Contributions to Several Complex Variables: In Honour of Wilhelm Stoll

Aspects of Mathematics Aspelde der Mathematik Editor: Klas Diederich Vol. El: G. Hector/U. Hirsch, Introduction to th...

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Aspects of Mathematics

Aspelde der Mathematik Editor: Klas Diederich

Vol. El: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part A Vol. E2: M. Knebusch/M. Kolster, Wittrings Vol. E3: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part B Vol. E4: M. Laska, Elliptic Curves over Number Fields with Prescribed Reduction Type Vol. E5: P. Stiller, Automorphic Forms and the Picard Number of an Elliptic Surface Vol. E6: G. Faltings/G. Wi.istholz et. al., Rational Points (A Publication of the Max-Pianck-lnstitut fiir Mathematik, Bonn)

Vol. E7: W. Stoll, Value Distribution Theory for Meromcirphic Maps Vol. E8:., w.:von Wahl, The Equations of Navier-Stokes and Abstract Parabo.IJc'Equations Vol. E9: A. Howard, P.-M. Wong (Eds.), Contributions to Several Complex Variables Band Dl: H. Kraft, Geometrische Methoden in der I nvariantentheorie

The texts published in this series are intended for graduate students and all mathematicians who wish to broaden their research horizons or who simply want to get a better idea of what is going on in a given field. They are introductions to areas close to modern research at a high level and prepare the reader for a better understanding of research papers. Many of the books can also be used to supplement graduate course programs. The series comprises two sub-series, one with English texts only and the other in German.

Alan Howard, Pit-Mann Wong (Eds.)

Contributions to Several Complex Variables In Honour of Wilhelm Stoll

Friedr. Vieweg & Sohn

Braunschweig I Wiesbaden

Professors Alan Howard and Pit-Mann Wong, Department of Mathematics, University of Notre Dame, Post Office Box 398, Notre Dame, Indiana 46556, USA.

AMS Subject Classification: 32 06

All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1986

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior permission of the copyright holder.

Produced by W. Langeliiddecke, Braunschweig Printed in Germany

ISSN ISBN

0179-2156 3-528-08964-4

Contents Picture of Wilhelm Stoll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VI

Foreword . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII

Group Picture of the Conference Participants and Explanations . . . . . . . . • . . . . . . . . . . . VIII Arithmetic Hilbert Modular Functions Ill Walter L. Bailey, Jr. The Heat Equation for the a-Neumann Problem on Strictly Pseudoconvex Domains Richard Beals and Nancy K. Stanton

41

Some Examples of the Twistor Construction Daniel Burns

51

Complete Kahler Domains. A Survey of some Recent Results Klas Diederich

69

On the Minimality of Hyperplane Sections of Gorenstein Threefolds Maria L. Fania and Andrew J. Sommese

89

On Meromorphic Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hans Grauert

115

Recent Developments in Homogeneous CR·Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . Alan T. Huckleberry and Wolfgang Richthofer

149

Problems of Value Distribution in Complex Analysis for Several Variables Pierre Lelong

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

179

On the Boundary Behavior of Holomorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . L/Jszl6 Lempert

193

Integral Geometry of the Monge·Ampere Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Robert Molzon Logarithmic Jet Spaces and Extensions of de Franchis' Theorem Junjiro Noguchi

227

Remarks on the Nakano Vanishing Theorem Bernard Shiffmann

251

Curvature of the Weii·Petersson Metric in the Moduli Space of Compact Kiihler·Einstein Manifolds of Negative First Chern Class . . . . . . . . . . . . . . . . . . . . . . . . 261 Yum·TongSiu Extension Problems and Positive Currents in Complex Analysis Henri Skoda

. . . . . . . . . . . . . . . . . . . 299

On the Uniformization of Parabolic Manifolds . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . 329 Pit-Mann Wong

Wilhelm Stoll October 1984

VI

Foreword

In 1960 Wilhelm Stoll joined the University of Notre Dame faculty as Professor of Mathematics, and in October, 1984 the university acknowledged his many years of distinguished service by holding a conference in complex analysis in his honour. This volume is the proceedings of that conference. It was our priviledge to serve, along with Nancy K. Stanton, as conference organizers. We are grateful to the College of Science of the University of Notre Dame and to the National Science Foundation for their support. In the course of a career that has included the publication of over sixty research articles and the supervision of eighteen doctoral students, Wilhelm Stoll has won the affection and respect of his colleagues for his diligence, integrity and humaneness. The influence of his ideas and insights and the subsequent investigations they have inspired is attested to by several of the articles in the volume. On behalf of the conference partipants and contributors to this volume, we wish Wilhelm Stoll many more years of happy and devoted service to mathematics.

Alan Howard Pit-Mann Wong

VII

VII I

"'

Cll

l6' "'Cll c.

E

CQ

z

l!!

....::I . !::!

Cl..

c.

::I

e

(.!) Cll

.s::.

....

.....0 Cll

E

Cll

.s::.

~

IX

Participants on the Group Picture 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

X

Qi-keng LU, Professor, Chinese Academy of Science, Peking, China. Pierre LELONG, Professor emeritus, University of Paris VI, France. Wilhelm STOLL, Professor, University of Notre Dame. Marilyn STOLL, Timothy O'MEARA, Provost and Professor of Mathematics, Univ. of Notre Dame. Mike SPURR, Professor, Rice University, Houston, Texas. B. A. TAYLOR, Professor, University of Michigan, Ann Arbor, Mich. Yi-Chuan PAN, Professor, Jackson State University, Jackson, Miss. David BARRETT, Professor, Princeton University, Princeton, N.J. Robert FOOTE, Professor, Texas Tech University, Lubbock, Texas. Weiqi GAO, Student, University of Notre Dame. Alan HOWARD, Professor, University of Notre Dame. Dennis SNOW, Professor, University of Notre Dame. Joanne SNOW, Professor, St. Mary's College, Notre Dame, Ind. Robert MOLZON, Professor, University of Kentucky, Lexington, Ky. Nancy STANTON, Professor, University of Notre Dame. Mary Jo KR EUZMAN, Student, University of Notre Dame. Paula A. RUSSO, Professor, Michigan State Univ., East Lansing, Mich. Eric BEDFORD, Professor, Indiana University, Bloomington, Ind. Zbigniew SLODKOWSKI, Visiting Professor, Univ. of California, Los Angeles. Hans GRAUERT, Professor, University of Gottingen, F.R.G. Giorgio PATRIZIO, Professor, University of Rome II, Italy. Alan HUCKLEBERRY, Professor, University of Bochum, F.R.G. J. RAMANATHAN, Professor, University of Chicago, Illinois. Henri SKODA, Professor, University of Paris VI, France. Wanxi CHEN,Student,Univ.of Notre Dame (from Univ.of Science and Techn.of China). Harry d'SOUZA, Professor, University of Michigan, Flint, Mich. Chong-Kyu HAN, Professor, University of Alabama-Tuscaloosa, Univ. Alabama. Stephen BELL, Professor, Princeton University, Princeton, N.J. Yum-Tong SIU, Professor, Harvard University, Cambridge, Mass. Klas DIEDERICH, Professor, University of Wuppertal, F.R.G. Y. KIM, Student, University of Michigan, Ann Arbor, Mich. Junjiro NOGUCHI, Professor, Tokyo Institute of Technology, Tokyo, Japan. Daniel BURNS, Professor, University of Michigan, Ann Arbor, Mich. Leonard SMILEY, Professor, University of Alaska, Anchorage, Alaska. Yong In KIM, Professor, Michigan State Univ., East Lansing, Mich. Pankaj TOPIWALA, Student, University of Michigan, Ann Arbor, Mich. Walter BAILEY, Professor, University nf Chicago, Chicago, Illinois. Herbert ALEXANDER, Professor, University of Illinois at Chicago, Chicago, Illinois. Pit-Mann WONG, Professor, University of Notre Dame. Kam Wing LEUNG, Professor, Chinese University of Hong Kong. Brian SMYTH, Professor, University of Notre Dame.

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Andrew SOMMESE, Professor, University of Notre Dame. Michael BUCHNER, Professor, University of New Mexico. Laszlo LEMPERT, Professor, Eotvos Lorimd University, Budapest, Hungary. Norman LEVENBERG, Professor, University of Kentucky, Lexington. Ky. Chia-Chi TUNG, Professor, Mankato State University, Mankato, Minn. Dan GROSS, Professor, Seton Hall University, South Orange, N.J. Steven ZUCKER, Professor, Johns Hopkins University, Baltimore, Md. Bernard SHIFFMAN, Professor, Johns Hopkins University, Baltimore, Md. John WERMER, Professor, Brown University, Providence, R.I. Gary HARRIS, Professor, Texas Tech University, Lubbock, Texas. James FARAN, Professor, Johns Hopkins University, Baltimore, Md. Thomas BLOOM, Professor, University of Toronto, Ontario, Canada. lan R. GRAHAM, Professor, University of Toronto, Ontario, Canada. Mark ANEMA, Student, University of Michigan, Ann Arbor, Mich. Patrick SMITH, Professor, University of Toronto, Ontario, Canada. Harold BOAS, Professor, Texas A. & M. Univ., College Station, Texas. Jay WOOD, Visiting Professor, University of Notre Dame. Bruce GILLIGAN, Professor, University of Regina, Saskatchewan, Canada.

XI

Arithmetic Hilbert Modular Functions Ill Walter L. Baily, Jr.* Department of Mathematics, University of Chicago 5734 S. University Ave., Chicago, IL 60637 U.S.A. INTRODUCTION The purpose of this paper, which is a continuation of [2,3,4,5], is to prove that the special values of Hilbert modular

functions of level 7L generate abelian extensions of

certain

CM fields, by using an essentially elementary theory

of arithmetic Hilbert modular functions, based on the theory of congruence Eisenstein series. The main results are generalizations of the main results of Heeke's thesis [10]. They are also subsumed in more far-reaching results of Shimura and Taniyama [15,13,14]. But our methods are quite different from the latter's and stem directly from Heeke's original ideas. We start from a totally real number field n

over

Q

k

of degree

and consider a family of arithmetic subgroups of

the projective group PGL 2+(k) acting on the product of n upper half planes. Letting K be a purely imaginary quadratic extension of

k , the various immersions of the multiplica-

tive group Kx in PGL 2 +(k) determine fixed points in certain copies of Hn , which are called special points, each of which is associated to a certain order in of integers of

k .

image of an element

K

containing the ring

~

Such a special point is realized as the T

of

K - k

under those immersions of

K in C for which the image of T has positive imaginary parts. This collection of n immersions of K in c as socia ted to T also determines a CM-type for K and a reflex field K*(T) Defining arithmetic Hilbert modular functions as in [3,4], we show here that i f f is an arithmetic Hilbert modular function of level TV and if (T) is a special point associated to the principal order of K and if K* (T) is the corresponding reflex field, then f((T)) generates an abelian extension of K*(T) ~moreover, if Ln. is the abelian extension of K*(T) generated by all such special values of arithmetic Hilbert modular functions of level ~ , then the Galois group of Ln(K*(T) is isomorphic to a sub*Support from NSF Grant No. DMS-8401708

2

group of a certain class group of K. Cf. [101, pp. 55-57 • The point of our developments here is to show much can be done without using the theory of families of abelian varieties and their moduli. Our treatment is based on [10, 3 , 4 , 5 1 and on a paper of Karel [111, in which the classical one variable case is similarly treated. contribution of [ 7 1

Of course, we must emphasize the

in characterizing the more basic geo-

metric and topological properties of the adelic space. We postpone to a subsequent publication consideration of Shimura's reciprocity law for the abelian extension Ln(K*{f) obtained from Kronecker type congruences for the special values.

0. NOTATION AND PRELIMINARY REMARKS. In this article we use the notation of [51.

As a con-

venience to the reader we summarize some commonly used notation.

Denote by

k

a totally real number field with

n > 1 , having ring of integers imaginary quadratic extension of

~

.

Let k . If

[k:Q1

be a purely a E k , a >> 0 K

will mean that a is totally positive. Let L: = {al, ... ,an) be the set of all immersions of k as a subfield of ffi , arranged in a definite order; in general,

0.1 L

=

'V

'V

{a 1 , ... ,an)

will be a "lifting" or set of extensions of

a 1 , ... ,an to immersions of K in C For a given lifting, d~fine [151 the reflex field K* { 't) by K* { 't) = Q { { L:6' E 't 11 I 11 E K}) Let A , A{k) , A{K) , I , I{k) and I(K) denote respectively the rings of adeles of

Q , of

k , of

K , and their groups

of units, the ideles, each supplied with its customary topology. The subscripts oo and f will denote the projections of an adelic object to its archimedean and non-archimedean components respectively, and the subscript + will indicate adelic objects with non-negative archimedean components. ~

resp. ~ will be the maximal compact subrings of Af and of A(k)f respectively. On the other hand, Q , k , and K are naturally imbedded in their rings of adeles and if a belongs to one of these fields, we use a to denote af , A

the non-archimedean projection of the image of

a

in the

3

adeles. Thus, if a E ~ , a E & , and the image of o under this mapping. We denote by by

Hn

its

(i, ... ,i)

H

&

the upper half complex plane

n-th cartesian power, and by

E

is the closure of

Hn, i = /-1

Moreover,

G1

Im z

> 0 ,

i.e , the point is the algebraic

group GL 2 defined over k and G = Rk/QG 1 • We have a cannonical isomorphism ¢ of G 1 (A (k)) onto G (A) such that with respect to sui table integral structures we have = G(~) G 1 and

P

If

~

f(~)

¢(G

1

(&) )

is the group of upper triangular matrices in R = k/Q P 1 . is an open compact subgroup of G(Af) , denote by P1

the corresponding arithmetic subgroup

G(Q)nG+(R)JK

of G+(~) , or its projection into G+(R) , according to the context. W~ denote by ~~ the isotropy group of i.e E Hn in

G+(R) For most other notation we refer to [ 5]

However,

some modifications of our earlier conventions are needed. The center of G is Z = Rk/QZ 1 center of G 1 • In [ 4] , the symbol

,

where Z 1 is the CA,CIK ,w) was used to

denote the space of modular forms of weight to

w

with respect

, and that space was defined as the complex vector

~

space of functions @ on G+(A) satisfying three conditions (i), (ii), and (iii) . In fact it turns out that (ii) is undesirable and unsuited for our purposes, and instead of replacing it with the condition that

@ transform according

to some finite character on the center Z(A) of G+(A) , we drop the condition (ii) altogether. (In fact, condition (i) alone implies that all

z

@(zg)

=~(g)

for all

g

E

G+(A)

belonging to a subgroup of finite index in

and for Z(A).

Indeed, the Eisenstein series (3), p. 596 of [ 3] does not satisfy (ii) unless the function s on double cosets does. Nevertheless, the further discussion of Eisenstein series, and in particular of their convergence, is not affected by this.) Secondly, while most of the open compact subgroups ~ considered in [ 4] were principal congruence subgroups of G 1 (o) of the form ~(n) = {k E G 1 (o) 1 k = 1 2 mod7L}, we want to consider here, as far as possible, an arbitrary open

4 ~

compact subgroup

of

G(Af)

In later sections we return

to the principal congruence subgroups as a practical consequence of their relationship with ray class groups. Returning to the purely imaginary quadratic extension K of k , let B = (6 1 , B2 ) be a k-basis of K . If a E K , there is a unique two by two matrix q B (a l

The mapping

~--~~

a

(

~ ~)

qB(a)

= (:

qB(a)

E

M2 (k)

such that

~ ~)

is called a regular imbedding of

K in M2 Ck) and may be extended in a natural way to an injection of A(k)-algebras of A(K) into M2 (A(k)) If B is replaced by another k-basis B' of K , qB is replaced -1

by qB' SqBS for some S E GL 2 (k) If T E K - k , we write qT in place of qB

B

=

(T

I

l)

1. SPECIAL POINTS AND RAY CLASS POLYNOMIALS 1.1 DEFINITIONS

be a set of reWe refer to §2 of [ 5] . Let @ C G (Af) presentatives for the following decomposition of G+(A) into double cosets:

= VBE@G+(Q)8G+(R)G(~)

G+(A) 'V

Let K and z be as in the introduction and T E K - k be 'V such that ~(T) = ~ , i.e., Im(T 0 j) > 0 , j = l, ... ,n . Then 'V

'V

I(k)f

and let

(T) = (Tol, ... ,Ton) E Hn and T defines an imbedding qT of K in M2 (k) which exin M2(A(k)) We tends naturally to an imbedding of A(K) also associate to any T E K- k its reflex field K*(T) K* (~h) ) , We may assume that each 8 c @ is of the form 8

=

(g'

ry

rJv~-r

~)

I

8'

be the

E

a-module

VT + o

v in

v8 K

id. (8')-l

Let

and let

If now we fix in K an order ~ conbe its order in K taining the integers ~ of k 1 we may define the subset :0: (~) = {T E K - k I!Rv h ) = 1 and a =v(~) of K by v

'V

subset

=vc~~z l

of

£ }

=v<~l

by

0

~v(tf<-,

I)

h

=

~v{d?.,) I f'(T)

E

I} ,

and ~ (~, ~) may be viewed as a subset of v mapping T - - > ( T) · E Hn . We have

e E

by the

= ~ E~G+{Q)f8G(~)

G(Af) For any

Hn

~ , define 8 G (IR) G ( ~ ) / 1K G ( ?z ) + 00

~e

and

~8

oo{(ft,

oo
I

~)

{g

=

g (i.e) =

E

(T)

I

T

E

}

I

I

~A(R.J

and then put

~oo (d{_) ~A (t!?,, ~)

We define § 2.)

G+ {Q)~ 00 (tR,)G(~)

=

v8

=

E

fj)

~8 1 oo (~) .e

~00 uR, ~)

and

analogously. ~ JK (/R)

We also carry over the notation

C~(~)

, etc., from

§2

, where

of loc. cit.

1.2 SPECIAL POINTS OF GIVEN TYPE We now prove analogs of certain propositions of Karel [11) for application in our more general situation. Lemma 1.2.6 of [11)

(so that e E fH'. and I; E ~e ((JG.J ~) '-J' 'V I 00 T E ~ (~, E) be such that ve Then each x E ~A(dG, ~) can be written in

Proposition 1.

e~ = (T)

t:e .

=

E

Analogous to

we have Let

~ l , and let 00

~(i.e)

the form

x = yq (a T

for some

y E G+{Q)

Proof. for some

)~ew

G>

I

where

(~, ~) Put vl = ve 1 I 00 1 id. (det(8))-l , and let Tl E K

id. (det(e 1 ))-l and v be such that (T 1 ) = t; 1 (i.e) , so that

0tT = are proper

VT +

0

(}{.-ideals of

x E G+(Q)t; 1 e 1 G(~)JK00

, we have

sl E ~e

=

T E ~v(~, ~)

w E ~ 00 .G(~)

, a E I(K)f , a~~

x E ~A(6G, ~)

If

el E

-1

,

1

'f(T)

'f
and K

T1 E ~v (~, ~)

Then

I

OlTl

,

and

=

VlTl +

hence, so is

0

Ol =aT -la Tl

6

since

OG

C:

e'

and

[K:k]

= 2

Oz. =

that

K fl A (K) oo a()(, .

ing Lemma 1 of [ 5] ,

j (g) preced-

which

g

belongs.

Let us

that context, T and ' l that ~(, 1 ) = ~(T) = ~ , it cit.)that 8 1 ~ 1 and

are in the same double coset with respect to

on the left and G(~) ~ 00 proof of Proposition 1. 1.3

such

should have been defined as the

double coset G+(Q)g~00 G(~) to make this correction. Then, in having already been chosen such follows from that Lemma 1 (lac.

q,(a- 1 )8~

is

a£ I(K)f

In the definition of

j(g)

Oo

Since, therefore,

everywhere locally pringipal, there exists

G+(Q)

on the right, which completes the

ACTION OF THE GALOIS GROUP AND OF

G(Af)

As in [ 4 ] we introduce the graded algebra

a(

1K ,Qab) of

Qab-arithmetic modular forms with respect to

IK

spect to a certain action of the Galois group

~ = Gal(~ab/0) ,

the graded O-subalgebra forms with respect to IK forms of weight

0

w

and, with re-

a(IK ,Oab)O

of O-rational modular The respective subspaces of modular

will be denoted by

{).
,w,(! b) a

and

0

{J,(IK,w,Oab) Denote by lfn.<~<, Oab) resp. IJn(IK , Oab) the algebra of Qab-rational meromorphic modular functions resp. the algebra of o-rational meromorphic modular functions with respect to

=

A (IK)

'in. (fK

I

IK •

Cl,(IK ,

As a convenient abbreviation, we also let

Oab) O

oab) 0 If M ~ G+(A)

A (I<

,

w)

a(l< , w , Oab) O , and

, we denote by

and left regular representations of tions on

G+(A)

, and for

a£~

R(M)

and

L(M)

M(IK)

the right

M on the space of funcwe denote by

a(IK

S(a)

the ac-

tion defined in [ 5 ] of a on elements of 1 Qab) According to §5.2 of [ 5] , we have R(M)S(a) = S(a)R(M) ME: G(Af) we define

I

etc. for

.,g

and a£ If cp £ a(IK, w) and M £ G(Af) ,
to Lemma 3.2 of [11] we then have Proposition 2. I f

=

and

1J1 £ A(l<)

, and if

M £

, then R(M)
7

Proof. The proof is analogous to that given in §3.2 of [ 11] • If

~

is an open compact subgroup of

G{Af)

, denote by

prK the canonical mapping of G+{A) onto its space of double cosets VI< = G+ {Q)\G+ {A)/IKJK00 1.4 OPERATION OF THE IDELES According to Proposition 1 , given any element

~e

E

:;;: {R,., 't> and T E :;;: {d?,, 'f) such that ~{i.e) = {T) , each ve can be written as yq {b)~ew in the notation X E :;;:A{~, 'f) 00

T

is a regular imbed-

there and with bE I{K)f • Then ding of K as a k-algebra into

We introduce for

constant use the following conventions and notation:

Suppose

e =(~' ~) I where e' E I{k)f I v = ve = id. {6')-l; then ~ is the order in K of the rank two o-module VT + o . Exinto M2 {A{k)) and define tend q to an imbedding of A{K) A simple calculation shows that q' C M2 {o) , q' = e -lqe where

,B vv

.0 is the closure of YV in

§3.3 of [Ill , we have: Proposition 3. If

a E I{K)f

A{K)f and

Analogous to

ME G{Af)

are such

that (A)

priK {a*x)

then for some

f3 E Kx

prl< {xM)

we have -1

{B)

MEw

Conversely,

{B)

{Remark.

q'{a-lS).IK

implies {A).

The action

x

~

a*x

of

a E I{K)f

on the

special points x E ::A (6{, '±'> is that described in § 2 of [ 5 ] • Cf. also [11], §1.2.5.) Proof. {A) holds if and only if for some y E G+{Q) , {*)

~q {a-lb) 6w E y~q {b) 6wMI
00

The archimedean component of this reads: ~ E Y 00 ~IK"" , which is equivalent to having {T) =~{i.e) = y 00 • {T) , and since q{Kx) is the stabilizer of {T) in G+{Q) , this means y = q(f3-l) for some f3 E Kx Therefore the non-archimedean component of {*) reads (**)

8

which is equivalent to having w-

1

A

q 1 (Ba-

1

M~

)w £

conversely,

(B)

we have

=

~

£

yoo~

(T)

IK 00

,

or

M

£

w

q(a -1

m

(!)

A

q 1 (Ba

-1

) 00

(T)

,

A-1

q(B

)8w £

-1

clearly implies (**)

q(B

, and for all

y

=

q(B

-1

B

of

K

Let

m

K

G ( ?l )

be a non-zero ideal in

Define the principal congruence subgroup = G I ( ; ) by IK (m.)

Then G (o) subgroup of 1

= {

g

= ~(1) 1K (1)

G1

£

•

(

o) I

g

=1 2

We shall let

containing some

K

Q.E.D.

be the maximal o'

be the smallest positive rational integer in m:

(m) .

£

• X

) , then

which, together with (**), gives (A). ~ in

or

)8wM~

which is (B)

)wiK

or if we put

Henceforth we let the order order

-1

K(~)

and let 2Z

nm of

m- }.

mod

~ denote a normal open IK (m.,) The imbedding

q' : I(K)f - - * G(Af) is a closed immersion which algebraically and topologically -1 identifies I(K)f with the group 8 T(Af)e for some Q-torus T in G Let H = q 1 -l(~) • This is an open compact subgroup of I(K)f . congruence subgroup mod~ medean idelic units of C ((J, If) operates on the set

If K = K(~) of the group

K

,

H

The class group

= I (K) /I (K) .H.Kx

~IK(~, ~)

is the principal

~x of non-archi-

, th: image of

~A(~,~)

in

VIK because q(Kx) C: G+(Q) , I(K) 00 acts trivially on ~A
= I< whenever x = yl;8w £ ::A($-) , and (T) = ~(i.e) so ,v that, according to the calculations of §2 of [5 ] , H acts trivially on :::~(~, ~) . Henceforth IK will denote a sub-

wiK

group of

G(Af) satisfying these conditions, and such a sub-

group will be called an n.o.c. subgroup of G(Af) . We let B be an orbit of C ( (J, H> in :::IK ( CJ, ~) Following the ideas of [ll] we wish to show that the 0-cycle B

on

VI<

is defined over the reflex field

1.5 RATIONAL STRUCTURES.

K*(~) .

ARITHMETIC MODULAR FUNCTIONS.

In the statements and results of [ 5 ] regarding the field of rationality for algebras of modular forms, the norm N~ of ~ may be replaced by m . For example, Prop. 2 of j5.2, loc. cit. may be strengthened to read: " (lciK (11t-) ) is the

9

e .v

integral closure of

in its graded ring of homogeneous

("111.)

quotients and we have (l(JK (m.) , Om) (f) 0 C , " m

and then Prop. 3 of loc. cit. applied to obtain immediately Proposition 4.

>0 =

A (I<

<ml > ~

For our given n.o.c. subgroup assume to contain Q-subalgebra of

{).
II< (17t)

of

we have that

,

G(Af)

Q..(IK)

II< =

belongs to for

A(IK(m,) ,w)

e;

(t
j = l, ... ,t

, which we

is a graded

and, by Prop. 4, we have

(J.ciK(l'Jt,))

Q.(K(m) ,w) = A(K(11l.) ,w)@ 0 c as a union of left cosets of ~(nL) , say w) C

Then if

om ,ttUK<~),Qm)

As a graded algebra over

lA

is generated by

'-;j:l

, and since - (L

1<

SjiK(11'\.)

, the average

and all

We may write

t

'!'

-1

t

.i:j=l R(Sj)q,

S(o)R(S.) = R(S.)S(o) J J 'I' e; A(IK, w)

a £J:l9, we have

Since the "average" of any

¢

A(IK,w) is just

e;

¢ , we have,

by linearization, Proposition 5.

C{(l<)

For a number field to

I<

is called

is spanned by

A (J<)

F , a modular function with respect

F-arithmetic if it is the quotient of two

F-arithmetic modular forms of the same weight. of such functions is denoted

IJTL
The

F-algebra

, and the subalgebra of

those which are "holomorphic at the cusps" is denoted by 'Ylt
•

The action

to an action on

elements invariant under~ 1.6

THE

13

of,/!

on

by

M(IK)

resp.

res9ect to its non-zero divisors) by Let on

extends

M(l<, {K})

•

JK-LEVEL CLASS POLYNOMIAL

Denote the algebra of quotients of K* <'fHHIK)

(}..<1<, Qab)

l)rt)IK , Qab) , and we denote the subalgebra of

MCIK>&oK* c'f>

(with

M(IK,K*('f))

and call this the compositum of

K* ('f)

and

or by M(JK).

e: K* C'fH1(1K)n'm,eiK, {K}) be such that ¢ is holomorphic =.IK((!)', 'f). (Cf. [11],§4.1.1.) Define the 9olynomial

H:(t) = lllz . . is Th 1s~analogous to t h e po 1 ynom1al

£

B ( t - .P(z))

TnL(t) introduced by Hasse [ 9 :p.78] and we shall prove here a result analogous to Satz

10

17, p. 84 of [ 9 ]; cf. also [11:§4.5.4]. We know from [ 5 ]

that

<&, 'f>

::: (l)

is a zero-cycle of

v(l) = v~(l) rational over K*('f) 0 • Now :::~(~ 'f> its pre-image under the natural (coset) mapping ~~ of onto V(l) A(~ ( 1) , w)

Let over

a3 (l)

= (b 0 , ••• ,bN(l)) be a basis of for some large w , and complement

(!

is v~

/3 (1 )

djiK

= (b 0 , ••• , bN (K)) of A (IK , w) over (! • If w is large enough, g,(l) resp. Q31K provides a set of projective coordinates for a projective imbedding of v(l) resp. of VIK as a reducible projective variety (of several componto a basis

ents), rational over

(!

as a cycle in projective space.

Then ~IK is merely the projection on the first N(l) + 1 coordinates and therefore is defined over (! • Hence,

:::K(~, 'f> is a zero cycle rational over K*('f) 0 also. (The components Vw of VIK are rational over Qm , and permutes them in a manner compatible with translation by the ele-

,£t

ments

]J(o) e: G(Af) for o A(~,w) , then by definition,

e:;if

"'ilxc I(k)f.

R(]J(O)-l)!fl 0

What we wish to show is that

~(cr,

=if>

If if> e: for all

oEJ/t .)

B , as a subcycle of

'f>

is rational over K*('f) Moreover, we can then see at the same time that for each of the narrow ideal class representatives

v , the set

of the components of

Bv =

V(l)

~IK

-1

(Vv)n B , where

, is also rational over

Vv

is one

K*('f) .

1.7 IK-LEVEL MODULAR CORRESPONDENCES AND THEIR FIXED POINTS Let

CIK(s 0 ) =

1Ks 0 ~

be the double coset determining a

modular correspondence eiK(SO) C: V~ X V~ , where SO belongs to G(Af) , and ~ is an n.o.c. subgroup of G(Af) • The projective coordinate ring of VK has a ~-structure A(K) which was described in the last section. Let us assume that s 0 e: M 2 (~) (within our usual identifications). Let if> be a homogeneous element of degree zero in the quotient algebra; that is, if> = a/b , where a , b e: A(R) and are homogeneous of the same weight. Then we form the transformation polynomial of

if>

with respect th

T<J>,s 0 ,1<{g)

<x>

By its choice,

N

~(s 0 )

llj=1 (xif>

:

ifl(gsj»

satisfies

S (a) if>

N

, SK<s 0 > =~=lsjlK. if>

for every

11

a £ ~ = Gal(~ab/0) . Moreover, l, ••• ,N • Therefore, if

(3(cr)R(S.)

J

then each coefficient is a ~m-arithmetic modular function with respect to ~ and belongs to M(~) • Moreover, ~ is also an element of M(~) • Let

FP(SK(S 0 ))

be the set of fixed or self-correspond-

ing points of V~ with respect to C~(s 0 ) • Essentially as remarked in § 2. 5 of [ 4 ] , FP
FP(~(s 0 ))

~<ss> = 1, .•. ,N

(1)

~(gS)

Therefore,

~(g)

~

(gSj)

for some

j =

and

~(g)N +

LV£N

av(g)~(g)v = T~,SO,JK(g) (~(g))=

0

while if the image of g' £ G+(A) is not such a fixed point, there exists ~ £ M(K) holomorphic at g' and at the images of all the elements T~,so,IK(g'l <~
g'S ,

t-o .

s £ SK<s 0 ) , such that Since (1) is an algebraic relation

among the values of elements of

M(IK)

at the pre-images

g

of points of FP (CIK (S 0 )) , and each av E H (lK), therefore is the quotient of homogeneous elements in the projective coordinates of the ~-structure, it follows that FP(CIK(S 0 )) is a zero cycle on v~ and rational over ~ . By the same kind of argument as in [11,§3.5], which we shall not repeat here, the special points of VK (that is, the images of the special points K - k ~ Hn ~ XI;J ~ v w in each component Vlei of VIK are the fixed points of all the modular correspondences ~1Kcs 0 ) , with s 0 £ G+(~)f , such that CIK(s 0 ) = IKS 01K does not contain a scalar multiple of the identity (in which case the set of fixed points of CIK(s 0 ) is finite). As before, let q = q, , e , e•, and v = v 8 be such

12 VT + d

that the order of

is the maximal order /'o

Then, as remarked earlier, q I ( ~) ql FP (SK (q I VIK a E Kx

=n

(Cf.

[11:§4.1.3].)

Since

a

kx

E

c

.

A

M2

(0')

(J'

of

K

Now define

(a»> implies that

q(a)

is

a scalar multiple of the identity, the essential terms on the right hand side are those for

a

K - k •

E

We have seen

that each of the terms on the right hand side is a cycle rational over

0 •

If

pal

prime ideal

then

S!<(q' (a))

a

'f =

E

K

is a prime element for a princi-

(a) = a.fY

of first degree in

cannot contain a scalar multiple of the

identity [11:§3.5].

Hence, for such 0

a finite zero cvcl_e rat:i,.onal over

[11:§4.2.2], we have q'

with

e

V~

Proposition Sa.

a , FP(SKfq 1 {a))) is Therefore, in analogy V~

is a finite zero cycle on

0

rational over

~ = B\Jt*B IS A CYCLE RATIONAL OVER K* (f) O

2. PROOF THAT

2.1 TOPOLOGICAL PROPERTIES OF CORRESPONDENCES AND THE ARTIN KERNEL be the closure k , let ~ in other words, is the

For the totally real field in

I(k)

of the group

kxkx oo+

kernel of the Artin map [18] I (k) - - > - Gal (kab/k)

Ol,:

Let ~ be the subgroup of GL 2 (A(k)) consisting of the matrices s-12 , s E~' • Then~ is a subgroup of the center Z(A(k)) of GL 2 +(A(k)) (whose center is the same as that of GL 2 (A (k))) . Proposition 6. G+((l)giK 00 ' } Proof. G ----> Ad(G) Ad(G) (A)

, and

, while

G+((l) .g~oo ;

in

~

c

g

Int(IK 00 )

E

G+(A)

, the set

Int(G+((l))

E

, then

Z(A)

and thus

.

T.~

Int:

is a compact subgroup of is discrete.

the image of this in

x E clos. (T.~) for some

For each

is closed in G+(A) . [11] We have the standard homomorphism Let

Ad(G) (A)

Int(x) E Int(T) We shall show that

, so that c

T

=

is closed.

If

x E c.T

may be chosen

is closed, which implies that

13

G+(Q)giKoo = T.~.g Of course, x

is closed.

= c.t

clos.

F

(T.~)

, where

may be written as t = y 0 .k 0 , Yo E G+(O) , Since x E clos. (T.~) , we may also write X

=

lim m

~

oo

y mk m1 C m

ym

,

E

G+(Q)

k

,

m

I

t

1

oo

T

and

E g~oo

k0

E giK

E

,

cm

E}• E

,

is the unique maximal compact subgroup of where IKI 00 (which is commutative), and E is the maximal finite subgroup of Z(R) , consisting of then-tuples of matrices (~1 2 , .•• ,~1 2 ) associated to n (all real) archimedean places

'J-·

of k , because IK 00 = Z (IR) • IK ! and Z (R) C E • By taking 1 a subsequence we may assume that lim km exists and equals k 1 E g IK 00 Since em are all central, we obtain ct = cy 0k 0 = limm~oo(ymcm)k 1 , so that limm~ oo(ymcm) = ctk 1 -l Now Int(ymcm) = Int(ym) for all m , therefore, Int(ym) has a limit and since Int(G+(O)) C Int(G) (0) is discrete, we is constant, say, Int(ym) = IntCy 1 ) may suppose Int(ym) for all Thus, since

y 1 z;m , sm E z (0) for all m , y 1 E G+ (0) y 1 - 1 ctk 1 -1 = cy 1 -1 y 0 k 0 k 1 -1 E Z(A) z = lim ( z;mcm) Z(A) is closed, and Yl-lYo = (c-lz). (kok~-1)-1 E m

and Ym

=

Looking at the non-archimedean part, this implies that k 0k 1

-1

= (c

-1

y1

-1

z). (yl

y0

-1

E

yO)

Z(O) E

=

Z+(O)

fl

Z (A)}

1

g

K00

Also we then have '2.._

Z (R)C:.. E. ([

'J.

Now em E .E and z;m E Z (!D) for all m . But Z (0) is contained in ~.E and '}- is the closure of k:+kx .1 2 hence is closed, and therefore ~-E , being the union of at most 2n cosets of ~ , is also closed, so that

Therefore c

z. Ck 0 k'- 1 l. Cy 1 - 1 y 0 J E ~-E , But since E C Z (R) C: IK 00 , and

=

and x E c.T T = G+ {Q). g )K 00 , 1 1 we see that E.T = T. Therefore, x E C .T, c E ~ is closed, which is what we wanted. and so T.j2.2 FIXED POINTS OF MODULAR CORRESPONDENCES Definition. For a given open compact subgroup

~

of

14 G(Af)

and for any element {z

FPIK(IT) Define

G+(A)

E

I

FP 00 (IT) = {z E G+(A) Now we have Z(IR)Z(Q)~

=E.~ C

IT

of

G(Af)

ziT

E

G+ (0) z 1<0011<

ZIT

E

G+(Q)zl< 00 ~

} •

}.

Z(IR)Z(O)Un.(O')C Z(IR)Z(Q)IK(n)

for any non-zero (integral) ideal I (k) f

define

7l..

c. cr ,

where ~::

is the group of finite idelic units

u

U7t(O')

1 mod 1L , be-

X cause k x koo+ n<~> is an open and hence closed neighborhood of kxkx oo+ for all such n . Therefore we have

FP 00 (IT) C FP n, (IT)

n

for all

where

f4.4.3 of [11]

Proposition 7. G' (cr)

FPn (IT)

FPIK (7t) (IT)

.

In analogy with

we have Let

n

be any compact open subgroup of

Then

(A) ~

where

runs over dll n.o.c. subgroups of

Remark.

G(Af)

.

The principal congruence subgroups of

K(7t)

IK

form a cofinal system among all open compact subgroups G(Af)

.

of

Therefore it amounts to the same thing to formulate

the statement (A) with this family of open compact subgroups. Proof.

We have just seen that the left side of the

equality in

(A) is contained in the right side.

it suffices to show that if

z

FP.71_,(IT)

E

.n

Therefore,

for all

n ,

then

Let Tn= z- 1 G+(Q)zlK.,.IK(7V) Then zT7l = G+(Q}zfK00 ~tK(7l) , and G+(Q}ziKoor} is closed by Prop. 6 , while IK (1Z.) is compact; hence, Tl£ is closed.

n

Let If

be the continuous map n : X XITX-l ; Q ----+ G+ (A}

FP n• (IT)

is non-empty, and IT

n- 1 (Tn1

Thus since

FPn, (IT) X

E

Q

z

E

E

I;

E

l;-lG+(Q}I;IK00 1K(n!)

•

is a closed subset of

)

FPn, (IT)Q

and

FP 7l' (IT)

X E Q

depend on

n'

n

, then

for every

~

, this means that there exist such that

z

=

Thus we have

l;x

-1

, where

n' ,

~

in and

and

15 -1

xt;

n (x) z -lG+ (()) z IKOO IK ( 1L') so that

x

n- 1 (T~).

E

G ((!) IK

+

1K (n,')

00

= T 1'1..'

In deriving the last sequence of

equalities, we have used the fact that K(n') is a normal 1 1 subgroup of G' (a) . Therefore n- (Tn ) is a closed nonempty subset of

n'

In..

for every

r.l

that is

I

T 71.

c

T1L'

n- 1 (Tn1 )

of the closed subsets are non-empty.

, while

"1L'

IK ( 17) C

of the compact group

Hence there exists a point -1

E

if

-1

Z

x0

r.l

common to all

n• :

of them and therefore for all x 0 ITxQ

K ( 71..')

so that the finite inter sections

I

G+(Q)ZIK00 IK(7V)

so that

The intersection on the right hand side is equal to G+(Q)ziK 00 ~

= D

r; = zxoiTXo -1 .

-n, ,

for each ideals lim

en

E

-n =

n

, say, which is closed by Prop. 6.

n rz.IK(n)=

such that I';

f

x 0 ES1 , and z E FPOO(IT).Q 2.3

{1}

clos. (D) = D •

,

8n = r;kll

Therefore

zx 0 IT E G+(Q)zx 0 1Koo;} • Q.E.D. RATIO~ALITY

-l

then

,

zx 0 ITx 0

so that

the

is rational over

D , and

be the involu-

Then for any n.o.c. subgroup VK

E

and

over

OF

])j

on

-1

zx 0 E FP 00 (IT)

Proposition 8. (§4.4.4 of [IJ ].) Let tory element defined in § 3 of [ 5 ] and put

])j

For if

D K (n.) I say 1;: = 8nkn I en. E D , kll- E K (1t-) where -n, tends "to oo " through a sequence of 7t.

IK

of

K*(~)

G(Af)

BV 1*B •

a-cycle

0

(Remark. To avoid complicating notation, we omit any label referring to Proof. If

IT

IK E

from

G(Af)

B .) and

det(IT)

E

det(G+(Q)f).det(K),

the set of fixed points of the modular correspondence

IK ITIK =

SK(IT)

we have

is the projection in

VIK

of the set

FPK(IT)

seen that this cycle, FPIK(~(IT)) , is (!-rational. Moreover, if IK' is another open compact subgroup of G(Af) and IK' C 1K , then FPIK, (IT) C FPIK (IT) hence FPJK, (IT) .$1 c

FPIK (IT) • n

16

and

n

1<' C. IK FPIK' (II) Q (\IK' :n.o.c. FPIK, (II) Q = so that for any open compact subgroup Q of IK we have

FP (II)Q 00

n

IK I c

I(

FPIK I (II) n

Therefore, priK(FP (II)!l) pr1Kcf)IK'C IK FPIK' (II)IK) and if we let n = ~ , this shows that 00

priK (FP oo (II) ) = priK cf\ IK, c IK FPIK, (II )JK) prlKcniK'c (since obviously

I<

G+(Q)FPIK, tK

00

lK)

G+ (Q) FPIK, (II) lK = FPIK, (II)) 00

('\ IK'C IK priK (G+ (Q) FPIK' (II) IKooiK) (because each double coset G+ (Q) a IK IK is the complete inverse image under priK of pr1K (a), so that "the intersection of the projections is the projection of the intersection"), and the last expression is finally equal to the intersection over lK' C: IK of the sets prl< (FPIK, (II)) , which intersection is defined over 0 . That is, we have: (*) priK(FPoo(II)) is a zero cycle rational over 0. Now we assume B to be the orbit under CIK(~ (by way of its action on :::IK (Cf, 'fl , the projection of :::A (t)', 'fl into V,K) of a point 8. (T) in the image of ::: (dr, 'fl . This may be done without loss of generality becaus: :=:Accr, 'tl = G+ (Q) :::oo Ce", 'fl G (lZ) IK 00 , so that each pre-image in G+ (A) of a point of :=:IK(~, 'tl contains an element of the form ~ew , where w E G(lZ) , and ~ E :=: 8 ,oo (~, 'tl , and if ~ is an F-arithrnetic modular form with respect to IK , then R(w)~ is, too, because w normalizes IK : thus, as remarked in [111 the family of "ray class" polynomials with respect to IK on an orbit B of CK(~ in :=:IK(~, 'tl is invariant under right translation by w E G(~) : hence, we may take a point of the orbit B to be the image e.(T) of a point ~e E :=: ((9', 't) . Suppose now that II= q'(TI) with 7T = Af for some A E K such that K = k(A) • We first show that tl C priK (FP (q'( rrl)) • On account of the results of §§2,3 of [ 5] , :::A ({J', 'fl is stable under the action (described there) of I(K)f and 00

00

00

17

of

on

B=

and

:":A((J', 'f>

I(K)f , a*e.(c) and

•

Now

ua*e,(,)

=

= q(f1.): 1 t;IK

=

VIK

of

G_(Q)q(1) 00 q(a

I(K)f }, and

E

If

l;IK

-1

00

-1

G+(Q)q(a

-1

)t;8IKIK00

a

E

)t;8IKIK00 Now

•

= G+(Q)q(A); 1 q(a- 1 )t;e IK

G+(Q)q(a 00

Ia

{a*8.(T)

is the image in , that of

G+(Q)q(a- 1 >t;eq 1 (rr) IK00

because

B

'13 C ::IK ( 8', 'f> , anyway.

B\...h *B ; hence

00

)t;81< 00 since

,

=

q

qT , with

T

as

above, while G_(Q)q(1) 00 q(a

=

-1

)t;6q 1 (1r) IK00

= G (Q)q(rr)q(1) 00 q(a

-1

G_(Q)q(A) 00 q(1) 00 q(a

G_(Q)q(1) 00 q(a

-1

-

-1

=

)q(A) 00 l;8IK 00

for the same reasons.

-1

H:eiKoo

)t;6IK00

=

G_(Q)q(1) 00 q(a

Hence, both

-1

a*e (T)

-1

and

)!;6JK 00 a*1*e (T)

1*a*8~}belong to the fixed points of the correspondence

~ C FP00 (q 1

SK(q 1 (rr)), i.e., y

normalizes

G+(Q)

(n))

= G+(Q) .Yx.y = G+(Q)yx

G+(Q)y

•

(N.B. For any

; therefore, if

x E G+(Q)

y

E

G_(Q)

,

, G_(Q)

.)

Suppose, conversely, that

z

belongs to the intersection

of and of such

If we show that

priK (z)

'V

E

B

for every

z , it will follow that

])3;::) :":1!<((9; 'f)f\ priK(FP (q 1 (n))) 00

because each element on the right hand side is the image under priK

of an element of

:":A(cr, 'f>rl FP 00 (q

will follow that 'V

B

=

::IK (CJ", 'f)

n

1

(rr))

, and thus it

priK (FP 00 (q I (rr)) )

I

and the right hand side is a zero cycle rational over K*('f>o· 'V 'f (In fact this will show that B = VI< q I f"'l .::IK (~I ) • ) But for a given such z I we may, by Prop. 1, write A

z = yt;q(a)Bn , y E G+(Q) , 6 £@ , a£ I(K)f, n E AG(?l) IK 00 and where !;(i.e) = (T) . Write n = n 1 u 1 , n 1 £ G(?l) ul £ IKOO . We now show that ene- 1 belongs either to q(I(K)f)IK 00 or to q(tfi(K)f)IK00 , and this says that z £ Yq(c)!;6(K 00 , where c is either in I(K)f or in I(K)f~f

, so that

z £ B~l*B

=

'V

B

, which is what we want.

,

18

In fact we have z = yE;q(a)en £ FP 00 (TI) c FPIK, for every n.o.c. subgroup ~~ of G(Af) Hence, zq 1 (n) £ G+(Q)Z~00 IK 1 and priK 1 (zq 1 (n)) = pr~ 1 (z) ; therefore, by Prop. 3 (with a= 1), there exists b~ 1 ,f £ ~x such that q 1 Cn>

£

n

-1 -1

e

q(b~~ ,f>en~~

-1 -1

so that q 1 (n) = n e q(b~ 1 ,f>enK 1 for some K 1 £ IK 1 • For the rest of this proof, we may assume ~~ = IK(~) for some integral ideal TL of tends to { 1} , then K1 = Kn ~ 1 , so that biK 1 , f = b7l, f must also tend to a limit b 0 which belongs to the closure of Kx in I(K)f (because the extension of q to I(K) is a closed immersion of I(K) into G+(A) , since the image q(I(K)) is equal to T(A(k)) for a certain maximal torus T of GL 2 > defined over k) . Let en = w = w 1 u 1 , so that w 1 = en 1 £ G(Af) , and put w" -1 -1 en 1 s Then q 1 (n) = w 1 q(b 0 )w 1 for some b 0 in the closure of Kx in I(K)f , or -1

-1

w 1 q 1 (n)w 1 = w"q(n)w" = q(b 0 ) Now K = k(A) , n = Af , and q = qT . We want to prove that AX b 0 £ K , too. For this purpose we use 2.3.1 A LEMMA BASED ON A RESULT OF E. ARTIN [1] Lemma 1. Let q be a regular imbedding of K in M2 (k) , extended to an imbedding of A(K) in M2 (A(k)) . Suppose that b 0 belonas to the closure of Kx in I(K) A AX X that c belongs to K , c £ K ,. such that K = k(c) , -1

and that for some w £ G(Af) we have w q(b 0 )w = q (~) • Then w normalizes the subalgebra q(A(K)) of M2 (A(k)) and b 0 = b for some b £ Kx. Proof. According to [18: p.62] , if 81 , 8 2 are a k-basis of K , there is a canonical identification of A(K) with A(k)8 1 + A(k)8 2 Since K is a eM-extension of the totally real field k , there exists o £ K such that K = k( 0 ) , o2 £ k , and o2 << 0 . Hence A (K) "' A (k) + A (k) o which is an isomorphism of A(k)-algebras. Hence we may write uniquely

19 A

y +

c b0 and

q(A(k))

=

c:x

with

y,~

~

+ SS ,with

c:x,S

~ A(k)

~o

and

M2 (A(k)) . q(c) we have

-1

w q(b 0 )w = is an algebra homomorphism) q(y) + q(~)wq(8) = q(c:x) + q(S)q(8)

q

~ ~ 0

(2)

,

is contained in the center of

Therefore from the relation (since

k ,

A

since K = k(c) . Therefore we have wq(;5) q(~)-l{q(c:x) + q(S)q(S)- q(y)}

That is, since

= A(k)f

A(K)f

~ q(A(K)) .

+ A(k)f8

wq (A (K) f) C q (A (K) f)

(3)

Let ~

be any finite place of q(K~)

wq(KI-) C

(4)

K~

k , and let

k)®kK = k~ + ks&

product

be the tensor

Then (3) implies that

= q(kj)

+

q(k~.)q(~)

Now q(K~) = q(k~) + q(~)q(o) contains the kj-rational elements T(k;> of a maximal torus in GL 2 (k;f) . Since ~ ~ GL 2 C~;t> , Inn(w.f) is an automorphism of M 2 (~) and (4) implies that ~q(K5 ) = q(~) , and hence w1

belongs to the normalizer of T(k1)

The normalizer of

in

T(k5 )

= q(K~)()GL(kJ)

GL 2 (~)

.

contains its central-

izer, namely T(k~) , as a subgroup of index at most two. The 1f-component of the (relatively) involutory element 1 also normalizes T(k~) , but q(l)q(o) = -q(o) . Hence, the normalizer of Therefore,

wt-

of

k .

~,

=+

T(k5) f.'

(wq(o) )~

Thus,

GL 2 (~)

in

is T(k$)~q(l)T(kl) . for all finite places

T(k1 )Vq(l)T(k~)

=::!:.

q(o)~

=

C:).q(o)~

1 for each ~ ; therefore, since 1 are :inearly i~dependent over kg> for each a= y and q(£).f = ::!:_ q(B~ = We have S-f ~ 0 for all ~.

q(C:~)q(B~

Call

C:

f.

, say, where and

;j ,

8

= 8~

we have

Since € ~ 0 , the idelic unit

~

:finite in I(K)f Clearly c: 2 = 1 . ~y hypothesis, there is a sequence {bn} C: Kx

b0

=lim

n~

an + Bno , with

oo

bn

in

I(K)f .

an , Bn

~

k , and

We may write each bn

~

topology (which is finer than the adelic). b -1

0 all

"

-1

exists and bn n , and if m , n

~

-1

b0

such that bn

in the idelic

Since

b 0 e I(K),

b0 , thus bn ~ 0 for almost are very large and m > n , then

20 b = E b , where E rn,n belongs to a small congruence subrn rn,n n group of the units of K • Since K is a pure imaginary quadratic extension of

k , the totally positive units of

are a subgroup of finite index in the units of Chevalley's theorem [ 6]

k

K , so by

on the units, we may assume each

Ern,n is a totally positive unit of k. Let n 1 , ... , nt be an independent set of generators of the totally positive units

k

of

(t = [k:Q] - 1)

w

Since

A

q<.Y> + q(~~>q<s> = q + q(S>q<s> we have in the limit

~

By discarding the early terms in the 1\

A

rn,no

(a

no

(since 13 -F 0

because 1 k: cf. (2)), where

over in

I(K)f , and each

of

k

Hence, if

•

+ 13

and

no

o)

o

Ern = £ is a Cauchy sequence x rn,no E ~ ~ + , the totally positive ~nits A rn 13m = Ern'~n ~ , we have 13m ~ B = E~ , 0

in the idelic topology.

Since

E

~~ ~

A

A-lA

E

E , Brn E k), a unit of

ern = urneO , where

e0 E

X

~

~ ~

13m

= + 1

13m = ern , where we may assume

K, hence (since Say

-F 0 '

are linearly independent

Therefore,

~-11'\

n0

A

in the idelic topology. place,

= ~ n + sn6 ,

q(a) + q(l3)q(S)

sequence, we have for some large enough

brn

, we have

bn

, and i f

q(;n) + q(Bn)q(;5) in the idelic topology.

A

q(~o)

q(o) =

and

=

(~)

,

for every finite A

ern

is a unit of

k

urn E

for all X

~

A xl rn A xt rn urn= n 1 ' .···.nt ' xj,rn E Z1: We may take a subsequence such that for each

+

rn

for all

rn ,

and

x.

J 'rn

of

Then

Z1: ) • ~

Since

r, 2

since

eo

that = Z1: •

ix.J

lim (ern) =

j = l, ..• , t ,

(the standard cornpactification

converges to

eolirn ~rn

1 , this gives

2

X

By a lemma of Artin [ 1 ] , this implies E 0 + = a. j = 1 , •.. , t , h ence -xj = aj /2 E .;; ,..., .!..., 2~ J

That is,

'

=• •

21

X.

J

~

so that

72:

is in fact equal to A

n

=l,,,.,t

j

I

n = + 1

for a unit

A

b

E K

Since a = y , we obtain + E E k , which proves the lemma.

2.3.2

COMPLETION OF THE PROOF OF PROPOSITION

8

B

Thus y"'

=

E

"'"'

~ Eo

En =

"'X

0

E

k .

=

Returning to the proof of Proposition 8, this shows that But w"q(n)w" thus

Int(w")

and since

-1

=

q(b 0 l

gives an automorphism of

=

A(K)

A(k) + A(k)A

automorphism extends to one of have

b0

=

n

or

b0

tralizer in GL 2 (k~) normalizer of T(k5 ) the identity or we have either

=

n

E

the case described in

,,

q(k)

,

, it follows that this A(K)

Clearly we must

=

T(kJ)

X

q(~)

is its own cen-

and N(~) = T(~)~1T(kj) is the It follows that either Int(w") is

Int(lw") w"

over

qCil

is the identity, and therefore

q(A(K)x)

or

lJ.J"

[11:~4.4.4].

£

Thus

q(A(K)x~ , as in

z =

y~q(a)w'u

1

belongs to G+ (Q) E, [q (I (K) f) V

"'B

the full pre-image of

which shows that 3

"-• B

.

Hence

is a zero-cycle rational over

THE RATIONALITY OF

Proposition 9.

C: ~K

q ( 1 fi (K) f)] 611< 00

B

The orbit

OVER B

of

K*cr) 0 .

