A', such that the following diagram commutes:

M

H

M'

h

We consider the classification of minimal degereating families of genus g , i.e., minimal in the sense that it does not contain any (-1)-curve, up to topological equivalence. Let us define the quotient set: S, = {degenerating families of Riemann surfaces of genus g } /

TZP

(3)

This set is considered as the set of all topological types of germs of (minimal) singular fibers of genus g. Our purpose is to understand this set. In his 1963 paper, Kodaira classified the set S1, the case of genus 1. Ten years later, Namikawa and Ueno lo classified the genus 2 case, S2. We study the general case g 2 2. We do not intend to enumerate the set S,, but only to interpret the set in terms of the topological monodromy around the singular fiber. Let S,' be a small circle (of radius E ) in A centered at the origin 0. Then to the restricted bundle, flf-'(S;) : f-'(S;) + Sz, which is a surface bundle over the circle S;, there corresponds as the characteristic map an orientation preserving homeomorphism cp : C, -+

c,.

(4)

Here C, is a closed surface of genus g. Once an identification of the reference fiber f - l ( ~ )and the standard surface C, is chosen, the homeomorphism cp is determined up to isotopy; if one changes this identification, cp changes to its conjugates. By slight abuse of language, we call the map cp the topological monodromy of the degenerating family f : M + A. The nature of the monodromy homeomorphism cp has been studied by many authors. In fact, in local case, the monodromy associated with the Milnor fibering at an isolated critical point of a complex hypersurface has been the central theme of active research of singularities. Notably, L6 proved that the monodromy is periodic ( having a finite order ) if the hypersurface f = 0 in C2 is analytically irreducible at the singular point. On the other hand, A'Campo showed that the monodromy is not necessarily

390

periodic in non-irreducible cases. As a decisive result, L6, Michel and Weber proved that the monodromy homeomorphism is “quasi finie”, in the sense that it has a finite order outside a certain system of disjoint simple closed curves on C,. The same nature of the monodromy homeomorphism has been observed in global case: As a consequence of the work of Imayoshi 3 , Shiga-Tanigawa l3 and Earle-Sipe 2 , it was clarified that the topological monodromy of a degenerating family of Riemann surfaces has the “quasi finie ”property, too. This quasi finie type of homeomorphisms had been introduced by Nielsen l2 (in 1944) under a long name as “surface transformation classes of algebraically finite type ” , which we called in our papers pseudo-periodic homeomorphisms. Let us recall an important algebraic invariant, called the screw number, introduced by Nielsen 12. In what follows we assume g 2 2. Suppose ‘p : C , -+ C, is a pseudo-periodic homeomorphism reduced by a system of simple closed curves C = C1 U C2 LI. . . U C,. We may (and will) assume that each connected component of C, - C has negative Euler characteristic. For each Ci,the screw number s(Ci) isdefine< as follows: + Let ai be the smallest positive integer satisfying ‘pai(Ci)= Ci, where Ci denotes the curve Ci with an arbitrary orientation given. Let Li be the smallest positive integer such that ‘pLi is a full integral (say ei E Z)Dehn twist about Ci, then

This rational number measures the amount of the Dehn twist performed by 'psi about Ci. By deleting the curve with si(Ci)= 0, we may (and will) assume s(Ci) # 0 for each i E {1,2,. . . ,r } . Shiga-Tanigawa l 3 and Earle-Sipe discovered a certain chirality on the screw numbers.

Lemma 1.1. Supose a pseudo-periodic homeomorphism cp is the monodromy homeomorphism of a degenerating family of genus g 2 2, then all screw numbers s(Ci), i E {1,2,. . . , r } , are negative. The sign convention of the Dehn twist depends on the authors. In our terminology, a Dehn twist is negative if it is a right handed twist. Now let P; denote the subset of the conjugacy classes of the mapping class group M,/{conjugate}, represented by pseudo-periodic homeomorphisms of negative twist (i.e. having the property of Lemma 1.1 ). The

391

monodromy defines a map p:

s,

-+

Pi.

(6)

Our main theorem is the following:

Theorem 1.1. If g

2 2,

the map p is bijective.

2. Generalized quotients The strategy of proving Theorem 1.1 is to construct the inverse map CJ:

Pi +s,.

(7)

In other words, given a pseudo-periodic homeomophism cp : C, + C, of negative twist, we construct a degenerating family of Ftiemann surfaces of genus g whose monodromy homeomorphism represents the same element of P; as cp. In the case where cp : C, ---f C, is a periodic map of period n, this is easy to prove. In fact, we have only to take the product A x C, and divide it by the action of a cyclic group of order n:

{: rotation of A} x cp.

(8)

Making use of Nielsen’s result 11, we can introduce a complex structure to C, such that cp : C, -+ C, is holomorphic with respect t o this complex structure. Then the quotient variety of the periodic map (8) becomes a complex variety. If this quotient variety has singularity, then resolve it to obtain a complex manifold M (of complex dimension 2). Note that the quotient space A/{: rotation} is isomorphic to a disk A. Thus a holomorphic map f : M + A is induced from the first projection A x C, + A. It is easy to see that this f : M --f A is a degenerating family of Ftiemann surfaces of genus g and has the monodromy cp. The central singular fiber f - l ( O ) is the quotient C,/cp or its blown up. In general case, we decompose C, as

C,=AUB

(9)

where A is an annular neighborhood of C and B is the closed complement. We will construct a certain “generalized quotient ”. On the part B , the homeomorphism cp has finite periods (period depending on the connected component of B). We take the quotient B/cp. On the annular part A, we construct certain “sausages”. Let Ai be an annular neighborhood of Ci.For explanation’s simplicity, we assume that

392 -+ -+ Ci is not amphidrome, that is, there is no integer cx such that cp*(Ci) = -Ci.

Let S1 and S 2 be the two boundary curves of Ai given the orientations as the boundary of B . Let (milA1,al) and (mi1Azla2) be the valencies of S1, S 2 , respectively. Namely, mi is the smallest positive integer --f -+ such that ‘pmi(Ci) = Ci. cpmi is the rotation of S1 through the angle 27~61/A1(0 < 61 < A l l gcd(A1,Sl) = l),and a1 is the integer determined by 6101 = l(modA1),0 < a1 < XI. The meaning of (mi1A2,a2) is the same. Cf. Nielsen 12. Let s(Ci) be the screw number of Ci. Then we have an elementary number theoretic lemma: Lemma 2.1. There exists uniquely a sequence of positive integers no, n1, .. . ,n k (k 2 1) satisfying the following conditions: (i) no = x 1 , n k = A,; (ii) n1 = c ~ 1(mod XI), nk-1 = 0 2 (mod A2); (iii) nj-1 nj+l = 0 (mod n j ) , j = 1 , 2 , . . . , k - 1; (zv) (nj-1 nj+l)/nj L 2, j = 1 , 2 , . . . , k - 1 ; and (v) l / n j n j + l = Is(Ci)l.

+ + ~3k_t

Let

D; u s; u . . . u s;c2-1 u D;c2

(10)

be a sequence of 2-disks and 2-spheres1where j-th member intersects only 1-th) member transversely in a point. We give the multiplicity minj t o the j-th member for each j. Then this “numerical sausage’lis the “generalized quotientl’of Ai U ’p(Ai) U . . . U ’pmi-’ (Ail by CPFinally we glue up all the components to obtain the generalized quotient S, of ‘p : C , --+ C,. There is naturally defined a projection map T : C, + S,. Take the mapping cylinder C, of this projection, and construct an LLopenbook”N with the page C, and the binding S,. It is proved that N has a complex structure. Blow up N , if it has singularity. Finally we obtain a smooth complex manifold M equipped with a holomorphic map f : M -+ A, which is a degenerating family of Riemann surfaces of genus g with the prescribed monodromy ‘p.

j - 1-th ( and j

+

References 1. N. A’Campo, Sur la monodromie des singularitts isole‘esd’hypersurfaces complexes, Inventiones math. 20 (1973) , 147-169. 2. C. J. Earle and P. L. Sipe, Families of Riemann surfaces over the punctured disk, Pacific J. Math. 150 (1991), 79-96.

393 3. Y. Imayoshi, Holomorphic families of Riemann surfaces and Teichimuller spaces, Ann. of Math. Stud., 97, Princeton Univ. Princeton, (1981), 277300. 4. K. Kodaira, O n compact analytic surfaces, 11,Ann. of Math. (2) 77 (1963), 563-626. 5. D. T. L6 , Sur les noeuds alge'briques, Comp. Math. 25 (1972), 281-321. 6. D. T. L6, F. Michel and C. Weber, Courbes polaires et topologie des courbes planes, Ann. scient. Ec. Norm. Sup. 24 (1991), 141-169. 7. Y. Matsumoto, Lefschetz fibrations of genus two - a topological approach, Proc. of the 37th Taniguchi Symposium (Kojima, et.al., eds.), World Scientific (1996), 123-148. 8. Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of Riemann surfaces, I, 11,preprint, Univ. of Tokyo and Univ. Complutense de Madrid, 1991/1992. 9. Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudoperiodic homeomorphisms and degeneration of Riemann surfaces, Bull. Amer. Math. SOC. 30 (1994), 70-75. 10. Y. Namikawa and K. Ueno, The complete classification offibers in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143-186. 11. J. Nielsen, Die Structur periodischer Transformationen uon Flachen, Math.Fys. Medd. Danske Vid. Selsk. 15 (1937); English transl. by J. Stillwell, The structure of periodic surface transformations, Collected Papers 2, Birkhauser, 1986. 12. J. Nielsen, Surface transformation classes of algebraically finite type, Mat.Fys. Medd. Danske Vid. Selsk. 21 (1944), Collected Papers 2, Birkhauser, 1986. 13. H. Shiga and H. Tanigawa, O n the Maskit coordinates of Teichmuller spaces and modular transformations, Kodai Math. J. 12 (1989), 437-443.

394

GRADED ROOTS AND SINGULARITIES ANDFL~SNEMETHP Department of Mathematics Ohio State University Columbus, OH 4321 0 E-mail: [email protected] Rdnyi Institute of Mathematics Budapest, Hungary E-mail: [email protected] The present article aims to discuss the graded roots introduced by the author in his study of the topology of normal surface singularities. In the body of the paper we emphasize two aspects of them: their potential role in the classification of normal surface singularities, and also their connections with the Seiberg-Witten (and Heegaard-Floer) theory of rational homology sphere 3-manifolds. The article contains many non-trivial applications and examples. Keywords: normal surface singularities, (Q-)Gorenstein singularities, rational and elliptic singularities, geometric genus, links, Zariski's conjecture, NeumannWahl conjecture, Seiberg-Witten invariants of Qhomology spheres, Heegaard Floer homology, graded roots, surgery 3-manifolds, unicuspidal rational projective plane curves

1. Introduction.

The present article has a two-fold goal. Firstly, it aims to advertise the graded roots introduced by the author in his study of the topology of normal surface singularities. In the body of the paper we emphasize two aspects of them: their potential role in the classification of normal surface singularities, and also their connections with the Seiberg-Witten (and Heegaard-Floer) theory of rational homology sphere 3-manifolds. In order to make the presentation more complete, we organized some of the sections in the spirit of a review article, listing some of the important 'The author is partially supported by NSF Grant, Marie Curie Grant and Hungarian OTKA Grants.

395

constructions and results relevant to the theory. In these sections there are less proofs, but we always provide the original sources. As a second goal of the article, we provide also a series of new results. For example, sections 4 and 5 consist of unpublished results (although they circulated in preprint form, cf. [35,36]).These sections contain all the proofs with all the necessary details. The main motivation of the author in the connection of singularity theory with the Seiberg-Witten invariant of 3-manifolds was born when the article I371 was written: this article formulates a conjecture connecting topological and analytical invariants (for its generalization, see section 4). For the ‘history’ and ‘realizations’ of this conjecture the reader is invited to read [33]. In the first verifications of the conjecture for different particular families, we used the realization of the Seiberg-Witten invariants based on the Turaev’s torsion. Later, the article [51] provided a different model how one can understand these invariants via Heegaard-Floer homology. It was a big surprise for the author that some parts of the computational technique described in [51] for the Heegaard-Floer homology resonated incredibly with the technique of computational sequences initiated by Laufer and S. S.-T. Yau (and used by author too) in the computation of different singularity invariants (like the geometric genus and Hilbert-Samuel function). This alloy lead to the definition of graded roots, and to the algorithm of its computations (for almost rational plumbing graphs, a family which is also a novelty of the theory). We definitely believe that the theory of graded roots will have many deep applications. In fact, we believe that it is the guiding structure in many phenomena. This is exemplified in 3.4, but also in the last section by its appearance in a different problem, which a priori sits rather far from the above circle of ideas, the classification of rational unicuspidal projective plane curves. For the organization of the article, see contents below. 1. Introduction. 2. Normal surface singularities. 2.1. The link. 2.2. The combinatorics of the link. 2.3. The topology of the link. The Heegaard Floer homology If&’+(-M). 2.4. Some analytic invariants of the singularity. 2.5. Rational singularities. 2.6. Weakly elliptic singularities. 2.7. Almost rational singularities.

396

3. Graded roots. 3.1. Graded roots. 3.2. The homology of a graded root. 3.3. Graded roots associated with plumbed graphs. 3.4. Characterization of rational and elliptic graphs via roots; classification. 3.5. Graded roots and Heegaard Floer homology of almost rational graphs. 3.6. Example. Lens spaces. 4. Line bundles associated with surface singularities.

4.1. Introduction. 4.2. Line bundles on X. 4.3. Some cohomological computations. 4.4. Main (conjectured) properties. 4.5. Example. The case of rational singularities. 5. The graded roots of S3pl,(K). 5.1. Introduction. 5.2. The manifold S?,/,(Kf). 5.3. The main invariants of S?,/,(Kf). 5.4 The first part of the proof of 5.3.2: k.r2 #J'. 5.5. The second part of the proof of 5.3.2: (RTfk,, xTfk1). 5.6. Examples. 5.7. S?pl,(K) as Kulikov graph-manifold.

+

6. Unicuspidal rational plane curves and S!,(K). 6.1. The semigroup distribution property.

6.2. The semigroup distribution property and surface singularities. 6.3. The semigroup distribution property and graded roots/Heegaard Floer homology.

2. Normal surface singularities. 2.1. The link. 2.1.1. Definition. The link. Let ( X , O ) be a complex analytic normal surface singularity embedded in (CN,O),and let B, be the €-ball in CN centered at the origin. Then, for E sufficiently small, the intersection M := X n dB, is a connected compact oriented 3-manifold, whose oriented C" type does not depend on the choice of the embedding and E . It is called the link of (X,0) [28]. Moreover, X n B, is homeomorphic to the cone over

397

M . In particular, M characterizes completely the local topological type of (X,0). Therefore, if an invariant of (X, 0) can be deduced from M , we say that it is a topological invariant. 2.1.2. The link as a plumbed manifold. Not any oriented 3-manifold can be realized as the link of a singularity. In order to see this, consider the following resolution procedure. Fix a sufficiently small Stein representative X of (X,O) (e.g. X n B, as above) and let r : X -+ X be a resolution of the singular point 0 E X. In particular, X is smooth, and r is a biholomorphic isomorphism above X \ ( 0 ) . We will assume that the exceptional 0 a ) normal crossing divisor with irreducible compodivisor E := ~ ~ ' ( is nents { E j } j C z . Such a resolution is called good. For a good resolution r,let r(r) be the dual resolution graph associated with r decorated with the self intersection numbers { ( E j ,Ej)}jand genera { g j } j G z (see [19]).Sometimes we write e j for ( E j , E j ) .Notice that H z ( X , Z ) is freely generated by the fundamental classes {[Ej]}j.Let I be the intersection matrix { ( E j ,E i ) } j , i . Since r identifies I ~ with X M , the graph r ( n ) can be regarded a plumbing graph, and M can be considered as an S1-plumbed manifold whose plumbing graph is r(r). The crucial point is that r(r)is connected and I is negative definite [29]. The converse is also true, it was proved by Grauert [12]: a connected plumbing graph can be realized as a resolution graph of a (complex analytic) normal surface singularity if and only if the associated intersection form I is negative definite. This gives a complete classification of the possible topological types of (analytic) normal surface singularities. 2.1.3. Assumption. We note that M is a rational homology sphere (QHS), i.e. H l ( M , Q ) = 0, if and only if r(r) is a tree and gj = 0 for all j E 3. In this article we will assume that M , or equivalently the corresponding plumbing graph, satisfies this additional property. We recall also that M is an integral homology sphere, i.e. H I ( M ,Z)= 0 , if and only if additionally I is unimodular, i.e. det(I) = f l . 2.1.4. As we already said, by the plumbing construction, any resolution graph r(r)determines the oriented 3-manifold M completely. The converse is also true in the following sense. We say that two graphs (with negative definite intersection forms) are equivalent if one of them can be obtained from the other by a finite sequence of blow-ups and/or blow-downs along rational (-1)-curves. Obviously, for a given (X,0), the resolution r, hence the graph r(r)too, is not unique. But different resolutions provide equivalent graphs. By a result of W. Neumann [41],the oriented diffeomorphism

398

type of M determines completely the equivalence class of qn).

2.1.5. In the sequel will denote either a good resolution graph, or a plumbing graph of M . Moreover, X denotes either the space of a good resolution, or the oriented 4-manifold obtained by plumbing disc-bundles corresponding to a plumbing graph. 2.2. The combinatorics of the link.

By the 'combinatorics of the link' we understand the combinatorial machinery related with a fixed resolution (plumbing) graph, or with different lattices associated with it. 2.2.1. Definition. The lattices L and L'. The exact sequence of Z modules

0 -+ L

5 L'

-+

H -+ 0

(1)

will stand for the homological exact sequence 0 4 H 2 ( X , Z ) 4 H z ( X , M , Z ) -% H l ( M , Z ) 4 0,

(2)

or, via Poincark duality, for 0 -3 H,"(X,Z) H 2 ( X ,Z)-+ H 2 ( M ,Z)+ 0.

(3)

Hence, L , considered as in (2), is freely generated by the homology classes {Ej}jGg (we prefer to use the same notation for the exceptional curves, their homology classes, and the bases of the lattice). For each j , consider a small transversal disc Dj in X with dDj c Then L' is freely generated by the (relative homology) classes { D j } j E g .Notice that the morphism i : L 4 L' can be identified with L -+ Hom(L,Z) given by 1 H (Z,.). The intersection form has a natural extension to LQ = L@Q,and it is convenient to identify L' with a sub-lattice of LQ: a E Hom(L,Z) corresponds with the unique 1, E LQ which satisfies a(Z) = (ZQ,Z) for any Z E L. By this identification, D j , considered in LQ (and written in the base { E j } j ) ,is the j t h column of I-l, and ( D j , E i ) = Sji. This shows that the exact sequence (1) can be deduced from r(n) - or, in view of 2.1.4, from M - i.e. L is the free Z-module generated b y the vertices of r(7r),the bilinear form is I , while L' is the dual of (L, ( , )).

ax.

2.2.2. Elements 2 = C j r j E j E LQ will be called (rational) cycles. If zi = C j r j , i E jfor i = 1 , 2 , then min{sl,sz} := &min{rj,l,rj~}Ej. x2 means ( x ,x ) . We define the support 1x1 of x by U E j , where the union runs

over { j : rj

# O}.

399

2.2.3. ‘Positive’ cones. One can consider two types of ‘positivity conditions’ for rational cycles. The first one is considered in L. A cycle x = C j rj Ej E LQ is called effective, denoted by x 2 0 , if rj 2 0 for all j. Their collection is denoted by LQ,,, while LL := LQ,,nL’ and L , := LQ,,nL. We write x 2 y if x - y 2 0. x > 0 means x 2 0 but x # 0. 2 provides a partial order of LQ. The second is the numerical effectiveness of the rational cycles, i.e. positivity considered in L’. We define LQ,,, := {x E LQ : (2, E j ) 2 0 for all j } . In fact, LQ,,, is the positive cone in LQ generated by {Dj}j, i.e. it is exactly {& r j D j , rj 2 0 for all j } . Since I is negative definite, all the entries of Dj are strictly negative. In particular, -LQ,,, c LQ,e.Similarly as above, write L,, := L n LQ,,,. Later, in 2.2.7, we will discuss a series of crucial elements of these cones. The cycle which stands as a model for all of them is the classical Artin’s cycle, introduced in [3,4]. Artin proved that, if X I , z2 are elements of -Lne \{O}, then min{xl, x2} is also an element of --L,,\{O}. In particular, -Lne\{O} has a unique minimal element, Zmin, the Artin’s fundamental (or minimal) cycle. 2.2.4. The canonical cycle. The canonical divisor K 2 of 2 numerically is

codified by the canonical cycle K E L’, defined by the system (of adjunction relations) ( K ,E j ) = - ( E j , E j ) - 2 for all j . Since I is non-degenerate, the system has a unique solution in L‘. Although the self-intersection K 2 depends on the choice of the resolution T , the rational number K 2 # J is independent of the choice of T , and it is an invariant of the link M . Another crucial importance of the canonical cycle K consists of its role in the Riemann-Roch formula. For this, we fix an integral cycle x E L , \ ( 0 ) . Although the individual cohomology dimensions hi(.) := hZ(2,0,) (i = 1 , 2 ) , in general, are not topological, the Euler characteristic x(x) := hO(x)- h’(x) depends only on r(T)by the Riemann-Roch theorem: x(x) = -(x, x K ) / 2 .

+

+

2.2.5. Definitions. We say that (X, 0 ) is numerically Gorenstein if K has integral coefficients, i.e., if K E L. (This is the ‘topological counterpart’ of Gorenstein property. Indeed, ( X , O ) is said to be Gorenstein if the line bundle a$,Io) is holomorphically trivial. The topological analogue of this fact is when this line bundle is topologically trivial - this happens if and only if K E L.)

400 2.2.6. Characteristic elements. Spinc-structures. The set of charac-

teristic elements are defined by

+

Char = Char(L) := {k E L’ : (k,x) (x,x)E 2 2 for any z E L } .

+

Notice that the canonical cycle (2.2.4) is in Char and Char = K 2L’. There is a natural action of L on Char by x * k := k 22 whose orbits are of type k 2L. Obviously, H acts freely and transitively on the set of orbits by [Z’] * (k 2L) := k 21’ 2L (in particular, they have the same cardinality). If 2 is a resolution as above, then the first Chern class (of the associated determinant line bundle) realizes an identification between the spinc-structures Spinc(X) on 2 and Char C L’ = H 2 ( X , Z ) (see e.g. [ l l ]2.4.16). , The restrictions to M defines an identification of the spincstructures SpinC(M)of M with the set of orbits of Char modulo 2L; and this identification is compatible with the action of H on both sets. In the sequel, we think about SpinC(M)by this identification, hence any spincstructure of M will be represented by an orbit [k]:= k 2L c Char. The canonical spinCstructure corresponds to [ K .]

+

+

+

+ +

+

2.2.7. Liftings. If H is not trivial, then the exact sequence (2.2.1)(1) does not split. Nevertheless, we will consider some ‘liftings’ (set theoretical sections) of the element of H into L’. They correspond to the positive cones in LQ considered in 2.2.3. More precisely, for any 1’ L = h E H , let ZL(h) E L’ be the unique minimal effective rational cycle in LQ,, whose class is h. Clearly, the set {ZL(h)}hEHis exactly Q := {&rjEj E L’; 0 5 rj < 1 ) (the intersection of L‘ with the closed/open unit rational ‘L-cube’). Similarly, for any h = I’ L , the intersection (1’ L ) n L Q , ~has , a unique maximal element ZLe(h),and the intersection (1’ L ) n ( - L Q , ~ , ) ) has a unique minimal element pne(h) (cf. [34],5.4). By their definitions Pne(h)= -Z;,(-h). The elements pn,(h) were introduced in [34] and were denoted there by l i k l , where [k]= K 2(1’ L ) . Using these elements, one defines the distinguished representative k, of [Ic] by k, := K 2pn,(h) ( h = I‘ L). Sometimes, in the body of the paper, we will use these notations as well. For some h, pn,(h) might be situated in Q, but, in general, this is not the case (cf. 4.5.3 and 4.5.4). In general, the characterization of all the elements Lk,(h) is not simple (see 3.6.2 when M is a lens space, or [34]for Seifert manifolds).

+

+

+

+

+

+

+

+

401

2.2.8. The X-functions (Riemann-Roch formula). For any characteristic element k E Char one defines X k : L’

+

Q by

x k ( Z ’ ) := -(l’,l’

+k)/2.

Clearly, x k ( L ) c Z. For k = K we recover the classical Riemann-Roch function (cf. 2.2.4). For the interpretation of X k in terms of (twisted) RiemannRoch, consider the following. Fix a line bundle C E P i c ( X ) ,and set c1(C) = 1’ E L’ (cf. 4 . 2 ) . Set k := K - 21’ E Char. For any 1 E L with 1 > 0 one defines the sheaf 01 := O x / O x ( - l ) supported by E. Consider the sheaf and let x ( C @ Ol) = ho(C@Ol)-hl (C@Ol)be its (holomorphic) Euler-characteristic. The Riemann-Roch theorem states that this can be computed combinatorially, namely

x(L @ Ol) = - ( l , 1

+k)/2 =xk(1).

2.3. The topology of the link. The Heegaard Floer homology

HF+(--M). 2.3.1. The list of invariants of connected (oriented) 3-manifolds is huge. Any of them can be considered as a topological invariant of the singularity. The invariant becomes really interesting (for algebraic geometers) if it can be related with the analytic structure of the germ. This list is separated into two, rather distinguishable parts. Some invariants are motivated by singularity theory, are described e.g. in terms of the lattice L , or their positive cones. The other part is produced by the classical 3-manifold invariants developed by topologists. Sometimes it is difficult to find a dictionary connecting them. For example, our (negative definite plumbed) three manifolds are completely characterized by their fundamental groups (here, in the QHS case, the lens spaces are exceptions, but they are also well understood). Then, one might ask, how can one understand the integer Z i i n (associated, say, with the minimal good resolution) in terms of 7r1(M)? In this article we will concentrate mainly on a rather new invariant, the Seiberg-Witten invariant of M. There are many ways to introduce it (here we will mention briefly three of them). One of them it realizes as the ‘euler-characteristic’ of Heegaard-Floer homology. 2.3.2. The Ozsv6th-Szab6 invariant. For any oriented rational homology 3-sphere M the Heegaard Floer homology H F + ( M ) was introduced by Ozsv6th and Szab6 in [50] (see also their long list of articles). In fact,

402

H F + ( M ) is a Z[U]-module with a @grading compatible with the Z[U]action, where deg(U) = -2. Additionally, H F + ( M ) also has an (absolute) &-grading; HF&,,(M), respectively HF$,( M ) , denote the part of H F + ( M ) with the corresponding parity. Moreover, H F + ( M ) has a natural direct sum decomposition of Z[U]modules (compatible with all the gradings) corresponding to the spincstructures of M : H F + ( M )= @0€SpinC(M) HF+(M,D). For any spinc-structure c,one has a graded Z[U]-module isomorphism

where H F A , ( M , e ) has a finite Crank and an induced (absolute) Zzgrading, %+ is an irreducible Z[U]-module of infinite %rank such that ker(UlT+) has rank one of degree d (for a more precise definition, see 3.2.1); in fact, d ( M ,c) can also be defined via this isomorphism. One also considers

x ( H F + ( M ,0)) := rankz HF,f,,,,,,,(M,

0)

-rank

HF,f,d,odd(M, a).

Then one recovers the Seiberg-Witten topological invariant of ( M ,c) (see [56]) via

With respect to the change of orientation the above invariants behave as follows: The spinc-structures SpinC(M)and Spinc(- M ) are canonically identified (where -M denotes M with the opposite orientation). Moreover,

d ( M , a )= - d ( - M , c )

and x ( H F + ( M , e ) )= - x ( H F + ( - M , a ) ) .

Notice also that one can recover H F + ( M ,a) from HF+(-M, a) via (7.3) [50] and (1.1)[52].

2.3.3. We will use the notation A ( M ) for the Casson-Walker invariant of M (normalized as in [23] (4.7)). If M is an integral homology sphere (i.e., if H is trivial), then there is only one spin' structure, the canonical one, c,,,, and by [49] (1.3): A ( M ) = S W O S Z ( M , cca,). 2.3.4. The 'original' definition of the 'topological' Seiberg-Witten invariant is the following.

403

For any fixed spinc-structure a E SpinC(M),one defines the modified Seiberg-Witten invariant swL(a) as the sum of the number of SeibergWitten monopoles and the Kreck-Stolz invariant, see [6,24,27] (for this notation, more discussions and references, and relevance with singularities, see [37]). In the present article we prefer to change its sign: we will write swo(M,a) := -sw”,a). In general it is very difficult to compute swo(M,a) using its analytic definition, therefore there is an intense activity to replace this definition with a different one. Besides the OzsvBth-Szab6 theory, this invariant can also be recovered by the sign refined ReidemeisterTzlraev torsion 7 ~ = ChEH , ~ 7 ~ , ~ ( hE) Q h [ H ](determined by the Euler structure) associated with a [62] via

+ X(M)/IHI.

S W T C W ( M ,a) := -7M,u(l)

Here 1 denotes the neutral element of the group H (with the multiplicative notation). For negative definite plumbed 3-manifolds swo(M,a) = swTCW( M ,a) = swosz ( M ,a) by [47,56].Nevertheless, different realizations might illuminate essentially different aspects of the theory. E.g., since C , 7 ~ , ~ (=1 0,) for any rational homology sphere one has

X ( M ) = CswTCW(M,a).

(1)

U

If we do not want to specify the source of the invariant, we just write sw(M, a). 2.4. Some analytic invariants of the singularity.

2.4.1. Definitions. The analytic type of (X,0) is characterized completely by its local analytic ring OX,Owhose maximal ideal will be denoted by mo c O X , ~Here . are some of its discrete invariants.

The Halbert-Samuel function is defined by

f ~ s ( k= ) dimc Ox,o/mt for any k 2 1. Then f ~ s ( 1=)1 and f ~ s ( 2 ) 1 = dimmo/m$ equals the minimal N for which some embedding (X,0) c (CN, 0) can be realized, hence is called = P ~ s ( l cfor ) some the embedding dimension of (X,O). For k >> 1, f~s(k) polynomial PHS (called the Hilbert-Samuel polynomial) P ~ s ( k=) mk2/2

+ a l k + a2.

The integer m above is called the multiplicity of (X,O), and it is denoted by muZt(X,O). It is not difficult to verify that if (X,O) c (CN,O) is an

404

arbitrary embedding and l a generic affine space of codimension 2 (close to the origin), then m u l t ( X , 0 ) = # X n !. The geometric genus is defined by p , := dime H1(*,O*), where X -+ X is any resolution as above. It can be recovered from the sheafcohomology of some one-dimensional spaces as well: by the theorem of formal functions, for x E L , x = C j E y m j E jwith mj >> 0 , one has p , = hl(x). For a generalization ofp,, where the structure sheaf 02 will be replaced by line bundles on X ,see 54.

2.4.2. Question: Are the analytic invariants topological? In the theory of surface singularities, one of the guiding questions is the following: is it possible t o recover (some o f ) the analytic invariants f r o m the link M , or equivalently, f r o m a resolution graph r(n) of (X,O)? Even more, is it possible t o characterize some important families of singularities (defined a priori via analytic terms) by their topology. For a rather detailed discussion of this problem see the review article [33]. But, in some sense, the whole philosophy of the present article, and some of its main results, are also motivated by these questions. The 'classical' models for such invariant and family-characterizations are the cyclic quotient, rational and (weakly) elliptic singularities. For a detailed discussion of their properties, see also [30,33],and references therein. 2.4.3. Example. Hirzebruch-Jung singularities, by definition, are characterized by the existence of a finite projection ( X ,0 ) + (C2,0 ) whose reduced discriminant space is included in the union of the coordinate axes of (C2,0 ) . On the other hand, one can show that ( X ,0 ) is Hirzebruch-Jung if and only if its link is a lens space L ( p , q ) with 0 < q < p and gcd(p, 4) = 1; or equivalently, if the minimal resolution graph is a straight line graph with all genera zero:

-k1 0

-kz

-

-k3

...

4

where -51,. . . ,-k, are given by the continued fraction [kl, kz, . . . ,k S ] :

1

P/Q = [ k l ,kz, . . . , k s ] = k l k2

-

, kl,...,ks>2.

1

.--1 ks

405

Notice also that a Hirzebruch-Jung singularity can also be realized as a cyclic quotient singularity X p , q:= (C2, 0 ) / Z p .Here the action is <* (u, v) = (
<

2.4.4. Example. Rational singularities were defined by Artin by the vanishing of the geometric genus p , = 0. The next subsection (2.5) is devoted to their topological characterization. Subsection (2.6) recalls the relevant facts about the elliptic singularities. 2.5. Rational singularities. 2.5.1. Recall that (X, 0) is rational if p , = 0. In the sequel we f k a resolution 7r of (X, 0). It is easy to see that p , = 0 if and only if h l ( z )= 0 for any z E L, z > 0. (In particular, all the genera g j should vanish, and r(7r) should be a tree.) The main point is that Artin succeeded in replacing the vanishing of h l ( z ) ' s by a criterion formulated in terms of x. In fact, it is enough to consider only one cycle, namely the fundamental cycle Zmin. It is instructive to recall that for any normal surface singularity ho(Zmin)= 1, hence X ( ~ m i n5 ) 1. 2.5.2. Topological characterizations of rational singularities [3,4].

p , = 0 H x(z) 2 1 for all cycles z > 0 H x(Zmin)= 1. Notice that these characterizations are independent of the choice of the resolution 7r. If a resolution graph satisfies the second (or equivalently, the third) property of the above equivalence, we say that it is a rational graph. Any rational graph is automatically good. 2.5.3. Examples. (a) The rational double points (RDP), (i.e. rational singularities with multiplicity two) are exactly the (simple) hypersurface singularities of type A-D-E. Topologically they are characterized by the fact that their minimal resolution graphs are the well-known A-D-E (negative definite) graphs. In fact, any connected, negative definite graph with g j = ej 2 = 0 for all j is the minimal resolution graph of some RDP (and it is of type A-D-E). (b) Let I? be an arbitrary tree. For any vertex j set Sj to be the number of edges with endpoint j . Consider the decorations gj = 0 and -Sj if Sj # 1 ej = for any j. Then the intersection matrix I is au-2 if Sj = 1

+

406

tomatically negative definite, hence I? is the minimal resolution graph of some singularity. One can show that Zmin = CjE j , x(Zmin) = 1, hence any singularity with minimal resolution graph is rational. Moreover, Zkin = - # { u : Sj = 1). (This also shows that the multiplicity and the embedded dimension of a rational singularity can be arbitrarily large, cf. 2.5.4 .) (c) The class of rational graphs is closed while taking subgraphs and decreasing self-intersections. In particular, in the above example (b), one can decrease any of the decorations e j , and still obtain a rational graph. For these modified graphs one still has Zmin = C j Ej. Rational surface singularities with reduced fundamental cycle (i.e. with Zman = C E j ) are also called minimal singularities. E.g. , the cyclic quotients are minimal. (d) Another subclass of rational singularities has the name of sandwiched singularities. A sandwiched singularity is, by definition, a normal surface singularity that admits a birational map to ((C2,0).If we consider a resolution X -+ X, then we get a diagram (R,E ) --+ (X,O) -+ ((C2,0), hence X is sandwiched between two smooth spaces via birational maps. They also can be characterized by their (minimal) resolution graphs as follows [57].Consider a plane curve singularity (C,0) C (C2,0), and let Y -+ C2 be a (in general, non-minimal) embedded resolution 4 of it. Consider the collection E of those irreducible exceptional divisors which are not (-1)-curves (and assume that they form a connected curve). If one contracts E then one gets a sandwiched singularity, and any of them can be obtained in this way (although the choice of (C,0) and q5 is not unique). Notice that this can be reformulated in terms of the combinatorics of the graph as well. The next result targets the analytic invariants introduced above. 2.5.4. Theorem. [3,4] Assume that ( X , O ) is rational and 5 2 1. Then fHS(k) = - 5 ( k - 1)/2 . Zkin 5. I n particular, mult(X,0) = -Zkin and e m b d i m ( X , O ) = -Zkin 1.

+

+

2.6. Weakly elliptic singvlarities. 2.6.1. Definition. A normal surface singularity ( X , O ) is called weakly elliptic, in short elliptic, if its graph r(7r)is elliptic. A graph r(n)is elliptic if min,>o x ( x ) = 0, or equivalently, x(Zmin)= 0. (Wagreich used the first vanishing [63],Laufer the second one [22];see also [30]for their equivalence.) The definition is independent of the choice of the resolution graph. 2.6.2. Remark. The set of elliptic singularities includes all the singularities

407

with p, = 1,and all the Gorenstein singularities with p , = 2. But an elliptic singularity might have arbitrary large p,. In general, the geometric genus and the other analytic invariants listed in 2.4.1 of an elliptic singularity are not topological (for discussion, see e.g. [30,33]).But Laufer, in [22], identified topologically a subclass of elliptic singularities for which p , is topological (in fact, p , = 1). For this characterization, it is convenient to consider the minimal resolution of (X,O) (i.e. a resolution which has no rational (-1) curve). 2.6.3. Theorem. Minimally elliptic singularities. [22] Consider the minimal resolution T of ( X ,0 ) . Then the following statements are equiva-

lent. If a singularity satisfies them, it is called minimally elliptic. (i) (X,O) is numerically Gorenstein and -K = Zmin. (ii) x(Zmin)= 0 and any proper subgraph of r(T)supports a rational singularity. (iii) p , = 1 and (X,O) is Gorenstein. Notice that (i-ii) give topological characterizations of Gorenstein singularities with p , = 1. For additional characterizations, see [22], or [30]. Here is an example when the link is QHS: 2.6.4. Example. ‘Polygonal’ singularities. Assume that the resolution graph r has the following form with s 1 vertices (with g j = 0 ) :

+

. . .‘

es

+

The intersection matrix is negative definite if and only if 2 - s Cj>ol l e j 0. If s > 3 then is minimal, and -K = Zmin = 2Eo Cj,o Ej. If s = 3, then Zmin # - K . But in this case r is not minimal: EO is a rational (-1)-curve, so it should be contracted. After blow down, in the minimal graph, -K = Zmin = C Ej again. If s = 3 then ( X ,0 ) is the Dolgachev’s triangle singularity Del--l,ez-l,es--l.

+

2.6.5. Remark. Laufer in [22] computed the Hilbert-Samuel function of minimally elliptic singularities from their graphs. This is generalized by

408

the author in [32] for Gorenstein singularities with bl(M) = 0. In this description, the crucial topological ingredient is the elliptic sequence of an elliptic singularity, introduced and intensively studied by S.S.-T. Yau, see [65,66]. We prefer to recall it only in the numerically Gorenstein case. We consider a minimal resolution 7r with E = 7 r - ' ( O ) , and we write ZK for - K . In such a case ZK 2 Zmin. 2.6.6. Definition. The elliptic sequence consists of a sequence { Z ~ ~ } j m _ ~ , where Z B is ~ the fundamental cycle of Bj c E. We define {Bj}j inductively as follows. For j = 0 take Bo = E , hence ZB, = Zmin. Then ZK 2 Z B ~ . If ZK > ZB, then we set B1 := ~ Z K - 2 ~ ~Similarly, 1. if Bi is already defined for any i L j,then it turns out that ZK 2 . . . Z B ~If. the inequality is strict then we define Bj+l := ~ ZKZB, .- Z B 1,~otherwise we stop. In particular, ZK = ~ ~ = Z B=~The ,. length of the elliptic sequence { Z ~ ~ } jismm, ~ 1. The case m = 0 (i.e. when the identity ZK = Zmin holds) corresponds exactly to the minimally elliptic singularities.

+ +

- a

+

It is worth mentioning that by Yau's inequality (cf. [66, (3.9)]), m + 1 is an optimal topological upper bound for p,: for a numerically Gorenstein elliptic singularity p , 5 m 1. This (and other partial results of Yau and Tomari [SO]) were the starting points of the following result:

+

2.6.7. Theorem. [32] Assume that the link of the ellzptic Gorenstein singularity ( X ,0 ) is a rational homology sphere. Then p , is a topological invariant: it equals the length of the elliptic sequence in the minimal resolution of ( X ,0 ) . Moreover, the following holds: (a) m u l t ( X , 0) = max(2, - Z i i n ) ; (b) emb d i m ( X ,0 ) = max(3, -Zkin); (c) If Zkin 5 -3 then dimm;/mk+' = -kZkin (k 2 1). 2.7. Almost rational singularities.

Recall that rational singularities can be characterized by their graphs; those graphs which satisfy the corresponding combinatorial properties are called rational graphs, cf. 2.5.2. The next definition is in the same spirit: using a combinatorial definition we enlarge the set of rational (plumbing) graphs: 2.7.1. Definition. Assume that the plumbing graph I? is a negative definite connected tree. We say that r is almost-rational (in short AR) if there exists a vertex j o E J of I? such that replacing ejo by some e;, 5 ej, we get a rational graph r'. In general, the choice of j o is not unique. Once the

409

distinguished vertex j o is fixed, we write 3 = (0) U J * such that the index 0 E J corresponds to this vertex. 2.7.2. Examples. The set of A R graphs is surprisingly large. 1) Obviously, all rational graphs are AR. 2) Any elliptic graph is A R (for a proof, see [34], (8.2)). 3) Any star-shaped graph is AR. Indeed, first blow down all the (-1)vertices different from the central vertex (this transformation preserves the A R graphs); let this new graph be T. Then take for j o the central vertex of and take for -eiO an integer larger than the number of adjacent vertices of the central vertex of Then the modified graph will become minimal rational, cf. 2.5.3. (In other words: all the Seifert 3-manifolds are plumbed manifolds associated with A R graphs.) 4) The class of A R graphs is closed while taking subgraphs and decreasing the Euler numbers e j (since the class of rational graphs is so). 5 ) The rational surgery 3-manifolds S!p,,(K) considered in $5 are A R (see 5.2.4). 6 ) Not any graph is AR. For example, if l? has two (or more) vertices, both with decoration -ej 5 Sj - 2, then r is not AR. E.g.:

r,

r.

-4

-4

-4

-4

This graph has the following property too: if we delete one of the (-2)vertices, then all the components of the remaining graph are rational. Still the graph itself is not AR. 2.7.3. In fact, all the main results of the present article are related with A R graphs. Their generalization for the general case looks rather difficult (and it can be a beautiful goal for interested reader). 3. Graded roots [34]. 3.1. Graded roots. 3.1.1. Preliminary remarks. The Z[U]-module H F + ( M , [k])of Ozsv6thSzab6 can be computed (for any plumbed 3-manifolds M associated with an A R graph, see below) in a combinatorial way from the corresponding plumbing graph r. Our goal is to define an intermediate object, a graded

410

root Rk associated with any negative definite plumbed graph I? and a characteristic element Ic. This will contain all the needed information to determine the homological object H F + , but it preserves also some additional, more subtle topological information about I? (or, about M ) . In this and next subsection we give the definition and first properties of abstract graded roots. Subsection 3.3 contains the construction of the graded roots Rk from the plumbing graphs I?. [Although both I? (the plumbing graph) and the constructed graded root Rk are "connected trees", they serve rather different roles. E.g., the edges of I? codify the corresponding gluings in the plumbing, while the edges of R k codify the Z[U]-action. We hope the terminology will not create any confusion.]

3.1.2. Definitions. (1) Let R be an infinite tree with vertices V and edges E . We denote by [u, v] the edge with end-points u and v. We say that R is a graded root with grading x : V -+ Z if (a) x(u)- x(v) = f l for any [u,v]E E ; (b) x(u)> min{x(v),x(w)} for any "1L,vI, [u,wl E E , 21 # w ; (c) x is bounded below, x-'(Ic) is finite for any Ic E Z, and #x-l(Ic) = 1 if Ic is sufficiently large. (2) v E V is a local minimum point of the graded root (R,x) if x(v) < x(w) for any edge [v,w]. Their set is denoted by Vl,. In fact, Vl, coincides with the set of vertices with adjacent degree one. (3) A geodesic path connecting two vertices is monotone if x restricted to the set of vertices on the path is strict monotone. If a vertex can be connected by another vertex w by a monotone geodesic and x(v) > ~ ( w ) , then we write v + w.+ is an ordering of V . For any pair v, w E V there is a unique +-minimal vertex sup(v, w)which dominates both. (4) If (R,x) is a graded root, and r E Z, then we denote by (R, x)[r]the same R with the new grading x [ r ] ( v := ) x(v) r . (This can be generalized for any r E Q as well.)

+

3.1.3. Examples. (1) For any integer n E Z, let R, be the tree with V = { w k } k l n and & = {[v', vk+l]}k>n. The grading is x(vk) = Ic. (2) Let I be a finite index set. For each i E I ~IXan integer ni E Z; and for each pair i ,j E I fix nij = nji E Z with the next properties: (i) nii = ni; (ii) nij 2 max{ni,nj}; and (iii) n j k 5 max{nij,nik} for any i , j , k E I . For any i E I consider Ri := Rni with vertices {w,"} and edges for any pair (i,j),iden{[v,",v;+']}, (Ic 2 ni). In the disjoint union

ui&,

411

tify vf and vjk, resp. [vf ,vf+'] and [wj",v;+'], whenever k nij. Write @ for the class of vf. Then Hi RiIN is a graded root with x(Vf) = k. It will be denoted by R = R({ni},{ n i j } ) . Clearly Vlm(R)is a subset of { u l i } i E ~ and , this last set can be identified with I . Vlm(R)= I if in (ii) all the inequalities are strict. Otherwise all the indices I \ Vlm(R)are superfluous, i.e. the corresponding Ri's produce no additional vertices. In fact, any graded root (R',x') is isomorphic (in a natural sense) with some R({ni},{nij}).Indeed, set I := Vlm(R'),nu := ~ ' ( v )and nuu := x'(sup(u,v)) for u,v E I . (3) Any map T : {0,1,. . . ,TO}--+ Z produces a starting data for construction (2). Indeed, set I = {0, ..., TO},ni := ~ ( i (i) E I), and nij := max{nr, : i 5 5 5 j} for i 5 j. Then UiRilN constructed in (2) using this data will be denoted by (R7,x7). For example, for TO= 4,take for the values of T : -3, -1, -2,O and -2 (respectively -3,0, -2, -1 and -2). Then the two graded roots are:

3.2. The homology of a graded root. 3.2.1. Z[U]-modules. We will use the following notations. Consider the graded Z[U]-module Z[U, U-l], and (following [51]) denote by its quotient by the submodule U . Z[ U] .This has a grading in such a way that deg(U-d) = 2d (d L 0). Similarly, for any n L 1, define the graded module '&(.) by the quotient of Z(U-("-l), U - ( n - 2 ) ,.. . ,1,LJ,...) by U . Z[U] (with the same grading). Hence, z ( n ) ,as a Z-module, is the free &module Z(1, U - l , . . . , U-("-')) (generated by 1,U-', . . . , U-("-')), and has finite Z-rank n. More generally, for any graded Z[U]-module P with d-homogeneous elements P d , and for any T E Q, we denote by P[T] the same module graded in := z ( n ) [ r ] . such a way that P[?-]d+r = P d . Then set I,+ := &+[TI and

16'

z(n)

3.2.2. Definition. The Z[U]-modules associated with graded roots. For any graded root ( R ,x), let W(R, x) (briefly W(R ) )be the set of functions q5 : V 4 with the following property: whenever [v,w] E I with x(v) <

412

x ( w > ,then

u.+(v) = q5(w). Or, equivalently, for any w + v one requires UX(+X(") . +(v) = q5(w).

(*> Clearly W ( R ) is a Z[U]-module via (Uq5)(v)= U.q5(v).Moreover, W(R)has a grading: q5 E W(R ) is homogeneous of degree d E Zif for each v E V with 4(v) # 0, q5(v) E %+ is homogeneous of degree d - 2x(v). Notice that in (*) one has 2x(v) deg$(v) = 2x(w) deg$(w), hence d is well-defined. Notice that any q5 as above is automatically finitely supported.

+

+

3.2.3. From the definitions, it is clear that MI((R,x)[r]) = W ( R , x ) [ 2 r for ] any r E Z. 3.2.4. Proposition. Let ( R , x ) be a graded root. W e order Vlm as follows. The first element u1 is an arbitrary vertex with x(vl) = min, ~ ( v ) . If 211,. . . ,vk as already determined, and Jk := (211,. . . ,vk} Vim, then let V k + l be an arbitrary vertex in Vlm\Jk with x ( v k + ~= ) minvEvl,\Jk ~ ( v )Let . Wk+l E V be the unique +-minimal vertex of R which dominates both V k + l , and at least one vertex from Jk. Then one has the following isomorphism of Z[U]-modules

W(R,x ) =

@ @ k 2 2 % x ( ~ ~( )x ( w k )- x ( v k ) ) .

I n particular, with the notations m := min, x ( v ) and

x ) := @ k > Z % x ( v k )

( x ( w k )- x ( v k ) ) , one has a canonical direct s u m decomposition of graded Z[U]-modules: W(R,X ) = & : @ W e d ( R, x). Wred(R,

3.2.5. Examples. (a) W(R,) = 5'&. (b) The graded roots R1 and R2 constructed in 3.1.3(3) are not isomorphic but their Z[U]-modules are isomorphic: W(R1)= W ( R 2 )= 7-$ @ Tm4(1)@ 7 - 4 ( 2 ) . Hence, in general, a graded root carries more information than its Z[U]-module. 3.2.6. Corollary. Let ( R r ,x T )be a graded root associated with some function r : N -+ Z, cf. 3.1.3(3), which satisfies r(1) > r ( 0 ) . Then the Z-rank of W r e d (R, 7 xT) is:

rankzWred(RT)= - r ( O ) + m i n ~ ( i ) + x max{r(i) - - r ( i + l ) , O } . 220

The summand

iZ0

12', of W ( T T ,xr) has index m = m i n i-x r ( i )= min,

xT(v).

413

3.3. Graded roots associated with plumbing graphs.

Fix a connected plumbing graph I? whose bilinear form is negative definite. In this section we will construct a graded root ( R k , x k ) associated with any characteristic element k . 3.3.1. The construction of ( & , X k ) . Fix any k E Char and define x k : L 4 z by X k ( x ) := - ( k ( x ) + ( x , x ) ) / 2 as in (2.2.8). For any n E z, we define a finite 1-dimensional simplicia1 complex Z k , < n as follows. Its 0-skeleton is L k , < n := { x E L : X k ( X ) 5 n}. For each x and J' E 3 with x , x Ej E L k , < n , we consider a unique 1-simplex with endpoints at x and x Ej (e.g., the segment [ x ,x + Ej] in L 8 R). We denote the set of connected components of &,
+

+

z,

I_

x k l ~ O ( ~ k , < n=) 72-

I f % E T O ( E k , < n ) , and vn+1 E r O ( E k , < n + l ) , and c u , c Cvn+l, then [vn,vn+l] is an edge of R k . All the edges E(&) of R k are obtained in this way. 3.3.2. Proposition. For any k E Char,

(&,Xk)

is a graded root.

3.3.3. Some of these graded roots are not very different. Indeed, assume that k and k' determine the same spinC structure, hence k' = k 21 for some 1 E L. Then X ~ / ( X- 1 ) = x k ( x ) - x k ( l ) for any x E L. This means that -the transformation x ++ x' := x - 1 realizes an identification of L k , S n and L k ' , I n - X k ( l ) . Hence, we get:

+

(&,

Xkl) = ( R k ,Xk)[-Xk(l)]

whenever k' = k

+ 21 for some 1 E L.

In fact, there is an easy way to choose a graded root from the multitude { ( R k , X k ) } k E [ k ] . For any k E Char we define m k :=

+

Since ( k 21, k [k]: m k = 0).

k2 - maxkqk](k')2 5 0. 8

+ 21) = (Ic, k ) - 8 x k ( l ) , m k is an integer. Set M [ k ] := { k E

3.3.4. Lemma. Fix a spin' structure [ k ] . Then ko E M [ k ] if and only if 5 0 for any 1 E L. Moreover, if ko and ko 21 E M p ] , then -Xk,,(l) = 0. In particular, any choice of ko E M [ k ] provides the same

+

414

graded root (Rko,Xko),which will be denoted by (R[k],x[k]). Moreover, f o r any k E [k]

(Rk,Xk)= (R[k],X[k])[mkl. The notation is compatible with 3.2.4: mk = minxk. In fact,

3.3.5. The graded root associated with the canonical spinCstructure sometimes is denoted by (R,,, ,X c a n ) . It has the following property: #x,-,', (n) = 1, provided that n 2 1. 3.3.6. Clearly, many different plumbing graphs can provide the same 3manifold M . But all these plumbing graphs can be connected by each other by a finite sequence of blowups/downs (-1)-vertices of degree 5 2. One can show that { ( R p ] x, [ k ] ) } [ k ]depends only on M , i.e. it is independent of the choice of the (negative definite) plumbing graph r which provides M . 3.4. Characterization of rational and elliptic graphs via roots; classification. We believe that the right object which guides the classification of normal surface singularities is the (canonical) graded root associated with the connected negative definite plumbing graphs. In order to exemplify and support this statement, we start with the following results. Recall that (Rcan, xcan) denotes the canonical graded root. We invite the reader to recall the definitions of the roots R , ( m E Z) in 3.1.3 as well.

3.4.1. Theorem. Characterization of rational graphs [34].Let I' be a connected, negative definite plumbing graph whose plumbed three-manifold is a rational homology sphere. Then the following facts are equivalent: (a) I? is rational; (a0 #X,-,l,(O) = 1; (b) Rcan = Ro; (c) R,,, = R, for some m E Z; (d) For all characteristic elements k E C h a r , Rk = R,, f o r some mk E z; (Hb) W(Rcan)= (Hc) W(Rc,,) = 2-2 for some m E Z; or equivalently, IHIred(Rcan)= 0; (Hd) For all k E C h a r , W(Rk)= f o r some mk E Z; or equivalently, @ [ k ] & d ( R [ k=] )0.

la';

415

Moreover, if r is rational and k = K

+ 21’, then

I n particular, if r is rational and k, = K

+ 21jk], then min Xkr = 0.

It is instructive to compare (a’) with the property #x;;!(n) for any r and n 2 1; cf. 3.3.5.

= 1, valid

The next result involves only the canonical graded root (the interested reader may clarify the other cases as well): 3.4.2. Theorem. Characterization of elliptic graphs [34]. Let be a connected, negative definite plumbing graph whose plumbed three-manifold is a rational homology sphere. Then the following facts are equivalent: (a) r is elliptic; (b) R,,, = R({ni},{nij}) f o r some index set I , # I = 1 1 2 2, and ni = 0 f o r any i E I , and nij = 1 for any i # j ;

+

@’

‘+ , @ ( ‘& (1) ) f o r some 1 2 1. (Hb) W(Rc,n) = & Moreover, 1 above can be identified with the length of the elliptic sequence. In particular, if the graph r is minimally elliptic then 1 = 1. 3.4.3. Remarks. (a) The results 3.4.1 and 3.4.2 can also be interpreted as follows: The grading xcanof R,,, satisfies minxcan 2 0 (or, equivalently minx,,, = 0) if and only if r is either rational or weakly elliptic. In this = 0 then I? is rational, otherwise it is weakly elliptic. situation, if WTe~(Rcan) is the optimal topological upper bound for p , . The rank of Wred(Rcan) (b) In both rational and weakly elliptic case, for any spinc-structure [k], one has minxk, = 0. (c) One can find elliptic graphs with 1 = 1 which are not numerically Gorenstein (i.e. K $ L),hence which are not minimally elliptic. E.g., the following graph l? has these properties:

r:

Rcan : -4

Here the two minimal points of R,,, Artin’s fundamental cycle.

. . . . .

correspond to the zero cycle and to

416 3.4.4. Classification. Continuation of the Artin-Laufer program. By the ‘Artin-Laufer program’ we understand results in the spirit of subsections 2.5 and 2.6 (Le., results which provide topologically, say, the HilbertSamuel function and the geometric genus for a topologically identified family). Although we know some obstructions to continue this type of results (see e.g. [33]), we still hope in its future. In order to continue this program, we first have to identify subfamilies for which one can show that they share ‘common properties’. We propose to identify these subfamilies by graded roots: For each fked possible graded root, we consider all the singularities which have their canonical graded root identical with the fixed one. We believe that this invariant is exactly the right object which (conjecturally) guides the topological classification of singularities. (As for starting point, see 3.4.1 and 3.4.2.) Let us formulate here a very precise, new situation/family. Let us say that a singularity is of ‘general type’ if it is not rational or elliptic. This happens if and only if x(Zmin) < 0. Now, the subfamily which collects the ’simplest’ general type singularities is characterized by the fact that they share the same canonical graded root. This is the following:

It is not difficult to see that for this class 2 5 p , 5 3, and p , = 3 if and only if the singularity is Gorenstein. We expect that all the results of Laufer for minimally elliptic singularities (or, those of [32]) will have their analogs in minimally general case as well. Notice that the class of ‘minimally general’ singularities is not empty, e.g. it contains z5+y5+z3. But in general, it is not clear at all what abstract graded roots can be realized as canonical graded roots associated with singularities (even when we consider only numerical Gorenstein singularities, when the canonical root has a Zz symmetry). 3.4.5. Problem. Determine all the possible canonical graded roots.

Notice that the possible resolution graphs are characterized by Grauert criterion, namely that they are negative definite. For each negative definite graph (tree) we construct the graded root. The problem is to find a combinatorial characterization of all of them.

417

3.5. Graded roots and Heegaard-Floer homology of almost rational graphs. 3.5.1. This subsection contains two important results. The first one says that for a plumbed 3-manifolds obtained from an A R plumbing graph, the associated Heegaard-Floer homologies can be determined purely combinatorially from the plumbing graph via the homology of the corresponding graded roots. More precisely, theorems (4.8) and (8.3) of [34]read as follows (we invite the reader to recall the definition of the distinguished representatives k, in 2.2.7): 3.5.2. Theorem. Assume that SpinC(M)

r

is an AR-graph. Then, for any [k]E

HF,+dd(-M, [kl) = 0, and

In particular,

d(-M, [k]) = - max

(k’)2+ #J- -

-e + #J-+

2 min xk,. 4 4 This is generalization of the main result of [51] (where the statement was proved for ‘almost minimal rational’ graphs). k’E[k]

3.5.3. Corollary. If I’ is an AR-graph, and [k]E SpinC(M),then:

3.5.4. We wish to emphasize that in most of the applications, the root (Rk,.,Xk,.) is not determined by its definition 3.3.1, but by an algorithm which will be described in the sequel. This algorithm, the second main result of the subsection, is motivated by singularity theory; its idea was suggested by different computational sequences (as generalizations of Laufer’s computation sequence whose output is the Artin’s minimal cycle Zmin), used by Laufer, Stephen Yau and others in the combinatorics of elliptic (or other) singularities. In order to\ present the algorithm, we distinguish one of vertices of our AR-graph r, which satisfies the definition of AR-graphs. By convention, it corresponds to the index 0 E J (fixed for ever), cf. 2.7.1. For a rational cycle z = Cjrj Ej E L’ we write prj (z) for the coefficient rj .

418

The definition of the sequence { x c [ k l ( i ) } i > o .We fix a class [k].Recall the definition of likl from 2.2.7. Then for any integer i 2 0 there exists a unique cycle zc[k](i) E L with the following properties: (4P r o ( q k ] ( i ) >= i; (b) (.[k](i) $ 4 , Ej) I 0 for any j E 3 \ (0); (c) z [ k ] ( i ) is minimal (with respect the partial ordering 5 ) with the properties (a) and (b).

+

(Part (b) for the canonical class [k] = [ K ]reads as (zcan(i),Ej) I0 for any j # 0; notice the similarity with the definition of Artin's minimal cycle.)

3.5.6. Theorem. ((9.3) of [34]) Fix a spinc-structure [k]. There exists an 2 xrc,(x[kl(i))f o r integer To (which depend on [k])such that x~,(z[k](i+l)) any i 2 TO.Moreover, the graded root associated with T [ k ] : (0,. . . ,TO}-+N given by 7 [ k l ( i ) := x k , ( x [ k ] ( i ) ) satisfies ( R k r 7 X k ~ )=

(RT[k]lXT[kj)'

3.5.7. Remarks. (a) In general, it is not easy to find the cycles z [ k ] ( i ) . Fortunately, one does not need all the coefficients of these cycles, only the values q k ] ( i ) = x k , ( z [ k ] ( i ) ) . In most of the cases they are computed inductively using the following equality (cf. (9.l)(c) [34]): Xk, b [ k ](i

+ 1)) = X k , b [ k ](4 + EO).

+

+

Since the right hand side is x k , ( z [ k ] ( i ) ) 1 - (Eo,Zik1 z [ k l ( i ) ) , basically one needs only the intersections (Eo,x [ k ] ( i )for ) any i. (b) Clearly 7 [ k ] ( O )= 0. One also shows that 7[k1(1) > 0, hence 3.2.6 can be applied: , - min xk, = C i >-o max ( 0 ~ p l ( i> q k 3 (i + 1)) rankz m r e d ( ~ k , xk,)

3.6. Example. Lens spaces. 3.6.1. Notations. In this subsection M is the lens space L ( p , q ) , for its plumbing graph see 2.4.3, whose notations we will preserve. M can also be obtained as a -p/q surgery on the unknot in S3. We invite the reader to refresh the notations of 2.2.1; in particular, we recall that { D j } j E 3denotes the dual base in L'. For any 1 5 i I s we write the continued fraction [ k i , . . . ,k,] as a rational number nis/dis with

419

ni, > 0 and gCd(ni,,dis) = 1. E.g., n1, = p and 712, = q. Consider also q’ defined by qq’ 3 1 modulo p and 0 < q’ < p. ({x} := z - [z] denotes the fractional part of the real number x.) 3.6.2. The group H , the spinCstructures, and the elements

+

Zikl. We

write [Dj] for the class Dj L of Dj in L’/L = H . In the present case, H = Z p , and [D,] is one of its generators. In fact, [ D j ]= [nj+l,,D,] in H (1 I j I s). Similarly, the set of spinc-structures on M is the set of orbits { [ - U D ~ ] }=~ {-nD, ~ ~ < ~ L } o l a < p (we prefer to use this sign, since -D, is effective). More precisely, this correspondence is [Ic] = K + 2(-aD, L), where a runs from 0 to p - 1. In order to emphasize the role of a, we also for Itkl. For any 0 5 a < p write use the notation

+

+

The next discussion clarifies the relation between the integer 0 5 a < p (which codifies SpinC(L(p,q))) and the system S(a) := ( a l ,. . . ,a,) (the coefficients of the corresponding minimal vectors Z[-aDsl). Using the definition of l’[-aDsl and the combinatorics of the continued fractions, one shows (for details, see [34], $10) that the entries of ( a l , . . . ,a,) satisfy the system of inequalities:

By this system one can identify the integers 0 5 a < p with the possible combinations ( a l ,. . . , a,) satisfying (SI). Indeed, the integer a can be recovered from ( a l ,. . . a,) by

And, any 0 formula

Ia
determines inductively the entries a l , . . . , a , by the

3.6.3. As a curiosity, we mention that the above system (SI) can also be interpreted in language of ‘generalized’ continued fractions. For any 1 5

420

i 5 s write T i := ni,/ni+l,,. Then

...+ a

a1

-=

+

a2

a,-1

+

a, +r,

... r3

7-2

P r1 The inequalities (SI) imply that all the possible fractions in the above expression are < 1; and this property guarantees the uniqness of the entries ( a l , . . . ,a,) in this continued fraction (for any given 0 5 a < p).

3.6.4. In concrete examples, the fastest way to determine S(a) = ( a l , . . . , a s ) ,0 5 a < p, in the order S(p-1), . . . ,S(O),is the following. Start with S(p- 1) = (kl - 1, k2 - 2,. . . ,k, - 2). Assume that S(a) = ( a l ,. . . ,a,) is already determined. Then, if a, > 0, then S(a-1) = ( a l ,. . . ,a , - l , ~ ~ - 1 ) . If ai = . . . = a, = 0, but ai-1 # 0, then S(U-1)

=(a1

,..., ~ i - 2 , ~ i _ l - l , k i - l , I c i + l - 2 ,...,k , - 2 ) .

3.6.5. The Ozsvdth-Szab6 Heegaard-Floer homology H F + ( f M ) . Since I? is rational, by 3.4.1 and 3.5.2 (see also 2.3.2), H F L d ( & M )= 0, and H F + ( f M ,[k])= 7Zd,and s w o S Z ( M ,[k])= -d/2, where k2+s K2+s d := d(M, [k])= P 4 = 4 Notice that (see e.g. (7.1) of [37]): ~

-

2X(lj,] 1.

where s(q,p) denotes the Dedekind sum

(In fact, this formula for K 2 of Hirzebruch.)

+ s for cyclic quotients goes back to the work

Finally, we determine X ( Z [ - ~ ~ ~ ~ ) .

3.6.6. Proposition. For any 0 5 a < p one has:

421

3.6.7. Clearly, with the choice D1 as a generator, one has

3.6.8. Remark. The Casson-Walker invariant is X ( L ( p , q ) ) = p s ( q , p ) / 2 (see e.g. [37](5.3)).It satisfies -X(M)=

c [ k ] E S p i n =( M )

k: + s 8 ’

or, equivalently:

4. Line bundles associated with surface singularities [35].

4.1. Introduction. In [37] L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (X, 0) (whose link M is a rational homology sphere) with the Seiberg-Witten invariant of M associated with the ‘canonical’spinc structure of M . This is a generalization of a conjecture of Neumann and Wahl [43] formulated for complete intersection singularities with integral homology sphere links. The interested reader can find in the articles [33,34,37-391 the verification of the conjecture in different cases; in [26]some counterexamples, showing that the original assumptions of [37] (namely, that (X,0) is eGorenstein with QHS link) are too weak (hence one needs to consider some additional restrictions in order to give a chance to the conjecture). For related constructions and results, see also the articles of Neumann and Wahl [44-461. Since the Seiberg-Witten theory of the link M provides a rational number for any spinC structure (which are classified by H l ( M , Z ) ) ,it was a natural challenge to search for a complete set of conjecturally valid identities, which involve all the Seiberg-Witten invariants (giving an analytic i.e. singularity theoretical - interpretation of them). The formulation of this set of identities/properties - conjecturally valid for some ‘nice’families of normal surface singularities - is one of the goals of the present section. In fact (similarly as in [37]), we formulate conjecturally valid inequalities which hopefully become equalities in special rigid

422

situations. In this way, the Seiberg-Witten invariants provide optimal topological upper bounds for the dimensions of the first sheaf-cohomology of some line bundles living on a resolution of ( X ,0). The first part of the section is devoted to the construction and study of these ‘natural’ holomorphic line bundles on the resolution X. This construction automatically provides a natural splitting of the exact sequence

0 + PicO(2)

-+

Pic@)

4P ( X ,Z) -+ 0.

The line-bundle construction is compatible with abelian covers. This allows us to reformulate the ‘conjecture’ in its second version, which relates the echivariant geometric genus of the universal (unbranched) abelian cover of ( X ,0) with the Seiberg-Witten invariants of the link. In the last subsection we verify the validity of the conjectured properties for rational singularities. The presentation is based on the author’s unpublished preprint [35]. Some hl-computations for the case of rational singularities were also found independently by T. Okuma [48]. 4.2. Line bundles on

2.

7r : (2, E) -+ ( X , O ) be a fixed good resolution of (X ,O ).Similarly as above, we assume that the link M is a QHS, i.e. H1(X,Z) = 0. Therefore, one has the exact sequence

4.2.1. Let

0 -+ PiCO(2) -+ P i c ( 2 ) 4L’

-+

0,

(1)

where PicO(2) = H’(X,O*), P i c ( X ) = H’(X,O>) (= isomorphism classes of holomorphic line bundles on X),and c1( L ) = ,°(L1Ej) Dj is the set of Chern numbers (multidegree) of C. In the sequel, we will use the same notation for 1 = C n j E j E L and the algebraic cycle CnjEj of X supported by E. E.g., for any 1 = C n j E j E L one can take the line bundle Ox(1):= O x ( C n j E j ) . Notice that cl(Og(1))= I , hence c1 admits a group-section s~ : L + P i c ( X ) above the subgroup L of L’. The main goal of this subsection is to construct a (natural) group section s : L’ -+ P i c (X ) of (1) which extends SL (and is compatible with abelian coverings). Clearly, if P i c o ( z ) = 0 (i.e. if ( X , O ) is rational, see [3],or $2.5 here) then there is nothing to construct: c1 is an isomorphism, and s := c l l identifies the lines bundles with their multidegree. Nevertheless, for nonrational singularities, even the existence of any kind of splitting of (1) is not so obvious.

423 4.2.2. The first construction. Notice that X \ E M X \ (0) has the homotopy type of M , hence the abelianization map T ~ ( \XE ) = x l ( M ) + H defines a regular Galois covering of X \ E. This has a unique extension c : 2 -+ 2 with 2 normal and c finite [13].The (reduced) branch locus of c is included in E , and the Galois action of H extends to 2 as well. Since E is a normal crossing divisor, the only singularities what 2 might have are cyclic quotient singularities (situated above Sing(E)).Let r : 2+ 2 be a resolution of these singular points of 2, such that (c o r ) - l ( E ) is a normal crossing divisor. Let (X,,O) denote the the universal (unbranched) abelian cover of (X,O), i.e. the unique normal singular germ corresponding to the regular covering of X \ (0) associated with T ~ ( X \ (0)) + H . Notice that 2 is a good resolution of (X,,O). Hence, one can consider the following commutative diagram:

O+

L

-+

L’

-+

H -+0

1 1.’ lPH

0 + L a -+ LL + Ha -+ 0 Here, the first, resp. second, horizontal line is the exact sequence 2.2.1(3) applied for the resolution X -+ X, resp. for 2 -+ X,. The vertical arrows (pull-back of cohomology classes) are induced by p = c o r. 4.2.3. Lemma. p~ = 0. I n particular, p’(L’) C La (i.e. any element p’(l‘), 1’ E L’, can be represented by a divisor supported by the exceptional divisor

in 2 ). Proof. Denote by Ma the link of ( X a ,0). The morphism p~ : H 2 ( M ,Z)+ H2(M,, Z) is dual to p* : H I ( M a ,Z)-+ H I ( M ,Z), which is zero since Hl(M,,Z) is the abelianization of the commutator subgroup of r l ( M ) . 0 This shows that for any 1’ E L’, one can take Oz(p’(1’))E P i c ( 2 ) . 4.2.4. Theorem. The line bundle Oz(p’(1’)) is a pull-back of a unique

element of ~ i c ( X ) . Proof. We break the proof into several steps. Let f : S + T be one of the maps r : 2 + 2, c : 2 -+ X or p : 2 + X . In each case one has a commutative diagram of type

424

Hl(T,Z) + PicO(T) --+ Pic(T) + H2(T,Z) -+ 0

Hl(S,Z)

-+

PiCO(S) -+ Pic(S) -+ H2(S,Z) --t 0

[I). Assume that f = c. Then: 0 c’ is injective since c’ g~ Q : P ( X , Q) M ~ ~ Q ( 2) ~t ~ , H ~ (Q) z , is so and H 2 ( X , Z )is free. 0 coi* is injective with image P i ~ ’ ( 2 ) Indeed ~. (see also the proof of 4.2.9), c*Oz has a direct sum decomposition e x L C ,into line bundles C x , where the sum runs over all the characters of H, and for the trivial character C1 = 02. Therefore, since c is finite, H1(Z, O z ) = H (X , c * O z ) = e X H 1 ( XCx), , whose H-invariant part is H1(X,O x ) . 0 Im(H1(Z,Z) -+ Pic’(2)) n Impoy* = 0. Indeed, since any element of Impo?*is H-invariant, the above intersection is in Im(H1(Z,Z)H + P i ~ ’ ( 2 ) ~But ) . H 1 ( Z , Z ) H embeds into H 1 ( Z , Q ) H = H 1 ( X , Q ) = 0, hence it is trivial. 0 c* is injective, a fact which follows from the previous three statements. (11). Assume that f = r . Then T” is an isomorphism (since such a resolution does not modify H I ( ., Z)of the exceptional divisors), To** is an isomorphism (since a quotient singularity has geometric genus zero), and r‘ is injective. Hence (by a diagram check) T * is also injective. (111). Assume that f = p = c o T . Then: 0 For any I’ E L’, O~(p’(1’)) is in the image of p * . Indeed, take a line bundle C E P i c ( X ) with clC = 1’. Then C’ := Oz(p’(Z’))@p*C-’ has trivial multidegree, and it is in P ~ c O ( Z )But ~ . using (1-11)poi* is onto on P ~ C O ( hence ~ ) ~ C’ , is a pull-back. Hence Oz(p’(Z’))itself is a pull-back. 0 Again, from (1-11),p* is injective; a fact which ends the proof of Theorem 4.2.4. 0 4.2.5. Notation. Write

0x.Z’)

for the unique line bundle C E P i c ( X )

with p*(C) = O~(p’(1’)). The proof of 4.2.4 also implies the following fact.

Pic(%) defined by 1’ H Ox(1’) is a group section of the exact sequence 4.2.l(l) which extends S L (cf. 4.2.1).

4.2.6. Corollary. s : L’

--+

4.2.7. Example. Consider an A2-singularity. Then the exceptional divisor

E of the minimal resolution X contains two components El and E2 which intersect each other transversely, both with self-intersection -2. H = Z3,

425

and Z also contains two divisors, but they intersect each other in a singular point whose resolution consists of a -3 curve. Hence 2 contains a string of three rational curves F1, F12 and F2 with self-intersections -1, -3 and -1. Then p’(E1) = 3F1 F12 and ~ ’ ( E z=) F12 3F2. Since -D1 = (2E1 & ) / 3 , one gets that p’(-D1) = 2F1 F12 F2, hence it is an integral cycle. In particular, cT1 (-01) is that line bundle whose p*-image is Og(2F1 F12 F2).

+

+

+

+

+

+ +

4.2.8. The second construction. The following result will illuminate a different aspect of the line bundles 02 (1’). In fact, the next proposition could also serve as the starting point of a different construction of these line bundles. Below we will write H for the Pontrjagin dual Hom(H, 5’’) of H . Recall that the natural map

0 : H -+ H, induced by [l’] H e27Fi(’’>’) is an isomorphism. 4.2.9. Theorem. Consider the finite covering c : 2 + X , and set Q := { C r j E j E L’ : 0 5 rj < 1) C L‘ (cf. 2.2.7) as above. Then the Heigenspace decomposition of c*Oz has the form:

c*Oz = CBXEHCX,

where

&(h)

= O*(-lL(h)) for any h E H . I n particular, c*Oz =

CBl’EQOX (4‘).

Proof. The proof is based on a similar statement of Koll&rvalid for cyclic coverings, see e.g. [18],$9. First notice that c*Cz is free. Indeed, since all the singularities of 2 are cyclic quotient singularities, this fact follows from the corresponding statement for cyclic Galois coverings, which was verified in [MI. Moreover, above X \ E the covering is regular (unbranched) corresponding to the regular representation of H . Therefore, rankc,Lz = )HI, and it has an eigenspace decomposition @,C,, where all the characters x E H appear, and C,lX \ E is a line bundle. By a similar reduction as above to the cyclic case, one gets that C, itself is a line bundle. Moreover, C1 (corresponding to the trivial character 1) equals 0 2 . Next we identify C, for any character. Fix xo E I?. It generates the subgroup (XO) in I?. Write k := H/(xo). The morphism 7r1(M) + H -+ (xo)^defines a well-defined normal space Y as a cyclic Galois (Xo)^-covering

426

-

x

of X. Clearly, one has the natural maps 2 f Y with e o f = c. By similar argument as above f*Oz = @ E E ~ & for some & E Pic(Y) and 2 1 = OY . Since e*& = @[,l=~L,,and [ X O ] = 0 in K ,one gets that L,, is one of the summands of e,,?1 = e,Oy. In particular, L,, can be recovered from the cyclic covering e as the Xo-eigenspace of e,Oy. Assume that the order of xo is n, i.e. ( X O ) = Zn, and we regard xo as the distinguished generator of Z., Write gj := [aDj]E H . Then for any j E 3,x o ( g j ) has the form eanimjln for some (unique) 0 5 mj < n. Using these integers, define the divisor B := C j mjEj. Since for any fixed i E 3 one has I

one gets that B / n E L'. Let h = [B/n]be its class in H . Clearly, 8(h) = xo since

On the other hand, by [18], 9.8 (and from the fact that all the coefficients of B / n are in the interval [0,1)), the Xo-eigenspace of e,Oy is some line bundle L-' with the properties (i) LBn = O g ( B ) and (ii) e*L = Oy(e'(B / n )). From (i) follows that c l ( L ) = B / n , hence (ii), via our definition 4.2.5, reads as L = O x ( B / n ) .Therefore, L-l = Og(-B/n). Since B / n is in the unit cube, B / n = l;(h). 0 4.3. Some cohomological computations.

x

4.3.1. Let (X,O) and 7r : -+ X be as above. In this section we analyse h l ( X , ~for ) any L E pic(%). First, recall the following general (Kodaira, or Grauert-Riemenschneider type) vanishing theorem (cf. [55],page 119, Ex. 15):

+

If c l ( L ) E K LQ,,~,then hl(1,Lll) = 0 for any 1 E L, 1 > 0, hence hl ( X ,L ) = 0. The next statement is an improvement of it, valid for rational singularities: 4.3.2. Assume that (x,o) iS a ratzonal SingulUdy. IfCl(L) E LQ,ne, then hl(l,LIl)= 0 for any 1 > 0 , 1 E L , hence h l ( X , L )= 0 too. Proof. For any 1 > 0 there exists Ej c 111 such that ( E j ,1+K) < 0. Indeed, (Ej,1+ K ) 2 0 for any j would imply x(1) = - (1,1+ K )/2 5 0, which would

427

contradict the rationality of ( X , O ) [3]. Then, from the cohomology exact sequence of 0 -+ L €3 o&(-l+ Ej) +

el1

-+

LI1-q

+0

D

one gets hl(C1l) = hl(CI1-Ej), hence by induction hl(L)l)= 0. We will generalize this proof as follows.

2 + X be a good resolution of a normal singularity ( X , O ) as above. (a) For any 1‘ E L’ there exists a unique minimal element E L, with e(1’) := 1’ - 111 E LQ,,,. (b) 11, can be found by the following (generalized Laufer’s) algorithm. One constructs a “Computation sequence” xo, X I , . . . ,xt E Le with xo = 0 and xi+l = xi Ej(i), where the index j ( i ) is determined by the following principle. Assume that x i is already constructed. Then, if 1’ - X i E LQ,net then one stops, and t = i. Otherwise, there exists at least one j with (1‘ xi, E j ) < 0. Take for j ( i ) one of these j ’s. Then this algorithm stops after a finitely many steps, and xt = 111. (c) For any C E P i c ( X ) with c1(L) = 1’ one has:

4.3.3. Proposition. Let

111

+

h l ( L ) = h l ( C €9 O*(-lit))

- (l‘,lit) - ~ ( 1 1 , ) .

I n particular (since c l ( C €9 O*(-lp)) E LQ,,,), the computation of any hl(,C) can be reduced (modulo the combinatorics of ( L ,(., .))) to the computation of some hl(L’) with cl(C’) E LQ,ne. Proof. (a) First notice that since B is negative definite, there exists at least one effective cycle 1 with 1’ - 1 E LQ,,, (take e.g. a large multiple of some x with ( x ,E j ) < 0 for any j ) . Next, we prove that if I’ - li E LQ,,,for li E L,, i = 1,2, and 1 := rnin{ll,lz}, then 1’ - 1 E L Q , , ~ as well. For this, write xi := li - 1 E L,. Then lzll n 1x21 = 0 , hence for any fixed j , Ej $ lxil for at least one of the i’s. Therefore, (I’ -1, E j ) = (1’ -li, E j ) ( x i ,Ej)2 0. (b) First we prove that xi 5 for any i. For i = 0 this is clear. Assume that it is true for some i but not for i+l, i.e. Ej(i)$ 11p-xil. But this would imply (1’ - xi, Ej(i)) = (I’ - l p , E j ( i ) ) (11, - xi, E j ( i ) )2 0 , a contradiction. The fact that xi 5 11) for any i implies that the algorithm must stop, and xt 5 111. But then by the minimality of 11, (part a) xt = 11,. (cf. [20].) (c) For any 0 5 i < t , consider the exact sequence

+

111

+

0

-+

c €3 o~(-xz+l)L €9 O*(-Xi) -+

+

c €9 0 E j @(-xi) )

+ 0.

428

Since deg(L 63 O ~ ~ ( ~ ) ( - x=i )(1’) - x i , E j ( i ) ) < 0, one gets ho(L €G OEj(i) (-xi)) = 0. Therefore

which equals -(V, zi+l -xi) induction. 4.3.4. Examples. If reads as

+ x(xi) - x(xi+l).Hence the result follows by 0

C = Og(1’) for some 1’ E L’ (cf. 4.2.5) then 4.3.3(c)

~‘(o*(z’))) = hl(O%(e(Z’))) - (Z’,Zp)

-~(ll,).

Additionally, if (X,O) is rational then hl(Ox(e(l’)))= 0 by 4.3.2, hence hl(O%(Z’))= -(l’,Zp) - x(Zp). In particular, for ( X , O ) rational, hl(,C) depends only on topological data and it is independent of the analytic structure of (X,O) . 4.3.5. Definition. We will distinguish the following set of rational cycles:

It,’

:= (1’ E

L’ : e(1‘) = Zk,(h) for some h E H } =

u

lk,(h)

+ L,.

heH

It is easy to verify the following inclusions: Lk

c Ul,EQ-1’

+ L, C IL’.

4.4. Main (conjectured) properties.

4.4.1. Preliminary words. The next ‘conjecture’/expected property is a generalization of the conjecture of [37], where only the case of canoni-

cal spinC structure was considered. The conjectured property provides an optimal upper bound for hl(L) (C E Pic(-%))in terms of the topology of ( X ,0) and c1 (C). The topological ingredient is provided by the SeibergWitten theory of the link. Now, the author knows that the conjecture of [37] is not true in the high generality in which it was formulated, see [26] and [33] for details. Nevertheless, we expect that it is true for a large class of normal surface singularities (subclass of QGorenstein singularities with QHS links). In this article we will not enter in the guess of the (largest) possible class for which the property is valid; we just formulate these conjectures as expected properties. But the reader might consider them as conjectures valid for, say (jokingly), N Y I (‘not-yet-identified’) singularities. We will present two versions (‘strong’ and ‘weak’), both of them having two parts. The first ones, (SIn) and (WIn), are inequalities:we expect their validity for a class of singularities (mainly) topologically identified. The

429

second parts, (SId) and (WId), are identities: we expect their validity under some additional analytic assumption. (In the last subsection we will verify all of them for rational singularities, - without any additional assumption.)

a normal surface singularity whose link M is a rational homology sphere. Let 7r : X + X be a f i e d good resolution and s := # J the number of irreducible exceptional divisors of 7r. Consider an arbitrary 1’ E IL’ and define the characteristic element k := K - 21’ E C h a r . Then, we say that ( X ,0 ) satisfies (SIn) or (SId) if: (SIn) f o r any line bundle C E Pic(%) with c1(C) = 1’ one has

4.4.2. Property (strong version). Let (X,O) be

+

k2 s h’(C) 5 -sw(M, [k])- 8 . (SId) f o r C = 02(1’)in (1) one has equality. (For a generalization of (SIn) see 4.4.3(d), where the restriction 1’ E IL’is dropped.) 4.4.3. Remarks. (a) If C = 02 then h l ( 0 2 ) is the geometric genus p g of (X,O). For

detailed historical remarks and list of cases for which (SIn) and (SId) are valid, see [26,33,34,37-391. (b) Notice that sw(M, [k])depends only on the class [k]of k. In particular, the right hand side of (1) consists of the ‘periodical’ term -sw(M, [k]) and the ‘quadratic’ term -(k2 s ) / 8 . (c) In order t o prove the property, it is enough to verify it for line bundles C with c l ( C ) = 1’ of type 1’ = lk,(h) (for some h E H ) . Indeed, write 1’ in the form 1’ = 1; 1 where 1; = e(1’) = lLe(l’ L ) and 1 E L,. Let RHS(l’), resp. RHS(l;),be the right hand side of (1) for l’, resp. 1;. Since [K - 21’1 = [K - 21;], the Seiberg-Witten invariants are the same, hence

+

+

+

( K - 21;)2 = -(l, 1’) - x(1). 8 This combined with 4.3.3(c) shows that the statements of 4.4.2 for C and 13 @I 02 (-1) are equivalent. (c’) Consider any set of representatives { Z ’ } I ~ ~ R ( R c IL’) of the classes H , i.e. (1’ L } 1 t E = ~ H . Then the comparison (c) can be also be done for any C with I’ := c l ( C ) E R and for C CXI Oz(-Z).Therefore, the validity of the property 4.4.2 follows from the verification of (1) for line bundles L with c l ( C ) E R. RHS(1’) - RHS(i’,) =

+

-(K

+

- 21’)2

430

E.g., one can take R = Q, or R = -Q. The importance of R = -Q is emphasized by 4.2.9. This fact is exploited in the second version of the Property. (d) Similarly, if one verifies the inequality (1) for any 1’ E IL’, then one gets automatically the inequality (1) for any 1’ E L’, hence for any C E P i c ( X ) . This statement follows by induction: if the inequality (1) is valid for some C,then it is valid for C @ O%(-Ej) (for any j E 3). Indeed, using the exact sequence 0 C @ O x ( - E j ) 4C -+ C ~ + E 0, ~ one gets that hl(C 8 O g ( - E j ) ) 5 h l ( C ) 1 (cl(C), E j ) . The proof ends with similar comparison as in (c). The point is that if 1’ $ IL’, then the inequality (l),in general, is not sharp (optimal). -+

+ +

Theorem 9.6 of [34] implies the following fact, emphasizing once again the importance of almost rational singularities (cf. 52.7): 4.4.4. Theorem. Part (SIn) of the strong version of the Property 4.4.2

(hence 4.4.3(d) too) is true for any almost rational singularity. 4.4.5. Discussion. (a) Let (X,, 0) be the universal abelian cover of ( X ,0) with its natural N-action. Obviously, if p : 2 -+ X , is a resolution of (X,, 0), then 2inherits a natural H-action. Recall that the geometric genus

p,(X,, 0) of (X,, 0) is h ’ ( 2 , O i ) . But one can define much finer invariants: consider the eigenspace decomposition @,,-fiH1(2, Oi), of H1(2,Oi), and take p,(X,, O), := dim@H1(Z,O i ) ,

(for any

x E I?).

(b) Obviously, we can repeat the above definition for any (unbranched) abelian cover of (X,O) . More precisely, for any epimorphism H -+ K one can take the composed map n l ( M ) -+ H -+ K which defines a Galois K-covering ( X K ,0) + ( X , 0) of ( X ,0) (with ( X K ,0) normal). Similarly as above, one can define p g ( X ~0),, for any x E K.But these invariants are not essentially new: all of them can be recovered from the corresponding invariants associated with the universal abelian cover. Indeed, consider x E H via I? H . Then p g ( X ~ , 0 )=, p,(X,, O),. In particular,

-

P,(XK,O)

=

c

Pg(Xa,O),.

X€K

(c) In the above definition (part (a)), p,(X,, 0), is independent of the choice of 2,in particular one can take 2 considered in the proof of 4.2.4.

431

Those facts, together with 4.2.9 show that

p,(X,,O)tJ(h) = h ' ( O k ( - l i ( h ) ) ) (for any h E H ) . Since the set { - l L ( h ) } h E H is a set of representatives (cf. 4.4.3(c')), the previous version 4.4.2, restricted to the set of line bundles of type Ok(l'), is completely equivalent with the following.

(X,O) be a normal surface singularity whose link M is a rational homology sphere. Let T : + X be a fixed good resolution and s := # J the number of irreducible exceptional divisors of T . For any h E H = H l ( M , Z ) consider the character x := 8(h) and the characteristic element k := K + 21L(h) E Char. (WIn) for any h E H 4.4.6. Property (weak version). Let

(Wld) in (2) one has equality. 4.4.7. Remark. (I). Notice that (WIn) (for all h ) and 2.3.4(1) imply

or

(11). Assume that for some singularity one can verify the inequalities (WIn) (see e.g. 4.4.4). Then equality in ( 3 ) implies 4.4.6 with equalities for all h, i.e. (WId).

+ +

4.4.8. Example. Assume that (X,O) = { x 2 y3 z12t+2 = 0). Here we assume that t 2 1 (if t = 0 then (X,O) is rational, a case which will be clarified in the next section.) The following invariants of ( X , O ) can be computed using [37],section 6. The minimal good resolution graph of M is star-shaped with three arms and (normalized) Seifert invariants ( a , w ) equal to (3, l ) ,(3, l ) ,( 6 t 1,2t), the self intersection of the central curve is -1, the orbifold euler number equals -1/(18t 3 ) , K 2 s = 2, X ( M ) = -(24t 1)/12, and H = Z3. Consider the two arms corresponding to Seifert invariants ( 3 , l ) . Both of them contain only one vertex with self intersection -3. We denote them

+

+

+

+

432

by E l , respectively Ez. Then Q contains exactly three element. They are 0, 1; := (El 2 & ) / 3 and 1; := (2E1 E 2 ) / 3 . One can rewrite

+

+

c

= IHI.

( K + 2Z’)2 8 +

K2+s -

1’EQ

c

x(l’)*

1’EQ

In our case, by an easy verification ~ ( 1 : ) = x ( l & )= 1 / 3 . Therefore, the right hand side of (Wa) is (24t 1 ) / 1 2 - 3 . 2 / 8 2 / 3 = 2t. On the other hand, the universal abelian cover of (X,O) is isomorphic to a Brieskorn singularity of type (X,,O) = {u3 v3 dt+’ = 0) (with the action E * (u,v,w)= ( [ u ,$J,w),E3 = 1). And one can verify easily that p,(X,, 0) = 2t. Hence, for ( X ,0) , ( 2 ) is valid with equalities. In fact, in this case p , ( X , 0) = 2t as well, hence p,(X,, 0), = 0 for any XZ1.

+

+

+ +

4.5. Example. The case of rational singularities.

4.5.1. Assume that (X,O) is rational. Then by 3.4.1 and 3.5.2 one has -sw(M, [ k ] )= (k,2

+ s)/8.

(1)

4.5.2. Theorem. Property 4.4.2 (hence 4.4.6 too) is true (with equality) f o r any rational singularity.

+

Proof. By 4.4.3(c), we can assume that 1’ = lk,(h) where h = I’ L. Using 4.3.2, and the fact that Z;,(h) E L Q , ~ ,one , gets that h’(Og(1’)) = 0. On the other hand, by ( l ) ,-sw(M, [ k ] )= (Ic: s)/8. Here (see also 2.2.7), k = K-22’, hence Ic, = K+2pn,(-l’+L) with pne(-Z’+L) = -lk,(-(-l’+ L ) ) = -Zk,(l’ L ) = 4’.Hence Ic, = k and the right hand side of 4.4.2(1) is also vanishing. 0

+

+

Theorem 4.5.2 (and its proof) and the identity 4.5.1(1) have the following consequence (whose statement is independent of the Seiberg-Witten theory): 4.5.3. Corollary. Assume that ( X , O ) is rational, and fix some h E H .

Then

I n particular, (X,,O) is rational if and only ifX(pn,(h))= X(Z;(h))f o r any h E H.

433

This emphasizes in an impressive way the differences between the two ‘liftings’ ZL(h) and Pne(h).Recall: for a class h = 1’ L, both lL(h) and pn,(h) are elements of 1’ + L, but the first is minimal in LQ,, (i.e. it is the representative in Q), while the second is minimal in -LQ,~,.Since -LQ,,, C Lq,,, one has ZL(h) <_ pne(h).In some cases they are not equal. For example, take the A4 singularity, where E has three components E l , E2, E3, all with self-intersection -2, and E2 intersecting the others transversely. Then -D2 = (1/2,1,1/2) = pn,(h) for some h, but it is not in Q: lL(h)= -D2 - E2 = (1/2,0,1/2). Nevertheless, in this case, their Euler characteristics are the same (corresponding to the fact that - A4 being a cyclic quotient singularity - the universal abelian cover is smooth).

+

4.5.4. Example. Assume that

(X,O) is a rational singularity with the

following dual resolution graph.

-3

Let X be its minimal resolution, we write EOfor the central exceptional component, the others are denoted by Ei, i = 1,2,3. Then by a computation 1’ := -(Dl D2 0 3 ) = (1,2/3,2/3,2/3). This element is minimal in but not in LQ,,. Its representative in Q is Z‘- Eo. The character x = @([/’])is x(g0) = 1 and x(gj) = e4xi/3for j = 1,2,3. By a computation K = -Z’, ( K 2Z’)2 = -2 and ( K +2(Z’ - E o ) )=~ -10. Therefore, pg(X,, 0), = (-2 10)/8 = 1. In particular, (X,,O) cannot be rational. Indeed, one can verify (using e.g. [40]),that the minimal resolution of ( X a , O ) contains exactly one irreducible exceptional curve of genus 1 and self-intersection -3. In particular, (X,, 0) is minimally elliptic with p,(X,,O) = 1. This also shows that the above eigenspace is the only nontrivial one. We reverify this last fact for the conjugate of x. In this case h = P1(2) is the class of - 0 0 = (1,1/3,1/3,1/3) = pne(h);and ZL(h)= -00- Eo. By a calculation K 2pn,(h) = EOand K 2ZL(h) = -Eo, hence their squares are the same. In particular, p g ( X a ,O)z = 0.

+

+

+

+

+

+

434

5. The graded roots of S 3 p , , ( K ) . 5.1. Introduction. In this section we determine the graded roots (in particular, the Heegaard Floer homology HF+( - M ) and the Seiberg-Witten invariants swoSZ(M,a)) for the oriented 3-manifold M = S?,/,(K) obtained by a negative rational surgery (with coefficient -p/q) along an algebraic knot K C S 3 . In this case, since H l ( M , Z ) = Z,, the spinc-structures { c T ~ of } ~ M can be parametrized by integers a = 0,1,. . . p - 1. The main result establishes the graded roots in terms of the integers p , q, a, and the Alexander polynomial A of K c S 3 . Notice that the Alexander polynomial of an algebraic (any) knot is well-understood, it can be easily computed from most of the other invariants of the knot (e.g., in the present algebraic case, from Puiseux or Newton pairs, or from the semigroup associated with the corresponding local analytic germ). In particular, the input of the theorem is the simplest that one can hope for. Since (in some sense) all the coefficients of A are effectively involved, in fact, the result is optimal. In the very recent manuscript [53], OzsvAth and Szab6 computed HF+ (S; ( K ) )- for any knot K and any integer surgery coefficient p - in terms of the filtered chain homotopy type of the Heegaard Floer complex associated with the pair ( S 3 ,K ) . Compared with this, our starting data, and also the description of H F + ( - M ) , are simpler, and totally elementary; as a price for this we have to impose the ‘negativity restrictions’ for the surgery coefficient and for K . The proof is based on the results and constructions of [34] valid for AR graphs. The needed facts are listed here in $3.5. Although [34], or $3.5 here, presents a precise algorithm how one should compute H F + , its implementations in different situations sometimes is not straightforward. In the present case too, the proof and additional constructions run over two subsections. In fact, with this presentation, we also wish to advertise the efficiency, novelty and the power of [34]. (For another example, see [34], 3 11, where the case of Seifert manifolds is treated.) Subsection 2 also recalls the classical invariants of algebraic knots and connects them with the plumbing of M . The last subsection contains some concrete examples as well. 5.2. The manifold S : , / , ( K f ) . 5.2.1. Review of algebraic knots. [5,7,14] Let K f c S3 be the link of an irreducible complex plane curve singularity f : ( C 2 , 0 ) -+ (C,O); i.e.

435

for E > 0 sufficiently small, write S3 = { z E C2 : llzll = E ) and take the transversal intersection K f := { f = O}nS3 c+ S3.The natural orientations of S3 and of the regular part of {f = 0) induces a natural orientation on K f . Since f is irreducible, K f S1. We will assume that {f = 0) is not smooth a t the origin, i.e. K f c S3 is not the unknot. The isotopy type of K f c S3 is completely characterized by any of the following invariants listed below. 0 The Newton pairs of f consist of g 2 1 pairs of integers { ( p i ,q i ) } b l l where pi 2 2, qi 2 1, q1 > pl and gcd(pi, qi) = 1. 0 In some topological constructions, it is preferable to replace the Newton pairs by the 'linking pairs' (or, the decorations of the splice diagram, cf. [7]) ( p i , C L ~ ) : =where ~,

0 Let A(t) be the Alexander polynomial of K f c S3, or equivalently, the characteristic polynomial of the algebraic monodromy acting on the first homology of the Milnor fiber of f. It is normalized by A ( 1 ) = 1. In terms of the ( p i , ai)i pairs it is

The degree of A , or equivalently, the first Betti number of the Milnor fiber of f , is the Milnor number p of f. 0 The semigroup S of the germ f is a sub-semigroup of N defined by S := { i o ( f ,h ) : h E O(@Z,O)}, where io( f,h ) denotes the local intersection multiplicity of f and h (which equals the codimension of the ideal (f,h) in Ocez,~,).For h invertible i o ( f , h) is zero, hence 0 E S. It is known that S is generated by the integers 00 = plpz . . 'p,, ,& = akpk+l . . ' p , for 1 5 k 5 g - 1, and = a,. Moreover, #(N \ s)is finite. Its cardinality is the delta-invariant b of f , which in this case also equals p / 2 , cf. 1281. It is also known that 6 = p / 2 equals the minimal Seifert genus of K f c S3. The largest element of N \ S is p - 1. In fact, for 0 5 k 5 p - 1 one has: k E S if and only if p - 1 - k # S. The embedded minimal good dual resolution graph of the germ f has the following shape

p,

436

I

I

I

I

6

6

6

6

Above we emphasize only the vertices of degree one and three. The dash. line between two such vertices replaces a string The number of vertices of degree three is exactly g. In general, any vertex j is decorated by the self-intersection of the corresponding irreducible exceptional divisor Ej. In the above diagram we put only the decoration of the vertex j o , which corresponds to the unique (-1)-curve, and which also supports the strict transform of {f = 0 ) . The other decorations are not really essential in our next discussions; for the description of the complete graph see e.g. [5],or 1311, section 4.1. The above diagram can also be identified with the plumbing graph of ( S 3 ,Kf). In this case the self-intersectionsare the corresponding Euler numbers of the S1-bundles, all the surfaces used in the plumbing construction are S2’s, and Kf is a generic fiber of the unique (-1)-bundle. The above resolution graph r(f)will be denoted by the following schematic diagram:

The polynomial A(t) can also be deduced from r(f)by A’Campo’s formula [l].Notice that (if we disregard the arrowhead) the graph can be blown down completely. We emphasize again, the information codified in the following objects isotopy class of Kf c S 3 ,set of Newton pairs, set of linking pairs, Alexander polynomial, semigroup or the embedded resolution graph - are completely equivalent. This means that the polynomial Q introduced below can be deduced from any of them. ~

5.2.2. Definition of the polynomial Q. One has the following identity

connecting the Alexander polynomial A and the semigroup S (probably

437

know already by Zariski and Teissier, see [14]):

Since A(1) = 1 and A’(1) = S (use e.g. the formula (1) of 5.2.1), one gets

+

A(t) = 1 6 ( t - 1)

+ (t - 1)2 . Q(t)

aitz of degree p - 2 with integral coeffifor some polynomial Q ( t ) = cients. In fact, all the coefficients { a i }are ~ ~strict ~ positive, and: 6 = Q g 2 a1 2

. * *

2 a p - 2 = 1.

+

Indeed, by the above identity (2), one has 6 (t - l)Q(t) = Q(t) = Clcgs(tk-’. . . t 1). This shows that

+ + +

s

a2 = #{k $ :ik

xlcsls t k ,or

> i}.

5.2.3. The surgery 3-manifold M = S!p,q(K). In the sequel we fix an oriented algebraic knot K f c S3 and we consider the oriented manifold M := S!p,q(Kf) obtained by -p/q-surgery along K f C S3, where p/q > 0 is a positive rational number (p > 0, gcd(p, q ) = 1). It is easy to see (e.g. by Kirby calculus, see e.g. [ll])that M can be represented by the following plumbing diagram r ( M ) (the symbols j o , . . . ,j s are the ‘names’ of the corresponding vertices, they are not really parts of the decoration of the diagram):

where k l 2 1 and Icj 2 2 (2 _< j 5 s) are integers determined by the continued fraction p/q = [Icl,I c 2 , . . . , Ic,] and m f = agp,. In terms of the embedded resolution graph of f (cf. 5.2.1), mf is the multiplicity of the pull back of the germ f along the (-1)-curve. Topologically, e.g. if one uses Kirby calculus and blows down all the vertices of f , m f is a sum C i m : of squares of a sequence of linking numbers mi. In fact, in the language of plane curve singularities, this sequence of mi’s is exactly the ‘multiplicity sequence’ of f . Notice that if we would start with the unknot K f c S3, then M = S!,,,(Kf) would be the lens space L(p,q) (in general, normalized with 0 < q < p, hence k l 2 2 as well). This case is completely solved in 83.6.

438

Therefore, in the sequel we will assume that K j is not the unknot; and we will allow the case 0 < p/q 5 1 (i.e. kl = 1) as well. Take a point * E S3 \ K j and identify S3 \ * with R3.Then a Kirby diagram of M is given by the knot K j c R3 with decoration (surgery coefficient) -p/q. In particular, the Kirby diagram of the manifold -M ( M with reversed orientation) - whose Heegaard Floer homology will be computed in the sequel - is given by the mirror image Kfm of K j (with respect t o any plane in R3)with surgery coefficient p/q; i.e. -M = S3 (Kfm). P/q.

5.2.4. Lemma. r ( M ) is an AR-graph.

Proof. We will show that if we modify the -1 decoration of j o in r ( M ) into -2 (let us call this new graph by I '( M ) - z ) , we get a sandwiched graph (cf. 2.5.3(d)). Indeed, consider the graph Blow up the -1 vertex j o . The new decoration of j , will be -2, while a new -1 vertex is created. Then blow up this new vertex ( m j 5 1 - 1) times. Then its new decoration will be -mf - 5 1 , while it has ( m j + kl - 1)neighbours which are all -1 curves. Fix one of them, and blow up kz - 1 times. If one continues this procedure, one gets a graph which contains r ( M ) - z as a subgraph.

r.

+

5.2.5. Remark. Using 3.4.1, the above proof also shows that the plumbed 3-manifold associated with I?( M )-2 has trivial reduced Heegaard-Floer homology (i.e., it is an L-space in the sense of OzsvBth-Szab6). 5.3. The main invariants of S E , , , ( K f ) . 5.3.1. Consider M = S!,,,(Kj) as above. Then H := H l ( M , Z ) = Z,. There is natural identification of the spinc-structures of M with the classes of Z,- or, with the integers a = 0,1,. . . , p - 1, a = 0 corresponding to the canonical spinc-structure. For the precise identification, see 2.2.6 and 5.4.7. Let (T, be the spinc-structure associated with a. The next theorem provides a purely combinatorial description of H F + ( - M ) from r ( M ) .More precisely, H F + ( - M I (T,) will be expressed in terms of the delta-invariant 6, the coefficients {ai}i of the polynomial Q (cf. 5.2.2), and the integers p, q and a. The symbol s(q,p) denotes the Dedekind sum (see e.g. 3.6). The integer q' is uniquely determined by 1 5 q' 5 p and qq' = 1 (mod p). 5.3.2. Theorem. For each fixed a = 0 , 1 , . . . , p - 1, - corresponding to the p different spinc-structures of M - one defines the following objects: t, :=

[(zs-l:-a-l

.

1 7

439

r, := 3 S ( q , p ) + 2 c ; = , 0

a function T, : {0,1,. . . ,2ta

J

7,(2t) = t ( 1 - S) Ta(2t

0

{+}+2)

Z by

[TI(t,0 , . . . ,t , + 1 ) ;

+

1) = Ta(2t

4

=

+ 2) +Q[(tp+a)/q],

(t = 0,. . . I t a ) .

and the graded root (R, , xTa) associated with T,.

T h e n (Rra, xTa)is the graded root of M associated with aa.Hence the following facts also hold: HF,+dJ-M,%)= 0,

-(I+:

+ # J ) / 4 = r,;

HG,,(-M,cJ,)

d(-M,a,) = 2 . m i n ~ , + r , ,

where

= ~(%dxTa)[r,1, and

min7, = 7 , ( 2 [ t , / 2 ] ) .

The proof is based on the theorems 3.5.2 and 3.5.6. The details will be provided in the subsections 5.4 and 5.5. We continue with some remarks and corollaries. 5.3.3. Remark. Notice that for any t E (0,. . . ,t,}, Q Y [ ( ~ ~ + ,is) / strict ~~ positive, hence ~ ~ (+21 )t > 7,(2t 2 ) . On the other hand, using properties of S (see e.g. 5.2.1, or 5.5.9(2)), one can verify that the following identity also holds:

+

7,(2t

+ 1 ) = 7,(2t) + #{y

E

S

:

y 5 (tp

+a)/q}.

+

Therefore, since 0 C S , one has 7,(2t 1 ) > 7,(2t). In particular, the above representation of the graded root is the most 'economical': all the values are essential, see also 3.1.3(2). This also shows that ( R , , xTa)has exactly t, 2 local minimum points, and they correspond to the values 7,(2t), t = 0 , 1 , . .. , t , 1.

+

+

5.3.2, 5.3.3 and 3.2.6 imply the next two corollaries. 5.3.4. Corollary.

Recall that H F + ( - M , a,) is a Z[U]-module. Denote by ker U(a,) (resp. by cokerU(a,)) the kernel (resp. the cokernel) of the U-action. They are finitely generated graded free Zmodule. Let Z(,.) denote a rank one free Zmodule whose grading is r.

440

5.3.5. Corollary.

kerU(oa) =

@

z(z,,(zt)+r,)*

O
I n particular, ker U ( U a ) depends only on the integers p , q a, and 6, but not on the coeficients { Q e } e . O n the other hand, cokerU(ca) =

@

z(2To(2t+l)+T,-z),

o
which depends essentially o n the coeficients {Qe}e of Q . Moreover:

c(ta +

P- 1

ranka: ker U =

2) = p

+ (26 - 1)q = p + ranka: cokerU.

a=O

5.3.6. Remark. If a 2 (26 - l ) q , then ta = -1, hence R,, = Ro (cf. 3.1.3(1)). In particular, for such a , WT,d(R,,) = 0. For all the other a's, WT,d(RT,) is not trivial. E.g., for the canonical spinc-structure corresponding to a = 0, W,,d(K0) is never trivial. In particular, -M is never an L-space. 5.4. The first part of the proof of 5.3.2: k:

+ #3.

5.4.1. The index set 3. According to the shape of the plumbing graph r ( M ) ,the index set of its vertices is 3 = Ju(1,. . . ,s}, where 3 is the index set of the vertices of and the indices { 1,. . . , s} correspond to j 1 , . . . ,j,. The distinguish vertex of r ( M ) is j o corresponding to 0 E 3 c 3. The base elements will also be denoted accordingly: Eo,E l , . . . , E,; or Ej for j E 3.

r,

r

5.4.2. The graph SI'. As we already mentioned, the sub-graph can be blown down. After this blow down, we get a graph which will be denoted as follows:

We can think about this graph as the dual graph of a minimal resolution (of a normal surface singularity) obtained from a resolution with dual graph r ( M ) . In this sense, the new vertex jl is a rational curve with a singular point which has delta-invariant 6; and the decorations -ki are the corresponding self-intersections. On the other hand, one can think about

441 this graph also in the language of Kirby diagrams: jl represents the knot K f c S 3 , the other vertices represent unknots, they are linked as usual with linking number one, and -ki are the corresponding surgery coefficients. The point is that a large number of numerical invariants of the graph l ? ( M )can already be determined from al? in terms of 6 and { k i } i (for this comparison, see the second part 5.4.6 of this subsection). The advantage of this is that the above graph ar is the same as the graph of a lens space L ( p , q ) , provided that we disregard the decoration [6]. In particular, its invariants can be computed by similar methods as those of lens spaces - in fact, in their computations we will even use the corresponding relations valid for lens spaces. This will be the subject of the first part of this subsection. Our model is subsection 3.6 where we run the algorithm for lens spaces. The reader is invited to consult this subsection and also the original source [34] and to verify that our present claims, verified in 3.6 for case q < p , are valid for q 2 p as well. In fact, any invariant of the lattice (which does not involve 6) equals the corresponding invariant of L(p, 4). But, formulas which involve 6 (like the ‘adjunction formula’ determining the canonical characteristic element, or the ‘Riemann-Roch’ formula for x k ) depend essentially on 6. 5.4.3. Notations. In order to make a distinction between invariants of the graphs r ( M ) and al?, the invariants of the second one will have an extra -decoration; e.g. L denotes the lattice of rank s with base elements {&}:=I, while {Bi}ixl E L’ are the dual bases, etc. The integers { n i s } l land have the same meaning as in 3.6.1. Similarly, q’ is the unique integer with p 2 q‘ 2 1 and qq’ = 1 (mod p ) (notice the small difference with 3.6.1 where q < p , but in the present case p < q is allowed too). 5.4.4. The group Z? and the elements

Z,,and [BS]= B, + L

tjE1.Clearly, cf. 3.6, fi = Z ’ / E =

is one of its generators. Then [Bj]= [nj+1,s13s]in the set of orbits {-d,

Z? (1 5 j I s). The set of spinc-structures is L}olaip. For any 0 5 a < p write

qaBs1

= -(u1l31+ a &

+

+. . . +a,&).

By 3.6.2, the system ( a l ,. . . ,a,)can be realized as the set of coefficients of a minimal vector $-a6,1 for some 0 5 a < p if and only if the entries satisfy the system of inequalities (SI) of 3.6.2.

ak denote the canonical characteristic element associated with the graph al? (see below). It is

5.4.5. The canonical characteristic element. Let

442

convenient to consider the canonical characteristic element K of the graph of L ( p , q ) (recall that this graph is obtained from 6r by omitting 6). The adjunction relations defining K are the usual ones, but those which iden- kj 2 should equal tify a K are the following (see also 5.4.9): (ak,,E?j) twice the delta-invariant of the vertex j, namely = 26 for j = 1 , and = 0 otherwise. Hence:

+

a K =K

+ 26D1.

In particular,

ak2+ S = k2+ S + 4 6 ( k , B1) + 4S2(fi1,d1). (R,dl)is the &-coefficient of I?, which equals - l + ( q + l ) / p , see e.g. [37], ( 5 . 2 ) . Similarly, d: = jfil= - n z s / p = - q / p . Also, by 3.6.5: K 2+ s = 2(p - l ) / p - 1 2 . s ( q , p ) . Therefore, - 4624 -.

1 2 s ( q , p ) - 46(1 - *)

gK- 2 +s=-- 2 ( p -

P

P

The distinguished characteristic element

P

kT of the orbit -ads + 2. is

Therefore,

By 3.6.6

On the other hand, by formula ( 1 ) of 3.6.2: S

(dl,q-aDsl) = -XUi(B1,Bi) i=l Summing all, one gets

S

= xuini+i

i=l

P

s

=a

P'

443

5.4.6. Back to the graph I’(M). Finally, we compare the invariants of the graphs r ( M ) and &I?. There are two natural morphisms connecting the corresponding lattices L and L. The first one, IT* : L -+ is defined by r * ( E j )= 0 for any j E 3, while 7r*(Ei)= & for i = 1,.. . , s. In order to define the second morphism, we need a n additional construction. Let Zf:= CjE3 mjEj be the cycle supported by F which satisfies

z

(2,+ E l , E j ) = 0 for j E

3.

Since det = f l , this system has a unique integral solution { m j } j .In fact, in terms of the diagram r(f)(cf. 5.2.1), mj is the vanishing order of the pull back of f along the corresponding irreducible exceptional divisor. E.g., mo = m f . Then one defines 7r* : L -+ L by 7r*(Ej)= Ej for j 2 2 and 7r*(&) = Zf E l . By the very definition follows the ‘projection formula’:

+

(7r*(i),l) = ( ~ , T * ( I ) )f o r d €

L, I

E L.

5.4.7. The group H . 7r4 has a natural extension to LQ 4 LQ, and 7r4(L’)c Therefore, 7r4 induces a group morphism H -+ R. Its surjectivity follows from 7r*7r* = 1. In order to prove injectivity, consider an I’ E L’ with 7 r 4 ( Z ’ ) E I?, Then r*7r4(l’) is an element of L. Since 7r*7r4(l‘) - 1’ is supported by and det = 4x1,one gets that 7r*r*(Z’) - 1’ E L as well. Hence 1’ E L , and IT* induces an isomorphism H 4 H = Z, and H is generated by D,. In the sequel, the set of spinc-structures of M will be identified with the set of orbits {-aDs L } o ~ ~We < denote ~ . by ga that spinc-structure which correspond to the orbit -aD, L.

z’.

r,

+

+

We claim that lipaDs1 = 7 r * ( { - a b s l ) .

5.4.8. The element

Indeed,

) E ( - L Q , ~by ~ )the projection formula. Next, we verify the [-absI minimality of 1’. Notice that any 1 E L with 1 2 0 can be written in the form 7r*(Z) +a:, where 2 2 0 and 3 is supported by I?. Assume that for such 7r*(Z) a: 2 0 one has:

I’

:= n*(?

+

(1’ - 7r*(Z)

-

a:, E j ) 5 0 for any j

E

3.

(*)

Since (n*(%),Ej) = 0 for any j E 3, one gets that (-a:,Ej) 5 0 for any j E 3. Since is negative definite, it follows that a: 5 0. Using this, (*) for j = 1,.. . , s gives

r

0 2 (I’

-

r*(Z), Ej)

+ (-a:,

E j ) 2 (I’

-

~ * ( 2E) j,) = ( T i - a f i s l - 5,E j )

444

hence, by the minimality of 3 5 0 implies 3 = 0 as well.

q-aB81,one has 5

= 0. But then Z

1 0 and

5.4.9. Claim: r * ( K )= 8 K . First notice that by the projection formula:

mjEj as above). The adjunction relations for r ( M ) can (where 2, = CjEs be rewritten as

K =-

C Ej - C(2- s j ) D j , j € 3

j€.?

where s j is the adjacent degree of the vertex j in I'(M). They also satisfy (cf. [l]):

C (2 - s j

)mj =

1 - p = 1 - 26.

j€3

+ 26Bl = aI?. 6k2+ s. Indeed, set K p := K - r*(ak).

Then, taking .rr* one gets .rr,(K) = I?

+

5.4.10. Claim: K 2 # J = Then Kp is supported by and satisfies the adjunction relations (Kp, E j ) = -ej - 2 €or any j E 3,hence it is the canonical cycle of F. Moreover, since can be blown down (and since by an elementary blow up of a smooth point K 2 decreases by l),one has K$ = -(#J - s). By projection formula .rr*(ak) is orthogonal to K p , hence

ak2+ K:

K 2 = (r*(aK)+ Kp)' =

=

aK2 - ( # J - s).

Finally, this claim, via the projection formula, shows for any spinc-structure that

k:

+ # J = &! + s.

5.4.11. Example. Integer surgery. Assume that q = 1, i.e. M S!,(K,). Then q' = 1 as well, and -

q+#J-- --=q + s

=

(p+26-2-242-p

4 4 4P 5.4.12. Example. (l/q)-surgery. Assume that p = 1, i.e. M = S?,,,(Kf). Then again q' = 1. There is only one spinc-structure corresponding to a = 0, and - K 2 + # J --

4

_-K 2 + S 4

- q6(6 - 1).