K*Cr) CIK ( C})

is rational over the reflex field

in

~ ( &, r)

K*Crl

In the statement and proof of this proposition, the notation retains the same meanings as in the statement and proof of Proposition 8. Proof.

We already know that

~ = B\/1*B

is rational

over the totally real subfield K*Crl 0 of K*Crl B and t*B are equal or they are disjoint; for if

Either a*B(T)

1*a'*8(T) a'*t*B(T) for some a , a' t: I(K)f, then clearly B = t*B , so that B = ~ , and B is rational even over

22 'U

K*(t:) 0 • Now suppose

B

and

l*B

are disjoint.

We want to in the composi tum M(IK, K*(I:')) of K*(I:') and M(IK) , such that ci> is constant cf>(B) 'i' cf>(uB) • on each of B and l*B and such that construct an arithmetic modular function

ci>

3.1 STABILIZER SUBGROUPS AND FIXED POINTS OF CORRESPONDENCES priK(~8)

We start from the chosen point

of the orbit

B , where 8 is an element of ~ and ~ £ ~ 8 ,oo <6r, I:') • If z £ ~A(cr, I:') , then by Prop. 1, z belongs to the set

G+(Q)q(I(K)f)~8WIK 00 so that ment of

8.(T)

for some

w

£

G(zl).

(T) =~(i.e),

Let

~e ({9-, I:')

£

For the present, z and not necessarily in B

~A(~, I:')

is any elePut n = 8w

and define

Jilczl g = z- 1 y'zk'00

Suppose that

z = Yq(a)~8Wk00 , Y the requirement g ~

-1

Y: 1 Yc!,Yoo

or

Yoo

-1

£ £

G (Q)

EIK 00

~IK 00 nG+(Q) 00

£

tJ?
,

kc!,

K00 and that

£

, a E I(K)f , and k 00 £ JK00 Then reads, for the archimedean part,

~(z)

Yc!,Y 00 ~

£

,

for some Y' £ G+ (~) = q(Kx) 00 , where q = q -1

'f1

and

q',

T

etc., will have their usual meanings. Thus, y Y'Y = q(c) where c £ Kx Since g £ G(Af) , the conditions on the non-archimedean part read g = n

-1

where

q(a)

-1

yf

c = cf •

-1

yf'yfq(a)n

Thus

0.

~(z)

If

=

-1

AX

q(K )Tl IK' is an n.o.c. subgroup of IK , let n

~IK , =

On the other hand, for a fixed element xq' = IntJK (q' (b)) = {kq' (b)k-l

~· Let and

(z) =

Jin (Xq,IK' l

, and

x~,

Ik

£

b

of

JJ cz >n fK • • Kx

let

IK },

(z) =

J

n xq,

YIK' (z) resp. Y;, (z) denote ~· (z) IK' resp. x~, (z)IK' ZIK, ( z) resp. Z;(, ( z) denote the images of these in

z;,

VIK, Clearly Y;, (z)C YIK, (z) and (z) C ~· (z) for all n.o.c. subgroups IK' of IK We claim that z~, (z) is always finite and that for sufficiently small JK' , we have

23

z;, (

z~,

z)

{z)

•

First of all, trivially,

Int IK (q' (b)) C

IKq' (b) IK

which is compact and open in contained in

=

cl< (q' (b))

G(Af) ; since

IK , this implies that

xq'

IK'

contains represen-

tatives of only finitely many left cosets of hence

~·

(z)

modulo

empty); therefore, ~·

(z)

IK'

for each k 'IK,

e:

IK'

IK'

G(Af)

IK'

of

are finite.

IK'

,

Suppose

that is

~ ¢

IK

where

'

w

= e-1 n Then

K

IK ,

there exist

A

IK'

in

such that -1 w q' (elK, )ul

(*) Let

z;, (z)

and

= x ,IK'nw- 1 q'(Kx)w q

for all n.o.c. subgroups

IK'

on the right is finite (possibly

ZIK, {z)

is non-empty for all such

xq ,IK•n..IJ

is open and

=

"shrink to {1}"

A

~,q' (b)~,

-1

k'IK'

through a linearly ordered sequence

of n.o.c. subgroups constituting a neighborhood basis for the identity in G(Af) , and for each ~· , let ~· , ~· , and k'IK' be chosen to satisfy (*) • Automatically k'IK' tends to the identity in the usual topology on G(Af) , and since IK is compact, by going to a subsequence we may assume limiK'

~

{1}~·

ko £ IK

lim IK'

~

A {1}'11<'

co

Then

.

We exists and belongs to the closure of Kx in I(K)f KX from takIn fact, we have claim that co belongs to 1\ -1 -1 ing the limit in (*) that w q' (c 0 )w = k 0 q I (b) k £ Xq' 0 ' -1 ~ with 1 or and applying Lemma nk 0 = q'( > ' Ax -1 as claimed. we see that co £ K w = nk 0 e ' '

.

Let G(Af)

z £~A(~,~)

, fix the n.o.c. subgroup

, and let IK' be an n.o.c. subgroup of K Proposition 10. For all sufficiently small ~· {z) C

~

Xq' {z) 1K

xq' (z) = x~,(z) = xq' n~ {z) • Proof. For any 1K' C. IK we have

,

IK

IK'

of we have

24

= .sS n xq ,IK • c xq ,IK = IKq • IK

x__ , -lK ~·

so that

(z)

contains representatives from only a fixed

finite set of left co sets of

JK

for all

IK'

.

IK' C IK ,

If

write where ~K"

~· (z)JK = \_)a E A(JK') aJK is a finite set. If IK"C: IK'

A(IK')

(z) C ~· (z)

,

and if we put such that

= niK

A (oo}

A(IK 0 ') = A(oo}

A(oo)IK = X , (·z)(K. If a q X exist 9K' E K , ~· E i.e., such that -1

n

giK I =

A

q ( CII( 1 )

and also such that limit as that

c0

I

then there exists

A

lim elK,

k'IK' €

IK'

-1 = kl[{ 1 q (b) kl( 1 ka( 1

giK,IK. g0

IK I

I

hence

aiK

Xi( 1

E

there

(

Z)

As before, passing to the

= n-1 q(cf\ 0 >n

k0

E:

IK , c 0

-1

E:

IK' we have for all sufficiently for all

g 01KC xq' (z)JK

Kx such

E:

A

k 0q(b)k 0

X , ( z) , and therefore for small enough q "' n-1 q (cl<,) n E: gr}K ; then a E: giK,JK = g 01J.<

small

I

satisfying (*),

A

and

IKO

JK' C IK

, then for every

and

IK ,

E:

A((K")C: A(IK') ,

It suffices to show that

A(oo)

E:

n

a

A (IK I )

I

.

shrinks to {1} we obtain

IK'

=

, then

hence we may assume that

a

E:

A(JK 0

'> ,

proving the result. 3.2 MULTIPLIER POLYNOMIALS The coset space

~·

(z}JK'/lK'

is finite, because [IK :IK']

XIK, (z)/~tK' (z) , where IK' , by means of the mapping of cosets

is finite, and may be identified with

~K'

(z) =

~(z)()

x.JIK' (z)

x, y

for if

/K

I

nli(z) Let

~K

,

pr IK

~·

E:

=

~IK'

priK'/IK

Then

and if

0.

YWJK' (z) C

xlK'

, then

x

-1

y

(z)

be the canonical projection of

be the canonical projection of

G+(Q)\G+(A)/lK~

vlK •

(z)

xiK'

onto

prlK(~,

VIK , and (z)) =

is a(finite)zero cycle on

priK

onto

V00

be that of

P:tjK'/lK(~,

VIK, G+(A)

onto

(z)MIK' (z))

VIK , as is also

pr(K(~,

(z)) which

E:

25 i t contains. Fix

1jJ E A (IK , w)

for some

zero at every point of for any

::A

=

ljJM(g) where

such that

If

M E G(Af)

c&, 'f>

t-

ljJ(g)

such that

g E G+(A)

w > 0

nomial in the indeterminate

t,

~

=

Pl)J,IK' ,b(t; z)

, define

'

Cl)J!IM) (g)/l)J (g)

is defined as in §6.1 of

l)JjjM

0

is non-

1jJ

[ 5 ]

Define a polyz E ::A(~,~), by

for given

E XIK' (z)IK'/IK'

( t - ljJM(z))

Similarly define P, 1, b(t.; z) with X co' (z) in place of X __ , (z) 'I'' co' q "11< For the time being, we fix b E K , such that K k (b) , and 1jJ , and omit the indices

M

runs over

XII<' (zw)IKYIK'. ion

1/JIIM

b

•

XIK' (z) IK' ;lK'

,

M'

1Jt.Cq'

Mw

(b ) )

runs over and the funct-

then

UCM)

=

for every M ,

w E IK , and

IK co

= since

ljJ(gw)

lj.J(g)

PIK' (t; zwK) so that the Pco(t;

,

ljJ(gK)

= =

N(7/l)w1jJ(zM)/1jJ(Z)

N(U)wljJ(zMwK)/ljJ(Zu.lKl

lj.JM(z)

= xw(K)l)J(g)

for all

g E G+(A)

=

PIK' (t; z)

and

Pco(t; ZulKl

(coefficients of the)

=

polynomials

Pco(t; z)

[11])

(practically as in

and

~

of

•

RATIONALITY OF THE MULTIPLIER POLYNOMIAL

Definition. VIK •

,

PIK,(t ;

may be viewed as functions on the subset

3.3

,

Therefore,

WEIK,KEIK co

on

IK

E

-1

UCM)

and we have for each such

N('M.,)wljJ(ZwKM')/ljJ(ZwK)

VIK

w

I f we define the ideal

Vl.CM') is the same and equal to

E

w

as in the reference cited above,

M E XIK' (z)JK K

I t is easy to see that

-1 = w XIK' (z)w

XIK' (zw) As

and

1jJ

If

A

inverse image in

Let

~

G+(A) v

P

exist functions

Q-algebra of modular functions

is a subset of

Jwith indeterminate we say that

be a

t)

t---~

and

A"' is· its complete

P(t; v)

is a polynomial-valued function on

is of type N' ••• ,

VIK

and i f

<1>

$"'[t] 0 E

$'

on

A

or on

A"''

A ,

i f there

such that for every

v

t: A

26 we have

=

P ( t ; v)

Ev

N

=0

In many cases, we shall take

t

v

.

~= M(~)

, the

is an n.o.c. subgroup of Lemma 2. If ~· each of the polynomial-valued functions --~ P~, ( t ;

v

is of type

[t]

M(~)

Proof.

~

on

(Cf.[ll].)

~-struc-

v~

ture on the algebra of modular functions on

IK , then

v)

.

the monic g.c.d. of two monic polynomials of type

'11<•

The double coset

(q' (b))

= ~'q'

(b)~'

*

v~,

x

e~, (q'

*

If

<&. E'>

=IK'

e~ 1 (q I

v

=IK'

.::. t

i

1

1

X

~£ •

I

X

I

E

s

=

Now if

££

x' , then

~

M , M'

=

{u

pr~,

3A(~, E'> ( z) : If

,

t

~

f 1 , ••• ,

S

Denote by x

E

VIK'

for some

u

, and

z

E

E

JK'

T(z;M,M')

or

,

u E

G+(A) , let

M'w

is unchanged if

for any

w

E ~·

•

M or

K

E ~oo

,

then

M'

Hence, there

z , and for each

the family of such sets depends only on w

o

is equal to and that

S

F (x') A

(b))

lj!M(z) + Fu (zM) - Fu (z) ~ lj!M' (z) } •

C£

JK'C IK, the set

Mw

U

are only finitely many such sets for each E

I

and let

xq' = Int(IK) (q' (b))

E E

F (x)

be

S

e~ 1 (q I

M(IK')

u 1 f 1 (x) + • • • + u£f£(x)

X

is replaced by z

(b))

generated by

~-algebra

a variable point of

I

T(z; M,M') With

be such that

separate the distinct points of

(u 1 , •.. ,u£) Fu(x)

If

e;_, (q'

We may assume that each is regular on

f 1 , ... ,f£

=

(ef, E'>.

Let

fOr SOme SUitably large

f 1 , ... ,f£ E M(IK')

the quotient algebra of the f£ •

associated

iS the Union Of all iterateS

(b))

eJKI (ql (b))

Let

V~~

by this correspondence.

A

the finite set

Of

v~,

denote the set of points of

(b))

in the usual way to

Where

* ,

is any subset of

A

as in [ 4 ] ,

,

~, (q' (b)) C. let

4, determines a modular correspondence

§ 2. v~,

u

MOO<) [t]

~ .

on

A.

is

P~,(t;v)

Onefirstshowsthat

27

I)!M(ZKw) + Fu(ZKwM) - Fu(zKw) - 1/!M I (ZKW) = I)! OJM

If T(v)

( Z) + F ( Z. wM) - F (z) - I)!

v E S , let

U

z E G+(A)

be such that

be the intersection of those sets T = ~

non-empty, and

V€

(#)

priK, (zM) = prlK' (z)

then

T(z;M,M')

8 T(v)

Let

s S , then for

uo

T(z;M,M') u

£

c~

)

(#) holds.

, set

priK, (z)

, we may replace

z

by a

is of type

[t] by the usual arguments on the symmetry of its coef-

while for given Xq 1 1K 1

M

E

::IK ((9',

'fl .

Now

~ E xq,IK' /IK' (t - I)!M (v')) ,

X ,IK'

, we have (##) for some

q

only if (#) holds because of the choice of

Z E ::A (CJ,

~~

and priK,(z)

(z) = I)!M' (z)

u E ~! , Q(t;u,v')

If

Q (t; O,v') =

of

is a zariski

xq,IK' /IK' (t - I)!M(z) - Fu (z) + Fu (zM) ).

E

ficients, and is regular on

ing

uo

z s G+(A)

•

that are

is Zariski open

z s G+(A)

If

is empty, i.e., only if

v' E VIK 1

v ,

, we have

Since this depends only on M(IK 1

T(z;M,M')

Therefore, T

Jl.

(zM) - F

and

(z)

If we have

~/IT.

u0 E

Q(t;u,z) =TIM

variable

pr~,

I)!M(z) = 1)J~1 , (z)

C .

M,M' s Xq'

I)!M (z) + F

If

wM'

is empty; otherwise, T(z;M,M')

and dense, too.

only if

.

and

open and dense subset of

(##)

(z)

u

M

'fl ,

Q(t;O,z) (z)

then

E~(z)IK'

n

'

priK, (zM) = priK' (z)

in If

is equivalent to hav-

by Prop. 3, and the monic polynomial g.c.d.

and of ME

M' u0 •

\<•

is, since xq,n<·n~Cz)

Q(t;u 0 ,z)

(z) /IK' (t - I)!M (z)) = PI<' (t; z)

.

As remarked in [11] in the classical case, it is easy also here to see that for every XI< 1

(a* z) =

XI< 1

(

every

~ .

z

such that

I(K)~1I(K)

and

z)

Therefore, the functions on

a s

v

t-----7>

one has

xq' (a*z) = xq' (z) PIK' (t;v) = PIK' (t; z)

pr~(z) = v ~ ~)

(for

are of constant degree

Hence we may use the theorem of Weber-Perron [9,17]

28 to calculate the polynomial

P~ 1

Q {t; u 0 , z)

{t; z)

Q {t; 0, z)

and

PlK 1 {t; z) age under

are indeed elements of pr~~ /IK

invariant

under on

M {IK) [t]

of

B

and belonging to

c

some

)

on the inverse im-

Since they are right

itself by Prop. 2.

~e,co
E

1

the polynomials are in fact of type This proves the lemma. K*{~)

IS RATIONAL OVER

We now suppose that prlK{z) orbit B to contain the image

~

M{~

V~ 1

in

~.

3.4 PROOF THAT

as the g.c.d. of

, so the coefficients of

B.

E

e.

We have chosen the of

{T)

z

Then

q

or to

{PROP.9)

~e

, where

G+ {Q) q

E

{c) ~ e JK IK00

for

I{K)ftf , so that we may

assume either {a)

z = yq {a)

{b)

z =

~enu

yq{atf)~8nu

where

y

E

xq 1 (z)

=

J{z){'\ X

{a)

x

reads

=

b

q {c)

=

or

c

wl

q {b)

= b,

or

11

{b) W

w

E ~

q{c) =

wl

u

,

To describe

E IK 00

If

If we put

and by Lemma 1

,

z

is as in

W1 =

8 w , this

this implies either E q(I(K)f)

W1

or

In case {b), the relation (c) is replaced ~

~

ElK

t*B

E

-1

1

by {c I)

prlK{z)

depending on whether

q{tfi(K)f)

E

Wq

and 1\

B , or

E

, then we have ~

=

{C)

E K~

c ~

1

{z)

in these two cases:

1

~

q

, if I{K)f ,

E

Xq 1 {z)

E

X

for some

W1

a

,

q

, and

{c)

c

G+{Q)

pr~

if

A

x = q 1 {c) = wq 1 {b)w ~

q(b)

-1

, leading to the same conclusion with the

alternatives reversed.

In either case,

X q

1

(z)

consists of

one or both elements in the set {q 1 {b)

But if for a single

,

q

1

(b)}

z , it contains both of them, a simple

calculation shows that

B

l*B , contrary to our assumptions.

Of course

X 1 (z) is not empty because in case (a) or in case q {b) we may take c = b respectively c = b and in either

case,

w

to be the identity, to obtain an element of

So let us suppose l*Z

E

t*B

pr~{z)

E

B

as in case {a) and let

X

q

Z1

1

(z)

=

29 When

card(X, (z)) = 1 , the proposition implies that q

for sufficiently small

K' , the polynomial

a single root 1/!H (z) , where Xq' (z) = {r-1 0 } • nomial function N v-----? PI<, (t; z) = (t - 1/!M (z))

°

N > 0 , is of type

M(IK) [t]

has

The polyv

0

for some integer

z)

P~,(t;

on

priK(z) ~ =

Bl/t*B , and this implies that the polynomial function

v

--~> t - 1/!M

Cz"l

t-1

0

is also of type M(~) [t] on ~ In other words, 1/!M coincides on ~ with the restriction of some function We now calculate 1/!M ( z) and 1/!M ( Z I) in M(ll<) 0 0 We have z = yq(a -1 li;:n I and without loss of generality -1 -1 ~ (instead of n q(b) : the second may assume Mo = n q (b) possibility is treated similarly). Now

.

.

°

~

1/!M (z) = (1J!IIMo) (z)/1/! (z) 0

and

( 1/!ll M0 ) ( z) = N 'ln-w1/! ( zM 0 ) , where w is the weight of 1/! • ~b -2 v, where ~is the ideal ~ v = det(MO) = NK/k(b) and ~-l is the d-ideal of all A E k such that AMO belongs We have w(i;:q(a

-1

lnn

-l -1

Since q = q q(b) 00 -11;: = !;:~ 00 1j.l(q(b) 00

-1

~

q(b)) = 1ji(i;:q(a

-1

~

)q(b)n)

-1

= 1j.l(q(b) 00 i;:q(a )n) , where (T) = !;: (i.e) , we further have Therefore for some k 00 E IK 00 i;:q(a

-1

)n)

1j.l(t;k 00 q(a

-1

)n)

-1

= xw(koo)1j.l(i;:q(a )n) = xw(koo)1j.l(z) So the problem is to evaluate xw(k 00 ) .

Xw(koo) = Xw(!;:

-1

-1

q(b)oo U = j (!;:

-1

-1

:

q(b)oo t;, i.e)

w

-1 w =j(q(b)oo ,(T)) by the cocycle relation. The point (T) of Hn is to be identified with the element

and

q(b)

is the matrix

defined by the

30

equations b.'L

en + 8

b

Y< + 0

Define the type norm

(~ ~)-l

•

as before and let

N~

( ya

8 1)

1

OI

I

Then we have

j ( q ( b} :1 '

= N~ ( (a

( "[" })

0I

I

8Iy

-

I }

(

y

I "["

+ 0I

} -

2}

=

= N'f((bb}-lb 2 } = N'f(bb-l}

-2

I)};

f/1- =fk

Moreover,

=

v

lations is equal to

and

1/J (zM 0 } , by these calcu-

N~ (bb -l} w 1/J ( z}

Therefore,

= NUtwN~(bb-l} wl/!(z} =

(l/JIIM 0 } (z} and

-2 bb ,

Cit

NOV- 2 wN'fb 2wljJ(z}

hen~e

1/JM (z} = N oz.- 2wN'fb 2 w 0 In the same way we may calculate zl

while

G (~}q(l}

l*Z E

= T)

M0

z 1 .M 0 E

-1

-

~q(a-l}T)

•

Since -1

G+(Q}q(l}f~q(a

}n

00

q(b}T)

, we have A

G+(Q}q(l}fq(b}~q(a

=

1/!M ( z 1 } 0

-

-1

G+(Q}q(b} 00

-1

~

G+(~}q(b}q(l}f~q(a

}

q(l}f~q(a

-1

-1

}n

}T)

Therefore

=

l/J(z 1 M0 }

l/J(q(b)

-1 00

q(l}f~q(a

-1

}T1}

=

1/!(~~

1/!(~q(l}fq(a-l}T)-1.~-lq(b}oo~}

-1

- -1 -1 q(b} 00 ~q(l}fq(a }n}

N~(bb-l}wl/!(zl)

according to the same calculations we have just completed for

1/!(zM 0 } • 1/!

Hence, (z 1} = N Q{,-2wN'llb2w

Mo

2.

By our earlier calculations, the polynomial function v M(~} [t]

is of type

t-----">

on

~ .

t - 1/JM (z}

z E prK

·o

-1

(v}

,

What we now see from our calcu-

lations is that the function N oz., -2wN-vb2w _ 1/J

E

which belongs to the compositum and

~

0

on

t*B

when

B

~

M0 M(~,K*(~}}

, is zero on

B

1*B , provided

(d)

To achieve (d) we choose, for example,

b

=~ ,

the generator

of a principal unrami·fied prime ideal of first degree in K , for which (d) must certainly be satisfied because the prime

31 N~b

factorizations of

and of

N~b

are disjoint.([ 5 ],§6.3)

It follows that B is a zero cycle on over the reflex field K*(~) 4.

v~

rational

ABELIAN EXTENSIONS GENERATED BY SPECIAL VALUES

Let K orbit under

=00 (t)', 't> ,

be an

n.o.c. subgroup of G(Af) and B be the I(K)f of the image in VI< of g = 1;8 e: Thus BC =IK ((J', ~) 8 E@ , ~; e =8 ,co (CSJ, ~>

.

Let ot,K be the field obtained by adjoining to the values {(z)

I

z E': B, 4>

£

M(IK)

, 4>

regular on

K*('f)

all

B }.

Since B is a 0-cycle on v~ invariant under Gal(Q/K*(~)), i.e., is a K*(~)-rational 0-cycle, ~IK is a finite normal extension of the reflex field K*(~) , and as in [11] is seen to be independent of B since R(w) .M(~) = M(IK) for every w £ G (:1Z) IK 00 is of the form Bw

,

and every orbit of I (K) f in =A((:), for some w e G ( :1Z) Kco by Prop. 1

't>

Let ~ 1 , ••. , ~N be a set of generators of M(IK) , as a direct sum of fields, which are all holomorphic at all points of

=K(cr, ~) and let u 1 , .•• ,uN be indeterminates. Lemma 3. (Cf. [11]§5.2.1) If 4> £ M(IK) and is regular

on

B , and if

b

£

I(K)f , then there is a rational function

Rb, 4> (uv) E K* ('i:;') (u 1 , ... ,uN) such that for each z 6 B , 4> ( b* z) = Rb 1 4> ( ~ l ( z) , •.• , ~ N ( z) ) , with the understanding of course that the denominator of ~,¢

is non-zero at the point (~ 1 (z), •.. , ~N(z)) • Proof. We may, without loss of generality, assume to begin with that the N-tuple ~v = (~ 1 , ••• ,~N) is also holomorphic at every point of

~ (CJ, 't> . elK (q.

where fore.

(i)) >

eiK(q' (b)) is the modular correspondence defined beWe may,and do,assume that for each i = l, ... ,N , ~i

is holomorphic on and separates the distinct points of the finite set S defined in §3.3 with IK in place of IK' • Moreover, it suffices to prove the lemma for each of a set of generators ~ 1 , •.. ,~N of the normal affine coordinate ring

32

K*(~)-open subset of

of an affine

V~

s .

containing

Therefore, we shall assume ~ is one of these, say ~ = '!' .• ~ Finally, it suffices to treat the case when ~ is a princiIK (1Z,)

pal congruence subgroup

The ray class modulo 7t ramified prime ideal

f

= K

n

"'

K00(9' rr

1?

of of

G (?l)

id. (b- 1 )

contains an un-

of first degree in

Th~n

K •

for some 1T

E:

(cf. [12], pp. 145-9). ~ (b* z) , we may assume

6-

nKCOX

Kxu1td9J

b -l

Since ¢(rr- 1 *z) is then equal to b = rr -l . We may also assume 1T'

belongs to the principal congruence subgroup modulo

~·

of

n (

I (K) f , since -:{? and b are prime to [12], Theorem 6, p. 317). This means that rr - 1 modxrv For an indeterminate t , we define the polynomial Trr,¢(t; z) =-~-~ME
av

of

as a polynomial in t are clearly right invar; therefore, those coefficients lie in M(~)

~

(as functions of z). Therefore, T rr,..,m(t; z) is a polynomial of type M(IK) [t] on all of VIK wherever its coefficients are holomorphic. We have defined the ray-class polynomial HB ¢ ( t)

which has coefficients in

=

-~-~ z ~ B

K*(~)

(t

-

¢ ( z) )

,

as a consequence of Prop.9.

We have assumed that B is the orbit I(K)f*so of so= rrl< (g) , g = 1;6 , where (T) = t; (i.e) , t; E :::: 6 ,co (C)-,~) T E K , and ~(T) = ~ • It follows immediately from the def-

inition of H8 ¢(t) and from the choice of ¢ , by which ~ takes distinct values at distinct points of B , that H8 ¢(t) has no double roots. Therefore, for any z such that

33

pr~(z) E B , the monic g.c.d. of

TTI,~(t; z)

and of

H 8 ~(t)

is the monic whose roots are simple and are the numbers ~(a*Z)

= ~(zM)

, with

a E I(K)f

and

M =MaE CIK(q'(n)) ~

Then we have necessarily (by the assumptions on p~K(a*z)

We may assume that

=

z

• l;q(b)8 , be I(K)f

)

= priK(zM)

Then, in the X

notation of Prop. 3, w = y = 1 and for some S £ K , we have M £ q' (a- 1 S) .JK Since ME IKq' (n)IK , this implies that q' (a- 1 8>

has adelic integral entries and hence that

-lA

~

a- 1 6

el7, while the divisor of a S has the same norm 'f to k as -lA the divisor -:(2 of TI Since q' (a S) E IKq' (n)~ and -lA

=

A

, it follows that a S TT mod 7ZCJ Then c E ( id. (c) 7t) = (1) , 1 1

8-

and

-f , c

=

=

id. (NK/k (c)) we have (c)

(n, n) =

since

~X

r...J

(1)

,

,

and

c

1.

So i f Suppose (c)

TT ( E - 1)

we have

Another choice of, say, same conditions as

=

f? .

-=f?

E€

En , where

id. (NK/k (~)) or

c1

E

=1

=

c1

=

::'

ElTT

Let c = TT mod 7Z (9

h

7l. fY' ; hence ,

c - TI € c

1

= id. (lT) = -:f = (TI) , (TI)

mod 1l and

in place of

with

• C

q' (E) E K (fl.,).

satisfying the

also gives

q' (E 1 ) E

-1

K{n) hence, i f M1 £ q' (c 1 )K(7Z-) , we h~~ M ~ 1 e K(n,) If (c) = = (n) , c = E'i with £' ~ ~, then since 'if

= TI

K (1Z.)

with

1?

=1

so that if

mod

-1

M Ml e. K (n,..) H 8 ~(t) and of

common roots of ((En) -1 *z)

we have

c

is defined in the same way with respect to

M1

c 1 , then ~

n..- ,

E ' _ 1 mod 7l.- and q ' ( E ' ) E and if in the same set-up we have c 1 in p·lace of (ii') c 1 = TiE" , then E I E II - l E K (1Z ) (c 1 ) = 1?

_ c

and

~ (

T TT,'I'"' ( t ; z)

are

Pl (z) =

(E 'iT)- 1 *z)

, and each of these is HB~ , but also a simple root of p 1 (z) -1 p 2 (z) . We have p2

(z) =

Therefore, the only possible

not only a simple root of TTT,~

(t; z) if p 1 ( z)

because

= ~ (TT-l* z)

q' (E)

and

q I (E I)

-?

and

p 2 ( z)

both belong to

~ (TI

-1

*Z)

K(n.)

.

"--{? and are in the same ray class modulo n. X AEC/,TT , a :: 1 modxn , a£K and hence 1T (al72 ' to ~x, belongs which imply (since .. 1 mod 7l) TT = 1T E Hence, E - 1 mod Suppose

'12 =

Then where

n.

34 71- 1 u

=

=

-1

q(a)q(a)oo ~q(71b)8Kn

where

K00 E IK00

since

t;q(nb)e

=

--1

(TI

= q(~lt;q(nb)eq'(E)

t;q(71b)e

*z)

=

= q(a)t;q(nb)8K 00 K~

Kn.. E lK (7l-)

and, of course,

7T -l *Z , this implies p 2 (z)

two polynomials

=

=

(71 -l*z)

<1>

Therefore, the monic g.c.d. of the

HB(t)

t - ((E71)-l*Z) which is of type

•

p 1 (z)

q (a) E G+ (Q)

=

T71 ,<1>(t ~ z)

and

is

t - <1>(71-l*Z)

K*(~)MUK) [t]

on

B

since both of the

first two polynomials are, on account of the Weber-Perron theorem. This proves Lemma 3 in case ::{? and -;2 are in the same ray class modulo ~ . Suppose on the other hand that :f! and --:;2 lie in distinct ray classes modulo ~ ~ then for every z F B we have p 1 (z) ~ p 2 (z) and the g.c.d.of HB (t) and Trr,(t ~ z) , which is of type K*(~)M(K) [t] , is equal to (t - pl (z)) (t - p 2 (z)) = L (t ~ 'l'v (z)) , say, provided q' (n) in fact belongs to IKq' (71)1K . In fact, this is automatically so, for det(q' (71)) is an idele of k generating a prime ideal

1

of first degree in

k , which

by elementary divisor theory over k~ implies that (* ) q ' (iT) = K1q ' (71 ) K2 , K1 , K2 E IK ( 1 )

A

=

G {Z:l )

and since (*) is a system of equations over the different finite places of

k , and

71

and

71

are units

=1

mod~~

at every place cti7L {where 7lr;- is the power of Cf dividing 71.,), we may assume for each such ff that Kl and K 2 ~ both belong to IK {7l)} , and for all other Cf- , betong to K {l)f" which is the same as K {7Z)'}- if_ 'f.../'Tl- • therefore, we may take Kl , K2 E K(n) = ~ and q' (rr) E IKq' {71)1K . Now each coefficient of L(t, '!'v{z)) is in K*{~)M{~) , and in particular we have {A)

(

71 -l*Z) + (n -l

for every z E N variables of coeffients in for every z E

*Z)

= R0 {'!'l (z) '• • • ''I'N (z))

B , where R0 is a fixed rational function in which the numerator and denominator D have K*{~) and such that D{'!' 1 {z) , •.. ,'!'N(z)) ~ 0 B •

35 [1~

We may now continue the argument of §5.2 of

to its

end in essentially the same way as there, but now for the case of Hilbert modular functions, as follows: Let

be a positive integer such that ~e

e

-:pe

principal ray mod7l...., i.e., Then we have

71

e E IT.

U n,k.1) ~

=II.{)'

1

II _ 1 modJl.

~ IT. '7J v..TL (v)

-e 71 E

and

lies in the

II E Kx

, where

"" = { E E ~I "x E ~(e) 1 mod7L}. Let ~ be an arithmetic modular form in A (IK , w) of weight w > 0 , which is nowhere

=

S

vanishing on the set the ideal where

For each

th..-=

and

M.

I(K)f , we define

are defined for

QUM)

in §6.1 of [5 ], the ideal of the entries of

E

~q'(a)-1)

'lJta

'U<M)

a

~(M)

M E G (Af)

as

being essentially the g.c.d.

Then for each

z E ~A(~,~)

put

¢a (z) = N'lltaw ¢ (a*z)/¢ (z) If

bE I(K)f , then

Setting

¢a (b*z) = N'Ua w ¢ ( (ab) *Z) /¢ (bH) b = ai , i = O, ••. ,e-1 in turn, we get, analogous-

ly to the classical case ¢ (z) ¢ (ad).···.¢ (ae-l*Z) = N?J ew ¢ (ae*Z) /¢ (z) a a a v'a. ae = a.E: with a £ Kx and E e U1?..
, then let

If

~-divisor,

no proper integral as ideals,if we take

a= 71-l

and

since

= NK/ka Therefore> v(M 0 )

-1

-:f!

e

M0 has

= NK/krr ,

= NK/Qnaw ¢(au)/¢ (z) if we take

a

=

71

-1

--1

resp. 71

and

II a =

since, for reasons explained earlier,

¢(E:*(~*Z)) = ¢(;*Z)

II

¢((~E:)*z) =

But now if we put

then

¢(~q(~-lb)6) = ¢(q(a) 00 ~q(b)6) = = while

¢(~kooq(b)6)

=

resp. II

xw
k 00

~-lq(a) 00 ~

36

and therefore

=

~ e(z)

a

with (B)

a

-1

=

w --1 w NK/QIT N~(ITIT )

= N~IT

, so that we have e-1 ~a(z)~a(a*z) .····~a(a *Z) =

2w

n

N~IT

2w

In like manner we obtain a similar equation with IT

a

and

replaced by their complex conjugates. -1

Now p 1 (z) = ~(n *Z) and p 2 (z) distinct and therefore simple roots of the polynomial in t given by P(t; 1;;)

=

L.

T TI'

M

CIK ( q

E

1 (

TI))

--1

=

~(n

T

TI,""

~ (t ~

;1J< t -

*z) are m(t; z). We form

; 1:;)

•

( t;:M)

~ ( rM)

~

for 1;: E G+(A) t;: E G+(A) , the

It is easy to verify that as functions of

coefficients of P(t; t;:) are in M(IK) and are regular on the set s. (Cf. Prop. 2.) Thus, as a polynomial in t I is of type

p ( t ; r;)

M(~)

•

We know

. _ ( t ; I:) , hence

T

Tit'¥

T n,~1 (t; s) with respect to t , has . coefficients in M(IK) . -1 --1 Again let a denote either TI or n , and let

also its derivative

be as before. If then we substitute
z E B , and this is the same as

(C)

<Pa (z)

N~.

~(zMa)

T

z

for

..,(t; z)

TI,'!'

t ,

P(~(a*Z)

; z) T n,..,.._ (
Ua

~~(a) and R1 (t; u"') is a rational function and uv = (u 1 , .•• ,uN) with coefficients in 0 , whose denominator does not vanish when we substitute

where in t

=

(~(a*z),

'1' 1 (z), •.. ,IJ'N(z))

for

(t,u 1 , ... ,uN)

By our choice of IT , we see that N~ITm " N~TI m for any non- zero m E :!Z • To deal with equation (A), which may be rewritten as (D)

~ ( TI

-1

* Z) =

RQ ( ljl

"

( Z) )

-

--1

~ ( TI

* z)

37 let

s

M{~)

,

Z{A)

F

, the center of

R{s)¢

~v

ional function in q{nn) £ Z{A)

•

M{R)

Then for any

¢ £

and therefore is a rat-

with rational coefficients.

Since s =

, this implies that we have

¢{n-l*Z)

{D')

G+{A)

also belongs to

= R 2 {~v{z), ~v{'IT*Z))

z £ :::A {(J', r-) ' -2 .,; -1 ¢{n *z) = R 2 {~ {n *Z),

or, for all {D")

~

v'

{z))

similarly, with the same rational function R2 , {D"') ¢(rr- 2 *z) R2 <~ v(;r -1 *Z) ' ~~~ Czl > Here R2 z by ai- 2 *z ¢ (n

-i

has for

coefficients in a-l ='IT

*Z) = R2 (~

and the same

with

v

n

(n

-i+l

or *z) ,

n' ~

K*(r-)

Replacing

we obtain for

i

> 2 '

v{n -i+2 *z))

in place of

n

Letting

¢ =

~i

,

i = l, •.• ,N, in turn, and iterating, we obtain rational mappings

~v('IT-i*Z) = R."(~v(z)

{E)

N '

~"(n*z)) ' i

>

1 '

Ri = {Ri 1 , ... ,RiN) and Rij is a rational function 2N variables with coefficients in K* cr-> , i, j = 1, ... ,

where in

1

II

such that the denominator of each

when the

2N

RiJ' (~

variables are replaced by

z £:::A(~ r-)

, and one obtains the same

in place of

'1T

result with

n

•

On the other hand, for each struct the polynomial in

l, ... ,N, we may con-

i

t

~

Pi{t; z)

is non-zero "' {z) , ~ v ('IT -l *z)) ,

= LM

Tn,¢(t;z) E:

c__(q'(TI))/IK -IK

t

-

(

zM)

-~i(zM)

which is in manner similar to that in a previous case seen to be of type

MID<) [t]

Then by the same kind of argument

as that just now used, we obtain a rational expression for ~i(a*z)

in terms of

¢(a*Z)

and

inK*(~) (for some fixed choice of a-l = 1T or 1T :

~v(z)

with coefficients

as earlier), valid for

{F) ~"' (a*z) ; ~ v(z)) Combining (B) , (C), (E), and (F), we obtain polynomials P{t; uv) amd Q(t; u~) with coefficients in K*(~) such that Q{
38 every

z £ B , such that

(G)

N'f (II a) 2 w =

Note that Put

P

and

Fa ( t ;

U

v

)

P ( ¢ (a* z) ;

\jl

v ( z) ) I Q ( ¢ (a *Z) ;

\jl

v ( z) )

are the same whether a = TI 2w - - v = N'f (II a) Q ( t ; u ) - P ( t ; u V)

Q

or

For every z £ B, t =
is not a root.

F TI-l (t;

every

1j1 11(z))

z £ B •

,

Therefore, the monic g.c.d.

L¢ (t; \j/ 11 (z))

and of

is

t -

¢

(b*z) for

It follows from the Weber-Perron Theorem

that for some rational function Rb,¢(u 11) have v z £ B ¢ ( b* z) = Rb I ¢ ( \jl ( z) ) as required. This proves Lemma 3.

t.

K*C'fl (ull)

we

Now by exactly the same argument as that used in proving of [111 one obtains at once:

Prop. 5.2.2

Proposition 11. of extensions

For dny

K*('f) (
z

t.

, with

B ,

~

¢ £

M(~)

we consider again the action of

I(K)

is the compositum regular on on

B

~~ccr, 'fl

•

If the double coset n<~el

belongs to have for

~~(~, 'fl

= G (Q)~elKIK

+ , with

. (Tl

co

=~(i.e)

and

q

, and

y

we

a£ I(K)f a*D(~el

if and only if for some

k

=

o<~el

£

IK ,

k

00

~

£

00

£

G+ (Q)

,

q(a)E,8 = y~ekkoo By looking at the archimedean component of this, one obtains as usual A

q(a)8k or or

y = q(a) A-1

q' Ca

a £ Kx~ , where

I{K)f

defined

for some

a) = k £

~

in §1.4.

~

a£ Kx ; hence,

q(a)e =

, which is to say that

A-1

a

!)_,

a F H,

is the open compact subgroup of Therefore, in the action of

I(K)

on the finite set ~IKe&, 'fl , the stabilizer of D(8~) is K 00 xKx~ Since I(K) is abelian, the stabilizer of any point z

of the orbit

the class group

B

of D(8~)is also

c (f)', H) = I (K) /KxK

simply transitively on the orbit Therefore, with the Gothic

K00 xKxft. x~ 00

Consequently

acts faithfully and

B

&,_ of [11] replaced by the

39

K*(~) , one may repeat the formal proof of

reflex field

Lemma 5.3 of [11] to obtain Lemma 4.

Given

is exactly one class ~

£

M(~)

regular on

z 0 E B and o AH(a) E C(~, H) B : ~(z 0 )

0

=

E

Gal(oC~/K*(~)) , there such that for every

~(Ag(o)*z 0 )

.

Similarly one obtains by the formal arguments of [11]: Lemma 5. o

£

In the same notation as above, given

Gal(O(oc/K*(~)) , we have ~(A~(o)*z)

for every

z

£

B

and

~

£

M(OC)

= ~(z) 0 regular on

B

In conclusion, one obtains as in [11], so in this case also: Theorem 2. from

The map

Gal (~/K* (~))

a r----> Ag(o)

into

is a monomorphism

c (&', H)

Corollary. ~ is an abelian normal extension of the reflex field K*(~) . The next problem is to recover by these and similar rather straightforward elementary considerations the known [13] reciprocity law for this extension.

We intend to

return to this question in a subsequent publication. REFERENCES 1. Artin, E., Representatives of the Connected Component of the Idele Class Group, Proc. International Symposium on Algebraic Number Theory, Tokyo(l955), pp. 51-54.

2. Baily, W.L.,Jr.,Galois action on Eisenstein series and certain abelian extensions of number fields, pre-print, 1980. 3.

Arithmet~c

567-605.

, On the Theory of Hilbert Modular Functions I. Groups and Eisenstein Series, J. Alg., 90(1984),

4. , Arithmetic Hilbert Modular Forms, Automorphic Forms of Several Variables, Taniguchi Symposium, Katata, 1983, Birkh~user, 1984. 5. , Arithmetic Hilbert Modular Functions II, Revista Matem&tica Iberoamericana, 1(1985), 85-119.

40 6. Chevalley, c., Deux Theoremes d'Arithmetigue, J. Math. Soc. Japan, 3(1951), 36-44. 7. Deligne, P., Travaux de Shimura, Seminaire BOURBAKI, 23(1970/71), No. 389, 123-165. 8. Hasse, H., Neue Begrtindung der komplexen Multiplikation, Jour. Reine Angew. Math. 157(1927), 115-139. 9. , Neue Begrtindung der komplexen Multiplication II, Jour. Reine Angew. Math. 165(1931), 64-88. 10. Heeke, E., HOhere Modulfunktionen und ihre Anwendung auf die Zahlentheorie, Math. Ann. 71(1912), 1-37. (=No. 1 in Mathematische Werke) . 11. Karel, M.L., Special Values of Elliptic Modular Functions, Proc. of International Symposium on Automorphic Forms of Several Variables, Katata 1983, Birkh~user, 1984. 12. Lang,

s.,

Algebraic Number Theory, Addison-Wesley, 1970.

13. Shimura, G., Construction of class fields and zeta functions of of algebraic curves, Ann. Math., 85(1967), 58-159. 14. , On canonical models of arithmetic quotients of bounded symmetric domains, 91(1970), 144-222; II, 92(1970), 528-549. 15. , and Taniyama, Y., Complex Multiplication of Abelian Varieties and its Applications to Number Theory, Publications of the Math. Soc. of Japan, No. 6, 1961/ 16. Taniyama, Y., Jacobian Varieties and Number Fields, The Complete Works of Yutaka Taniyama, Yutaka Taniyama Complete Works Publication Society, 1961, 1-68 (originally mimeographed notes, University of Tokyo, September 1955). 17. Weber, H., Lehrbuch der Algebra (Kleine Ausgabe), Friedrich Vieweg & Sohn, Braunschweig, 1912. 18. Weil, A., Basic Number Theory, Springer-Verlag 1967.

The Heat Equation for the a-Neumann Problem on Strictly Pseudoconvex Domains Richard Beals * Yale University New Haven, Connecticut 06520 and Nancy K. Stanton ** University of Notre Dame Notre Dame, Indiana 46556

1.

BACKGROUND

The heat equation for the a-Neumann problem on strictly pseudoconvex domains is a complex analogue of a classical problem in Riemannian geometry. In this section, we will describe some of the classical Riemannian results. To keep things simple, we will only talk about domains. Let

be a bounded domain in

g

~n

with smooth boundary

powerful method of relating geometry and analysis in heat equation. vanish on tion

Let

M,

6

Q

M.

is to study the

denote the Laplacian on functions on

Q

i.e., which satisfy Dirichlet boundary conditions. 2 -

f(x,t) E C (Q x

+ ) (x E Q,t E R+ )

~

One

which A func-

solves the heat equation with

Dirichlet boundary conditions if

a

6) f

fiMXR+

=

=o (1.1)

O.

The initial value problem for the heat equation with Dirichlet boundary conditions is the following.

Given

f0 E

c 0 (Q),

find a solution

f(x,t)

of the heat equation satisfying lim f(x,t)

t-+o

=

f 0 (x).

(1. 2)

The solution of the initial value problem is given by applying the t6 e the semigroup generated by ~. to the initial value,

heat semigroup

*

Research supported in part by NSF grant DMS8402637.

**Research

supported in part by NSF grant DMS8200442-0l and the Alfred P. Sloan Foundation.

42 so

Let

f

= ~e

tr e t!J.

0 > A. 1 ':: A. 2 ':: • • •

be the spectrum of

Then

fJ..

t\. 1

(1.3)

McKean and Singer [9] proved the existence of an asymptotic expansion of this trace, tr e

t!J.

where

(4TTt) -n/ 2 ~ c.tj/ 2 j'::O J

Q

is the volume of

as

t -+ 0,

and for

j ::: 1,

M of a universal polynomial (depending only on

M as a submanifold of

invariants of

Rn.

(1.4) is the integral over c. J j and n) in geometric is equi-

The calculation of

valent, via an Abelian theorem and Karamata

1s

Tauberian Theorem, to an old

result of Weyl's {12].

THEOREM 1.5 (Weyl).

Let

N(A.)

(counted with multiplicities) of N(A.) ~

Volume (Q)

A.n/2

(4rr) n/ 2r (n/2+1)

denote the number of eigenvalues

-fJ.

which are less than

as

\-+ ~.

For the special case of a plane domain, (1. 4)

n

=

A..

Then (1.6)

2,

the first few terms of

are

tr e

where

x

_l_ 12TT

where

tfJ.

h

Area Q

~ ~ -

Length M

sv'Tit

is the curvature of

f

M

X =

+

1 l2iT fMx

M.

.r. + 0 ( vt) ,

(1.7)

By the Gauss-Bonnet Theorem

l(l-h) 6

(1.8)

is the number of holes.

The heat semigroup is given by integration against the heat kernel p(x,y,t) E C~(Q x

Qx

R+)

and

p(x,x,t)dx.

(1.9)

The proof of the asymptotic expansion (1.4) comes down to a detailed construction and analysis of a good enough approximation to approximation 1 )n/2 -lx-yl 2/4t

p(x,y,t) ~ (4TTt

e

'

p.

To first

43 the Euclidean heat kernel.

This is Kac' s "principle of not feeling the

boundary" [8] and immediately gives

c0•

McKean and Singer worked in the more general context of Riemannian manifolds with boundary.

Their work was generalized to elliptic boundary

value problems by Greiner [6] and Seeley [11].

2.

STRICTLY PSEUDOCONVEX DOMAINS

One analogous problem in several complex variables is the heat equation for the a-Neumann problem.

~n+l, n ~

domain in

(0,1)

2,

A form

forms.

Let

Q

be a bounded strictly pseudoconvex

with smooth boundary. For simplicity, we work on u E c 1 (A 0 ' 1 (Q)) satisfies a-Neumann boundary condi-

tions if u

au

0

norm

norm

on

(2.1)

M.

Let AO,q

u

b

o}

norm

(2.2)

and let (2.3)

denote contraction with the vector field is the inward unit normal vector field on plex structure on

~n+l

N"

= .!_(N v2

M and

+ iJ 0 N)

J0

where

N

is the almost com-

Then we can rewrite the a-Neumann boundary con-

ditions (2.1) as Vu

=

vau.

0 =

(2.4)

The a-Laplacian

D

is defined by vu

--* aa

0

If

u

+

-i = :2:u.dz l. ou

-

o} (2.5)

-*a a E Dom o,

then

2

a ui -i 2 :2:. (:2: --.--. dz ). azJa~J i j

(2.6)

44 1 n+l (z , ••• , z )

Here A

form

(0,1)

are the complex coordinates on ,.n+l. ~ 2 0 1 F(z,t) E c (A ' (Q x R+)) solves the heat equation for

the a-Neumann problem if for fixed

t,

F(·,t) E Dom o (2. 7)

a

+

(at

O)F = 0.

The initial value problem for the heat equation for the a-Neumann problem is the following. Given f E c 0 (A 0 ' 1 (Q)), find a solution F of the heat equation (2.7) satisfying (2.8)

lim F ( ·, t) = f. ~

The solution to the initial value problem is given by applying the semigroup generated by

-o,

to the initial value.

given by integration against a smooth kernel

c=
®

Al,o <S2

p(z,w,t)

e

-to

This semigroup is which belongs to

x 52 x R+ll,

F(z,t) =(e-t 0 f) (z) = fr:2 p(z,w,t)A*f(w).

(2.9)

Now, tr e

-to

L:e

-tA (2.10)

fg, tr p(z,z,t)dV.

The sum in the first line of (2.10) is over all eigenvalues of

o

(counted

with multiplicity). In the second line, dV is Lebesgue measure and the local trace tr:/\ 0 ' 1 ® A1 ' 0 ~ G: is the linear map satisfying tr ~ ® v = ~A*V/dV.

To study the asymptotic behavior of tion of

p.

tr e-to

we need a good descrip-

The difficulty is that a-Neumann boundary conditions are non-

elliptic so the methods used by McKean-Singer, Greiner and Seeley do not apply.

We will describe our methods.

The details are in [2].

use the classical method of reduction to the boundary.

Greiner and Stein !7] in their work on the a-Neumann problem. it reduces the construction of

p

First we

This was used by In our case,

to the inversion of a first order clas-

sical parabolic pseudodifferential operator on

M.

To carry out this reduction, we write e

-to

= P

G + H

(2.11)

45 where

G

is the Green's operator for the heat equation, the fundamental

solution of the initial value problem with boundary value correction term.

For

with initial value heat equation Hf

0.

f E c 0 (A 0 ' 1 (Q)),

Hf

=

0

vg so

H

is a

satisfies the heat equation

g

=

HfiM' (2.12)

Jg.

The first of the a-Neumann boundary conditions is vGf

and

Thus it is given by the Poisson operator for the

applied to its boundary values

J

0,

vpf

hence, by (2.11),

= vHf

g E Ab0,1 •

VPf - VGf = 0,

(2.13)

Also by (2.11) and the second a-Neumann boundary condition, -vaGf.

(2.14)

Let (2.15) acting on sections of

A0,1 b

over

Then, we can rewrite (2.14) as

-vaGf.

+

The operator

oh

(2.16) is a classical first order parabolic pseudodifferential

operator on the space

A~' 1 (M x ~+).

If it is invertible, combining (2.16)

with (2.11) and (2.12) gives g

(2.17)

p

(2.18)

and

If we had an elliptic boundary condition, the operator analogous to

0~

would be invertible in the classical parabolic pseudodifferential cal-

culus.

Because of the non-ellipticity of the boundary condition, we can-

not invert

o~

classically.

However, following Greiner and Stein [7], we

can find another first order classical parabolic operator

oh

such that (2.19)

46 Because

n :::: 2,

is invertible in the Heisenberg parabolic calculus

of [ 1] and (2.20) Using a mixture of classical and Heisenberg parabolic calculus, we can calculate a full local asymptotic expansion of the symbol of

+-1

oh

is well known by [6], [9] -to Thus, to know the asymptotic behavior of the trace of e by

The asymptotic behavior of the trace of and [11].

G

(2.18) it suffices to study the asymptotic behavior of the trace of (2.21)

H

Now,

H

vanishes to infinite order in

Q,

subsets of near

Q

as

~

t

0,

uniformly on compact

so the asymptotic behavior of its trace is concentrated

M.

{z E Q:r(z) S 13}. For M 13 We denote points of M by (x,r), 13 X E M, r E [ 0, 13]. Let be the volume element on M. Then, in M 13' dV = g(x,r)dr dV(x). We write the kernel h of H on M as 13 h(x,r,x' ,r' ,t), x,x• E M, r,r' E [0,13]. Then as t ~ 0 Let

13

r

denote distance to

sufficiently small,

f

tr H ~

M 13 d'll

""'

M,

and let

M X [ 0,13].

tr h(x,r,x,r,t)g(x,r)dr dv(x).

(2.22)

MX[0,13) To analyze this integral, we introduce an auxiliary operator

A0 ' 1 (M

x

~+)

with kernel

k(x,y,t) = Then as

t

~

k

K

on

given by

J 130 h(x,r,y,r,t)g(x,r)dr.

(2.23)

0

tr H ~ tr K = !Mtr k(x,x,t)dv(x).

(2.24)

To analyze the right side of (2.24), we work locally, and now let denote the local coordinates of points in an open set variables dual to k(x,y,t) = where

(x,r,t).

is

x,y

(s,P,~)

the

Then

13 v

is the symbol of

H

(2 .25)

in

the inverse Fourier transform in the variables K

M,

J0 q (x,r,x-y,-r,t)g(x,r)dr

q(x,r,s,p,~)

symbol of

uc

u x [0,13] (s,P,~).

and

V

denotes

Thus, the local

47 (2.26) denotes the inverse Fourier transform in the variable dual to

where

symbol calculus shows that

q1

r.

has an asymptotic expansion as a sum of

matrices where each matrix entry is the product of a symbol with classical parabolic homogeneity and a symbol with Heisenberg parabolic homogeneity. The kernel of the operator corresponding to such a product is then the convolution of two functions, each with a different kind of homogeneity.

of course, if we had an elliptic boundary condition, the kernel would be homogeneous and would contribute one term to the asymptotic expansion of the trace.

Here we have the following theorem.

(We have suppressed the

first manifold variable in the kernel.) Suppose

THEOREM 1.

(i)

a(x,t), b(x,t) E

(ii)

for some

m E

~.

c""uR2 n+l m

~

X

-2n-3,

~)\0)

and vanish for

and all

t < 0;

A > 0

2 a(Ax,A t) if

m

=

(2.27)

-2n-3,

(iii)

a

for some 2

b(A•x,A t) where

is a principal value distribution;

E

m'

~. m' ~ -2n-2,

and all

A >

0,

m' A b(x,tl

=

(2.28)

x

f

a(x,s)b(-x,t-s)ds dx

R2n+2

(2.29)

t (2n+m+m' +4) /2

l: {k. + t( 2 n+m+ 2 l 12 (k! + k'! log t)]tj/ 2 • j=O

Here,

m

k., k'

and

j

J

= -2n-3,

m'

vergent for

=

k" j

-2n-2,

m > -2n-3

J

J

J

are constants determined by k

0 = o.

a

and

b,

and if

The integral in (2.29) is absolutely con-

and is a principal value integral for

m

=

-2n-3.

From this, we immediately obtain the following theorem. THEOREM 2.

tr e~to

As

t

~

0,

t-(n+ll (c 0 +

l: (c.+ c! log t)tj/ 2 ). j~l

J

(2.30)

J

We have not yet worked out examples, but the proof of Theorem 1 leads

48 us to believe that the

log t

terms do occur.

If this is correct, then

the expansion (2.30) has two features not present for elliptic boundary value problems.

First,

log t

enters the expansion.

Logarithmic terms

have been encountered recently in problems with singularities, for example, in the work of Cheeger [5], Callias and Uhlmann [4] and Bruning and Seeley

[3].

Second, the coefficients in the expansion do not behave nicely with

respect to scaling of the domain.

tr e

In the Riemannian case, it is easy to read off the leading term of t6 t6 from the construction of the symbol of e Here it is not immec0•

diate, but we have calculated

It is the sum of two terms - the volume

term coming from the trace of the Green's operator and the integral over the boundary of a positive function of the Levi form. of not feeling the boundary" does not hold. we need some notation.

Let

tr L

denote the Levi form of

Q,

Q

=

=

on

M.

be the sum of the negative parts of the eigenvalues of

L.

We

r

of

N

denote the volume of a manifold

by

{r < 0}, with

M calculated 1

using a defining function Let

L

Thus, Kac's "principle

To state the result about

jdrj

Vol N.

THEOREM 3.

- 11 { (n+l)Vol Q (2rr)n+

fMf~(tr COROLLARY 4.

+ (2. 31)

e-aL)e-(tr-L)adet(aL(l-e-aL)- 1 )da dv}.

Let

Q

= Bn+l,

the unit ball.

Then (2.32)

By Karamata's Tauberian Theorem, we have a new proof of a result of

M~tivier I1oJ.

M~tivier used completely different methods to prove his

result. THEOREM 5

of

o

(.M~ti vier) •

Let

N(~)

(counted with multiplicity) which are less than or equal to

Then as where

denote the number of eigenvalues

is the constant of Theorem 3.

~.

49 All of the results we have described generalize to

(p,q)

forms on

compact complex Hermitian manifolds with boundary if the boundary satisfies Kohn's condition

Y(q)

{2].

BIBLIOGRAPHY 1.

R. Beals, P.C. Greiner and N.K. Stanton, The heat equation on a CR manifold, J. Differential Geom., to appear.

2.

R. Beals and N.K. Stanton, The heat equation for the a-Neumann problem, in preparation.

3.

J. Bruning and R. Seeley, Regular singular asymptotics, Advances in Math., to appear.

4.

C.J. Callias and G.A. Uhlmann, Singular asymptotics approach to partial differential equations with isolated singularities in the coefficients, Bull. A.M.S. 11 (1984), 172-176.

5.

J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575-657.

6.

P. Greiner, An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal. 41 (1971), 163-218.

7.

P.C. Greiner and E.M. Stein, Estimates for the a-Neumann problem, Math. Notes 19, Princeton Univ. Press, Princeton, N.J. 1977.

8.

M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), No.4, Part II, 1-23.

9.

H.P. McKean Jr. and I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 43-69.

10.

G. M~tivier, Spectral asymptotics of the a-Neumann problem, Duke Math J. 48 (1981), 779-806.

11.

R. Seeley, Analytic extension of the trace associated with elliptic boundary value problems, Amer. J. Math. 91 (1969), 963-983.

12.

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), 441-479.

Some Examples of the Twistor Construction D. Burns Department of Mathematics, University of Michigan Ann Arbor, Michigan 48109 U.S.A. To Professor Wilhelm Stoll 1.

INTRODUCTION

A hyperkahler manifold X is a kahler manifold of even dimension with a holomorphic 2-form

IJJ,

everywhere non-

singular, and covariant constant. In [2] Calabi constructed such metrics on the cotangent bundle of IPn, with w equal to the canonical symplectic 2-form wcan on T*(IPn), and gave an analytic construction of an associated complex manifold of dimension 2n+l.

This construction, for n=l, is equivalent to

the Atiyah-Hitchin-Singer version of Penrose's curved twistor space ([£], [4], [6]). Calabi asked for an algebraic or geometric description of this twistor space. On the other hand, it is widely known that the construction is reversible, i.e., there is an inverse reconstruction of a hyperkahler metric from geometric data.

In examining the question of the

geometric construction of the twistor space for Calabi's examples, it appeared that a simple enough procedure emerged to enable one to construct new hyperkahler metrics where Calabi's method no longer worked.

The metrics constructed are

on cotangent bundles again, and the constant, holomorphic, non-degenerate 2-form w is the canonical symplectic form. They are local only, i.e., defined on a neighborhood of the zero section in T*(M).

To make the necessary calculations

feasible we assume M is a generalized flag manifold, i.e., a compact, simply-connected homogeneous kahler manifold. Theorem:

Let M be a generalized flag manifold, U its holo-

morphic isometry group.

There exists a U-invariant hyperkahler

metric on a neighborhood of the a-section in T*(M) such that wean is covariant constant.

52 In case M in the theorem is symmetric, the general construction can be made more explicit and the metrics found above can be extended to be complete on all of T*(M). There is much contact and overlap between the present paper and work of N. Hitchin and M.

Ro~ek,

cf.

[5].

I would

like to thank B. Lawson for pointing this out, and especially Martin Ro~ek for his explanation of some of their very interesting ideas. The paper is arranged as follows.

§1 sets down the

twister notation, much along the lines of [6], and gives a proof of the inversion theorem for the twister construction in the hyperkahler case.

In §2 we make our global geometric con-

struction but verify the hypotheses of the inversion theorem only in a neighborhood of the 0-section.

In §3 this is

globalized for M a symmetric space, and a few questions/remarks ·are added in §4. I would to thank E. Calabi for very useful discussions, and for drawing my interest to this problem. §1.

The Inverse Twister Transform We start by reviewing the characteristic properties of

the twister spaces associated to hyperkahler manifolds, cf. [2],

[6].

For X a hyperkahler manifold of complex dimension

2n, Z(X) is a complex manifold of dimension 2n+l with i)

a holomorphic map 1T: Z (X) + IP 1 and smooth map p: Z(X) +X such that Z(x) (1T,p) IPl x

x

is a differmorphism. ii)

iii)

an anti-holomorphic involution o: Z(X) + Z(X) . . 1 ut1on . . by cover1ng t h e 1nvo o 0 : IPl + IP 1 , g1ven [z,w] + [w,-z] in homogeneous coordinates. The fiber IP 1 • s of the map p in i) are holomorphic curves in Z(X), stable by a.

53 -1

iv)

For a fiber Cx=p (x), the holomorphic normal bundle N of C in Z(X) is isomorphic to 2 @ . H n-cop~es x x where His the line bundle of degree +1 on C =IP 1 . X

v)

There exists a nowhere degenerate holomorphic section ~of n2 l ® n*H®2 . Z (x) /IP _1 ~ restricted to a fiber n (t) of n is identified, up to a constant, with a closed 2-form on n- 1 (t).

Finally, ¢ must satisfy a reality condition with respect to a.

a acts on relative differential forms by w

+

a*(w),

where the bar is complex conjugation on the coefficients of a differential form. a also acts on n* H®K, any k, compatibly via n with the action of a 0 on H®k over IP 1 . We fix homogeneous coordinates [z,w] on IP 1 so that, on H0 (IP l ,0 (H)), a 0 acts conjugate linearly and a 0 (z)=w, a 0 (w)=-z. For each

x € X,

induces a non-degenerate skew-form on

~

H0 (C , 0 (N X

above.

X

® n*

H*)), as follows from

iv

and v

The reality conditions on ¢ are: vi)

¢(v 1 ,v 2 ) = ¢(a(v1 ) ,a(v 2 )), and ¢(v 1 ,a(v 1 )) > 0, if v 1 "I 0,

where v 1 ,v 2 are sections in H0 (Cx,O(Nx ® n* H*)). We remark that these conditions are formally the same as those encountered in the case n=l (X a kahler surface) except that in higher dimensions one has to check that ¢ is closed on n- 1 (t) in (v), and that ¢(v 1 ,acv 1 )) is (positive) definite in (vi).

These last facts are automatic for n=l.

Conversely, one has the following inversion theorem, due originally to Penrose, in a Minkowski form, for n=l, [4]. Theorem: Let z be a complex 2n+l manifold admitting maps n, p, a and differential ¢ as above, verifying (i) to (vi). Then there exists a unique hyperkahler metric on X such that Z=Z(Xl, the twister space for that metric.

54 Proof: By a standard deformation theory argument (cf. [7], for example), one can identify X with a component of the oreal points in the space of all sections of the map n. ~

this identification, we can identify T (X) H0 (C ,O(N )) , where C =p X

X

X

-1

¢

Under

with

X

(x), and T (X) with the o-real X

vectors in H0 (Cx,O(Nx)). For x € X, let Ex=H 0 (Cx,O(Nx ~ n* H*)). By condition (iv), dimE =n, and E=U E is a smooth vector bundle. Denote X x€X X H0 (IP 1 ,0(H)), by V and identify V with the sections in H0 (Z,O(n* H)) pulled-up from IP 1 • above, T (X)~¢

E

®

By condition (iv), and the

Let w0 denote the skew-form on V such The form¢ in (v) defines a non-degenerate V.

that w0 (z,w)=l. skew-form on E, which we continue to call just ¢, and is a non-degenerate symmetric form on T(X)~.

G.9

w0 By condition ¢

(vi), ¢~ w0 is a real symmetric form on T(Xl~¢, i.e., is real on T(X), and is positive definite on T(X). Denote this metric by g. Using this metric, we identify T(X) with T*(X). Then ~ 2 (El&s 2 (V) + s 2 (E)~~ 2 (V). Under these identifications, we have a distinguished three dimensional space of

~ 2 T*(Xl®¢ =

2

2-forms on X, of the form ¢ ® S, for S € S (V). Such a 2-form is real if and only if S is real with respect to o 0 . LetS be real € s 2 (v), with jsj 2 =2: such an S defines an almost complex structure on X, via the 2-form ¢@ S and the metric constructed on X.

More explicitly, fort € IP 1 , let

st be a section in V such that st(t)=O, and 2

jstj =w 0 (st,o 0 (st))=l. uniquely as s = st

-i(st de f.

Then every such Scan be written

® o 0 (stl

+ o 0 (stl

® stl

for some t e IP 1 . Writing¢~ st in terms of the metric g and a skew-symmetric transformation Jt on T(X)g¢, one checks that Jt is +ion the subspace € ~¢·o 0 (st)} and-ion E

® {¢ •St}.

55

Lemma:

The map p, restricted to Zt=n

-1

(t), is holomorphic to

X with the almost complex structure Jt. In particular, Jt is integrable, and ~ ~· st is its kahler form for the metric g. We postpone briefly the proof of the lemma. ~®(o 0 (st)~o 0 (st))

The complex 2-form represented by

is of

type (2,0) for the structure Jt, and its pull-back via p to Zt agrees with the restriction of

~

on Z to Zt.

Since

~

is closed

on zt by condition (v), ~t is closed on X. Since the 2-forms ~t' t€IP 1 , on X span the same three-dimensional space as the forms ~ (iP St, we conclude that ~ ~ St is closed. metric g is kahler for each Jt, and

~t

Hence, our

is covariant constant,

as a linear combination of kahler forms for various of the 1 structures Jt'' t' e P Thus, g is hyperkahler. Finally, the twister space of g is the manifold Z(X)=X x IP 1 , where the complex structure on X x {t} is given by Jt. By the lemma, Z ~ Z(X) by p x cluding the proof of the theorem.

TI

is biholomorphic, con-

Proof of the Lemma: We have only to make explicit the identification of T(X) and the a-invariants in r(c ,N ) to evaluate X

X

X

the differential of p. Fix q € Z such that p(q)=x, n(q)=t. We identify geometrically T(Zt) and the real normal vectors IR IR q N to C at q, and then N with the complex vector space N q

at q.

q

X

X

The o-real holomorphic sections of Nx are nowhere zero

on ex' explicitly, they are of the form s = v®o 0 (st)- o(v) for v € Ex.

®

st

The differential of p sends s(t) to

s € H0 (Cx,O(Nx)) = T(X) ®¢.

So, dp*(v(t)®J 0 (st) (t)) ~ 0 (st) - o(v)~t' and dp*(i v(t)~ 0 (st) (t)) = iv ® a 0 (st)+io (v)~t· On the other hand, since 2

wo(st,o 0 (st)l=lstl =1, we calculate

56

and

Thus, p restricted to zt is holomorphic. §2.

The Geometric Construction Let us fix some notation for describing the generalized

flag manifolds.

G is a complex semi-simple lie group,

e

an

anti-holomorphic Cartan involution fixing u, a maximal compact subgroup of G. algebras.

Gothic letters denote corresponding Lie

Tis a maximal torus in U,

respect to T, and Xa. €

'J- a

~the

roots of 'with

non-zero root vector for a..

For

z 0 € it. c ,, ~ + ( z 0 ) = { a.€~ I .:!:_a. ( z 0 ) > 0 } , and fl.+= (j) ¢ • X+a.. - a.e~+(z 0 J Let C:(z 0 ) be the centralizer of z 0 in1f,1'+=()t(z 0 )tti'tL+' and P + the normalizer off+ in G under the adjoint representation. Our flag manifold M is-the quotient space G/P+.

Since

9(z 0 )=-z 0 , S(P+)=P_, and G/P+ is conjugate biholomorphic to G/P_ via e. The holomorphic tangent bundle to M is the quotient of -1

G x(11f+l by the action of P+' p• (g,X modl"+)=(gp ,Ad(p)X modtJtl• Let B be the Killing form of,. Under B, the dual of~,+ is ~+'

and the cotangent bundle of M is G

of P+, p• (g,Y)=(gp- 1 , Ad(p)Y). P+.)

x~+

modulo the action

(Note that~+ is normalized by

The tangent space to T*(M) at (g,Y)mod P+

is~xft,+

mod

vectors of the form (X,-[X,Y]), X €'!"+ (these are the tangents to the P+-orbit through (g,Y)).

Thus, each tangent vector to

T*(M) at (g,Y) has a unique representation as (X_,X+), with X+ € 11. +.

The canonical symplectic form w on T* (M) is given at

(g,Y) by w( (X_,X+), (Y_,Y+) )=B(X_,Y+)-B(Y_,X+)-B( [X_,Y_] ,Y). Finally, let 1>lct®¢ be the center of ct(z 0 ). The u-invariant kahler metrics on M are given, at e mod P+ € M, by (X,X)=-B(X,ad(z)S(X)), where z € &Lsatisfies a.(z)>O, all

a € ~+ (zo).

Here,

X

e Vr+·

57 Note that the same constructions, mutatis mutandis, work for the homogeneous space G/P_ and its cotangent bundle. We want to build a twistor space in two patches over IP 1 Let u 0 =IP 1 -{oo}, Ua=IP 1 -{0}, t, the affine parameter on u 0 , r; the affine parameter on U00 , r;=l/t on Ua f\ Uoo . We want fiber spaces over u 0 , U00 such that the fibers over 0 and oo will be T*(M) and its conjugate respectively. We view T*(M) as T*(G/P_).

The problem will be to patch these two spaces over

uo" uoo. Let us fix z in the center of

a

e

6+(z 0 ).

Consider the space

P+ acts as above. (g,tz +

X+l~t

e

~(z 0 )

z0 =G

x {¢•z +~+}/P+, where

Since z is central in

which factors through

~(z 0 ),

z0 :

¢=u 0 is well-defined on

map rr is a submersion.

such that a(z)>O, all the map

call it rr.

The

We also have another map

z0 .

We have:

q

ZO

-+lx¢

+rr

+p2

uo

¢

On rr -1 (t), tiO, q I rr -1 (t) is biholomorphic.

The map q gives a

resolution of singularities of the variety in ~x ¢ given as

z

the Zariski closure

x ¢*1 X=Ad(g) (tz), 3ome 0 of {(X,t) € (Note that qlrr- 1 (0) is the moment map of the action

g € G.}.

of G on M.) rr:

Over Ua' we construct Z00 =G x {¢z +~_}/P_, and define (g, r;z + X_) ~ r; € U00 , and q :

(g

r; z + X-)

I

The image q(Z ) is ·ra-x

~

'~-*

•

00

-+

Z. 00

(Ad (g) ( r; z +

X-)

I

r; ) € ~

X

¢.

Consider the transformation f of -2 -1 X, t ) . This takes z 0 to Z00

send1ng (X,t) to (t

over ¢*=u 0 ~ U00 ,

if we set r;=t- 1 .

obtained by gluing

Denote by

z 0 ,zoo in this way.

z

the manifold

The map q is globally

58 defined on Z to

'fJ- ® H® 2

over IP 1 .

Note that Z has a G-action

induced from g• (g' ,X) = (gg' ,X) on G x,., and q is equivariant G acting on '0-<29 ~2 via the adjoint representation. Z will be our twister space. We must next construct a U-equivariant conjugation on Z, covering t

-1/t on IP 1 •

+

On

z0 ,

a is defined as

a((g, tz +X+) mod P+) = (6(g),-tz + 6(X+)) mod P_, and similarly from that TI(a(TI

-1

z 00

to -

(t))) = -1/t.

z0 •

For t#O, one checks directly

(Note that 6(z)=-z.)

Since

6(ug) = u 6(g), for u € U, a is U-equivariant. The main question will hereafter be the existence of a-real sections over IP 1 in Z.

We compute this on Z c

~H(})j 2 ,

Thus, over u 0 , a secis written as x 0 + x 1 t + x 2 t 2 , xi € , , and the reality conditions are

where a acts 6xa 0 (a 0 as in §2 above). tion of

®H~

8(X 0 )=X 2 8(X 1 )=-X 1 . On the other hand, for any u € U, x 0 =x 2 =0, x 1 =Ad(u)z defines such a real curve. Its lift to z0 is represented as (u,tz)mod P+.

At t=O, these sections pass through the zero

section of T*(M)=TI-l(O), and likewise at t=oo, or ~=0 in U00 •

Before calculating the normal bundle to these sections,

we describe the twisted two form ~ on Z. In ~x ¢*,note that q(t- 1 (t)), t#O, is G-orbit (Ad(g) (tz) ,t) ,g € G, of (tz,t). A

by B, and on Zt=q(TI

-1

We've identified~ with~*

(t)) we are going to set

the negative of the Kiriilov 2-form.

~O

equal to

More specifically, at

s € Zt' a holomorphic tangent vector is a vector [X,s] €~, for X € ~(since Zt is G-homogeneous), and for X,Y€,, set

59 4>0, ~;( [X,I;]

I

-B(I;, [X,Y]).

[Y,I;])

B([X,I;],Y) = -B(X,[Y,I;]). It is clear that 4>0,I; is well-defined, i.e., the right hand side is 0 if [X,I;]=O, that 4>0,I; varies holomorphically with 1; in~, is G invariant of type

(2,0) on Zt' and is readily

checked to be closed on each zt. To represent this on z 0 , take (g,tz + Y0 ) e z 0 , when Y0 € ~+' g € G, tiO, and consider Lie alge~ra elements X_, X' __ in.,._, Y+' Y~ in"+" Let Y+ €f1+ verify [Y+, tz + Y0 ] = Y+: Y+ exists, since tiO and ad(z) is invertible on~+" Under q, the tangent vectors (X_,O) and (O,Y+) to z 0 are repre~ented by [Ad(g) (X_),Ad(g) (tz + Y0 )] and Ad(g) Y+ = [Ad(g) Y+' Ad(g) (tz + Y0 )J at I; = Ad(g) (tz + Y0 ) in similarly for X~, Y~. Hence, on z 0 , we have: 1)

4>0,I;((O,Y+),

(O,Y~))

-B (Ad (g) (Y+) 2)

4>0,I;((X_,O),

4>0,I;((X_,O),

Ad (g)

I

(Y~))

0.

(O,Y+)

= -B(Ad(g) (X_) 3)

=

I

Ad(g) (Y+))

(X~,O))

=

-B(Ad(g)(tz + Y0 ), -B(tz + Y0 ,

[X_,

[Ad(g)(X_),

Ad(g)(X~)])

X~])

From these formulas it is clear that 4> 0 extends holomorphically across t=O in z 0 , and agrees with the canonical 2-form -1 on n- 1 (0) = T*(M). One constructs 4> 00 on Z00 -TI (oo), and extends across Z00 ,

in a completely similar fashion.

Under the patching

map f, we have f*(j> 00 so that 4> 0 , 4> 00 patch to give a section of ~ 2 ®n*(~2 ) Z/Pl

60 globally.

The reality condition (vi) of §1 follows directly

from the fact B(8(X),9(Y)) = B(X,Y), X,Y € Next we calculate the normal bundles of the real curves

e u

(u, tz), u

z0 ),

(represented in

tivity condition in (vi), §1. assume u=e.

and to verify the posi-

Without loss of generality,

By a theorem of Grothendieck, the normal bundle

to our section, call it C, splits as a sum of line bundles:

~

n* H@ii = N.

i=l By the non-degeneracy of Nc

~

n*(H*) =

N~

~

constructed above,

® n*(H).

Since the integers d. are uniquely determined, we see that 1

each di is 0, 1 or 2, and the number of O's is equal to the number of 2's.

To show that all di=l, it suffices to show

that for every vector v in the fiber of N at t=O, there exists a global section V of N over C such that V(O)=v, and V(oo)=O. (This will guarantee that no di=O, and hence no di=2.) X

€,,

let VX denote the vector field on Z, the derivative of

the action of exp(sX) € G on Nc.

For

z.

VX along C gives a section of

If X=X_ €1,_, VX_(O) = (X_,O) €11._

x~+

= T(T*(M)) at

(e,O) = N at (e,O). On the other hand, if X=X+ €~+' c 1 VX (0) = (X+,O) = 0 inN at (e,O). Thus, -t VX defines a + c + holomorphic section of Nc' which vanishes at t=oo, and whose value at t=O can be computed at lim t+O - lim t+O

1 d

t ds(exp(s X+), t z)

Is=O

1 d

t ds(e, Ad(s X+) (t z)) ls=O

(O,[X+' z]) Since adz is invertible on~+' this is an arbitrary vector in 0 Xfl_ c. Nc at (e,O).

Finally, for X_ € '1.-C~-' VX- is 0

61

.

in N at t=co, since in zoo' G X fl_ is divided by p c The preceeding argument shows that the (nowhere vanishing)

® TI* (H*)

holomorphic sections of Nc

over u 0 , equivalent to

1)

t1 VX

2)

are spanned by 1

~

vx_ over 0

over u 0 , equivalent to VX

+

00 ,

over U00 ,

+

The action of a on global sections is given by

for X+ int\.+. 1

tve(x_l'

over u 0 • Thus, ~ 0 (vx , a(VX )) is a constant, which we evaluate pointwise over u 0 as

- t1 B(t z, [X_, 8(X_)]) B(X_, 8([z,x_])) > 0, if X Similarly,

'/ 0.

~ 0 (~ VX ,a(~ VX )) +

+

-B(X+' 8([z,X+])) > 0, if X+'/ 0. Finally, ~ 0 (~ VX ,a(VX ))=0.

Since~ is everywhere non-

+

degenerate, this suffices to verify the positivity of ~(v,a(v))

· §3.

everywhere, concluding the proof of the main theorem.

Global Calculations The metrics constructed in the preceeding

algebraic in nature.

§

are rather

Thus, although the existence proof is

local, one expects these metrics to have global extensions to T*(M), to be complete there, perhaps asymptotically locally

62 flat at infinity, and unique subject to these conditions.

In

this § we work out the global extension in the case where M is a compact hermitian symmetric space.

Without loss of general-

ity, we assume M is irreducible, and G is Sl(n,¢), Sp(n,¢) or SO(n,¢).

1)

We fix the notation briefly.

G = Sl(n,¢).

P+ is given by block upper triangular

matrices p

(: I :}

q

p+q=n

z 0 is of the form~ :pb

:q ') , a,

b real, where

pa + qb = 0 and a-b>O. 2)

G = Sp(n,¢) preserves the skew-form represented by

P+ are as in 1), if p=q, restricted to the subgroup Sp (n, ¢l . 3)

G = SO(n,¢) preserves the symmetric form represen ted by

Q

G

I

n-2

0

p+ is of the form

0

a I 0

n }1

0

n-2

}n-2

}1

-:')

I

where a' denotes transpose.

a

e

zo

¢n,

is of the form

63

0

' ->..)

, A.>O.

For all three cases 8(A)=-A'.

For symmetric spaces

-1\.+ is abelian. We will describe the real curves as in §2 above, but take advantage of the following ansatz, which is immediately valid only in the case of symmetric M: given Y0 € rL +, we seek tC C=C(Y 0 ) e ~- so that (e , tz 0 + Y0 ) mod P+' or equivalently

describes a real section of Z passing through (e,Y 0 ) at t=O. We, of course, want C to vary real analytically with Y0 , etc., all of which will be clear by the construction. Note that (adc) 3 =0 in our cases, so that Ad (etc) e

(tz 0 + Y0 )

t adC

(tz 0 + Yol

= tz 0 + Yo + t[C,Y 0 ] + t

2

t2 [c,z 0 J + - (ad C)2
which is automatically holomorphic at

For such a section

to be real, one needs

and

In our specific cases, this means [C,Y 0 ] should be hermitian and ( 3 .1)

For Y0 close to zero, this last equation clearly has a unique solution, C(Y 0 ), such that C(O)=O. This verifies the

64

hermitian condition as well, and is the solution described in the previous §.

It thus suffices to continue the solution to

3.1 to all Y0 en+. We'll treat the cases of Sl(n,~), Sp(n,¢), which are formally identical, and describe the corresponding result for SO(n,¢).

0,

We'll seek a solution to 3.1 of the form C=DY

c

0 are

and Y

where

viewed as q x p matrices, and where D is a q x q

0

hermitian matrix, a spectral function f(Y Y0 ) of the nonnegative hermitian matrix Y Y0 • (3.1) becomes

0

(a-b)DY' - DY' Y D Y' = -Y' 0 0 0 0 0 ((a-b)D-

o2 y•y )Y' 0 0 0

0

The solution is given by D=f(Y Y0 ), where f of the real variable

~

is

4~

/(a-b) 2 +

(a-b) -

2~

which is real analytic f(0)=-1/(a-b) and

for~>- i(a-b) 2 , and for which

1 -f(O) < f(~) < 0, for ~>0, and so -b a-

D=f(Y Y0 ) 2 o. We use the notation of the proof of the local existence

0

theorem in §2.

We first note that along any of the real sec-

tions found above, the vector fields VX , resp. VX , as normal + vectors along the section, have zeroes at t=O, t=oo, respectively. bundle.

1

Hence, t VX

is a holomorphic section of the normal + So, the bundle is trivial with sections as in

Elc

§2,

over u 0 , 1 t~

over uoo

1)

vx

2)

1 t~ + over UO' vx + over u 1

Over u 0 , o(VX ) the value of

f

1

00

-v 8 (X )" We calculate + at t=O, taking the limit as t+O along our

t ve(x_) ,o(t vx+l VX +

65

curve, represented by (e 1

lim t VX t+O +

lim~

t+O

tc 0

,tz 0 + Y0 ) mod P+.

-tc (Ad(e

Thus,

0 )x+' 0) mod 'P+

2

-t [ad(C 0 ) (X+) ,z 0 J) mod (0, [X+,z 0 ] + ad(Y 0 )ad(C 0 ) (X+)) mod f+·

Setting wx =(b-a)X+ + ad([Y 0 ,c 0 ]) (X+), we conclude that our

+

metric, at (e,Y 0 )mod P+ in T*(M), is given by

f't-

66

B(X_,WS(X_)). -(a-b)B(X_,S(X_))

+ B(X_,ad([Y 0 ,c 0 J) (8(X_))). From the properties above, i t is easy to see that T*(M) is complete in this metric.

Presumably, the sectional curvative

decays to zero at infinity, but I have not verified this. The same calculations are valid for the quadrics, i.e., G=SO(n,¢).

Here, for the real curve (e

tc

0, tz 0 + Y0 ) through

(e,Y 0 ) with

-;.)

a 0 0

one solves for

where

(Recall that the eigen-values of

]J

2la12 ad z 0 are 0, 2A>0 and -2A). 'i'he reader can check that the same arguments as above treat, less explicitly, the two exceptional compact hermitian symmetric spaces.

One uses a matrix representation of G in

which z 0 is real and diagonal, the root ordering in ~is compatible with the usual one in s.€(n, ¢l , and 8 (A) =-A'. §4.

Final Remarks The general result of §2 should be globalized as in §3.

Pursuing the present method, this might mean a more

67

sophisticated ansatz than that used in §3.

A twistor approach

would almost necessarily involve higher order algebraic equations to solve for real sections of n. If these metrics are globalized in the case P+ is a Borel subgroup, one should try to find embedded twister subspaces representing the Ricci flat Einstein metrics on the resolutions of 2-dimensional rational double points. REFERENCES [1]

Atiyah, M., N. Hitchin, I. Singer, Self-duality in fourdimensional Riemannian geometry, Proc. R. Soc. Lond. A 362, 425-461 (1978).

[2]

Calabi, E., Metriques kahleriennes ~t,f~bres holomorphes, Ann. scient. Ec. Norm, Sup., 4 sen.e, t. 12, 269-294 (1979).

[3]

Hitchin, N., Polygons and gravitons, Math. Proc. Comb. Phil. Soc. 85, 465-476 (1979).

[4]

Penrose, R., Non-linear gravitons and curved twister theory, Gen. Rel. Grav. 7, 31-52 (1976).

[5]

Ro~ek, M., Supersymmetry and non-linear a-models, to

appear, proceedings of the conference "Supersymmetry in Physics," Los Alamos, NM, Dec. 1983. [6]

Salamon, s., Quaternionic kahler manifolds, Inv. Math. 67, 143-171 (1982).

[7]

Burns, D., Some background and examples in deformation theory, in D. Lerner, P. Sommers (eds.), Complex Manifold Techniques in Theoretical Physics, London 197 •

Complete Kahler Domains. A Survey of Some Recent Results Klas Diederich

1.

Introduction

One of the rrajor aspects of canplex analysis consists in the investigation of the implications betvreen geometric properties of complex analytic rranifolds (or complex spaces) and the nature of certain complex analytic objects on them. Again, an important part of this program is the farrous Levi problem, a general somewhat imprecise version of which can be fonnulated in the following way: Generalized Levi problem:

Characterize Stein rranifolds or, rrore generally,

rranifolds satisfying certain cohorrology vanishing or finiteness theorems by gec:metric conditions.

(Here the word "geometric" is meant in a very

broad sense. ) The following, nowadays classical solution of this problem in the Stein case is

vrell-known:

'Iheoren 1

(Docquier -Grauert 1960 [} ij)

A complex rranifold M is Stein

if and only if there is a strictly plurisubharmonic

Coo function

~

on M

such that Me

for all

= { x EM

c E 1R (i.e.

~

~

(x) < c } cc M

is exhaustive fran arove).

The questions dealt with in this survey article immediately arise from

the following observation: let

~

on M be as in the theorem. We rray suppose that

Then the fonn (JJ

= i

aa ~

2

~ -- 0

on M .

70 induces a camplete Kahler metric on M • COnsequently, we have

Proposition.

lmy Stein manifold

M carries a complete Kahler metric.

This fact was observed for the first time by H. Grauert in his thesis, published in 1956

[HJ.

At the same time, H. Grauert asked whether the

converse also holds: Question.

Let M be a camplete non-carpact Kahler manifold. Is M

necessarily Stein? There are several reasons why the answer to this question is, in general, negative. In order to eliminate some of them from the beginning we will specialize our situation somewhat nore by considering at first

only the Question. Suppose

Let M be a Stein manifold and D is complete Kahler. Is

D c M an open subdornain.

D necessarily Stein?

The answer to this question is, in general, still "no". The following class of counterexamples is, essentially, already contained in Theorem 2.

Let M be a Stein manifold and

analytic subvariety. Then

codmu:

A~

2

A c :.1

0 '[! :

a closed canplex

D : = M' A is camplete Kahler.

If

D is, of course, not Stein.)

A small variation of the proof of H. Grauert [2] goes as follows. Put fort>

0 t

p (t)

J

1:

( 1 ) \ ~n"

2

( tn ] tn(,/e)

0

1)'

d'

and

t a (t)

J 0

Then the function

1:

p(1:) d1:

for

1 > t > 0

( 1 )

71

t

(z) : = cr (z

z)

is subhannonic and continuous on the unit disc hamOIJ.ic and

Coo

the Kahler IlEtric details see

[s]).

on

ds 2 = d d ct on After changing

subharnonic function

a:• . f

m

6* : = 6' {0}

1Jr

on

0::

( 2 )

6 c 0:: , strictly sub-

and a sinple calculation shCMS that is canplete at

f..*

t

slightly near

which is

Coo

a6

0

(Fbr rrore it extends to a

and strictly subharrronic on

Finally, in the situation of thrn. 2 we choose functions

E ll>(M)

A = {x E M : f 1 (x) = · • • d cr2 on M and r;ut on D

with

:Kahler !1Etric

= f m(x)

f 1 , ... ,

= 0} , a conplete

This metric has the desired properties. Remark.

Notice that

1Jr

is continuous across

A .

'lheorem 2 shows that the property of being canplete Kahler does not inply holanorphic convexity . .Additional assunptions have to be made. They can go into two different directions: A)

Curvature assunptions on the canplete Kahler IlEtric.

B)

Regularity assunptions on the boundary of

D .

we will discuss sane recent results concerning A) . Section 3 then deals with B) . In section 4 we will study the weak 1-canpleteness

In section 2

of certain canplete Kahler danains

2.

D cc M where M is not Stein.

CUrvature conditions

Let M be a canplex manifold and

D cc M an (open) danain. We can

ask the following questions: a)

Which curvature conditions on a canplete Kahler metric

guarantee or even characterize that

D is locally Stein in M

ds 2 (and,

on D

72 therefore, globally Stein if M is Stein)? b)

Let M be non-compact. Which curvature conditions on a corrplete ds 2

Kahler metric

on M guarantee or even characterize that M is Stein~

Let us first consider the question a). It has been clarified very much

[3]

by the work of Cheng and Yau

0![1

In

Theorem 3.

and Mok and Yau

it is shown: Let

D cc M be a (non-ccmpact) danain on a ccrnplex manifold M.

D such that

Suppose there is a complete hermitian metric on - C then

0 ![! .

~

Ricci curvature

~

0 ,

is locally Stein (and, hence, Stein, if M

D

The same statement also holds, if complex manifold M, such that

rr : D

-+

is Stein or M

=

lPn).

M is a Riemann dorrain over a

rr(D) cc M and there is a metric on

D

as above. Remark.

Since a Stein Dorrain

D always can be .imbedded into some

~ ,

it always carries a corrplete Kahler metric with non-positive Ricci curvature (see f. i. Kobayashi, Nanizu

0 ?J ,

prop. 9. 4) .

In generalizing previous work of Griffiths also was proved in

0 ![!

0~

and Shiffman [?~ it

that local Steinness follows fran a

condition on

the holomorphic sectional curvature: Theorem 4. M with on

Let

rr : D

M be a Riemann dorrain over a complex manifold

-+

rr(D) cc M . Suppose, there exists a ccrnplete Kahler metric

ds 2

D which satisfies the following condition : There is a function

A : lR

-+

lR

with

sectional curvature

J-~

A(t) = 0

K of K( T)

for all

ds 2 ~

A (d(q,'1a))

q E D , T E T~ 0 M , where

distance with respect to locally Stein.

such that one has for the holarorphic

ds 2

d ( · , '1al

denotes the geodesic

from a fixed point

'1a

E D . Then

D is

73 Remark.

Again, since a Stein domain

D can be imbedded, it always carries

a complete Kahler metric with non-positive holamorphic sectional curvature. The proofs of both theorems 3 and Kontinuitatssatz of Hartogs holds on

4 are given by showing that the D as a consequence of a Schwarz

lemna for volume fonns (in the case of the Ricci oondition) due to Yau. The existence statement in the remark after theorem 3 can considerably by sharpened for Riemann domains over Stein manifolds. Cheng, Mok and Yau

proved in Theorem 5. such that

[3] Let

and Q~ together:

11

D

--+

M be a Riemann danain over a Stein manifold :vi

rr(D) cc M . Then

D carries a complete Kahler-Einstein metric.

(It automatically has negative Ricci curvature.) Fbr the highly conplicated proof of this theorem, which starts with the solution of the Monge-Ampere equation det (

a2 u

l

with boundary values

~

az. · az.

=

e(n+1)u

J

on strictly pseudoconvex danains in

af1 ,

the

reader is refered to the original articles. VE now

c<J.Ire

to the discussion of question

b) from above. We only can

nention a few results in this direction. Elencwaig considered in situation of a pseudoconvex domain Theorem 6.

If

D.:?] the

D in a Kahler manifold M and showed:

D cc M is a pseudoconvex domain in a Kahler manifold M

with positive holornorphic bisectional curvature, then

D is holomorphically

convex. ~·

Of course,

R.E. Greene and

M need not be holornorphically convex. H.H. Wu [}5~ gave already in 1974 the following list

of different properties which imply the property of being Stein:

74 Theorem 7.

let M be a corcplete Kahler manifold. Then M is Stein, if

it satisfies one of the following properties: is sinply connected and its sectional curvature is

1)

l\1

2)

M is noncc.upact and its sectional curvature is

;;;; 0

~

0 ;

and strictly

positive outside a cc.upact set; 3)

M is noncorcpact, its sectional curvature is

,;; 0

and its holonorphic

bisectia1al curvature is strictly positive. 4)

M is noncc.upact, its Ricci curvature is strictly positive, its sectional curvature is

~

0

and its canonical bundle is trivial.

In all these cases the hypotheses are essentially made such that it is

possible to construct strictly plurisubharrronic exhaustion functions on M using the geodesic distance with respect to the given Kahler metric.

3.

ve

The boundary of a::mplete Kahler domains

now care back to the approach of the thesis of H. Grauert

considering case

[1-[]

by

B) of the introduction. More precisely:

let M be an open carplex manifold and D cc M a carpleteKahler danain. We want

to investigate the nature of the obstructions in

aD

that might prevent

D from being locally Stein. H. Grauert proved in [}-[] in this direction: Theorem 8.

If

D as above has real-analytic srrooth boundary, then

D is

locally Stein. This shows that the obstruction can, indeed, be ruled out by regularity assurrptions on

aD • In

fact, the nature of the counterexanq:>les of

theorem 2 might be considered as a I!Otivation for the question whether all obstructions are located on the so-called "thin carplement" of

D ,

namely the set A

( 1 )

75 This is indeed the case. one has Theorem 9.

D cc M be complete Kahler and suppose that

[4] Let

locally Stein near all points

x E A defined as in ( 1 ) • Then

D is is

D

locally Stein everywhere. Remark. D cc M

As a consequence one obtains that 0 any complete Kahler danain

which is topologically fat, i.e.

i5

= D , is everywhere locally

Stein. This result was first proved by T. Ohsawa

aD

assl.liTption that

is

the same line as T. Ohsawa as the manifolds is the main tool. In

[?'[! (

of H. Skoda fran Proposition.

Let

D cc

[?fl

under the additional

c 1-srrooth. The proof of [4] follows in so- far L 2 -theor:y of

[4]

a

on canplete Kahler

this is combined with the techniques

8~ might also be used) in order to prove:

afl

Then there are holorrorphic functions

n l:

o

hJ. (z) (zJ. -zJ.)

j =1

h 1 , ••• , hn

=1

on

on

D

f i5

z0

be a complete Kahler danain and

such that

D.

Theorem 9 is then a simple consequence of this proposition. The next question which now, obviously, has to be asked is: Question:

What is the nature of thin canplerrents

Kahler domains

2. A =- D 'D

of complete

D cc M ?

In particular, because of theorem 2 one might ask: Are all thin complements

A as al:ove complex analytic? I t turns out that this is

indeed the case under certain additional regularity assumptions on

At first, T. Ohsawa proved 1980 in Theorem 10.

Let

and suppose that

A

[?:[]:

A be a thin complement of a complete K4}ler domain A is a

codinension 2. Then

c

1 -smooth

real submanifold of

i5

of real

A is a complex-analytic hypersurface in

0

i5 •

Ohsawa's method of proof, which, again, makes essential use of the L2 -theory of

a

for solving

a-closed (n, 1) -fonns on complete Kahler

D

76 domains, does not seem to be generalisible to the cases of higher (real) codirnension of

A . Therefore, new methods had to be used in order to show:

[s]

Theorem 11 .

let

D and suppose that codimension

0

A is a real-analytic subvariety of

3 . Then

~

1)

Remarks.

A be a thin corrplement of a canplete Kahler danain

i5

of (real)

A is camplex-analytic.

Notioe, that srroothness of

A is not assumed. This does,

however, not cause essential new difficulties, since it is not difficult to prove (see [5]) that real-analytic subvarieties which are complexanalytic at all regular points, are necessarily corrplex-analytic everywhere. 2)

Theorem 11 says, in particular, that real-analytic subvarieties of

odd codimension

1

>

never occur as thin canplements of complete Kahler

domains.

Because of remark 1 one may assume in the proof of theorem 11 that is srrooth. Then the idea of the proof is as follows: complex-analytic at a point

If

A

A is not

z 0 E A , a small neighborhood

U can be

chosen and a closed 2-dimensional canplex submanifold X c U such that y : =X

nA

is a real-analytic curve. Then the restriction of the given ds 2

Kahler metric

to X ' y

in such a way that there is a plurisubharmonic function is

C~

on

!;,

fact that

n

t;,r

dcr 2 • Furthenrore,

y • The corrplexification of

regular disc t;,r \ y

and induces

X, y

corrplete along

y

da2

~

y

on X which

can be kept

now contains a complex

!;, n X = !;,r n y • let 1Jr : = r is at infinite distance with respect to

c X with

r

dcr2

is changed into a new Kahler metric

I

!;,

•

r d de 1Jr

Then the on

gives an :immediate contradiction to the other fact, that the

Laplacian of

1Jr

on

t;,r

is locally integrable.

This proof is very simple. But canpared to Ohsawa' s theorem 11 the result also seems to be much weaker because of the assumption that

A is

Cw . One has to ask whether Ohsawa's result also holds in higher codirnension for real

c1

subrnanifolds. Surprisingly enough this is not

the case. Namely, one has: Theorem 12. [s] Fbr each integer k ,;;; 3 c~-submanifold A of pure codimension k

there is a closed real in a ball B which is nowhere

77 complex-analytic and such that

B' A is, nevertheless, complete Kahler.

The construction of such manifolds

d;)

technical, since they may not be

Ac B

is naturally rather

anywhere because of theorem 11 . Let

us indicate sorre basic ideas of it for the case of a complex Here

Coo-submanifold

A c B c ~3

A is constructed as a graph over the

coo-function

The function

polynomials

2-dirrensional non-

(i.e. the case of real codimension 4. (x 1 ,x 2 )-plane of a

G:- valued

f :

f

is, of course, constructed by an approximation by canplex

Fk on

~ 2 • But in order to avoid that

A has a kind of

"camplexification"

which 111'0\lld, indeed, prevent make sure that the sequence

but that

B' A Fk I

from being complete Kahler, one has to

JR2

ronverges in the

Coo- sense near

0 ,

diverges for all (z 1 , z 2 ) E C2 ' lR 2 near 0 . The canplete Kahler rretric ds2 on B ' A is obtained by constructing a Fk (z 1 , z 2 )

continuous plurisubharrronic function that

ds 2 :

= d de
han;_onic function ~' ~

is canplete on

';ji' of ( 2 )

on

B , Coo

on

B' A and such

B' A . Fbr this the continuous sub-

and the canplete Kahler metric

ds~ on

constructed from it in the proof of theorem 2 is used

~ = {(z 1 ,z 2 ,Fk(z 1 ,z 2 ))

: (z 1 ,z 2 ) E C2 } • In ~act, suitable srroothing of the corresponding potential functions across ~ multiplied with small

constants can essentially be added up. The difference between the situation of then complerrents

A of real

eodirrension 2 (Ohsawa's theorem 10) and the case of real oodimensions

t2

(theorem 11 and 12) , which just becarre clear, is nevertheless quite unsatisfying. It might be considered as a contradiction to "the beauty of

the mathematical univers", unless it can be better understood. But this can be achieved if one uses the following notion:

Definition. '!hen

Let

A be a real

c 1-submanifold of an open subset u c ~ .

A is called (linearly) generating at a point

p E A if C ® T A = ~ p

78 T A is the real tangent space to

(here

A at

p

Remark.

p ).

Notice that the property of being linearly generating is an open

condition on

A .

Since any real sul:Inanifold A c U , for which T A is carplex linear p

for all

p E A , is complex-analytic, one has the following simple

observation:

Lemma.

c1-sul:Inanifold

A real

A c U of (real) codimension 2 is complex-

analytic if it is nowhere linearly generating. With it the theorem 10 of that any thin complement

~hsawa

can now be refonnulated as saying,

D which

Ac

is a real

c 1-submanifold

of

codimension 2 is nowhere linearly gE:!llerating. Although this fonnulation is not as nice as the original one, it is in a certain sense the right one, because we can now say that Ohsawa 1 s theorem mlds in all real codimensions .:; 2

despite of the "counterexamples" of theorem 12. Namely, one has: 0

Theorem 13. domain

[9]

D . If A

Let A c D be a thin canplement of a complete Kahler is a

C~ real sul:Inanifold of codimension

~ 2 , it is

nowhere linearly generating. Remark. The "counterexall1?les" constructed for the proof of theorem 12 are also c~ Again, we will not go into the technical details of the proof; but we want to indicate at least its main ideas. Namely, we assmne that

A is

sanewhere (and we may, therefore, assume everywhere) linearly generating and that the codimension of

A is

3 . It then follows from results

~

of Harvey ( [] ~) and El-Mir ( [] :[]) (see also Sibony ~~) that the Kahler fonn

w

OOod

~

~ c

ds 2

of a complete Kahler metric 0

positive ( 1 , 1) -current to o

D

on

~

= w

cannot be

precisely, one has: There is a

D continues as a closed

Therefore, any point

with a plurisubhar.monic function

I ~ 'A is c~ and iaa~

this function

D•

on

~

on

~

such that

~' A . - In a next step one shows that

on a large part of c E lR

the m-dimensional Hausdorff measure

p E A has a neighbor-

such that for

A one has m

A1 = A n m :

=

Q

•

More

d~ A

and

79 (jl(Z) > C } > 0 .

Am ({z E A'

This follows by foliating

A'

locally by arcs of fixed length on the

[}fl contained in

bOundaries of holOIIOrphic Sadullaev discs

5"1 '

(C.. up to the boundary) and using the ~rean value property for discs. -

The real tangent space

T A to

A at

p

p

A' on these

has the following

structure after a suitable affine coordinate change: T A= lRk p

(because

X

(iO)k

X

afl-k

A is linearly generating). Therefore, one can construct a nice

SITOOth real (2n-2) -dilrensional family of holOIIOrphic discs which intersect A'

transversally at their centers and are pairwise disjoint on sectors of

fixed size. We then can apply the following unifonn estimate for subhanronic

A to the restriction of

functions on.the unit disc

the family for which the value of

at the center is

to those discs of > c

with the

c

from the arove lemma:

Isma.

Fbr any

e > 0

subhannonic function

there exists an 1jr < 0

r0

on the unit disc

,

0 < r0 < A

with

such that any \jr(O) > - 1 satisfies:

The set 1jr (r e

P : = {8 E @,2rr)

has linear measure By this

curves

y

~rethod

~

ie

) < - 1000

for sone

0 < r

~

r0

}

e •

we have ensured that there are many real disjoint

running in

A'

5"1 '

towards

A'

on which

-

1CXXJ

Ic I

every-

where. In fact, the union

W of these curves has positive ~reasure. The

plurisubhannonic function

A

open neighborhood of

W

in

5"1 '

=- sup { -2CXXJ Ic I ,
agrees with

on an

A' . Since it is bounded one has according

to Bedford and Taylor [1]

( 2 )

~

the other hand, by using the fact, that all the above curves

y

have

80 infinite length with respect to the Kahler metric induced by

/1

on

W

one obtains

v'Iil(Z}

f

( 3 )

k 1 d vz dist - (z,A)

w where at

m(z)

is the maximal eigenvalue of the Levi form

C > 0

the curves that on

/

a2a \ n

-a a- ) 2i

2j

By putting together ( 2),

following contradiction

JYmfZf

dv

dk-1 (z)

;:;;

remains bounded and plurisubharrconic,

(I

1

~

~

( ) _m_,_z~=--=-----.---

c

( 3) and ( 4), one, finally, arrives at the

J

z

dv z

VriiTZT' d (k-1) (1-e)/2 (z)

d(k-1) (1+e)/2(z)

w

l (k-~,(1+<) l d

m(')

wd(k-1)(1-t::)(z)

c

c-1

I

( 0 -1

r

J w

( 4 )

dist(k- 1 ) ( 1-e) (z,A)

(d (z) : = dist (z ,A)) :

dv z

d

)

( ddc a)n

1/2

(,)

dv

MA(a) (z) dv z

w

;:;;

by

still have infinite length, but one can shov.r in addition

y

\

=

0

of

W

MA(a) : = det :

<

a

very large. Then

(jJ

0

z E W . In a final step, one essentially replaces

with

L11 (z)

)

1/2

)

1/2

d (k-1),(1+,) (')

dv

J d(k Q'

A

<

1)z(1+e) (z) ,

~

81

where the last integral is finite if

E <

1

, because codim A = k .

k-1

D

The last theorem obviously can be reformulated in the following stronger fo:rm: Theorem 14. D . If

£ A c D be a thin complement of a ccrnplete Kahler danain

Let

A is a

Coo real submanifold of codimension

then there is no neighborhood M c u , p E M , such that

U = U(p)

~ 2

and

p EA ,

with a complex submanifold

A n M c u is a real submanifold,

A n M

c

t- M ,

but lt®T(MnA) p

Remark.

=TM p

This refonnulation now also contains the staterrEnt of theorem 11

for real-analytic thin ccrnplements. Namely, in this case we nay take near any regular point

p E A as M the ccrnplexification of

not be complex-analytic near

A . If

A -v.ould

p , we V<.Quld, indeed, have on a neighborhood

U = U(p) An U~M

and

lt®T A =T M. p

p

The examples constructed for the proof of theorem 12 show that the structure of thin complements can be very complicated even if they are Coo real submanifolds. Therefore, it is important to know more examples

and to study their properties. A whole class of examples arises in the

following way: Proposition. ([6])

Let

A c S2

of the pseudoconvex danain plurisubharrnonic function

A

Then the danain

·~· we
I

D

is

={

z

be a closed complete pluripolar subset

S2 c a;fl , such that there is a continuous

<' S2 :

= - oo

with

} •

D : = S2 ' A is complete Kahler.

may assume that Coo ( ~f[)

.

is in fact strictly plurisubharmonic and

Furthenrore, we choose a complete Kahler-metric

82 d a 2 on

12 . Then

is canplete Kahler on

D .

o

Furthenrore, one has the following Theorem 15. ( [9]) Any real closed

coo

CR-manifold

A c 12 c

afl ,

which

is complete pluripolar, is maximally foliated b¥ camplex submanifolds. Proof.

This follows again from the existence of certain holarorphic discs.

Namely, if of

A is not maximally foliated, then the vector-valued Levifo:rm

=0

A is not

Polking [2] a

. Therefore, there exists according to Boggess and

Coo -map

h :

li

-+

12 , hI I!.

holarorphic, with

h(al!.) c A

h(!!.) c 12 'A . This gives a contradiction if one considers for a

and

plurisubhamonic function property for

h

on

li

on 12

with

A = {

oo }

the sul::neanvalue

o

Because of the above proposition the last theorem might be considered as an indication of an additional property of thin camplements are

Coo

real

A which

CR-sul:manifolds. At the same tine one might ask whether all

thin oamplements are

automatically complete pluripolar.

Unfortunately, the answer to both questions is "no", because one has: Theorem 16.

[9]

There exists a real closed

of the unit ball in

afl ,

n ~ 5 , such that

coo

CR-subnanifold

Ac B

A is not maximally foliated

py oamplex sul::manifolds, but, nevertheless, B' A admits a canplete Kahler metric. In particular,

A is not complete pluripolar in

B .

The construction of these examples is a variation of the construction in the proof of theorem 12. In order to avoid the existence of the foliation, A is constructed as a CR-graph over a part of the unit s 2n-? c afl- 3 c afl . The manifold A then has real dimension

sphere

and the rank of its holarorphic tangent bundle is

[9].

2n-7

n- 4. Ebr details see

83 ~·

Whether it is possible, to give a canplete characterization of

arbitrary thin canplerrents (without additional regularity assumptions) by simple geometric properties, is an open question.

3.

Pseudooonvexi ty and

weak 1-convexity

let M be a carplex manifold and cLboundary. I f

aD

D cc M a danain with SIIDOth

is strictly pseudoconvex, then

D

is (strictly)

c"" function

1-oonvex (as is well-known), i.e., there is a

D

-+

lR

which is exhaustive from above and strictly plurisubhann::1nic outside a oompact subset

K c D . Furthenrore, as is also well-known ( [1]),

strictly 1-convex manifolds are holcm::>rphically oonvex and, therefore, proper rrodifications of Stein spaces. It is, therefore, a natural question to ask, what can be said about danains D cc M with SIIDOth

c2 -boundaries

which are only known to be

weakly pseudooonvex. Of oourse, they are always locally Stein and, therefore, even Stein, if M is Stein. But it is well-known fran exarrq:iles of H. Grauert ~q} that without this hypothesis

D need not be

1-convex. One, therefore, has to ask, whether such

strictly

D are at least weakly

1-convex in the sense of the following Definition. if there is a

A non-canpact canplex manifold M is called weakly 1-convex,

c"" plurisubharrronic function

M

-+

lR which is

exhaustive from above. Such weakly 1-corrplete canplex manifolds r-1

still have interesting

analytic properties as can, for instance, be seen fran the following

theorem. · 'lbeorem 17.

(Nakano

0 !D)

let M be a weakly 1-convex manifold,

dim M = n , and (E,h) a hennitian vector bundle over M which is Nakanopositive. Then one has Hq (M, fJ.n (E) ) = 0

V q ~ 1 •

84 With respect to the last question it must, tmforttmately, be said that pseudoconvexity of a domain

D cc M is, tmforttmately, in general not a

sufficient condition for its weak 1-convexity, even not, if

dJJ- s:rrooth.

[7]

'l'heorern 18.

H given by H = ( C2 '

{0})

I

(z ~ 2 z)

,

D cannot be exhausted by an increasing sequence of relatively

compact pseudoconvex danains. In particular, Remarks.

is

There exists a locally trivial holarorphic disc bundle

D over the Hopf surface

such that

aD

Narrely, one has

1)

JPl-bundle M

D -+

-+

D is not weakly 1-convex.

H obviously extends to a locally trivial holarorphic

H and

D cc M has a

Cw-s:rrooth boundary and is pseudo-

convex. 2)

Such cotmterexamples are impossible of

pseudoconvex with aD

aD

dim M = 2 , D cc M is

Cw- s:rrooth and if on each connected ccrnponent of

there is at least one strictly pseudoconvex point [8]. In this case

D is even holorrorphically convex. In the direction of the questions dealt with in this article it has to be asked whether a domain

D

cc M with

aD

C -SITDOth (or only 00

D =

0

D)

is weakly 1-canplete if it is ccmplete Kahler (notice that it is then autanatically pseudoconvex according to theorem 9). 'l'he answer to this question is, in general, not known. It might, however, be considered as a hint in the positive direction that one has the following result: Theorem 19. [] <2]

Any locally trivial holarorphic disc btmdle

a canpact Kahler manifold M is weakly 1-convex.