445

5.5. The second part of the proof of 5.3.2: ( R 7 1 k l , ~ T [ k l ) .

The construction of the cycles z [ k ] ( iis ) given in two steps. The first step provides similar cycles associated with the graph f , and it is based on the combinatorial properties of the graph r(f)(involving also some techniques of plane curve singularities). In particular, we prefer to think about as the dual graph of irreducible exceptional curves obtained by repeated blow ups of (C2,0).The lattice of f is denoted by L, and we write L,, := {y E L : ( Y , E j )2 0 , j E 3). 5.5.1. The cycles {y(i)}i>o. - Since

IT is negative definite, (7.6) of [34]guar-

antees, for any i 2 0, the existence of a positive cycle y(i) 2 0 (supported by f ) with the following properties: ( 4 Pro(Y(4)= i; (b) (y(i),Ej) 5 0 for any j E 3 \ ( 0 ) ; (c) y(i) is minimal (with respect the partial ordering 5 ) with the properties (a) and (b). E.g., y(0) = 0. Although there is very precise algorithm which determines all the cycles y(i) (see e.g. the proof of 5.5.3, or [34]), we are not interested in all the coefficients of y(i). Instead, what we really have to know is the set of intersection numbers (y(i), Eo) (cf. 3.5.7). Let Z f be the divisorial part (supported by f ) of the germ f , cf. 5.4.6, which satisfies

o ifjE3\(0}; -1 if j = 0. Recall also that S

c N denotes the semigroup o f f , see 5.2.1.

5.5.2. Proposition. (u) I f i = tmf +io with t

1 0 and 0 I io < m f , then y(i) = t Z f (b) For any i < m f one has 1 if i

+ y(io);

# S;

0 if i E S.

+

Proof. (u) Clearly, y’(i) := t Z f y ( i 0 ) satisfies (5.5.1)(a)-(b), we need to verify (c). But if for some y 1 0, y’(i) - y satisfies (a)-(b), then y(i0) - y would also satisfy (a)-(b) for io, hence y = 0 by minimality of y(i0). Part ( b ) of (5.5.2) will follow from the next sequence of lemmas. 0

The first lemma was proved and used intensively in [34] as a general principle of rational graphs. For the convenience of the reader, we sketch its proof, for more details, see [loc. cit.].

446

5.5.3. Lemma. (y(i), Eo)

5 1 for any i 2 0.

r,

Proof. We denote by Zmin Artin’s minimal cycle associated with i.e. the minimal nonzero cycle in The cycle Zmin can be determined by the following (Laufer’s) algorithm (see [20]): Construct a ‘computation’ sequence of cycles { & } l as follows. Set 20 := 0, 2 1 := bo; if 2 l is already constructed, but for some j E 3 one has ( Z l , E j ) > 0, then take Zl+1 := 2, b j . If Zl E then stop and 2~= Zmin. Now, by a geometric genus computation, one gets for all rational graphs that whenever ( Z l ,E j ) > 0 in the above algorithm, in fact one has the equality (t): ( Z l ,E j ) = 1(cf. [20]). Since can be blown down, it is rational, hence (t) works. The point is that y(i) can also be computed by a similar algorithm: Assume that y(i) is already determined. Then construct the sequence of cycles {Zl}l as follows. 20 := y(i), 21 := y(i) Eo, and if (for I 2 1) (Zl,E j ) > 0 for some j E 3 \ {0}, then take Z ~ + := I 2l Ej, otherwise stop and write 2l = y(i -t 1). Then one can verify that any sequence connecting y(i) with y(i 1) can be considered as part of a computation sequence associated with Zmin, 0 hence lemma follows by the above property (t).

-zne.

+

(-zne)

r

+

+

+

5.5.4. Lemma. Fix an arbitrary i 2 0 . Then (y(i), Eo) 5 0 if and only if

i E s.

(-zne).

Proof. ‘+’If (y(i), Eo) 5 0 then y(i) E Since is the dual graph of a modification of (C2,0), the cycle y(i) is the divisorial part 2, of a holomorphic germ 9 E O(~z,o).Since the intersection multiplicity i o ( f , g) of the germs f and g a t 0 E C2 is the multiplicity of 2, along the exceptional divisor supporting the arrow of f ,one gets i o ( f , g) = p r o ( y ( i ) ) = i. But, by definition, i o ( f , 9) E S. ‘+=’ If i E S, then there exists g E O ( C Z , ~with ) i o ( f , g ) = i; i.e. with pro(2,) = i (see above) and 2, E (-En,) (a general property of divisorial parts of holomorphic germs). Then the minimality of y(i) implies y(i) 5 2,. This, and p r o ( y ( i ) ) = pro(Z,), and the fact that I off the diagonal has positive entries imply that (Eo,y(i)) 5 ( E o , ~ , )But, . since 2, E ( - L n e ) , (E0,Zg)5 0. 0 Lemma 5.5.4 can be improved for i 5.5.5. Lemma. Assume that i

i E s.

<mf:

< m f . Then (y(i),Eo)

=0

zf and only if

447

Proof. Via 5.5.4, it is enough to prove that ( y ( i ) ,Eo) < 0 is not possible for i < m j . Indeed, assume that ( y ( i ) ,Eo) < 0 for some i. Then, by a verification y ( i ) - Z j E hence (since is negative definite) y ( i ) Z j 2 0. In particular, i - m j = p r o ( y ( i ) - 2,) 2 0. 0

r

(-zne),

5.5.6. Back to r ( M ) .We consider again the graph r ( M ) .We fix an orbit

[k]= -aDs

+ L for some 0 5 a < p

(corresponding to the spinc-structure (T, of M ) . In the sequel we will use freely the notations of the previous subsection. Additionally, we write for any j = 1,. .. ,s:

a; := nj+l,,aj

+ nj+z,saj+l + . . . + a,.

E.g., by 3.6.2, a; = a . Consider now the sub-graph of r ( M ) with vertices j o , . . . , j , and the s edges connecting them. We wish to identify the cycle z ( i ) = iEo xi==,U j E j which has the properties of x ( i ) ‘restricted on this sub-graph’. More precisely:

+

5 a
5.5.7. Proposition. Fix i 2 0 and 0

(a) For j

=

T h e n the cycle z ( i ) := iEo + Cg,,

+

( ~ ( i )Ij-,D,I,

ujEj

is 2 0 and satisfies

E j ) 5 O for any j = 1,.. . ,S.

(b) Assume that the cycle E ( i ) := iE0

+ C,”=, iijEj is 2 0 and satisfies

+

( ~ ( i )Zf-,D,j, E j ) 5 O for any j = I , . . . , S.

Then

Proof. This property was used in a similar situation for the plumbing graph of a Seifert manifolds, applied for its ‘legs’, cf. 11.11 of [34]. For the convenience of the reader we sketch the proof. (a) By the definition of uj one has ujnj, 2 uj-lnj+l,s - a$. This is equivalent, via the identities njs = Icjnj+l,, - nj+z+, resp. a$ = nj+l,,aj + a$+l, with

448

+

Using the definition of uj+l, one gets u j k j - uj-1 aj 2 uj+l, which is exactly ( z ( ~ ) + Z ~ - , ~ ~ ~ , E j ) 5 0.(b) can be proved by a descending induction, using the above sequence of arguments in reversed order. 0

5.5.8. Corollary. Fix 0 5 a and 0 5 io < m f . Then

< p and write i = tmf + io for some t 2 0

I n particular,

Moreover:

Proof. Since the subgraphs I? and its complement are connected in r ( M ) only through j o , the two parts of x[k](i) - one supported by the other by its complement - have independent lives. But, for any j E 3,( q k l ( i ) ljkl,Ej)= (zp] 7r*(iik1),Ej) = (x[kl,Ej). Hence zp](i)restricted on should be exactly ~ ( i )Clearly, . the restriction on the complement of I? should be z [ k l ( i ) . Hence the first two statements follow. The last is the consequence of 3.5.7 together with the identity xk,(Eo)= 1, for (Zikl, Eo) =

r,

(..*(i[k]),

+

r

+

Eo) = 0.

0

5.5.9. 7,. Corresponding to this and 3.5.6, we write -ra(i):= xk,(z[kl(i)). With the notations of 5.5.8, notice that

iq - a It+l, Qmf+ P

+

hence ~ , ( i 1) - ~,(i) 2 -1 for any i, and = -1 only in very special situations, namely if

+

(tmf i 0 ) q - a > t and io @ S. 4mf + P In order to analyze when is this possible, we will consider sequences S e q ( t ) := {tmf io : 0 I io < m f } for fixed t 2 0. In such a sequence, notice that the very last element of N \ S , namely p - 1 = 26 - 1 is strict

+

449

smaller than mf- 1 (a fact, which can be proved by induction over the number of Newton pairs), hence the complete set N \ S sits in (0,. . . ,m f - 1). Therefore, there exists in Seq(t) an io satisfying (1) if and only if

+

(tmf 26 - 1)q - a > t. 4mf + P This is equivalent to t I t,, for t , defined in 5.3.2. In other words, if i 2 TO:= (to+l)mf,then ~ , ( i + l2) ~,(i), hence those values of 7, provide no contribution in the graded root (cf. 3.1.3). Moreover, for t E (0,. . . ,t,}, in Seq(t) one has:

( 0 if i o +1 if io A(&) := T,(tmf+io+l)-T,(tmf+io) = -1 if io 0 if io

5 ( t p + a ) / q , and i o # S ; 5 ( t p + a ) / q , and io E S ;

> ( t p + a ) / q , and io 6S; > ( t p + a ) / q , and io E S.

+ a ) / q with exactly E S :Y I (tp+a)/q}

In particular, A(i0) 2 0 for 0 I io 5 ( t p

At

:= #{Y

times taking the value +1, otherwise zero; and A(i0) with exactly

I 0 for io > ( t p + a ) / q

Bt := #{y $2 s : y > ( t p + a ) / q } times taking the value -1, otherwise zero. Notice that both At and Bt are strict positive (since 0 E S, respectively 26- 1 $Z S and 26- 1 > ( t p + a ) / q ) . This shows that Mt := max

O
+

+

~ , ( t m f io) = ~ , ( t m f ) At = T,((t

+ 1)mf) + B t ,

(2)

and

Mt > m a { Ta(tmf), Ta(tmf

m f )}. Therefore, the graded root associated with the values { ~ , ( i _) _} ~ < ~ < ( ~ , + is the same as the graded root associated with the values 7a(o),Mo,7,(mf),M1,7,(2mf),M2,. . . , T , ( t a m f ) , M t ~ , T , ( t , m f +mf). Finally notice, since #{y ( t p a ) / q } ,hence 6 - Bt

+

# S } = 6, one has 6 - Bt

=

#{y

# S :y 5

+ At = [ ( t p+ a ) / q ] + 1. Hence, by (2),

Since ~ ~ ( = 0 )0, this gives ~ , ( t m f inductively. ) Notice also that Bt 9(tP+4/91.

=

450

Clearly, the graded root associated with ra is the same as the graded root associated with ?, : {0,1,2,, . . ,2ta +2} -+ Z, where ?,(2t) := ra(trnf) and ?,(2t 1) := Mt. This is the tau-function of 5.3.2 if we delete -.

+

5.6. Examples. 5.6.1. Example. Assume p = q = 1. In this case M is integral homology sphere; a = 0 and t o = 26 - 2 = p - 2. In particular, the rank of ker U is p. Moreover, TO = S(6 - 1) and 7-0(2t) = t(t - 26 1)/2. The reader is invited to draw the graded root and verify that

+

6-1

H F + ( - M ) = I,f @ Z T O W - ~ @) Z(i+l)(ai-1+6

le2;

i=l 6-1

ker U = @ Z?iz+i). i=O

5.6.2. Example. More generally, assume only that p Floer homology of the unique spinc-structure is:

HF+(-M) =

la'

=

1. The Heegaard

(&--l)q

@

~ ( ~ a - @ 1 ) ~ ~?i1i/~j+i)(ji/~}~+i)(a6--1 i=l

5.6.3. Remark. Apparently in 5.6.2, H F + ( - M ) contains less information than the polynomial P , the above formula involves only the coefficients at for l > 6-1. But this is not the case. Indeed, since the Alexander polynomial A(t) is symmetric, one gets that

tP-ZP(l/t) - P ( t ) =

6(1+ t p - 1 )

+t + * . .+tp-1)

- (1

t-1

7

hence ~ ~ - 2 --i ai is a universal number. In particular, from { a e } e 2 6 - 1 one can recover P . This shows that from HF+(-S?,,,(Kf)) one can recover both the integer q and the isotopy type of K f c S3. It looks that similar result is valid for general surgery coefficients as well.

5.6.4. Remark. In the above example it is striking a &-symmetry of the graded root and of H F + ( - M ) . We will explain this for a = 0. The point is that if the canonical characteristic element has only integral coefficients (i.e., if the graph is numerical Gorenstein), then all the theory associated with the integral cycles has a duality. This happens for example if

45 1 p = 1;or if q = 1 and 26-2 is divisible by p. In our case, in these situations, the function 70 is stable with respect to the 22 action i H 2to 2 - i , i.e. 70(i) = 70(2t0 2 - i). This induces a symmetry of the root and of the

+

+

Heegaard Floer homology. But, for general p/q, the graphs are not 'numerical Gorenstein'. Even if they are, for general a, the symmetry may fail. 5.6.5. Example. Assume that K f c S3 is the torus knot (4,5) (i.e. g = 1 and (pl, q l ) = (4,5)).In this case 6 = 6, p = 12, S is generated by 4 and 5, hence

P ( t ) = 6 + 5t + 4t2 + 3t3 + 3t4 + 3t5 + 2t6 + t7 + t8 + t9+ tlO. Then for some (p/q)-surgeries the corresponding graded roots are presented in the next Figure. [Here we did not draw all the vertices, only those ones which are either local minimums or supremums of pairs of local minimums; in fact, exactly these ones are given by the function 7,. Also, the roots should be continued upward with one vertex in x-'(n) for each n 2 1.1 Recall that when we compute the grading of H F + , the value of T, is doubled, and then shifted by r,. The corresponding values of r , in these five cases are: 30, 71/4, 49/4, (P + 10)' 4P

-

P

, 60.

In particular, in the first and fifth case (i.e. when p = 1) one has d(-M,ao) = 0, as we expected: the Heegaard Floer homology are written in 5.6.1, resp. 5.6.2. In the second and third cases (i.e. when p/q = 2 and a = 0 , l ) the Z[U]-modules are:

In the forth case it is

452 ........

Ta (i)

0 ‘

..... .....

.

..

-10.

. .

-5

. . .

..

-15

.

-20

.................................

q=l p/q= 1 a=O

5.7. S”,,(K)

p/q=2 a=O

p/q = 2 a=l

P L 11 a=O

p / q = 1/2 a=O

as Kulikov graph-manifold.

5.7.1. Notice that 5.2.4 and 4.4.4 imply that the geometric genus p, of any normal surface singularity ( X ,0), whose link is diffeomorphic to M = S!,,,(K), satisfies the inequality (SIn) of 4.4.2, namely pg

I -sw(M,acan)

- (KZ f #J)/8.

This, via 5.3.2 and 5.3.4 reads as follows:

453 5.7.2. Theorem. Assume that the link M of ( X ,0 ) is S ~ p l q ( K T) h. e n

t2o

In the spirit of the second part (SId) of 4.4.2, (see also [37]), we expect that the above upper bound for p , is an optimal topological upper bound (when we run all the possible analytic structures on ( X ,0)). In order to be more cautious (for some warnings see [26]), we formulate it as a question:

5.7.3. Question. Is it possible to find a special analytic singularity ( X ,0 ) with M as its link, such that t>o

We invite the reader to formulate the similar inequalitylidentity for twisted line bundles, respectively for the other spin‘ structures. 5.7.4. The above Question 5.7.3 can be made even more precise; i.e., we even propose a candidate for an analytic structure (X,O). The point is that there is a natural family of normal surface singularities, the so-called Kulikov singularities, with link M . In general, the fact, that on the topological type of a singularity one can put the analytic structure of a Kulikov singularity, can be determined topologically from the combinatorics of the plumbing graphs. The corresponding graphs are called ‘Kulikov graphs’ (Karras called them ‘Kodaira graphs’). These particular graphs were studied intensively by Kulikov [16], Karras [15], Stevens [58] and others. In the sequel, we exemplify the construction via M (in fact, proving that r ( M ) is a Kulikov graph). The construction also provides a possible analytic germ ( X ,0) with special features. Consider the plumbing graph:

Clearly, the corresponding plumbed 3-manifold is 5‘2 ( K ) (the manifold obtained by 0 surgery along K ) . The point is that this graph is negative semi-definite, hence by a theorem of Winters [67], there exists a holomorphic proper map h : y -+ D (where D is a small disc, and y is a smooth surface) such that h-’(O) is a normal crossing divisor whose dual graph is rO(M),

454

and the generic fiber of h is a smooth curve. Then one blows up Icl generic points of the irreducible component of h-'(O) corresponding to the vertex j + . In this way one creates Icl (-1)-curves. Blow up k2 - 1 generic points of one of these new curves, and keep the other k~ - 1 unmodified; and continue this procedure. Then, the strict transform of h-'(O) together with all the exceptional divisors form a curve. Notice that the dual graph of all the irreducible components which are not (-1)-curves is exactly I'(M). These curves can be contracted providing a normal surface singularity (X, 0) with dual resolution graph I ' ( M ) . The analytic structure of (X, 0) has some nice properties (e.g. h induces a nice function on it, for details see e.g. [58]). A singularity constructed in this way is called Kulikov singularity.

5.7.5. Question. Is it true that p , = &,,a[tp/ql for a Kulikov singularity ( X ,0 ) with minimal good resolution graph G ( M ) ? 6. Unicuspidal rational plane curves and S E , ( K ) .

6.1. The semigroup distribution property. 6.1.1. The main objects of this section are irreducible rational projective plane curves in the complex projective space P2. It is a very difficult open problem to characterize the local embedded topological types of local singular germs which can be realized as the singularities of such a projective curve C of degree d. This problem has a long and rich history providing many interesting compatibility properties connecting local invariants of the singular germs { ( C , p i ) } i with some global invariants of C - like its degree, or the log-Kodaira dimension of P2 \ C , etc. For a (non-complete) list of some of these restrictions, see e.g. [8-101 and the references therein. The simplest one is the genus formula: the sum of the Milnor numbers of the local plane curve singularities {(C,pi)}ishould be ( d - l ) ( d - 2). (But, this is far to be enough for the characterization.) The article [8] proposes a new compatibility property - valid for rational cuspidal curves C. (A 'cusp' means a locally irreducible singularity, C is 'cuspidal' if it has only cusps.) Its formulation is the following. Consider a collection (C,pi):==, of cusps, let A i ( t ) be their characteristic polynomials, and set A ( t ) := Ai(t).Its degree is 26, where 6 is the sum of the deltainvariants of the singular points. (By the above mentioned genus formula: 26 = ( d - l ) ( d - 2).) Write A(t) as 1 ( t - 1)6 ( t - l)'&(t) for some polynomial &(t),cf. 5.2.2. Let cl be the coefficient of t(d-3-l)din & ( t )for any 1 = 0 , . . . , d - 3. (In the notations of 5.2.2 one has cl = ad(d-3-1).)

ni

+

+

455 6.1.2. Conjecture A. 181 Let (C,pi)y='=,be a collection of cusps, such that 26 = (d - l ) ( d - 2 ) for some integer d . If (C,pi)y='=,can be realized as the local singularities of a degree d (automatically rational and cuspidal) projective plane curve, then

c1 5 (1

+ 1)(1+ 2 ) / 2

f o r all 1 = 0,. . . ,d - 3.

+

In fact, the integers nl := c1 - ( I 1)(1+ 2 ) / 2 are symmetric: nl = nd-3-1; and no = n d - 3 = 0 automatically. We also mention that strict inequality (with v > 1 ) may occur, cf. [8]. Let k(P2\ C) denote the logarithmic Kodaira dimension of IF2 \ C. The main result of [8] is: 6.1.3. Theorem. [8] If k(P2\ C ) 5 1, then the above conjecture A is true (with n1 = 0).

On the other hand, rather surprisingly, in the unicuspidal case one can show the following. 6.1.4. Proposition. [8] If v = 1 then c1 2 ( I d-3.

+ 1)(1+ 2 ) / 2

for 0 5 1 5

Therefore, conjecture 6.1.2 in this case can be reformulated as follows: 6.1.5. Conjecture B1. With the notations of 6.1.2, i f v = 1 , then

c1

=

+

( 1 + 1)(1 2 ) / 2 for all 1 = 0,. . . ,d - 3.

If v = 1,the characteristic polynomial A of (C,p ) c (P"p ) is a complete embedded topological invariant of this germ, similarly as the semigroup S c N. Hence, one can reformulate conjecture B 1 in terms of S and d. It turns out that the collection of vanishings of all the coefficients nl is replaced by a very precise and mysterious distribution of the elements of the semigroup with respect to the (half-open) intervals Il := ( (1 - l ) d ,Id]: 6.1.6. Conjecture B2. Assume that v = 1. T h e n f o r any 1 > 0, the interval I1 contains exactly min(1 1,d } elements f r o m the semigroup.

+

In other words, for every rational unicuspidal plane curve C of degree d , the above conjecture is equivalent with the conjectural identity D ( t ) 3 0, where:

trk'q - ( 1 + 2t +. . . + ( d - l)td-2+ d(td-'

D ( t ) := kES

+ td+ td+' +. . .

456

For an explanation of the equivalences of conjectures B1 and B2, see 6.2.4. Here we only recall the key connection between the coefficients c1 and the semigroup S: ~1

= #{k E

S ; k 5 Id}.

This follows from 5.2.2 and from the fact that for 0 5 k if and only if p - k - 1 @ S.

< p one has: k E S

6.2. The semigroup distribution property and surface singularities. 6.2.1. Superisolated singularities. The theory of isolated hypersurface surface singularities ‘contains’ in a canonical way the theory of complex

projective plane curves via the family of superisolated singularities. These singularities were introduced by I. Luengo in [25]. A hypersurface singularity f : (C3,0 ) -+ (C, 0), f = fd +Id+’ (where fd is homogeneous of degree d and 1 is linear) is superisolated if the projective plane curve C := {fd = 0 ) c P2 is reduced, and its singularities are not situated on (1 = 0 ) . The equisingular type of f depends only on fd, i.e. only on the projective curve C c P2. In particular, all the invariants (of the equisingular type) of f can be determined from the invariants of the pair (p2,C). 6.2.2. A s s u m e in the sequel that C i s irreducible, rational with exactly one singular point (C,p ) . In the next discussion we follow [8,26]. Let p = 26, S and A be the local invariants of ( C , p ) .Set A ( t ) := tV6A(t). Let K c S3 be the local embedded link of ( C , p ) .Then, one shows that M = S!,(K); in particular, Hl(M,Z) = Z d . In fact, if we blow up the maximal ideal of the singular point of {f = 0) we get its minimal resolution, which contains one irreducible exceptional divisor, which is isomorphic to C and has self-intersection -d. From this picture the above identification of M also follows. But from this fact one also has that the invariant K 2 #J’ (of M ) equals 1 - d ( d - 2)2, hence it depends only on d. One also computes p g = d(d - l ) ( d - 2)/6. Before we continue our discussion, we recall that the Property (SId) (conjectured for ‘nice’ singularities), applied for the germ f and the canonical spin‘ structure reads as follows:

+

6.2.3. Property SWC. [37]

+

-sw(M,a,,) - ( K 2 # J ’ ) / 8

=p,.

457

+

In our case, since p , and K 2 # J depend only on d , the validity of SWC (6.2.3) would impose serious restriction on the link M = S?,(K), or equivalently on the local knot K c S 3 , which can be measured by the local invariants A or S. In this subsection we will explain this fact via the interpretation of sw(M,ocan)by the Reidemeister-Turaev torsion, cf. 2.3.4. One shows, cf. [26], that

1

7 M , u c a f i ( 1 )=

;i

c

P=1#E

A(E) and -

(I-

X(M) = - A(t)”(l) 2

+

(d - l ) ( d - 2 ) 24

Consider also (motivated partly by the above formula of the torsion):

1 R(t):= d

c p=1

A(@) - 1 -td2 (1- @)2 ( 1 - td)3’

Similarly,

Notice that this D ( t ) agrees with the one defined in subsection. In [8] the following facts are verified: 0 0

(DP)in the previous

R(t)= D ( t d ) / ( l - t d ) = N ( t d ) ; N ( t ) (hence R(t) too) has non-negative coefficients;

+

R ( 1 ) = -pg - SW(M,can) - ( K 2 # J ) / 8 . In particular, we have the equivalence of the ‘Seiberg-Witten invariant conjecture’ with the ‘semigroup distribution property’: 6.2.4. Theorem. Under the assumptions (6.2.2) the following facts are equivalent: (a) R ( l ) = 0, i.e. (the conjectured) Property SWC (6.2.3) is true (for the above germ f); (b) R(t)3 0; (c) N ( t ) = 0, i.e. Conjecture B 1 (6.1.5) is true; (d) D ( t ) = 0, i.e. Conjecture B2 (6.1.6) is true. 6.3. The semigroup distribution property and graded roots/Heegaard Floer homology. 6.3.1. Assume that we are in the situation of 6.2.2. In this subsection we will compare the invariants of the link M with the corresponding invariants of the Seifert 3-manifold C ( d ,d, d+ 1 ) - this is the link of the hypersurface

458

Brieskorn singularity x d + yd + z d f l = 0. This connection is very surprising, the possible conceptual relationship between these two spaces (together, hopefully, with the proof of conjectures listed in 6.2.4) will be explained in a forthcoming manuscript. 6.3.2. Theorem. [lo] With the notation sw(M) := sw(M, (T), the following facts are equivalent: (a) Conjecture B1 (6.1.5) is true, (b) The canonical graded roots of s!d(K) and C ( d , d , d 1) are the same. (c) The canonical Heegaard-Floer homologies of -s?d(K) and -C(d, d , d 1 ) are the same modulo a grading shift, namely:

+

+

H F + ( - S : d ( K ) ) [ l - d(d - 2)'] = H F + ( - C ( d , d, d

+ 1))[-d(d - l ) ( d - 3 ) ] .

(d)

The proof is given in several steps. The main point is that both 3manifolds s?d(K) and C ( d , d , d 1 ) are almost rational, cf. (2.7). In particular, in both cases, the canonical graded root can be determined via the 'T-construction' 3.5.6. In the first case this is done explicitly in 5.3.2, while for the second case, see [34],last section.

+

6.3.3. Example. Let us rewrite 5.3.2 for the case M = S?,(K) and for the canonical spinC structure: a = 0, q = q' = 1, p = d. Set c1 := CY(d-3-l)d and define T : { 0 , 1 , . . , ,2d - 4) + Z by

d - 1(6 - I ) ,

T(21) = v

Then

(&an, X c a n )

= (&,

T(21

+ 1)= T(21 + 2 ) f cd-3-1.

x7).

6.3.4. Example. Consider the Seifert manifold C ( d ,d , d+ 1). Its canonical graded root is the following (cf. [34]).For any 0 5 1 5 d - 3 write cy := (1 1)(1+2)/2, and26:= ( d - l ) ( d - 2 ) . ThendefineT": { O , l , ..., 2d-4} +Z bY

+

T"(21)

Then

(&an

=

V

d - Z(6 - l ) ,

T"(21+ 1 ) = T"(21+ 2)

+ c;-34.

, X c a n ) = (RT" ,X7" 1.

Notice the shocking similarities of 6.3.3 and 6.3.4: the graded roots associated with S?,(K) and C ( d , d , d 1 ) coincide exactly when c1 = cy for all

+

459

I , a system of equalities subject of Conjecture B1. This proves ( a ) @ (b). ( b ) + (c) follows from 3.5.2, since in both cases

+

where the shift in the case of M = S?,(K) is K 2 # J = 1 - d(d - 2)2, while for M = C ( d ,d, d 1) is K 2 # J = -d(d - l ) ( d - 3). This, and the corresponding definitions show (c) + ( d ) . Since the Property SWC (6.2.3) is true for the Brieskorn singularity ~ B R := xd yd zdS1 (cf. [38]), and the geometric genus of the superisolated singularity f equals the geometric genus of ~ B (both R equal d(d - l ) ( d - 2)/6), this last identity (d) is also equivalent with the validity of the Property SWC (6.2.3) for f - a fact, which is equivalent with (a) by 6.2.4.

+

+

+ +

6.3.5. Remark. For ( d ) + (b) one can also argue as follows. Regarding the Seiberg-Witten invariant of M = S!,(K), by 5.3.2 and 5.3.4 one has

and there is a similar formula for M = C ( d , d , d rep lacements.

+ 1) with the obvious

6.3.6. Example. Assume that d = 5 and C is unicuspidal whose singular point has only one Puiseux pair ( a ,b) with a < b. Then by the genus formula the possible values of (a,b) are (4,5), (3,7) and (2,13). It turns out that the first and the third cases can be realized, while the second not. The corresponding canonical graded roots (together with the root of C(5,5,6)) are drawn in the next picture.

0 -1

-2 -3 -4 -5 s 3 5 ( T 2 , 1 3 ) , s35(T4,5)

.V5,5,6)

s35(T3,7)

460

References 1. N. A’Campo, La fonction zeta d’une monodromy, Com. Math. Helvetici, 50 (1975), 233-248. 2. V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable maps, Volume 1 and 2, Monographs Math., 82-83,Birkhauser, Boston (1988). 3. M. Artin, Some numerical criteria for contractibility of curves on algebraic surfaces. Amer. J. of Math., 84,485-496 (1962). 4. M. Artin, On isolated rational singularities of surfaces. Amer. J. of Math., 88,129-136 (1966). 5. E. Brieskorn, H. Knorrer, Plane Algebraic Curves, Birkhauser, Boston (1986). 6. W. Chen, Casson invariant and Seiberg-Witten gauge theory, Turkish J. Math., 21 (1997), 61-81. 7. D. Eisenbud, W. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Studies 110,Princeton University Press (1985). 8. J. Fernhdez de Bobadilla, I. Luengo-Velasco, A. Melle-Hernkndez, A. NBmethi, On rational cuspidal projective plane curves, Proc. of London Math. SOC. 92 (3) (2006), 99-138. 9. J. Fernhdez de Bobadilla, I. Luengo-Velasco, A. Melle-Hernindez, A. NBmethi, Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair, to appear in the Proccedings of the Singularity Conference (San Carlos-Luminy, 2005). 10. J . Ferntindez de Bobadilla, I. Luengo-Velasco, A. Melle-Hernhndez, A. NBmethi, On rational cuspidal curves, open surfaces and local singularities, submitted. 11. R.E. Gompf, I.A. Stipsicz, An Introduction to 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20, Amer. Math. SOC.(1999). 12. H. Grauert, Uber Modifikationen und exceptinelle analytische Mengen, Math. Annalen, 146 (1962), 331-368. 13. H. Grauert, R. Remmert, Komplexe Riiume, Math. Ann. 136 (1958), 245318. 14. S.M. Gusein-Zade, F. Delgado, A. Campillo, On the monodromy of a plane curve singularity and the PoincarB series of the ring of functions on the curve, Functional Analysis and its Applications, 33(1) (1999), 5667. 15. U. Karras, On Pencils of Curves and Deformations of Minimally Elliptic Singularities, Math. Ann., 247 (1980), 43-65. 16. V.S. Kulikov, Degenerate elliptic curves and resolution of unimodal and bimodal singularities, Functional Anal. Appl. 9 (1975), 69-70. 17. M. Kreck, S. Stolz, Nonconnected moduli spaces of positive sectional curvature metrics, J. of the AMS., 6 (1993), 825-850. 18. J. Kollk, Shafarevich Maps and Automorphic Forms, Princeton University Press, Princeton (1995). 19. H.B. Laufer, Normal two4imensional singularities. Annals of Math. Studies, 71,Princeton University Press (1971). 20. H.B. Laufer, On rational singularities, Amer. J. of Math., 94 (1972), 597-608.

461 21. H.B. Laufer, Taut two-dimensional singularities, Math. Ann., 205 (1973), 131-164. 22. H.B. Laufer, On minimally elliptic singularities, Amer. J. of Math., 99 (1977), 1257-1295. 23. C.Lescop, Global Surgery Formula for the Casson-Walker Invariant, Annals of Math. Studies, vol. 140,Princeton University Press (1996). 24. Y. Lim, Seiberg-Witten invariants for 3-manifolds in the case bl = 0 or 1, Pacific J . of Math., 195 (2000), 179-204. 25. I. Luengo, The p-constant stratum is not smooth, Invent. Math., 90 (1) (1987), 139-152. 26. I. Luengo-Valesco, A. Melle-Hernindez, A. NBmethi, Links and analytic invariants of superisolated singularities, Journal of Algebraic Geometry, 14 (2005), 543-565. 27. M. Marcolli, B.L. Wang, Seiberg-Witten invariant and the Casson-Walker invariant for rational homology 3-spheres1 math.DG/0101127, Geometrice Dedicata, to appear. 28. J. Milnor, Singular points of complex hypersurfaces, Annals of Math. Studies 61,Princeton University Press (1968). 29. D. Mumford, The topology of normal singularities of an algebraic surface and criterion for simplicity, IHES Publ. Math. 9,5-22 (1961). 30. A. NBmethi, Five lectures on normal surface singularities, lectures delivered at the Summer School in Low dimensional topology Budapest, Hungary, 1998; Bolyai Society Math. Studies 8 (1999), 269-351. 31. A. NBmethi, Dedekind sums and the signature of f(rc,y) zN,II., Selecta Mathematica, New series, 5 (1999), 161-179. 32. A. NBmethi, “Weakly” Elliptic Gorenstein Singularities of Surfaces, Inventiones math., 137 (1999), 145-167. 33. A. NBmethi, A Invariants of normal surface singularities, Contemporary Mathematics, 354 (2004), 161-208. 34. A. NCmethi, On the Ozsvith-Szab6 invariant of negative definite plumbed 3-manifolds, Geometry and Topology 9 (2005), 991-1042. 35. A. NBmethi, Line bundles associated with normal surface singularities,

+

arXiv:math.AG/0310084.

36. A.

NBmethi,

On

the

Heegaard

Floer

homology

of

S?,,,(K),

arXiv:math.GT/0410570.

37. A. NBmethi, L.I. Nicolaescu, Seiberg-Witten invariants and surface singularities, Geometry and Topology, Volume 6 (2002), 269-328. 38. A. NBmethi, L.I. Nicolaescu, Seiberg-Witten invariants and surface singularities I1 (singularities with good C*-action), Journal of London Math. SOC.(2) 69 (2004), 593-607. 39. A. Ndmethi, L.I. Nicolaescu, Seiberg-Witten invariants and surface singularities: Splicings and cyclic covers, Selecta Mathematica, New series, Vol. 11, Nr. 3-4 (2005), 399-451. 40. W. Neumann, Abelian covers of quasihomogeneous surface singularities, Proc. of Symposia in Pure Mathematics, vol. 40, Part 2, 233-244. 41. W.D. Neumann, A calculus for plumbing applied to the topology of complex

462

surface singularities and degenerating complex curves. Pansactions of the A M S , 268 Number 2 (1981), 299-344. 42. W.D. Neumann, F. Raymond, Seifert manifolds, plumbing, p-invariant and orientation reserving maps, Algebraic and Geometric Topology, (Proceedings, Santa Barbara 1977), Lecture Notes in Math. 664,161-196. 43. W. Neumann, J. Wahl, Casson invariant of links of singularities, Comment. Math. Helv. 65 (1991), 58-78. 44. W. Neumann, J. Wahl, Universal abelian covers of surface singularities, ’T+-endsin singularities, 181-190, Trends Math., Birkhuser, Base1 (2002). 45. W. Neumann, J. Wahl, Universal abelian covers of quotient-cusps, Math. Ann., 326 (2003), no. 1, 75-93. 46. W. Neumann, J. Wahl, Complex surface singularities with integral homology sphere links, Geom. Topol., 9 (2005), 757-811 (electronic). 47. L.I. Nicolaescu, Seiberg-Witten invariants of rational homology spheres, Commun. Contemp. Math., 6 (2004), no. 6, 833-866. 48. T. Okuma, Universal abelian covers of rational surface singularities, J . London Math. SOC.,(2) 70 (2004), 307-324. 49. P.S. OzsvLth, Z. Szab6, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundaries, Adw. Math., 173 (2003), no. 2, 179-261. 50. P.S. OzsvLth, Z. Szab6, Holomorphic disks and topological invariants for closed three-manifolds, A n n . of Math., (2) 159 (2004), no. 3, 1027-1158. 51. P.S. OzsvLth, Z. Szabb, On the Floer homology of plumbed three-manifolds, Geom. Topol., 7 (2003), 185-224 (electronic). 52. P.S. OzsvLth, Z. Szabb, Holomorphic triangle invariants and the topology of symplectic four-manifolds, Duke Math. J., 121 (2004), no. 1, 1-34. 53. P. OzsvLth, Z. Szabb, Knot Floer homology and rational surgeries, math.GT/0504404. 54. H. Rademacher, E. Grosswald, Dedekind Sums, The Carus Math. Monographs, MAA (1972). 55. M. Reid, Chapters on Algebraic Surfaces. In: Complex Algebraic Geometry, IAS/Park City Mathematical Series, Volume 3 (1997) (J. KollLr editor), 3159. 56. R. Rustamov, A surgery formula for renormalized Euler characteristic of Heegaard Floer homology, math.GT/0409294. 57. M. Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Annuls of Math., 131 (1990), 411-491. 58. J. Stevens, Elliptic Surface Singularities and Smoothings of Curves, Math. Ann., 267 (1984), 239-247. 59. J. Stevens, Kulikov singularities; Thesis, Leiden (1985). 60. M. Tomari, A pg-formula and elliptic singularities, Publ. R. I. M. S. Kyoto University, 21 (1985), 297-354. 61. M. Tomari, Kei-ichi Watanabe, Filtered rings, Filtered Blowing-Ups, Normal Two-Dimensional Singularities with “Star-Shaped” Resolution, Publ. R. I. M. S. Kyoto University, 25 (1989), 681-740. 62. V.G. Turaev, Torsion invariants of Spinc-structures on 3-manifolds, Math.

463 Res. Letters, 4(1997), 679-695. 63. Ph. Wagreich, Elliptic singularities of surfaces. Amer. J. of Math., 92 (1970), 419-454. 64. K. Walker, An extension of the Casson’s invariant, Ann. of Math. Studies 126,Princeton University Press (1992). 65. S.S.-T. Yau, On almost minimally elliptic singularities, Bulletin of the AMS, 83 Number 3 (1977), 362-364. 66. S.S.-T. Yau, On maximally elliptic singularities, Transactions of the AMS, 257 Number 2 (1980) , 269-329. 67. G.B. Winters, On the existence of certain families of curves, Am. Journal of Math. 96(2) (1974), 215-228.

464

Chern classes and Thom polynomials Toru Ohmoto

Department of Mathematics, Hokkaido University Sapporo, 060-0810, Japan E-mail: [email protected] This is an exposition on an equivariant theory of Chern-Schwartz-MacPherson classes for singular algebraic varieties and related topics.

1. Introduction Following Jean-Paul Brasselet's lecture2 on Chern classes of singular varieties in this ICTP summer school, I introduce the theory of equivariant Chern-Schwartz-MacPherson classes and show two different types of applications. We work in the complex algebraic context for simplicity; for a variety X (an irreducible and reduced scheme), let H , ( X ) denote the Borel-Moore homology group of the underlying analytic space. In an algebraic (quasiprojective) context over a field of characteristic 0, the homology should be the Chow group A , ( X ) of algebraic cycles under rational equivalence. Our main theorem is

Theorem 1.1 (Equivariant Chern class transform22). Let G be a complex algebraic group. For the category of complex algebraic G-varieties X and proper G-morphisms, there is a natural transformation f r o m the equivariant constructible function functor to the equivariant homology functor

c,": P ( X ) -+ H,G(X)

-

such that if X is non-singular, then CF(llx) = c G ( T X ) [ X ] G where c G ( T X ) is G-equivariant total Chern class of the tangent bundle of X . CF is unique in a certain sense. I n particular, if G = { e } , then CF coincides with ordinary C, .

465

Remark that in general the quotient X/G does not make sense as a variety or a scheme, but the quotient stack [X/G] exists (see 3.5 below). Thus the above theorem may be regarded as an extension of original MacPherson’s transformation C, to a wider category of spaces, quotient stacks. This equivariant setting is based on Totaro-Edidin-Graham’s “algebraic Bore1 construction” of classifying s p a c e ~ ~so~ first ? ~ , I will talk about the basic idea of this construction ($3). Second, I will show some applications of C,“. For a G-variety X, the top term of our Chern class coincides with the equivariant fundamental class, and the degree (if it makes sense) equals the Euler characteristic:

c,”(n,)=x(x)[pt] +.-+[x]G. So, in principle, when we are given some formulae of fundamental classes or Euler characteristics, we may expect similar type formulae for the total Chern classes, because of the additivity of the Chern transformation. In 54 and $5 below we outline about such generalization^^^-^^. The theory of “Thom polynomials (Tp)” has been newly developed since the mid of go’s, see Kazarianl3>l4,Feh6r-Rim6nyigy30.Given a nonsingular G-variety V and an invariant subvariety r], the T h o m polynomial Tp(r])of in V is defined to be the G-equivariant Poincare dual to L * [ ~ ] ] G E HF(V), where L is the inclusion. In particular, if V is a G-affine space, then the Thom polynomial of r] is expressed by an universal polynomial in G-characteristic classes:

T p ( v ) := h a l ~ L , [ r ] E] ~Hz(v)11 H E ( p t ) N H*(BG).

I propose in 54 a “total class version” of Tp, that is the “Segre class transformation” corresponding to C,“ (Subsection 4.2): SSM

: Fgv(v)

H*(BG).

Our s S M does not hold the naturality for pushforward but has a good property for pullback, like as T p does: in fact the lowest degree homogeneous term of S S M ( n 1 7 ) is just ~ ( 7 ) . Historically, T p has appeared in a modern enumerative theory of singularities of complex analytic or real smooth maps. In this case, roughly, V is the space of map-germs and G is the group of some equivalence relations (R, d,K: etc). For instance, the degeneracy loci of vector bundle morphisms belong to the case of linear maps with the action of general linear groups (see Example 4.1), and a prototype of sSM has been studied in Parusiri~ki-Pragacz~~. For mapping cases, I will talk a bit about s S M ( p ) where p. is the Milnor number constructible function associated to complex

466

maps (in negative codimension); such “local invariants” of map-germs have been missed in the recent T p theory so far. In 55 we deal with a typical case, the symmetric product SnX of a complex (possibly singular) variety X . As a “total class version” of a wellknown formula for orbifold Euler characteristics of S n X (Hirzebruch-Hofer 12), the generating function of orbifold Chern homology classes of S n X is given by

n=O

k=l

c,”==,

in the Q-algebra znH, (S”X;Q) of the formal power series whose coefficients are total homology classes. Here D is a letter indicating a diagonal operator. This (ob) is reduced from the “Dey-Wohlfahrt formula” for equivariant Chern classes associated to Sn -representations of a given group A (Theorem 5.2). If X is a point, the formula coincides with a classical enumerative formula in group t h e ~ r y ~ ~ This ~ ~direction ’. tends to the further theory of Chern classes for more complicated graded spaces, such as Hilbert schemes, configuration spaces, moduli spaces of stable maps (curves), etc. I should mention other characteristic classes and natural transformations: Brasselet-Schurmann-Yokura3have introduced the theory of motivic Chern classes and Hirtzeburch classes, which unifies the Chern-SchwartzMacPherson class and Baum-Fulton-MacPherson’s singular Todd class (see also a comprehensive survey32). So, it would be a promising task to look for a similar type formulae in singular Todd classes as in (ob) above by passing through their theory.

Acknowledgments I would much like to thank organizers and people whom I met in this ICTP summer school and workshop; especially thank Jorg Schurmann for valuable remarks, and the referee for his comments. 2. A quick review on Chern classes 2.1. Chern classes of vector bundles

For a compact complex manifold M of dimension n and for 1 5 i 5 n, the i-th Chern class c i ( T M ) is given by the obstruction class for the existence of ( n- i 1)-frames over M : roughly speaking, let s be a generic collection of n - i 1 vector fields over M , then the singular set ~ ( s ) ,at which s is

+ +

467

linearly dependent, represents the obstruction class

where L is the inclusion (cf. Example 4.1). The Poincar6-Hopf theorem for vector fields is the special case of i = n (then the RHS equals x ( X ) ) .The total Chern class c ( T M ) means the sum 1 c l ( T M ) . . . k ( T M ) E H * ( M ) .Later we will define Thom polynomials as this kind of obstruction classes. Chern classes are actually defined for (topological) complex vector bundles, not only tangent bundles T M : to a complex vector bundle E -+ M the function c assigns a total class c ( E ) = C c i ( E )where ci(E) E H Z i ( M ) so that it satisfies the following a x i ~ m ’ ~ > l ~ :

+

0

0 0

0

+ +

co(E) = 1 and ci(E) = 0 for i > rank E ; c ( E ) = c(E’)c(E’’) for any exact sequence 0 -+ E’ -+ E -+E” -+ 0; c ( f * E ) = f * c ( E ) where f * E is the pullback bundle via a base change f ; cl(yl) ,-.. [CP1]= 1, for the canonical line bundle ? I ( = O(1)) over the projective space CP1.

An important fact in topology is that any rank n vector bundle can be obtained from a universal vector bundle & over the classifying space BGL(n). Here BGL(n) is given by the inductive limit (Ic -+ m) of the Grassmanian of n-dimensional subspaces in Cn+k,and Cn is the limit of tautological vector bundles of the Grassmanianns. For any E -+ M , there is a classifying map (unique up to homotopy) p : M -+ BGL(n) so that E = p*&. Note that H * ( B G L ( n ) )= Z [ c l , .. . ,G](the degree of ci(= ci(&)) is 2 i ) and q ( E ) coincides with p*ci. Also in algebraic geometry, the classifying space and a variant of classifying maps are available35.

2.2. Chern class f o r singular varieties For a singular algebraic variety X , the tangent bundle does not exist. To define reasonable “Chern classes” for X , one needs “substitutes” for frames or tangent bundles by some manipulations. The most particular feature is that those “Chern classes” are no longer cohomology classes of X ,but are homology classes, because of the lack of the Poincar6 duality. M. S ~ h w a r t z ~ ~ defined a certain obstruction class for “radial vector fields (frames)” on X , which is today called the Chern-Schwartz-MacPherson class C * ( X ) E

468

H,(X): the degree C o ( X ) is equal to x(X) and the top component Cn(X) is the fundamental class [XI. The axiomatic description is due to MacPherson'*: there exists a unique natural transformation C, : 9 ( X ) -+ H,(X), where F ( X ) is the group of constructible functions over X , so that 0

0

(natrural transform) C, is a homomorphism of additive groups, and f, o C, = C, of, for proper morphisms f : X -+ Y ; if X is non-singular, then C,(Ix) = c ( T M ) [XI.

-

In fact it is shown4 that the above two classes coincide: C,(X) = C*(llx). In a purely algebraic context (char(k) = 0 ) , Kennedy15 reformulated MacPherson's transformation, that is, C, : F ( X ) + A , ( X ) , through the Lagrange cycle approach. 3. Equivariant theory

A G-action on a variety X is a morphism G x X -+ X, ( g , x ) -+ g.x, with properties h.(g.x) = (hg).x ( h , g E G) and e.x = z (e is the identity element of G). We call X a G-variety for short. A morphism f : X -+ Y between G-varieties is called G-equivariant if f (9.z) = g.f (x)for any g E G, z E X . (Precisely,those identities mean the commutativity of corresponding diagrams of morphisms.)

3.1. Totaro's construction of BG Let G be a complex linear algebraic group of dimension g. Take an 1dimensional linear representation V of G and a G-invariant Zariski closed subset S in V so that G acts on U := V-S freely. Let I(G) denote the collection of such U (that is, all pairs (V,S ) ) .We say U < U' (where U = V - S , U' = V' - S') if there is a representation V1 satisfying V @ V1 = V', U @ V1 C U' and codim v S < codim vt S'. Then ( I ( G ) <) , is a directed set (in fact U, U' < U @ U'). All quotients U -+ U/G form an inductive system over I(G) (via inclusion maps). The inductive limit is the algebraic counterpart of the universal principal bundle E G -+ BG (Tota1-0~~). Let X be a G-variety. For any U E I(G), the diagonal action of G on X x U , which is always a free action, gives a principal bundle X x U + X X G U = (XxU)/G, and thus the equivariant projection X X U + U serves the fibre bundle X X G U --+ U/G with fibre X . We denote Xu := X X G U for short. Roughly saying, the universal fibre bundle X X G E G + BG is approximated by those mixed quotients Xu t U/G (Edidin-Graham7).

469

3.2. G -equivariant (co) homology

Let X be a G-variety having pure dimensional n. The i-th equivariant cohomology of X is given as the projective limit

H&(x) = limHi(XU). e We denote H$(X) = limH*(Xu). This becomes a contravariant functor *(the pullback of a G-morphism f is denoted by Let be a G-equivariant vector bundle E --+ X (i.e., E, X are G-varieties and the projection is G-equivariant so that the action of g sends each fibre E, to Eg.zlinearly). Then we have a vector bundle EU -+ XU over the mixed quotient for each U , denoted by &I, and define the G-equivariant Chern class cG(C) E HZ(X) to be the projective limit of Chern classes c(€&). When X = { p t } , an equivariant vector bundle is a representation V(-+ { p t } ) ; its equivariant Chern class is denoted by c G ( V ) E H & ( V ) N H*(BG). Next, let us define the homology as follows: a key point is shafting dimensions via pullback. For any pair U < U', we have a diagram of natural projection and inclusion

fz).

<

xu P XU@V*5 xu,

(1)

Put g = dimG, 1 = dim U , 'I = dim U' and s = codimv S (U = V - 5'). Note that L is an open embedding, so the pullback L* is defined and is in fact isomorphic if 2(n - s ) < i 5 2n because of dimensional reason. We prefer to denote its inverse by L, := (i*)-l,and then the diagram (1) induces the following diagram for (Borel-Moore) homology groups

We then define the i-th equivariant homology group to be

H?(x)= Hi+2(l-g)(xu) for U with codimS high enough. This group is trivial for i > 2n and nontrivial for any negative i in general. The direct sum is denoted by H,"(X) = $Hy(X). For a proper G-morphism f : X -t Y , the pushforwad f," is defined by taking limit of (fu)* : XU -+ Yu; thus H," becomes a covariant functor. For any U ,the ordinary (Borel-Moore) fundamental class [XU]tends to a unique element of H g ( X ) , denoted by [X]G, which is called the Gequivariant fundamental class of X. It induces a homomorphism

-

[ x ] G : H ~ - Z ( X ) + H?(x),

a ++ ru(a)

-

[XU]

470

where ru denotes the restriction. If X is nonsingular, this is isomorphic, called the G-equivariant Poincare' dual.

Example 3.1. G = G L ( 1 ) , X = { p t } and Um = Cm+' - (0) in I ( G ) : P

p m + 1 - {Pt} L

\ ...

... J

... c

pm

C

pm+l

c ... c p w

Then H_GzL,")(pt) 21 H g ( l ) ( p t )= Z for n 2 0, and trivial otherwise. 3.3. Equivariant constructible functions

Let F ( X ) denote the Abelian group consisting of all constructible functions over X. The subgroup of G-invariant constructible functions is denoted by

FZ,(X):= { Q E F(X) 1 a ( g ( 2 ) ) = a ( z ) ,1~ E X,gE G }. For any U < U' with the projection p : V' = V @ V1 -+ V, we have p* : Fgv(Xx V ) -+ F&(X x V ' ) ( a H a o (id x p ) ) . We define the group of G-equivariant constructible functions associated to X t o be the inductive limit

For a proper G-morphism f , the pushforward :f is defined in an obvious way, so FG becomes a covariant functor. There is a canonical inclusion, denoted by 40,

F:,(X) c F ~ ( x ) ,a0 H lim(a0 x

nv).

Note that F$,(pt) = Z, but F G ( p t ) consists of functions over representations. Now let us think of the group F ( X u ) of (ordinary) constructible functions over mixed spaces. The previous diagram (1) induces

F ( X U ) 5 F(XU@V1)5 F ( X U , )

(3)

Note that F ( X u )is canonically identified with F&(X x V ) .Then the above composed map ~ , p *commutes with p* : F$,(X x V ) -+ Fg,(X x V ' ) via restrictions caused by U c V and U' c V'.

471 3.4. Equivariant natrual transformation

The space X XGEGis the inductive limit (via inclusions) of mixed quotients X u , while the definitions of HF and 3Ginvolve a contravariant operation p*. Roughly saying, C$(It,) is the limit class “C,(X X G EG)” divided that is precisely defined below in terms of limits of Radon by “~*c(TBG)”, transforms. Recall that the Verdier-Riemann-Rochtheorem (VRR theorem) refers to C, with the contravariant operation induced by f : X -+ Y11737!31:

Theorem 3.1. For a smooth morphism f : X -+ Y , let c ( f ) be the Chern class of the relative tangent bundle. Then the following diagram commutes:

F ( Y )5 H * ( Y )

f* 1

1 c(f)/-.f*

F ( X ) 5 H,(X) Now we put

where Htr means the direct sum of Hiover 2(n-s+Z-g) < i _< 2 ( n f l - g ) (1 = dim U ,s = codim v S ) . Note that cG(V)is the Chern class of the vector bundle X X G TU -+X U . a We apply the VRR theorem for p : X u B v l X u , then it turns out that CUBV, )* 0 P* = P* 0 cup. a For an open embedding L : Xuev1 X u ! , it holds that L* 0 Cuev,,* = Cut,, o L* in this range of dimension. Consequently, “Radon transforms” (3) and (2) commute as follows’: -+

-+

Since p* : 3$,(X x V ) -+ F g w ( X x V’) commutes with the left ~ * p *(as noted in the end of 3.3) , we can take the limit homomorphism

This is our equivariant Chern-MacPherson transformation in Theorem 1.1.

472

3.5. Quotient stacks

By definition, a quotient stack X = [X/G] is a category itself, whose objects are principal G-bundles p : P + B together with G-equivariant morphism cp : P --+ X and its arrows are morphisms between principal bundles which make their equivariant morphisms to X commute. In EdidinGraham7 (Proposition 16) the integral Chow groups of a quotient stack X = [X/G] is introduced by A,(X) := AF+:+,(X)where g = dimG; in fact it is independent from the choice of the presentation X and G. In exactly the same way, 3 ( X ) := 3 G ( X )is well-defined. “Arrows” f : X + Y should be suitably chosen so that they admit pushforwards for the above 3 and A,, e.g., assume they have presentations X = [X/G], Y = [Y/H] and a proper morphism f : X --+ Y, together with an (injective) homomorphism p : G + H and the mixed space Y x p H being proper over Y. Then pushforwards FG(X)+ 3 H ( Y ) and AY+:+,(X)-+ AF+:,(Y) are defined via the Bore1 construction, and they commute with C,” and CF22723. Thus Theorem 1.1 can be translated as follows: For the (above) category of quotient stacks X, there is a natural transformation C, : 3 ( X ) -+ A,(X) so that for any nonsingular varieties X = X (with G = { e } ) , it holds that C,(llx) = c ( T X ) [XI.

-

Remark 3.1. An object of [X/G] is thought of as “a family of G-orbits in X which is parametrized by B” . Here is another useful interpretation: [X/G] can also be regarded as the universal space for sections of associated bundles with jibre X and structure group G. In fact, t o an object B 8 P 3 X we associate E = X X G P (as an algebraic space) together with a section s : B + E so that s ( B ) = (graphcp)/G. This interpretation leads us to T p in the next section. 4. Thom polynomials 4.1. Universality

Let V be a G-afine space and r] an invariant subvariety. Let i : r] + V be the inclusion map. The T h o m polynomial Tp(r]) E H*(BG)(=H;(V)) is defined to be the G-PoincarB dual to i f [ r ] ]E~HF(V). By definition, Tp(r]) satisfies the following universality: (u): For any bundle E + M with fibre V and the structure group G over a manifold M of dimension m, let E,, --+ M be the associated fibre bundle with fibre r]. For a “generic” section s : M --+ E (e.g., s is transverse to the

473

subvariety E,, in E) , we define the singular set of type

v

v ( s ) := s - l ( E v ) , which has the expected codimension 1 = codimv. Let i : v ( s ) -+ M be the inclusion. Then, after substituting ci(E) to universal classes ci, it holds that

i*[v(s)l= TP(rl)(C(E)) [MI E H2(m-I)(M). We may drop the condition in (u) (“genericity” of the section and/or “smoothness” of the base space), if we replace i,[v(s)] in the formula by a certain localized class (a residue class) having expected dimension.

Example 4.1 (Thorn-Porteous formula2g).Let V := H o m ( C m , Cn) o n which the group G = G L ( m , C ) x GL(n,C) acts as linear coordinate changes. Take := D k , the closure of the orbit of linear maps with dim ker = k . Then the Thom polyomial is given by a certain Schur polydet[ck+l-i+j]). Namely, f o r a suitably nomial: T p ( D k ) ( c ) = A,+,(c)(:= (k) generic vector bundle map f : E + F (i.e., a section f : M + Hom(E, F ) ) , where E and F are vector bundles over a manifold M of rank m and n, respectively, then the degeneracy loci D k ( ( f ) ( ={x E M,dimker fz 2 k}) is expressed by

b[Dk(f)= ] det[ck+i-i+j(F - E ) ] l < i , j < k

[MI.

w h e r e c ( F - E ) = (l+~l(F)+...)(l+cl(E)+..,)l . There are some ways to correct the formula without the genericity/smoothness condition. For instance, in the context of algebraic geometry Fultonlo defines the degeneracy loci class by the pushforward of the localized top Chern class of a bundle over the Granssmanian bundle Gr(Ic; E ) . I n complex analytic geometry, there is a different way of localizations, see e.g. S U W ~ ~ ~ . 4.2. Segre- Sc hwartz-MacPherson class

For a subvariety 2 in a non-singular variety M , the Segre-SchwartzMacPherson class1>22of 2 in M is defined by s S M ( 2 , M ) := c ( T M l z ) - 1

-

C*(Z)E H*(Z),

which is an analogy to the relation between F’ulton’s canonical class C F(2) and the Segre covariance class s ( 2 , M ) : in fact C F ( Z )is defined to be c ( T M l z ) n s(2, M)?

474

Now, for an invariant subvariety v in a non-singular G-variety V, we define the equivariant Segre-Schwartz-MacPherson class by

sZM(v,v) := c ~ ( T v I , ) - ~

c,G(n,)E H,G(Q).

In particular, suppose V is a G-affine space, then we have an additive homomorphism

sSM : Fg,(v)

-+ H*(BG), S S M ( n , )

:= DualG L * s ~ , ( v ,

v).

Here is an immediate consequence of the definition:

Theorem 4.1. Let V and 71 be as above. Then (1) sSM(r])(:=s S M ( l , ) ) = Tp(v) higher degree terms; (2) For G-morphisms L : V’ + V being transverse to q , it holds that

+

SSM(L*n,)

(3) (universality) Let E then we have

2,s

SM

.+

=

L*SSM(ns).

M and a generic section s be as in (u) of 4.1,

-

( v ( s )M , ) = s S M ( v ) ( c ( E ) ) [MI E H * ( M ) .

In particular, the Euler characteristic x ( q ( s ) ) admits a universal expression, that is, the degree c(TM)sSM(v)(c(E)).

s,

As a remark on (3), the base space M can be possibly singular: under suitable “genericity” of s, the formula is replaced by i*C*(v(s))= s S M ( v ) ( c ( E ) ) C * ( M ) E H*(M). 4.3. Singularities of maps Let k = C or R. In the T p theory in Singularity Theory of Differentiable Mappings, V is the space E(m,n) of map-germs ( k m , 0) --+ ( k n , 0) and G is the d-equivalence group or the K-equivalence group Km,n. More precisely, we think of their jet spaces which are finite dimensional. In case that G is the K-equivalence group, we can take a stabilization of K-orbits via embeddings -+ E(m,n) .+ E(m + 1,n 1 ) .+ defined by trivial unfoldings. It turns out that for any K-invariant subvariety v, Tp(v) is a universal polynomial in the Chern classes of the virtual normal bundle ~ ( f=)ci(f*TN - TM)1379921. We also denote G(f) = q ( T M - f * T N ) . In the above setting, the Segre class sSM can also be defined: sSM(r]) is a formal sum of polynomials in ci = ~ ( f (i) _> 0), whose leading term

+

475

is Tp(r]).Let us see an example in case that 1 := m - n 2 0. For the Icinvariant “Milnor number constructible function” p : E(m,n) + Z (off the germs whose Milnor number is not defined), it can be shown23 that

Theorem 4.2. s S M ( p )= (-l)’+’ ‘&>o

ciCl+j+l.

This is a universal expression arising from the following simple e q ~ a l i t y ~ ~ ? ~ f*(nM

-k (-l)m-n+lp(f)) = X ( F ) ’ I N

for f : M + N betwen connected complex manifolds with generic fibre F and the Milnor number p ( f ) ( x )< 00 for each 2 E M . Further, this relates and suggests a connection to a %elative version” of the Milnor class25~26*28~5 between the recent T p theory and the geometry of polar varieties (e.g., L 8 Teissier 16).

Remark 4.1. In real analytic case (k = R), C, should be replaced by the Stiefel-Whitney homology class W,: Sullivan’s definition of Wi(X)of a real analytic variety X is the sum of all i-simplices in the barycentric subdivision of a subanalytic triangulation of X . Let r] be a Ginvariant subvariety in V and f : M + N a suitably generic map between real analytic manifolds. Then the fundamental Z2-cycle of r](f) is expressed by a universal polynomial with Z2-coefficients Tp(77)2 in wi(f) = wi(f * T N T M ) , and LLSegre-version” is also:

i*(w(TM)-l

-

-

W * ( r ] ( f )=) )sSM(77)2(w(f)) [M]2.

Remark 4.2. Multiple point formulae are also included into the T p theory: M. Kazarian14 established the T p theory for multi-singularity g e m s ( k m , S ) 4 (kn,0) (S: finite, m < n) using classical cobordism theory. His theorem says that the dual to the locus of a multi-singularity type is expressed by a universal polynomial in Chern classes ci(f) and NovikovLandweber classes f*f* (cr(f)). Those characteristic classes come from the L‘classifying space BKmulti for multi-singularities” : the space admits a stratification by classifying spaces Br] for individual singularity types r] at least up to homotopy (cf. Thom-Pontragin-Szucs construction for singular maps30). The counterpart in algebraic geometry has not yet been obtained: presumably LLBKmulti” would be realized as an LArtin stack admitting a stratification by global quotients’ together with a certain intersection theory of it. In the study on multiple points, the action of symmetry group S, comes up. In the next section we will see a general theory of C 3 .

476

5. Orbifold Chern classes

5.1. Canonical constructible functions

At first, recall that for a quotient variety (an orbifold) X = X/G of a possibly singular variety X with an action of a finite group G, there are two kinds of Euler characteristics (ordinary one and physicist's oneI2):

Here Xg is the set of fixed points of g and Xh>gI: X h nxg, and the second sum runs over all pairs (h,g) E G x G such that gh = hg. Now let G be an algebraic group and X a G-variety. Let A be a group so that Hom(A, G) is a scheme, on which G acts by (g.p)(a) := p(a)g. Put 2 := { (2,p) E X x Hom(A, G) I p(a).x = 2 (Va E A) }

(with the diagonal action of G). The projection to the first factor is denoted by 7r : 2 + X , which is a G-morphism. Assume that 7r is proper. We then define the integer canonical constructible functions over X associated to a group A by (A) G QX;G := T* nZred E F ; W ( ~ ) *

If G is a finite group, then we define the rational canonical function to be the average Itly",L = &aly",L in 3 g w ( X )@ Q. If A = Z m , we denote the canonical funciton simply by 1%;.Roughly, the canonical constructible function measures how 'large' (with respect to a group A) the stabilizer group of each point is. the integer (resp. ratioWe call CF(o$!-J E H F ( X ) (resp. @(Illy",',)) nal) canonical quotient Chern class associated to A. In the case that G is a finite group and X/G is a variety, it immediately follows that

Furthermore, in this case, it holds that HF(X;Q) 21 H,(X/G;Q) (e.g., Theorem 3 in Edidin-Graham7). Through this isomorphism, the canonical Chern class CF(il$iG)is identified with the ordinary (rational) Chern-SM class c,(X/G) of the quotient variety. The corresponding class to c,"(n$iG) is denoted by CYb(X/G) E H,(X/G; Q), which is the class appearing in the

477

formula (ob) in Introduction. In a standard wayla, we have an alternative expression of the class:

where the sum runs over the set of all conjugacy classes in G, g is a representative in each conjugacy class, C(g) is the centralizer of g , and L~ : X g / C ( g ) .+ X / G is the canonical inclusion. 5.2. Symmetric products

E o m now on, we concentrate the case that G = S,, the n-th symmetric group acting on the n-th Cartesian product X" of a variety X as o ( z ~ , . *, * X n ) := ( Z ~ - I ( ~ ] , X + ~- -. I ( ~ ) ) F . and H, are assumed to have rational coefficients and omit the notation 8 Q. For cy E FE;(Xm) and p E F 2 u ( X n ) (or corresponding homologies), we define the product 0 by

where u* is the pushforward induced by u : Xm+n -+ Xm+n. This produces commutative and associative graded Q-algebras of formal power series

n=O

n=O

-

We denote a@. .ocy (c times) by aCor @aC.For a proper morphism f : X 4 Y , the n-th Cartesian product f" : X" -+ Y n induces a homomorphism of algebras (similarly in the homology case)"':f : F~,sym[[~]] + F~,sym[[~]] given by f."Ym(Can~n) := C f p a n Z n . Theorem 5.1. The following

CZymgives a natural transformation (as a

Q-algebra homomorphism) M

M

n=O

n=O

5.3. Partitions

A collection c = [cl,...,%I of non-negative integers satisfying IcI := C:=, ici = n is called a type of weight n (i.e., a partition of n). We put flc := n!/lC1c1!2"2c2!~~ . ncn%!,which is equal to the number of elements of a conjugacy class in Sn (i.e., permutations having ci cycles of length i).

478

For a group A , we denote by & ( A ) the number of the subgroups of A with index r. We denote the (smallest) diagonal in X" by A X n . Here is the "Dey-type formula" on canonical constructible functions: Lemma 5.1. The canonical function

lyi;snis equal to the s u m

5.4. Diagonal operators

The standard n-th diagonal operator D n (n = 0,1, . - ) is defined to be the pushforward homomorphisms: Do = 1, D1 = D = i d , (id : X -+ X ) and

D" := (An),: F ( X ) -+ F%v(Xn) (similarly in the homology case) where An : X -+ X n is the diagonal inclusion map, An(z) := (z,... ,z). We call U := Cn,lanznDn (a, E Q) a standard diagonal operator, in particular, we put-Log(1 z D ) := En21 $D,. The mixed n-th diagonal operator of type c = [Q,... ,%I means the maps Dc : 3 ( X ) 3 2 v ( X n ) (similarly in the homology case) given by

+

(D'(Q))"'0 (D2(a))Q0 . . . 0 (D"(Q))"".

~ " = l * " ' ~ c n ]:= ( ~ )

We also define a formal diagonal operator as a formal series T = C,"==, znT, of linear combinations T, = v , P . Every formal operator T operates on F ( X ) and H , ( X ) . By using 0 we define e z p ( T ) := C,"==, $Ton for T with zero constant term. As a convention of notation, we put (1

+ U ) a := ezp(Log(1+ U)(Q)),for standard operators U ,

where Q E

F(X)or H , ( X ) .

5.5. Exponential formula For a group A , let O A ( r ) (resp. O A ) denote the set of all subgroups B in A of index ( A: BI = r (resp. subgroups of finite index) and &(A) := ( O A ( T ) ( . Lemma 5.1 together with a simple computation implies

Proposition 5.1. I t holds that

479

Apply CZymto both sides of this equality of constructible functions. Since it holds that T o C, = CZymo T for any T , we obtain the following theorem:

Theorem 5.2 (Dey-Wohlfahrt formula for Chern classes23). Assume that $.(A) < 00 for any r. Then it holds that

-

There is also an equivariant version of this theorem23 (for the action of wreath products G Sn on X" where X is a G-variety).

Remark 5.1. When X = p t , this theorem gives the classical DeyWohlfahrt f ~ r m u l aon ~ the ~ ? enumeration ~~ of S,-representations of a group

5

IHom(A, sn)I zn = exp n!

n=O

( c -)

zlA:BI

J A : BI

.

BERA

5.6. Examples

(1) A = Z (Hom(Z, Sn) = Sn). Our Chern class formula (Theorem 5.2) is

n=O

which generalizes Macdonald's well-known formula of Euler characteristics of SnX18: M

n=O

(2) A = Z" (Hom(Zm,Sn) = the set of mutually commuting m-tuples of elements in Sn): m

nZ0

r=l

The formula (ob) in §1corresponds to the case of m = 2, which generalizes the generating function of xoTb(Xn;Sn)12. For general m, the degree part of the above formula gives the generating function of generalized orbifold Euler characterisitics in Bryan-Fulman6: a3

480

(3) A = Z/dZ, t h e cyclic group of order d:

(4) A = Z - limZ/pkZ as a n additive group, where p is a prime number. p + We have t h e “Artin-Hesse exponential” for the Chern class of X :

References 1. P. Aluffi, Inclusion-Exclusion and Segre classes, Communications in Algebra 31 (2003) 3619-3630. 2. J. P. Brasselet, lecture note in this volume. 3. J. P. Brasselet, J. Schiirmann and S. Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, preprint, math.AG/0503492. 4. J. P. Brasselet and M. H. Schwartz, Sur les classes de Chern d’une ensemble analytique complexe, Asthrisque 82-83 (1981) 93-148. 5. J. P. Brasselet, D. Lehmann, J. Seade and T. Suwa, Milnor classes of local complete intersections, Trans. A. M. S. 354 (2002) 1351-1371. 6. J. Bryan and J. Fulman, Orbifold Euler characteristics and the number of commuting m-tuples in the symmetric groups, Ann. Combinatorics 2 (1998) 1-6. 7. D. Edidin and W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998) 595-634. 8. L. Ernstrom, T. Ohmoto and S. Yokura, Topological Radon transformations, Jour. Pure and Applied Algebra 120 (1997) 235-254. 9. L. Fehhr and R. R i m h y i , Calculation of Thom polynomials and other cohomological obstructions for group actions, Real and Complex Singularities (Sm Carlos, 2002), Contemp. Math. 354 (2004) 69-93. 10. W.F‘ulton, Intersection Theory, Springer-Verlag (1984). 11. W. F‘ulton and R. MacPherson, Categorical framework for the study of singular spaces, Memoirs A. M. S. 243 (1981). 12. F. Hirzebruch and T. Hofer, On the Euler number of an orbifold, Math. Ann. 286 (1990) 255-260. 13. M. E. Kazarian, Characteristic Classes of Singularity Theory, ArnoldGelfand Seminars, Birkhauser, (1997) 325-340. 14. M. E. Kazarian, Multisingularities, cobordisms and enumerative geometry, Russian Math. Survey 58 (2003) 665-724. 15. G. Kennedy, MacPherson ’s Chern classes of singular algebraic varieties, Comm. Algebra 18 (1990) 2821-2839. 16. D.-T. L6 and B. Teissier, Vari6t6s polaires locales et classes d e Chern des vari6t6s singuli&es, Ann. Math. 114 (1981) 457-491.

48 1 17. I. G. Macdonald, The Poincare'polynomial of a symmetric product, Proc. Camb. Phil. SOC.58 (1962) 563-568. 18. R. MacPherson, Chern classes for singular algebraic varieties, Ann. Math. 100 (1974) 421-432. 19. J. Milnor and J. Stasheff, Characteristic classes, Ann. Math. Studies 76, Princeton Univ. Press (1974). 20. I. Nakai, Elementary topology of stratified mappings, Singularities (Sapporo 1998), Adv. Stud. Pure Math. 29 (2000) 221-243. 21. T. Ohmoto, Vassiliev complex for contact classes of real smooth mapgerms, Rep. Fac. Sci. Kagoshima Univ. (1994) 1-12. 22. T. Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions, Math. Proc. Cambridge Phil. SOC.140 (2006) 115-134. 23. T. Ohmoto, Generating functions of orbifold Chern classes I : Symmetric Products, preprint, math.AG/0604583. 24. T. Ohmoto, Thorn polynomial expression for Segre type classes of singularity loci of maps, preprint (2006) 25. T. Ohmoto, S.Yokura, Product formula of Milnor class, Bull. Polish Academy of Sciences 48 (2000) 388-401 26. T . Ohmoto, T. Suwa and S. Yokura, A Remark on Chern Classes of Complete Intersections, Proc. Japan Acad. Ser. A 73 (1997) 93-95 27. A. Parusiriski and P. Pragacz, Chern-Schwartz-MacPhersonclasses and the Euler characteristic of degeneracy loci and special divisors, Jour. A.M.S. 8 (1995) 793-817. 28. A. Parusiriski and P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Alg. Geom. 10 (2001) 63-79. 29. I. R. Porteous, Simple singularities of maps, Proceedings of Liverpool Singularities I, Springer Lect. Notes Math. 192 (1971) 286-307. 30. R. Rimfinyi, Thorn polynomials, symmetries and incidences of singularities, Invent. Math. 143 (2001) 499-521. 31. J . Schiirmann, A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes, preprint, math.AG/0202175. 32. J. Schiirmann and S. Yokura, A survey on characteristic classes of singular spaces, preprint, math.AG/0511175. 33. M. H. Schwartz, Classes caracte'ristiques dkfinies par une stratification d'une variktk analytique complexe, C. R. Acad. Sci. Paris 260 (1965) 3262-3264, 3535-3537 34. T. Suwa, Indices of Vector Fields and Residues of Singular Holomorphic Foliations, Actualitks Mathkmatiques, Hermann (1998). 35. B. Totaro, The Chow Ring of a Classifying Space, Proc. Symp. Pure Math. (K-Theory, 1997), A.M. S. 67 (1999) 249-281. 36. K. Wohlfahrt, Uber einen Satz von Dey und die Modulgruppe, Arch. math. (Basel) 29 (1977) 455-457. 37. S. Yokura, On a Verdier-type Riemann-Roch for Chern-SchwartzMacPherson class, Topology and Its Application 94 (1999) 315-327 38. S. Yokura, Chern classes of proalgebraic varieties and motivic measures, preprint, math.AG/0407237.

482 39. Y. Yomdin, The structure of strata p = const in a critical set of a complete intersction singularity, Proc. Symp. Pure Math. (Singularities, Arcata, 1981), A.M. S. 40 Part 2 (1983) 663-666. 40. T. Yoshida, Classical Problems in Group Theory (I): Enumerating s u b groups and homomorphisms, Sugaku Expositions A. M. S. 9 (1996) 169188.

483

MCKAY CORRESPONDENCE FOR QUOTIENT SURFACE SINGULARITIES 0. RIEMENSCHNEIDER Department of Mathematics, University of Hamburg, D-20146 Hamburg, Bundesstrasse 55, Germany *E-mail: riemenschneiderOmath.uni-hamburg.de These notes are aimed at giving an introduction to the subject - including the original MCKAYcorrespondence which applies to KLEINsingularities by means of numerous examples and only few proofs. They are based on my manuscript [lo] and a book project in progress and are partially revised and expanded compared to the slides I presented at the workshop in Trieste. Keywords: McKay correspondence; quotient surface singularities; Klein singularities; special reflexive modules.

Table of contents 1. Platonic triples and simple objects

2. KLEINsingularities

3. MCKAY’Sconstruction

4. The phenomenological MCKAYcorrespondence Appendix: KLEINsingularities and reflection groups

5. The geometric MCKAYcorrespondence 6. The construction of ITO and NAKAMURA

7. Cyclic quotient surface singularities 8. Interpretation in terms of derived categories and A. ISHII’Sresult

9. What remains to do?

1. Platonic triples and simple objects

The Platonic triples consist of integers p , q, r ( p , q , r 2 2 ) satisfying

(*>

1 1 1 - + - + - > l . P q r

484

Normalizing to r = 2 , we get the following pairs ( p , q ) and their well known relation to the (symmetry groups of the) Platonic solids or to regular tesselations of the sphere S2 (by regular pgons, q of them meeting at each corner). (generalized) Platonic solid 2

n

n

2

Hosohedron Dihedron

3

3

Tetrahedron

4

3

Hexahedron (Cube)

3

4

Octahedron

5

3

Dodecahedron

3

5

Icosahedron

In the following, we do not distinguish between the pairs ( p , q ) and (q, p ) ; so we find unordered pairs of numbers attached to the corresponding symmetry groups in SO (3, R) .

Theorem 1.1. Besides the cyclic groups, there exist - up to conjugacy - only 4 (classes of) finite subgroups of SO (3, W) . They are (as abstract groups) presented by generators a, p and relations aP

=

pQ = ( a ~= )e .~

Under the group isomorphism PSU (2, C ) 2 SO (3, R) the finite subgroups of SO (3, R) can be lifted to subgroups of doubled order in SU(2, C )

c SL(2, C )

a

They are called the binary polyhedral groups.

Theorem 1.2. Besides the cyclic groups of odd order, there exist - up to conjugacy - only the binary polyhedral groups as finite subgroups of SL (2, C ) . They have (as abstract groups) a presentation by generators A, B and C and relations governed again by the associated Platonic triple (P, Q, r = 2 ) , namely

AP = BQ = C2 = A B C . Condition (*) also appears in other contexts for certain quadratic forms to being positive or negative definite, in particular in LIE the0y.

485

Theorem 1.3. The simply laced simply connected simple complex LIE groups are classified by their DYNKIN-diagrams of type ADE which, in the cases D and E, are also bijection to the (unordered) Platonic triples. Thus, there seems to be a close connection between finite groups and LIE groups, and already FELIX KLEINhas speculated about this interaction of discrete mathematics and geometry. The main tool used nowadays for understanding the relation is contained in his famous book on the Equation of the fifth degree and the Icosahedron of 1884 where he introduced of course not under this name - the KLEINsingularities (also known as Rational Double Points, Simple Surface Singularities, etc.). In fact one can find any KLEINsingularity (and even its complete deformation theory together with a simultaneous resolution) inside the geometry of the corresponding simple LIE group (work of GROTHENDIECK, BRIESKORN and SLODOWY, see e. g [l]).

2. Klein singularities

The theory of KLEINsingularities establishes a (formal) one-to-one correspondence between the conjugacy classes of finite subgroups of SL (2, e) and the COXETER-DYNKIN-WITT (CDW) diagrams (or DYNKIN-diagrams, as they are usually called) of type ADE via the following scheme that uses their resolutions (for more details, see Section 5 ) :

{ finite subgroups r

c SL (2, C ) } / conjugacy

I { KLEINsingularities

-

{ CDW-diagrams of type ADE }

I { minimal resolutions

Here, the symbol in the lower line denotes complex-analytic equivalence. The right arrow is given in the upward direction by associating to a minimal resolution IT : X r + X r

-

486

Figure 1 the dual graph of its exceptional set E := r-l(O) c -%r (with the irreducible components E l , Ez, . . . , all equivalent to the projective line P I ) :

Figure 2 3. McKay’s construction

In 1979, MCKAY[2] observed the possibility to construct directly via representation theory the resulting bijection in the upper line of this diagram. In particular, according to this so called MCKAY correspondence, each (nontrivial) irreducible complex representation of r corresponds uniquely to an irreducible component of the exceptional set E . We recall the formation of the MCKAY quiver associated to a binary polyhedral group and - more generally - to a finite small subgroup

r c GL(2, C) (A finite subgroup of GL (2, C) is called small, if it acts freely on Cz\ (0) or equivalently, if it does not contain any (pseudo-) reflections).

,

487

Let

denote the set of irreducible complex representations of one, and

r , po

the trivial

Irr 0 r := { p l , ... , p r } the set of its nontrivial irreducible complex representations. Let further c denote the natural representation on C2 given by the inclusion I’ C GL (2, C ) . Then,

where c* denotes the dual representation of c (of course, c* = c for

I’c SL (2, C ) ). The MCKAYquiver is then formed in the following way: Associate to each representation a vertex and join the ith vertex with the jth vertex by aij arrows. - MCKAY’Sobservation may be formulated in the following way:

Theorem 3.1 (McKay). For every finite subgroup r C SL (2, C ) , one has aji = aij E { 0, 1} . Replacing each double arrow by a line, one finds exactly the (extended) CD W diagrams of correct type. I n particular, there is a canonical bijection IrrOr

-

I r r E = { E ~... , ,E,.}.

Let us check this for the binary tetrahedral group T ,that is the preimage of the symmetry group T c SO (3, R) of a regular tetrahedron under the SO (3, R) . The irreducible canonical group epimorphism SU (2, C ) complex representations are easily calculated. The isometry group T E U 4 in SO (3, R) has obviously an (irreducible) representation of order 3 . The group 2 l 4 of order 12 contains the KLEIN“Viererguppe” as a normal subgroup, and the quotient of U 4 by this subgroup is just the cyclic group Z3 of order 3 . This leads to 3 further representations of order 1 of T . These 4 representations induce via the epimorphism T -+ T representations po (= trivial representation), p4, pz ( 1-dimensional) and p2 ( 3-dimensional) of T . Of course, by our realization of T as a subgroup of SL (2, C) , there is a canonical 2-dimensional representation which we call c = p1 . Finally, we define p3 := c 8 p4 and p: = c 8 p z .

-

488

Using the (obvious) character table of U4, one can construct the character table for T (see also the next section where we determine the conjugacy classes for the binary groups T, 0 and I):

%-

1

c =

1

2-2

P1

3

1

1

1

1

1

0 -1

-1

1

1

0

0

0

0

3-1

2 -2

0

-4-3

-4-:

4-3

1:

2 -2

0

-4-:

-4-3

4-:

4-3

1

1

1

6 4-,”

4-3

4-3”

1

1

1

c:

4-:

4-3

c3

Since the orthogonality relations are satisfied, we have indeed found all irreducible representations of T . The resulting quiver looks as below where a subgraph

-

-

stands for a double arrow, i.e. two arrows in opposite direction. Replac, forgetting po and ing such subgraphs by a simple line inserting the ranks of the corresponding representations, yields the other diagram below which, in fact, is not only the CDW diagram of type E6 but also represents the fundamental cycle 2 of the singularity C 2 /T .

2

P4

P3

P2

P:

Pt

I 1 2 3 2 1

Remark. For general small subgroups in GL (2, C ) the situation is much more complicated. An example is given in the last section.

489

4. The phenomenological McKay correspondence

In the following, we reproduce MCKAY'Sresult by explicit calculations as in the section before. For this, we list all elements of T , 0 and 1 using the results (and symbols) of FELIX KLEIN(the calculations in the completely trivial cases of cyclic subgroups and for the dihedral cases D, are left to the reader). We then determine the irreducible representations of these groups, starting from the representation theory of the symmetry groups of the regular polyhedra. Finally, we construct the corresponding MCKAYquiver and -graph. Placing the regular polyhedra conveniently in S2 c R3,one can compute explicitly all elements of the binary groups. According to a variant of Theorem 1.2, it is enough to find - in each case ( p , q , r ) - matrices

A, B

E

SU (2, C ) such that AP = Bq = (AB)'

=

-E .

We list such generating matrices in the following tables: Binary dihedral groups D, :

B =

1

s).

(g

Binary tetrahedral group T :

)

A='(

& &3

fi

7

B = - (1

Jz

& E 7

&

&

),

&=
&3 &7

Binary octahedral group 0 :

A =

(:

&=[a.

) :&

Binary icosahedral group II :

B=&3

- &4

&4

- &3

7

&

= c5.

Under the above realization one has the following chains of normal subgroups (the symbol C, stands for the cyclic group of order n generated by the diagonal matrix diag E SL (2, C ) ):

(en,el1)

c C2k c C2k c Dk c b c k

CZ

CzCD2CTC0,

k

CZCII.

490

Moreover, no other (nontrivial) chains exist; in particular, this means that SO (3, R) - the icosahedral group is the only one (besides the cyclic groups of prime order) having the property to be simple. We now go over to other generators of the binary polyhedral groups. ( T ) We put - among the finite groups in

which are elements of SU (2, C) of order 4, 4, 6 , resp. Then,

U = A, S = BA, T = AB and

A = U , B = U5T = - U 2 T . Hence, the group T is also generated by S, T , U ; it consists in fact of the 24 elements

U p , S U P , TU’”, STU’”, p = 0 ,... , 5 , which can be seen from the relations:

S2 = T 2 , The group

ID2

u3 = - E

, US

=

TU

UT

=

STU, TS = S T .

of order 8 is isomorphic to the normal subgroup

{ U p , S U P , TU’”, STU’” : p = 0 , 3 } of T,and T/D2 is isomorphic to the cyclic group 2 3 of order 3 . Finally, T = (D2, U ) . T decomposes into the following 7 conjugacy classes (in the parantheses, “order” means the order of the elements): C1 = {Uo = E } C2

= {U3 =

-E}

C3 = {kS,f T , & S T } C4 =

{ U 2 , SU2, T U 2 ,STU2}

(order 1) (order 2) (order 4) (order 3)

Cs = {- U, SU, TU, STU}

(order 3)

Cs = {- U 2 , - SU2, -T U 2 , - STU2}

(order 6)

C7 = {U, - SU, - T U , - STU}

(order 6) .

491

If we reduce modulo U 3 =

-E

, we get by the identifications

a := UmodU3 g (123) E 214

p

:= U2Tmod U3 =

the following images in c 1=

-

c 3

SU2mod U 3

(124) E

2l4

Uq :

C2 = ((1)) = {(12)(34), (13)(24), (14)(23))

Cq = Cs = {(124), (132), (143), (234)) C5 = C7 = {(123), (134), (142), (243))

which together precisely establish the decomposition of T/( - E ) into conjugacy classes.

U4

(0)In addition to the matrices S, T , U of the last paragraph, we introduce

of order 8 . Then the generating matrices A, B for the binary octahedral group can be written as

A=V,

B=U5

Since one checks that

VS = SV,

V T = STU3V = - S T V ,

VU = SU5V = - S U 2 V ,

it follows

0 = ( A ,B ) = ( S , T , U,V ) = ( T , V ) . Moreover, T is a normal subgroup of 0 satisfying O / T g V = S U 3 , we get

Z2.

Since

0 = {U’V”, SU’V”, TU’V”, STU’V” : p = 0 , . . . , 5 , v = 0 , l ) .

492

The 48 elements of 0 decompose into 8 conjugacy classes: c 1

= {UO = E }

(order 1)

c 2

= {U3 = - E }

(order 2)

C3

==

{ & S ,f T , +ST}

(order 4)

C4

5

{&TV, &STV, +SUV, &TUV, f U 2 V , ztTU2V}

(order 4)

C5 = {- U, U 2 , SU, SU2, TU, TU2, STU, STU2}

(order 3)

Cs = {U, - U2,- SU, - SU2, -TU, - TU2, - STU, - STU2}(order 6) c 7

= {&V, * s v ,

kU}

(order 8 ) (order 8) .

Cs = {zt STUV, f S U 2 V , &STU2V} Again by identifying

U3 2 (1234) E 6

Q

:= Vmod

P

:= U 5 mod U 3 E (143) E 6 4

we get the following 5 classes of

4

,

6 4:

For G4,we have 5 irreducible representations of order 1, 1, 2, 3, 3 , , resp. which we call after lifting to 0 : po (trivial representation), p5 , p4 , p2 and p a . If we denote by c = p1 the canonical 2-dimensional representation of 0 and define

we obtain the following character table:

493

1

Po

1

1

1

1

1

1

1

Jz Jz-Jz

c = PI

2-2

0

0-1

1-Jz

PT

2-2

0

0-1

1

P2

3

3-1-1

PZ

3

3-1

P3

4-4

0

0

P4

2

2

2

0-1-1

P5

1

1

1-1

1

1

0

0

1 0

0

-1

-1

1-1

0

0

0

0

1 -1

-1

1

Thus, we have found all irreducible representations of 0. The graphs constructed in the same manner as above look like P4

2

and

(I) Following F . KLEIN,we introduce new matrices ( E E3

s= (o

0 & 2 ) ’

= (5

E4 - &

T =

&’-&’)#, & - & 4

$ ( & 2 - & 3

U =

):

(;

4)

1

which lie in SU (2, C) and are of order 5, 4 and 4 , resp. Here (and in other computations) we need the following relations for E = ( 5 (which indicate the close connection between the regular 5-gon and the golden ratio):

&1+&4

A - 1 2 ,

= ___

&2+&3

d3+1

= -~

,

E1+E2+E3+E4=-l.

494

In terms of preceding notations, we have A = - S , B = S4T such = - E and S = - A , T = - A B . Finally, that A5 = B3 = U = S2TS3TS2T.Therefore, II is also generated by S, T , U or by S, T alone. Explicit calculations (or better: geometric arguments as in KLEIN’S book on the icosahedron) show that I[ consists of the 120 elements f S P , f S P U , fSPTS”,f S P S T ” U ,

p, u =

O,... , 4 .

They constitute the following 9 classes: c 1 =

cz

{SO = E }

= {-SO =

(order 1)

-E}

C3 = { f S ’ L U , fS-PTSP, fSC”TS”U}

(order 2) (order 4)

C4 = {S1-PTSP , - S 4 - P T S P , S2-PTS-PU

3)

C5 = {- S1-PTSP

6)

c 6 =

, -S3-PTS-PU} (order , S4-PTSP , - S2-PTS-PU , S3-PTS-PU}(order

{ s, s4, - S2-PTSP, S3-PTS’}

, - S4, Sz-fiTSP , - S3-pTS’”} Cs = {S2, S3, S1-’”TS-’”U, - S4-’TS-’U} Cg = {- S2, - S3, - S‘-PTS-PU , S4-PTS-PU) C7 =

{- S

(order 5) (order 10) (order 5) (order 10) .

Reducing modulo fE gives the following 5 classes in ‘U5: -

C1 UCz = { E } C3

= 15 elements of order 2 in U5

-

C4 U Cg = 20 elements of order 3 in 9l5

c 6

u 777 = 12 elements of order 5 in 2 5 representing rotations through an angle 2 ~ / 5

Cs u c

g =

12 elements of order 5 in U5 representing rotations through an angle 4n/5 .

Using the character table for U5 , we find 9 representations of II with the following character table:

495

I

1

1

1

1

1

1

1

a' - a

P1

2-2

0-1

P2

4

0

1

1-1-1

P3

6-6

0

0

0

1-1

P4

3

3-1

0

0

a' a'

P5

5

5

P6

4-4

p7

3

c = p 8

4

0

1-1-1

0

3-1

2-2

In this table, the number

1-a'

1 - 1 -1

0

0-1 and

a

0

1

1

Q

-1-1 1-1

a

Q

0

0

1-1

1

0

a

a' a'

a-a'

-a!

1

a'

are defined by

a' = 1 (1 - &), 2 P O , p 2 , p 4 , p 5 , p7 are representations of %5, lifted to 8, c = p8 is the canonical representation of II in SU (2, C ) , and p1 , p 3 , p6 are determined bY

1 a = -(1 2

cB.7

= c@p6

7

c @ p 4 = p3

,

c @ p 2 = p1 @ p 3 .

The two MCKAY-diagrams are: P4

P1

p2

p5

p3

ps

p7

p8

3 and

the last one being again identical with the fundamental cycle of C 2 /II .

496

Appendix: Klein singularities and reflection groups The regular polyhedral groups can be embedded into wflection groups with index 2 by considering the full symmetry group of a regular solid in 0 (3, JR) . A similar statement is valid for the binary groups, as has been remarked and strongly used for explicit calculations in [3].'In this Appendix, we carry out the construction in some of the cases by a geometric consideration (leaving the remaining ones to the reader), and at the end, we give a conceptual group-theoretical argument. The additional involution T that we have to attach to the group should operate on the quotient C 2 / r , I? c SL (2, C ) , which can analytically be described by an equation of the form z2 = g (x,y) , in such a way that it identifies the two sheets of the branched covering given by this equation over the (z, y)-plane, i.e. that it acts by z H - z . Using this idea, it is in fact possible to find these reflection groups 3 I' for all finite subgroups r c SL (2, C ) . In the cyclic case, the situation is very simple. One has to add to the cyclic group C, with generator

r

the element

r= such that

En = (C,,

(: :)

r ) consists of 2n elements of which the n elements

are reflections of order 2 generating gebra C (u, u ) ~ are % apparently:

{

En.Generators for the invariant al-

2 = 20

+ 52 = un - v y

y =21

= 2121

On the function z = xo - x2 = u" - un (which satisfies the relation 2 2 = 22 - 4 y") , T operates by T ( z ) = - z . For the group T , we have to find an element T E GL (2, C ) of order 2 modulo T such that 7 (Xi)

= Xi,

7 (X2)

=

52, 7 (23)

= -2 3

,

497

where XI

= u8 23

+ 14u4v4 + V’

=

2112

- 33u8v4

, -

=

22

- v4) ,

UV(U~

33u4v8

+ 2112 .

Such an element is given by

One checks that the group = ( T , T ) has 48 elements and that the following 12 reflections generate the full group:

it is again the element T = In the octahedral case 0 , does the job. The 18 reflections of order 2 generating

(i:) -

which

6 = (0,T ) are:

and

L(1

Jz

),

1

1-1

L(-1-1),

L(

Jz

Jz

-1

1

L(-;-;), L Jz Jz (-2-1 a ) ,

l-l),

$(-; ;),

-1-1

$(

1-2) i-1

L Jz (-1 -2

i

).

1

498

We close this Appendix with a concrete description of the covering SU(2, C ) -+ SO(3, R) and some more conceptual remarks on the embedding of binary polyhedral groups into reflection groupsa. As we will see in a moment, we can derive all of this from the existence of a natural epimorphism

(**I

GL(2, C )

-

CO(3, C )

with kernel {fE } . Here, 0 (3, C ) denotes the group of linear automorphisms of C3 leaving the quadratic form

+ Y22 + Y3”

Y

C3 1 fixed, and CO (3, C ) is the larger group of conformal automorphisms which yi y?j fixed up to a nontrivial factor c E C* . Hence, C E leave yi GL (3, C ) is in CO (3, C ) , if and only if

YT

7

= (Yl,9 2 , Y3) E

+ +

t C * C= c ’ E ,

CE

C*

,

and we have the following diagram of subgroups

SO(3, R)

c O(3, R) c CO(3, R)

n

n

n

SO(3, C ) c O(3, C ) c CO(3, C ) . Clearly, 0 (3, C ) and SO (3, C ) are connected complex Lie groups of dimension 3 , and CO (3, C ) is connected of dimension 4 . In order to obtain (w), we remark that every action of a group G on a vector space V induces actions on the dual vector space V * and its symmetric powers Se(V*).In particular, the (right) action of GL (2, C ) on V = C2 gives rise to an action on the vector space S2(V*) which we identify with the symmetric 2 x 2-matrices

-

The (right) action of GL (2, C ) on SZ(V*) is then explicitly described by

( S ,A )

A-l . t S . A-l .

On S2(V*) there exists a (canonical) quadratic form, the determinant:

( S ,S ) = a y

-p2.

_ _ _ _ ~

aThis construction has been communicated to me by PETERSLODOWY in Hamburg some years ago.

499

By polarization, we get a symmetric bilinear form on S2(V*): 1 ( S 1 , s 2 ) = 2 (a172 + 71a2 - 2 P l P 2 ) 1

(:y)

and it is easily checked that the elements

O ) , E3 = ( O i, 0 -2 i0 form an orthonormal basis of S2(V*) with respect to (., .) such that ( S z ( V * ) , ( ..)) , can be identified with C3 endowed with the symmetric bilinear form obtained by polarizing the quadratic form y: + yg -t- y i . Using the decomposition El =

,

E2 = ( 2

one deduces with the aid of a trivial computation that, for instance,

+ d2 - (ab + cd) b2

- (a2 + b2 1 +2i (b2 + d2

where A =

(1:)

-

- (ab + cd) a2

+ c2

+ c2 + d 2 ) El

(a2

+ c')) E2 + i (ab + cd) E3

1

,

. Computing also A - l . E 2 .tA-l and A - l . E 3 .tA-l ,

we get the foilowing concrete representation of G L ( 2 , @) on S2(V*) by associating t o a matrix A E GL ( 2 , C ) the 3 x %matrix 1 1 - (a2 b2 c2 d 2 ) - (b2 d2 - a2 - c2) i(ab cd) 2 2i i 1 - ( d 2 - b2 c2 - a 2 ) - (d2 - b2 - c2 a 2 ) ab - cd 2 2

I

+ + +

+

+

-i

(UC

+ bd)

+

+

a c - bd

multiplied by the factor 1/ det A 2 . Since det (A-' . S . t A - l ) = det S/ det A2 ,

ad

+ bc

500

the image lies automatically in CO (3, C ) . By computation of the Jacobian of the part 1 + b2 + c2 + d 2 ) (b2 + d2 - a2 - c2) 22 - i (ac + bd) uc - bd

(a2

at the identity E , one sees immediately that the (holomorphic) map (**) is locally biholomorphic at E . Both Lie groups being connected and of the same dimension, this implies the surjectivity. We leave it as an exercise to show that only diagonal matrices and matrices of type

are mapped to diagonal matrices. In particular, the kernel is just the group {3= E } and (**) is a two-sheeted covering. By the same reasoning, we infer from (**) the existence of two-sheeted coverings

---

{ A E GL (2, C ) : det A = f 1) SL (2, C )

0 (3, C )

so (3, C )

Restricting (**) to the unitary group U (2, C ) , the image consists of real matrices. In fact, we get successively the following coverings:

u (2, C ) { A E U(2, C ) : det A = z t l }

su (2, C )

co (3, W)

-

0 ( 3 , W)

so (3, R).

r

-

Now, let I' be a binary polyhedral group, and let be its image in SO (3, W) . By geometrical reasons, F is contained in a reflection group F C 0 (3, R) , the full symmetry group - of the corresponding Platonic solid. We claim that the preimage of f; in U (2, C ) is also generated by reflections. For this to be true, it suffices to show that reflections in 0 ( 3 , W) lift to reflections in U (2, C ) . So, let R be such a preimage. Then we can find an element A E U (2, C ) such that = A-' R A maps to

(' : :) 0

0 -1

50 1

If 3 has coefficients a , b, c, d as usual, we can conclude from the same calculations as mentioned above that the case a = d = 0 is impossible and that for b = c = 0 we must have 1

(ad)-2(a2 + d 2 ) = 1 , (ad)-l 2 which is only satisfied for the matrices f

= -1

(:-:) *

5. The geometric McKay correspondence for quotient

surface singularities Of course, geometers wanted t o understand the phenomenological McKay correspondence geometrically, and the first who succeeded in this attempt were GONZALES-SPRINBERG and VERDIERin 1983 [3]. They associated to each nontrivial irreducible representation of J? a vector bundle 3 on the of Xr whose first CHERNclass c l ( 3 ) hits precisely one resolution component of E transversally. Their proof was not completely satisfying since they had to check the details case by case. But in 1985, ARTINand VERDIERgave in [4]a conceptual proof using only standard facts on rational singularities, and in combination with the so called multiplication fornula contained in the paper of HBLENEESNAULT and KNORRER151 from the same year it became clear how to understand the full strength of the correspondence, i.e. how to reconstruct the dual graph of E c Z r from the representations of I' completely in geometrical terms. I would like to discuss this construction from the beginning in the more general setting of quotient surface singularities, or in other terms: for small finite subgroups of GL (2, C ) (instead of finite subgroups of SL (2, C ) ) in WUNRAM, RIEMENSCHNEIDER). more detail (work of ESNAULT, By a well known result of GOTTSCHLING and PRILL, every quotient (C2/I' is complex-analytically isomorphic to a quotient by a small group, and two quotients by small subgroups are complex-analytically isomorphic if and only if the subgroups are conjugate in GL (2, C ) . Hence, the classification of quotient surface singularities consists in the determination of the conjugacy classes of finite small subgroups in GL (2, @) . This classification has been carried out by BRIESKORN(see also [6]). Recall that cyclic quotient surface singularities of C2 are determined by two natural numbers n, q with 1 5 q < n and gcd (n,q) = 1 . The

Zr

502

cyclic group

acting is generated by the linear map with matrix

Lemma 5.1. Two cyclic quotients C2/Cn,g and @ 2 / C n , , gare , isomorphic i f and only i f

n1 = n

and q l = q or qq’

E

lmodn.

For the general case, notice that we have a surjective group homomorphism $J : ZGLz x SL(2, C)

+ GL(2,

C)

(ZGL2 denoting the center of GL (2, @) consisting of all multiples a E , a # 0 , of the unit matrix E ) defined by multiplication. It is not difficult to convince oneself that the following is true:

Lemma 5.2. Each noncyclic finite subgroup I’ of GL (2, C) may be obtained f r o m a quadruple ( G I ,N I ;G z , N z ), where (a) G I c ZGLz and Gz c SL (2, C) are finite subgroups, G2 not cyclic, (b) N I c G I and N Z c G2 are normal subgroups such that there exists an isomorphism cp : Gz/Nz

2 Gi/Ni

by the following construction :

r

:= + ( G ~ xV

,

where

G I X V GZ :=

((91, 92) E

G I x GZ : 91 = ~ ( 9 2 ); )

(here, iji denotes the residue class of gi in GiINi, i

=

1, 2).

Remark. The conjugacy class of r in GL (2, C) does not depend on the specific isomorphism cp . Therefore, we use the symbol

( G I ,N i ; Gz, Nz) also as a name for the conjugacy class containing the groups

+ (GIx ~ G z ) .

BRIESKORN’S classification can now be given in form of the following table:

503

(6; 2, 1, 3, 2, 3, 2) (b; 2, 1, 3, 1, 3, 1) m = 6(b - 2) (b; 2, 1, 3, 1, 3, 2)

+

' 1 7 11 13 17 19 23 , 29

Here, Ze denotes the group (CeE),and the dual resolution graph

-2

0

. . . m-• . . .

is encoded by the symbol ( b ; 721, q 1 , nz, 4 2 , 723, for the notation concerning continued fractions)

q3)

where (see Section 7

504

(Of course, the pair n1 = 2 , q1 = 1 belongs to the upper (short) arm). Remarks. 1. The quotient surface singularities are characterized by several finiteness conditions. They are the only surface singularities having a finite fundamental group, carrying only finitely many isomorphism classes of indecomposable

reflexive modules. 2. It is conjectured that they are also exactly the deformation finite surface singularities. A proof of this would imply a positive answer to the old conjecture that there are rigid normal surface singularities. Next, we associate to each representation of a finite small subgroup a geometric objects on the quotient surface singularity X r and on its minimal resolution Let p be such a representation on the vector space V = Vp. r operates on (C2 \ (0)) x V via the natural representation c and p , and the quotient is a vector bundle on (C2\ { O } ) / r whose (locally free) sheaf of holomorphic sections extends to a reflexive sheaf M p on

Zr.

C2/ r = X r : Mp := p*(Oc2 €4 where p denotes the canonical projection C2 -+ X r and p* is the dual representation. M is indecomposable if and only if p is irreducible. - In fact, one gets all reflexive modules M on X r in this manner (see [7]):

Theorem 5.1 (Esnault). There exists a one-to-one correspondence between

{ (indecomposable) reflexive modules M o n ( X , 0 ) }

{ (r-indecomposable) free modules

M^

o n (C2, 0 ) with a r-action }

{ (irreducible) representations p of I' ) Sketch of proof. We start with a reflexive module M on X , form the pull back p* M with respect to the finite covering p : C2 + X = X r and denote its reflexive hull, i.e. the double dual (p*M)**, by M^. This is a (locally) free sheaf on C2 which carries a natural r-action: Starting with a local presentation

0;

- - 0%

M

0,

505

- - - - -

we get the exact sequence

0g2

and, by dualizing,

0

0g2

(p*M)*

p*M

0E2

0

0g2.

In particular, the sheaf (p*M)* is already reflexive on C2 and thus locally free. The map 0g2+ 0g2is r-equivariant since it is defined by a matrix with entries in the invariant ring under I?. Thus, the sheaf (p* M ) * and its dual M carry canonical I?-actions. h

For a locally free I?-module M^ , the group I? acts naturally on the germ M^o at the origin and therefore on the finite dimensional vector space mMo , where m := ~ 2 , denotes o the maximal ideal sheaf of 0 ~ at2 the origin. We then associate the dual of this representation to M since this is the representation of I? on the fiber over the origin of the vector bundle underlying the free sheaf MI. 0 A

GO/

h

For the step invoking the minimal resolution, we can study more generally any rational surface singularity X and an arbitrary reflexive module M on it. Let 7~ : 2 X be a minimal resolution, and put M := T * M / torsion. Such sheaves on 2 were baptized full sheaves by ESNAULT [7]. By local duality, one has the following

-

-

Theorem 5.2 (Esnault). A sheaf 3 on )7: is full zf and only if the following conditions are satisfied : 1. 3 is locally free, i.e. (the sheaf of holomorphic sections in) a vector bundle, 2. 3 is generated by global sections, in particular, H 1 ( - f ,3)= 0, 3. H 1 ( Z , .F*8 wx) = 0 , where wx denotes the canonical sheaf o n

-

X .

-

Under these assumptions, M = IT*3 is reflexive and 3 = M . Moreover, M * = .rr,(3*) (but 3* is, in general, not a full sheaf). Remark. For quotient surface singularities X r which are not KLEINsingularities one has always #IrrEr < #Irror .

506

So we can not expect MCKAY'Scorrespondence literally true in this situation.

Definition. A full sheaf G/ a reflexive module M / a representation p is called special (perhaps better exceptional), if and only if W(2,(G)*)= 0 (where M := M p in case of a representation p ) . Special full sheaves have been characterized by WUNRAM [8], special re[9,101. Notice flexive modules and representations by RIEMENSCHNEIDER that in former articles, we associated the module p*(O@z@ Vp)r to a representation p instead of p*(O,p @ Dealing with the dual representations fits better into the framework of the ITO-NAKAMURA construction to be discussed later. Theorem 5.3. 1)

G special

the canonical map

G@wX

-+

[(M@wx)**]-

is a n isomorphism. 2 ) M special M 18 w x / torsion is reflexive. 3) p special the canonical map

(f&o)r@J (O@Z,O

@J

-

vp*y

(J2;2,0

@

vp*y

is surjective. Here, of course, two stars denote again the reflexive hull of a coherent analytic sheaf, fl? is the sheaf of KAHLERm-forms and w x := (fig)** the dualizing sheaf o n a complex analytic surface X .

As a Corollary to the next Theorem of WUNRAM[8], one obtains the MCKAYcorrespondence since for the KLEINsingularities one has w x OX (GORENSTEIN property) and wx 2 Ox.Or in other words: For KLEIN singularities all reflexive modules etc. are special. Let 3 = G be of rank r and full. Then, r generically chosen sections in G define an exact sequence 0

-

0;

-+

3

-

N

-+

0

with D := supp N a divisor in a neighborhood of the exceptional set E which cuts E transversally at regular points only. We call D the CHERN divisor c1(F) .

507

Theorem 5.4 (Wunram, 1987/1988). There is a bijection (special nontrivial indecomposable reflexive modules)

T

J.

{irreducible components Ej of E ) via

-

t--f

cl(M)Ek =

The rank of Mj equals the multiplicity mental cycle 2 = C rj Ej .

djk

.

of the curve Ej in the funda-

rj

Proof. 1. We first show that a special reflexive module M is determined up to trivial summands by its Chern divisor D . For doing this, we start with a defining exact sequence

(+>

0

--+

0;

- -M

0.

OD

By the special properties of D , it is clear that

XomOx(OD,0,) = 0 and that, moreover, & x t b x ( 6 D , 0,)

OD.

(Just take the locally free resolution 0 + O x ( - D ) + 0, 00 ). Hence, dualizing (+) leads to the exact sequence

(++I

0

--+

-

(G)*

0;

------)

OD

--f

-t

OD + 0 of

0

and, after application of the long exact direct image sequence, due to the definition of special reflexive modules to

(+++) 0

+ M* =

-

7r,(M)* --+

05

-+

T*Og

+

-

R1?'r,(hf)* = 0 .

Consequently, M* is a zeroth syzygy module of T * O D ,and as such, it is uniquely determined up to trivial summands. 2. In order to understand the correspondence it is necessary to specify the construction in the upward direction. i.e. to prove existence. For this, we

508

choose a divisor D intersecting E j transversely at one point, not meeting any other component of E . Since the cardinality of a minimal set of generators of T*OD is equal to dimcn,0D/rnz,n,0~ = (D.2) = rj

,

we find r = rj global sections in H o ( z , O D ) generating there is an exact sequence

(++++)

0

--+ R -+

0;

--+

T * ~ D Hence, .

00 --+0 .

Using the same considerations as in l.,we see by dualizing twice that R R**, such that R is a locally free sheaf on 2 ,necessarily of rank r .

=

Dualizing only once and forming global sections yields, by the rationality of ( X , XO) , the exact sequence

0

--i

H o ( z ,Ox)@'

-

H o ( z ,R*)

-

H o ( z , 00)-+ 0

which immediately implies that the dual module R* is generated by global sections. Moreover, cl(R*)= D , such that R* can only have trivial summands, and the exact sequence 0

--+

T*R --+

05

-

T*OD 4R'IT*R+ R ' T , ~ ; = 0

gives, because of the surjectivity of

05

-+ T*OD,the

relations

R'T,R = 0 and then R%,(R@w,) = 0 . In conclusion, we see that D.

1M

=

R* is a full vector bundle with cl(%) =

3. It remains to check that % has no trivial summands. So, suppose that $ @ 0,. Then, from

=

-

0 = R1n*(M>* = R'n*(fi)*CB R1n*OX,

we deduce an exact sequence

which contradicts the minimality of r .

0

509 6. The construction of Ito and Nakamura

In 1996 YUKARIITO and IKUNAKAMURA [ll, 121 constructed in joint work the minimal resolution .%r in the case of finite subgroups of the special linear group SL (2, C ) by invariant theory of I' acting on a certain HILBERT scheme. They were able, again by checking case by case, t o produce the r (and correct representations from the irreducible components of E c Z even more). Two years later, NAKAMURA lectured on this topic in Hamburg; I soon became aware of how one should generalize the statement to (small) subgroups of the general linear group GL (2, C ) and developed some vague ideas how to prove this without too many calculations. This conjecture could be checked in the case of cyclic quotients by a simple computation which depended on the concrete results in the doctoral thesis of RIE KIDOH[13], written in Sapporo under the supervision of NAKAMURA. I gave some lectures on this topic in Japan during September 1999 and learned from AKIRAISHIIin August 2000 that he succeeded in proving the conjecture via rephrasing the multiplication formula of WUNRAM in terms of a functor between certain derived categories. Besides the general proof of A. ISHII[14] which uses much heavier machinery there exists now another independend proof in the cyclic case via toric geometry by Y. ITO [15]; she doesn't use KIDOH'Sexplicit construction but the characterization of special representations in Theorem 5.3. Let Hilbn(C2) be the HILBERTscheme of all 0-dimensional subschemes on C2 of length n , i.e. the scheme of all coherent ideals 1 C Ocz with dim@Op/I = n . It is well known that the canonical HILBERT-CHOW morphism Hilb"(C2)

-

Sym"(C2) = (C2)"/6,

is a resolution of singularities (FOGARTY), and Hilb"(C2) carries a holomorphic symplectic structure (BEAUVILLE). Let I? c GL (2, C ) be a finite small subgroup of order n = o r d r , and take the invariant part of the natural action of r on Hilb"(C2). The resulting space Hilb"(C2)r is smooth and maps under the canonical mapping Hilb"((C2)r -+Symn((C2)I' 2' C2/F to X r . It may a priori have several components, but there is exactly one which maps onto Xr and thus constitutes a resolution of X r which will be denoted by Yr = Hilbr(C2). In fact, more is true:

510

Theorem 6.1 (A. Ishii). Hilbr(C2) is equal to the open subset of socalled l?-invariant n-clusters in C2 ( i.e. r acts o n 0 @ 2 / I via the regular representation ), and the resolution

Yr = Hilb'(C2)

-

C2/r = X r

is minimal. Remarks. 1. Theorem 6.1 has been checked case by case for subgroups l? c SL(2, C) by ITO-NAKAMURA [ll, 121; a conceptual proof via preprojective algebras was given by CRAWLEY-BOEVEY [16]. 2. For cyclic subgroups of GL(2, C) this has been shown via direct calculation by KIDOH[13]; see Section 7. It has been conjectured for general finite small subgroups l? c GL (2, C) by GINZBURG-KAPRANOV [17].

3. We will sketch of proof for the minimality of resolution in Section 8.

4. A. ISHIIeven proves a posteriori that the r-invariant Hilbert scheme Hilbn(C2)r is irreducible. Hence Hilbr(C2) = Hilbn(C2)r . In particular, a point on the exceptional set E of Yr may be regarded as a r-invariant ideal I c 0 ~ with 2 support in 0 . Now, let m be the maximal ideal of 0 ~ 2 , 0 ,m x that of OX,' = 0g2,' and n = rnxOc2,o. Put

V ( I ) := I / ( m I

+ n).

This is a (finite-dimensional) I'-module. For a (nontrivial) irreducible repwith representation space V, put resentation p E Irr'

E, := { I : V ( I ) contains V, } .

In the case of KLEINsingularities, i.e. for finite subgroups l? C SL (2, C) , one has the following beautiful result of ITO and NAKAMURA which opened up a new way to understand the MCKAYcorrespondence completely in terms of the binary polyhedral group l? . Theorem 6.2 (It0 - Nakamura). Let I' be a finite subgroup of SL ( 2 , C) . For p E Irr'I', E, 2 P I . Moreover, E, n Eft is empty or consists of exactly one point for p # p' , and

E=

u

E,.

parrOr

More precisely, V ( I ) = V, for the ideals I E E, corresponding to smooth points of E , and I E E, nEft for p # p' i f and only i f V ( I ) = V, @V,t .

511

7. Cyclic quotient surface singularities The cyclic group Cn,, with gcd(n, q) = 1 operates on the polynomial ring C [ u , w] by (u, w) H (&u,Ciw). A monomial u * d is invariant under this action if and only if a+qp=Omodn, e. g. for ( a ,P ) = (n, 0) 7 (n - Q, 1)7 (0, n) * The HIRZEBRUCH-JUNG continued fraction n n-q

= a1 -

a2

1

=a1-lJ--..-lJ-z

- 1/.**

with a, 2 2 gives a strictly decreasing sequence a0 =

stopping with

n

> a1

am+l

po

= n -q

>

a2

=

alal

- a0

>

..*

= 0 , and a strictly increasing sequence

<

=0

p1 =

1<

p2

= alp1

- po <

* * *

stopping with &+I = n . It is well known that the monomials u * f i w ~, ~ p = 0,.

.. ,m + I ,

generate the invariant algebra

+

minimally. In particular, embdim A,,, = m 2 , hence, mult An,, = m 1 . The numbers a, are exponents in canonical equations for A n , , . On the other hand, the continued fraction expansion

+

gives invariants for the minimal resolution of C2/ C,,q whose exceptional divisor consists of a string of rational curves with selfintersection numbers -bk.

Define correspondingly the decreasing sequence

io = n > and

il

=q

> i2

=

blil

- 20

> ... > i,

= 1>

i,+l

= 0

512

Theorem 7.1 (Kidoh). Let (n,q ) be given. Then, Hilbcn#q(C2)consists of the Cn,q-invariantideals

Ik(sk,tk) of length n = ord Cn,q,which are generated by the elements -

Uik-1

,

~ ~ u j k - 1

v j k

- tkUik ,

Uik-l-ik

Here, 1 I k Ir -t 1, and the parameters

-

,$k-jk-l

(Sk, t k )

.

E C2 are arbitrary.

Remarks. 1. These are in fact C,,,-invariant ideals, since ik and the functions z ~ ~ k - l - v~ j kk - j k - 1 are invariant.

+

Sktk

= q j k mod n

2. The ( r 1) copies of C2 patch together to form the minimal resolution of C2/Cn,q i.e. Ik ( S k y t k ) = Ik+1 ( S k + l , t k + l ) 1 and Sk+ltk

*

tk+l = t p sk. 3. The exceptional divisor E equals

u r

I1 (0, t l ) u

{Ik (sky tk) : sk

tk =

0) u Ir+1 (%+I,

0) .

k=2

4. It is not difficult to deduce KIDOH’Sresult by induction using the well known partial resolution of cyclic quotient singularities constructed by FUJIKI.