D over

85 References

[1]

BEDFORD, E., TAYLOR, B.A.:

Invent. Math. 37 (1976), 1-44

Monge-Ampere equation.

[2]

BCGGESS, A., POLKIN3, J.: Duke

[3]

Holarorphic extension of

CR-functions.

J. 49 (1982), 757-784

~ath.

CHEN:;,

The Dirichlet problem for the complex

S-Y, YAU, s.-T.:

On the existence of a canplete Kahler

metric on non-compact complex manifolds and the regularity of fefferman 1 S equation. [4]

DIEDERICH, K., PFLill, P.: metrik.

[5]

DIEDERICH, K., FORNAESS, J.E.:

DIEDERICH, K., OHSAWA, T.:

A srrooth pseudoconvex domain without

Manuscripta math. 39 (1982), 119-123 A Levi problem on tv.D-dimensional

Math. Ann. 261 (1 9ti2), 255-261

DIEDERICH, K., FORNAESS, J.E.: ccmplete Kahler metrics.

0~

Thin complements of complete Kahler

Manuscripta math. 37 (1982), 121-125

corrplex manifolds. [9]

tiber Gebiete mit vollstandiger Kahler-

DIEDERICH, K., FORNAESS, J.E.: Srrooth, butnotcorrplex-analyticpluri-

pseudoconvex exhaustion. [8]

507-544

Math.Ann. 259 (1982), 331-341

polar sets. [7]

1

i"iath.Ann. 257 (1981), 191-198

DIEDERICH, K., FORNAESS, J.E.: domains.

[6]

COmn.Pure Appl.Math. 33 (1980)

DIEDERICH, K., OHSAWA, T.: compact Kahler manifolds.

On the nature of thin ccmplements of

Math. Ann. 268 (1984), 475-495 Harrronic mappings and disc bundles over Publ.RIMS, Kyoto Univ., 21 (1985),

819-833

0 fJ

r:x:x:J:)UIER, F., SRAlERI', H.:

Levisches Problem und Rungescher Satz

fiir Teilgebiete Steinscher Mannigfaltigkeiten.

94-123

Math.Ann. 140 (1960),

86 [12] ELE.'ITCWAJG, G.:

Pseudoconvexite locale dans les varietes ki:ihleriennes.

Ann.Inst.Fourier 25 (1975), 295-314 [13] EL-MIR, M.H.: Acta

Sur le prolongement des CO
math.~

[14] GRAUERI', H.:

f~s.

(1984), 1-45

Charakterisierung der Holonorphiegebiete durch die

vollstandige Kahlersche Metrik. [15] GRIFFITHS, P.A.:

Math.Ann . .111_ (1956), 3.3-75

TwO theorems on extension of holanorphic mappings.

Invent. Math. 14 (1971), 27-62

[}5
Rem:Jvable singularities for positive currents.

J\m.J.

Math. 96 (1974), 67-73 [17] KOBAYASHI, S., NCMIZU, K.:

Foundations of differential geanetry II.

New York 1969. Interscience Publishers [18] MJK, N., YAU, S.-T.:

Corrpleteness of the Kahler-Einstein metric on

bounded danains and the characterization of danains of holaTPrphy by curvature conditions. [19] NAKANO,

s.:

Proc.Syrrp,Pure Math. 39 (1983), 41-59

Vanishing theorems for weakly 1-conplete manifolds.

Number theory, algebraic geanetry and ccmnutative algebra, Kinokuniya, Tokyo ( 1973) [20]

NARASIMHAt~,

R.:

The Levi problem in the theory of functions of

several canplex variables.

Proc. Int. Congr. Math. (Stockholm 1962) ,

385-388 [21

J

OHSAWA, T. : RIMS, Kyoto

[22] OOSAWA, T.:

On canplete Kahler danains with ~'

C1-boundary.

Publ.

929-940 (1980)

Analyticity of canplements of canplete Kahler danains.

Proc. Japan Acad. 56, Ser. A, 484-487 (1980)

87

8fl

RIOffiERG, R.:

Stetige streng pswdokonvexe Funktionen.

.Math.Arm.

175 (1968), 257-286

8-!1

SAOOLLAEV, A. :

A l:x>undary uniqueness theorem in

tfl .

Math. USSR

Sb. 30 (1976), 501-514 8~

SHIFFMAN, B.: manifolds.

8~

Extension of holOIIPrphic maps into hermitian

Math.Arm.

S:rBC:NY, N. :

SKODA, H. :

(1971), 249-258

Quelques problerres de prolongement de courants en

analyse corrplexe.

~'[I

12i

D.lke .Math.J. 52 (1985), 157-197

Application des techniques

L2

a la

theorie des ideaux

d'une algebre de fonctions holamorphes avec poids. Nonm. Sup. ~![!

SKODA, H.:

~

Arm.Sci.Ecole

(1972), 545-580

MJrphismes surjectifs de fibres vectoriels semi-positifs.

Arm. Sci. Ecole Norm. Sup.

11

(1978), 577-611

Klas Diederich Bergische Universitat GHS Wuppertal Mathematik GauBstr. 20 D-5600 Wuppertal Fffi

On the Minimality of Hyperplane Sections of Gorenstein Threefolds Maria Lucia Fania and Andrew John Sommese To Wilhelm Stoll on his Sixtieth Birthday Let

X

be a normal irreducible three dimensional projec-

tive variety whose local rings are Cohen Macaulay and whose dualizing sheaf,

KX

is invertible (see §0 for more details).

We will call such a variety a Gorenstein threefold throughout this article. We say that a pair

(X,L)

with L an ample line bundle on

a Gorenstein threefold has non negative logarithmic Kodaira dimension [I] if there is some integer 0 h (x, (~ ~ Llnl > o . Assume that

n > 0

such that

(X,L)

is such a pair and that there is at This conleast one smooth element of the linear system ILl dition of course implies that X has at most isolated singularities. The logarithmic Kodaira dimension condition on (X,L) implies that all smooth elements of Kodaira dimension (see §2).

ILl have non negative

The main theorem of this paper (which generalizes [So4], [SoS J,

[So6 J) is the following.

Main Theorem.

Let (X,L) be as above.

There exists a pair

(X',L') with L' an ample line bundle on a Gorenstein threefold X

such that there is a holomorphic surjection n: X---+ X', X as X' with a finite set F c X' reg blown up, Which satisfies the following conditions: expressing

a) given a smooth n(S) is the map of S

s E ILl I TI s : s---+ S' onto its minimal model,

b) L' = [n(S)J for smooth S E ILl and there is a one to one correspondence between smooth S' E IL'-FI and smooth S E ILl gotten by sending such S' to their proper transforms in X , c)

Kx,~

L' is numerically effective.

90 ~11

the corollaries of the smooth version of this result

from [So4J, [SoSJ, [So6], and [So7J carry over with little work; we will discuss these results in another place. ~

remarked earlier, the smooth version of this result was

proved by the second author [So4], [SoSJ. classified all the pairs

(X~)

In [So6], he further

with L an ample line bundle on

a smooth threefold X and the logarithmic Kodaira dimension of (X

~)

negative.

These last results were never published be-

cause N. Shepherd-Barron showed that they were easy Gonsequences of Mori's Theory of extremal rays (cf. [Mo], [Ka 2 J); for explicit details on the use of Mori's Theory see [Be+Pa].

We needed (e.g.

[Fa]) the results of [So4], [SoSJ, and [So6] for local complete intersections with isolated singularities.

In this case Mori's

theory does not apply but the methods of [So6J combined with the results of [L+So]

(see (0.9)) work.

We use these methods to prove the theorem stated above. substantial part of this paper is identical with [So6]. a sequel we will deal with the pairs (X

~)

A

In

where the logarithmic

Kodaira dimension is negative. Let us give a detailed description of this paper. In §0 we give background material and results for which we don't know a good reference.

In §1 we recall the basic results

on the Fano-Morin 3 dimensional adjunction process.

We work in

more generality than needed since we will use the results in a sequel to classify pairs with log

(X~)

< 0.

In §2 we prove the main theorem. The authors would like to thank the Max Planck Institut fur Mathematik for its support.

The second author would like to thank

the University of Notre Dame, the National Science Foundation [NSF Grants #M<:SB0-03257 and #MCS82-00629J, and the Sloan Foundation fo~ their support. §0

NOTATION AND BACKGROUND MATERIAL

For the convenience of the reader, we will review our notation, which agrees with that of [So 3 J and [So 4 J. We suggest that the reader look over (0.1) to (0.4) and (0.7) and then go to §1 returning to this section when needed. (0.1) Given a sheaf 5 of abelian groups on a topological space X , we denote the global sections of 5 over X by r (5)' or, when some confusion can result, by r (X , 5).

91 {0.2)

All spaces and manifolds are complex analytic unless

otherwise specified; all dimensions are over abreviate complex analytic to analytic. space

~.

We often ab-

Given an analytic

X , we denote its structure sheaf by @X and its smooth

points by Xreg We do not distinguish between a holomorphic vector bundle E on a complex space, X , and its sheaf of germs of holomorphic sections.

Thus when

a coherent analytic

sheaf and a holomorphic vector bundle are being tensored together, the meaning is clear; the appropriate sheaves are being tensored over mx. Given a coherent analytic sheaf, S, on an analytic space, X, we let x{X,S), or X{S) for short denote its Euler characteri i istic. We let h {X,S), or h {S) for short, denote dim Hi{X,S). We let hO,i{X) = hi{mx>. If X is in addition a complex manifold, then hp,q{X) tangent bundle of Let

X

hq{APT;) where T; is the holomorphic coX

be a normal complex analytic space.

denote the dualizing sheaf on X • By definition if X is smooth and i*{KX ) where i: X ---+X is the reg reg inclusion, in general. A normal complex analytic space X is called Gorenstein if its local rings of holomorphic functions are Cohen Macaulay and KX is invertible. The main example for us of Gorenstein varieties are local complete intersections. Serre duality in its usual form holds for Gorenstein varieties. The following corollary {see [Sh+So]) of the Kawamata-ViehwegKodaira-Ramanujam vanishing theorem is very useful. {0.2.1) Kawamata-Viehweg-Kodaira-Ramanujam Vanishing Theorem. Let X be an n dimensional normal irreducible Gorenstein projective variety. line bundle on on X If X then

Let

X, i.e., L •

L

c

be a numerically effective

> 0 for all effective curves

has isolated singularities and

Hi{X,L-l) = Hn-i{X,KX ~ L)

!£E.

=

L · L····L > 0 L n copies -I

0

i < dim X

(0.3) Let X be an irreducible projective variety. Let D be an effective Cartier divisor on X • Denote by [DJ the holomorphic line bundle associated to D. If L is a holomorphic

92

line bundle on

X , let

ILl

Cartier divisors associated to

denote the linear system of all L •

Given a finite set F

~

X,

IL-FI denotes the set {DElLI jF ~ D}. We often let L denote either the homology class associated to a DelLI or its Poincare dual c 1 {L), e.g. given a curve C EX, L • C = D • c = c 1 {L)[CJ. In particular if dim X= 2 and D E ILl is a smooth connected curve, then the genus of D is given by

~{D • KX + D • D + 2) = ~{L · KX + L · L + 2 ). We denote the restriction of a vector bundle E on an analytic space X to an analytic subspace S , by E8 One slight exception is when E is the normal bundle NT of a submanifold of X and S c T , then E8 is denoted NT,S" tve denote the restriction of a holomorphic map n:X ~ Y where Y is a complex analytic space, by n 8 : S ~ Y. We often denote isomorphism of two vector bundles E and F on an analytic space {0.4)

X

by

E

~

F.

We often denote complex projective space by

F~

when its

exact dimension is irrelevant. A line bundle L on a projective variety, X, is very ample if r{L) spans Land the map ~:X----+ F~ associated to r(L) is an embedding. A line bundle L on a projective variety X is ample if Ln is very ample for some n > 0. Good references for the standard facts about ample line bundles are [Ha 1 , Ha 2 J. {0.5) A smooth connected projective surface S' is said to be a minimal model or a minimal surface if S' has no smooth rational curves, E , withE · E = -1. It is well known [Z 1 , cf. B+H also] that given any smooth connected projective surface S, there is a holomorphic birational map n: S --+ S' from S onto a minimal modelS'. If S is of non-negative Kodaira dimension then S' is uniquely determined by the function field of S , and the map n is uniquely determined by S For us there are two important consequences of this for a surface of non-negative Kodaira dimension, S . The first is that two distinct smooth rational curves E and F on S with E • E = -1 = F • F are disjoint. The second is that there are only a finite number of distinct smooth rational curves on S with self-intersection equal to -1. A minimal surface, s , of non-negative Kodaira dimension has the important property [cf. Mu, B+HJ that K~ is spanned by

93

global sections for some n > 0. for all effective divisors lowing important lemma. (0.5.1)

Lemma.

Let

S

C

on

This implies that K8 • C ~ 0 S . This implies the fol-

be a connected smooth projective sur-

face of nDn=negative Kodaira dimension. for some t > 0 or (Kg + L)

(Kg + L)

~

Then either

K;

IDs

L · L + 1 •

L = 0 then since some power K~ with t > 0 of Ks t Ks has a non-trivial section, Ks = IDs· Therefore it can be assumed without loss of generality that Ks L > 0 Proof.

If

.

Let n:

S

Then Kg

----+

S' be the map of

S

.

onto its minimal model.

n*Ks• + P where Pis an effective divisor satisfying

-k = P • P =Kg Ks- Kg, · K8 ,. Since S is obtained from S' by a sequence of blowups, we see that k' = c 2 (s) - c 2 (S') equals the number of reduced and irreducible components of P. By the invariance transformations k' Therefore:

of K5 , · Kg, + c 2 (S') under birational k . By the ampleness of L, L · P > k'.

(Kg+L)

(Kg+L) = (n*Ks, + P+L)

(n*Ks, + P+L)

Kg'· Kg, + P • P + L · L + 2 (n*Kg,) > -k+L · L + 2(n*Ks,)

• L + L · P + k'

L

L + 2 (n*Ks,

· L)

with equality in the last inequality only L + L • P > L · L if L • P 0 and (n*K8 , • L) = 0 The former equality implies that S S' and the latter inequality combined with this implies that lemma.

K5 • L = 0 •

This contradiction proves the

0

(0.6) The Hirzebruch Surfaces [Ha 2 , pg. 369ff; Nag]: By Fr with r ~ 0 we denote the rth Hirzebruch surface. Fr is the unique holomorphic lP ~ bundle over lP ~ with a section E satisfying E • E = -r. Let n: Fr--+ lP~ denote the bundle projection. In the case r = 0, F 0 is simply lP ~ x lP ~ • In the cases r ~ 1, E is the unique irreducible curve on Fr with negative self intersection. By Fr for r ~ 1, we denote the normal surface obtained from Fr by blowing down E. In case 2 r = 1, F 1 is lP~. A basis for the second integral homology of Fr is given by E and f, a fibre of n; of course f · f 0 and f • E = 1. The line bundles on F are given by [E]a ® [f]b r and the latter i; ample if and only if it is very ample, and it

94 a > 0

is very ample if and only if [E]a ~ [f]b and

b > ar

to F

a > 0 •

Given a line bundle L on F , the pullback

is of the form {[EJ

r

b > ar + 1 •

and

is spanned by global sections if a~d only if

r A

~

[fJ }

r

for some integer

A

we will use the following generalization of a result of Kobayashi-Ochiai [K+OJ many times.

{0.6.1}

Theorem.

Let

X

be an n dimensional connected normal

irreducible Gorenstein projective variety with isolated singularities.

Let

L

be an ample line bundle on

X

~

Assume that:

K~ ~ Lb :: (!)X

where

b < a < 0 •

Then there is an ample line bundle M on X such that Mt = K-Xl , and Mq = L wh ere b q = t a. In par t'~cu 1 ar t > q > 1 n n, {X, L} "' {JP <£, (!) n {1} } • If t JP(£ n+l and then X is biholomorphic to an irreducible quadratic in JP (£ Lis isomorphic to the restriction of(!) n+l{l} JP(£ Proof. First note that H1 {X,?l} = 0. To see this, we use a

and t < n + 1

If t = n + 1,

pretty argument that the second author has heard attributed to Kobayashi. finite

k

Assume that H1 {X,?l} ~ 0 and let X be a connected sheeted cover of X Since K~a"' Lb, it follows that

K~l and hence K~l are ample. *} Therefore X{
=

1

By the Kodaira vanishing theorem:

=

HO,i{X}

=

x{
hO,i{~}

=

0

for

i > 0

But by functoriality of Chern

classes and the Hirzebruch-Riemann Roch it follows that: 1

=

x<

=

kx<

=

k

implying that H1 {X,?l} = 0 By the above, *},and the exponential sequence, it follows that: Pic{X}Z H2 {X,?l} is torsion free. Therefore M

E

Pic{x} exists such that: Mt

= K~l

and Mq

=

L where bq

=

ta.

The rest of the theorem follows along the lines of [K+O J. {0.7)

We need a few facts from deformation theory.

D

95

The next lemma is a special case of lemma (0.7.2) of [S0 4 ].

(0.7.1)

Lemma.

Let E be a smooth rational curve contained in

the smooth points of an irreducible projective threefold, x·. If f(NE) spans NE, the normal bundle of E in X, then the union of the deformations of E in X is dense in x. (0.7.2)

Lemma.

Let D be a smooth surface contained in the

smooth po~of an irreducible projective threefold X. be a smooth rational curve on D.

Let E

Let ND be the normal bundle

of D in X. If E · E > 0 in and the degree of ND,E is ~ 0, then the union of the deformations of E in X is dense in X. Proof. Let ND/E be the normal bundle of E in D, and let NE denote the normal bundle of E in X. Note that by hypothesis NJl/E and ND,E are spanned by global sections. It, therefore, follows from: 0

-

that r(NE) spans NE. (0.7.3)

Lemma.

ND/E

---4

NE---+ ND,E ___,. 0

Use (0.7.1)

D

LetS be a smooth ample divisor on a connected

three dimensiOnal normal projective variety, X. Let E be a smooth rational curve on S satisfying E · E = -1 on s. If the union of all smooth deformations of E is not dense in X, then there exists a divisor D S transversely in E. p:

~X

with sing (D)

c

sing(X), meeting

Let D denote the normalization of D,

D----+ D.

k r, p*[S]D ~ [E] ~ [f] for k > r + 1 in the notation of (0.6), or, 2 b) U is biholomorphic to lPa:,p*[S]!I ~ (f) 2 (1); lPa: deg(ND IE) = -1 . reg ~The proof of theorem (1.2) of [So 4 J minus the last sentence of that proof with lemma (0.7.1) above replacing (0.8) of [So 4 J, shows that there exists a divisor D with the normalization D biholomorphic to F, lP; or; r in the notation of (0.6). r ... The non-normal points of D must belong to sing(X) since sing(D) in an analytic set in x-s. a)

Dis biholomorphic to F

Since K

s

• E = -1 it follows that:

96

A simple argument based on (0.6) shows that given two Cartier divisors A and B on Fr' it follows that A • B is a r. Therefore *) precludes r > 2.

D being

m~ltiple

of

biholomorphic to Fr with

Since E = D n S is smooth, rational, and an ample divisor on D, it follows from (0.6) that

a) is true if

D is

biholomor-

phic to Fr.

D is biholomorphic to

If where ND

E • E

=

reg

lP

~'

is the normal bundle of D reg

-1 on

s.

I

E) = -1, reg in X, follows from

then deg (ND

=

This implies that p*[SJD

(!)lP 2 (1)

D

The following is a simple but important notion. (0.7.4)

Definition.

Let L be a holomorphic line bundle on an

irreducible Gorenstein projective threefold, X.

A reduction

(X',L') of (X,L) is a pair consisting of a line bundle, L', on irreducible Gorenstein projective threefold, X', such that: a)

~=

X is the blowup,

X

--+

X' of X' at a finite set

of smooth points, F, b)

~*(KX' ®

L' 2 )

~

KX ® L 2

Let us make some observations. (0.7.4.1)

b)

is equivalent to: L ~ ~*L' ® [~-l(F)]-l •

(0.7.4.2)

If S is a smooth element of ILl, then ~(S) is smooth

and [~(S)] ~ L'.

Further sending smoothS'

E

IL'-FI to their

proper transforms on X, sets up a one to one correspondence between smoothS'

E

IL'-FI and smoothS

ILl.

E

(0.7.4.3) Let X and L be as in the definition of (0.7.4). such that: Let {Dl, ••. ,Dn} be a set of disjoint divisors c: X - reg 2 a) each D. is biholomorphic to ]p !J: 1 1

b)

Ln. 1

~

(!)

2(1) and ND.

lPa:

1

~

(!) 2 lPa:

(-1) where ND.

is the normal bundle of n.1 in X.

1

97 Let X' be the threefold gotten by blowing down the. Di.

Let L'

be the unique line bundle on X' such that b) of (0.7.4) holds. Then (X',L') is a reduction of (X,L). (0.7.4.4)

Let (X',L') be a reduction of (X,L).

then so is L'.

If Lis ample,

This is an almost immediate consequence of

[z 2 , theorem (6.2)]; cf. [F 1 , lemma (5.7)] for a proof of a more general fact.

(0.7.5)

Lemma.

Let X be a normal irreducible Gorenstein pro-

jective threefold.

Let L be an ample line bundle on X such

that there is a smooth S

ILl and the pair (X,L) has non-nega-

E

tive logarithmic Kodaira dimension. curve on

s

=

satisfying E · E

Let E be a smooth rational

s.

-1 on

Then the union of all

the smooth deformations of E is not dense in X and the linear

D of lemma (0.7.3) satisfies (0.7.3b). Proof.

If the union of all the deformations of E were dense in

X it would follow that (KX+S) fact that -1

=

degree(KS,E)

To see that

C

c

D

E > 0. (KX+S)

This contradicts the

· E.

a) is not possible, choose a smooth curve

with homology class E + (k-l)f in the notation of (0.7.3).

Since in homology

E · and deg(ND

E)

=

=

C+

f

-1 it follows that either:

reg,

1)

C)

degree(ND

~

0

reg, or

f) ~ 0 • reg, 1) happened then by (0.7.2) the union of the deforma2)

If

degree (ND

tion of C are dense in X and thus (KX+S) -2

= KX

· C + degree(ND

(KX+S)

reg,

· C + degree(ND

which is > -1 by 1) and (Kx+S) tion happens in case 2) •

·

c

·c

~

0

C) + C · C

But: on D

C) - C • f reg' > 0 . A similar contradic0

98

(0.8}

Theorem.

Let L be an ample line bundle on a smooth con-

nected projective surface, S.

Assume either that L is very

ample or that f(L} spans L, h (L} ~ 4, and L • L > 5. 0

Ks

€1

Then

L is spanned by global sections unless:

=

2 (lP «:'Q) 2 (e)) where e lPa: 1

a)

(S,L}

b)

S is a lP a: bundle and LF "' ID

1 or 2,

1 (l)where F is a fibre

lPa:

of the bundle. This second condition of

=

b) is equivalent to smooth C

E

ILl

being sections of the bundle. Proof.

The above is proved for L very ample by Sommese [So 3 J, and Van de Ven [VdV]. The argument in [VdV] has two parts. The first part shows that L • L ~ 5, h 0 (L) ~ 4, and the two connectedness of C E ILl imply f(K 5 €1 L) spans K5 €1 L. This part of the argument, which is based on a method of Bombieri [BJ applies with no change to the situation of the theorem. The second part of Van de Ven's argument shows that, with the exceptions listed in b), C E ILl are two connected for all very ample L satisfying L L > 5. It is a straightforward check that the second part of Van de Ven's argument works for L ample, spanned by global sections, and satisfying D

L • L > 5.

(0.8.1)

Lemma.

Let L be an ample line bundle on a smooth

connected projective surfaceS. Assume that f(K~ €1 LN) spans A N K5 €1 L where A and N are positive integers.

Let~: S ~ lPa: be the map associated to f(K~ a)

If dim

~(S)

€1

LN).

= 2, then any connected component, E,

of a positive dimensional fibre of

~

is a smooth

rational curve satisfying E · E = -1 on S and: A = N(L E) • *) In particular for large enough n, the map~·: S ~ ~· (S) associated to f(K~n €1 LNn) expresses S as a smooth surface,¢' (S), with a finite set blown up. b)

If dim ~(S) = 1, then a connected component F ~ general fibre of ¢ is a smooth rational curve and:

99 **)

2A = N (L · F)

For large enough n, the maps

<1> 1 :

s -

<1> 1

(S) associated to

(K~n ® LNn) has connected fibres and maps S onto a smooth curve,

<1> 1

(S).

If dim <j>(S) = 0, i.e. K~ ~ LN ~ @S, then either A~ N, or

c)

(S,L) is as in

a) of (0.8), or S is biholomorphic to a

smooth quadratic

in-;~

and

L~s

isomorphic to the restric-

tion of @ 3 (1).

lPa: ~·

In case

a), the assertions about care immediate con-

sequences of lemma (2.3.3) of [So 3 J; the reader can check that the proof of (2.3.3) of [So 3 J still holds under the hypotheses of (0.8.2) above. In case fibre of <j>.

The rest of part

a) is standard.

b), let F be a connected component of general Since KS,F

KF it follows that:

z

A deg(KF) + N deg(LF)

=

0 •

Since LF is ample and {A,N} are positive, this implies that -1 1 KF is ample. Therefore F z lPa: and deg KF = -2 giving**) of b).

The rest of part b) is standard.

Assume that K~ ® LN

z

@s.

If N >A, then by (0.6.1),

we are done. The following is a very slight modification of [L

+ So, (2.3) ].

(0.9)

Theorem (Lipman and Sommese).

Let V be a three dimen-

sional irreducible normal Gorenstein variety. ample line bundles in P

~

Rv , P

v.

V which is biholomorphic tow; "' @ 2 (-2). lP

Let L and H be

Assume that there is a subvariety and assume that*

Assume that there exists a map p: V -

A where:

a)

A is affine and V* is a Zariski neighborhood of P,

b)

P: V* - P

--7

A - p(P) is a biholomorphism.

Assume that there is a smoothS < ILl and H is spanned by glo~al sections in a neighborhood of s u P and Hp ~ @ 2 ( 1) • Assume that s + n E !HI where D ~ P. Then P does Wot meet the Lingular set of

v.

100

Proof.

This will follow from the proof of [L +So,

for any x

E

with isolated singularities such that x Let IH-xl denote the set of D x

P n Sing(V).

E

meet

s.

(2.3)] if

P n Sing(V) we can find an irreducible D E

E

!HI

D.

!HI that contains

E

Note that the base locus of IH-xl does not

Indeed if it did then since H is spanned in a neigh-

borhood of S, it would follow that IH-xl = IH-yi where y Since S + D

l

y but not x by D

Bertini's theorem there is

D

P this is absurd.

c

(0.10)

Lemma.

~(A))+l

~

Let

< dim A.

~(A)

Let a= (dim

~(A)

0

Let A be an effective ample divisor on a con-

nected projective manifold X.

dim

Thus D is irreducible and

V-S which implies that Sing(D) is finite.

phic map with dim Proof.

Thus by

IH-xl with D meetingS trans-

E

versely in a smooth ample curve. Sing(D)

S.

E

: X ---+- IP a: be a holomor-

Then

~(A)

=~(X).

and assume that

~(A)

~~(X).

Then

< a.and:

*)

L •

•

0 in homology

• L • A

La timesJ where L L

~*@

(1). L is spanned by global sections and thus IPII: L can be represented by an effective cycle D that meets

A in a cycle representing D · A. Further D is a non-trivial union of dim X- a > 1 dimensional analytic sets since dim

~(X)

+ 2.

~dim ~(A)

Therefore Lemma.

(0.1)

+ 1 =a

and dim X= dim A+ 1 >dim

*) contradicts the ampleness of A.

~(A)

0

Let A be an ample divisor on an irreducible pro-

jective local complete intersection X.

Assume that there is a

continuous map r: X ---+-A such that rA:A ---+-A is a homotopy equivalence. Proof.

Then dim X < 2.

Assume that dim X > 3.

Then by the first Lefschetz

theorem:

*) Since rA is a homotopy equivalence: **)

r~: H2 (A,II:)

--+

H2 (A,II:) is an isomorphism.

Combining*) and**), it follows that:

101

r

-

*

H 2 (x,~) is

an isomorphism.

Therefore a Kaehler class won X can be written r*n where 2 a+l a+l n E H (A,~). This implies that w r*(n ) = 0 where a

= dim

A.

This is absurd since w raised to the dimension of

X must be non-trivial.

0

The Adjunction Process

§1

Throughout this section L is an ample line bundle on

(1.0)

an irreducible three dimensional normal Gorenstein projective variety, X. smooth S

E

It is further assumed that there is at least one

IL I .

The adjunction process that we use in this paper is a modification of the adjunction process Morin [Ro, pg. 66] used to reprove Fano 1 s classification of threefolds with rational hyperplane sections.

This process used by Morin was based on

the Castelnuovo-Enriques adjunction process for surfaces [C+E]. The following lemma is at the heart of the process. Lemma.

(1.0.1)

Let £ be a holomorphic line bundle on a smooth,

connected, projective threefold, X. divisor on X.

Let S be a smooth ample

K~ ® [S]d ® !.

Let !(d) denote

Assume that:

(1.0.1.1)

£ is spanned by g~_~.e_~~--~!::~!:i:_ons,

(1.0.1.2)

£(d) S is spanned by g_lobal s-~~-"!:_io:t_:l_~-~C?E_ __~ < d < N.

Then there is an integer N 1 > 0 such that either:

a)

(!(N))

Nl

is spanned by global sections, or,

b)

(!(d 1 ))N 1 is spanned by global sections for some nonnegative d

and the map associated to r((C(d 1 ))N 1 )

< N,

1

has an image of less than 3 dimensions. ~·

Let d

1

be the largest integer less than or equal to N

such that there is an N 1 > 0 such that: r (( Since £ is spanned d

1

c< d

> 0.

I )

)

NI

If d

spans ( ( ( d

)

1

Therefore it can be assumed that d

= 1

)

NI

N there is nothing to prove. <

N.

that the map associated to f({C(d 1 ))N 1 ) image, or else the statement

I )

It can be assumed also has a three dimensional

b) would be true.

Note that this

102

implies by the Kodaira vanishing theorem, e.g. [(0.9.1) or Mu 1 J that: 1

~

H (X,KX ® £(d 1 ) )

*)

0 .

Consider the residue sequence for S tensored with £(d 1 ) : o

Kx ® £(d 1 )

--+

£(d 1 +ll

--+

--+

£(d 1 +ll 8

--+

o •

By assumption £(d 1 +1) 8 is spanned by global sections. By *) and the above exact sequence, it follows that:

r (( (d

**)

1

+1)) spans '(d 1 +1) s.

**) implies that the set

Since S is ample the statement

where £(d 1 +1) is not spanned is finite. Therefore by a theorem of Zariski

rz 2 ,

theorem (6.2)], there is anN"> 0 such that: (£(d 1 +1))

N"

is spanned by global sections.

This last statement contradicts my choice of d

1 •

This absurdity

0

proves the lemma. (1.0.2)

Remark.

The previous proof shows a little more:

r (£ (d 1 ) ) spans £ (d 1 ) 8 • This observation has one very important consequence. that a smooth S 1

£

assigned point x

£

(cf.

Assume

I [SJI can be chosen to pass through a pre~ssume th~t

X; e.g:

(0.7.4)) of a pair (X,L) where L

(X,[SJ) is a reduction i~

dle on a smooth, projective threefold, X. is true for all smooth S

1

£

a very ample line bunThen, if (1.0.1.2)

!CsJI, it follows that N 1 can be

chosen to be 1 in the above lemma. To utilize the above lemma, we need criteria for (1.0.1.2). The next theorem does this.

We are more general than necessary

because we will use this result in a sequel. (1.1)

Theorem.

Let L be an ample line bundle on a smooth,

connected, projective surface, s.

There exist arbitrarily

large N with the property that f(K~ ® LN) spans K~ ® LN ~ all non-negative integers n < A for some A > 0 where:

a)

A > N and KA ® LN is not ample,

s

or, b)

s is a lP 1 bundle r : s

a:

--+

c over a smooth curve C,

103

@ 1 (1) for any fibre, F, of r; in this case

lPa::

2A

N £E_,

c) Proof.

K; ~ LN is trivial for some a > 0. We aJopt the notation !(n) ~ K~ ~ LN where N is a

positive integer to be specified.

Note that if s has negative

Kodaira dimension, then h 0 (((n))

0 for all large n, [B+H;

If s is not a minimal model, then

Proposition (3.3), pg. 371].

there is a smooth rational curve E such that K8 · I = -1 and hence regardless of the Kodaira dimension of S, r(£(n)) can't span !(n) for n > N(L ·E)

•

Next assume that S has non-negative Kodaira dimension. We have the following lemma. (1.1.1)

Lemma.

Let £ be an ample line bundle on s, a smooth

connected projective surface on non-negative Kodaira dimension. Assume that ( i s spanned by global sections, h 0 £ • £ > 7 .

Then h (K8

~

£)

~

4 and (K 8 +£)

(()

~ 4, and

· (K 8 +£) > 7.

By (0.5.1), (K 8 +£) · (K 8 +£) ~ 7. By the Kodaira vanishing theorem and the Riemann-Roch theorem for K8 ~ £:

Proof.

h (Ks

~

' ' = xC@s) + i(Ks+£)

. '·

Since sis of non-negative Kodaira dimension, x(@ 8 ) Thus h 0 (K 8 ~ £) > 4.

(0.5) K8 · £ > 0.

~

0 and by

0

Choose N large enough so that LN is very ample, 2

N

o

N L • L ~ 7, and h (L) ~ 4. Theorem (0.8) and lemma (1.1.1) immediately yield that either f(!(n)) spans !(n) for all n > 0 or there is a finite smallest non-negative integer, A, such that r(!(A+l)) doesn't span !(A+l), and !(A) is not ample. By

*) of

a) of lemma (0.8.1), it follos that there is a smooth

rational curve E on

s

such that:

E · E = -1 on S and A= N(L • T:). Therefore by the first equality S is not a minimal model and 'by the second A~ N. Therefore the theorem is proved if S has non-negative Kodaira dimension. Next assume that S is rational.

Choose for N an even

104

number that is large enough so that LN is very ample.

We have

the following lemma. ( 1.1. 2)

Lemma.

Let !. be a very ample line bundle on a smooth connected surfaceS surface satisfying h 1 ' 0 (S) = 0. Assume that K 8 ® !. is ample and spanned by global sections. K 8 ® !. is very ample unless: a)

....~ = K-2 s ,

Then

. K2 l.e. s ""~ ....~ ~ or

b) Proof.

In [So 3 , §3], the second author studied the mapping associated to f(K 8 ® !.) . He showed that under the hypotheses of the lemma, K 8 ® !. is very ample except in the two cases given by (2.5.1) and (2.5.2) of [So 3 J. In the first case S is a two sheeted branched cover of A direct computation using the description in (2.5.1) of -2 shows that K 8 ~ !.. In the second case S is a two sheeted branched cover of a singular quadratic.

A direct computation using the description -3 in (2.5.2) of [so 3 J shows that K 8 ~ !.. D Choose the smallest non-negative integer A such that !(A)

is not very ample; by the first paragraph of this proof such a finite A exists. (1.1.3)

We claim that either the theorem

is true or:

!(A) is spanned by global sections.

To see this note that since !.(A-1) is very ample, it follows from theorem (0.8) that (1.1.3) can fail only if: a.)

s

~

lP

~ and !.(A-1) ~

Q)

2 Ce) for e

1 or 2,

lPa:

or, 8)

1

S is a lP C bundle r: S !.(A-l)F

~

---+-

lP

1

a: and

ID 1 (1) where F is a fibre of r. lPa:

If case a.) occurs then (S,L) is as in case c) of the conclusions of the theorem. If case S) occurs then since

105

K S,F

~

KF

~

@ l (-2) : lPC

-2(A-l) + N(L · F)

=

l

which contradicts the fact that N is even. We claim that either the theorem is true or: (1.1.4)

((A) is ample.

To see this assume that ((A) is not ample. Let~: S -+lP~ be the map associated to r (((A)). Since according to (1.1.3),

r (((A) ) spans ((A) , we can use ( 0. 8 .1) . If dim ~(S) = 2, then exactly as in the case when Shad non-negative Kodaira dimension, we can use (0.8.1) to conclude that A > Nand therefore (S,L) is as in part sions of the theorem that we are proving. conclude from part

a) of the conclu-

If dim

~(S)

= l, we

b) of (0.8.1) that (S,L) is as in part

of the conclusions of the theorem that we are proving.

b)

If

dim ~(S) = 0, the we conclude from part c) of (0.8.1), that {S,L} is as in part c) of the conclusion of the theorem we are proving.

Therefore without loss of generality, we can

assume that {1.1.4} is true. But by lemma {1.1.2), we conclude that either (S,L) is as in part c) of the conclusions of the theorem we are proving or ({A} is very ample.

Since by

the choice of A, ({A) is not very ample, it follows that the theorem is proven if S is rational. Finally assume that S is birationally ruled and .h 1 ' 0 {s}

> 0.

2

Choose N = {12!} · 4 • d · N' where d and N' is chosen large enough so that: {1.1.5}

n.

LN is very ample, N2 (L · L) > 5, and h {LN) :::_ 4 0

By the first paragraph h 0 (((n)) = 0 for all large enough Choose the largest non-negative integer A such that: f(((A-1)) spans

(A-1),

(A-1)

(A-1) > 5,

(1.1.6)

h 0 (((A-l) :::_ 4, and C(A-1} is ample. By the same argument as in the case of rational

s, it can

106

be assumed that: (1.1.7)

r(!(A)) spans !(A) and !(A) is ample.

Since S is birationally ruled: h 2 ' 0 (S) = 0.

(1.1.8)

By (1.1.6) and the Kodaira vanishing theorem for !(A-1): hi(!(A)) = 0 fori> 0.

(1.1.9)

Using (1.1.8),

c

E

(1.1.9) and the residue sequence for smooth

I!(A-1) I: 0

---+

KS

--+

!(A)

--+

KC

--+

0 .

We conclude that: (1.1.10) where g(C) is the genus of C. Since S is birationally ruled and satisfies h 1 ' 0 (s) > 0 it follows that: (1.1.11)

KS · KS ~ 8- 8hl,O(S) < 0 •

Combining (1.1.11) with (1.1.10) we get: !(A)

·!(A) = (KS+!(A-1)) (KS+!(A-1)) = KS · KS + 4g(C) - 4

- !(A-1)

· !(A-1) ~ 8- 8h 1 ' 0 (s) + 4g(C) - 4- !(A-1)

· !(A-1)

4 + 4h 0 (!(A))- 4h1 ' 0 ( s ) - !(A-1) · !(A-1) By (1.1.5) and the fact that h 1 ' 0 (s) > 1, the last inequality becomes:

j.c£ (A)

• !(A)) + 1 < h o (!(A))

Therefore if I show that: (1.1.12)

! (A)

• ! (A)

> 13.

We will conclude that h 0 (!(A)) ~ 4. This combined with (1.1.7) would show the contradiction that (1.1.6) is true with A+l in the place of A.

Therefore to prove the theorem, we must

only show that (1.1.12) is true. Therefore assume that !(A) · !(A) < 12. Let: x =!(A) · !(A) = (AKS+NL) • (AK 8 +NL) =-A 2 K8 · K8 +NT for some integer T. If A = 0 or K . K = O, then N divides x

s

s

107

and therefore x > N > 13.

Therefore we can assume that:

A · KS · KS :} 0 Recalling that Ks · KS

~

0, and using the definition of N, we

conclude that: y = -A 2 + N • T

*) where x = YIKs

.

Ksl, and N = N

IKs

Ks I.

I f Y_~ 12, then from the form of *) and the fact that (12!) 2 2 divides N,

we conclude that y is of the form z

write

.

We can re-

*) :

**)

N'T

where zA =A, and

z 2 ~•

N.

Since 4 divides N', we get a contradiction from**). Therefore (1.1.12) is true. §2

0

Proof of the Main Theorem

Assume that L is an ample line bundle on a normal irreducible Gorenstein projective threefold, X. a smoothS

E

Assume that there is

ILl and that log (X,L) ~ 0

It is easy to see that such S have non-negative Kodaira dimension by means of the argument used in [So4]. through it.

D

E

Let us go

Since log (X,L) > 0, there is an effective

IN(KX+L) I for sure N > 0

If S is not a component of D

we are done by the adjunction formula.

Therefore we can assume

without loss of generality that D r S + ~ where r > 0 and S . N r ~s not a component of E. Thus Ks = Ls ® [f. n S]. Since some power of KS is a product of an ample divisor and an effective divisor, it follows that s is of non-negative Kodaira dimension. By theorem (1.1) and the fact that s has non-negative Kodaira dimension, it is true for all sufficiently large N > 0 that (Kn ® LN) spans Kn ® LN for:

s

s

s

a)

s

all n if S is minimal or

b)

for 0 < n·< A where A> Nand KA ® LN s s is not ample.

108

Since log (X,L) ~ 0 it follows that if r(K~ ® Lb) spans

K~ ® Lb with b

> a then the map associated to

K~ ® Lb has a 3

dimensional image. Therefore by lemma (1.0.1) there is for each n > 0 an integer N' such that r(KnN'® L(N+n)N'> span X KnN'® L(N+n)N' X

In case

a) we see that KX ® L is numerically effective.

Indeed let C be an effective curve.

Letting n go to

oo

We have

· C + ~ n L · C > 0 gives the result.

Therefore the main theorem is proven if all smooth S E ILl are minimal and therefore we can assume without loss of generality that we are in case b). By (0.8.1),

{

*)

(0.7.5) and the fact that A> N we conclude:

For each n > 0 and < N there exists an N' such that r(K~N'® L(n+N)N') s;ans K~N'® L(n+N)N' and the assorted

map ¢ n, N' has a 3 dimensional image. When n map has some positive dimensional fibres.

= N,

the

We can choose the N' large enough so that the maps ¢n,N' have normal images and connected fibres.

Denote this map

from X to ¢ n, N' (X) by ¢ n . We claim that ¢ : X X' = ¢N(X) is the reduction n with the properties of the main theorem. We proceed by ana~

lyzing the positive dimensional fibres of ¢N. ( 2 .1) Lemma. Proof.

By

-

.

Hl (Ka ® Lb) = 0 i f b > 2a 2 > 0 X *) there is an N" > 0 such that K~a-l)N" ® LbN"

spanned by global sections and has a 3 dimensional image. Therefore by (0.2.1) the lemma is proven.

D

From this we see that r(Ka ® Lb) X

for b > 2a > 0.

--+

r(Ka ® Lb-a) ~ 0 S

S

Therefore by choosing the N' large enough it

can be assumed that

109 ~n(S)

has connected fibres with

normal.

Further

~

n,s

is an

embedding for 0 < n < N . By (0.7.5) there are Weil divisors u1 , ••• ,Dk on X such that each Di meets S transversely in a smooth rational curve E.1 such that E.1 · E.1

-1 on S and these E1. are precisely the fibre of ~N,S" By the last paragraph ~N(Eil ~ ~N(Ejl for i ~ j implies that the n. are disjoint. 1

Note K~-l ® LN-l ® LN is spanned in a neighborhood of S. This implies that r(K~-l ® L 2N-l) is spanned in a neighborhood 2

of each ni. Indeed let pi: W ---+D. be the map from the normalization of Di to Di. Since p~(K~-l ® L 2N-ll ~@ 2 (1)

w

1

and since K~-l ® L2N-l is spanned by global sections in a neighborhood of S, it suffices to find a global section of KN-1 ® L2N-l whose restriction to Di vanishes only on X

S n Di = Ei. Choose any section t of K~-l ® L2N- 2 which does not vanish identically on S n Di and let s be the tautological section of S vaishing on S. Then t ® s is the desired section. . N-1 S1nce KX borhood

2N-l . 1s spanned by global sections in a neighof Di and PfCK~-l® L2 N-ll is @ 2 Cll we easily see that ~

L

w

the pi are biholomorphisms. (2.2) Lemma.

There exist large t such that K~ ® L~-l® ~ [Ii]-l

is spanned by global sections. ~Consider ~N,S: S--+ S' ~N,S(S~. Note that there is an ample line bundle £on S' such that ~N,S'£ = K8 ® L8 . Thus by (0.8.1) and since:

Kt ® Lt-1® ® [E.]-1 ~ ~* S

S

i

N,S'

1

the lemma is clear.

(K

S'

® £t-l) D

We claim that ~N has only the Di as positive dimensional fibres. If we show this then by theorem (0.9) ~N: X --+ X' = ~N(X) is a reduction. If there was any other irreducible positive dimensional Variety V such that ~N(V) is a point, then V n s is non-trivial. We claim V n S belongs to uE 1.• Indeed this will follow from i

110 d~N

being of rank 3 on T

d~N

is of rank 2 in TS,x for x

X,x

for x E

F

S-uE.. i

By construction

1

Therefore d~N will be

S-uii.

if we can produce a section of KNN' ® L 2 NN' ,x X for some N' of the form t ® s where t is a section of NN' 2NN'-l KX ® L with t(x) ~ 0 and s is the tautological section of rank 3 on TX

of [S] = L·

This is clear by lemma (2.2) and lemma (2.1).

Further by lemma (2.2) and (2.1) given x E Ert can be chosen so that ts vanishes only to the first order on Ei in a neighborhood of x on s.

Therefore by the implicit function theorem

t vanishes only on a manifold in a neighborhood of x on X. Assume x was chosen so that V n

s 7 x.

Since s ® t vanishes

on V it follows that V c S u Di which implies that the only positive dimensional fibres of ~N are the Di. Let L' =

[~N(S)]

¢N(~Di).

and F =

l

We claim that S' is minimal for all smooth S'

E

IL'-FI

Indeed if not then there is by repeating the whole argument we find a smooth rational curve E in S' E · E

=

-1 on S', L'E

~ ~

lP

=

~N(S)

1 (1) and ¢N(Ii)

This is absurd since L{

~

~i

E

such that E for some E.

where E is the proper trans-

form of S'. This proves that the smoothS' E IL'-FI are minimal. The argument at the beginning of the proof shows KX, ® L' is numerically effective. 0 (2.3)

Corollary.

Let L be an ample line bundle on an irredu-

cible projective Gorenstein threefold, X. Assume there is a smooth s ILl· There is a polynomial p(n) such that:

"

Proof.

h o (Kn ® Ln) = p(n) for n > 0 . X If log (X,L) < 0 the corollary is trivial with p(n) = 0.

If log (X,L) > 0 let 7T : X ---+ X' be the reduction of the main theorem. Since KX' ® L' is numerically effective it foln-1 n l~ws that KX ® L is ample for n > 0. By (0.2.1) H1 (Kn ® Ln) = 0 for i > 0 and therefore by the Riemann-Roch X

theorem the above is true.

0

In a sequel we will give a detailed structure theorem for (X' ,L') in terms of the degree of p(n). The reader can consult [So5], [So6] for a description of these results in the case

111

References [Bo]

E. Bombieri, canonical models of surfaces of general type, Pub!. Math. I.H.E.S. 42 {1973), 171-219.

[Be+Pa]

M. Beltrametti, M. Palleshi, On threefolds with low sectional genus, preprint.

[B+H]

E. Bombieri and D. Husemoller, Classification and embeddings of surfaces, Proc. Symp. Pure Math. 29, ed. by R. Hartshorne, {1975), 329-420.

[ C+E]

G. castelnuovo and F. Enriques, Sur quelques resultats nouveaux dans la theorie des surfaces algebrique, Note V in P+S below. T. Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan 32 {1980), 153-169. T. Fujita, On the structure on polarized manifolds of total deficiency, I , J. Math. Soc. Japan 32 {1980), 709-725. R. Hartshorne, Pmple Subvarieties of ~lgebraic Varieties, Lecture Notes in Math. 156 {1970), SpringerVerlag, New York {1977). R. Hartshorne, Plgebraic Geometry, New York {1977).

[Hi]

Springer~Verlag,

F. Hirzebruch, New Topological Methods in Plgebraic Geometry, 3rd Edition, Springer-Verlag, Berlin {1966). S. Iitaka, On D dimensions of algebraic varieties, J. Math. Soc. Japan 23 {1971), 356-373. S. Iitaka, On logarithmic Kodaira dimension of algebraic varieties, Cbmplex ffialysis and ~lg·ebraic Geometry, ed. by W.L. Baily, Jr. and T. Shioda, (1977), 175-189, Iwanami Shoten. Iskovskih and v.v. Sokurov, Biregular theory of Fano 3-folds, Proceedings lllgebraic Geometry Cbnference, Cbpenhagen, 1978, edited by K. Lonsted, Lect. Notes in Math. 732 {1979), 171-182.

[I+SJ

v.~~

[Ka]

Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Jnn. 261 {1982}, 43-46. Y. Kawamata, Elementary contractions of algebraic 3-folds, Jnn. of Math 119 {1984) 95-110.

[Kl]

S. Kleiman, Towards a numerical theory of arnpleness, Jnn. of Math. 84 {1966), 293-344.

[K+OJ

S. Kobayashi and T. Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math Kyoto Univ. 13 {1972), 31-47.

112

[Ko]

K. Kodaira, Pluricanonical systems on algebraic surfaces of general type, J. Math. Soc. Japan, 20 (1968)' 170-192.

[L+So]

J. Lipman and A.J. Sommese, On the contraction of projective spaces on singular varieties, preprint.

[Mo]

S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math 116 (1982), 133176.

[J.I.1u]

D. Mumford, The canonical rings of an algebraic surface, Appendix in z2 below.

[Nag]

M. Nagata, On rational srufaces I, Mem. Call. Sci. Kyoto (A) 32 (1960), 351-370.

[Nak]

s.

[P+S]

E. Picard and G. Simart, Theories des Fonctions Algebriques de Deux Variables Independantes, Chelsea Publ. Co., Bronx, New York (1971).

[RaJ

C.P. Ramanujam, Remarks on the Kodaira Vanishing theorem, Jour. of the Indian Math. Soc. 36 (1972), 41-51; Supplement to the article "Remarks on the Kodaira vanishing theorem," J. Indian Math. Soc., 38 (1974), 121-124.

[Ro]

L. Roth, Algebraic Threefolds, Springer-Verlag, Heidelberg, (1953).

Nakano, On the inverse of monoidal transformation, Publ. R.I.M.S. Kyoto Univ. 6 (1971), 483502.

F. Sakai, Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254, 89-120 (1980). F. Sakai, D-dimensions of algebraic surfaces and numerically effective divisors, preprint. [Sh+So]

B. Shiffman and A.J. Sommese, Vanishing Theorems on complex manifolds, to appear. A.J. Sommese, On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), 55-72. A.J. Sommese, Non-smoothable varieties, Com. Math. Helv. 54 (1979), 140-146. A.J. Sommese, Hyperplane sections of projective surfaces: !-the adjunction mapping, Duke Math. J. 46 (1979)' 377-401.

[So 4 J

A.J. Sommese, On the minimality of hyperplane sections of projective threefolds, Journal fur die reine und angewandte Mathematik, 329 (1981), 16-41.

113

A.J. Sommese, Ample divisors on 3-folds, Algebraic Threefolds, Springer Lecture Notes in Math 947 (1982) I 229-240. [So 6 J

A.J. Sommese, On the birational theory of hyperplane sections of projective threefolds, unpublished 1981 manuscript. A.J. Sommese, Configurations of -2 rational curves on hyperplane sections of projective threefolds, Classification of Algebraic and Analytic Manifolds, ed. by K. Ueno, Progress in Mathematics 39 (1983) Birkhauser Boston.

[VdV]

A. Van De Ven, On the 2 connectedness of very ample divisors on a surface, Duke Math. J. 46 (1979), 403-407.

[Vi]

E. Viehweg, Vanishing theorems J. reine angew. Math. 335 (1982), 1-8. 0. Zariski, Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces, Pub. Math. Soc. of Japan 4 (1958). 0. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor of an algebraic surface, Ann. of Math. 76 (1962), 560615.

Department of Mathematics University of L'Aquila L'Aquila, Italy Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556

On Meromorphlc Equivalence Relations Hans Grauert Mathematisches Institut, Universitat Gottingen, Bunsenstr. 3-5, D-34oo Gottingen, West Germany Introduction. 1. We denote by

X

a weakly normal (see§ 2.3.) complex

space with countable topology and by

R

c

x

X,

X

X

x

an analytic

set with the following two properties: 1)

R

contains the diagonal

2)

R

is mapped by the reflexion

X x X~ X x X

D

X

c

(x 1 ,x 2 )

through

x €. X

is defined as

denotes the projection (as XX

p2

(x 2 ,x 1 l

onto itself.

Such an analytic set defines a fibration in Xx

~

X

X

x

~

X

p 1 (R !"'\ (X

X. x

The fibre x))

where

onto the first component

will denote the projection onto the second).

Here,

always is considered as a set, not as a complex subspace

with a nilpotent structure.

Definition: i~

R

i~

a

no~mat

an equivalenee

~elation

1)

R

2)

the eodimen~ion o6 the equal :to c ;;: o.

3)

the

p~ojeetion~

pi

eomple~

6ib."Ct..~

R

-+

X

equivalenee 4elation i6: in i~

a~e

X, eon~tant

eve~ywhe~e,

open.

We shall prove in § 5 that under this assumption the .Quotient space dimension

c •

X/R

is a weakly normal complex space of pure

116

2. But the main purpose of this paper is to prove something for meromorphic equivalence relations in normal complex spaces:

~odime~~io~

~et),

a

~owhe~e

that

~uQh

RIX' P

2)

me~omo~phi~

equivate~~e ~etatio~ i~

o6

X

i6:

c

the~e i~

1)

a

i~

Definition: R

=

de~~e a~atyti~ i~

R n (XxP)

~et

P c

~owhe~e

x

(pota~

de~~e ~~

R,

R n ((X,P) x (X'~)) i~ a ~o~mal Qomptex equio6 QOdime~~io~ c i~ X' t.

vate~Qe ~etatio~

We denote by the holomorphic

n :

R~

map

~

has pure codimension

c

R

the normalization of

R

p2 ~ R ~ X.

R

We have

The analytic set

The normal complex space

R

X

decom-

poses into connected components Xi of dimension ni. For xEXi the generic fibre ~- 1 (x) has codimension c+n. We look at the degeneration set E := {(x1,x2) E ~

-1

(x2) < c + ni'

'R

set of

E

X•

J.

~· = ~IR'. The set

'E =

R' = 'R ' E

We put ~

-1

:

codim( x ,x ) 1 2 which is a nowhere dense analytic

Xi},

(see [Re]).