What about the representations of Cn,q on the V (Ik)? For 1 1 (0, t l ) the first generator uzo= un is an invariant. The third is such in all cases anyway. So, Cn,q acts on V (I1 (0, t l ) )2 C as the one dimensional representation xil where xi

: z

-

(A2

(recall that q j k E ‘&mod?%).This remains automatically true for I2 ( 5 2 , t2) with t 2 = 0 , s2 # 0 . The first normal crossing point of the exceptional set is the ideal I2 (0, 0) which is generated by u i l , vjz and an invariant. Therefore, the corresponding representation is the sum Xi1

CB X q j 2 =

Xi1

@Xi2

*

The ideal I2 (0, t 2 ) , t 2 # 0 , is generated by uil , invariant uil-+ vjz-51 . Now,

t2uil

’

=. j i

(uii--iz

v j ~ - j i )-

uil-iz

(2132

- t2 2

vj2

2 )

- t 2 uiz

and the

E m12 (0, t2) .

513

Therefore, the representation is just the one-dimensional xi2

=

XQjZ

*

It should be clear how this game goes on: We get precisely the r representations x i k , k = 1 , . . . ,r , resp. the correct sum of two of them at the intersection points. Due to a result of WUNRAM[18] these are precisely the special representations of the group C n , q .He even computed the CHERNdivisors explicitly.

Lemma 7.1. For a given number i E N between 0 and n - 1 , there exist uniquely determined nonnegative integers d l , . . . ,d, with

i = dl il -I-t i , tk

= dk+l

ik+l

-b

tk+l,

0 5 tl <

21,

0 5

< ik+l , 1 5 k 5

tk+l

T

- 1

.

Then, the CHERNdivisor of the full sheaf associated to the one-dimensional representation xi is

I n particular, if i =

ik

k=l , then this CHERN divisor is equal to 1 . E k

.

This gives a hint how the result of ITO-NAKAMURA generalizes to arbitrary quotient surface singularities (see Theorem 8.3). 8. Interpretation in terms of derived categories and A. Ishii’s result

MCKAYcorrespondence may also be understood as an equivalence of derived categories. This has been worked out by KAPRANOV and VASSEROT [19] for SL(2, @) and by BRIDGELAND, KING,REID[20] in dimension 3. The last paper led A. ISHII[14] to study more closely the canonical functor

9 :D,r(@2)

-

D,(Yr)

where D;(C2) denotes the derived category of I?-equivariant coherent analytic sheaves with compact support on C 2 , r a finite small subgroup of GL (2, C ) , and D,(Yr) the derived category of coherent analytic sheaves on Yr = Hilbr(C2) with compact support. The main ingredient of his considerations is WUNRAM’S multiplication formula [8] which generalizes the one of ESNAULT and KNORRER[5]. We

514

denote by M a reflexive module on X = C2/r, its AUSLANDER-REITEN translate, i.e. the module ( M I8 w x ) * * , by T ( M ), and finally, we write N M = ( M R k ) * * . Then we have:

Theorem 8.1 (Wunram).

i

E j , M = M j special, j

C~(%M)

-

Cl(%i)

-

=

Ci('T(M))

#

0,

2 , M = Mo := O x ,

0 , M nonspecial .

Here, Z denotes the fundamental cycle of the minimal resolution of X

.

A. ISHIIfirst restates and proves once more WUNRAMS'S multiplication formula in the following form. Theorem 8.2 (A. Ishii). Let p be a n irreducible representation of r c GL (2, C) and put 00 = O@Z,O/ m , where m denotes the maximal ideal of C2 at the origin. Then O ~ ~ ( - l ) [,l p] = Q ( 0 0 I8 VP*)=

oz, 0,

pj

special, j

#0,

p=po,

p nonspecial .

He then explicitly constructs a right adjoint @ to Q . The resulting isomorphism Horny,

(Q

(A), V)

Homq(c2) (A, @ (V))

leads to the desired result when applied to A := 00@I V,.

,V

:= 0, , y E

Yr : Theorem 8.3 (A. Ishii). The ITO-NAKAMURA construction yields mutatis mutantur the same result as in Theorem 6.2 for finite small subgroups C GL (2, C) i f the set Irr'r of all nontrivial irreducible representations is replaced by the subset Irrspec'r c Irr'r of nontrivial special ones.

Remark. This statement includes the fact that the canonical mapping Yr = Hilbr((C2) -+ C 2 / r = X r in Theorem 6.1 is indeed the mini- arguments here. It is sufficient mal resolution. We want to sketch A. ISHII'S to show that the minimal resolution 7r : X -+ X = X r factorizes over Hilbr(C2) (and is bijective outside the exceptional set E which is trivially true). For this, he identifies C2 with the graph of the projection map

515 + X , hence with a subvariety of C2 x X such that p*O@z is canonically isomorphic via the second projection to the quotient of the Ox-algebra OX[u, ,u] OX @@ C [u,w] by a r-invariant ideal sheaf. From this, he deduces that the torsionfree preimage ( p * O p ) on 2 is also the quotient of Ox[u, w] by a r-invariant ideal. But by the so called normal basis theorem, to each irreducible representation p of r of rank rp there exist (indecomposable reflexive) Ox-submodules Mjl) g ... 2 M p ) of p*O@z such that

p : C2

(XI

p*O,z

( M pf 3 f . . @ MjTp)) .

@

E!

pEIrr r

This leads to a decomposition (p*O@Z) 2

@ (MP(l) @

* * *

a3

Mpy

pEIrr r

into vector bundles and consequently to a flat family of r-invariant clusters on 2 which induces the canonical map 2 -+ Hilbr(C2). The considerations above also show that it is possible - at least in principle - to construct the indecomposable full vector bundles by invariant theory explicitly! Of course, one has to determine Hilbr(C2) and to identify this r-invariant Hilbert scheme with the minimal resolution Zr , which might be tedious, but not so difficult after all. Due to the construction, Hilbr(C2) carries a natural tautological bundle T with fibers (xx)

Hilbr(C2) 3 I

-

TI := O@2/1= p*O,p/p,I

on which the group r acts through the regular representation. Decomposing the regular representation Preg =

C

n p

P

pEIrrr

yields a decomposition

-

This, of course, is - via ( x X ) - compatible with the decomposition ( x). Moreover the mapping 0' defining the Chern divisor is in principle describable by invariant theory of the group r . - So we can formulate:

x

516

-

Corollary 8.1. One can describe the indecomposable full vector bundles M p via invariant theory as subbundles of T o n Hilbr(C2). I n particular, the CHERNdivisor of Gpcan, in principle, be computed in these terms. 9. What remains to do?

Most questions being answered in a perfect conceptual manner: Why does there remain anything to be done? The answer is quite simple and lies in the words “in principle” used several times before. For instance, to determine the special representations of a small finite subgroup of GL(2, C) would require an effective method to find the CHERNnumbers of its irreducible representations. E. g., mathematical physicists are interested in “non-supersymmetric configurations of D-branes and their evolution via tachyon condensation” (c.f. YANG-HUIHE: Closed String Tachyons, NonSupersymmetric Orbifolds and Generalized McKay Correspondence, hep-th or Adv. Theor. Math. Phys 2, 2003). In the Abelian case the special representations are associated to “D-brane charges sitting on the HIGGSbranch”. They would therefore be interested in a solution of the following concrete

Problem. Determine explicitly the special representations for a given small subgroup r c GL(2, C) as in BRIESKORN’S list in Section 5 and attach them to the vertices in the dual resolution graph of ZI-. Or more generally: Determine the CHERNdivisor for any given irreducible representation of

r. WUNRAM has this task carried out in full detail only for cyclic quotient singularities (see Lemma 7.1); his result can easily be rediscovered by the invariant theoretical method described above. For the remaining cases, however, he sketched only a quasi-algorithmic method to compute the CHERN divisors and to detect the special representations in the MCKAY-quiver in each particular case. He finds, e. g., for the group J? = ( 2 1 4 , 214;1,I), i.e. for the quotient surface singularity with resolution graph -2

-

-

-

-

-2

-2

-2

-2

-3

517

the MCKAY-quiver and the CHERN“numbers” as indicated in the diagram on the next page. (The Chern numbers are the intersection numbers of the Chern divisor with the various exceptional curves). For technical reasons the resolution graph has been reflected horizontally. The numbers at the bottom denote the ranks of the representations which sit in the quiver over that number. The arrow on the right side gives the direction (downward) for all the arrows in the quiver. The first and the last “line” must be glued together.

518

Hence the open task consists in deriving the complete information encoded in this diagram by the invariant theory of the group I?.

Remark. This example shows that the irreducible reflexive modules are not determined by their CHERNdivisor and their rank; see the two representations indicated by black squares. This, however, is always true for the special objects (ESNAULT [7]; see also part 2 of the proof of Theorem 5.6) shown as small black disks (with the exception of the trivial representation). Notice that the fundamental cycle in this example is the following:

2

Bibliography 1. P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics 815 (Springer, Berlin-Heidelberg-New York, 1980). 2. J. McKay, Graphs, singularities and finite groups, in Finite groups, Santa C m z 1997, Proc. Symp. Pure Math. 37,183-186 (1980). 3. G. Gonzales-Sprinberg, J.-L. Verdier, Construction ge'ometrique de la cowespondance de McKay, Ann. Sci. Ec. Norm. Super. 16,409-449 (1983). 4. M. Artin, J.-L. Verdier, Reflexive modules over rational double points, Math. Ann. 270,79-82 (1985). 5. H. Esnault, H. Knorrer, Reflexive modules over rational double points, Math. Ann. 272,545-548 (1985). 6. K. Behnke, 0. Riemenschneider, Quotient surface singularities and their deformations, pp. 1-54, in: Singularity Theory (Eds.: Le, Saito, Teissier), Proceedings of the summer school o n singularities held at Trieste 1991 (World Scientific, Singapore-New Jersey-London-Hong Kong, 1995). 7. H. Esnault, Reflexive modules on quotient surface singularities, J. Reine Angew. Math. 362,63-71 (1985). 8. J. Wunram, Reflexive modules o n quotient surface singularities, Math. Ann. 279,583-598 (1988). 9. 0. Riemenschneider, Characterization and application of special reflexive modules o n rational surface singularities, Institut Mittag-Leffler Report No. 3 (1987).

519 10. 0. Riemenschneider, Special representations and the two-dimensional McKay correspondence, Hokkaido Mathematical Journal XXXII, 317-333 (2003). 11. Y. Ito, I. Nakamura, McKay correspondence and Hilbert schemes, Proc. Japan Acad., Ser. A 72,No. 7, 135-138 (1996). 12. Y . Ito, I. Nakamura, Hilbert schemes and simple singularities, in: New trends in algebraic geometry. Selected papers presented at the Euro conference, Warwick 1996, Hulek, K., et al. (eds.). London Math. SOC.Lect. Note Ser. 264, pp. 151-233 (Cambridge University Press, Cambridge, 1999). 13. R. Kidoh, Hilbert schemes and cyclic quotient singularities. Hokkaido Math. Journal XXX, 91-103 (2001). 14. A. Ishii, O n the McKay correspondence f o r a finite small subgroup of GL (2, C) , Journal fur die Reine und Angewandte Mathematik 549, 221233 (2002). 15. Y. Ito, Y. Special McKay correspondence, in: Proceedings of the summer school on toric varieties, Grenoble 2000, organized by Laurent Bonavero and Michel Brion, SBminaires & Congrhs 6,213-225 (2002). 16. W. Crawley-Boevey, O n the exceptional fibres of Kleinian singularities, Am. J. Math. 122 (5), 1027-1037 (2000). 17. V. Ginzburg, M. Kapranov, Hilbert schemes and Nakajima’s quiver varieties, Preprint (unpublished). 18. J. Wunram, Reflexive modules on cyclic quotient surface singularities, in: Singularities, representations of algebras, and vector bundles, Greuel, G.M., llautmann, G. (eds.), Lect. Notes Math. Val. 1273 (Springer, BerlinHeidelberg-New York, 1987). 19. M. Kapranov, E. Vasserot, Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316 (3), 565-576 (2000). 20. T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. SOC.14 (3), 535-554 (2001). 21. M. Reid, La correspondence de McKay, SBminaire Bourbaki, Volume 1999/2000, ExposBs 865-879, Asterisque 276,53-72 (2002). 22. H. C. Pinkham, Deformations of quotient surface singularities, in: Symposia in Pure Mathematics, Vol. 10 (AMS, Providence, 1976). 23. K. Behnke, C. Kahn, 0. Riemenschneider, Infinitesimal deformations of quotient singularities, pp. 31-66, in: Singularities, Banach Center Publications 20 (Polish Scientific Publishers, Warsaw, 1988). 24. 0. Riemenschneider, Cyclic quotient surface singularities: Constructing the Artin component via the McKay-quiver, in: Singularities and Complex Analytic Geometry, Sunrikaiseki Kenkyuosho Kokyuroku (RIMS Symposium Report) Nr. 1033,163-171 (1998).

520

LECTURES ON THE TOPOLOGY OF POLYNOMIAL FUNCTIONS AND SINGULARITIES AT INFINITY DIRK SIERSMA Mathematisch Instituut, Universiteit Utrecht PO Box 80010, 3508 TA Utrecht The Netherlands. email: siersmaOmath.uu.nl

MIHAI

TIBAR

Mathkmatiques, Universitk des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq, France. email: tibarOmath.univ-lillel.fr In the last 15 years there has been an increasing interest for the study of the global topology of polynomial functions, especially in connection with the asymptotic behaviour of fibres. This stream of research is closely related to the affine geometry and to dynamical systems on non-compact spaces. We present here several aspects of this topic. Keywords: affine hypersurfaces, singularities at infinity, exchange of singularities, real and complex curves

1. Introduction One of the first authors who studied the topology of polynomial functions was Broughton2. In the same time Pham31 was interested in regularity conditions under which a polynomial has good behaviour at infinity, and HA and Lel’ proved a criterion for detecting atypical values in two complex variables. Some evidence for the crucial importance of singularities at infinity in understanding the behaviour of polynomials is the Jacobian Conjecture. In C2, an equivalent formulation of this conjecture is the following, see L b Weber21 and the authors33: I f f : C2 -+ C has n o crttical points but has singularities at infinity then, f o r any polynomial h : C2 + C , the critical locus Z(Jac(f, h ) ) is not empty. Indeed, if the polynomial f has no critical points and no singularities at infinity then all the fibres of f are CWcomplexes with trivial homotopy groups, hence contractible (cf. Corollary

521 2.1). In this case the Abhyankar-Moh theorem tells that f is linearisable. The open case is therefore the one of singularities at infinity.

In general, let f : K” -+ K be a polynomial function, where K is CC or R. We assume once and for all in this paper that n 2 2 and that the polynomial is not a constant. A value a E K is called typical for f if the fibre f-’(u) is nonsingular and the function f is a locally trivial topological fibration at a (i.e. over some neighbourhood of a). Otherwise a is called atypical. The set of atypical values is known to be finitea, cf. Thom41. The first natural problem is to identify the set of noncritical atypical values of f and to describe how the topology of fibres changes at such a value. There is yet no solution in whole generality. One can solve the problem in case n = 2 over the complex, cf. H&L612 and over the reals, cf. Tibk-Zaharia48,but in case n > 2 there have been proved such criteria only when the singularities which occur are in some sense isolated and K = CC, see Sier~ma-Tibgr~~ , P a r u s i r i ~ k iTib2ir44. ~ ~ ~ ~ ~This , includes the situation of “no singularity at infinity” cf. Broughton3 and NBmethi-Zaharia26, which has been treated earlier. There are very interesting studies of the topology of fibres in different degrees of generality, by Neumann-Norbury, PgunescuZaharia, Gusein-Zade-Luengo-Melle. These notes are not a survey. They are only intended a s an introduction to “singularities at infinity” and their behaviour, and to give several steps to the understanding of the problems which occur in this topic. They are essentially collected from the following papers and manuscripts of the authors: S i e r ~ m a - T i b k ~T ~i b?g~~~-Tibk-Zaharia48, ~ ~ ,~ , Bodin-Tibk4. The notes do not cover topics like monodromy, Hodge theory, equisingularity, curvature etc. In order to keep a reasonable length, a number of proofs are skipped; on the other hand we give many examples and some of them are largely commented. The forthcoming book47 will include all this material and much more on polynomials and their vanishing cycles. The contents of these notes are as follows. We discuss in 92 the topology of the fibres of polynomials f : Cn -+ @. We first define W-singularities at infinity, where W refers to a certain Whitney stratification on the space X which is the union of the compactified fibres of f . We show that the general fibre of a polynomial with isolated W-singularities at infinity has the homotopy type of a bouquet of spheres and that the number of these afar a proof, see e.g. Pham31 , which uses resolution of singularities, or Verdier50, Sier~rna-Tibk-~~ and Corollary 3.1 in this paper, where stratification theory is used.

522

spheres is pf +Xf, where pf is the total Milnor number off (i.e. the sum of Milnor numbers of the &ne singularities) and Xf is the sum of Le-Milnor numbers at infinity. We study or recall in 53 several regularity conditions at infinity which insure topological triviality and comment their mutual relations. Let us show below, in advance, their determination scheme: Malgrange regularity H t-regularity tameness

ps-regularity,

+ quasi-tameness + pE-regularity (M-tame) + top. triviality.

We introduce in 54 the polar curve, a key tool in exploring topological properties of polynomial functions, and explain its relation to regularity conditions. We look closer in 55 to the case of two variables, in the complex and in the real setting, and completely characterise the topological triviality of fibres. In 56 we introduce the problem of global topological equivalence of polynomial functions. We explain a recent result which gives a numerical criterion for topological triviality in a family of polynomial functions. A special phenomenon which one encounters in families of polynomial functions is the “singularity exchange at infinity”, when singular points escape from the space Cn and produce “virtual” singularities (i.e. singularities at infinity) of the limit polynomial. The total quantity of singularity involved in this phenomenon may not be conserved, as it is in the case of local deformations of holomorphic functions with isolated singularities. We discuss in 57 semi-continuity results which enable us to find rules of the exchange phenomenon. This material was the text support of the two lectures delivered at the Singularities School of the ICTP Trieste, in August 2005. We own special thanks to the organisers and to Ld D h g T r h g , head of the Mathematics section at ICTP. 2. Topology of the fibres of a polynomial 2.1. Singularities at infinity

Let f : C” --+ C be a polynomial of degree d and let ~ ( z , z o )be the homogenized of f by the new variable ZO. One replaces f : C” --+ C by a proper mapping T : X --+ C which depends on the chosen system of coordinates on C”, as follows (see Broughton3). Consider the closure in

523

Pn x @ of the graph of f , that is the hypersurface

X := {((x;~ o )t,) E Pn x @ I F

d

:= J(x, XO) - t ~ = o 0},

which fits into the commuting diagram

cn 2 f\

X J

T

7

@ where i denotes the inclusion x H ([x: 11,f(x)) and T is the projection on the second factor. This proper extension of f has been used several times in the study of polynomial functions at infinity: Broughton2, Pham31, ~~~~~, P a r u s i f ~ s k,iS~ i~e r ~ m a - T i b ? i r etc. Let H" denote the hyperplane at infinity {xo = 0) c Pn. Let f k denote the degree lc homogeneous part of f. The singularities of X are contained in the divisor "at infinity" X" := X n (H" x C), namely:

The singular set of X" is:

Xrng := W x C, where W

afd

:= { - = . . . =

ax1

afd = 0} c H*.

axn

We have C C W . The singularities of f , i.e. the affine set Sing f := Z( ., may be identified, by the above diagram, with the singularities of T on X \ X". One can prove, by an easy computation, that Sing f n H" c C, where Sing f denotes the closure of Sing f in I F. In particular we get dim Sing f 5 1 dim C.

g,.g),

+

Definition 2.1. (i) f is a F-type polynomial if its compactified fibres and their restrictions to the hyperplane at infinity have at most isolated singularities. (ii) f is a B-tgpe polynomial if its compactified fibres have at most isolated singularities. (iii) f is a W - t y p e polynomial if its proper extension T has only isolated singularities in a stratified sense, as defined below. Let us recall that, for an analytic function .1c, on a complex space Y endowed with a complex Whitney stratification C, one has the well defined notion of stratified singularity of $ (alternatively: singularity of $ with

524

respect to C). Namely, the stratified singularities of $, denoted by Sing $, are the union UciEcSing$ 1 ~ (see ~ e.g. Goresky-MacPherson", LQ19).We introduce the following definitions: Definition 2.2. (Canonical stratification at infinity) Let W be the least fine Whitney stratification of X that contains the strata X \ X" and X" \ Xsing. This is a canonical Whitney stratification (see Mather23 ) with two imposed strata instead of the smooth open X \ Xsing. We may call W the canonical Whitney stratification at infinity of X. Let Sing T be the singularities of r : X C with respect to the canonical Whitney stratification at infinity and denote Sing O0 f := Sing rnX". Using 52.1, one then gets the following equality: Sing r = Sing f U Sing O0 f. Let us also remark that Sing T n (XM \ Xsing) = 8 and that dim Sing "f 5 dim C. The class of polynomials we want to focus at is defined as follows, by making use of LB's definitionlg of isolated singularities. Definition 2.3. (Isolated W-singularities at infinity) We say that the polynomial f : @" -+ C has isolated W-singularities at infinity if the projection 7 : X t CC has isolated singularities with respect to the stratification W (equivalently: dim Sing T 5 0). We have: F-class c B-class c W-class.

(1)

The first inclusion is clear from the definition and the second one is proved in S i e r ~ m a - T i b kIn ~ ~2. variables, if f has isolated singularities in C 2 ,then it is automatically of F-type. Broughton3 considered for the first time B-type polynomials and studied the topology of their general fibers. R e m a r k 2.1. From the above definition and the expressions of the singular loci we have the following characterisation: (i) f is a B-type polynomial H dim Sing f 5 0 and dim C 5 0, (ii) f is a F-type polynomial H dim Sing f 5 0 and dim W 5 0. &mark 2.2. "Isolated W-singularities at infinity" implies that the polynomial f has isolated singularities in Cn (in the usual sense), which fact does not depend on coordinates.

525

Example 2.1.

(a) If the polynomial f has isolated singularities in C" and dimC 5 0 then f has isolated W-singularities at infinity. This is the case for all reduced plane curves. (b) h = z 3 y z z2 : (C3 4 C has isolated W-singularities at infinity (with respect to W') but dim C = 1. (c) The polynomial g := z 2 y z : C3 -+ C has non-isolated W-singularities at infinity (namely in the fibre g-l(O)). It turns out that it has nonisolated W-singularities at infinity in any coordinates, see also our next Remark.

+ +

+

2.2. A bouquet theorem

The following result may be viewed as a global version of the local bouquet ~ ~ Le D.T.19 . Such a global result has been theorems of J. M i l n ~ r and first proved by B r ~ u g h t o nfor ~ > B-type ~ polynomials, see also Note 2.2 for comments on related results. We give here the version by Sier~ma-TibZr~~, concerning the W-type polynomials. Theorem 2.1. Let f : C" + C be a polynomial with isolated Wsingularities at infinity. Then the general fibre o f f is homotopy equivalent to a bouquet of spheres of real dimension n - 1. Remark 2.3. As in the local case, if f has nonisolated W-singularities at infinity, then one cannot expect to get a bouquet of the type in the Theorem above. The Example 2.l(c) shows that even if the polynomial has smooth fibres (but some nonisolated W-singularity contained in Xm), the result above is no more true: the general fibre of g is a circle (whereas it should have been a bouquet of spheres of dimension 2). Proof of Theorem 2.1. Step 1. We prove first that the reduced homology of a general fibre is concentrated in dimension n - 1. Relatively to the stratification W, the projection T : X + CC has only isolated singularities, namely a finite number of points situated on Xm and another finite set on X\Xw which corresponds to the set Sing f. Let R be the set of critical values of T . For each b E R, let bb be a small enough disc centred at b. Then T : X n~ - l ( ( L 1 \ U b E R 6 b ) + C \ U b E R b b is a stratified topological fibration (with respect to W), hence its restriction to f-l(CC \ U b E R b b ) is a locally trivial topological fibration, by Thom first isotopy lemma.

526

Let Xs := r - ' ( S ) , Fs := f - l ( S ) , for some S c C. Let us fix c E C \ U b E R S b and Cb E d 6 b . We get, as usually by deformation retraction and excision, the following splitting:

which actually holds in full generality, for any polynomial f, whatever its singularities are. We stick to such a term Hi+l(Fbb, Fcb).For simplicity of notations, let D be one of the discs b b and fix some d E dD. We have, according to Broughton3 :

H.(FD, Fd) 2 H2"-'(X~,Xd). It remains to prove that H ' ( X D , Xd)is concentrated in dimension n. Let b be the centre of D. The singularity of rlx, are on X b , let those be denoted by a l , . . . , a k . We may choose a good neighbourhood of ail say of the form Bi n XD, where Bi is a small enough closed ball in some local chart and also suppose D small enough such that the restriction r : Bi n X D -+D is a Milnor representative of the germ r : (XD, a i ) + ( C ,b). Since this germ is an isolated singularity with respect to the induced stratification, it follows that the fibres r - l ( u ) , Vu E D ,are transversal to a certain semi-algebraic Whitney stratification of dBi n XD, constructed as in the proof of Le'sl' Theorem 1.1, or LG'sl' Theorem 1.3. Thus there is a trivial topological fibration: 7

: xo

\ Ui=l,kBi+ D

and, by an excision, we get the isomorphism: H'(xD,x~)

= @i=l,kH'(Bi

n xD,

~i

n~

d =) @i=l,kH'-l(Bi

n~

d ) ,

where Bin Xd is the local Milnor fibre of the germ of T at ai. We may conclude our proof by applying a theorem due to LG D.T. (see e.g. LG1', Theorem 5.1, for a more general result) which says that the Milnor fibre of an isolated singularity function germ on a hypersurface of pure dimension n has the homotopy type of a bouquet of spheres of dimension n - 1.

Step 2. To get the homotopy result, we focus on the local situation. Let first ai E X" be a singular point of r. This fits into a statement due to Hamm and LZ,14, Theorem 4.2.1, Corollary 4.2.2: the conditions are obviously fulfilled, namely r has isolated singularities with respect to W and rHd(X \ Xm) 2 n (since X \ Xoo is smooth). The ingredients in the proof are homotopy excision (Blakers-Massey theorem) and stratified Morse theory.

527

Now by a very slight modification of the above cited result of Hamm and L& (i.e. by using cylindrical neighbourhoods, which are conical by GoreskyMacPhersonlo, p. 165), we get that the pair

(Bin XD \ X”,Bi nxd \ Xm) is (n - 1)-connected. We may apply S w i t z e r ’ ~Prop. ~ ~ , 6.13, to conclude that Bi n XD \ X” is obtained from Bi n X d \ X” by attaching cells of dimensions 2 n. A similar (actually better) situation is encountered on the affine piece: if aj E Xh \ Xm is a singularity of rlxD then it is well known that Bj n FD is obtained from Bj f l Fd by attaching n-cells, by L6’s result 19. For global fibres, it follows that FD is obtained (up to homotopy) by attaching a finite number of cells of dimension 2 n to Fd. Finally, the whole space C” = Fc is obtained, up to homotopy, by attaching a finite number of cells of dimensions 2 n to a general fibre F,. Since F, has the homotopy type of a n-dimensional CW-complex, we get ht F, N VySn-’, by Whitehead’s theorem. 0

Note. Improvements of Theorem 2.1, by considering a sharper type of singularities, were proved in P a r u s i ~ i s kand i ~ ~T i b 5 1 - ~see ~ ; also the forthcoming book47 for an update. The above bouquet statement applies to complex polynomials of two variables with irreducible generic fibre and to n-variables complex polynomials with isolated singularities such that are pregular at all points y E X”. The latter fact follows from the definition of pregularity, see $3, and its proof is left to the reader. Theorem 2.1 also extends the similar results for some classes of polynomials with “good behaviour at infinity”: tame 3, quasi-tame 25, M-tame 26 (which coincides with pE-regular, see $3). We recall that “tame” implies “quasi-tame”, which implies in its turn “M-tame” cf. N6methiZ5,NBmethi-Zaharia26.It turns out from Corollary 2.l(a) that if such a polynomial has isolated W-singularities at infinity then actually it has no W-singularities at infinity. The homology counterpart of Theorem 2.1 (i.e. Step 1 of its proof) was proved under the additional hypothesis dimC = 0, by Broughton3, Theorem 5.2; he also got the particular case of point (a) of the following Corollary 2.1. More recently, Hamm13 studied the cohomology of fibres. He proves the semicontinuity of the ranks of certain cohomology groups and of stalks of direct image sheaves. This contributes to understanding why and how occur the ‘?jumps” in the topology of the fibres.

528

Definition 2.4. Let a E X* and let us denote by A, the number of spheres in the Milnor fibre of the germ T : (X, a ) --t (C,b ) and call it the Milnor number at infinity, at a E Xoo. Corollary 2.1. Let f be a polynomial with isolated W-singularities at infinity. Then:

(a) The number y of spheres in a general fibre is equal to the s u m pf + Xf, where pf is the total Milnor number o f f and Xf is the s u m of the Milnor numbers at infinity. I n particular Xf is invariant under digeomorphisms of C". (b) Let p~~ denote the sum of the Milnor numbers of all the singularities of the fibre Fb and let X F ~ denote the sum of all Milnor numbers at infinity at X b f l Xoo. Then x(FU)

- X(Fb) = ( - l ) " - l ( X F b

where F, is a general fibre o f f .

+ PFb), 0

Corollary 2.2. (General connectivity estimation) Let f : C" + C be any polynomial. Then its generalfibre Fu is at least (n-fL-dim(CUSing f))connected. 0 Note. For the proof we refer to Sier~ma-TibGr~~. This result improves, in case dim Sing f 5 dim C the connectivity estimations by Kato and by Dimca. While the above result holds for general fibres, the level of connectivity of atypical fibres may be at most 1 less, cf. Tibk44 (see for instance the example f = z 2 y z : C2 --t C). An improved connectivity estimation was shown more recently by Libgober and TibGrZ2,which superseded another improvement by Dimca and PGunescu5.

+

3. Regularity conditions at infinity

We consider here polynomial functions of real or of complex variables, and we denote K = R or @. For proving topological triviality at infinity (i.e. on Kn \ K , where K is some large compact) one would try to produce a foliation which is transversal to the fibres of f. A natural attempt is to integrate the vector field grad f (or grad f/ll grad f 112, which is a lift by f of the vector field d / d t on K). The resulting foliation may have leaves that "disappear" at infinity (since grad f may tend to 0 along some nonbounded sequence of points), hence it is not of the kind we want. Such a foliation was used by many authors, for instance by L. Fourrier8v9 in order

529

to characterise, in two complex variables, the topological right-equivalence at infinity of polynomials. See also $6 for the general problem of global topological triviality of families of polynomials. In order to construct a “good” foliation, one needs some regularity conditions on the asymptotic behaviour of the fibres of f. Keeping in mind the idea of using controlled vector fields (rather than just gradf), we introduce two regularity conditions at infinity: t-regularity and p-regularity. The former depends on the compactification of f , but allows one to apply algebro-geometric tools, more effectively in the complex case. The latter condition does not depend on any extension, but on the choice of a proper non negative C1-function p which defines a codimension one foliation. These regularity conditions correspond to the two main strategies used up to now by many authors in the study of polynomial functions, and which were regarded as parallel methods. One of the methods is to “compactify” in some way the function (i.e. to extend it to a proper one), as we have seen in the sections before. This gives the advantage of having the “infinity” as a subspace of the total space Y , but in the same time creates the problem of getting rid of it in the end. The space Y has singularities (usually nonisolated ones) exactly “at infinity”; one may either resolve those singularities or endow Y with a stratification. This strategy is used in different ways by several authors, like Broughton, Neumann, Oka, Hamm, LB and Weber, Libgober and Wood, Durfee, Zaharia, Parusiriski, etc. The second main method is to stay within the affine and to use for instance Milnor-type methods. We may refer to Broughton, HA, Nemethi and Zaharia, and more recently Gaffney, Jelonek, Kurdyka, D’Acunto, Grandjean. Showing that t-regularity implies pregularity will provide a link between the two aforementioned methods, notably in the real case, which is more and more explored in the last time (Kurdyka, Szafraniek, Parusinski, D’Acunto, Grandjean, Jelonek and others). 3.1. p-regularity and t-regularity

For a real or complex polynomial f : K” X K := {.f(z, zo)

-

-+

K we consider as in 2.1 the set:

tz: = 0)

c

x K,

which is the closure in P K x K of its graph. Let us recall that the projection T : XK --+ K is a proper extension of f and that Xg := XK n (20= 0) the part at infinity (which is a divisor in case K = C). We shall suppress the lowercase K when the setting is clear from the context.

530

Definition 3.1. We say that f is topologically trivial at infinity at to if there is a neighbourhood D of t o E K and a large closed ball B c K” centered at 0 such that the restriction f l : ( X \ B ) n f-’(D) -+ D is a topologically trivial fibration. We say that f is locally trivial at y E X” if there is a fundamental system of neighbourhoods Ui of y in X and, for each i , some small enough neighbourhood Di of ~ ( y )such , that the restriction f , : n (X\ X-) n f - l ( D i ) 4 D~ is a topologically trivial fibration.

ui

Remark 3.1. Local triviality at all points y E Xmn7-l(t0)does not imply topological triviality at infinity at to. To be able to glue together a finite number of locally trivial nonproper fibrations, one needs a global control over these, which is not available in general. See Example 3.2. This should be contrasted to the regularity conditions we define in the next. Definition 3.2. (pregularity) Let p : Kn \ K 4 R>o - be a proper C1submersion, where K c K” is some compact set. We say that f is pregular at y E X” if there is a neighbourhood U of y in X \ K such that f is transversal to p at all points of U n Kn. We say that the fibre f - l ( t o ) is p-regular at infinity if f is pregular at all points y E X” n T - l ( t 0 ) . Remark 3.2. The definition of pregularity at infinity of a fibre f - l ( t o ) does not depend on any proper extension o f f , since it is equivalent to the following: for any sequence ( z k ) k E ~c Kn, l z k l + 00, f ( z k ) -+ t o , there exists some ko = k 0 ( ( z k ) k E ~ ) such that, if k 2 ko then f is transversal to p at zk. It also follows from the definition that if f - ’ ( t o ) is pregular at infinity then this fibre has at most isolated singularities. The transversality of the fibres of f to the levels of p is a “Milnor type” condition. In case p is the Euclidean norm, denoted in this paper by p ~ , this condition has been used by John Milnor in the local study of singular functions MilnorZ4,§4,5. For complex polynomial functions, transversality to big spheres (i.e. pE-regularity, in our definition) was used in Broughton3, pag. 229, and later in NQmethi-ZahariaZ6,where it is called M-tameness.

(c:=,

Example 3.1. p : Kn --t R>o, p(z) = 1zil2P’* ) lI2P, where (w1, ...,w,) E N”,p = lcm(w1, ...,w,} and wipi = p , Vi. This function is “adapted” to polynomials which are quasihomogeneous of type (w1,. . . ,w,). By using it, one can show that a value c E K is atypical for such a polynomial if and only if c is a critical value of f (hence only the value 0 can be atypical). Namely, let E,. := { x E Kn I p ( z ) < r } for some r > 0. Then the local Milnor fibre o f f at 0 E Kn (i.e. f-l(c) nE,, for some

531

small enough E and 0 < IcI << E ) is diffeomorphic to the global fibre f-'(c), since f - I ( c ) is transversal to a%, Vr 2 e. The first pleasant property of pregularity is that it implies topological triviality, more precisely we prove the following:

Proposition 3.1. 45 If the fibre f-'(to) topologically trivial at infinity at to.

is p-regular at infinity then f is

Proof. If f is pregular a t y E F ' ( t 0 ) n X" then one can lift the (real or complex) vector field slat defined in a neighbourhood D of to E K to a (real or complex) vector field on UnK" tangent to the levels p =constant. If f -l(to) is pregular at infinity then we may glue these local vector fields by a partition of unity and get a vector field defined in some neighbourhood of X" without X". This is a controlled vector field which can be integrated to yield a topologically trivial fibration fl : (K" \ K ) n f - ' ( D ) -+ D , as shown by Verdier5' in his proof of Theorem 4.14 of Thom-Mather isotopy t h e ~ r e m ~ (Notice l > ~ ~ .however that Thom-Mather theorem does not directly apply since f is not proper.) We use here p as "fonction tapissante" , a notion introduced by Thom41. In case n = 2, a similar procedure was used by HB H.V. and L6 D.T12. Some preliminaries are needed in order to introduce the second regularity condition. First, let us define the relative conormal, following Teissier40 and Henry-Merle-Sabbah15, then we state some technical results which we need. Let X c KN be a K-analytic variety. In the real case, assume that X contains at least a regular point. Let U c KN be an open set and let g : X n U -+ K be K-analytic and nonconstant. The relative conormal Tllxn, is a subspace of T*(KN)lxnudefined as follows:

q,,, := closure{(y,5) E T * ( K N )I Y E X0n u7S(Ty(g-l(9(Y)))

= 017

where X o c X is the open dense subset of regular points of X where g is a submersion. The relative conormal is conical (i.e. (y,c) E T'lxn, + (y, A t ) E Tg*(,,,, VA E K*>.The canonical projection T'lxnu + X n U will be denoted by 7r.

Lemma 3.1. 44 Let (X,x) c (IKN,x)be a germ of an analytic space and let g : (X, x) -+ (K, 0 ) be a nonconstant analytic function. Let y : X + K be analytic such that y(x) # 0 and denote by W a neighbourhood o f x in K N - Then (T;l,y,w)x = (T;gl,ynw)x.

532

Definition 3.3. Let Ui = {xi # 0}, for 0 < i 5 n, be an affine chart of I P and let y E (Uix K) n X n {ICO = 0). The relative conormal T~olxnvixK is well defined. We denote by (Cg),the fibre n-l(y) and call it the space of characteristic covectors at infinity, at y. It follows from Lemma 3.1 that (CM), does not depend on the choice of afine chart Ui. Definition 3.4. (t-regularity) We say that f (or that the fibre f-'(to)) is t-regular at y E X" n 7-l(t0) if (y, dt) 4 (Cg),. We also say that f-'(to) is ®ular at infinity if this fibre is t-regular at all its points at infinity.

Proposition 3.2. 45 Iff is t-regular at y E X" then f is pE-regular at y, where p~ is the Euclidean norm.

Proof. The mapping d" : XK 4 R, defined by: d'(x, f (x))= l/p'$(x), for x E K" drn(Y)= 0 , for y E Xw

(Tzo12,

is analytic and defines Xw. In the real case we have (T&,Ix)y = by Lemma 3.1, and the latter is in turn equal to (T&lx)y= (C,"),. The pE-regularity at y E XM is certainly implied by (y, dt) 6(T&,Ix)y, which is just t-regularity, by the above equalities. This finishes the proof in the real case. Now the complex case. Let us first introduce the map L : PT*(R2n) --+ PT*(en) between the real and the complex projectivised conormal boundles (where is the real underlying space of C") defined as follows: if E is conormal to a hyperplane H c R2n then L( [El) is the conormal to the unique complex hyperplane included in H . This is clearly a continuous map. We then have the following equality: W@M)Y

= L(wq*.o121x)Y)7

since the complex tangent space Tz{xo =constant} is exactly the unique complex hyperplane contained into the real tangent space T,(IZO~~ =constant}. The equality follows by the fact that L commutes with taking limits. Now (y,dt) 6 (C,"), implies (y, ~-l([dt])@ P(~zo121x)y, which in turn implies pE-regularity at y since, as above in the real case, we 0 = (~I:o121x)y. still have (T'mIx)y Corollary 3.1. 45 Let f : C" -+ C be a complex polynomial. Then the set of values to such that f -'(to) is not pE-regular at infinity is a finite set. In particular, the set of atypical values o f f is finite.

533

Proof. Take a Whitney stratification W = {Wi}i of & with a finite number of strata and with CCn as a stratum. It turns out from S i e r ~ m a - T i b k ~ ~ , Lemma 4.2, or Tib5r44, Theorem 2.9, that any pair of strata (Cn, Wi)with Wi E X" has the Thom property with respect to the function 20, in any local chart. If -r-l(to) is transversal to a stratum Wi c X", then f - l ( t 0 ) is t-regular at infinity. Now the restriction of the projection T : Xc -+ C to a stratum contained in X" has a finite number of critical values. These imply that the values to such that f-'(to) is not t-regular at infinity are finitely many. The conclusion follows by Proposition 3.2. 0 Remark 3.3. The t-regularity at some point y E X" is implied by the transversality of {t = t o } to the strata of W . In particular, if the polynomial f has isolated W-singularities, then these singularities are the only points where the corresponding fibres of f might be not t-regular. Indeed, this follows from the proof of Corollary 3.1 above. Nevertheless, the pE-regularity is really weaker than t-regularity. We show by the next example that the converse of Proposition 3.2 is not true. This makes the pregularity interesting and raises questions concerning real methods and their interplay with the complex ones.

Example 3.2. Let f be the following complex polynomial function f := x+x2y : C3 CC in 3 variables x, y, z . We show below that f is not t-regular at a whole line L := {XO = x = t = 0 ) within q, hence has a nonisolated t-singularity. On the other hand, there is a single point in Xp at which f is not PEregular. Therefore the singularity is isolated in this sense. Here follow the computations. Let F := xxi+x2y-txi. The t-regularity in the chart y # 0 is equivalent to IyI IIglIf , 0, as Ilx, yII -+ 00, cf. Sier~ma-TibEr~~, pag. 780. But in our example IyI. 111+2xyll tends to 0, for instance if y -+ 00, x = l/y3 - 1/(2y), z = ay. The limit points in X" are the 1-dimensional set L. We now find the set of points ( p ,t ) E X" where f is not pE-regular. This amounts to finding the solutions of the equation grad f = (Ax,Xy, Xz). One m assume X # 0. It follows 11x112%= ~+2%11y11~, z = 0. The solution is a 2-dimensional real algebraic set A and the set we are looking for is An X" . If we work in two variables instead, i.e. with g := x x2y : C2 -+ C, then the single point where g is not pE-regular is xo = x = t = 0. In our case (3 variables), one intersects with z = 0 , hence A n X" consists of a single point. ---f

+

534

3.2. The relation t o Malgrange condition

In the complex case, F. Pham formulates a regularity condition which had been found by B. Malgrange, cf. Pham31, 52.1. We give below a definition which works also in the real case, together with a localised version at infinity.

Definition 3.5. We consider sequences of points xi E K" and the following properties:

(LI) (Lz)

IIxill -+ xi + y

00

E

and f(xi) 4 to, as i --+00.

Xg,as i -+ 00.

One says that the fibre f - l ( t 0 ) verifies Malgrange condition if there is S > 0 such that, for any sequence of points with property (L1) one has

(MI

Ilxill . I1 gradf(xi)ll > 8.

We say that f verifies Malgrange condition at y E Xg if there is S > 0 such that one has (M), for any sequence of points with property (Lz).

Note. It clearly follows from the definition that f - ' ( t o ) verifies Malgrange condition if and only if f verifies Malgrange condition at any y = (z,to) E Xg.We have proved in S i e r ~ m a - T i b k ~ Proposition ~, 5.5, that t-regularity implies the Malgrange condition, both at a point or at a fibre (the proof works in the complex case as well as in the real one). Reciprocally, Malgrange condition implies t-regularity, as more recently proved in the complex case by P a r u s i ~ i s k i Theorem ~~, 1.3. In fact the same proof works over the reals. Briefly, we have the following relations: Malgrange condition

t-regularity

& pE-regularity.

(3)

The Malgrange condition at y E Xg is equivalent to saying that the Lojasiewicz number Ly(f) at y is 2 -1, where Ly(f) is defined as the smallest exponent 0 E R such that, for some neighbourhood U of y and some constant C > 0 one has:

1 gradI ) f. (

2 Clxle, Vx E U n C".

There are two other regularity conditions used in the literature which are similar to Malgrange condition but clearly stronger than that: Fedoryuk's c o n d i t i ~ n ~(or ? ~ tameness, ?~ see Proposition 5.1) and Parusiriski's condition2'.

535

4. Polar curves and regularity conditions

We further explain, in the complex case, the relation between t-regularity and affine polar curves associated to f. Local-at-infinity and affine polar curves were used in several papers: S i e r ~ m a - T i b k ~ Tib&r44, ~ - ~ ~ , to get information on the topology of the fibres of f. Given a polynomial f : Kn -+ K and a linear function 1 : Kn -+ K, one denotes by r(Z,f) the closure in KN of the set Crt(1, f ) \ Crtf, where Crt(l, f ) is the critical locus of the map (1, f) : KN K2.A basic and useful result is that r(l,f) is a curve (or empty) if 1 is general enough. We give below the precise statement (a particular case of Lemma 1.444).For some hyperplane H E Ik-l, one denotes by ZH : Kn -+ K the unique linear form (up to multiplication by a constant) which defines N. -+

Theorem 4.1. 44 There exists an open dense set Rf C Pn-l (Zariski-open in the complex case) such that, for any H E R f , the critical set r ( l H , f) is 0 a curve or it is empty. Definition 4.1. For H E Rf,we call r ( l H , f ) the afine polar curve of f with respect to 1 ~ A. system of coordinates ( X I , . ..,x,) in Kn is called generic with respect to f iff {xi = 0) E R f , V i . It follows from Theorem 4.1 that such systems are generic among the systems of coordinates. We first relate the affine polar curves to the local polar curves at infinity, as follows. Let us consider the map germ: (7,XO) : ( K P )--$ W 2 , 0 ) , for some p E Xm, with critical locus denoted by Crt,(T,xo). The local (nongeneric) polar locus rp(T, 20)is defined as the closure of Crtp(T,xo)\X" in W. Let p = ([0 : p l : ... : p , ] , ~ ) where , pi # 0, for some fixed i. In the chart Ui x K,the polar locus rp(T,xo) is the germ at p of the analytic set Gi c X,where

and f ( i ) := f(x0, X I , . , . ,xi-1,1, xi+l,. . . , xn). On the intersection of charts (Uo n Ui) x K,the function from

g ,by a nowhere zero factor, for

j

# 0, a.

differs

536

Thus the germ of subset of Pn x K:

at p is the germ at p of the following algebraic

which is equal to T ( x i ,f), where: -

r(l,f) := closure{(z, t)

E

Kn x K I x E F ( l , f ) , t = f(x)} c Pn x K.

One may now easily prove the following finiteness result: Lemma 4.1. If the system of coordinates (XI,.. . ,xn)is generic with respect to f, then there is a finite number of points p E X" for which the polar locus rp(7, xo) is non-empty and at such a point I?,(T, 50)is a curue. Proof. Fix a generic system of coordinates. By Theorem 4.1, the set r ( x i ,f) is a curve (or empty) and therefore F(xi,f) c X is a curve too (or empty), 'di E (1,. . . ,n}. We have shown above that, for a point p E X" with pi # 0, one has the equality of germs rp(r,xo)= F ( x i ,f),. Then the assertion follows by the fact that the intersection X" n (Uy=li=(xi, f ) is a finite set. 0

In the complex case, the affine polar curves are closely related to the isolated t-singularities, defined as follows. Definition 4.2. We say that f has isolated t-singulaxities at the fibre f-'(to) if this fibre has isolated singularities and the set {p E X" I f-'(to) is not t-regular at p} is a finite set.

Let us also remind that if a complex polynomial f has no W-singularity at some point p E X" then f is t-regular at p. This was indicated in the proof of Corollary 3.1. Proposition 4.1. Let f : K" 4 K be a polynomial function and let p E X" fl r-l(t0). Then the following are equivalent:

(a) rp(t,xO) # 0. (b) r ( x i ,f) 3 p for some i.

If IK = C and f has isolated t-singularities at f-'(to) moreover equivalent to the following ones: (c) p is a t-singularity.

then these conditions are

537

(d) A, # 0, where A, is the Milnor number at infinity at p , as defined in Definition 3.4 of Siersma- Taba"733. Proof. The equivalence (a) e (b) follows from the already proved equality &$, .o) = q . i , f .), The equivalence (a) e (c) is Proposition 5.3 in Sier~ma-Tib5.r~~ and (c) w (d) is a consequence of Proposition 4.5 in 1oc.cit. and Tibk44, see Remarks 4 bellow. 0 In the case of a nonisolated t-singularity at p E Xm, the general affine polar curve might be empty at p . See Example 3.2 for a discussion of such a case.

On the Milnor number at infinity In case of a complex polynomial with isolated t-singularities, one proves the following formula, which shows in particular that the fibre Fa := f - ' ( a ) is atypical if and only if its Euler characteristic is different from the one of a general fibre F, := f - ' ( u ) :

where p ~ is, the sum of the Milnor numbers at the isolated singularities on the fibre Fa and XF, is the sum of the so-called Milnor numbers at infinity at the t-singularities on X" n .-'(a). In the more general case of a polynomial with isolated singularities in the affine space (but no conditions at infinity), the t-singularities at infinity may be non-isolated and one does not have numbers A, anymore. However, one can still give a meaning to the number AF, by taking the relation (4) as its definition: Definition 4.3. Let f : @" 4 C be a polynomial with isolated singularities. We call Euler-Milnor characteristic at infinity of the fibre Fa the following number:

If AF, = 0 then Fa is a typical fibre, whereas the converse is not true in general. However, one has the following interesting statement, which gives an extension, at the Euler characteristics level, of S i e r ~ m a - T i b k ~ ~ , Corollary 3.5(a):

538

Proposition 4.2. Iff : C" then

-+

X(E) = 1

C is a polynomial with isolated singularities

+ (-1)"-l C(X, +/%I, ach

where A is the set of atypical values off, and 'u. E A.

Proof. Let 6, be a small disc centered at a E C, which does not contain any other atypical value. Let "la, a E A, be suitable paths (non self-intersecting, etc.) from a to a typical value u.Then 1 = x(Cn) = ~ ( u , , ~ ( f - ~ ( S , )U f-l(Ta)) = X(&) C a E ~ [ X ( F a-) x(FU)], by retraction and by a standard Mayer-Vietoris argument. 0

+

5. T h e c a s e n = 2

5.1. The complex setting Let us first remark that any polynomial function f : C2 --f C has isolated t-singularities at its reduced fibres. Not only that Proposition 4.1 applies in this case, but one has further relations. See Remark 5.2, and SiersmaT i b 5 1 - ~P~a,r u ~ i h s k i Tib51-'~ ~~, for other results.

Proposition 5.1. Let f : C2 -+ C be a polynomial. For to E C, the following are equivalent: (a) f is topologically trivial at infinity at to. (b) f-'(to) is pE-regular at infinity. (c) f-l(to) is t-regular at infinity. (d) f is tame at to (i.e. there is no sequence of points xi E C2 with property (L1) and such that 1 grad f(xi)l -+0 as i 00.) -+

Proof. The proof of (a) =$ (c) runs as follows. By absurd, suppose that f-l(to) is not t-regular at infinity. Its t-singularities are isolated, as remarked above, and we may apply Proposition 4.1 together with Corollary 3.5 in S i e r ~ m a - T i b g to r~~ deduce that there is a nonzero Milnor number at infinity X ~ - I ( ~ which ~) gives a jump "at infinity" in the Euler characteristic of the fibres at f-'(to). Therefore f is not topologically trivial at infinity at to (cf. Definition 3.1). We get a contradiction. Now (c) + (b) is Proposition 3.2 and (b) + (a) is Proposition 3.1 (both valid for n >_ 2). The equivalence (a) H (d) was proved by H&H.V.ll, see also Durfee'. The implication (d) + (a) is an easy exercise. 0

539

Note. The above result shows that, given a polynomial function f : C2 ---$ C and some point p E Xm, if the Lojasiewicz number L , ( f ) is 2 0 then L , ( f ) 2 -1. This is not anymore true for n 2 3 (see Example 3.2). Moreover, in the following example f (z, y , z ) = z z 2 y z 4 y z taken from Parisiriski3’, f is not tame at t o , for any t o E C, whereas all except a finite number of fibres are pE-regular at infinity (cf. Corollary 3.1).

+

+

Remark 5.1. If we consider the “tameness” in the sense of Broughton3 (i.e. 3K > 0 such that, for any sequence (xi(-+ 00, one has 1 grad f (xi)(2 K ) then “f is tame” obviously implies “f is tame at t , for any t E C ” , whereas the reciprocal is not true, see Nr5methi-Zaharia26for examples. 5.2. The real setting Our aim is to characterise those atypical values which are not critical values, in the real case K = R.The exposition follows the paper48, where we refer the reader for more details.

Theorem 5.1. 48 Let { X t ) t E R be an algebraic family of real curves and let t o be a regular value of T : X -+ R. Assume that the total space X is nonsingular. Then the curve X t , is typical i f and only i f the Euler characteristic x ( X t ) is constant when t varies within some neighbourhood of t o and there is no component of X t which vanishes at infinity as t tends to t o . The following simple example shows why we need X be nonsingular.

+

Example 5.1. Take X := {z2 y2 - z2 = 0) U { z = 0) c R3, the union of a cone and a plane through the vertex of the cone. Take T the projection on a line L through the origin. Then, for an adequate choice of L , the curve X t , for t # 0, is the disjoint union of a line with an oval, but X O is just a line. When t tends to 0 , the oval is “vanishing” in the origin, nevertheless the Euler characteristic is constant. Remark 5.2. The criterion in our Theorem 5.1 looks natural, since the constancy of the Euler characteristic and the non-vanishing are necessary conditions. Moreover, it has a striking similarity to certain criteria in the complex case, as we explain in the following. For a family X t = f-’(t) given by the fibres of a complex polynomial function f : C2 -+ C, it has been proved by H B H.V. and L8 D.T.12 that: A reduced fibre X t of a two variable complex polynomial f is typical if and

540

only if its Euler characteristic x ( X t ) i s equal to the Euler characteristic of a general fibre o f f . This gives the answer (within the considered class of families) to the problem stated above, i.e. a criterion for a fibre f - ’ ( t ) to be atypical, since it is known that a critical fibre is atypical (by a monodromy argument due to L6 D.T.). An equivalent form of the H&Le criterion is the following, see S i e r ~ m a - T i b :k ~ ~

A regular fibre X , of a complex polynomial function is typical i f and only i f there are n o vanishing cycles at infinity corresponding to this fibre. As we see, a common idea of “non-vanishing” appears in both real and complex case. In contrast to the complex, in the real case the two conditions (i.e. constancy of Euler characteristic, respectively non-vanishing condition) have to be considered together: neither of them implies that Xt is atypical. We show this by Example 5.2 (constancy of Euler characteristic holds but non-vanishing condition fails), respectively Example 5.3 (non-vanishing and “non-splitting” condition both hold but constancy of Euler characteristic fails). We consider polynomials f : R2 + R of the following type

f (Xt.,Y):= 4 Y ) X 2 + V I

:=I

( Y b+ Y ( Y ) .

Let A := { y E R I a ( y ) = 0). We assume that - E , E [ contains only regular values of f and that

(5) E

> 0 is such that

for any y E A we have: p ( y ) = 0 and Ir(y)\ 2 E .

(6)

Then for any t E I , the equation f = t in the variable x has two complex . A ( y , t ) = P 2 ( y ) - a ( y ) ~ ( y ) + t c r ( y )and let us denote solutions z ~ , z ( y , t )Let

D

:= { ( y , t ) E

R2 1 A ( y , t ) 3 0) , IC

L ( s ) := { ( y , t ) E R2 I t

=s}

:= { ( y , t ) E

R2 1 A ( y , t ) = 0)

and A := { ( y , t ) E R2 I y E A }

,

.

Then x1,2(y,t) are real numbers if and only if ( y , t ) E D.It is easy to see that if ( y , t ) E D and y tends to a point in A , then Ixl,z(y,t)l tends to dA infinity. Note also that A IC and that IC\A C d D because - = a ( y ) # 0 at for y # A.

541

For t o E I , the topology of the fibre f-'(to) projections

{(x,y,t) E It3 I f(x,y) = t ) J R2

-+

R

7

can be described using the

b,Y,t>

2 - b

(Y,t> t .

More precisely, the connected components of the sets F(to) := D n L(to) and F(t0)\ A are segments and isolated points. By (2), if P is an isolated point of F(t0) such that P E A, then 7r-'(P) = 0. Moreover, if Q is an isolated point of F(t0) such that Q @ A, then 7r-'(Q) is an isolated point of f - l ( t o ) , hence a critical point of f ;but to is a regular value of f . Now, consider the one-dimensional connected components of F(t0)\ A. Let .7 be such a segment, let 7be its closure in [-m, m] x R and let n ( J ) be the number of endpoints of 7 which are contained in K: \ A. If (y, t o ) E .7 \ 8.7, then 7r-' (y, t o ) consists of two distinct points. Now assume that (y, t o ) E .7\ 8.7 tends to an endpoint Q of J . There are three possibilities. If Q E K: \ A, then the two points in 7r-'(y,to) tend to the (unique!) point 7r-' (Q). If Q E A, then the two distinct points in 7r-'(y, t o ) tend to infinity, because their x-coordinate will be unbounded. If Q = ( f c o , t ~ )then , the two distinct points in x-'(y,to) tend to infinity, because their y-coordinate will be unbounded. These imply the following. (i) If n ( J ) = 2, then 7r-'(y)is diffeomorphic to a circle. (ii) If n ( J )= 1, then 7r-'(7)is diffeomorphic to a line. (iii) If n ( 3 ) = 0, then ~'(7) is diffeomorphic to a disjoint union of two lines. Thus, for t E I , we can read the topology of the fibre f - ' ( t ) from the pictures of D,K: and A. Moreover, using our main result, we can decide if 0 is an atypical value of f or not. Our first three examples use these considerations. We leave the details to the reader. Example 5.2. The polynomial

f (z, y)

2 3

2

+2xy(y2- 25)(y+25) - (y4+y3 -50y2-51y +575)

:= x y (y - 25)2

has the property that 0 is an atypical value, but the Betti numbers of the fibres f-'(t) are constant, for It[small enough. Namely, all these fibers have 5 non-compact connected components. For this polynomial, 0 is a regular value and condition (6) is satisfied. Besides the lines in A, the set K: contains

542

-

also the graph of the function Y

cp(Y) :=

+

-(y2 - 25)2(y 1) Y

It is easily seen that this graph has two connected components, separated by the vertical asymptote {y = 0). The set 23 consists of the lines in A and the region of the plane situated between the two connected components of the graph of cp. The only local extrema of the function cp are two local maximums, for y = f 5 , and a local minimum, between -5 and -1. For It1 sufficiently small, the equation cp(y) = t has five (complex) solutions, say a j ( t ) , j = 1,.. . ,5. One of these solutions, say a3(t), is a real one, for all t , while the other four are real if and only if t 5 0. Assume that lim a l ( t ) = lim a2(t) = -5 and lim a 4 ( t ) = lim a5(t) = 5 . t+O

t+O

t+O

t+0

For It1 sufficiently small and t < 0, the set F ( t ) \ A has 5 connected components and each of them corresponds to a line component in f-’(t). Namely, we have:

F ( t )\ A = ( [ a l ( t ) 7

-5) x { t ) )U ((-5,a2(t)l x { t ) )U

We also have:

F‘(0)\ A = ([-170) x (0)) u ((075) x (01) u ((5,001 x (01)

’

Therefore, when t < 0 tends to 0, the line components in f - l ( t ) corresponding to the segments ([al(t),-5) x { t } )U ((-5,az(t)] x { t } )will “vanish” at infinity since limt+oal(t) = limt+oaz(t) = -5 E A. Also, each of the line components in f-’(t) corresponding to the segments ( ( 0 7 a 4 ( t )x] { t } )u ([a5(t),00) x { t } ) will “split” in two line components for t = 0 since limt-,o a4(t) = limt+o a 5 ( t ) = 5 E A. For It1 sufficiently small and t 2 0, the set F ( t )\ A has 3 connected components: one corresponds to a line component in f-’(t) and each of the other two corresponds to two line components in f-’(t). Namely, we have:

F(t)\ A = ([as(t),O)x { t } )U ((075) x

{tl) U ((5700) x

{t}) .

Thus, for It1 sufficiently small, f-l(t) is a disjoint union of 5 line components. This means that the Betti numbers of f-’(t) do not depend on t , if It1 is sufficiently small.

543

On the other hand, for E > 0 sufficiently small, the restrictions f : 4 (--E,O) and f : f - l [ O , ~ ) [O,E) are easily seen to be Cm trivial fibrations, while f : f - l ( - ~ , ~-+) ( - E , C ) is not a topological fibration. f-l(-~,O)

+

+

Example 5.3. The polynomial f(z,y) := z2y2 2xy (y2 - 1)2 has the property that the conditions 'hen-vanishing" and "non-splitting" are satisfied at 0, but 0 is an atypical value. Besides the line in A, the set K: contains also the graph of the function cp(y) := y4 - 2y2. This function has a local maximum, for y = 0, and two local minimums, for y = f l . The set V consists of the line in A and the region of the plane situated above the graph of cp. For t < 0 with Jtl sufficiently small, the curve f-'(t) has two circle components. For t 2 0 sufficiently small, the curve f - l ( t ) has two line components. Example 5.4. The polynomial

f(x,y) := z2y3(9-y2)2

+2 4 9 -y2)(y3

+y+6) +2(y5 - 6y3 +6y2 + 2 5 ~ + 6 )

has the property that the conditions non-splitting at 0 and constancy of the Euler characteristic are satisfied, but 0 is an atypical value. Besides the lines in A, the set K: contains also the graph of the function

This graph has two connected components, separated by the vertical asymptote {y = 0). For It1 sufficiently small, the equation cp(y) = t has six real solutions. There exists a ~ ] 1 , 2 and [ b E]2,3[ such that the local maxima of cp are -b and a, and the local minima of cp are -a and b. The set V consists of the lines in A and the region of the plane situated between the two connected components of the graph of cp. For It1 # 0 sufficiently small, the curve f-' ( t )has a circle component and 4 line components. The curve f-l(O) has only 4 line components. The next example shows a polynomial with line-fibres having a big perturbation (in the shape of an "S') which disappears to infinity as the value of t tends to zero. This phenomenon implies that the polar curve r(x0,f) is non empty at the fibre f - ' ( O ) .

Example 5.5. Let f : R2 -+ R be defined by f ( z , y ) := 2z2y3-9zy2+12y. Then f is a trivial C" fibration because the map F : R2 + R2,

544

is a diffeomorphism of order two and F-' = F . The polynomial h : R2 -+ R, defined by h(z,y ) := f (z y , y ) = 2z2y3 4xy4 - 9 z y 2 2y5 - 9y3 12y shows that the non-emptiness of the polar curve I'(z0, f ) at f - ' ( O ) does not imply that the value 0 is atypical. This is only true in the complex setting, as we have seen in Proposition 4.1.

+

+

+

+

6. Families of complex polynomials with singularities at infinity Two polynomial functions f , g : Cn + C are said to be topologically equivalent if there exist homeomorphisms : C" -+ C" and Q : C -+ C such that Q o f = g o a. A challenging natural question is: under what conditions this topological equivalence is controlled by numerical invariants? We consider a one-parameter family of polynomials f 3 ( x ) = P ( z , s ) , where P : C" x [0,1] -+ C is continuous in s and such that deg f 3 = d , for all s E [O, 11. We assume that the a f i n e singularities of f s are isolated, i.e. dimsing f 3 5 0 for all s, where Sing f 3 = {z E C" I gradf,(z) = 0). The set of a f i n e critical values of f3 is a finite set and we denote it by B a ~ ( s=) { t E C I p t ( s ) > 0}, where pt(s) is the sum of the local Milnor numbers at the singular points of the fibre f;'(t); remark that we also have B a ~ ( s=) f3(Sing f 3 ) . The total Milnor number is p(s) = CtEhfl pt(s). If our f and g are topologically equivalent then clearly their corresponding fibres (general or atypical) are homeomorphic. In particular the Euler characteristics of the general fibres of f and g and the numbers of atypical values of f and g coincide respectively. Let us denote by G(s) the general fibre, and by B ( s ) the atypical set of the polynomial f 3 .

Theorem 6.1. Let (fs)sEIO,l~ be a continuous family of complex polynomials of F-type in n # 3 variables. If the Euler characteristic x(G(s)),the number of atypical values # B ( s ) and the degree deg f 3 are independent of s E [0,1],then the polynomials f o and f 1 are topologically equivalent. In case of a smooth family of germs of holomorphic functions with isolated singularity gs : (Cn,O) -+ (C,O), a famous result by L6 D.T. and C.P. Ramanujam'O says that the constancy of the local Milnor number (equivalently, of the Euler characteristic of the general fibre in the local Milnor fibrationZ4)implies that the hypersurface germs go1 (0) and gT'(0) have the same topological type whenever n # 3. J.G. Timourian4' and

545

H. King" showed moreover the topological triviality of the family of function germs over a small enough interval. The techniques which are by now available for proving the LB-Ramanujam-Timourian-King theorem do not work beyond the case of isolated singularities. In other words, the topological equivalence problem is still unsolved for local non-isolated singularities. The global setting poses new problems since one has to deal in the same time with several singular points and atypical values. Singularities at infinity introduce an essential difficulty since they are of a different type than the critical points of holomorphic germs. Let us remark that Theorem 6.1 only requires the continuity of the family instead of the smoothness in L8-Ramanujamz0 and T i m ~ u r i a n ~ ~ . It is also worth to point out that the finiteness of topological types in a family does not hold for the equivalence up to diffeomorphisms, as already remarked by T. Fukuda. For example, the family f,(x,y) = xy(x-y)(x-sy) provides infinitely many classes for this equivalence, because of the variation of the cross-ratio of the 4 lines. Example 6.1. Let f,(x,y,z,w) = ~ 2 y 2 + z 2 + w 2 + x y + ( 1 + s ) x 2 + x .For s E C \ {-2, -1) we have B ( s ) = ( 0 , p(s) = 2 and X(s) = 1. It follows that x(G,)= -2 and # B ( s ) = 3. Since f, is not of F-type, one has to consider fi(x,y) := x2y2 zy (1 s ) x 2 x. We then have fs = fj @ ( z 2 w2).By applying Theorem 6.1 to the family f: and then coming back to f,, we get that fo is topologically equivalent to fs if and only if s E C \ {-2, --I}, since #B(-l) = #B(-2) = 2.

-a, -as},

+ + +

+

+

7. Singularity exchange at infinity

Let P be a deformation of fo, i.e. P : Cn x C k + C is a family of polynomial functions P ( z , s ) = f,(x) such that fo = f . We assume that degf, is independent on s, for s in some neighbourhood of 0 , and we denote it by d. We attach to P the following hypersurface:

Y = {(I.:

ZO],S,t)

E

EDn

x

C k x C I P(x,xo,s) -tx;

=O},

where P denotes the homogenized of P by the variable 20,considering s as parameter varying in a small neighbourhood of 0 E Ck.The method we use impose us to assume that the deformation depend holomorphically on the parameter s E Ck,however in certain results one only needs the continuity. The projection r : Y + C to the t-coordinate extends the map P in the sense that Cnx C kis embedded in Y (via the graph of P ) and rlcnx @ k = P. Let u : Y --f C k denote the projection to the s-coordinates.

546

Notations. Y,,* := Y n n-'(s), Y,,t := Y n T - ' ( t ) and Y,,t := Y,,* fl T - ' ( t ) = Y*,t n n-'($). Note that Y,,t is the closure in P" of the affine hypersurface f;'(t) c C". Let Y" := Y f l (20= 0) = {Pd(x, s) = 0 ) x C be the hyperplane at infinity of Y, where Pd is the degree d homogeneous part of P in variables 1c E Cn. Remark that for any fixed s, YTt := f l YOo does not depend on t . For a single polynomial f , we shall use the notation r : X -+C; in this case r is a proper extension of f . 7.1. Local semi-continuity 7.1.1. Deformations in general It is well known that the ( n - 1)th Betti number of the Milnor fibre of a holomorphic function germ is upper semi-continuous, i.e. it does not decrease under specialisation. In case of a polynomial f, : C" 4 C, the role of the Milnor fibre is played by the general fibre G, of f,. This is a Stein manifold of dimension n - 1 and therefore it has the homotopy type of a CW-complex of dimension 5 n - 1, which is also finite, since G, is algebraic. Moreover, the ( n - 1)th homology group with integer coefficients is free. We have the following general specialisation result, which contrasts to the one in the local case.

Proposition 7.1. 37 Let P : C" x C k 4 C be any holomorphic deformation of a polynomial fo := P(.,0 ) : C" 4 C. Then the general fibre Go of fo can be naturally embedded into the general fibre G, of f,, for s # 0 close enough Hn-l(GS) to 0. The embedding Go c G, induces an inclusion Hn-I(Go) which is compatible with the intersection form. L-)

Proposition 7.1 will actually be exploited through the following semicontinuity of highest Betti number, as a consequence of the inclusion of homology groups:

bn-l(G,)

2 b,-l(Go), for s # 0 close enough to 0.

(7)

7.1.2. Semi-continuity at infinity Let P be a deformation of f o such that f, is of W-type, for s # 0. From 52 we know that the reduced homology of the general fiber G, is concentrated in dimension n- 1 and the vanishing cycles of f , are quantified

547

by two well defined, non-negative integers: p(s) = the total Milnor number, respectively A(s) = the total Milnor-L2 number at infinity. It is shown in $2 that, for a W-type polynomial f , , the general fiber G, is a bouquet of n - 1 spheres and that bn-I(Gs) = p(s) A(s). It is shown in 52 that the vanishing cycles are localized at certain points, either in the aEne space or at infinity, and which we shall call p-singularities and A-singularities respectively. To such a singular point p E Ys,* one associates its local Milnor number denoted p p( s ) or its Milnor-LG number A, ( s ) . As we have seen in $2, the atypical fibers of a W-type polynomial f s are exactly those fibers which contain p or A-singularities; equivalently, those of which the Euler characteristic is different from x(G,). We denote by A( f , ) the set of atypical values of f,. Let us assume that f o is itself a W-type polynomial. Then the above cited facts together with our semi-continuity result (7) show that, for s close to 0 we have:

+

P(S)

+ A(s)

2 PU(0)+ Y O ) .

Remark 7.1. The total Milnor number p(s) is lower semi-continuous under specialization s -+ 0. In case p(s) decreases, we say that there is loss of p at infinity, since this may only happen when one of the two following phenomena occur: (a) the modulus of some critical point tends to infinity and the corresponding critical value is bounded ( S i e r ~ m a - T i b kExample ~~, 8.1); (b) the modulus of some critical value tends to infinity ( S i e r ~ m a - T i b k ~ ~ , Examples (8.2) and (8.3)). In contrast to p(s), it turns out that X(s) is not semi-continuous; under specialization, it can increase or decrease (Example 7.1, 7.3). Moreover, the A-values may behave like the critical values in case (b) above, see Example 7.6. In the B-class one has the following more precise result. Theorem 7.1. (Lower semi-continuity at A - ~ i n g u l a r i t i e s ) ~ ~ Let P be a constant degree one-parameter deformation inside the B-class. Then, locally at any A-singularity p E Uo,t of f o , we have:

I

c

APi (s)

i

+

c

PP3

(s),

j

where pi are the A-singularities and p j are the p-singularities of f , which tend to the point p as s + 0. 0

548

7.2. Persistence of A-singularities In order to get further information on the p H A exchanges described by the formula in Theorem 7.1, we focus on two sub-classes of the B-class. In this section we define cgst-type deformations and in the next section we study deformation with constant p A. Let us first remark that for a deformation { f s } s inside the B-class the compactified fibres of f, have only isolated singularities. When s + 0 these singularities can split or disappear. Let x(0) E Co. Take t 6R(f0) and assume without dropping generality that t $! A(fs) for all small enough s. The non-splitting argument from A’Campo’ and LB1’ tells us that the Milnor number of Uo,t at (x(O),t ) is larger than the sum of the Milnor numbers of Y,,t at all points ( x ( s ) t, ) E C, x { t } such that xi(s) -+ x(O),unless there is only one such singular point x ( s ) and the Milnor number of at ( x ( s ) t, ) is independent on s. In the latter case we say that the cgst assumption holds.

+

Definition 7.1. We say that a constant degree deformation inside the Bclass has constant generic singularity type at infinity at some point x ( 0 ) E COif the cgst assumption holds. If the cgst assumption holds at all points in CO,then we simply say constant generic singularity type at infinity. Note that in the B-class the cgst assumption does not imply that bn-l(Gs) is constant, see Example 7.6. We send t o Remark 7.3 for further comments on cgst.

Theorem 7.2. 37 Let P be a constant degree deformation, inside the Bclass, with constant generic singularity type at infinity. Then:

(a) A-singularities of fo are locally persistent in f,. (b) a A-singularity of fo cannot split such that two or more A-singularities belong to the same fiber.

Remark 7.2. Part (a) means that a A-singularities cannot completely disappear when deforming fo, but of course it may split or its type may change, see 57.4. The case which is not covered by part (b) can indeed occur, i.e. that some A splits into A’s along a line { x ( s ) }x C, see Example 7.2.

549

7.3. Conservation in p -l-X constant deformations

In 57.4 we comment a couple of examples where the inequality of Theorem 7.1 is strict. Under certain conditions the equality holds, like in the following.

Corollary 7.1. 37 Let P be a constant degree deformation inside the Bclass such that p(s) + X(s) is constant. Then: (a) As s -+ 0, there cannot be loss of p or of X with corresponding atypical values tending to infinity. (b) X is upper semi-continuous, i.e. X(s) 5 X(0). (c) there is local conservation of p X at any A-singularity of f o .

+

Proof. (a). If there is loss of p or of A, then this must necessarily be compensated by increase of X at some singularity at infinity of f o . But Theorem 7.1 shows that the local p X cannot increase in the limit. (b). is clear since p(s) f X(s) is constant and p(s) can only decrease when s -+ 0. (c). Global conservation of p X together with local semi-continuity (by Theorem 7.1) imply local conservation. 0

+

+

Remark 7.3. It is interesting to point out that within the class of B-type polynomials there is no inclusion relation between the properties “constant generic singularity type” and “p(s) A(.) constant”, see Examples 7.4, 7.6. We shall see in the following that in the F-class the two conditions are equivalent because of the relation (9) below.

+

7.3.1. Rigidity in deformations with constant p

+X

For B-type polynomials, one has the formula: b n - l ( ~ s )= P ( ~ ) + x ( s )=

(-l)n-l(~nld-l)-

C

p~,gen(S)-(-l)n-lxoo(S),

XEC,

(8) where x ~ =’ x(Vgne;,d) ~ = n 1( - l ) n ( d - l)n+l} is the Euler characteristic of the smooth hypersurface Vsn,l$ of degree d in P and xoo(s) := x({fd(lc, s) = 0)). We denote by pz,gen(s)the Milnor number of the singularity of U,,t at the point ( z , t ) E C, x C, for a generic value of t. The change in bn-l(Gs) can be described in terms of change in

+

i{l+

550

and ~ " ( s ) . Since the latter is not necessarily semi-continuous (cf. Examples 7.4-7.6), we may expect interesting exchange of data between the two types of contributions.

Proposition 7.2.

37

Let Ax" denote (-l)n(X"(s)

-~"(0))

(a) If Ax" < 0 then the deformation is not cgst. (b) l f Ax" = 0 and the deformation has constant p+X then, for all x E C,, is constant. (c) If Ax" > 0 then the deformation cannot have constant p + A.

0 For F-type polynomials, formula (8) takes the following form:

where p F ( s ) denotes the Milnor number of the singularity of Y,,t n H" at the point ( x , t ) E W, x C, which is actually independent on the value o f t . Note that in the F-class we have Ax" 2 0. The relation (9) shows that the change in the Betti number bn-l(G,) can be described in terms of change in the px,gen(~) and change in p r ( s ) . Both are semi-continuous, so they are forced to be constant in p+X constant families. Consequently, the class of F-type polynomials such that p+X is constant verifies the hypotheses of Theorem 7.2. It has been noticed by the first named author that in the deformations with constant p X which occur in Siersma-Smeltink's lists32 the value of X cannot be dropped to 0. Since these deformations are in the F-class and in view of the above observation, this behaviour is now completely explained by Theorem 7.2(a). More precisely, we have proved:

+

Corollary 7.2. 37 Inside the F-class, a A-singularity cannot be deformed into only p-singularities by a constant degree deformation with constant

0

p++.

7.4. Examples 7.4.1. F-class examples; behaviour of X

+

+

Example 7.1. f, = ( x Y ) ~s x y x , see Figure l(a). This is a deformation inside the F-class, with constant p A, where X increases. For s # 0: X = 1 + 1 and p = 1. For s = 0: X = 3 and p = 0.

+

551

+

+

Example 7.2. fs = ( ~ y ) ~~ ( z y ) ~z, see Figure l(b). This deformation has constant p = 0, X(0) = 2 at one point and X(s) = 1 1 at two points at infinity which differ by the value of t only, namely ([0: l],s,O) and ( [ 0 : l],s,-s2/4).

+

+

+

Example 7.3. fs = zy4 ~ ( z y ) y, ~ see Figure l(c). Here X decreases. For s # 0: X = 2 and p = 5. For s = 0: X = 1 and p = 0.

(a)

A increases

@)

A is constant

(c)

A decreases

Fig. 1. Mixed splitting in (a) and (c); pure A-splitting in (b).

7.4.2. B-class examples We use in this section formula (8). We pay special attention to the sign of Axoo and illustrate the difference between cgst-type deformations and ( p + A)-constant deformations.

+

+ +

Example 7.4. fs = z4 sz4 z3 y. This is a deformation inside the B-class with constant p A, which is not cgst at infinity (Definition 7.1). We have X = p = 0 for all s. Next, Y,,t is singular only at p := ([0 : 1 : O],O) and the singularities of YFt change from a single smooth line {z4 = 0) with a special point p on it into the isolated point p which is a I?$ singularity of Y r t . We use the notation @ for the Thom-Sebastiani sum of two types of singularities in separate variables. We have: s = 0: the generic type of Yo,t at infinity is A3 @ E7 with Milnor number 21, and x(Yct) = 2. s # 0: the generic type of Y,,t at infinity is A3 @ EG with Milnor number 18, and x(Yrt) = 5. The jumps of +3 and -3 compensate each other in the formula (8).

+

552

+ + +

Example 7.5. fs = z4 sz4 z2y z. This is a p X constant B-type family, with two different singular points of Uo,t at infinity, and where the change in one point interacts with the other. It is locally cgst in one point, but not in the other. We have that X = 3 and p = 0 for all s, YS,t is singular in p := ([0 : 1 : O ] , O ) for all s (see types below) and in q := ([l : 0 : O],O) for s = 0 with type As. The singularities of Y r 5 change from a single smooth line {z4 = 0) into the isolated point p with E7 singularity. For the point p we have for all s the generic type A3 @ 0 5 if t # 0, which jumps to A3 @ 0 6 if t = 0. This causes X = 3. At q , the A3-singularity for s = 0 gets smoothed (independently of t ) and here the deformation is not locally cgst. The change on the level of x(UTt) is from 2 t o 5, so Ax" = -3, which compensates the disappearance of the As-singularity from Uo,t to

+

+ + +

Example 7.6. fS = x2y z z2 sz3. This is a cgst B-type family, where p X is not constant. Notice that fS is F-type for all s # 0, whereas fo is not F-type (but still B-type). The generic type at infinity is 0 4 for all s and there is a jump 0 4 ---f D5 for t = 0 and all s. For s # 0 a second jump 0 4 -+ 0 5 occurs for t = c/s2, for some constant c. There are no affine critical points, i.e. p ( s ) = 0 for all s, but X(s) = 2 if s # 0 and X(0) = 1. We have that A(fs) = (0, c / s 2 } for all s # 0, and that xDochanges from 3 if s = 0 t o 2 if s # 0, so Ax" = +l. There is a persistent A-singularity in the fibre over t = 0.

+

References 1. N. A'Campo, Le nombre de Lefschetz d'une monodromie, Indag. Math. 35 (1973), 113-118. 2. S.A. Broughton, On the topology of polynomial hypersurfaces, in Proceedings A.M.S. Symp. in Pure. Math., vol. 40, I (1983), 165-178. 3. S.A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math. 92, 2 (1988), 217-241. 4. A. Bodin, M. Tibk, Topological triviality of families of complex polynomials, Adv. Math. 199,1 (2006), 136-150. 5. A. Dimca, L. Phnescu, On the connectivity of complex affine hypersurfaces, Topology 39 (2000), no. 5, 1035-1043. 6. A.H. Durfee, Five definitions of critical point at infinity, in Singularities (Oberwolfach, 1996), 345-360, Progr. Math. 162, Birkhuser, Basel, 1998. 7. M.V. Fedoryuk, The asymptotics of the Fourier transform of the exponential

553

8. 9. 10. 11.

12.

13. 14.

15.

16. 17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

function of a polynomial, Docl. Acad. Nauk 227 (1976), 580-583; Soviet Math. Dokl. (2) 17 (1976), 486-490. L. Fourrier, Classification topologique B l’infini des polyndmes de deux variables complexes, C.R. Math. Acad. Sci. Paris, t. 318, I (1994), 461-466. L. Fourrier, Topologie d’un polyndme de deux variables complexes au voisinage de l’infini, Ann. Inst. Fourier, Grenoble 46,3 (1996), 645-687. M. Goresky, R. MacPherson, Stratified Morse Theory, Springer-Verlag Berlin Heidelberg New-York, 1987. HB H.V., Nombre de Lojasiewicz et singularites B l’infini des polyndmes de deux variables complexes, C.R. Math. Acad. Sci. Paris t. 311, I (1990), 429-432. HA H.V., L6 D.T., Sur la topologie des polyndmes complexes, Acta Math. Vietnamica 9 (1984), pp. 21-32. H. Hamm, On the cohomology of fibres of polynomial maps, in Trends in singularities, 99-1 13, Trends Math., Birkhuser, Basel, 2002. H. Hamm, L6 D.T., Rectified homotopical depth and Grothendieck conjectures, in . Cartier et all. (eds) Grothendieck Festschrift 11, pp. 311-351, Birkhauser 1991. J.P. Henry, M. Merle, C. Sabbah, Sur la condition de Thom stricte pour un morphisme analytique complexe, Ann. Scient. Ec. Norm. Sup. 4e sBrie, t. 17 (1984), 227-268. H.C. King, Topological type in families of germs, Invent. Math. 62 (1980/81), 1-13. H. Kraft, Challenging problems on affine n-space, Siminaire Bourbaki, Vol. 1994/95. AstBrisque, 237 (1996), Exp. No. 802, 5, 295-317. LG D.T., Some remarks on the relative monodromy, in Real and Complex Singularities, Oslo 1976, Sijhoff en Noordhoff, Alphen a.d. Rijn 1977, pp. 397-403. LG D.T., Complex analytic functions with isolated singularities, J. Algebraic Geometry 1(1992), pp. 83-100. L6 D.T., C.P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), 67-78. L6 D.T., C. Weber, PolynBmes B fibres rationelles et conjecture Jacobienne B 2 variables, C.R. Math. Acad. Sci. Paris, t. 320, s6rie I (1995), pp. 581-584. A. Libgober, M. Tibiir, Homotopy groups of complements and nonisolated singularities, Int. Math. Res. Not. 2002,no. 17, 871-888. J. Mather, Notes on topological stability, Harvard University (1979). J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Studies 61,Princeton 1968. A. NBmethi, ThBorie de Lefschetz pour les variBtBs algebriques affines, C.R. Acad. Sci. Paris S i r . I Math., 303,no. 12 (1986), 567-570. A. NBmethi, A. Zaharia, On the bifurcation set of a polynomial and Newton boundary, Publ. Res. Inst. Math. Sci. 26 (1990), 681-689. W. D. Neumann, P. Norbury, Vanishing cycles and monodromy of complex polynomials, Duke Math. J. 1011,no. 3 (2000), 487-497. W. D. Neumann, P. Norbury, Unfolding polynomial maps at infinity, Math.

554

Ann. 318,no. 1 (2000), 149-180. 29. A. Parusiriski, On the bifurcation set of a complex polynomial with isolated singularities at infinity, Compositio Math. 97 (1995), 369-384. 30. A. Parusiriski, A note on singularities at infinity of complex polynomials, in Simplectic singularities and geometry of gauge fields, Banach Center Publ. V O ~ .39 (1997), 131-141. 31. F. Pham, Vanishing homologies and the n variable saddlepoint method, in Arcata Proc. of Symp. in Pure Math., vol. 40, I1 (1983), 319-333. 32. D. Siersma, J. Smeltink, Classification of singularities at infinity of polynomials of degree 4 in two variables, Georgian Math. J. 7 (2000), no. 1, 179-190. 33. D. Siersma, M. Tibgr, Singularities at infinity and their vanishing cycles, Duke Math. J . 80,3 (1995), 771-783. 34. D. Siersma, M. Tibir, Singularities at infinity and their vanishing cycles, 11. Monodromy, Publ. Res. Inst. Math. Sci. 36 (2000), 659-679. 35. D. Siersma, M. T i b k , Deformations of polynomials, boundary singularities and monodromy, MOSC.Math. J. 3 (2) (2003), 661-679. 36. D. Siersma, M. T i b k , On the vanishing cycles of a meromorphic function on the complement of its poles, in Real and complex singularities, 277-289, Contemp. Math., 354, Amer. Math. SOC.,Providence, RI, 2004. 37. D. Siersma, M. T i b k , Singularity exchange at the frontier of the space, math.AG/0401396. 38. R. Switzer, Algebraic Topology - Homotopy and Homology, Springer Verlag, Berlin-Heidelberg-New York 1975. 39. B. Teissier, Cycles Bvanescents, sections planes et conditions de Whitney, in Singularit& ci Cargesse, AstLrisque, 7-8 (1973). 40. B. Teissier, VarietBs polaires 2: MultiplicitBs polaires, sections planes et conditions de Whitney, in Ge'ome'trie Algbbrique ci la Rabida, Springer Lecture Notes in Math., 961 (1981), pp. 314-491. 41. R. Thom, Ensembles et morphismes stratifibs, Bull. Amer. Math. SOC.75 (1969), 249-312. 42. M. T i b k , Bouquet decomposition of the Milnor fibre, Topology 35, no.1 (1996), 227-242. 43. M. T i b k , On the monodromy fibration of polynomial functions with singularities at infinity, C. R. Acad. Sci. Paris SLr. I Math. 324 (1997), no. 9, 1031-1035. 44. M. T i b k , Topology at infinity of polynomial maps and Thom regularity condition, Compositio Math., 111, 1 (1998), 89-109. 45. M. T i b k , Regularity at infinity of real and complex polynomial functions, in Singularity theory (Liverpool, 1996), xx, 249-264, London Math. SOC. Lecture Note Ser. 263, Cambridge Univ. Press, Cambridge, 1999. 46. M. T i b k , Asymptotic Equisingularity and Topology of Complex Hypersurfaces, Int. Math. Research Notices, 18 (1998), 979-990. 47. M. T i b k , Polynomials and vanishing cycles, monograph in preparation. 48. M. T i b k , A. Zaharia, Asymptotic behaviour of families of real curves, Munwcripta Math. 99 (1999), no. 3, 383-393.

555 49. J.G. Timourian, The invariance of Milnor’s number implies topological triviality, Amer. J. Math. 99 (1977), 437-446. 50. J.-L. Verdier, Stratifications de Whitney et th6orBme de Bertini-Sard, Inwentiones Math. 36 (1976), 295-312.

556

Hodge Theory: the search for purity C.A.M. Peters

Department of Mathematics, University of Grenoble UMR 5582 CNRS- UJF, 38402-Saint-Martin d'Htres, fiance E-mail: [email protected] J.H.M. Steenbrink

Institute for Mathematics, Astrophysics and Particle Physics Radboud University, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands E-mail: j . steenbrinkomath. ru.nl These notes aim at providing a summary of mixed Hodge theory, starting with the origin of weights on the cohomology of algebraic varieties in etale cohomology and ending with the discussion of mixed Hodge modules. Big parts of the text are extracted from the book [25] in preparation.

Keywords: mixed Hodge structure, mixed Hodge module

1. Introduction

The second author spent the academic year 1974-1975 at the Institut des Hautes Etudes Scientifiques in France. Many things were discussed there which occur in these notes. Goresky and MacPherson already had the basic ideas of intersection homology. With Shi-Wei-Shu we had a mini-seminar on D-modules, discussing the Bernstein polynomial. And of course, Deligne was there, the founder of mixed Hodge theory, and John Morgan, who extended this theory to homotopy groups. On hindsight one can say that all ingredients of mixed Hodge modules were present at the time. However, these ingredients seemed totally unrelated, and it required the joint effort of many people to achieve the complete picture of Hodge theory which we have now: Deligne, proving the Weil conjectures, and providing a sheaf-theoretic framework for intersection homology; Kashiwara and Mebkhout connecting D-modules and constructible sheaves, Malgrange who discovered how to formulate the nearby and vanishing cycle functors on the level of D-modules. But the great synthesis was

557

accomplished by Morihiko Saito in the years 1981-1988. His work is very complicated, but the results are very powerful. These notes contain these ingredients of mixed Hodge modules, starting with the notion of weights. Then pure and mixed Hodge structures are defined. We proceed with perverse complexes, deal with the foundations of D-module theory and formulate the Riemann-Hilbert correspondence. Finally an axiomatic treatment of mixed Hodge modules is given and some applications are discussed. We hope that these notes will be a useful introduction to this fascinating but until now rather inaccessible topic. We presuppose a knowledge of algebraic geometry including sheaf theory; in the last three sections knowledge of derived categories is very useful. See e.g. 1121. 2. Weights in t-adic cohomology

Let X O be an algebraic variety over a finite field k with q elements and let X = X O x k where 1is the algebraic closure of k. . The zeta function of X O is given by

Z ( X 0 ,t ) =

n

det(1 - F * t , H : ( X , Qe))(-l)i+l

i

where C is a prime not dividing q and F is the F'robenius morphism on X, given in coordinates by (21,. . . ,x,) H (x;,.. . ,x:). This definition involves the etale cohomology groups of X with values in Qe. However, the zeta function can also be obtained by point counting:

where a, is the number of points of Xo with values in IFqn. The equality of (1) and (2) is a consequence of the Lefschetz fixed point theorem in Ladic cohomology, applied to the action of Fn on X. Deligne showed [7, Thm. (141

Theorem 2.1. Suppose that X O is smooth projective. Then for each i, the characteristic polynomial det(t - F*, H : ( X , Qe)) has integer coefficients independent of l . The complex roots cx of this polynomial (the complex conjugates of the eigenvalues of F * ) have absolute value la1 = qiI2. This result, one of the Weil Conjectures, can be reformulated as: the i-th Gadic cohomology group of a smooth projective variety over a finite field is pure of weight i as a module over the Galois group of the ground field.

558

Next consider a complex algebraic variety X . It is defined by a finite number of polynomial equations. The coefficients of these polynomials generate a subalgebra R of @, of finite type over Z,and X is obtained from a scheme X over R by extension of scalars to C. Take a maximal ideal m of R, let k = R/m and let q = Ikl. Write X , = X X R k and let F, denote its Frobenius endomorphism.

Theorem 2.2 ( [8, Thm. 141). There exists non-zero f E R such that for all m with m c R - f , and all primes C 6m the eigenvalues of Fm o n Ha(X,,Qe) are algebraic integers, and for each eigenvalue a there exists an integer w ( a ) such that all complex conjugates of a have absolute value q v .

One has a natural identification of H i ( X , Q ) @ Qe with Hi(X,,Qe), which gives a rational structure to the latter vector space. If ,Wj denotes the sum of the generalized eigenspaces corresponding to the eigenvalues a with w ( a ) _< j , then the increasing filtration m W is defined over Q and its intersection W with H i ( X ,0 ) is independent of C and m. 3. Mixed Hodge theory

Let us again consider a complex algebraic variety. The filtration W on the cohomology of X obtained from Theorem 2.2 is Deligne’s weight filtration, and is one of the main ingredients for his theory of mixed Hodge structures on the cohomology of complex algebraic varieties. For the i-th cohomology group of a smooth projective variety X the weight filtration is trivial:

W i - l H i ( X ) = 0 , W i H i ( X )= H i ( X ) which we rephrase as: H i ( X ) is pure of weight i. The other ingredient of Deligne’s mixed Hodge theory is the Hodge filtration. This is not defined on the cohomology with rational coefficients, but one needs to pass to complex coefficients to see it. For a smooth projective variety we can choose a Kahler metric so that we are in the realm of compact Kahler manifolds. The complex cohomology of such a compact Kahler manifold admits a Hodge decomposition

P ( X , C )=

@ HP74(X)

(3)

p+q=i

where we consider the cohomology as de Rham cohomology, i.e. as the space of (complex valued) closed differential forms modulo the exact forms.

559

The Kahler metric provides us with an adjoint d* of the operator d , and the Laplacian b is defined as 6 = dd* d*d. A form w is called harmonic if bw = 0. Each cohomology class is represented by a unique harmonic form. On the other hand, complex valued differential forms on X admit a decomposition according to type (a form is of type ( p , q ) if in local holomorphic coordinates it is given by an expression containing p factors dzi and q factors d z j ) . A cohomology class is called of type ( p ,q ) if its harmonic representative is of type (p,q). The Kahler identities imply that a form is harmonic if and only if all of its (p,q)-components are, hence we have the Hodge decomposition. One can show that a cohomology class is of type (p,q) if and only if it can be represented by a closed form of type (p, q ) . Hence the Hodge decomposition does not depend on the choice of Kahler metric. The Hodge filtration on Hi(X,@) is defined in this case by

+

FpHi(X,@)

=

@ HT7i-T(X).

(4)

T>P

Note that it is a decreasing filtration and that one needs the de Rham complex (differential forms) to define it. It reflects the analytic structure in contrast with the weight filtration, which is rather reflecting the topology. Two reasons to work with the Hodge filtration rather than with the Hodge decomposition is that the Hodge filtration has better behaviour in families of varieties (it varies holomorphically) and that it has a good generalization to arbitrary complex algebraic varieties. The Hodge decomposition of the cohomology of a compact Kahler manifold gives rise to the following concept. Definition 3.1. A pure weight i (rational) Hodge structure on a finite dimensional rational vector space V consists of a direct sum decomposition Vc

=

@ VplQ,WithVPyQ

= I/g,p

p+q=i on its complexification Vc = V @ @. The numbers hpiQ(V):= dimVpSQ are the Hodge numbers of the Hodge structure. The polynomial Phn(V) =

c c

(5)

hp,q(V)UPvQ

P4EZ

=

hp-yV)upvi-p E

Z [ U , 21,

u-1,21-11

560

is its associated Hodge number polynomial. The corresponding Hodge filtmtion is given by

p ( V ) = @ vr+r. r2p

The classical example of a weight i Hodge structure is furnished by the rank i (singular) cohomology group H i ( X ) (with Q-coefficients) of a compact Kahler manifold X.

Definition 3.2. A polarization o n a pure Hodge structure V of weight k is a bilinear form S on V which is (-l)'-symmetric with the property that for all p , q with p q = i the hermitian form

+

(2,y

)

ip-qs(2,

g)

is positive definite on VPiQ, Various multilinear algebra operations can be applied to Hodge structures. Suppose that V and W are two real vector spaces with a Hodge structure of weight k and l respectively. Then: (1)

V

@I

W has a Hodge structure of weight k

FP(V €3 W ) , =

c

+ l given by

F"(Vc) €3 FP-"(Wc) c vc EJcWe

m

and with Hodge number polynomial given by

h n ( V €3 W )= h n ( v ) f i n ( W ) .

(6)

(2) On Hom(V, W ) we have a Hodge structure of weight l - k:

FPHom(V,W)c = {f : VC -+ We I fFn(V@)c F"+"(Wc) V n } with Hodge number polynomial Phn(Hom(V,W ) ) ( U ,u)= &(V)(u-l, V-l)Ptm(W)(% u).

(7)

In particular, taking W = Q with WC= W0>O we get a Hodge structure of weight -k on the dual V * of V with Hodge number polynomial Phn(V*)(U,u)= Phn(V)(U-', u-l).

(8)

The Hodge structure Q ( l ) of Tate is the Q-vector space (27ri)'Q C CC with Hodge structure of pure type (-l,-l).This Hodge structure can be used to reduce weights of a given pure Hodge structure V to 0 or 1 by Tate twisting:

561

if V has weight 21, V(1):= V @ Q(1) is a Hodge structure of weight 0 and if V has weight 21 1, its twist V ( l )has weight 1. Note also

+

ehn ( ~ ( 1 = ) )ehn (v) (u,)-'.

(9)

Definition 3.3. Let HS be the category of pure Q-Hodge structures (of varying weights). The Grothendieck ring Ko(HS) is the free group on the isomorphism classes [V]of Hodge structures V modulo the subgroup generated by [V]- [Vl]- [V"]where 0 t V'

t

v

t

V"

t

0

is an exact sequence of pure Hodge structures where the complexified maps preserve the Hodge decompositions. Because the Hodge number polynomial ( 5 ) is clearly additive and by ( 6 ) behaves well on products, we have:

Lemma 3.1. The Hodge number polynomial defines a ring homomorphism Phn : &(€IS)

-i

Z[U,V,U-~,V-'].

Pure Hodge structures in algebraic geometry arise as the cohomology groups of smooth projective varieties: for such X the cohomology group H k ( X ,Q) carries a Hodge structure of weight k. We put

ehn(X) := Phn(XHdg(X))= C ( - l ) k P h n ( H k ( X ) )E z[U,21, U - l ,

U-l]

which we call the Hodge-Grothendieck class and the Hodge-Euler polynomial of X respectively. These characteristic classes can also be defined for arbitrary complex algebraic varieties, basically because they are additive and algebraic varieties can be cut and pasted together from smooth projective ones as now will be explained. The crucial concept is that of the naive Grothendieck group Ko(Var) of (complex) algebraic varieties. It is the quotient of the free abelian group on isomorphism classes [XI of algebraic varieties over C with relations [XI = [ X - Y ] [Y]for Y c X a closed subvariety. Bittner has shown that there is a simpler description, using only blowing-ups.

+

Theorem 3.1 ( [2, Theorem 3.11). The group Ko(Var) i s isomorphic to the free abelian group generated by the isomorphism classes of smooth complex projective varieties subject to the relations [@I = 0 and [Z] - [El = [XI - [Y]where X , Y,2,E are as in Lemma 3.2.

562

Now indeed, this makes calculations in Var easy in view of

Lemma 3.2. Suppose that X is a smooth projective variety and Y c X is a smooth closed subvariety. Let T : 2 + X be the blowing-up with center Y and let E = T-'(Y) be the exceptional divisor. Then XHdg

(XI- XHdg ( y )= XHdg (2)- XHdg ( E );

e h n ( X ) - ehn(Y) = ehn(Z) - ehn(E). Proof. By [14, p. 6051 0 + H"X) -+ H y z ) @ H " Y ) -+ H k ( E ) + 0

is exact.

0

Hence, as claimed, the previous two characteristic classes can be extended to the category Var since for every complex variety X there exist projective smooth varieties X I , .. . ,X,,Y1,.. . ,Y, such that

[XI = c [ X i ]- c [ y 3 in]Ko(Var). j

i

For instance, for a compact X, the construction of cubical hyperresolutions (XI)D+ICA of X from [15] leads to such an expression: [XI =

c

(-l)"-l[x,].

0#ICA

The upshot is:

Corollary 3.1. The Hodge-Euler polynomial extends to a group homomorphism

ehn : Ko(Var) + Z[U,V , u - ~v,- l ] For further details and applications of this motivic point of view, we refer to [24]. Coming back to Hodge structures themselves, one cannot however expect a pure Hodge structure on the cohomology of singular or non-compact algebraic varieties. For example, if a vector space V carries a Hodge structure of odd weight, then its dimension must be even. So if X is an algebraic variety such that H 1 ( X ,Q) carries is Hodge structure of weight one, then the first Betti number of X had better be even. However, if X is an irreducible algebraic curve with one node, then the first Betti number is odd.

563

Deligne's first discovery is, that the graded parts for the weight filtration GrEHi(X, Q) underly pure Hodge structures of weight m. His second discovery is, that the Hodge filtrations F'GrEHi(X, C ) on all the graded quotients are induced by a canonical Hodge filtration F' on Hi(X,@).The data of the weight and Hodge filtrations on the cchomology of X constitute what has been called a mixed Hodge structure, constructed by Deligne in [5], [6]: Definition 3.4. We let V be a finite dimensional Q-vector space. A mixed Hodge structure on V consists of two filtrations, an increasing filtration on V, the weight filtration W. and a decreasing filtration F' on Vc = V C3 @, the Hodge filtration which has the additional property that it induces a pure Hodge structure of weight k on each graded piece Grr(V) = Wk/Wk-l. Mixed Hodge structures turns out to form an abelian category. Every morphism of mixed Hodge structures is strictly compatible with the Hodge and weight filtrations. As a consequence, an exact sequence of mixed Hodge structures remains exact if at each place one applies the functor V H G r r ( V ) . Deligne's main result is: Theorem 3.2 ( [5,6]). Homology, cohomology, Borel-Moore homology and cohomology groups with compact supports of algebraic varieties carry functorial mixed Hodge structures. Virtually all natural maps like cup product and Poincare' morphisms are morphisms of mixed Hodge sructures. The following properties are noteworthy: 0

The weight filtration has the following properties: -

0

If X is compact, then W i H i ( X ) = HZ(X); . if X is smooth, then Wi-lHi(X) = 0.

The mixed Hodge structures on H i ( X ) and on Hi(X) are dual to each other, and the same holds for H : ( X ) and HFM(X).

It is clear that the Grothendieck ring of the category of mixed Hodge structures is the same as for Hodge structures. Moreover, Lemma 3.2 has the following generalization to the context of singular varieties:

564

Lemma 3.3. Let f : X + X be a proper modification with discriminant D . Put E = f - ' ( D ) . Let g : f l E : E + D and let i : D + X a n d ; : E 4 X denote the inclusions. Then one has a long exact sequence of mixed Hodge structures

.. . -+ H k ( X )-+ H k ( X )@ H k ( D )+ H k ( ( E )+ H"'(X)

+

...

It is called the Mayer-Vietoris sequence for the discriminant square associated to f . One has XHdg(X) = X H d g ( X )

+ X H d g ( E ) - XHdg(D).

Here the discriminant D is the minimal closed subset of X with the property that f is an isomorphism when restricted to the inverse image complement of D. 4. From groups to sheaves

In this section we describe Deligne's construction of the Hodge and weight filtrations on the cohomology of smooth algebraic varieties. Let U be a smooth complex algebraic variety. By [23] U is Zariski open in some compact algebraic variety X , which by [16] one can assume to be to be smooth and for which D = X - U locally looks like the crossing of coordinate hyperplanes. It is called a normal crossing divisor. If the irreducible components Dk of D are smooth, we say that D has simple or strict normal crossings.

Definition 4.1. We say that X is a good compactification of U = X - D if X is smooth and D is a simple normal crossing divisor. We return for the moment to the situation where D c X is a hypersurface (possibly with singularities and reducible) inside a smooth ndimensional complex manifold X and we set

j:U=X-D-X Recall that a holomorphic differential form w on U is said to have logarithmic singularities along D if w and dw have at most a pole of order one along D. It follows that these holomorphic differential forms constitute a subcomplex n>(log D ) C j&j. Suppose now that D has simple normal crossings, p E D and V c X is an open neighbourhood with coordinates ( z , . . . ,z,) in which D has

565

equation z1 . . - z k = 0. On can show [14, p. 4491

P

n,.

(log qp =

A c& (log qp.

An essential ingredient in the proof of the following theorem is the residue map which is defined as follows. We set x k = { Z k = 0 ) and we let D’be the divisor on x k traced out by D.Then writing w = V A ( d z k / z k ) with v, 1;1‘ not containing d Z k , the residue map can be defined as

+$

res : a$ (log D)+ 0%;’(log 0’)

*

‘1Xk.

As a special case we have the Poincar6 residues Rk :

(log 0)+ Ox,.

Theorem 4.1. Let U be a complex algebraic manifold and let X be a good compactification, i.e. D = X - U is a divisor with simple normal crossings. Then the following is true. (1) H k ( U ;@) = W“X, Q>(logD)); (2) The trivial filtration F o n the complex R>(log D ) given by

FPQ>(logD) = [0 --t

+ Cl$(logD) + S2$+’(lOgD)

--t

*

a

*

]

together with the filtration W defined b y W&2pX(logD)

=

for m < 0 R$(logD) for m 2 P { Ofigvrn A RF(log D)i f 0 5 m 5 p .

induce in cohomology two filtrations FPH~(U C) , = Im (~‘(x, FPn>(logD))

~ ( uc)) ; , w,H~(u;C ) = Im (W‘((X, wm-kn>(lOgD)) -+ H’((u; CC)). +H

which put a mixed Hodge structure on H k ( U ) . A few words about the symbol W occurring in this theorem. It stands for hypercohomology, of which we give here a simple treatment. See also [12, Definition 2.1.41. The book [12] is also a good reference for derived categories, constructible sheaves, perverse sheaves and intersection cohomology. Let F be a sheaf on a topological space X . We view it as the presheaf U H F(U):= r ( U , F ) ,where r is the functor “taking global continuous sections”. If instead, we consider 3z, the “discontinuous sections”

nsEU

566

over U we obtain a presheaf C $ d m 3 which is in fact a sheaf. By definition it comes equipped with an injective homomorphism 3 LS c,%drn3.Following [ll,11.4.31 one inductively defines

2'3 = 3 := C & k 3 / 2 P - 1 3 C G d m 3 := c;dm (c&;F/2p-13) 2 p 3

.

1

( Godement's resolution)

The sheaves c , ? d m 3 are flabby, i.e. any section over an open set extends to the entire space X. The natural maps

d : c&,3

* ZP"3

Lt

Cp+l Gdrn 3

fit into an exact complex whose global sections by definition yields the cohomology groups of 3

wp(x,F) := Hp(l?(x,cGdrnF)). From the definition of the Godement resolution it follows that any morphism of sheaves f : F 6 induces a morphism of complexes C j d m ( f ) between the respective Godement resolutions. Moreover, for two such morphisms f and g, we have: -+

c&dm

(fog>= C&dm (f)' cGdm(g)*

Iff : 3 -+ 6 is a morphism of sheaves on X, the induced morphism C&dm(f) induce maps H q ( f ) : Hq(X,3)-+ H Q ( X G , ) which therefore behave functorially. Seiondly, any exact sequence of sheaves of R-modules

0 -+ F1-+ 3 -F1I i -+ 0 induces short exact complexes on the level of their Godement resolutions and hence long exact sequences

. . . W(X,FI)

--f

HQ(X,3)-+ P ( X ,3")-+ HQ+l(X,FI). .

Next, we pass to a bounded below complex of sheaves 3' on X. For every P take its Godement resolution C&,,FP. The derivatives dp : 3 P 4 FP+' induce maps of complexes d p : C6,,3P -+ C&,,.F'+l and by functoriality, dP+lodP = 0 so that we have a double complex C & , F * . Since the Godement sheaves are flabby, the associated simple complex S(C&,F') gives a flabby resolution of 3'. The hypercohomology groups Wk(X,3*)are now defined as the cohomology groups of the complex of global sections of S(C&rn3').

567

5. Intersection homology and perverse sheaves Consider the cohomology of a possibly singular projective variety X which is irreducible of dimension d. Suppose that the cup product pairing

H i ( X , Q ) 63 H y X , Q) --+ H y X , Q) II Q(-d) is a perfect pairing. (Note that for any resolution T : X + X the induced map T * : H 2 d ( X , Q )+ H2d(%,Q) is an isomorphism of mixed Hodge structures, so H 2 d ( X )is isomorphic to Q(-d).) This pairing is compatible with the filtration W . On the other hand, H i ( X ) has weights 5 i and ~ 2 d - (i X ) has weights 5 2d - i because X is compact. This implies that the image of W i - l H i ( X ) @ H 2 d - i ( X ) under cup product is contained in W ~ ~ - I H ~=~0.( Hence X ) we obtain that W i - l H i ( X ) = 0 i.e. H i ( X ) is pure of weight i. So we see that purity is a consequence of Poincar6 duality. The search for purity is therefore intimately connected with the search for Poincare duality. In this direction, the main development of the previous decades is the discovery of intersection homology by Goresky and MacPherson [13]. Its definition involves the choice of a perversity, but for complex analytic spaces, which admit a stratification with only even-dimensional strata, there is a canonical "middle" perversity which is commonly used. Intersection homology has as its input furthermore a local system V of rational or complex vector spaces on a dense open subset of the regular part of X. From these data a sheaf complex "V is constructed, the minimal perverse extension of V.Assuming X compact of dimension dx, we have

I H , ( X , V) = W d x - y X , V). The characterization of "V involves the notion of perverse sheaf on X , which we now introduce.

Definition 5.1. Let R be a commutative ring with 1 and let 3be a sheaf of R-modules of finite rank on a Whitney-stratifiable analytic space X. We say that 3 is (analytically) constructible if X admits an analytic stratification such that F restricts to a locally constant sheaf on any of the strata. We say that a bounded complex 3' of sheaves of R-modules is (analytically) cohomologically constructible if the sheaves H q ( 3 ' ) have finite rank and are analytically constructible. We set derived category of bounded cohomologically constructible complexes of sheaves of R-modules on X

568

On this category one has the Verdier duality functor D x . Definition 5.2. A bounded analytically cohomologically constructible complex 3' of sheaves of R-vector spaces is perverse if the following two conditions hold:

dim@Supp H j ( 3 . ) 5 - j dim@Supp Hj(iTD(3')) 5 -j

V j < d x (support condition); V j < d x (cosupport condition).

In the derived category, perverse complexes make up a subcategory

The category PervR(X) is abelian. This is a non-trivial assertion and perhaps the most elegant way of proving it is by identifying this category as the core of the triangulated category of D:(X; R). At this point one could consult [12, Sect. 51. Note that a local system V on a complex manifold, considered as a complex concentrated in degree zero, is not perverse, because its Verdier dual is V"[ 2 d x ] .However, V [ d x ]is perverse. Definition 5.3. Let X be a complex variety of pure dimension d x and V a local system over a dense open subset of X . With 7~ the middle perversity, the (analytic) intersection complex for V is the unique perverse compZez V on X which restricts to V [ d x ]on U and has no sub- or quotients object in PervR(X) supported on X - U .

These complexes do compute intersection (co)homology: Lemma 5.1.

Remark 5.1. (1) Note that the intersection (co)homology is nonzero only in the interval [ 0 , 2 d x ] as it should. ( 2 ) The complex Qx [ d x ]is perverse on any complete intersection variety X. However, it need not be the minimal extension of its restriction to the regular locus.

569

6. D-modules

Let X be an n-dimensional complex manifold. A germ at 2 E X of a holomorphic vector field on X is the same thing as a C-linear derivation D : Ox,x -+ Ox,x. As such it is an example of a germ of a differential operator of order 1.Germs of functions acting by multiplication on the left give germs of differential operators of order 0. Together they generate a subalgebra of germs of C-linear endomorphisms of Ox,x.This is the germ at x of the sheaf of diflerential operators on X , denoted by

Ox). VX c Horn@,(Ox, The order filtration FEd,m = 0,1,. . . is defined recursively as follows. One sets F F d V x = OX and for any open set U C X , one sets F:dvx(u)

={PE

VX(U)I P f - f P E F:~,vx(u)vf

E Ox(U)}. (11)

This defines a presheaf on X which then needs to be sheafified to obtain FgdVx.To see this concretely, let (U, ( ~ 1 , .. . ,zn)) be a holomorphic chart. Putting & = d / d z i , i = 1,.. . ,n and using multi-index notation I = (il, . . . ,in), 111 = x k ik, sections P of F g d V x over U can be uniquely written as

P=

C PIal,

pI E

oX(u), d1 = a:. . .a$

IIllm

This shows that the sheaves of m-th order operators are locally free of finite rank and that GrLordVx2 Sym"(T(X)). In this way we may compare the non-commutative algebra commutative graded, the symmetric algebra on T ( X ) :

V X with its

00

GrDx := @ GrLordVx2 Sym(T(X)).

(12)

m=O

This algebra has a familiar geometric interpretation:

Lemma 6.1. The sheaf GrDx can be identified with the sheaf of holomorphic functions a:TVX+C

which restrict polynomially to each cotangent space.

570

A sheaf M of left Dx-modules is called a Dx-module, or, if no confusion is possible, a D-module.

Definition 6.1. A Dx-module M is coherent if it is first of all locally finitely generated, i.e. every point has a neighbourhood U over which there exists a surjection 2); -+

MIU

-t

and secondly if every homomorphism D$ locally finitely generated.

0,

-, MIU

has a kernel which is

From the fact that Ox is coherent it is not hard to see [3, 11.531 that V X is coherent (as a left-DX-module) and from this one deduces the following test for coherence:

Lemma 6.2. A D-module is coherent if and only if it is locally finitely presented: locally over an open subset U c X we have an exact sequence of V(U)-modules

D ( U ) p-, D ( U ) p-+M ( U ) -t 0. Examples (1) The structure sheaf OX is a left Dx-module, generated globally by the section 1. In local coordinates ( 2 1 , . . . ,zn) on an open set U c X the kernel of the sheaf homomorphism ev : D X -, O X given by P w P(1) is generated by the vector fields 81,. .. Hence OX is a Vx-module locally of finite presentation, and therefore a coherent Dx-module. A coordinate invariant description of ker(ev) can be given as follows. The sheaf T ( X )of germs of holomorphic tangent vectors is locally free of rank n over Ox.Hence the tensor product D X @ox T(X)is a locally free left Dx-module. The map P @ O H PO defines a homomorphism of left Dxmodules Vx @ T ( X ) -t DX and it represents ker(ev). This shows that OX is a coherent Dx-module. (2) Every locally free Dx-module is coherent. (3) Every Ox-coherent Dx-module M is locally free as an Ox-module. To see this, it suffices to show that M , is a free Ox,,-module for any x E X . Let m, denote the maximal ideal of OX,, and choose elements e l , . . . ,e, in M , which map to a C-basis of the fibre

,an.

M ( x ) := M x / m x M x .

571

By Nakayama's lemma, M, is generated by el,. . . ,e,. These generators form a free basis. Indeed, if not there would be a relation fiei = 0 such that not all the fi are zero. Let k to be the minimum of the orders of vanishing at x of f i . We call it the order of the relation. For simplicity, assume that f l realizes this minimum. We cannot have k = 0, since in that case the classes of the ei in M (x)become dependent. But if k > 0, we can reduce order of the relation: choose i such that in local coordinates, f l vanishes to order k at x. Then, writing aiej = bjkek, we find

XI=,

ai

S

S

S

S

which is a relation of lower order. This contradiction indeed shows that M is locally free as an Ox-module. (4) Let Mx denote the sheaf of germs of meromorphic functions on X. This is a Dx-module which is not locally of finite type. 6.1. Good Filtrations and Characteristic Varieties

Let X be a complex manifold of dimension n and M be a Dx-module.

Definition 6.2. A filtration on M is an increasing and exhaustive ( M = U pFpM) sequence of submodules ( F p M ) p Esuch ~ that F,"'dV~[FpM] c Fp+,M 'dr,s E Z. It is called a good filtration if moreover (1) locally on X this filtration is bounded below (in the sense that locally FpM = 0 for p E Z small enough) and above (in the sense that locally for r E Z large enough we have F,"'dDx[FpM] = Fp+,M; (2) each FpM is a coherent Ox-module.

A filtered D-module is a D-module equipped with a good filtration. For every coherent DX-module M such a good filtration exists locally on X: starting from a local presentation ~ " D X @DX 3 M 4 o one can put FpM := u ( ~ F , O ' ~ D X for) p 2 0 while FpM = 0 for p < 0. Then F F d D x[FpM]= Fp+,M for all r, p E Z. Conversely, if M locally possesses a good filtration, M is coherent. To test if a given filtration is good, the following Lemma is useful (see [3, 11.41. To state it, recall (12) that GrDx is the graded module associated to D x with respect to the order filtration. Similarly, we set GrFM

= @Gr$M. k

572

Lemma 6.3. Let (M,F ) be a Dx-module equipped with a filtration. Then F is good precisely when GrFM is coherent as a GrDx-module. It is also important (and easy to show) that any two good filtrations F and G on a given Dx-module M are locally commensurable in the sense that there exist two integers a and b such that locally for all p E Z we have Fp-,M c G p M c Fp+bM. Using this, one proves

Proposition 6.1. Let Z(M, F ) be the annihilator of GrFM, i.e. the ideal of GrVx consisting of w with w m = 0 for all ? Ei i GrFM. Put JZ(M, F ) = { a E Gr(Dx) I 3k E N,ak E Z ( M , F ) } .

Then JZ(M, F ) does not depend o n the choice of the good filtration F o n

M. Since locally good filtrations F exist, we deduce from this that there exists a globally defined sheaf of ideals JZ(M) c Gr(Dx) which locally coincides with JZ(M, F). Recall (Lemma 6.1) that Gr(Dx) consists of the sheaf of functions on the total space T V X of the cotangent bundle of X which are polynomial on each fibre T J X . The ideal JZ(M) thus defines a subvariety of T V X , the characteristic variety of M, which in each fibre T J X is a cone. It will be denoted Char(M) :=

u V(&)

C T(X)'.

(13)

XEX

We finally remark that if we have a good filtration F on M, the characteristic variety can also be seen as the support of the ideal GrF(M) c Gr(Dx) (inside the cotangent bundle).

Examples (1) Let M = OX.Then a good filtration is given by FpM = 0 for p < 0 and FpM = M for p 2 0. The same procedure works if M is a Dxmodule which is coherent as an Ox-module. The characteristic variety of such a Dx-module is the zero section of the cotangent bundle. Conversely, suppose that the characteristic variety of M consists of the zero section. Then for local coordinates (21,.. . , z,) on U c X , considering the differentials dzj as local functions wj on the total space of the cotangent bundle, (z1, . . . ,z,, w1,. . . ,w,) give a set of local coordinates on T V ( U )s! U x P . Then JZ(M) is generated by (w1,. . . ,wn).This means that GrFM is killed by a power of the ideal (w1,. . . ,w,) and hence is

573

a finitely generated Ou-module. Hence M is itself a finitely generated Ox-module i.e. M is a coherent Ox-module and hence free. (2) Let D c X be a submanifold of codimension one. Recall that

Ox(*D) := U O x ( m D ) , m

the sheaf of meromorphic functions on X , holomorphic on X - D and having a pole along D. Let M = Ox(*D)/Ox and put FpM = 0 for p < 0 and FpM = Ox(pD)/Ox if p 2 0. This defines a good filtration on M . If N D ~ = X Ox(D)/Ox is the normal bundle of D in X , then GrF(M)= 0 for p 5 0 and Gr:(M) _N N!& for p > 0. Let ( ~ 1 , .. . ,zn) be local coordinates on X such that D is given by z1 = 0. Let b(z1) be the class of 2;' modulo Ox. Then ~(zI) locally generates G r F ( M ) over Gr(Dx) N Ox[wl,. . . ,tun]. The annihilator ideal of this generator is generated by 21, w2,.. . , wn.Hence Char(M) is the conormal bundle of D in X , i.e. the subspace of T V ( X )consisting of pairs ( z , a )such that the covector a vanishes on tangent vectors to D. (3) Let 0 + M' + M

--f

M"

--f

0

be an exact sequence of Dx-modules. If two of these are coherent, the third one is coherent too. In that case, we have Char(M) = Char(M') U Char(M"). Applying this to the defining sequence for O x ( * D ) / O x it follows that the characteristic variety of Ox(*D)is the union of the zero section and the conormal bundle of D. (4) The order filtration on DX is a good filtration. We see that I ( M ,F ) is the zero ideal, so the characteristic variety of JUis the whole cotangent bundle. 6.2. Basics on Holonomic D-Modules

The cotangent bundle T * ( X )of any complex manifold X is a symplectic manifold: its tangent bundle carries a canonical symplectic form. A linear subspace A of a vector space endowed with an alterating nondegenerate bilinear form is called isotropic if A c A'- and involutive if A 3 A l . If A = A l then A is called Lagrangean. These definitions have their analogs for subspaces C c T ( X ) :it is called isotropic iff each tangent space to C at a regular point is isotropic, and similarly for involutive and Lagrangean.

574

It is a deep theorem that for a coherent Dx-module M the characteristic variety Char(M) C T ( X ) " is involutive. See [20].Hence the following definition makes sense. Definition 6.3. A coherent Dx-module is holonomic if its characteristic variety is Lagrangian, or, equivalently if dimChar(M) = d x . In that case Char(M) consists of the union of closures of normal bundles to regular loci of irreducible subvarieties of X. Definition 6.4. A coherent Vx-module M is called regular if it has a global good filtration whose annihilator ideal is equal to its radical, i.e. such that the components of the characteristic variety have multiplicity one. Note that this is not the usual definition of regularity (see [12, p. 1441, but it is equivalent to it by a result of Kashiwara [17]. 6.3. De R h a m f i n c t o r and Riemann-Hilbert correspondence

For a coherent Dx-module M we define its de Rham complex as DRx(M) = [(M

+

fl; @ox M

+

-+

fl? @ox M)] [dx],

(14)

where the derivatives in the complex are given in local coordinates by n

d(w @ m) = dw @ m - c ( d z i A w ) @ aim. i=l

The link between D-modules and perverse complexes is given by Kashiwara's theorem, one of the central ingredients of the Riemann-Hilbert correspondence: [17] Theorem 6.1. Let X be a complex analytic manifold. The de R h a m complex of a holonomic Vx-module is a perverse complex. We let D,b(Dx) denote the derived category of bounded complexes of Dxmodules whose cohomology sheaves are regular holonomic. The de Rham complex can equally be defined for complexes of Dx-modules and we have the celebrated

575

Theorem 6.2 (Riemann-Hilbert correspondence). Let X be a complex algebraic manifold. The de R h a m functor establishes a n equivalence of categories

-

It induces a n equivalence of categories {regular holonomic Vx -modules}

{perverse complexes o n X } ,

i. e. the cohomology sheaves of a regular holonomic complex M' is concentrated in degree 0 if and only if DRx(M') is perverse. See [18,21,22].

7. Mixed Hodge modules 7.1. Motivating example Let X be a projective manifold of dimension n and let Y C X be a smooth hypersurface. We are going to define some sheaf data on X whose ingredients are a perverse sheaves and a D-module, both filtered, which together determine the mixed Hodge structure on the cohomology of U = X - Y . Let j : U + X and i : Y -+ X denote the inclusion maps. First we consider the derived direct image K ' = Rj,Qu[n]. Its cohomology sheaves are given by H-"(K') = Qx and H-nil(Ke) = i*Qy(-l) where (-1) refers to a Tate twist. As Qu[n]is self-dual the Verdier dual of K' is j ! Q u [ n ]and we see that K' is an object of Pervx(Q). It carries a weight filtration, given by

0 c WnK' = T<-"K' -

c Wn+lK'

= K'.

Here T k. This filtered complex will take care of the rational structure, and of the weight filtration which is therefore defined over Q. On the other hand we have the logarithmic de Rham complex 52; (log Y ) with its filtrations W and F . Consider the Vx-module M = OX(*Y)whose sections are meromorphic functions on X with only poles along Y . It has the submodule WmM = O x , and we put Wm-1M = 0, Wm+lM = M. This filtration is a filtration by Vx-modules.

Lemma 7.1. W e have a n isomorphism of filtered objects in Pervx(C):

(KO,W) 8 C x

DR(M, W).

576

This isomorphism provides a rational structure on the right hand side. Then, the weight filtration on the right hand side gives the weight filtration on the cohomology of U . We also have a good filtration by Ox-submodules given by F,M = O x ( p Y ) ,the sheaf of germs of meromorphic functions on X which are holomorphic on X - Y and have a pole of order at most k along Y . It induces a filtration on the de Rham complex, which in turn gives the Hodge filtration on the cohomology of U . This is the simplest non-trivial example of a mixed Hodge module we know of. At this point this cannot be proved; all we need to understand here is how the above procedure provides the ingredients for a mixed Hodge module on a smooth projective variety X: (1) an object K' of Pervx(Q) equipped with an increasing filtration W ; (2) a regular holonomic Vx-module M equipped with two filtrations: its weight filtration, which is a filtration by Vx-submodules, and a Hodge filtration, which is a good filtration in the usual sense. (3) an isomorphism of filtered objects in Pervx(C):

(K', W )8 C x

DR(M, W ) .

The task to formulate the right conditions for these ingredients guaranteeing that their hypercohomology produces mixed Hodge structures is indeed formidable and has been accomplished by Morihiko Saito in the eighties of the past century [26,27]. 7 . 2 . Axioms for mixed Hodge modules

As we just observed, the definition of mixed Hodge modules is very involved. For this reason it is more suitable to start with an axiomatic introduction. This makes it possible to deduce important results rather painlessly, such as the existence of pure Hodge structures on the intersection cohomology groups. It is not really necessary to understand the intricacies of the construction of mixed Hodge modules in order to be able to relate mixed Hodge modules to mixed Hodge structures. The principal reason is that mixed Hodge modules on points are just mixed Hodge structures. To start, we recall some notation. For any complex algebraic variety X the derived category of bounded complexes of cohomologically constructible complexes of sheaves of Q-vector spaces on X was denoted @ ( X ; Q); it contains as a full subcategory the category Pervx(Q) of perverse Q-complexes.

577

The Verdier duality operator D is an involution on D,b(X;Q) preserving Pervx Associated to a morphism f : X --f Y between complex algebraic varieties, there are pairs of adjoint functors (f-l,Rf,) and (f!,Rf!) between the respective derived categories of cohomologicallyconstructible complexes which are interchanged by Verdier duality. Let us now state the axioms:

(a).

For each complex algebraic variety X there exists an abelian category M H M ( X ) , the category of mixed Hodge modules on X, together with a faithful functor

a).

ratx : D ~ M H M ( X -+ )D,~,(x;

(15)

such that MHM(X) corresponds to Perv(X; Q). We say that r a t x M is the underlying rational perverse sheaf of M. Moreover, we say that

M E MHM(X) is supported on 2 e r a t x M i s supported on Z . The category of mixed Hodge modules supported on a point is the category of polarizable mixed Hodge structures H for which the graded pieces Grw H are polarizable; ; the functor rat associates to the mixed Hodge structure the underlying rational vector space. Each object M in MHM(X) admits a weight filtration W such that morphisms preserve the weight filtration; the object G r r M is semisimple in MHM(X); the functors M H WkM, M H G r r M are exact; if X is a point the W-filtration is the usual weight filtration for the mixed Hodge structure. Since MHM(X) is an abelian category, the cohomology groups of any complex of mixed Hodge modules on X are again mixed Hodge modules on X . With this in mind, we say that for complex M' E DbMHM(X) the weight satisfies weight[M']

{Ln

G r y @(Me) = 0

n7

fori>j+n for i < j n.

+

The duality functor IDx of Verdier lifts to MHM(X) as an involution, also denoted Dx, in the sense that Dxoratx = ratxoDx. For each morphism f : X Y between complex algebraic varieties, there are induced functors f*,f! : DbMHM(X) -+ DbMHM(Y), f*,f! : DbMHM(Y) -+ DbMHM(X) exchanged under Dx and which lift the analogous functors on the level of constructible complexes. -+

578

F) The functors f!,f * do not increase weights in the sense that if M a has weights 5 n, the same is true for f!M' and f*M'. G) The functors f!,f* do not decrease weights in the sense that if M' has weights 2 n, the same is true for f'M' and f,M'. By way of terminology, we say that M a E D b M H M ( X ) is pure of weight n if it has weight 2 n and weight 5 n. We say that a morphism preserves weights, if it neither decreases or increases weights. Since for a proper map f* = f! axioms F) and G) entail: H) Proper maps between complex algebraic varieties preserve weights. 7.3. Some Consequences of the Axioms

From axiom A) and B) we see that there is a unique element QHdg E

MHM(pt ),

rat(QHdg)= Q(O),

(16)

the unique Hodge structure on Q of type (0,O). The next lemma explains how the various cohomology groups can be expressed using direct and inverse functors. On the level of mixed Hodge modules this then naturally leads to mixed Hodge structures. We have the following purely topological result (see 112, Sect. 21):

Lemma 7.2. Let a x : X + p t be the constant map to the point. Then we have: Hk(X;Q) = H k ( p t ,(ax)*o;LQ) H-k(X; Q) = H k ( P t ,( a X ) ! a X Q ) H_BY(X; = H W , (.x)*akQ>).

a)

Moreover, if i : Z L) X is a closed subvariety, we have H ~ ( x ; Q ) = H k ( p t ,(ux)*i*i!u>Q). Motivated by Lemma 7.2, using axiom D) and E) we do the same for the complex of mixed Hodge modules Q H d g (16): ~~~g

:= .;~Hdg

g9

1

E D~ MHM( X)

:= akQHdg E D~ MHM ( X).

(17)

By axiom E), applying the direct image functors associated to a x produces complexes of mixed Hodge modules on the point p , hence, by axiom B), their cohomology groups have mixed Hodge structures. We deduce:

Corollary 7.1. Let X be a complex algebraic variety and i : Z -+ X a subvariety. The complexes of mixed Hodge modules ( a x ) * Q y ,(ax)!DQy,

579

( a ~ ) * l D Q yrespectively , i,i!Qy put mixed Hodge structures on cohomology, homology, Borel-Moore homology, and cohomology with support an Z respectively. These mixed Hodge structures coincide with the ones constructed by Deligne. 8. Some Applications

Pure Hodge Structure on Intersection Homology Let X be an irreducible projective variety, with regular locus U . The hypercohomology groups of the minimal perverse extension

are the Intersection Homology groups of X. One has ICxQ = Im(j!Qv[dxl + j*Qv[dx]). This construction makes sense for mixed Hodge modules, and the resulting object

is simple and hence pure (of weight dx). Therefore its hypercohomology groups are pure Hodge structures! This puts a pure Hodge structure on the intersection homology groups of any irreducible projective variety. Cf. [26, Cor. 5.4.71for a more general statement, where Qu is replaced by a variation of Hodge structure V of weight k on U . In that case, V[dx] underlies a pure Hodge module of weight k + d x on U . By Axiom F), the mixed Hodge module j!V[dx] has weights 5 k d x and by Axiom G), the mixed Hodge module j,V[dx] has weights 2 k d x ; this implies that IC x V is pure of weight k t d x . The case of a variation of Hodge structure on the complement of a normal crossing divisor in a projective smooth variety was also treated by Cattani, Kaplan and Schmid [4] and by Kashiwara-Kawai [19].

+ +

Mixed Hodge Structure on Vanishing Cohomology Let f : (Cn+', 0) 4 (C,0) be an isolated hypersurface singularity with Milnor fibre Xf.The second author [29] has put a mixed Hodge structure on the cohomology group & ( X f ) with the following properties. Let T : f i n ( X f ) H n ( X f ) be the monodromy operator. Write T = T,T, = T,T, with T, semisimple and T, unipotent, and let N = logT,. Then T, is an automorphism of mixed Hodge structures, and N maps wk to wk-2 and FP to --f

580

Moreover, N determines the weight filtration completely. Using the automorphism T, for each a E Q one defines a non-negative integer d f ( a ) by Fp-l.

d f ( a ) = dimker(T, - exp(-2nia); Grk+l-al H n (Xf,C ) ) and subsequently one defines the spectrum o f f (in Saito's sense) by ~ p ( f := ) C d f ( a ) t a E ~ : = l i+m ~ [ t i , t - l ] . a

This mixed Hodge structure has also been constructed by Varchenko, whose construction has been translated into the language of D-modules by Saito and Scherk-Steenbrink. A first product of this description is a proof of the following Thom-Sebastiani-type result for the spectrum: Theorem 8.1 ( [28,31]). Consider holomorphic germs f : (cCn+l,O) -+ (C,O) and g : (Cm+',O) -+ (C,O) with isolated singularity. T h e n the g e m f @g : x Cm+l, (0,O)) -, C with (f @ g ) ( x ,y ) := f ( x )+ g ( y ) has also a n isolated singularity, and

The Thom-Sebastiani property has been a crucial ingredient in the proof of the following Theorem 8.2 (Semicontinuity of the Spectrum [30,32]). Suppose that f deforms into a function f' with critical points an the same fiber. T h e n f o r all b E Q

XI,.

. . ,x k

The theory of mixed Hodge modules has been used t o extend these results (with the appropriate notion of spectrum) t o non-isolated singularities (Saito) and to isolated complete intersection singularities [lo]. References 1. Beilinson, A., J. Bernstein and P. Deligne: Faisceaux pervers, in: Analyse et topologie sur les espaces singuliers I, Asterisque 100 (1982). 2. Bittner, F.: The universal Euler characteristic for varieties of characteristic zero, Comp. Math. 140(2004), 1011-1032. 3. Borel, A. et al.: Algebraic D-modules, Perspectives in Math. 2, Academic Press, Boston, etc (1987).

581 4. Cattani, E., Kaplan, A., Schmid, W.: L2 and intersection cohomologies for a polarizable variation of Hodge structure. Inv. Math. 87 (1987), 217-252. 5. Deligne, P.: ThBorie de Hodge 11, Publ. Math. I.H.E.S. 40 (1971), 5-58. 6. Deligne, P.: ThBorie de Hodge 111, Publ. Math. I.H.E.S. 44 (1974), 5-77. 7. Deligne, P.: La conjecture de Weil, Publ. Math. I.H.E.S. 43 (1974), 273307. 8. Deligne, P.: Poids dans la cohomologie des varikt6s alghbriques, Proc. ICM Vancouver 1974, Vol. 1, pp. 79-85. Canad. Math. Congress, Montreal, Que. (1975). 9. Deligne, P.: La conjecture de Weil 11, Publ. Math., I.H.E.S. 52 (1980), 137-252. 10. Ebeling, W., Steenbrink, J.H.M.: Spectral pairs for isolated complete intersection singularities. J. Alg. Geom. 7 (1998), 55-76. 11. Godement, R.: Thkorie des faisceaux, Hermann, Paris (1964). 12. Dimca, A.: Sheaves in Topology, Universitext, Springer-Verlag, Berlin etc. (2004). 13. Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19 (1980), no. 2, 135-162. 14. Griffiths, P. and J. Harris: Principles of algebraic geometry, John Wiley & Sons, New York etc. (1978). 15. GuillBn, F., Navarro Aznar, V., Pascual-Gainza, P., Puerta, F.: Hyperrbolutions cubiques et descente cohomologique, Springer Lecture Notes in Math. 1335 (1988). 16. Hironaka, H.: Resolution of singularities of an algebraic variety of characteristic zero, Ann. Math. 79 (1964), 109-326. 17. Kashiwara, M.: On the maximally overdetermined system of linear differential equations I. Publ. Res. Inst. Math. Kyoto Univ. 10 (1974/75), 563-579. 18. Kashiwara, M.: The Riemann-Hilbert problem for holonomic systems. Publ. RIMS, Kyoto Univ. 20 (1984), no. 2, 319-365. 19. Kashiwara, M., Kawai, T.: The PoincarB lemma for variations of polarized Hodge structures. Publ. RIMS, Kyoto Univ. 23 (1987) 345-407. B.: 20. Malgrange, L’involutivitB des caractBristiques des systkmes diffbrentiels et microdiffkrentiels. SBm. Bourbaki 522 (1977-1978), Lect. Notes in Math. 710, 277-289, Springer Verlag, Berlin etc. (1979). 21. Mebkhout, Z.: Une Bquivalence de catkgories. Compositio Math. 51 (1984), no. 1, 51-62. 22. Mebkhout, Z.: Une autre Bquivalence de cathgories. Compositio Math. 51 (1984), no. 1, 63-88. 23. Nagata, M.: Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2(1962), 1-10. 24. Peters, C.A.M., Steenbrink, J.H.M.: Hodge Number Polynomials for Nearby and Vanishing Cohomology, t o appear in Annual EAGER conference and workshop on the occasion of J. Murre’s 75-th Birthday on Algebraic Cycles and Motives, August 3(r September 3, 2004, Leiden

582

25. 26. 27. 28. 29.

30. 31.

32.

University, the Netherlands, editors J. Nagel and C. Peters, Cambridge University Press. Peters, C.A.M., Steenbrink, J.H.M.: Mixed Hodge Structures, Book in preparation. Saito, M.: Modules de Hodge polarisables. Publ. RIMS, Kyoto Univ. 24 (1988), 849-995. Saito, M.: Mixed Hodge Modules, Publ. RIMS, Kyoto Univ. 26 (1990), 221-333. Scherk, J. and Steenbrink, J.H.M.: On the Mixed Hodge Structure on the Cohomology of the Milnor Fibre, Math. Annalen 271 (1985), 641-665. Steenbrink, J.H.M.: Mixed Hodge structures on the vanishing cohomology, in Real and Complex Singularities, Oslo, 1976, Sijthoff-Noordhoff, Alphen a/d Rijn, 525-563 (1977). Steenbrink, J.H.M.: Semicontinuity of the singularity spectrum. Inv. math. 79 (1985), 557-565. Varchenko, A.N.: Asymptotic mixed Hodge structure in the vanishing cohomology. Izv. Akad. Nauk SSSR, Ser. Mat. 45, 540-591 (1981) (in Russian). [English transl.: Math. USSR Izvestija, 18:3,469-512 (1982)l Varchenko, A.N.: Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface. Soviet Math. Dokl. 27 (1983), 735-739.

PART I11

Norkshop on Singularities in Geometry and Topology

This page intentionally left blank

585

On sufficiency of jets Hans Brodersen Department of Mathematics, University of Oslo P.B. 1053, 0316 Oslo, Norway 'E-mail: [email protected] www.math.uio.no In this paper we will report on some new results in the theory of sufficiency of jets. We consider a certain class of real jets from the plane t o the plane, and state necessary and sufficient conditions for such jets t o be sufficient with respect t o topological right left -equivalence. We also consider jets into R having singularities along a line. We state necessary and sufficient conditions for sufficiency of such jets with respect to topological right-equivalence leaving the singular line invarient. Proofs and technical details will appear elsewhere. Keywords: Sufficiency of jets, finite determinacy, jets from the plane to the plane, jets with line singularities.

1. Introduction In this paper we will report on some new results in the theory of sufficiency of jets. Proofs and technical details will appear in [l]and [2] . Before we state our main results, we will however put these results in relation with now classical theory about sufficiency of jets and determinacy of mappings. Let z be a jet in J T ( n , p )We . can identify z with a polynomial map z : (Rn, 0) -+ (RP, 0). Let €[,l(n,p)be the set of CTgerms f : (W, 0) -+( W p ,0), and let j Tf (0) be the Taylor polynomial of f at 0 of order r. Consider an equivalence relation R on €[,.I (n,p ) and a subset E of €[,I (n,p ) .

Definition 1.1. z is R-sufficient in E if any two f , g E E with j r f ( 0 ) = j r g ( 0 ) = z are R-equivalent. The study of sufficiency of jets started with now classical papers of Kuiper, Kuo and Bochnak and Lojasiewicz. In these papers sufficiency 1) = €[,I and €[,+I] with respect t o topological rightof r-jets in €[,](n, equivalence and sufficiency of r-jets in € [ T + l l ( n , p )with respect t o V equivalence are studied, and necessary and sufficient conditions for suffi-

586

ciency are given. (Two map-germs f and g are topologically right-equivalent if there exists a germ of homeomomorphism h such that f = g o h, and they are V-equivalent if f -'(O) and g-l(O) are homeomorphic.) In these cases a necessary and sufficient condition is formulated in terms of a Lojasiewicz inequality which has to be satisfied.

Theorem 1.1 (Bochnak and Lojasiewicz, Kuiper and Kuo [3-51). Let z E J T ( n ,1).z i s topologically right-sufficient in &IT] zf and only i f there is a neighbourhood U of 0 and a constant C > 0 such that IIVz(x)ll 2 CIIxll"--l

(L)

for x E U . z is topologically right-suficient in €[,+11 if and only i f there exist U , C and 6 > 0 such that

IIVz(x)II 2 c11x11'-6

(L')

for x E U .

The Lojasiewicz inequality (L) (resp. (L')) in the theorem above implies that every &[,I (resp. €1,+11)-realization admits 0 as an isolated singularity. germ Consider a family f(z,t ) = z ( x ) + t p ( x ) , where p is a €[,I (resp. with j " p ( 0 ) = 0. Then the Lojasiewiwicz inequality admits the construction of a germ of a vectorfield on R" x R,which is tangent to the level-sets of f and which has a continous flow, giving a one-parameter family of homeomorphisms ht such that f t o ht = fo = z (where ft(x) = f ( x , t)). On the other hand, if no such inequality holds, then it is possible to construct two realizations of the given jet one which is non-singular outside 0 and another realization which has a sequence of Morse-type singularities tending t o 0. Hence the jet is not sufficient since two such germs cannot be topologically right-equivalent. If we consider topological right-sufficiency for jets in J ' ( n , p ) with p > 1, a similar theorem is valid, but the expression IIVz(x)II in the Lojasiewicz inequality has to be replaced by the expression d(Vzl(z), , . .Vzp(x)) where z1, . . . ,zp are the components of z and

) mpdist(Vzi(x),span{Vzj(x), 1 I j d(Vzl(z), . . . V z p ( z )=

Ip , j # i } )

(see [6] or [7]). Most theorems about topological sufficiency of jets are in this spirit, where a necessary and sufficient conditon for sufficiency is formulated in terms of a (or several) Lojasiewicz inequality(ies) which has (have) to be

587

satisfied. This Lojasiewicz inequality implies that every realization of the jet is, in some sense, non-singular outside 0. If we consider an unfolding (z(x) t p ( x ) , t ) of the given jet with all levels of the unfolding €[,I (or €[,+11) realizations of this jet, the Lojasiewicz inequality will be "stable" in the sense that each t-level f t ( x ) = z(x)+ t p ( z ) satisfies a similar inequality. This will imply that we can define a vectorfield, which has a continous flow, trivializing the unfolding. On the other hand, if the Lojasiewicz inequality does not hold for the jet z , we can find two realizations of z which cannot be equivalent, one which has some sort of singularities outside 0 and with 0 in the closure of this singular set, and one which is non-singular outside 0. Here are two such theorems characterizing U-sufficiency and topological left-sufficiency (two map-germs f and g are topologically left-equivalent if there exists a germ of homeomomorphism h such that f = hog) respectively:

+

Theorem 1.2 (Kuo, [ S ] ) . z E J T ( n , p ) is V-suficient in €[,+ll(n,p) if and only i f the following condition is satisfied: For any polynomial map g of degree r 1 realizing z , we can find positive constants C , b and a neighbourhood U such that

+

d ( V z l ( x ) ,. . .VZ,(Z))2 C11~11'-~

(L)

f o r x E U n H ( g ) where H ( g ) = {z : Ig(z)l 5 IIzllT+l}. Theorem 1.3 ( [9]). z E J T ( n , p ) is suficient in €[,](n,p) with respect to topological left-equivalence i f and only if the following condition is satisfied: There exists C > 0 and a neighbourhood U such that

f o r x, y E U . In the first theorem the condition (L) is equivalent to the condition that for every realization f of z in €[,+1](n,p), f - ' ( O ) admits 0 as isolated singularity. In the second theorem, (L) is equivalent to the condition that every realization f of z in €[,I (n,p ) is a one-to-one immersion outside 0. In general, given a topological relation on €[,](n,p) and E a subset of E[,](n,p),one can hope that a necessary and sufficient condition for sufficiency will be one or several Lojasiewicz inequalities, and these inequalities will be satisfied if and only if every realization of the jet in E has a sort of nice non-singular behaviour (or well-behavied singularities) outside 0. So

588

in each particular case one needs to find out what is the relevant nice nonsingular behaviour outside 0, and then formulate Lojasiewicz inequalities according to this. Searching for the relevant notion of non-singularity, one can consider an analogy between the theory of sufficiency and the theory of finite determinacy or infinite determinacy of map-germs. Let K be either C or R. Let R denote the group of germs of analytic or C" diffeomorphisms (in the case K = R) of (K",O) and let C denote the corresponding group of diffeomorphisms of (lKP,O). R or C acts on the set of analytic or C" germs (P, 0) + (KP, 0) by composition on the right or the left respectively. Let A = R x C. A acts on the the space of germs by composition on the right and the left. Finally let K denote the group of germs of diffeomorphisms of (K" x KP,O) of the form H(z,y) = ( h ( z )K , ( z ,y)) where K ( z ,0) = 0, acting on germs f : (K", 0) + (KP, 0) by the formula ( H . f ) ( z , y ) = K(h-'(z), f ( h - l ( z ) ) .R, C and K-equivalences (defined by these group actions) can, in some sense, be considered to be the smooth or analytic counterpart to topological right-equivalence, topological left-equivalence and V-equivalence respectively. Let G be any of the C" or analytic groups R , K , C and A. We say that a germ f is finitely determined with respect to 4 , if there exists an integer k < 00 such that any germ g with j ' g ( 0 ) = jkf (0) is in f ' s G-orbit. Letting K = IR and putting k = 00 (and letting j""f(0) denotes f ' s formal Taylor series at 0), we get the analogous notion of infinite determinacy for C" germs. From the theory of finite G-determinacy with respect to any of these groups, we have necessary and sufficient geometrical conditions for finite determinacy. For any of the groups &7 acting on a germ f, we can define the extended G-tangent space of the G-orbit of f, T , G f , which is equal (in Mather's notation ) to t f ( e ( n ) ) , tf(O(n)) + f*(m,)e(f), w f ( e ( P ) ) and t f ( e ( n > ) w f ( e ( P > >in the cases R, Ic, C and A respectively (see [lo] or [ll](using a slightly different notation) for precise definitions of these spaces) . In the cases C and A we can define corresponding spaces for multigerms. We define a germ (or multi-germ) to be G-stable if T,Gf = e(f). In the case R, this means geometrically that f is a submersion. A germ f is K-stable at a if either a is a non-singular point or f ( a ) # 0. A germ (multigerm) at a (or a set S) is C-stable if it is an immersion at a (or a one-to-one immersion at S ). The following theorem characterize finite G-determinacy in terms of G-stability.

+

589

Theorem 1.4 (GaRney, Mather [lo]). A complex analytic germ is finitely G -determined in the case R or K (L or A) i f and only if it has a representative whose germ (or multigerm) at any point (set of points) outside 0 is G-stable. W h e n G = A, we must also suppose that the germ is K-finitely determined to get the conclusion above.

If f is real-analytic f is finitely G-determined if and only if its complexification satisfies the criterion in the above theorem. In the cases R, K and C,we find that the geometrical conditions the finitely G-determined complex analytic germs have to satisfy, are exactly the same as the conditions we characterized as nice non-singular behaviour in the various cases of sufficiency. So if we want to study notions of sufficiency with respect to other topological equivalence relations, a reasonable strategy can be to look into corresponding smooth or analytic equivalence relations and seek for geometrical conditions characterizing finite determinacy for these relations. Then one should formulate Lojsiewicz inequalities which should be equivalent with the condition that every representative of the jet satifies these geometrical conditions. We will now follow this strategy in two cases. Firstly, we will look into the case of topological right left-sufficiency for jets from the plane to the plane. Secondly, we will consider jets with line singularities (jets with a singular set which contains a one-dimensional manifold) and characterize sufficiency of such jets with respect t o topological right-equivalences which leaves the singular line invariant. 2. Right left-sufficiency of jets from the plane to the plane

The discussion in section 1 indicates that if one wants to prove theorems about topological right left-sufficiency one should formulate Lojasiewicz inequalties implying d-stability outside 0. Since, however, to be finitely d determined is a property not holding in general (see [12] for a discussion), but finitely right-left topological determinacy holds in general (see [13]),we expect that the geometrical conditions for finitely d determinacy are only relevant in the so-called semi-nice dimensions (where finitely d-determinacy holds in general). In the papers [14] and [15], Lojasiewicz inequalities implying d-stability outside 0 are given in general, and it is (among other things) proven that they are equivalent with infinite d-determinacy. In this case the exponents of the inequalities are of no importance. In the case of

590

sufficiency of jets, finding the correct definition of the Lojasiewicz inequalities will be much more delicate, and the exponents in the inequalities must be adjusted such that they imply stability outside 0 for the class of representatives we consider. (See for example Theorem 1.1 and the difference of the exponent considering representatives in 1€. or &[,+ 11 respectively.) This indicate that for each pair of dimensions ( n , p ) ,one much identify the stable singularities, and put up inequalities, with properly adjusted exponents, implying that all €[,] (or perhaps €[,+l~)representatives only have these singularities outside 0. We will now follow this strategy for a class of jets from the plane to the plane. Proofs and more technical details will appear in [2]. Apart from regular points, the stable singularities for plane-to-plane mapgerms are folds and cusps and also pair of folds intersecting transversally in the target. Cusps and pair of folds are stable singularities which are isolated. So they cannot have 0 in its closure for an analytic germ which is stable outside 0. Therefore we expect that the only singular points that can occur for a sufficient jet are fold points, and we must put up Lojasiewicz inequalities avoiding non-fold singularities and singular double points outside 0. We will now formulate such inequalities and state results on sufficiency of jets from the plane to the plane. Let J2(2, 2) be the set of 2-jets (R2,0) -+ (It2,0). An element z E J2(2,2) can be identified with a polynomial map

+ + ex2 + 2fzy + gy2,cx + dy + hx2 + 2izy + j y 2 ) .

z ( z ,y) = (ax by

Now J 2 ( 2 , 2 ) can be identified with RIO by identifying z with the tuple ( a ,b, . . . , j ) . Given a jet z E J 2 ( 2 , 2), we also consider the splitting

and thus identify z with a point in R4 x R6 via this splitting. Consider €[,1(2,2), the set of CT-germs f : (R2,0) -+ (R2,0). Let T 2 2 and let f : U -+ R2 be a representative of a germ in &[,.I (2,2). For p E U we can define j 2 f ( p ) E J2(2, 2) as the 2-jet of f((z,y) p ) - f(p) at (z,y) = 0. For any f E CT(2,2)we can consequently define a germ j2f : (It2,0) 4 J2(2,2), and thus define the germ ( L f , H f )by ( L f , H f ) ( p )=

+

( L P f ( P )H , Pfd Consider a jet z = z ( z ,y) = (ax + . . . , cz + . . .). Then the Jacobian determinant of z is equal to J z ( z ,y) = ad - bc 2(ai - bh - cf de)z 2(aj - bi - cg df)y + terms of degree 2. Assume ad - bc = 0. Then j l z is

+

+

+

+

591

transverse to the set of singular jets if V J z ( 0 )= 2

ai - bh - cf a j - bi - cg

+ de + df

and then

+

a j - bi - cg df bh cf - de -ai

+ +

is a tangent vector to the singular set C ( z ) at 0. Let r bY

r = { ( a , . . . , j ) Ia d - b c = O ,

c J 2 ( 2 ,2) be defined

+

a j - bi - cg df -ai bh c f - de =

+ +

) (:)

via our identifications. It is then clear that I' is the set of singular jets which are not folds. Let z E J"(2,2) be a singular jet which we identify with a polynomial map z : (R2,0) --i (R2, 0) of degree 5 r. The singular set of z is an algebraic set. Let C ( z ) be the germ of this singular set. Assume C ( z )-{O) # 0. Hence, C ( z )- (0) has only finitely many topological components which we label C1, C2,.. . , C N .If z is not the zero jet, the Ci's are 1-dimensional and the Curve Selection Lemma implies that all these components have a well defined tangent direction at 0. Let J h ( 2 , 2 ) be the subset of J'(2,2) consisting of the jets for which all these tangent directions are distinct. Suppose z E J h ( 2 , 2 ) . Then we can find a neighbourhood U of 0 and constant 6 > 0 such that the sets Ui = { p E U I dist(Ci,p) 5 6 JJpll}satisfies Uin Uj = (0) when i # j , so the Ui's split C1, C2,. . . , CN,and whenever p E Ui, q E Uj for some i # j, Ilp - qll 2 Srnax(llpll ,11q11). For each c > 0 define

where J z is the Jacobian determinant and Dz is the Jacobian matrix of z and 11.1) is the usual Euclidean norm. Put Hi = H, n Ui, i = 1,.. . ,N . We have defined a hornshaped neighbourhood H , of C ( z ) - (0) in a way such that the origin is contained in H, and Ci c Hi, but Cj n Hi = 0 for i # j. Our main result is now: Theorem 2.1. Let r and H , be as above and let 11. 1 1 denote the Euclidean norm. If r > 2, then the following condition is necessary and suficient for a jet z E J h ( 2 , 2 ) to be topologically right left-suficient in €[,.1(2,2):

592

There is a neighbourhood U of the origin and constants C > 0 and such that if p E U and ( L ,H ) E I?, then

€0

>0

c

IILAP) - LII + IIHz(P> - HI1IlPll 2 IIpllr-l 7 (1) andifc
+ llYllr-l)

1 1 2 - YII

.

(11)

+

Example 2.1. Let z(z, y) = ( IC, zy yk ) for some integer k > 3. (For k = 3 this is the normal form of a cusp). We claim that this jet satisfies the conditions in Theorem 2.1. To show this we need a lemma:

Lemma 2.1. Let z E JT(2,2). Then the Lojasiewicz inequality (I) of 2.1 is implied by the following Lojasiewicz inequality:

There is a neighbourhood U of 0 in R2 and a real number C > 0 such that for all p E U ,

A proof of this Iemma will appear in [2]. Now a computation gives the following.

and J z ( z , y) = z

+ kyk-'.

It is clear that the singular set is a single curve tangent to the y-axis at the origin. This means that z E Jk(2,2). Further computation shows that

Hence, z satisfies (1'). By lemma 2.1, z satisfies (I). It is more cumbersome to verify (11). Notice that if (z, y) is close enough to the origin, then

1 11% +

He = { (GY) IJz(z,Y)) 5 EIIDZ(z,Y)\\l\(%Y)Ilk-' c { (z,y) Icyk-'[ I 2 ~ 1 y l ~ -=: ~ }H,*.

1

593

Let

H& = H,* n {(GY) IY 2 0 ) be the two components of If,*\ { O } . It is clear that if (2, y) E If,* is sufficienly close t o the origin and e is small, then 1x1 L ((k - l)lyl"-') and 1x1 5 ((k l)Iylk-'). From this follows that l[(x,y)llk-' 5 2lyl"l 5 1x1. If k is an even number and p = (2, y) E H+ and q = (x',y') E H-, then certainly the first component of z becomes dominating and since, in this case, x and x' have different signs we will get

+

IIZ ( P > - z(q) II = 11 ( 2 - 2' 2

7

ZY + Yk - Z'Y'

- Y'k

) I(

1 2 - x'l

= 1x1

+ lx'l

+ 11411k-1) 2 (IIPllk-l + 11411k-1) IlP - 411 2 (IIPllk-l

as long as llpll, llqll and 6 are chosen small enough. If k is odd, then p = (x,y) E H+ and q = (x',y') E H- can have nearly equal first components, but in this case z separates these points in the second component if the first components are getting very close. More precisely we get that

We may assume that

llpll L Ilqll. If llqll L (1/2) l l ~ l l then ,

llPllk + llQllk L (llPllk-l

and if llqll

I (1/2)

+ 11411k-1)

1

llqll L ~ ( l l P / l k - '+ IIqllk-l) IlP - 411 7

IlPll, then

llPllk + llQllk 2 llPllk 2

z1 IIPllk-l

IlP - Qll

1

2 ,(llPIlk-l + IIqllk-l)

IlP - Qll

'

So we get that 1 IMP) - z(q)II 1 s(llPllk-l

+ IIqllk-')

IlP - 441

in this case, and it will follow that (11) holds in any case. This shows that z is topologically right left-sufficient in &[k] (2,2) for every integer k 2 3.

594

The techniques developed in the proof of Theorem 2.1 will also give us another theorem about topological right-sufficiency for plane to plane jets:

Theorem 2.2. Let z = (f,g) E JT(2,2) be a singular jet and assume C ( z ) = (0). Then z is topologically right-suficient in €1.1(2,2) if and only if z satisfies the inequality (I) in Theorem 2.1. Remark 2.1. Note that to be topologically right-sufficient is by [6] (or [7]) equivalent with an inequality d(Vf ( p ) ,Vg(p)) 2 CI Ipl and this inequality is trivially equivalent with the inequality dist(jlz(p), C) 2 Cllpl/"-l (where C denotes the singular jets and C is perhaps another positive constant). The left hand side of the inequality (I) in Theorem 2.1 measures the distance from the 2-jet j 2 z ( p )to the set of singular jets which are not folds. So apriori this is a much weaker inequality than the inequality dist(j'z(p),C) 2 CIlplIT-l, but we can actually show that these two inequalities are equivalent for jets z with C ( z ) = (0), proving Theorem 2.2. Remark 2.2. So far, we have not been able to prove a version of Theorem 2.1 for general jets in J2(2,2). For jets z with singular sets C ( z ) - (0) = ClU. . .UC, such that two components Ci and Cj are tangential the distance between points in Ci and Cj can be small compared to the distance to 0. This makes pertubation arguments complicated since a jet is close to its representatives compared with the distance to 0. 3. Sufficiency of jets with line singularities We now want to look at jets of functions with line singularities, and sufficiency of such jets inside the class of realizations having line singularities. Let z : (Wn+l,0) + (R, 0) be an r-jet in J'(n 1 , l ) identified with a polynomial of degree r with T > 2. Assume that its singular set, C ( z ) , contains a 1-dimensional manifold L. After a change in coordinates, we may assume that L = R x (0) c R x Rn. We say that z is a jet with line be the set of C' germs whose singular set contains singularities. Let L. Let 726 be the set of homeomorphism germs h : (RnS1, 0) + (Rn+l,0) leaving L invariant.

+

€il

+

Definition 3.1. We say that a jet z E J " ( n 1,l) is sufficient in €{I if any two f , g in €{] with j"f(0) = j"g(0) = z are 72:-equivalent. From our discussion above we conclude that we must find out what we mean by nice non-singular behaviour in this case, and formulate Lojasiewicz

595

inequalties implying that every realizations have this non-singular behaviour outside 0. Complex analytic functions with line singularities have been initially studied by Siersma [16]. Let RL be the group of germs of analytic coordinate transformations leaving L invariant. One can define the notion of RL-finite determinacy inside the class of germs having a singular set containing L. Among other things, Siersma gives necessary and sufficient conditions for functions to be RL-finitely determined. The geometrical condition they will and must satisfy is that they have isolated line singularities. This means that they are non-singular outside L. On L outside 0 they have Morsesingularities in a direction transverse to L. In [17], Sun and Wilson study infinite determinacy of C" germs with line singularities (among C" germs having line singularities) and they obtain necessary and sufficient conditions for this in terms of Lojasiewicz inequalities. In the case of &;]-sufficiency, we will need Lojasiewicz inequalities that are equivalent with the property that every &;]- realization of the jet has an isolated line singularity. Let us denote the coordinates in Rn+' = R x R" by (x,y) = (x,y1, ...,y,). Since L then becomes the x-axis, and the partial derivatives of z vanish along L , z must have the form Z(Z,Y) =

c

YiYjzij(x,Y)

l
where zij is a polynomial of degree r - 2 and zij = zji. Let S y m ( n ) denote the symmetric n x n matrices, and let R ( n ) c Sym(n) denote the subset of singular matrices. Let D%z(x)E Sym(n) denote the n x n matrix (zij(x,0)). (The zij(x, y ) ' ~are not uniquely determined but the matrix Diz(x) is determined, beeing the Hessian matrix of z in the y-direction.) The following theorem gives necessary and sufficient conditions for a jet to be sufficient in &.];

Theorem 3.1. The conditions (1) and (2) below are equivalent (1) z is suficient in (2) There exists a constant C > 0 and a neighborhood U of 0 such that

&El.

f o r (x,y) E U , and dist(Diz(x),R(n)) 2 CIIxllr-2

f o r x E U n L.

(ii)

596

Remark 3.1. (i) is the relevant inequality implying that z is non-singular outside L and (ii) implies that z has Morse-singularities in the y-direction on L outside 0. The proof of Theorem 3.1 and other details of this will appear in [l].

Example 3.1. z ( z , y) = (y;

21~111Y11211(~~Y)ll + 2(lYll

+ yi)(z2 + y? + y;).

We have

+ lY2l)(Z2 + 211Y112)11Yll

2 211Y11211(~c,~)112,

so (i) holds since z is a 4-jet. Furthermore we have

and it is easy t o see that dist(D;z(z),R(l)) = z2, so (ii) holds and z is sufficient.

Example 3.2. Let z(z,yl,y2) = y ; + 2 ( 1 + z ) y l y 2 + ( 1 + 2 z ) y ~ + y ~ . Then it is not hard t o see that z has isolated line-singularities along the z-axis. We have

and since

(A

1+2x+x2

) EA(~),

dist(D;z(z,O),R(2)) 5 z2. So, since z E J3(2,2), (ii) does not hold. It is h(z,y) = however easy t o see directly that z cannot be sufficient in z ( z , y) +z2y2 is a C3(actually analytic)-realization of z in When z # 0 , the germ of {y I z ( z , y) = 0 ) at y = 0 is always homeomorphic with a pair of lines, but {y I h(z,y) = 0 ) is a cusp, and it is therefore clear that these two maps cannot be %$-equivalent.

&il. &il.

References 1. Brodersen, H., Suficiency of jets with line singularities. In preparation. 2. Brodersen, H., Skutlaberg, O., do suficiency of j e t s from R2 to R2. In prepa-

ration. 3. Bochnak, J., Lojasiewicz, S., A converse o f t h e Kuzper-Kuo Theorem, Proceedings of the Liverpool Singularities Symposium 192, 254-261. Springer 1971.

597 4. Kuiper, N. H, C'-equivalence of functions near isolated critical points Proc. Symp. in Infinite Dimensional Topology (Baton Rouge 1967), Annals of Math. Studies 1967, (Princeton, 1968) 199-218 5. Kuo, T.C., O n Co-suficiency ofjets of potential functions, Topology 8 (1969), 167-171. 6. Bochnak, J., Kucharz, W., Sur les germes d'applications differentiables a singularites isolees, Trans. Amer. Math. SOC.252 (1979), 115-131. 7. Trotman, D., Wilson, L., Stratifications and finite determinacy, Proc. London Math. SOC.(3), 78, 1999, 334-368. 8. Kuo, T.C., Characterizations of V-suficiency of jets, Topology 11 (1972), 115131. 9. Brodersen, H., Suficiency of jets with respect to C-equivalence, Real analytic and algebraic singularities, Pitman Research Notes in Mathematics Series 381, Longman, 1988, 78-83. 10. GafTney, T., Properties of finitely determined germs, Thesis, Brandeis University, Waltham, Massachusetts, 1975. 11. Wall, C.T.C., Finite determinacy of smooth map-germs, Bull. London Math. SOC13, 1981, 481-539. 12. du Plessis, A.A., Genericity and smooth finite determinacy, Proceedings of symposia in pure mathematics, Vol. 40, Part 1, AMS, Providence, Rhode Island, 1983, 295-313. 13. du Plessis, A.A., O n the genericity of topologically-finite determined mapgerms, Topology 21 (1982), no.2, 131-156 14. Brodersen, H . , O n finite and infinite Ck - A determinacy, Proc. London Math. SOC.75 (1997),369-435. 15. Wilson, L., Mapgerms infinitely determined with respect to right-left equivalence, Pasific J. Math, 102 (I), 1982, 235-245. 16. Siersma, D. Isolated line singularities , Proceedings of symposia in pure mathematics, Vol. 40, Part 2, AMS, Providence, m o d e Island, 1983, 485-496 17. Sun, B., Wilson L.C. Determinacy of smooth germs with real isolated line singularities , Proc. of the American Math. SOC.129 (2001), 2789-2797

598

SINGULARITY THEORY APPROACH TO TIME AVERAGED OPTIMIZATION A. DAVYDOV * and H. MENA-MATOS**

* Vladimir State University, Russia; IIASA, Laxenburg, Austria E-mail: [email protected], [email protected]. at ** Faculdade de Cigncias and CMUP Universidade do PoTto PoTto, Portugal E-mail:[email protected] Using singularity theory tools we study the time averaged problem for control systems and profit densities on the circle. We show that the optimal strategy always can be selected within periodic cyclic motions and stationary strategies. When the problem depends additionally on a one dimensional parameter, we analyze generic transitions between these two types of strategies under change of the parameter and classify all generic singularities of the averaged profit as a function of the parameter.

Keywords: Singularities; Averaged Optimization; Control Systems.

1. Introduction

Consider the following smooth control system on the circle S1:

i = w(2,u) where x is an angle on the circle and u is a control parameter belonging to the control space U , which is a smooth closed manifold or a disjoint union of smooth closed manifolds with at least two different points. An admissible motion of the control system is an absolutely continuous map x : t H z ( t ) from a time interval I to the system phase space S1 for which the velocity of motion (at each moment of differentiability of the map) belongs to the convex hull of the admissible velocities of the system, more precisely i ( t )E [w,i,(~(t)),w,,,(2(t))] for a.e. t E I , where w,in(x) = min w(x,u)and wmaz(x) = maxv(z, u)denote the minimum and UEU

maximum admissible velocities at

UEU 2, respectively.

599

Remark 1.1. Any admissible motion can be defined for all t E IR because the phase space is compact. Together with a smooth profit density f : S14 IR on the circle, the control system gives rise to the following optimal control problem:

To maximize the averaged profit o n the infinite time horizon

over all the system’s admissible motions o n the positive semiaxis. Remark 1.2. If the last limit does not exist one must take the upper limit. This optimization problem is an important problem of control theory and includes in particular the optimization of the averaged profit of periodic processes when the phase space is the circle. Such problems are well known and were treated in different ways 1-3. In this work we look to this problem through singularity theory. When the problem depends on parameters, that is when both the control system and the profit density depend additionally on parameters, then the optimal strategy can vary with the parameters and the optimal averaged profit on the infinite time horizon, as a function of the parameters, can have singularities (points where it is not smooth). We are so led to the problem of classifying such singularities. This singularity theory approach was firstly proposed by V.I. Arnold in Refs. 4, 5 and 6. V.I.Arnold showed that between the motions providing the maximum averaged profit on the infinite time horizon there can appear 0

0

a level cycle: motion using the maximum and minimum velocities when the profit density is less or greater, respectively, than a certain constant, or a stationary strategy (equilibrium point): the staying at a point with admissible zero velocity.

He also studied some of the respective generic singularities of the maximum averaged profit as a function of the parameter in the case of 1-dimensional parameter Here the analysis of the one-parametric Arnold’s model is continued. We prove that the strategy providing the maximal averaged profit on the infinite horizon always can be found inside these two kinds of motions. But due to the larger concept of admissible motion, our notion of equilibrium

‘.

600

point is wider, namely, such a point is any one where the convex hull of the admissible velocities contains the zero velocity. Additionally we analyze transitions between optimal strategies of these two types and prove that only two forms of them can appear in a generic case under change of the one-dimensional parameter. The respective singularities of the maximum averaged profit on the infinite horizon as a function of the parameter are found, and their stability, up to a small perturbation of a generic family of pairs of control systems and profit densities, is proved. Notice that, in the case of a one dimensional parameter, the obtained results are true for a generic family of control systems (profit densities) when a generic family of profit densities (control systems, respectively) is fixed. 2. Optimal strategies: level motions and equilibrium points

Here we state and prove the selection theorem for the optimal strategy. 2.1. Stationary strategies

A control system has the local transitivity property at a point of the phase space if for any neighbourhood V of this point there exist a time T and another neighbourhood such that any two points from can be joined through an admissible trajectory lying inside V and in time less than T .

v

v

Example 2.1. Let in the flat sea which is represented by the real plane Oxy the water stream be defined by a continuous vector field (211,112). A swimmer staying in the sea is uninertial and can swim in any direction with a velocity less or equal 1. The respective control system is defined by the dynamic inequality (k- u1(x,y))’ (0 - 1 / 2 ( 5 , Y ) ) ~5 1. The system has the local transitivity property at any point of the domain where the modulus of the drift is less than 1 7-g. Generically for a smooth drift the system also has the local transitivity property at a typical point of the boundary of this domain 9,10.

+

When a control system on a smooth manifold has the local transitivity property at a point, then the upper limit of the maximum averaged profit is not less than the value of the profit density at this point. To show that, it is sufficient to construct an admissible motion near such a point which tends to it when time goes to infinity. It is clear that the averaged profit for such a motion on the infinite time horizon is the value of the profit density at this point.

601

Such a motion can be defined in any sufficiently small neighbourhood V of the point. Staying within such a neighbourhood one can think without loss of generality that we are in an arithmetical space and that the point is the origin. For a natural number i we define the neighbourhood V, which is obtained from V by a homotety with centre at the origin and coefficient l/i, that is V, = (2111:= y / i , y E V } . Due to the local transitivity property of the system at the origin, there exists a sequence of neighbourhoods such that any two points from the neighbourhood can be joined in a time less than some fixed time T > 0 by an admissible motion whose trajectory lies in the neighbourhood V , (Fig. 1).

v,

Fig. 1. Local transitivity

Now consider a point y E V1 (=V) different from the origin and glue the admissible motions from the origin to y and back. If TI is the duration of the glued cyclic motion we repeat such a motion n1 times such that nlT1 2 1. For i 2 2, the point yi = y / i and the neighborhoods V, and we do the same and get a glued motion mi for a time Ti repeated ni times. Finally we define the motion m gluing all these motions (= nimi) sequentially. The motion m is defined on the positive time semi-axis and lies in the neighbourhood V , for time t 2 nlTl+ n2T2. . . ni-1Ti-1. It is clear that the averaged profit along it tends to the value of the profit density at the point under consideration (=origin). If the phase space is a smooth connected compact manifold and a control system has transitivity at each of its points then any two points of the space can be joined within a finite time, as it is easy to see. In such a case any admissible motion, that converges t o the point where the profit density attains its global maximum, provides the maximum averaged profit on the infinite horizon. For the parameter depending case, when such situation occurs for any value of the parameter then the maximum averaged profit

+

602

A on the infinite horizon is defined as A = m={f (z, P ) b E MI,

(1)

where M is the phase space, p is a point in the parameter space and f is the family of profit densities. The classification of generic singularities of the maximum of a family of functions depending on parameters (=family of profit densities) up to dimension 5 6 of the parameter were obtained by L. Bryzgalova l 1 , l 2 . These singularities are stable up to small perturbations of the family of functions (profit densities) (see also Refs. 13, 14, 15, and 16). For example, for one and two dimensional parameter the result is the following

Theorem 2.1. For a generic family of profit densities the germ of the solution of equation (1) at any point of the parameter space is R+-equivalent to the germ at the origin of

either (1)0

or

(2) lpll

when the parameter is one-dimensional and in addition either (3) m a x { ) p l l ,p2}

or else

(4) max{-w4 +plw2 +p2w I w E

R}

in the two-dimensional case. Remark 2.1. A property holds for a generic object if it holds for any object belonging to some open everywhere dense subset in the space of objects with the chosen topology. In our case we work with the fine smooth or sufficiently smooth topology. In Theorem 2.1, pl and p2 are appropriate local coordinates in the parameter space near the point under consideration; the R+-equivalence admits diffeomorphisms of the function’s domain and the adding of a smooth function.

Fig. 2. Graph of the maximum function (singularities (2), (3) and (4)respectively) and its projection 7r on the parameter space

603

Remark 2.2. The singularities in Theorem 2.1 appear below from our studies. For the singularities of the minimum function we just have to change the sign in the normal forms of Theorem 2.1. For that case they can also be observed as singularities of the distance function in a typical lake with “three” ends (Fig. 3); the lake and beach are white and grey, respectively, and the points with numbers (1) - (4)nearby provide the respective singularities (with sign minus) of Theorem 2.1 for the distance function. These singularities are connected with the singularities of conflict sets studied in Refs. 17 and 18.

Fig. 3.

Singularities of the distance function

Thus the case of a family of completely controllable systems is reduced to a well known problem in singularity theory. We define an equilibrium point as a point where the convex hull of the admissible velocities contains the zero velocity. Notice that our definition of admissible motion permits directly the staying at such a point. The union of such points is called stationary domain. When the control system is not completely controllable but has equilibrium points and permits only stationary strategies due to some constraints, the maximum averaged profit is also the solution of an extremal problem like (1). But in that case the maximum should be taken among the equilibrium points only. The optimal strategy in such a case is a stationary one, e.c. the staying at the equilibrium point providing the maximum of the profit density among such points. The classification of the respective generic singularities for a one-dimensional parameter is presented in Section 4.

604

2.2. Cyclic motion Many natural periodic processes with control can be described by a control system on the circle with positive velocities only. In such a system there is no other possibility than to rotate along the circle selecting admissible velocities. For a value c of the profit density f , we define the c-level motion as the one using the maximum and minimum admissible velocities at points where the profit density is not greater and greater than c respectively. The c-level motion is so the motion that uses the following velocity: maxv(z, u ) if f (z) I c minv(z,u) if f(z) > c

A value of the profit density is called cyclic if for all nearby values, the respective level motions provide rotation along the circle. For example, for a system with positive velocities only, all values of the profit density are cyclic. For a cyclic value c we call its level motion c-level cycle or just level cycle. The period of a level cycle is its smaller period. Theorem 2.2. For continuous control system and profit density o n the circle, the best averaged profit o n the infinite time horizon always can be provided by a level cycle or by a stationary strategy. Remark 2.3. Notice that there can exist a lot of different optimal motions. For example, the change of an optimal motion on any finite interval of time preserves its optimality. To prove Theorem 2.2 the following statements are useful.

Lemma 2.1. If an admissible motion of a dierentiable control system comes in (or outcomes from) a point in finite time, then at that point the maximum velocity in the direction of the motion is positive. Lemma 2.2 ( 1 9 ) . If for a continuous control system and profit density, an admissible motion x ( t ) , 0 I t < 00, provides an averaged profit A o n the infinite horizon and some arc of its trajectory between points z ( t l ) ,x ( t 2 ) , with f ( z ( t ) )less (greater) than A for all t E [ t l , t z ] ,can be crossed faster (slower, respectively) then the averaged profit of the modified motion with the faster or lower crossing, respectively, provides no worse averaged profit on the infinite horizon. Firstly we prove Theorem 2.2 and then the lemmas. The proofs are those of Ref. 20.

605

Proof of Theorem 2.2. Denote by f and A the profit density and the maximum averaged profit on the infinite horizon, respectively. When the domain {x : f (x)2 A} (really the level {x : f(x) = A}) contains equilibrium points then the maximum averaged profit can be provided by one of this points through a stationary strategy. If it does not, then for sufficiently small E > 0 the closed domain D = {x : f(x) 2 A - e} also does not contain equilibrium points due to continuity of the profit density and of the maximum and minimum velocities. In particular, in this domain these velocities have the same sign and are separated from zero. So, any admissible motion on a connected component of the domain D has to leave it in finite time and can income back to it only after a complete rotation along the circle. Hence the number of rotations along the circle of an admissible motion providing the maximum averaged profit has to go to infinity as time goes to infinity. Otherwise this motion would have to spend a finite time in the domain D and an infinite one in the rest part of the circle, where the density is less than A - E. Consequently, the averaged profit of the motion on the infinite horizon would be not greater than A - E , what contradicts the optimality of the motion. On each cycle of the rotation the motion under consideration can be improved using Lemma 2.2. In fact, the existence of such rotation permits us to conclude by Lemma 2.1 that the maximum velocity in the direction of its motion is always positive. In particular, it is separated from zero because of its continuity and compactness of the circle. The A-level cycle uses it on the set f(x) I A. On the rest part f(x) > A of the circle the improved motion has to use the minimum velocity which is also separated from zero on this part due to the absence of equilibrium points in its closure. Thus the modified motion is the A-level cycle. Its averaged profit on the infinite horizon is not less than the one for the initial motion, e.c. no less than the value A, but it is also not greater than this value due to the optimality of A. So Theorem 2.2 is valid. 0 Proof of Lemma 2.1. It is clear that at a point 20,the maximum velocity in the direction of the entrance or outcome of the motion has to be nonnegative, due to continuity of the maximum velocity. But the maximum (minimum) velocity V of a differentiable control system, having a closed smooth manifold or a disjoint union of closed smooth manifolds as the control parameter’s set, is a Lipschitz function. So if V vanishes at the point xo then near this point IV(x) - V(x0)l 5 Clx - 201 with some positive

606

constant C. Therefore it is not possible to come in or walk out the point in

s dxlx and s d x l x are not finite. That 0

finite time, because the integrals

0

-€*

contradicts the assumptions, and so the value V(x0) must be positive. Proof of Lemma 2.2. We analyse only the case of the faster crossing. The other case can be studied in the same way . Without loss of generality we can assume that the profit density is positive, because the addition of any constant to it leads to the addition of the same constant to the averaged profit. Denote C = max{f(x(t)), t E [tl,tz]} and A(Ti)the averaged profit of the considered admissible motion on the interval [0,Ti]. Due to the definition of averaged profit on the infinite horizon there exists a sequence of times {Ti}such that Ti + 00 and A(Ti)--t A when i + cm.Let us take i so big that Ti > t 2 and A(Ti)> C. Denote by A, 0 < A < T ,the economy of time when we pass the arc between x(t1) and x(t2) faster. Calculating now the difference D between the averaged profits of the initial motion on the interval [0,Ti]and the modified one on the interval [0,Ti - A]we get

D 5 A(Ti)- TiA(Ti)- CA Ti - A

A Ti - A (C- A(Ti))< 0.

--

So the motion with the faster crossing of the arc between x(t1) and z(t2) provides no worse averaged profit on the infinite horizon than the initial one and Lemma 2.2 is proved. 0 Remark 2.4. When we have a family of control systems and profit densities, the optimal strategy can vary depending on the parameters and the best averaged profit, as a function of the parameters, can have singularities. For example, this profit can be discontinuous, even when the families of control systems and densities are smooth 6 . Theorem 2.2 gives us the possibility to subdivide these singularities into three groups in order to analyze them, namely, the singularities for stationary strategies, for level cycles and for transitions between stationary strategies and level cycles. We formulate the respective results in the following three sections. 3. Averaged Profit singularities for level cycles

We will now present some general properties concerning the profit, the period and the averaged profit for level cycles. All the results are true locally near a cyclic level of the profit density. For the case of a control

607

system with positive velocities only, all the levels of the profit density are cyclic and the results are valid for all values of the profit density. 3.1. Monotonicity and continuity of the profit and the period for level cycles Proposition 3.1 (19). The period of a level cycle decreases continuously as a function of the level near any cyclic value of the density i f the differ-

entiable profit density has only a finite number of critical points (=zeros of its derivative) and the maximum and minimum velocities of the continuous control system coincide at isolated points only. Proof. Under increasing of the cyclic value of the profit density, the mea-

sure of the domain where the minimum velocity is used decreases. This measure decreases continuously because the profit density has only a finite number of critical points. Consequently the period of the cyclic motion decreases continuously because the minimum and maximum velocities are continuous and coincide at isolated points only. To prove the next statement one can use the same arguments but must take into account the sign of the cyclic value. (19). The profit along a level cycle decreases or increases continuously as a function of the level when this level is positive or negative, respectively, if the differentiable profit density has a finite number of critical points and the maximum and minimum velocities of the continuous control system coincide at isolated points only.

Proposition 3.2

Remark 3.1. In these forms the statements of Propositions 3.1 and 3.2 were proved by E.O.Kukshina (2004). In general they are not true when there are intervals of coincidence of the maximum and minimum

velocities or intervals of critical points of the profit density. Under more strong assumptions these statements actually are presented in the paper of V.I.Amold '. E.O. Kukshina also showed that under the conditions of Propositions 3.1 and 3.2, the profit along a level cycle is a two times differentiable function of the cycle period. Remark 3.2. The statements of Propositions 3.1 and 3.2 are true for a generic smooth pair of families of control systems and profit densities with a finite dimensional parameter, because a generic family of profit densities on the circle has only points of finite multiplicity, and so a finite number of

608

critical points. Besides, the maximum and minimum velocities of a generic smooth family of control systems with a finite dimensional parameter never coincide if the number of different values of the control is at least the dimension of the parameter plus two. For such a family and any fixed value of the parameter these velocities always coincide in no more than a finite number of points. 3.2. Diflerentiabilitg of the averaged profit for level cycles

We denote T(c) and P(c) the period and the profit respectively of a c-level cycle. Its averaged profit is so given by P(c)/T(c).

Theorem 3.1 (19). W h e n the maximum averaged profit is provided by a cg-level cycle then the respective cyclic value cg is the unique solution of equation

if the dijjerentiable profit density has a finite number of critical points and the maximum and minimum velocities of the continuous control system are equal at isolated points only.

Proof. Firstly we show that the optimal cyclic value cg must be a solution of equation (2). Let A(c) be the averaged profit of the c-level cycle. Suppose that A(%) < cg. Let E > 0 be small enough so that 6 < cg - A ( @ )and the (cg - €)-level cycle is well defined. Then A = T(cg - E ) - T(cg) is positive due to Proposition 3.1, and (T(cg) A)A(cg - E ) - T(cg)A(cg) > (cg -€)A.

+

so

what contradicts the fact that cg is the optimal cyclic value. Analogously one can show that A(%) can not be greater than cg. So we conclude that the optimal cyclic level must be a solution of equation (2). Now we will show that there exists at most one level cycle providing the maximum averaged profit. Assume the opposite, that there exist two cyclic values c1 and cg , c1 < c2. providing the maximum averaged profit. All values between them are also cyclic as it is easy to see. Subtracting the constant c1 from the profit density we reduce to zero both the cyclic value c1 and the profit along the cl-level cycle. After that subtraction the profit along the c-level cycle, c = ( c ~ cl), has to be zero too. But that contradicts the strict monotonicity of the profit

609

for positive cyclic values which was stated in Proposition 3.2. So there can exist only one level cycle providing the maximum averaged profit. 0

Remark 3.3. Actually equation (2) was found by V.I.Arnold in Ref. 6 under construction of an optimal distribution, for example, for the efforts under the retrieval of an object with a given distribution in such a way to provide the maximum averaged effect which is the part of the detection probability corresponding to the unit of the efforts applied. However, in Ref. 6 , uniqueness of solution can not take place because for uniqueness, conditions of or similar to those of Theorem 3.1 are essencial. For example, under the appearance of an interval of critical points of the profit density with a critical value Q providing a level cycle, the period function T = T ( c ) gets a discontinuity at the point Q. Besides the selection of the motion’s velocity on this interval has no influence on the averaged profit if G, is the maximum one, as it easy to see. So in such a case there are a lot of other cyclic motions providing the maximum averaged profit. That corresponds to the nonuniqueness of the optimal distribution studied in Ref. 6 . When the density has a finite number of critical points then each one of its levels consists of a finite number of points too, and intervals where it is constant are absent. Under the presence of intervals of coincidence of maximum and minimum velocities the uniqueness of solution of equation (2) also can be broken. Theorem 3.2. The averaged profit along a level cycle is a differentiable function of the level near the cyclic value providing the maximum averaged profit, if the control system is continuous and the differentiable profit density has a finite number of critical points. Proof. Outside the critical values of the profit density, the differentiability was proved by V.I.Arnold 6 . As the number of critical values is finite, to prove the statement in general situation, it is sufficient to show the differentiability at the cyclic value % providing the maximum averaged profit. Subtracting this value from the profit density we do not change the differentiability of the averaged profit but reduce these value and the profit to zero. For a differentiable density with a finite number of critical points the measure of the domain where the maximum (minimum) velocity is used changes continuously. Besides after the subtraction of ~0 the absolute value of the density where this domain changes by the varying of the level near zero is no more then the modulo of this varying. Consequently now the

610

averaged profit has zero derivative at zero (= the cyclic level providing the maximum averaged profit). 0

Remark 3.4. We notice that even in a generic situation both the period of a level cycle and the profit along it can be non-differentiable at the level value providing the maximum averaged profit. For example, that is so when the maximum averaged profit crosses the level of a non-degenerate local extremum of the profit density in a generic case.

Remark 3.5. Theorems 3.1 and 3.2 play an important role in the classification of generic singularities of the maximum averaged profit for level cycles. The derivative has to be zero at a level providing the maximum averaged profit, and the derivative of the left hand side of equation (2) at this level is equal to 1. By the implicit function theorem that gives us the possibility t o recalculate the singularities of the best averaged profit through the ones of the period of the level cycles and the profit along them. 3.3. Singularities of the averaged profit f o r optimal level cycles

In this subsection we present the generic singularities of the maximum averaged profit for level cycles in the case of a one parameter family of pairs (f,C) of profit densities and control systems on the circle. By generic singularities we mean those that arise for a generic smooth one parameter family of such pairs. The equivalence relations involved in the classification of the optimal averaged profit are the following: r-equivalence, Fa-equivalence and R+equivalence. We recall that two germs of functions are l7-equivalent if their graphs are equivalent, by a smooth diffeomorphism preserving the natural foliation over the function’s domain. The diffeomorphism carrying one graph into the other has the form ( p , a ) H (cp(p),h(p, a ) ) where p belongs to the function’s domain and a E R.F,-equivalence is a r-equivalence which is affine along the function’s target and R+-equivalence is the particular case of r-equivalence when the second component h of the diffeomorphism is of the form a $ ( p ) , where 1c, is a smooth function. It is clear that the germ of a smooth function at a point is R+-equivalent to the germ of the zero function at the origin. The germ of the maximum averaged profit A at a point po corresponding to a generic one parameter family of pairs (f,C) is called r-stable, if there exists a neighborhood of po, such that for any one parameter family of pairs

+

611

sufficiently close to ( f , C), the graph of the respective maximum averaged profit in that neighborhood is reduced to the graph of A by I?-equivalence (close to identity). Theorem 3.3 (19). O n the circle f o r a generic smooth one parameter f a m i l y of pairs of profit densities and control systems with positive velocities only, the germ of the ma&mum averaged pmfit at any value of the parameter, is equivalent to the germ at the origin of

where g ( p ) is one of the functions in the second column of Table 1 and the equivalence is the one pointed out in the third column. Besides, all these singularities are r-stable. Table 1.

-P

r 5 6 7

PL P3 -p7/2

R+ R+

r

#U>2 #U 2 2, transition through a local minimum of the profit density #U 2 2, transition through a local maximum of the profit density #U 2 2, transition through a tangent double point of the velocity used #U 2 3, transition through a triple point #U 2 2, switching a t a regular double point dim U > 0, transition through a swallow point ~

Remark 3.6. This classification also works in a generic case (not assuming the positiveness of the velocities) near a cyclic value of the parameter providing the maximum averaged profit. In Ref. 19 the sign"-" in the last singularity is missing. To make the statement more clear we will give some explanations. The maximum velocity of a one parameter family of control systems on the circle with a smooth closed manifold U of control parameter values is either smooth or can have generic singularities of three types, namely, its germ at any point is R+-equivalent to the germ at the origin of one of the following three functions:

1x1

max{lxl,p)

max{-w4 + p w 2

+ xw I w E R)

(3)

612

which are just the last three singularities of Theorem 2.1. For the minimum we have t o change the sign of these functions. Notice that these functions are not the normal forms for the germs of the vector fields themselves. We will call a point with one of these singularities, double, triple or swallow point, respectively. Transversality theorems imply that in a generic case the closure of the set where the minimum or maximum of a family of functions is not differentiable (=Mumell set) is either empty or is - a smooth curve when the number #U of different control parameter values is equal to 2, or - a smooth curve with triple points when #U = 3, and triple points for minimum and maximum velocities axe the same, or else - a smooth curve with triple (and swallow) points with transversal selfintersection outside them when #U > 3 (dimU 2 1, respectively), and triple and swallow points for minimum and maximum velocities are different. Besides, this set is stable, namely, for a generic family of control systems and any one sufficiently close to it such sets can be reduced one to another by a smooth diffeomorphism close to the identity (see Refs. 14, 11, 12 and 21). Hence in a generic case this set has a typical replacement with respect to the natural foliation of the ambient space over the family’s parameter space. Consequently the Maxwell set of a generic family of control systems can be tangent to fibers of this foliation only at points where it is smooth and exactly with first tangency order. Besides for any value of the parameter, at the phase space there is no more then one double point with such tangency (=tangent double point), triple or swallow point, or else a point of selfintersection of this set. All that follows from multijet transversality theorem. A point where the Maxwell set is smooth and which is not a tangent double point, is called regular (Fig. 4). The last column of Table 1 provides information about the cause implying the respective singularity. By Theorem 3.1 the optimal profit for level cycles is the solution with respect to c of equation (2) that in parametric case has the form:

Due to the implicit function theorem, for a value of the parameter, the solution near the level of the optimal cycle has not lesser class of differentiability than the functions of the cycle period and the profit along it. But in a generic case these functions can lose differentiability only in the following three cases:

613

fibers p = const Fig. 4. The Maxwell set ((1)triple point; (2)swallow point; (3)tangent double point; (4) regular point; ( 5 ) point of selfintersection)

A) the optimal level is critical for the profit density, and that implies the nondifferentiability in some integral limits for the profit and the period; transition means the crossing of this level by the maximum averaged profit under change of the parameter; in a generic case such transition gives singularities 2 or 3 of Table 1; B) the maximum or minimum velocity has a singular point of tangent double, triple or swallow type inside the domain where it is used; transition here means the crossing (under change of the parameter) of a parameter value providing such a singular point; generically that implies singularities 4, 5 and 6 of Table 1; the analogous situation of the appearence of a regular double point has no influence on the differentiability of the functions, as it is easy to see; C) the switching between maximum and minimum velocities on the optimal level cycle takes place at a point of the Maxwell set; in a generic one parameter case that can happen only at regular points of this set, and that provides singularity 6 of Table 1. After these remarks we outline the proof of Theorem 3.3: Firstly we analyze “point singularities” , namely, the conditions which define locally each one of these three situations. After that, using the theory of local normal forms and transversality theorems, we classify them assuming the absence of any additional phenomena of this type for a given value of the parameter. Finally, using multijet transversality theory, small perturbations of the family of densities and continuous dependness of the period of the level cycle and the profit along it, we show that in a generic one dimensional parameter case it is not possible to have any two phenomena described in

614

A), B) and C) appearing for the same parameter value. We illustrate now how this procedure works for the transition through a swallow point. All other cases can be treated in an analogous way. 3.4. Transition through a swallow point We will follow mainly the proof given in Ref. 19. For a parameter value PO consider the level cycle that provides the optimal averaged profit A. For this level motion suppose that the maximum velocity has a singular point ( z 0 , p o ) of swallow type inside the domain where it is used (the case of a swallow point of the minimum velocity is proved analogously). We shift the point ( z 0 , p o ) t o the origin. After an appropriate choice of a new smooth control parameter smoothly depending on z and p , the inverse value w of the maximum velocity near the origin takes the form

where a , p and y are smooth and a(0,O)= 0 = p(0,O) < y(0,O). In fact W h P )=

1 ?Jm&,

-

P)

1

maxu€u 4x7 P ,).

= min

1

4 2 , P , ).

and using the versa1 deformation for the A3 singularity l4 we get the previous normal form. Because the velocity that is used near the origin is the maximum velocity, f(0,O) 5 A. Due to a transversality theorem, in a generic case (pzap- azpp)(0,O) # 0 (because (0,O) is a swallow point) and the Maxwell set (in particular, the set where the function w is not smooth) is not tangent to the natural fibering over the parameter. This implies ,&(O,O) # 0. Due to Mather theorem we can write /3 on the form p ( x , p ) = (x b ( p ) ) B ( z , p ) .By the coordinate change 5 = x b(p) we reduce, near the origin, w to the form (tilde is omitted):

+

+

w(x, P ) = uER

+ a ( x ,P ) U 2 + xcp(x,P ) U ) + Y(Z,P )

with some new functions a , p and y satisfying a(0,O) = 0 # ap(O,O), p(0,O) # 0 and y(0,O) > 0. We can suppose p(0,O) > 0 because if this is not the case we just have firstly to change u into -u.Due to Mather division theorem the new function a near the origin can be presented as a ( x ,p ) = a ( p ) x G ( z , p ) with some smooth functions a and 1c, with a(0) = 0 # ap(0).Choosing a new smooth coordinate fi = - a ( p ) / 2 we can present near the origin the function w in the following form (tilde is omitted):

+

615

x=a Fig. 5. swallow point

Next we will localize the influence of the maximum velocity’s singularity on the averaged profit in order to analyze the singularity of the profit which appears under transition through the swallow point. For a small positive value a the level cycle providing the maximum averaged profit uses the velocity with the inverse value defined by (4) in the strip 1x1 5 a for small parameter values p . Near the point (c = A , p = 0) the time of the motion and respective profit along a c-level cycle are given by +a

+a

T(a,c,p) + / w ( x , P ) d Z -a

and P(a,c,p)

+/w(x,P)f(x,ddx,

(5)

-a

respectively, where T and P are the time and profit of the motion along the c-level cycle outside the strip 1x1 5 a. To get the optimal motion inside this strip one must use the controls leading to the minimum of u4+(xa(x,p)-2p)u2+~~(x,p)u. Those controls are the maximal and minimal controls uma2,umin satisfying equation 4u3

+ 2 ( x a ( z , p )- 2p)u + xcp(z,p) = 0

(6)

for z negative and positive, respectively (Fig. 5 ) . Due to the implicit function theorem, equation (6) has a unique smooth solution near the origin:

616

+

where the “dots” stand for terms ukpl with k 1 2 4. Including in T and P of (5) the smooth parts of the integrals corresponding to the term y in the expression of w and substituting the variable x by u in the remaining integrals we get the following expressions for the time and profit of the motion along a c-level cycle: Urnin(a,p)

J

T ( a ,C , P ) +

Urnaz(0,~)

G(u,p)du+

Urnin (

Urnas(-a,p)

= &(36u6

J

- 36pu4

G(u,p)du

0,~)

(7)

+ 8p2u2)+ . . .

and

F(’ZL,P)= f(X(u,p),p)G(’LL,p) = W ( 3 6 u 6- 36pu4

(8)

+ 8p2u2)+ . . . , +

where the “dots” represent the terms ukpz even on u with k 21 > 6 and the odd terms on u. /, and The limits umaz(O,p) and umin(O,p) are equal respectively to @ -4for a small p 2 0 . It is easy to see that the functions

urnaz(-a,p)

urnar(-a,~)

do not depend on a and have the same class of smoothness of the functions T and P, respectively. We denote these new functions by the same letters T and P. So the expressions for the time and the profit along a c-level cycle take the forms 4 fi

+

T(c,P)

1

-&

+

G(u,p)du and P(c,P)

1

-4

F(u,p)d~,

617

for (c,p) near (A,O).Notice that the integrals are present only for p 2 0. In a generic case the functions T and P are smooth. In fact, for that it is sufficient that the profit density’s level A is not critical. That could be achieved by small perturbations of a generic family, namely, of its densities. After that this property is preserved under new small perturbations of the family due to stability of the set of critical values of a generic family of densities 22 and due to stability of the Maxwell set for a generic family of systems (see, for example, Refs. 14 or 21). Consequently in a generic case the new functions T and P are smooth near the origin. In the same way one can show that in a generic case at a swallow point the value of the profit density is different from the maximum averaged profit for the respective value of the parameter. In particular f(0) < A for the case under consideration. So after omitting the integrals in the last expressions of the period and the profit, the respective maximum averaged profit A*(p) is also smooth near the origin due to Theorem 3.1 and to the implicit function theorem. After subtracting A*(p) from the profit density we reduce to zero the optimal cyclic levels for p near the origin. So we get near the origin P(0,p ) = 0 and also P,(O,p) = 0 because of the optimality of the zero level cycles, namely due to (TP, - T,P)IC=o= 0. Consequently now due to Hadamard lemma the profit P can be presented in the form c2D(c,p)near the origin with some smooth function D. By Theorem 3.1 the maximum averaged profit for small positive parameter values is the unique solution c, c(0) = 0, of equation

C=

+

Writing F and G in the forms F ( u , p ) = F1(u2,p) uF2(u2,p) and G(u,p ) = GI(u2,p ) uG2(u2,p ) respectively and rearranging the integrals we get equation

+

fi

2

J [ W ( 3 6 u 6 - 36pu4 + 8p2u2)+. . .]du 0

C=

+

T(c,p) 2

v5

J (&(36u6 0

-

36pu4

+ 8p2u2)i-. . .)du - cD(c,p)

where the “dots” represent even terms ukpz on u with k

+ 21 > 6. After

618

integration we get equation

with “dots” standing for smooth functions vanishing at the origin. As we mentioned above f(0)< A and so after subtracting A*(p) from the profit density we get f(0) < 0. The last equation has a unique solution near the origin of the form c(p) = -p712a(p) + p 7 b ( p ) with smooth functions a and b and a(0) = - l o ~ ~ ~ > ~ 0. ) To ( osee ) that it is sufficient to consider --p7j2 as a new variable q and apply the implicit function theorem near the origin. That gives a solution of the form c = C ( q , p )that is zero when q = 0. So due to Hadamard lemma it can be rewritten in the form c = q c ( q , p ) near the origin with some smooth function Writing now = qC1 ( q 2 , p ) + C z ( q 2 , p )near the origin, we get the solution c = qC2(q2,p) q 2 C l ( q 2 , p ) , that is the solution c = -p7/2C~(p7,p)+p7C1(p7,p). It has the form stated above with a ( p ) = C 2 ( p 7 , p ) and b(p) = C l ( p 7 , p ) . Now choosing a new coordinate Z such that c = &(p) Z2b(p), that preserves the zero level of the maximum averaged profit for small negative values of p , we reduce this profit for small positive p to the form Z = -p712. Thus at the point under consideration the profit has the last singularity from Table 1.

c.

c

+

+

4. Singularities of stationary strategies

We recall that for a family of control systems on the circle, an equilibrium point is a point where the convex hull of the admissible velocities contains the zero velocity. For each equilibrium point ( z 0 , p O ) there exists an admissible motion which tends to it when time goes to infinity. So that motion provides an averaged profit on the infinite horizon that equals the value of the profit density at ( 2 0 P, O ) . The maximum averaged profit A, over the stationary strategies is therefore the solution of the following extremal problem A,(P) = max{f(GP) I x E SPI,

(9)

where Sp = {x I ( z , p ) E S } and the stationary domain S is the union of all equilibrium points. Notice that for a continuous system this domain is closed. The classification of generic singularities of the maximum averaged profit over the stationary strategies can be done in two steps. At first we classify the generic singularities of the stationary domain and then those

619

of the solution of problem (9). For the first classification we use smooth diffeomorphisms preserving the natural foliation over the parameter. It is clear that on the first step we should work with a generic family of control systems and on the second one we can treat a generic family of profit densities when a generic family of control systems is fixed. That gives the following results:

Theorem 4.1. For a generic one-parameter family of control systems on the circle, the germ of the stationary domain at any of its boundary points is equivalent to the germ at the origin of one of the seven sets in Table 2. Besides, all these singularities are stable and the number of different values of the control parameter must be not less than 2 f o r singularities 1 and 2, not less than 3 f o r singularities 3 and 4 and equal to 2 f o r singularity 5. Table 2. 1 2 5 0

4

3

P i 1x1

x

2 --/PI

The equivalence relation underlying the classification and stability of the singularities in this theorem is of course the one using smooth diffeomorphisms preserving the natural foliation over the parameter. We omit the proof of this theorem. It is based on transversality theorems and simple calculations.

Remark 4.1. In a generic case the maximum and minimum admissible velocities have different signs inside the stationary domain, and on the boundary of this domain at least one of them vanishes. Besides, outside singularities of type 2* the vanishing velocity always has both one sided derivatives and they are different from zero. For the case of finite dimensional families of polidynamical systems (when the number of different values of the control parameter is finite) Theorem 4.1 was proved by CBlia Moreira 23.

620

Theorem 4.2. For a generic smooth one-parameter family of pairs of control systems and profit densities o n the circle, the germ of the optimal averaged profit over the stationary strategies at a value of the parameter admitting equilibrium points, is equivalent to the germ at the origin of one of the five functions in the second row of Table 3 up to the equivalence pointed out in the third one. Besides, all these singularities are r-stable.

Table 3. TYPe Singularity Equivalence

1

o

2 Jpl

3 plpl

@,P 2 o

R+

R+

R+

R+

4

5

max {0,1+

r

Remark 4.2. All singularities from Table 3 are well known in the theory of parametric optimization In the theory of time averaged optimization they were found by V.I. Arnold '. Here all of them appear as generic singularities of the maximum profit for stationary strategies only. All of them appear already generically in the case of bidynamical (polydynamical) systems on the circle. In this case they were studied by C6lia Moreira 26 who used the same definition of equilibrium point as in this paper. 11,21724925.

To simplify the proof of Theorem 4.2 we introduce firstly some auxiliary definitions and statements. A boundary point of the stationary domain is called regular if at that point the domain has singularity 1 of Theorem 4.1. Otherwise we call it singular. As above S, is the set of stationary strategies for a value p of the family's parameter.

Lemma 4.1. For a generic one parameter family of pairs of control systems and profit densities o n the circle

(a) at any point at least one of the derivatives f x , fxx and fxIx of the profit density family f does not vanish, (b) at any singular boundary point of the stationary domain, the derivative f x does not vanish, and at a regular one at least one of the derivatives f x and f x x does not vanish, ( c ) for any parameter value p with S, # 8 the maximum averaged profit A,(p) over the stationary strategies can be attained at most at two different points.

621

These statements follow easily from Thom transversality theorem in cases (a) and (b) and from multijet transversality theorem in case (c). So we omit their proofs.

Proof of Theorem 4.2. Choose a generic family of pairs of systems and profit densities (C,f) for which the statements of Theorem 4.1 and Lemma 4.1 are true. Suppose firstly that at a parameter value po the profit A, is provided by one equilibrium point Q = ( z 0 , p o ) only. If Q is an interior point of the stationary domain, then f z ( Q ) = 0 and (a) implies that fzz(Q) # 0. As Q is a local maximum of the density f (.,Po) we have fzz(Q) < 0. Hence A,(p) = f (x(p),p)near po where x(p) is the curve of local maxima of the profit density, namely, the solution of equation fz = 0 with respect to x near the point Q. Because fzz(Q) < 0, the implicit function theorem guarantees that this solution is unique and smooth. Thus the maximum averaged profit is a smooth function near the point po and so it has the first singularity of Table 3 at it. If Q is a regular boundary point of the stationary domain, then the derivative f z ( Q ) is either zero or not. In the second case near the point po the function A, is smooth because it coincides with the restriction of the smooth family f to the boundary of the stationary domain which is the graph of a smooth function z = z ( p ) for a regular point. Thus we get singularity 1 from Table 3. When f z ( Q ) = 0 , we have f z z ( Q ) # 0 by (b) and in addition f z z ( Q ) < 0, because flsPoattains the maximum at Q. Generically due to Thom transversality theorem the differential of the restriction of the derivative fz to the boundary of the stationary domain does not vanish at Q. Hence at the point Q this boundary is not tangent to the curve of local maxima of the family f with respect to x. So near the point Q the profit A, is the maximum of the restriction of the family of densities to the part of this curve lying inside the stationary domain and to the boundary of this domain. This maximum as function of the parameter is smooth near the point PO except the point itself where it is differentiable but has the simple discontinuity of the second derivative. Consequently the profit A , has the third singularity of Table 3 at the point PO. If the point Q is a singular boundary point of the stationary domain, then f z ( Q ) # 0 due to statement (b) of Lemma 4.1. At such a point the stationary domain can have only one of the singularities 2+, 4 or 5- of Theorem 4.1. Indeed in the case of singularity 2- or 5+ the point Q is an

622

interior point of S,, and it can not provide the maximum of the averaged profit because fZ(Q) # 0. It is clear that the maximum averaged profit A, has at the point po the second singularity of Table 3 when at Q the stationary domain has singularities 4 or 5-. When at the point Q the stationary domain has the singularity 2+, then the singularity type of A, at the point po depends on whether S,, contains points different from Q or not. When the set S ,, \ {Q} is empty we get the fourth singularity of Table 3. If it is not empty then due to the multijet transversality theorem any other local maximum o f f ( . , P O ) in this set is either an interior point or a regular boundary point of the stationary domain with nonzero derivative fz. In both these cases the value of this maximum is less then f (Q). Hence the maximum averaged profit has the fifth singularity of Table 3 at the point PO. Due to (c) of Lemma 4.1 to complete the proof we only have t o analyze the case of the maximum averaged profit provided by two different points of Spa. The multijet transversality theorem implies that in a generic case there can be only competition of two local maxima of f ( . , p ) inside S, with singularities of type 1 of Table 3, and these maxima have different derivatives at po (but their values have to be equal due to the presence of competition). Hence the profit A, has at the point po the second singularity from Table 3. For a generic pair, a singularity from Table 3 is defined by transversality of jets or multijet extensions to submanifolds in jet or multijet spaces, respectively. That implies the stability of singularities, namely, the last statement of Theorem 4.2. 0

5. Transition singularities 5.1. Transition between optimal strategies

A parameter value is called a transition value if in any neighborhood of it, the maximum averaged profit can not be provided by one and only one type of strategy, namely, either by level cycles or by equilibrium points. Theorem 5.1. For a generic smooth one parameterfamily of pairs of control systems and profit densities on the circle, the germ of the maximum averaged profit at a transition parameter value is R+ -equivalent to the g e m at the origin of one of the two functions in Table 4. Besides those singularities are stable.

623

N

Singularity

1

IPl

2

max (0, -&(l+

Type

H))

Stop at an interior point of the stationary domain with f, = 0 Stop at a regular boundary point of the stationary domain with fZ # 0

&,w)

In Table 4, H = h(p, where h is a smooth function of its variables with h(p,O,O) = 0. Actually the function c(p) = -&(l H) is given implicitly by an equation of the form clnc = F(c,p), with F smooth. The word “stop” means the switch between the optimal level cycle strategy and the stationary strategy of the given type at the point under consideration. We will give a clear description of these phenomena, but will not give proofs for all our statements because they are rather technical. The complete proof can be found in Ref. 20.

+

Proof of Theorem 5.1. It is clear that for a transition value po, the set S,, is not empty and therefore the best averaged profit A,(po) among all stationary strategies is well defined. Indeed if S,, = 0 then by continuity of the control system we conclude that S, = 8 for p in a sufficiently small neighborhood of po and so po could not be a transition value. Also any neighborhood of po contains values p for which there exist level cycles. So let A l ( p 0 ) be the upper limit of the averaged profit provided by level cycles when p --f PO. Due to stability of the Maxwell set of a generic family of control systems one can fix this set and the respective stationary domain and make only perturbations of the profit density family. Again using tranversality theorems and small perturbations of the profit density family one can show that in a generic case for any transition parameter value po: (A) The fiber p = po contains only regular points of the Maxwell set. (B) The profit A,(po) is provided by only one equilibrium point which is either an interior point of the stationary domain with f x = 0 > fZZ or a regular boundary point of the stationary domain with f x # 0. In particular, the function A, is smooth near the point po. ( C ) The value A,(po) of this profit is less than the maximum m(p0) of the density f (.,p o ) on the circle. (D) If the profit A,(po) is a critical value of the profit density f ( . , P O ) , then the level f ( . , p o ) = A,(po) contains only one critical point of the density and this point belongs to the interior of the stationary domain, and else

624

it is exactly the point providing the maximum averaged profit among stationary strategies. Thus due to (B) one must analyze two types of transition, namely, to a stationary strategy at an equilibrium point Q either inside the stationary domain or at a regular boundary point of the stationary domain. Without loss of generality we shift the point Q to the origin (0,O) and consider these two cases subsequently. In the first case, minimum and maximum velocities have different signs near the point Q (see Remark 4.1). Consequently the c-level motion with c L A,(p) is well defined for a value p close enough to zero (=PO) because it uses the maximum velocity near the point Q. So if we do not permit any switching between maximum and minimum velocities near this point we can extend the computation of the period T and the profit P to all ( c , p ) close enough to the point R = (A,(O),O).These new functions and the respective averaged profit A are smooth near R due to statements (A) and (D). Besides A ( R ) = Al(0) = A,(O) because po = 0 is a transition value, (cT - P)(R) = 0, and also (P/T),(R) = 0 due to the optimality of the level cycle provided by R. Consequently equation (2) has a unique smooth solution c = C(p) near the origin with C(0)= A,(O) due to the implicit function theorem. So near the origin the maximum averaged profit A is defined as the maximum of the functions A, and C. The coincidence of derivatives of these functions at the origin gives a new independent condition on the transition and so it does not take place in a generic case. Consequently, these derivatives are different and the maximum averaged profit has at the point po the singularity 1from Table 4 up to R+-equivalence. In the second case, changing the 2-coordinate but preserving its orientation, we reduce the stationary domain’s boundary near the point Q to the form x = 0 near the origin. Near this point the minimum velocity w is smooth, negative in the stationary domain (see Remark 4.1) and has a nondegenerate singular point at x = 0. Consequently by a new smooth change of the x-coordinate near the point Q = 0, which preserves the form x = 0 of the boundary and the orientation of the x-axis, we reduce this velocity to the form w ( z , p ) = xy(p) near the origin with some smooth function y with y(0) # 0 27. In addition to simplify calculations we subtract the profit A, from the profit density near p = 0. That reduces to zero the restriction of the profit density f to the boundary x = 0 of the stationary domain. Now due to Hadamard lemma, near the origin the density can be presented in the

625

form f ( z , p ) = z h ( z , p ) with some smooth function h; h(0,O) # 0 because f z ( Q ) # 0. Besides h(O,O)y(O) > 0 due to the existence of c-level cycles for small positive c for zero parameter value. Consider the case of positive values of both h(0,O) and y(0). The case of negative values can be treated in the same manner. Due to the implicit function theorem, equation c = zh(x,p ) has a unique smooth solution z = cX(c,p) near the origin with X(0,O) > 0. Take positive values a and 6 such that in the rectangle [-a, a] x [-q E] the forms f(z) = z h ( z , p ) of the profit density and v ( z , p ) = zy(p) of the minimum velocity take place. Then the period of a c-level cycle with 0 < c < a and and the profit along it have the forms

respectively, where T* and P* correspond to the time of motion and the profit outside the interval [c,a]. Denoting T(c,p) = T*(c,p,a) Wy (ap ))

+

ln(X(c,p)) 7( P )

and P(c, p ) = P* (c, P,a)

+

Jc> ( c , p ) r(p) h ( Z ' p ) d we z,

arrive to the forms:

for the period and the profit, respectively. Due to statement (D) the transition level A,(O) is not a critical value of the profit density. And by statement (A) in the fiber p = 0 there is no double tangent (triple or swallow) point or point of selfintersection of the Maxwell set, or else double point in the level As(po) of the profit density. Consequently the functions P and T are smooth near the origin if a and E are sufficiently small. Now we just have to analyse for parameter values near zero which level cycles provide a positive averaged profit. The respective levels must be the solutions of equation (2) , that is

in our case. Simple transformations reduce it to the form clnc = F(c,p),

(10)

where F ( c , p ) = y(p)(cT(c,p) - P(c,p)) is smooth near the origin and F ( 0 , p ) = -y(p)P(O,p). Using Hadamard lemma we write the function F in the form F(c,p) = cH(c,p) + p B ( p ) near the origin with some smooth functions H and B , B ( 0 ) = -y(0)Pp(O,0). The vanishing of the derivative

626

Pp at the origin gives an excessive independent condition on the transition and so it does not take place in a generic case. Consequently B(0) # 0 in a generic case. Changing eventually the sign of p we always can get B(0) < 0. Now near the origin equation (10) has no solution for negative parameter values but for positive ones its unique solution can be found in the form c = In P with some function z of p. Substituting this form and the form of the function F in equation (10) we get equation

-=

-pz In(--) Pz

lnp

lnp

Pz PZ = --~(--,p) lnp

lnp

+p~(p).

Dividing by p and making simple transformations we reduce it to the form

Now it is easy to see that the value z ( 0 ) must be equal to -B(O), and so it is positive. Introducing new variables T and s such that T = 1 / h p and s = lnIlnpl/lnp we get that the left hand side of last equation is a smooth function of p , z , T and s near the point (0, -B(O), 0,O). Besides its derivative at this point with respect to z is equal to -1. Consequently near this point this equation has a unique smooth solution z = Z(p, T , s) with Z(O,O,0) = -B(O). That implies that equation (10) has the solution

The change of c, E = c/Z(p, O,O), reduces the function Z to the form Z = = 0. Hence the maximum 1 H (p, n lnp with H 0, averaged profit has at the point po singularity 2 from Table 4 up to R+0 equivalence.

+

&,!&$)

( &,,

)

Acknowledgments This work was supported by grants RFBR 03-01-00140 (Russia) and FCT XXI/BBC/22225/99(Portugal), as well as by Centro de Matemfitica da Universidade do Porto (CMUP) financed by FCT (Portugal) and was partially done during the visit of the first author to Departamento de Matemfitica Aplicada e Centro de Matemfitica da Universidade do Porto. He is very thankful to the staff of the department for a good scientific atmosphere and the nice working conditions.

627

References 1. H. Maurer, C. Buskens and G. Feichtinger, Optim. Control Appl. Meth. 19, 185 (1998). 2. A. Zevin, Autom. Remote Control 41, 304 (1980). 3. A. Tsirlin, Methods of averaging optimization and their applications (Nauka. Fizmatlit., Moskva, 1997). 4. V. Arnold, Sib. Math. J. 28, 540 (1987). 5. V. Arnold, O n a Variational Problem Connected with Phase Transitions of Means in Controllable Dynamical Systems, in Nonlinear Problems in Mathematical Physics and Related Topics I , ed. M. B. et al. (Kluwer/Plenum Publishers, 2002). 6. V. Arnold, Funct. Anal. and its Appl. 36,83 (2002). 7. N. Petrov, Differ. Equations 4, 311 (1968). 8. N. Petrov, Differ. Equations 4, 632 (1968). 9. A. Davydov, Qualitative theory of control systems (AMS, Providence, 1994). 10. A. Davydov and J. Basto-Goncalves, J. Dyn. Control Syst. 7,77 (2001). 11. L. Bryzgalova, Funct. Anal. Appl. 11, 49 (1977). 12. L. Bryzgalova, Funct. Anal. Appl. 12, 50 (1978). 13. V. Arnol’d, A. Varchenko and S. Gusein-Sade, Singularities of differentiable maps, vol.1 (Birkhauser, Boston, 1985). 14. V. Arnol’d, Catastrophe theory (Springer-Verlag, Berlin, 1984). 15. V. Arnol’d, A. Davydov, V. Vasiliev and V.M.Zakalyukin, Mathematical models and control of catastrophic processes, in UNESCO Encyclopedia of Life Support systems, (EOLSS Publishers Co.Ltd. 16. V. Matov, Usp. Mat. Nauk 37,167 (1982). 17. D. Siersma, Properties of conflict sets in the plane, in Geometry and topology of caustics - CAUSTICS ’98. Proceedings of the Banach Center symposium, eds. Janeczko and S. et al. (Warsaw, Poland, 1999). 18. D. Siersma and R. Garcia, Curvatures of conflict surfaces in euclidean 3space, in Geometry and topology of caustics - CAUSTICS ’98. Proceedings of the Banach Center symposium, eds. Janeczko and S. et al. (Warsaw, Poland, 1999). 19. A. Davydov, Proceedings of the Steklov Institute of Mathematics 250, 70 (2005). 20. A. Davydov and H. Mena-Matos, Generic phase transition and profit singularities in arnold’s model, Submitted to Math. Sbornik, (2006). 21. A. Davydov, Singularities of the maximum function over a preimage, in Geometry in nonlinear control and differential inclusions. Proceedings of the workshop o n differential inclusions held May 17-29, 1993 and the workshop o n geometry in nonlinear control, May 31-June 25, 1993 in Warsaw, Poland, eds. Jakubczyk and B. et al. (Poland, 1995). 22. J. Mather, Ann. of Math.(2) 98, 226 (1973). 23. C. Moreira, Singularities of the stationary domain for polidynamical systems, in Proc. 4th Junior European Meeting on Control and Optimization, (Poland, 2005). 24. A. Davydov and V. Zakalyukin, J . Math. Sci., New York 103, 709 (2001).

628 25. H.Th.Jongen, P. Jonker and E.Twilt, Math. Program. 34, 333 (1986). 26. C. Moreira, Singularidades do proveito mBdio cjptimo para estrathgias estacionkias, Master’s thesis, Universidade of Porto (2005). 27. V. Arnol‘d, V. Afrajmovich, Y. Il’yashenko and L.P.Shil’nikov, Bifurcation theory and catastrophe theory (Springer, Berlin, 1994).

629

INDICES OF COLLECTIONS OF 1-FORMS W. EBELING' Znstitut fur Algebraische Geometrie, Universitat Hannover, Postfach 6009, 0-30060 Hannover, Germany *E-mail: ebelingOmath.uni-hannover.de

S. M. GUSEIN-ZADE Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia E-mail: sabirQmccme.m An isolated zero of a vector field or of a 1-form on a smooth manifold M has a well known invariant - the index. By the PoincarkHopf theorem it corresponds to the Euler characteristic x ( M ) of the manifold M . For an ndimensional complex analytic manifold M the Euler characteristic x ( M ) is the characteristic number ( k ( T M ) ,[MI)where & ( T M ) is the top Chern class of the manifold M . Indices of collections of 1-forms correspond t o other Chern numbers. We discuss generalizations of the notion of the index of a collection of 1-forms to singular varieties. Keywords: singular variety, 1-form, index. Mathematics Subject Classification: 14B05, 58A10, 55399.

1. Introduction

An isolated singular point (zero) of a (continuous) vector field on a smooth manifold has a well known invariant - the index. A neighbourhood of a point P in a manifold M n of dimension n can be identified with a neighbourhood of the origin in the affine (coordinate) space R". A germ v of a n

vector field on (Rn,O) can be written as v(x) =

vi(x)&.

Let BF(0) be

i=l

the ball of radius e centred at the origin in R" such that the vector field v is defined in a neighbourhood of this ball and has no zeros in it except at the origin. Let S:-'(O) be the ( n - 1)-dimensional sphere dBp(0). The

630

vector field v defines a map

v

-:

llvll

s;-l(o) sy-1. 4

The index indp v of the vector field v at the point P is defined as the degree of the map x. lbll One of the most important properties of the index of a vector field is the Poincar&Hopf theorem. Suppose that the manifold M is closed, i.e. compact without boundary, and that the vector field v has finitely many singular points on it. Theorem 1.1 (Poincar&Hopf). The sum

indpv PESing w

of indices of singular points of the vector field v is equal to the Euler characteristic x ( M ) of the manifold M .

This notion of the index of an isolated singular point of a vector field (on a smooth manifold) may in a natural way be adapted to 1-forms. There are a number of generalizations of the notion of the index to vector fields and/or 1-forms on singular varieties. As an example, there are notions of the so called GSV index of a vector field and of a 1-form on a (complex) isolated complete intersection singularity (ICIS) (see In the complex analytic setting, i.e. if M is an n-dimensional complex analytic (compact) manifold, the Euler characteristic x ( M ) of M is equal to the characteristic number ( % ( T M ) ,[ M I ) ,where % ( M ) is the top Chern class of the manifold M . The Euler characteristic (or the top Chern number ( % ( T M ) ,[MI))of a complex manifold is counted by zeros of a (generic) vector field and/or a 1-form. Other Chern characteristic numbers are counted by points where several sets of vector fields or 1-forms fail to be linear independent. This leads to the idea to try to formulate and to study analogues of indices of vector fields or of 1-forms on singular varieties for collections of vector fields or of 1-forms. One can say that the corresponding notions of indices of collections of vector fields or of 1-forms correspond to Chern numbers different from the top one. Here we shall discuss some notions of indices of collections of 1-forms on singular varieties constructed in 8,14t4.

517.

631

2. The GSV index

Let 7r : E + M be a complex vector bundle of rank m over a complex analytic manifold M of dimension n. It is known that the (2(n--k)-dimensional) cycle Poincare dual to the characteristic class ck(E) (Ic = 1,.. .,m) is represented by the set of points of the manifold M where m - k 1 generic sections of the vector bundle E are linearly dependent (cf., e.g., ). For natural numbers p and q with p 2 q, let M ( p , q ) be the space of p x q matrices with complex entries and let Dp,q be the subspace of M ( p ,q ) consisting of matrices of rank less than q. The complement Wp,q= M ( p , q )\ Dp,q is the Stiefel manifold of q-frames (collections of q linearly independent vectors) in CP. The subset Dp,qis an irreducible subvariety of M ( p ,q ) of codimension p - q 1. Therefore Wp,qis (2p - 2q)connected, H Z ~ - Z ~ + ~ ( ZWand ~ , the ~ ) latter homology group has a natural generator: the boundary of a small ball in a smooth complex analytic slice to Dp,qat a non-singular point (cf., e.g., l o ) . Let k = ( k l , . . . ,ks)be a sequence of positive integers with C:==, ki = k. Consider the space M m , k = M ( m , m - ki 1) and the subvariety Dm,k = Dm,m-ki+l in it. The variety Dm,k consists of sets {Ai} of m x ( m - k i + l ) matrices such that rk Ai < m - k i + 1 for each i = 1 , . . . ,s. Since Dm,k is irreducible of codimension k , its complement Wm,k = Mm,k\ Dm,k is (2k-2)-connected, Hz&l(Wm,k) e!Z, and there is a natural choice of a generator of the latter group. This choice defines a degree (an integer) of a map from an oriented manifold of dimension 2k - 1 to the manifold

+

+

n:=,

n;=,

+

Wm,k. 0) be an n-dimensional isolated complete intersection Let ( X ,0) c (CN, singularity (ICIS) defined by equations fl = . . . = f ~ - ,= 0 (fr E C?CN,O, r = 1 , . . . ,N - n). Let f be the analytic map (fl, . . . ,f~-,) : ( C N ,0 ) -+ (CN-n,0 ) ( X = f - l ( O ) ) . Let {u:)} be a collection of 1-forms on a neighbourhood of the origin in (CN, 0) with i = 1,.. . ,s, j = 1 , . . . ,n - ki 1, ki = n. We say that a point P E X \ {0} is non-singular for the collection LO:^'} on X if at least for some i the restrictions of the 1-forms uji’(P), j = 1,, . . , n - ki 1, to the tangent space T p X are linearly independent. Suppose that the collection {u:’} has no singular points on X outside of the origin in a neighbourhood of it. Let U be a neighbourhood of the origin in CN where all the functions fr ( r = 1 , . . . ,N - n) and the 1-forms u:) are defined and such that the collection {uji)}has no singular points on ( X n U ) \ (0). Let SS C U be a sufficiently small sphere around the origin which intersects X transversally and denote by K = X n Ss the link of the

+

+

632

ICIS ( X ,0). The manifold K has a natural orientation as the boundary of a complex analytic manifold. Let Q’xbe the mapping from X f l U to Mn,k which sends a point z E X nU to the collection of N x ( N - lci 1)-matrices

+

Its restriction

I)X

to the link K maps K to the subset

WN,k.

Definition 2.1. The (GSV) index indx,o{uji)} of the collection of 1-forms {uy’}on the ICIS X is the degree of the mapping $x : K -+ W N , k . One can easily see that the index indx,o{ujZ)} is equal to the intersection number of the germ of the image of the mapping Q X with the variety D N , k . If all the 1-forms uy)are complex analytic, the mapping Q’xis complex analytic as well. For s = 1, kl = n, this index is the GSV index of a 1-form (see, e.g., 4). Let X c CP” be an n-dimensional complete intersection with isolated singular points, X = {Fl = . . . = F N - ~ = 0) where FT, T = 1,.. . ,N - n, are homogeneous functions in ( N 1) variables. Let L be a complex line bundle on X and let {wji)} be a collection of continuous 1-forms on X with values in L. Here this means that the forms uf)are continuous sections of the vector bundle T * X @ L outside of the singular points of X . Since, in a neighbourhood of each point P , the vector bundle L is trivial, one can define the index indp{uji)} of the collection of 1-forms {uji’}at the point P just in the same way as in the local setting above. Let 2 be a smoothing of the complete intersection X, i.e. )7 is defined by N - n equations F1 = . . . = F N - ~= 0 where the homogeneous functions F,. are small perturbations of the functions F,. and 2 is smooth. One can consider L as a line bundle on the smoothing )7 of the complete intersection X as well (e.g., using the pull back along a projection of 2 to X ) . The collection {wji)} of 1-forms can also be extended to a neighbourhood of X in such a way that it will define a collection of 1-forms on the smoothing 2 (also denoted by {uji)}) with isolated singular points. The sum of the indices of the collection { u j Z )on } the smoothing 2 of X in a neighbourhood of the point P is equal to the index indp{uji)}. The usual description of Chern classes of a vector bundle as obstructions to existence of several linear independent sections of the bundle implies the following statement.

+

-

-

633

Theorem 2.1. One has indp{uJi)} = ( n c k i ( T * j ?@ L ) , [j?]), PEX

where

i= 1

2 is a smoothing of the complete intersection X .

As above, let ( X , O ) c (CN,O be the ICIS defined by the equations f1 = ... = fN-n = 0 and let {aj } (i = 1,...,s; j = 1,...,n - ki 1) be a collection of 1-forms on a neighbourhood of the origin in CN without singular points on X \ { 0 } in a neighbourhood of the origin. If all the 1forms uf) are complex analytic, there exists an algebraic formula for the index indx,o{wj(i)}. Let I x , ( wbe ~ )the l ideal in the ring Q Ngenerated ,~

02)

+

+

by the functions f l , . . . , fN-n and by the ( N - ki 1) x ( N - ki of all the matrices (4 (41 . . . ,dfN-n(x), ~1(4(x), * .,wn-ki+l (.I)

+ 1) minors

-

for all i = 1,.. . ,s.

Theorem 2.2. (see 5 , indx,o{wji)} = dim@U C N , O / I ~ , ~ + ) ~ .

Remark. Let { u p ) } be a collection of vector fields on a neighbourhood of the origin in (C13, 0 ) (i = 1 , . . . , s; j = 1,.. . ,n - ki 1; C ki = n) which 0 ) at nonare tangent to the ICIS (x,0 ) = {fi = . . . = fN-n = 0 } c (P, singular points of X . One can define the index indx,o{v,(9} as the degree of the mapping K 4 W N , k which sends a point x E K to the collection of N x ( N - ki 1) matrices

+

+

{(gradfl(z), . . . ,gradfnr-n(x),wii)(z),. . . , ~(i) ~ + + ~ ( x ) ) }i, = 1,.. . ,s. Here

where z is the complex conjugate of the complex number z. For s = 1, k l = n, this definition coincides with the definition of the index of a vector field on an ICIS from 8,14.For vector fields the analogue of Theorem 2.1 holds with the only difference that the sum of the indices is equal to the characteristic number cki(T2 @ L ) @I). , However, a formula similar to that of Theorem 2.2 does not exist. A reason is that in this case the index is the intersection number with D N , k of the image of the ICIS ( X ,0 )

(nE,

634

under a map which is not complex analytic. Moreover, in some cases this index can be negative (see e.g. ). 3. Local Chern obstructions

Another generalization of the notion of the index of a vector field or of a 1-form to singular varieties (generally speaking with non-isolated singularities) is the, so called, Euler obstruction: see There exists a generalization of the notion of the Euler obstruction to collections of I-forms corresponding to different Chern numbers. Let (Xn,O) c (CN,O)be the germ of a purely n-dimensional reduced complex analytic variety at the origin (generally speaking with a nonisolated singularity). Let k = {ki}, i = 1,.. . ,s, be a fmed partition 1,2y6.

6

ki = n). Let {w,!")} (i = 1 , . . . ,s, i=l j = 1,. . . , n - ki 1) be a collection of germs of 1-forms on ( C N ,0) (not necessarily complex analytic; it suffices that the forms w,!") are complex linear functions continuously depending on a point of C N ) .

of n (i.e., ki are positive integers,

+

Definition 3.1. A point P E X is called a special point of the collection {wji)} of 1-forms on the variety X if there exists a sequence {P,} of points from the non-singular part Xregof the variety X such that the sequence TpmXregof the tangent spaces at the points P, has a limit L (in G(n, N ) ) and the restrictions of the 1-forms w r ) , . . . , wn-ki+l (4 to the subspace L c TpCN are linearly dependent for each i = 1 , . . . ,s. The collection {w,!"'} of 1-forms has an isolated special point on (X,0 ) if it has no special points on X in a punctured neighbourhood of the origin.

Remark 3.1. For the case s = 1 (and therefore k l = n), one has a notion of a singular point of the 1-form w on X : a point P E X is called a singular point of the 1-form w on X if the restriction of w to the stratum of the (minimal) Whitney stratification of the variety X vanishes at the point P. One can easily see that a special point of the 1-form w on X is singular, but not vice versa. (E.g. the origin is a singular point of the 1-form dx on the cone {x2 + y2 z2 = 0 } , but not a special one.) On a smooth variety these two notions coincide.

+

Definition 3.2. A special (singular) point of a collection {wji)} of germs of 1-forms on a smooth n-dimensional variety X is non-degenerate if the map \kx : X n U + Mn,k desribed above is transversal to Dn,k C Mn,k (at a non-singular point of it).

635

Let s n-k;+l i=l

j=1

be the space of collections of linear functions on CN ( considered as 1-forms with constant coefficients). The following statement holds.

u

Proposition 3.1. There exists an open and dense set C & such that each collection {ly’}E U has only isolated special points on X and, moreover, all these points belong to the smooth part Xreg of the variety X and are non-degenerate. Corollary 3.1. Let { w y ’ } be a collection of 1-forms o n X with an isolated

special point at the origin. Then there exists a deformation {Zjz’} of the collection {uji)}whose special points lie in Xreg and are non-degenerate. Moreover, as such a deformation one can use {wja) M;’} with a generic collection { l y ’ }E &. The set of collections of holomorphic 1-forms with a non-isolated special point at the origin has infinite codimension in the space of all holomorphic collections.

+

Let u : 2 4 X be the Nash transformation of the variety X defined as follows. Let G(n,N ) be the Grassmann manifold of n-dimensional vector subspaces of CN.For a suitable neighbourhood U of the origin in CN, there is a natural map CJ : Xregn U -+ U x G(n,N ) which sends a point z to ( x , T z X r % )(Xreg is the non-singular part of the variety X ) . The Nash transform X of X is the closure of the image I m g of the map u in U x G(n,N ) . The Nash bundle T^ over 2 is a vector bundle of rank n which is the pullback of the tautological bundle on the Grassmann manifold G(n,N ) . Let T(CNIxbe the restriction to X of the tangent bundle T C N of CN. There is a natural bundle map from the Nash bundle T^ t o TCNIx which is an embedding on fibres:

This is an isomorphism of part Xreg of X .

T^ and T X r e g c TCNIx over the non-singular

636

Let {u:’} be a collection of germs of 1-forms on ( X ,0) with an isolated special point at the origin. Let E > 0 be small enough so that there is a representative X of the germ ( X , O ) and representatives uji’ of the germs of 1-forms inside the ball BE(0)C section i3 of the bundle

CN.The collection {u:’} gives rise to a s

n-lci+1

i=l

j=1

?Cj

where are copies of the dual Nash bundle T^* over the Nash transform r? numbered by indices i and j . Let 6 c ? be the set of pairs (z,{a:’}) where z E r? and the collection {a:’} of elements of ? ; (i.e. of linear

a2iki+l

functions on T,) is such that a?’, . . . , are linearly dependent for each i = 1,.. . s. The image of the section i3 does not intersect 6outside of the preimage v-’(O) C r? of the origin. The map ? \ 6+ 2 is a fibre bundle. The fibre W, = T \ D of it is (2n-2)-connected, its homology group H2n-1(W,; Z) is isomorphic to Z and has a natural generator (see above). The latter fact implies that the fibre bundle ? \ fi -+ r? is homotopically simple in dimension 271 - 1, i.e. the fundamental group nl(r?) of the base acts trivially on the homotopy group ~ 2 ~ - 1 ( of W the ~ ) fibre, the latter one being isomorphic to the homology group H2,-1(Wz): see, e.g., 16. 6

-

A

Definition 3.3. The local Chern obstruction Chx,o {u:’} of the collections of germs of 1-forms {uji’} on ( X , O ) at the origin is the (primary) obstruction to the extension of the section i3 of the fibre bundle T \ D + r? from the preimage of a neighbourhood of the sphere S, = CIB, to 2,more - l v(- l ~ (Xn z)) on precisely its value (as an element of ~ ~ ~ ( v BE), the fundamental class of the pair ( v - ’ ( X n B,), v - l ( X n S,)). A

-

ns,);

The definition of the local Chern obstruction Chx,o {uji’} can be reformulated in the following way. Let D$ c CN x Ck be the closure of the set of pairs (2, {l:’})such that z E Xreg and the restrictions of the linear

l!lki+l

functions l?’,. . . to TzXregc CN are linearly dependent for each i = 1 , . . . ,s. (For s = 1, k = {n},D$ is the (non-projectivized) conormal space of X 1 7 . ) The collection {uji’} of germs of 1-forms on (CN, 0) defines a section 5 of the trivial fibre bundle CN x Ck -+ CN.Then Chx,o {uji’}= ( G ( C N )o Df;)o where (. o -)o is the intersection number at the origin in C N x Ck.This description can be considered as a generalization of an expression of the

637

local Euler obstruction as a microlocal intersection number defined in see also l3 .