'R

~(E)

-

~-1 (P)

is not analytic in

and

'R,

in

general. But it is a countable union of local nowhere dense analytic subsets like (x 1 ,x 2 ) E

If ~

D ~

R

R

in

~(E)

~(0)

2)

~- 1 (E)

3)

~(D' {0}) c

is.

is a point there are many holomorphic maps D c ~

of the unit disc

1)

X

around

0 E ~

(x 1 ,x 2 ),

is countable,

R'

We consider the (set theoretic) fibred product

Rx

D

with:

and take the union

z

Rx

X D

c

of all irreducible components

'l
D , completely. All

117

zt := z n (R x {t})

fibres

points over

Xj

~- 1 (~(t)).

x

Xi.

If

have dimension t

So we call the

R

fibres in

of the fibres of

J

is generic we have

R:

6-ib'l..if..¢ .<.n

zt

6-<.b~at.<.on

is a

-

n.

.<.n

R.

may cross.

.

c

in all

zt The set of the

We denote it by

All

Some

are contained

S E Jll~

~-1 (x).

in a 3.

t.<.on

The set of images of

Jll

X

(p 1 o n) (S), S E

Jll~

gives a

6-<.b~a

in pure c-codimensional analytic sets. This is

considered to be the fibration given by the meromorphic equivalence relation X

by

R.

R •

We shall construct a quotient space of

The points of this will be the fibres

S E Jll.

In

general, probably, it is only a weakly normal complex space. But, probably, the quotient space will not exist without further assumption. We require a condition which can be verified before constructing the quotient space. Definition: R K c X

~egula~

.<.6

6o~

eve~y

a ~elat.<.vely compact open ¢ub¢et

that 6o~ all S E

.{.¢ called

Jll~

x E K

ente~

.<.n

B,

compact ¢Ub¢et

B cc R

all .<.~~educ.<.ble component¢

.<.6

S'

ex.<.¢t¢, ¢uch

s 1 o6

6-<.b~e¢

.{.¢ completely conta.<.ned .<.n

~- 1 (x) n E. It follows directly that any equivalence relation nents in

R

S

to a regular meromorphic

has finitely many irreducible campo-

E, only.

In order to obtain the quotient space construct a proper modification thereafter in

X.

Jll

n :

X

X/R

~ X

of

we have to X, such that

becomes a normal complex equivalence relation

By this we prove in this paper:

118

Main Theorem: 16 tio~

codime~¢io~

o6

catio~

q :

X

n :

~ Q

¢pace

R

Q

X~ x

i~

c

n(Sl

The map

on

X

s

Q = X/R

We put

u~ique p~ope~

modi6i-

with a~ ope~ holomo~phic map

¢uch that:

map¢ the 6ib~e¢ cally onto s =

2)

i¢ a

equivale~ce ~ela-

o~to a c-dimen¢ional weakly no~mal complex

n

1)

the~e

X

togethe~

X

o6

~egula~ me~omo~phic

i¢ a

by

~

i

X topologically holomo~phi

in E ~.

s i¢ a bijection

and call

Q

the

Q ~ ~.

(gene~alized)

quotient ¢pace

R.

In §

we prove a crucial lemma, in § 2

we prove an im-

portant proposition and in § 3 the Main theorem, in § 4

we

prove the existence of complex quotient spaces of normal complex spaces by an analytically closed Lie group action. Finally, § 5

brings the proof of the result on normal complex equiva-

lence relations stated in the first paragraph of the introduction. The main result is a solution of a problem on the existence of m-bases stated by K.Stein in

[S~].

Group quotients were con-

sidered in the proper case already in [Li]

and [Fu]

and the

Crucial Lemma of§ 1, however for the proper case only, is contained in Horonakas Flattening Theorem (see [Hi]). We use the notations of [STCER].

So

ft

always stands for

the simple complex equivalence relation to a complex equivalence relation

R

and

~,

R~

etc.

relation to holomorphic maps

denotes the complex equivalence F,

~'

etc.

119 The Crucial Lemma

§ 1.

1.

R~

We prove that

leads to a proper (in general only

R

weakly normal) modification of of

X

x

x.

and the second factor

X

Since this proposition is local with respect to

the second factor we may assume that this has pure dimension n and is an analytic subset of a domain ~.,

F : R

holomorphic map

n + c

ic fibre is

-+

X

m

-+

«: .

U c{m.

So we have the

The codimension of the gener-

and the degeneration set of

F

is

E.

We just may consider the following general situation: is a normal complex space, of

Y

the degeneration set

codimy F- 1 F(y) < n} the

fib~ation

subsets given by Lemma (n):

6o4 1)

-+

~m

is a holomorphic map

onto a pure n-dimensional normal analytic subset

A c U c ~m,

~

F : Y

Y

16

of

Y

RF

(like ~¢

Y. -

mo~ph~e

o6

We denote by

into pure n-codimensional analytic R~

eompaet

~~).

gives

the~e ~¢

a

We prove: 6~n~te

6am~ly

o6

w~th the 6ollow~ng p~ope~t~e¢:

a4e tiC1tmal c.omple.x. ¢paee.¢~ nA, qA,

YA, QA 6~b~e

is nowhere dense in

K c Y

A= 1, ... ,1

E := {y EY :

map¢.

!A

The._d1me.n¢~on

~¢

o6

o-d~men¢~onal.

QA

~¢

n,

!\ the

hologene~~e

120

-1

AU 6-<.bJte-6

2)

YA.t := qA. (t)

c:

YA.

w.<.:th

t E QA. ha.ve pulte

c.od.<.meM.<.on n . They a.Jte ma.pped by nA. onto a.n open -6 ub-6 e.t o6 a. 6.{.bJte s E JJ!.F , .ta.in.<.ng s n K. s E JJ!.F

FOJt

3)

a.

.the in.teJt-6e.c..t.<.on

6inLte. open .the -6 ubH.t c.o n-

s n K i-6 in .the. ima.ge o6

YA.t

4)

TheJte a.Jte Jteta..tive.ty c.ompa.c..t open -6ub-6e..t-6 0~ c:c: QA. -6uc.h .tha..t a.6.teJt Jte-6.tJtic..tion .to Q~ .the p!topeJt.ty 3) -6.titt hotd-6 .tJtue.

5)

16 Qi c:c: QA. p!topelt.

.the ma.p

nA.

We shall deduce a proposition from Lemma (n) in § 2:

A-6-6ume .tha..t

Proposition (n):

X, Y, Z

a.Jte noJtma.t c.ompte.x

a. -6uJtjec..t.<.ve hotomo!tphic. ma.p who-6e geneJt.<.c. 6ibJte ha.-6 c.od.<.men-6ion n, -6uc.h .tha..t

a.nd .tha..t

hotomo!tph.<.c. ma.p. MoJte.oveJt, we a.-6-6ume a. hotomo!tphic. ma.p ~

: z

~

x with

1)

: y

~

o p. n :

n- 1 ~(E),

it- 11EI

ou.t-6-<.de.

~

y ~ Y,

it

di6ic.a..t.<.on-6 ~

~

Then .the.Jte. a.Jte. unique p!tope.Jt mo-

X_.

X,

whic.h a.Jte. bihotomo!tphic.

.toge.the.Jt with a. hotomo!tphic ma.p.t

X -6UC.h .tha..t

.the dia.gJta.m

y --

A

~y

~ Xl __~n_..., 1~ X

i-6 c.ommu.ta..tive., 2)

.the 6.{.blte-6 o 6

~

a.Jte o 6 pulte c.odimeMion

n,

121

each 6ib~e "' S = --1 ~ (x) € ~~ i~ mapped by nA topologically onto a 6ib~e S € ~~ ~uch that p o ~ i~ a bijection ~ -+ p ~~, whe~e p ~~ i~ the ~et {p(S)): s € ~~} (attention: the 6ib~e~ a~e equipped with multiplicity o6 thei~ i~~educible component~: ~ee § 2. 1).

3)

We prove the Lemmas (n)

2.

n =

case

the degeneration set equal

YA. -+ QA.

by an induction on

y

-+ A

n > 1

We assume

linear projection

F

A

mens ion 1. p

a:m -+ a:n

such that

A -+ a:m -+ a:n

p : a:m

of the composition

v

Hence, the image

F.

coincides

and that Lemma (n-1) holds true. We take a

=

E

and hence nowhere dense in

...

Y.

E*

of

is discrete

a:n-+ a:n- 1 . crn-1

= p(A) c a:n-1

The degeneration set

F*

Q

~F

So nothing has to be proved.

and compose this with a linear projection fibres in

In the

is empty. We simply put

A = F(Y) c U c a:m.

with

with the set of fibres of

E

.

n

F*

The

are of pure diis open. We put is contained in

This means we are with

F*

in the case of Lemma (n-1). We obtain a finite family of commutative diagrams:

* qA.

* nA.

r

y

*

!';>..

* QA.

with

A.= 1, ... ,1*

l

a:

F* n-1

and the properties 1) - 5).

We may cut

into smaller pieces. Hence, we may assume that all lytic subsets of domains and

F~ = (fA.,Ffl,

where

u fA.

c

a:m.

We put

Ef

of

are ana-

Y* ~;t Qr ~d u A.

is the m-tupel of holomorphic

functions defining the holomorphic map tion set

F* A.

QA.*

Q~

F o nr.

The degenera-

is the union of those irreducible compo-

122

nents of the fibres of

F~

where

tained in the inverse image of

3.

fA

is constant.

and nowhere dense in

E

t E

Now, we take an arbitrary point

Y~tx

~~tx

~~tx

components.

Y~t

Y~tx"

components of

q~-1

=

(t)

and the other irreducible

is an open subset

0

yx E

points

Y~tx

the difference of

Y~.

and denote by

Q~

the irreducible components of the fibre

and by

It is con-

Y~tx"

of

We take fixed

We restrict ourselves to those irreducible

YAt'

whose

enter in

~~-images

This is a

K .

finite number. We always represent a neighbourhood an analytic covering of

w

-K

c

Q~,

~d+ 1 ,

such

d = d

b/bK,

Y~~~

Yx

positive integer divides

dim

~

~hat

b ,

with a connected open n+1,

Wz

W=

and a fixed open

b

such that the number of sheets

always. If we consider

we denote by

as

V(t) cc

is the only point over its image. We take a

the coverings

WK

have

b

WK

bx

of

z E W ,

WlzxV

(we have to ob-

and obtain a b-sheeted normal analytic

V .

We denote by

I

w·,

I

W~

the symmetric power of

is a normal analytic covering of sheets. There are

b

Wx V

resp.

canonical projections

W, Wz' V

I

w·

~

which

with b!

w,

I

W~ ~

Wz

w X V, take a -K component f of fA and lift flq*-1(v) and fiW to A zx The normal complex y~ X v WZK and take the difference g We put

(zx,t)

for the image of

in

Yx

.

space

Y~

q~- 1 (V).

Wx

with multiplicity

sheets always. If

the normalization of

serve multiplicity!) covering of

Wx V

Y~

WK = WK(yx) oc

x

If

V Wzx

is a b-sheeted analytic covering of

Yx E E~

the function

g

vanishes on the fibre

123

of

w}t

X

tions

over

v wz }t

g 1,

0

0

0,

w! -+ w }t x"

tions

on

gb

For

lytic covering of !(z)

v

t

,

identically.

w}t1

v wz

x

~x

z E

with

g1,

0

0

0

w·zI

each

wz

x

(b!)

b

V

is a normal ana-

sheets. We denote by

w· z

spanned by the

v wz }t

X

I

w· z

I v wz }t -+ w·

X

which we have to take for all components b ·

to func-

}t

I

under

,gb

g

using the canonical projec}t

the coherent ideal sheaf on

inverse images of

We lift

f

of

}t

The

f';v

(b!)-symmetric functions of the direct image of span a coherent ideal sheaf

replaced

V

z 1 , ... ,zr E

I(x,z 1 ) + ... + I(x,zr) Yx (

on

V.

!(z)

on

After having

by a relatively compact open neighbourhood of

we take several points

If

I(x,z)

v wz ,

X

E~

~K

I

we get

(if

= 1

X

I }t :=

such that

is maximal on

t

V.

V

is small enough)

and

nothing will happen later on. Therefore, we omit these irreducible components in our consideration. 4.

We do the monoidal transformations of

sheaves

I 1 , ... ,Ik

defined to

ducible components of

Y~t

V

(to normal spaces!).

one local cross-section! ary symmetric function of to

V),

where

g

by the ideal

running through the irreE~.

completely contained in

obtain a proper modification to

yx

V

V of v.

We lift everything

Now, locally

We denote by g

on

Y~

We

Ix

.9:x 1•

(=

is spanned by the b-th element-

lifting of the old

is the holomorphic function on

Y~

Yt x V Wz X

defined by the v-th component

f

of

fA..

IX

are locally principal in each point of

f

:=

X"

ab 1;I

-'JUJ

X

Since the ideals

V,

the quotient

can be locally along the fibres of

considered as a meromorphic function. If

t

Yf-+ "'

V

stands for any

124

V,

point out of

which is mapped onto the old

t

,

these

0

functions are holomorphic in a neighbourhood of

Yltx'

which

is defined as the inverse image of the old object. But now 0

Yltx

will not be irreducible any longer in general. It can be

seen rather easily that not all the

f

vanish identically

KV

0

on

Yltx . 5.

Over a neighbourhood

over to the graph of

(fKY :

proper

Y*A.

modification

wK

verse image of codimension ation set

n-1, (over

in

G

v

of

V we pass

of any point of K,V)

Y!IG.

and obtain over We put

wK

Y*A. ...

has

G

everywhere. However, there may be a degenerqX(E1)).

By Proposition

~ : w~

. . GK

with

(n-1)

there are proper modifications morphic map

a

for the in-

The generic fibre of

'Y!.

G

z

=

wK

and a holo-

such that the diagram

commutes. All the fibres of

are of pure codimension

n-1. We apply this to all of

G.

~ : Y1

everywhere now,

holomorphic maps Yltx

and obtain a proper modification

We lift everything to

sult is a map n-1

K

(here

t

Y1

-+

~

~

,

~

have codimension

and the meromorphic functions -+ ~ 1

The re-

starting from

The fibres of

fxv

are

Not all of these are constant on

any point over the old

t) .

~

125

6.

We wish to arrive at the case where not all

constant on each inneducible decompose

* YAtX

componen~

of

Y1tx

f

are

X~

So we have to

into irreducible components and to take a new

enumeration of the irreducible components of

Y1t

and do all

of the construction again. But now the maximal number of sheets of the

wX

(without multiplicity)

has decreased. So by an in-

duction on this number we come to the desired case. (fx~'q)

The generic fibre of the map now.

The codimension is also

n

has codimension

in each point of

The degeneration set is closed. By making ler we obtain that the codimension is over, that for fied

V

the properties

n

V(t)

Ytt over K.

and

Y*A

smal-

everywhere and, more2)

and

5)

are satis-

(for the case of Lemma (n-1)).

7. ~A

~

Yt

A

A

n ,

of

The fibration Yt,

~F

of

Y

can be lifted to a fibration

since all maps are constant on the fibres of

~F.

However, we had to pass over to normalization several times. This may lead to analytic coverings. The holomorphic map nA : YA = Yt

~

subset of each containing

Y

induces an analytic covering map of an open

S

€ ~A

onto an open subset of a fibre

S € ~F

s n K.

and A) xv'q coarser than the simple equivalence relation to this fibration. ~~

A

is finer than the fibration defined by

Over a dense set of

YA

the fibres of

~A

R(f

coincide with those

of RF

(by construction). Thus, over the inverse image of on A and ~A are the same. By taking the closY'E the fibres of RF on A -1 ure of RFonAIYA,nA (E) in YAxYA we obtain a complex equivalence relation

RA in YA whose fibre set is

~A.

The projec-

126

tions

RA ~ YA are open (since

locally irreducible). Hence,

A c U c ~m RA

was assumed to be

is a normal complex equiva-

lence relation in YA. We denote the normal quotient space of YA by QA. It has pure dimension n

Since

§ 5. We have the quotient map

YA

since

F o rrA

those of

~

~s

qA

~

We obtain

!A

qA, which are

:m:A.

We may replace the

Q*A

QA:

constant on the fibres of

Qt

we started with by a relatively

compact open subset. Thus, we can cover over

YA is normal, by

and

K

91

and the fibration

by the images of a finite number of

* A YA->V.

I\

We even may shrink the

V .

So we arrive at a finite family

y

The conditions 2), 3), 5)

are trivial from construction. Since

A

we may even shrink the rna (n) has been proved.

V,

also 4) is satisfied. Thus, the Lem-

127

§ 2.

Proof of the Proposition (n)

1. ~

We prove the Proposition (n) from§ 1.1. The map ~

: Y

space

X X

is surjective and holomorphic, the normal complex

n •

has codimension

The fibration

G

B cc

z,

is regular. We take

G oc X

a relatively compact open subset subset

and to the compact

B c Y, B

a relatively compact open subset

such that for

of fibres

s

tained in

~

E Jl1.

~

-1

X E G

enter in

(x) n E.

B

=

p

-1

all irreducible components

,

if

S'

(~),

S'

is completely con-

We apply Lemma (n)

to

K

=B

c a:m

u

can be locally embedded in an open subset

X

~

is pure n-dimensional and the generic fibre of

Since we ob-

tain a finite family of commutative diagrams: n>.. q>..

r· >..

=

~

X

1, ... ,1

p(S), S E Jl1.

* >.. 2

1

.EA.

Q>..

for

y

~

with the properties 1) - 5).

may be the image of fibres

*

Y,

~,t1

Any fibre , Y,

~2t2

with

p o n>.. (Y>.. t n K) = 1 1 1 p o n>.. (Y>.. t n K). In such a case we call points y 1 E Y>.. t , 2 2 2 1 1 y 2 E Y>.. t equivalent if they are over the same point of Y. 2 2 We also consider the points t 1 ,t 2 to be equivalent. By this >.. 1

or

t 1

t 2,

i.e.

such that

we obtain an equivalence relation and an equivalence relation A

Y*

the part of

"

y

R

lying over

in

1\

in

d=

B* = B n

y

:=

LJ

y>..

A.= I, .. ,1

0 Q>... ~

-1

We denote by

(G),

by

128 A

the part of

tv

...

R*

Proposition:

an.d

over

Q

G

and prove:

,.

/1

= R\Q*

RIY*, R*

=

a~e compl~x,

holomo~phic

.6emip~ope~.

Proof: If

S E

~~

is a fibre over

n ~*, B* := ~ n ~- 1 (G)

.. ,zr E p(S)

G,

we take points

on the p-image of the var-

ious images sets of those irreducible components of are completely contained in local complex subspaces

E .

z 1 , ..

S

which

We take pure n-dimensional

Ai c Z, i=1, ... ,r

with the following

properties: 1)

Ai cUi cc Z

are n-dimensional complete intersections

(relatively to U), Ai n p(S) = {zi} .

2)

A neighbourhood

s1

E

over

~~

S n B*

G

= 11

~

is defined as the set of all

~

s1

such that

n B*

can be connected with

by a chain of holomorphic 1-parameter families

Sx(t) n B* , I tl < 1 X

c

-V(S)

•

•

•

over

G

with

p Sx(t) n

i)

A.

1

*¢

and

,k •

We denote by Clearly,

A

v

"' v

v

the inverse image of

is open and its inverse image

again. By projection we obtain subsets

A

Q

in

u" c

"

y

l

A.= I

Q;v

is open

1\

u c Y*/'R*, v

0

:=

c:

II

Q*/R*

which are open in the quotient topology. There is a multiplicity for the irreducible components of the fibres bourhood of

S E S.

~~

If

coming from the generic fibres of a neighS

is generic this multiplicity always is

one. Otherwise it can be a higher integer. It might happen

129

that a settheoretic

S

has various multiplicities coming from

different neighbourhoods. Then these have to be considered as different fibres. Thus, a fibre is the set

S

equipped with

multiplicity of the irreducible components. -The multiplicity carries over to p

s1, s1

y

E

p(S).

We take this multiplicity! Then each

has the same intersection number with

multipli~ity

We use the multiplicity to define the

s 1 n Ai, s 1

intersection points function on

Ai'

y.

If

g

for the

is a holomorphic

we take the elementary symmetric_polynomials

of the values of functions on

E

Ai.

g

s 1 n Ai.

in

~

v .

These all lead to holomorphic

We may assume that an embedding of

in

a complex number space is given by finitely many holomorphic functions. We take these for g. A

We define

V

so small that it is over an open subset of

X,

which is isomorphic to an analytic subset of a domain in the complex number space. We also take the holomorphic functions on ~

V coming from the finite number of coordinate functions on this open subset.

H

Altogether, we obtain a finite set A

v

tions on A

(t,T) E V

The differences A

x

V

I

p(S)

n

~*

n

p E,

E

a neighbourhood of

on

A

V

~

x

V.

which is somewhere dense on each p-

We take the S

I

f E H,

Firstly, we take an infinite set of

image of any irreducible component of contained in

for

maximal.

We proceed as follows: on

- f(T)

define a coherent ideal sheaf

We wish to have the

zi

f(t)

of holomorphic func-

Ai c z

S ,

which is completely

such that

V

is still

This is possible! We just have to use

130

the parametrization of fibres given by Lemma (n) and that the ln-

s

verse image of the "point" make sheaf

e I

smaller. The set

H

a

is closed in

is infinite now. But the ideal

is still coherent. The functions of

different fibres of

irreducible component of

Q

separate the t

):

each

and

this is a well-known statement. smaller we find finitely many

V

points among our infinite set

{zi}

such that the ideal

is spanned by the functions to these become finite and the functions

H

zi

already. Now

So

Qi cc QA..

-1 -

-1

nA. : qA. (Qil n nA. (B*)

follows immediately that

R*

and

H

R*, R*

holomorphic equivalence relations. - We may replace relatively compact open subset

I

has

separate the fibres out of V. R( H,nA. ) .

pact and the maps

t0

~

converges against some irredu-

qA.(t 0

After having made

H

We have to use that for

V :

cible components of

and that we can

R*

Since ~

B*

Qi

QA.

are by a

is com-

are proper, it

are semiproper. So the

Proposition is proved. 2.

Hence

~* = Y*/R*, X*

have holomorphic maps

0•/R*

are complex spaces.

We

~* : Y* ~X*, ~*

such that the diagram Y*

l

·Q>*

,..., X*

~*

B*

1~

rt*

G

is commutative. All fibres of They are mapped by By

,.,

R*

tt*

~*

are of pure codimension

finite open onto an

S n B*, S € N~

this map is bijective. Hence, it is topological. The

n.

131

composition maps

p o ~*

n , "*

~~ ~ p(~~ n B*}.

gives a bijection

The

are proper. Since the inverse image of a generic

point consists of one point only, they are proper modifications. 3. X*

is defined as a set independently of the choice of B.

Since every irreducible component

G with pS,

S' c E

S E Jltcp

enters in

over

G

B*

S'

of a fibre

the set

X*

S €

By making

B

o :

X* ~ G ,

X** ~ X*

over

,..

X*

might

larger it might be-

come finer. We would obtain a proper modification stead of

cp

is just the set of

But already the topology of

depend on the choice of B.

~

X** ~ G

in-

but also a bijective holomorphic map

such that

~**

X

61·~

~ ~

G

commutes.

X*

In general, they are

·x*, X**

qua¢i no4mal:

will not be normal complex spaces. But any local continuous complex function

which is holomorphic outside a nowhere dense analytic set, is holomorphic. When we pass over to the normalization of 6

the map

is still bijective and holomorphic: This follows from the

modification properties. By a well-known theorem (see [CAS)) it is biholomorphic then. This implies that already the old is biholomorphic. Thus, the proper modification only on

X* ~ G

depends

G .

The same holds true for have a commutative diagram

"' Y*

~

B*.

If

B**

6

is larger we

132

B*

Y:•• ---•

6

where of

B**

is a biholomorphic map of

onto an open subset

·"'Y**· 4.

Now we exhaust

open subsets Bv cc

El.'+1

=

Y.

Y* -+Y:+ 1 1B~ : "' ~""

l

X

Y

by a similar sequence

We construct the commutative diagrams

and get isomorphisms

rv

and

G*y

X*v

·"" y

by a sequence of relatively compact

r

l

fV

" v cations

X

GV cc Gv+l cc X

"'Y*...

~

Y*

A

n

I

.

o"

=

x;::: x;+ 1 1G~,

We glue together and obtain proper modifi-

Y-Y, n : X

-.x

such that the diagram

y

l

X

is commutative. The properties 2) and 3) of Proposition (n) are satisfied. Its proof is completed!

133

§ 3.

The Proof of the Main Theorem

1.

Assume now that

X

is a normal complex space and that R

is a regular meromorphic equivalence relation in sion

We denote by

c

R~

n :

ization

R

P c X

is the degeneration set of X.

connected components

~

"" R

:

~X

X

and

X

Xi

with X

The set

E

decomposes into

The codimension c + ni.

is

p2 : X

X

~

X

The space

over

equal

4.1

.

(c + ni)

We apply Proposition

.....

: R

p2 o n

of dimension

1

of the generic fibre of

n

~

of codimen-

a polar set. We take the normal-

~

and put

X

Y

~

~

X. We obtain a com-

X

Z

equal

mutative diagram 1\

n

N

y

"" ~

1 X

image of

1 ~

n

X

,...

R

....,

1"-

where

1'-

X

is a proper modification of .'V

in

R

X

X

X.

"' X under the map

We take the inverse ,..-

~

X

ducible components which are completely in with fibre

'P = n- 1 (P*) S E

~~

,P*

=

p

u

({I {E) •

is contained in E

n

and omit all irre-

....., p

X

"' X

and

,...

X

X

"" p

then a full irreducible compo-

p~ojection

,... P c X

1"-

I f a point of a

We note:

nent through this point is contained in E. Since it follows that the

or in

E

~

i~

X

R

is regular

~emip~ope~.

So

are nowhere dense analytic sets.

Thus, we obtain an analytic set

R0 c

X x X,

which contains

the diagonal and is invariant under reflexion. The blowing-up to.ok place in the restriction

~

(E) only. So we have R0

1X'P

-

....

X' p

X ' P* . Hence,

is a normal complex equivalence rela-

134

R0

tion.

R n (X,P*) x (X,P*))

is simply the closure of

By a generic fibre xx' x E X'P*

S €

such that

~

Xx '

:= p 1 P*

we understand a fibre

~~

is dense in

each generic fibre has pure codimension

,...

point

x E X

,..,

We call a map into with

x

S

-+

S = p1 n n

the fibre

eon¢tant to

X

continuation to

...,_,

ni

S

: s

n

in

6~b~e¢

P*.

< ni

A

X=

s

P* x X

R

-x

:

through an

s

,..,

-+

€

~

Hence

Tifs "' X

=

X € X '

s -+

n

s

-+

~~ 1 it

X

the map

·n- 1

is

has a unique

have codimension

always: We put

s

€

J\

R = (p, x

and

lt Tt' ,

g...,_ 1 n

S -+>

S

c: X

~

.....-1

n

~

c:

are generic,

s

N

-+

X

which

can be approximated

iPJ (YJ

c:

X

X

x.

is contained in an

~

Then

is constant on each (x)

is a topological map. Its inverse

is a finite open map. We obtain a fibration

X two different

X

are approxima-

P*

X E X '

into pure c-codimensional analytic sets

tion of

is attached.

So we have a fibre constant

P*.

__ ,

~.

X

XX, X € X' P* '~

through an

is the fibre over

c: X

S E

X

in

the set of points, where this

is constant to fibres. Now, every

s,

To every

S e: ~

ted by such fibres. Thus, we also have a

by

X

(degeneration set) . The image

is nowhere dense. All fibres

"'n _,

in

x.

-+

all other fibres

Clearly,

which is also constant to fibres. We denote

We denote by

codimension is

xx.

if the composition

N

S ' P*. Since

The fibres of the projection ~

~

is constant. For a generic

constant to fibres in

this by

c

,.._, (x)

...

in

....S E "' ~

S . So

~

~

~

S • By the

"' ~ in

construe~

never intersect. Each point of ~

is an equivalence relation in x.

135

i

We wish to prove that R0

know that x € "' X

R0

1x 'P

rJ

sX

=

s

€

X

i

the fibre through

always converges against

5ii.jx ' P.

we put

"" X .

jection onto X

If

n °!

X

0

X

€

S'

0

R -t .R

always converge against Ro.

is nowhere dense in p2: Ro

tion set of

N

,..., -+

X

sX

X -+ X

be open. So

Ro

0

X-+X

is the

Jfl.

cpo

Then we have for

sX

and so the S' xv ,.., ,_ Ro n (P X X) u X

v

<x

sX

has to be empty and we get

"' Jfl.

= Jl['=

always against

is normal and

Q

Q

0

=SI xo

Ro. c

S'

and

X

con-

S' X has to p2 : Ro -+ "' xo ' a weakly normal comX/R0 In our case

Q

are the fibres of

is a usual

"" Jfl.

is topologically mapped onto a fibre in

Each X

The

map is a bijection of fibres. That completes the proof of the Main Theorem.

0

This all implies that the de genera-

quotient and the fibres over

"' X

For

0

plex space. We have to use § 5.6.

fibre in

If

given by the pro-

Since the fibres are of pure codimension verges for

X .

there is a sequence of generic

S' x., Moreover,

That means

xo € X.

X

where

,

cp 0

the equation

€ Jfl.'

XV

for all

"' 0

converging against

v

the fibres

,...

We have

sxo

Jfl. 1 = p 1n° Jfl.

fibration in the normalization

points

Then we also

•

is a complex equivalence relation in

we denote by

the fibre

R0

is given by

Pl

136

§ 4.

Quotients by Lie Groups. Meromorphic Reduction

1.

We assume that

space and that on

L

X

is ann-dimensional normal complex

is a complex Lie group acting holomorphically

We assume that there is no smaller union of connected

X

components of

X , on which

of the generic orbit by

L

d

acts. We denote the dimension

and put

= n-d.

c

We define

RL

as

the graph of orbits:

In general, the closure

R =

~

is not an analytic set. But it

contains the diagonal and is invariant under reflexion X

X ~ X

X

X.

X

act~

analytically

Definition:

L

lytic

dimen~ian

a6

~et

and

n+d

an X i6 R

cla~ed

R

an ana-

i~

negulan in the

i~

~en~e

a6

panagnaph 3 o6 the intnoductian. In the case that an algebraic group acts algebraically on an algebraic space n + d .

X , the set

If, moreover,

closed on

X

R

is always analytic of dimension

is complete

L

acts analytically

X.

We prove: Theorem:

une

16 i~

R

X/L := X/R We call group 2.

L

L

act~

analytically

clo~ed

on

X ,

a menomonphic equivalence nelation on i~

a weakly nanmal complex

X/L

the quotient of

R

is the closure of

~pace

X

the gnaph X

o6 pune

clo~

and hence dimen~ian

by the complex Lie

.

Since

~

each fibre Rx

R n(xxx)

c.

137

is closed against the action of

x

L

on

X .

The set of points

where the vector space of infinitesimal transformations

E X ,

has dimension less than

d , is a nowhere dense analytic sub-

set

all orbits have pure dimension

P c X .

In

so no fibre of

X '

P

p2 : R

-+

X

has dimension less than

the generic case is that the fibre dimension is by

the degeneration set of

E

p2 : R '

p2 : R

-+

X

.

d

and

We denote

d

The map

is open.

E -+ X

,.._

We define the fibration

Jf[ (j)

in the normalization

obtain by projection the fibration

R ,

d .

the fibration

in

Jfl.

X

in

All fibres

R

of

R

and by

p1

are pure d-di-

mensional and invariant against group action. Hence, they are singularity-free in

X' P.

But in

X' P

they may consist

of an at most countable number of connected components. The group

acts on such a component always transitively. - We

L

S E Jfl.

call the fibres

3. Proof of the Theorem: tion

E

p2 : E

X

-+

p 2 (E) c X

L.

generalized orbits of Since

R

is regular the projec-

is semiproper (see § 3.1). So

is a nowhere dense analytic set. We put

0

X

X'\.P'\.E.

We take a point R

XC

and

c

n X. U(y 0

t

x0 E X

and a connected component

v(x 0

There are connected open neighbourhoods C·

)

cc X

to

y 0 E R1 ,

such that

Rx n (U

x

R1 )

of ' c X

V), xEV

is a regular family of connected d-dimensional submanifolds At

for

the set

t

in an analytic covering

B1 c

B

of points

t E B

B

with

of

v.

At c

Because of the continuity of the action of

L

Since R =

~

~

is dense.

it is open.

138

We denote by R1 n

(U

E V1 .

X

in

v

The set

is in the accumulation set of the orbits Lx,

xo)

X

a,

the open image of

v,

Since

L

c

acts on

transitively and acts on XxV

R1

this holds also true for the whole set

R1 .

We take finitely many connected components 0

of

Rx0 n X

and take a fixed

v

and put

in the accumulation set of the orbits

.,

V* = v 1 n ... nv 1 . R1 u ... u R1

V. The set

This set is dense and open in

R1 , ... ,R1

Lx, x

0

e:

is

V* . This im-

0

n X = R n X i f x, e: R f) X. So the transitive X x, 0 0 ' is a complex law for the relation RIX is valid, i.e., R I X ,. t• c equivalence relation. Moreover RIX is normal and Rn(XxX)

plies

R

X

0

is dense in 4.

R .

That completes the proof of the Theorem.

We assume that

X

is a compact connected n-dimensional

normal complex space. We denote by morphic functions on of

X ,

of

X , such that

X.

M = M(X)

the field of mero-

We wish to construct a reduction

X

which is a Moishezon space and a biregular invariant

We denote by

M c

is the field of meromorphic functions on X.

the degree of transcendency of

meromorphic functions

f 1 , ... ,fc

M and take

which are algebraically (and

analytically) independent, and a nowhere dense analytic set P c X

which contains the polar sets of

f 1 , ... ,fc. f' : X

gives a meromorphic map

f'

x ~, c-times. The graph

G c X x !Pc

G

-+

X

GIX'P

"' R c G x G

x,p

is biholomorphic the restriction

plex equivalence relation.

X

• • •

G

-+

!Pc

defines

which projects by

to a meromorphic equivalence relation -+

!PC = IP

is an n-dimensional

analytic subset. The holomorphic projection a complex equivalence relation

-+

The c-tupel

R c X x X . Since RIX'P

is a com-

139

From now on we assume that

P

is so large that it contains

also the image of the degeneration set

G

X.

~

For

RIX'P

there is the well defined simple equivalence relation

RlX'P

which is the finest complex equivalence relation, whose fibres are locally the same as those of

RTX,p

closure of

in

X

lence relation in R n ((X,P) c+n

(X,P))

x

~

R.

RIX,P.

ft

Then

P

of

f 1 , ••• ,fc'

P

The map

has fibres of pure codimension

~

and is open. It is immediate that

the choice of

is independent of

and moreover it does not depend on the choice it is an invariant of

X

The meromorphic equivalence relation

R

and defines the unique proper modification quotient space sion

c .

the

is a meromorphic equiva-

with polar set (X,P)

ft

We denote by

Q

is trivially regular

"' X

of

X

and the

which is a weakly normal space of dimen-

We have the commutative diagram

,._

X~

I

X

Q

Every meromorphic function

f

function

on

"' X

f E M

lifts to a meromorphic

and hence comes from a meromorphic function degree of transcendency of Hence, Theorem:

a

'

Q

"' X

which is constant on the fibres

Q

f

on

Q .

~

Q The

(and of its normalization) is c.

is a Moishezon space.

Fa~

eve~y compac~ connec~ed

un~que Mo~~hezon ~pace

~~ve d~ag~am

complex

Q , connec~ed w~~h

~pace

X

X

by a

~he~e ~~

commu~a

140

whe~e

q dimen~ional

6unction~

We call

6ib~e~.

M{Q) Q

=

~uch

M{X )

a

i~

n

a

p~ope~

~u~jective

that

=

6o~

the

6ib~e

modi6ication and map with

6ield~

o6

M{X).

the meromorphic reduction of

X .

pu~e

(n-d)-

me~omo~phic

141

Appendix

§ 5.

1.

Assume that

is a normal complex space and that

Q

Q

a semiproper complete equivalence relation in

R

is

with the fol-

lowing properties: 1) If

R1 c R

is an irreducible component of

different from the diagonal

D c Q x Q ,

then

R , which is

p 2 (R 1 ) c Q

is

nowhere dense. 2) R

is discrete, i.e.,

Since

R

Q

~

Q/R

point of

U(t 0

Since

in

)

cc Q ,

p2 : R

finitely many points in

t

is a discrete map.

t E U .

If

U

Q

~

t0 E Q

V(!ol

of the image

is discrete, there are only

which are equivalent with

t1 E Q '

U

t

, for

is a point equivalent to

, then the quotient image of an open neighbourhood V .

there is an

which is mapped by the quotient

onto an open neighbourhood

t0

any point

Q

~

is semiproper, for each point

open neighbourhood map

p2 : R

W(t 1 )

is

But because of property 1) there would be points in

not equivalent to points of

W

U . This is a contradiction. It

follows: R

-1

n p2

(U)

R

n (U x u)

Thus, the projection

p2 : R

n

-1 p 2 (U)

~

the union of irreducible components of diagonal, the set set of

~ =

p 2 (B)

is proper. If

B

R

different from the

is

is a nowhere dense analytic sub-

Q

The complex equivalence relation the set

U

~

R

sews

Q

together on

: always finitely many points are identified. How-

ever, probably, the quotient

Q/R

will no longer be a complex

142

space in general. There will be not enough local holomorphic functions. We define: Definition: A ~u~u~ed complex ~pace ~~ a no~mal complex ~pace

a

~oge~he~ w~~h

the

~em~p~ope~

7) and

p~ope~~~e~

2.

complex

equ~valence ~ela~~on w~th

2).

We prove the following:

I6

Theorem: no~rnal

complex

p~ope~

and

~pace.

16 X

a weakly

~~

X

complex

no~mal

equ~valence ~elat~on

holorno~ph~c.

So X/R

~~

~~ no~mal then al~o

~n

then

X

a weakly

x/R

and

~pace

~~

R

T : "' X

~

X

~~ no~mal.

of

X

R

with

and lift

N

X •

~em~-

complex

no~rnal

Proof: We denote the codimension of the fibres to We take the normalization

a

R

We obtain a complex equivalence relation

R

R "-

in

c. to

with

X

pure c-codimensional fibres. We pass over to the simple equi-

" R

valence relation

belonging to

"' R

Then

normal complex space of pure dimension

c

,._

A

Y = X/R

is a

. The equivalence re-

lation

R

is obtained from a discrete complex equivalence re-

lation

R

on

We have

Y

X/R

by lifting. Y/~

.

But in general

We shall define the open part

R*

of

R

g

will not be open. We denote by

g ,

just the union of those irreducible components of ~

Y

which

have dimension

c

Then the projection

The rest of

R

is

mapped onto a nowhere dense subset of

The set

contains the diagonal, it is invariant under re-

R*

p 2 : R*

R*

is open. Y.

flexion. We have to prove that it is an equivalence relation. For every

y0 E Y

we have

143 p

lim

y-+yo . y = gener1c since .Sy

p 2 : R* -+ Y

and since

1 R;

R

1 R;

is open. If

y

is generic, we have

R* y

is an equivalence relation,for each

R* = p 1 R* . So we get also for y 1 E p R* y y1 1 Yo the equation p R* = p R* That means the transitive law 1 Yo 1 y1 and that R* is an equivalence relation.

p

3.

1

We prove that Yo E y

note by If

p

also

y1 E y

is holomorphic and semiproper. We de-

a point and by

an open neighbourhood.

W(yo)

y E w

is a point equivalent with

(y1,y) E R*.

aueh

R*

There are open neighbourhoods

R* n (UxV)

~ha~

U' (y 1 ) c U

an analytic covering of

q(W) c Y/R*

Q

R*

morphic function

of points in Y equivalent with

w cc

Y .

Then

w

is compact and

is semiproper. - To prove that

f

in

V

=

V(y 0

)

=

U

V , lift it to

to the b-th elementary symmetric function is constant on the fibres in V

=

U'.

q(W) c

R* is holomorWe take a holo-

R n (UxV) and pass over f

on

U . This

is isomorphic to an analytic set in a domain of hole-

an f . There are finitely many

! =

f

U .

morphy we can separate any two equivalence classes in

of

is

is open.

phic, we consider the case

If

R* n (U'xV)

such that

is open. By definition of the quotient topology

We may assume q(W). Thus,

w

c

U'. So the points of U' are equivalent

with points of W: The set W

u (y 1)' V(y)

is an analytic covering of V. There is

an open neighbourhood

points of

we have

'

C£ 1 , ... ,f1 >

£1 , ... ,!1

have dimension

is holomorphic on the open set

by

such that the fibres

0 . Now,

u := q- 1

U

q(U)

F =

(~oq

-1

)oq

and we have

144

" ~

D

.

c

R*

That means that R* c RF is a complex space.

Q = Y/R*

Q

We prove that function in

q(U) g

analytic set,

A

' 0

is normal. If

q

I

where

g

is holomorphic. So

is a bounded holomorphic

A c q(U)

is holomorphic in

is a nowhere dense

b' 1\

u

is a nowhere dense analytic set in 1\

open. Since goq

U

q- 1 (A). The set because

is normal, the bounded holomorphic function g"'

has a unique analytic extension

i\

u

to

which is

I

fibre constant: there is a unique complex function with

g o q

=

is

~

.

We have

the complex structure on

glq(U)'-A q(U)

this

=

on

~

U

g

By definition of

~

is holomorphic. So

the first Riemann Extension Theorem is valid:

Q

is a normal

complex space. 4. R

~

If

X, X

i.e.

X

X ~

Y

the projection

p2 : ~ ~ Y

are open, then. So we have

Q

is the usual quotient space of ..... case we have a holomorphic map X ~ Q .

"' • : X

X

~

XxQ

R*= R

X~

XxQ

X . In the general The normalization map

is finite. Hence, the image

X*

~

Q

(x 1 ,x0

We can also prove that x0 E X , )

""X

of

X*

and a holomorphic map

,...

This is finite since we have the factorization

point

Q = X/R,

is an analytic subset and a reduced complex space.

We have a projection

5.

and

gives a commutative diagram:

The map in

is open, since

E Rx

R

~

X*

~

X. X

is semiproper. We take a

an open neighbourhood and an analytic covering

0

X

~

X*

V(x 0

)

c

X , a point

A c p -1 2 (V) n R

with

145

Then bourhood of

S

with

=

S

-1 p 1 (p 2 (V)

p 1 (Rx).

n

R)

is an open neigh-

It is saturated by fibres.

0

So it is the inverse image of an open neighbourhood The imageK~p 1 (A) c X

q(S) E X/R

point have

q (K) ::> W .

is compact and we

This proves the statement.

We get immediately , that

R

is semiproper.

by lifting a semiproper equivalence relation Y .

B=

We have

R* u A,

where

A

components of dimension less than Q x Q

under

less than

Y x Y

Q x Q

~

The dimension of R'

B

X

.

all fibres of

Since

.....

c . The image

~

Q

"' X

-+

Q

'

Q

to

B

of

A

in

is an analytic covering.

c

=

We have

dim Q

:

R

is mapped by

Q

X~

Pz

from

nowhere dense.

is a sutured complex space.

The generic fibre of fibre of

Y

is less than

(Q,R')

R'

is obtained

is the union of irreducible

is discrete andJi·:p 2 (B) c Q

That proves that

R

is an analytic set of dimension

c , since locally

R' = D u B,

W of the

~

X

N

X

-+

X

onto

is open, this holds true for

hence it is true for the fibres

but these are analytic subsets of

a

X

So the map

stricted to fibres is biholomorphic. Outside

B

X*

n*:X*~x

the map

-+

Q,

ren*

is bijective. Since

n*

is finite it is topological there. If

local holomorphic function there,

f

o n*

-1

f

is a

is a local con-

tinuous function which is holomorphic outside a nowhere dense analytic set. Because So

X

is weakly normal, it is holomorphic.

n* : X*IO'~ ~ n*(X*IQ,~))

is a biholomorphic map. The

rest is a nowhere dense analytic set. That means that a proper modification.

n*

is

146

We call 6.

n* : X*

~

X

In the case of the Main Theorem each fibre in

X

(there denoted by

·x)

the

no4mal~at~on

in

Q

of

R

has a well defined multiplicity on its

irreducible components. We can use like in§ 2.1 complete intersections Q/R'

Ai

to construct local ho·lomorphic functions in

which separate the fibres of

is holomorphic and that

X .

This proves that

R'

Q/R 1 is a weakly normal complex space.

So, in our case we can divide by

R'

spaces are not necessary: The quotient

and the sutured complex X/R

is a weakly nor-

mal complex space. But this is also like this in general. It can be seen that a multiplicity can be locally fined in all of the cases of our Theorem. So

resp. X/R X/R

de-

always is

a weakly normal complex space. This proves our Theorem.

147

Bibliography

[Fu]

Fujiki,A.: On Automorphism Groups of Compact Manifolds. Invent.Math.i!, 225-258 (1978).

K~hler

[STCER] Grauert,H.: Set Theoretic Complex Equivalence Relations. Math.Annalen 265, 137-148 (1983) [CAS]

Grauert,H. and R.Remmert: Coherent Analytic Sheaves. Springer Heidelberg 1984.

[Hi]

Hironaka,H.: Flattening Theorem in Complex-Analytic Geometry Amer.J.Math.97, 503-547 (1975).

[Li]

Lieberman: Compactness of the Chow Scheme: Fonctions de Plusieurs Variables Complexes III (Seminaire Norguet). Lecture Notes in Mathematics 670 (1978).

[Re]

Remmert,R.: Holomorphe und meromorphe Abbildungen komplexer R~ume. Math.Ann.133, 328-370 (1957) .•

[St]

Stein,K.: Maximale holomorphe und meromorphe Abbildungen I,II. Ann.J.85, 298-315 (1963); 86, 823-869 (1964).

Recent Developments in Homogeneous CR-Hypersurfaces A. Huckleberry and W. Richthofer Dedicated to Wilhelm Stoll*

1. INTRODUCTION

Let G be a connected complex Lie group and X

=

G/H

a com-

plex manifold homogeneous under a holomorphic G-action. In order to understand X, e.g. how it fits into a fine classification, details of its function theory, etc., one should use as much

Lie theoretic information about

G as

is possible. In

particular it is often useful to study the orbit structure of real subgroups of

G.

Such orbits are usually not complex sub-

manifolds of X. Conversely, if G is a connected real Lie group and one wishes to understand G and its representations, then one naturally looks for orbits M

G/H where strong analytic or

algebraic tools are at hand. A very interesting setting is where X is a complex manifold where G is acting as a group of holomorphic transformations and M

= G/H

is a G-orbit in X. If

1is the Lie algebra of G, then the complex Lie algebra ~

=~

+ i

AJ

(not necessarily a direct sum) is represented as

an algebra of holomorphic vector fields on X. Ideally the vector fields in

0 are

integrable so that we have an induced

holomorphic action of the associated complex Lie group

e.

In

this situation we have the inclusion of orbits G/H ~G/H and it is possible to derive information about M from the complex homogeneous space If M a

~

=

G/H.

G/H is an orbit in X as above, then we refer to X as

-complexification of M. If in addition the Lie algebra

of vector fields induces a G-action, then we refer to X as a G-complexification of M. In either case M inherits a G-invariant Cauchy-Riemann structure from X (see [G],[AHR] for generalities). With this in mind we can refine our line of * Stoll's beautiful results on parabolic spaces were one of the strong motivating forces for the research described in the present paper.

150

questioning: Given an orbit M

=

G/H of a real Lie group, what

are the possible G-invariant CR-structures on M? What are the possible~

-complexifications X so that M - x i s CR-embedded?

What are the possible G-complexifications? Is it possible to find X

= G/H

complex homogeneous so that M - x is CR-embedded

as a G-orbit? If we reach the ideal situation, i.e. X

G/H

and M

=

G/H

is CR-embedded as a G-orbit, then there is so much structure at hand that it is often possible to make very strong statements. These statements can go both ways: The structure of M yields information about X or vice versa. Example 1: Let V := F 1 x Fn be embedded via the Segre embedding inFm' m = 2m-1. This is equivariant with respect to the usual

=

SL 2 (C) x SLn+ 1 (C) action on V. Note that G acts transitively on the complement F 'V =: X = G/H. Let K = su 2 x SU 1 . · m n+ It turns out that the CR-structure of the minimal K-orbit M in G

X reflects the fact that V is not a complete intersection in Fm (see [BFS]).

D

1\

Example 2: Let G be a complex semi-simple group and A 1\

Ha

para-

bolic subgroup, i.e. G/H is a homogeneous rational manifold. Let 1\

G be a non-compact real form of G. Then the generic orbit of G is open and G has a unique orbit M of minimal dimension ([WO]). Theorems from complex analysis lead one to consider certain CR-cohomology spaces on M.

They

are interesting G-moduls,

because they have stable Hilbert subspaces which realize interesting representation theory for G (see [RP] for a special case).

D

We began studying the interplay between CR-orbits and homogeneous spaces from the point of view of the homogeneous space, i.e. given X

1\

1\

= G/H,

what influence do the natural CR-

orbits M4 X have on the classification theory for X? A most striking application arose in the classification of homogeneous surfaces, where Tomilieri's classification of invariant CRstructures on Heisenberg groups ([TO]), the Andreotti-Fredricks embedding theorem ( [AF]) , and •ranaka' s extension theorem for CR-maps of real quadrics ([TA]) provided a fundamental step in the case of solvable groups (see [OR], [H]).

151 On the other

hand, while working in the complex

homogeneous situation, we realized that the CR-manifolds M

= G/H

are important in their own right and therefore began

an organized study of the subject. The purpose of the present paper is to outline the recent developments in this direction so that the reader has a guide to the somewhat technical details in [AHR and R]. Up to this point most of the fine classification results have been proved for compact homogeneous CR-hypersurfaces. However, the basic tools are available for the higher codimensional case. As far as we know, the first results in the classification theory for compact homogeneous CR-hypersurfaces are due to Morimoto and Nagano ([MN]). They considered the situation where M

= G/H

is CR-embedded in a Stein manifold X. In this case one

immediately sees that M

= an,

where Q is a strongly pseudo-

convex open subset of X. In particular the Levi-form of M is everywhere positive dfinite. Since

n

can be recovered as

the envelope of holomorphy of the CR-functions on M, it follows that G acts as a group of holomorphic transformations on

n.

In order to go ahead with their classification theory, Morimoto and Nagano needed the theory of compact groups. Thus they assumed that rr 1 (M) is finite and as a result any maximal compact subgroup K of G acts transitively on M ([S]). Under these assumptions they proved that either n ~ mn' the ball in Cn, and M is CR-equivalent to s 2 n- 1 with its induced structure, or M is a sphere bundle in the tangent bundle of a compact symmetric space of rank 1 and

n

is the tube around

the 0-section with this sphere bundle as boundary. Since the symmetric spaces of rank 1 are completely classified, i.e. spheres,projective spaces over

m,

C, orR, and the Cayley

projective plane, this is certainly a fine classification theorem. Rossi ([R1]) extended the work of Morimoto and Nagano in ···two ways. First, he used the theory of "filling in holes" ([R2]) to show that if M

= G/H

is an abstract strongly pseudo-

convex compact CR-hypersurface with di~M ~ 5, then M is G-equivariantly the boundary of a domain n in a Stein space X. For the same reason as above G acts as a group of holomorphic

152

transformations

on~-

Rossi's methods also require that a

maximal compact subgroup K of G acts transitively on M. Thus he also assumed rr 1 (M) to be finite. His classification goes as follows. If

~is

smooth, then the results of Morimoto and

Nagano may be applied. If not, then it is easy to see that it has exactly one singular point x 0 . Taking a K-equivariant ~ -1 minimal desingularization rr: ~ ~ ~, it follows that Q:=rr (x 0 ) is a K-orbit and Q can be identified with a K-invariant tube neighborhood of Q in this normal bundle. Ampleness criteria show that Q is homogeneous rational and the normal bundle is very ample. Thus the singular cases in the classification arise as follows: Let G be a semi-simple complex Lie group and P a parabolic subgroup. Let Q := G/P and recall that any very ample principal ~*-bundle

over Q is G-homogeneous, i.e. the bundle can be

described by a homogeneous fibration G/H

~*G/P.

Attaching

thew-section and blowing it down to a point x 0 , we obtain A A an affine cone C with vertex x 0 so that C'{x 0 }= G/H. Let K be a maximal compact subgroup of G. Then any K-orbit M

=

K/L

in C'{x 0 } is a strongly pseudoconvex CR-hypersurface. Any two K-orbits are equivalent under the right C*-action and any two maximal compact subgroups are conjugate. Thus the construction only depends on the ample bundle. The only possibility for a non-singular vertex is when the bundle is the hyperplane section bundle overFn, i.e. C = Cn+ 1 ([HO]). Using non-trivial analytic methods and a complicated check of cases in the spherical case, Burns and Shnider ([BS]) showed that rr 1 (M) is finite if M is a strongly pseudoconvex compact homogeneous CR-hypersurface with di~M ~ 5. Thus, except for possibilities in the 3-dimensional case, the fine classification of Morimoto-Nagano-Rossi is indeed complete.

o

In [AHR] we began a study of abstract homogeneous CRmanifolds and applied our methods to the classification problem for compact homogeneous CR-hypersurfaces. One should note that if X is an arbitrary compact complex homogeneous manifold, then M = s 1 x X is a homogeneous CR-hypersurface. Thus a classification of homogeneous CR-hypersurfaces would naturally contain a classification of homogeneous compact complex

153

manifolds. This is of course not possible at the present time, and thus it is necessary to make restrictions. The general theory in fact shows that problems with Levi-flatness, e.g. S

1

x

X, are critical. Thus we began by considering homogeneous

compact CR-hypersurfaces with non-degenrate Levi-form. Of course this contains the strongly pseudoconvex case discussed above. A fine classification is proved in ([AHR]) and described in § 4 of the present paper. There are no restrictions on dimension. Since one can't always "fill in the holes", there are certain cases where there are no G-complexifications (see

§ 3 for examples). However we give an exact description of these cases. The fine classification of compact homogeneous CR-hypersurfaces M with non-degenerate Levi-form is given in terms of a root theoretic description of a canonical fibration of M which has a strongly pseudoconvex fiber and a compact complex homogeneous rational base. The simplest case of this fibration is the s 1 -fibration of M induced by the cone fibration 1\

1\

G/H

~

1\

1\

G/P in the case discussed above. The Levi-form of M,

C* which is itself an interesting invariant of the group theory, can be explicitely calculated ([AZ]). In fact the resulting characteristic polynomials have a very simple form. In § 5 of the present paper we give a complete description of the compact homogeneous CR-hypersurfaces M

=

G/H which

possess a Kahler structure, i.e. there is a CR-embedding M

'"-X in a Kahler manifold. On the one hand, this may be

thought of as a continuation of the classification theory of Borel-Remmert ([BR]) and Matsushima ([M]) in the complex Kahler case. On the other hand, we hope that this is the beginning of a project which will significantly aid in understanding complex homogeneous manifolds. For example, if X

=

e1~ is Kahler, e.g. Stein or quasi-projective and M

=

G/H

is a compact orbit of some real subgroup of ~, then M inherits this Kahler structure. In fact one only needs for some neighborhood of M to be Kahler. For example this is guaranteed if there are sufficiently many holomorphic or meromorphic

154

functions on X. The results in the hypersurface case indicate that the existence of a Kahler structure on M is very restrictive. This then imposes strong funtion-theoretic conditions on the ambient space.

c

2. SOME BASIC METHODS Let M

= G/H

be a homogeneous real analytic generic CR-

submanifold of a complex manifold X with complex structure J. If R is the maximal J-invariant subbundle of TM, then we refer to (R,J) as the induced CR-structure on M. We wish to study how properties of X affect M and vice versa. The "vice versa" needs to be clarified, because only the germ of X along M can be determined by M. However, in many situations involving group actions there is a global relationship, e.g. if X is complex homogeneous and G is a subgroup of

= ~~~

a having compact

hypersurface orbits in X. If X is not complex homogeneous as above, it is often possible to construct an embedding M

=

G/H

'-+

X

1\

'-+

1\

G/H

=

1\

M ,

1\

1\

where M ~M is equivariant. In this case M is called a Gcomplexification of M. There are at least two different ways of approaching the classification of (M,X): 1) Use complex analytic methods and concentrate on X(extrinsic); 2) Forget X and study Mas a real manifold with some additional structure (intrinsic) . One of the basic ingredients of the following is the interaction between intrinsic and extrinsic methods. We begin by giving an intrinsic characterization of a homogeneous CR-manifold. Proposition 1:

Let M

=

G/H

be~

real-homogneneous manifold.