11,

Remark 3.2. On a smooth manifold X the local Chern obstruction Chx,o {uy’} coincides with the index indx,o {wji’} of the collection {wji)} defined above. Being a (primary) obstruction, the local Chern obstruction satisfies the law of conservation of number, i.e. if a collection of 1-forms {Gji’} is a deformation of the collection {wf’} and has isolated special points on X, then Chx,o {wji)} =

Chx,o {Zy’}

where the sum on the right hand side is over all special points Q of the collection {G;)} on X in a neighbourhood of the origin. With Corollary 3.1 this implies the following statements.

Proposition 3.2. The local Chern obstruction Chx,o { w y ’ } of a collection {wf)} of germs of holomorphic 1-forms is equal to the number of special points o n X of a generic (holomorphic) deformation of the collection. This statement is an analogue of Proposition 2.3 in

15.

Proposition 3.3. ~f a collection {wji’} of 1-forms on a compact (say, projective) variety X has only isolated special points, then the s u m of local Chern obstructions of the collection {wja’} at these points does not depend o n the collection and therefore is a n invariant of the variety. It is possible to consider this sum multiplied by (-l)n as a version of the corresponding Chern number of the singular variety X (or, more accurately, taking into account the similarity with Mather classes 12, Mather-Chern number). Let ( X ,0) be an isolated complete intersection singularity. The fact that of both the Chern obstruction and the (GSV) index of a collection {uy’} 1-forms satisfy the law of conservation of number and they coincide on a smooth manifold yields the following statement.

Proposition 3.4. For a collection {wf)} of germs of I-forms o n a n isolated complete intersection singularity ( X ,0 ) the difference indx,o {uji’ -}Chx,o { w j a ) }

does not depend o n the collection and therefore is a n invariant of the germ of the variety.

638

A similar statement about the difference between the GSV index and the local Euler obstruction of a 1-form obtained by radial extension on certain varieties (possibly with non-isolated singularities) can be derived from 3 . By Proposition 3.1, for a generic collection {l?’}of linear functions on

C N ,one has Chx,o {ty)}= 0. This implies that Chx,o {w:’} for a generic collection

{a:))

= indx,o { w y ’ } - indx,o {l:’} of linear functions on

CN

Acknowledgments This research was partially supported by the DFG-programme ”Global methods in complex geometry” (Eb 102/4-3). The second author also was partially supported by the grants RFBR-04-01-00762 and NWO-RFBR 047.011.2004.026.

References 1. J.-P. Brasselet, L6 Diing TrAng, J. Seade: Euler obstruction and indices of vector fields. Topology 39,1193-1208 (2000). 2. J.-P. Brasselet, D. Massey, A. J. Parameswaran, J. Seade: Euler obstruction and defects of functions on singular varieties. J. London Math. SOC.(2) 70, 59-76 (2004). 3. J.-P. Brasselet, J. Seade, T. Suwa: Proportionality of indices of 1-forms on singular varieties. math.AG/0503428, to appear in the Proceedings of the Third Franco-Japanese Symposium on Singularities. 4. W. Ebeling, S. M. Gusein-Zade: Indices of 1-forms on an isolated complete intersection singularity. Moscow Math. J. 3,439-455 (2003). 5. W. Ebeling, S. M. Gusein-Zade: Indices of vector fields or 1-forms and characteristic numbers. Bull. London Math. SOC.37,747-754 (2005). 6. W. Ebeling, S. M. Gusein-Zade: Radial index and Euler obstruction of a I-form on a singular variety. Geometriae Dedicata, 113,231-241 (2005). 7. W. Ebeling, S. M. Gusein-Zade: Chern obstructions for collections of 1-forms on singular varieties. math.AG/0503422. To appear in Proceedings of the 2005 meeting on singularities in Marseille. 8. X. G6mez-Mont, J. Seade, A. Verjovsky: The index of a holomorphic flow with an isolated singularity. Math. Ann. 291,737-751 (1991). 9. Ph. Griffiths, J. Harris: Principles of Algebraic Geometry. John Wiley & Sons, New York etc., 1978. 10. D. Husemoller: Fibre Bundles. Second Edition. Graduate Texts in Math. 20, Springer-Verlag, New York Heidelberg Berlin (1975). 11. M. Kashiwara, P. Schapira: Sheaves on Manifolds. Springer-Verlag, 1990.

639 12. R. MacPherson: Chern classes for singular varieties. Annals of Math. 100, 423-432 (1974). 13. J. Schiirmann: Topology of Singular Spaces and Constructible Sheaves. Birkhauser, 2003. 14. J. A. Seade, T. Suwa: A residue formula for the index of a holomorphic flow. Math. Ann. 304, 621-634 (1996). 15. J. Seade, M. TibZlr, A. Verjovsky: Milnor numbers and Euler obstruction. Bull. B r a . Math. SOC.(N.S.) 36,no.2, 275-283 (2005). 16. N. Steenrod: The Topology of Fibre Bundles. Princeton Math. Series, Vol. 14, Princeton University Press, Princeton, N. J., 1951. 17. B. Teissier: Varibtbs polaires. 11. Multiplicitks polaires, sections planes, et conditions de Whitney. In: Algebraic geometry (La Rhbida, 1981), Lecture Notes in Math., Vol. 961,Springer, Berlin, 1982, pp. 314-491.

640

A LEFSCHETZ THEOREM ON THE PICARD GROUP OF COMPLEX PROJECTIVE VARIETIES

HELMUT A. HAMM Mathematisches Institut , Uniuersitat Miinster Miinster, 0-48149,BRD E-mai1:hammQmath. uni-muenster. de LE DONG TRANG Mathematics Section, The Abdus Salam I C T P Trieste, 1-34014, Italy E-mail: [email protected]

Introduction Let V be an algebraic variety over an arbitrary field k, i.e. in the language of Grothendieck, an irreducible reduced k-scheme of finite type. We can define the Picard group PicV of V as the group of isomorphism classes of invertible Ov-modules. In this paper a theorem of Lefschetz type is a theorem which compares geometric invariants of a subvariety of the projective space and the ones of a hyperplane section of this subvariety. The prototype is the theorem of S. Lefschetz which states that the k - th homology of a non-singular complex projective variety of dimension n equals the Ic - th homology of a hyperplane section for Ic I n - 2. This type of results was generalized to homotopy groups by R. Bott ([B]) using the Morse theory approach of A. Andreotti and T. Frankel ([AF]). In parallel with the Lefschetz theorem which compares the (algebraic) fundamental group of a non-singular projective algebraic variety of dimension 2 3 and the one of a hyperplane section of this variety, A. Grothendieck [G2] considered cases when a theorem of Lefschetz type for the Picard group could be obtained. The purpose of this paper is to give new cases when a theorem of Lefschetz type is true for the Picard group of a complex projective algebraic

641

variety under hypotheses on the singularities of this variety.

1. The main result

A Lefschetz theorem for the Picard group of projective varieties over an arbitrary field has been proved by A. Grothendieck [G2] under hypotheses of vanishing of some cohomology groups associated to the variety (see [G2] Corollaire 3.6 Exp. XII). In the case of complete intersections, these hypotheses are satisfied, so, for complete intersection projective algebraic varieties over an arbitrary field, a Lefschetz type theorem is true for the Picard group (see [G2] Corollaire 3.7 Exp. XII). Over the field of complex numbers, we have a Lefschetz type theorem for the Picard group for complex projective algebraic varieties which are also more general than complete intersections, but for which the hypotheses differ in nature from those of Grothendieck’s theorem. Since we shall only consider varieties over the field of complex numbers, we have other techniques available: analytic and topological methods. In particular we shall use the notion of rectified depth introduced by A. Grothendieck in [G2], Expose XIII,$6 DQfinition2 (see also [HLl]) or Theorems of Lefschetz type for the homology of singular varieties or non-complete varieties as considered by M. Goresky and R. MacPherson in [G-MI or by Lk Diing TrQng and H.A. Hamm in [HL4]. We will also work in the category of complex analytic spaces. If X is a complex algebraic variety with structural sheaf OX,there is an associated complex analytic space X = Xan.For instance, = depthOx,a: if 2 is a closed point of X, i.e. a point of X. On the other hand, the corresponding Picard groups PicX N H1(X, Og), defined with algebraic sheaves, and PicanX N H1(X, O;), defined with analytic sheaves, may be different in general (e.g. see [HL2]). For simplicity we will write PicX instead of PicX and speak of the algebraic Picard group of X ,in contrast to the analytic Picard group PicanX. We shall state a theorem which can be proved by three different methods. Each of them allows some generalization in a different direction.

642 In order to state the theorem we shall adapt to the complex case definitions and concepts which are due to A. Grothendieck ([G2] Exp.111, see e.g. [BS] Theorem 11.3.6):

Definition 1.1. Let X be a complex space, 3 a coherent Ox-module. a) Sn(7):= {z E X 1 d e p t h F , 5 n}. b) d e p t h A F 2 n :@ dim(A n S Z + ~ ( F5) I) for all I E Z. c) d e p t h 3 2 n :@ d e ~ t h 2 3 ~n for all x E X . Here, dim0 := -m. Note that d e p t h o x , , 5 dim,X, so codimA 2 n if d e p t h ~ 2 3 n. Also, d e p t h { , ) 3 = depth3,. Furthermore we need the notion of "rectified cohomological depth" that we shall abreviate by rcd:

Definition 1.2. Let X be a complex space. Then r c d ( X ) 2 n if and only if for all k E N we have dim{z E X I Hk(X,X \ {z};Z) # 0 ) 5 k - n. This notion of rcd is the cohomological analogue of the rectified homotopical depth of A. Grothendieck (see [G2] Exp. XI11 56 D6f. 2) that we have considered in [HLl] (Def. 1.1 and Theorem 1.4) in relation with theorems of Lefschetz type for homotopy groups of complex varieties (see e.g. [HLl] 3.4). Observe that using some Whitney regular stratification of X we can see that the dimension is well-defined because the set

.{ E

x I H " 4 X \ {z};Z) # 0 )

is constructible (cf. [HLl] 1.1.2). Finally S x will denote the singular locus of

X

Now let X be a closed complex subspace of P,.(C) and H be a hyperplane in P,.(@).Denote Y := X n H . We make the following assumption:

(L) codimxY = 1.

643

We take Y with the induced complex structure. Notice that assumption (L)is satisfied with a general hyperplane H . By the theorem of Chow (see [Se2] Proposition 13), the complex spaces X and Y come from closed subschemes of the projective scheme P r o j C[Xo,. . . , X,]. As said before, let P i c X (resp. Pic"" X) denote the algebraic (resp. analytic) Picard group of X . But P i c X 2~ P i c a n X by GAGA because X is compact (consequence of [Se2], see e.g. [Ha] p.440). The main result is the following:

Theorem 1.1. Assume (L) and that depthOx\y 2 3, rcd(X\Y) depthsv Oy 2 3. Then Pic X 21 Pic Y.

> 4 and

Comments: Notice that the hypotheses imply that dimX 2 4 and that X is normal (see also Lemma 2.2 and Lemma 2.10 below). Furthermore these hypotheses are automatically satisfied if X is locally a complete intersection of dimension 4 and the codimension of S y in Y is 3, by [HLl] Theorem 3.2.1 and [BS] Prop. 1.1.18.

>

>

As mentioned before we will present three different proofs of Theorem 1.1. The first is by reduction to the smooth non-compact case which has been treated in [HL2], see also [RS], the second one uses a generalized Kodaira vanishing theorem [AJ], the third one is based on the approach of Grothendieck [G2]. In fact, we shall mainly concentrate on the comparison of P i c Y with lim P i c U,where U runs through the set of Zariski-open neighbourhoods U 4

U

of Y in X . Note that X \ U is always a finite set of points. This procedure is motivated by [G2]. Because of this we shall first compare P i c X with PicU, U being the complementary of finitely many points. Here the notion of parafactoriality is useful.

2. Analytic parafactoriality

I. First suppose that X is a complex analytic space with critical locus S X .

644

We begin with some preliminaries.

Lemma 2.1. Let 3 be a locally free Ox-module of finite type and suppose that X is purely of dimension N . The following conditions are equivalent: a) d e p t h s x 3 2 n, b) codimxSx 2 n, and dim Sl+n(Ox) 5 1 f o r all 1 < N - n.

Proof. We may assume 3 = OX. We have S ~ ( 0 x = ) X , and

for 1 < N . Then, we apply Definition 1.1.

Lemma 2.2. Let 3 be a locally free Ox-module of finite type. Then: a ) d e p t h s x 3 2 1 if and only i f X is reduced, b) depthsx.F 2 2 af and only i f X is normal.

Proof. We may assume 3 = Ox. a) +: Suppose that x E X, f E Ox,z, d > 0 with f d = 0. Then f is defined on some neighbourhood V of x. Since f IV \ Sx = 0 we get f IV = 0 because H o (V, O X ) H o (V \ SX,OX) is injective by [Sch], [T2],see [BS] Theorem 11.3.6. -e:Let IC and V be as before. Iff E Ho(V,Ox), f IV\Sx = 0 , we have that the continuous function defined by f is 0 because S ( X ) is nowhere dense in X . Since X is reduced we get f = 0. So Ho(V,O x ) H o ( V \ Sx, Ox) is injective, which implies our assertion because of [BS] 1oc.cit. b) By [KK] (Prop. 74.7) X is normal if and only if, for every open subset U of X , the mapping Ho(U,Ox)t Ho(U \ Sx, Ox)is bijective. If U is (Ox)= 0, k = 0,1, Stein, H1(U,Ox)= 0, so if X is normal, we have therefore depths,Ox 2 2, by [BS] Theorem 11.3.6. Conversely d e p t h s x O x 2 2 implies that Ho(V,Ox) Ho(U \ Sx, O x ) is bijective, by [BS] loc. cit.

-

-

Recall that "normal" implies that the connected components of X are reduced and irreducible. Therefore, in Theorem 1.1we can reduce ourselves

645

to complex projective varieties X. Now let us compare Pican X with Pic"" U provided that U is open in X and A := X \ U is discrete. Remember that Pican(X)N H1(X,Ofu).

Lemma 2.3. If depth Ox,, 2 2, x E A , the mapping PicanX is injective.

-

Pican U

Proof. By [BS] (Theorem 11.3.6) we have H i ( X , O x ) = 0 , k = 0 , l . Using the exact sequence of support cohomology (see e.g. [G2] Exp. I Corollaire 2.9), it is sufficient to show that HA(X,O;) = 0. Since we have that A is discrete, we have

H M ,0):

=

n

Hiz> (X, 0 ):

xEA

and, by the excision formula for cohomology with support in a closed subset (see Proposition 1.3 of [GH]), we may assume that X is Stein and contractible, A = { x } . Then, since now X is Stein, we have

H 1 ( X , O x ) = H y X ; Z) = 0, so H ' ( X , O > ) = 0 by the exponential sequence. Using again the exact sequence of support cohomology, to prove H f z > ( X0;) , = 0, it is sufficient to show that

-

H O ( X 0:) , is surjective. But H o ( X ,O x ) H:,(X, O x ) = 0 , so

HOP \ { x } ,0 ):

H o ( X \ { x } ,Ox) is surjective, because

Hi,} ( X ,Ox)= 0. So f E H o ( X \ {z},O>) c H o ( X \ {.},OX) has an inverse image f in P ( X , O x ) . Assume f(z)= 0. Since f does not vanish on X \ {z}, we must have dim, X i 1, so depthox,, 5 1, which is a contradiction. So f E HO(X,0;).

We could argue in a different way, as in the algebraic case; see end of section I1 below.

646

Now let us recall the notion of parafactoriality which is due to Grothendieck [G2] XI 3.1 (compare with the remark of A. Grothendieck after XI Lemme 3.4 of [G2]);first, we use the analytic analogue :

Definition 2.1. X is called analytically parafactorial at x E X if

depthox,, 2 2 and if for every open neighbourhood V of x in X and every invertible sheaf 3 on V \ {x} there is an extension 3 to an invertible sheaf on V.

Because of Lemma 2.3 the extension is unique up to isomorphism. In particular with U := X \ A, where A is a discrete set of points, we have:

Lemma 2.4. Suppose that X i s analytically parafactorial at e v e y point of the discrete set A, the natural morphism

Pic""

x + Pic"" u

is a n isomorphism. Lemma 2.5. Suppose that d e p t h o x , , 2 2 and j : X \ {x} inclusion. T h e n the following conditions are equivalent: a) X is analytically parafactorial at x, b) j , 3 i s invertible i f 3 is a n invertible sheaf on X \ {x}.

-

X is the

Proof. a) + b): Let be an extension of 3 to an invertible sheaf on X. Since 3 is invertible, we also have d e p t h 3 , 2 2, i.e. depth{,) 3 2 2. Then, on a small neighbourhood U of x in X, we shall show that 11

f

j * j * 3 l U 2i j*FlU. By the exact sequence of cohomology with supports, it is sufficient to prove that for any sufficiently small neighbourhood V of x in X , Hixl(V,3) = 0 for i 5 1. Then it is a consequence of [BS] (Theorem 11.3.6) since depth?z 2 2. b)

+ a): trivial. 0

647

Lemma 2.6. Let x be a point of X. a) Suppose that H3(X,X \ {x};Z) = 0 and d e p t h O ~ 2 , ~3. T h e n X i s analytically parafactorial at x. b) Conversely, suppose that X is analytically parafactorial at x and that depth 2 4. T h e n H3(X, X \ {x};Z)= 0. c) Suppose that X is, locally at x, a complete intersection of dimension 2 4. T h e n X is analytically parafactorial at x. Proof. a) Let 7 be an invertible sheaf on X \ {x}. Since we have to consider an extension at a point x of X, we may restrict ourselves to a Stein open small neighbourhood of x in X, i.e. assume that X is Stein and X \ {x};Z) = 0 , the contractible. In particular H 2 ( X ,Z) = 0. Since H3(X, exact cohomology sequence of the pair (X, X\{x}) gives H2(X\{x}; Z) = 0. Since by assumption depth Ox,x 2 3, Theorem 11.3.6 of [BS] gives that H1(X\ {x}, Ox) 21 H1(X, Ox) which is trivial since X is assumed to be Stein. Now Cl(7)

E

H2(X\ {x}; Z) = 0 ,

so F is trivial because of the exponential exact sequence and the fact that H1(X\{x}, Ox)21 H1(X, OX)= 0. So we may suppose that F = then our assertion is trivial.

b) Choose X Stein and contractible as in a) and, similarly as above, note that

H2(X \ {x}, Ox) = 0. Then, from the exact cohomology sequence of the pair (X,X \ {x}) and the contractibility of X ,we have that H2(X\ {x};Z) N H3(X, X \ {x};Z). The exponential sequence gives the exact sequence

H1(X\ { X I , 0;) H 2 ( X \ (2);Z) H2(X\ {x}, Ox). We have a natural map PicanX + Pican (X\ {x}) N H'(X \ {x},O;). +

-+

Since we assume that x is analytically parafactorial at x,this map is surjective, but X being Stein Pican X = 0 , so H1(X\ {x}, 0;) = 0. As we saw that H2(X\{x}, Ox) = 0 , we have H2(X\{x}; Z)N H3(X, X\{x};Z)= 0. c) follows from a) because of the local Lefschetz theorem, see [HLl], 3.3 and the fact that, for local complete intersections X at a point x, d e p t h O x , , = dim X (see e.g. [BS] Prop. 1.1.18). 0

648

11. Let X be a compact complex space coming from a proper C-scheme X. Let U be a Zariski-open subset of X with A := X \ U finite. Call j the (analytic) inclusion of U into X and j the (algebraic) inclusion of U into X. Let us compare PicX,PicanX ,PicU and PicanU . Of course, PicX 2~ Pica"X by GAGA, see [Se2] and [G3] Corollaire 1 p.2-09 and Thkorkme 6 p.2-14, or [G5].

First we have

Lemma 2.7. If d e p t h o x , , 2 2 , x E A, the mapping PicX injective.

-+

PicU is

Proof. This follows from Lemma 2.3, by looking at the composition

Pic x

+ Pic

which is injective because PicX

u

-+

picanu,

PicanX.

I I

For PicU and Pican U , we have:

Lemma 2.8. a) Suppose that d e p t h o u , , 2 2 for all x except for some compact set. Then PicU -+ Pican U is injective. 2 3 for all x except for some compact set. Then b) Suppose that depth OU,, Pic U + PicanU is bijective.

-

Proof. a) Suppose that 3 represents an element of PicU and that its image in PicanU is trivial, then we have an isomorphism Tan Ou. It extends to an isomorphism &Tan+ j*O". Let us show that these sheaves are coherent. First, since T is algebraic, a result of A. Grothendieck (see [G4] Cor. 9.4.3 or [GD] I Corollaire 6.9.5) shows that there is a coherent extension 3 of T on X. We have the following exact sequence of sheaves -+ 9 n

+ j*j**n

4

a;(*")

-

0

Since dimA = 0 and depth(3uan,, 2 2 for all x in V \ A , where V is an open neighbourhood of A , Theorem 11.4.1 of [BS] (see also [Tl],[Sill) shows that Xi(@") and Xi(@") are coherent on V ,hence on X ,because @n is

649 coherent. By [Sill or [Si2] Theorem B, these sheaves are isomorphic to the analytic sheaves associated with the sheaves .Hi(.?) and .Hi(.?) which are coherent on X. Therefore j*j*$ = j , 3 is coherent on X, and the associated analytic sheaf is isomorphic to j*j*@" = j*3"" . Since X is a proper Cscheme, by GAGA (see [Se2] and [G3] Corollaire 1 p.2-09 and Thkoriime 6 p.2-14), the isomorphism j,Fan + j*Ov comes from an isomorphism jJ 4 j*Ou, therefore 3 N 0 ~ . b) Surjectivity: Let 3 be an invertible sheaf on U . Then depth3, 2 3 outside some compact subset of U , so j,S is coherent on X , see [T2] (4.2), [FG] Cor. VII.5 or [Si3] Theorem 2. By GAGA, j * 3 comes from an algebraic coherent sheaf G on X, therefore 3 N (j*g)an. Since the stalks of j*G are free of rank 1 we obtain that this coherent algebraic sheaf on U is invertible. 0

Corollary 2.1. Suppose that A = X \ U is finite and that X is analytically parafactorial at all x E A, then we have Pic X N Pic U N Pic"" U .

Proof. Since X comes from a proper C-scheme, by GAGA we saw that PicX 11 Pic"" X . By Lemma 2.4 we have that Pic"" X 21 Pic"" U . Note that the hypothesis implies that depthox,, 2 2 except for some compact subset of U , since depthox,, 2 2 for x E A implies depthox,, 2 2 for all points x in an open neighbourhood of A , by using a theorem of Scheja (see Theorem I1 2.1 of [BS]). Now, PicU + Pic"" U is injective by Lemma 2.8. Looking at PicX + PicU + Pic"" U , from the isomorphism Pic X N Pic"" U , we get the assertion. 0

Following Grothendieck [G2] XI 3.1 we may define: Definition 2.2. X is called parafactorial at x E X if depthox,, 2 2 and if for every Zariski open neighbourhood V of x in X and every invertible (algebraic) sheaf 3 on V \ {x} there is an extension $ to an invertible sheaf on V.

Of course, analytic parafactoriality implies parafactoriality. Note that a coherent sheaf on V is algebraic, i.e. comes from a coherent sheaf on V,

650

as soon as it admits a coherent extension to X .

Proposition 2.1. Suppose that A = X \ U is finite. Then we have the implications a ) 3 b) + c ) + d ) , where: a ) H 3 ( X , X \ (2); Z)= 0 and depth U X , 2~ 3 for all x E A, b) X is analytically parafactorial at all x E A, c) X is parafactorial at all x E A , d ) P i c X N Pic U.

Proof. a ) + b): see Lemma 2.6 a). b) =+ c): see above. c) + d ) : This follows from Lemma 2.7 and the definition of parafactoriality. 0

Now assume that Y = X n H , where X is a closed complex subspace of PT(C)and H is a hyperplane in P,.(C) such that Y satisfies assumption (L) of Section 1:

Theorem 2.1. Under assumption L, we have the implications a ) =+ b) + c) + d ) , where: 2 3 f o r all x E X \ Y , a) H 3 ( X ,X \ (2); Z)= 0 and depth b) X is analytically parafactorial at all x E X \ Y , c ) X is parafactorial at all x E X \ Y , d) P i c X N lim P i c U , where U runs through the set of Zarislci-open neigh+ U

bourhoods U of Y in X

Proof. See proof of Proposition 2.1. 0

Remark: We may also use arguments which do not use analytic methods and can be transferred to algebraic varieties over any field. For instance, in a more special situation P i c X and P i c U can be compared directly: Let CaCl X and Cl X be the Cartier and the Weil divisor class groups of a scheme X. If X is normal the canonical mapping CaCl X -+ C1 X is

651

injective, see [Gl] Prop. 21.3.4b), Cor. 21.6.10. Lemma 2.9. Suppose that X is normal. Let U be Zarislci-open in X, and assume that codimx(X \ U) 2 2, X \ U not necessarily finite. Then Pic X + Pic U is injective.

Proof. We have PicX 21 CaCl X C l X N C l U , cf. [Ha] I1 Prop. 6.5.

c

C1 X, Pic U

N

CaCl U

c C1U

and 0

Also, we have an algebraic proof of Lemma 2.7, (exceptionally) working in the category of algebraic varieties: Suppose that 3 is invertible on X and its restriction on U is trivial, i.e. 3 ) U 21 OU.As in Lemma 2.5 we can prove algebraically (by using the algebraic analogue, Corollaire 3.5 Exp. I11 of [G2], of Theorem I1 3.6 of [BS]) that 3 N j * j * F and OX N j*j*Ox, so we obtain 3 Y OX.

111. Let us still suppose that assumption (L)holds. It is useful to compare depth conditions for X and Y : Lemma 2.10. Let n 2 2 and assume that X \ Y has no isolated points. a) Suppose that depths,Oy 2 n. Then: (a) depthsxOx 2 n along Y ; (ii) depths, O X 2 n i f and only if depth Ox\y 2 n. b) Conversely, suppose that depths,Ox L n. Then depthsy@ 2 n provided that H has been chosen to be general enough.

Proof. It is sufficient to treat the case where X is connected. a) We use Lemma 2.1. Since S X n Y C S y we have that w d i m x S x 2 codimySy L n. Also, dim S,+l(Oy) 5 1 for all 1 < dim Y - n. This means that dim H n S,+l+l(Ox) 5 1 for all 1 < dim Y - n. In particular, H n S n ( O x ) = 0, and dim S,+l+l(Ox) I 1 1 for -1 5 1 < dim Y - n, i.e. dim S,+l(Ox) 5 1 for 0 5 1 < dim X - n. This implies (i). In order to have depths, O x 2 n we need moreover that dim Sn+l ( O X )5 1 for 1 = -1, i.e. depth Ox\y 2 n (we know already that H n & ( O X ) = 0). b) If H is general enough we have that S y = S x nY , dim S y = dim S X - 1,

+

652

3. Reduction to the smooth quasi-projective case

Here we will use our Lefschetz theorem for the Picard group of smooth quasi-projective varieties [HL2]. We still suppose that assumption (L) is satisfied (see Section 1). Moreover, since we only consider algebraic varieties in this section, an algebraic variety X will be also denoted X .

Proposition 3.1. Suppose that X is normal along Y and that Y is normal, S a closed subset of X such that X \ S and Y \ S are smooth, dim X 2 2, codimxS 2 2, codimyS n Y 2 2, and that H j ( X \ S ;Z) + H j ( Y \ S ;Z) is bijective, j = 1 , 2 . Then the mapping lim Pic U + lJ

-

Pic Y

is bijective.

Proof. Similarly as in the case of the Picard group let us write C1X instead of C1 X and CaCl X instad of CaCl X if X = Xan. Suppose that U is a Zariski-open neighbourhood of Y in X which is normal; then we have that C1 U C1 Y is bijective: In fact, under the hypotheses of our proposition, by [HL2] Theorem 1.4, we have that the natural map Pic(U \ S) -+ Pic(Y \ S ) is an isomorphism, therefore the natural map Cl (U \ S ) C1 (Y \ S ) is an isomorphism, since U \ S and Y \ S are non-singular. Since U n S is codimension 2 2 in U and Y n S of codimension 2 2 in Y , by Proposition 11.6.5 of [Ha] , we have the natural isomorphisms C1 (U \ S ) N Cl U and C1 (Y \ S ) N C1 Y . Therefore we obtain the isomorphism above. See also [RS]. This implies that CaCl U + CaClY is injective, because, since U and Y are normal, by [Gl] Prop. 21.3.4b), Cor. 21.6.10, the natural maps CaCl U + C1 U and CaCl Y --+ Cl Y are injective. Then, we consider the commutative diagram

-

-+

CaCl ( U ) -+ CaCl ( Y )

1 Cl(U)

1 N

Cl(Y)

653

Therefore lim Pic U 4

+ Pic Y

U

is injective, too. It remains to show the surjectivity. Let DObe a Cartier divisor on Y , then there is a Weil divisor D on X such that [D]H [DO].We may assume that D is mapped to DO. We must show that there is a Zariski-open neighbourhood U of Y such that DIU is a Cartier divisor, i.e. that Ox(D)IU is invertible. First, let us show that the stalks Ox(D),, II: closed point of Y, are free of rank 1. D )iso~ Let Z be the ideal sheaf of Y . We know that O X ( D ) ~ / Z ~ O X ( is morphic to Oy(D0),, hence a free Oy,,-module of rank 1. Let SO be a basis element of this module and s an inverse image in Ox(D),. Put M := Ox(D),/(s). Then M/Z,M = Oy(Do),/(so) = 0, hence we have Z,M = M . Since M is a Ox,,-module of finite type, because Ox(D) is coherent, by Nakayama’s lemma, M = 0, i.e. s generates Ox(D),. Now Ox(D), is contained in the ring of germs of meromorphic functions Mx,, of X at z. Since X is normal at II:, Mx,, is a field, hence O x ( D ) , is torsion free. This implies that the map OX,, Ox(D), defined by f w f s is an isomorphism, hence Ox(D), is free of rank 1. Therefore the set U of closed points z such that Ox(D), is free of rank 1 is a Zariski-open subset of X which contains Y , and DIU is a Cartier divisor.

-

0

Proposition 3.2. Suppose that there is a Zariski-closed subset C of Y with codimxC 2 4 such that H intersects X transversally outside C an the stratified sense, dim,X 2 4 at every x E X. Put S := SXU C. Then H j ( X \ S ;Z) Hj(Y \ S; Z) is bijective, j = 1,2.

-

Proof. Recall that X is a complex subspace of Pr(C).Let L be a linear subspace of H with dim L dim C = r - 1 which is general enough. Then, L n C = 0, and L intersects X \ C transversally, so

+

dimL nx = dimX - dimC - 1 2 3. Also, SY\ C = Y nSX\ C. By a theorem of Zariski-Lefschetz type of [HL4] (see Theorem 1.1.3), we have H j ( X \ S ;Z) N H j ( L n X \ S ;Z) as well as Hj(Y\S;Z)?Hj(LnX\S;Z),j=1,2. 0

654

Altogether we obtain:

Theorem 3.1. Suppose that X is normal along Y and that Y is normal, C a Zariski-closed subset of Y with codimxC 2 4 such that H intersects X transversally outside C in the stratified sense, dim, X 2 4 at every x E X . Then the mapping lim Pic U + U

-

Pic Y

is bijective.

-

Proof. By Proposition 3.2 we have that H j ( X \ S;Z) H j ( Y \ S;Z) is bijective for j = 1,2, where S := CUSx. Proposition 3.1 gives the bijection lim Pic U P i c Y.

-

U

0

In particular, we obtain:

First proof of Theorem 1.1 Since we assume depths,@ 2 3, Lemma 2.10 tells us that depthOx\y L 3 is equivalent to depths,Ox 2 3. By Lemma 2.2 this implies that both X and Y are normal. Now choose C := Sy. By Lemma 2.1 the hypothesis depths,@ 2 3 implies codimySy 2 3, so codimxC 2 4. Since C = Sy, H intersects X transversally at any point of X \ C. The condition rcd(X \ Y ) 2 4 implies that dim, X 2 4 at every x E X. We have all the hypotheses of Theorem 3.1, so lim Pic U P i c Y is an isomorphism.

-

U

-

PicU. It remains to show that P i c X -+ lim + U

Here we use Theorem 2.1. Since rcd(X \ Y) 2 4, Definition 1.2 shows that H3(X,X \ {x}; Z)= 0, for all x E X \ Y. Now Theorem 1.1 assumes that depthox,, 2 3, for all x E X \ Y . This proves that X is analytically parafactorial at every point of X \ Y .

0 4. Use of the exponential sequence

Here we need a generalization of the classical Kodaira vanishing theorem (see [AJ] Prop. 1.1 and Lemma 3.3):

655

Theorem 4.1. (Generalized Kodaira vanishing theorem): Suppose that X is a compact complex analytic space, dim X 2 n, depthsxOx 2 n, and let C be an ample line bundle on X . Then

H k ( X ,Lc-')= 0 , k

< n.

Now let us return to the assumption (L) of $1. Let ZH be the ideal sheaf of H in 0 p r ( @ )and Z H ~ Xthe analytic restriction to X . A priori it does not need to coincide with the ideal sheaf Z of Y in X . We need:

Lemma 4.1. Assume that depthsxOx 2 1, x E X . Then the canonical (whose image i s TZ)i s injective. mapping ( Z H ( X )+ ~

Proof. This follows from [ST] Corollary 1.18 (see [BS] Cor. 11.3.8). 0

Corollary 4.1. Suppose that dim X 2 n and depthsxOx 2 n. Then

H k ( X ,0x1 + H k ( Y ,O Y )

is bijective for k < n - 1 and injective for k = n

- 1.

Proof. We may assume n L 1. Note that Z H ~ Xis invertible and that (ZHIX)-' is ample. By Lemma 4.1, Z H ~ XN Z. The rest follows from Theorem 4.1 and the cohomology sequence of

Here i : Y

-

0 -+I+

ox + i,oy + 0.

X is the inclusion.

Theorem 4.2. Assume that depthsxL?x 2 3, and rcd(X P i c X 21 P i c Y .

-

\ Y ) 2 4.

Then

Proof. We compare the exponential sequences for X and Y . By Corollary 4.1 we have that H j ( X , O x ) H j ( Y , O y ) is bijective for j = 1 and injective for j = 2. By the Lefschetz theorem with integer coefficients [HLl] (3.4), H j ( X ;Z) H j ( Y ;Z) is bijective, j = 1,2. So P i c X 21 P i c Y .

-

656

Obviously we get a second proof of Theorem 1.1: note that depth Ox\y and depthsvOy 2 3 implies that depths,Ox 2 3 (see Lemma 2.10).

>3

Now let U be a Zariski-open subspace of X whose complement is finite. In fact later on, we shall consider Zariski open neighbourhoods of Y in X. In this case, we need a slightly more general statement. First we can prove:

Theorem 4.3. Suppose that U is a Zariski open subset A := X \ U is finite. Let 3 be a coherent analytic sheaf on depth3lU 2 n and {z E U I depthT, = n} is compact. Let L invertible sheaf on X . Then Hq(U,3 8 C k = ) 0 for q < n,k

of X , then X such that be an ample

>> 0.

Proof. We may suppose X = P,.(C). Then there is a topological duality between the separated spaces associated with Hq(U,T 8 L T k ) and the hypercohomology ExtE-q(U, 3 8 C kw) , of X for the complex j !j*Rhom(3 8 C-'", w),where j is the inclusion of U into X and w := Clk, see [BS] Theorem VII.4.1. Moreover in view of [BS] 1oc.cit. it is sufficient to show that Ext:(U,T 8 C W kw, ) = 0 for q > r - n, k >> 0 t o obtain our statement. Using an adequate spectral sequence it is sufficient to show that

H,P(U,Estq(T,w)8 ck)= 0, for p + q > r - n, k >> 0. So we have to show that

+

+

is bijective for p q > r - n, k >> 0, and surjective for p q = r - n, Ic >> 0. But both groups are 0 for p > 0, k >> 0 (the first one because of the cohomological criterion for ampleness, see Proposition I11 5.3 of [Ha], the second one because A has dimension 0 ) . We have an isomorphism for p = 0,q > r - n since E z t Q ( F , w )is conn. For centrated on A then because of the hypothesis depth31U p = 0 , q = r - n, we have an epimorphism because Eztr-n(.F, w)is concentrated on the disjoint union of A and another compact set.

>

0

Now, we can prove a modified version of Corollary 4.1:

657

Theorem 4.4. Suppose that X is a closed complex subspace of PT(C)and dim, X 2 n+ 1 f o r all x E X . Suppose that U is a Zariski open neighbourhood of Y , depths,Ou 2 n. Then H k ( U , O u ) H k ( Y , O y ) is bijective for k < n - 1 and injective for k = n - 1.

-

Proof. We may assume n 2 1. We have to prove that Hq(U,Z) = 0 for q < n. For this purpose we can assume that H is chosen to be general, because Hq((U,Z)21 Hq(U,O(-l)). Now depthsuOu 2 n; since dim,X 2 n 1 for all z E X we have dim S,(Ou) 5 0, so depth Ou 2 n 1 outside some finite set, by Lemma 2.1. By Theorem 4.3, H q ( U , Z k )= 0 for q < n, k >> 0; note that Z k l X N Z k by Lemma 4.1. Therefore, (Zk/Zk+l)lX N Zk/Zk+l, too. On the other hand, Hq(U,(Z$/ZL+l)IX) = 0 for q < n , k > 0 , by Theorem 4.1 observing that Zis L-'. By descending induction we get that Hq(U,Zk) = 0 for q < n, k > 0, in particular for k = 1.

+

+

0

Now we get

Theorem 4.5. Assume that depthsxOx 2 3 near Y and rcd(X \ Y ) 2 4 near Y . Then lim Pic U

N

4

Pic Y.

U

Proof. Let U be a Zariski-open neighbourhood of Y in X such that

-

and rcd(U \ Y ) 2 4. We compare the exponential sequences for U and Y. By Theorem 4.4 we have that H j ( U , O x ) H j ( Y , O y ) is bijective

-

for j = 1 and injective for j = 2. By the Lefschetz theorem with integer coefficients [HLl] (3.4), H j ( U ; Z ) H j ( Y ; Z ) is bijective, j = 1,2. So Pic"" U _N P i c Y . By Lemma 2.8 b), PicU _N Pic"" U . 0

We can deduce Theorem 4.2 again, using Theorem 2.1.

658

5. Grothendieck’s approach Finally, we may apply the result of A.Grothendieck in [G2] Exp. XII, Cor. 3.6. We still assume (L). Let us apply the Kodaira vanishing theorem mentioned before t o Y instead of X . This is just the missing link in order to get:

Theorem 5.1. Assume that X is a compact subvariety of Pr(C), depth O x 2 2, depths, O y 2 3 and dim X 2 4. Then: lim Pic U c

N

Pic Y.

U

Proof. From the result of A. Grothendieck, the only statement to be verified is that H k ( Y , O y ( - s ) ) = 0 for k = 1 , 2 and all s > 0 , which follows from Theorem 4.1 applied to Y . 0

In particular, we obtain a third proof of Theorem 1.1, using Theorem 2.1, because our hypotheses imply the analytic parafactoriality of X at every point x E X \ Y .

Remark: Theorem 5.1 is in fact a consequence of Theorem 3.1. In Theorem 3.1, put C := S y . Since depthsyOy 2 3, we have codimxC 4. F’urthermore, H is transversal to X \ C. Because of Lemma 2.2, Y is normal, and it is easy to see from Lemma 2.10 that we must have depthsxOx L 3 along Y , so X is normal along Y , too. Bibliography: [AF] A. Andreotti-T. Frankel: The Lefschetz theorem on hyperplane sections, Ann. Math. 69 (1959), 713-717. [AJ] D.Anapura, D.B. JafFe: On Kodaira Vanishing for Singular Varieties. Proc. AMS 105 (1989), 911-916. [B] R.Bott: On a theorem of Lefschetz, Mich. J. Math. 6 (1959), 211-216.

659 [BS] C. Bhicii, 0. St&nin$il%, Algebraic methods in the Global Theory of Complex Spaces, John Wiley, London, N.Y., Toronto, 1976. [FG]J. Frisch, J. Guenot: Prolongement de faisceaux analytiques cohkrents. Invent. Math. 7,321-343 (1969). [G-MI M. Goresky, R. MacPherson: Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 14. Springer-Verlag, Berlin (1988). [Gl] A. Grothendieck: Elkments de la GQom6trie IV, 4me partie. Publ. Math. IHES 32 (1967). [G2] A. Grothendieck: Cohomologie locale des faisceaux cohkrents et thkorkmes de Lefschetz locaux et globaux (SGA 11). North-Holland: Amsterdam 1968. [G3] A. Grothendieck: Sur les faisceaux algbbriques et les faisceaux analytiques cohkrents (Exp. 2). Skm. H. Cartan 9 (1956/57). [G4] A. Grothendieck: Elkments de la GkomQtrieI. Publ. Math. IHES 4 (1960). [G5] A. Grothendieck: RevBtements Qtaleset groupe fondamental, exp. XI1 (Mme M. Raynaud). Springer Lect. Notes in Math. 224 (1971), Springer Verlag, Berlin-Heidelberg-New York, or: SOC.Math. France 2003. [GD] A. Grothendieck - J.A. Dieudonnk: Elkments de la GkomQtrie AlgQbriqueI. Springer-Verlag: Berlin-Heidelberg-New York 1971. [GH] A. Grothendieck - R. Hartshorne, Local cohomology, Springer Lect. Notes in Math. 41 (1967), Springer Verlag, Berlin-Heidelberg-New York. [HLl] H.A.Hamm, LB D.T.: Rectified homotopical depth and Grothendieck conjectures. In: The Grothendieck Festschrift, vol. 11, pp. 311-351. Birkhauser: Boston 1990. [HL2] H.A.Hamm, LB D. T.: On the Picard group for non-complete algebraic varieties. Singularits Franco-Japonaises, 71-86, Smin. Congr., 10, SOC.Math. France, Paris, 2005.

660

[HL4] H.A. Hamm, L6 D.T.: Lefschetz theorems on quasi-projective varieties. Bull. SOC.Math. France 113, 123-142 (1985). [Ha] R. Hartshorne, Algebraic Geometry, GTM 52 (1977), Springer Verlag, Berlin-Heidelberg-New York. [KK] L.Kaup, B.Kaup: Holomorphic functions of several variables. De Gruyter: Berlin 1983. [RS] G.V.Ravindra, V.Srinivas: The Grothendieck-Lefschetz theorem for To appear in normal projective varieties (arXiv:math.AG/O511134). J.Algebraic Geometry. [Sch] GScheja: Riemannsche Hebbarkeitssatze fur Cohomologieklassen. Math. Ann. 144, 345-360 (1961). [Se2] J.-P.Serre: GBometrie algebrique et gkomktrie analytique, Ann. Institut Fourier 6 (1956), 1-42. [Sill Y.T.Siu: Analytic sheaves of local cohomology. Trans. Amer. Math. SOC.148, 347-366 (1970). [Si2] Y.T.Siu: Analytic sheaves of local cohomology. Bull. Amer. Math. SOC.7 5, 1011-1012 (1969). [Si3] Y.T.Siu: Extending coherent analytic sheaves. Ann. of Math. (2) 90, 108-143 (1969).

[ST] Y.T.Siu, G.Trautmann: Gap-sheaves and extensions of coherent analytic subsheaves. Springer Lect. Notes in Math. 172 (1971). [Tl]G.Trautmann: Ein Endlichkeitsatz in der analytischen Geometrie. Invent. Math. 8, 143-174 (1969). [T2] G.Trautmann: Ein Kontinuitatssatz fur die Fortsetzung koharenter analytischer Garben. Archiv der Math. 18, 188-196 (1967).

661

Braid monodromy and

7r1

of discriminant complements

Michael Lonne * Our aim is to introduce the braid monodromy of discriminants of hypersurface singularities. We present our result on the braid monodromy of hypersurface singularities of Brieskorn-Pham type and show, how this leads to presentations of fundamental groups of the corresponding discriminant complements. These are shown t o be straightforward generalisations of the presentations of Artin groups associated t o the standard Dynkin diagrams of ADE type singularities.

1. Introduction

In the case of simple hypersurface singularities of type ADE discriminant complements in a semi-universal unfolding are identified as spaces of regular orbits for the Weyl group of the same type and are shown to be aspherical. Their fundamental groups are given by the Artin-Brieskorn groups of the same type with a natural presentation encoded by the corresponding Dynkin diagram. Though already thirty years ago Brieskorn [3] asked explicitly for the fundamental group and suggested to obtain such groups from a generic plane section using the theorem of Zariski and van Kampen, only a few results have been obtained since recently. In fact our recent achievment [S] in case of Brieskorn-Pham polynomials is based on his idea, but relies heavily on the investigation of non-generic plane sections and their relation to generic ones. singularity theory

Let us first briefly review some basic notions of singularity theory. Def. : A holomorphic function f defined close to 0 E C n defines a singu'Institut fur Mathematik, Universitat Hannover, Am Welfengarten 1, 30167 Hannover, now at: Mathematisches Institut, Lehrstuhl Mathematik VIII, UniversitatsstraOe 30,D95447 Bayreuth

662

larity, if 0 is a critical value. f(0) = &f(O)

= 0.

We will naively speak of polynomials, functions and affine space, instead of using the appropriate language of germs.

Def. : Singular functions are considered equivalent, if they differ by a change of coordinates only. With the concept of unfoldings we get hold of all slight perturbations of f , at least up to equivalence. In case of isolated singularities, which we investigate, the equivalence class [f] of f determines - up to non-canonical isomorphism - a pair of functions F, F' such that

F' : C" x Ck-l 4 C ,F'(z,O) = f(z),F'(0,u) = 0 , F : C" x C k -+ C , F ( z , z , u )= F ' ( x , u ) - z . The existence of the following diagram is essential to our approach:

z,z,u

C"+k

1

19

z,u

Ck

3

X

3

v

I 1P u

Ck-l

323

In this diagram we placed some emphasis on of F , on the discriminant set

X

:= F-l(O), the zero set

V :={ (2,u)IFZ,+: z H F ( z ,u, 2) has singular 0-levelset} and on the bifurcation set

B := {u I FL

:z

H

F ' ( z , u)is not a Morse function}

We note the following features: 0

0

0

X intersected with the preimage of the complement of V is the total space of the Milnor fibration which surjects onto C k - V with fibre M . .rrl(Ck- V )acts on H,-l(M) and preserves the intersection product, V 4 C"' is finite with branch locus B,

663 0

V is the zero set of a monic polynomial P of degree k in . coefficients in C[u]

braid monodromy and

z with

?rl

-

By the last two properties we get up to conjugation

h : 7r1(Ck-'- L?)

Brk,

the braid monodromy, cf. [9], with im h the braid monodromy group. We can deduce h from the k-punctured disc bundle structure with strucBrk, e.g. closely following the approach of the theory of ture group G polynomial coverings by Hansen [ 5 ] . Alternatively we may interpret the braid monodromy using the coefficient map to Ck-l2 C"'/Sk, the Lyashko-Looijenga map. Then h should be understood as the induced map on fundamental group with recourse to some isomorphism

rI(Ck-'/Sk- A)

Brk,

where A is the discriminant in the space of monic polynomials of degree k with zero coefficient of the monomial of degree d - 1. We want to use the knowledge of the braid monodromy group on the fundamental group. To this end the result of Zariski and van Kampen, cf. [7], has to be employed, which states -

q e! (tl, ...,t k I ti = @,)*(ti))

where { p s }is a set of generators for im h and each (p,), is the automorphism of the free group induced by the Hurwitz action of Brk.

Remark: Suppose a braid p, is a conjugate of a power of u1,then all relations tr'p, ( t i ) are actually the consequence of a single relation tL1/3,(ti,) for a suitable 2,. As we will see the braid monodromy group is generated by such elements and therefore the number of relations drops considerably. ADE-case

Let us recall that for a finite Euclidian reflection group W of type A , D ,E there are Coxeter-Dynkin diagrams encoding a presentation of W as a Coxeter group: A cc.. . . -D ...

>

E6

' 7 E7 ?*

-

E8- *

664

(

W % ti, i a vertex1

t:, (titj)3,subgraph i - -. (titj)2,subgraph i -

.j’) j

W acts on C kby complexification with trivial stabilisers in the complement of ‘H, the union of reflection hyperplanes. The quotient space by W is

(Ck, 7-l)I)IW (Ck, D), (Dthe discriminant). The complement C k-D is aspherical (7ri trivial, i tal group B w ,the Artin group associated to W .

> 1) with fundamen-

titjti = tjtitj, subgraph titj = t j t i , subgraph

i - -. i-

.j’) j

In fact using the corresponding singular functions f,

zk+’(Ak), x2Y

+ yk-’(Dk), x 3 + y4(&), x 3 + Zy3(E7),

Ic3

f Y5(E8),

we can recover: 0 0

0

0

the orbit space C k , D as the unfolding space and the discriminant, Bw 7r1(Ck-D) with the standard presentation from braid monodromy, for a set of suitably chosen pa, W as the image of 7r1(Ck- D)in Aut(H,-I(M)) provided f has been stabilised to an odd number n of variables, the Coxeter Dynkin diagram as the intersection diagram of suitably chosen elements of H,- 1 (M).

results for Brieskorn-Pham polynomials First we have to introduce some more notions and to cite another result:

Def. : f E C[xl, ..., x,] is a Brieskorn-Pham polynomial if

f (51,..., 2,)

= ZF+’

+ .. . + &+I,

la E Z’O.

By the combined efforts of Pham [lo],Hefez-Lazzeri [6] and Gabrielov [4] it is known that the quadratic form (cf. [l,p. 611) of a Brieskorn-Pham polynomial f is given by a geometric basis {wi} indexed by a lexicographically ordered set of sequences of length n

I = {ili 2...i, 11 5 i, 5 1, for 1 <_ v I n }

665

such that the intersection product is -2 ifi=j if li, -j,l > 1 for some v 0 if (i,, - j,,)(i,,! - j v t ) < 1 for some v,v' the corresponding diagrams in low dimension are e.g.

.

=

=

-

G

o

/

/

/

/

We will state the result in terms of the band generators ai,j of the braid group, cf. [2], which are conjugates of the standard generators. Every strand in ai,j is straight except for the i-th and j-th, which are crossed in front of all intermediate ones. Now the generators can be given in terms of the band generators and in terms of properties of the subgraphs supported on the corresponding indices: In fact the generators of the braid monodromy group are determined by the subgraphs supported on two or three vertices:

+

Theorem 4. Given a Brieskorn-Pham polynomial f = x?+l+. . . xk+' the braid monodromy group is generated by a set of braids a?. w1

a&, a? w .a2 z,k 0 Z ~J

for i . .j for i - - - j , or i . - - ' j , 2 for , i'\T 7 . k or i',; 7 . k j'

j'

The condition on the indices can also be expressed as (vi,vj) = 0 in the first, (vi,vj) # 0 in the second and (vi,vj)(vi,v~)(vj,vk) < 0 in the last row. A straightforward computation using the result of Zariski and van Kampen yields: Theorem 5. Given a Brieskorn-Pham polynomial f = x?+l+. . the group r1(Ck - 2)) has a finite presentation

titj = tjti, for i ' .j titjti = tjtitj, for i . - . j , i . titjtkti = tjtktitj, for i ' < ?'k,i'\; 3'

-.j,

7'k

3'

+ xk+'

666

The appealing aspects of this assertion are i) that it obviously generalises the corresponding claim for simple hypersurface singularities with respect to the tree intersection diagrams for their quadratic forms, where the last case is void, ii) and that it rises immediate hopes that its validity extends to even more genera1 singularities.

+

In fact we have found an extension to singularities in the series x 2 y y k , but in general one has to be very cautious, since the ’correct’ Dynkin diagram has to be found, though this may be related to the polar approach of Gabrielov to intersection diagrams.

Thorn-Sebastiani property We can not but give a very rough outline of the ingredients of the proof: i) a genericity argument showing how the braid monodromy group can be determined from appropriate non-versal unfoldings, ii) setting up an induction over the number n of variables with n = 1 (easy) and n = 2 (hard) as the basic cases. iii) a proof of the induction step based on a Thom-Sebastiani principle for discriminants of sums of singularities (see below). iv) a lot of braid group calculations to keep the occurring braid under control. Actually the only topic we want to say something about is the ThomSebastiani principle which we consider a most important guideline of our approach: Suppose p ( u , z ) , q(w, z ) are polynomials monic in z , and denote by P * 4(u,w, z ) = rest (P(%t

+ z ) ,Q(V’ -t))

7

where rest is the resultant with respect to the variable t of its two arguments. p * q is then the monic polynomial, such that for (u, w) fixed the zeroes of p * q(u,w ) E C [ z ] are all sums of a zero of p ( u ) E C [ z ]with a zero of q(w) E C [ z ] . In case p , q are the polynomials of discriminants of functions f,g we get

Theorem 6. Suppose the braid monodromy groups of p , q are generated by (each conjugate t o a: o r a:), then the braid monodromy group of

pi, yj

667

a slight perturbation of p of Pi, ~ j .

* q is generated

b y braids given explicity in t e r m s

4j(Pi

if rj &(as,t) if rj 4 7 $&4 if rj 4, lclrj(4if rj 4. &J&)

-

&, gz,t, as,t,a:,a:, @- universal objects depending o n l y o n t h e degrees of P, 4. T h i s is in fact t h e key to t h e induction step since i) t h e Brieskorn-Pham polynomials can be inductively defined as sums of simple Ali singularities, ii) and t h e restricted unfolding of which p q describes t h e discriminant is appropriate to find the braid monodromy group of t h e s u m according to our genericity argument.

*

Bibliography 1. V.I. Arnol’d, S.M. Guseh-Zade, A.N. Varchenko: Singularities of differentiable maps, Vol. 11, Monographs in Mathematics, 83. Birkhauser Boston, Inc., Boston (1988), 2. J. Birman, K. KO, S.-J. Lee: A new approach to the word and the conjugacy problems in the braid groups, Adv. Math. 139 (1998), 322-353 3. E. Brieskorn: Vue d’ensemble s u r les probhnes de monodromie, Singularit& A Cargkse (Rencontre sur les Singularit& en GBom6trie Analytique, Inst. Etudes sci. de Cargkse, 1972), pp. 393-413, Asterisque Nos. 7 et 8, SOC. Math. France, Paris, 1973 4. A. M. Gabrielov: Intersection matrices for certain singularities, Funct. Anal. Appl. 7 (1973), 182-193 5. V. L. Hansen: Braids and Coverings, Cambridge Univ. Press (1989), London Math. SOC.Student Texts 18 6. A. Hefez, F. Lazzeri: The intersection matrix of Brieskorn singularities, Invent. Math. 25 (1974), 143-157 7. E. R. van Kampen: On the fundamental group of an algebraic plane curve, Amer. J. Math. 55 (1933), 255-260 8. M. Lonne: Braid monodromy of hypersurface singularities, Habilitationsschrift (2003), Hannover, mathAG/0602371 9. B. Moishezon: Stable branch curves and braid monodromies, In: Algebraic Geometry (Chicago, 1980), Lect. Notes in Math. 78, Amer. Math. SOC. (1988), 425-555

668 10. F. Pham: Formules de Picard Lefschetz gkn6raliskes et ramification des intkgrales, Bull. SOC. Math. France 93 (1965), 333-367

669

TANGENTIAL ALEXANDER POLYNOMIALS AND NON-REDUCED DEGENERATION

M. OKA Department of Mathematics, Tokyo University of Science 26 Wakamiya-cho, Shinjuku-ku, Tokyo 162-8601 E-mail: [email protected]. tus. ac.jp

Dedicated to Professor Kyoji Saito for his 60th birthday We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with &Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration Ct,-+ Do jL. We show that there exists a certain surjectivity of the fundamental groups and divisibility among their Alexander polynomials.

+

1. Introduction

Let C be a plane curve. We are interested in the geometry of plane curves. Choose a line L c P2 and put := P2 - L. As geometrical invariants, we consider (a) Fundamental groups: 7r1(P2 - C) and 7rl(Ci - C) (b) Alexander polynomial A, (t; L ). Zariski studied nl(P2 - C) systematically [37] and further developments have been made by many authors. To compute the Alexander polynomial, we need to choose a line at infinity L. However for a generic L , the Alexander polynomial has too much restrictions and we have often the trivial case Ac(t; L ) = (t - l)r-lwhere T is the number of the irreducible components. In our previous paper [25], we have introduced the notion of &Alexander polynomials. This gives more informations for certain reducible curves but it does not give any further information for irreducible curves. The purpose of this paper is to introduce the notion of the tangential Alexander polynomials. Namely we consider all tangent lines TpC for the line at infinity. It turns out that tangential Alexander polynomials are related to @Alexander polynomials. We apply these polynomials to study non-reduced degenerations. Let Ct, E A be an analytic family of reduced

Cci

670

+

curves for t # 0 such that Ct 5Co = DO j L where L is a line. The case j 2 2 is a typical non-reduced degeneration. In this situation we study the geometry of DOusing that of Ct. One of our results is the surjectivity assertion of the natural homomorphism: f$ : nl(C; - Do) -+ nl(C; - Ct)

Here the point is that L is the line component of the limit curve CO ( Theorem 14, 55). This paper consists of the following sections. $2 Fundamental groups $3 Alexander polynomial 54 Dual stratification and tangential fundamental groups $5 Degeneration into non-reduced curves with a multiple line 2. Fundamenta1 groups

Let L be a fixed line and put Ci := P2- L. We say L is generic with respect to C if L and C intersects transversely. The topology of - C does not depend on L if L is generic and we call it the generic afine complement and we often write as C2 instead of Ci. The following Lemma describes the relation of two fundamental groups.

Cci

Lemma 1. ( [ l g ] ) Let w be a lasso for L and N ( w ) be the subgroup normally generated by w. (1) The following sequence is exact. 1 -.+ N ( w ) -.+ 7Tl(@i-

c,bo)

-.+

n1(P2

-

c,bo)

-.+

1

(2) Assume that L is generic. Then (i) w is in the center of n l ( C i - C ) and N ( w ) 2 Z. (ii) W e have the equality D(nl(C2- C ) )= D(nl(P2- C ) ) among their commutator groups. Thus nl(P2 - C ) is abelian if and only if nl(C2- C ) is abelian.

For non-generic line L , n l ( C i - C) may be non-abelian even if nl(P2 - C ) is abelian. For example, let C = { Y 2 Z - X 3 = 0) and take L = ( 2 = 0). Then nl(C.2, - C) B3 where B3 is the braid group of three strings and we recall that B3 ( a ,b I aba = bab) ( [4]). 2.1. First homology group H1(P2 - C ) .

Assume that C is a projective curve with r irreducible components C1,. . . ,C, of degree d l , . . . ,d, respectively. By Lefschetz duality, we have the following.

671

Proposition 2. H1 (P2- C, Z) is isomorphic to Z'-' do = gcd(d1,. . . , dv). I n particular, H1(C; - C ) Z'.

x (Z/doZ) where

Take a lasso gi for each component Ci of C for i = 1 , . . . ,r. Then the corresponding homology classes { [gi],i = 1 , . . .,r } give free abelian generators of H$; - C). 2.2. Degenerations and fundamental groups

Let C be a reduced plane curve. The total Milnor number p ( C ) is defined by the sum of the local Milnor numbers p(C,P ) at the singular points P of C. Let A := {< E C I I<] 5 1) the unit disk. We consider an analytic family of projective curves Ct = { F t ( X ,Y,2 ) = 0 } , t E A where F t ( X ,Y,2 ) are reduced homogeneous polynomial of degree d for any t. We call {Ct;t E A} a reduced degeneration. We assume that Ct, t # 0 have the same configuration of singularities so that they are topologically equivalent but COmay obtain more singularities, i.e., p(Ct) 5 ~ ( C O ) .

Theorem 3. For a given reduced degeneration {Ct;t E A}, there is a canonical surjective homomorphism for t # 0: cp : 7rl(P2 - CO) -+ 7r@

I n particular, if 7r1 (a2- CO)is abelian, so is

- Ct) XI (P2- Ct).

See for example, [25] and also Theorem 14 of $5 for another simple proof. 2.3. Product formula

Assume that Ci is a curve of degree d i , i = 1 , 2 which are intersecting transversely at dld2 distinct points. We denote the transversality as C1 h C2. Take a line L such that L n C1 n C2 = 0. Note that L need not be generic for CI or C2.

Theorem 4. (Oka-Sakamoto 1,291) Under the above assumption, we have

For further information about fundamental groups, we refer to [4,13,19,21,33].

672 2.4. Example

2.4.1. Abelian cases A curve C with small singularities has often commutative fundamental group 7r1 (P2- C). Some examples are here: - C is a smooth irreducible curve. - Irreducible curves with only A1-singularities (i.e., nodes) by [ 37,10,9,11,17,30], or irreducible curve of degree d with a nodes and b cusps (i.e., A2) with 6b 2a < d2 ( [ 1 7 ] ) . - 7rl(P2 - C) (respectively TI(@: - C)) is abelian for any irreducible curve of degree d if it has a flex of order 2 d - 3 in Ci (resp. of order d - 2 ) ( 1371). Let f : C2 -+ C be a polynomial mapping. Recall that a is a atypical value at infinity if the topological triviality at infinity fails at t = a for the family of curves Ct := f-l(t) (see [35]).

+

Proposition 5. ( / . I ] ) Let f : C2 + C be a polynomial mapping and assume that 0 is not an atypical value at infinity and C = f-'(O) is smooth in C2. T h e n 7r1(C2 - C ) S Z.

2.4.2. Non-abelian case Assume that p , q are positive integers greater than 1and consider the curve

c,,, :

f P ( X ,y,21,

+ f,V, y,ZIP = 0

where f p , f, are poIynomials of degree p , q respectively. CP,, is called a curve of ( p , q)-torus type. Assume that two curves {fp = 0 } and {f, = 0 } intersect transversely and there is no other singularities of Cp.q. Then 7r1 (P2 - CP,,) S G(p, q, q) and 7r1 (C2 - C ) E G(p, q). In particular, if p , q Z,* Z,. For the definition of G(p,q) and are coprime, 7rl(P2 - C,,,) G ( p ,q, r ) , we refer to [19]. 2.5. Class formula and flex formula

For the study of curves of low degree, it is often important to know the existence of flex points. Let d = degree(C), d' be the degree of the dual curve C , let C(C) be the singular points of C and let a(C) be the number of the flex points. Then d and a ( C ) are given by the formula:

d' = d(d - 1 ) - C P E C ( C ) ( P ( C ,P ) + m(C,P ) - 1 ) a!(C)= 3d(d - 2) - C P E C ( C )Y ( G P )

673

where m(C,p) is the multiplicity of C at P and r(C,P ) is the flex defect of the singularity (C,P ) [16,22]. (In [22], we have denoted y(C,P ) as b ( c ,P ) . To distinguish with &genus of the singularity, we change the notation.) 3. Alexander polynomial 3.1. General definition

Let X be a finite connected CW-complex and let 4 : m ( X ) -+ Z be a surjective homomorphism. We fix a generator t of the infinite cyclic group Z. Let A is the group ring of Z. Then A is isomorphic to the Laurent polynomial ring C [ t ,t-'1 and A is a principal ideal domain. Consider an infinite cyclic covering p : 2 -+ X such that p # ( n l ( j i f ) ) = Ker 4. Then H1 ( g ,C ) has a structure of A-module where t acts as the canonical covering transformation. Thus by the structure theorem of modules over a principal ideal domain, we have an identification:

Hl(z,C) r A / A l @ . . . @ A / A , as A-modules. We normalize the denominators so that X i is a polynomial in

t with Xi(0) # 0 for each i = 1,. . . ,n. The Alexander polynomial associated to 4 is defined (see [12]) by the product A+(t):= Ai(t).

nzl

3.2. Alexander polynomials of plane curves

In our situation, we consider a plane curve C = C1 U - . U C, where C1,. . . ,C, are irreducible components of degree d l , . . . ,d, respectively. Take a line L as the line at infinity and let $0 be the composition $0 :

0

7rl(Ci - C ) L H l ( C i - C,Z) 2 Z '4Z

where B is a surjective homomorphism. Recall that 9 is determined by giving an integer ni to each component Ci such that gcd(n1,. . . ,n,) = 1. We call ni the weight for Ci. The Alexander polynomial of C with respect to (L,8) is defined by A+,(t) and we denote it as Ac(t; L,8). (1) (Generic case) Assume that L to be generic and 0 = Osum where 9,,, is defined by the canonical summation 9 s u m ( a l r.. . , a,) = XI=, ai ( weight 1 for each component.) In this case, we simply write as Ac(t) and call it the generic Alexander polynomial of C , as it does not depend on the choice of a generic L. (2) If 9 is the canonical summation Osum but L is not generic, we denote it as A c ( t ; L ) ,omitting 0. In particular, when L is the tangent line of a

674

smooth point P E C,we call A c ( t ;L ) the tangential Alexander polynomial at P and we also use the notation A c ( t ;P ) . (3) If L is generic but 0 is not OsUm, we called A c ( t ;L , 0) the &Alexander polynomial and we denote it as A c ( t ;0). In [25] we denoted it by A , e ( t ) , but for the consistency of the notation with (2), we use the notation Ac(t; 0). Recall that (t - l)'-' I A c ( t ) ( [25]). Thus this is also the case for A c ( t ;L ) with any line L , as A c ( t )I A c ( t ;L). We say that A c ( t : L ) is trivial if A c ( t ;L ) = (t - .'-)I

3.3. Fox calculus Suppose that G is a group and q!I : G -+ Z is a given surjective homomorphism. Assume that G has a finite presentation as

G

(51,.

. . ,x, I R1,.. . , &)

This corresponds to a surjective homomorphism : F ( n ) -+ G so that Ker $J is normally generated by the words R1 , . . . ,& where F ( n ) is a free group of rank n, generated by X I , . . . ,xn. Consider the group ring @ ( F ( n ) ) of F ( n ) with @-coefficients.The Fox differentials :@ ( F ( n )4 ) @(F(n)) for j = 1 , . . . ,n, are additive homomorphisms which are characterized by the following properties. (1) &xi = & , j and (2) (Leibniz rule) Ga( u w ) = u, w E @ ( F ( n ) )The . composition + o $ J : F ( n ) + Z gives a ring y : @ ( F ( n ) )-+ @[t, t-l]. The Alexander matrix A is an homomorphism rn x n matrix with coefficients in @[t, t-'] and its ( 2 , j)-component is given by y ( e ) . Then the Alexander polynomial A,(t) is defined by the greatest common divisor of ( n- 1) x (n- 1)-minors of A. In the case of G = r l ( X ) for some connected topological space X , this definition coincides with the previous one (Fox [S]). $J

&

&+tie,

3.3.1. Examples We gives several examples. 1. Consider the trivial case: G = Z' and q!I = Osum, the canonical one. Then 1-1. G Z 2 ( X I ) . Then A ( t ) = 1. 1-2. If G = Z' (xi,.. . , x r I RiJ. = x 2. x3 . %~ . - ~ x j 1 - ~5, i < j 5 r ) , we

675

have A(t)= (t - l ) v - l .This follows from the Fox derivation:

2. Let C = { Y 2 Z - X 3 = 0) and Lgen = { Z = Y ) ,L = (2 = 0). Note that ( 0 , 1 , 0 ) is a flex point of C and L is the flex tangent. Then TI

(@tgen -C)

2, AC(GL g e n ) = 1

I

T~(@L - C ) = ( 2 1 , ~x1x2x1 ~ = ~ 2 ~ 1 x 2 B3, )

Ac(t;L ) = t2 - t + 1

3. Let us consider the curve C = { Y 2 Z 3 - X 5 = 0) c (C2 and L = { Z = 0 } , A4 = {Y = 0). Then n1(P2 - C) E 2 / 5 2 and T ~ ( C ;- C) E G(2,5) and

TI

(@& - C ) E G(3,5). In this case, we get

3.4. Weakness of the generic Alexander polynomial A c ( t ) The following Lemma describes the relation between the Alexander polynomial and local singularities.

Lemma 6. (Libgober [12]) Let P I ,. . . ,Pk be the singular points of c (including those at infinity) and let Ai(t) be the characteristic polynomial of the Milnor fibration of the germ (C,Pi). Then the generic Alexander polynomial satisfies the divisibility: Ac(t;L ) I Ai(t).

nb,

Lemma 7. (Libgober 1121) Let d be the degree of C . Then the Alexander polynomial Ac(t;L,) divides the Alexander polynomial at infinity A,(t). If L , is generic, A,(t) is given by (td- l)d-2(t- 1 ) . I n particular, the roots of the generic Alexander polynomial are d-th roots of unity. Corollary 8. ( [12], See also [37]) Assume that C is an irreducible curve of degree d and assume that the singularities are either nodes (i.e., A l ) or ordinary cusp singularities (i.e., A2). If d is not divisible by 6, the generic Alexander polynomial Ac(t)is trivial. This implies that there does not exist any non-trivial generic Alexander polynomials of degree n with n $ 0 mod 6, for example, this is the case for cubic, quartic and quintic curves, whose singularities are copies of A1 or A2. However even though there does exist interesting geometry on these curves. We will show by examples that certain tangential Alexander polynomial

676

gives non-trivial Alexander invariants and we will give an explanation from viewpoint of non-reduced degeneration in 55. Another weakness of generic Alexander polynomials is for reducible curves. Let C1 and CZ be curves which intersect transversely each other. We take a line L so that L does not contain any points of C1 n Cz. Note that L need not be generic for C1 U CZ.Theorem 4 says that 7rl((cI;

-c 1 UCZ) 2 7r&

- Cl) x 7rl(cCi - CZ)

However the Alexander polynomial A,, uc2(t;L ) loses these informations. In fact, we have

Theorem 9. ( (251) Assume that C1 and CZ intersect transversely and let C = C1 U C2. Let L be a line such that L n C1 n CZ = 0. Then Ac(t;L ) = (t - l)”-’ where r is the number of irreducible components. For further information about Alexander polynomials, we refer to

[7,12,14,15,32]. 4. Dual stratification and tangential fundamental groups 4.1. Dual stratification of curves

Let C be a finite set of topological equivalent class of curve singularities and let M ( C ,d ) be the configuration space of plane curves of degree d with a fixed singularity configuration C. Take two curves C, C’ E M ( C , d ) in the same connected component and two smooth points P E C and Q E C’. We consider their tangent lines L = TpC, L’ = TQC.Though the topology of (P2,C) and (P2,C’)are topologically equivalent, this may not the case for (P2,C U L ) and (P2,C’ U L’). To analyze this, we introduce the dual stratification S ( C ) for C E S ( M ( C , d ) )and S ( M ( C , d ) )of M ( X , d ) as follows. Let P2 be the dual projective space. Recall that a point a = (a1 : az : a3) E P2 (resp. a point P = (p1 : p2 :p 3 ) E Pz)can be considered as a line La = (a1 X a2Y a3Z = O} in P2 (resp. a line L p = {pl U p z V p 3 W = 0) in Pz). First take C E M ( C , d ) . Let C(C) = { P I , .. . , P k } be the singular points of C. Let P(d) be the set of partition of the integer d. We consider the mappings : C --+ P(d) and : C 4 P(d) defined as follows. Let a E C (resp. P E C ) and let La n C = { R l , .. . ,R,} (resp. L p n C = { S l , .. . ,S&}). We define $(a) = { I ( C , L a&), ; i = 1 , . . .,v } where I(C,La;&) is the local intersection multiplicity. Respectively we define &P) = { I ( C ,Lp; S j ) ;j = 1,. . . , p } . Note that for a generic line

+

+

+

+

4

+

677

a E C, L, is a simple tangent line and therefore $(a) = { 2 , 1 , . . . ,1}. For a generic flex point P , the tangent line L = TpC gives the partition $ ( L ) = { 3 , 1 , . . .,1}. A line a E C is called an multi-tangent line if $ ( a ) has at least two members 2 2. A simple bi-tangent line is a typical such line which is simply tangent at two smooth points. A smooth point P E C is called tangentially generic if it is smooth and the tangent line TpC gives the partition { 2 , 1 , . . .,1}. Recall that the Gauss map associated with C, denoted as G c : C --+ C, is defined by G c ( P ) = TpC. Let Cntg(C) = {Pk+l,. . . ,Pk+t} be smooth points which are not tangentially generic and put g(C) = C ( C ) U P g ( C ) = {PI,. . . ,Pk+t}. The dual stratification S(C) of C is by definition, S := { C - g ( C ) , g(C)}. Thus if C is irreducible, S(C) has one open dense stratum made of tangentially generic points and k + t starata made of isolated points. 4.2. Dual stratification of the configuration space

M ( X ,d )

Now we consider the dual stratification of M ( C ,d). To distinguish a point in M ( C ,d ) and the corresponding curve, we denote points in M ( C ,d ) by (Y E M ( C ,d ) and the corresponding curve by C,. The configuration of the singularities of the dual curve C(C,) is not unique for a E M ( C ,d ) but it has only finite possible types, say Cy, . . . ,Cz when we fix the configuration space M ( C ,d ) . We consider the partition of the configuration space by the following sets: {a E M ( C , d ) ;(C(Cff),S(Cff),S(C,),&,&) are constant}

The dual stratification S ( M ( C , d ) )is defined by the strata which are the connected components of these partitions. Thus for a stratum M E S ( M ( C ,d ) ) , each C, and C,, a E M have constant dual stratifications. For a stratum M E S ( M ( C , d ) ) we , can associate a family of plane curves C,, (Y E M such that the dual family of curves C,, a E M is a family in M(Cj*,d)for some j . Observe that any a E M , the dual stratification S(C,)and S(Ca)are constant for a by definition. Thus for two a,/3 E M , C,, C, are homeomorphic as a stratified sets. More precisely we have Proposition 10. Take a0 E M and take a point L,, E Cao.This induces a continuous family of lines L , E C, such that &(La) is constant. Then the topology of the afine pair (cCia,C) does not depend on a E M . In

particular the fundamental group

( C i a - C ) does not depend on a E M .

ProoJ Recall that the local topology of C, U L, at an intersection point P is determined by the local Milnor number p(C, U L,,P) and this is

678

determined by p(C,, P ) and the local intersection multiplicity I ( C a ,L,; P ) . The definition of the dual stratification of S ( M ( C , d ) )guarantees the pconstancy of the family of plane curves C, U L,, cy E M of degree d 1. qed.

+

4.3. Tangential fundamental group and Tangential

Alexander polynomial For a line L E C, we call T I ( ( C ~ - C) the tangential fundamental group and A c ( t ;L ) the tangential Alexander polynomial. If L = TpC for some simple point P E C , we also use the notation A c ( t ;P ) for Ac(t;TpC). We also define Ic-fold Alexander polynomial Ac(t;P I ,. . . , Pk) by A c U L ~ U . . . U L ~ - ~ with ( ~ ; PLj ~ ) = TpjC. It is easy to observe that T I ( C~ C) and A c ( t ;P ) are constant on the open (dense if C is irreducible) strata of the dual stratification S(C). However in general it may give a different polynomial for singular lines L E C (they are the images of isolated strata of S(C) by the Gauss map). We will see some examples later. Thus the tangential Alexander polynomials altogether contain more geometrical informations than the generic Alexander polynomials. The main purpose of this paper is to investigate the property of the tangential Alexander polynomials. Note that if C is irreducible, there is only one choice of 0 (up to *) but there are many choices for L , even for irreducible

C. 4.4. Alexander spectrum

We also consider the set of tangential Alexander polynomials t-AS (C) := {Ac(t;P ) ;P E C } and we call t-AS (C) the tangential Alexander spectrum of C. There exist at most finite number of polynomials in the spectrum. In fact, it is bounded by the number of strata of S(C). We can also define the k-fold tangential Alexander spectrum of C by t-AS(”(C) := { A c ( t ;P I , .. . ,P k ) ; Pj E C} It often happens that even when the Alexander spectrum t-AS(C) is trivial, 2-fold Alexander spectrum t-AS(2)(C) (or higher one) is not trivial. 4.5. Example

+

We consider M(2A2 Al, 4) and M(Eg,4). By class formula, the dual curve C of a generic member C of M(2Az Al, 4) or M(E6,4)is a quartic

+

679

+

with 2A2 A1 in both cases. (C has generically 2 flex points.) In both configuration spaces M (2A2 A1 ,4) and M (&, 4), there are strata which correspond to degenerated members, namely curves with one flex of order 2. This implies that the dual curve has an E6 singularity. For these configuration spaces, there is a beautiful work by C.T.C.Wal1 [36]. Consider the subsets:

+

We can easily see that {MI, M2), { N1, N2) are respective dual stratifications of the configuration spaces M(2A2+Al, 4) and M(E6,4). We observe that under the Gauss map, -MI and N2 are self-dual and -M2 and N1 are dual each other. We observe also that M2 C aM1 and N2 C aN1.

+

(1-1) We consider quartic C1 E MI with C(C1) = 2Az A1 with two flexes. By the class formula, such a curve has a bi-tangent line. As an example, we take:

C1 :

17 4

- y4

+ 8 y3 + 1/4 + 7/2 y2 - 7/2 y2z2 + 1/4x4 - 1/2 z2 = 0

(1)

C1 has two cusps at PI = ( l , O ) , Pz = (-1,O) and one A1 at (0,-1). Two flexes are at Q1 = ( l O m , 9 ) , QZ = ( - 1 O m , 9 ) . We have a bitangent line y = 1 which are tangent at B1 = (2& l ) , QZ = ( - 2 f i , 1). For the dual stratification S(Cl), we have to take two more points S1 = (25/7,8/7), SZ= (-25/7,8/7) whose tangent lines pass through P2 and PI respectively. Using Zariski-van Kampen pencil method, we can compute T ~ ( C ; - C1) as r l ( c i - C l ) = ( < 1 , 6 , 6I & t 2 &

= &66, < 2 6 < 2 = 66253, <1<3 = t3<1, < 1 < 2 6 = <3<2<1)

where L = {y = 1). This gives &,(t;L) = t2 - t -t 1 by Fox calculus. Other tangent lines give the trivial Alexander polynomial. We leave the proof of this assertion as an exercise. Thus t-AS(C1) = (1,t2 - t 1).

+

680

(1-2) We consider the following quartic C 2 E M Zwith C(C2) = 2A2 with a degenerated flex of order 2 at infinity L = { Z = 0 ) :

CZ: 1- 12 y + 36 y2 - 32 y3 - 2 2’

+ A1

+ 12 x2y + x4 = 0

nl(Ci - c 2 ) = (&,&!,&[&~3El = 6 < 1 6 , 6 t Z b = tZ63&, b E 1 = <1<2) We can also see that A,, (t;L) = t2 - t 1. Note that two A2 singularities are at ( f l , 0) and one A1 is at (0,112). In the dual stratification S ( C z ) , there are two more ’singular’ points R 1 = (9’8) and Rz = (-9,8) whose tangent line pass through the cusps. However these tangent lines give trivial tangential Alexander polynomials. (2-1) Consider a quartic C3 E N 1 , defined by y3 x4 - xzy2 = 0 which has one E,+ngularity at 0. Two flex points are at (f6&/5,36/5). The bi-tangent line is given by y = 4. By an easy computation, we observe that ; = t2 - t 1. Other L = { y = 4 } g ives n l ( C i - C3) E B3 and A c S ( tL) tangent Alexander spectra are trivial. So t-AS(C3) = (1,t2 - t + 1). The dual stratification has 5 isolated points. (2-2) Consider the following quartic C4 E Nz, f ( 2 , y ) = y3 x4 = 0, with E6 and one flex of order 2 at P = (0,1,O). Take the flex line 2 = 0 as L. Then

+

+

+

+

r l ( G- C )

G(3,4) = ( Q , & , E ~ I E o = w E ~ w - ~t ,i =wEzw-’) and Ac,(t;L) = (t2- t l)(t4 - t2 1) where w = E 2 6 Q . 1 1 3 ( 3 ) L e t C : f(x,y) = - - y 4 - - - 3 x z y 2 + y 2 + - x 4 - 4 x 3 + 3 x 2 =Obea3 2 2 2 cuspidal quartic. As is well-known [37, 201, the generic affine fundamental group is a finite group of order 12, with presentation

+

+

m(C2 - C ) = (E,CIECE = C E C ,

E2 = C2).

Though the fundamental group is not abelian, the generic Alexander polynomial is trivial. For L = {y = 0) (this is the tangent cone of a cusp and L corresponds to a flex of C ) ,

m(@2,- C ) = ( E ’ C I E C E = C E C ) = B3. By the class formula, the dual curve C is a cubic curve with a node. S ( C ) has 3 singular points from 3Az and two ’singular points’ from the bi-tangent line. We can also see that n l ( C i - C) 2 B3 for an arbitrary tangent line L except the bitangent line Lb. The bi-tangent line is given by x = 213 in this example. By an easy computation, we see that T 1 ( C i b-

C)

(Eo,E1,J2,C1E05150

=GEOEl,

ElE2El

=EZGEZ,

E2Ct

= CE2C,

C =E1lEob)

68 1

and we have

+

Proposition 11. For the bitangent line Lb, Ac(t;Lb) = (t’ - t 1)’. For any other tangent line L, Ac(t; L ) = ’t - t + 1. In particular, this implies that t-AS(C)= { t 2 - t + 1, ( t 2- t 1)’).

+

4.5.1. Further example In the above examples of quartics, the geometry of CUTpC does not change for flexes of the same order. However this is not the case in general. Consider a fixed mark point P E C. We call (C,P ) a curve with a marked point P. Two curves with marked points (C,P ) and (C’, P’) are called a marked Zariski pair if {C U TpC, C’U TplC’} is a Zariski pair. For further information about Zariski pairs, see [1,2,3,25,26]. In [27] we have shown that for any quintic B5 with configuration in the next list, there exist two different flex points P, P’ such that (B5,P ) and (B5,P’) are marked Zariski pairs.

(tt)

{

4A2, 4A2

+ A1 , A5 + 2 4 , A5 + 2A2 + A1, G3 + 2Az

E6+A57 2 A 5 , A 8 + A Z , A S + A Z + A l , A l l

+

In fact their generic Alexander polynomials are given as ’t - t 1, 1 respectively. This implies that the among flexes of these quintics, there are two classes of different topological nature: one class which does not contribute the tangential Alexander spectrum and the other which contributes by (t2- t+ 1). We give one example. The following quintic B5 : f(x,y ) = 0 has A11 singularity at the origin and 9 flex points. Among them, the flex at P = ( 0 , l ) is different from others (a flex of torus type). In fact, B5 UTpB5 is a sextic of torus type [27]. All other flex points gives trivial tangential Alexander polynomial.

13

4.6. 8-Alexander polynomials

To cover the weakness of Alexander polynomials for irreducible curves, we have proposed &Alexander polynomials in [25]. First recall that the

682

, Figure 1.

Quintic with

A11

radical 4(t)'ed of a polynomial 4 ( t ) = H:=,(t - &)pi is defined by

:=

nbl(t-ti). Here pi 2 1,Vi. The following theorem shows the importance of &Alexander polynomial.

Theorem 12. (1251) Assume that C, C' be reduced curves and we assume further C' is irreducible. For a given integer n, suppose that the surjective homomorphism 4" : 7ri(C: - C U C') + Z which has weight 1 on each component of C and weight n on C'. Then (1) Acuct(t;L , 4") is divisible by g c d ( A c ( t ;L ) x (t - l),(t" - 1))for n # 0. Suppose further that C h C' and 7rl((c; - C') 2 Z.Then (2) Ac(t;L ) x (t - 1) is divisible by A c u c t ( t ; L , 4"). red (3) A C u c ~ ( t ; L , 4 n ) - gcd(Ac(t;L) x (t - 11, (t" - 1)). (4) I n particular, if g c d ( A c ( t ;L ) x (t - l),(t" - 1))= &(t; L ) , we have

Acuct(t;L, 4") = Ac(t;L ) x (t - 1). Proof. The proof goes exactly as in [25]. Consider the canonical surjective homomorphism: h : 7rl((c; - C U C') +. 7r1(@; - C) x Z. Consider the presentation. TI(@:

- C) = (gi,. . . ,gs I RI,. . . ,Rk),s I degreeC

The homomorphism homomorphism

4" factors as &

=

1c, o h where 1c, is the surjective

683

$ : TI(@; - C ) x H I ( @ ; - C') 7r1 (Cc; - C ) x Z -+ Z where the weight of the second factor is n. Note that Acuct (t;L , 4") = Ab,, ( t )in the notation Now of $3.1. By the above factorization, we have the divisibility: A, the calculation of A, is done in the exact same manner as in [25]. We use the presentation:

7rl(ci c)X Z = ( g i , . . . , g s , < I R 1 ,... ,&,Ti, -

15 i 5 S)

where Ti = giEgzF1[-l. The key point of the calculation is the following:

Let M be the Alexander matrix of 8,, : 7rl(Ci - C ) -+ Z and let M' be the Alexander matrix of : .rrl(CL - C ) x Z -+ Z. Then M' is written as $J

MI=( MO'

)

Nl Nz

6 is a zero vector and N1 is a s x s-matrix which is explicitly given as (1 - P ) E s where E, is the identity matrix of rank s. The vector Nz takes the form ' ( t - 1,.. . ,t - 1) where t - 1 is repeated s times. For any (s - 1) x (s - 1)-minor B of MI let B be the s x s-minor adding (k 1)-th row and the last column. Then det B = det B x (t - 1). Thus we have A c ( t ; L )x (t - 1) as the common divisor of such det B's. Also we get

where

+

(t" - l)s by taking a minor from N1. We observe also that any determinant of a s x s-minor which contains at least two rows of (N1,N z ) is divisible by (t" - 1). Thus we observe two divisibilities: A,(t)

I gcd(Ac(t;L ) x

!t - 11, (t" gcd(Ac(t;L ) x (t - 11,(t" - 1)) I A&)

Note that A,(t) I Acucr(t; &, L ) , by the usual degeneration argument [25]. Thus the first assertion is immediate from the last divisibility. Suppose further that C h C' and 7rl(C; - C') E Z. Then h is an isomorphism and therefore Acuct(t; &, L ) = A+(t). Thus the assertions (2), (3) follow immediately. The assertion (4)is a result of (1) and (2). qed. 4.7. Relations between the tangential Alexander

polynomials and 8-Alexander polynomials Let C be a plane curve of degree d and let P E C and let L = TpC. We consider the tangential Alexander polynomial &(t; L ) . Let

~i(@i--2 C )( 9 1 , . . . , 9 d I R l , . . . , R e )

684

be a presentation of 7r1 (Ci- C)by generators and relations. Take a generic line L , for C U L and put C2 = P2 - L , as usual. Then by Theorem 4, we have

7r1(C2-CUL)= 7rl(c:-cuL,)

= 7rl((c:-C)xz

7r1(c:-C)x7rl(c:-L,)

and it has a presentation:

m(c2- c U L ) = (91,.. . ,Sd, h, h, I Ri,. . . ,Re,Ti,.. . , T d , s) (gi,...,gd,h,IRi,...,%,Ti,...,Td)

=

where Tj = h, gj h&' gj' and S = h h , gd . . .g1. Now the tangential Alexander polynomial A, (t;L ) is associated to the surjective homomorphism

esUm: 7 r l ( ~ :

-

c)-,z= ( t ) ,gi H t

Let On be the surjective homomorphism with weight n on L

On : 7r1(C2- CuL)-Z

= ( t ) ,gi

H

t, h H tn

Now taking 91,. . . ,grc, h, as generators of 7r1 (C2 - C U L ) = 7r1 (Ci- C U L,), 8, corresponds to the homomorphism: qn+d : 7r1 ((ci-

c U L,)

+

z, gi ++

t , h,

I+

t-n-d

The last property is the result of the observation: 1 = O,(S) = td+n8n(h,). Notation. Hereafter we mainly consider the weight like On which has weight one except a line component L in consideration. So we introduce the following notation which is easier to be understood:

; A c u L ~ ( ~:=) A c u L ( ~en) The upper index n implies that L has weight n. Using this notation, we also write ACUL, (t;7 %+d, L ) = Acu~;n-d(t; L ) . Thus combining the above argument with Theorem 12 we have shown the following. Theorem 13. For a n y integer n, = AcuLGn-d(t;L) and we have the divisibility: AcuLGm-d(t;L)lAc(t; L ) x (t- 1) and gcd(Ac(t; L ) x (t - l),(tn+d- l))lAcuLLn-d(t; L ) . Furthemore if gcd(Ac(t; L ) x (t l),( t n + d - 1)) = A c ( t; L ) , we have the equality:

685

4.7.1. Examples

+

Let C be a quartic with either 2A2 A1 or E6 and one flex of order 2 . Let L be the flex tangent line. Then we have shown that A c ( t ;L ) = t2 - t 1 and (t2- t l)(t4- t2 1) respectively. We can compute generic Alexander polynomials AcU,y( t )of C u L as follows. Take a generic line L,.

+

+

+

( 1 ) C is a quartic with 2A2+A1 and L is the flex tangent line. As &(t; L ) = + 1, we take weight n = 2 on L and by Theorem 12, we get

t2 - t

AcuL-6(t; L ) = Acu,y(t) = (t2 - t 00

+ l)(t - 1)

+

Let Ct be a family of quartics with 2A2 A1 with two flex points for t # 0 and CO= C. Let L1, L2 be two tangent limes at the flex points of Ct.Then Ct L1 L2 4 C 2L. Thus the weight 2 on L is canonical. See $5.7. ( 2 ) C is a quartic with E6 and L is the flex tangent line at a flex of order 2 . AS Ac(t; L ) = (t2-t+l)(t4-t2+1), gcd(Ac(t; L)red,(t12-1)) = Ac(t; L ) . Thus we take n = 8.

+ +

+

ACULs(t) = ACULG12(t; L ) = (t - l)(t2 - t

+ +

+ l)(t4 - t2 + 1)

This can be interpreted as Y 3 Z X4 = 0 is a line degeneration of (3,4)torus curve of degree 12 as ( Y 3 Z X4)Zs = ( Y Z 3 ) 3 ( X Z 2 ) 3 ,See $5.4. Note also that

+

AcuLz(t) = AcuLGe(t; L ) = (t - l)(t2 - t

+ 1).

We can also interpret this equality as a result of a line degeneration of (2,3)-sextics of torus type as ( Y 3 Z X4)Z2= ( Y Z ) 3 ( X 2 ~ ) 2 .

+

+

5. Degeneration into non-reduced curves with a multiple line In this section, we study an analytic family of curves Ct,t E A such that COis not reduced but it has a line component with multiplicity. 5 .I. Admissible polydisk

Consider a reduced curve C C P2 which is defined by a polynomial f (x,y) = 0 in the affine space Ci := P2 - L where L = { Z = 0 ) . (We do not assume the genericity of the line L . ) We assume that f(z,y) is a polynomial in y of degree n. The base point of the pencil {L,,,7 E C} where L, := { X - 7 2 = 0 ) is given by B = (0,1,0) in the homogeneous coordinates. Note that n < d if and only if B E C. We say the pencil {x = 7 , 7 E @} is base point

686

admissible (respectively base point non-admissible) if n = d (resp. n < d ) . Recall that L, is a singular line for C if L, n C n CcE contains some nontransverse intersection point. For the case n < d, we also call L , singular if the number of the points L, n C n Ci counted with multiplicity is strictly less than n. In this case, we say L, a singular line with disappeared points at infinity. Using Zariski-van Kampen pencil method with respect to the pencil lines x = r], r] E C, we get a presentation ~

I - C( ) = ~(gl,.~* . 9gn

I R1,.. .,Rm)

We consider the polydisk A,,p := Aa x Ap where A, := {x 1 1x1 5 a } and Ap := {y I IyJ 5 p } and we consider the following conditions.

(1) For any 7 E A,, L, n C c { r ] } x Aplz and L, does not have any disappeared points at infinity and (2) For any r] E dA,, L, is not a singular line. (3) For any singular line L,, r] E A,. We say that the polydisk A,,@ is admissible for C with respect to L if it satisfies (1) and (2). Furthermore, we say that the polydisk A,,@ is topologically presenting for C with respect to L if it satisfies (l),(2) and (3). Observe that if A a , p is topologically presenting for C with respect to L , the inclusion Aa,p - C n Aa,p L) Ci - C is a homotopy equivalence.

5.2. Two non-reduced degenerations In this section, we focus the following two types of non-reduced degenerations. Type 1: Line Degenerations. {Ct, t E A} is an analytic family of irreducible curves of degree d and it degenerate into CO= DO j L , , j 2 0 where DOis an irreducible curve of degree d - j and L is a line. We assume also (#) there is a point Q E L - L n DOsuch that Q E Ct, V t # 0 and the multiplicity of (Ct,Q) is j. We call such a degeneration a line degeneration of order j . L and Q are called the limit line and the base point of the degeneration respectively. Here j is a non-negative integer. (We are mainly interested in the case j 2 2.) The condition (#) can be weakened as (#’) there is an analytic family of points Qt E L n Ct such that QO E L - L n D Oand the multiplicity of (Ct,Q t ) is j. In fact, under (#’), we may assume that L = { Z = 0) and Qt = (a(t),1 , O ) .

+

687

+

Then taking a linear change of coordinates (z,y) (z,y a(t)z),we can assume that Qt = (0,1,0). We recall that 0. Zariski observed that 3 cuspidal quartic is a non-reduced line degeneration of order 2 from a family of sextics of torus type [37]. Type 2: Flex Degenerations. First we have a family of reduced curves {Ct, t E A} of the same degree. COcan be a degeneration if p(C0) > p(Ct), t # 0 or in the same configuration space if p(Ct) = p(C0) but in this case, COis in a different stratum of the dual stratification. On Ct, we are given flex lines L l ( t ) ,. . . ,Lk(t),k 2 2 such that the family Li(t) is an analytic family with Li(0) = L for i = 1,.. . ,k . We associate a non-reduced degeneration Ct+Ll(t)+.. . + L k ( t ) + Co+kL and we call this family a f l e z degeneration. 5.3. Surjectivity Theorem f o r line degenerations

Assume first that we have an analytic family of curves {Ct, It1 5 1) such that Ct is an irreducible curve of degree d and it degenerates into CO= Do j L where DOis an irreducible curve of degree d - j and L is a line. We assume that {Ct,t # 0) has Q = ( O , l , O ) as the base point and the multiplicity of C, at Q is constant equal to j. Let F ( X , Y , Z , t ) = 0 be the defining homogeneous polynomial of degree d. We assume that L = { Z = 0). By the assumption, we have F ( X ,Y, 2,0) = ZjG(X, Y, 2 ) where degreeG(X, Y, 2 ) = d - j. Put f(z,y, t ) := F ( z ,y, 1,t). This is the affine equation of Ct. Note that

+

(1) degree{z,,}f(z, Y,t ) = d for t # 0 (2) degree{,,,}f(z, y, 0) = d - j and degree, f(z,y, t ) = d - j for any t. The second assertion follows from Bkzout theorem and the assumption that Ct has multiplicity j at Q. Theorem (Surjectivity) 14. Under the above assumption, there is a canonical surjection

4:

'rrl(~i DO)

4

.rrl(~i - c,),

T

# 0, sufficiently small

Proof. Note that the linear system L, = {z = q z ) , q E C has the base point Q = (0, 1 , O ) . Suppose that we have chosen a topologically presenting polydisk Aa,p for DO. Let f ( z , y , ~ )be the defining affine polynomial for C,. As the effect of the non-reducedness disappears when we put z = 1 in F ( X ,Y, 2,t ) , f(z, y, t ) is a analytic family. Write f ( z ,y, t ) as

f(z,y,t ) = ad-j(z,t ) y d - j

+ + ao(z,t ) * * *

688

By continuity, we can assume that this polydisk is also admissible for C, for some S and 171 I 6. Let 71,. . . ,qv be the parameters corresponding to the singular lines for Do. We take a small positive nuber E so that the disks AE(qi) := {qllr] - vil I E } , i = 1,.. . ,v are disjoint each other and A,(vi) c Aa. By the assumption, we have 1qjl < a for each j. We choose a generic pencil line L,, and generators 91,. . . ,Qd in this pencil so that we have a presentation: Tl(CE-DO)z(gl,.*.,gd

I Rl,*..,Rm)

(2)

For sufficiently small T , the original singular pencil LVi for DOmay splits into several singular lines for the curves C,, T # 0 but they are inside AE(vi). Put the corresponding parameters v i , l , .. . ,r]i,vi. Note that the monodromy relations around LVi for DO is nothing but the product of the monodromy relations around L,,,, for s = 1 , . . . , vi under a suitable ordering. This is immediate from the topological stability of the pencil restricted on the circle aAE(vi). Note also that C, may have a singular line L, such that 1771 -+ 00 when r -+ 0. Anyway we can get a presentation to (2) (using the same by adding several more relations & + I , . . .,&+n generators 91, . . . ,g d ) : ~ I ( ~ ~ - =C( gTl , *) . . r g dI

Rl,...,Rm,&+l,...,~+n)

This and (2) implies that there is a canonical surjection q5 : TI (Ci- DO) -+ n l ( C i - C,.). qed. R e m a r k 15. Though we are mainly concerned with the case j 2 2, the assertion for j = 0 gives another proof of Theorem 3.

Corollary 16. Under the same assumption, we have the divisibility ACT(t;L ) I ADO@; L). Taking a generic line L , for C,, 13 to obtain:

T

small and DOU L , we apply Theorem

Corollary 17. Under the same assumption,

Proof. First note that AcT( t )I AcT(t;L ) and Acr (t;L ) 1 AD, (t;L ) by (t;L ) = ADoULj( t )by Theorem 13. Lastly Corollary 15. Secondly ADouLGd we have ADouLGd(t; L)red = gcd(AD,(t; L ) x (t - l),td - 1)

689

The conclusion is now immediate from these observation, as the factor (t-1) does not appear in Ac7( t )by the irreducibility [25].’qed. 5.3.1. Examples of lane degenerations of order 1 In [24], we have classified configurations of reduced sextics of torus type. Among them, there are sextics of torus type with components B5 L where B5 is an irreducible quintic. In fact, each of them is a degeneration of irreducible sextics of torus type. We give one such example. D O = B5 has one A11 singularity at 0 = (0,O) and L = ( 2 = 0) and it is a flex tangent. A generic irreducible sextic Ct has [All As] as singularities. By Theorem 14, we know that Ac,(t; L ) I Ao,(t; L ) . As Ac,(t; L) is divisible by the generic Alexander polynomial, which is t2- t 1 by [23], we conclude t2 - t 1I A,,(t; L ) .

+

+

+

+

5.3.2. Line degenerations of order 2 We consider quartics which are non-reduced line degenerations of sextics. As a quartic D , we can take quartics with configuration (a) C(D)= 2A2, or (b) C ( D ) = 2A2 A l , 3A2 (with an outer singularity A1 or Az) or (c) C ( D ) = AS, E6 or C(D)= A2 + A3, A6 (with a wild inner singularity). We only explain here the case ( b ) with C ( D ) = 3A2. The other case will be explained systematically in 55.4. Three cuspidal quartic is very special. As the degenerated line, we can take any one of a simple tangent line or a tangent cone at a cusp or a unique bi-tangent line. This is not the case for other quartics listed above. The following family gives line degeneration of order 2 of sextics of torus type defined by gz = 0 into a quartic with 3 A2 and a line (2= 0) with multiplicity 2.

+

fz +

(a-1) L is a simple tangent line. (Cs, s E A} is a family with an outer A2 singularity at Q = (0,1,0) and L = ( 2 = 0) is a simple tangent line of the quartic DO.This degeneration is not a line-degeneration of torus curve which we study in the next section.

+ +

Do : y4 - 2 y2 + 3 zy2 - 3/422y2 1- 3 x + 3 5 2 - 2 3 = 0 Ci’” : ( y - 3 / 2 x y - y 3 + s z 3 )2 ( - y 2 + 1 - 2 ) 3 = O

690

(a-2) L is a tangent cone of an AP. {C:2), s E A} is a family with an A1 singularity at Q = (0,1,0) and L = { Z = 0) is a tangent cone of an A2 singularity.

+ + 3 + 1+ 3 x2y2- 5/4 x3y + 3/16 x4 - 3/4 x2 - 2 y2 = 0 fi(x,y, S) = 1+ (--1/4 + S) x 2 + - y2 9 3 ( ~y,, S) = -y3 + 3/2 xy2 + (-3/4 x2 + 1)y + 1/8 x3

Do : - 3 % ~ y4 ~

XY

XY

(a-3) L is the bi-tangent line. As is observed in Proposition 11, the Alexander polynomial has multiplicity 2 for (t2- t 1). Thus this case is exceptional for the tangential Alexander polynomial of quartic with 3A2. To expalin this we consider a family of sextics of torus type {Ci3),s E A} with 8A2, where two A2 are outer singularities and they are located at at R = (0,5/2,1) and R’ = (-2,1/2,1). An inner A2 is at Q = (0,7/2,1). The line L = {X = 0) is a bi-tangent line and the limit line degeneration. In an affine equation, we can define them as

+

1

Do : xy3 + - ( 1 9 -~ 12y2 ~ + 7 5 +~75 + ~ x Y )Ci3) ~ , = {f,”+ 932 = 0) 256 fi(X,Y,S) =

1

- 21 s2 +21 s3 - 4 xy -4

-48 sx

+ 18

g3(’, Y,

S ~ X -

(20 ysx

+ 8 s2xy - 8 s3xy - 70 s + 4 s3x2- 20 s3y

+

s2x2 20 s2y -8 y2s-4 s2y2+4 s3y2+48 ys -8 sx2

18 s ~ x ) . 1

= 16(s - 1 ) 2 ) (552 ysx+36 xy2s5-90 ys2x2+138ysx2-42 ys3x2

+36 ~ ~ s ~ x y2xs4+36 -84 ~ ~ ~ ~ ~~ ~ - ~ 1 ~ 0~- ~83 ~ 3y 6~- 9~ 6~ y ~- +7 551 6X S ~ Y +120x2s4y-228 s5xy-48 s5x2y-525 s+24 y3s+333 s5x-75x2+117s3x2

+18 s3y+42O s2- 161s3+21 s5x3-741 xs4+12 xy2- 13sx3-518

s4+259 s5 +16 s4y3-270s5y-354 s4x2+177s2x2-288 s2y-8 s5y3-8 s3y3-168s4y2 - 6 x2y

+ 540 s4y - 19 x3 + 8 s2x3- 204 y2s + 48 s2y2+ 36 s3y2 + 144 s5x2

+570 y s - 2 1 3 s ~ ~ - 6 4 5 s x + 1 5 s ~ x + 5 4 3 s ~ x + 3 6 s ~ x ~ - 5 7 s ~y2s5) x~+84 As the Alexander polynomial of sextics of torus type with 8A2 is given ( t 2- t 1)2([23]), this explains Ao,,(t;L ) = (t2- t + 1)2. Four inner A2’s are not visible in the Figure 2. It is quite interesting to study how the family degenerates into DO 2L. Observe that two cusps of DO are not real points and L = {x = 0) is the bitangent line of DO.Note that (a-1)

+

+

691

and (a-2) can not be a line-degenerated torus curve in the sense of the next subsection.

Figure 2.

Ci3’, s = -1/3,sextics with 8A2

5.4. Line degeneration of curves of torus type

We consider a pair of coprime positive integers p , q curves of ( p , 9)-torus curve:

> 1 and consider the

f p ( X ,y, Z ) q+ gq ( X ,y, 2)” = 0 where f,, gq are polynomials of degree p , q respectively. Consider the special case that cp,q

fp(x,

:

2 ) = fp-a(X, y, 2 ) x

z”,gq(x,

2 ) = gq-b(x, y, 2 ) x Z b

where f p - a , gq-b are homogeneous polynomials of degree p - a , q - b respectively and 0 < a < p and 0 < b < g . Assume for example that aq > bp and factoring ZbP from f , we have a curve

D

:

g ( X ,Y,2 ) = f p w a ( XY, , 2 ) q 2qa--pb

+ 9q--b(X, y,ZIP = 0

(3)

We call a curve D a line-degenerated torus curve of type ( p , q ) and we call the line L = ( 2 = 0 ) the limit line of the degeneration. Note that the

692

degree of D is pq - bp. The simplest case is a = b = 1 and

D

g ( X , Y , 2 ) = f p - ~ ( X , Y , Z ) q Z q -+p 9 q - ~ ( X , Y , Z ) = p 0, p < q

:

Theorem 18. For a degenerated torus curve D of type ( p , q ) defined (3), there is a family of line degeneration f ( X , Y , Z , t ) of order bp such that f o ( X , Y,2 ) = g ( X ,Y,2 )ZbP and each curve Ct: f ( X ,Y,2,t ) = 0 passes through a f i e d point Q E L and the. multiplicity of Ct at Q is pb for each t # 0.

Proof. We may assume that Q = (0, 1,O) is not on D. Let h ( X ,Y ) be a homogeneous polynomial of degree q - b with h ( 0 , Y ) # 0. We put f p ( X ,Y,2 ) = f p - a ( X , Y,z)z“ and 9q(X,Y,2,t) = gq-b(X, Y,Z ) Z b -k t X b h ( X ,Y )and put Ct : f ( X ,Y,2,t ) := f p ( X ,Y,Z ) q + g q ( X ,Y,2,t)” = 0. We can easily see that Ct : f ( X ,Y,2,t ) = 0 passes through Q and the multiplicity of (Ct,Q) is pb, as the local equation at Q is given by P

f (x’,1, z’, t ) = f p - - a ( d , 1, Z ’ ) ~ Z ’ ~ + ” (gq-b(x’, 1,z ’ ) z ‘ ~ 4- t dbh(x’,1 ) ) = 0 where x’ = X / Y , z’ = Z / Y . The affine equation of Ct in Ci is given by

f(x, y , 1 , t ) = f p - - a ( x : Y , 1)‘

+ (gq-b(x,Y , 1) -k t z b h ( x ,Y))” = 0.

where 3: = X / Z , y = Y / Z . We see that degree, f (2, y , 1, t ) = pq - pb. qed. Put s = gcd(p, q). As the generic Alexander polynomial of (p,q)-torus curve of degree pq is divisible by A,,,(t) := (tpq’’ - l)’(t - l ) / ( ( t P - l ) ( t q I ) ) , we get

Corollary 19. Let D be as above and let L = ( 2 = 0 ) be the limit line of the degeneration. Then Ao(t;L ) is divisible by A,,,(t).

5.4.1. Singularities of line-degenerated torus curves We consider the curve defined by (3)

D

:

g ( X ,Y,2 ) = f p - a ( X ,Y,2)‘ Zqa-pb

+ gq-b(X, y,2)” = 0

Suppose that P E D is a singular point. P is called an inner singularity (respectively outer) if f p - a ( P ) = g q - b ( P ) = O (resp. g q - b ( P ) # 0). P is called wild if P is also on the limit line Z = 0. The following describes the type of inner non-wild singularities.

Lemma 20. ([5]) Let C be a curve of torus type

c:

f ( x , Y ) p+ 9 ( x , Y ) q = 0,P < Q

693

and assume that f(0,O) = g(0,O) = 0 and the curves f(z,y ) = 0 is smooth at 0. Let v be the local intersection number o f f (x,y ) = g(x,y ) = 0 at 0. Then the singularity (C,0 ) is topologically isomorphic to the Brieskorn singularity

Bp,pv: yp

+ zpv

=0

The singularity is more complicated when f(z,y ) = 0 is singular at 0. The description of wild singularities is also more complicated in general. 5.5. Examples of line degeneration of torus curves

5.5.1. Cubic A cuspidal cubic Q can be understood as a line-degenerated (2,3)-torus curve of order 3 by taking

fz(z,Y ) = Y X , 93(z,Y ) = gi(z,y ) x 2 , Q : y3

+ gi(zly)'a:

= 0.

The limit line of the degeneration is L = {x = 0 ) . This explains that A,@; L ) = t2 - t 1.

+

5.5.2. Quartic We give further quartics which can be a line-degenerated (2, 3)-torus curve. We consider the quartics of the form:

.I2 + (Y .I3 = 0 D : g ( z , Y ) ) = f&, Y I 2 + Y3 x = 0

co :

(fz(Z,Y)

(4) (5)

where f ~ ( zy ,) is a polynomial of degree 2 and the limit line of degeneration is chosen to be {z = 0 ) . In general, D has two inner AZ singularities at y = fz(z,y ) = 0. If y = 0 is tangent to the conic CZ := {fz(z,y ) = 0 ) , the singularity is an Ag. If moreover CZ degenerates into two lines, the singularity is an Es-singularity. The limit line L is a bi-tangent line at {x = 0 ) ncz.If L is tangent to the conic CZ, D obtains a flex of order 2 and L is the flex tangent line. Further more we can put one outer singularity, either A1 or Az. There are two more configurations which can be a line degeneration of torus curves: A2 A3 and As. For these singularities, we have to consider wild inner singularities. We have already studied most of these quartic and their Alexander invariants in $4.5. Theorem 14 and Theorem 18 explain our previous computations.

+

694

5.5.3. A quartic with 2Az For a generic quartic D with C ( D ) = 2A2, its dual curve D is a sextic with 8A2 + A l , in particular, D has one bi-tangent line.

D :g(x,y)=(y2-1+x2)2 + y 3 x = 0 The bi-tangent line can degenerate into a flex tangent line of order 2 so that D has 6Az E6.

+

D :g(x,y)= ( y 2 - 2 2 y - 2 x 2 + x + 1 )

2+ y 3 x = ~

See Figure 4, 35.9 for graphs of these quartics. 5.5.4. A quartic with 2A2

+ AI +

A generic quartic D with C(D)= 2A2 A1 has two flexes and one bitangent line (i.e., the configuration space is self dual). Degenerated quartic D' has one flexes of order 2 and no bi-tangent line as we have seen before.

D' :

(y2

+ :(

x-

g)

y

+ 23 x2 3

+ y3x = 0

See Figure 5, $5.9 for graphs of these quartics. 5.5.5. A quartic with A5

A generic quartic D with C(D)= {As} has 6 flexes and one bi-tangent line, thus the dual curve is a sextic with 6Az +As A1 . Degenerated quartic D' has 4 flexes and one flex of order 2 and thus the dual curve is a sextic with 4A2 E6 As. Tokunaga has studied a certain dihedral covers branched along these quartics [34]

+

+ +

2

D : ( y 2 - y ~ - ~ 2 + 2 x - 1 ) +y3z=0 2 D': ( y 2 - 2 2 + x 2 + 2 x + 1 ) + y 3 x = ~ See Figure 6, 35.9 for graphs of these quartics.

695

5.5.6. A quartic with E6

A generic quartic D with C ( D ) = { E s } has two flexes and one bi-tangent line. Degenerated one D’has one flexes of order 2 and no bi-tangent line as we have seen before. D : (y2-l-z2+2z)

2

fy3z=OD’: ( z - y + l ) 4 + y 3 z = 0

Note that the last quartic D’ with degeneration of (3,4)-torus curves as

((.

-

+ 114 +

y3z)

can be also considered as a line

E6

-

x8 = ((.

+ q z 2 ) 4 + (yz3)3

+

This explains that A o / ( t L ; ) = (t2- t l)(t4 - t2 See Figure 7, $5.9 for graphs of these quartics. 5.5.7. quartics with A2

(6)

+ 1) with L = {z = 0).

+ A3 and A6

+

We start the general form: C : f ~ ( z , y ) ~ y 3 x = 0. We assume that 0 = (0,O) is a wild inner singularity. Thus f i ( 0 , O ) = 0. The configuration A2 A3 is obtained when fz(0,O) = 0 and f2(zly) = 0 is not tangent to y = 0 as (C,O) A3. The limit line is tangent at one smooth point and also passing at 0. For example, C : (-y 2 y - z2 z)2 y3z = 0. If we make fz(z,y) = 0 is tangent to y = 0 at 0, (C,O) E As. An example 2 is given as follows: C : (y2 y z2) y3z = 0. See Figure 8, 55.9 for graphs of these quartics.

+

+

+ +

+

+

+

Remark 21. Degtyarev classified quintics with non-abelian fundamental groups [8]. The above quartics C are in his list as quintics C U L with L being the axis of degeneration. 5.5.8. Quintics as line-degenerations

+

We consider (2,5)-torus curve C : f ( z , y ) = f 5 ( z , ~ ) ~g2(z,y)5 = 0 of degree 10 which is degenerated as f5(z,y) = f2(z,y)z3, g z ( z , y ) = yz. Then we get a quintic

D:

f 2 ( ~ y)’ ,

+ y5 = 0

(7)

In general, D has 2 A4 singularities at f~(z, y) = y = 0 and it has a flex of order 3 at 0 with the tangent line L = {x = 0). As a special case where the conic fz(z,y) = 0 is tangent to y = 0, we get one As singularity:

D:

(y2+zY+z2-22+1)

2

z + y 5 = ~

696

If f2(z,y) = 0 is two lines intersecting on y = 0, the singularity is locally topologically isomorphic to C5,5 in the notation [31]: c5,5 :

y5

+ x2y2 + x5 = 0

If fi = 0 is a line with multiplicity 2, the singularity becomes B4,5 singularity which is locally defined as y5 z4 = 0. There are two other possibilities. A quintic as a line degeneration of torus curves of type (3,5): take g3 = yz2, f5 = f~(z,y)z~. Then we get a quintic

+

Q : y5

+

2’

fi(z,y)3 = O

The quintic Q has one E8 singularity and a A4 singularity on the limit line z = 0. The limit line is also the tangent line of the singularity Aq. Another possibility is as a line degeneration of torus curves of type (4,5): take 94 = yz3, f5 = g1(z,y)z4. Then we get a quintic

Q’ :

y5

+

2 gi(z, y)4

=0

Q’ has one B5,4-singularity and the limit line is z = 0. This quintic can be considered as a degeneration of (7) when f2(z,y) --+ g1(z, Y)~. 5.6. Sextics as line degenerations Sextics as line degenerations can be either from (2,5)-torus curves, or from (3,5)-torus curves or from (3,4)-torus curves. The sextics from (2,5)-torus curves take the form:

c : g(z, y) = f 3 ( z ,y)2 + y 5 z = 0.

(8)

Generically C has 3 A4 singularities and the degeneration line L = {z = 0) is a tri-tangent line. By the degeneration of the intersection y = f 3 ( z ,y) = 0, we may have also either A4 Ag or A14. If the cubic f3(z,y) = 0 has a node or cusp, the singularity becomes more complicated. Sextics from (3,5) torus curves take the form:

+

Generically C‘ has 2 B3,5 singularities and degeneration line is a bi-tangent line at two flex points. If y = 0 is tangent to the conic f2(z,y) = 0, the singularity is &,lo. The curves C : f 2 ( z ,y)3 y4 z2 = 0 can be considered as a line degeneration of (3,4)-torus curves but at the same time, it is a torus curve of type (2,3). Generically C has 2Es 2A2. Thus by Corollary 19, the

+

+

697

Alexander polynomial Ac(t;L ) is divisible by (t2- t that Ac(t)= t2 - t + 1 by [28].

+ l)(t4- t2+ 1). Note

5.7. Flex degenerations Let us consider C, is a family of irreducible curves in the configuration space M ( C ;d ) with two marked flex points P,, Q7 E C, of order 1 for r # 0 and assume that (a) P,, Q, -+ PO when r -+ 0 and PO is a flex point of order 2 of Co E M ( C ; d ) . (b) The intersection Tp,C, n T Q ~ Cn, C, is empty for r # 0.

+

+

Theorem 22. Consider the degeneration: C, Ll,, Lz,, Then we have the divisibility of Alexander polynomials:

Ac,u.,,,u.,,,(t)

I Ac,,.z(t>

4

Co

+ 2L.

x (t - 1).

Here Ll,, = Tp,C,, Lz,, = TQ,C, and L = Tp,Co.

Proof. First we may assume that C, = { f (z, y, r ) = 0}, PO= (0,O) and the tangent line of Coat POis defined by y = 0. Take a generic line L , and we work in (C2 =. ,@ : Taking a presenting polydisk Aa,p for COU L with pencil line L,, r] E @, let r]l ,. . . ,vrn be parameters corresponding to the singular pencil lines with 111 = 0. Fix a small E > 0 to see the monodromy relations along Ir] - r]il = E . We take generators in a fixed generic line L,, , (7701 = E (see Appendix). Then we get a presentation: TI(@' -

co u L ) = ( g 1 , . .

,gd,h I R1,. . . , R k )

(10)

We consider the Alexander matrix MO with respect to the weight function 02 which has weight 2 for L. Take a positive number S so that

C,hL,,

V r , ) 7 ) I S , V ~ , ) r ] - r ] ~ l = & , i =..., l, m

We may also assume that Aa,p is admissible for C,, 171 5 S. Next, we consider C, U Ll,, U L2,, for sufficiently small r in the above sense. In the generic fiber L,, the intersection L, nL are bifurcated into two points L , n Ll,, and L, n L2,,, but they are observed only with a microscope and they move exactly as a twin satellite along Ir] - r]il = E. For the presentation of (C2- C, UL2,, ULz,,), we need two generators h l , h2 presented by lassos for the lines L2,, and L2,, instead of one h. However we can understand as h = hlh2. For the further detail about the choice of generators, see

698

Appendix. This implies that there are canonical homomorphisms 11,, 9 which make the next diagram commutative.

1Q

F ( d + 1)

F(d

+ 2)

-

-

c, u L )

0

2z

111, 7rl(cC2 - c, u Ll,, u L2,,)

1

id

z

B-

Here Q is defined on generators as gi ++ g i , h H hlh2 and 11, is canonically induced by 9. The monodromy relations R1,. . . ,R k remains the same along 177 - qjJ = E . This means that the relation Ri remains true where R$ is obtained simply substituting the letter h by hlh2. To get a complete relations, we have to add some more relations, say S1,. . . , Se along the singular pencil lines. Among them, we can assume that -1 s 1 = gdh2gd

-1

h2

which is the relation at the transverse intersection L2,, n C, near PO. (In the appendix, we will explain this situation more.) Thus the presentation is given as r i ( c 2- C T

u

L ~ , T

u L2,T)

= (91,. . . , Q d , hl,h2

I R;,.. ,Rk, s1,.. st) . l

Let G,

be the subgroup of G, := r1((C2 - C, U L1,T U L2,T) generated by 11, is a surjection on G, c G,. For C, U Ll,, U Lz,,, we consider the summation homomorphism 0,,,. Let y2, y, be the ring homomorphisms corresponding to 0 2 , OsUm: 91,. . . ,g d and the product hlh2. Then

72

:

+

+

C ( F ( d 1 ) ) + q t , t-11, y, : C ( F ( d 2 ) ) --$ q t , t-11

Note that the following diagrams are commutative.

+ 1))

7 2 : (C(F(d

’)’T

19*+

: (C(F(d

2))

(C(T1((C2-

c, u L ) )

-c(rl(c2c,I+*u -

Ll,, u L2,r))

0

-%

e(z) = A

1

id

eaumr 4

c(z) =

Now we consider the Alexander matrix M, of C,LJL2,, U L2,,. We consider the row corresponding to the relation Ri. It is a word of g1 , . . .,g d and hlh2. By the definition of 0 2 , we can see easily that

The (d+ 1)-th column split into two columns, which correspond to the Fox i = 1,2. As h = hlh2, we obtain differentials

&-,

699

Note also that

+

Let M: be the matrix obtained by adding ( 4 )x (d 1)-th column to (d 2)-th column so that the last column is zero up to k-th row. M: is written as

+

Mo

;(

0’ (til))

where MO is the Alexander matrix of COU L and w comes from the differential of S1, and the other terms N1, v’ are coming from S j , j > 2. Thus for any ( d - 1) x (d - 1)-minor A of Mo, we associate d x d-minor A’ of M:, by adding (k 1)-row and the last column. Then the corresponding determinant is equal to det(A) x (t - 1). Thus the assertion follows from the Fox calculus definition of the Alexander polynomial.qed.

+

5.8. Appendix

In this appendix, we will explain the existence of the relation S1 in the proof of Theorem 22. First we may assume that C, = {f(x,y , ~ )= O } , PO= (0,O) and the tangent line of COat PO is defined by y = 0. Changing the scale and using Implicit function theorem, we may assume that COis defined by y = &,(x) where &(x) = x4 higher terms in the polydisk A ~ J= ((2, y); 1x1, IyI 5 1). This follows from the assumption that 0 is a flex of order 2 of CO. We consider the pencil lines x = q. Now we consider C,. Assume that C, is defined by y = &(x) in A I , ~ .Write &(x) = C y c v ( ~ ) x yFirst . observe that Icq(7) - 11 << 1 by continuity. In the parametrization, the flex points are defined by {(a1p)l4:(a)= 0). Thus by RouchB’s principle, we see that there are two flex points which bifurcate from PO.They corresponds to the roots of $:(x) = 0 in 1x1 5 1, say x = a l ( ~ ) , a 2 ( Thus ~ ) . Ll,,, L2,, corresponds to the tangent line at these flex points. By Bkzout’s theorem and the local stability of intersection numbers, there is one transverse intersection point of Li,, n C, and we put them Qi = (pil&(/3i)) for i = 1,2. Note that pi --+ 0 when T --+ 0. Thus the local singular pencils for 8, = C, U Ll,, U Lz,, is bifurcated in four points x = ai,Pi,i = 1,2. We consider the local geometry of p : ( A I J ,A1,l n 8,) --+ A. Let &(x) be the polynomial of degree 4 which is the Taylor expansion of a,(%)modulo x5. First we observe that this

+

700

branched covering p : ( A ~ JA, ~ n J 6,) -+ A is topologically equivalent to Next, the the one where we replace C, by the curve C, = {y = &(x)}. J equivalent to the situation for C, and its two flex tangents inside A ~ is following explicit one:

c: :y = J,(Z),

&(z)

:= x4 - 6

~

~

2

~

+

For this, we consider simply a homotopy E.t(z) = t & ( z ) (1 - t)&(z). Except a finite number of t = t l , . . . ,t,, this family of curves defines equivalent covering over A. In this model &(z)) we have ~ 1 a2 , = &T and PI, P 2 = ~ 3 7 We . choose {z = 1) as the fixed generic fiber i.e., r ] ~= 1. On the fiber z = 1, P, Q1,Q2 are the intersections of the line z = 1 and C, L1, L2 respectively and we choose the generators as in Figure 7. The base point B is chosen on the circle 1yI = 1. The other d - 1 intersection points of C, n {z = 1) are outside of the unit disk and the generators 91, . . . ,gd-1 are omitted in the figure. The loops are oriented counterclockwise. Now we consider the loop in the base space e o w o I-1 where C is the line segment from z = 1 to z = 0 2 E', E' << (1 - Pz) and w is the loop Ir] - P2l = E ' . It is now easy to see that the monodromy relation along this as is expected. loop is nothing but S1 : gdh2g;lh;'

+

Figure 3.

Choice of generators g d , h l , h2

5.9- Graphs of various quartics

We put the graphs of various line-degenerated quartics of torus type.

701

Figure 4.

Quartic with 2A2, bi-tangent limit line (left), flex tangent limit line (right)

Figure 5. Quartic with 2Az (right)

+ A1, bi-tangent limit line (left),

flex tangent limit line

Figure 6. Quartic with As , bi-tangent limit line (left), flex tangent limit'line (right)

702

Figure 7.

Quartic with

&,

bi-tangent limit line (left), flex tangent limit line (right)

...

7

02

04

0.B

0.8

1

,

0.5

1.5

-2

Figure 8. Quartic with A3

+ A2 left,

with A6 right

References 1. E. Artal Bartolo. Sur les couples des Zariski. J. Algebraic Geometry, 3:223247, 1994. 2. E. Artal Bartolo and J. Carmona Ruber. Zariski pairs, fundamental groups and Alexander polynomials. J. Math. SOC.Japan, 50(3):521-543, 1998.

703 3. E. Artal Bartolo and H.-o. Tokunaga. Zariski pairs of index 19 and MordellWeil groups of K 3 surfaces. Proc. London Math. SOC. (3), 80(1):127-144, 2000. 4. E. Artin. Theory of braids. Ann. of Math. (2), 48:lOl-126, 1947. 5. B. Audoubert, C. Nguyen, and M. Oka. On alexander polynomials of torus curves. J. Math. SOC.Japan, 57(4):935-957, 2005. 6. R. H. Crowell and R. H. Fox. Introduction to Knot Theory. Ginn and Co., Boston, Mass., 1963. 7. A. I. Degtyarev. Alexander polynomial of a curve of degree six. J. Knot Theory Ramifications, 3:439-454, 1994. 8. A. I. Degtyarev. Quintics in CP2 with non-abelian fundamental group. St. Petersburg Math. J., 11(5):809-826, 2000. 9. P. Deligne. Le groupe fondamental du compl6ment d’une courbe plane n’ayant que des points doubles ordinaires est ab6lien (d’aprhs W. Fulton). In Bourbaki Seminar, Vol. 1979/80, volume 842 of Lecture Notes in Math., pages 1-10. Springer, Berlin, 1981. 10. W. Fulton. On the fundamental group of the complement of a node curve. Ann. of Math. (2), 111(2):407-409, 1980. 11. J. Harris. On the Severi problem. Invent. Math., 84(3):445-461, 1986. 12. A. Libgober. Alexander polynomial of plane algebraic curves and cyclic multiple planes. Duke Math. J., 49(4):833-851, 1982. 13. A. Libgober. Fundamental groups of the complements to plane singular curves. In Algebraic geometry, Bowdoin, 1985, volume 46, Part I1 of Proc. Symp. Pure Math., pages 29-45. Amer. Math. SOC.,Providence, RI, 1987. 14. A. Libgober. Characteristic varieties of algebraic curves. In Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), volume 36 of N A T O Sci. Ser. 11 Math. Phys. Chem., pages 215-254. Kluwer Acad. Publ., Dordrecht, 2001. 15. F. Loeser and M. Vaqui6. Le polyn6me d’Alexander d’une courbe plane projective. Topology, 29(2):163-173, 1990. 16. M. Namba. Geometry ofprojective algebraic curwes. Decker, New York, 1984. 17. M. V. Nori. Zariski’s conjecture and related problems. Ann. Sci. Ecole Norm. SUP.(4), 16(2):305-344, 1983. 18. M. Oka. The monodromy of a curve with ordinary double points. Invent. Math., 27:157-164, 1974. 19. M. Oka. On the fundamental group of the complement of certain plane curves. J. Math. SOC.Japan, 30(4):579-597, 1978. 20. M. Oka. Symmetric plane curves with nodes and cusps. J. Math. SOC.Japan, 44 (3):376-4 14, 1992. 21. M. Oka. Two transforms of plane curves and their fundamental groups. J. Math. Sci. Univ. Tokyo, 3:399-443, 1996. 22. M. Oka. Geometry of cuspidal sextics and their dual curves. In SingularitiesSapporo 1998, pages 245-277. Kinokuniya, Tokyo, 2000. 23. M. Oka. Alexander polynomial of sextics. J. Knot Theory Ramifications, 12(5) :619-636, 2003. 24. M. Oka. Geometry of reduced sextics of torus type. Tokyo J. Math.,

704 26(2):301-327, 2003. 25. M. Oka. A survey on Alexander polynomials of plane curves. Singularitks Franco- Japonaise, Skminaire et congrhs, 10:209-232, 2005. 26. M. Oka. Zariski pairs on sextics I. Vietnam J. Math., 33: SI, 81-92, 2005 27. M. Oka. Zariski pairs on sextics 11. math.AG/0507052, 2005 28. M. Oka and D. Pho. Fundamental group of sextic of torus type. In A. Libgober and M. Tibar, editors, Trends in Singularities, pages 151-180. Birkhauser, Basel, 2002. 29. M. Oka and K. Sakamoto. Product theorem of the fundamental group of a reducible curve. J . Math. SOC.Japan, 30(4):599-602, 1978. 30. S. Y. Orevkov. The commutant of the fundamental group of the complement of a plane algebraic curve. Uspekhi Mat. Nauk, 45(1(271)):183-184, 1990. 31. D. T. Pho. Classification of singularities on torus curves of type (2,3). Kodai Math. J., 24(2):259-284, 2001. 32. R. Randell. Milnor fibers and Alexander polynomials of plane curves. In Singularities, Part 2 (Arcata, Calif., 1981), pages 415-419. Amer. Math. SOC.,Providence, RI, 1983. 33. I. Shimada. Fundamental groups of complements t o singular plane curves. Amer. J. Math., 119(1):127-157, 1997. 34. H.-o. Tokunaga. Dihedral covers and an elementary arithmetic on elliptic surfaces. J. Math. Kyoto Uniu., 44(2):255-270, 2004. 35. H. H. Vui and L. D. Trfing. Sur la topologie des polynbmes complexes. Acta Math. Vietnam., 9(1):21-32 (1985), 1984. 36. C. Wall. Geometry of quartic curves. Math. Proc. Camb. Phil. SOC.,117:415423, 1995. 37. 0. Zariski. On the problem of existence of algebraic functions of two variables possessing a given branch curve. Amer. J . Math., 51:305-328, 1929.

705

On rigidity of germs of holomorphic dicritic foliations and formal normal forms. Rosales-GonzBlez, E. * Instituo de Matemdticas Uniuersidad Nacional Autbnoma de Mkxico, C.P. 04510, Mkxico City, Mhico e-mail: [email protected]

A bstact: In this work we explain the rigidity of generic germs of dicritic foliations in a neighborhood of the origin and give their normal forms. Keywords: dicritic foliations, dicritic vector fields, rigidity, formal equivalence, analytic equivalence, formal normal forms, analytic normal forms. 2000 Mathematics Subject Classification: 32870, 32805, 32830, 34A25, 34C20. 57R30

1. Introduction.

The problem of the analytic classification of germs of holomorphic vector fields goes back to Poincark and it depends, in the generic situation, on the eigenvalues of the linear part of the vector field at its singular point: A germ of holomorphic vector field v in (Cn,0 ) with isolated singularity is said to be resonant if the spectrum X := {Xi,. . ,An} of its linear part at the singular point 0 satisfies at least one relation klX1 ... knXn = X j , (ki,. . . , kn) E ( N U { O } ) , , kl . * kn 2 2, j = 1, * ,n. Moreover, given a germ of holomorphic vector field v in (Cn,O) with isolated singularity, the spectrum { A l , . . . ,A}, of its linear part at the singular point is said to belong to the Poincark domain if its convex hull does not contain the origin inside or in the boundary. The Siege1 domain is the complement of the Poincark domain in C". The well known theorem due to Poincark states that any germ of a nonresonant analytic vector field in the Poincark domain is linearizable by an analytic change of coordinates.

+. +

+

*This work was supported by the grants CONACyT 41300, PAPIIT-UNAM IN114703 and ICTP.

706

The presence of resonances gives formal obstructions for the linear part of a vector field. Nevertheless, in the Poincark domain, even in the resonant case, the formal and analytic classifications coincide. In the Siege1 domain, for the nonresonant case, the presence of small divisors produces the non coincidence of the formal and analytic classifications (for example, formal linearizing changes of coordinates may diverge). For the resonant case in the Sigel domain, normalizing series also, as rule, diverge. In particular, for resonant saddles (germs at the origin of holomorphic vector fields in (C2,0) having negative rational ratio of eigen= - F) the formal orbital classification ( for a definition see next values section) is given by the values m,n, Ic, Q arising in the formal normal form:

2

X = X

L= y(-;+uql+Quy) for Ic E N, Q E C, where u = xmyn is the resonant monomial (see [Bo], [Br], [Ill], [I13]).While the formal orbital classification is rather simple for resonant saddles, the orbital analytical classification gives rise to functional moduli: the Martinet-Ramis moduli (see [Ma,Ra2], [Ill], [El,Il]);the only exception are those fields whose formal orbital normal form is already linear [Br]. The same happens for the analytic (non-orbital) classification of resonant saddle vector fields. In fact, the classification of the moduli of resonant saddles relies, by means of the monodromy map, on the classification of germs of analytic diffeomorphisms, f ( z ) = e - 2 . 1 r m / n ~ + . . . Hence, it relies on the Ecalle-Voronin moduli [Ma,Ral], [Ill], [Ma,Mo].The same happens for the analytical (non-orbital) classification [Vo,Gr]. The so-called saddle-node vector fields (one eigenvalue is equal to zero) give another example of a simple formal orbital classification,

X

= x P + l ( l + Xxp)-l

Y=Y, where X E C and p E N,whose orbital analytic classification is given by the Martinet-Ramis functional moduli [Ma,Ral], [Ill], [Vo,Me], [El. In more degenerated cases such as the degenerated non dicritical vector fields [Vol]even the formal orbital classification becomes rather complicate: functional moduli arises. The same happens for the nilpotent vector fields [Ta], [EISV], [Loll, [St,Zl]. Surprisingly, is in these complicated cases, where the rigidity takes place: analytic and formal classifications coincide again (for the nilpotent case see [Ta], [MI, [Ce,Mo], [EISV], [Loll, [St,Zl]; for the orbital analytic

707

classification of nondicritical degenerated germs of vector fields see [Vol] and [ORVl] for the classical (non orbital) classification). In this work, the orbital rigidity is explained for degenerated generic dicritical vector fields. The full proof of this theorem, the non orbital rigidity, the formal classification and the realization theorems are given in [ORV2]. A near result (for families of foliations) is given in [MI and a topological classification of dicritic foliations is considered in [Kl], [K12]. Recently, the orbital analytic classification was obtained in [Ca] for dicritic vector fields with higher degeneracies. The problem on the analiticity of the formal normal forms had an overturn after the singular and beautiful works of F. Loray [Lo21 and L LO^]. In a forthcoming paper [ORV3] we prove that the analyticity of the formal normal forms given in [ORVZ] is a fact in the case n = 1 (when there is only one singular separatrix). In this case as in the formal ones, there is (for the strict analytic classification) one analytic functional parameter and one complex non zero parameter c (see Theorem 2.5). And in the non strict equivalence the parameter c can be taken as 1.

Acknowledgements. I profit this space to acknowledge J-P Brasselet, J. Damon, M. Lejeune-Jalabert, M. Oka and L. Dung-nang for the invitation and organization of the meeting and to the referee for the commentaries and remarks to this work. I profit to acknowledge as well my closest collaborators and friends S.Voronin and L.Ortiz: all the results presented here have been done in collaboration with them. 2. Basic notation and statements.

(1) We denote by V the set of germs of holomorphic vector fields with isolated singularity at the origin. (2) Given v E V , we denote by FV the germ of foliation generated by v. By cpv : (CC, 0) x (C2,(0,O)) -+(C2, (0,O)) we denote the flow of v in a neighborhood of the origin. (3) We recall that two vector fields v , w E V are analytically (formally) equivalent if there exists an analytic (formal) change of coordinates H : (C2,(0,0)) --+ (CC2,(0,0)) such that H,v = w o H (i.e. H(cpv(t,(2,y))) = cpw(t,H ( z ,y)) in a neighborhood of the origin. The analytic (formal) equivalence is strict if the linear part of the germ H is the identity.

708

Fig. 1.

analytic equivalence of vector fields v and w

(4) The foliations FV, 3, generated by the germs of vector fields v ,w E V , respectively, are called analytically (formally) equivalent if there exist an analytic (formal) change of coordinates H : (C’, (0,O)) .+ (C2,(0,O)) and an analytic function (formal series) K : (C2,(0,O)) .+ C*, K(0,O) # 0 such that H,v = (Kw) o H.If the foliations FV, 3, are analytically (formally) equivalent we say that the vector fields v ,w are analytically (formally) orbitally equivalent. If the linear part of the germ H is the identity and K(0,O) = 1then we say that the vector fields v ,w (the foliations FV,3, ) are strictly orbitally equivalent (strictly equivalent).

Fig. 2. analytic equivalence of foliations Fv and Fw

( 5 ) Let Vn+l denote the subset of V of holomorphic germs of vector fields with zero n-jet and non zero (n 1)-jet. (6) V,”+l denote the subset of Vn+l of holomorphic germs whose first homogeneous Taylor series term (Pn+l,Qn+l) at the origin satisfies the identity xQn+l - yPn+I 0. (7) Let VfT:n desote the subset of V,”+l of holomorphic germs v of the

+

709

form:

such that Pn+l has only simple factors and has not common factors with the polynomial xQn+2 -yPn+z. The elements of this set are called generic dicritic germs. A geometric definition of our generic family is given below in section 4.2 there is shown that this generic conditions implies that the blow up of the foliation generated by v does not have singular points and has exactly n leaves which are tangent to the divisor.

Remark. Consider a generic germ v E V$". A n y linear change of coordinates in ( C 2 ,(0,O)) acts in the linear factors of the first terms of the germ v. A s consequence, it is easy to show that in the analytic equivalence class of v there is at least one representative which has the following additional property: all the factors of the polynomial Pn+l are different to the polynomial y . This family of representatives will be denoted by 8. We finish this section by stating the principal results obtained for generic germs in V:;":". In the next section a sketch of the proof of theorem 2.1 is given. For more details see [ORV2]

Theorem 2.1. The formal orbital equivalence of generic germs in V:,yn implies its analytic orbital equivalence. Theorem 2.2. The formal equivalence of generic germs in V:;":" its analytic equivalence. Let R(x,y ) = n j " = l ( y - u j z ) , ui # of germs v E Vf+l having the form

uj,i

implies

# j . We denote by V R the set

710

where a j , bj, j = 1, ...,n are formal power series with zero 1-jet and b is a homogeneus polynomial of variables x , y of degree n. Moreover, i f the germ 00 03 v is generic and a j ( x ) = C a j , n xk , b j ( x ) = C b j , k X k , then aj++2 = bj,n+2, k=2 k=2 and a j , 2 # b j , 2 , j = 1, ...,n. With this generic condition the formal normal f o r m is unique. Theorem 2.4. A n y generic germ v E V R is strictly formally orbitally equivalent to a formal vector field va,c,a = ( a l ,...,a,), c =

...+*)(

&+Y%)+ r

l

\

\j=1, ...,n and where

Cj,k

E C , cj,2 # 0, j = 1,..., n.

Er stress that the analiticity of the formal normal forms is a fact in the case n = 1 (when there is only one singular separatrix). In this case as in the formal ones, there is (for the strict analytic classification) one analytic functional parameter and one complex non zero parameter c. This is shown in the next result which is in preparation [ORV3]: Theorem 2.5. [ORV3] A n y generic dicritic g e m v E V z R is strictly analytically orbitally equivalent to a vector field of the f o r m

Remark 2.1. If the equivalence considered is not strict then the analytic normal form stays similar with the constant c = 1.

3. Basic tools. In this section we introduce some basic tools which are necessary for the proof of Theorem 2.1.

(C2,( 0 , O ) ) Consider the map A : C2\ ((0,O)) 3.1. Blow-up of

-+ C P 1 , A : ( x , y ) H ( x : y)(:=line generated by ( x , y ) ) . Let M be the closure, in C2 x CP1, of the graphic g r a p h ( A ) of the map A. The set M endowed with the projection T : M +

71 1

( C 2 ,( O , O ) ) , ((z, y), C) H (2, y), is called the blow-up of ( C 2 ,(0,O)).The sphere L := 7r-'(O, 0 ) is called the exceptional divisor. Properties. (1) M is a complex 2-dimensional manifold and M = gruph(A) U 7r-' ( 0 ) (2) The map 7r is holomorphic and its restriction 7rlM,c to the set M \ L

. is a biholomorphism whose inverse is denoted by u := 7rlM,c >-l (3) The complex 2-manifold M is biholomorphic to the paste of two copies of C2with coordinates (called standard charts) (z,u)and (w, y), where the point (x,u), z # 0 is pasted with the point (w, y) iff w = y = xu.

(

i,

Fig. 3.

Blow-up of (C2, (0,O))

3.2. Blow-up of g e r m s of vector fields in Vn+l. Consider a germ v in

Vn+l, of

the form

where Pk, Q k are the homogeneous terms of order k of the Taylor's expansion of v at the origin and dots stand for higher order terms. Denote by the analytic extension of the lifting to ( M ,L)of the vector field u*v.We stress that ir), = 0. Moreover, there exist two germs of holomorphic vector fields ++, in the standard charts (2, u),(w,y) (respectively) in a neighborhood of L,such that

+

+-

712

where the natural number m is equal to n if the polynomial zQ,+~ yP,+1 # 0 (nondicritic case) and equal to n 1 if the polynomial zQ,+~yP,+1 E 0 (dicritic case). The pair of vector fields 3+ and G- generates a holomorphic foliation F,, in a neighborhood ( M ,C)called the blow up of the vector field v. The foliation Fvhas a finite set of singularities and in a neighborhood of the the sphere L all of them are on C.

+

In the standard charts the vector fields 3+ and 3- may be written as follows: a) Non dicritic case

Fig. 4.

b) Dicritic case

Blow-up of a nondicritic vector field in (Cz, (0,O))

713

'

(0,O)) Fig. 5. Blow-up of a dicritic vector field in (C2,

The set .Eyv of singularities of the blow up &, on L: which lies on the standard chart ( x ,u)is, in the nondicritic case, the set:

3.3. Properties of the blow-up of dicritic germs of vector fields in Vt+l. Consider now a germ of vector field v E V,d+l,

v = (pn+1+pn+2

a + ...)-ddX + (Qn+l+ Qn+2 + dY a*.)--,

where Pk, Q k are the homogeneous terms of order k of the Taylor expansion of v at 0. The dicritic condition ZQ,+I - yPn+l = 0 implies that there exists a homogeneous polynomial R(x,y) of degree n, such that Pn+l = x R , Qn+l = y R . This fact allows to express the blow-up Fvof v in the standard chart (2,u)as the foliation generated by the vector field:

+

a

+

V + ( X u) , = (R(1,u) O((x1)) ~ + ( Q n + 2 ( 1 7 u)- upn+2(1,u) O(Ix1))

d *

F'rom now on we denote by r(u) the polynomial R(1,u) (whose degree is I n) and by p(u) the polynomial Qn+2(1,u)- uPn+2(1,u)(whose degree

714

+

is 5 n 3). Thus, the singular points of the foliation Fvin the standard chart ( x , u ) are given by the set {(O,u) : r ( u )= 0 = p(u)}. Lemma 3.1. In the chart ( x , u ) the point (0,ii) is a singular point of the foliation F,, if and only if the linear polynomial y - iix is a common factor of the homogeneous polynomials R ( x ,y ) , and X Q n + Z ( Z , y ) - yPn+2(x,y ) . Definition 3.134. We denote by 7, the set of points on L where the leaves of the foliation ? ., at such points are nonsingular and tangent to C. This kind of points will be called tangency points on C. Lemma 3.2. The set 17; is finite and those points in 7, lying on the standard chart ( x ,u ) are ( ( 0 ,u) : r(u) = 0 , and p(u) # 0 ) . Y- ujx = o

Fig. 6 . Tangency points on

L

4. Properties of generic dicritic germs of vector fields in

V,”p:”. 4.1. Tangency points on L.

First we give some consequences of given definitions for generic dicritic germs v E Vi$’yn of the form (2). By the remark in section 2, we may suppose, by performing a linear change of coordinates if necessary, that v is in the subfamily 6. In this case the polynomial r has exactly n simple roots u1,+. . ,u,,all different from zero. Moreover, R(z,y ) = (y-ujx), P ~ + I ( x ,= Y )x C ; = l ( ~ - u j x ) ,Qn+l(x,y)= Y ~ ; = ~ ( Y - U ~ X ) . A S Y - ~ can not be a factor of xQ,+2 - yP,+2, then p ( u j ) # 0. The following lemma is a straight forward consequence of the above considerations:

n;=l

Lemma 4.1. Consider a germ of vector field v E of the form (2). In a neighborhood of C the foliation Fvhas not singular point, there are exactly

715

n tangency points o n .C, all the tangencies o n C are simple, lie in the standard chart (z, u ) , and the set I, coincides with the set ( ( 0 ,ul), . . , (0, u n ) } . 4.2. On the generic set

Definition 4.1. Consider the set C’ of vector fields v E the following properties:

Vi+l

which has

(1) The blow-up has not singular points in a neighborhood of C. (2) All the tangency points on C are simply and are exactly n

A simple consequence of this definition is the following remark: Remark 4.1. The set V$,n and C’ coincide. Indeed, since for germs in C’ the action of a linear change of coordinates of the domain in the blow up induces an automorphism of the sphere L.So, we can choose the linear change of coordinates in such a way that all the singular points and all the tangency points on C lie in the open set with the standard chart (z,u). 5. Preliminary normalization for generic dicritic germs of vector fields in Vn+l d,gen.

Let v E Vify be a generic germ of vector field of the form (2). From section 3.1 follows that the leaves of the foliation Fvat the nontangency points of C are transversal to C.In particular, we stress that the leaves of the foliation passing through the point (0,O) in the standard charts ( v ,y ) and can be parametrized through the point (0,O) in the standard charts (z,u) as graphic of some function: (p(y),y ) , and (z,a(.)) respectively, where the functions p, a are holomorphic germs from (C, 0 ) to (C,0). This fact provides an analytic change of coordinates from (C2, (0,O)) to itself with identical linear part and such that the original vector field v is transformed into a vector field v1 E V dn f en l where the germs of lines {y = 0 ) , {z = 0 ) in a neighborhood of the origin are leaves of the foliation Fv1. From now on, we will assume that our generic vector field v is a representative of its analytic class, belongs to the set 6 and may be written as v(z,y) = (zf (z, y), y g ( z , y)). This set of germs will be denoted by 61. Lemma 5.1. From the above considerations we can assume that i f v and w in Vtf;n are formally orbitally equivalent, then, up to an analytic change of coordinates, the sets I, and I, coincide and the foliations 3; and 3, have as leaves the germs at the origin of the lines {y = 0 } , {x = 0 ) .

716 6. Geometric and analytic elements associated to the

analytic orbital class of v.

6.1. Tangency curves. Consider the auxiliary foliation FO = {z = const} in Cz generated by the vector field (0,l). Let YObe its blow-up in M . In the standard charts (2, u), Yo coincides with the foliation {x = const}. In charts (w,y ) , = {wy = const}. Thus, the leaf {w = 0) is a leaf of PO. Consider the generic representative of the analytic class v E GI.We denote by FV the set of points of tangency among the leaves of the foliations PO and Yv.This set consists of exactly n 1 germs of curves ro := ({w =O},(O,O)), (Fl,(O,ul)), (rn+l,(O,un)), where each curve rj contains the tangency point on C:p j := (0,uj). Since the tangency at this points is simple then rj,j = 1,...,n 1, has transversal intersection with C. This set of curves is called the polar curves of the foliation Pv relative to the direction (0,l).

+

. . a ,

+

Lemma 6.1. If v, and w in Vt$,n are formally orbitally equivalent there exist representatives v1, and w1 of the analytical orbital class of v and W, respectively, such that their respective sets: Fv1, FW, coincide.

Fig. 7.

Tangency curves

6.2. Involutions.

Let v be a generic vector field in the family 91 of the form (2). In a neighborhood of each point p j = (0, u j ) in the charts (z,u),each leaf of 3vis either tangent to L (and in this case it passes through p j ) or it intersects

717

L in exactly two points: (0,u) and ( 0 ,I j ( u ) ) .This fact defines uniquely a holomorphic germ Ij,,, j = 1,.. . ,n such that (1) Ij,v(uj) = uj (2) I,,, 0 Ij,v = id (3) I,,, is an invariant for the vector field under the action of change of coordinates from (Cz,0 ) to itself with identical linear part.

Fig. 8. Involutions.

We denote by Zvthe set {Ij,v,j= 1,.. .,n } .

Lemma 6.2. If v, and w in are formally orbitally equivalent, there exist representatives v1, and w1 of the analytical orbital class - of -v and w, respectively, such that their respective sets: lvl and I,,, rvl,rwl,Zv,, ,Z , coincide. 6.3. Local first integral.

Let v be a generic vector field in the family 81. In a neighborhood of each point p j = (0,u j ) in the charts (5,u), the germ Ij,vdefines the holomorphic germ: gj,v = (uj - u)(Ij,,,(u)- uj).This germ satisfies the equality gj,vo = gj,,,. In other words, the germ gj,. takes the same value in u , G if and only if the leaf of the foliation FV passing through (0, u)also intersects L in the point (0,C). Thus, in a neighborhood of each p j = ( 0 , u j ) there exists a unique first integral Jj,v such that Jj v (0, u)= gj,v(u) (2) %(O,uj) = 0 and %(O,uj)

(1)

#0

718

Moreover the restriction of Jj,+, to the polar curve rj,+, of v passing through pj provides a good parameter zj,+,such that, if is parametrized as graph: { ( z , a ( z ) ) } ,E z (C,O), then zj = zj,+,(z) = Jj,,(z,a(z)). This parameter is called the standard parameter of the curve rj.

Fig. 9. Parameter on tangency curves

6.4. Local normalizations and end of the proof.

Let v be a generic vector field. Without loss of generality, suppose that v is the representative of its analytic orbital class in the family 61. The following lemma gives a local normalization of v via its blow-up:

Lemma 6.3. For any point p E L, in a neighborhood ( C 2 , p ) there exists a holomorphic change of coordinates HP,+,: ( C 2 , p )-+ ( C 2 , p ) such that a) the restriction to a neighborhood (L,p) of HP,+,is the identity; b) HP,+,leaves invariant a neighborhood of each leaf of the auxiliary foliation (%,,p) and c ) HP,+,applies the foliation yv to the foliation:

(I) {u= const} i f p = po (2) {v = const} i f p = p , (3) {zj,+,

:= (O,u),u $! { u j , j = 1,..., n } := (0,O) in charts (v,y ) :

in charts

+ gj,+,(u)= const} if p = p j := (0,uj)in charts

(2,

(2,~).

u):

Finally, let v, w E V:,, be formally orbitally equivalent. According to the above reasonings we may suppose, without loss of generality, that v, w are representatives of their respective analytical orbital class such that their respective sets: I,,I,, f'+,, ,,?I Z,,Z , and the respectives families of functions {gj := gj,+,,j = 1,. . . , n } and {gj,,, j = 1,. . . ,n} coincide. Then the holomorphic map:

719

e m st .

r.-& .. . .7 ....i ... i

:

:

!

Fig. 10. Local normalizations

fi : ( M , L )-+ ( M , C )

fi((z, u ) ) =& ! o fiE,v(O,u ) in charts (2,u) fi((v, Y)) = H;2,w 0 Hp,,v(v, Y) in charts (v,Y). is well defined. This biholomorphism is the identity on C and sends the foliation .&, to the foliation FW.The map fi is in fact the lifting of a biholomorphism H : (C2,0) -+ (C2,0) such that H*Fv = 3,. This finishes the sketch of the proof of the Theorem 2.1

720

Fig. 11. Construction of the conjugating map H

References Ar.

Arnold,V.I., Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, Berlin and New York, 1988. Bo. Bogdanov, R.I. Local orbital normal f o r m s of vector fields o n the plane. (Russian) Trudy Sem. Petrovsk. No. 5 (1979), 51-84. Br. Brjuno, A.D. Analytic f o r m of differential equations. I, II. (Russian) Trudy Moskov. Mat. Obshch, 25 (1971), 119-262; 26 (1972), 199-239. English transl. in Trans. Moscow Math.Soc. 25 (1971), 26 (1972). Ca. Calsamiglia, G, Singularidades Dicriticas e Vzdnhancas Folheada de Superficies de Riemann.; Ph.D.Dissertation, Instituto de Matematica Pura e Aplicada 2003. Ce,Mo. Cerveau, D.; Moussu, R. Groupes d’automorphismes de (C,O) . . et e‘quations diffdrentielles y d y . . . = 0. (French) [Groups of automorphisms of (C,O) and differential equations of the form y d y .. . = 01. Bull. SOC.Math. France 116 (1988), no. 4, 459-488 (1989). E1,Il. Elizarov, P. M.; Ilyashenko, Yu. S. Remarks o n orbital analytic classification of germs of vector fields. (Russian) Mat. Sb. (N.S.) 121(163) (1983), no. 1, 111-126. Ecalle, Jean. Les fonctions rdsurgentes. Tome I, 11. (French) [Resurgent 8. functions. Vol. 1,111. Publications Mathbmatiques d’Orsay 81 [Mathematical Publications of Orsay 811, 5. UniversitB de Paris-Sud, DBpartement de Mathkmatique, Orsay, 1981. EISV. Elizarov, P.M; Ilyashenko, Yu.S; Shcherbakov, A.A. and Voronin, S.M. Finitely generated groups of germs of one-dimensional conformal map-

+

+

721

pings, and invariants f o r complex singular points of analytic foliations of the complex plane, In Adv. Soviet Math., 14,Amer. Math. SOC.,Providence, RI, 1993. pp. 57-105. Ilyashenko,Yu.S., Nonlinear Stokes phenomena In Adv. Soviet Math., 14, Ill. Amer. Math. SOC.,Providence, RI, 1993. pp 1-55 Ilyashenko,Yu.S., Singular points and limit cycles of differential equations 112. o n the real and complex plane. Preprint, Computer Center AN SSSR, Pushchino, Moscow, Region, 1982. (Russian). Ilyashenko, Yu. S. Dulac's memoir " O n limit cycles" and related questions 113. of the local theory of differential equations. (Russian) Uspekhi Mat. Nauk 40 (1985), no. 6(246), 41-78, 199. Klughertz, M., Feuilletages holomorphes iL singularite' isole'e ayant une K1. infinite' des courbes int e'grales. V01.111, ThBse, Toulouse, 1988. K12. Klughertz, M., Existence d'une intBgrale premibre mhromorphe pour des germes de feuilletages h feuilles fermkes du plan complexe. Topology 31 (1992), no. 2, 255-269. Loray, F. Rbduction formalle des singularities cuspidales de champs de Lol. vecteurs analytiques, J.Diff.Equations 158 (1999), 152-173. L02. Loray, Frank Versa1 deformation of the analgtic saddle-node. Analyse complexe, systmes dynamiques, sommabilit des sries divergentes et thories galoisiennes. 11. Astrisque No. 297 (2004), 167-187. Lo3. Loray, F. A preparation theorem f o r codimension-one foliations. (English) Ann. of Math. (2) . . 163 (2006), no. 2, 709-722. Ma,Ral. Martinet,J; Ramis, J.P. Classification analytique des e'quations diffkrentielles n o n lindaires rdsonnantes du premier ordre. (French) [Analytic classification of first-order resonant nonlinear differential equations] Ann. Sci. Ecole Norm. Sup. (4) 16 (1983), no. 4, 571421 (1984). Ma,Ra2. Martinet, J; Ramis, J.P. Problbmes de modules pour des e'quations diffdrentielles n o n liniaires du premier ordre. (French) [Moduli problems f o r first-order nonlinear differential equations] Inst. Hautes Etudes Sci. Publ. Math. No, 55 (1982), 63-164. M. Mattei, J.F. Modules de feuilletages holomorphes singuliers. I. dquisingularite'. (French) [Moduli of singular holomorphic foliations. I. Equisingularity] Invent. Math. 103 (1991), no. 2, 297-325. Ma,Mo. Mattei, J.-F.; Moussu, R. Holonornie et inte'grales premibres. (French) [Holonomy and first integrals] Ann. Sci. Ecole Norm. Sup. (4) 13 (1980), no. 4,469-523. MS. Mattei, J.F; Salem, 8. Classification fomnelle de feuilletages singuliers de (C2, 0 ) . (French) [Formal classification of singular foliations of (C2, O)] C. R. Acad. Sci. Paris SBr. I Math. 325 (1997), no. 7, 773-778. ORV1. L.Ortiz-Bobadilla, E.Rosales-Gonzalez, S.M.Voronin. Rigidity theorem f o r degenerated singular points of germs of holomorphic vector fields in the complex plane. J. Dynam. Control Systems 7 (2001), no. 4, pp. 553-599. ORV2. L.Ortiz-Bobadilla, E.Rosales-Gonzalez, S.M.Voronin. Rigidity theorem for degenerated singular points of germs of dicritic holomorphic vector fields in the complex plane. Moscow Math.J. (2005), no. 1, 171-206.

722 ORV3. L.Ortiz-Bobadilla, E.Rosales-Gonzalez, S.M.Voronin. Analytic normal forms of germs of holomorphic dicritic foliations.. Preprint to appear in Preliminares del Instituto de Matem6ticas de la UNAM. St,Z1. Strozyna, E; Zoladek,H; The analytic and formal normal forms for the nilpotent singularity, J.Diff. Equation 179 (2002) 479-537. St,Z2. Strozyna, E; Zoladek,H; Orbital formal normal forms for general Bogdanov- Takens singularity, J.Diff.Equations 193 (2003) 239-259. Ta. Takens, F. Normal forms for certain singularities of vector fields. Colloque International sur 1’Analyse et la Topologie Diffbrentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972). Ann. Inst. Fourier (Grenoble) 23 (1973), no. 2, 163-195. Ta2. Takens, F. Singularities of vector fields. Inst. Hautes Etudes Sci. Publ. Math. No. 43 (19741, 47-100. Teyssier, L. Analytical classification of singular saddle-node vector fields. Te. J. Dynam. Control Systems 10 (2004), no. 4, 577-605. Vol. Voronin, S.M. Orbital analytic equivalence of degenerate singular points of holomorphic vector fields on the complex plane., Tr. Mat. Inst. Steklova 213 (1997), Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 35-55 (Russian). V02. Voronin, S. M. Analytic classification of germs of conformal mappings (C, 0 ) + (C, 0). (Russian) Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 1-17. Vo,Gr. Voronin, S.M.; Grintchy, A.A. A n analytic classification of saddle resonant singular points of holomorphic vector fields in the complex plane. English (English summary) J. Dynam. Control Systems 2 (1996), no. 1, 21-53. Vo,Me. Voronin, S. M.; Meshcheryakova, Yu. I. Analytic classification of generic degenerate elementary singular points of germs of holomorphic vector fields on the complex plane. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 46, 2002, no. 1, 13-16; translation in Russian Math. (Is. VUZ) 46 (2002), no. 1, 11-14.

723

Polar Multiplicities and Euler obstruction of the stable types in weighted homogeneous map germs from Cn to C3, n 2 3 E. C. Rizziolli

Departamento de Matemdtica, IGCE, UNESP Rio Claro, S.P., 13506-700,Brazil E-mail: [email protected]. br www. rc.unesp. br/igce/matematica M. J. Saia * Departamento de Matemdtica, ICMC, USP SBo Carlos, SP, 13560-970,Brazil E-mail: [email protected]. br www.icmc.usp. br In this article we show that for corank 1, quasi-homogeneous and finitely d+ termined map germs f : (Cn, 0) -+ (C3, 0 ) , n 2 3 one can obtain formulae for the polar multiplicities defined on the following stable types o f f , f ( A ( f ) and f(Cn-2>1(f), in terms of the weights and degrees of f . As a consequence we show how to compute the Euler obstruction of such stable types, also in terms of the weights and degrees o f f .

Keywords: polar multiplicities; quasi-homogeneous map germs; Euler obstruction of stable types.

Teissier introduced in some key papers in singularity theory ( [22], [23]) the notions of polar varieties and polar multiplicities. His work was taken up by several authors which show that the polar varieties and its multiplicities are powerful tools for solving some problems in singularity theory. Gaffney in [4]states that, for an important class of finitely determined map germs (Cn,0) + (CP, 0), if all the polar multiplicities of the strata in the source and target of a well chosen stratification are constant along a family of such germs, then this family is Whitney equisingular. With the aid of GonzalesSprinberg’s purely algebraic interpretation of the local Euler obstruction for singular varieties, Li3, and Teissier proved a formula showing that the local Euler obstruction is an alternate sum of the polar multiplicities of the local polar varieties. However these invariants have not been used extensively. This could be due to the fact that they are difficult to compute in practise.

724

In this paper we show that for, quasi-homogeneous and finitely determined map germs f : (Cn,0 ) + ( C 3 ,0) with n > 3, of corank 1, one can obtain formulae for the polar multiplicities defined on the stable types of a generic unfolding of f . In addition, we apply the results shown in [12] to show how to compute the Local Euler obstruction of such stable types in terms of the weights and degrees. 1. Stable types and polar multiplicities

We denote by O(n,p) the set of origin preserving germs of holomorphic mappings from Cn to CP, O,(n,p) denotes the set of germs at the origin but not necessarily origin preserving. For a germ f E Oe(n,p),J ( f ) denotes the ideal generated by the set of p x p minors of the derivative o f f . The critical set C(f) of f is the set of points z E Cn such that J ( f ) ( z )= 0. The discriminant A(f) o f f is the image of C(f) by f . The determinant of the derivative of f E O,(n,n) is denoted by J [ f ] . 1.1. The Stable types

A map-germ f : (Cn,S) -+ (CP,0) is stable in a finite set S if, by composition with families of holomorphic diffeomorphisms in source and target, every deformation is d-trivial, where d denotes the usual Mather group of germs of holomorphic difeomorphisms in the source and in the target. We call stable type, an equivalence class of stable map germs.

If f : (Cn,S) --t (CP,y) is a stable multi germ, we say y E CP is of stable type Q i f f is a representative of this stable type, we denote the set of stable points in CP of type Q by Q(f). The set f-'(Q(f)) n C(f) is denoted by &(f) and &(f) denotes the set f-'(Q(f)) - &(f), where C (f) is the critical set of f . Our interest in this article is to compute the polar multiplicities which appear in a versa1 unfolding of a A-finitely determined weighted homogeneous map-germ f : (Cn,O) -+ (C3,0),with n 2 3.

A germ f in O(n,p)is k-A-determined if any g E O(n,p)with the same k-jet as f, i.e. jkg = j k f , is d-equivalent t o f . The germ f is said to be finitely A-determined if it is k-d-determined for some k. Mather and Gaffney showed the characterization of finitely determined map germs in terms of stable germs.

725

Proposition 1.1. Suppose f E U ( n , p ) . Then f is finitely determined if, and only if, for each representative f of f , there exist neighborhoods of the origin U c C", V c CP such that f-'(O) n U n C(f) = 0 and for each y E V , y # 0 , the germ f, : (C",S,) -+ (CP,y) is stable, where S, = f - l ( y ) n U n E(f) and C(f)denotes the critical set of f.

See [5] for a proof.

A finitely determined germ f has discrete stable type if there exist a versal unfolding of f in which only a finite number of stable types occur. If the numbers ( n , p ) are in Mather's "nice dimensions" (which is our focus here) or on the boundary thereof, then every finitely determined germ f E O ( n , p ) has discrete stable type. For finitely determined map germs, we need to consider the stable types which appear in a versd unfolding of such germs. Denote by G : (@" x C", 0) -+ (C" x CP, 0) a versal unfolding of f . Definition 1.1. A stable type Q appears in G if for any representative G = (id(u),gu(x)) of G, there exists a point (u,y ) E C" x CP such that the multi germ gu : (C", S,) + (CP,y ) is a stable multi germ of type Q, here S, c C" denotes the image of the restriction to S := G-l(u,y) n C(G) of the projection of C x Cn on the second factor C".

-

-

--

I f f is finitely determined, we write Q(f ) = ((0) x CP)n Q(G),Qs(f ) = ((0) x C") n&s o and ec(f)= ((0) x en)n Qc(G),where the bar over a set means the closure of this set. Suppose Q(G) = { & I , , . . , Q,)(G) is the set of points of a 0-dimensional stable singularity type of a versal unfolding G of f . Definition 1.2. The 0-stable invariant of type & of f , denoted by m(f , Q ) is the multiplicity of the ideal m , O m , ( o , oin ) Om,(o,o)

This multiplicity is just the degree of the map obtained by projecting

&(G) onto C" using the projection of @" x CP on the first factor C". This projection gives a finite map because the fiber over zero is just Q(f), which is either the origin or empty, since f is finitely determined and Q is a zero dimensional singularity type. See [20] p. 121, for details on the relationship between degree and multiplicity. Since Q(G) lies in G ( C ( G ) )if, Q(G) involves r algebras, GlQc(G) dominates Q(G) and is generically r to 1. Them, r.m(f , Q) is the multiplicity of m , O m When O Q c ( ~is)CohenMacaulay for a stable map G , r-m(f , Q) is the length of m / m , % . The multiplicity m(f , Q ) is independent of the versal unfolding of f , and is invariant under coordinate changes on f , see Proposition 3.4 of [4].

726

We remark that we can deal with the entire class of cases f : (C", 0) 4 (C3,0),for all n 2 3, in consequence of the fact that for all n 2 3, the stable singularities are precisely the same type, namely double points curve, cuspidal edge, triple point, swallowtail, cuspidal edge crossing transversally with a plane and the regular part. We can also use the Thom-Boardman stratification in the source to describe the stratification by stable types which appear in the discriminant A(f) of f. For any Boardman symbol i = ( 2 1 , . . . , ir), we denote by Ci(f) the set of points in C(f) of type i. Below we show the r-dimensional stables types, for r = 1 , 2 , which appear in any finitely determined map germ germ f : (Cn,O) + (C3,0). These stable types are in A(f) and are formed by the smooth parts of the following sets. (1) The discriminant A(f) = f ( E (f)), which is 2-dimensional; (2) The 1-dimensional set f (E("-2i1)(f)); (3) The image of the double points o f f , which is 1-dimensional and denoted by f (0X.fIW))). The first purpose of this article is to describe formulae to compute the polar multiplicities, which we describe next, of the stable types A(f) = f (C( f) ) and f(IY2>'(f)), in terms of the weights and degrees of weighted homogeneous map germ f of corank 1. We remark that in the case of corank one map germs, the stable types which appear in A(f) and in f(En-2t'(f)) are related to ICIS which are in C", therefore the following results form the key tool t o obtain relations between the polar multiplicities.

Theorem 1.1. (LG-Greuel, [15], page 47) Let X I be a n ICIS, with a singularity at 0 E C". Let X be a n ICIS defined in X I by f k = 0, and let f l , ..., f k - 1 be the generators of the ideal that defines X1 at 0 in C". Then

where p ( h ) denotes the Milnor number of h. In the case of a zero-dimensional ICIS we can use the following simpler formula.

Proposition 1.2. ([17]p . 78) Let f : Ck,O 4 Ck,O be a germ such that X = f - l ( O ) is an ICIS. Then p ( X , O ) = d e g ( f ) - 1, where d e g ( f ) denotes the degree o f f .

727

Another elementary result that we use here is the following. Let f : C",O -+ C",O be a finitely determined germ. Then f : C(f) c C",O -+ A(f) C C", 0 is bimeromorphic (see [5] p.154, or [17]). 1.2. Polar multiplicities

The polar multiplicities are the multiplicities of the polar varieties, notion developed as a means of studying the singularities of any pure d-dimensional analytic germ ( X ,z) c (C", z). The kth, with 0 5 k 5 d - 1, polar varieties of X were defined by Teissier in [23], using the Nash blowup and by Henry and Merle in [9] and [lo], using the conormal modification. These kth polar varieties of X are obtained by taking the closure of the critical set of the restriction of a generic projection p : C" --+ Cd-k+l to the regular part of X. Roughly speaking, "generic projection" means that the projections which define the polar varieties are the least singular of projections . We denote the kth polar variety of X by Pk(x,x,p). The key invariant of P k ( X ,x,p), for k = 0, . . . ,d - 1 is its multiplicity at 0, called Ic-polar multiplicity, which we denote by m k ( X , p ) . Since it is constant for an open set of projections, we denote it by m o ( P k ( X ) )or m k ( X ) . Gaffney in [4] p. 195, also defines a d-polar multiplicity associated to a stable set Q ( f ) of a finitely determined map germ f, which should be given by the "polar variety of greatest codimension". But as this variety is zero-dimensional, it is not even well defined, Gaf€ney defines the d-polar multiplicity (of codimension d ) in each stratum as following: Take a versal unfolding G : (C" x C", (0,O)) -+ (C" x CP,(0,O)) of f. Specify a stratum Q = &(G) in target (or Q = &(G) or Qs(G)in source), such that dim Q(f) 2 1. Select D a linear subspace of (CP, 0) (or of (Cn, 0)) of dimension 1 and form the relative polar variety on Q, denoted 7rITs), where 7rs is the projection to C" and d = dim@) - s. Definition 1.3. The dth-stable multiplicity of f of type Q ( f ) , denoted md(f,

is the multiplicity Of mSoPd(G(G),~,),(O,o) in opd(G(G),~,),(O,O).

In [12] it is shown the relation between the polar multiplicities of A(f) in terms of the Milnor number of the singular set.

728

Theorem 1.2. ( [12], p. 05) Let f E O(n,3), map germ. Then:

n

2 3 be afinitely determined

m d A ( f ) )- m 1 ( A ( f )+ ) m o ( A ( f ) = P(C(f)) + 1

(4

As some results shown in the proof of this result are important to prove the formulae which allous us to compute the polar multiplicites, see 3.1, we show it here.

Proof Since the discriminant is 2-dimensional, there are 3 polar multiplicities. To describe these polar multiplicities we follow the definition done by Teissier in [22]. To compute rnl(A(f)) we choose a generic projection p l : C3 -+ C2 such that the degree of p l l A ( f ) is the the multiplicity of A ( f ) at 0 and also the polar variety P l ( A ( f ) ) is C ( p l l a o ( f ) ) ;denote by m , ( A ( f ) ) its multiplicity. To compute mz(A(f)), choose another linear generic projection p2 : C2 C such that the degree of (p2 O p l ) I p l ( A ( f ) )is m l ( A ( f ) ) and we also require (p2 o p l ) to be a generic projection which gives m z ( A ( f ) ) . To obtain the multiplicity m o ( A ( f ) ) ,consider the following diagram: ---f

C(f)

c C n L A ( f ) c C3

vof

JPl

(C" 0)

From the choice of p l , we see that m o ( A ( f ) ) = d e g ( p l l A ( f ) ) ,where d e g ( G ) denotes the degree of a map germ G : C k-+ Ck.

To show the relation between the polar multiplicities of A ( f ) in terms of the Milnor number of the singular set we consider the following diagram:

c Cn A ( f ) c C3 3 C2 2 CC Now call X Z = V ( p 2o p l 0 f,J ( f ) ) and X I = V ( p 1o f , J ( f ) ) . As X 1 and C(f)

X , are ICIS and subsets of V ( J ( f ) )= C ( f ) ,from the Theorem 1.1 we get: p(X2)

+ p ( X 1 ) = dim@(P2

on 0 P l 0 fl

J(f), Jbl 0 f,J(f)l)

(1)

Since C(f) is also an ICIS we apply again the Theorem 1.1 to C ( f ) = M

729

But, X I is 0-dimensional then from the Proposition 1.2 we see that

4x1) = deg(p1 f,J ( f > >- 1 O

(4)

From the equation ( 5 ) we shall obtain the equation (I) which gives the relationship between the polar multiplicities of the discriminant. In fact these multiplicities are implicitly described in the equation ( 5 ) , as we shall show now. Since f : C(f) -+ A ( f ) is finite and bimeromorphic, deg(pllA(f))= d e g h 0 f l W ) = d e g h 0 f,J ( f ) ) . Therefore

d4Pl

O

f,J ( f ) ) = m o ( A ( f > ) .

(2)

Now we find m l ( A ( f ) ) let , V’ = V ( J ( f )J, b l o f,J ( f ) ] )and call g : @ the defining equation of A ( f ) , that is g - l ( O ) = A ( f ) .As flv, is finite and bimeromorphic we can also obtain P l ( A ( f ) )= V ( J [ g , p l ]J,( f ) ) .

C3 -+

From this definition of P l ( A ( f ) )since , V‘ = V ( J ( f )Jbl , o f , J ( f ) ] )we have that the projection of p2 o p l o f ( V ’ ) in @ gives the same image than the projection of p s ( P l ( A ( f ) )in ) @.

730

(ii) On

Now we shall show that m z ( A ( f ) )= dim@

(J(f>,Jb2O P l f,J(f)l).

-

-

O

Since this multiplicity involves the stable types, choose an s parameters A ( F ) c C" x C 3 , versal unfolding F o f f , to get F : C(F)c CC" x Cn (z, u) (u, fub)). From the fact that p2 is generic and linear, we have

c(((n,,P20pl>oF)IC(F)) = V(J[F], J((~,,P20Pl)OF, J [ F ] )= ) v

c cnxcC3.

We remark that m z ( A ( f ) >is controlled by the degree of the projection o p l ) l v , or in other words, by the length eJ(f) of the maximal ideal 0, m, in O,, then e J ( f )= dim@ (7r,,p2

( J ( f >J,( P 2

OPl O

f , J(f))>'

The possible components of V are the closure of the sets: F-'(P2(A(F>,n,)),F-'(A3), F-l(A(1,z))and F - ' ( A ( l , l , l ) ) ,therefore we need to count the contribution for the degree of the projection (7r,,pp o p l ) restrict to each one of these components. To do this we choose a generic parameter u and neighborhoods U2 c Cs x C n ,U1 C C"such that for all point in U1 there exist eJ(f) pr6-images in V n U2, counting its multiplicities. Then we obtain

where S = n,-'(O) n V. Since the parameter u is generic we can consider that fu is stable. Then t o count the contribution of these components we need to use the normal forms of the stable types which appear in the dimensions (n,3).

73 1

We present here an explicit description of the normal forms of all corank one stable map germs in O ( n , 3 ) , n > 3, we remember that they are suspensions of the stable map germs which appear in O(3,3). First we describe the mono germs. Stable Germs: 1. Submersion Co, (Ao): 2. Fold

. . ,z,) = (21, 2 2 , 2 , )

f(sl,22,.

(Ai): f ( 2 1 , 2 2 , .

. . ,z,)

= (21, 2 2 , f~: f . . . fZ;-I

3. Cuspidal edge CnP2v1,(A2): f(Z1,22,.

4. Swallowtail Cn-'>lJ f(21, 2 2 , .

f ... f 2;-l f 2;

. . , X n ) = (21,22, 1

+

f2;).

212,).

(A3):

. . ,2,) = (21, 2 2 , &2: f . . . f xip1 f2: f 212, f2 2 2 3 .

To describe the stable multigerms we consider the normal crossing between the stable germs. Stable multigerms: 1. Double points, (A(1,l)): { ( 2 1 , 2 2 , * 4 f * . . fz;-, & I : )(; 2 1 , f z ; f2; f . . . *s;-, f 2 i , 2 3 ) } . 2. Plane with a cuspidal edge, (21,fz;

f** f 2* i-1

(A(1,21):

f 2 ; , 2 3 ) ; ( Z 1 , 2 2 , f Z $ f . ..fz;-1f%:+zlx,)}.

Now we return to count the contribution of the stable types: we remark that these points can or not appear in V, depending of the type of the singularity. 0

Singularity of type A3: Here we obtain that there is no contribution of this stable type, since

dim@ (23,.

On,z

. . , 2,-1,42,3

+ 2222, +

21,.

. . , 1)

= 0,

732

and there is no contribution of the stable type A(l,2). 0

Singularity of type

( J ( f u ) ,J ( P 2

Pl

A(1,1,1):J ( f u ) = ( 2 1 , X Z , 2 3 , .

fui J ( f u ) ) )= ( 2 1 , 2 2 7 2 3 , .

Therefore dima:

7

zn-1, Zn, 1)

-

On,z

( J ( f u ) ,J ( P 2 0 Pl 0 fu, J ( f u ) ) ) Qn,x

= dim@ (23,

..

. . , x n - l , x n ) and

* * * 7

2 1 7 227 2 3 ,

*

Zn-1, Z n , 1)

= 0.

Again we obtain that there is no contribution of the stable type A ( l , I , l ) . We conclude that the components of V are in the closure of F-l (p2 (A (F , > n3)) > then

Now we use (i), (ii) and (iii)in the equation (5) to obtain (I) and the 0 Theorem 1.2 is proved. The relation between the polar multiplicities of f(Cn-2y1(f)) is given in terms of the Milnor number of the set Cn-2i1(f) and also of the number of singularities of type A3, denoted dA3 which appears in a generic unfolding of a finitely determined corank one map germ in O ( n ,3) with n 2 3. Theorem 1.3. ( [12], p. 09) Let f E O(n,3), n > 3 be a finitely determined corank one map germ. Then:

mo(f(cn-2J(f))) - m l ( f ( C n - 2 3 1 ( f ) ) ) = #A3 - P(Cn-271(f))+ 1

(11)

We also include the proof of this result, since it clarifies some aspects of the proof of the Theorem 4.1.

733

Proof

The ideal that defines Cn-2*1(f) is defined as

(f)= I n ( d ( f ,13(d(f)))>,

J(n-2~)

where d ( h ) denotes the Jacobian matrix of a map germ h and I s ( M ) denotes the ideal generated by the s minors of some matrix M . Then for any corank one map germ f(z1,. . .,xn)= (xl3x2,g(x1,.. .)xn))we obtain J(n-2,1)(f) = (gz3,gz4,. . . ,gz,, ,M ) , where gxi denotes the partial derivagxz . . gz3Xn gx4x3 . . * Qx4xn tive of g in the variable xi,M is the determinant and gxixj

I

I

denotes the partial derivative of gxi in the variable xj. Since f is finitely determined, l Y 2 ? l ( f )has reduced structure, from the fact that f is of corank one, En-2>1(f)= V(J(n-2,1)(f)) is an ICIS, then to get the equation (11) we apply again the Theorem 1.1. Choose a generic linear projection p : C3 + C such that X := Cn-2i1(f) n ( p o f-'(O)) is an ICIS and m ~ ( f ( C ~ - ~ > ' ( f )=) ) deg(pl(f(Cn-2,1(f))))= V(J(n-z,l)(f),P0 f). We apply the Theorem 1.1 for the sets Cn-2j1(f) and

X to obtain

From this equation we obtain the equation (11). First we remark that X is an O-dimensional ICIS, then again from the Proposition 1.2 we see that

p ( X ) = dim@

on

(J(n-2,1)(fLP

O

f)

- 1 = deg((p o

f)l~"-~J(f)).

734

0,

Then we have m ~ ( f ( C ' + ~ ? l ( f ) )=) dime

(J(n-2,1)(f),P

O

f)

(9 *

To get the equation (11) we work with ml(f(C"-2J(f))), since dim(f(Cn-2y1(f))) = 1 we need to consider all stable types which appear here and count the contribution of each one of the 0-stable types in ml(f(Cn-291(f))). Let F a s-parameters versa1 unfolding of f: F:Cn+'(F) c c"x C"+ F(Cn--2J(F))c c"x c3, (u,2) ++ ( u , f u ( 2 ) ) . From the linearity of the generic projection p ,

c(((r",P)oF)IC("-2'1)(F)) = v(J(n-2,1)p), J [ ( r s , p ) o F ,J(n-2,1)(F)I)

=

v

and we conclude that the ml(f(Cn-2>1(f))) is controlled by the degree of the projection r, restrict to V c C" x en,that is, by the length eJ(f) of the maximal ideal m, in the source 0,. Then

Since the possible components of V are the closure of the sets F - l ( A 3 ) , F-1(A(1,2)),FP1(A(1,I,1))and F - 1 ( P l ( F ( C ( n - 2 ~ 1 ) ( Fr,)), ) ) , we need to count the contribution for the degree of the projection ( r s , p ) restrict to each one of these components. To do this we choose a generic parameter u and neighborhoods U2 c C" x Cn, Ul c C" such that for each point in U1 there exist eJ(f) pre images in V n U2, counting its multiplicities. Therefore, for S = r , - l ( O ) n V : =

c

Os+n,x

dime

XES

0n,z

dime

Jb

(J(,-2,1)(fu),

XES

-

J ( n - 2 , 1 ) ( F ) ,J [ ( T " , P ) 0 F, J ( n - 2 , 1 ) ( F l )

(7%

O

fu,J(n-2,1)(fu)1)'

From the genericity of the parameter u we suppose that fu is stable and to count the contribution of the components we use the normal forms.

Contribution of the stable types: Type A3: J(n-2,1)(fu) = ( 2 3 , 2 4 , .

Jb

(J(n-2,1)(fu),

12zn2

O

%,-I,

fu,J(n-2,1)(fu)I)

+ 222, J(22, 2 3 , . . . , ~

+ 222) and ,2n-l, + 2222, + 21, 122n2 + 222)).

+ 2222, + = .. + 2222, +

45n3

~ - 42n3 1 ,

(23,.

21,

42n3

21,

735

dimc

on,x

= dim@

(J(n-2,1)(fu),J ( P 0 fu,J(n-Z,l)(fu)))

on,x

(22,

1

7

X n , 1)

= 0.

Therefore there is no contribution of the stable type A(1,z). Type A(i,i,i): Let fi(21,. ..,2,) = (xi,xz,ft5; f ... f 2X-l f xi), fz(x1,. . . ,x,) = (21,f x i fxi f . . . fxi-l fxi,x3) and f3(21,. . . , 2,) = ( f ~ : f ~ i f . . . f ~ if -~ 1i , ~ 2 , ~ 3 ) . Here we use the results done for the ideal J(n-2,1)(f1) in the case A(1,z) to conclude also that there is no contribution of the stable type A(1,1,1). Therefore

= ml(f(C(n-2J)(f))

+ #A3

Using (i) and (ii) in the equation (1) we obtain the equation (11)and the Theorem 1.3 is proved 0

736

We remark that it is possible to compute the number #A3 in consequence of the results [4.6-item(6)], 4.3 and 2.5 (in this order) of [3]. Then we have

0" (91,, 914, . . . 7 91, 7 M , Mx3 ,wc,,. . * ,MX,) ' where M is the determinant given in the proof of the Theorem above and MZi denotes the partial derivative of M in the variable xi.

#A3 = dime

As a consequence of the above results, in [12] it is shown a Corollary which involves the polar multiplicity m l ( A ( f ) )with the polar multiplicity mO(f(Cn-2J(f)>).

Corollary 1.1.

m ( A ( f >=>m o ( f ( C n - 2 w ) ) ) .

(111)

2. Weighted homogeneous map germs

In order to compute the polar multiplicities of the stable types we need to know the Milnor numbers p ( C ( f ) ) , and p ( C n - 2 ' 1 ) ( f ) ,which are ICIS. We describe how t o compute these numbers. An analytic map-germ f : (C", 0 ) -+ ( C P , O ) , f = ( f l , .. . ,fp) is said to be quasi-homogeneous, or weighted homogeneous, of the type ( w 1 , . . . ,w,;dl,. . . , d p ) if there are positive integers w 1 , . . . ,w n , called weights, and positive integers d l , . . . ,d p , called weighted degrees, such that fi(Xwlzl,.. . , XWnz,)= Xd*ffi for all i = 1,.. . , p , z E 6"and X E C.

A map germ f : (Cn,O)

( C P , O ) , f = ( f 1 , ...,f,), is said to be semi quasi homogeneous of the type ( ~ 1 , ... ,w,; d l , . . . ,d p ) if f can be expressed as a sum f = fo f', where fo = (f,",. . . , is a finitely determined weighted homogeneous map germ of weights w1,. . . ,w , and weighted degrees d l , . . . , d p ; f' = (fi, . . . , and each fi is a germ of function such that w l k l . . . w,k, > di for all Ic = (Icl,. . . ,Icn) of any monomial zk in the Taylor series of f'. The germ fo is called the initial part of f . 4

fi)

+

+

fb)

First we see that we only need to compute the Milnor number of the initial parts.

Theorem 2.1. ( [8], p. 86) Let f : (Cn,O) -+ (CP,O) be a semi quasihomogeneous germ that defines a complete intersection isolated singularity. T h e n the space f - l ( O ) is a complete intersection isolated singularity of dimension ( n - p ) and p(f) = p( f').

737

We shall need then use the following result of Greuel and Hamm.

Theorem 2.2. ( [8], p. 77) Let f : (Cn,O) -+ (CP,O), n 2 p, f = ( f l , . . . ,f p ) be a quasi-homogeneous g e r m that defines a complete intersection with isolated singularity of weights w1, . . . ,w, and weighted degrees d l , . . . ,d p . Then (a) If d l = . . . = d p = d : p(f)=(-l)"-P+l+ ( - 1 ) n,..- P L C O < l < n - p (-1)' (1 .

y.,

(b) If di

l
nLi

9)

# d i , when i # j :

To compute the polar multiplicities we also need to apply the Theorem 2.3. ( [l])Let h : (Cn,O)-+ (Cn,O), h = ( h l , . . . ,h,) be a semi quasi-homogeneous m a p germ of weights w1, . , . ,fun and weighted degrees d l , . . . ,dn. Suppose that h l , . . . ,h, is a system of generators of an ideal I On d l ...dn of finite codimension, then dimc - = -. I w1 ...wn 3. Polar multiplicities in A ( f )

In this section we compute the polar multiplicities of the strata in the target associated to a finitely determined, quasi-homogeneous, corank one germ f E O ( n ,3), n 2 3. We start dealing the set A(f). Theorem 3.1. Let f ( z l , z ~ ..,z,) ,. = ( z 1 , z z , g ( q , z 2 , .. . , z n ) ) be a finitely determined, quasi-homogeneous, corank one map g e r m with weights w1,. . . ,wn, where w1 < w2 and w3 < . . . < W n and let d be the weighted degree of g , with d > w1 and d > w2. Then,

mo(A(f)) =

nn(9) ;

j=3

wj

738

with D1 = w1, D2 = ( n - 2 ) d - 2Cy=3wi, Dz = d - wl,3 5 15 n. We remark that when n = 3 this result was done by Jorge-Perez in the following

Theorem 3.2. [ll]Let f = (z, y , g ( z , y, 2)) E 0 ( 3 , 3 ) be a finitely determined, quasi-homogeneous, corank 1 map germ with weights w1, w2, w3 and d denotes the degree of g . Then,

(d -~ m 2 ( A ( f )= )

3 ) (d 2W3)

w2*w3

+ (d - ~

3 ) (d 2W3)(d - ~1 - ~2 - ~

3

Wl.W2.W3

For the proof of this result, Jorge PBrez made a direct use of the formula mo(A(f))= d e g ( f ) - 1, which is obtained from the Proposition 1.2. However we cannot apply these results here, since we also consider here the case n > 3. For the particular case n = 4 and p = 3 we derive

Corollary 3.1. Let f ( Z l r z 2 , 2 3 , z 4 ) = (21,22,9(21,22,23,24)) be a finitely determined, quasi-homogeneous, corank one map germ with weights w l , w2, w3, wn, where w1 < w2 and w3 < w4 and let d be the degree of g , d > w1 and d > w2. Then,

)

739

-

(d-wl-w4)(d-w2-w4)(d-W3-W4)(d-2w4)

(d-W3)(d-W4)

w1 W Z W 3 W 4 ( w 4-w3)

W3W4

t

(2d-wz-2~3-2w4)(2~-3w3-2w4)(2d-2w3-3w4)(d-w3)(d-w4) wzw3w4(d-2w3-w4)(d-w3-2w4)

+

+ + 2.

(d-~2-~3)(2d-2~3-2~4)(d-w3-~q)(d--2~3)(d-~~)

wzw~w4(-d+w3+2w4)(w4-w3)

(d-w2-w4)(2d-2w3-2w4)(d-w3-w4)(d-w3)(d-2w4~ wzw3~4(-d+2w3+w4)(w4-w3)

Proof of Theorem 3.1

The first polar multiplicity that we consider is

mo(A(f)). Using the relation mo(A(f)) = deg(p1 o f , J ( f ) ) given in the Theorem 1.2, where pl is a linear generic projection pl : (C3 --t (C2 and without loss of generality we may assume that this projection pl is given ) ( a l z + a 2 Y + a 3 z , b 2 Y + b 3 Z ) ) and that J ( f ) = (gz,, . . . ,gz,). by P I ( Z , Y , ~= Then, mo(A(f)) = deg(alzl+a2z2+a3g(zlr.. .

Xn)l

b2z2+b3g(z1,.

. .,zn),gX3, g X 4 . . .,gx,,) =

0

dim@( a 1 2 1 +azsz+a3g(zi ,...,i,),b*~2nfb3g(i ~,...,z,),gZ3,gZ4...,gZ,)

'

(*I

+

+

Note that the n-generators of the ideal I = (a121 a222 b 3g(~1,...~2,),g,~,g,~...,g~,) form a semi quasi homogeneous map germ whose weighted degrees are D1 = w1, D2 = w 2 , D ~= d - w3, D4 = d - w4,. . . ,D, = d - W n , respectively. Therefore. from the Theorem 2.3 it follows that the dimension f*) is DlD2.. . 0, , but D1 = w1 and D2 = w2, so this dimension is given by w1w2.. . wn 0 3 0 4 . . . 0, 0 3 0 4 . . .On equal to . Consequently, mo(A(f)) = w3w4.. . w, 203304.. .w, a3g(zl,...,%),b2~2 +

\

I

Now, we exhibit ml(A(f)) in terms of the weights and the weighted degrees of f . We have the equality (ii) given in the proof of the Theorem 1.2 m1(A(.f)) = dim@

on

(P2 0 Pl

0

f,J(fL JIPl f,J(f)l). O

Now we obtain again from the Proposition 1.2 that

We can suppose, without loss of generality, that P20Pl(S,Y,Z) = c l ~ + C 2 Y + c 3 z .

p2 o p1

is given by

740

The next step is t o show that the ideal = (P2 O P l

O

f,J ( fJbl l O f,J(m

is semi quasi homogeneous and apply the Theorem 2.1. To get this it is necessary t o study each generator of this ideal. First we note that (P2°P10f)(21rZ2,...,Zn)

= P 2 o P 1 ( ~ 1 , 2 2 , g ( ~ 1 , ~ 2 ,%)) ,. =c121+c 2 x 2 + a .

. . ,x,) and J ( f ) is generated by { g x 3 , . . . ,g x n } .

C 3 g ( q ,2 2 , .

For the ideal Jlpl of, J ( f ) ] ,we remark that that (a121

+ a222 + a g g ( z 1 , . . . ,

Then,

+

X n ) , b 2 ~ 2 b3g(Z1,..

Jbi

0

f,J ( f ) ]

=

. . . ,zn) =

(p1 o f ) ( z ~ ,

. ,S n ) ) .

+ a222 + a 3 g ( z l , . . . , 2 n ) ,b 2 ~ 2+

J[alzi

b 3 g ( Z l , * . .,zn),gx3,...,gx,]*

This last ideal is generated by determinant of the following matrix of the order n:

Consequently,

+

terms with upper degree) and J[PI 0 f,J(f)] = (alb2gx;gx2 . . .gx; K = ( ~ 1 x 1 ~ 2 x 2 c3g(Z1,22,...,2n),9x3,...,gxn,alb2gx~gx2...gx; +

+

+

terms with upper degree). Now we consider the ideal KO = ( 2 1 , a1bzg,;g,2 . . .gx;, g X 3 , .. . ,g z n ) , whose weighted degrees of the generators are respectively n

D1 = w l ,

0 2

= ( n- 2)d - 2 c

Dl = d - w1,

wi,

with 3 5 1 5 n.

i=3

By hypothesis, we suppose w1 < w2 and w 3 < . . . < wn, SO that K is semi quasi homogeneous. Moreover as p l is generic alb2 # 0, hence the initial part is equal t o KO. From the Theorem 2.1, it follows that p ( K ) = p ( K 0 ) and we obtain m l ( A ( f ) ) = p ( K ) 1 = p(&) 1.

+

+

We use the Theorem 2.2-(b) to calculate p ( K 0 ) in terms of the weights and the weighted degrees of the map germ f, namely:

741

Hence,

where D1 = w1,

0 2 =

(n-2)d-2Cy=3 wi,

Dl = d-w1, with 3

I1 I n.

Now, it remains to express mz(A(f>) in terms of the weights and the weighted degrees of the map germ f. For this, note that from the Theorem 1.2 we obtain the equality m2(A(f))-m1(A(f))+mo(A(.f)) = p(C(f))+l. Therefore mz(A(f)) = p(C(f)) - mo(A(f)>

+ m i ( A ( f > >+ 1.

(**>

We observe that C(f) = V(g,,, . . . ,gz,) and each generator gx, has weighted degree D1 = d - wl, 3 5 I I n, then from the Theorem 2.1 we get

Finally, it follows by replacing p ( C ( f ) ) , m o ( A ( f ) ) ,ml(A(f)) by their values in the relation above, the last desired equality

with D1 = wl, D2 = (n - 2)d - 2 Cz3wi, Di = d - wi, 3 5 I

I n.

4. polar multiplicities in f ( ~ ( ~ - ~ ? ' ) ( f ) )

We now deal with the polar multiplicities of the set f(C(n-211)(f)).

0

742

Theorem 4.1. Let f (xi,2 2 , . . . ,Zn) = (21, x2, g ( z l , z ~ .,..,x,)) be a finitely determined, quasi homogeneous, corank one map germ with weights w1, . . . ,w,, where w1 < w2 and wg < . . . < w, and let d be the weighted degree of g , with d > w1 and d > w2. Then:

n

with D1 = w1, D 2 = ( n - 2 ) d - 2 C i = 3 w i , D1 = d - w1,3 51 5 n. For a quasi-homogeneous map germ in 0 ( 3 , 3 ) , Jorge PBrez in [ll] showed how t o compute the polar multiplicities in terms of the weights and weighted degrees. Theorem 4.2. (111 Let f = ( x ,y , g ( z , y, z ) ) E 0 ( 3 , 3 ) be a finitely determined, quasi-homogeneous, corank 1 germ with weights w1, w2, w3 and d the weighted degree of g . Then, mo(f(Wf))) =

, Iw3

d-w3) d-2w3 (

) l

For the demonstration of this theorem the author used strongly the fact that n = p = 3. To quasi-homogeneous germs in 0 ( n , n ) there is a important result, due to Marar, Montaldi and Ruas, which shows the number of singularities of type A k in terms of the weight and weighted degree of f , namely: Theorem 4.3. (191 Let f : (Cn,O) + (Cn,O) be a corank 1 weightedhomogeneous A-finite map-germ with weights and degrees as above. For any stabilization of f , and any partition P of n,

nyi;

where 1 is the length of P, w, = w t ( f n ) ,d = degree(f,), w = N ( P ) define the order of the sub group of Si which fixes P . Here o n R1 by permuting the coordinates.

Si

and acts

743

Again, for the particular case n = 4 and p = 3 we get

Corollary 4.1. Let f ( Z 1 , 2 2 , 2 3 , Z 4 ) = ( 2 i 1 2 z , g ( 2 i , 2 z , 2 3 , 2 4 ) ) be a finitely determined, quasi-homogeneous, corank one map germ with weights w1, wz, w3, wn, where w1 < w2 and w3 < w4 and let d be the weighted degree of g , with d > w1 and d > W Z . Then,

with

D1= w l ,

Dz = 2(d - w3 - w4),

D3

= d - w3 and D4 = d - w4.

Proof of Theorem 4.1 First, since f has corank one, it follows by the Corolary 1.1 that rnl(A(f)) = m~(f(C("-~~l)(f))). Thus, by this equality and by item (i) from the Theorem 3.1, we obtain n

n

p=l v=1

n

with D1 = w1, DZ= ( n - 2)d - 2 C i = 3 w i ,Di = d - w l r 3 I1 5 n. It remains t o show that rnl(f(C(n-2i1)(f))) also can be expressed in terms of the weights and the weighted degree. From the Theorem 1.3 we have the equality

Equivalently,

In order t o compute the Milnor number p(C(n-z31)(f)) we write c(n--2,1)(f) = V ( J ( n - Z , l ) ( f ) )hence, , C ( n - 2 J ) ( f )= V(M,gz31gz'i,* - ., g z n ) , wi, D l= and each generator has weighted degree DZ= ( n - 2)d - 2 Cy=3 d - wl, 3 5 1 5 n, respectively. Therefore, we can apply the Theorem

2.2,

to obtain

744

n and D2 = ( n - 2 ) d - 2 C i = 3 ~ i ,D1 = d - w1,3 5 15 n.

Now if we replace (1) and ( 3 ) in ( 2 ) we obtain

and D1 = 201, D2 = ( n - 2 ) d -2C:=3wi, D1 = d - wi,3 5 1 I n Since D1 = w1 we have that

2

-

1 = 0, so

p = l v=l

Observe that for the equation (4),if we put the factor

n:=,

(D, n"n=,( in evidence ) we get w, 6 1 - 1)

K#P

-1

with D1 = w1,D2 = ( n - 2)d - 2 c y = 3 w j , D1 = d - wi,3 51 5 n.

0

5. The local Euler obstruction of the stable types The local Euler obstruction for varieties, introduced in [18] by R. MacPherson in a purely obstructional way, is an invariant that is also associated to the polar multiplicities. Li! and Teissier in [16], with the aid of Gonzales-Sprinberg's purely algebraic interpretation of the local Euler obstruction, showed that the local

745

Euler obstruction is an alternate sum of the multiplicity of the local polar varieties. The autors in [12] apply these results to obtain explicit and algebraic formulae for the Euler obstruction of the stable types which appear in mappings from C, to C 3 . In this section we apply these results to show how to compute the Local Euler obstruction of A(f) and f(Cn-2?1(f)), in terms of the weights and degrees, of any finitely determined quasi homogeneous map germ in O(n,3) with n 2 3. First we recover the basic definitions and results.

Suppose that X c Cnis an analytic space of dimension d , v the transformation of Nash of X . Let p E X and z = ( ~ 1 , .+ .,z,) be local coordinates in Cn such that z i ( p ) = 0. 2 2 Let 11 z 11 = C z i z . Since 11 z 112 is a real-valued function, dll z 11 may be considered as a section of (TCn)*where * denotes the real dual bundle retaining only its orientation from the complex structure. We can also consider dll z 112 as a restriction to a section r of ( T X ) * .In [2] it is showed that for small E , the section T is non zero over Y-' where 0 5 11 z 5 E . Therefore let B, = { z / l l z 11 5 E } and S, = { z / " z 11 = E } . The obstruction to extendwhich we deing r as a non zero section of T X * from v-'(Se) to v-l(BE), note by E u ( T X * ,r ) , lies in Hd(v-'(BE), v-'(&); Z). If O(v-l~~~),v-i(s denotes the orientation class in Hd(v-l(B,), v - l ( S E ) Z), ; then we define the local Euler obstruction of X at p to be E u ( T X * , r ) evaluated on O(v-1 ( B ~ ) , ~ (s,)) -I or symbolically

I(

= ( E u ( T X * r, ) ,o(v-l(Be),v-l(sL)))

to .-'(Be) (see [18] or [7] for the definition and more details). The following result shows how the local Euler obstruction and the polar multiplicities are related. Theorem 5.1. (L6 Diing Trang et Teissier, [16], p. 476, Corollary 5.1.2.) Let X be a reduced analytic space at 0 E Cn+l of dimension d . T h e n

Euo(X ) = Cfz;

(-l)d-l--imd-l-i(

X)

where md-l-i(X) denotes the absolute polar multiplicity of the polar variety Pd- 1--i (X) . When X is the 2-dimensional set X = A(f), in order to apply this result we need to use the polar multiplicities mo(A(f)) which is the multiplicity of A(f)) at 0 and ml(A(f)) which is the multiplicity at 0 of the 1-dimensional polar variety PI (A(f)). Therefore this formula becomes:

746

or

E u o ( A ( f ) )= - W ( A ( f ) )

+ mo(A(f)).

When X is the 1-dimensional set X = f f)),to apply this result we need to use only the polar multiplicity m~(f(C"-~~~(f))) which is the multiplicity of f ( C n - ' > ' ( f ) ) at 0 and the Euler obstruction is given as:

E u o ( f ( C " - 2 w ) )= mo(f(C"-2J (f))). Therefore, we can apply directly the Theorems 3.1 and 4.1 to compute the Euler obstruction at 0 of these sets. 6. Euler obstruction of simple germs f : (C3,0) + (C3,0)

A classification of the d-simple germs (C3,0)

+

(C3,0) is given in [14].

Also, in that paper there is shown a list of invariants associated to these germs. V. H. Jorge Perez in [ll]increases this list by computing the polar multiplicities of the discriminant and of the image of the cuspidal edge curve. As a direct consequence of the Theorems 3.2 and 4.2 we obtain the following:

Corollary 6.1. The local Euler obstruction of A(f), where f is one of the A-simple germs below is as follows:

747

Corollary 6.2. The local Euler obstruction of f(E'>'(f)), where f is one of the A-simple germs below is as follows:

Normal Form (z,y, z3 (z2 y"')z), k 2 0 (x,y, z3 (z2y y"')~), k 2 4 (2, Y,z3 + (z3 Y 4 ) 4 (z, y, z3 ( x 3 zy3)z)

(Wf)) (Wf))

(2,

(Wf))

+ + + + + + + (z, Y,z3 + (z3+ Y 5 ) 4 (z,y, z4 + z z + ykz2),k 2 1 (z,y,2.4+(y2+zk)2.+zz2),k y, z5 + z z + yz2) (z,y, z5 + z z + y2z2 + yz3) (z, y, z5 + z z + yz3)

Euler Obstruction Euo(f =2 Euo(f (E19'(f)) = k - 1 Euo(f =3 E u o ( f ( W f )= ) 3 E u o ( f ( W f ))) = 3 Euof(C'J( f ) ) = 2 22 E u o ( f ( C ' J ( f ) )=) 3 Euo(f =3 Euo(f =3 Euo(f(E'J(f)) = 3

(.wf))

We remark that the polar multiplicities used t o compute these Euler obstructions are done in the Propositions 4.2, page 254 and 4.3, page 256 of [ll].

Acknowledgement: This work is part of the Ph.D. Thesis of Eliris Cristina Rizziolli, supported by CAPES, under supervision of M. J. Saia and V.H. Jorge Perez. The authors thank USP and CAPES for this support. The authors also thank the referee for his several suggestions which helped t o clarify this article.

References 1. V.I. Arnold, A. Varchenko and S. Goussein-Zadk, Singularitks des applications diffbrentiablesI. Classification des points critiques, des caustiques et des fronts de ondes, Editions MIR, Moscow (1982). 2. J.P. Brasselet and M. H. Schwartz, Sur les classes de Chern d'un ensemble analytique complexe, Astbrisque 82-83,93-147, ed. J.L. Verdier, SOC.Math. France, Paris, (1981). 3. T. Fukui, J. J. Nufio Balesteros and M. J. Saia, On the number of singularities of generic deformations of map germs, J. of The London Mathematical Society, London (2) 58, 141-152, (1998). 4. T. Gaffney, Polar Multiplicities and Equisingularity of Map Germs, Topology, 32,1, 185-223, (1993). 5. T. Gaffney, Properties of Finitely Determined Germs, Ph.D Thesis, Brandeis University, (1975). 6. T. Gaffney and R. Vohra, A numerical characterization of equisingularity for map germs from n-space, (n 2 3), to the plane. J. Dyn. Syst. Geom. Theor. 2, 43-55, (2004).

748 7. G. Gonzalez-Sprinberg, L’obstruction locale d’Euler et le thborhme de MacPherson, AstBrisque 8 2 et 83, ed. J.L. Verdier, SOC.Math. France, Paris, 7-33, (1981). 8. G. M. Greuel and H. A. Hamm, Invarianten quasihomogener vollstiindiger Durchschnitte, Invent. Math. 49, 67-86, (1978). 9. J. P. G. Henry and M. Merle, Limites d’espaces tangents et transversalith, Actes de la conference de gBom6trie algebrique 18 Rabida, Springer Lecture Notes. 961, 189-199, (1981). 10. ;I.P. G. Henry and M. Merle, Limites de normales, conditions de Whitney et kcatlement d’Hironaka, Singularities, AMS Symposia in Pure Math. 40-1, 575-584, (1983). 11. V.H. Jorge PBrez, Weighted homogeneous map germs ofcorank one from c 3 to C 3 and polar multiplicities, Proyecciones, 21 n: 03, 245-259, (2002). 12. V. H. J . Jorge PBrez, E. C. Rizziolli and M. J. Saia. Whitney equisingularity, Euler obstruction and invariants of map germs from c n to C3, n > 3. To appear: Proceedings of The VIII International Workshop on Real and Complex Singularities, Ed. J. P. Brasselet and M. A. S. Ruas, Trends in Mathematics, Birkhauser, (2006). 13. G. Kennedy, MacPherson ’s Chern classes of singular algebraic varieties, Comm. in Algebra 18, 2821-2839, (1990). 14. W.L. Marar and F. Tari, On the geometry of simple germs of corank 1 maps from R 3 + R3. Proc. Camb. Phil. SOC. 119 (1996) 469-481. 15. D. T. L6, Calculation of Milnor number of isolated singularity of complete intersection, F‘unktsional’ny: Analizi Ego Prilozheniya, vo1.8, no.2,45-49, (1974). 16. D. T. L6 and B. Teissier, VariBt& polaires locales et classes de Chern de variBt6s singulhres, Annals of Matematics 114, 457-491, (1981). 17. E.J.N. Looijenga, Isolated singular points on complete intersections, London Mathematical SOC.Lecture Note Series 77,Cambridge University Press, Cambridge, (1984). 18. R.D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 2, 423-432, (1974). 19. W . L. Marar, J. A. Montaldi and M. A. S. Ruas, Multiplicities of zeroschemes in quasi-homogeneous corank-1 singularities C -+ C n, Singularity Theory (Liverpool 1996), London Math. SOC.Lecture Note Ser., 263, pp. 353-367, Cambrigde Univ. Press, Cambrigde, (1999). 20. D. Mumford, Introduction to Algebraic Geometry, Springer-Verlag, (1976). 21. J. J. Nuiio Ballesteros and M. J. Saia, Multiplicity of Boardman strata and deformations of map germs, Glasgow Mathematical Journal, 40 , 21-32, (1998). 22. B. Teissier, Variktb Polaires 2: Multiplicit6 Polaires, Sections Planes, et Conditions de Whitney, Actes de la conference de geometrie algkbrique &la Mbida, Sringer Lecture Notes, 961, 314-491, (1981). 23. B. Teissier, Cycles Bvanescents, sections planes et conditions de Whitney, Singularitb 8 Carggse 1972, Asterisque, 7-8, 285-362, (1973).

749

Logarithmic vector fields and multiplication table. Susumu TANABE Department of Mathematics, Kumamoto University, Kumamoto, 860-8555, Japan *E-mail: stanabeC2kumamoto-u.ac.jp, tanabeC2mccme.m This is a review article on the Gauss-Manin system associated t o the complete intersection singularities of projection. We show how the logarithmic vector fields appear as coefficients t o the Gauss-Manin system (Theorem 2.3). We examine further how the multiplication table on the Jacobian quotient module calculates the logarithmic vector fields tangent to the discriminant and the bifurcation set (Proposition 3.1,Proposition 5.2). As applications, we establish signature formulae for Euler characteristics of real hypersurfaces (Theorem 4.1) and real complete intersections (Theorem 5.1) by means of these fields. Keywords: complete intersections, Gauss-Manin system, real algebraic sets.

1. Introduction

This is a review article on the Gauss-Manin system associated to the isolated complete intersection singularities (i.c.i.s.) of projection and objects tightly related with them. The notion of i.c.i.s. of projection has been picked up among general i.c.i.s. by Viktor Goryunov [1,2]as good models to which many arguments on the hypersurface singularities can be applied (see for example Theorem 2.1, Lemma 2.1). All isolated hypersurface singularities can be considered as a special case of the i.c.i.s. of projection. Many of important quasihomogeneous i.c.i.s. are also i.c.i.s. of projection. The main aim of this article is to transmit the message that the multiplication tables defined on different quotient rings calculate important data both on analytic and topological characterisation of the i.c.i.s. of projection. We show that the multiplication table on the Jacobian quotient module in (O,,,)k calculates the logarithmic vector fields (i.e. the coefficients to the Gauss-Manin system defined for the period integrals) tangent to the discriminant and the bifurcation set (Proposition 3.1, Proposition 5.2) of the i.c.i.s. of projection. This idea is present already in the works by Kyoji Saito [3] and James William Bruce [4]for the case of hypersurface singu-

750

larities (i.e. k = 1). On the other hand, as applications, we establish signature formulae for Euler characteristics of real hypersurfaces (Theorem 4.1) and real complete intersections (Theorem 5.1) by means of logarithmic vector fields. These are paraphrase of results established by Zbigniew Szafraniec [5]. It is well known in the study of real algebraic geometry, Oleg Viro’s patchworking method ( [S]) furnishes us with a relatively simple and effective method to construct various nonsingular real plane projective algebraic curves of a given degree m with different isotopy types. As this method is based on perturbations of singular curves with quasihomogeneous singularities, our study on the versa1 deformation of hypersurface singularities fits into the context of real algebraic geometry. We shall notice that Viro’s patchworking method does not describe all possible curves corresponding to the full deformation parameter values outside the real discriminant. The deformation parameter values s E Rpthat can be treated by Viro’s method are located (on a quasihomogeneous curve) in certain specially selected real components of the complement to the discriminant. This situation is explained by the essential use of regular triangulation of the Newton polyhedron of the defining equation F ( z ,s) in his construction. At the end of 56, Example 2 , we indicate cases of real curves with different Euler characteristics that are impossible to distinguish after patchworking method. We hope that this approach would give a new complementary tool to the topological study of real algebraic curves. The author expresses his gratitude to Aleksandr Esterov who drew his attention to the utility of multiplication table and proposed the first version of Theorem 5.1. The main part of this work has been accomplished during author’s stay at the International Centre for Theoretical Physics (Trieste) and Hokkaido University where the author enjoyed fruitful working condition. The author expresses his deep gratitude to the concerned institutions and to Prof.Toru Ohmoto who gave him an occasion to report part of results at RIMS (Kyoto) conference.

2. Complete intersection of projection Let us consider a k-tuple of holomorphic germs

(Wk

&, ). = (fib,.), . . . 7 fk(2, .)) E (2.1) in the neighbourhood of the origin for X = (Cn+’, 0). This is a 1- parameter deformation of the germ (2.2)

.?’(,)

= (fl(z,01, ’ ’ ‘ 7 fk(z,0)) E

(ox)k

751

for X = (Cn,0). After [l]we introduce the notion of R+equivalence of projection. Let p : Cn+l C be a nondegenerate linear projection i.e. d p # 0.

-

Definition 2.1. We call the diagram

y

L-f

cn+l+ p

c,

the projection of the variety Y ~f Cn+l on the line. Two varieties Yl,Y2 belong to the same R+ equivalence class of projection if there exists a biholomorphic mapping from Cn+l to Cn+l that preseves the projection and induces a translation p -+ p const on the line.

+

In this way, we are led to the definition of an equivalence class up to the following ideal,

and the subring,

that is nothing but the tangent space to the germ of R+ equivalence class of projection. We introduce the spaces := ( O x ) ' / T f ,

(2.5)

Qf

(2.6)

Q: := ( o ~ ) ~ / T , f .

We remark that though T,f is not necessarily an ideal the quotien Q; can make sense. Assume that Q f is a finite dimensional C vector space. In this case, we call the number r := d i m c Q f the R+- codimension of projection. We denote by (.i(z, u), , &(z, u))the basis of the C-vector space If T < 00, it is easy to see that f ( z , u ) = 0 (resp. f(z,O)= 0) has isolated singularity at 0 E X (resp. 0 E 3). We shall denote by p the multiplicity of the critical point (z, u)= 0 of the height function u on X O := { ( z , ~ )E x ; f i ( z , u ) = ... = f k ( 2 , u ) = 0). Let us consider a R+versa1 deformation of flo)(z) e .

QT.

(2.7)

F(x,U , t ) = f7O'(~)

+ &(z, + tlZi(z, + . . . + tTZT(z,u ) , U)

U)

with &(z, u) = f(z, u)- f(z, 0). We consider the deformation of X O as follows (2.8)

x,:= {(z, u)E x;F ( z ,u,t ) = 6},

752

+

that is also a (T 1)-dimensional deformation of the germ XO := {z E 2;f l ( z ,0 ) = . . . = f k ( z , 0) = 0 ) . The following fact is crucial for further arguments.

Theorem 2.1. ( [ l ] ,Theorem 2.1) For the k-tuple of holomorphic germs (2.1) with 0 < p < +co, we have the equality p = T 1.

+

As a consequence we have p = d i m c Q f . Recently a conceptual understanding in terms of homological algebra of this phenomenon appeared. See [7], $3. F'uther, in view of the Theorem 2.1 we make use of the notation, S = (C'+',O) = (C., O),s = ( u ,t ) E S, so = u , si = ti, 1 5 i 5 T. We will denote the deformation parameter space t E T = (CT,0 ) . Let Ic,, c OX be the ideal generated by k x k minors of the marix (aT(W) ... a A z , q az, ' ' az, . Proposition 2.1. ( [l],Proposition 1.2 ) We have the equality

Let us denote by Cr(@)the set of critical locus of the projection --f S. That is to say

7r

:

UtETXt

(2.9) Cr(@)= { (z, u , t ) ;(x,u ) E X t , rank( a @ ( x , s) 1 ... 7

d F ( z , s)

8x1

ax,

) < k).

We denote by D c S the image of projection r ( C r ( @ )which ) is usually called discriminant set of the deformation X t of projection. It is known that for the R+-versa1 deformation, D is defined by a principal ideal in 0 s generated by a single defining function A(s) [8]. Under this situation we define 0 s - module of vector fields tangent to the discriminant D which is a sub-module of Ders the vector fields on S with coefficients from 0 s .

Definition 2.2. We define the logarithmic vector fields associated to D as follows,

Ders(Zog D ) = {.'E

Ders;{(A)

E 0 s . A}.

We call that a meromorphic p-form w with a simple pole along D belongs to the Osmodule of the logarithmic differential forms Rs(log D) associated to D iff the following two conditions are satisfied l ) A . W E 05,

753

2 ) d A w E R",: or equivalently

A . dw E O;+'.

For the 0s-module of the logarithmic differential forms the following fact is known.

Theorem 2.2. (See [9] for the case k = 1 , [8,10] for the case k general) The module Ders(1og D ) is a free 0s-module of rank ,u. Furthermore there exists a p-tuple of vectors GI,+ . ,Gp E Ders(1og D ) such that

A(s) = d e t ( i i l , - . . ,Gp),

Proposition 2.2. (see 1111for the case k = 1 , [I] for general k) For e v e y Zj E Ders(1og D ) , 1 5 j 5 p, there exists its lifting Gj E D e r z x s tangent to the critical set Cr($). More precisely, the following decomposition holds,

+

for some hs,j(x,s) tion,

E

Oz,,,

bj,q(x,s,F ) E Ozxs

m i . I n this nota-

Conversely, to eve y vector field Gj E Der*xs tangent to the critical set C r ( F ) we can associate a vector field Gj E Ders(1og 0)as its push down. This is a direct consequence of the preparation theorem (see [12]).Further on in this article we denote by Z(F(x,5 ) ) the action of a vector field ii E D e r z x s on a function F ( z ,s).

Lemma 2.1. ((11) The discriminant A ( s ) defined in Theorem 2.2 can be expressed by a Weierstrass polynomial,

A(s) = U' ~ i t h d 1 ( 0 ) = * * * = d , ( O=)O .

+ d l ( t ) d ' - l + . . . + d,(t),

754

This can be deduced by another way by making use of (5.5) for the case of CI (5.1). Namely we have A(s) = detP(s). From this lemma we deduce immediately the existence of an “Euler” vector field even for nonquasihomogeneous f(z,u) that plays essential r61e in the construction of the higher residue pairing by K.Saito [13].

Lemma 2.2. (For k = 1, see [13] (1.7.5)) There is a vector field (u ny(t))& CIFla!(t)& E Ders(Zog 0 )such that

+

+

$1

=

.i(A(s)) = P A ( S ) .

Proof. It is clear that for a vector field $1 E Ders(1og 0 )with the component (u a:(t))& whose existence is guaranteed by Theorem 3 , l [l] , the expression .i(A(s)) must be divisible by A(s). In calculating the term of .i(A(s)) that may contain the factor u p , we see that

+

.i(A(s)) = p u p

+ Jl(t)~’-’ + + J p ( t ) . *. *

Thus we conclude that &(t)= p d i ( t ) , 1 5 i 5 p. Now we introduce the filtered+ 0s-module of fibre integrals multi-index of negative integers X = (XI,. . . , Xk) E ( Z < O ) ~ .

0

7-6’) for a

for +(z,s) E O*z-xs. Let us denote by X ( q ) := {z E X ? ; F , ( z , s )= 0) a smooth hypersurface defined for s # D. In this situation we define the Leray’s tube operation isomorphism (see [14,15]),

t : Hn-k(nt=lx(q))-+ H,(X \ u,k,,~(q)), 7

H

t(y)*

The concrete construction of the operation t can be described as follows. First we consider the coboundary isomorphism of the homology groups,

6 : iYn-k(n,k=lx(q)) -+ ~ ~ - ~ + ~ ( n \ ~, (k~=1 )~. x ( ~ ) A cycle y in n&,X(q) is mapped onto a cycle 6(y) of one higher dimension that is obtained as a S1 bundle over y. Repeated application of 6 yields an interated coboundary homomorphism, i ~ ~ - ~ ( n , k = , x ( q ) ) - + ~ i ~ ~\ x - ~( +~~)()n. + .~ =~~. ~ ( ~ )

755

. . . --'%,-1(X(k)

\ u$:;x(q))46Hn(2 \ u;=lx(q)).

The Leray's tube operation is a k-time iterated 6 homomorphism i.e. t = 6". The Froissart decomposition theorem ( [14], 56-3) shows that the collection of all cycles of H n ( X \ U:=,X(q)) are obtained by the application of iterated 6 homomorphism operations to the cycles from H ~ - ~ n( ~X( 4 1 n) ~ ( 9 2 .) . . n ~ ( q p ) ) p, = 0 , . . . , k. Let us denote by @ the C vector space whose C - dimension is equal t o p after the Proposition 2.1. d e denote its basis by ($o(x,u),. * . 7 $7b, Now let us introduce a notation of the multi-index -1 = (-1, .. . 1) E ( Z < O ) ~ .We consider a vector of fibre integrals I@ :=t (&')(s),.:. ,Iky')(s)). The following theorem for k = 1 has been anounced in [3] (4.14) without proof.

2

Theorem 2.3. 1. For every v' E Ders(1og D ) , we have the following in-

clusion relation v ' : "-1)

L)

.I'-(%

That is to say for every i7j E Ders(1og D ) , there exzsts a p x p matrix with holomorphic entries Bj(s)E E n d ( C h ) 8 0 s such that

Gj(I@)= Bj(S)I@, 1 I j 5 p. 2. The vector of fibre integrals I@satisfies the following Pfaff system of Fuchsian type

dI@= R . I+, for some R E E n d ( C p ) 8 o s R&(log D ) .

Proof. As for the proof of 1, we remark the following equality that yields from Proposition 2.2,

756

which evidently belongs to 7d-l).The last equality can be explained by the vanishing of the integral

because of the lack of the residue along F,(z,s) = 0 and

in view of the lack of at least one of residues either along F,, = 0 or along Fq2= 0. These equalities are derived from the property of the Leray’s tube t ( y ) which needs codimension Ic residue to give rise to a non-zero integral. 2. Let us rewrite the relations obtained in 1. into the form,

r=l

for some w , , ~ E R i ( - D ) meromorphic 1-forms with poles along D. These w , , ~ satisfy the following relations,

r=l

If (v’j, w,,,) E 0 s for all 3 ’ E Ders(1og D ) 1 5 j 5 /A then W q , r E Rk(Zog D) 0 in view of the Theorem 2.2. Let us introduce a filtration as follows ‘Id(’) = @X1+,..+Xk=X 7i(x).For this rough filtration we have the following generalisation of the Griffiths’ transversality theorem ( [16] Theorem 3.1).

Corollary 2.1. For every v‘ E Ders(1og D ) , we have the following inclusion relation

v’ : 1’“

+

7.p).

757

Proof. For

I+ E H ( - k - l ) and $ E Ders(log D ) we have

+ &,G(la) = [fle,&,IIa + &,(Be(s)l+)= [Ce,&,]Ia + (&,Be(s))I++ Be(s)(&,Ia). fle(&,I+) = [G, &,]Ia

As the commutator [i7e,i9a.j]is a first order operator, the term above [ G ~ , a , , ] belongs l~ to H(-"'). The term d,,Be(s)l+ E again belongs to 7-l(-k-1).Thus we see 3e(i3a.,Ia)E H ( - k - l ) . In an inductive way, 0 for any X 5 -k we prove the statement. 3. Multiplication table and the logarithmic vector fields

Po)

We consider a versal deformation of a mapping (x)which can be written down in the following special form for s = (u,t)E S ,

[ Y; Fl(2,t ) -

7

(3.1)

F(x,S) = FO'(Z)+ C e= 1 teZe(x) + uZ'(X)

=

1

7

for

{Zo(x),. . . ,Z T ( ~ ) E)

&f,

t (-1,O,...

where
The functions T < ~ ( s )E 0 s exist due to the versality of the deformation @(x, s). We denote by

ktj

(3.3) Tj(4 = (4) o<j,e<7 7 a p x p matrix which is called the matrix of multiplication table. We denote the discriminant associated to this deformation by D c S.

758

w,

Further on we will make use of the abbreviation mod(d,@(z, s ) ) instead of making use of the expression mod(O%xs( . , *)). After Proposition 2.2 the vector field $1 constructed in Lemma 2.2 has its lifting $1 E Der%xxS. Let us denote by $1 = 31 - 31 E O%xs@ Derg.

&(@(z, s ) ) . q52(.)

=

T

= ~ ~ l ( s e ) z e ( z ) 4 i (mod(dz9(z,s)). z) e=o Lemma 3.1. There exists a vector valued function M ( x , @ ( x , s ) )E OX^^^)' such that

&(F(z,s)) = M ( z ,F ( x ,s)) mod(d,@(z, s ) ) , with

M ( x ,9 ( x ,s ) ) = M 0 * @(z,s)

+ M1(x, 9(x,s)),

where M o E G L ( k , C ) : a non-degenerate matrix and M1(x,F(x,s)) E (0% @ Especially the first row of M o = (1,0,. . . ,O).

mi)k.

Proof. First of all we remember a theorem due to [17] 81.1, [3] Proposititon 2.3.2 which states that the Krull dimension of the ring of holomorphic functions on the critical set Cr(@)is equal to p- 1 and this ring is a CohenMacaulay ring. Let us denote by L = rick. We have (k L)tuple of k x kminors jk+l(z,s) . . j k + L ( x , s) of the matrix ( & ~ ( z ,s ) , . . . , & ~ ( z , s ) ) such that

+

--A

4

+

c r ( @ )= v ( ( ~ l (sx) ,,... ,~ k ( zs),jlc+l(z, , s ) , . , j k + L ( z , .I)). + .

The lemma 2.2 yields that the lifting relations,

(Fl(z, s)~

' ' ' 7

= (&(Fl(Z,s ) ) , . * .

21 of the vector field $1

satisfies the

Fk(z,S),jk+l(z,S), ' '. , j k + L ( x ~ , S ) )

,&(&(x,

S)),&(jk+l(2, s)),

- .. ,& ( j k + L ( x ,

s))).

As it has been seen from the above Proposition 2.2, the vector 31 is tangent to C r ( 9 ) .If the above equality does not hold, it would entail the relation {S E

S;A(s) = 0 )

759

ST

(v(($1 (Fl(z, s ))

7 ' '

.

7

(4(5, s ) )

$1

7

$1 ( j k + 1 (z, s ) )

,

* * *

$1 ( j k + L

(z, s ) ) )) ) ,

after elimination theoretical consideration. This yields k

$i(Fq(z,

s)) =

k+L

C C,Fe(z, + m,(z, F)+ C S)

e=i k+L

&(jp(z, s)) =

C,je(z, s), 1 I qI k,

e=k+l

C

C;je(z,S)

+ mP(z,$), k + 1 I p I k + L,

e=k+l

+

for m,(z,$) E Ox 8 m i , 1 I r 5 k L and some constants C:, 1 I l! 5 k. First we see that the expression Z 1 ( j p ( z , s ) ) cannot contain terms s) like Fq(O, s) in view of the situation that the versality of the of Fq(z, deformation makes all linear in z variable terms dependent on some of deformation parameters. Secondly the non-degeneracy of the matrix M o := (C:)l
z ,s) &(F(z,s)) = M o . @(z,S) + M1(z,$(z,s)) + C hl,j(z,S)a qaxj ' j=1

with Ml(z,$(z,S)) = t ( m l ( Z , @ ) , . * * , m k ( z , $ ) ) E (02 More precisely we can state that C: = 1, Cf= 0 , 2 I lI k. The dependence of some cofficients of $1 on Fi(z,t) is necessary so that Cf # 0 for some 2 5 l! 5 k . But this is impossible because if not it would mean that some of the coefficients of $1 contains factor Fz(z,s),-.,Fk(z,s) that contradicts the construction of $1 in Proposition 2.2. This can be seen from the fact that the expressions , aFi(x,S) ax, , aFi(x,s) , .. . , do not contain the deformation parameters present in the polynoas, mials F2(z, s), . . . ,F k ( 2 , s). 0

v,

Lemma 3.2. A basis of logarithmic vector fields GO,. . ,v', E Ders(1og D ) can be produced f r o m the functions I Y ; ( S ) defined as follows,

7

E

C u ~ ( s ) Zm~o d ( d x F ( z ,s ) ) , e=o

760

where the vector valued fucntion M ( x , @(x, s ) ) denotes the one defined in the Lemma 3.1 and 8j = hjlp(x,s ) a is a certain vector field with 8% holomorphic coefficients.

c;==,

Proof. We remark the following relation, W ( Z , S))+i(Z)

7

fl1(sj)Z’(x)+i(x) mod(d,@(x, s)).

E

j=O

The relation (3.2) above entails, 7

M ( x ,F(z, s)) *

+i(~)

7

x V ; ( S j ) T t j ( S ) Z t ( z ) m o d ( d z F ( ~s),). e=o j = o

As +i(z) can be considered to be a basis of 0 s module @(s) above (see Proposition 5.1), vectors (ap(s),... ,a,7(~)),0 5 i 5 T are 0 s linearly independent at each generic point S \ D . If we put

j=O

then the vector field & E Der*xs

is tangent to C r ( 2 ) .The only non-trivial relations that may arise between and v’i, i # i‘ is

si

+i(z)sz/= +i’(Z)5i.

These vectors give rise to the same push down vector field in Ders(log 0 ) . Namely,

761

for the coefficients Rf,i,,j(s) determined by

t=O

j=O

j=O

This means that $0,. . . ,Gi- form a free basis of D e r a x s ( C r ( F ) ) hence 0 $0,. . . , V; that of the module Ders(log D ) . This lemma gives us a correspondence between $i(z) E CP and G E Ders(1og D),therefore it is quite natural to expect that the mixed Hodge structure on @ would induce that on Ders(1og D), and would hence contribute to describe & ( s ) of Theorem 2.3, 1 in a precise manner. A good understanding of this situation is indispensable to characterize the rational monodromy of solutions to the Gauss-Manin system in terms of the mixed Hodge structure on CP. Confer to Proposition 5.3 below. We formulate the lemma 3.2 into the following form (see [4] Theorems A2, A4, [9] (3.19), [3] (4.5.3) Corollary 2 for Ic = 1 and [8] (6.13), [l] Theorem 3.2 for Ic general). Proposition 3.1. There exist holomorphic functions w j ( s ) E Os, 0 5 j 7 such that the components of the matrix

I

i-

(3.4)

C(S) := C W j ( S ) T j ( S ) , j=O

give rise t o a basis of logarithmic vector fields 50,+ . ,v', E Ders(1og 0 ) . Namely, if we write C(s) = ( ~ Z " ( S ) ) e
i-

(3.5)

-

v'i = C.p(s)-,

a

ase e=o consists a base element of the 0 s module Ders(1og D ) .

Especially in the case of quasihomogeneous singularity f(z, u ) we have the following simple description of the vector field that can be deduced from Lemma 3.2. To do this, it is enough to remark that the vector field 5 1 is the Euler vector field by definition and . i ( s r ) = s s , . , where w(sj) denotes the quasihomoeneous weight of the variable s j . Proposition 3.2. ( [2] Theorem 2.4) I n the case of quasihomogeneous singularity (2.1), the basis (3.5) of Ders(1og D ) can be calculated by

c 7

af(s) =

W(Sj)SjT&(S).

j=O

762

Furthermore, the vector valued function M ( x , $(x, s ) ) of Lemma 3.1 has the expression,

M ( x ,$(x, s ) ) = M o . $(x, s ) = diag ( w ( f ~ ).,

. ,w(fk)) . $(x, s).

4. Multiplication table and the topology of real

hypersurfaces

+

In this section we continue to consider the situation where p = r 1 for k = 1 in (2.5). We associate to the versa1 deformation of the hypersurface singularity

c T

F ( z ,s ) = f (x)+

(4.1)

siei(z),

i=O

the following matrix C(s) = ( c ~ ( s ) ) O ~ i after , e ~ Tthe model (3.2), 7

(4.2)

F ( x ,s)ei(x)=

c,"(s)ee(z)mod(dxF(x,s)). e=o

Further on we make use of the convention eo(x) = 1 and s = ( s 0 , t ) . We denote the deformation parameter space t E T = (CT, 0). We recall the Milnor ring for k = 1 whose analogy has been introduced in (2.5) (and in the case k general, @(s) will be introduced in Proposition 5.119 %xs

Q F :=

aF x , s

~~xs(+y

.. . 7

-)*ax,

We introduce the Bezoutian matrix B F ( s )whose ( & j ) element is defined by the trace of the multiplication action F ( z ,s)ei(z)ej(z).on the Milnor ring Q F , 7

~ ( xs)ei(x>ej(z) , =( C e ~ ( s ~ z ) ) e j ( x ) c=o T

7

c=o

r=o

763

For the sake of simplicity we will use the following notation, (4.4)

7T(t)

= (Tl,b(t))O
To clarify the structure of the Bezoutian matrix B F ( s ) we introduce a matrix (4.5)

with the notation 7

~ ( t= )tr(eT(z>.)= CT:,e(t). (4.6) e=o The ( i , j ) element of the matrix T ( t )(4.5) equals to tr(ei(z)ej(z).)on the Milnor ring Q F . It is possible to show that {t E T ;det(T(t))= 0) coincides with the bifurcation set of F ( z ,s) outside the Maxwell set (see Proposition 5.2 below). Thus we get the Bezoutian matrix

B F ( s ) = C(s) . T ( t ) .

(4.7)

Following statement is a simple application of Morse theory to the multiplication table see [5] Theorem 2.1. From here on we assume that Is( is small enough and denote by X = {z E C"; 1x1 5 S} a closed ball such that all critical points of F ( z ,s) are located inside

x.

Proposition 4.1. sign C(s) T ( t ) ={ number of real critical points in ~ ( zs>, > 0, z E X n R") -{ number of real critical points in ~ ( zs), < 0, z E X nR"}.Here sign(A) denotes the signature of a symmetric matrix A i.e. the difference between the number of positive and negative eigenvalues. Let us denote by h ( z ,t ) the determinant of the Hessian

h ( z ,t ) := det

(

Ili,jln

.

We associate the foilowing p holomorphic functions ho(t),. . . ,h T ( t )E 0 s to the function h ( z ,t ) , (4.8)

h ( z ,t ) =

7

he(t)ee(z) rnod(d,F(z, s ) ) .

e=o Further by means of (4.7) we introduce the matrix (4.9)

e=o

764

where

We consider the matrix B H F ( s )= (.)osa,bsT whose (a,b)-element is defined by the trace of the following expression on the Milnor ring Q F , (4.10)

(c 7-

h(x,t)F(x,s)e,(x)eb(x)

e=o

T

e=o

T

7

7

hl(t)ee(x))(c c:(s) c=o

Tz(t>em(x>) m=O

T

c=o m=O

If we take the trace of this, we get 7

c=o

7

m=O l = O

r=O

After (4.8) and (4.9) this matrix has the following expression, (4.11)

B H F ( s ) = C ( S ) .B H ( t ) .