The G-invariant CR-structures (R,J) dence with the pairs

~Mare

in 1-1 correspon-

(R,J) which satisfy the following condi-

tions: (0)

is ~ subspace of 11 with ~1 c an endomorphism;

R

Rc

·'J

and

J

R -+ R

is

155

( 1) JX = 0 if and only i f X E ,_; ~2

~

l..

~

(2) J X + X E ·L for all X E R; (3) Ad X E R and JAd X-Ad JX E ~ for all g E H, X E R; g g g (4) [X,Y] - [JX,JY] E Rand J ([X, Y] - [JX,JY]) - [JX, Y] - [X ,JY] E It for all X, Y E R. Two pairs (R,J) and (R' ,J') are equivalent if and only if

1

R = R' and JX - J'X E for all X E R then (3) may be replaced !?.Y_ (3) • [X, Y] E R and J[X, Y] for all X E

1,

R'. If H is connected,

[X,JY] E ~

Y E R.

c

The proof of Prop. 1 is a straightforward consequence of the definitions (see [R], p. 17). Corollary 2: Every G-invariant CR-structure on G/H is analytic. The following results of Andreotti-Fredricks provide the first steps toward connecting the intrinsic and extrinsic points of view. ~~~~~~~=~

([AF]). Let M be an analytic CR-manifold of type A Then M admits ~generic complexification (M,T). Given a generic complexification (M,T) of M, there are open neigh(m,~).

borhoods U of T(M)

U such

map f : U -+

in~

and

U of

that f · T

T(M) in Manda biholomorphic

= T.

A

Remark: If M is a generic complexification of M and M is of type (m,~) them dim~~