We consider the following closures of semi-algebraic sets,

WZ0:= { x E X

n R";F ( z ,S) = 0 } ,

Theorem 4.1. The following expression of the Euler characteristics for W, holds,

X(W - x(W=o) = (-1)

,sign(BH(t)) - sign(BHF(s)) 2

765

Proof. After Szafraniec [ 5 ] , or simply applying Morse theory to the real fibres of F ( z ,s), we have the following equalities,

c

(sgn h(z7 t ) )

%€criticalpoints ofF(z,s)

c

= sign(tr(h(z,t)ei(z). ej(z).))lli,jln=

(-

l)X(").

zEcritica1 points ofF(z,s)

Here we denoted by tr(h(z,t)ei(z) ej(z).) the trace of a matrix defined by the multiplication by the element h(z,t)ei(z). e j ( z ) considered rnod(d,F(z, s)) for the basis ei(z),1 5 i 5 p.

c

(sgn h ( z ,t ) ) ( s g n F ( z ,s))

%€criticalpoints ofF(z,s)

c

-

(-l)X(z)(sgn F ( z ,s)).

%€criticalpoints ofF(z,s)

We denoted by tr(h(z,t ) F ( z ,s)ei(z). ej(z).)the trace of a matrix defined by the multiplication by the element h(z,t ) F ( z ,s)ei(z) . ej(z) considered rnod(d,F(z, s)) for the basis ei(z), 1 5 i 5 p. The exponent X(z) is the Morse index of the function F ( z , s) at z and sgn h ( z ,t ) = (-l)A(z). CI 5. Topology of real complete intersections

Let us reconsider the situation (3.1) for the deformation of the CI,

Fl(2,t ) - u

(5.1)

with s = ( u , t ) E S. Define the ideal Ico(t)c O x x s generated by Ic x k

w).

minors of the marix , We have the following isomorphisms (*,..a

a=

OX Q x ( f 1 k )- u , f z ( z ) , * ,f* k(.)).

+kO(O)

766

where Ico(0)is the corresponding ideal in 02. The dimension of this space is equal to p introduced in Proposition 2.1. As for this number we remember that it can be expressed by means of the Milnor number of the singularity X I := {z E X ; f 2 ( s ) = = f k ( z ) = 0) and the Milnor number of the function f 1 restricted on X1 i.e. that of the singularity ZO:= {z E x;fl(.) = f2(.) = . . * = f k ( Z ) = O } ,

+

P = PL(X1) p(J70).

This formula is known under the name of Le-Greuel formula [17,18]. Let us denote by +i(x) E @, 1 5 i 5 p a basis of a.

Proposition 5.1. We have the following free 0 s module of rank p,

Proof. We reproduce the argument by [4],Lemma A 1. First of all we see that the module @(s) is a finitely generated 0 s module. This can be shown by a combination of the Weierstrafl-Malgrange preparation theorem and the fact that for each fixed s E S the space (5.3) is a finite dimensional (Ip ) C vector space (see [5]). The above space (5.3) is isomorphic to the direct sum of C vector spaces,

Since this direct sum has dimension p = the multiplicity of the critical point ( z , u ) = 0 of the height function on Xo,as mentioned at the very beginning of the paper, it follows that (+i(z)}o
(Fl(2, 4 - .)4i(.)

767

Thus the matrix

is defined. In analogy with (3.3), we define another multiplication table

7

(5.8)

~ c ( t := ) t.(+c(x>*)

= Cwt,e(t)-

e=o

Thus 7

(5.9)

td(Fl(2,t ) - u)+a(x)+b(x)*) =

CPm e=o

7

pJec,b(t)CC@). c=o

We introduce the notation, 7

T ( t )=

(5.10)

C Cc(t)WC(t). c=o

jF'rom here on we assume that Is1 is small enough and denote by = {x E C";1x1 5 6 ) a closed ball such that all critical- points of Fl(x,t ) - u on F2(x,t ) = . . * = Fk(x,t ) = 0 are located inside X. In combining the results of [5], Thoerem 2.1, Theorem 3.1, with our above arguments we get the following. Theorem 5.1. 1. The discriminant set of the deformation of projection X t is given by the matrix (5.5), (5.11)

D

= {s E

5';d e t ( P ( s ) )= 0 ) .

768

2. { number of positive critical points of Fl (x,t ) - u o n

. . . = Fk(x,t ) = 0, x

F~(x, t) =

2 n R"} - { number of negative critical points of Fl(x,t ) - u on F2(z, t ) = . . = Fk(x,t ) = 0, x E X n R" } E

= sign(P(s). T ( t ) ) .

In opposition to the case k = 1, we cannot write down a simple formula for Euler characterisic of closures of semi-algebraic sets,

W, = {Z E X n R";Fi(z,t) - 2~ * 0 , Fz(x,t) = * * . = F k ( 2 , t ) = 0 } , with * =>, 5 ,=. As a matter of fact, it is quite easy to establish an analogous theorem to [5]Theorem 3.3 on x(W>o)&x(Wg) by the aid of matrices introduced above. We leave this task as an exercise in view of complicated form of the analogy to the Hessian. is defined as B F ~ := { t E T ; number of critical The bifurcation set points of FI(s,t ) - u on Fz(x,t ) = . . = Fk(x,t) = 0 is strictly less than p } \ B M . Here BM denotes the Maxwell set of Fl(x, t ) - u, namely BM := {t E T ; two critical values of Fl(x, t ) - u on F2(z,t ) = . = Fk(z,t ) = 0 coincides } . +

Proposition 5.2. The bifircation set has the following expression

(5.12)

BF~ = { t E T;det T ( t )= 0).

Proof. We consider the critical set

Co(t) : = { ~ € X ; d F ~ ( ~ , t ) A d F 2 ( ~ , t ) A . . . A 1d 0F ~, ( ~ , t )

F2(z,t ) = * .

= Fk(S,t ) = 0).

Here we remark that the critical set Co(t) has codimension n in 2 for a fixed generic value t and it is a set of points. After [5] Corollary 2.5, the rank of T ( t ) is equal to the number of points { p E Co(t)}.Therefore T ( t ) 0 degenerates if and only if ICo(t)l < p which means our statement. Regretfully, to the moment we cannot state how to deduce the basis of Ders(log 0 )from the matrix P ( s ) . Consequently we cannot establish the relationship between the Gauss-Manin system and the topology of the real algebraic sets. This fact is due to the situation mentioned in the Remark 5.1 below.

769

To remedy the situation, we state a proposition on the multiplication table and the coefficients to the Gauss-Manin system. Let us consider the multiplication between q& and i7j by the following way, (5.13) 7

Here 8j = Cr==, h j , p ( x ,s)& in Lemma 3.2.

denotes the vector field that has been defined

Proposition 5.3. The Gauss-Manin system for the period integrals IhL')(s) introduced in the Theorem 2.3 is expressed by means of multiplication tables (5.6) and (5.13) as follows,

c c1

~j($')(s)) =

((tr M O ). w&(s>

+ ~ f , ~ (1:1l)(s) s))

1

I j , q I p.

e=i

Here tr M o stands for the trace of the non-degenerate matrix M o defined in Lemma 9.1. Proof. First of all we remark the following chain of equalities,

Here we remember Lemmata 3.1, 3.2 and see that the above expression equals to

I,,

k

k

C,eFl(Z,s) k

+ mq(z,@(a:,s>>>

770

As the terms with Ci, l # q (resp. terms with m,(z, $(z, s)) E 02 @ m: ) vanish because of the lack of residues along &(z, s) = 0 (resp. some other F,(z,s) = 0 ) , the last expression in its turn equals to

Remark 5.1. The rank of C-module of Leray coboundaries t ( y ) E Hn(X\ Ut=l{z E %; Fq(z, s) = 0)) is equal to ~ ( 2 0 ) :the Milnor number of the singularity 20due to the tube operation isomorphism t : defined in Lemma 2.2. In view of the LB-Greuel formula mentioned in connection with (5.2), the dimension p of the space CP is bigger than p(X0) as it represents the sum of the ranks of (n - k)- dimensional cycles and ( n- k + 1)-dimensional cycles. Thus we have no exact duality between the integrands and the integration cycles. This means that the Gauss-Manin system of the above Proposition 5.3 is defined only for the Riemann period matrix of size p x ~ ( 2 0 ) . To get the the Gauss-Manin system defined for the Riemann period matrix of size p(X0) x p(&,), one need to consider the multiplication table on the Brieskorn-Greuel lattice 51; XI!:= dFl(Z,S)A . . . A d F k ( Z , S )

(Fl(Z,S),.** ,Fk(X,S))O2'

that is known to be a 0 s free module of rank p(&). This procedure can be done in an analogous way to that in Proposition 5.3. For the case of quasihomogeneous i.c.i.s., the concrete calculus of the the Gauss-Manin system is done by means of Brieskorn-Greuel lattice in 1191. 6. Examples

1. Let us consider the simplest example of the Pham-Brieskorn singularity, F(z1,22)

= z;

+ z; + + 1 '1

bz1z2

+ + a 1

dz2,

771

with deformation parameters s = (u, t ) = (u, b, c, d). We calculate the data (4.4), (2.4), (4.10), (4.11) as follows.

rI =

[,

0

0

0

-1L3d

0

1/9bc]

-1/3 c 1/9 bd

10 1/9bc 1/9bd 1/9dc]

'0

1

0

0

-

1

0

0

1/9b2

0

0

-1/3b-1/3~

, O 1/9 b2 -1/3 c 1/9 bd -

1

0

0 -1/3b

0

-1/3d

1

0

1/9b2

0 3 7 -

0

0

-

0 -1/3 d 1/9 b2 1/9 bc -

r4 =

000

1 -

001

0

010

0

1 0 0 1/9b2-

-

+ 1/9 b2c c2 + 1/9 b2d

-2/3 d2

-

-2/3

-

2d

3u

519 bcd

3u

+ 1/9 b3 -bc

-2/3 c2

2c

b

-bd

2c

+ 1/9 b3 d2 + 1/3

3u

+ 113 b2d -2/3

2d b2C

3u

+ 1/9 b3

772

B H =

8 b2

16 bc

16 bc

-8 b2d

b4

16 bd

b4

b4

16 bd

b4

+ 16dc

+ 16dc

-8 b2c

+ 16dc

-

813 b3c - 1613bd2 813 b3d - 1613bC2

+ 16dc 813 b3c - 16/3 bd2 813 b3d - 1613bC2

+

b2dc 119 b6)

-

rbl.l b1.2 b1.3 b1.41 b2.1 b2.2 b2.3 b2.4

B H F ( s= ) C(S) . B H =

b3.1 b3.2 b3.3 b3.4 Lb4.1 b4.2 b4.3 b4.41

where bi.1 = 24ub2+80bcd+b5, b1.2 = b2.1 =

b1.3 = b3.1 =

b1.4 = b4.1 =

152 9

64 C2b2 3

--

64 --d2b2+14/3b4~+48~b~+32d~2, 3

+ 1413b4d + 48ubd + 32 d2C,

32 3

32 3

-b3dc - - bc3 - - bd3 + 3 ub4 + 48 udc + 119 b7, 112 bcd 3

b2.2 = --

64 17 +b3c2 - 24 b2du - - b5d, 9 9

152 32 32 dc - - bc3 - - bd3 9 3 3

b2.3 = b3.2 = -b

b2.4 = b4.2

32 3

- -106 c2 b4 --c3d+-b6d+-b 17

27

27

112 bdc2 3

b3.3 = --

+ 3 ub4 + 48 UdC + 119b7,

= b3.4 = b4.3

176 2 cd2 +8ub3d-16ubc2, 9

64 17 +b3d2 - - b5c - 24 b 2 a 9 9

245 32 32 56 1 16 bC2d2- - c3b3- -b d - ub2dc+ 113Ub6 - b9. 81 9 9 3 81 After Theorem 4.1 the signature of this matrix gives us the Euler characteristic of real algebraic sets defined by F ( z ,s ) >, 5 ,= 0. We calculate the determinants of these matrices. b4.4 = -b5cd

+

+

+

+

det(BH) = 1/9 (256 b2d3 768 d2c2 96 b4dc - b8 det(C(s)) =

+

+ 256c3b2)’ ,

773

1 23 4 2 2 11 8/3 b2c4d- 243 bscd+8/3 d4cb2+- b d c +32 ubc2d2- - ub5cd-30 u2b2dc 27 9 --

243

1 b6c3 - 32 d3c3 b6d3 - 243 9

1 ub9 + 24u2d3+ 1/3u2b6+ 9u3b3 + 243

20 16 16 _-20 uc3b3 - ub3d3 + 24 c3u2 + 81 u4 + - d6 + -c6

9 9 9 9 The discriminant of the polynomial det(C)(s) with respect to the variable u is calculated as follows, Dscrirn(det(C), u ) = 27 (d - c)'(d2

+ dc + c2)'(256 b2d3+ 768 d2C2+ 96 b4dc- bs + 256 ~ ~ b ' ) ~ .

These results combined with the Proposition 5.2 calculate the Maxwell set,

M

= {S E

C3; (d - c)' (d2

+ dc + c2)2 = 0).

Example 2. The versa1 deformation of the singularity We consider the following deformation, F(z,y,t)

E6.

+ u = z3 + y4 + g q 2 + dy2 + a y + by + a x +u.

with t = (a, b, c, d, 9). As F ( z ,y, 0) is a quasihomogeneous polynomial in (2, y), we attribute to the deofromation parameters (u, t ) E S corresponding quasihomogeneous weights. This means that there is a C* action on the space of deformation parameters S. This allows us to consider 2 = C2 , S = C6 in the arguments of §4. Thus we deal with the global parameter values t E C5.Essentially all the informations on the multiplication table (4.3) are contained in the following equivalence relations, 2' E -1/3gy2 - 1/3 CY - 1 / 3 ~ mod(d,F(z, 9, t), d,F(z, y, t)) y3 E ( - 1 / 2 g y - l / 4 c ) z - 1 / 2 d y - l / 4 b s 2 y = (1/12gc+ 1/6g2y)z+ 1/12gb- 1/3cy2 (1/6dg- 1/3a)y z2y2 E (1/6g2y2 1 / 4 g ~ y +1/12 C')Z (1/6 dg - 1/3 a)y2 (1/12gb+ 1/6 cd)y 1/12 cb zy3 E ((-1/2d- l/12g3)y- 1/24g2c-1/4b)z+1/4gcy2+(1/6ga+ 1/12 c2 - 1/12 g2d)y - 1/24g2b 1/12 ca z2y3 E (1/4gcy2 ( 1 / 3 g ~- 1/3g2d 1/12c2 - 1/36g5)y - & g 4 c + 1/6 ca - 1/ 12 g2b- 1/ 12 dgc)z (1/ 12 gb+ 1/6 cd 1/ 12g3c)y2 ( -1/36 g4d 1/18g3a+1/3ad+1/36g2c2 - 1/6d2g+1/12cb)y+1/36g2ca- 1/12dgbg4b 1/6 ab

+

+

+

&

+

+

+

+

+

+

+

+

+

+

774

zy4 3 ((-l/12g3-l/2 d)y2+(-1/4 b-1/6g2c)y-1/16gc2)z+(1/12 1/12g2d+1/6ga)y2+ (-1/24g2b- 1/8dgc+1/12ca)y- 1/16gcb z2y4 G ((1/12c2

c2-

+ 1/3ga - 1/36g5 - 1/3g2d)y2+

11 (--g4c144

1/8g2b+1/6ca-

7 -dgc)y 24

-1/12 gcb - 1/32 g3c2 - 1/24 c2d)z

+(1/3 ad - 1/6 d2g +(-1/12c&

13 + 1/18g3a - 1/36g4d + 1/12 cb + g2c2)p2 144

- 1/16g3cd+ 1/6ab- 1/8dgb+ 1/48gc3+

-1/48 gb2 - 1/24 cdb

5 72

1 72

- g2ca - - g4b)y

+ 1/48 gc2a - 1/32 g3cb.

We can write down these results in the form of matrices (4.4) and the polynomials &(t),1 I k 5 6, (4.6), 1 0

0

0-" 3 0 0

0 0

0 % 0 0

0 -b

0 h 12 0

4 12 G-b? 12 12 24

obc

r2 -

p1

bc 12

--4b

12

..-Lid

0

0 0 12

Pl

12 24

0 -& 16

bcd 24

-!!9 16 & I ac2g 48 48

bcg3 48

775

73

0 0 1 =

1 0 0 0 0 - $ + $ 0 $+kJ 0 o - - d2

0 -f

0 O - 3" + & 6 o 0 0 --d2

?+%

7-1

7-3

7-1

--

7-2

7-3

7-2

7-4

o 3 + +j 7-1

:

where c2 + ag - dg2 =-

12

7-2

bc

7-3 = -

12

ac 12

6

12 '

cdg 8

bg2 24 '

= - - -- -

ad - d2g +-c2g2 + -ag3 +--

6

12

r 4 = -a-b- cd2 + -c3g -12 12 48

36

72

dg4 72 l

bdg 12

r4

0 75

0

0 = 0 0

-f

1

0

0 -2

0 0 1 0 0

0 --C 3

0

-$+$ 9

0

1 0

0 % 7 !

4

--d

2

Y

cd+!Y+d 6 12 24 cZ+%L&L 12

6

95

12

776

where b c a d d2g 11c2g2 q5 = - + - - -+-+--12 6 12 144 <5(t) =

6

0 0 0 0

0

0 0 0 0 0 0 0 1

1 0

-2d-

6

1

c 0

0 0 1 % 0 0 1 0 0 1 ho92-d-C

?

-;

4

2

-4 - L 13 6

2+EL&L-L. 6

1212

5c2 + 2ag - dg2 - g5 C6(t)= 12

dg4 72 '

-.g3

7 -

6

ag3 36

3

2

Finally we get the matrix (4.5) as follows.

36'

5

72

777

It is a conceptually easy exercise to calculate further B H(s) and B H F(s) to establish correspondence between parameter value s = (a,b, c, d, g, u ) and the Euler characteristic of a semi-algebraic set defined by F ( z ,y,t ) u. For instance, for the values

+

-0.6

5 a 5 1,( b , c , d , g , u ) = (-0.4,0.1,0.1, -0.1, -lo),

we calculate with computer (Mathematica computation achieved by Galina = x(Wo) -

-1 < a < -0.8,(b,~,d,g,~)=(-0.4,0.1,0.1,-0.1,-10), we have x(W20) = l,x(W50) = -1. For the values -1

< a < -0.8,

( b , c , d , g , u ) = (-0.4,0.1,0.1, -O.l,8.5),

we have x(W20)= - 1 , x ( W g ) = 1, and -0.6 5 a 5 1,( b , c , d , g , u ) = (-0.4,0.1,0.1, -0.1,8.5),

we have x(W20) = O,x(W
+, +,

+, +,

References 1. V.V.GORYUNOV, Projections and vector fields that are tangent to the discriminant of a complete intersection, Funct. Anal. Appl. 22 (1988), No.2, pp. 104-113. 2. V.V.GORYUNOV, Vector fields and functions o n the discriminants of complete intersections and bifurcation diagrams of projections, J . Soviet Math. 52 (1990), no. 4,pp.3231-3245. 3. K . S A I T O , O n the periods of primitive integrals I, RIMS preprint 412, K y o t o U n i v . 1982. 4. J.W.BRUCE,Functions o n discriminants,J.London Math.Soc. 30 (1984), pp.551-567. 5. ZSZAFRANIEC,O n topological invariants of real analytic singularities,. Math. Proc. Camb. PhilSoc., 130 (2001), pp.13-24. 6. O.YA.VIRO,Real plane algebraic curves: constructions with controlled topology, Leningrad Math. J . 1 (1990), no. 5, 1059-1134 7. I. DE GREGORIO, Deformations of functions and F-manifolds, to appear i n Bull.London MathSoc.

778 8. E.LOOIJENGA, Isolated singular points o n complete intersections, London Math. SOC.Lect. Notes Ser., 1984, No. 77, 200pp. 9. K.SAITO, Theory of logarithmic differential forms and logarithmic vector f i e l d s , J.Fac.Sci. Tokyo, Sec. I A Math.,27 (1980), pp.265-291. Nonisolated Suito singularities,. Math.USSR Sbornik 10. A.G.ALEKSANDROV, 65 (1990), No.2, pp.561-574. 11. H.TERAO,The bifurcation set and logarithmic vector fields, Math. Ann., 263 (1983), No.3, pp.313-321. Ideals of differentiable functions, Tata Institute of Funda12. B.MALGRANGE, mental Research Studies in Mathematics, No. 3 Tata Institute of Fundamental Research, Bombay; Oxford University Press, London 1967, vii 107pp. 13. K.SAITO,Period mapping associated t o a primitive f o r m ,Publ. RIMS Kyoto Univ.,lS (1983), pp.1231-1264. 14. R.C.HWA and V.L.TEPLITZ,Homology and Feynman integrals, W.A.Benjamin, Inc., 1966. 15. V.A.VASSILIEV, Ramified integrals, singularities and Lacunas, Kluwer Academic Publishers, Dordrecht, 1995. 16. PH.GRIFFITHS and W.SCHMID, Recent develpments in Hodge theory, in Discrete subgroups of Lie groups and Applications to Moduli, Oxford University Press, 1973, pp.31-127. 17. G.-M.GREuEL, Der Gaug-Manin-Zusammenhang isolierter Singularitaten von vollstandigen Durchschnitten, Math. Ann. 214 no.3, (1975), pp. 235-266. 18. LB DUNG TRANG, Calculation of Milnor number of isolated singularity of complete intersection,. Funkts. Anal. Appl. 8 (1974), No.2, pp.45-52. 19. S . T A N A BTransforme'e ~, de Mellin des inte'gmles- fibres associe'es aux singulurite's isole'es d'intersection compldte quasihomogdnes, Compositio Math. 130(2002), no.2, pp. 119-160.

+

779

Some surface singularities obtained via Lie algebras K. Nakamoto* and M. Tosun University of Yamanashi, Yamanashi, Japan *E-mail: [email protected] M. Tosun Yildiz Technical University and Feza Gursey Institute, Istanbul, Turkey E-mail: [email protected] In this work, we present a relation between a special six dimensional Lie algebra and the simple elliptic singularities of type 85.

Keywords: Simple elliptic singularities, Lie algebras.

1. Introduction

The very well known connection between simple singularities of surfaces (in the literature, also called rational double points, Du Val singularities or Kleinian singularities) and the nilpotent varieties of simple Lie algebras was conjectured by A. Grothendieck 3 and solved by E. Brieskorn 1. A very nice consequence of that connection is to construct semi-universal deformations of simple singularities by using the corresponding Lie algebras (cf. 1, 11). To get a similar relation for the simple elliptic singularities of surfaces, defined in 8, K. Saito constructed the elliptic root system (see ?. Since then, the simple elliptic singularities of type I&, E, and 1!?8 and, their semi-universal deformations are studied by many mathematicians. Here we establish a relation between a special six dimensional Lie algebra and the simple elliptic singularities of type & and construct semiuniversal deformation of this singularity by using the geometry of the corresponding Lie algebra.

780

2. Simple and simple elliptic singularities

A surface singularity is a germ ( S ,0) defined as (S,O) = ({z E

-

cNI fl(.)

=

. . . = fk(Z)= O } , O )

where, for each i, fi : CN C is a germ of a holomorphic function. Assume that (S,O) is a normal surface singularity (means the local ring OS,Ois normal). A resolution of ( S ,0) is a map 7r : X ( S ,0) such that (2) the surface X is nonsingular, (ii) the map 7r is proper, (iii) the restriction of 7r to n-l(S - 0) is an isomorphism. A resolution is called minimal if any other resolution of (S,O) factorizes via this resolution. Minimal resolution exists and unique. The fiber 7 r - ' ( O ) is called the exceptional divisor of IT which is, by Zariski's Main theorem, connected and 1-dimensional. The singularity of (S,O) is called simple singularity if and only if it is defined by one of the following equations:

-

+ y2 + z2 = 0, + zy2 + z2 = 0, X4 + y3 + =0 x3y + y3 + z 2 = 0 z5 + y3 + z2 = 0

A , zn+' D, xn-l E6 E7 Es

2 1) ( n 2 4)

(TI.

Z2

By 2, the exceptional divisor is, in each of cases above, a union of projective lines intersecting each other transversally and the dual graph of the minimal resolution is the Dynkin diagram having the same name. The second class of singularities of surfaces which are relatively simple among other singularities than those given above, is the simple elliptic singularities defined by K. Saito in 8. A singularity of (S,O) is called simple elliptic singularity if and only if it is defined by one of the following equations:

EB

+ + z2 + xzyz = 0 + + z2 + xsyz = 0 x3 + y2 + xzyz = 0

&

x2+y2+xzw=o

E6 x6 y3 E7 s4 y4

zy

+ z2 + w2 = 0

where A E @. such that these equations define an isolated singularity. For

781

each type, the exceptional divisor is a nonsingular curve with genus 1 and, it has self-intersection -1, -2, -3 and -4 for &, E 7 , E g and D5 respectively. Let us construct semi-universal deformations of the singularities given above: A deformation of a variety (V,v) is a map q5 : ( X , p ) (T,t ) such that q5 is a flat morphism with (+-'(t),p) (V,w). A deformation q5 of (V,v) is called semi-universal deformation if and only if any deformation 4' : ( X ' , p ' ) (T',t') of (V,u ) can be induced, up to isomorphism, from q5 by a base change p : (T',t') -+ (T,t ) , and the tangent map dp is uniquely determined.

-

-

Theorem 2.1. (see 10) For any isolated singularity, there exits a semi-

universal deformation. Let us first construct semi-universal deformations of an isolated hypersurface singularity ( S ,0): The vector space

T 1 := C(z1,. . . , zn}/ < f,a f / a z l , .. . , af/azn>

,

is finite dimensional. Let

91,. . . g k

x = {(zl,. * . ,z n ) x (tl?

* 7

be a basis over

C of this space. Then k

*

tk)

E CnX(Ck 1

f ( 2 1 , . . z n ) + c tigi(z1,* . 7

.zn)}

i=l

is the semi-universal deformation of ( S ,0). Now we want to construct semi-universal deformations of a singularity of type 0 5 . More generally, we assume that (S,O) is an r-dimensional isolated complete intersection singularity defined by n - r equations. Hence we consider a free module C { q , . . . ,zn}n--rover C{z1, . . . ,zn}with the basis el = ( l , O , . . . , O ) t , . . .en+ = (O,O, . . . , l)t and, the basis over C of the finite dimensional vector space

T 1 := C{z1,. . . , x ~ } ~ - ~ / I where

I =< fiej, ( a f i / a z l , . . , afn--r/dz1)t7.. . , ( a f l / a z n , . .. , with 1 5 i , j I n - r. It can be easily proved that

af,-,/a~,)~ >

Proposition 2.1. (see 12) For each D5-singularity, dimT1 = 7.

In the next sections, we will obtain semi-universal deformations of the simple and simple elliptic singularities by using the corresponding Lie algebras.

782

3. Semi-universal deformations via Lie algebras The reader can find the details on the results of this section in the excellent book of P. Slodowy (11). Let G be a semisimple complex Lie group and g be the corresponding Lie algebra over C, i.e. g = T,G with e E G neutral element. Let fj be a Cartan subalgebra of g of dimension T . On fj, we have a natural action of the Weil group W . We have the following result due to Chevalley:

--fjw

Theorem 3.1. (cf. 11) The homomorphism of restriction C[g] induces a n isomorphism of the algebras C[gIG C[fjIw. Hence we have the embedding C[fj]"

7:

-

C[b]

C[g], so a natural morphism

This map is called the adjoint quotient map. Moreover, C[gIG is generated by r homogeneous G-invariant polynomials 71,. . . ,yr. Then we obtain a polynomial map g 5

++

C' (71(5)>..-,yr(X))

Theorem 3.2. (see 4 , l l ) W i t h the preceding notation, each fiber of y (i) consists of finitely many orbits. (ii) has codimension r in g. (iii) contains a unique regular orbit which is dense in the fibre. Note that an element of g is called regular if its orbit dimension is maximal among all G-orbits. This is equivalent to say that the centralizer of a regular element in g is of minimal dimension. This gives that the regular orbits have codimension 2 r+2 in g. The elements in g whose G-orbits is of codimension T 2 are called subregular.

+

Definition 3.1. The fiber y-l(y(0)) is called the nilpotent variety of g. We will denote it by n/(g). For example, when g = sl(2,C), the nilpotent variety N(g)is isomorphic to { ( a ,b,c) E C3 I a2 bc = 0).

+

+

Definition 3.2. A local submanifold S c g of dimension T 2 is called a transversal slice if S intersects N(g)transversally at the subregular element of g.

783

Theorem 3.3. (see 1) Let g be a Lie algebra of type A,, D,, E6, E7 or Eg . With preceding notation, (i) The intersection N(g)n S is a surface singularity of the same type as 8.

(ii) T h e restriction of y t o S is a semi-universal deformation of the singularity of N(g)n S.

A detailed proof of the theorem can be found in 11. Then the natural question arising from Brieskorn’s theorem is whether N(g)nS for a finite dimensional Lie algebra g has always a nice singularity or simply, when it has a singularity. For example, when g = d ( 2 , C ) ,the nilpotent variety N(g)has an Al-singularity at 0 or, when g = d ( 2 , C )@C, N(g)is isomorphic to { ( a ,b,c) E C 3 I a2 bc = 0 ) x C, so it has a cDVsingularity at (0,o) , i.e. there exists an hyperplane section having a simple singularity at (0,O) (see 7). Here we will deal with the six dimensional Lie algebra g = d ( 2 , C ) @ d ( 2 , C ) :The nilpotent variety N :=N(g)of g is

+

N = { ( ~ ~ a ) ( a 2 + b c = O x}

{ (af_p;l)/d2+ef

1

=o .

The Lie group G = SL(2, C)x SL(2, C ) acts on N and N can be decomposed into G-orbits in the following way:

For p E N , let us take a 4-dimensional affine subspace S of g passing through p . We say that S is a generic slice at p if its intersection with N at p gives an isolated singularity (see 6 for a detailed definition of genericity). In the sequel, we denote by S a generic slice at the given point. It is obvious that if p E O:egx O:eg,then the surface N n S is non-singular at p and, if p E O:egx ( 0 )or p E ( 0 )x O:eg,then N n S has an Al-singularity at p . Proposition 3.1. (see 6) With preceding notation, let p = (0,O). The surface singularity ( X ,0) := ( N n S , 0 ) i s a simple elliptic singularity.

Proof. Let transform

3 be

the blowing up of S

C4 at 0. By taking the strict

x of X , we have

P3CS+S

u

u

u

Ec%-,X,

784

where E is the exceptional curve. Since X is defined by two quadratic equations in S , the exceptional curve E will be defined by two generic quadratic equations in P3. Hence E is an elliptic curve. Moreover, the pullback by p : + X of the divisor divcp of a generic linear function cp on S can be written as p*(divcp) = E C , where C is the strict transform of divcp. Since (p*(divcp) . E ) = 0 we obtain E 2 = -(C . E ) . By considering the intersection of E and the hyperplane associated t o C in P3, we obtain that E . C = 4. Therefore E 2 = -4. 0 Now let us consider a special transversal slice SO:= {c = d e , f = a b } c g and denote ( X O0) , := ( N n SO, 0).

+

+

+

Remark 3.1. It can be easily seen that the singularity (X0,O) is of type Ds. Now we want to construct semi-universal deformations of ( X O 0): , A Cartan subalgebra g of g is defined as

h : = { ( a 0 -ao ) } @ { ( d 0 -dO

) } .

and the adjoint quotient can be regarded as 4

lJ/w c2

where W is isomorphic to 2/22@ 2/22.Let us deform the adjoint quotient x by ( a , @E) C2. So, we define f(a,o)as

When ( a ,0) = (O,O), we have f(0,o) = x. Now let us deform the slice SOby ( y , b , ~E) C3, we define the slice S(Y,+) as S(?,6,€):= { c = d

+ e + y,f = a + b + 6e +

E}.

For (y, 6 , ~ = ) (O,O, 0), we have S(O,O,O) = SO.

Theorem 3.4. (see 6) With preceding notation, consider

s := c2x c3x g/w = { ( a , @E) CZ} x { ( y , 6 , e ) E C 3 } x { ( X , p ) E Q/W}.

785

Let X be the family of surfaces on S defined as

X

:= { ( X ,a,P, Y,6, € 7

A P) E B

x

s I f ( a , P ) ( X )= (4P ) , x E S(y,S,,)}.

Set o := (O,O,O,O,O,O,O) E S and q := ( 0 , o ) E X . Then the morphism of germs ( X ,q ) 4 (S,o) gives a semi-universal deformation of ( X 0 , p ) . Proof. Since f ( a , p ) ( X = ) (-a2-bc-ae, -d2-ef -pb) and c = d+e+y, f = a b be E for X E S(r,6,E), the family X is defined by

+ + +

+ bd + be+yb+ ae + X = 0 := d2 + ae + be + be2 +Ee +,Bb + p = 0. f 1 := a2

f2

The coordinate ring O X , of ( X O , ~is)isomorphic to C{a,b,d,e)/(gl,gz), where g1 = a2 bd be and g2 = d2 ae be. The (C-vector space T 1 of ( X 0 , p ) is defined as T1 = O$,/M, where M is the Ox,-submodule of

+ +

+ +

O$, generated by the 4 vectors: (-,a91 -), ag2 892 (-a g 1 -). de ’ de

da

da

(-,dgl -), dg2 db db

(-a91 -) dg2 and d d ’ dd

Note that we have

( f i ,f 2 ) = (91,g2)+a(e7O)+P(O,

b ) + ~ ( bO)+b(O, , e2)+E(0,e ) + W , o)+P(o, 1).

We can verify that 7 vectors appeared over ( e ,0), (0, b), (b, 0 ) , (0, e 2 ) ,(0, e ) , (1,0), and ( 0 , l ) form a basis for T1. Hence ( X ,q) 4 ( S ,o) is isomorphic to a semi-universal deformation of (X O p, ) . 0 Now we want to construct semi-universal deformation spaces for a general transversal slice: For this, consider the space Aff (g,4) of all 4dimensional affine subspaces of g. Since any 4-dimensional affine subspace of g can be described by two linear equations, Aff (g,4) is embedded in the Grassmann variety Grass(dimg 1,2) = Grass(7,2). The space of all 4-dimensional linear subspaces Grass(g,4) of g is a closed subvariety of Aff(074). By Proposition 3.1, N n S gives us an &-singularity for a general S in Grass(g, 4). Then we obtain:

+

Theorem 3.5. (see 6) Let S be a general element of Grass(g,4). Let S, be a “general” 3-dimensional subvariety passing S of Aff(g, 4). Set

s := (C2 x s, x lj/w = {(a,/?) E (C2} and

x { 7 E S,} x { ( X , p ) E b/W

=

(C2}.

786

Then the m o r p h i s m of g e r m s ( X , q ) + (S,o) gives us a semi-universal deformation of ( N n S,p), where o = (0,0 , S , 0,O)and q = (0,o).

Prooj The condition that any 7 vectors are linearly independent in T 1 = O%ns/Mis open for S E Grass(g, 4). Since the condition that a given family becomes a semi-universal deformation is open, we can choose a suitable 0 3-dimensional subvariety of Aff ( g , 4 ) .

References 1. E. Brieskorn, Singular elements of semisimple algebraic groups Actes Congr6.s Int. Math., 2, 279-284, 1970. 2. P. DuVal, O n isolated singularities which do not affect the conditions of adjunction Proc. Cambridge Phil. SOC.30, 453-459, 1934. 3. A. Grothendieck, Seminaire C. Chevally: Anneaux de Chow et Applications (Sec. Mathematique, Paris). 4. B. Konstant, Lie group representations o n polynomial rings Amer. J. Math. 85, 327-404, 1963. 5. E.J.N.Looijenga, Isolated singular points o n complete intersections London Math.Soc.Lect. Note Series 77, 1984. 6. K. Nakamoto-M. Tosun, Semi-universal deformation spaces of some simple elliptic singularities submitted for publication, 2005. 7. M. Reid, Canonical 3-folds In GhmBtrie Algebrique Angers 1979 (A. Beauville, ed.) Sijthof - Noordhoff, 273-310, 1980. 8. K.Saito, Einfach-elliptische singularitaten Invent. Math. 23, 289-325, 1974. 9. K. Saito, Extended a f i n e root systems I( Coxeter Dansformations) Publ. RIMS, Kyoto Univ., 26 (1990), 15-78. 10. M. Schlessinger, Functors of Artin rings Trans. Amer. Math. SOC. 130, 208222. 11. P. Slodowy, Simple singularities and simple algebraic groups Springer Lecture Notes in Math. 815, 1980. 12. G. N. Tjurina, Locally semiuniversal flat deformations of isolated singularities of complex spaces Izv. Akad. Nauk SSSR, Ser. Mat. 33, No. 5, 967-999, 1970.

787

MAXWELL STRATA AND CAUSTICS

M. VAN MANEN* Department of Mathematics, Hokkaido University Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan Email: manenOmath.sci. hokudai.ac.jp

We review the notions of cut-locus/medial axis and conjugate locus/caustic. An effective way of computing Jacobi fields is presented. We show that the combination of caustic and symmetry set can be approximated by what we call a weighted symmetry set.

1. Introduction

This is a review on related notions, that are often treated separately.

I Cut-locus and conjugate locus

II Medial axis and caustic For I start out with a compact manifold M , a point p E M , and a Riemannian metric g on M . Each q # p defines a path space Rpq. It consists of the mappings

y: [0,1] -+ M that are continuous, piecewise smooth, and that have y(0) = p , y(1) = q. On R,, we have a length, namely the length of the path with respect to the metric g on M . We also have for each q an energy function ( the squared length ) :

E, : R,,

-+

R

The closure of those q E M for which E, has a nonunique global minimum is the cut-locus Cut(M,p) of p E M . There are also q for which Eq has a degenerate critical point. Such q are called conjugate points. *Work partially supported by JSPS Grant-in-aid for young scientists B nr. 17740028

788

For II fix an embedding $ of a manifold M in R". The function fz: M

--+

R

sH

112

- $(s)112

family of squared distance functions to M , parameterized by a point x in R". Given M , the closure of the set of those x in R" for which fz has a nonunique global minimum is called the medial axis. Those x for which fz has a degenerate critical point make up the caustic. The general scheme is as follows. You have a family of functions, parameterized by some manifold B. For most b E B the function fb is Morse. A function is Morse when it satisfies two conditions: all of its critical values are distinct and all of its critical points are nondegenerate. The closure in B of those b for which global minima are not unique is in the setting I the cut-locus, and in the setting II the medial axis. Such strata are known as Maxwell strata. Those b E B for which a critical value is degenerate lead in setting I to the conjugate locus, and in setting II they form the caustic. In this review article we pay a lot of attention to computational issues and to the properties of these objects when the metric g is generic, or, in the case of medial axes, when the embedding $ is generic. New results are the method to calculate Jacobi fields in section 6.2 and the notion of a weighted symmetry set in the last section. D. Siersma first asked what a weighted symmetry set would look like and the author gave the answer. The weighted symmetry set gives more insight into the relation between caustics and Maxwell strata. The cut and conjugate locus are treated in sections 2 to 11. From section 12 onwards the topics dealt with are the medial axis, the caustic and what treasures are hidden in 2. The energy functional on the space of paths

Let (M,g) be an m-dimensional Riemannian manifold. Geodesics in M are locally shortest paths between two points. They are the solutions of a system of differential equations that we shall now derive. Write g = (gij(x))l
So the energy is half of the squared length of a path: paths of minimal energy are also paths of minimal length. One finds the paths of minimal

789

length using the Euler Lagrange equations. These are

Instead of directly using this equation we will ponder a little over how it is derived in this special case. We view E as a function on the space of paths with fixed end points y(0) = p and y(1) = q. So all the variations of y will have the same end points p and q . Minima are critical points. We need to find the critical “points” of the energy functional on the space of paths. The LLcoordinates’’ on the space of paths are y and ;U. The first variation 6E is

One calculates the second term separately using partial integration:

Hence 6E = 0 is equivalent to:

With the usual definition of the Christoffel symbols

the geodesic equation is:

-&

=-

C rFj?ii.jfor IC = 1,.. . ,m

(2)

i,j

From an example it is clear that a locally shortest path is not necessarily globally a shortest path. The simplest example is of course the sphere. But on any closed surface there are many geodesics, that are not shortest. A minimizing geodesic on a Riemannian manifold M is a geodesic fro= p to

790

q such that there is no geodesic that is shorter, or equivalently whose value of the energy functional is lower. Geodesics are critical points of the energy functional on the space of paths. Minimizing geodesics are global minima of the energy functional. The space of paths is not a manifold. However, our reasoning is allowed because S E is well-defined. In fact there is a well defined tangent space T,f12,, along a path y : [0,1] + M . The tangent space consists of piecewise smooth - also called “broken” - vector fields W along y, that satisfy W ( 0 )= W(l) = 0. These vector fields are infinitesimal perturbations of the path y. The first derivative 6E only depends on these infinitesimal perturbations.

3. The cut-locus

It might happen that for some p , q E M there are two or more minimizing geodesics from p to q. Here the simplest example is again the sphere. Between the north pole and the south pole there are infinitely many minimizing geodesics. For a Riemannian manifold M and a point p E M the cut-locus Cut(M,p) of p in M consists of the closure in M of those points q E M such that there are at least two minimizing geodesics from p to q. Examples: (1) The cut-locus of any point on a sphere is just the opposite point. (2) The cut-locus of a point on a torus with a generic metric consists of two loops, that form a figure eight. See figure 1. (3) The cut-locus of a point that is not an umbilic on an ellipsoid is a line segment.

Figure 1. A cut-locus on a torus.

Cut-loci are important in topology because of the following theorem, see Klingenberg2’ :

79 I

Theorem 3.1. If M is a compact manifold then C u t ( M , p ) is a strong deformation retract of M \ { p } . Cut-loci are notoriously hard to calculate: the geodesic equation can only be solved explicitly in some special cases. Of these we mention the surfaces of revolution. These are surface of the form 2 = f ( r )cos(8)

y = f ( r )sin(8) z = g ( r )

In most cases though we can only use a computer. Here a warning is in place. The geodesic equation is solvable for some interval t E [0, €1, according to the general existence theorem for ordinary differential equations. But it is by no means assured that the solutions exist for all t E R. If they do then we say that the manifold M is geodesically complete. Recall that a metric space is complete if every Cauchy sequence converges. The manifold M is a metric space if we put the distance between two points to be with E the energy of a minimizing geodesic. Here's the Hopf-Rinow theorem, see also S ~ i v a k ~ ~ .

dw),

Theorem 3.2. If the solutions of the geodesic equation at one point in M exist f o r all t , then they exist at all points in M f o r all t. A Riemannian manifold is geodesically complete i f and only zf it is complete as a metric space. We will assume from here that all manifolds we consider are compact. As metric spaces compact manifolds are complete. The Hopf-Rinow theorem tells us that they are also geodesically complete. The numerical approximation of cut-loci on 2-dimensional surfaces was carried out in Itoh and Sinclair17. The cut-locus C u t ( M , p ) can also be considered as a subset of T,M. The closure of the set {U E

T p M Iy(0) = p y(0) = u y is a nonunique minimizing geodesic from p to y( 1))

is called the tangential cut-locus. 4. The geodesic flow

For computer implementation the geodesic equation can not be applied directly: there is no coordinate patch that covers the whole manifold. A technique of Perelomov can be used to calculate geodesics in an important special case.

792

Assume that the manifold M is given as the zero set of a Coo function F : R" -+ R. In TR" the equations for a geodesic in R" with the Euclidean metric are xi:=u ir=o

(3)

where u are the coordinates in the fiber of TR" --+ R". produces curves that have

Equation (3)

1 1 ~ 1 31 ~ Constant

&

(ll~11~)= 2(u,ir) = 0. However they do not remain on the because manifold F ( x ) = 0. To get the right equations we have to use a Lagrange multiplier technique. The functional that has to be minimized is not

but

where we vary over paths with fixed end points.

Theorem 4.1. Geodesics on a manifold R" 3 M = {x I F ( x ) = 0 } , where M has the metric induced from the Euclidean one on Rn, are found by integrating:

Remark 4.1. From this point on we will use the notation

a2F

d2F

uiuj

and similar ones, in order t o avoid a large number of indices.

Proof. [Proof of theorem 4.11 We compute

b

(IL 0

1"

(

(vb(%)

~ - X F~( x ) ) d~t )

lL

=u

(~611~ ~

~

793

So we get

It remains to determine A. We have

F(x) = O

+- -ddXFX =

O+

d2F dF = 0 + -VV+ 8x2 dX

-V

aF =O dX

-V

from which we derive that equation (6) becomes as in PerelomovZ6,and in

(5).

0

Remark 4.2. Theorem 4.1 is more or less due to Perelomov. The proof is ours, although we do not exclude that this type of reasoning can be found elsewhere. Equation (5) has the advantage that it uses “global coordinates”. So it can be used to calculate geodesics without having to change coordinates when the geodesics walk off a coordinate patch, or it can be used to calculate geodesics on a surface, whose implicit equations we have, but whose parameterization is not provided. Equation (5) can be derived in another way. We know that geodesics on an embedded manifold - with the metric induced from the ambient space - are the curves whose geodesic curvature is zero. So the curvature of a geodesic is the normal curvature, and thus the second derivative with respect to t is everywhere orthogonal to the embedded manifold. In other words, V is a multiple of the gradient of F . For those who remain sceptical about (5) there is an easy way of checking the correctness. Namely, using the program supplied with Gray13, the geodesic equation (2) is quickly calculated. One can then compare it with (5) if one has both the implicit equation as well as the embedding of a manifold available. We can generalize this framework rather easily. In the most general situation we have the manifold as a zero set of a function F : R” --f Rk. A theorem of Nash says that any Riemannian manifold can be realized as such. We minimize the functional

and we put x = v. Finding the X i = & ( x , v) ( 1 5 i 5 k ) is done as in the previous case by differentiating Fi(x) = 0 ( 1 5 i 5 k ) with respect to t

794

twice. Solving the resulting equations involves inverting the matrix:

-dFdFT ax ax

(

Earn ... 2ELm a x : a x ... a x ! a x )

mEa ...

ax

ax

mm ax ax

It can easily be proved that this matrix is invertible. But there seems t o be no nice expression for the inverse. So, the formulas for X i = Xi(z, v) are rather ugly. We only want to work with manageable formulas, so we restrict to the above case, in theorem 4.1. 5. Covariant differentiation and curvature Recall the basic theorems about the Levi-Civita connection. In each point p E M a vector field V determines a geodesic yv by putting qv(0) = V(p), and yv(0) = p . Covariant differentiation is the operation that assigns to two vector fields V and W in T M a new vector field VvW according to the rule:

Using covariant differentiation define the Riemann curvature tensor R, see S ~ i v a kchapter ~ ~ , 6:

R(V,W ) = [VV, VWI - V[V,W] (8) The Ftiemann curvature tensor is a (3,l)-tensor. It is a called a tensor because it is linear over the Cw functions on M . It is called a (3,1)-tensor because we have to feed it 3 vectors Vl, V2 and V3 to get R(V1,V2)V3which is an element of the dual of the tangent space. Its fundamental significance is twofold: (1) It is preserved under isometries. (2) If R = 0 then the manifold is isometric to Euclidean space.

If we have a geodesic y c M we can define the covariant derivative of a vector field along a curve using the formula (5). Theorem 5.1. The covariant derivative of a vector field in T M along a geodesic is

795

Proof. First of all we need to show that D V/ d t is again a vector field in T M . For this we need to prove that BFDV =o ax d t For that we differentiate the identity with respect to t: c -

whereas

So that (11) gives zero because of (10). To prove that this is really the Levi-Civita connection, we recall that the Levi-Civita connection is the unique operation satisfying: D af D -(fV) = -V+f-v dt at dt D D D -(V W) = -v --w dt dt dt a D D -(VW) = (-v, W) (,,W,V) at dt and such that the covariant derivative of the tangent along a geodesic is zero. One readily verifies that these properties hold, so that equation (9) indeed represents the Levi-Civita connection on T M , in our special case.D

+

+

+

Though the author found (9) independently it can be found in almost the same form in Linn6rZ1. There the expression (9) is used to get an expression for the base of the tangent space to the unit tangent bundle. Suppose that the vector fields V and W in equation (8) are tangent vectors to some patch of a surface embedded in M at p by a map s : R2 -+ M , s(0,O) = p . Let (tl,t,) be coordinates on the left hand side. They correspond to vector fields d/&l and d/dtz. We have

0 = s*

[-a

-1

d

at, ' at2

= [.*-,s*-

at, a

7

at,

So that

a

a

at,

at,

R(s*-,s*-)V=

--

The right hand side of equation (12) can be calculated using the extrinsic coordinates, when M is given as the zero set of a function F: R" 4 R.

796

6. Jacobi fields

This section contains some horrible calculations. We quote from the book Bergerl, page 204: “DOnot despair if the curvature tensor does not appeal to you. It is frightening for everybody. We hope that after a while you will enjoy it.”. Along a geodesic there exist so called Jacobi vector fields. A Jacobi vector field W along a geodesic y has initial values

DW

W(y(0)) = WOand - = W, d t t=o and its values for other points on the geodesic are determined by the differential equation

The Jacobi vector fields form a 2 dim(M) dimensional vector space, because they are completely determined by their initial values. There is also a symplectic structure WJ on the space of Jacobi vector fields:

wJ((WO,Wl), (WA, W:))= (WO,Wi) - (wl,WA) where (., .) is the inner product coming from the Riemannian metric on M . Next, remark that if W(t) is a solution to the equation (13) then also (u bt)? W(t) is a solution. Indeed,

+

+

+

D2 W - D2(W (u + bt)j) = R ( j ,W (U bt)?)? = R(?,W)? d t2 d t2 On a 2-dimensional manifold M the Jacobi equation becomes much simpler. The only interesting Jacobi fields are orthogonal to the geodesic. When the tangent space T p M is two dimensional there is thus only one interesting Jacobi field. For a geodesic the vector field orthogonal to i. and ? itself form a basis of T p M . We have --

+ +

so the only interesting solution is the one for which

and that look like f (t)W. Then we find the differential equation

797

where K(y(t)) is the Gaussian curvature at y ( t ) . In accordance with what we have been doing before we want to describe the differential equation (13) using the coordinates on R".

Theorem 6.1. Jacobi fields o n a manifold { F = 0) in R" can be obtained by solving the ordinary differential equation:

d2F

dF

da:

+

d2F

(w) zT2w)

Proof. For the left hand side of (13) we apply D / d t - equation (9) - twice to a vector field. For the right hand side of (13) we use the expression for the Ftiemann curvature tensor (12). Start of the calculation with the left hand side:

= I1

+ + I3 + I4 + 12

We calculate the terms I I to

d (".w) at ax2

= d3FVVW ax3

a3F

= -vvw ax3

15

separately. Start with

+ -irw a2F 8x2

-

(16)

15

I2.

+82FVW 8x2 d2F .

798

For

13:

So that

The next step is to apply the formula (12) to compute the right hand side of (13):

Consider I;, in which we find the terms:

d3F ds ds

d 2 F d2s d 2 F dV ds V+--V+--

So that in I; only the terms d 2 F dV ds - --d 2 F dV ds --ax2 a t 1 a t 2

8x2

at2 at1

remain. But this is exactly -I;. So we find I;

)

___ 11yJ12

+ I; = 0. We focus on I;.

799

To calculate it we use again equation (18).

Next we need to calculate the Jacobi equation:

D2 W

--

dt2

-

R(v,W ) v

We use equations (20) and (16), (17), (19). In equation (20), we replace,

to obtain:

and in equation (13): '$I

p.

0

The geodesic flow is a map q$ from T p M to M . What we now contend is that Jacobi fields are the flow invariant vector fields along a geodesic. Theorem 6.2. The exponential map T T p M 3 TIM corresponding to the system (15) is the derivative of the geodesic flow T p M + M , i.e.

800

Proof. All we need to do is to find the total differential of

with respect to the x and v variables and replace in that expression d x by W and d v by p. It is straightforward to verify that this leads to the equations (15). 0

Remark 6.1. With this theorem, it becomes theoretically possible to also calculate Jacobi fields in the more general setting sketched in section 4, equation (7). In its, more general, intrinsic form theorem 6.2 is due to Cartan. It can be found in Cartan”, paragraph 160.

7. Conjugate locus and Jacobi fields Now we can define the conjugate locus exactly. Suppose we have a geodesic. It is determined completely by its initial value y(0) = p , T(0) = VO. Suppose there exists a time T and a vector po E T,M, orthogonal to V O , such that the Jacobi field defined by the initial conditions W ( 0 )= 0 and p(0) = po satisfies W ( T )= 0, then y(T) is a called a conjugate point. Such a point need not exist. For instance if we have a 2-dimensional manifold with K < 0 everywhere then we can see easily from equation (14) that there are never conjugate points. The multiplicity of the conjugate point q is the number of linearly independent vectors po that satisfy the above condition. The conjugate locus Conj(M,p) consists of the first conjugate points along geodesics from p . Each q E Conj(M,p) comes from one or more unit speed geodesics y,with y(0) = p , y(1) = q and v = y(0). Plotting those u in T,M results in a curve called the tangential conjugate locus. From theorem 6.2 we conclude that at a conjugate point the corank of the geodesic flow 4t is the multiplicity of the conjugate point. This means that the image of q5t “turns vertical”, at a conjugate point. The k-th conjugate locus consists Conj’(M,p) of the k-th zeroes of the Jacobi field with initial values (O,p(O)) # (O,O), with g (p (O ),j (O )) = 0. When dim(M) > 2 the k-th conjugate locus Conj’(M,p) needs to be defined using the Morse index theorem, see section 17.

801 8. The symplectic side

The geodesic flow on a complete Riemannian manifold forms a ray system. The Riemannian metric gij becomes a Hamiltonian

on the cotangent bundle with the zero section removed: TToM. Tangent vectors v are mapped to covectors by means of the Legendre transform : v H <, where
The flow-out of TT0,,M n {H(x,<) = 1) is a Lagrangian manifold L, c TT0M. The rays are the image of a point ( p , J ) E TTOlpMn{H(z,<)= 1). The conjugate locus is contained in the image of the singular points of the projection map T*M 3 L, -+ M. The singular values of the map L, -+ M are either self-intersections of wavefront from p , or points where the rank of the differential of L, 4 M is less than dim M. The cut-locus is part of the set sweeped out by self-intersections of wavefronts emanating from p. By theorem 6.2, points where the rank of the differential is not maximal are conjugate points. 9. Known theorems about cut and conjugate loci

Before we continue we list some known theorems about cut and conjugate loci. There are relatively few results, but for a more complete list we refer to the book Berger4. The bare necessities for dealing with cut and conjugate loci are explained in S ~ i v a kchapter ~ ~ , 8. One of the first questions one can ask is: do the ( tangential ) conjugate locus and the ( tangential ) cut-locus always meet? Very surprisingly, the answer is “no” in general. The sphere S2 is exceptional because of the following, see Weinstein*O:

Theorem 9.1. Let M be a compact smooth manifold, not diffeomorphic t o S 2 , T h e n there i s a Riemannian metric o n M and a point p E M f o r which the tangential conjugate locus and the tangential cut-locus are disjoint. Weinstein gives some examples of cut and conjugate loci. The projective plane can be given a metric of constant curvature 1. Weinstein states

802

that on this projective plane the tangential cut and conjugate loci are two disjoint circles. These statements are proved in Besse5, section 3.34. In complex projective space the conjugate locus and the cut locus coincide. Weinstein’s theorem is complemented by the following theorem of Klingenberg Theorem 9.2. For a compact simply connected even dimensional manifold M all whose sectional curvatures are positive, there is a point p E M f o r which the tangential cut-locus and the tangential conjugate locus intersect. About the conjugate locus little is known. The reference 39, though forty years old, is still the standard one. Another nice result is the following. Weinstein’s theorem seems to suggest that things are still quite intuitive on spheres S2. But Margerin proves in 22 that there are metrics on S2 for which the conjugate locus is not a closed curve. In his examples the tangential cut-locus escapes to infinity. Those examples are surfaces of revolution, just as the examples of Gluck and Singer below. We can ask whether the cut-locus is always a triangulable set. If the manifold is real analytic then this is the case, see Buchner8. In the smooth case the answer is “no”. Intuitively, a counter-example is easy to construct. Take a wavefront at some distance d from a point p. Then arrange the metric such that a wavefront from another point q touches the wavefront in infinitely many points, as indicated in figure 2. The point q lies in the cutlocus Cut(M,p). However if we remove q from Cut(M,p) the complement of the cut-locus is no longer locally finite. Any triangulable set is locally finite so we have a counter example. Again, this is intuitive reasoning, for the details see Singer and G1uck3’.

r Figure 2.

A wavefront from p at distance d.

Non-triangulable cut-locus example.

Wall has proved in Wall38 for a generic metric the triangulability of the cut-locus. The example of Gluck and Singer is thus an example of a

803

non-generic metric on a surface of revolution. A much stronger statement proved in Buchnerg is true in low dimensions dim(M) 5 6: for a generic metric on a compact manifold of dimension I 6 there is a finite list of singularities of the cut-locus. Combining that with theorem 3.1 we get a result of Myersz5.

Theorem 9.3. O n a simply connected 2-dimensional manifold the cutlocus is a tree. The distance from a point q E Cut(M,p) to p is called the injectivity radius. The length of a geodesic to the cut-locus Cut(M,p) is a function L:

(T,M n { g p ( v , v )= 1))

---$

iR

Itoh and Tanah'* prove that this function is not just continuous ( as is proved in most differential geometry books ) but that it is actually Lipschitz. On a surface, that is a 2-dimensional manifold, if M has negative curvature everywhere then the function f in equation (14) is monotonely increasing. Therefore if K < 0 everywhere then the conjugate locus is empty. In higher dimensions if the sectional curvatures k i j < 0 everywhere, then the conjugate locus is empty everywhere. We have the following related statements, for complete Riemannian manifolds, see S ~ i v a kchapter ~ ~ , 8.

Proposition 9.1. If a geodesic from p has a conjugate point q then either q is also a cut point for p or there is a cut point for p on the geodesic between p and q. Every geodesic from p E M has a cut point if and only i f M is compact. A further special case is that of symmetric spaces. On a symmetric space cut and conjugate loci can be calculated explicitly, see Crittenden", or Besse5. 10. A conjecture of Arnol'd

A - now disproved - conjecture of Arnol'd concerns caustics of ray systems close to the ray system of points on a sphere. The conjecture was formulated in Arnol'd3, but the reference Arnol'd' is more readily available. In T*W"consider the ray system that is the flow out of

We get the Lagrangian cylinder L,:

804

The Lagrangian cylinder is a local model for the simplest of conjugate loci: the conjugate locus that you find at the south pole when the rays emanate from the North pole. If you perturb the metric on the sphere a little bit you get a Lagrangian manifold of rays that is close t o the Lagrangian cylinder. Arnol’d proved that for not too big symplectic perturbations E ( those done with some Hamiltonian ) the perturbed manifold E ( L ~has ) a caustic with at least four cusps. He conjectured that those cusps cannot be removed by Hamiltonian isotopies. This was disproved by Entov12. There is another paper where Arnol’d states the following far more reasonable conjecture.

Conjecture 10.1. See Arnol’d2 For a point o n a generic compact strictly convex surface in R3 each conjugate locus Conjk((M,p) has at least four cusps. Compact strictly convex surfaces are characterized by their metric. A compact 2 -dimensional Riemannian manifold whose Gaussian curvature is ev~~. erywhere > 0 can be isometrically embedded in R3,see S ~ i v a k Moreover, it was proved in Itoh and TanakalS that the distance to the k-th conjugate locus is a Lipschitz continuous function, and therefore on a generic strictly convex surface Conj‘(M,p) is a closed curve of finite length. 11. The examples of Markatis

In 1980, a student of Ian Porteous, Stelios Markatis, studied a number of convex surfaces in R3. His main interest in these surfaces was to study how the pattern of umbilics changes as we pass from one surface to the other. His surfaces are also interesting for cut and conjugate loci. From Markatis’ thesis we already know a lot about these surfaces so they provide nice examples to test conjectures, such as Arnol’ds conjecture in the previous section. The examples Markatis studied were the bumpy spheres 0

0 0

The bumpy The bumpy The bumpy The bumpy

+ + + + + + + + + + + +

cube: 2: xi 23 sz11c2q = 1. tennisball: 2: x$ x i ix1x$ = 1. sphere of revolution: x: x; x: ix: = 1. orange: xp 2; xi $(x! - 3x12;) = 1.

If E is small enough then one connected component of these surfaces is still convex. For an analysis of the bumpy spheres, we refer to chapter 16 in Porteous2’, or to the papers P o r t e o u ~and ~ ~ Porteous28. The analysis of the ellipsoids is classical and it can be found in Klingenberg2’, section 3.5. Their cut and conjugate loci have been determined by Itoh and Kiyohara“.

805

When a point on the ellipsoids is not an umbilic, the cut-locus is a line segment, and the conjugate locus is a curve with four cusps. Very nice pictures of the conjugate loci, also of higher order order are in Sinclair31. The methods used to compute those conjugate loci are similar to the ones we propose here, but the formula (5) is not explicitly mentioned. Here we show one example, the cube 6 = &. We also show the conjugate

Figure 3. A conjugate locus on Markatis’ cube. On the left we see the conjugate locus alone. On the right we see the complete picture, where we have drawn the geodesics starting from t = 2.

locus in the tangent space, and we plot the squared length of a Jacobi field. To produce figure 3 we look for the zeroes of the Jacobi field. The length of a Jacobi field is (W,W).The derivative of that function is 2(W,p). In figure 4 we used the results of numerically integrating the system (15). We plotted both (W,W ) and 2(W, p ) directly from the results, so the graph we see is also a verification of our results. Checking that everywhere (W,E ) = 0 and ( p , W ) = 0, and (E, E ) = 1 also gives good results. For a geodesic of We took length 4 times the injectivity radius the numerical error is 5 the cube as an example because the cube is the worst example. There are always at least six cusps on the first conjugate locus. So the perturbation we chose is not generic. Indeed it isn’t, the surface has some symmetries, whereas a generic perturbation of the sphere should have no symmetries at all. For numerical purposes, one finds the kth conjugate locus by looking for the 2kth zero of the function ( p , W ) .

806

&.

Figure 4. Again the Markatis' cube with E = On the left we see the squared length of a Jacobi field, together with its derivative. On the right we see the conjugate locus in the tangent space.

It is also possible, though much more cumbersome, to visualize the cutlocus. One needs to find the intersections of the wavefronts, though not all. We will come back to this problem in section 17. When significant parts of a surface are almost flat we can see from equation (14) that the differential equation for the Jacobi fields becomes rather unstable, a fact that can be confirmed experimentally. 12. Questions asked by Thorn We have not yet studied the medial axis, mentioned in the introduction: we studied Maxwell strata and caustics using the intrinsic geometry of a surface, even though, for computational reasons, we used the embedding of a manifold. We will turn to the same sort of questions using extrinsic geometry proper. Recall that the intrinsic geometry of a Riemannian manifold studies the properties of the manifold itself, whilst the extrinsic geometry studies the properties of the embedding of a manifold. The intrinsic geometry is completely determined by the metric, the extrinsic geometry is determined by the second fundamental form. For the convenience of the reader, let us also repeat the stanza in the introduction. Let M be a compact manifold without boundary, embedded in Rn as a hypersurface, through an embedding $: M + Rn. A function f E C"(M) has a critical point at so E M when the derivative Df(s0) = 0. When in addition, the second derivative D2 f ( s 0 ) is a nondegenerate matrix then we say that f has a nondegenerate critical point at SO. When a function

807

only has nondegenerate critical points and all its critical values are distinct, then we say that f is a Morse function. Some authors use only the first condition and call a function that satisfies both conditions an excellent Morse function. The index of a nondegenerate critical point is the dimension of the maximal subspace for which the second derivative is strictly negative definite. So a nondegenerate minimum has index 0 and a nondegenerate maximum has index dim(M) Generic functions on a manifold are Morse. However in a family of functions one might encounter other than Morse functions. Thom proposed to construct a family of functions as follows. Take any C" map $ from M to R", where dim(M) = rn < n. The family of functions to study is

We can view F as a map R" -+ C"(M). It associates to x E R" the function fz(s)= F ( z , s) E C"(M). Because M is compact all nonconstant functions of C"(M) have at least two critical points. In the space C"(M) consider the functions that have at least two identical critical values or one degenerate critical value. If we perturb such functions a little bit around one of the critical points we get a Morse function, because Morse functions are dense in C"(1M). It would be very nice if for the perturbations of a non-Morse function in the family (21) we can take the family itself. But when can this be done? Thom conjectured that for generic embeddings of M in R" the family of functions would in fact be representative for most perturbations, i.e. the family of functions is topologically stable. Topological stability is sometimes called structural stability. An interesting case is where the map $ is an embedding. In that case the values of z for which fz(s) has a degenerate critical value form the caustic. The values of z for which fz(s) has a multiple critical value form the symmetry s et Syrn(4) . The values of x for which fz(s) has a nonunique global minimum form the medial axis Cut($). The medial axis, the caustic, and the symmetry set have become widely studied notions. So much so that the original questions of Thom have almost been forgotten. Denote S the set of functions in C"(1M) such that f is not Morse. One can view S as a "hypersurface" in C"(M). Inside the set S we have the subsets b

f

E

S has a degenerate critical value + caustic.

808 0 0

0

f has a multiple critical value + Sym($). f has a nonunique global minimum + Cut($). Si,j: f has a unique multiple critical value f(s1) = f ( s 2 ) for which we have indexf(s1) = i and indexf(s1) = j

Thom proposed to study the topological and geometrical properties of these subsets. The medial axis corresponds to a subset of the closure of &,o. In the correspond to case of the ellipse ( figure 9 ) all points of the closure of SO,O a point on the medial axis. In general this will not be the case. The reader is encouraged t o draw a counterexample. Thom proposes to study the other Si,j as well. Moreover most of the attention since the publication of has been directed at the case where q5 is an embedding of a codimension one manifold, whereas Thom put forth a more general problem, where $ is a smooth map, but not necessarily an embedding. Denote DifFOO(M) the group of smooth diffeomorphisms of M . Denote DifY(R) the group of smooth diffeomorphisms of R. There is an action of DifFOO(M) x DifP'(R) on Cm(M). If a E D i f P ( M ) , and ,B E DifP(R), then (a,,B)o f = ,B(f(a)).The subsets of Cm(M)that we are interested in are invariant under this action. 13. Answers given by Looijenga

In the thesis of Looijenga the main conjecture of Thom stated in his article was found to be true. Let us explain this result and the conjecture. We copy the definition of stratifications from Gibson et. all4. A stratification W of a closed subset X of a manifold is a subdivision of the set into subsets - called strata - such that (1) Each stratum of W is locally closed (2) The subdivision is locally finite

A Whitney stratification is a stratification that satisfies conditions A and B of Whitney. Condition A . Let Wl E W be a stratum and let { ~ i be} a ~sequence ~ that converges to some xo in Wz E W and such that xi E Wl for all i. The sequence Tzi W1 converges to r and T,, W2 C T . Condition B. Let {xi}& be a converging sequence, with all elements again in some Wl and {yi}&

a sequence of points in Wz, such that both

809

sequences converge to a point zo E W1. If the lines Ci = z;yi converge, they W1. converge to some limit C c lim TZi We refer to Gibson et. all4, chapter 1, for an exact definition of canonical stratification. Informally speaking: a canonical stratification is the coarsest subdivision of X in strata such that the stratification is Whitney. The results of Looijenga concern a stratification of C"(M, R). Of course, C"(M,R)is an infinite dimensional "manifold". Note that it is not a Banach manifold. It is only a F'rechet space: the space has a countable family of semi-norms, but there is no norm that defines the topology. For details see Hirsch15. The notion of a derivative of a map C"(M, R) + RN is well-defined though. A weak codimension k submanifold in a F'r6chet space is a subset V of a F'r6chet space such that for all f E V there exists an open neighborhood U of f and a submersion .1c,: U + Rk such that $-'(O) n U = V n U.A weak stratification of a subset X of a Fr6chet space 3 is a partition of X into weak codimension k submanifolds. A weak Whitney stratification is a weak stratification of X such that any map 9 from RN H 3 that is transversal to all strata of X pulls back to a Whitney stratification of 9 - ' ( X ) . A family of functions F E C"(M x Rn)is called topologically stable ( resp. smoothly stable ) if for any F' sufficiently near to F , there are homeomorphisms (resp. Coo-diffeomorphisms) h, h', h" such that the diagram (22) commutes. Thom conjectured that for a generic embedding 4 : M an the family of functions F ( z ,s) = 112 - q5(s)1I2 is topologically stable.

-

M x

FxId

u+ax

U ' d U

Theorem 13.1. Looijenga .There exists a subset W ( M ,R) of C"(M, R) such that the complement of W ( M ,R) in C"(M, R) has a weak stratification

S . The weak stratification has the following properties: If a map F : Rn + C"(M,R) is transversal to S and the image of F is disjoint from W ( M , R ) then (1) F : Rn x M + R defines a topologically stable family of functions.

(2) The weak stratification of the image of F in C"(M,R) pulls back to a canonical stratification of R". (3) Mappings transversal to S form an open and dense subset of C"(RN, C"(M, R)).

810

(4) The strata of S are invariant under the action of the group DifFOO(M) x DifP(IW) o n Cw(M,R). (5) If S' is any other weak stratification of C w ( M , R) \ W ( M ,R) having properties 1, 2, 3 and 4 then it is a refinement of S.

Remark 13.1. In the way it was originally formulated this theorem only holds for families of functions parameterized, not by z E R", but by 2 E N , where N is a compact manifold. Therefore it also holds for a compactification of R", say RP". In the case of the distance function it follows that we can formulate the theorem as in the above. Remark 13.2. Many authors refer t o the thesis of Looijenga. It is often

used to prove that in low dimensions a family of functions is smoothly stable. Of course this is a corollary of theorem 13.1, but the theorem is much stronger than that. It can be used to prove topological stability. Smooth stability of generic families of functions under study usually follows from the results of Mather. The theorem of Looijenga simplifies the study of singularities as suggested by Thom significantly. It is fairly easy to prove that for a residual set of embeddings the family of functions I(z - 4(s)(I2 is transversal to the canonical stratification. Hence that family of functions is topologically stable. 14. The caustic

Let us give the reader some intuition for the different strata that we study. If the function s -+ 112 - $(s)1I2has a critical point then z lies on the normal pointing out from $(s). In three dimensions the typical picture is as in figure 5 . The caustic is the conjugate locus of an embedded manifold. If for some z E R" the function s -, 112 - $(s)1I2 has a degenerate critical value at SO then infinitesimally near normals along the line of curvature from the critical point SO) intersect at IC. The same happened with the conjugate locus: geodesics are normals to a point. The envelope of the geodesics is the conjugate locus, the envelope of the normals is the caustic. At a point where the hypersurface has n-1 different principal curvatures kl < kz < . . . < kn-l, the normal intersects the caustic at n - 1 different points. At these intersection points the caustic is tangent to the normal, as shown in figure 5 . When two of the principal curvatures coincide, two or more of the sheets of the caustic coincide and the caustic has a singularity.

811

Minimum SO The normal ....

.

Figure 5.

,...

Critical points of the distance function.

Starting from a point p = $(SO) on the hypersurface and moving z along the normal 6 , at first the function s 5 llz - $(s)l12has a minimum. Then as we pass k:’, the first curvature radius, the index of the critical point is no longer zero, but becomes one. Then, as we pass kzl the index becomes two, which in the above picture corresponds to a local maximum. The singularities of the caustic are well-known for n 5 5 . They are the ADE-singularities, see Arnol’dl . In higher dimensions moduli appear, but we still have topological stability, according to the theorem of Looijenga. In dimensions 2 and 3 a beautiful analysis of the interplay between the geometry of the caustic, and that of the curve or surface is contained in Porteo~s~~.

15. Singularities of the medial axis The medial axis is in fact, as Thom remarked, the cut-locus of an embedded manifold. It is not surprising that their generic singularities coincide. The list of local normal forms of the medial axis for a compact submanifold of dimension n is the same as the list of normal forms of cut-loci of points in an n 1 dimensional manifold. Indeed both are singularities of a minimal distance function. The generic singularities of the medial axis of a compact submanifold were classified in two papers, Mather23 and Yomdin41. The generic singu-

+

812

larities of cut-loci were classified in Buchnerg. Let us discuss the theorem of Mather. Mather discusses the singularities of the minimal distance function:

where 4 is an embedding of M in R".

Theorem 15.1. For a residual set of embeddings of a compact manifold of codimension 1 in R",with n 5 7 there exists a finite set EM C Rn such ) be reduced to one of the that outside EM the distance function p ~ ( x can following normal forms. 0 0

A0 : 2 1 , A1 = min(z1, z ~ ). ,. . , A7 = min(z1,. . . , z7), B2 = minsE~(s4 21s' x2s z3), B3 = m i n ( B ~ ( ~ l , x 2 , ~ 3 ) ,.2. 4. ,) , B7 = m i n ( B 2 ( ~ 1 , z z , ~ 3 ) , 2 4 , 2 4 r 2 6 , 2 7 ) , min(B2(x1,m,~3), B2(24,25,26)),

+

+

+

min(Bn(zl,zz,~3),B2(~4,~5,~6),~7)r

0

=

+

+

+

+

+

+

minSER(s6 z1s4 z2s3 23s' min(C4(~1,...,z5),z6)r m i n ( C 4 ( ~ , . . ,x5),z6,z7), . & = minsEa(s8 z1s6 z2s5 x3s4 z4s3 55s' c 4

+

+

+

z4s

+

26s

+

z5),

+ z7)

The theorem of Mather thus excludes certain points, the set E M , on the medial axis. In the planar case his theorem is almost void. He excludes the only two singularities, the end point and the trivalent vertex. The singularities of generic medial axes and singularities we get in R3 are exhibited in figure 6. Those that are not on the finite set of Mather

Figure 6. Generic singularities of the medial axis of a compact submanifold of dimension 2 in It3, and also generic singularities of a cut-locus in a 3-dimensional manifold.

are A3 and B2. Let us construct the pictures associated to AS. The germ associated to the distance function is locally min(zl,z2,x3). Thus the

813

nonunique minima are located on (21

=2 2

5 2 3 ) u (21 = 2 3

5.2)

u(22

=23

5 21)

These are parameterized by

{(t,t,t+s) I s L O } U { ( t , t + s , t )

I

sLO}U{(t+s,t,t)

I

s L O } (23)

and if we plot equation (23) we get the second picture from the rhs. in figure 6. To find the picture belonging to Mather’s B 2 singularity, we consider the polynomial p ( s ) = s4 z 1 s 2 2 2 s 2 3 . We have to find the values of ( 2 1 , 2 2 , 2 3 ) for which the polynomial p ( s ) has a nonunique global minimum. In that case p’(s) = 4s3 2 2 1 s 2 2 has three real roots. Hence, we have D = 82: 272; 5 0. If we plot this we see that we deal with a region that is the LLinterior” of a cuspidal edge, and thus Mather’s B 2 is equal to the first picture on the right side in figure 6. The restriction in the theorem of Mather can be removed. To find all normal forms of the minimal distance functions in dimensions 5 6 without excluding a finite set of points take transversal sections of the normal forms one dimension higher. For instance, Mather’s normal forms of the medial axis give for an embedding of a curve in R2 only the smooth part: all the singularities are part of the excluded set. But in R3 we get exactly the two normal forms on the left hand side in figure 6. If we take a transversal intersection of these stratified sets with a plane we get the end point and the trivalent vertex. These are all generic singularities of the medial axis in the plane. Similarly to get the four-valent vertex, second from left in figure 6, consider the A4 germ min(z1, 2 2 , 2 3 , 2 4 ) . The nonunique minima are located on the union of the six manifolds with corners:

+

+

+

+

+

{zci=zj <min(zk,zl)

+

I

{z,j,k,l}={1121314}}

The intersection of this stratified set with ~ 1 + ~ 2 + 2 3 + 2 4= 0 is transversal and in the plane 2 1 + 2 2 + 2 3 + 2 4 = 0 it is the four-valent vertex. Similarly, to get something locally diffeomorphic to the picture on the left hand side of figure 6 intersect the B 3 normal form with x1 2 2 2 3 2 4 = 0. It appears that in general one can take the plane C:=,xi = 0 in the formulae of Mather t o get all local normal forms of the generic medial axes in dimensions n I 6 . The normal forms for n 5 4 coincide with singularities of generic cut-loci when dimM 5 4, compare the lists in Yomdin41 and Buchnerg.

+ + +

814

16. Two lesser known theorems in In this section we present some aspects of the work of Thom that appear to have been largely forgotten, but that we still find interesting. In the previous two sections 14 and 15 we treated the case where $ is an embedding. Interesting theorems can also be proved if we drop that condition.

Lemma 16.1. Let $: M + R" be a smooth map. Let $(so) = xo E R". The matrix D $ has maximal rank in SO if and only if the second derivative of s + (1zo - $(s))I2is a nondegenerate matrix in s = so. Proof. It is no restriction to assume that xo = 0, so $(so) = 0. Then we see that

This matrix is nonsingular if and only if

has maximal rank.

0

This elementary lemma has a very nice corollary. Denote Cut($) set of z o E Rn for which s

-+

q $ , z o ) = 11x0 - $ ( s ) [I2

does not have a unique global minimum. Denote Emb($) open subset of M for which

c R" the (24)

c M the largest

$IEmb(+)

is an embedding.

Proposition 16.1. The sets Cut($>

n $(MI

and $(Emb($))

are disjoint. Their union is $ ( M ) . Proof. Let z o E $ ( M ) . Suppose that zo # Cut($). The map (24) has a unique global minimum at SO, with SO being the unique element of M for which $ ( s o ) = zo. According to the previous lemma it follows that the 0 differential of $ has maximal rank at SO.

815

We now come to the analogue of theorem 3.1.

Theorem 16.1. T h e $(Emb($)) image is a deformation retract o f t h e complement R" \ Cut($). Let $ be an embedding of S"-' in R". According to the previous lemma Cut($) does not intersect $(S"-'). In the interior of 4(S"-') there lies a point of Cut($). Thus the interior medial axis IntCut($) is well-defined. From theorem 16.1 we see IntCut(4) is contractible, so it is a tree. That assertion is the analogue of the theorem of Myers, section 9.3. Figure 7 illustrates this. One views a cut-locus as the interior medial axis of a wavefront from p. Clearly this can only be done if the manifold M \ { p } is some R", that is M is a sphere S".

Figure 7. Relation between the theorem of Myers on cut-loci and results on the medial axis

In the comments after theorem 16.1 Thom states: "A simple adaptation of a theorem of Weinstein ( see theorem 9.1) shows that there exist embeddings of S2 in R3 such that the interior medial axis does not meet the caustic". It is unclear to the author what Thom exactly means. On S3 there exists according to the theorem of Weinstein a metric and a point p such that the tangential cut-locus and the tangential conjugate locus are disjoint. An explicit example was constructed in Itohlg. Thom suggests that these examples can be transposed to R3 to give examples of embeddings of S2 in R3 where the medial axis does not meet the caustic. A sufficiently small wavefront from such a point p E S3 is homeomorphic to S2. The wavefront separates S3 in two domains: the inner one containing p and the outer one not containing p. The outer domain can be viewed as the interior of a domain bounded by the image of an embedding 4: S2 4 EX3. The interior medial axis is the cut-locus of the point p , if that diffeomorphism from the outer domain on S3 to the domain bounded by $ can be made to be an isometry.

816

In fact, with the Euclidean metric on R3 the caustic and the medial axis cannot be separated: there are always end points. Is a similar phenomenon true for the special cases of cut and conjugate loci embedded as hypersurfaces in R"? We suspect that in those cases, with the metric induced from an embedding into R", cut and conjugate loci can not be separated for convex surfaces. Another stratum is the maximal medial axis MaxCut($). The maximal medial axis consists of those z E R" for which F ( $ , z ) ( s )E Cm(M) has a nonunique global maximum.

Theorem 16.2. If $: S"-l IntCut($) # 0.

-+

Rn is a n embedding then MaxCut(4) n

Proof. We will establish a contradiction, so assume MaxCut(4) n IntCut(q5) = 8. On the complement of the medial axis Cut($) we have a map

The map Gmin associates to z the unique minimum of (111: - $(s)//'. If then Gmin(zo)= SO. On the complement of the maximal medial axis MaxCut(q5) we have a map 50 = $(SO)

Gmax:Rn \ MaxCut(4)

+ S"-l

The map G,, associates to z the unique maximum of 112 - $ ( ~ ) 1 1 ~ . Take a small neighborhood U of IntCut($), that is disjoint from MaxCut(4), and that is contained in the domain bounded by $(Sn-l). On the boundary of U - denoted dU - we have two maps to Sn-': G,, and Gmin. Suppose that for some z E dU we have G,in(z) = Gmax(z). It would follow that for that z the function 1111: - $(s)1I2 is constant, so we would have that z lies on IntCut($), which is contrary to our assumption. Hence the graphs of Gminand G,, over dU are disjoint. and Gmincan be seen as sections of the fiber bundle The graphs of G,, sn- 1 x dU H dU. The section G,, is homotopic to the constant section of that bundle because U n MaxCut(q5) = 8. Remove the section G,, from that fiber bundle. Because the section G,, is homotopic to the constant

817

Int C ut (4)

Figure 8.

Proof of theorem 16.2.

section we get a homotopy:

The image of the section Gmin through this homotopy is a section of (Sn-' \ { p } ) x d U , which is a bundle with contractible fibers. It follows that the map Gmin is also homotopic to the constant section. As a consequence the map Gmin is homotopic to the constant map, but from the picture 8 it is immediately clear that the map Gmin is also homotopic to a homeomorphism dU -, S"-l. We have established a contradiction and the proof is complete. 17. Singularities of the symmetry set

The references for this section are Bruce et. al.7 and Bruce and Giblin6. Again we have a family of functions: by x E R"

F(x,

= 112 - +(s)Il2

As with cut-loci we are looking to minimize the distance. In this case we want to minimize the distance between the point x E X and M . A nonunique global minimum defines the medial axis Cut(+), whilst multiple critical values define the symmetry set Syrn(q5). The symmetry set contains both MaxCut(4) and Cut(+). Confirm figure 9. Singularities of the symmetry set are classified using the canonical stratification of Looijenga. Thus

818

Figure 9. Medial axis and symmetry set

the symmetry set Syrn(4) is topologically stable in all dimensions. Its singularities can be derived from the singularities of caustics, medial axis and wavefronts. Let us do this for the plane case. The trivalent vertex of the medial axis becomes a triple crossing as on the right hand side of figure 10. All strata on the triple crossing are SO,O, but the parts that follow after the vertex corresponds to minima that are no longer global. The end points of the medial axis are unchanged. New end points show up because also the S1,1 stratum has end points at singularities of the caustic. Contrary to the medial axis the symmetry set has cusps, because it can meet the caustic. At such points the type of the stratum changes, because one of the critical points becomes degenerate. There can also be transversal intersections of strata. s0,o

S0,l

I

Figure 10. Other strata we find on the symmetry set. The dashed line is the caustic.

In the case of cut and conjugate loci a similar analysis can be made. The energy function on the infinite dimensional manifold R , , also has a second derivative. It is an invariantly defined bilinear symmetric form

S2E: T,Rp, x T,Rp,

-+

R

on the tangent space T,Rp, - where y is a geodesic - defined in section 2.

819

At a conjugate point q E Conj(M,p) the second derivative is degenerate. The dimension of the maximal subspace of T,R,, on which S2E is negative definite is called the index of the geodesic. If q is a conjugate point along a geodesic y, with y(0) = p , y(1) = q then the dimension of the space of Jacobi vector fields J along the geodesic, that satisfy J ( 0 ) = J(1) = 0 is called the multiplicity of the geodesic. Along a geodesic from p to q we might meet several conjugate points. The Morse index theorem, see MilnorZ4 815, states that the sum of their multiplicities is equal to the index of the geodesic. Now look at picture 12. It is the same example as

(1)

Conjugate points P = Y(0) Figure 11. The Morse index theorem. At y(1) the index of the geodesic is the sum of the multiplicities of the conjugate points 41, 42 and 43.

the one used in section 11. The thick black curves are traced out by selfintersections of wavefronts. The “strata” are defined by the indices of the two geodesics of equal length that meet. They are strata Si,j just as the strata of the symmetry set. However, now there is no a priori limit on the indices i and j. Again, these strata have never been studied. Clearly the cut-locus is a subset of the closure of &,o. Is any Si,+a tree on a strictly convex compact surface? 18. A weighted symmetry set

The symmetry set is the closure of the set of points x where the distance function

f d s ) = 1111: - 4(4112 has two critical points with equal critical value. Take a ratio [XI; Xz] E P1 with A1 # 0 # X2. Define the weighted symmetry set Syrnx(q5) to be the closure of the set of x E R” such that there exist two critical values s1 and s2 of fz(s) = llz - +(s)1I2 such that Xlf,(sl) = X2fz(s2). D. Siersma asked what the weighted symmetry set looks like. Here is the answer:

820

Figure 12. Intersections of wavefronts on a convex surface.

Theorem 18.1. Let q5 be a n embedding of a circle in the plane, whose image is M c R2. Fix a compact subset D of R2. For any E > 0 there is a 6 and a X = [XI; A], E P1 with d p ( X , [I;11) < 6 such that for any point x E D o n the smooth part of the symmetry set there are two points x1 and x2 in Symx(q5) such that 11x-x111 < E and 11x-x211 < E . Furthermore for any point x E D on the regular part of the focal set there is one x1 E Symx(q5) such that llx - x111 < E . Proof. Theorem 18.1 is best understood in pictures, see figure 13. Near a smooth point 20 E R2of the symmetry set the family of functions is generic. So the unfolding of the multi-germ of the distance function s + (Ixoq5(s)1I2 at (xo,s0 (1) ) and (xo,so (2) ) is smoothly equivalent to (f(s(')

+

prl(x - x o ) , f ( d 2 ) pr, xo). Here pr, xo denotes the projection onto the first coordinate of R2. For simplicity's sake let us assume we stratum. Then the unfolding is smoothly equivalent to are on the S1,l ( F ( l )F , ( 2 ) )=(-(&I - s : ) ) ~ - pr,(z - xo),-(d2)- SF))^ pr, xo). The left upper corner of figure 13 shows such a family of functions as x varies. Points on the weighted symmetry set are described by

+

821

We get: ~ ~ ( - ( s( lsF))2 ) - prl(z - q > = >~ z ( - ( s ( ~) s1,2) J1) = (1) SO

2

+ prl zo)

,(2) = (2)

SO

For values of X = [Al : A,] close enough to [l : 11 we get two nearby sheets of the weighted symmetry set, indicated by the arrows. Near a smooth point of the caustic the family of functions IIz - q5(s)1I2 is generic as well. Hence the germ distance function is smoothly equivalent (s - so) prl(z - zo). Such an unfolding is shown in the right to (s upper corner of figure 13. When pr,(z - zo) < 0 we get the dashed graph, which gives the point on the weighted symmetry set. When prl(z -zo) > 0 we get the solid graph which does not give a point on Symx(q5).

+

Unfolding near the symmetry set

__________

Unfolding near the caustic

_ _ _ _ _ ____

__.____ __.__. _ _____ .._.__.____. ___.

i

I

i One nearby sheet Another nearby sheet

I

/

One nearby sheet

Figure 13. T h e unfolding gives two sheets of Syrnx(q5) near the symmetry set and one sheet near the caustic. 0

A higher dimensional analogue of this theorem is also true. For values of X close to [l : 11 the weighted symmetry set Syrnx(q5) is a double cover of the symmetry set, and near to each point on the caustic there is always another sheet of the weighted symmetry set. Moreover, the weighted symmetry set is for a generic embedding the projection of a smooth conic Lagrangian

822 manifold in T\$Rw". T h a t Iast fact can be proved using the techniques developed in van Manen3'.

References 1. V. I. Arnol'd, Singularities of caustics and wave fronts, Mathematics and its Applications (Soviet Series), vol. 62, Kluwer Academic Publishers Group, Dordrecht, 1990. 2. -, Topological invariants of plane curves and caustics, University Lecture Series, vol. 5, American Mathematical Society, Providence, RI, 1994, Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. 3. -, Sur les proprie'te's topologiques des projections lagrangiennes e n ge'ome'trie symplectique des caustiques, Rev. Mat. Univ. Complut. Madrid 8 (1995), no. 1, 109-119. 4. Marcel Berger, A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003. 5 . Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin, 1978, With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bbrard-Bergery, M. Berger and J. L. Kazdan. 6. J . W. Bruce and P. J. Giblin, Growth, motion and 1-parameter families of symmetry sets, Proc. Roy. SOC.Edinburgh Sect. A 104 (1986), no. 3-4, 179204. 7. J . W. Bruce, P. J. Giblin, and C. G. Gibson, Symmetry sets, Proc. Roy. SOC. Edinburgh Sect. A 101 (1985), no. 1-2, 163-186. 8. Michael A. Buchner, Simplicia1 structure of the real analytic cut locus, Proc. Amer. Math. SOC.64 (1977), no. 1, 118-121. T h e structure of the cut locus in dimension less than or equal t o six, 9. -, Compositio Math. 37 (1978), no. 1, 103-119. 10. Elie Cartan, LeCons sur la ge'ome'trie des espaces de Riemann, LesGrands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Editions Jacques Gabay, Sceaux, 1988, Reprint of the second (1946) edition. 11. Richard J. Crittenden, M i n i m u m and conjugate points in symmetric spaces, Canad. J. Math. 14 (1962), 320-328. 12. M. Entov, Surgery o n Lagrangian and Legendrian singularities, Geom. F'unct. Anal. 9 (1999), no. 2, 298-352. 13. Alfred Gray, Modern differential geometry of curves and surfaces with Mathematica, second ed., CRC Press, Boca Raton, FL, 1998. 14. Christopher G. Gibson, Klaus Wirthmuller, Andrew A. du Plessis, and Eduard J. N. Looijenga, Topological stability of smooth mappings, SpringerVerlag, Berlin, 1976, Lecture Notes in Mathematics, Vol. 552. 15. Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994, Corrected reprint of the 1976 original.

823 16. Jin-ichi Itoh and Kazuyoshi Kiyohara, T h e cut loci and the conjugate loci o n ellipsoids, Manuscripta Math. 114 (2004), no. 2, 247-264. 17. Jin-ichi Itoh and Robert Sinclair, Thaw: a tool f o r approximating cut loci o n a triangulation of a surface, Experiment. Math. 13 (2004), no. 3, 309-325. 18. Jin-ichi Itoh and Minoru Tanaka, T h e Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. SOC.353 (2001), no. 1, 21-40. 19. Jin-ichi Itoh, Some considerations o n the cut locus of a Riemannian manifold, Geometry of geodesics and related topics (Tokyo, 1982), Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1984, pp. 29-46. 20. Wilhelm P. A. Klingenberg, Riemannian geometry, second ed., de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter Az Co., Berlin, 1995. 21. Anders LinnBr, Periodic geodesics generator, Experiment. Math. 13 (2004), no. 2, 199-206. 22. Christophe M. Margerin, General conjugate loci are not closed, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. SOC.,Providence, RI, 1993, pp. 465-478. 23. John N. Mather, Distance f r o m a submanifold in Euclidean space, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. SOC.,Providence, RI, 1983, pp. 199-216. 24. J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. 25. S. B. Myers, Connections between diflerential geometry and topology i: Simply connected surfaces., Duke Math. J. 1 (1935), 376-391. 26. A. M. Perelomov, A note o n geodesics o n ellipsoid, Regul. Chaotic Dyn. 5 (2000), no. 1, 89-94, With comments by A. V. Borisov and I. S. Mamaev, Sophia Kovalevskaya to the 150th anniversary. 27. Ian R. Porteous, T h e normal singularities of surfaces in R3, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. SOC.,Providence, RI, 1983, pp. 379-393. 28. I. R. Porteous, Ridges and umbilics of surfaces, The mathematics of surfaces, I1 (Cardiff, 1986), Inst. Math. Appl. Conf. Ser. New Ser., vol. 11, Oxford Univ. Press, New York, 1987, pp. 447-458. Geometric differentiation f o r the intelligence of curves and surfaces, 29. -, Cambridge University Press, Cambridge, 1994. 30. David Singer and Herman Gluck, T h e existence of nontriangulable cut loci, Bull. Amer. Math. SOC.82 (1976), no. 4, 599-602. 31. R. Sinclair, O n the last geometric statement of Jacobi, Experiment. Math. 12 (2003), no. 4, 477-485. 32. Michael Spivak, A comprehensive introduction t o differential geometry. Vol. I, second ed., Publish or Perish Inc., Wilmington, Del., 1979. A comprehensive introduction to differential geometry. Vol. 11, sec33. -, ond ed., Publish or Perish Inc., Wilmington, Del., 1979. 34. ___ , A comprehensive introduction t o differential geometry. Vol. IV, second ed., Publish or Perish Inc., Wilmington, Del., 1979. A comprehensive introduction to differential geometry. Vol. V , sec35. -,

824

ond ed., Publish or Perish Inc., Wilmington, Del., 1979. 36. R. Thom, Sur le cut-locus d'une varie'te' plonge'e, J. Differential Geometry 6 (1972), 577-586, Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays. 37. Martijn van Manen, The geometry of conflict sets, Rijksuniversiteit te Utrecht, Utrecht, 2003, Dissertation, Universiteit Utrecht, Utrecht, 2003. 38. C. T. C. Wall, Geometric properties of generic differentiable manifolds, Geometry and topology (Proc. I11 Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Springer, Berlin, 1977, pp. 707-774. Lecture Notes in Math., Vol. 597. 39. Frank W. Warner, The conjugate locus of a Riemannian manifold, Amer. J. Math. 87 (1965), 575404. 40. Alan D. Weinstein, The cut Locus and conjugate Locus of a riemannian manifold, Ann. of Math. (2) 87 (1968), 29-41. 41. Yosef Yomdin, O n the local structure of a generic central set, Compositio Math. 43 (1981), no. 2, 225-238.

825

COMPLEX CURVE SINGULARITIES: A BIASED INTRODUCTION Bernard TEISSIER Institut mathkmatique de Jussieu, U M R 7586 du C.N.R.S., Case 7012, 2 Place Jussieu, 75251 Paris Cedex 05, France E-mail: [email protected] The goal of this text is to provide an introduction to the local study of singular curves in complex analytic geometry. It contains resolution of singularities, the Newton polygon and the Newton parametrization, the classical NewtonPuiseux invariants, the semigroup associated to a branch as well as the specialization to the corresponding monomial curve, and a proof of BBzout’s theorem. It ends with the computation of the class of a projective plane curve. Some familiarity with the basic concepts of commutative algebra and complex analytic geometry is assumed. Keywords: curve singularities, Newton-Puiseux, monomial curves, BBzout Theorem, Plucker formula.

embedded

resolution,

1. What is a curve? In these lecturesa I will discuss singular points of complex analytic curves. A complex curve may locally be regarded as a family of points in complex affine space Ad(C)depending algebraically or analytically on one complex parameter. The dependence may be explicit, which means that the coordinates of our points depend explicitely on one parameter, as in: = Zl(t) z2 = Z 2 ( t ) 21

aThis text is an expansion of the notes of a course given at the CIMPA-LEBANON school in Beyrouth in July 2004 and of lectures at the ICTP College on Singularities.

826

where the zi(t)may be any functions C -+C although here, since we said we would consider families of points depending analytically on the parameter, we shall consider only polynomials or convergent power series. In this case the functions zi(t) may be defined only in a neighborhood of some point, which we will usually assume to be the origin t = 0 and it is convenient to assume that Z i ( 0 ) = 0 for all i ; one may reduce to this case by a translation on the coordinates zi and t. The curves which appear naturally are usually finite unions of parametrized curves as above. The parametrized curves are then the irreducible components or branches of the union. A curve is non singular at the origin if and only if it has only one component, for which the minimum of the t-adic orders of the zi(t) is equal to one. By the implicit function theorem, this means that the curve is locally analytically isomorphic to the complex line.

A germ of curve at a point (which we take to be the origin) is an equivalence class of curves given parametrically or by equations in an open neighborhood of the origin. Two such objects defined respectively on U and U' are equivalent if their restrictions to a third neighborhood U" c U n U' of the origin coincide. Of course when we talk of germs we think of representatives in some "sufficiently small" neighborhood of 0. Because of analyticity, to give a germ is equivalent to giving the convergent power series parametrizing the branches of the curve at the origin in some coordinate system. A curve may also be given implicitely, which means that it is given by equations. Here the fact that we are dealing with a curve should manifest itself in the fact that we have d - 1 equations in d variables, so that implicitely all the variables depend on one of them. However, the situation is not so simple in general, and a curve may need more than d - 1 equations. It is a fundamental fact of the theory of analytic curves that each germ can be decomposed uniquely as a union of irreducible germs, and that on an irreducible germ all coordinates can be expressed as convergent power series in the n-th root of one of them, for some integer m. This is often called the Newton-Puiseux theorem. The simplest case is that of a plane algebraic curve in the 2-dimensional af€ine space A2(C),defined by an equation f(z,y ) = 0 , where f E C [ z , y ] is a polynomial:

+

f(z,y ) = ao(z)yn a&)y"-l

+ * . . + a,(z),

with a i ( z ) E C [ z ] ,uo(z)# 0. Recall that C [ z l , .. . , z d ] , C ( z 1 , . . . , zd} and

827

C[[zl,. . . , z d ] ] denote respectively the ring of polynomials in d variables with complex coefficients, the ring of convergent complex power series in d variables and the ring of formal power series in d variables with complex coefficients. The degree n in y of the polynomial is the number of solutions in y (counted with multiplicities) for any fixed value zo of z which is "sufficiently general" in the sense that uo(z0) # 0. The curve is non-singular at the origin if at least one of the derivatives does not vanish at the origin. All this remains valid locally if f ( z ,y) is a convergent power series in z, y such that f (0,O) = 0, thanks to the Weierstrass preparation theorem which asserts that, if f(0,y) is not identically zero, which we can always assume at the cost of a linear change of variables, then f ( z ,y) can be written

2,

f ( z , y ) = u(z,y)(yn

+ al(z)yn-l + ... + a,(z)),

with ai(z) E C{z} where u ( z ,y) is a convergent series which does not vanish in a neighborhood of the origin. This means that u(z,y) is an invertible element in the ring C{z,y} and also that the curve defined by f ( z , y ) = 0 is just as well defined by the polynomial in y, in a sufficiently small neighborhood of the origin. We shall see later an interpretation for the total degree max(i j ) for monomials ziyj appearing in a polynomial f ( z ,y)). To a parametrized plane curve z ( t ) ,y ( t ) , one can associate a series f(z,y) such that f ( z ( t ) y, ( t ) ) = 0, by a process of elimination, to which we shall come back later. If the exponents of t appearing in the two series have no common divisor, this series is an irreducible element in the ring C{z,y}; it is not a product of two other series vanishing at the origin. Conversely the set of zeroes f(z,y) = 0 of a convergent power series in two variables defines coincides with a finite number of parametrized analytic curves in a small neighborhood of the origin of the plane, (the branches of the curve defined by the equation). The construction will be detailed below. The number r of these branches is given by the decomposition of f ( z ,y) into a product . . . f , " ~ of powers of irreducible elements in the ring C { x ,y}. We shall deal here with reduced curves, which means that all the ai are equal to one or, as we shall see that the exponents appearing in the power series z ( t ) ,y ( t ) of a parametrization of each branch, taken all together, have no non trivial common divisor. The description of a curve in Ad(C)for d _> 3 is much more complicated in general. One can prove that it requires at least d - 1 equations, but it may require much more, and then what is important is the ideal I which

+

fr'

828

these equations generate in the ring C[zl,. . . ,zd] or the ring C(z1,. . . ,zd}. In fact, it is the quotient ring C[zl,. . . ,zd]/I, or C(z1,. . . ,zd}/I which we consider, and the fact that we a re dealing with reduced complex curves is translated into the fact that we suppose that it is a reduced C-algebra with Krull dimension 1. In the case of plane curves, this ideal is a principal ideal so we have one equation only; more precisely, in the case of polynomials, this equation is unique up t o multiplication by a nonvanishing polynomial, i.e., a nonzero constant. In fact, it is the quotient ring C[zl,. . . ,zd]/I, or C(z1,. . . , z d } / I which we consider. Assuming that we are in the analytic (as opposed to algebraic) case and the ring 0 = C(z1, . . . ,zd}/I is an integral domain of dimension one, we can bring together the equational and parametric representations together in the following diagram of C-algebras: C(z1,. . . ,Zd}

-+

0 c C(t},

where the first map is the surjection with kernel I describing 0 as a quoiI d. tient, and the second one is determined by: zi H zi(t) 1 I The conclusion is that a complex curve, locally, should be thought of as a one-dimensional reduced local C-algebra 0, which is the localization of a C-algebra of finite type (algebraic case), or is a quotient of a convergent power series ring C(z1, . . . ,z d } (analytic case) or of a formal power series ring C[[z1,.. . ,zd]] (formal, or algebroid, case). Then we can consider it as given by an ideal in a regular local ring, namely the kernel of the map defining it as a quotient of a localization of a polynomial ring, or as a subring of a regular one-dimensional semi-local ring (its normalization) 8,which in the analytic or formal case corresponds to the parametrization. It is a theorem that in the cases considered here, the normalization B is a finitely generated 0-module. 2. What does one do with curves?

Real curves (often analytic) appear everywhere in mechanics as trajectories, and complex curves appear everywhere in Mathematics as soon as points depend on one parameter; for example given a family of square complex matrices depending on a complex parameter, the family of the eigenvalues lies on a complex curve. In order to study a real analytic curve, it is often useful to look at its complexification.

829

If a complex algebraic group G acts algebraically on a variety X , one may study the action by restricting it to one-parameter subgroups C* c G and the the orbit of each point x E X is a curve. Its closure in X is in general singular. More generally, the closure of a non singular curve will in general have singularities. We have seen also non singular algebraic curves which are themselves algebraic groups; elliptic curves. However, if we want to understand the totality of elliptic curves, we must also consider their singular limits. Curves also appear naturally in inductive steps in algebraic geometry: non-singular surfaces are studied in large part through the families of curves which they contain, and singular curves must appear in these families. To study the geometry of a non singular surface S, it is natural to project it to a non-singular space of the same dimension. The set of points of S where the projection p : S -+ P2 is not a local isomorphisme, the critical locus of the projection, is a curve (if it is not empty) which contains much information about the geometry of S. The critical locus may be a non singular curve, but its image under projection p will in general be singular. More generally, even if one is interested in non singular curves only, their plane projections will in general be singular, having at least ordinary double points or nodes In these lectures, I will mostly study different ways to transform curves into other curves, by deformation either of the parametric representation or of the equations, by taking the transforms of curves in A: under maps 2 -+ A:, and by projection (in the case of non plane curves). Since the passage from the implicit presentation t o the parametrization uses some form of the implicit function theorem, I will mostly work in the context of complex analytic functions, for which there is an implicit function theorem which does not exist for polynomials. If one wishes to work over a field different from C, one could replace convergent power series with formal power series with coefficients in a field, keeping most of the algebra but losing a lot of geometry or one could work in the henselizations of the polynomial ring and its quotients, keeping just about everything. If the field has positive characteristic new phenomena appear: Newton-Puiseux’s theorem is no longer true (see section 3 below). Let us call analytic algebra any C-algebra which is a quotient of a convergent power series ring C(z1,. . . , z d } . To any localization R, of a finitely generated local algebra R over the field of complex numbers at one of its

830

maximal ideals is associated in a unique way (up t o unique isomorphism) an analytic algebra RL, which has the property that any C-algebra morphism R, 4 A from R, to an analytic algebra A factors in a unique manner R, + l3: + A where RL + A is a morphism of analytic algebras. The candidate for Rh is simple to see: write a presentation R = C[T1,. . . ,T,]/I, where I is generated by finitely many polynomials. The maximal ideal m of R is the image of a maximal ideal iiz of C[Tl,. . . ,T d ] . By the nullstellensatz, the ideal iiz corresponds to a point ( a l , . . . , a d ) of the affine space Ad(C). Set Rh = C{Tl - a l , . . . , Td - ad}/Ih,where Ih is the ideal generated in C(T1 - a l , . . . ,Td - a d } by the polynomials which generate I . In the case of curves, the first serious difficulty comes from the fact that an irreducible polynomial P ( x , y ) E C[x,y] may well become reducible in C{x, y } = C [ x y]pz,yl. , In other words, the analytization of a local integral C-algebra may not be integral. Consider for example the nodal cubic with equation

y2 - x

2

-x3=0;

it is an irreducible affine plane curve, but the image of its equation in C{x, y } by the natural injection C[x,y ] C C{x, y } factors as

y2

-

x2 - x3

=

(y

+x

G ) ( y-x G ) .

The interaction between the global invariants of a plane projective curve and its singularities is also an important theme requiring the local study of singularities: - We know how to compute the class and the genus of a non singular plane projective curve of degree d. Assume now that it is singular; how does it affect the formulas for the class and genus? - The single most important result about plane curves is BQzout's theorem, which is the generalization of the fundamental theorem of algebra: Given two plane projective curves C and C' of degrees d and d' having n o common component, the number of their points of intersection counted with multiplicity is the product dd'. At the points of intersection where both curves are not regular and meeting transversally, how does one properly count the intersection multiplicity? can one effectively count it, given the equations of the two curves? In fact, as we shall see, the best situation to compute the intersection multiplicity is to have one curve given parametrically and the other given implicitely, although if both are given parametrically, there is a formula, due to Max

831

Noether (see [MI). What is the geometric meaning of intersection multiplicity? We know that we can normalize an algebraic curve to obtain a non singular algebraic curve, and it can be shown that the same is true in complex analytic geometry. The parametrization of an analytic curve mentioned above is in fact its normalization, provided we take care that the powers o f t appearing in the series are coprime; if this is the case, the inclusion 0 C C{t} induces an isomorphism of fraction fields and is therefore the normalization of 0. The normalization, however, is a priori difficult to compute from the equations of our curve. An algorithm to do precisely this, in fact to compute a parametrization from an equation of a plane curve, was given by Newton; it is based on the Newton polygon. 3. Newton’s study of plane curve singularities

Let us recall that the ring of power series in fractional powers with fixed denominator m of a variable 2 with fixed denominator m is defined as C[[zk]]= C[[z]][T]/(Tm - z), and similarly for C{zf}. The notation zf describes a multi-valued function of z, defined as a function not on C but on an m-fold ramified covering of C. The various ”determinations” are exchanged by multiplying one of them by an m-th root of unity. The work of Newton and Puiseux shows that functions on a branch can be viewed as functions of z f for some m. Let f(z,y) E C[[z,y]] be a formal power series without constant term. We seek series y(z) without constant term such that f(z,y(z)) = 0. Let us first eliminate a marginal case; if f(0,y) = 0, it means that f(z,y) is divisible by some power of z; let a be the maximum power of z dividing f(z, y), and let us set f(z,y) = zaf’(z,y). Geometrically, the equality f(0,y) = 0 means that the curve f ( z , y ) = 0 contains the y-axis, and the equality above means that this axis should be counted a times in the curve. This component may be parametrized by z = 0, y = t and we are left with the problem of parametrizing the rest of the curve, which is defined by f’(z,y) = 0. We now have f’(0, y) # 0, and we may thus reduce to the case f(0, y) # 0. From now on we shall assume that f(0, y) # 0. We may then write, since f(0,y) is a formal power series in y, f(0, y) = Y n d Y ) , with d o ) # 0. The proof of the existence of parametrizations proceeds by induction on the integer n. If n = 1, we have g(0,O) # 0, and by the implicit function

832

theorem there exists a unique formal power series y(x) E C[[z]]such that y(0) = 0 and f(x,y(z)) = 0. We now assume that n > 1. Considering series f(x,y) of the form yn - x‘J with n,q > 1 and (n,q ) = 1 shows that one cannot hope to find series in powers of x. Newton’s idea is to seek solutions which are fractional power series in x,that is, he seeks series in xh for some integer m, say $(xk)E C[[xh;l] such that f(z,$(.A)) = 0. More precisely he seeks solutions of the form: y = x’(Q

with

Q

# 0,

Y E

+ $o(.h))

&+, $0 without constant term. If we write

and substitute, we get Cai,jxi+uj(Q

+4o(xh))j

id

and we seek v , Q # 0 and a series $o(z&) such that this series is zero. In particular, its lowest order terms in x must be zero. Since $0 has no constant term, if we denote by p the minimum value of i v j for (2, j ) such that a i j # 0, we have

+

where h has no constant term. So Q must satisfy i+uj=p

For this equation to have a non-zero root in C , it is necessary and sufficient that the sum has more than one term. Let us consider in the (&$-plane the set of points ( i , j ) such that 0. It is a subset N ( f )of the first quadrant

Rg =

{ ( i , j ) /a 2 0, j 2

aid

#

O},

called the Newton cloud of the series f(x,y). Any two subset A and B of Rd can be added coordinate-wise, to give the Minkowski sum A+B = {a+b, a E A, b E B } o f A and B. Let us consider the subset N+(f) = N(f) Ri of R:; its boundary is a sort of staircase with possibly infinite vertical or horizontal parts. The Newton polygon P(f) of f ( x , y ) is defined as the boundary of the convex hull of N+(f). It is a broken line with infinite horizontal and vertical sides, possibly different from the coordinate axis.

+

833

I hl I I

1

i axis

Just above is a picture of a Newton polygon in the case where the infinite sides do coincide with the coordinate axis, or equivalently where the area bounded by the polygon is finite. Recall that the convex hull of a subset of Rd can be defined as the intersection of the half-spaces which contain it. A half-space is the set of points situatedi on one side of an f i n e hyperplane. Thus, the number p = min,,,j+o{i

+ vj}

is the minimal abscissa of the intersection points with the z-axis of the lines meeting N+(f).Let us denote by L, the line which gives with slope this minimum; an example in drawn on the picture. So the polynomial

2

i+uj=p

corresponds to the sum of the terms ai,jziyj such that (z,j)lies on the intersection of the line L, with the Newton polygon. A necessary and sufficient condition for this polynomial t o have more than one term is that $ is the slope of one of the sides of the Newton polygon. For simplicity of notation, let us call v the inclination of the line of slope Let us denote by the inclination of the "first side" of the Newton polygon of f , that is, the side with the smallest inclination. Let co be a non zero root of the corresponding equation, and let us make the change of

3.

834

variables 1 = z1

x

+Y1)

Y = (:.Q

The substitution in f (2, y ) gives h

e

f ( z l , z l ( Q+

~ l )= ) C

By definition of p, for each ai,j the series above as

z y h f l ( z l y1) , ,where We remark that

hi+ej ai,jzi (Q

# 0, we have hi + l j 2 p h , so we may factor

f l ( z 1 ,y1)

f l ( o ,y i )

+~1)’ .

=

= Cai,iz:i+ei-ph (cg

C

ai,j(Q

+Y l ) j .

+~1)’ ,

i+vj=p

and since ao,k # 0 by definition of n, the order in y1 of f l ( O , y l ) is 5 n. Since co has been chosen as a root of the polynomial C i + v j = p a i r j ~ , this order is 2 1. We remark that The order in y1 of f l ( 0 , y l ) is equal to n if and only if Q is a root of multiplicity n of the polynomial Ci+vj=p ai,jTj = 0 But then we must have an equality

which implies by the binomial formula and since C is a field of characteristic zero, that the term in T”-l is not zero; this is possible only if v is an integer and then the equality above shows that the ”first side of the Newton polygon” meets the horizontal axis at the point (vn,0), which corresponds to the monomial xun, which has the non zero coefficient (-l)nao,ncg, so it is actually the only finite side of the Newton polygon of f (z, y ) , which means that we may write in this case

f ( x ,y ) = ao,,(y

-

+

ai,jxiyi

with v E N, p = vn.

i+uj>p

Making the change of variables z =21 Y = Y 1 +cozy

the series f (z, y ) becomes

f’(z,y ) = ao,kyy

+ C

i+vj>p

ai,jzZ;(yl+~

0 z ~ ) j

835

The monomials which appear are of the form z;+”’yf-’, so that they all satisfy i vl v ( j - 1 ) = i vj > 1/12. This means that i f the order of fl(0, y1) is n, the Newton polygon of fl(z1,yl) still contains the point (0,n ) and the inclination u1 of its first side is strictly greater than v. The proof now proceeds as follows, : a) If the order in y1 of fi (0, y1) is less than n, by the induction hypothesis,

+ +

+

1

there exist an integer ml and a series $l(z?)

E C[[z$]] such that

By the definition of fl, this implies that

+

If we set m = mlh and +(zk) = zk(c0 + 1 ( z k ) ) E C[[zh]], we have f(z,+(zk))= 0 and the result in this case. b) If the order in y1 of f(0, y1) is still equal to n, we saw that v is an integer and the inclination of the first side of the Newton polygon of the function fl(z1,y1) obtained from f(z, y) as above is strictly greater than v. We now set vo = u E N and repeat the same analysis for f 1, defining a function f2(z2,y2). If again the order of fi(0, y2) is n, the slope of the first side of the Newton polygon of f l ( q , y l ) is an integer vl > vo and after the change of variables z = z2, y = y2 ~ z f ; o clzvl the slope of the Newton polygon has become greater than v1. There are two possibilities; - either after a finite number of such steps we get a function fp(zp, yp) such that f (0, yp) is of order < n, and by the induction hypothesis we have a

+

1

series $,(zG)

+

1

E

C[[zG]] such that

1

fP(%, $,(zG))

= 0, and so a series

such that f(z, y(z)) = 0; Or the order remains indefinitely equal to n and we have an infinite increasing sequence of integers

vo


<...
...

and a formal power series +,(z)

+ clzV’+ * . +

= cozvo

*

cp2”p

+ . . . E C[[z]]

such that the Newton polygon of the function foo(z,,y,) f(z, y) by the change of variables z = IC, , y = yoo $,(z)

+

obtained from has a Newton

836

polygon containing the point (0, n) and with inclination 0. This means that fco(zco, ym) is divisible by y&, so we may write fco(zco,yco) = Y&g(zcorYco)

This implies that the order of g(0, yco) is zero, so g(0,O) # 0. Geometrically, our curve is the non singular curve y = q5M(z) counted n times. Indeed, for each integer p , we have f(z,QZVO + clzyl + . . . + c p z ” P )

= 2”o+vl+-+vP

f P h O),

so that by Taylor’s expansion theorem, f(z,4(z)) = 0. This completes in the formal case the proof of the existence of a fractional power series such y(z)) = 0. that f(s,

In order to describe all the solutions of the equation f ( z , y ) = 0, it is convenient to develop a little more the formalism of the Newton polygon. Let P and P‘ be two Newton polygons; we can define their sum P P‘ as the boundary of the convex hull of the Minkowski sum of the convex domains in R t bounded by P and PI respectively. It is easy to verify that the following equality holds for f , f’ E C[[z,y]]

+

P(ff0 = P(f)+ P(f’). Any Newton polygon has a length and an height which are the length of the horizontal and vertical projections of its finite part, respectively. We say that a Newton polygon is elernentay if it has only one finite side. If it bounds a finite area, it is then uniquely determined by its length and height. We use the following notation for such an elementary Newton polygon.

1

837

We also need a little more algebra, beginning with the following fundamental theorem: One says that a holomorphic function f ( z 1 , .. . ,zd, y ) defined on a neighborhood of 0 in Cd x C is y-regular (of order n ) if f (0, y ) has a zero of finite order n at 0 E (0) x C . Geometrically this means that if we consider the germ of hypersurface (W,0) c C d x c defined by f ( z 1 , . . . ,xd, y ) = 0 and the first projection p : W ---t C d ,then for a small enough representative, if W is not empty (i.e., n 2 l),the fiber p-l(O) is the single point 0. In other words, the fiber is a finite subset of (0) x C . The general idea of the avatars of the Weierstrass preparation theorem is that finiteness of the fiber over one point x in an analytic map implies finiteness of the fibers above points sufficiently close to z. Theorem 3.1. (Weierstrass preparation Theorem) If f ( 2 1 , . . . ,X d , y ) i s regular of order n in y, there exist a unique polynomial of the form

. . ,a, y ) = y"

~ ( z l , .

+ al(z1,.. . ,zd)yn-' + + a n ( z l , .. . ,

zd)

with ai E C(x1,. . . , z d } and a convergent series u(z1,.. . ,xd) with u(0) # 0, i.e., invertible in C(x1,.. . , zd} such that we have the equality of convergent series

f( E l , .