=m

- ~.

~~~~~~~=~ ~

([AF]). Let f ; M-+ M' be an analytic CR-map between A A analytic generic CR-manifolds M c M and M' c M'. Then

~are open neighborhoods U of M in~ and U' of M'~~· and~ holomorphic map ~ : U-+ U' suc~that ~IM =-;. As a result we have the following simple but useful

Corollary 5: Let M be ~analytic generic CR-submanifold of A complex manifold M. Then for every analytic CR-vector field

~

156 1\

X E rCR(M,TM) there is an open neighborhood U of M in M and ~

holomorphic

vector field Z on U such that

=

X

Z)

(Z +

IM.

Given a homogeneous CR-manifold M

=

G/H, Prop.1-Cor.5

show that M is an analytic generic CR-submanifold of a complex 1\

manifold M. For every g E G there are open neighborhoods U , g

1\

vg of M in M and a biholomorphic map f f

g

=

IM

: U ~ V such that g g g g. Taking M to be smaller if necessary, we may assume 1\

that every

~-vector

field on M is the "restriction" of a holo/\

morphic vector field on Mas in Cor. 5. If G acts almost effectively on M, i.e. the ineffectivity of the G-action is discrete, we have an of~

embedding~~

1\

1\

r 0 (M,TM). The complex hull

with r;spect to this embedding is denoted by

called the M-complexification 1\

M is refered to a

a~

of~.

1\

~and

is

Any such complex manifold

-complexification of M

=

G/H.

One of the main methods for studying M is to find G-equivariant EibrationsM

= G/H

~

G/I such that G/I and I/H are

known. We begin by considering the normalizer fibration G/H

~

G/NG(H), where NG(H)

:= {g E GlgHg

-1

= H}.

In some sense

this fibration factors out the group manifolds which are equivariantly contained in G/H. Of course we must take the CRstructure into consideration. This is done at the Lie algebra level and therefore the fiber is only locally a group. With Prop.1 in mind it is natural to define NCR(H), the CR-normalizer of H, as NCR(H) := {g E GlAd X E R and Ad JX- JAd X g g g

c't

for all X E R}.

Using Prop.1, a direct calculation shows that NCR(H) consists of the g E NG(H 0 G/H 0

~

G/H 0

,

)

such that the right-translation aH 0

~

agH 0

,

are CR-mappings with respect to the CR-structure

on G/H 0 which is induced by the covering map G/H 0

~

G/H.

We are now in a position to give intrinsically defined fibrations of G/H which are reasonable in the category of CRmanifolds.

157 ~Q~~~~~=g·

Let M

=

G/H be a homogeneous CR-manifold and let

L c NCR(H) be a closed subgroup such that H c L and

J ( R, n R)

c R,

n R. Then there is ~ unique G-invariant ~

CR-structure on G/L such that G/H

G/L is a CR-submersion. []

J R and

Using the properties of

J ('1! CR n R)

that

1icR n

c

Corollary 7. There is G/NCR(H) such that G/H

described in Prop.1, it follows thus we have

~unique

~

G-invariant CR-structure on ~

G/NCR(H) is

CR-submersion. []

If we look at NCR(H)/H as a submanifold of G/H with the induced CR-structure, then NCR (H) "normaJ.izes" the CR-structure. As a result we have Proposition 8.

The fibers of the fibration G/H

~

G/NCR(H) are

Levi-flat. Moreover the distribution generated £y the maximal complex subspaces tangent to the fibers of G/H

~

G/NCR(H) is

contained in the Levi kernel of M. []

·'1

We now assume M is embedded in a Let

n : G

~

1\

-complexification M.

G/H denote the projection and 0 : = n (e) . It is

useful to note that the isotropy Lie algebra 1\

~ :=

{Z E

1\

1

I Z(O) = 0} is directly connected with the CR-

structure of G/H. To see this ;:ecall that J

1\

~~

= A•J

+

Jx1,

where

is the complex structure on M and l'J is regarded as a real 1\

subalgebra of

JX(O)

=

1\

r 0 (M,TM). Hence Z =X+ JY E

(\

1

if and only if

Y(O). This means that, considered as elements in

TGe, X and Y are contained in Rand JX

=

Y (mod :. ) . Elementary

calculations then show that NCR(H) where Ad : G ~

= {g/\E Aut(~·:)

1\

1\

G I Adg h( f,( }, 1\ denotes the adjoint action of G on '''

which is induced from the usual adjoint action of G on tj. This (extrinsic) argument shows that G/NCR(H) naturally inherits a G-invariant CR-structure from the Grassmannian 1\ defined by the complex subspaces of ,.. which have the 1\

I

same dimension as :.1 : NCR (H) is just the isotropy N at \ the "point" i'l· Using PlUcker-coordinates we therefore have a G-equivariant map

158

G/H -+ G/ N-+ lPk («::) . This map is also given by the holomorphic sections of the A

anti-canonical bundle of M which are generated by the

A

~-sections

and is therefore a CR-map. We refer to it as the o:-anticanonical fibration of G/H. Since N

= NCR(H)

and G/H-+ G/HCR(H) is a

CR-submersion, the map G/NCR(H) -+ G/N is CR, but in general its inverse is not. However, if G/H is a CR-hypersurface, is totally real, or is a complex manifold, the map G/NCR(H) -+ G/N is a CR-isomorphism. The equivariant embedding G/N

~

lPk(«::) yields a representation

of Gin PSLk+ 1 (C). Let G denote the smallest complex Lie subgroup which contains the image of G (Recall that G is connected!), and let N be the G-isotropy group at the point N. Consequently we have G/H -+ G/N

'-+

G/N

'-+

lPk ( «::) ,

where G/N is a generic CR-submanifold of the complex homogeneous manifold G/N. Since N c: NG (H 0 ) , the fiber N/H 0 of the ';'f -anticanonical fibration can be written as the quotient A/f, where A = N/H 0 and r := H/H 0 is a discrete subgroup in A. Moreover the ~-anti canonical fibration of N/H is degenerate, i.e. the base is a point. We call such a manifold flat. Indeed flat implies Levi-flat, but not vice versa. From now on we focus our attention on homogeneous compact CR-hypersurfaces. In this case either G/N is projective rational and N/H is

~

compact flat CR-hypersurface or G/N

is ~ compact hypersurface in

G/N

and N/H is ~ compact

parallelizable complex manifold. In the latter case there is not always a G-complexification of G/H (see

§ 3). However,

in the first case we have ~~~~~~~=~·

Let G/H be

~compact

homogeneous CR-hypersurface

where G is acting effectively. If the base G/N of the

tj-

anticanonical fibration of G/H is projective rational, then there is

~

""

G-complexification G/H of G/H and

equivariant diagram of CR-maps

~

commutative

159

-

G/H

l

1\

1 1\

G/N 1\

where N

1\Q

= N&(H

--+

1\

G/H

1\

G/N

).

c

For details of the results in this section see [R].

3. THE BASE OF THE 1-ANTICANONICAL FIBRATION OF A COMPACT HOMOGENEOUS CR-HYPERSURFACE. Recall that if M

=

G/H is a homogeneous compact CR-hyper-

surface, then we have the

~-anticanonical

fibration

G/H -+ G/ N -+ lPJC In this section we make some detailed remarks about the base. Thus we assume that G is a real group of linear transformations 1\

acting onlPk' G is the smallest complex Lie group in PSLk+ 1 (C) which contains G, and that the G-orbit M = G(x) = G/H is a 1\

real hypersurface in the G-orbit X

1\

= G(x) =

1\

1\

G/H. Without loss

of generality we may assume that G is acting almost effectively on M. Note that the base of the

.lJ -anticanonical

fibration

either satisfies these conditions or is a compact homogeneous rational complex manifold. The latter situation is well-understood. The following is the fundamental for the classification of the linear situation described above.

E~~g~g~~~~~=l· ~

Either M is the equivariant product s 1 X Q of circle and a complex homogeneous rational manifold or the

~-simple

part Kss of any maximal compact subgroup K of G

~transitively

on M. In the latter case rr 1 (M) is finite.

The proof of this fact goes by induction on dimension and uses fibrations of the complex homogeneous space X

1\

1\

G/H.

We assume that the Levi-flat case M = s 1 x Q is wellunderstood and only consider the latter case in the above proposition. Hence we assume that G = K is a semi-simple compact linear group. Of course AutCR(M) may be much larger

160

= alBn,

e.g. M

but going to the possibly smaller compact group

has distinct advantages. In the usual way we have M = K/L

4

S/H =: X

4

]pk

I

where M is a hypersurface in X. Since linear semi-simple groups are algebraic, H is an algebraic subgroup of S, and X is Zariski open in its closure inlPk. It is therefore enough to understand algebraic homogeneous spaces S/H of complex semi-simple linear groups where the general orbit of is

~

~

maximal compact subgroup

real hypersurface.

If X

= S/H

is Stein, then we are in the case handled by

Morimoto and Nagano which was described in§ 1. In fact X is then the tangent bundle of a symmetric space of rank 1. If X is not Stein, i.e. H is not reductive, then H is contained in a proper parabolic subgroup of S. Let P be a minimal such subgroup. Since KP has hypersurface orbits in the fiber P/H of S/H

~

S/P, we are in a good inductive situation and it is

relatively easy to prove E~~g~g~~~~~=~·

Let S be a linear semi-simple· complex group and

H an algebraic subgroup. Suppose that the generic orbit of

~

maximal compact subgroup K of S in the homogeneous space X

=

S/H is

~

real hypersurface. Let P be

subgroup of S containing H (P

=

~

minimal parabolic

S is allowed) . Then P/H is

Stein. This result had already been used in the classification of certain almost homogeneous spaces

([HS],

[A]). In [AHR] it is

pointed out that the "Stein-Rational" fibration S/H

~

S/P is

essentially unique and exactly reflects the structure of

M

=

K/L.

The only cases where P is not unique arise from building the following type of example inside of

s.

Therefore all possible

such minimal parabolic groups can be described (see [AHR]). -t ~~giDgl~· Let S = SLn+l (~) act on 1Pn x 1Pn by A(p,q) = (Ap , Aq). There are exactly twoS-orbits inlPn X

=

x

1Pn' i.e. an open one

S/H which is the complement of a divisor orbit defined

by { (v,w)

I vt

· w

=

0}. There are exactly two minimal parabolic

161

groups, P 1 and P 2 , which contain H. They just arise by projecting on the respective factors of lPn x lPn: en S/H -+ S/Pi = lPn , i = 1, 2 . Given that M

=

c

K/L is described by its complexification

S/H and that S/H breaks into Stein and rational parts, i.e. P/H and S/P, it only remains to describe the possible pairs (H;P) coming from a given S. The situation is so concrete that in principle one could calculate virtually anything one needs, e.g. analytic cohomology as in [BFS]. So far calculations have been carried out in two directions: 1) A detailed root-

,

theoretic description of all pairs (H,P)

([AHR]); 2) An

explicit calculation of the Levi form of M ([AZ]). Since these results are a bit technical, we only discuss here an important case of 1). Consider the class of algebraic manifolds X

S/H as above.

Assume further that a Stein-rational fibration has the form S/H-+ S/P with P/H

=

en, i.e. a homogeneous affine bundle

where the generic Kp-orbit in en is a hypersurface.·~nder these assumptions it is easy to see that P is

repre~ented

in

the complex affine group, that Ru(P) is represented as the full group of translations en, and the reductive part of Pis represented as GLn(C), SLn(C) or Spn in the standard_ way. The latter case can only occur when n is even. Since Kp has a fixed point in P/H, we may take the reductive parts of P and H in a Levi-decomposition to be the same, i.e. P

=

L

~

Ru(P) and H

=

L

~

Ru(H), and Ru(H) is obtained

from Ru(P) by removing n 1-dimensional root groups: Now L is determined by a set TI of simple roots, i.e. the simple factors in L are determined by orthogonal connected chains in the Dynkin diagram for S. We write n as a disjoint union n 1 U n 2 , where the simple factors coming from n 1 yields the factor which is either represented as SLn or Spn. It is not difficult to show that P contains exactly one simple root group which is not in H. Let a be the root corresponding to this group. Then Ru (P) and

=

Ru (PTI U {a}) Ru (Pn 2 U {a})

162

Ru(H) = Ru(Pll U {a}) Ru(Pll U {a})', where P 0 is the parabolic group c~rresponding to the set a of simple roots. Of course n 1 must be orthogonal to n 2 U {a}. Given the above information one can write down the possible diagram. Since n 1 is orthogonal to the connected chain n 2 U {a}, it is enough to describe the latter. The following is a list of all possibilities where the "white" circles represent the roots in n 2 . 0

--

---

0

0- -

a

0

0

a o- --o_:( o

• ••

-

0

>-. a

--e

a

~ ~~g~g~~· Here we discuss G-invariant CR-structures on s 3 . The first such structure which comes to mind is M = am 2 , the c2 with the induced structure. Let

boundary of the ball in K

= su 2

act linearly on ~ 2 as usual. In this way K acts freely

and transitively on M and therefore inherits a left-invariant A

CR-structure. Since K

=

SL 2 (c), one easily finds a G-complexi-

fication, S

3

A A 2 = M = K/L ~ K/L = C' {(0,0)}

,

and the k-anticanonical fibration is just the restriction of the Hopf fibration,

£2 '

A

{ (o,o)}

A

K/L

c1

A

K/B

= lP 1 ,

to M. Now K is not the full group of CR-automorphisms of M. However, this group G is easy to describe: By analytic continuation G = AutCR(am 2 ) ~ Aut 0 (1B 2 ) = PSU(3,1). It should be remarked that the induced Hopf fibration of M is not G-equivariant. The •1-anticanonical fibration is just the embedding inlP 2 induced by the standard inclusion lli 2 ~ lP 2 . In fact G = PSL 3 (~) and we again have a G-complexification, am 2 = M = G/H

A A

<-+

G/H = lP 2 .

We now construct a family {Mr I r > 0} of su 2 -invariant CR-structures on s 3 which are pairwise inequivalent and which are not equivalent to the above structure M. For this let X:= Q( 2 ) be the affine quadric which is

so 3 (~)-homogeneous

163

via the standard representation of 8L 2 (~)

80 3 (~). Recall that the universal cover of 80 3 (~). Hence we can

G is

=:

" " where H " is the group of diagonal matrices write X = Q( 2 ) = G/H,

G.

in

1 r

~

Let gr := ( 0 1 ), r

" -1 in G/H. " " grHgr Then {Nr

orbits of K

=

8U 2 . N0 manifold, and Nr ~ JP 3 r

>

0, and Nr be the K-orbit of the point

Ir

=

~ 0} is a parameterization of the

8 2 is embedded as totally real sub-

(IR)

as a real analytic manifold. For

0 let Mr be the universal cover of Nr. Equip Mr with the

CR-structure corning from the covering Mr

~

Nr and lift the

K-action. It follows that M is K-hornogeneous, is ~-analytically 3 r equivalent to 8 , and K acts freely and transitively. Thus, for each r > 0 we have a left-invariant CR-structure on K. Identifying Mr with K ~-analytic

~~wmg=l·

= 8U 2 , for every g

automorphism int(g), h

Int(g)

: Mr

~

is~

Ms

~

ghg

-1

E K we have the

, of Mr.

CR-rnap if and only if r

If int(g) is~ CR-rnap and g is not the identity, then r 1 0 -1 0 1 1 -1 -1 and g E { ( 0 1 ) , ( 0 _ 1 ) , ( _ 1 1 ) ' ( 1 -1 ) } ·

= s. = s

a -5 = {(b a:l, Ia 12 + Ib 12 = 1}, and q> := Int(g). Then CR-rnap if and only if q>* : TCR(M )~ TCR(M ) and q> is a e r e s \P* o Jr = Js o \P*. In this case q>* = Ad(g), Proof. Let g

TCR(M) = (( e r and Jr(v) = wr.

(0-1), 1 0

(-ir-1 i ) i ir-1

))JR.=:

The condition q>* : TCR(M )~~ TCR(M ) can be formulated as e r e s follows: ( 1) 2s Irn{ab} = -Irn{b 2 + 2 }

a

so that q>*(v) = Re{z} · v 2 -2 where z = b + a ,

+

Irn{z}

ws'

and

-2 2 -1Re{a -b -2r ab} so that where w

q>*(wr) Irn{w} •V + Re{w} ws' = a2 - b 2 - 2r- 1 ab.

164

The condition that

~*

commutes with the J-operators can be

spelled out as follows: - Im{z} · v + Re{z}·w

=- Im{w} · v + Re{w} · ws. s · r = - ab. Hence b = 0 or rb = - a. If b = 0, then (1) implies that Im{a 2 } 0. Since lal 2 +lbl 2 1,

Equivalently, w = z or b

2

it follows that a = ± 1 • Consequently g

±I and r = s. -2

If rb = - a, then ( 1 ) again implies that Im{a } = 0. Thus a, b E lR. Applying ( 1 ) and (2)' we see that IP*(v) = v and ~*(wr)

= ws.

Now~*=

Ad(g) and therefore application of IP* a -:6 amounts to conjugation with the matrix g = (b a). In particular,

det(wr) = det(ws)' i.e. r = s. Finally, a concrete calculation of centralizers shows that unless Int(g) = I we have r = 1 and g is in the group of order 4 in the statement of the proposition. ~:

Since a CR-isomorphism

Mr

~

[]

Ms induces an automorphism

of AutCR(Mr)' it is useful to note ~~miDg=~·

Let SU 2 act~ Mr as above. Then AutCR(Mr)

Proof. Set G := AutCR(Mr)

0

and consider

0

=

su 2 •

the~-anticanonical

fibration M = G/H

~

G/N.

The fiber is infinitesimally defined by the isotropy algebra 1\ 'II.. :

=

'YI-1

1\

1\

('t,) • Since ~ => sl 2 (C)

and the anticanonical fibration

for the affine quadric is finite G/H

~

(in fact 2-1), the fibration 1\

1\

G/N is likewise finite. Thus G/N is a homogeneous

surface on which SL 2 (c) acts and has an open orbit. Fer the same reason as above, the sl 2 (C)-anticanonical bundle of this orbit is finite. The classification of homogeneous surfaces 1\

then shows that G = SL 2 (c) (see [HL], [OR], [H]). Thus, since 1\ G => su 2 and G is a real form of G (Mr is strongly pseudoconvex!), [] it follows that G = SU 2 . Remarks (1) Since

su 2

acts transitively on Mr' no other Lie

group of CR-transformations acts transitively. (2) Even though Mr is strongly pseudoconvex, one can not "fill in the hole"

(see [AHR]).

(3) There is no G-complexification of Mr' because Mr G/H 1\ 1\ 1\ 1\ would be contained in G/H, G = SL 2 (c) and dimcH = 1. A simple

165

"

check shows that, no matter which 1-parameter group H is chosen, the generic G-orbit is not simply-connected. However, 0

~~QQQ§itiQD_J.

;=~~~i;~~~~lly N = M = am 2 and

Let N be

CR-hypersurface which is 3 the 3-sphere s • Let G = AutCR{N) 0 . Then either ~homogeneous

G = PSL(3,1) or N = Mr for some r > 0. In the

latter case G = su 2 . The CR-manifolds M, Mr' r

>

0, are pair-

wise different. Proof. Since the su 2 -anticanonical fibration of M has positive

*

dimensional fiber, M Mr' r > 0. Analytic continuation arguments show that AutCR(M) 0 = Aut 0 0B 2 ). Thus it remains to discuss the manifolds Mr. Suppose that

~

: Mr

~

Ms is a CR-isomorphism. We may assume

that ~(e) = e, where Mr and Ms are E-analytically identified with K = su 2 and e is the identity inK. Define~ E Aut(K) by g ~ ~g~-1. Now Aut(su 2 ) = Int(SU 2 ) ~ ~ 2 , where ~ 2 is generated by complex conjugation b. Thus ~ = int(h) or ~ = int(h) o b for CR some h E K. Note that b stabilizes Te (Mr) for all r. Since ~(g)· f = ~(g · ~- 1 (f)) for all f E K and ~(e) = e, it follows that ~(g) = ~(g) · e = ~(g) for all g E K. We have already shown that~= int(h) implies that r s (Lemma 1). Thus it remains to handle the case where

~(g)

= int(h) (g). We do

this by carrying out essentially the same calculations as in the proof of Lemma 1. Using the fact that ~*: T;R(Mr)~~ T;R(Ms) we obtain exactly the same equations as in Lemma 1, i.e.

(1) and {2). Further, ~*Jr = Js~* yields z = -w. Thus in this case ra 2 = ab and arguing along the same lines as in Lemma 1 yields r = s. o

166

4. THE FIBER OF THE1-ANTICANONICAL FIBRATION; CLASSIFICATION AND THE CASE OF NON-DEGENERATE LEVI FORM. The aim of this section is to understand the fiber N/H of the 2r -anticanonical fibration M

=

G/H

-+

G/N. Without imposing

further conditions it is impossible to say more than we already know, i.e. N/H is either complex parallelizable or a flat hypersurface. Thus we will impose two kinds of further conditions which we believe to be quite natural from the complex analytic viewpoint: We first consider the case where the Levi form of M

=

G/H is non-degenerate. Secondly, without

any Levi condition, we assume that G/H admits a Kahlerian complexification, i.e. a Kahler manifold X which contains G/H as a real analytic compact hypersurface. In this case we refer to M

=

G/H as a Kahlerian homogeneous CR-hypersurface.

Kahlerian structures on CR-manifolds have been primarily studied by methods

coming from Riemannian geometry (see [KY]) .

It is however our aim to handle this situation by purely group theoretic and complex analytic techniques. The first part of this section is devoted to the study of a compact homogeneous CR-hypersurface M

=

G/H where the Levi

form of M is non-degenerate, i.e. no zero eigenvalues. The basic tools for this are Prop. 2. 8, Thm. 2. 9 and the discussion just prior to Thm. 2. 9. So let us have a look at the )-J-anticanonical fibration G/H

-+

G/N. If G/N is a CR-hypersurface, then N/H is a complex

manifold and 3. 8 (the "moreover" part) implies that N/H is finite. If G/N is a complex manifold, then there is a 1\

1\

G-complexification G/H of G/H as described in 2. 9. In this case N/H is a flat CR-hypersurface. Again by 2. 8 it follows that N/H

= s1

(Note that N is connected, because G/N is simply-

connected.). From the proof of 2. 9 (see [AHR] for details) '

1\

1\

it is clear that G,H can be chosen such that N/H

= C*.

Summing up we have the following ~Q~~~~W=l· Let G/H be ~compact homogeneous CR-hypersurface with non-degenerate Levi form. Then either G/N is projective

167

rational and G/H ~ G/N is an ~

equivariantly embedded in ~

pal bundle over G/N,

s 1 -cR-principal ~

G/H

bundle which is ~*-princi

homogeneous non-trivial G/N is finite.

o

Remark. Together with the results in 3., Theorem

gives a

complete fine classification of homogeneous compact CRhypersurfaces with non-degenerate Levi form. Now assume that M

=

o

G/H is a Kahlerian homogeneous CR-

hypersurface. If the base G/N of

the~

-anticanonical

Fibration of G/H is a hypersurface, then N/H is obviously a compact complex homogeneous parallelizable manifold, i.e. every connected component of N/H is a compact torus [W]. If the base G/N is not a CR-hypersurface, then it is projective rational. In this case by 2. 9 we have a G-complexification G/H so that G/H

·--~

A A

G/H

1 C5 l

G/N

--->

e~~ A A

where N/H is a flat CR-hypersurface in N/H. Since G/H lives in a Kahler tube X and since the germ of a generic complexification A A

is unique, there is an open neighborhood U of G/H in G/H which is Kahlerian. Thus the same is true for N/H A A

=

N/H

A A

A

A/r , where A

=

A AQ

N/H

A

and r

stand the following situation:

Gbe

Let

a complex Lie group,

~

A A

N/H. Now

A AQ

H/H . Hence we must under-

A

r

A

< G a discrete subgroup

A

A

and G a Lie subgroup of G such that the G-orbit G/r of r A A

in G/r is a compact hypersurface admitting a Kahlerian A A

neighborhood in G/r. Before stating the result in this setting, we want to A A

*

discuss two special cases. First, assume that O(G/rl Let

11:

Now :::

:

= 'Y n i ''I

A ' · <1 ''r and, since

dimensional in A

A

"I. '

A

.

G is a hypersurface in G, •:· is 1-co-

Let G , denote the normal complex subgroup "'·

of G which is generated by m. Since G A

C.

be the maximal complex subspace of Jl-'f in ·,,

f>L

< G, the G -orbit of ''>t

r is contained in the compact hypersurface G/r . Let A A f E 0 (G/H) be non-constant and note that f 1 G/r is bounded. Consequently the restriction of f to a G~,L-orbit in G/r is constant, and, since the fibers of f and the G,,-orbi ts are

168

1-codimensional, it follows that the G_-orbits are closed.Hence 1\

1\

*

if O(G/H)

"'-

~,

then we have 1\

1\

G/f----+ G/f

l

J

(J

1\

G/G,nf- G/G r -~ 1\ /\"'" It follows that G/G_f = 81 ~ G/G'l'ltf = C*. Thus G/f is a torus ··o bundle over 8 1 which is equivariantly embedded in a torus 1\

bundle over C*. In particular G is abelian or 1-step solvable. The second special case is where G is simply-connected and 1\

abelian. In this case G

=

n 1\ C and r is a discrete subgroup of

rank 2n or 2n-1 and r has rank 2n-1. We then have

= E2n-1;r2n-1 ~

G/f

Cn/r2n-1

=

G/r

1\

1\

and there is always a fibration 1\

G/f

-+

1

G/f

/\t ,

G/L ..-.:'--+ G/L where 8 1

= L/f ~ ~/r

G/L G/L is a torus (see and e.g. [V]). Thus if G is abelian, we always have an 8 1 -CR1\

/\

fibration of G/f onto a compact torus. Note that if r is of 1\

1\

rank 2n and G/f is an irreducible torus, then this fibration 1\

1\

is not induced by a fibration of G/f. 1\

1\

1\

If G is abelian and simply-connected and O(G/f) 1\

rank r

=

*

C, then

2n - 1 and the above shows that

=

G/f 1\

I f 0 (G/f)

=

8

1

X

~

G/L

C*

1\

X

1\

G/L

1\

1\

G/f.

1\

C, then G/f is called a Cousin group.

With these preparations we can.now state the main results in the flat Kahler case. For this it is convenient to refer to 1\

1\

(G, G, r, f), G/r

1\

-+

1\

G/r as above, as the data of a "flat Kahler

CR-hypersurface, FKH". 1\

~g~~~~m=~·

1\

1\

Let (G, G, f, f) be the data of FKH. Then G is A 1\ solvable and there is a closed normal complex subgroup I c G 1\

such that I

1\

0

is contained in G, is abelian, with I

a•~ c~If G~ admits -1\ 1\ = G r.

can choose I

a

non-con~ant CR-fun~n,

=

1\

I 0 f and

then one

"ltl_.

We now state the main structure theorem for flat Kahler CR-hypersurfaces.

0

169 ~Q~~~~~=~·

~ fol~owing

statements are equivalent: (G, G, f, f) is the data of FKH;

i)

G~~

ii) GnLis abelian and either a-is abelian or the

------_, ,

orbits in G/f are closed and the image of the

representation of p : f-> Aut (G_,J, p (y) (g):= ygy is finite. A

Remarks:

A

(1) If (G, G, f, f) is the data of a FKH and

A

O(G/f)=~

A

then Theorem 3 shows that G is abelian. A

A

(2) If G is non-abelian, then 0 (G/r) *C. The holomorphic reA

A

duction G/f -> G/G,.,J is a torus bundle over C* which admits a finite covering which is trivial. Conversely, given a homogeneous torus bundle which is trivial after going to a finite cover, the preimage of s 1 is a FKH. D

Sketch of the Proof of Theorem 3. Using Theorem 2 and the fact that there is a G-invariant measure on G/f, standard integration arguments yield a neighborhood of G in

e which

is right

G-invariant and admits a right G-invariant Kahler metric. Explicit calculations show that nilradical of

e and

e is

abelian or Gm is the

G. In the latter case the Gm-orbits in G/f

are closed ([MOS]). Furthermore, direct calculations show that ad(~)

has purely imaginary spectrum and, since p(f) stabilizes

a full lattice, it consists of torsion elements and is therefore finite.

D

The proof of Theorem 2 is carried out by induction on di~G/H

(see [R]). To give a taste of what is done, we give a

sketch of the proof of one step in the solvable case. Proposition 4. Let

e, G ,~and

f be given as in Theorem 2.

Assume the following: A

1)

G and G are solvable and simply-connected;

2)

O(e/~)

=

C;

3) The center of

e is

discrete;

4) There is an abelian connected complex normal subgroup ~ of

e such

that ~~ is closed in A

A

e and e•c ~A

A

Then there is a closed complex subgroup I c G such that f c I,

170

A A A I/f is connected and I 0 is

~

non-trivial complex Lie subgroup

of G'~~',.: Sketch of the Proof of Proposition 4. Since~ is an abelian normal subgroup of a simply-connected solvable group, it can be identified with its Lie algebra. Consider the map A A A -1 A int : G ~ Aut(N) ~ GL(~), int(g) (n) = gng • Since N is abelian thisfactors through the abelian quotient e;~ and induces arepresentation A A p : G/N

~

A Aut(N).

A

If N c G, then there is nothing to prove. Thus we assume that A

A

N ¢ G. In this case N = N n G is an abelian normal subgroup of A A A A G (Note that G' c G~~c G!) and G/N = G/N. Let H := Nf/N. Since H is a lattice in G/N

= e;~,

it follows from 3) that

A

p (H) c Aut (N) is no·i:: unipotent. Hence there exists h E: H such that adh

=

p(h) -

id~

is not nilpotent.

Now consider the following chains of subspaces in ~: A A Ak Ak+1 No N, N adh(N ), k ;;: 0 k No N, Nk+1 adh (N ) , k ;;: 0 No

.....

N"'l.

=

N n G ' Nk+1 •n

.,_

k adh (N~, k ;;: 0.

It follows that k c Nk c •.•.. Nk+1 c NAk+1 c N.,_ Ak Now adh is not nilpotent. Thus the chain (N ) kEN becomes · f or some k 0 . Th erefore Mko ~s · a complex L~e · sub group stat~onary A ko A of G""'1. and one can show that I = N · r does the job. D

The proof of Prop. 4 shows one of the main difficulties in studying homogeneous CR-manifolds: One can rather easily find fibrations by real groups, but it is much more difficult to find the appropriate complex subgroups. D

171 5. CLASSIFICATION OF

K~HLERIAN

CR-HYPERSURFACES

In this section we describe a fine classification of homogeneous compact Kahlerian CR-hypersurfaces M 1\

a Kahlerian tube with M

->

A

= G/H.

Let M be

M and look at the 1-anticanonical

fibration G/H ... G/N of G/H. We first consider the case where G/N is projective rational. In this case it follows from 1\

1\

Thm. 2.9 that there is a G-complexification G/H with an open 1\

1\

Kahlerian neighborhood U of G/H in G/H and we have the diagram 1\

1\

G/H -+ G/H

+

1\

G/N 1\

where N

+1\

G/N

1\

1\

NG(H 0

).

1\

By Thm. 4.2 we know that N/H 0 is solvable

and that we have fibrations 1\

N/H

--+

+ N/I +S1

1\

-->

1\

N/H

+1\

N/I

N/L where N/I

=

1\

1\

1\

1\

N/I is a torus, I/H is an abelian group, and N/L

is a torus. Thus we have the following picture: 1\

G/H + G/I +S1

1\

-->

G/H

-

+ 1\ 1\ G/I

G/L + G/N

-

1\

1\

G/N.

The most striking question at this point is whether or not G/L is Kahler. A theorem of Blanchard ([BL]) implies that G/I 1\

1\

1

is a Kahlerian hypersurface in G/I. So we have an S -CR-principal fibration of G/I onto G/L which extends to a holomorphic 1\

1\

submersion on some Kahlerian open neighborhood U of G/I in G/I. This does not in general imply that the base of the s 1 -CRprincipal bundle is Kahler.

172

Example (F. Lescure). Take a E ~*with lal< 1 and let

r :=[(:nan

a~)

fibrations:

E GL 3 (C) In E

3

c . . . {0}

J.

'l

Consider the following

1\

GL 3 (~)/H

+C*

T 1\

GL 3 (C)/N

JP2 (C)

where

and 1\

N 1\

1\

={(~

aAb)

\ E ~*, a,b E: C, A E GL 2 1\

(c)} ~

1\

fH/H ~ 7L and T N/fH = ~*I 7L is a compact complex torus. Now'let E = c 3 ..... { o} xJP (~)x denote the fiber product of 2

Obviously

the bundles ~ 3 ..... {0} ~JP 2 (~) and X ~JP 2 (~). Thus we have the following diagram of holomorphic fiber bundles:

=0, the bundle E

~ c 3,{0}

is

~*

triviaL Thus E is Kahlerian. The restriction of E ~ T to T x s 5 c: E is an s'-CR-principal bundle over X·. But X is not Kahler.

a

The critical point in the above example is that the s 1 is fibered out in the wrong direction. In order to make this precise we need to introduce some notation. Let X be a Kahlerian complex manifold containing a compact hypersurface M. Let J be the complex structure on X, g a J-invariant (i.e. Hermitian) Riemannian metric, and w(X,Y) =g(X,JY) its associated Kahler-form. For p E M Ep : = {X E ™p I w (X, Y) = 0

v

y E ™p}

173

=

is 1-dimensional and transversal to RM , i.e. TM p p Ele~entary

RM e E . p p

differential geometric techniques lead to the

following result. Proposition 1. Let M ~

be~

in~

real hypersurface

Kahler mani-

holomorphic map onto ~ complex manifold Y, and assume that rrlM is an s 1 -CR-principal bundle. fold X. Let rr : X

Y be

~

If one of the following conditions hold, then X is Kahler: 1) M is Levi-flat; 2) ker(rriM)*

=

E

for all pEM.

p-

[J

The above shows that in order to show that G/L is Kahler s 1 -fibration in the "E direction". By

we need to find an

p

1\

1\

induction this is reduced to the case where N/H 0 is abelian. 1\

In this situation one has

th~

N/H-CR(resp. N/H-holomorphic)1\

principal bundle G/H

~

1\

1\

G/N(resp. G/H

1\

~

1\

G/N). Since G/N is

projective rational, a maximal semi-simple compact subgroup K of G acts transitively on G/N. Thus the group A

=

N/H

x

K acts

transitively as a group of CR-transformations on G/H and we 1\

may write G/H

1\

A/D. Since A is compact and also acts on G/H,

we may assume that the Kahlerian neighborhood U and the Kahler-metric w are A-invariant. Let rr :

A~

A/D denote the projection and let 0 := rr(e). -1

~

For a subspace v c T (A/D) 0 let V denote the space rr* (V) c Since the Kahler-farro w is A-invariant, it determines an element F c H2 (.ct, m) given by F = rr* (w JA/D) JN'(. Note that C

= N/H

is the connected component of the center of A, and

= -t

~

e ,1;0 • Let p 1 :a~ ,c and it. denote the projections. Now [.k, ,l] = k, H2 (./i,m) = 0,

=

0 for

thus the Lie algebra splits, P2 :/.'If, and d~

/.Jt..

/./t

all~ EC 2 (~,m). Hence, elementary calculations

Pi

2

show that F = p1(F 1 )+ (dF 2 ), where F 1 E H (..c,m) and F 2 E C 1 (k,m) . By the non-degeneracy of the Killing form B on k, we may write F 2 (X) F(X,Y)

=

= B(X,W)

for some WE.fv. Thus we have

F 1 (p 1 X,p 1 Y) + B([p 2 X,p 2 Y],W).

It follows that E0 = p 1 (E 0 ) + p 2 (E 0 ). Furthermore, since the C-orbits in A/D carry a CR-hypersurface structure we obtain E'0 = p 1 (E0 ) + d . The above formula also implies that E'0 n k is

174

the centralizer of W E ~ and D is contained in the centralizer of the torus in K which is generated by W. Now, excluding the Levi-flat case in which we show A/D CR N/H

x

G/N and in which G/L is Kahler (Prop.1), we show

E0

= p 1 (d-) + rL. From this and Prop.1 it follows that the fibration with the group < exp (E 0 n k) > • D yields an S 1 -CR-principal fibration G/H ~ G/L where G/L is Kahlerian. that

It is now possible to state the first result in the classification of G/H. ~Q~~~~ID=~·

Let G/H be

wh~

surface,

~

homogeneous compact Kahlerian CR-hyper-

'/1

G is connected and the base G/N of the

-anti-

canonical fibration of G/H is projective rational. Then there

Gwith

exists a complex Lie group one has ---

G/H -1-

G/N

-

G~

N/H 0 is solvable, and

a;~ -1-

a;~

as in Thm. 2.9. Moreover there is a closed complex Lie group

7\ A

A

I c N such that I

~compact

Hc G n

~

/\--/\0

c N · H0

,

7\

-A

A

A

--

1\7\

N'•H c I, I 0 /H 0 is abelian, I/H is

torus, and there is a closed subgroup L c G such that ~ ~ G/L-is an S 1 -cR-principal b~e over

c L c N and G/G n

$

the homogeneous compact Kahler manifold G/L. If

c

1\1\1\1\-

a is

a

7\7\

closed complex subgroup such that IcJ, J/HcG/H, and J=J 0 1\

1\

1\

1\

•

I',

1\

then J c N and J has the same properties as I. c

The remaining case is that in which G/N is a CR-hypersurface. s 1 xQ (Q projective

In this situation we know that either G/N =

rational) or a maximal compact semi-simple subgroup K of G acts transitively on G/N (see Prop.3.1). In the latter case rr 1 (G/N) = 1, ~ 2 (see [AHR]). In both cases the components of N/H are compact complex tori. The fine classification in this case is handled by methods similar to those in the proof of Thm. 2. For this consider the bundle G/N° n H torus principal bundle. Again A:= N°/N° n H

x K

K

-+

G/N°, i.e. a

acts transitively on G/N° and

acts transitively on G/N° n H. Using F as in

the proof of Thm. 2 and observing that the orbits of

175

c

N° /N° n H are complex, one shows that

We close by stating the main result on the bundle structure of a compact homogeneous Kahlerian CR-hypersurface. ~g~~~~ID=~·

Let M be a compact homogeneous connected Kahlerian CR-hypersurface. Then either 1) M is~ torus bundle over an s 1 -CR-principal bundle over

~

homogeneous complex Kahler manifold T

x

Q

(T a complex torus, Q projective rational), or 2) either M or a 2-1 covering of M is T

x

is

~

CR-product

M' , where T is ~ compact compact torus and M1 ~

simply-connected compact homogeneous Kahlerian

CR-hypersurface which either lies 2-1 covering

of~

CR-hypersurface

in~

-

in~

-

n n

(~)

or is a

-

(~). []

Remarks.

(1) The hypersurfaces M1 in (2) above are classified

via the Stein-Rational fibration (see § 3). (2) The reader should note that by using Thm.4.3 one can say much more about the structure of Min case (1) above. In particular the fiber of the ~ -anticanonical fibration M is either an s 1 -bundle over a torus or vice versa.

~

Q

For detailed proofs of the results in this section see [R].

References [AHR]

Azad,H., Huckleberry,A., Richthofer,W.: Homogeneous CR-Manifolds, Crelles J. (to appear)

[AF]

Andreotti,A.,Fredricks,G.A.: Embeddability of real analytic Cauchy-Riemann Manifolds, Ann. Scuola Norm. Pisa 6, 285-304 (1979)

[Az]

Azad,H.: Levi-Curvature of Manifolds with a SteinRational Fibration, Manuscripta Math. (1985)

[BFS]

Buchner,M., Fritsche,K., Sakai,T.: Geometry and cohomology of certain domains in the complex projective space, J.reine ang. Math. 323, 1-52 (1981)

[BL]

Blanchard,A.: Surles varietes analytiques complexes, Ann. Sci. Ec. Norm. Sup. 73, 157-202 (1956)

176

[BR]

Borel,A., Remmert,R.: Uber kompakte homogene Kahlersche Mannigfaltigkeiten, Math. Ann. 145, 429-439 (1962)

[BS]

Burns,D., Shnider,S.: Spherical hypersurfaces in complex manifolds, Inv. Math. 33, 223-246 (1976)

[G]

Greenfield, S.J.: Cauchy-Riemann equations in several complex variables, Ann. Scuola Norm. Pisa, 257-314 (1968)

[H]

Huckleberry, A.: Homogeneous Surfaces,

[HL]

Huckleberry, A.T., Livorni, E.L.: A classification of homogeneous surfaces, Can. J. Math., Vol.XXXIII, No.5, 1097-1110 (1981)

(to appear)

[HO]

Huckleberry, A.T., Oeljeklaus, E.: A characterization of complex homogeneous cones, Math. Z. 170, 181-194 (1980)

[HS]

Huckleberry, A.T., Snow, D.: Almost-homogeneous Kahler manifolds with hypersurface orbits, Osaka J. Math. 19, 763-786 (1982)

[KY]

Kon, Masahiro, Yano, Kentaro: CR Submanifolds of Kahlerian and Sasakian Manifolds, Progress in Math. v.30, Birkhauser, (1983)

[M]

Mats~shima,

[MN]

Morimoto. Y., Nagano, T.: On pseudo-conformal transformation of hypersurfaces, J. Math. Soc. Japan 15, 289-300 ( 1963)

Y.: Surles espaces homogenes kahleriens d'un groupe de Lie reductif, Nagoya Math. J. 11, 53-60 ( 1957)

[MOS] Mostow, G.D.: Some applications of representative functions to solv-manifolds, Am. J. Math. 93, 11-32 ( 1 971) [OR]

Oeljeklaus, K., Richthofer, W.: Homogeneous Complex Surfaces, Math. Ann. 268, 273-292 (1984)

[R]

Richthofer, W.: Homogene CR-Mannigfaltigkeiten, Dissertation, Ruhr-Universitat Bochum (1985)

[R1]

Rossi, H.: Homogeneous strongly pseudoconvex hypersurfaces, Rice Studies 59 (3), 131-145 (1973)

[R2]

Rossi, H.: Attaching analytic spaces to an analytic space along a smooth pseudoconcave boundary, Proc. of the Conf. on Complex Analysis, Minneapolis 1964, Springer Verlag (1965)

[RP]

Rossi, H., Patton, C.H.: Unitary Structures on Cohomology (to appear)

[S]

Samelson, H.: A class of complex-analytic manifolds, Portugaliae Math. 12, 129-132 (1953)

[TA]

Tanaka, N.: On the pseudoconformal geometry of hypersurfaces of the space of » complex variables, J. Math. Soc. Japan 14, 397-429 (1962)

[TO]

Tolimieri, R.: Heisenberg manifolds and theta functions, Trans. Am. Math. Soc. 239, 293-319 (1978)

177

[V]

Vogt, C.: Line bundles on toroidal group, J.reine ang. Math. 335, 197-215 (1982)

[W]

Wang, H.C.: Complex parallelisable manifolds. Proc. Am. Math. Soc. 5, 771-776 (1954)

[WO] Wolf, J.A.: The action of a real semi-simple group on a complex flag manifold. I: Orbit structure and holomorphic arc components. Bull. Amer. Math. Soc. 6, 1121-1237 (1969)

A. Huckleberry and \v. Richthofer Fakultat und Institut fur !1athematik Ruhr-Universitat Bochum UniversitatsstraBe 150 D-4630 Bochum 1, FRG

Problems of Value Distribution in Complex Analysis for Several Variables P. Ielong Universite de Paris VI Ma.thematiques 4, Place Jussieu F-75005 Paris dedicated to Wilhelm Stoll

INTROOOCTICN

Value distribution theory for holarorphic functions and holarorphic mappings is an irrportant part of CC!lplex analysis theory was established only for mappings

a:

For a long tilre, the

ii: , i.e. ,

+

for

entire or merarorphic functions defined in the CC!lplex plane ranges in

a:

or

ii:

the

a: ,

values of with

(identified with the Riemann sphere).

We are indebted to Wilhelm Stoll for giving beautiful extensions of the theory to CC!lplex manifolds, opening ways to new problems of value distrihltion theory, and giving to the theory a larger danain of research. Haw does one generalize to

ly for holarorphic mappings the image

f(C)

en

a:

f :

-+

gives a covering of

the general situation of

f : X

the classical results obtained fonre-

0: ? ii:

is not a rational function, of unboW1ded degree. 'Ib consider If

f

Y , we must assume that an exhaustion

+

_ {X:i"~)o of X is given , Xr=> Xr•' for r > r', and r~ Xr =X. We suppose Xr is a relatively CC!lpact danain in X , with image f (Xr) c Y of bounded degree

n (r) , defined by

n(r,a) =card [Xr n f and

n(r) = sup n(r,a) <

-1 oo

(a)]

for

a EY ,

for

0 < r <

oo

a E Y

am

the problem is to discover situations in which

"asymptotic growth" as

r

+

+

oo

,

for alm::>st all

n(r,a)

has the same

a E Y . Then the theory

Will consist in giving connections between types of asymptotic growth and classes of "exceptional sets". Historically the theory was built after the

fai!DUS theorem of E.Picard (1883), to obtain a more canprehensive view of Picard's result ; the irrportance of the growth indicators in this problem appeared in the works of E.Borel, R.Nevanlinna (see [ 12]) • On the other

hand much of the theory was

proved

to be a consequence of particular

180

properties of confo:rma.l mappings (or rrore generally of sane classes of quasi-confo:rma.l mappings

a:

-+

ii: )

according to the paper of

The general problem was to define a class class

<.p of mappings

r

L.Ahfors [ 1].

of manifolds and a

X and Y in r , with the fol-

f : X -+ Y , for

lCMing properties : (I) - The fiber for "alrrost all"

f- 1 (a)

is discrete or of the sazre pure dim:msion

a E Y •

(II) - There exists an exhaustion {Xr} subsets of

X such that

n(r,a)

of

X by relatively ccrrpa.ct f- 1 (a) n Xr has the

or sane 'area" of

sazre asymptotic growth for "alrrost all"

a E Y •

Results for this very general problem were given by M.H.Schwartz (see [ 14] and [ 15] ) for differentiable manifolds in

Ifl

en . She ex-

and

tended Nevanli.TIDa' s theory using the Gauss-Bonnet fonnula in the sazre way that Ahlfors in [ 1 ] used the Euler-Poincare characteristic of the Riemann sphere

=C •

Y

In the follCMing we suppose rieties in q

~

p .$: n ;

ilnage in

en

X and Y are caJq?lex analytic subva-

of (caJq?lex) pure dim:msion

f : X

p

for

X and q

for

Y ,

Y will be a holarorphic mapping X -+ Y with dense Y • The fiber f- 1 (a) is an analytic subvariety in X , but it -+

is not of pure dim:msion p-q

for all

a E Y . Thus,the class of the

"exceptional sets" we have to introduce in the theory rrust contain the class of the analytic subvarieties of

Y •

Comparison with results of the classical case gives another indication on hCM one should define this class. For and

X =

a:

and

Y =

a: ,

f : X + Y an entire function, we have a general result. Let be 13 J2n logif(rei0 ) - aide , such that n(r,a) = ;---M(r,a); 1 no oogr T(r) = - 21 J2n M(r,ei<.p)d<.p is an indicator of growth of f for TT 0

M(r,a) = - 21

r + + ... Then (see [ 12 ]

, p. 264): llin M(r,a) = 1 for all a with r="' T(r) the possible exception of a polar set (i.e. a set of null capacity) in C. This result is not the deepest of the theory, but it proves that the set A of the Picard's or Nevanlinna' s exceptionnal values (we have card A~ 1 in the first case, A is at rrost countable in the second case) belongs

to the class of polar sets in

c .

Indeed the preceding result is a conse-

quence of the property of that non-polar subsets of

a:

are control sets

(see later the definition) in the foll<Ming situations : (I) subhanronic functions in a danai.n G =>A

- (II)

for negative

for the class of

181

subhanconic functions

c;,

V(z) ~ r >0

V(z)

fixed and for In

in

a:

satisfying the estiitlate

+ log[1 + lzll • Obviously

c? ,

a €

M(r,a)

is in the second class for

a: •

the sets which are not pluripolar will be control sets for

plurisubharnonic functions. For the general problem of bolarorphic mappings F: X+ Y

with X , Y canplex analytic subvarieties in

c? ,

of pure dimension

= [z

and q , the exhaustion will be given by the.balls B(r) and

= B (r) n X

Xr

the area

crx(a,r)

p

€ en ;llzi!
• A general result of L.GRIJMI\N [ 7 ] gives a bound of of the fiber Xr n F- 1 (a) , with the possible exception

of a pluripolar subset of

Y. Thus, for a wide class of problems in value

distribution theory in en , or in carplex spaces, we will probably find

r of sets satisfying both of the properties : (I) r is a subclass of the class of pluripolar sets. (II) - r contains

for exceptional sets classes

the class of analytic subvarieties. M:>st results given here will appear in the forthcaning book [ 10]. First we recall the property that non-pluripolar sets are control sets (see

[ 9b ]) and consequences of the works of E.Bedford and B.A.Taylor [2]

and of J.Siciak [16]. In the second part, a short survey of the results of

L.Gruman [ 7 ]

is given. In the last part of this lecture, I will consider

a particular problem of value distribution theory with

F

of finite order.

'!hen it is possible to obtain (using elementary properties of holarorphic

functions) classes of analytic sets as exceptional sets. The result is a generalization of a result [9a] , I have given fonrer. It contains results given recently in [15] by N.Sibony and Pitt I-lan Wong . I. PLURIPOIJIR SETS AND CCNI'ROL SETS (see [9b] ) I recall : a set

A in a oamplex manifold

in X if there exists a function

X is called pluripolar

V , plurisubharmonic in

A c A' = [z € X ; V(z) = _,] • We denote by of plurisubharmonic functions defined in

PSH(X)

X . If

X , and

the convex cone on

X is a danain of

en

or a Stein manifold, locally pluripolar sets in X are pluripolar in We denote by

g

= liln

sup f

, n n i above, we have

g * (z)

X

= liln

sup g(y) the upper regularization of g • I f y + z , where f € PSH(X) is a sequence locally bounded fran

*

g E: PSH

n * and the set [z E: X ; g(z) < 9 (z)] is kn<:Mn to

he locally pluripolar (see [2 l ). Given a set A in a d.anain G of a:n we denote by PSH(A,G) the convex cone of functions v E: PSH(G) with

R+

182

conditions V(z)

~

0

for

Definition •

z € G and

A set A

a control set for the class

~

V(z)

-1

for

z € A •

cfl

in a bounded danain G of PSH(G)

is called

if

= [sup V(z) ; V € PSH(A,G)]

gA G(z)

,

v

!)as a strictly_ n~?S~tiye +W'Ularization

*

gA,G (z)

in G •

* G(z) = lim sup gA G(y) € PSH(G) and gA* G(z) ~ 0 • Then gA , y .... z ' * ' * we have only two possibilities : (I) gA,G (z) = 0 , or (II) gA,G (z) < 0 Indeed,

for all

z E G

, by the maximum principle. Then :

Propositim 1 • A set A set

(for

if and only if it is not contained in the

PSH (G) )

negative function If

_..,

of a

V € PSH(G) •

= [z

~A'

A

in a bounded danain G of en is a control

= _..,]

€ G ; V(z)

and

V(z)

~

0 , V € PSH(G) , there

E G , such that V (z ) > - .., • Then for the sequence 0 1 0 Vq = su~(-1, V) € PSH(G) , we have Vq € PSH(A,G) , lim Vq(z0 ) = 0 exists

z

q

g~,G(z0 ) = gA,G(z0 ) = 0. Then g~,G(z) s 0 : therefore A trol set. conv;rsely, if set

= gA (z)

gA (z)

such that such

I: IV (z ) I <

q

q

is a dense subset in

gA,G(z0

0

oo •

= g~,G(z0 ) = 0

)

=0

lim Vq(z0 )

Then

* gA,G(z) ;r 0 • The

G ; then there exists

• We consider a Se:;IUence

has value

S(z) =I: Vq(z) € PSH(G) is contained in the

A

hannonic function

S (z) •

* gA,G(z)

call

c:cn-

z0 € G

vq € PSH(A,G)

• We can suppose, taking a subse:;IUence, that

which proves that We

A is not a control set, then

is not a

and

[sup

the extreraal function-of

*

oo

on

A ,

of the negative plurisub-

-oo

V(z) , V € PSH(A,G)]

-

A with respect

•

Proposition 2 (Inverse function theorem for plurisubhanronic func~)

(~

[9b] and [10]) •

Given a dcmain

G~

cfl

and

V(z,u) € PSH(G x C) , z € G , u €

a: ,

we consider M(z,r) and for

m > M(z,1)

= sup

V(z,u)

(respectively for

for

~

€

lui ~ r, u €

a:, 1~1

a: ,

< e-M(z, 1 ) )

,

~

183

[sup r , r > o, M(z,r) < m] ;

6 (z,m)

o' (z,r;;)

(1)

u1 (z,m)

= - lCXJ 6 (z,m)

6(z, -lCXJir;;l ) =[sup r, M(z,r) + lCXJir;;l <0]

u2 (z,r;;)

=-lCXJ6 1 (z,r;;).

u1 is a negative plurisul:hanronic function d£,fined for z E Gq = [z E G; M(z,1) < q] and m > q; u2 (z,r;;) is a negative pluriThen

subhal:noni.c function in ll = risu.bha.r!lonic for functions

r;; = 0

I~; I < e-M (z' 1 )

[z E G ,

• The theorem replaces an u.nbouxrled family of

z + M(z,r)

, for

r

+ + oo , by a family of negative pluri-

su.bha.r!lonic functions. For the proof see Remarks. (I)

If a set

~

0

in If

(II)

G1

• Hence, i f

Gc X

in G it is a control set in V(z)

[ 9b] and [ 10].

A is pluripolar in a manifold

pluripolar in each subdanain condition

which remains plu-

G1

:J

G

A

for the class

PSH (G 1 )

G = en ,

A

with

•

G is an unbounded danain in

f!l ,

is not pluripolar i f and only i f there exists a carpact bouxrled danain

X , it is

is a control set

G 1 c G such that

is not pluripolar in

K

a set

K c A

G1

Ac G ,

and a •

For

is not pluripolar i f and only i f it is a control set for

the class of the functions

v E PSH(cfl)

with condition

V(z) ~cv + lCXJ(1 + llzlll

(2)

VEPSH(en)

y(z) = [sup V(z)] *

A is not pluripolar i f and only if conditions (2)

,

, for

v

is plurisul:hanronic in

cfl

V with

and not the constant +

oo

(see [2]) •

Proposition 3. (I) Given a bounded danain ~

t

~

A c G , not pluripolar, and

G

in en

ljl ( t) , a function of

t

and a

can-

defined for

t , with lim ljl (t) = co , we - - t=oo x E G , u E IC , a function VE PSH(G ~< II:) ,

0 , which is increasing and convex of

consider in

G

~

=

M(x,r)

~<

IC , for

sup lul~r

V-(x,u)

for

x E G , r > 0

•

We suppose a bound is given M(x,r)

'!ben i f M(x,1)

~

~

ljl(lCXJ r) m0

oA,G (x) E PSH (G) ,

for

x E A

is bounded in

,

r > r0 > 0 •

G , there exists a control function

not depending on

V

but only on

A, G and

oA,G (z) ~ 1,

184 such that we have the bound M(x,r) ~

(3) with

r1

= sup

[r0

,

~[oA,G(x)

~- 1 (rn0 )]

log r]

for

x € G , r > r1

•

(II) If we do not suppose M(x, 1) the same assunptions on

-1

l~==up[~(oA,G(x) log r)]

For the proof of (I) , see is defined by

oA,G(x)

[ 9b ]

or

* -1 = [- gA,G(x)]

q

>%

,

the c:crrpact set A

: the function

€ PSH(G)

Gq where

q €N , q + +

q,

oA,G(x)

•

G by danains

<

an open canponent of the set M(x, 1)

M(x,r) ~ 1 •

[10]

For (II) we take an exhaustion of

For

bounded in G, by

G, A , we have

is contained in Gq

oo

Gq

is

•

and by

(I)

we

obtain a bound

(4) rq

= sup

[r0

,

~ - 1 (q)] • For q

+ + oo

x fixed, the sequence of the control

,

* G which is defined for q >q(x) is decreasing and has a gA I q limit gA,G E PSH(G) • Then on a canpact set Kc:G , oA G (x) is de' q fined and decreasing in q > q(K) oA G (x) has a limit _oA,G (x) uni' q for.mly on oarnpact subsets of G • functions

*

Remark. If

-----

exists sets

A is not a pluripolar canpact subset of

x0 € A , such that

oA,G (x) < 1 +

£

*

gA,G (x0 )

=-

are not atpty for

NaN we consider the order

define for holaoorphic functions

1 , and £

p (x)

in

(5)

p (x)

= lim

• The open

x

of

u + V(x,u)

and we can

and for V(x,u) E PSH(G x a:)

F(x,u)

M' (x,r) = sup [1, M(x,r)]

=1

> 0 (see [2] ) •

class of functions of finite order with respect to we define

oA,G (x0 )

G , there

the

u • For V € PSH (G x

and

sup (log r) - 1 log

M' (x,r)

r=oo

or For

equivalently, we consider

V'

= sup(V,O)

•

F E H (G x G) , an holaoorphic function,we define the order

as the order of

v = log IFI in

Proposition 4. Given

V E PSH(G x

wing possibilities for the ordre

p (x)

x E G •

p(x)

a:)

of

there exists only the follo-

,

u

+

V(x,u) :

a:) ,

185 p (x) :. 0

(I) -

(II) - p * (x)

= +"'

p * (x)

(III) I
p (x)

and

E PSH(G)

p (x) < p * (x)

and the set

V(x,u) E PSH(Cn x C)

then in (III), p* (x)

is a pluripolar set

< "'

is pluripolar!

is of finite tctal order in

en x

c ,

is a constant p •

For the proof, one applies Proposition 2 with o(x,rn)

= [sup

r , r > 0,

M(x,r) < rn]

- log o(x,rn) ~ 0 log m '1:

- _1_ = lim su p (x) p="' p

p*~x)

We pbtain W(x) =p * (x)

then and

= p(x) = 0

(II) i f

E PSH(G)

or

W(x) : - " ' • I f W(x)

=-co,

W(x) < 0 ,

W(x) E PSH(G) , we have (III) if

• If

=0

\'l(x)

and

•

Thus, we can give a definition : Definition. A function

V(x,u) E PSH(G,C) ,

of finite ordre with respect to finite for

x E A , and A

Proposition 5. If (x1ul E G its order with

x

C

x E G

1

u

i f the order

u E

of

a:

is called

u + V(x,u)

is

G •

V(x,u) , a plurisubhanlnnic function of u E

1

a:

and the set

p E PSH(G)

p (x)

a not pluripolar subset in

1

is of finite order with respect to

is bounderl on each canpact in

p (x)

x E G ,

p (x)

<

*

p (x)

G

p (x) ,s p (x)

and

1

-

is pluripolar in

u 1

G •

It is a consequence of Proposition 4 and of the property that a "negligible" set p (x)

<

p * (x))

(i.e.

the set of all points

1

is pluripolar

II. THE GENERAL RESULTS OF L.GRUMAN

A -

x E G

such that

(see [2], [3])) (see [7] ) •

A first result of L.Gruman is for the holarorphic mappings F :X+ Y

where

X and

d.inension

X and

for

in X is given by balls Theorem 1. Given Ecy

1

q

for

B(r)

Y

=[ z

p-q

for

1

E ~ ; liz II ~r ] •

s >0 , a > 1

a E Y

:r n F (a)

area of

B(r)

area of

B(r) n X

1

1

with pure (canplex)

1

q ~ p (. n • The exhaustion

1

there exists a pluripolar set F- 1 (a)

with foll<JW"ing properties : (I) -

dimension

~

Y are analytic subvarieties in

p

a f. E • (II)

n X and r:JX(r)

and Mp(r) = sup.

is of canplex

is the the (p dimensionnal)

II F(z) II

If

r:JX(a 1r)

far

liz I ~ r

1

then

1

186

foraEY,aflE, r

(6)

2q ax(a,r)

=0

lim r=o

An important step in the proof of (8) V

>0

, Mv(r)

.

var~ety

X •

I

(7)

=

sup V(z) and

is the following : let be

V E PSH(Cn),

[X] the current of integration on the sub-

. z
G~ven

.

.

y , 0 < y < 1 , there ex1.sts a boun:i

[X] A 13p-q A (iaav)

~ c(y,n) [ Mv(r) ]q ax(a,r)r2q

B' (r)

in a ball

B' (r) = B(yqr) , for a constant

=i

and y ; 13

depending only on

n

aaiJ z1J2 • The proof uses the r.bnge-~re operator defined for

n L~oc;

V E PSH

c(y,n)

E.Bedford and

by

B.A. Taylor (see [2])

am

the property that the extremal function of a non-pluripolar set is a control function. The dena:ninator of (6) can be replaced by iaa log[1 + IFI 2 lq A[X] Al3

(log r>ai B(r+er) in

( 13s

is the volume element

p-q

C8 ) • ·

B -

A second result of L.Gruman in [7] gives an upper bound and a lower

bound of the area R. of

X

n

R.

n B(r)

for

"alnost" all linear subspaces

~. Let be

q ~ n-p

X c en

an analytic subvariety of pure dimension

• We denote by Gq (~)

of diinension

q

in

~ ,

am

p

, and

the Grassmann manifold of the subspaces

by

nx

0R. the current of integration on R.

Let be ax(R.,r)

z E ~ , we denote

For

L

= I

B(r)

z =[ R.E Gq (~)

an irreducible analytic variety in a danain z E ~ , we have

each

we say that

Q

c Gq (~)

-

Q c

Gq {~) , such that for

n Lz) = dim Y - (n-q)

If

or

Y n Lz

= cp

,

Y is a non-degenerate irreducible subvariety in a

and

E c Y is a non-pluripolar set in

ls an analytic subvariety of ?Jre dimension (I)

z E R. ] • Then i f Y is

Y is non.degenerate.

Theorem 2. danain

dim (Y

eR. A 13p+q-n

For

E > 0 , 13 > 1

1

n-q

there exists

in

~ ,

R. E E

Y , and i f X then. :

such that

lim [(log r) 13 ax(r + E1)]- 1 ax{R.,r)r2 (n-q) = o. r=c:o (II) Given an increasing sequence rm , lim rm = + "", there exists

187 t >0

R. E E

and

such that

l~f[ax(trm)l- 1

(8)

a (R. r )r2 (n-q) =

.... -

' m m

X

o.

SUch general results have iirq;:lortant consequences for the develO];ll\eilt of value distribution theory : future researchs may focus attention on exhibiting situations -perhaps very particular- in which we can assert that the "exceptional" set has rrore precise properties, for example that it is an analytic set or an algebraic set. In the folla.ving we consider an example where the mawings is of finite order. III. SPEX::IAL PIDPERI'IES OF A MAPPING WI'IH FIBER OF FINITE ORDER.

Let be n = G x

a: , n c: r:fl

and

G a pseudo-convex danain in cn- 1

arxi

F(x,u) E H( ) , holarorphic of

for

x E G . We denote by E6 c: G the set of the points

Theorem.

Let

that u-+ F(x,u) be

E..,

has at rrost

= Es

s

equation

q>v (x,u)

~

s

zeros

0

G of the

E'

E~

G , and

variety Ms c: G such

x

U E

co

q>v

and there exists

q> :

r >0

u

E~

exists

ex> 0

such

that

u

llx-x0

11 < ex ; the ball

for

aurn

E~

> s isolated zeros , Iu I < r

lui = r

IF(x,u) I ~ ~

n(x,r) =.f-.. for

s'

has

such that the disk

zeros of lP arxi IF(x0 ,u) I ~ c > 0

n

affiF = - (x,u)

~

is open. Let x 0 E G , x 0 f/.

F(x0 ,u)

+

is defined in

is a polynanial with constant coeffi-

cients of a finite number of the variables

the entire function

(with multiplicity). Let

= G or there exists an analytic sub-

= 0 , where

G'

u

x E G , such

such u + F(x,u) ;;; 0 • 'fuen

E~ c: Ms • Moreover Ms = Ms x c

First we prove that

a: •

be of finite order with respect to

F E H (r!)

the analytic set in

is a closed set in

(x,u) , xEG , u E

contains

s'

• By continuity, there

arxi

~F (x,u)F- 1 (x,u)du = s'

f lal=r

B(x0 ,cx)

u

does not belong to

E~ ; E~

is

closed. Now we prove that E~ is contained in an analytic subvariety Min a acmpact ball B c: G , or (equivalently) that E' = E x C is contained s s in an analytic subvariety M of 'B. We give first the equations of M

188 ~

~c:B

in a danain ~

n w =

cl>

~

1

Let be

•

xo

(x01 u0

F(x,u) =

(9)

G (x1 u)

with

lPx (u)

=

G(x,u) 1

= aq(xluo) = q!

s

n

-1

n [II X

= ~

~c:B

1

such that

such that 1

= q, •

~nw

F(xoluo) "f 0

Then

(x 1 u) - xo

II

<

X

• Then

1

II?~ (u)

u •

q

0

0

aq -1 aF auq [ F au lu=uo

The coefficients

c

uo E

J

p (x)

a (x 1 u ) (u-u )q

0

r0

(u - u.) exp P (u)

1

xEB 1 aP - - ] = ~ R (u) u-uj u x

apx s -a- [ I: u 1

of degree s in

Let be aq

= sup

Px(u) ~ p

with degreee

(x 1 u) F

X E Es

and for

I

lu- u0 l <

1

such that

)

aF = au

G(x 1 u)

~

< a'

E Es : there exists

and we define ~ of center

is holO!!Orphic in

II

llx - x 0

:

for

(x 1 u) E

, with,

•

are holarorphic of

aq(x 1 u 0 )

~

By an easy calculation we obtain a polyncmial

(x,u0 )

in

Pq(~ 0 . . . . , ~q+ 1 )

Q' W •

such that

-q-1 F (x 1 u 0 )aq(x 1 u0 ) aq+1 and aq' (x 1 u0 ) = Pq [ F(z,u) 1 • • • --:;:r F(xlu0 ) ] = is a holarorphic auq u uo function in all o. Fran (9) we deduce 1 for llx - x < a' , x E Es 0 aP X (11) apG(x,u) lPX(u) = Rx(u) • (10)

=

aq(x,u0 )

II

au-

Then We write

aux

ll>x (u)

s n [llx - x 0

x E E (12)

Rx (u)

=

is a polyncmial of

s

I: b. (u-u ) J 0

J

G(x 1 u)

1

(11)

of degree at I!Dst p+s-1 •

1

with

bJ.

depending on

x

for

0

I
which is oonvergent for fran (9)

u

. , and

~

aj(xlu0 ) (u-u0 )q

lu - u 0

1

•

Then for

x E ns

1

we obtain

and (12) the equations

s I: a (x 1 u ) b = 0 for all v >,.s+p and x E ns· p=o v-p o p The polyrDII lP(U) is not the oonstant null, the coefficients bp are (13)

not all zero, then the detenninant (14)

I

189

IIUlSt vanish for

and

x E ns

sion remains i f we replace

u0

v

and

by u'0

~

s+p • The same conclu-

such lu0 - u'0

I

r

°

< 2

~ then

=E

G(x,u)

G (x,u') (u-u')q q 0 0 llx- x 11~ a.' , lu- u' I < £2 , and 0

is convergent for aq(x,u0') D)x,u) · for

0

a conse;IUence the detenninants q! auq au u=u~ are holanorphic functiors of (x,u) E /:;. , which vanish i f v ~ s+p

(x,U:) E (E x s

=

with o

a:> n

• As

a (x,u) by aq' (x,u) , we obtain q D (x,u) = F- 0 (x,u)D' (x,u) v \) (s+1) (v+1) and D~ (x,y) is obtained writing a' . instead a .

t:;. ·• If we replace

v-J

D~

in (13) • Then of

0

~ [F- 1 aF]

= - 1-

(x,u) is holarorphic of (x,u) En

.- - F(x u) "o ' ' · ••

v-J

and is a polynanial

."v+s+ 1

aF (x u) with constant coefficients. v+s+ 1 ' au a/ We suppose that the equations D'\) (x, u) = 0 , for v 'l s+p , define an analytic sUbvariety

Ms

in

B x C • Cbviously,

Ms

is invariant by the

(x,u) + (x,u + v) for v E a: • Moreover Ms contains a: ; E's in contained in B in the analytic sUbvariety Ms n [u=O].

translations E.. X

E' s

'!ben

b/ If

is contained in an analytic sUbvariety of

=0

D'\) (x,u)

in

Q

for v 'l s+p , and i f

E.. x C are analytic sUbvarieties in a neighboorhood

II

/:;. : llx- x0

then there exists solutions ~ (u)

= b0

+ b 1 (u-u0 )

and fran (14)

,

•• •

b0

,

(E

co

X

F(x0 ,u0 ) C)

u (Es

X

for

F(x,u) ¢ 0 , both W and (x0 ,u0 ) f/. W , we construct

lu - u 0 l < r~ , such that /:;. n W = cp ~ bs , not all zeros, and

••• ,

G(x0 ,u)

is a polynan of degree at rrost is a rational function of

poles (with Im.lltiplicity). Then

zeros. The conclusion holds for all with

~

+ bs (u-u0 ) s

,

we deduce that

with at rrost s

Q

# 0 , i.e. C)

= !)

for each

u

+

u

F(x0 ,u) has at rrost

x E G such that we can choose x E G

which is not in

s

s

u0

E... Then

•

Corollary. 1/ I f p* (x)

is a constant (in particular for

then the system of e;IUations

tic sUbvariety Ms

G •

in

Q x

D~ (x,u)

C , and

=0

for

(Es U E.. )

v

~

= Ms

s+p

Q =

cn- 1 ) ,

definies an analy-

190

2/ I f

X

is Cousin data of zeros of finite order

such that the origin is not in

Ixi

p

in cfl

Jxl) , the set card(tnJxJ) ~ s

(the support of

of the oamp1ex line through the origin such that

Es is

a closed set and i f it is not contained in the zeros of a haoogeneous polynanial

(ll(z)

=0

in

JxJ

cfl ,

For the proof consider

= F(z 1u , ••• ,znu) boorhood of the origin z

F 1 (z,u)

F(z)

for

=0

are haoogeneous polynani.als of

z , u

is an algebraic set • such that

X

= (z 1 , ... ,zn) = 0 in which

z • The set

= [z

E en , F(z)

we take for

1:1

t- 0 ;

the

E.., is empty , arrl

Es

;

F 1 (z,u)

an analytic subvariety in the projective space

P(cfl) or

v ~ s+p • Then

vx (r)

(see

Es

= P (Cn)

and the indicatrix

D' (z,u)

of

= 0]

an

a neighaq(z) is

=0

for

X is l:x:lunded

[ 10]), X is algebraic giving the result of [15] BIBLIOGRAPHY

[1] L.AHLFORS : Zur Theorie der uberlagerungsflachen •

Acta Math., t.65,

(1935), p. 157-191. [2] E.BEDFORD et B.A.TAYLOR : A new capacity for plurisubhal::nonic functions. Acta Math., t. 149, (1982), p. 1-40.

[3] E.BEDFORD : The operator

(ddc)n

on oamplex spaces (preprint).

[ 4] J .CARISCN : A rroving lemna for the transcendental Bezout problan. Ann. of Math., t. 103, (1976), p. 305-330. [ 5] L.GRUMAN

The area of analytic subvarieties in

cfl • Math. Scand., t. 41,

p. 365-397. [6] L.GRUMAN

La geanetrie globa.le des ensembles analytiques dans

cfl •

Lecture Notes n° 822, Springer, 1980, p. 90-99. [ 7] L.GRUMAN

Ensembles exceptionnels pour les applications holarorphes dans

cfl. Lecture Notes Springer, n° 1028, (1983), p.125-162.

[8] C.O.KISEr.MAN : The grcwth of restrictions of plurisul::hanronic functions (preprint) • [ 9a] P.LE:I.CNG

Sur les valeurs lacuna.ires d 'une relation

a deux

variables.

Bull. Sc. Math., t. 56, p. 103-112. [ 9b] P.I..EI.lNG

A class of Frechet spaces in which the bounded sets are C-polar sets. Holarorphy arrl approximation theory : Math. Studies, North-Hollarrl, vol. 71, p. 253-272.

191 [ 10] P. I..E:J:.(N; et L. GRUMAN : Entire functions of several CCI!plex variables. Grundlehren der l.fath. , n ° 282, Springer, 1985. [11] R.IDLZCN, B.SCHIFEMAN et N.SIBCNY- Average grcMth- est:iJnates for hyperplane sections of entire analytic sets • r-lath. Arm.

t. 257, 1981, p. 43-59. [ 12] R.NE.VANLINNA : Eirrleutige analytische Funktionen. Grundlehren der Math., 46, Springer, (1936).

[13] M.-H.SCHWARI'Z : Fonnules apparentees a la formule de Gauss -Bonnet. Acta Math., t. 91, (1954) 1 P• 189-244. [14] M.-H.SCHWARI'Z : Fonnules apparentees

a celles

de Nevanlinna -Ahlfors

pour certaineS applicatiOns d I Uile Variete

a

n-dirrensi0ns

dans une autre. Bull. S.M.F., t. 84, (1954), p. 317-359. [15] N.SIBCNY et PITI' MAN

~G

: Sane results on global analytic sets.

Lecture Notes n° 822, p. 221-238.

[ 16] J. SICIAK : Extranal plurisubhanronic functions in ~ • Proceedings of the first Finnish. Polish SUnl!!er School in ccnplex Analysis, p. 115-152.

17 J .A. TAYLOR :

An estimate for an extremal plurisubhal:rronic function

on ~ • Lecture Notes n° 1028, p. 318-328.

On the Boundary Behavior of Holomorphic Mappings Laszlo Lempert

1

E6tv6s L6rand University, Dept. of. Math., Budapest, Muzeum Krt. 6-8,

1088 Hungary

0. Introduction

Let

01 , 02

be l:x:>unded danains in

afl

and let

"'

be a"biholo-

norphic mapping between the two. Under what conditions is it possible to conclude that

cf>

extends (with some regularity) to the

ao 1

I t is natural to require that the l:x:>undaries

,

o1 of o1?

clos~e

ao2

be regular to

some extent; and based on the negative evidence of the lack of counterexamples, it is generally believed that no further assumptions are needed. However, the existing theorems do use further assumptions, partly of

geanetric, partly of analytic nature. Rather than listing

a~

these

theorems, we shall restrict ourselves to describe those that are strongest in the sense that they contain the others as special cases. Moreover, we

c""-

shall always assurre that the danains in question have at least l:x:>undaries. I f l:x:>th danains are such that their Bergman projections map

functions into

C"" functions (this is the so-called condition

S. Bell and E. Ligocka proved that

between the closures

51

52

,

, see

cf>

extends to a

~-lJ.

c""

R ) , then

C""-diffearorphism

D. Catlin proved tret pseudo-

corwex dcrnains of "finite type" satisfy condition

R , see

[c].

These

danains include strictly pseudoconvex danains and also pseudoconvex danains with analytic l:x:>undaries. When the danains are analytically l:x:>unded and the Bergman projection

satisfies a "condition Q ", analogous to condition

R

but pertaining to

analytic functions, then

¢

is known to extend to a biholanorphism

between neighl:x:>rhoods of

51

,

52

(see

s.

Bell ~]>. The inconvenient

feature of this theorem is, :tx:Mever, that apparently the only general gecmetric condition which implies a>ndition

Q

is strict pseudoconvexity

1SJrooorted in part by NSF grant MSF grant M:::S 82-13077.

194

(D. Tartakoff [!~, G. Komatsu

ffi
M. Derridj - D. Tartakoff

IQ-11.

On

the other hand, for strictly pseudoconvex domains a si.rrpler approach also

[!.e],

yields the analytic continuation of biholcm:>rphic mappings (H. Iewy S. Pinchuk [P]).

In [p-F~ K. Diederich and J .E. Fornaess consider analytically bounded a:2 such that (on both boundaries) the set of pseudoconvex

domains in

points is separated fran the set of pseudoconcave points by a totally real surfaces. Under that condition they show that

¢

extends to a horceo-

norphism between the closures.

Further results treat exceptional situations such as Reinhardt danains (see W. Kaup

ffia],

S. Bell and H. Boas (!!-~[)), or, slightly nore generally,

domains with many syrrmetries (D. Barrett Hartogs domains in

llf

f!laU),

and analytically bounded

(K. Diederich - J .E. Fornaess [p-F:[j). The aim of

this paper is to show that a gearetric condition totally different from pseudoconvexity (in fact, one bearing on the first derivative of a defining function rather than on the second) also ensures the extendibility of a biholcnorphism to the boundaries. r-Dre precisely, we shall prove

Let D1 , D2 be strictly starshaped, bounded danains in a;ll having real analytic boundaries, ¢ a biholcnorphic mapping from D1 onto D2 . Then ¢ extends to a Holder-continuous homeom:>rphism between 51

Theorem 0.1

and

D2 •

Strictly starshaped here means that there is a point Oi that

Di

is starshaped with respect to

angle between the radius vector in

z

oi z

oi

and for any

in

Di

z E aDi

and the tangent hyperplane to

such

the aoi

is nonzero.

On certain parts of the boundaries the mapping can be shown to have

better regularity than just Holder-continuity. Indeed, that once the continuous extension of displayed in 1) Points on on

ao 2

¢

s.

Bell observed

is derronstrated the ideas

(!leJ prove that: ao 1

near which

ao 1

is pseudoconvex are mapped to points

with the same property, and near these points the extension of

195

q,

is

c""-SltDOth; and

2) Strictly pseudoconvex points on

to strictly pseudoconvex points on

aD2

aD 1 are mapped

q,

and the mapping

analytically

extends across such points. One of the important ingredients of our proof is a theorem by D. Barrett about a weak regularity property of the Bergman projection in strictly starshaped danains ( [ Ba:[j ) . In the sarre paper he proves stronger regularity results, too, provided the domain in question satisfies more restringent conditions. It should be pointed out that when these regularity results become sufficiently strong, they can be coupled with ideas fran

@-:Jd ,

to prove better SltDOthness of biholorrorphic maps. The

results thus obtained have the following flavor. Suppose

D1 , D2 are smoothly bounded and very strictly starshaped in the following sense:

If

Oi

is the center of

the radius

0i z

Di , then for any

z E aDi

and the tangent hyperplane to

aD i

the angle betv..een in

z

should be

greater than 89° . Then any biholorrorphic mapping q, : D1 - D2 extends to a c1 diffeorrorphism betv..een the closures, provided the d:ir!Ension n is low. If one wants to have the sarre regularity in sorrewhat higher d:ir!Ensions, or get better regularity in low d:ir!Ensions, it suffices to replace

89°

in the assunptions by

89°59' , and so on. However, in the

case treated in this paper, only the v..eakest regularity result of D. Barrett is available and this requires and approach entirely different fran that of

@-lJ ,

Acknowledgenent.

as will be explained in the next chapter.

A part of this work was done while the author was a

visiting lecturer at Princeton University. I wish to express my gratitude to this institution. Thanks are also due to D. Barrett who kindly showed me a version of his manuscript

@a:[] .

196

1 . On the proof of Theorem 0. 1

Our proof consists of two parts. In the first we prove that our

biholorrorphic mapping does not change the distance to the boundaries very radically. l'Dre precisely, there are positive constants ( 1 • 1)

c dist(z,ao 1 )

T]

-1

< dist(¢(z),ao 2 ) < c

c , TJ

dist(z,ao 1 )

such that

1/T]

for z E o 1 . This is achieved as follows. ( 1 . 1) is easily seen to be equivalent to a lower estimate on the Jacobian determinant of ¢ (and on its inverse) by a high power of the boundary distance. NJw a lower estimate of

det ¢ 1

is equivalent to an upper estimate for

det (¢ - 1 ) 1

A suitable upper estimate is obtained by using D. Barrett 1 s theorem in

@afl ,

whence ( 1 . 1 ) will follow.

Before proceeding to the second part of the proof, we note that if is a defining function of

o 2 , (1.1) implies that

r o ¢

r

is Holder-

continuous. Then we face the problem whether the regularity of a canposed function implies the regularity of the inner function. When this question is considered in the greatest possible generality, the answer is clearly no. Indeed,