*

I

zd,

Y ) = u(z1,*

* 7

.

zd, !/)p(xl,.

7

zd,

!/).

The polynomial P is said to be distinguished in y , or to be a Weierstrass polynomial. If we start with any power series f , we have the same result but in the ring of formal power series. It can be shown that, given a function f , for almost every choice of coordinates in C" x C , the function f is distinguished with respect to the last coordinate.

It follows from the Weierstrass preparation theorem that provided we have chosen coordinates such that f ( 0 ,y ) # 0 , say f ( 0 ,y ) = aqyq+- . . with aq # 0, it is equivalent to seek solutions of f ( z ,y) = 0 and of P ( z ,y ) = 0, where P ( x ,y ) is the Weierstrass polynomial u - ' ( z , y ) f ( z , y )= yq+a1(z)yq-'

+ . . a +

a,(z) = O withai(z) E C [ [ z ] ]

Now from an algebraic point of view, we must consider the field of fractions C ( ( z ) )of the integral domain C [ [ z ]the ] ; irreducible polynomial T" - z E C ( ( z ) ) [ Tdefines ] an algebraic extension of degree m of C((z)), denoted by C((zA))which is a Galois extension with Galois group equal to the group

838

p m of m-th roots of unity in C . The action of p m is exactly the change in determination of xh,determined by xh H w x k for w E p m . A series of the form y = C a i x k such that the greatest common divisor of m and all the exponents i which effectively appear is 1 gives m different series as w runs through p m .

Suppose now that our function f is an irreducible element of C [ [ xy,] ] ,and that the order in y of f (0, y ) is n < 00. Then the construction described above provides a series y ( x k ) with m 5 n such that f ( x ,y ( x k ) ) = 0. In fact m = n since f is irreducible.The product

is a polynomial Q ( x ,y ) E C [ [ z ] ] [ which, y] by the algorithm of division of polynomials in C ( ( x ) ) [ y ]divides , P ( x ,y ) ; the rest of the division of P by Q is a polynomial of degree < n - 1 with n different roots; it is zero. We have therefore Q ( z , y ) = P ( x , y ) and m = n in this case. We remark that the expansions y ( w x A ) all have the same initial exponent here and by the definition of Q ( x , y ) , only monomials xiyj with $ 2 appear, and the monomial xh actually appears. So we have verified:

i, +

i, 0

Proposition.- The Newton polygon of an irreducible series is elementary, and of the form {+}, where n is the order of f(0, y ) .

Now it is known that rings such as k [ [ xy, ] ] where , k is a field, or C { x ,y } are factorial ; each element has a decomposition f = . . . f,". where fi is irreducible, which means that it cannot be factored again as a product f i = gh in a non trivial way, that is, without g or h being an invertible element in k [ [ xy, ] ] ,(= a series with a non zero constant term). My aim now is to prove the following

fr'

Theorem 3.2. .- a) Let k be an algebraically closed field of characteristic zero, and let f E k [ [ x , y ] ]be a power series without constant term such that f (0, y ) # 0. Consider the decomposition f = uf,"' . . . f,". of f into irreducible Weierstrass polynomials f:i, with a factor u which is invertible in k [ [ z , y ] ]For . each index i, 1 5 i 5 r , there are power series without constant term zi(t),y i ( t ) E k [ [ t ] ]such that f ( z i ( t )y, i ( t ) ) = 0; we m a y choose xi(t) = tmi where mi is the degree of the Weierstrass polynomial f i , and y i ( t ) is then uniquely determined. Moreover i f we then write y i ( t ) = Gtli + . . . w i t h ci E k*, then the Newton polygon of f in the coordinates

839

(x,y)is the sum

Here we have to allow the case where for some i, y i ( t ) = 0, that is li = CQ. b) If k = C and f E C{x,y} is a convergent power series, the series xi(t) and y i ( t ) are also Convergent. Remark: if we do not assume f (0, y ) # 0 , a similar result holds, but we may no longer apply Weiertrass’ theorem and we have to allow expansions of the form x = 0, y = t and the corresponding Newton polygons appears as summands in N (f). The geometric interpretation of this result is that if we take any reduced analytic plane curve f = ufl . . . f T with f i irreducible, i.e., all ai = 1, the curve defined by f = 0 is a sufficiently small neighborhood of the origin is the analytic image of a representative of a complex-analytic map-germ

u(C, T

0)i

i= 1

-

(C2,0)

which we can explicitely build by using Newton’s method. Remark The Newton polygon depends upon the coordinates. One usually chooses the coordinates (x,y)in such a way that the degree of the Weierstrass polynomial is equal to the order of the equation f (2,y). I leave it as an exercize to show that if one writes the series f as a sum of homogeneous polynomials

f (x,Y) = f n ( G Y) 4-fn+l(X, Y) +

* * * 7

where fi is homogeneous of degree i, this condition is equivalent to: fn(0,Y) # 0. Conversely, given two power series x ( t ) ,y ( t ) E k[[t]] without constant term, one may eliminate t between them to produce an equation f (2,y) = 0 with the property that f(x(t),y(t) = 0. Indeed, by using the ”natural” elimination process (see[Tl])we may do this in such a way that eliminating t beween x(tq),y(tq) produces the equation fq(x,y ) , so that we may even represent parametrically a non-reduced equation. There are several ways to prove this theorem; one is to prove the convergence first, either directly by providing bounds for the coefficient of the series produced by Newton’s method, which works but is inelegant, or by

840

considering the analytic curve f(t", y) = 0, and proving that it is a ramified analytic covering of the t-axis; it is also the union of m non singular curves, so each of them is analytic, and this proves the convergence of the series. (see [L], 11.6). These proofs give no basis for generalizations to higher dimension, so I chose to present a geometric method of constructing the analytic map

r

i=1

This method was perfected by Hironaka and is the basis for his method of resolution in all dimensions over a field of characteristic zero. Remark In the study of analytic functions of one variable near one of their zeroes, a basic fact is that given two monomials xa, xb,one must divide or even C[x]. This allows us to write any series the other in C { x } , C[[z]], f(x) = xau(x) with u(0) # 0, in C{x},and the local behavior of f is determined by the integer a. It is no longer true that given two monomials in ( x , ~ )one , must divide the other; the typical example is the pair of monomials yn, xq. In particular, the ideal of C{z,y} generated by all the monomials appearing in the expansion f ( z , y ) = x a i j x i y j is no longer principal. However, since C{x, y} is a noetherian ring, this ideal is finitely generated. If we plot the quadrant Rij = ( i , j ) R: for each monomial xiyi appearing in our series, and observe that the integral points in this quadrant correspond to the monomials which are multiples of xi$, we have a graphic way of representing the generators of the ideal generated by all the monomials appearing in the series f: Consider the union of all the Rij for (z,j)/uij # 0; its boundary is a sort of staircase. Our generators correspond to the insteps of the staircase. The convex hull of the union is the Newton polygon. Note finally that from the viewpoint of considering lines L, : i jv = c as above, and where they meet the staircase, it is the convex hull which is relevant.

+

+

841

4. Puiseux exponents

Let f (x, y) E C{x, y} be such that f (0, y) = y"u(y) with u E C{y}, o(0) # 0. As we have just seen, it is equivalent to find solutions y(x) for f and to find roots of the Weierstass polynomial

P(x,y) = yn

+ u1(x)yn-' + + u,(z) * * *

corresponding to f . If the element f (2, y) E C{x, y} is irreducible, so is the Weierstrass polynomial in C{z}[y]. Newton's theorem tells us that such an irreducible polynomial has all its roots of the form

i=l

where w runs through the n-th roots of unity in C .

842

This is equivalent to the statement that an analytically irreducible curve as above can be parametrized in the following manner: 2 = tn

y=

CEl aiti

In particular, this shows that the polynomial P determines a Galois extension of the field of fractions C{{z}} of C { x } , and of the field of fractions C((z)) of C[[z]],with Galois group pn. A direct consequence of this is the: Theorem 4.1. (Newton-Puiseux Theorem). The algebraic closure of the field C{{z}} (resp. C((2)) ) is the field Un,lC{{z$}} (resp. Un21 C((x6))

1

This result is the algebraic counterpart of the fact that the fundamental group of a punctured disk is Z. A linear projection onto (C,O) of small representative of a germ of complex analytic curve ( X ,0) c (C2,0) which is finite (the curve does not contain the kernel of the linear projection as one of its components) can be restricted over a small punctured disk DQ= D, \ {0} to give a finite covering of DQ,which is connected for all sufficiently small q if and only if (X,O) is analytically irreducible at 0. It can be shown that conversely the connected coverings of a punctured disk correspond to irreducible curves as above, and that in this correspondence, the Galois group of the covering, which is of the form Z / k Z since the fundamental group of the disk is Z, corresponds to the Galois group of the extension of the field C{{z}} or of the field C ( ( 2 ) )defined as above by the curve. Let

be an equation for a branch ( X ,0) C (C2,0), which means that the series f is an irreducible element of C{x, y}. As we saw, we may assume thanks to the Weierstrass preparation theorem that f is of the form f ( x , y ) =yn+a1(z)yn-l

+...+ a,(z)

where n is the intersection multiplicity at 0 of the branch C with the axis 2 = 0.

843

As we saw, after possibly a change of coordinates to achieve that x = 0 is transversal to it at 0,the branch X can be parametrized near 0 as follows x ( t ) = tn y(t) = a,tm

+ am+ltm+l + . . . + a j t j + .

with m >. n

Let us now consider the following grouping of the terms of the series y ( t ) : set PO = n and let p1 be the smallest exponent appearing in y(t) which is not divisible by PO.If no such exponent exists, it means that y is a power series in z, so that our branch is analytically isomorphic to C , hence non singular. Let us suppose that this is not the case, and set e l = (n,,&), the greatest common divisor of these two integers. Now define ,& as the smallest exponent appearing in y(t) which is not divisible by e l . Define e2 = ( e l ,P 2 ) ; we have e2 < e l , and we continue in this manner. Having defined ei = (ei-I,&), we define pi+l as the smallest exponent appearing in y ( t ) which is not divisible by ei. Since the sequence of integers

n > el > e2 > . . . > ei > . . is strictly decreasing, there is an integer g such that e, = 1. At this point, we have structured our parametric representation as follows:

z ( t )= t n

+ + .. . + + apl+elt'l+el + + ~ p ~ + k ~ e ~ t ' l + ~ l ~ l +a&t'z + a&+ezt'z+e2 + . . . + ap,t'q +

y(t) = antn

akntkn

a2nt2n

+aplt'1

+apgtpg

*

+ apofltPg+l + * . .

q+eq-l

tPs+e,-l

+ ...

where by construction the coefficients cf the tpi; i 2 1 are not zero. Let us define integers ni and mi by the equalities ei-1

= niei,

,& = m i e i

forllilg

and note that we may rewrite the expansion of y into powers of t as an expansion of y into fractional powers of x as follows: y = an2

+ a2nx2 + . . . + akn,xk+ +...

The set of pairs of coprime integers (mi,ni) are sometimes also called the Puiseux characteristic pairs. Their datum is obviously equivalent to that of the characteristic exponents Pi. The sequence of integers B ( X ) =

844

(PO,01,. . . ,P,),

where PO = n, may be characterized algebraically as follows: let p n denote the group of n-th roots of unity. For w E p,, let us 2mik compute the order in t of the series y(t) - y(wt). If we write w = e n , we have y(wt) = anwntn

+ . . + aplwP1tP1 + . .. +

and we see that multiplying t by w does not affect the terms in t j n . The term in to1 is unchanged if and only if wP1 = 1, that is is an integer, i.e., kPl = In or kml = In1 with the notations introduced above. Since n1 and ml are coprime, this means that k is a multiple of n1, which is equivalent to saying that w belongs to the subgroup p~ of p,, consisting of nl n -- 722. . n,-th roots of unity. If this is the case, then the coefficients of all

%

+

71.1

the terms of the form tPl+jel in the Puiseux expansion are also unchanged when t is multiplied by w, and the first term which may change is ap,t&. An argument similar to the previous one shows that if w E p ~ then , wP2 = 1 nl if and only if w E p-, and so on. n1n2 Finally, if we denote by w the order in t of an element of C{t},we see that v(y(t) - y ( w t ) ) = pi if and only if w E p- nl"'ni-l

\ *p

for 1 5 i 5 g

This provides an algebraic characterization, and a sequence of cyclic subextensions C{z} c C { & }

c C{&T}

c . . . c c{5"I"1..."i } c ...c C{5+}

corresponding to the nested subgroups *p

of the group pn.

This shows that the sequence (PO,PI,. . . P,) depends only upon the ring inclusion C{x}c OX,^. We shall see later in a different way that this sequence does not depend upon the choice of coordinates (x,y) in which we write the Puiseux expansion as long as the curve x = 0 is transversal t o X. If this is not the case, one still obtains other characteristic exponents, which are related to the transversal ones by the inversion formula which I leave as an exercise (or see [PP] and [GP]). As an example consider the curve with equation y3 - x2 = 0.

Remark The Newton-Puiseux theorem is strictly a characteristic zero fact. It implies in particular that if a fractional power series in II: is a solution of an algebraic equation with coefficients in C[z] or C{II:} or C[[z]],the denominators of the exponents of x appearing in that power series are

845

bounded. Let k be a field of characteristic p, and consider the series where the exponents have unbounded denominators:

i= 1

it is a solution of the algebraic equation

as one can check directly. It is an Artin-Schreier equation.

The corresponding result in positive characteristic has been proved recently by Kiran S. Kedlaya (see [Ke]) and is quite a bit more delicate. 5. From parametrizations to equations

We have just seen an algorithm t o produce local parametrizations of the branches of a complex analytic plane curve from its equation. To go in the other direction is to eliminate for each parametrized branch the variable t between the equations 2 - z ( t ) = 0, y - y(t) = 0, and then make the product of the equations obtained. Elimination is in general computationally arduous. Is this special case, we have a direct method as follows: write our parametrization in the form x = tn,y = [ ( t )= a#. The product

xi

J L E h (Y - J(wt))

is invariant under the action of p n by t H w t ; it is a series f ( t ” ,y)) = f(x,y) which has the property that f(x,y) = 0 is an equation for our curve. However this method does not work for curves in 3-dimensional space, or in positive characteristic. Here is my favourite method (see [T5]) to compute images, explained in this special case.

5.1. Fitting ideals Let M be a finitely generated module over a commutative ncztherian ring A; then we have a presentation, which is an exact sequence of A-modules AQ-+ A P

4

M +0

The map A-linear map AQ-+ AP is represented, in the canonical basis, by a matrix with entries in A. For each integer j , consider the ideal F j ( M ) of A generated by the ( p - j ) x ( p - j ) minors of that matrix. Note that if j 2 p ,

846

then Fj ( M ) = A (the empty determinant is equal to l),and if p - j > q, then F j ( M ) = 0 (the ideal generated by the empty set is (0)). It is not very difficult to check that Fj(M) depends only on the A-module M, and not on the choice of presentation. Moreover, if A 4 B is a morphism of commutative rings, the sequence

Bq -+ BP

-+

M @ A B+ 0

is a presentation of the B-module B@AM , with the same matrix; therefore Fj(M @ A B) = Fj(M).B. One says that The formation of Fitting ideals commutes with base change. The most important feature of Fitting ideals, is as follows:

Proposition 5.1. A maximal ideal m of A contains Fj(M) if and only if dimAl,M

@A

A/m

>j.

Proof. Tensoring with A/m the presentation of M gives for each maximal ideal of A an exact sequence of A/m-vector spaces (A/m)q + (A/m)p -+ M @ A A/m

-+

0.

the dimension of the cokernel is > j if and only if the rank of the matrix describing the map (A/m)q + (A/m)p is < p - j , which means that all the p - j minors are 0 modulo m, which means that Fj(M) c m. Let me explain what this has to do with elimination: suppose that we have a branch parametrized by z(t),y(t). This gives a map C{z,y} -+ C{t}. Observe that this map of C-algebras gives C{t} the structure of a finitely generated C{z,y}-module. Indeed, since x H tn say, it is even a finitely generated C{x}-module, generated by ( l , t , . . . ,tn-’). We can therefore write a presentation of C{t} as C{z, y}-module:

C{z, y}q

-+ C{z, y}”

-+

C{t} -+ 0.

Now it is a theorem of commutative algebra that since C{z,y} is a 2dimensional regular local ring, for every finitely generated C{z, y}-module M , if we begin t o write a free resolution by writing M as a quotient of a finitely generated free C{z, y}-module, C{z, y}” -+ M , then writing the kernel of that map as a quotient of a finitely generated free C{x, y}-module, and so on, this has t o stop after 2 steps. This means that the kernel above is already free, so that in fact, in our case, we have an exact sequence 0 -+ C{X, y}q

4C{z,y}”

-+

C{t} -+ 0.

847

This immediately implies that we have q 5 p. On the other hand, the C{z, y}-module C{t} must be a torsion module, which means that it must be annihilated by some element of C{x, y}; intuitively this means that the image of our parametrization has an equation: an element f E C{z,y} such that fC{t} = f ( x ( t ) , y ( t ) )= 0. If this was not the case, there would be an ideal T c C{t) consisting of the elements which are annihilated by multiplication by some non zero element of C{z, y}. If we assume that T # C{z,y} and remark that by construction our map of algebras C{z,y} .+ C{t} induces an injection C{z,y} C C{t}/T, then either T # 0 and we have an injection of C{z,y} in a finite-dimensional vector space over C, which is absurd, or T = 0 and we have an injection C{z,y} c C{t}, but since C{t} is a finitely generated C{x,y}-module, the two rings should have the same dimension, by the third axiom of dimension theory (see [Ei], 8.1)l which is absurd. So C{t} is a torsion C{z,y}-module, which implies that the map induced by q5 after tensorization of our exact sequence by the field of fractions C{{z,y}} of C{z, y} is surjective, hence q 2 p and finally 4 = P. Now we know that q must be equal top, and that we have an exact sequence

0 .+ C{z,y}P

L C{x,y}P --+

C{t}

-+

0.

Proposition 5.2. The 0-th Fitting ideal of the C{z,y}-module C{t) is principal and generated by the determinant of the matrix encoding the homomorphism q5 in the canonical basis. Example 5.1. Consider the parametrization z = t 2 ,y = t3;it makes C{t} into a C{z,y}-module generated by (eo = 1 , e l = t ) . The relations are -ye0 z e l , x2eo - yel. In this case, p = 2 and the matrix q5 has entries

+

(2:J Exercise: 1) For any integer k, consider the curve parametrized by z = t2k,y= t3k.Show that the Fitting ideal is generated by (y2 - z3)lC. 2) Consider the curve in C3 parametrized by z = t3,y = t4,z = 0. In this case, the C{z, y, z}-module C{t} is generated by 1,t ,t 2 .Of course we can no longer hope to have q = p in its presentation, but one can compute a presentation ( see [T5],3.5.2) C{X,

y, z } -+ ~ C{Z,9, z } .~+ c{t}

.+

0.

and find that the Fitting ideal Fo(c{t})

=

(y3 - z4,z 3 ,2z2,zy2,z2y, Z~Z)C{Z, y, .}

848

It defines the plane curve y3 - x4 = 0, z = 0, plus a O-dimensional (embedded) component sticking out of the z = 0 plane. The appearance of this embedded component corresponds to the fact that we can embed our plane curve singularity in a family of space curve singularities, say x = t3, y = t4, z = vt5 for example, where u is a deformation parameter. Now if we compute the Fitting ideal of the image of the C3 x C given by x = t3,y = t 4 , z = ut5,v = u,and then map C x C set v = 0, we must find the Fitting ideal defining the image of our original map C -+ C 3 .This is one of the basic properties of Fitting ideals. On the other hand, by a classical result, the embedding dimension in a family of singularities can only increase by specialization. Since for u # 0 the image computed by the Fitting ideal has embedding dimension 3, it must also be 3 for v = 0, so the Fitting ideal must define something which contains our plane curve but has embedding dimension 3. Our zero-dimensional component increases the embedding dimension from 2 to 3. -+

Now we must prove that the generator of the O-th Fitting ideal is an acceptable equation of the image of our parametrization. Given our map 7r: C{x,y} -+ C{t} corresponding to the curve parametrization, we could say that the equation of the parametrized curve is given simply by the kernel K of this map of algebras. We are going to prove that the kernel K and the Fitting ideal have the same radical, and so define the same underlying set, but they are not equal in general, and the formation of the kernel does not commute with base extension while the formation of the Fitting ideal does. First, we must check that, with the notations of the definition of Fitting ideals, we have Fo(M).M = 0, which means that the Fitting ideal is contained in the kernel. In our case, where q = p , it follows directly from Cramer's rule if you interpret the statement as: det$.C{s, y}P E Image($). Note that this is true in the general situation of a finitely generated A-module; the Fitting ideal is contained in the annihilator of M . Secondly we must prove that K is contained in the radical of Fo(M). Take a non zero element h E K ; we have hM = 0, so that applying the rule of base extension to the map A -+ A[h-l], with A = C{x, y} in this case, we get Fo(M)A[h-'] = A[h-'], and since Fo(M) is finitely generated, this implies that there exists an integer s such that hs E Fo(M), and the result. So we have proved the inclusions

849 5.2.

A proof of Btkout's theorem (after [T5],31)

We begin with a Fitting definition of the resultant of two polynomials in one variable. Let A be a commutative ring and

P = po+p1x+*..+p,xn

Q = qo

+ qix + ... + qmXm

be two polynomials in A[X] Let us assume that p , and qm are invertible in A. The natural ring in which to treat the resultant is

considering the two polynomials

P = po + p 1 X + Q = 40 qlX

+

* * *

+pnX"

+ + qmXm

now with coefficients in A. The difference is that now the p i , qj have become indeterminates. Given any ring A and two polynomials P, Q as above with coefficients in A and highest coefficients invertible in A, there is a unique homomorphism ev: d + A such that ev(P) = P, ev(Q) = Q; it sends the indeterminate p i (resp. q j ) to the coefficient of X iin P (resp. Xj in 9). The d-module d[X]/Pis a free A-module of rank n, and multiplication by Q (which is injective since the p i , q j are indeterminates, gives us an exact sequence

0

-+

d[X]/P 3 A[X]/P -+ A[X]/(P, Q ) -+ 0.

This allows us to compute the O-th Fitting ideal of the A-module d [ X ] / ( P , Q ) as the determinant of the matrix of 4.

Definition 5.1. A universal resultant R ( P , Q ) of the universal polynomials P and Q is a generator with coprime integer coefficients of the 0th Fitting ideal of the d-module A[X]/(P,Q). Given a ring A and two polynomials as above, the resultant of P and Q is the image ev(R(P, Q) E A. It may be the zero element. Note that the ring A has a grading given by degpi = n - i, degqj = m - j. If we give X the degree 1, the polynomials P and Q are homogeneous of degree n and m respectively for the corresponding grading of d [ X ] . In order to deal with graded free modules, it is convenient t o introduce the following notation: If A is a graded ring, for any integer e, denote by A(e) the free graded A-module of rank one consisting of the ring A where

850

+

the degree of an element of degree i in A is of degree i e in A ( e ) . Any free graded A-module is a sum of A ( e i ) . The proof of BBzout’s theorem relies on the Fitting definition of the resultant and the following two lemmas:

Lemma 5.1. Let A be a graded ring; f o r any homogeneous homomorphism of degree zero between free graded A-modules P Q:

P

@A(ei)

--t

i= 1

@A(fj), j=1

setting M = cokerQ, the Fitting ideals F k ( M ) are homogeneous and moreover degFo(M) = deg(detQ) =

C ei - C i=l

fj.

j=1

Proof. To say that the morphism is of degree zero means that it sends an homogeneous element t o an homogeneous element of the same degree. This implies that the entries of the matrix of Q satisfy degQij = ei

- fj,

and this suffices to make the minors homogeneous; let us check it for the determinant. Each term in its expansion is a product $i,jl . . . $ipj,, where each i and j appear exactly once. It is homogeneous of degree

We can now compute the degree of R ( P ,Q ) E A. If we use the presentation given above, we find that the homomorphism q5 is of degree zero if we give each X i in the first copy of A [ X ] / Pthe degree i+m and keep X j of degree j in the second. Thus we find

Lemma 5.2. W e have the equality degR(P, Q) =

n

n

i=l

j=1

C(m+ i ) - Cj = mn.

Remark 5.1. 1) There are other presentations for the A-module A [ X ] / ( PQ). , For example

851

or 0 -+ A[Xl/(P.Q) a

+

AIXI/(Xn) @dJXl/(Xm)

+

H

(z,z)

d[XI/(P, Q)

+

0

(z - 6)

(Ti, b) H

+

The first of these two gives the usual Sylvester determinant, of size m n. The second follows from from the Chinese remainder theorem. 2) The total degree of a polynomial defining an affine plane curve is equal to the degree of the homogeneous polynomial in three variables defining the projective plane curve defining the closure of the affine curve in projective space; it is the degree of the curve. The other lemma is of the same nature and shows that the Fitting ideal locally computes the image of an intersection of two curve with a multiplicity equal to the intersection multiplicity of the two curves.

Lemma 5.3. Let R be a discrete valuation ring containing a representative of its residue field k, and let v be its valuation. Let

9 :RP -+ Rp be an homomorphism of free R-modules whose cokernel M is of finite length, i e . , a finite-dimensional vector space over k. Then we have the equality v(det9) = dimkM.

Proof. A discrete valuation ring is a principal ideal domain. By the main theorem on principal ideal domains we can find bases for both Rp such that the matrix representing 9 is diagonal, with entries a l , . . . , a p on the diagonal, say. Then clearly v(det9) = Cr=,v(ai)and dimkM = Cr=’=, dimkR/aiR. Thus it suffices to consider the case where p = 1. Then we have a = u7rs where 7r is a generator of the maximal ideal of R, and u is invertible in R. Then .(a) = s while R/aR is the k-vector space freely generated by the images of 1 , ~. . ,7rS-l. . 0 This applies to R = C { t } or R = C [ t ] ( t )the , valuation being the t-adic order. Now let us begin the proof of B6zout’s theorem. Let A be the graded ring C [ z l , z z ]and let P,Q E C[zo,z1,z2]be two homogeneous polynomials defining the curves C and D in the complex projective plane, of respective degrees m and n. We can write

P

=

Cr=lp i ( z 1 , x ~ ) x ;

degpi

=n

-

i

Q = Cj”==, qj(z1,z2)zj0 degqj = m - j .

852

After a change of coordinates, we may assume that the constants p , and qm are non zero, hence invertible in A. Geometrically this means that the point with homogeneous coordinates (1,0,0) does not lie on either of the curves C and D. As we saw above, there exists a homogeneous morphism of degree zero e v : A -+ A such that e v ( P ) = P, e v ( Q ) = Q, and if the resultant R ( P ,Q) is not zero, it is of degree mn by Lemma 5.2. I leave it as an exercise to check, using the factoriality of polynomial rings over C, that R(P,Q) = 0 if and only if C and D have a common component. Let us now consider the projection 7r: P2(C)\ (1,0,0) -+ P’(C) given by (20, IC~,ICH ~ ) ( I C ~ , I C ~It) .induces a well defined projection on C and on D since meither of them contains (1,0,0). For each point z E P1(C)there are ) IC. By the definition of the finitely many points y E C n D such that ~ ( y = resultant and the fact that the formation of the Fitting ideal commutes in particular with localization, we have the following equality:

R ( P ,Q)OP~,X= J’o(

@

Opz,y/(P, Q)Opz,y).

*(y)=x

vx(R(P7Q ) ) =

C

d i m c O ~ z , ~ / (QP), O P ~ , ~ . *(Y)=X

Since, if we assume that C and D have no common component, the resultant R(P,Q) is a homogeneous polynomial of degree mn in ( 2 1 , zz), it follows from the fundamental theorem of algebra applied to the homogeneous polynomial R that mn =

C vx(R(P, Q ) )= C XEPl

d i m c O ~ z , ~ / (QP ), O P ~ , ~ .

yECnD

This is BQzout’stheorem if we agree that the intersection multiplicity of C and D at y is equal to

(C, D)y = d i m c O ~ z , ~ / (Q P ,) O P ~ , ~ . We shall see later that there are many reasons to do that. Finally we get

Theorem 5.1. (BOout) For closed algebraic curves in P2(C)without comm o n component, we have degC.degD =

C yECnD

(C,D)y.

853 6. Resolution of plane curves

Let us consider the projective space P"(C) as the space of lines through the origin in Cn+'. If we choose coordinates X O , . . . ,x, on Cn+l the projective space is covered by affine charts Ui, the points of each Ui corresponding to the lines contained in the open set xi # 0. It is customary to take homogeneous coordinates ( U O : . : u,) on the projective space, corresponding to the lines given parametrically by xi = uit, or by the equations xiuj - xjui = 0 , where it is enough to take the n equations for which j = i 1 and i < n. The term "homogeneous coordinates " means that for any X E k* the coordinates ( U O : . .. : u,) and ( X u 0 : ... : Xu,) define the same point. Now consider the subvariety 2 of the product space Cn+l x P" defined by these n equations. It is a nonsingular algebraic variety of dimension n+ 1 and the first projection induces an algebraic morphism Bo : 2 t Cn+'. The fiber B t l ( 0 ) is the entire projective space P n ( k ) since when all xi are zero, all the equations between the uj vanish,while the fiber BC1(x) for a point x # 0 consists of a unique point because then the coordinates xi determine uniquely the ratios of the uj which means a point of P"(k). Blowing up a point "replaces the observer at the point by what he sees", because the observer essentially sees a projective space (in fact a sphere, if we think of a real observer, but this is just a metaphor). A basic properties of blowing up is that it separates lines: in fact consider the algebraic map 6: Cn+l \ (0) + Pn which to a point outside the origin associates the line joining the origin to this point. Of course we cannot extend the definition of this map through the origin; The graph of 6 however, is an algebraic subvariety of (Cnfl \ (0)) x Pn,and we may take the closure (for the strong topology if k = C, or for the Zariski topology) of this graph. It is a good exercise to check that this closure coincides with 2 as defined above. A point of Bt1(O)is precisely a direction of line, so the map 6 o Bo can be defined there as the map which to this point associates the direction: in 2 we have separated all the lines meeting at the origin. Let us consider in more detail the case n = 1. Then 2 is a surface covered by two affine charts corresponding to the charts of the projective space: for convenience of notation set uo = u, u1 = w , xo = x, x1 = y so that 2 is defined by vx - uy = 0. On the open set U of Z where u # 0 we may taxe as coordinates x1 = x , y1 = and then the map induced by Bo on U is described in these coordinates by

+

854

and similarly on the open set V defined by u # 0, we take as coordinates x1 = ,: y1 = y and the map Bo is described by

Bo = z1y1 YoBo= Y1 5

0

Remark that in the first chart the projective space BF1(0) is defined by x1 = 0 and in the second by y1 = 0 (remember that they are coordinates on two distinct charts, and on the intersection of the two charts they define the same subspace). It is a crucial property of blowing up that it transforms the blown-up subspace (here the origin) into a subspace defined locally by one equation (called a divisor); it is a good exercise to check that this is the case in any dimension. The space BF1(O) is called the exceptionnal divisor. We are now able to study the effect on a function f ( z , y ) (formal or convergent) of its composition with the map Bo. Consider the expansion of f as a sum of homogeneous polynomials

f(Z,Y) = f m ( x , Y ) + f m + l ( ~ , Y )+ . . . + f m + k ( x , Y ) + . . . where

fj

9

is homogeneous of degree j. In the chart U ,we may write

f 0 Bo = f ( z 1 , W l ) = x;" ( f m(1,Y1) w f m + l ( 1 , Y1)

+

+ . . + $fm+k *

(1,Y1)

+

* *

.)

and there is a similar expansion in the other chart. Now if we look at the zero set of f o Bo we see that in each chart it contains the exceptionnal divisor counted m times. If we remove this exceptionnal divisor as many times as possible, i.e., divide f o Bo by x;" in the first chart and by y;" in the second, we obtain the equation of a curve on the surface 2, either formal or defined near BF1(O),which no longer contains the exceptionnal divisor. This curve is called the strict transform of the original curve. We also say that the equation obtained in this way is the strict transform o f f ) . In the first chart it is xTmf(x1,x1yl), and in the second y ; m f ( z l y l , y l ) . By construction, the strict transform meets the exceptionnal divisor only in finitely many points; let us determine them: in the first chart they are given by f m ( l , y1) = 0 and in the second, by fm(l, y1) = 0. By construction of the projective space the points we seek are therefore the points in the projective line defined by the homogeneous equation f m ( u , u ) = 0. The homogeneous polynomial f m of lowest degree appearing in f ( x ,y) is called the initial form and f m ( a , y ) = 0 is a union of m lines (counted with multiplicity) called the tangent cone of f at the point 0. So we see that the strict transform of f meets the exceptional divisor at the points in this

855

projective space corresponding to the lines which are in the tangent cone at 0 of our curve. In particular, if OUT curve has two components with tangent cones meeting only at the origin, their strict transforms are disjoint. Consider for example f ( ~ , y = ) (y2 - z3)(y3- z2). In order to analyze in more detail what goes on, we have to assume that k is algebraically closed, which we will do f r o m now on, and introduce the concept of intersection number of two curves at a point. The simplest definition (but not the most useful for computations) is the following: Let f , h E k[[z,y]] be series without constant term and without common irreducible factor. Let (f,h) denote the ideal generated by f and h in k [ [ ~y ,] ] . Then the dimension dim"z,

Y I l / ( f7 h)

is finite and is by definition the intersection number of the two curves at 0. If k = C and f , h are in C{X,Y}, then the dimension above is also dimC{z, Y}/(f, h) where now (f,h ) is the ideal generated in C{X,y}. To prove the finiteness we first remark that it is sufficient to prove it after replacing k by its algebraic closure and then we may use the Hilbert nullstellensatz which tells us that since f = 0, h = 0 meet only at the origin, the ideal (f,h ) contains a power of the maximal ideal m = (z,y ) say mN. This implies the finiteness since k[[z,y]]/(f,h ) is then a quotient vector space of k [ [ z ,y]]/mN and also shows that we may without changing the ideal assume that f , h are polynomials of degree < N , so that for example if f,h are convergent power series the vector spaces C{X,y}/( f , h) and C[[X, y ] ] / ( fh) , are equal. The definition of intersection multiplicity at the point 0, of the two curves f = 0, h = 0, say in the analytic case is then

(f, qo= dimcC{z,y}/(f,

h).

Note that we use large parentheses for the intersection number, small ones for the ideal generated by f,g. In any case this definition of the intersection multiplicity has the advantage to suggest the following intuitive interpretation : Consider a 1-parameter deformation of one of the two functions, say f E; it is possible to show that if f,h converge in a nice neighborhood U of 0, for small enough c, then the two curves h = 0, f + E = 0 meet in U transversally at points which are non-singular on each. Moreover, these points tend t o

+

856

0 as E tends to 0, and the number of these points is dimC{x, y}/(f,h). So this number may be thought of as the number of ordinary intersections (i.e., transverse intersection of non-singular curves) which are concentrated at 0. There is another way to present this intersection number, which is very useful for computations: Suppose that h(z,y) = uh:' . . . h: with u(0)# 0. For each i, 1 5 i 5 T , let us parametrize the curve hi(^, y) = 0 by x(ti), y(ti). Now substitute these power series in f(z,y); we get a series in ti, the order of which we denote by Ii. Then we have

Ii = dimcC{z, Y}/(f, hi) , and

Remark: Given a germ of curve f

= 0,

where

f = fm + fm+z +

* * *

1

its multiplicity a t the origin may be defined as the smallest degree m of a monomial appearing in the series f . A better definition is to say that the multiplicity is the intersection number (f,t)ofor a sufficiently general linear form l . In fact, we have

with equality if and only if the line t(x,y) = 0 is not in the tangent cone defined by fm(x,y) = 0. Indeed, we may parametrize the line e = 0 by x = cut, y = Pt; then we substitute in f: f(Qt, P t ) = fm(% P)t"

+ fm+l(Q, P Y + l + . .

'

is of order 2 m, and of order m exactly if and only if fm(cu,p)# 0. It is convenient, given a curve f(z,y) and a point z in the plane, to define y , ) centered at z , the multiplicity o f f at z as follows: take coordinates (d, which means that they vanish at z ; if z = (a,b) we may take x' = x-a, y' = y - b. Then expand f in those coordinates (of course we assume that z is in the domain of convergence of f ) . We get f'(d,y') = f ( a + x', b + y'). Then we compute the lowest degree terms appearing in the expansion of f' and denote this by m,(f) or, if X is

857

the curve f ( x , y ) = 0 , by m , ( X ) . We see that m , ( f ) = 0 unless f (2) = 0, and that if e is a line through z , we have m , ( X ) 5 ( X , e ) , with equality except if C is in the tangent cone of X at z. Let us apply this, in our blowing up as described above, to the line x1 = 0 (the exceptionnal divisor) and the strict transform f 1 ( x l , y l ) = 0, at a point x' with coordinates x1 = 0,yl = t l ) where f m ( l , t l ) = 0 i.e., a point of intersection of the strict transform with the exceptionnal divisor. We have fl

= f m ( k Y 1 )+z1fm+1(l,y1)

+ ..

and if we denote by ex) the multiplicity of tl as a root of the polynomial f m ( l , Y ) ,it follows from what we saw above that we have

with equality unless the curve f l ( x 1 , y l ) = 0 is tangent to the exceptionnal divisor at the point x', in the sense that the tangent at x' to the exceptionnal divisor is in the tangent cone of f l = 0 at the point x'. Since the multiplicity of f l is zero at points where fm(l, y 1 ) does not vanish, we see that if we look at all the points x' in the blown up surface 2 which are mapped to our origin by the projection 2 + C2,which we denote by x' + 0, we have

so that in particular, if there is a point x' of the strict transform X ' of X which is mapped to 0 and is of multiplicity m on f l = 0, then it is the only point of X' mapped to 0 and X' is transversal to the exceptionnal divisor at x'. This fact and its generalizations play a crucial role in Hironaka's proof of the resolution of singularities. In order to show that the situation which we have just described cannot persist indefinitely in a sequence of blowing ups, we have to use the intersection number in another manner, according to Hironaka: Given a germ of a plane curve ( X ,x ) with r branches ( X i ,x ) l l i l r and a nonsingular curve W through the point x , define the contact exponent of W with X at x as follows:

858

and the contact exponent of X at x as followsb

& ( X ) = maxwb,(W, X ) , where W runs through the set of germs at z of non-singular curves.

Lemma 6.1. Let f ( x ,y ) = 0 be an equation f o r X . If the coordinates (z, y ) are chosen in such a way that x = 0 is not tangent to X at x and W is defined by y = 0, the rational number S z ( W , X ) is the inclination of the first side of the Newton polygon o f f (x,y). By definition of b z ( W , X ) is enough to prove that for an irreducible f, the inclination of the only side of it Newton polygon is $ (:?, but if we parametrize X by x = tm, y = t q . , we find that the transversality condition implies m 5 q, and we have ( X ,W ) , = q; the result follows.

+ --

Lemma 6.2. Assume that W is the curve y = 0 and that f ( x , y ) is in Weierstrass form, i. e.,

f(.,

Y ) = Yn

+ al(x)yn-l + . . . + a,(z)

ai(z)E C{z},

then the inclination of the first side of its Newton polygon is

Here as usual vo(a(x)) denotes the order of vanishing at the origin of the series a(.). Indeed, the point (0, n) is a vertex of the first side of the Newton polygon, and the lemma is just the observation that if we write a i ( z ) = aizcl+. . . , the other vertices of the Newton polygon are among the points (q,n - i ) , which follows directly from the definition.

A nonsingular curve W such that & ( W , X ) = S,(X) is said to have maximal contact at x . non singular curves with maximal contact are the nonsingular curves which it is hardest to separate from X by a succession of blowing ups (in the sense of separating strict transforms), and so when they eventually separate, something nice should happen; indeed once they separate, there is no point of multiplicity m , ( X ) in the iterated strict transform mapping to z. As one says, ”the multiplicity has dropped”. Hironaka’s bThis & ( X ) is not related to the 6 invariant of a singularity which measures the diminution of genus due to the presence of the singularity, and which is denoted in the same way. These are the classical notations, however.

859

approach t o resolution uses the existence of varieties with maximal contact to build an induction on the dimension. The next step is to prove the existence of curves with maximal contact. Assume that a non singular curve W defined by y = 0 does not have maximal contact with X at x. We way assume that the curve x = 0 is transversal to f ( x , y) = 0, which means that f(0, y) = aO,mym . . , where m is the multiplicity of f at 0. By a change of variable y = (ao,m)ky’, which does not change the contacts, we may assume that ao,m = 1. To say that &(W,X) < S,(X) means that there is a series A(x) such that the contact of the curve f(x,y) = 0 with y - A(z) = 0 is greater than its contact with y = 0. By a change of the coordinate x which does not affect the contacts, we may assume that A(%)= <xd for some integer d and E C * .Let us now compute the power series expansion in the coordinates X’ = X,y’ = y - A(%);

+.

<

f’(z,y’) =

C

ai,jxi(y’

+ <xd)j=

C

ak,exky/e.

$+elm

$+j?m

+

By expanding the powers of y‘ Exd we get, for each ( i , j ) , and k 5 j the inequality j - k 2 m but we know that $ +j 2 m. From this follows the inequality d 2 6 . Isolating the terms which lie on the first side of the Newton polygon, we get:

+

$+j=m

$+j>m

$+e>m

and the slope of the first side of the Newton polygon of the right-hand side > 6. Let us first assume that 6 = 1. Remark that all the terms xkyte with $ ‘k 2 m except ylm are in the ideal (2, Y ’ ) ~ + ’ . Therefore we must have the equality is 6’

+

C

ai,jxiyj = ylm mod.(x, y)m+l

$+j=m

so that the left hand side is the m-th power of y - <xd. This implies that d = 1 = 6 since the left hand side is homogeneous. If 6 > 1we follow the same method. Since we know that d 2 6, it is easy to check that the ideal of k[[x,y]] generated by the monomials zkyte , $ e 2 m , k # 0 is contained in the ideal I generated by the monomials

+

860

xiyj ,

+j

> m. Looking at the equation (*) modulo I gives us . .

ai,jxzy3= ytm m0d.I i+j=m

which again by homogeneity shows that d = S and the sum on the left hand side is (y - exd)". Note that this argument also works if 6,(X) = 00. So there are two possibilities: 1) We have S,(W, X ) < S,(X); in this case the sum of the terms of f(x,y) lying on the first side of the Newton polygon is of the form ( y - Exd)". 2) The sum of the terms of f(x,y) lying on the first side of the Newton polygon is not of the form (y - Exd)". In the first case, as we have seen, d = S,(W,X). We make the change of variables xt = x ; yt = y - Exd and in the new coordinates xt,y', if W' is the curve yt = 0, we have S,(W', X ) > &(W,X ) . This follows easily from the computation we have just made; an effect of the change of variables is that all the terms lying on the first side of the Newton polygon, of inclination d, are transformed into the single term monomial ytm. So the inclination of the new Newton polygon has to be > d;but we know this inclination to be S,(Wt,X ) . If we have not reached S,(X), we continue the same procedure, and after possibly infinitely many steps, i.e., after a change of variables of the form

x' = x ;

yt

=y

- ElXd1

- JZXdZ - . . . -

ST&

- . ..

we reach the stage where the sum of terms on the first side of the Newton polygon is not a m - th power, so S,(W,, X ) = S,(X), with s possibly infinite. Since the denominators of the S,(W,X)'s are bounded, the series is infinite only in the case where S,(X) = co. At least formally this series converges, since we have dl > dz > . . > d, > . . . , but we can omit the proof of convergence if we work in C{x,y} since the equality S,(X) = 00 means that in some coordinates f(x,y) is of the form u(x,y)y" where u is an invertible element in Ic[[x,y]]; indeed for any other case, we see from the definition that S,(X) < 00. But the Weierstrass preparation theorem tells us that if such a presentation exists with formal power series, it also exists with convergent power series, so that the series defining our final coordinates converges. So in all cases, we can find a nonsingular curve W which has maximal contact with X at x, i.e., such that S,(W,X) = S,(X). Remark that all the discussion above is valid on a germ of a non singular surface, since

86 1

it is analytically isomorphic to the plane. The definition of the blowing up is independant of the choice of coordinates, and makes sense on any nonsingular surface. The next step is to study the behavior of the contact under blowing up of the origin. I will leave the proof of this as an exercise, since it is a direct application of what we have just seen and the definition of blowing up:

Theorem 6.1. (Hironaka) Let m be a n integer, let f ( x , y ) = 0 define a germ of a plane curve, ( X ,0 ) c (C,O) of multiplicity m and let (W,O)c ( C ,0 ) be a non singular curve with maximal contact with X at 0. If, after blowing up the point 0 by the map Bo: Z C2, there is a point x' E X' of multiplicity m in the strict transform X' c 2 of X , then 1 ) The point x' is the only point of X ' mapped to 0 by Bo, 2) The strict transform W' of W by Bo contains the point x', and W' has maximal contact with X ' at x', 3) W e have the equality S,! (W',X I ) = S, (W,X ) - 1. -+

Corollary 6.1. The maximal length of a sequence of infinitely. near points of multiplicity m o n the strict transforms of X , each mapping to its predecessor in successive blowing ups

...

---f

Z(T) -+

z(v-1) +

. . . -+ 2 ( 2 )

-+ z(1)+ c 2

is equal to the integral part [d,(X)]. This suffices to show that unless the curve is of the form ym = 0, the multiplicity of its strict transform in the sequence of blowing ups obtained by blowing up at each step the points of maximal multiplicity drops after a finite number of steps. By induction on the multiplicity, this proves the resolution of the singularity of X at 0 by a finite number of blowing ups of points on non singular surfaces.We should remark that the map X' X --f

of the strict transform of X to X is defined by itself, without any reference to an embedding (X,O) c (C2,0)(see [K]). We have proved a local result, but if now we consider any algebraic or analytic curve, it has finitely many singular points, and the local resolution processes at each point are independant, so we have:

Theorem 6.2. Given a n algebraic or analytic plane curve X there exists a finite sequence of point blowing ups such that in the composed map X' -+ X the curve X' has n o singularities.

862

Actually we can get, by the same method, a better result, known as embedded resolution and originally due to Max Nether, as follows:

Theorem 6.3. Given a curve X on a non singular surface S , there exists a finite sequence of blowing ups of points

S their compositum, then the insuch that if we denote by 7r: S(")= S' verse image of the singular points of X (the exceptionnal divisor) is a union of non singular curves ( each isomorphic to P'(C)) meeting transversally o n the non singular surface S', and the strict transform X' of X by IT is a n o n singular curve meeting transversally these curves. In analytic terms, if f ( x ,y) = 0 is a local equation for X in S , then f o 7r is, at every point x' of S', of the form (f o = uavbfor suitable local coordinates of S' at 2'. Of course a and b will be zero unless we have x' E x-l(X).The induced map 7r: X' --+ X is a resolution of singularities of X. If we fix a singular point x E X , let r be the number of analytically irreducible components of the germ ( X ,x). The number of points in 7r-'(x) is equal to r and for a small enough representative X , of the germ (X,x), the part 7r-'(Xz) of X' lying over X , consists of r non singular curves Di, each marked with one of the points of 7r-'(x). The image by 7r of each of these non singular curves Di is one of the irreducible components of X,. If we choose for each non singular curve Di a coordinate ti vanishing at the only point zi of Di lying over x , then Di is described parametrically, in local coordinates (u, v) on S' centered at zi, by convergent power series u(ti),v(ti),because of the implicit function theorem. Since the map 7r: S' -+ S is a composition of algebraic maps, x o 7r and y 07r are at worst convergent power series in (u,v), so when we restrict them to Di, we get convergent power series in ti. This shows that each branch of our curve has a convergent parametrization, and from this we deduce that the formal parametrization constructed by Newton's method converges. Note that this convergence argument works equally well with the first resolution theorem. The new fact in the resolution result above with respect to the resolution theorem is the transversality of the strict transform with the exceptionnal divisor, which is not part of the resolution theorem as we have stated it above. The proof of this improvement is not difficult: it amounts to resolving singularities, by a sequence of points blowing up, of the union of the strict transform and the exceptional divisor of the map which resolves the singularities of X .

863

As an example, given an integer m > 1, after one blowing up the strict transform of a curve with equation ym - zmfl = 0, is non singular, but it is not transversal to the exceptionnal divisor. It is the first example of a fundamental fact of analytic or algebraic geometry: you can make spaces (in fact, their strict transforms) transversal by well chosen sequences of blowing ups. 7. Resolution of space curves 7.1. Integral dependance

To prove a resolution theorem for space curves, one meets the difficulty that their equations may be complicated (for example to define a curve in C" one may need more that n - 1 equations; those for which n - 1 equations suffice are called complete intersections, and also that rather different looking sets of equations may generate the same ideal in C(z1,. . . ,z,} and therefore define the same curve. In the proofs above we have used constructions which depend heavily on the equation. Moreover, even to show that a germ of a complex curve in C d has a finite number of irreducible components, which are analytic germs, is not completely trivial (see [L 1, 11.5). There are two possibilities: we can conceptualize and abstract the proof for plane curves to make it less dependent on the equation, or try to reduce to the plane curve case. As it happens, the two methods are not so different, at least for one of the ways of abstracting the ideas. To reduce to the plane curve case, the natural idea is to project the space curve X to a plane curve X I . One can then show that a resolution of X 1 has to map to X , and that this map is a resolution of the singularities of X!. The key idea is that of normalization. The Italian geometers called normal a projective variety 2 E P" having the property that any map 2' -+ 2 presenting 2 as a "general" projection by a linear map P"' \ L 4 P" of an algebraic variety 2' c P"' had to be an isomorphism. A typical non normal surface in P3 is therefore a general projection of a non singular surface in P4 which cannot be embedded in P3;such a projection has a curve of double points, on which are finitely many more complicated singular points, the "pinch points". Here the meaning of "general" has to be made precise; The variety 2 is normal if any map 7r: 2' 4 2 which a) is finite-to-one and b) induces an isomorphisme 2' \ z - ' ( U ) -+ U,over the complement U of a closed algebraic or analytic subset of 2 of smaller dimension,

864

is an isomorphism. The resolution theorem we saw above shows that a singular curve in P2 cannot be normal. The concept of normalization was ”localized” and transfigured into a concept of commutative algebra as follows: Recall that the total ring of quotients of a ring A is the ring of equivalence classes of couples ( a ,b) of elements of A , where b is not a zero divisor in A with addition ( a , b) (a’, b‘) = (ab’ ba’, bb’) and component wise multiplication, the equivalence being ( a ,b) E (a’, b’) when ab’ - ba’ = 0. The map a H ( a , 1) induces an injection of A in F and we indentify A with its image in F . If A is an integral domain, F is its field of fractions.

+

+

Definition Let A be a commutative ring without nilpotent elements, and let F be its total ring of quotients. Definition.- An element h E F is integral over A if it satisfies an equation hk

+ alhk-l + . . . + U k = 0

with ai E A .

Example.- Consider the germ of plane curve X in C2 defined by the equation y p - xq = 0. The quotient 0 of the ring C { x ,y} by the ideal generated by y* - xq is the ring of germs of analytic functions on the germ X (the restrictions to X of two analytic functions on C2 coincide if and only if their difference is in the ideal). The ring 0 is an integral domain; let K be its field of fractions. If we keep the notations x,y, etc.. for the restrictions t o X of functions on C2,we have f E F . I claim that if p 5 q , it is integral over 0 ; indeed, we have the relation

We can remark that the function f is defined and analytic on the strict transform of X by the blowing up of the origin for any sufficiently small representative of the germ X. We remark also that the condition p 5 q is equivalent to saying that the meromorphic function remains bounded on X for any small representative.

Proposition 7.1. Given a ring A without nilpotent elements, let F be its total ring of fractions; the set of elements of F integral over A is a ring for the operations induced by those of F . This ring is called the normalization of A (or the integral closure of A in F ) and often denoted by 2.Of course we have A c 2;a ring such that

865

A = 3 is said to be integrally closed. It is not difficult to check that 2 is integrally closed. If A is ncetherian and integrally closed, any injective map A + B to a subring B of the total ring of fractions of A which makes B into a finite Amodule is an isomorphism; this is the translation of the original definition of normality. To prove it, check that if h is an element of B , the powers of h cannot all be linearly independant over A, so h satisfies an integral dependance relation, and if A is normal, it is in A! An important theorem is that if A is an analytic algebra, i.e., a quotient of a convergent power series ring by some ideal, then 2 is a finite sum of integrally closed analytic algebras, and moreover that the injection A -+ 3 makes 2 into a finitely generated A-module. Taking a common denominator (in F ) for a finite set of generators of the A-module 2,we see that the ("conductor") ideal C = { d E 71, d . 3 c A } is not zero. Another important fact is that if the analytic algebra of germs of functions on a curve at a point is normal, the point is non singular on the curve, and the analytic algebra is isomorphic to a convergent power series ring in one variable C { t } . ([L 1, VI.3, Thm.2) 7.2. The 6 invariant of a plane curve singularity

Let 0 be the analytic algebra of a germ of curve (X, 0), plane or not, and let 0 be its normalization. Since it is an 0-module of finite type with the same total ring of quotients, a version of the Hilbert Nullstellensatz shows that the quotient vector space over C is finite dimensional. So we may define an invariant to measure how far 0 is from being integrally closed, i.e., regular:

-

-

0 6x,0 = dimc0 In the case of plane curves, this invariant is known as the diminution of genus which the presence of the singularity imposes on the curve: Take a projective plane curve C of degree d . If it is non singular, it is topologically a differentiable surface, classified by its genus. This genus is equal to ( d - l ) ( d - 2) g(C) = 2 If the curve has singularities, then its normalization is topologically a differentiable surface, and its genus is

( d - l ) ( d - 2) =

2

- CX€C~C,X,

866

where the sum on the right is finite since b ~ is ,nonzero ~ only at singular points, and those are finite in number. The genus of the normalization of C is traditionally called the geometric genus of C. Moreover, the local invariant b ~ has, a~local geometrical interpretation, (see [T4], [T6]) which I will describe only in the case of a branch, for simplicity: Let t n , y ( t )be a parametrization of our branch X . Consider the product of the normalization of X with itself, with coordinates (t't') and the two curves in (C2,0) = x 0) defined by

(x x,

The intersection number of these two curves at the origin is equal to 2 6 ~ ; if now we perturb slightly the parametrization of X by tn avt, y ( t ) pvt with two "general " complex numbers a, p, we can see that the two curves now have deformed and for small v they now meet transversally _ equations _ ~ in X x X . this means that the curve defined parametrically by in 2 6 points tn vat, y(t) wpt has bx ordinary double points (two branches meeting transversally), which tend to 0 as v tends to 0. So we can view 6x as the number of ordinary double points which have coalesced to form the singuIarity of X at the origin. Of course, for an ordinary double point b=l.

+

+

+

+

In fact this geometric interpretation follows from the fact that the b invariant plays a key role in understanding which deformations of curves come from deforming the parametrization. If a germ of plane curve is given parametrically by z ( t ) , y ( t ) ,we can define (one parameter) deformations of the parametrization as follows:

ci

z ( t ;v) = z ( t )+ Ui(V)ti, y ( t ; v )= y ( t )+ Cjbj(v)tj,

uz E C{v}, Ui(0) = 0 E C { V } ,bj(0) = 0.

bj

If on the other hand our curve is given implicitely by an equation f(z, y) = 0, then we can define a deformation as (i>j)#(O,O)

The elimination process can be performed over C{v} to show that a deformation of the parametrization always give a deformation of the equation (again this follows from the fact that the formation of Fitting ideals commutes with base extension).

867

Is the converse true in the sense that any deformation of the equation can be represented by a deformation of the parametrization? The answer is NO! In order to understand what happens, we must reinterpret the problem. To say that a family of curves is obtained by deforming a parametrization is to say that they all have “the same normalization” in some sense. Thus we are led to study how the normalizations vary in an analytic family of reduced plane curves. Definition 7.1. Let (C,O) be a germ of a reduced analytic curve, or let C be a closed reduced analytic curve in a suitable open polycylinder of C d , or an af€ine curve, with ring Oc.Then its b invariant is defined by & ( C )= d i m c ( z ) , or 6(C) = d i m c ( g ) . Since normalization of a sheaf of algebras form a sheaf, we have XEC

where the sum on the right is finite since b is nonzero only at singular points. Now let f : (S,O) + (C,O) be a germ of a flat morphism such that f-’(O) is a germ of a reduced analytic curve. Here flatness means that no element of is annihilated by multiplication by an element of C{v} where v is a local coordinate on (C,0). Let n : -+ S be the normalization of the surface S (a small representative of the germ), and let

s

p=fon:

(S,n-l(o)) (s,o). -+

Let us denote p-l(O) by (S)O,and to write b((S”)o) = C2En-l(0) b((S)o,z). similarly, write SO) for S ( f - ’ ( O ) , O ) and b ( S y ) for b ( f - l ( y ) when y E C \ (0) in a small enbough representative of f , so that all the singular points of f - ’ ( y ) tend to 0 when y + 0, and 0 is the only singular point of f - l ( O ) . Note that 6(Sy) = CzES, 6(Sy, z ) . Then we have:

Proposition 7.2. (see [T6], and [CH-L] for a beautiful generalization) a) The morphism p = f o n : (S,n-’(O))+ ( C ,0 ) is a multigerm of a flat mapping. b) W e have the equality m 3 0 )

for y

# 0 suficiently small.

=

q s o ) - W,),

868

To say that the normalizations of the various fibers f-l(y) glue up into a non singular surface is therefore equivalent to saying that p-l(O) is non singular and this is equivalent to saying that "the 6 invariant of the fibers S, is constant as y varies in C near 0. Note that the fiber f-'(y) will in general have several singular points, at which it is not necessarily analytically irreducible even if f-l(O) is irreducible.

+

This explains what happens when we deform the parametrization by ~ ( t ) awt, y ( t ) f ,But; since it is a deformation of the parametrization, the sum of the S invariants must be the same for all values of w, while for w # 0 the curve has only ordinary double points, whose 6 invariant is one. 7.3. projections of space curves

So this abstract idea, normalization, provides us with a proof of the resolution of singularities of space curves: given (C,0) E (Cd,O),the normalization 0 -+ of the (reduced) analytic algebra of germs of functions on C is an analytic algebra which is a product II~=.=,C{ti} of a finite number of convergent power series rings in one variable. If 21,. . . ,xd generate the maximal ideal of 0 , we get T d-uples of convergent power series expansions z j ( t i ) , which are our Newton series in this case. They geometrically correspond to a map T

U(C,0)i

(C,0) i=l which is our resolution of singularities. However, normalization is geometrically subtle in general, and the finiteness of normalization for the rings one meets in Geometry is a fairly deep theorem of commutative algebra; in addition, we may seek a more geometric proof, as follows +

We now turn to the definition of plane projections of a space curve. Let (C,0) E ( C d ,0) be a germ of a (reduced) space curve defined by an ideal I c C{zl,.. . ,zd}. Let us choose a linear projection p : C d --+ C2. Let M denote the space of all such projections; think of it as a set of d x 2 matrices of rank 2. We endow M with the topology (complex or Zariski) induced by that of the space of matrices. We wish to consider only the projections such that PIC:C + p ( C ) is finite to one. If that is not the case, the kernel of p , which is a linear subspace of codimension 2 of C d , contains one of the irreducible components of the curve C ;the intersection is analytic, so it is either of dimension 0 or 1. By looking at the equations of C , it is not too

869

difficult to check that the projections which do not contain a component of C form a dense open set of M . The fact that they are those which induce a finite map C -+ p ( C ) is a consequence of the Weierstrass preparation theorem. Assume now that the map C -+ p ( C ) is finite. Again by the Weiertrass theorem, it means that the map of analytic algebras C{z,y} --+ 0 defined by f H ( f o p ) l C makes 0 a C{z, 9)-module of finite type. Since C{z, y} is ncetherian, as we saw in a preceding section, it means we have a presentation by an exact sequence of C{z, y}-modules: C{z,y}Q --+ C{z,y}P

--+

0

-+

0

An argument which we have seen above shows that since C is of dimension 1, we must have q = p , so the first map is described by a square matrix with entries in C{z,y}. Let $(z,y) be the determinant of that matrix. This determinant is, up to an invertible factor, independant of the choice of the presentation. Then the image p ( C ) is the plane curve with equation 4 ( Z , Y ) = 0. On the other hand, let us say that a linear plane projection p : C d -+ C2 c C d at the point 0 E C if it has the following property: For any sequence of couples of points (ai,bi) E (C\ ( 0 ) ) x (C\ ( 0 ) ) tending to 0, the limit direction of the secant line ai, bi (for any subsequence) is not contained in the kernel of p. We will see in the next paragraph that all general projections of a given germ (C,O) of space curve are topologically indistinguishable as germs of plane curves in C2. In [T2] it is shown that if p is general for (C,O),then the inclusion of the ring 0 1 = 0x1,0of the image X1 = p ( X ) as defined above into the ring 0 = 0c,o (induced by the composition of functions with p ) induces an isomorphism of the total rings of fractions of these two rings, and because 0 is a finite 01-module, every element of 0 is integral over 01, as we saw above. Therefore 0 is contained in the normalization 0 1 of 01. Therefore 5 is also the normalization of 0, and it is a finite 0-module for general reasons (see [K]; it suffices to know that the integral closure of 01 is a finite 01-module). Now we can use the universal property of blowing ups: in 01all ideals become principal and generated by a non zero divisor in each C { t i } .By the universal property of blowing up ([L 1, VII.5) if we blow up the origin in 0 , the resulting algebra is still contained in 01, and as we repeat blowing up points, we get an increasing sequence is general for the curve C

870

n,

of 0-algebras contained in all having the same total ring of fractions. Since 01is a finite 0-module, this sequence stabilizes after finitely many steps. We have to show that this limit algebra is 01.But if this were not the case, the maximal ideal of one of the component local algebras would not be principal, so we could blow it up and get a strictly bigger algebra, contradicting the stability. In conclusion, we have shown that any space curve singularity can also be desingularized by a finite sequence of point blowing ups. One can also prove embedded resolution for space curves; it is not much more difficult than in the plane curve case. 8. The semigroup of a branch

There is another natural object associated to the inclusion 0 --t 8;again I will decribe it only in the case of a branch. Let 0 be the analytic algebra of a germ of analytically irreducible curve X, and let 8 be its normalization; we have an injection 0 --+ 8 which makes an O-module of finite type and 8 is a subalgebra of the fraction field of 0. Since 8is isomorphic to C { t } ,the order in t of the series defines a mapping v : C { t } \ 0 N which satisfies i) v(a(t)b(t))= v ( a ( t ) ) v(b(t))and ii) v(a(t) b ( t ) ) 2 min(v(a(t)),v(b(t)))with equality if v(a (t ))# u(b(t)); in other words, u is a valuation of the ring C { t } . We consider the valuations of the elements of the subring 0, i.e., the image r of 0 \ {0} by v ; in view of i), it is a semigroup contained in N. The fact that is a finite 0-module implies that N \ r is finite, and in fact (see [Z]) we have for the 6 invariant of C the equality

+

+

AX = #(N

\ r)

Now we seek a minimal set of generators of I' as a semigroup: be the smallest element be the smallest non zero element in r, let Let be the smallest element of which of I' which is not a multiple of E, let is not a combination with non - _negative integral coefficients of E and E, i.e., is not in the semigroup (Po, PI), and so on. Finally, since N \ r is finite, we find in this way a minimal set of generators:

This set is uniquely determined by the semigroup r, and of course determines it.

871

By a theorem of ApBry and Zariski (see [Z]), if (X, 0) is a plane branch, the datum of these generators, or of the semigroup, is equivalent to the datum of the Puiseux characteristic of (X, 0), or of its topological type. Let us take the notations introduced for the Puiseux pairs; it is-easy to check that if we set PO = n, the multiplicity, then = PO = n,P1 = PI. After that it becomes more complicated. Zariski ([Z], Th. 3.9) proved the following formula for q = 2, . . . ,g:

-

Pq = ( n l

- l)n2 . . . nq-101 + (n2 - 1)ns... n q - l P 3

+ . . . + (nq-1

- I)&-1

+

pq,

which can be summarized in the following recursive formula:

Pq

-

= nq-1Pq-1

- Pq-1+

Pq

The proof relies on a formula of Max Noether which computes the contact exponent of two analytic branches at the origin in terms of the coincidence of their Puiseux expansions in fractional powers of x. This fact leads to a very interesting constatation: Consider the Puiseux expansion of a root y(x) of the Weierstass polynomial defining an analytically irreducible plane curve near the origin, assuming that x = 0 is not in the tangent cone of that curve:

%

mg+l

mq

Eg

=Y-

nln2...n 9

+ apgfl2nln2"'n9 + . . . )

That is, the sequence of truncations of the Puiseux series just before the appearance of a new Puiseux exponent. Each C j , 0 5 j 5 g is a root of a Weierstrass polynomial Qj defining a branch Cj. Note that we have QO= x and that Q 1 = y if y = 0 has maximal contact with C. Proposition 8.1. (Apbry-Zariski), see also [PP] For the semigroup r = - (Po,P1,. . . ,P,) associated to a plane branch and the curves Cj just defined, we have the equalities Pj

= (C,Cj)O.

872

In fact, this equality remains true if we replace the expansions by any series which coincide with the series y(z) until just before the j-th Puiseux exponent; see [PP]. It follows easily from this that the datum of the semigroup is equivalent to the datum of the multiplicity n and the Puiseux exponents Pi of the curve. The semigroups coming from plane branches are characterized among all semigroups of analytically irreducible germs of curves by the following two properties:

That the semigroups of plane branches have these properties follows from the induction formula and the inequalities Pi < Pi+l. The converse can be proved by the construction outlined below (see [Z], appendix). Conversely, given a semigroup I' in N with finite complement, we can associate to it an analytic (in fact algebraic) curve, called the monomial -curve associated to r. If r = (PO,P I , . . . , P,), the monomial curve Cr is described parametrically by uo = t P o u1 = to1

If the semigroup r comes from a plane branch, the relations 1) above mean that there exist natural numbers l!j) such that we have

873

Theserelations translate into equations for the curve Cr ui = toi, our curve satisfies the g equations

c Cg+l; since

and it can be shown that they actually define Cr C Cg+l, so that if I? is the semigroup of a plane branch, Cr is a complete intersection. Remark that if we give to ui the weight the i-th equation is homogeneous of degree n&. The connection between a plane curve X having semigroup I? and the monomial curve is much more precise and interesting than the formal relation we have just seen; by small deformations of the monomial curve one obtains all the branches with the same semigroup. In fact the best way to understand all branches with semigroup I? is to consider the not necessarily plane curve Cr (Cr is plane if and only if C has only one characteristic exponent). By definition of r, there are elements Q E 0 with v ( Q ) = We can write these elements in C { t } as

E,

K.

Let us consider the one-parameter family of parametrizations ug

= tm

The reader can check that for v # 0, the curve thus described is isomorphic to our original curve C. (hint: make - the change of parameter t = wt’ and the change of coordinates uj = v-ojvi, and remember the definition of the <j). For v = 0, we have the parametric description of the monomial curve. So we have in fact described a map

c x c +c g f l x c which induces the identity on the second factors (with coordinate v). The image of this map is a surface, which is the total space of a deformation of the monomial curve, all of its fibers except the one for v = 0 being isomorphic to our plane curve C.

874

So the monomial curve is a specialization, in this family, of our plane curve. In this specialization the multiplicity and the semigroup remain constant; in a rather precise sense it is an equisingular specialization, or one may say that the plane curve is a n equisingular deformation of the monomial curve with the same semigroup. The same phenomenon can be also observed in the language of equations rather than parametrizations. Let us consider a one parameter family of equations for curves in Cg+l, of the form

For w = o we get the equations of the monomial curve, and for w # 0 we get a curve which has semigroup r; this is a general heuristic principle of equisingularity: we have added to each equation of the monomial curve, homogeneous of degree niE, a perturbation of degree > n&, and this should not change the equisingularity class (the perturbation is ”small” compared to the equation). Notice that for each fixed w # 0 the curve described by the above equations is a plane curve: for simplicity take w = l; then use the first equation

eg)

to compute u2 = uyl - uo , substitute this in the next equation, and use this t o compute u3 as a function of U O , U ~and , so on. Finally the last equation gives us the equation of a plane curve of the form

The first consequence (see the appendix to [Z]) is that we can produce explicitely the equation of a plane curve with given characteristic exponents: compute the semigroup and its generators, and then write the equation above. A more important fact is that one can show (see [loc. cit) that any plane curve with a given semigroup appears up to isomorphism as a fiber in a deformation depending on a finite number of parameters: it is a deformation of the monomial curve obtained by adding to the j-th equation a polynomial in the ui’s of order > njg, and these polynomials can in principle be explicitely computed.

875

In fact it is shown in [G-T] that we can in this manner produce equations for all branches having the same semigroup (or equisingularity type) up to an analytic isomorphism. The fact that the curve is plane corresponds to the condition that uj+l appears linearly in the deformation of the j-th equation, for 1 5 j 5 g - 1. Finally, all the plane branches with the same semigroup have ”the same” process of resolution of singularities: you have to blow up points according to the same rules, the multiplicities of the strict transforms are the same, and so on. So the resolution of the plane curve described above shows the structure of the resolutions of all the curves with the same semigroup. First you resolve the curve uyl - u$) = 0; when its strict transform is non singular (after a number of blowing ups which depends on the continued (1)

fraction expansion of the ratio $, you take it as a coordinate axis: then you have one parenthesis less in the equation above (the point is that the form of the equation does not change), and you proceed like this. After g such steps the branch is resolved. There is however another way to use the structure given by the description of our branch as a deformation of the monomial curve to get embedded resolution; it is the subject of the next paragraph. 9. Resolution of binomials

Let u1,u2 be two integral vectors in the first quadrant of R2, and assume that their determinant is kl.Then they are primitive vectors and they generate the integral lattice Z2 of R2. Consider the cone 0 = (+2)

of their positive linear combinations. It is a rational convex cone (= a convex cone which is the intersection of finitely many half spaces determined by hyperplanes with rational -even integral- equations). Because it is generated by integral vectors which form a basis of the integral lattice Z2, we say tha it is a regular cone. Since it is convex it has a convex dual which is a rational convex cone in R2: b = { m E R2/m(C) 2 0 V C E a}.

The cone 5 is also generated by two vectors with determinant kl, which therefore generate the integral lattice Z2 of R2.If we interpret each integral point of 5 as a (Laurent) monomial (here “Laurent” means that negative

876

exponents are allowed) in variables ( u I , u ~ )the , algebra C[6 n Z2] is a polynomial algebra in two variables, say C[yl, y ~ ] . Since a is contained in the first quadrant, its dual 6 contains the dual of the first quadrant, which is the first quadrant of R2. If we remark that the integral points of the first quadrant correspond exactly to the polynomial algebra C [ U ~ , U we~ see ] , that there is therefore an inclusion C[Ul,U21

-

c C[Yl,Y21

and it is an interesting exercise to check that it is given by U1 U2

a: a t

Y1 Y2 a' a: YI2Y2

where a! is the i-th coordinate of the vector a j . The transform of a monomial urn = uy'uT2 is, if we write m = (m1, m2): U y ' U y 2 ++

(a1,rn) ( a 2 , m )

y1

Y2

,

so that the transform of a binomial urn- Amnun is Urn

- Amnun H y1( a ' d Y2( a

2 d

.

- XmnY1 ( a ' , n ) y2( a 2 , n )

Now the key observation is that if (a', m - n) and (a2,m - n) are both non zero, they have the same sign, which means that the two vectors a1 and a2 are in the same half space determined the hyperplane Hrn.-n dual to the vector m - n, or equivalently that the cone a is compatible with Hm+ in the sense that (T n HmPn is a face of a , then we can factor the transform of the binomial. Assume that (ai,m-n) 2 0. We have non negative exponents in the identity ( a ' d ( a 2 d - XrnnY1 (al,n)y2 ( a 2 , n )Y1 Y2

( a ' , n ) ( a 2 , n ) (a',m-n) (a2,rn-n) Y1 Y2 (Y1 Y2 - Xmn)

Now we have an exceptional divisor defined by

and a strict transform defined by (a',m-n)

Y1

(a2,m-n) Y2

-Xmn=0

8

877

The next observation is that the strict transform is non singular, and meets the exceptional divisor if and only if (T n HmPn is an edge of (T,i.e., is not (0). Say that a' is in Hm.-n; the strict transform is then (a',m-n) Y1 - Am,

= 0.

Now if we assume that the binomial urn- Amnun is irreducible in C [ u l ,U Z ] , which is equivalent since C is algebraically closed to the fact that the vector m - n is primitive, in the sense that it is not an integral multiple of an integral vector, then it is not difficult to check (see [Tl],Proposition 6.2) that (a', m - n) = 1, so that finally our strict transform in this case is y 1 Am, = 0 , which is indeed non singular and transversal to the exceptional divisor. Actually the same proof works if the binomial is reducible but there are then several points above the origin in the strict transform of the curve. The next observation is that in two variables our binomial has to be of the form uy - Xu; unless the curve contains a coordinate axis, which we exclude in the irreducible case. By a change of variable we may assume X = 1 and by irreducibility, we have ( m , n )= 1. Now to study the strict transform under one of our monomial maps ~ ( owe ) have seen that the only interesting case is when one of the generating vectors of (T,say a l , is the vector (n,m).Let us assume that n < m. Set a2 = (a,b) and say that am - b n = 1 (we know it has to be f l ) .The transform of a monomial u";; n i f m j ai+bj is y1 yz . om this follows that if we consider a curve with equation

(*)

u r -ug

+ C

aijuiuj2

=

o

ni+mj>mn

it transforms into ni+mj-mn

ai+bj-am

Yz

)

ni+mj>mn

and one checks that all exponents are positive. The strict transform of our curve is still non singular in a neighborhood of the exceptional divisor, and transversal to the exceptional divisor at the point y1 = 0, y z = 1. If we consider the other cone (T'having the vector (n,m) as an edge, we find that the point where the strict transform meets the exceptional divisor lies in the open set of the corresponding chart Z ( d ) which is identified with an open set of Z((T);we are looking at the same object in two charts. This shows that the toric maps which provides an embedded resolution for the binomial u;"- u; = 0 in fact also gives an embedded resolution for all the

878

curves of the type (*), where one deforms the binomial by adding terms of higher weight, where the weight of u1 is n and the weight of u2 is m. Now by a general combinatorial result (see [Ew]), for any integer d 2 2, given a finite collection of hyperplanes whose equation has integral coefficients in the first quadrant Rd>oof Rd, it is possible to find a regular fan with support Rd>o, that is a finite collection C of regular rational cones such as our CT above (but now with d generating vectors of determinant &l)and its faces, whose union is Rd>o,and such that if CT E C its faces are also in C, and for any CT, CT’ E C,the intersection CT n CT’ is a face of each. To each CT of dimension d corresponds a polynomial ring C[6 n Z d ] and therefore an affine space Ad(C)with a birational map ~ ( c T ) Ad(C) : --+ Ad(C)

generalizing the map u1 u2

a: a: Y1 Y2 a: a: Y1 Y2

which we have seen above in the case d = 2. The sources of all these maps can be glued up together (see [Ew]) to form a nonsingular algebraic rational variety Z ( C ) in such a way that the maps n ( a ) glue up into a proper and birational (hence surjective) map n ( C ) : Z ( C ) --+ Ad(C). Coming back to the case d = 2 and a binomial, this gives us the existence of a regular fan (= a fan made of regular cones) with support R;,, and compatible with the line Hm-,, which means that this line in R?; - is the common edge of two cones of the fan. In fact in this case there is a minimal such fan, obtained as follows: Consider the set H- of integral points of R?o which are below the line H,-,, and the set H+ of the integral points which are above. The boundaries of the convex hulls of H- and H+ contain parts of coordinates axes, and they meet at the extremity of the primitive integral vector contained in Hm-,. Drawing lines connecting the origin to all the integral points which are on these boundaries defines a fan which has the required properties and is the coarsest such fan. It is closely connected with the continued fraction expansion of the slope of the line Hm-n. To this fan C is associated a proper birational map n ( C ) : Z ( C ) -+ A2(C)

879

which is an isomorphism outside of the origin and provides an embedded resolution of singularities for all plane branches which have an equation of the form

C

.

.

u~-uy+ aijuiui &+$>1

=o.

as one verifies by checking in each of the charts Z(a) 21 A2(C). Since we saw that every plane branch is similarly a deformation of the monomial curve with the same semi-group, which is defined by g binomial equations in variables U O ,. . . ,us, adding to each binomial only monomials of higher weight, one is ready to believe that similarly, a regular fan in R g2f0l which is compatible with the g hyperplanes corresponding t o the g binomials will provide a toric map

r ( C ) :Z ( C ) -+ Ag+l(C) which is an embedded resolution not only for the monomial curve, but also for our original plane curve re-embedded in AS+' (C) as was explained above. This is described in detail in [G-T] and generalized in [GP] to a much larger class of singularities. This method of embedded resolution is quite different from the resolution by point blowing ups explained above, but it assumes that one knows the existence of a parametrization. The connection between the toric map and the sequence of point blowing ups is rather subtle (see [GP]);in the case g = 1 it is equivalent to the relation between finding approximations of a rational number by the reduced fractions of its continued fraction expansion and finding approximations by Farey series. So the deformation to the monomial curve also explains to us how to resolve the singularities, and it is perhaps the best description. Can we generalize it to higher dimensions?

10. Relation with topology

I refer to the lectures of Lt! and to [B-K] for the Burau-Zariski topological interpretation of the characteristic sequence

(Po, P l , . . . P g ) as a characteristic of the iterated torus knot that one obtains upon intersecting the branch X with a sufficiently small sphere in C2 centered at the origin.

880

Given a germ of a reduced plane curve X , it has a decomposition X = X i into branches; each branch has its characteristic sequence B ( X i ) , and as numerical characters of X , we have also the intersection numbers ( X i ,X j ) o of distinct branches at 0. If we remember that these intersection numbers are equal to the linking numbers in S 3 of the knots corresponding to Xi and X j and are therefore topological characters of the link X n S:, since Milnor proved (see L6's lectures) that the curve X is homeomorphic to the cone with vertex 0 drawn on this link, we expect that the collection of the characteristic sequences of the branches and their intersection numbers may be a topological invariant of the curve X . Let us define the local topological type of a germ of subspace of C N as follows: Definition.- Two subspaces X1 and X Z of C N are topologically equivalent at 0 if there exist neighbourhoods U and V of 0 in C N and an homeomorphism $: U 4 V such that $ ( X I n U ) = X Z n V . Two germs at 0 of subspaces are topologically equivalent if they have representatives which are topologically equivalent at 0.

Theorem 10.1. (Zariski, Lejeune-Jalabert). Two germs of plane curves X = UiE:rXi and XI = U i E p X i are topologically equivalent i f and only i f there exists a bijection 4 : I 4 I' between their branches which preserves characteristics and intersection numbers, that is, satisfies B ( X i ( i ) )= B ( X i ) for i E 1,

(xici),xicj,)~= ( X i ,

xj)~ for

i Z j.

Topological equivalence is less strict a relation than analytic (or even C') equivalence. Let X 1 and X2 each consist of four lines through the origin in C 2 . According to the previous theorem, these two germs are topologically equivalent. However, if there was a germ et 0 of a C1 (and in particular analytic) isomorphism of C 2 to itself, sending X I to X2, its tangent linear map at 0 would have to send X1 onto X2. But two quadruplets of lines through 0 are linearly equivalent if and only if they have the same cross-ratio. If the slopes of the lines of X 1 are a l , b l , c l , d l , and similarly for X Z , the cross ratios are

and the numbers obtained by permutation. It is therefore easy to find examples where X 1 and X z are not C1-equivalent.

881

In particular, in an analytic family of curves such as the surface in C3 with equation

(Y - X ) ( Y

+ Z ) ( Y - 2X)(Y + t x ) = 0

for small values of t , the fibers are all analytically inequivalent but topologically equivalent.

Theorem 10.2. Given two reduced germs of plane curves ( X ,0 ) c (C2,0 ) and (X',O) c ( C 2 , 0 )the following conditions are equivalent: 1) X and XI are topologically equivalent, 2) There exists an integer d , a germ of curve (C,O) C (Cd,O)and two linear projections p,pI: Cd + C2, both general for C at 0 , and such that p ( C ) = x, p ' ( C ) = X ' , 5') There exists a one-parameter family of germs of plane curves that is a germ along (0) x U of a surface in C2x U , where U is a disk in C , say with equation f ( x ,y, u)= 0 and v,v' E U such that the germs of plane curve f ( x , y , v ) = 0 , f ( x , y , v ' ) = 0 are isomorphic to X , X' respectively and all the germs f ( x ,y , t ) = 0 have the same topological type for t E U. 4) There exists a bijection from the set of branches of ( X ,0 ) to the set of branches of (XI, 0 ) which preserves characteristic (Puiseux) exponents and intersection numbers. 5) The minimal embeded resolution processes of ( X ,0 ) and ( X I ,0 ) are "the same" in the sense that one blows up at each step points with the same multiplicity. In fact, the theory of Lipschitz saturation, summarized in [T4], shows that, given the topological type of a germ of plane curve ( X ,0), there exists a germ of a space curve (X",O) c ( C N , O ) ,unique up to isomorphism, such that the germs of plane curves having the same topological type as (X,O) are exactly, up t o isomorphism, the images of (X",O) by the linear projections ( C N , O )+ ( C 2 , 0 )which are general for (X",O). 11. Duality

A line in the projective space P2 is by definition a point in the dual projective space P 2 . Poncelet saw that given a nondegenerate conic Q, to any point P E P2, one can associate the polar curve of P with respect to Q, which is the line joining the points of contact with Q of the tangents to Q passing through P. We get in this way an isomorphism between P2 and its dual P2.

882

We shall, however, refrain from identifying P2 and its dual in this way, as was done at that time. The collection of points of P2 corresponding to the lines in P2 tangent to an algebraic curve C is an algebraic curve C c P2.Here we use the fact that one can say that a line is tangent to a curve C at a singular point x if its direction is a limit direction of tangents t o C at non singular points tending to x. A point x in P2 corresponds to a line j: in P2;each point of this line represents a line in P2which contains x, and the lines through x tangent to C correspond to the intersection points in P2 of the curve C and the line 2 . So the class of the curve C , defined as the number of lines tangent to C at non singular points and passing through a given general point of P2,is the degree fh of C. Let us compute it: Poncelet considered, following Monge, the polar curme (the terminology is his): Let

where f is a homogeneous polynomial of degree m, be an equation for C. The points of C where the tangent goes though the point of P2(C) with coordinates (E, r ] , C) are on C and on the curve of degree m- 1 with equation

obtained by polarizing the polynomial f with respect to the point

(E, r], C).

If C is non singular, the points we seek are all the intersection points of C and P ( C , ~ , C ) (By C )Bkzout’s . theorem, the number of these points counted with multiplicity is m ( m - l ) , for every point ( E , r ] , C) it is equal to m(m-1) if C has no singularities. It is “geometrically obvious” that C = C ; this is called biduality (it is completely wrong if we do geometry over a field of positive characteristic, so beware of what is “geometrically obvious”). If the curve C had no singularities as well, the computation of degrees would give m(m- 1)(m2- m - 1) = m, which holds only for m = 2. So if m > 2 the dual of a non singular curve has singularities ; for a general non singular curve, double points (a.k.a. nodes) of C correspond to double tangents of C and cusps correspond to its inflexion points. To understand biduality better, it becomes important to find the class of a projective plane curve with singularities, at least when these singularities

883

are the simplest: nodes and cusps. This was done by Plucker and the formula for a curve with 6 nodes and K cusps is

riz = m(m - 1) - 26 - 36. One said that“ a node decreases the class by two, and a cusp by three” This is perhaps the first example of a search of numerical invariants of singularities. One can compute the diminution of class provoked by an arbitrary plane curve singularity as follows: let f ( r c , y ) = 0 be a local equation for the singular curve at a singular point z which we take as origin. The ideal j(f) = g ) C { z , y } defines set-theoretically the origin, which is the only singular point (locally). Therefore it contains a power of the maximal ideal and it is a vector space of finite codimension p(C,z) in C{z, y}. It is called the Milnor number of the singularity. Let m(C,z) be the multiplicity of C at 0, the order of the equation f (z,y). Then (see [T6]) the diminution of class due to the singularity is

(2,

+

A c , ~= p(C, z) m(C,X ) - 1. This means that for an arbitrary reduced projctive plane curve C of degree m we have the equality (generalized Plucker formula)

riz = m(m - 1) - C x , z ~ A ~ , z . This number is also the local intersection number at z of the curve C and one of its “general local polar curves”, defined by a% = 0 for geneal values of a , ,B. In this way, one can see that if we locally deform our singular curve to a non singular one, say by taking the equation f (z, y) = A, the number of points where the tangent to this non singular curve has a fixed direction and which coalesce to 0 as X -+ 0 is AcYx;its is the number of possible tangents that are LLabsorbed’l by the singular point. It is a remarkable fact that the “diminution of class” A c , ~depends only on the topological type of the germ of plane curve (C,z). In fact there is a formula to express it in terms of the Puiseux exponents of the branches of (C,z) and their local intersection numbers. The fact that the degree of the dual of a projective variety depends only on global characters like the degree and the “topology” of its singularities extends to an arbitrary singular projective variety if one makes the notion of topological type somewhat more stringent. In the theory of algebraic curves, an important formula states that given an algebraic map f : C -+ C’ between non singular algebraic curves, which is of degree degf = d (meaning that for a general point c’ E C’, f-l(c‘)

+,Bg

884

consists of d points, and is ramified at the points xi E C, 1 5 i 5 r , which means that near xi, in suitable local coordinates on C and C’, the map f is of the form t tea+’ with ei E N, ei 2 1. The integer ei is theramification index of f at xi. Then we have the Riemann-Hurwitz formula relating the genus of C and the genus of C’ via d and the ramification indices:

-

2g(C) - 2 = d(2g(C’) - 2)

+ C ei i

C‘ between possibly singular curves, it If we have a finite map f : C extends in a unique manner to a finite map of the same degree between their normalizations, to which we can apply the Riemann-Hurwitz formula to get a relation between the geometric genera of C and C’. If we apply this formula to the case where C is non singular and C’ = P’, knowing that any compact algebraic curve is a finite ramified covering of P’, we find that we can calculate the genus of C from any linear system of points made of the fibers of a map C -+ P1if we know its degree and its singularities: we get -+

2g(C) = 2 - 2d

+c

e i

The ramification points xi can be computed as the so-called jacobian divisor of the linear system, which consists of the singular points, properly counted, of the singular members of the linear system. In particular if C is a plane curve and the linear system is the system of its plane sections by lines through a general point x = (E : 7 : C ) of P2,the map f is the projection from C to P’ from x; its degree is the degree m of C and its ramification points are exactly the points where the line from x is tangent to C. Since x is general, these are simple tangency points, so the ei are equal to 1, and their number is equal to the class m of C ; the formula gives 2g(C) - 2 = - 2 m + m , thus giving for the genus an expression which is linear in the degree and the class, whereas our expression in terms of the degree alone is quadratic. This is the first example of the relation between the “characteristic classes” (in this case only the genus) and the polar classes; in this case the curve itself, of degree m and the degree of the polar locus, or apparent contour from 2,i.e. in this case the class m. The extension to a non singular projective algebraic variety in characteristic zero is due to Todd.

12. The polar curve The (general) polar curve plays a much more important role in the study of the plane curve singularities than just giving the diminution of class by

885 its intersection number with the curve a t the singular point. Given an equation f (z, y) = 0 for a germ reduced plane curve in (C2,0), let us denote by l ( z ,y) a homogeneous linear form, i.e., the equation of a line through the origin. The we define a map

Fe: ( C 2 , 0 )4 ( C 2 , 0 ) by t o = f (2, !I> t l =,t ( z ,Y). The critical locus Pe of this map is the local avatar of the polar curve which we saw in the previous section. More precisely if our germ of plane curve comes from a projective plane curve, then Pe is the germ at 0 of the polar curve in the projective plane corresponding t o the point at infinity in the direction o f t . Remark now that the image of Pe by the map Fe is a plane curve whose equations is given by the Fitting ideal of the algebra of Pe as a C{to,t 1 ) module, at least when the map (Pe,O) --$ ( C 2 , 0 )induced by Fe is finite, which is the case when .t is general. This image De = Fe(Pe) is the discriminant of Fe, and it lives in a plane with given coordinates to, tl. The Newton polygon of De in the coordinates to,tl for a general choice of the linear form l is independant of l and is called the jacobian Newton polygon of f . M. Merle proved that if f(z, y) is irreducibule, the jacobian Newton polygon is a complete invariant of equisingularity of the curve f (z, y) = 0; it can be computed from the Puiseux exponents and determines them. The extension to the reducible case is due t o E. Garcia Barroso (for all this, see [GI). The jacobian Newton polygon encodes the essence of the dynamics as X goes t o zero of the points of the non-singular curve f (2, y) = X where the tangent is parallel t o 4!(z,y) = 0. Those are the points counting for the class of a non singular curve degenerating t o our singular curve which are "absorbed" by the singularity and so decrease the class. Bibliography

B. B-K. Ca.

N. Bourbaki, Alg6bre commutative, Chap. I-IX, Masson. E. Brieskorn, H. Knorrer, Plane algebraic curves, Birkhaiiser, 1986. A. Campillo, Algebroid curves in positive characteristic, Springer Lecture Notes in Mathematics, No. 813 (1980). CH-L. H.-J. Chiang-Hsieh and J. Lipman, A numerical condition for simultaneous normalization. To appear in Duke Math. J. Ei. D. Eisenbud, Commutative algebra with a view towards Algebraic Geometry, Graduate Texts in Math., Springer 1995.

886

Ew .

G. Ewald, Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Math. No. 168, Springer 1996. G. E.R. Garcia Barroso, Sur les courbes polaires d’une courbe plane re‘duite, Proc. London Math. SOC.,81, 3 (2000), 1-28. GP. P. Gonzfilez PBrez, Toric embedded resolution of quasi-ordinary singularities, Annales de 1’Institut Fourier, 2003. G-T. R. Goldin and B. Teissier, Resolving plane branch singularities with one toric morphism, in “Resolution of Singularities, a research textbook in tribute to Oscar Zariski”, Birkhauser, Progress in Math. No. 181, 2000, 3 15-340. GP-T. P. Gonztilez PBrez and B. Teissier, Embedded resolutions of n o n necessarily normal a f i n e toric varieties, Comptes Rendus Acad. Sci. Paris, Ser.1, 334,2002, 379-382. H. H. Hironaka, Resolution of singularities of a n algebraic variety over a field of characteristic zero I, 11, Annals of Math., 79,No.1 i964, 109-203 and 79,No.2, 205-326. K. B. Kaup, L. Kaup, and G. Barthel, Holomorphic functions of several variables, De Gruyter Studies in Math., 003, W. de Gruyter 1983. K. Kedlaya The algebraic closure of the power series field in positive Ke. characteristic. Proc. Amer. Math. SOC. 129 (2001), no. 12, 3461-3470 (electronic). L. S. Lojasiewicz, Introduction to complex analytic Geometry, , Birkhauser Verlag, 1991. M. Merle, Invariants polaires des courbes planes, , Inventiones Math., 41, M. 103-111 (1977). M-T. M. Merle and B. Teissier, Conditions d’adjonction, d’aprb Du Val, SBminaire sur les singularit& des surfaces, Palaiseau 1976-77, M. Demazure, H. Pinkham, B. Teissier, Editeurs, Springer Lecture Notes in Math., No. 777, 229-245. Ok. M. Oka, Non-Degenerate complete intersection singularity, Actualit& mathbmatiques, Hermann, Paris 1997. PP. P. Popescu-Pampu, Approximate roots, Proceedings of the Saskatoon Conference and Workshop on valuation theory (second volume), F-V. Kuhlmann, S. Kuhlmann, M. Marshall, editors, Fields Institute Communications, 2003, 285-321. M. Saia, The integral closure of ideals and the Newton filtration, J. AlS. gebraic Geometry, 5 , 1996, 1-11. H.J.S. Smith, O n the higher singularities of plane curues, Proc. London Sm. Math. SOC.,1, t. VI, 1873, p. 153. B. Teissier, Valuations, deformations, and toric geometry, Proceedings T1. of the Saskatoon Conference and Workshop on valuation theory (second volume), F-V. Kuhlmann, S. Kuhlmann, M. Marshall, editors, Fields Institute Communications, 2003. B. Teissier, MonBmes, volumes et multaplicite‘s, in Introduction B la T2. thBorie des singularitks, 11, p. 127-141, Travaux en Cours No. 37, Hermann, Paris.

887 T3. T4.

T5.

T6.

TE. Z.

B. Teissier, Cycles iuanescents, sections planes, et conditions de Whitney. Singularit& A CargBse, Astbrisque No. 7-8, SMF Paris 1973. B. Teissier, Multiplicite's polaires, sections planes, et conditions de W h i t ney, Algebraic Geometry, La RBbida 1981, J-M. Aroca, R. Buchweitz, M. Giusti and M. Merle, editors, Springer Lect. Notes in Math., No. 961, 1982. B. Teissier, T h e hunting of invariants in the geometry of discriminants, Real and complex singularities, Proc. Nordic Summer School Oslo 1976, Per Holm ed., Sijthoff and Noordhof 1977, 555-677. B. Teissier, Risolution simultane'e, I et II. SBminaire sur les singularitks des surfaces, Palaiseau 1976-77, Demazure, H. Pinkham, B. Teissier, Editeurs, Springer Lecture Notes in Math., No. 777, Kempf, Knudsen, Mumford, St. Donat, Toroidal embeddings, Springer Lect. Notes in Math., No. 339, 1973. 0. Zariski, Le problbme des modules pour les branches planes, (avec un appendice de B. Teissier), Hermann, Paris, 1986. Translation by the A.M.S. to appear.

This page intentionally left blank

889

Programme

Monday, 15 August 2005 14:OO - 14:15, Opening Ceremony K.R. Sreenivasan, ICTP Director, Trieste. 14:15 - 15:15, LB Diing Trbng and David Massey, Geometry and Topology of Singularities 15:30 - 17:00, Jean-Paul Brasselet, Poincar&Hopf Type Theorems on Singular Varieties - Characteristic Classes

Tuesday, 16 August 2005 10:30 - 11:30, Tatsuo Suwa, Introduction to Complex Geometry 11:30 - 12:30, LB Diing Trbng and David Massey, Geometry and Topology of Singularities 14:OO - 15:00, Michel Coste, Introduction to Real Geometry 15:15 - 16:15, Karl Fieseler and Gottfried Barthel, Introduction to Basic Toric Geometry 16:30 - 17:00, Working Groups - Seminars Wednesday, 17 August 2005

08:30 - 09:45 Jean-Paul Brasselet, Poincard-Hopf Type Theorems on Singular Varieties 1O:OO - 11:OO Tatsuo Suwa, Introduction to Complex Geometry 11:15 - 12:15 Lii Diing Trbng and David Massey, Geometry and Topology of Singularities 14:OO - 15:OO Karl Fieseler and Gottfried Barthel, Introduction to Basic Toric Geometry 15:15 - 16:15 Michel Coste, Introduction to Real Geometry 16:30 - 17:OO Working Groups - Seminars

Thursday, 18 August 2005 08:30 - 09:45 Jean-Paul Brasselet, Poincard-Hopf Type Theorems on Singular Varieties - Characteristic Classes

890

1O:OO - 11:OO Tatsuo Suwa, Introduction to Complex Geometry 11:15 - 12:15 Jose Seade, On Milnor’s Fibration Theorem 14:OO - 15:OO Michel Coste, Introduction to Real Geometry 15:15 - 16:15 Karl Fieseler and Gottfried Barthel, Introduction to Basic Toric Geometry 16:30 - 17:OO Working Groups - Seminars Friday, 19 August 2005 08:30

- 09:30 Jose Seade,

On Milnor’s Fibration Theorem 09:45 - 10:45 Tatsuo Suwa, Introduction to Complex Geometry 11:OO - 12:OO Le Dihg TrAng and David Massey, Geometry and Topology of Singularities 14:OO - 15:OO Karl Fieseler and Gottfried Barthel, Introduction to Basic Toric Geometry 15:15 - 16:15 Michel Coste, Introduction to Real Geometry 16:30 - 17:OO Working Groups - Seminars Monday, 22 August 2005 08:45 - 09:45 Bernard Teissier, Equisingularity theory 1O:OO - 11:OO Anatoly Libgober, Topology of the complements to curves and hypersurfaces 11:15 - 12:15 Herwig Hauser, The proof of resolution of singulaxities in characteristic zero 14:OO - 15:OO Walter Neumann, Surface singularities of splice type 15:15 - 16:15 Andras Nemethi, Invariants of normal surface singularities 15:15 - 16:15 Victor Goryunov and Vladimir Zakalyukin, Lagrangian and legendrian singularities 16:15 - 16:45 Working groups - Seminars

891

Tuesday, 23 August 2005

08:45 - 09:45 Bernard Teissier, Eq uisingularity theory 1O:OO - 11:OO Anatoly Libgober, Topology of the complements to curves and hypersurfaces 11:15 - 12:15 Herwig Hauser, The proof of resolution of singularities in characteristic zero 14:OO - 15:OO Joseph Steenbrink, Hodge mixed structures 14:OO - 15:OO Sabir Gusein-Zade, Monodromy of isolated singularities 15:15 - 16:15 Victor Goryunov and Vladimir Zakalyukin, Lagrangian and legendrian singularities 15:15 - 16:15 Andras Nemethi, Invariants of normal surface singularities 16:30 - 17:OO Working groups - Seminars Wednesday, 24 August 2005

08:45 - 09:45 Toru Ohmoto, Characteristic classes and Thom polynomials 1O:OO - 11:OO Bernard Teissier, Eq uisingulaxity theory 11:15 - 12:15 Walter Neumann, Surface singularities of splice type 14:OO - 15:OO Joseph Steenbrink, Hodge mixed structures 14:OO - 15:OO Sabir Gusein-Zade, Monodromy of isolated singularities 15:15 - 16:15 Shihoko Ishii, A resolution of singularities of a toric variety and a nondegenerate hypersurface singularities 16:15 - 16:45 Working groups - Seminars Thursday, 25 August 2005

08:45 - 09:45 Anne Friihbis-Kriiger, Computational approach to singularities 1O:OO - 11:OO Anne Friihbis-Kriiger, Computational approach to singulari ties

892

11:15 - 12:15 Lev Birbrair, Metric theory of singularities 11:15 - 12:15 Yukio Matsumoto, Topology of degeneration of Riemann surfaces 14:OO - 15:OO Oswald Riemenschneider, McKay correspondence for quotient surface singularities 14:OO - 15:OO Franqoise Michel, Polar quotients and Jacobiens quotients 15:15 - 16:15 Ragnar Buchweitz, Deformation theory and Quiver representation 15:15 - 16:15 Dirk Siersma and Michai Tibar, Singularities at infinity 16:30 - 17:OO Working groups - Seminars

Friday, 26 August 2005 08:45 - 09:45 Bernard Teissier Equisingularity theory 1O:OO - 11:OO Shihoko Ishii, A resolution of singularities of a toric variety and a nondegenerate hypersurface singularities 11:15 - 12:15 Yukio Matsumoto, Topology of degeneration of Riemann surfaces 11:15 - 12:15 Lev Birbrair, Metric theory of singularities 14:OO - 15:OO Franqoise Michel, Polar quotients and Jacobiens quotients 14:OO - 15:OO Oswald Riemenschneider, McKay correspondence for quotient surface singularities 15:15 - 16:15 Ragnar Buchweitz, Deformation theory and Quiver representation 15:15 - 16:15 Dirk Siersma and Michai Tibar, Singularities at infinity 16:30 - 17:OO - Working Groups 17:15 - 18:15 Social gathering Monday, 29 August 2005

09:OO - 1O:OO David Massey, The Perverse Structure of the Vanishing Cycles and Milnor Eq uisingularity

893

10:15 - 11:15 James Damon, A resolution for the geometric complexity of regions in R3 and its topological consequences 11:30 - 12:OO Ana Claudia Nabarro, Index of a vector field in R2 and the pencil’s geometry of its principal part 14:OO - 15:OO Helmut Hamm, Lefschetz theorem for the Picard group of singular quasiprojective varieties 15:15 - 16:15 Fouad El Zein, Lefschetz Theorems and Hodge theory 16:30 - 17:OO Michael Lonne, Braid Monodromy and Hypersurface singularities Tuesday, 30 August 2005 09:OO - 1O:OO Maria Ruas,

Topological triviality of families of singular surfaces 10:15 - 11:15 Marcel0 Saia, Whitney equisingularity of map germs, polar multiplicities and Euler obstruction of stable types 11:30 - 12:OO Raimundo dos Santos, Real Milnor fibration and C-regularity 14:OO - 15:OO Farid Tari, Bifurcations in implicit differential equations 15:15 - 16:15 David Trotman, Real equisingularity and tame geometry 16:30 - 17:OO Rogdrio Mol, Polar classes associated to singular holomorphic foliations Wednesday, 31 August 2005 09:OO - 1O:OO Terry Gaffhey,

Counting singularities, multiplicities, and pairs of modules 10:15 - 11:15 Frangoise Michel, The boundary of the Milnor Fiber for germs of surfaces in C3 with non isolated singular locus 11:30 - 12:OO Victor Hugo Jorge Perez, Multigraded rings and Buchsbaum-Rim multiplicities

894

14:OO - 15:OO Sabir Gusein-Zade, Chern obstructions for collections of I-forms on singular varieties 15:15 - 16:15 Jorg Schurmann, Indices for 1-forms on singular spaces and characteristic cycles 16:30 - 17:OO Ernest0 Rosales-Gonzalez, Rigidity of Germs of HolomorphicFoliations with Dicritic Singularity Thursday, 1 September 2005 09:OO - 1O:OO Claude Weber,

Four physicists at the origin of Knot theory in 19th. Century 10:15 - 11:15 Victor Goryunov, Logarithmic vector fields for discriminants of composite functions 11:30 - 12:OO Meral Tosun, Simple elliptic singularity and the corresponding Lie algebra 14:OO - 15:OO Susumu Tanabe, On Horn-Kapranov uniformisation of the discriminantal loci 15:15 - 16:15 Mutsuo Oka, Tangential Alexander Polynomial of plane curves 16:30 - 17:OO Pho Duc Tai, Plane curve Singularities: from local to global F'riday, 2 September 2005 09:OO - 1O:OO Aleksey Davydov,

Dansition between optimal strategies in Arnold's model 10:15 - 11:15 Masaaki Yoshida, Automorphic functions for the Whitehead-link-complement group 11:30 - 12:OO Martijn van Manen, Weighted Voronoi diagrams, discrete Morse theory and tropical geometry 14:OO - 15:OO Hans Brodersen, Sufficienty of jets with line singularities 15:15 - 16:15 Bernard Teissier, Valuations 16:30 - 17:OO Closure ceremony

895

LIST OF PARTICIPANTS ABRIANI Devis

S.I.S.S.A. - Trieste, Italy [email protected]

AGAFONOV Sergey Ivanovich

Martin-Luther-Universitiit, Germany sergey.agafonovOmathematik.uni-halle.de

AGUILAR Haydee

UNAM - Unidad Cuernavaca, Mexico [email protected]

AHMAD Khader Hamed

University of Damascus, Syria

AHMED Ahmed Mohamed

Al-Azhar University, Egypt ahmedelkbQ yahoo.com

ALTINTAS Ayse

Yildiz Technical University, Turkey ayseaQyildiz.edu. tr

APRODU Marian

Romanian Academy, Romania Marian.AproduOimar .ro

APRODU (n6e DUTU) Monica Alice

University of Galati, Romania Monica.AproduQuga1.ro

BAEZ SANCHEZ Andres David

Universidad Nacional de Colombia, Colombia [email protected] .co

BANERJEE Dipti

Rishi Bankim Chandra College, India deepbancuOhotmail.com

BARBOSA Gracielle Feliciani

Universidade de S b Paulo, Brazil [email protected]

BARTHEL Gottfried

Universit at Konst anz , Germany Gottfried.Barthe1Quni-konstanz.de

BELKHIRAT Abdelhadi

King Fahd Univ. of Petroleum & Minerals, Saudi A: belkhiratOhcc. kfupm.edu.sa

BIRBRAIR Lev

Universidade Federal do Ceara, Brazil levOmat .ufc.br

BOUZIANE Tmufik

tbouzianQictp.trieste.it

BRASSELET Jean Paul

Institut de Mathkmatiques de Luminy, France [email protected],s.fr

BRODERSEN Hans Christian Stang

University of Oslo, Norway broderseQmath.uio.no

BUCHWEITZ Ragnar-Olaf

University of Toronto, Canada [email protected]

CHALLAPA Lizandro Sanchez

Universidade de Sb Paulo, Brazil C hallapaOicmc .usp br

.

896 CIMPOEAS Mircea

Romanian Academy, Romania mircea.cimpoeasOimar.ro

CISNEROS-MOLINA Jose Luis

UNAM - Unidad Cuernavaca, Mexico jlcmQmatcuer.unam.mx

COB0 Helena

Universidad Complutense de Madrid, Spain hcobopabOmat.ucm.es

COSTA Joao Carlos Ferreira

Universidade de Siio Paulo, Brazil joaocarOicmc.usp.br

COSTE Michel

Universit6 de Rennes I, France michel.coste@univ-rennesl .fr

DAFOUNANSOU Ousmanou

University of Douala, Cameroon daf.ousmanouOcaramail.com

DAMON James

University of North Carolina, USA jndamonQmath.unc.edu

DARWISH Mohamed Abdalla

Alexandria University, Egypt darwishmaQyahoo.com

DATTA Mahuya

Indian Statistical Institute, India mahuyaQisica1.ac.in

DAVYDOV Aleksey Alexandrovich

Vladimir State Technical University, Russia

DINAR Yassir Ibrahim Yousif Ali

S.I.S.S.A. - Trieste, Italy dinarOsissa.it

DIVAANI-AAZAR Kamran

Az-Zahra University, Iran kdivaaniQipm.ir

DOAN Cuong Trung

Academy of Sciences of Vietnam, Viet Nam dtcuongQmath.ac.vn

EBELING Wolfgang Ditmar

Universitat Hannover, Germany ebelingOmath.uni-hannover.de

EL ZEIN Fouad

Universitk de Nantes, France elzeinQmath.univ-nantes.fr

ERDOGAN Sultan

Bilkent University, Thrkey erdoganQfen.bilkent .edu.tr

EYRAL Christophe

Universit6 de Provence, France eyralchrQyahoo.com

FIESELER Karl-Heinz

University of Uppsala, Sweden [email protected]

FIOROT Luisa

Universiti degli Studi di Padova, Italy fiorotOmath.unipd.it

897 FLORES BAZAN Fabian

Universidad de Concepcion, Chile ffloresQing-mat .udec.cl

FRUHBIS-KRUGER Anne

Universitat Kaiserslautern, Germany [email protected]

GAFFNEY Terence James

Northeastern University, USA gaffOneu.edu

GATEVA-IVANOVA Tatiana

American University, Bulgaria tatianagatevaOyahoo.com

GILES Arturo Enrique

UNAM - Unidad Cuernavaca, Mexico arturoOmatcuer.unam.mx

GONZALEZ VILLA Manuel

Universidad Complutense de Madrid, Spain mgvOmat.ucm.es

GORYUNOV Victor Vladimirovich

University of Liverpool, UK goryunovOliv.ac.uk

GRULHA JUNIOR Nivaldo de Goes

Universidade de S k Paulo, Brazil [email protected]

GUSEIN-ZADE Sabir Medgidovich

Moscow State University Moscow, Russia [email protected]

HAMM Helmut Arend

Westfalische Wilhelms-Univ. Miinster, Germany [email protected]

HASSAN Samia Zaki

Mansoura University, Egypt [email protected]

HAUSER Herwig

Universitat Innsbruck, Austria herwig.hauserOuibk.ac.at

HAYELAPETYAN Armen Garniki

Yerevan State University, Armenia ahayrOfreenet .am

HEGAZI Ahmed Sadek

Mansoura University, Egypt hegaziOmum.mans.eun.eg

HERNANDES Marcel0 Escudeiro

Universidade Estadual de Maringa, Brazil mehernandes@uem. br

HERNANDES Maria Elenice

Universidade de S k Paulo, Brazil eleniceQicmc.usp. br

IBRAHIM ARBAB Arbab

Comboni College for Computer Science, Sudan arbab640yahoo.com

ISHII Shihoko

Tokyo Institute of Technology, Japan [email protected]

ISHIKAWA Masaharu

Tokyo Institute of Technology, Japan ishikawaOmath.titech.ac.jp

898 IZADI Mohammad Ali

Shahre-Kord University, Iran m~izadi780yahoo.com

JORGE PEREZ Victor Hugo

Universidade de Sk Paulo, Brazil vhjperezQicmc.usp.br

JUNIATI Dwi

Pasca Sarjana Universitas Negeri Surabaya, Indianesia [email protected]

KAUP Ludger

Universitat Konstanz, Germany [email protected]

KEEM Changho

Seoul National University, Korea ckeemQmath.snu.ac.kr

LE Thi Thanh Nhan

Thai Nguyen Pedagogical University, Viet Nam trtrnhanOyahoo.com

LE Thuong Quy

Vietnam National University, Viet Nam leqthuongOyahoo.com

LEMAHIEU Lucienne August Ann

Katholieke Universiteit Leuven, Belgium [email protected]

LIBGOBER Anatoly

University of Illinois at Chicago, USA libgoberOmath.uic.edu

LICANIC Sergio Mariano

Universidade Federal Fluminense, Brazil serOimpa. br

LONNE Michael

Universitat Hannover, Germany [email protected]

LOPEZ HERNANZ Lorena

Universidad de Valladolid, Spain lorenalopez [email protected]

LU Guangcun

Capital Normal University, China gcluObnu.edu.cn

LUCHESI Vanda Maria

Universidade de SBo Paulo, Brazil vmlucQicmc.usp.br

LUDWIG Ursula Beate

Albert-Ludwigs-Universitat F'reiburg, Germany [email protected]

MARTINEZ Sergio

Universidad de Zaragoza, Spain sergiomjQunizar.es

MARTINEZ-OJEDA Emigdio

ICTP, Trieste, Italy emarti n e Oict p. i t

MARZOUGUI Habib

Universit6 7 Novembre de Carthage, Tunisia habib.marzoukiQfsb.rnu.tn

MASSEY David

Northeastern University, USA dmasseyQneu.edu

899 MATSUMOTO Yukio

University of Tokyo, Japan [email protected]

MAZIN Mikhail

Moscow State University, Russia [email protected]

ME1 Jiaqiang

Nanjing University, China [email protected]

MEZRAG Lahcene

Universit6 de M’Sila, Algeria

MICHEL Franqoise

Universitb Paul Sabatier, France [email protected]

MIRANDA Aldicio Jose

Universidade de Siio Paulo, Brazil

1mezragQyahoo.fr

[email protected]

MOL Rogbrio Santos

Universidade Federal de Minas Gerais, Brazil [email protected]

NABARRO Ana Claudia

Universidade de SLo Paulo, Brazil anac1anaQicmc.usp.br

NANG Philibert

Universitb de Masuku, Gabon [email protected]

NEMETHI Andras

Hungarian Academy of Sciences Alfred Renyi, Hungar nemethiQrenyi.hu

NEUMANN Walter

Columbia University, USA [email protected]

NGAKEU Ferdinand

Universitk de Douala, Cameroon

NGUYEN Huu Viet Khue

Vietnam National University, Viet Nam

[email protected] [email protected]

NGUYEN Thang Tat

Hanoi University of Science, Viet Nam [email protected]

NGUYEN Tu Cuong

Academy of Sciences of Vietnam, Viet Nam cuongntQhn.vnn.vn

NGUYEN Viet Anh

ICTP, Trieste, Italy [email protected]

NUNO-BALLESTEROS Juan Jose

Universitat de Valencia, Spain juan.nunoQuv.es

OBLANCA-GONZALO Oliver

Universidad de Valladolid, Spain

OHMOTO Toru

Hokkaido University, Japan [email protected]

oblancaOagt.uva.es

900 OKA Mutsuo

Tokyo University of Science, Japan okaQrs.kagu.tus.ac.jp

OUARO Stanislas

Universith de Ouagadougou, Burkina Faso souaroOuniv-ouaga.bf

P E PEREIRA Maria

Universidad Complutense de Madrid, Spain [email protected]

PEDERSEN Helge Moller

University of Aarhus, Denmark [email protected]

PERALTA-SALAS Daniel

Universidad Complutense de Madrid, Spain [email protected]

PHO Duc Tai

Hanoi University, Viet Nam phoductaiQyahoo.com

PHUNG Ho Hai

Academy of Sciences of Vietnam, Viet Nam phungQmath.ac.vn

PIREDDU Marina

Universitti degli Studi di Udine, Italy [email protected]

PRIMULANDO Reinard

Bandung Institute of Technology, India reinard-pQyahoo.com

RASTEGAR Arash

Sharif University of Technology, Iran [email protected]

RIBEIRO TEIXEIRA Ana Carolina

Universidade Federal do Rio Grande do Sul, Brazil anacarolQif.ufrgs.br

RIEMENSCHNEIDER Oswald

Universitat Hamburg, Germany riemenschneiderOmath.uni-hamburg.de

ROSALES GONZALEZ Ernesto

Universidad Nxional Autonoma de Mexico, Mexico ernestoQmatem.unam.mx

RUAS Maria Aparecida

Universidade de S5o Paulo, Brazil maasruasQicmc usp br

SAIA Marcelo Jose

Universidade de Siio Paulo, Brazil mjsaiaOicmc.usp.br

SALEHYAN Parham

Universidade Estadual Paulista, Brazil [email protected]. br

SANTOS Raimundo Nonato A.

Universidade de Sao Paulo, Brazil [email protected]

SCHEPERS Jan August Wim

Katholieke Universiteit Leuven, Belgim [email protected]. be

SCHURMANN Jorg

Westfaelische Wilhelms-Univ. Munster, Germany jschuermQmath.uni-muenster.de

. .

901 SCHWIERING Marco

Universitat Hannover, Germany [email protected]

SEADE KURI Jose A.

Universidad Nacional Autonoma de Mexico, Mexico [email protected]

SETAYESH Iman

Sharif University, Iran [email protected]

SIERSMA Dirk

University of Utrecht, Netherlands [email protected]

SIMSIR Fatma Muazzez

Baskent University, Turkey fsimsir@baskent .edu.tr

SKUTLABERG Olav

University of Oslo, Norway oskut1abQmath.uio.no

STAMATE Dumitru Ioan

Romanian Academy, Romania [email protected]

STEENBRINK Joseph Henri Maria

Radboud University Nijmegen, Netherlands [email protected]

SUWA Tatsuo

Niigata University, Japan [email protected]

TANABE Susumu

Independent University of Moscow, Russia [email protected]

TARI Farid

University of Durham, UK [email protected]

TEISSIER Bernard

Institut de Mathkmatiques de Jussieu, France [email protected]

TIBAR Mihai

Universite de Lille 111, France mihai.tibarQmath.univ-lillel.fr

TOMAZELLA Joao Nivaldo

Universidade Federal de S k Carlos, Brazil [email protected]

TOSUN Meral

Yildiz Technical University, Turkey [email protected]

TRINH Duc Tai

Pedagogical College, Viet Nam [email protected]

TROTMAN David John Angelo

Universitk de Provence Aix-Marseille I, France trot manQcmi univ-mrs fr

VAN MANEN Martijn

Hokkaido University, Japan [email protected]

VU Khoi The

Academy of Sciences of Vietnam, Viet Nam [email protected]

.

.

902 WANG Zhiren

Ecole Polytechnique, France zhiren.wang@polytechnique. fr

WEBER Claude

Universith de GenBve, Switzerland [email protected]

WU Zhen

Shandong University, China [email protected]

YOSHIDA Masaaki

Kyushu University, Japan myoshidaQmath.kyushu-u.ac.jp

ZAKALYUKIN Vladimir Mikhailovich

Moscow Aviation Institute, Russia [email protected]

This page intentionally left blank