¢

may map into a level set of

srrooth no matter how badly nevertheless, when

¢

¢

r

and then

r o ¢

will be

behaves. It is a rrost remarkable fact that,

is assuned to be holCl!IOrphic and

r

is supposed

to have certain properties to be specified in a nr::ment, then the Holdercontinuity of

r o ¢

does imply that of

¢ .

The simplest instance of this principle is when strictly plurisubharrronic and srrooth. Then the r o $

implies the

a/2-Holder-continuity of

r

is supposed to be

a-Holder-continuity of

¢

(a I Z) • For

a

= ~

this was first observed by N. Kerzman (see [!<~). Another instance of the above principle is when real analytic. 'Ib get a precise theorem

to be imposed on

r

and

SCXIe

r

is supposed to be

additional conditions have

¢ ; hJwever, these conditions are autaratically

met in the con=ete situation considered. Thus in the second part of our proof we start with the Holder-continuity of Holder-continuity of

r o $

and conclude the

¢ . This part does not use the fact that

¢

is

197

biholooorphic. The idea here is to show that the situation which gave rise to the counterexample above (narrely

mapping into a level set of

happen, not even approximately. On the contrary,

q,

has to be transversal

in sorre sense and to a certain extent to the level sets of exactly if

r ) cannot r . M:>re

is restricted to camplex curves, the images of these curves

cannot have arbitrarily large order of contact with the level hypersurfaces of

r . This part of the v.ork is based on results by K. Diederich

and J.E. Fornaess, and J. D'Angelo. A quantitative statement about the

contact between holooorphic curves and level hypersurfaces will then prepare the ground for the final step of the proof, which is the estimation of

q,' • The estimate we thus obtain. is equivalent to the

Holder-continuity of

q, •

Finally a 'lt.Ord about the notation. IbweVer, the occurrence.

c' s

c,c',...

will stand for positive

D1 , D2 and q, , unless stated otherwise). will not necessarily rrean the sarre constants on each

constants (depending only on

198

2. Estimating the boundary distance

Let

o 1 , o2

and

be as in Theorem 0. 1 • The aim of this chapter is

to prove

Lamia

2.1:

for all

There are positive constants TJ , c , such that

z E o1 .

dist( ·, ao 1 ) stands for distance to the boundary statement of LEmna 2.1 is equivalent to

Here

ao1 . The

such that we have the following estimate for the Jacobian determinants of ¢ and t = ¢-1 :

Lamia

2.2: ·There are positive constants

(2.2)

ldet ¢' (z)

(2.3)

ldet

I

c , TJ

> c dist(z,ao 1 )TJ ,

t' (w) I

> c dist(w,ao 2 )Tl

We shall first derive Lamia 2.2 and then show its equivalence with

LEmna 2.1.

Proof of Lernna 2.2: is,

P.

1

Let

Pi

denote the Bergman projection in Di • That L2 (D. ) to its subspace

is the orthogonal projection of

consisting of holorrorphic functions. In a canpactly supported

c;(o2 )

1

@eJ S. Bell showed that there is

such that

h E L2 . In other words,

1 = P2 1 = P 2cp • Using the transformation formula for the Bergman projection, this iltplies for all holCll!Drphic

det ¢'

= P1 (det

¢')

= P 1 ((det

¢ 1 )(cpo ¢)).

199

we now

invoke

o.

Barrett 1 s theorem (see @a:~:l>

;

according to which on

snooth strictly starshaped dana.ins the Bergman projection maps the Sobolev space w112 intoitself.Since (det¢ 1 )(q>o¢) isin CQ(D 1 ) , P 1 maps it into w112 (o 1) • In particular, there is an e > 0 such that det ¢ 1 = P 1 ((det ¢ 1 )q>O¢) E L2 +e, Hence "">

I

/det ¢ 1 12 +e

=

01

I

1-e

/det \jl 1

02

Applying now the sul::Iooan value property to the subharnonic function /det \jl 1 1-e

on balls centered at w E o2 and of radius dist(w,ao2 ) , the second inequality in I.ernna 2.2 is obtained. Since the role of o 1 and

o2 is synmetric, the first inequality holds as well. Proof of the equivalence of I.ernnas 2.1 and 2.2:

Suppose first that (2.1)

oolds. Then using the cauchy estimates and the boundedness of

4>

and

t ,

we have /det q, I (z)

I=

/det o/ 1 (¢ (z))

r1

>

Q)nversely, suppose that (2.2) and (2.3) l:x>ld.

the length of

q, 1 (z)v , where v

we

first want to estimate

is an arbitrary unit vector in

«fl .

Of course we have an upper estimate

(2.4)

/¢ 1 (z)v/ < c'dist(z,ao1 )-1

b.lt we shall need an estimate from below. TO this end, let v1

= v, v 2 , .•• , v n

be an ortl:x>nonnal basis in

«fl .

Then by cramer 1 s rule

and (2.4) (applied to v = v 2 ,v3 , ••. ) c dist(z,aD 1l'TJ < ldet ¢ 1 (z)

I<

n

n! j1I 11<1> 1 (z)vj I <

< nll¢ 1 (z)v1 1·(c 1 )n- 1dist(z,ao1 ) 1 -n

,

200 so that with sane M > 0

(2.5)

1<1> 1 (z)vl > c dist(z,ao 1 )M

let now to

cf>

(z)

w and

=

v E <J:I1

w . Chcx:>se a unit vector

direction as the vector 0 < t < dist(z,ao 1 )/2

(2.6)

WW 1

the point which lies the nearest

E ao2

W1

•

so that

f (t) =

Put

I!t 0

f (0) + f 1 (0) t = w +

at distance

> ct dist(z,ao1 )M

distance be

2lw-w 1

(2.7)

t < 2c f (O) + f 1 (O) t

and

I=

distance of

-1

cf>

cf>

1

have the scure

(z)v

(z +tv) • Then for

we have

lf(t) - f(O) - f 1 (Oltl

The point

•

cf>

1

(z) vt

2

f"(u) (t-u)dul < C 1 t dist(z,ao1 ) lies on the half line

from w • Choose

t > 0

WW 1

-2

,

so that this

2 dist(w,ao2 ) ; then

lw-w 1 ldist(z,ao1 )

-M

, o 2 . M:>reover, the lw-w 1 I (if z , and

lies in the C<XIq?lement of

f (O) + f 1 (0)

to

o2

will be

therefore w , is sufficiently near to the boundary

ao 1

Now there are two possibilities. Either the value of

res. t

aP 2 ) .

selected

t < dist(z,ao 1)/2 , or t ~ dist(z,ao 1 )/2 . In the first case (2.6) shows that the distance of f(O) + f 1 (O)t to 2 -2 f(t) = cf>(z+tv) E o 2 is less than c 1 t dist(z,ao 1 ) , hence in view of above satisfies

(2.7) lw-w 1

I<

4c 1 c- 2 1w-w 1 12dist(z,ao 1 )-2M- 2 • This implies the first inequality

in (2.1) since

lw-w 1 I= dist(w,ao2 )

= dist(¢(z),ao2 )

•

If, however, the second case arises, then, again by (2. 7), we have dist(z,ao1 )/2 < 2c- 1 lw-w 1 ldist(z,ao 1 )~ whence the first inequality on (2. 1 ) again follows. The second inequality in (2.1) is obtained by interchanging the roles

of

o 1, o2 •

201

A consequence of Lerrrna. 2 • 1 is the following. Let real analytic defining f1.mction of

o2

, i.e.,

o2

r :

afl

-+

lR

= {w : r(w) < 0}

be a and

dr # o on ao2 • LEmna 2. 3 :

The f1.mction

r o¢

is Holder continuous on

o1

z 1 ,z 2 E o1 . We have to estimate r(¢(z 1 ))-r(¢(z 2 )) in tenns of som= power of lz 1-z 2 1 • Tw cases will be distinguished according to whether I z 1-z 2 1 1/ 2 < max (dist (z 1 , ao 1 ) , dist(z 2 ,ao1 )) or not. Choose~

Proof:

points

In the first case we also have

for any

z

on the segment

near to

ao 1

z 1 z 2 , at least if

. Hence

In the second case we use Lemna 2. 1 :

<

-

clz -z 1

2

1

1/(2rj)

This proves Lemma 2.3.

z 1 ,z 2

are sufficiently

202 3. The relative position of a real analytic hypersurface and a holorrorphic curve (Review of results of J. D'Angelo and K. Dierlerich-J.E. Fomaess)

The result of K. Diederich- J .E. Fomaess that we are going to need

is the following (see @-F'Q, Theoran 4).

'I'heorem 3.1:

Suppose

s

crfl . Then

S

is an

arbitr~.J

corrq;>act real analytic variety in

does not contain any nontrivial genns of canple.x varieties.

Using the tools developed. by J. D1Angelo in [p'Q and [p~ , this theorem can be given a quantitative fonn. We are now going to review some of the concepts he introduced. and theorems he proved.. Let

~*

stand for the set of nonconstant genns of holcrnorphic

(0::,0) ~ (crfl,p) ,

mappings

p

being a fixed point in

(eventually vector valued.) soooth function near order in

0, i.e., the largest

v

such that

crfl . If

0 E G:: , v (g) !: ..

g(~) - g(O)

is an

g

is its

= O(~v)

(~~0).

Let nOW' of

r

be a real analytic function definerl in a neighborhood

p E crfl , dr (p) t-

0 . We put

ll(r,p)

sup

v (r o f) /v (f) !: ..

f E ~*

and call it the maximal order of contact of the hypersurface

{ z : r (z)

=r

(p)}

with holorrorphic curves.

The order of contact of a proper ideal

I

the ring of genns of holcrnorphic functions at

contained in

~

p

=

(!} ,

p , is an analogous concept,

defined by sup

f E ~*

of

-.llf (I)

By

the Nullstellensatz

I

consists of the single point

inf v (h of) /v (f) h E I

< ..

is finite i f and only if the variety

p , and in this case

I

contains

203 some power of the maximal ideal

(!) •

One also has the estimate

(3. 1)

(see

@i]

In

1

Theorem 2. 7).

@;U

D'Angelo shows how to reduce the carputation of

ll(r 1 p)

to

the carputation of 1:*(I) for certain ideals I c (!) • The way he p achieves that is the following. For the sake of simplicity we shall

assurre

r (p) = 0 •

First D'Angelo shows that there exists a separable Hilbert-space which we are going to identify with

£2 -

"if, -

and three genns of holorro:rphic

mappings H :

(!~flip) ~ (4:10)

I

(ttfllpl ~ (illo)

--+ ('ie

(ttfl 1 p)

1

0)

such that r(z) = 2 Re H(z)

+

=2ReH(z)

IIF(z) 11 2 - IIG(z) 11 2 =

+

Z(IF.(zli 2 -IG.(z)l 2 l

J

Next for any unitary transformation the ideals i.e.

I (U 1 p) c

F. - Z u .. G. J i Jl l Then he proves 1

(3.2)

(!)

p

1

U :

1l ---+ til

he introduces

generated by H and the components of u.. Jl

F - UG

being the entries of (the matrix of)

sup1:* (I(U 1 p)) ~ ll(r 1 p) ~ 2 sup 1:*(I(U 1 p))

u Where

J

U runs over all

u unitary mappings

ie. --+ -df

Theoran 10) • He also proves the following important theoran:

(see @~ 1

U .

1

204

Theorem 3.2:

Suppose

bounded for

p

near

ll(r,p0 ) is finite for some p0 . Then p0 • (See Theorem 4. 11 • )

ll(r,p)

is

@TI ,

Using these tools, it will be easy to prove the following theorem, which essentially says that analytically bounded domains are of finite type (in the sense of D'Angelo) • This theorem has also been found by D'Angelo and possibly by others, too, but it does not seem to have ever been published.

Theorem 3.3: r :

Gfl

~

Let

Dc

afl

be an analytically bourrled domain,

a real analytic defining function of

D • Then there is

a neighborhood

0

such that for

p E0

ll(r,p)

JR.

we have

Proof:

of

aD

and a positive integer

By Theorem 3.2 and tne canpactness of

ll(r,p) <

for

®

is a sequence

k

s k •

p E aD • Suppose that Uv = (uji>

aD

ll(r,p) =

it suffices to prove ®

•

Then by (3.2) there

of unitary matrices such that (v ._._.

(3.3) It can be assumed that for each

.

Then the operator nom of uji need not be unitary.

i,j

I

®

v uji has a limit s 1 ; however, u

the sequence

U= (uji)

will be

Let us now define the ideal J= (H,F.- 6 u .. G. J i Jl l We claim that the variety of

Indeed, let

z

u*

(uij)

I

the point

J

j

1 12, • • •)

consists of the single point

F(z) = UG(z)

is the adjoint of

no:rm at rrost 1, this implies variety of

I

. p .

be a point in this variety. Then

H(z) = 0 where

J

G.- 6 \.1."':' F. J i lJ l

is contained in

I

G(z) = u*F(z)

U • Since both

u

IIF(z) II = IIG(z) II , whence

I

and

u* are of

z E aD • Thus the

a D , so that by Theorem 3.1 it reduces to

p • By the Nullstellensatz we have then

205 d~ I!J/J =

Na.v by the Noether property of

d <

l!>p

00

there is an

N such that 1, .•. ,N) •

j

Let (H,FJ.- 1: u~ .G. i J1 1

j

1, ••• ,N) •

By one half of the Banach-Steinhaus theorem (the trivial half), there

is a neighborhood of

p

on which

F.- 1: u~.G.--+ F. J i ]1 1 J

--+

uniformly, as

v

continuity of

dime 0/I

oo

•

1: u .. G. i ]1 1

Apply.ing the theorem about the upper semias a function of

Proposition 5. 3) , we deduce that for

v

I

( !Jo], Chap. 11,

large

01. the other hand

J

since

v

(Uv)*

c (H,Fj

=

1: u~.G. i J1 1

G

(Uv)- 1 . Therefore

~

v

j - ~ uij

&;'1.

"1

dim ID/I(Uv,p)

(3.3). This contradiction proves Theorem 3.3.

1,2, ... )

j

s

I(Uv,p) ,

d , contradicting

206 4. More on the relative position of a real analytic hypersurface and a holarorphic curve

Theorem (3.3) implies in particular that if

v (f) = 1)

(afl, p)

is a holOIIOrphic gem with

f' (0) t- 0

Taylor polynomial of

(about 0) will contain at least one non-

r of

(i.e.,

f : (0::,0) ---7

then the

k'th

vanishing nonconstant tem. Sane general properties of real polynomials and analytic functions pemit us to estimate this nonconstant tem from

below. Lamna 4.1:

let

W: G x

for all

Let

JFil

K be a CC~Tpact subset of G , and

Gc

Iff

be open, let

~

JR

be a positive real analytic function such that

x E G the function

y

~

W(x,y)

at most. Then there are positive numbers

a , b

d

such that

2 -b

W(x,y) > a(1 + IYI )

(4. 1)

Proof:

is a polynomial of degree

Let

y 1 , .•• ,yM denote the cxx:mlinates of

to prove (4.1) for

y's

1 ~ IY 1 1 =max IYil • In the sequel

such that

i

we shall ass\.Ulle that this is the case. Introduce the new variables

y • It will be enough

TJ 1

1/y1 ,

TJi = Y/Y 1

(i=2, ••• M) .

Then

y~(x,y) =

V(X,T])

will be a polynanial in TJ , analytic in

x, TJ,

TJ 1 t- 0 • Hence the

f.ojasiewicz inequality (see

with same positive

a, b,

for

x E K,

and positive when

[!.o]

yields

ITJ 1 1, •.• ,1TJMI ~ 1 •

This gives

(4.1) in the case considered.

we r of

are now ready to estimate a nonvanishing Taylor coefficient of from below. We shall use the notation of Theorem 3 • 3.

207

Lemma 4.2: a , b with

There are a neighborhood

such that whenever

p EL

am

/f' (O)

I

L of

ao

f : (1!:,0) ~ (~,p) = 1

and positive numbers

is a holcm:>rphic gem

then

lf{ll.) (0) 1)-b

L

a (

(4.2)

O<j.J. :::k

~:

Let

L

be any c::arpa.ct neighborhood of

Theorem 3. 3. For L

X

x = (x' x") E

s2n-1 Ct.: ....n x q ...n; ,

1f1

x

~

ao

contained in

of

0

in a neighborhood of

y= (y 2 ,y3' .•• ,yk) E ... "- (k-1) n

(YIJ. E ...n 2 ••• , k) q; ,!J.=,

define

W(x,y)

r(x' +x"~ +

W is then a real analytic function, polynanial in

kL y ~IJ./J..L!) 2

12

IJ.

y , and Theorem

3.3 implies that it vanishes nowhere. Hence Lernna 4.1 applies and we

obtain (4.2).

208 5. Proof of Theorem 0. 1

Lemna 5.1:

Given a positive integer q(t,t) =

following property. If

polynomial of degree at rrost

k , there is a

c > 0

with the

L

2q .. ti tj/i!j! O:::i+j :::k 1 J

k2

such that

jq(t)

1- min Itis 1

is a

(5.1)

then (5.2)

max ltl:::1

jq(t)

I>

c

Proof:

It will be enough to prove (5.2) for polynomials q satisfying q(O) = 0 and 6 lq .. 12 = 1 • The set of such polynanials is, however, 1]

a ccrnpact set, and (5.2) follows by Weierstrass' theorem.

Proof of Theorem 0.1:

Let

Choose

so that (4.2) of Lerrma 4.2 be satisfied with o 2

a< 1 , b , k

substituted for z E o1

(5. 1)

where

r

be a real analytic defining function of

sufficiently near

II cp'

(z)

II

< dist(z,ao 1 )

such that for

o-1

,

stands for operator norm.

> 0 • Then there is a unit vector

(5.2)

> 0

ao 1

Suppose that (5. 1) does not hold for sorre

o

o

o . We claim that there is a

o2.

Icp' (z)vj

~

v E ~

dist(z,ao 1 )

z

near

ao 1

and some

o

is small

such that

o-1

We are going to show that this cannot be the case if enough. For brevity,

'We

introduce positive numbers

left hand side of (5.2) is

;>, ,

fj,

1/}, and the right hand side is

(5.2), together with the cauchy estimate, yields

so that the 6°- 1 . Then

209 (5.3)

Fbr

I; E

a:, Is/<

/f'

(5.4) for

~ t-. 1- 5

ct-. < "A

6/"A {::: t-. 5 )

put

f{l;) = q,{z+"Avl;)

• We have then

= 1 ,

j = 2,3, ... , by the Cauchy estimates and the boundedness of

Let furthermore 0 , R denote the

F

denote the

k'th Taylor polynomial of

k' th Taylor polynanial of

r

about

f

tfJ •

about

z = f (0) , and

let Q{l;) = R(F(t;))

Then

r of

and

Q agree at

(5.4) we have for

Put

I ' I /.,~ 2Q .. sl:'i s;::j /'~.]. OS i + j Sk ~]

a = 25k (b + 1)

z

near

to order

0

k • Hence by Leima 4. 2 and

ao 1

(any small

a

greater than that would also do) and

Then

L

0< i+j Sk

so that I.errma 5. 1 can be applied and we obtain

max

it/ s 1 which means

q(t) -min

/tl s 1

q(t) >

c

210

max

(5.5)

/1;

I:::

A1 + a/X

Q(l;) ~ c Ak(1 +a +be) x-k

min

Q(U

~

J!;J::: A1 +a/X

> c Ak(bo +o +a) = c Aa(k+1/2)

we

rJ.CM

wish to show that a similar

replaced by

r of • To see this, for

If(!;) - F(l;) I :::

Since

f

max II; I I ::: A1 + a/X

is bounded on the disc

J~;l

estimate holds with Q = R o F J!; J:O A1 +a/X 'lfie estimate

If

+ 1) (1;)(A1+a/X)k+1 I I (k+ 1)!

(k

so that

/f(I;)-F(I;) I ::: c Aa(k+ 1 ) , whence

(5.6)

lr(f(l;))-r(F(I;)) I < c Aa(k+ 1)

Furthenrore, since for

IF(!;)-F(O) I :::

'lfie

also have lr(F(I;))-R(F(I;)) I ::: ciF(I;)-F(O) 1k+1 < c Aa(k+1 ) •

(5.7)

(5.5), (5.6) and (5.7) together imply

max

r(f(l;))

II; /::: A1 + a/X

Let

1; 1

resp.

1; 2

min

r(f(l;)) > c Aa(k+1/2) •

]!; 1::: A1 + a/X

denote the two points where the maximum resp. minim.lm

is attained. Then for the points

z 1 = z + Xvl;i

(i = 1 ,2)

we have

211

r(q,(z 1 )) - r(q,(z 2 )) > c lla(k+ 1/ 2 )

(5.8)

On the other hand,

oo that if

e denotes the exponent for which r o q,

is

e-Holder-

continuous (see Iam1a 2. 3) , then

This,

~ed

with (5.8), implies

a> e/(k+1) • But then we proved that

if (5.2) holds then

o

z E o1 sufficiently small.

in a neighl:x:>rhood of

to hold for

> e/{2(k+1)k(b+1)}; in other words (5.1) is proved

ao1

, provided

o

is

This finishes the proof, since by a theorem of Hardy an4 Littlewood (5.1) is equivalent to the

a-Holder-continuity of

q, , see [G].

(Although Hardy and Littlewood proved their theorem for holari::>rphic functions defined in the unit disc, their proof with obvious rrodifications applies to holanorphic mappings defined in smx>thly bounded domains in 1/l-1

Gfl •)

Exchanging the roles of

is HOlder continuous.

o1

and

o2

, we obtain that aloo

212

6. Concluding Remarks

Our proof actually proved slightly stronger theorems than Theorem 0.1.

For instance, we have

Theorem 6.1:

Let

n1

be strictly star-shaped and

analytically bounded. Then any biholarrorphic map

to a Holder-continuous map

i5 1

~

i52

c2-bounded,

q, : n 1

~

n2

n2 extends

•

Strict star-shapedness can also be replaced by an analytic condition.

Theorem 6.2:

CU (D 1 ) q,

and

Let n 1 be such that its Bergman projection maps into L 2 + e (D1 ) with scrne e > o , and an 1 is c2 • I f n 2 are as in Theorem 6. 1 , then the conclusion of Theorem 6. 1 still

holds. In @a~ D. Barrett constructed a

C""-bounded danain

the assumption of Theorem 6. 2 fails for every

e

n1

for which

e > 0 , and also for every

he constructed analytically bounded domains for which the assumption

of Theorem 6.2 fails for this particular that for an arbitrary domain e > 0

n1

e • It is,

h~er,

feasible

with analytic boundary there is an

for which the assumption of Theorem 6. 2 holds.

In yet another version the essential condition is imposed on the mapping

q,

Theorem 6. 3:

rather than on

r.et n1 be

n1 C2

bounded I

and assume in addition that with sorre

D2 and q, as in Theorem 6 . 1 , c > 0 , e > 0

dist(q,(z),aD 2 ) < c dist(z,aD 1 )e , Then, again,

q,

z E n1 •

is Holder continuous.

Indeed, this is what we prove in Chapters 3, 4, 5. The condition of an2 being analytic is very heavily used. In the proof presented above, we relied on it on three independent occasions.

213

In LEmna 2.2 the function

3.3 fails to hold for just srooth dcrrains. Finally, and sarewhat surprisingly, I..am1a. 4. 1 also needs that the function Othawise it may fail, even if x # 0, e(O) = 0. Then if

N

=M =

W(x,y) = (1-xy)

K contains a neighborhood of

W be real

1 • Indeed, let

2

+ e(x)

e (x)

an~ytic.

= e-x

if

does not satisfy (4.1)

0 • Indeed

-1 - 2 min W(x,y) ::: W(y ,y) = e ':! xEK

214

References

Regularity of the Bergman projection on domains with

D. Barrett:

transverse synrnetries.

[~

Irregularity of the Bergman projection on a smooth bounded domain in a: 2 • Ann. of Math. 119 (1984) 431-436.

D. Barrett:

Regularity of the Bergman projection and local

D. Barrett:

geanetry of domains.

[!le]

Math. Ann. 258 (1982) 441-446.

S. Bell:

TO appear

Analytic hypoellipticity of the

the extendability of holonorphic mappings.

a-Neumann problem and Acta Math., Vol. 147

( 1981 ) 109-11 6.

s.

Bell, H. Boas:

Regularity of the Bergman projection in weakly

pseudoconvex domains.

(!l-:g

s.

Bell, E. Ligocka:

Math.Ann. 257 (1981) 23-30. A s.i.Irg;:>lification and extension of

Fefferman's theorem on biholonorphic mappings.

Invent. Math. 57

(1980) 283-289. [CJ

D. Catlin:

Global regularity of the a-Neumann problem.

Proc. Syrrp. Pure Math. Vol. 41, Atrer. Math. Soc. R.1, 1984. rPTI

J. D'Angelo:

applications. [p~

J. D'Angelo:

Real hypersurfaces, orders of contact and Ann. of Math. 115 (1982) 615-637. Intersection theory and the a-Neumann problem.

Proc. Symp. Pure Math. Vol. 41, .Amer. Math. Soc. R.1, 1984. [p-FTI

K. Diederich, J.E. Fornaess: analytic boundary.

Pseudoconvex domains with real

Ann. of Math. 107 (1978) 371-384.

K. Diederich, J .E. Fornaess:

Biholom:>rphic mappings between a:: 2 • Math. Ann. 245 (1979)

certain real analytic domains in 255-272.

215

@-Ffl

K. Diederich, J.E. Fornaess:

BiholO!!Orphic mappings between tY.U-

dimensional Hartogs domains with real-analytic boundaries. Recent Developrents in Several Corplex Variables, Princeton University Press, 1981.

[p-'fl

M. Derridj, D. Tartakoff: solutions to the

On the global real analyticity of

a-Neumann problem.

Contn. Partial Diff. Equ.

(1976) 435-601. [G]

G.M. Golusin: variable.

[!
W. Kaup:

Gearetric theory of functions of a corrplex

1\bscow-I.eningrad, GITI'L, 1952 (in Russian) •

tiber das Randverhalten von holO!!Orphen AutarorphiSI!EI1

beschrankter Gebiete. [!<~

N. Kerzma.nn:

Manuscripta Math. 3 (1970) 257-270.

A 1\bnge-Arrp'ke equation in COitplex analysis.

Proc. Symp. Pure Math., 30 (1977) 161-167.

[!
[!.e]

G. Kanatsu:

Global analytic-hypoellipticity of the a-Neumann

problem.

Tbhoku Math. J., Ser. 2, 28 (1976) 145-156.

H. l£M:f:

On the boundary behavior of holO!!Orphic mappings.

Att. Acad. Naz. deiLincei 35 (1977).

(!DJ

s.

fojasiewicz:

Sur le problema de la division.

Studia Math. 18

( 1959) 87-136. [P]

s.r.

Pinchuk:

mappings. IT~

D. Tartakoff: ob.

[Tb]

J

.c.

1972.

On the analytic continuation of holorrorphic

Math. Sb. Vol. 27 (1975) 375-392. On the global real analyticity of solutions to

Oomm. Partial Diff. Equ. 1 (1976) 283-311. Tbugeron:

Ideaux de fonctions differentiables.

Springer

Integral Geometry of the Monge-Ampere Operator R. Molzon Department of Mathematics University of Kentucky Lexington, Kentucky 40506

1. INTRODUCTION In 1971 W. Stoll proved an average Bezout estimate for complete intersections of analytic varieties. terms, the results may be described as follows.

In simplified Suppose

XcCJ:n is an analytic variety of pure dimension p and q <:: n-p. Let G(q,n) denote the Grassmannian of q-dimensional linear subspaces of a:n.

We measure the "growth" of a variety Y of

dimension p by computing vol 2 p(YnBn(r)) where vo1 2 P denotes the 2p-Hausdorff measure. Stoll's result roughly says that one can compute the growth of X from the average of the growth of

X•~,

iEG(q,n) if we average over all of G(q,n).

Stoll's theorem is an example of a result in integral geometry. could fix

Note that instead of averaging over G(q,n) we ~ 0 EG(q,n),

and let r be the group of complex affine

transformations on a:n, and then average the growth of X•y~ 0 as y ranges over

r.

This is the manner in which integral

geometry results are often stated. The intersection of two analytic varieties, X andY, can be reduced to studying intersection with a linear space by considering xxy and intersecting with the diagonal. One can ask if averaging is really necessary to determining growth.

For example, could one compute the growth of X

from the growth of

X•~

for some fixed

~EG(q,n)?

This question

is answered in the negative by an example of Cornalba and Shiffman. They produced two curves in a: 2 with order zero

growth whose common zero locus is zero dimensional but of infinite order. In 1976 J. Carlson considered the problem of intersection in a different way. In a: 2 suppose we have the families of varieties Xa={f(z)=a} and Y8 ={g(z)=S}. For a particular

218

a and 8 nothing can be said about the growth of the intersection Xa•Ys in terms of the growth of Xa and Y8 . Carlson considered the growth of the intersection XaoY 8 averaged over a set of

{a,8)EEc~ 2 which has positive Newtonian 2+£ capacity.

This

capacity is defined in terms of a kernel function K{z,w).

This

concept of positive capacity is closely related to the Hausdorff measure of a set. Carlson's result is an integral geometry result in which the averaging is done over a much smaller set of linear spaces than G{q,n) and is even "smaller" than any open subset of G{q,n).

This method works since we already have quite a bit

of structure on the Xa and

YS~

that is, they are analytic

varieties. More recently L. Gruman has obtained results on average growth in which the integration is done over sets even smaller than those considered by Carlson.

These sets are

pluripolar~

a property which is detected by a capacity associated with the Mange-Ampere operator.

We will first look at some properties

of this capacity and then the methods for applying it to integral geometry. It is clear that the entire subject of integral geometry in several complex variables owes a great deal to the work of

Professor

Stoll~

my work in this area is no exception.

In

addition, his support and encouragement have been extremely valuable to me and I am grateful for the opportunity to participate in this conference in his honor.

2. CAPACITY OF THE MONGE-AMPERE OPERATOR If Qc~n is open, a set EcQ is called pluripolar if it is the

-oo

set of a plurisubharmonic function.

denote the plurisubharmonic functions on function which detects pluripolar

sets~

n.

We let PSH{Q) There is a capacity

it is defined via the

nonlinear complex Mange-Ampere operator. If EcQ is compact, one associates to E an L~oc plurisubharmonic function uE as follows.

Let

219 Let iJE(z;;)=sup{v(z;;): VEPSH(n), vs;O onE and v
•

1;;-+-Z

The Mange-Ampere operator acts on ~E to define a nonnegative measure which is supported on E, d~E

c

= (dd uE)

n

•

The following result is basic [1]. Theorem (Bedford and Taylor).

J d~E

>0

if and only if E is not pluripolar.

a:n If Ecrlc
~E

so it" is a probability measure.

The method may be ex-

tended to define the property of positive capacity on a complex manifold. In contrast to polar sets and Newtonian capacity, there is no nontrivial relationship between positive Mange-Ampere capacity and Hausdorff dimension. This fact is demonstrated by the following result which extends an example of Diederich and Fornaess. Theorem.

The idea is the same as in their paper [4]. There is a C00 , real, closed curve y in a:N+l which is

not pluripolar. Proof.

The construction of the curve proceeds as follows.

First construct N uncountable families of disjoint Cantor sets in the unit interval, each having positive logarithmic capacity. The Cantor sets will be indexed as Ca = Ca a where a. EJ. 1'"""' N ~ ~ with J.~ an uncountable set. Next construct N smooth functions f 1 , •.• ,fN onm such that the following two conditions hold. (i)

fiiCa a =ca. is a real constant depending only 1, ••• , N ~ on the index a .. ~

(ii)

The set {cai= aiEJi} is a Cantor set of positive capacity.

The curve y may then be given in parametric form as:

220 y={zEa:

N+l

: Im z 1 =0, Re z 1 =t, z 2 =f 1 (t), ••• ,zn+l=fN(t)}

We first show that y is not a pluripolar set in ~N+l and then construct the Cantor sets and the functions f .• ~

Suppose

~

is a plurisubharmonic function, thus not identi-

cally -oo by definition, and

~I

y:: -oo.

Consider

~(t,ca

~(r.;,ca

1

, ••• ,co.)

, •.. ,co.) ::-oo for 1 N. which is a set of positive capacity. It follows

for fixed a=(a 1 , ... ,aN). t"C

Then

al' • • • ,aN that ~(r.;,ca 1 , •.• ,co.Nl:: -oo as a function of r.;.

N

Next consider

~(z 1 ,r.;,ca 2 ,

••• ,co.Nl for fixed a 2 , .•. ,o.N. As a function of r.;, ~(zl'r.;,ca 2 ,. • .,co.Nl ::-oo if r.;dca 1 : a 1 EJ 1 } which is a set of positive capacity and thus ~(z 1 ,r.;,ca , ••• ,co. ) ::-co. Continuing in this manner gives

~(z 1 ,z 2 ,

2

•.• ,zN+l)::

N

-oo

which is a contra-

diction and thus y must be non-pluripolar. Construction of the Cantor Sets.

The construction of the

Cantor sets uses a recursive process in the usual manner.

If

I denotes a closed interval inm, denote by (*) the following process. (*)

Remove the open middle 1/3 of I, the open middle 1/3

of the remaining closed intervals, and continue removing middle l/3's until 2N+l closed intervals remain, each having length (l/3N+l) III where III denotes the length of I. We may label these intervals with the lexicographic ordering on N-tuples of elements of z 2 • The first 2N intervals are labeled left to right as (0,0, ••• ,0), (0,0, ••• ,0,1), ••• , (0,1,1, ••• ,1),

(1,1,1, ••• ,1).

The second 2N intervals are

labeled in the same way so there are pairs of intervals with the same label.

To distinguish these pairs, label them Ib,l

and Ib 2 where bEZ~ = z 2 x ••• xz 2 (N times). Now define intervals ' . i N _ N Ib1 b2 bn k w~th b EZ 2 and k-1, ••• ,2 recursively by apply' I • • • I I ing (*) n times. The index k is used to distinguish intervals with identical lexicographic label. If the Ib1, •.• ,bn,k are defined, define the Ib 1 bn+l k- by applying (*) to each of 1' · k. • · ' There ' the intervals Ib 1 bn are thus (2 N+l ) n intervals 1' · · ·' ' N+l n Ib1, •.• ,bn,k' each of length (1/3 ) We now define the Cantor sets. The index set will be J=J 1 x • • • xJN where Ji consists of all possible infinite sequences

221

in z 2 •. If. a=(a 1 ,_ ••• ,aN)EJ write a.={a~}._ 1 2 where a~Ez 2 ~ ~ ]- I I • • • ~ and aJ=(ai, ..• ,a~). Now suppose a is fixed. Define C 1 to be 1 1 a, those intervals Ib 1 k with b =a , k=l,2. c is defined to be , . · a, n the intervals Ib 1 bn k with bJ=aJ, j=l, .•. ,n and I • • • I I k=l, ••• ,2n. Therefore C consists of 2n intervals each of N 1 a,n length (1/3 + )n. Furthermore, the minimum distance between the intervals is (2·3N-l) •(l/3N+l)n. Let C = n C Then C a n a,n a is a generalized Cantor set in the sense of Tsuji [3] or Carleson [1] and has positive logarithmic capacity. (See for example Tsuji, Theorem III.6.3 or Carleson, Theorem IV.3.) Construction of the Parameter Functions.

First define

Yb1, ..• ,bnk to be the initial points of the intervals Ib1 bn k' k=l, •.. ,2n. Define smooth functions X; onm as I • • • I I ..a.. follows. 1 xi(tl

1

0

if if

tEib1 = I(b11 I tEib1

with

•• • I

b1) N

where b~=l ~

b~=O ~

Further, we require 0 ::;; xi (t) ::;; 1 everywhere and x. (t) :: 0 out~ N side an E-neighborhood of [0,1]. Note that there are 2 intervals on which xi(t):: 1.

Now define the functions fi as:

First note that fi(t) is C00 since for any fixed n, the derivatives up to order ! of the nth term of the series are bounded by ( 2 N+l)n( 3N+l)!•N

• sup tEm O:s;j:s;!

Now fix a=(a 1 , .•• ,aN) and suppose tEe a Then tEI a 1 , ... ,a n , k for all nand some kE{l,2, ••• ,2n} where k of course depends on n. Consequently, if tEI 1 n+l k with a~+l=l(O), then N+l n a , •.• , a , ~ 1 n X;((3 ) (t-ya1 an k)) =1(0). For all (b , •.• ,b)" .&.1 I • • • I I (a , ••• ,an), Xi((3N+l)n(t-ya 1 , • • • ,an,k))=O. Therefore

222 00

L _1_. a~+l ~ n=O 10 n 2 ~oo n2 n+l cai= Ln=O(l/10 ) •ai • It remains to show that the set {ca.: aiEJi} has positive capacity. To do this we exhibit the ~ set as a generalized Cantor set. Let C0 =[0,2]. Suppose en has been defined as a union of 2n disjoint closed intervals. Apply

.

Def~ne

to each of these closed intervals, I, the following procedure. 2n+l . Divide I into 2·10 ~ntervals of equal length and keep the first two intervals and the 10 2 n+l+l 8 t and 10 2 n+l+2nd intervals (closed) .

Let Cn+l be the set of all subintervals so obtained

from the intervals of Cn.

(i) (ii)

(iii)

(iv)

Then Cn has the following properties:

cn+l c en; en consists of 2n closed subintervals of length 2/lOn ; The minimum distance between any two of the intervals is (1/10 (n-l) 2 ) • (1- (2/l0 2 n-l)); If C =

n

n

C

n

then by Theorem IV. 3. of Carleson [1],

C has positive logarithmic capacity. Now suppose cJ.E J is fixed and c

=

oo Ln_0 ( 1/10 n 2 ) ·an +l"

(We drop k2 the subscript i for simplicity.) Let ca,n = L~=O (1/10 ) •ak+l" Then c lies on the initial point of some closed interval in a,n and c ECn for all n,£. Since Cn. Furthermore, c =lim c a n~oo a,n a,n ~ c is a compact set caEC. Conversely suppose dEC. Then dECn for all n and it belongs to a unique maximal subinterval of a

en. Call this interval In. Since dECn+l' dEin+l and In+lcin. Let d denote the initial point of I • Then d = lim d where n k2 n n~oo n dn= L~=~ (1/10 )•ak+l with akEZ 2 . Therefore dE{ca: aEJ} and C={c : aEJ} which concludes the construction. a

3. INTEGRAL GEOMETRY The integral geometry problems I want to consider involve averaging over sets of linear spaces which are not pluripolar. The sets of linear spaces are sets in G(k,n), k-dimensional

223 planes in ~n.

If Xc~n is a pure p-dimensional variety, let

q = n-p and consider intersection with the q-dimensional linear subspaces of ~n.

The idea is to make an assumption on the

growth properties of

X•~

locally pluripolar.

One then tries to recover information

for all

~EEcG(q,n)

where E is not

about the growth of X from the growth on the average of

Xo~.

The growth of X is measured by ox(r)

(3 .1)

=

n

vol 2 p(XnB (r))

where Bn(r) is the ball of radius r centered at the origin in ~n.

We say that the function x(r) has finite order growth p

if

lim sup log X(r) log r r-+oo

(3.2)

X is said to be of finite order p if r

-2p

ox(r) is of finite

order p. Following Gruman [5], we say that p(r) is a precise order if: (i)

lim p (r) r-+oo

and

p

(3. 3) (ii)

lim r log r • p' (r) r-+oo

0 .

We say that x(r) is of normal type A with respect to the precise order p(r) provided A = lim sup x(r) r-+oo rP (r)

(3. 4)

If

~EG(q,n)

let

(3. 5)

Note that except for a pluripolar set of ~

in a discrete set of points.

~EG(q,n),

X intersects

The idea now is to obtain esti-

mates above and below for ox(~,r) by ox(r) on the average. L. Gruman has obtained very interesting and strong results in this direction [5] and so I will describe some of his .work. My aim is to indicate how the Mange-Ampere operator

224

comes into the picture.

The upper estimate referred to above

is easier, so I will describe it first. Theorem (Gruman). (3. 6)

Given £>0

E = {R.EG(q,n): lim sup r+co

is locally pluripolar in G(q,n). The two following results are basic starting points for proving the above result and for the integral geometry of the Mange-Ampere operator in general. Basic Integration by Parts Result. current of bidegree (p,p) with 2 8 = r-I a a I z I • Then

I

(3.8)

Let e be a closed positive

c"" and let

VEPSH(~n).

Let

v-eAsn-p-lAalzl 2

aBn(r)

=I

V•eASn-p Bn(r)

+I

(r 2 -lzl 2 >r-I aaveASn-p-l. Bn(r)

Basic Inequality for the Mange-Ampere Operator. Let ~-(r) - sup V(z) if V>O Let VEPSH(~n) and ex be the current -~

-lzl~r

- ·

of integration over X.

Then, if y
constant C(y) such that

(3.9)

IexA8p-qA(.ci Bn(yqr)

The upper estimate,

(3.6), is obtained by the following

argument. Let EcG(q,n) be a non-locally pluripolar set. Let uE be the associated potential function. One then has an estimate for luEI" Let 8 be a smoothing of ex, the current of integration over X. Extend S to a current on Ux~n where UcG(q,n). Then apply the basic inequality for the Mange-Ampere operator where one takes V=uE. This in turn gives an estimate

225 above on

The upper estimate holds without any assumption on the

Remark.

growth of X. We now consider the lower estimate for the average growth of

crx(~,r).

Let EcG(q,n) be non-locally pluripolar.

The

lower estimate is given by the following inequality. exist constants c 1 and c 2 and a probability measure on E such that if t>O, then (3.10)

There ~

supported

J crx(~,r(t+l))d~(~) E

Again this estimate holds without any assumption on the growth of X. If the analytic variety X has finite order growth, then it is possible to obtain a nicer form for the right side of the above inequality. If the function r- 2 pcrx(r) is of finite order p and of normal type A, and if {rm} is a sequence increasing to oo, then there exist constants M, T, and C such that if m~M

then

Corollary.

Suppose Xc~n is of pure dimension p, of finite

order p and normal type A.

Then if there exists a set EcG(q,n)

which is non-locally pluripolar such that ~EE,

Xo~

is finite for all

then X is algebraic.

During the conference in honor of Professor Stoll, Professor Lelong informed me that L. Gruman has obtained an estimate which would imply the above corollary .without the assumptions on growth of X~ one would need only that X•~ be finite for a non-pluripolar set of ~EG(q,n),

226 I would like to end with the following question. converse to the above result hold?

Does a

Given EcG(q,n), which is

locally pluripolar, can one construct Xc~n of dimension p=n-q such that X•1 is finite for all 1EE but X is transcendental?

REFERENCES 1.

E. Bedford and B.A. Taylor, Potential theoretia properties

of plurisubharmonia funations, Acta Math. 149 (1982), pp. 1-40. 2.

J. Carlson, A moving lemma for the transaendental Bezout

problem, Annals of Math. 130 (1976), pp. 305-330. 3. M. Cornalba and B. Shiffman, A aounterexample to the transaendental Bezout problem, Annals of Math. 96 (1972), pp. 402406.

4. K. Diederich and J.E. Fornaess, A smooth aurve in ~ 2 whiah is not a pluripolar set, Duke Math. J. 49 (1982), pp. 931-936. 5. L. Gruman, Ensembles exaeptionnels pour les appliaations holomorphes dans ~n, Lecture Notes in Math. 1028, Seminaire d'Analyse, P. Lelong - P. Dolbeault - H. Skoda, pp. 126-162. 6.

W. Stoll, A Bezout estimate for aomplete interseations,

Annals of Math. 96 (1972), pp. 361-401.

Logarithmic Jet Spaces and Extensions of de Franchis' Theorem Dedicated to Professor Wilhelm Stoll on the occasion of his sixtieth birthday Junjiro Noguchi University of Notre Dame, Department of Mathematics Notre Dame, Indiana 46556, U.S.A. Tokyo Institute of Technology, Department of Mathematics, Oh-okayama, Meguro, Tokyo 152, ,T.APAN. Jet bundles and logarithmic 1-forms play an important role in the study of the value distribution of meromorphic mappings into algebraic varieties (cf. [01] [G-GlJ and

[Nl~4].

On the other hand, logarithmic vector fields as well as logarithmic 1-forms have been used in the study of Gauss-Manin connection and singularities (cf. [Sl]).

In this paper we

define the sheaf jk(X;logD) of germs of logarithmic jet fields along

~

hypersurface D over a complex space X, which induces lo~

the logarithmic jet space Jk(Xj

D) along D, if D has only

normal crossings (see section 1 for details).

Then we inves-

tigate their functorial properties. In section 2 we apply the losarithmic jet space to prove the following theorem which generalizes de Franchis' Theorem to the higher dimensional case: (2. 5)

Theorem.

Let A be a

quasi-.~belian

variety of general type of A. of a compact complex space

Y.

many holomorphic mappings

f: Y

with respect to A(X)

(c£.

variety and X a sub-

Let Y be a Zariski open subset Then there are only finitely ~

X which are non-degenerate

(2. 3)).

Here A(X) is, roughly speaking, the family of the supports of logarithmic cannonical divisors on X and the above non-degeneracy of f means that f(Y) ¢ Z !or all ~ € A(X). It is noted that the non-degeneracy condition for f (see Example (2.14)).

is necessary

Since a compact smooth algebraic curve

C with genus > 2 is imbedded into the Jacobian variety, Theorem

228

(2.5) implies that there are only finitely many non-constant rational mappings from an algebraic variety into C (de Franchis' Theorem). H. Fujimoto [Fl] obtained a finiteness theorem for a family of linearly non-degenerate meromorphic mappings from the m-dimensional complex vector space ~m into the n-dimensional projective space Fn(~).

We will give a remark on the

relationship between Fujimoto's result and Theorem (2.5)

(see

Remark (2 .15)). In section 3, we will prove another extension of de Franchis' Theorem, which generalizes the result of [N-Sl , Main ~.

Theorem (1.2)] in the case of the complex number field

Let

V be a complete smooth al~ebraic variety, D a hypersurface of V with normal crossings and V

=

v-o.

Let T(V;log D) be the

vector bundle of logarithmic vector fields alon3 D and Tq(V; log D) denote the q-th exterior powerAqT(V; log D). say that the vector bundle Tq(V; lo~ D) --+ tive over V if there is a proper

V is

morphism~:

We

quasi-nega-

q-

T (V; log

D)~~

N

into the complex affine N-space ~N such that the restriction of ~ over Tq(V; log D)

lv

phism onto its image.

minus the zero section is an isomor-

Let W be another algebraic variety and

Fq(W,V) the set of proper rational mappings f: W

~

V with

rank f .:_ q. (3.1)

Theorem.

If Tq(V; log D) is quasi-negative over V,

then F (W,V) is finite. q

Acknowledgement.

The main part of this paper was written

during the author's visit to the University of Notre Dame, 1984/85.

He expresses his sincere gratitude to the Department

of Mathematics of the University of Notre Dame for the hospitality, and especially to Professor

w.

Stoll for numerous dis-

cussions on the subjects of this paper and related topics. §1 a)

Jet bundle.

LOGARITHMIC JET

f,g:

{~ 1 0)

-+

~

Let

standard coordinate z.

SP~CES

be the Gaussian plane with the

Let M be a complex manifold and

(M;x) germs of holomorphic mappings from nei9h-

borhoods of the origin 0 For a positive integer k

E

~

E

~

into* with f(O) = 9(0) = x E M. we write f ~ g if f and g have

229 the same Taylor expansions in z up to order k for some holomorphic local coordinate system around x. "~"

checked that the relation,

Then it is easily

is independent of the choice

of the holomorphic local coordinate system around x and defines an equivalence relation on the set {f:

(~;0)

~

(M;x)}.

Let jk (f) denote the equivalence class of f. and set Jk(M)x = {jk(f); f:

(~;0)

~

(M;x)},

(1.1)

Then Jk(M) naturally carries the structure of a holomorphic fibre bundle over M with the canonical projection ~=

Jk(M)

--+

M (cf.

[01] and [G-Gl]). The bundle (Jk(M),

is called the jet bundle of order k and jk(f)

~,

M)

Jk(M) is

E

It is noted that J 1 (M) is isomorphic to the holomorphic tangent bundle T(M) over M, and that Jk(H)

called a jet of order k.

has a structure of flag with the natural

~rejection

Jk(M) ~ Jk-l(M) (1.2) Jk (M) -----+ Jk-l (H) --- • • • - - Jl (M) :. T(M) ___, M. such that for a holomorphic sections E f(U,Jk_ 1 (M)) over an

open subset U

M, the restriction Jk(M) ls(U) of

c

Jk(M)--+ Jk_ 1 (M) over s(U) is isomorphic to J 1 (H) lu; T(M) lu. If M is a complex algebraic manifold, then (Jk(M), ~, M) is also a complex algebraic fibre bundle over M. open subset of

~

and G: W

-+

Let W be an

M a holomorphic mapping.

Then G naturally induces the lifting (1. 3)

such that Let over M.

~oJk(G)=

G (cf. [01 , p. 86]).

1

~M

denote the sheaf of germs of holomorphic 1-forms Take a holomorphic section wE f(U, ~ 1 ) over an open

subset U of M.

M

For jk(f)

f*w

Jk(M) lu, put

E

=

A(z)dz .

Then the derivatives djA/dzj (0), 0 ~ j < k - 1, are welldefined, independently of the representative Hence we have a mapping

f for jk(f).

230 Let w1 , ... ,wm with m = dim ~1 be holomorphic 1-forms on U such that w1 A ••• AWm does not vanish any-

which is holomorphic. where.

Then we have a biholomorphic mapping 1Tx (w 1 , .•• ,wm): Jk(Mllu~ux

(1.5)

(~klm

which we call the trivialization associated with {w 1 , ••• ,w m} .

=~ denote the sheaf of germs of meromorphic 1-forms and take~ E r(u, =~). Then as in (1.4), ~induces a meromorphic

Let

vector function

~: Jk (Ml lu

(1.6) b)

a:k •

-+

Jet spaces.

Let X be a complex space with structure

sheaf @X, which is, in this paper, always assumed to be irreducible and reduced unless otherwise mentioned.

We assume for

a while that X is biholomorphically imbedded into a complex manifold M.

Let J(X) denote the ideal sheaf of X.

that a jet jk (f)

E

we say

Jk(M)x with x e: X is tangent to X if Pof have

zero of order > kat 0 for all p set of all jets jk(f)

E

J(X)

E

X

.

Let Jk(X)

X

be the

Jk(M)x which are tangent to X and set Jk(X)

= xWx

Jk(X)x'

1T: Jk(X) - X , where 1T denotes the natural projection. It follows from the coherence of the ideal sheaf J(X)

([Cl]) that Jk(X) is a com-

plex subspace of Jk(M) and 1T: Jk(X) --+X is a holomorphic .fibre space.

Let X

complex manifold M'.

--+

M' be another imbedding of X into a

Then,in the same way as above, we have

another holomorphic fibre space 1T': Jk(X)'--+ X, which is however isomorphic to 1T: Jk(X) --+X as fibre space.

Hence

for general X, we define the holomorphic fibre space 1T: Jk(X) --..X by making use of local imbeddings of X into open subsets of complex vector spaces. the jet space of order k over X.

We call (Jk(X) ,1T,X)

The jet space J 1 (X) of order

1 is isomorphic to the Zariski tangent space

8(X) •

As in

(1.2), Jk(X) carries a sequence of fibrations. (1. 7)

Jk (X) --+ Jk-l (X)

~

•.. --+ J l (X)

5!

8 (X) -----+" X.

such that for a holomorphic sections s e: r(u,Jk-l(X)) on an

231

open subset U of X Jk (X)

Is (U)

;

H

(X)

Iu

.

Let G: W --+ X be a holomorphic mapping from an open sub~

set W of

Then as in (1.3), we have the lifting of G

into X.

(1. 8)

Jk(G): W - Jk(X)

such that

c) folds.

-1--

7ToJk(G) =F. Sheaf and space of logarithmic jet fields over mani-

Let D be a hypersurface of a complex manifold M and

fJM(log D) the sheaf of germs of logarithmic 1-forms alon9 D (cf. [Dl], [Il] and [I2]). For convenience, we recall the 1

1

For x e M-D, the stalk fJM(log D)

definition of fJM(log D).

1

1

is identical to the stalk fJM,x of fJM at x.

For x

X

E

D, take a

neighborhood U of x and irred11cible holomorphic functions a 1 , ... ,a 1 on U such that

u n D Then we define 1 (1. 9) fJM(log D) x

0}

1

L j=l

dcr. C>M

•

1

_ J + fJM

,x crj

,x

1

If D has only normal crossings, then fJM(log D) is locally free.

If M is bimeromorphic to a compact Kahler manifold, the 1

global sections of

fJM(log D) are d-closed (cf. [Dl]).

Let N

be another complex manifold, E a hyFersurface of N and

~: N--+ M a holomorphic mapping such that ~-l(D) ~

c

E.

Then

naturally induces a sheaf morphism

~*:n~(log D)

(1.10)

--+

n~(log E).

If D and E have only normal crossings, then for a mer6morphic

mapping~: N--* M with ~-l(D) lfi*: r (M, ~~(log D))

(1.11) Lets

E

r(u,

c

Ewe have

--+

r (N ,n~ (log E)).

Jk(M)) be a holomorphic section on an open

subset U of M. (1.12)

Definition.

along D if ~

1

wos J"u :

ruE r(U', fJM(log D)),

We say that s is a logarithmic jet field U --+ ~k are holomorphic for all

where U' is an arbitrary open subset of u (see(l.6) for the definition of w).

232 The sets of logarithmic jet fields along D over open subsets of M form a complete presheaf which defines a sheaf

Jk

We call /k (M~ log D) the sheaf of germs

log D) over M.

(H;

of logarithmic jet fields along D over M. Assume that D has only normal crossings. x

E

0

Take a point

D and a holomorphic local coordinate neighborhood

1

m

U(x , ... ,x) around x 0 so that xo = D n

u

(0, ... ,0), 1 R, = {x , ••• ,x = O}

(1 < R, :_ m).

Then any losarithmic 1-form w along D on an open subset U' of U is written as dx 1

w = al-l-+ X

where a. are holomorphic functions on U'. Jk(M) luJ: U x {dx 1 , ... ,dxm}.

(~k)m

be the trivialization associated with

Then a sections

s (x) = (x,Z (x)): U

---+

E

r(U, Jk(M)) is given by

(~k)m ,.,ith

U x

( zi <_x)

Z(x) i where Z. (x) J

Z~(x)

Z~(x)

are holomorphic functions on U and the indices j

f(z) = x

of 0

E

E

1

Put 1

+IT z1 (x)z + ..• + k!Zk(x)z

k

u, which is a holomorphic mapping from a neighborhood ~ into u. Then jk(f) s(x). Let x e U' and put

f*w = A(z)dz.

We have R,

A(z) =

I;'

L

i=l

aizli

iii os

.l1

+

X

It follows that (1.13)

)

Z~(x)

correspond to the orders of derivatives.

for x

Let

R,

(

I i=l R,

I i=l

i aizl i aizk

1

+

1

+

1X

---r X

m

I i=R-+1

.

m

I i=Hl

i aizl i aizk

)

233 Therefore one sees that s

(1.14)

fk (M;

r (U,

E

log D)) if and only i f

z~J = x iz,ij f.or 1

Z'~

where

J

<

~

i <

and 1 < j

are holomorphic functions on u.

~

k,

Hence we have the

following: (1.15)

Proposition. i)

If D has only normal crossings, then

there is a subbundle rr:

Jk(M; log D)

--+ M

of Jk(M) such that the sheaf of germs of holomorphic sections of Jk (M; log D) is isomorphic to

jk (M;

log D) •

ii) Let N be another complex manifold and E a hypersurface of N with normal crossings. ~

mapping such that

-1

(D)

~

Let c

E.

: N Then

--+ H ~

be a holomorphic

induces a bundle mor-

phism ~*:

Jk(N; log E)

--+

Jk(M; log D).

The assertion ii) follows from (1.10). We call the above Jk(M; log D) the logarithmic jet bundle of order k along D.

In the case of k

J 1 (M; los D)

=

=

1, we also write

T(M; log D),

which is called the vector bundle of logarithmic vector fields along D (cf. [Sl]).

As in (1.2), the fibre bundle Jk(M:log D)

has a structure of flag: (1.16)

Jk(M; log D)

--+

•••

--+

Jl (M; log D) :

such that for a holomorphic sections open subset U

c

E

T(M;log

D)~ M

f(U, Jk-l (M)) over an

M Jk(M; log D) is(U) ~ T(M: log D) iu.

We continue to assume that D has only normal crossings. 1

By definition, every wE r(u, nM(log D))

defines a holomor-

phic vector function (1.17)

w :

Jk(H;

log D)

lu--+

cck

E r(u, n~(log D)) are logarithmic 1-forms along D such that w1 A, •• A wm does not vanish anywhere 1 then

and if w1 , ... ,wm

234 (1.18)

-1 , •.. , w -m) rr x ( w

is a biholomorphism, which we call the trivialization of Jk(M; log D) over U associated with {w 1 , .•. ,wm}. d)

Logarithmic 1-forms and logarithmic jet fields over

complex spaces:

Imbedded case.

Let X be a complex subspace

of M. Then Jk(X) is a complex subspace of Jk(X). Let D be a ·hypersurface of M such that D ¢X. Let~: X--+ X x X be the J(~(X))

dia3onal imbeddins and

the ideal sheaf of the diagonal

1

~(Xl.

The sheaf nx of germs of holomorphic 1-forms is defined

by

Let MX denote the sheaf of germs of meromorphic functions over X and set

=~

=

Mx

®

ll>x

n~

which is called the sheaf of germs of meromorphic 1-forms over X.

By definitions,

n~ and B~ are defined for general complex

spaces which are not necessarily imbedded into complex manifolds.

The inclusion mapping iX: X --+ M induces ix* ix*

We set

n l - nl M X 1 nM (log D)

-

-

0, =~;

1 ·* 1 nx(log D) = 1X(nM(log D)) '

(1.19)

which is a coherent sheaf over X. 1

Let U be an open subset of X and wE r(u, nx(log D)). Then as in (1.6), w induces a meromorphic vector function (1. 20) Lets E r(u', Jk(X)) be a holomorphic section on an open subset U' D if

c

X.

We say that s is a logarithmic jet field along

wosJU: U--+ ~k are holomorphic for all

WE r(U, n~(log D)) with U c U'. We denote by JCk(X; log D) the sheaf of germs of logarithmic jet fields along D over X. Assume that D has only normal crossings. Let

235 (1. 21)

which we call the logarithmic jet space of order k along D over X with imbedding X (1.22)

Proposition. i)

M.

c

The sheaf of germs of holomorphic

sections of Jk(X; log D) is isomorphic tojfk(X; log D). ii) Let Y be a complex subspace of a complex manifold N and

E a hypersurface with normal crossings such that E

~

Y.

f: N - H be a holomorphic mapping such that f(Y)

c

X and

f- 1 oc E.

Then f induces a fibre morphism f*: Jk(Y; log E)

Proof.

Let

--+

Take an arbitrary sections

an open subset U of X.

For any x

E

Jk(X; log D). E

f(U, Jk(X; log D)) over

U and w

with a small open neighborhood U' of x in neighborhood V of x in M and n

E

€

u,

1 f(U', nx(log D))

we may choose a

r(V, n~(log D)) such that

* n =w . v n X = U' and ix n~

It follows from (1.17) and (1.21) that

U'.

Therefore s

take any s s

E

r(u,

E

Jk(X))

Jk (X; log D)

~rre

f(U, Jk(X; log D)).

E

r(u, c

Jk(X; log D)).

r(u,

E

r(u,

On the other hand,

Then, of course, By the definition of

have

s Hence, s

Jk(M)).

is holomorphic on

E

r(u,

Jk(M; loq D)).

Jk(X; log D)) by (1.21).

The second part ii)

follows from Proposition (1.15).

Q.E.D. e)

Logarithmic forms and logarithmic jet fields over

complex spaces.

For the purpose here, it suffices to consider

the case of a complex space X imbedded into a complex manifold M.

However, it is worthwhile to give the intrinsic definitions

of logarithmic forms and logarithmic jet fields over general complex spaces. Let X be a complex space and A a proper analytic subset of X. R(X)

A= that

Let S(X) denote the set of singular points of X and put

=

X-S(X). Let A: X--+ X be a blowing-up of X such that A-lA is of codimension 1 and has only normal crossings and

236 (1.23)

A.!\i{.:.A.-l(A u S(X)))!: X-A-l(A u S(X))-+ X-(A u S(X)

is biholomorphic (Hironaka's theorem). 1

1

nx(log A)

(1.24)

R,

nx(lo~

Then we set

-

A.* nx
~ ni
A)

(t

E 2Z, > O)

which are coherent analytic sheaves over X. We see by (1.11) that n 1 (log A) is independent of the choice of X and so is R, . X nx(log A)· Let Y be another complex space, B a proper analytic subset of Y and f: Y f-lA

c

B.

--+

X a holomorphic mapping such that

Then it follows from (1.11) that f induces a sheaf

morphism (1.25)

n~(lo3

f*

A) -

n;(1o3 B).

Take a section wE r(u, !l~(lo9 A)) over an open subset u of X. Then by (1.23) w induces a holomorphic vector function (1.26)

J k (X)

I (u- (A

s (X) ) )

u

-

a:k

Lets E f(U', Jk(X)) be a section over an open subset U' of X.

We call s a logarithmic jet field along A over U' if

wos!U-(A u S(X)) WE f(U,

!l~(lo3

have holomorphic extensions over U for all A)) with U

c

U',

We denote by

k(X; log A)

the sheaf of germs of logarithmic jet fields along A over X. We consider the case where X is imbedded into a complex manifold M so that A is the intersection of a hypersurface D of

~~

with X.

It immediately follows that R,

llx(log D)

c

~k(X; log A) (1.27)

Proposition.

c

~(X; log D).

If X is normal, then

~k (X; log A) Proof.

R,

!lX(log A),

=

fk (X;

log D).

Let s E r (U, ~ (Xj log D)) be a section over an open 1

subset of U and WE f(U', !lX(log A)) • Then is holomorphic. wos!U' n R(X) ; U' n R(X) - - - + a:k is normal,

Since X

woslu• n R(X) extends holomorphically over U'.

Hence it follows that s E r(u, ~k(X; log A)).

Q.E.D.

237

Assume that X is normal and A is a hypersurface such that for every point x

E

A, there are neighborhood u of x, an imbed~

ding of U into a domain M of some

N

and a

hy~ersurface

with normal crossings satisfyin9 D n X= A n U. tions (1.22) and (1.27) (1.28)

Corollary.

i)

D of M

Then Proposi-

imply the following:

Let the notation be as above.

Then

there is a complex fibre subspace Jk(X; log A) of

Jk(X) such that /k(X; log A) is isomorphic to the sheaf of germs of holomorphic sections of Jk(X; log A) over X: ii)

Let Y be another normal complex space and B a hyper-

surface of Y satisfying the same condition as A. f: Y

--+

X be a holomorphic mapping such that f

-1

Let A

c

B.

Then

we have ~

f*: Jk(Y; log B)

We call the above Jk(X; log A) the

Jk(X; log A). ~ogarithmic

jet space

of order k along A over X. §2

HOLOMORPHIC MAPPINGS INTO SUBVARIETIES

OF GENERAL TYPE OF QUASI-ABELIAN VARIETIES Let A be a quasi-Abelian variety; i.e., A is a complex abelian algebraic group carrying the exact sequence: 0 ~

where

~*

( ~*)

~

-- A -

A0

-

0

(~

E

2Z,

:: 0) ,

denotes the multiplicative group of non-zero complex

numbers and A0 is an Abelian variety (cf. [I2]). irreducible algebraic subvariety of A. (2 .1)

Let X be an

We note that

X is of general type i f and only i f the group {a € A; a + X = X} is finite (cf. [ Il] and [ I2]) .

Taking the compactification (lPl) ~

:J

A=AX

(~*) ~, we have (JPl)R.

(~*) ~ which gives a smooth compactification of A such that D A-A is a hypersurface with normal crossings. Let X be the Zariski closure of X in A. Then we have logarithmic jet spaces along D Jk (X; log D)

c

Jk (A; log D) .

Let oo 1 , •.. ,oo N (N =dim A) be the linearly independent invariant 1-forms on A. Then {ooj}~=l is a basis of H 0 (X, nk(log D)).

238

wj are d-closed and w1 A ... AwN does not vanish anywhere

All on

A(cf. [Il] and [I2]).

Therefore we have the global trivi-

ality of Jk(A; log D): 7T

w:

A. x (CCk)N,

Jk (A; log D) -

: Jk(A; log D) ~A is the projection and 1 (w , ... ,wN) (cf. (1.18)). It follows that

where

w

w

x

7T

Jk(A; log D)

--+

(CCk)N is a proper morphism and so is the

restriction (2. 2)

Set n =dim X.

Let S(X) be the set of singular points of X

and R(X) = X-S(X). mapping.

Le; H0

b~

Let iR(X)

: R(X) --+A be the inclusion

the linear subspace

of f(R(X), KR(X))

Jn), 1 .::_ j 1 < ••• < jn .::_ N over a:, where KR(X) is the canonical bundle over R(X). We denote by spanned by iR(X) (c}l, . . .

Z(n,X) th~ closure in X of the zero locus of n n "I

E

H0

If

o, then Z (n ,X) is a proper algebraic subset of X. A(X) = {S(X)} u {Z(n,X); n

(2.3)

Definition.

E

set

H0 -{o}} •

A holomorphic mapping f: Y

---+

X from a

complex space Y into X is said to be non-degenerate with respect to i\(X) if f(Y) By [N4], Lemroa(2.4) (2.4)

Lemma.

¢ Z for any Z

E

i\(X).

we have the following:

Assume that X is of general type.

Then the

differential dJtt of

has maximal rank at some points. Moreover, if a holomorphic mapping F : W --+ X from a domain W of a: into X is non-degenerate with respect to i\(X), then the lifting Jn(p): W--+ Jn(X)

c

Jn(X; log D) generically takes values at which din has

maximal rank. (2.5)

Theorem.

Assume that X is of general type.

a Zariski open subset of a compact complex spaceY.

Let Y be Then

there are only finitely many holomorphic mappings f: Y

--+

which are non-generate with respect to i\(X). Proof.

Taking a desingularization of

Y,

we may assume

X

239 that Y is smooth.

It follows from [N3, Corollary (4.7)] that

any f: Y --+ X, non-degenerate with respect to A(X) can be

Y into X. Then Y is projective, so

meromorphically extended from

by [N-Sl, Lemma

(2.3)] we may assume that

that f: Y--+ X

is a

(rational) morphism from Y into X. Assume that there are infinitely many such f's; to say,

f 1 ,f 2 , . . . . By Lemma (2.12) proved later, we take an algebraic irreducible curve C in Y such that the restrictions f.!c : C--+ X, j=l, 2, •.. , are distinct and non-degenerate J with respect to A(X). Therefore we may assume that Y is a smooth curve.

Put E = Y-Y.

Taking out several points of Y if

necessary, we may assume that there is a rational function t on Y such that t is regular on Y and the differential dt does not vanish anywhere on Y.

Let

(2.6) be the trivialization associated with dt (cf. n = dim X.

(1.5)), where

By Eroposition (1.22), any morphism f: Y --+

induces f*: J

n

(Y; log £-lD)

--+

J

n

X

(X; lo~ D).

By using (2.6) we have J

n

(Y· log f- 1 DJ! (Y-f- 1 D): (Y-f- 1 D) x a:n . '

Put (2.7)

Jn(f!Y-f

-1

D): Y-f

-1

D ~ y ~ f*(y,

(1)°b

)

E

J

n

(iC· log D). '

We use t as holomorphic local coordinates on Y-f-lD which yields the lifting of f!Y-f-lD by (1.3); this lifting coincides with J to

(fiY-f-lD).

A(X~,

If fiY-f-lD is non-degenerate with respect it follows from Lemma (2.4) that J (fiY-f- 1 DJ generin

cally takes values at which di -

n N

I n : J n (X; loa D) ~ (CC ) v factorization to I : n J

has maximal rank. n

.

is proper, we apply the Ste1n

(X, log D)

~!z ~n I

Where

z

Since

I

n

is a normal algebraic variety,

~

is a surjective mor-

240 phism with connected fibres, I' is a finite morphism and

=

In

n

I~ o].l.

Hence Z is affine algebraic. I~,

rational functions on Z to
Adding several

we have a proper imbedding

z --+ erN'

:

Put ( 2. 9)

J

We note that the fibres

of~

n

(X·'

log D)

---+

a: N'

are connected. By (2.7) and(2.9) ~oJn(f\Y-f- 1 D)

we have a rational vector function n(f) = on

Y for

E u f

-1

any f: Y--+ X, such that n(f) may have poles on

D and the orders of poles of n(f) are bounded from above

by an integer t

0

> 0, independent of f.

Let L(t 0 E) be the finite dimensional vector space of rational functions on

Y with

polar divisors ~ t 0 E, and put N'

H=

JIL(R.E)·

l 0 Let F be the family of all morphism f: Y

with respect to A(X). (2.10)

G

=

X, non-degenerate

Then we see that the mapping F 3

is injective.

~

f

f.---+

T)

(f)

E

H

Put

{(y, w, a)

Y x Jn(X; log D) x H; a(y)

E

=

~(w)}

which is an algebraic subset of Y x Jn(X; log D) x H. ~

Since

is proper, the projection : G 3 (y,w,a.) .....-

p

is proper. Y x H. at which

Put G'

=

(y,a.)

" Y x H

p(G), which is an algebraic subset of

LetS be the algebraic subset of points of Jn(X; log D) d~

does not have maximal rank, and put Z = H'

{a" H

H"

{a

E

H'; a.(Y)

~

Z}. Z~iski

For a. " H" we put a.#

w- 1 oa.: 'lfoO.#!

which are rational mappings.

Put

Y x {a} c G'}.

Then H' is an algebraic subset of H and H" is a subset of H'.

~(S).

y---+ Jn(X; Y-

log D),

X

Consider the equation

open

241 (2.11)

Denote by H"' the algebraic subset of H" defined by (2.11). Then n (f)

E H"'

for f E F and F 3 f

tive by (2.10).

1--->- n

(f)

E H'"

We are goin<J to show that dim H"'

Assume that dim H"'

> 0.

is injec= 0.

Take an irreducible curve R

and the normalization~: R' ~ R of R.

H'"

c

Then we have a

rational mappin9 'I': R' x Y 3(a,y) ............. ab(y)

Taking a smooth compactification

R'

EX.

of R', we have the rational

extension ¥of '1':

if:

R'

X.

Y -

X

Since the indeterminanc~ locus I(if) of¥ is a finite set of points, I('!') ~{a} x

V for

all a E

R',

so that Wl{a} x

For generic ~

fine morvhisms from y to X.

E

R'

Y

de-

we have

n(ifl{a}xY)EH

and hence a morphism y: R';;.~ ~-->- n(iiJifa} x Y)

R'

Since stant.

is compact and H is an affine space, y must be conThis contradicts (2.11).

Now we show the (2.12)

Lemma.

fi; Y--+ are

e: H.

x,

followin~

Q.E.D.

lemma used in the above proof:

Let Y be an alsebraic variety and

i = 1,2 ••. ,distinct holomorphic mappin5s which

non-de~enerate

with respect to A(X).

Then there is an

irreducible alsebraic curve c in Y such that f. I C: C - x ~

are all distinct and non-degenerate with respect to A(x). ?roof.

Put Z. . ~J

{y

=

Y; f. (y) ~ .

E

=

for i < j.

f. (y)} J

Then Z .. are proper algebraic subsets of Y. ~J

Z =

u

i<j

Z .. ~J

u

u

i

f~ 1 s (X) ~

Put

uS (Y).

Then the subset Z of Y does not contain a non-empty open set of Y.

Take a point y 0

hood U

c

R(Y)

E

Y-Z and a local coordinate neighbor-

such that u is biholomorphic to the polydisc

lim '"' { (;l' ... 'ym) ; Y0

.,.

(O, .. ,O)

Iyj I

=o,

< l} c

a:m,

242 where m =dim Y.

By [N4, Remark (2.3)] we have an analytically

non-degenerate holomorphic mapping ¢: ~ --+ ~m with ¢(0) = 0. Here ¢ is said to be analytically non-degenerate if the imase ¢(~)

is not contained in any proper analytic subset of ~m.

Let {cr 1 , ... ,cr} be . the maximal linearly independent system of . q {'* ( Jn · · ~ N} . h f i are 1R(X) w A ••• A wJn) ; 1 ~ Jl < ••• < Jn Ten non-degenerate with respect to A(X) if and only if -1 -1 -1 fi cr 1 , ... ,fi crq E r(u, £i KR(X)) are linearly independent. Therefore f. are non-degenerate with respect to A(X) if and 1 -1 -1 -1 only if (fio¢) crl, ... ,(fio¢) aq_ E ret:., (fio¢) KR(X)) are linearly independent. for every i. Tij on U.

f~ 1 KR(X)

Fix a trivialization

-1

Then f.

=~

x

~

a. are given by holomorphic functions

1 J Put Tfj = Tij(¢(z)).

Then it follows that

Ti 1 , ... ,Tiq are linearly independent if and only if the Wronskians

d. (z) J

Tfl (z)

T~

Tfi (z)

T~

"t' iiq-1) (z)

T

1q

( z)

1 (

1q_

Z)

;k(q-1) (z) 1q

* 0;

Note that ¢- 1 ( )

that is, the sets {d. (z) = O} are discrete. J

is the countable union of discrete subsets. We take a point z 0 E ll-(uj {dj(z) = O}u ¢- 1 (Z)) and put y~ = ¢(z 0 ). Set ¢ (z) = (¢

1

m (z), ... ,¢ (z)), 00

L

k=O

y' = 0

k

\)

ak (z - z 0

)

,

243 Now we may assume th a t y l

1 ••• 1

. t'1.ons o.f y Ill are t he r e s t r1c

affine coordinate functions over an affine open subset w of R (Y) with U c W c a:£. Let y 1 1 • • • 1 Yf/., be the affine coordinate functions of

a:f/.,.

Consider the affine equations: a

(2.13) y

Ill

a

1

1

. + al

0

Ill

0

1 + a q-1

z +

...

Ill + al z +

Z

Ill

+ a q-1

Z

q-1 I

q-1 I

where z is considered as the affine coordinate function over

a:.

Let V be the intersection of W x

a:£

set of

x a: defined by (2.13)

1

a:

and the algebraic sub-

and V' the irreducible com-

ponent of V containing the set of points defined by (2.13) with z E ~and (y 1 1 .• 1yrn) E ~rn ~ U. Let p 1 : V' ~ (y 1 Z) ~ y E Wand p 2 : V' 3 (y 1 z) ~ z E a: be the projections. Then p 1 and p 2 are locally biholornorphic around (y~ 1 0). Let C be the Zariski closure of p 1 (V') in Y. Then Cis smooth at y'0 1 . around which z gives the local coordinate. of

T1.. 1

di(z 0 T. l.q

1c

lc

n U1

... 1 T.l.q

lc

n

u at

by the construction.

)

n

u are

y'0

E

C with respect. to z are

Since di(z 0

linearly independent.

The Wronskians

)

'I

01 ti 1 1Cnul•••l

Therefore f. l

lc:

C--+ X

are distinct and non-degenerate with respect to A(X). (2.14)

Example.

Q.E.D.

Let C be a compact smooth curve with genus

3 and a: C-+ A the Albanese mapping.

Put

X = a.(C) + a(C) c A. Then X is an algebraic surface of general type.

For y

E C1

we have rnorphisrns: a

y

C 3 x r-+ a(x) + a(y)

EX.

Moving y E C 1 we have infinitely many rnorphisrns ay which are degenerate with respect to A(X). (2.15) position

C- X

Remark.

ox

Let H11 ••. 1Hn+ 2 be hyperplanes in general the n-dirnensional complex projective space Fn(a:)

z1 1 ... 1 Zn+ 2 divisors on a:rn which may be transcendental. Then FujiMoto [Fl] proved that there are only finitel~ many

and

linearly non-desenerate rnerornorphic rnappin~s f:
such that f*H. al9ebraic.

J

= Z. for

244

implies that f are necessarily rational. Put n+2. X = JPn ( C) - U H . j=l J Then X is of general type. Take a homogeneous coordinate system (u 1 , ... , un+l) of JPn { €) so that { ,n+l j Lj=lu

::._ j ::._ n+l, Hn+ 2 Set

O}.

= 1, 2, ... , n, and dzj . --.- , 1 < J ::._ n, ZJ

-

dz 1 + ... + dzn

n+l

w

1 n 1 + z + ... + z n+2 . of r (JP n (C) ,s-2 1 Then {w 1 , ••• , wn+l} forms a baslS (log u H.)) JPn( C) j=l J

and the quasi-Albanese mapping a:

x 1

X :3 x --+- ( J w , ••• ,

Jx wn+l } E ( C *) n+l

We identify X with a(X) C (C*)n+l.

is an imbedding.

By [Nl,

Proposition 4.1] a holomorphic mapping f into X is non-degenerate with respect to A(X) if and only if f is a linearly nondegenerate holomorphic mappings into XC Wn(C).

Therefore

Theorem (2.5) implies that there are only finitely many linearly non-degenerate holomorphic mappings f: em- uj:~supp zj X C Wn ( C) . This is the same as the above theorem of Fujimoto in the case where z. are algebraic. J

§3

RATIONAL MAPPINGS INTO ALGEBRAIC VARIETIES WITH QUASI-NEGATIVE Tq(V; log D)

In this section, we prove a finiteness theorem of another type, which generalizes the Main Theorem (1.2) of [N-Sl] in the case of the complex number field.

Let

v

be a compact

complex algebraic variety and V a Zariski open subset of V. Put D

V - V.

Let F

(3.1) Definition.

--+-

V be a vector bundle over V.

We say that F is quasi-negative over V if

there is a proper morphism

F

--+-

CN

such that -0):

(F!V)-0--+-
245

Remark.

If D =

~.

then the above quasi-negativity is

equivalent to the weak negativity of Grauert [Gl].

V is

Assume that mal crossings.

smooth and D is a hypersurface with nor-

Set Tq(V; log D) =

A

T(V; log D)

1

which is the dual of the locally free sheaf n~(log D). W be another complex algebraic variety and rational mapping.

Let

W--+ v a

f~

Then f is said to be proper if the first

projection of the graph G(f)

c

W x V onto

rank f = sup {dim W - dimxf

-1

w is

f (x); x

proper. E

Set

~7}.

We denote by F (W,V) the set of all proper rational mappings q f: W --+ V with rank f > q. (3.2)

Theorem.

over V.

Assume that Tq(V; log D) is quasi-negative

Then F (W,V) is finite. q

Proof.

We may assume that W is a Zariski open subset .of

a compact algebraic variety

Wsuch

that E = W~~ is a hyper-

surface with normal crossings; moreover we may assume that there are rational functions x 1 , ... ,xm (m=dim W) over w such that xj are regular on Wand dx 1 A ••• Adxm does not vanish anywhere on W.

-

-

Every f

E

F (W,V) defines a rational mapping from

q

W into V which is denoted by the same f. indeterminacy locus of f:

w--+

V.

Let I(f) denote the

Then f

E

F (W,V) induces q

the non-trivial bundle morphism. (3.3) If f,g

E

F (W,V) are distinct, then (f! (W-(I(f) ui(g)))* q

(g!W-(I(g)ui(f)))*.

Since Tq(V; log D) is quasi-n~gative over

V, we have a proper morphism ~: Tq(V; log D) --+ ~ nition (3.1).

as in Defi-

As in the proof of Theorem (2.5), we may assume

that ~-l~(y) are connected for y E Tq(V; log D). m we have a trivialization of x 1 , ... ,x, (3.4)

t

q-

T (W; log E) !W

=w x

By making use

(~)

~

.

By using (3.3), (3.4) and~: Tq(V; log D)--+ ~N, we apply the same arguments as in the proofs of Theorem (2.5) in section 2

246

and the Main Theorem (1.2) of [N-Sl] to complete the proof. Q.E.D. Example. Let H0 1 ... 1 H4 be 5 hyperplanes in general position of JP 2 (
H.

{uj

H3

{a 0 u 0 + a 1 u 1 + a 2 u 2

0}

I

H4

= {b 0 u 0 + b 1 u 1 + b 2 u 2

0}

1

J

where ai1bi 2 1

r

(JP

(
E

r2

I

JP

Set

2

Z

=

0}

0 ~ j

I

i __ ui/uo.

-< 2

Then a basis of

(log A)) is given by (
1 2 ~1j ' _ dzl = dz ' _ ' ~2 j'_ 2

z

z

ao + alz b 1 dz

1

1

+ a2z

+ b 2 dz

b0 + b1z

1

2

2

+ b2z

2

The vector fields z 1 3/3z 1 and z 2 a;az 2 form a local frame of T(JP

2

(
q, (V)

for v

l

v~

=

A) around (0 1 0). (¢

1

(v)

1

••• 1

cjJ

1 --3 + v2 2 2 --CJ 3zl az 2

Put

4

E

(v))

T(JP 2 (
Then the

morphism (3.6)

¢:

T(JP 2(a:); log A)-->-

a: 4

is proper.

We calculqte the Jacobian of q, with respect to the coordinate 1 2 1 2 2 system (z 1 z 1 V 1 V ) of T (JP (
247 (3. 7)

1 2 1 2 :J(z,z,v,v)

+ {b 1 v

1

+ (v

1

2 2 2 -v lb 1 b 2 z } v a 2

Now take 6 hyperplanes H0 , ••• ,H 5 in general position of F 2 (t) so that {u 2 =0}, 0 ~ j < 2 H. J ' 0 + ul + 2 {u u 0}' H3 {u 0 + a 11 u 1 + a 12 u 2 0}, H4 H5

2 1 2 2 {u 0 + a,u + a 2u .L

1/Jl

dz 1

Put

dz

1/!3

dz 2 --2 ' z

1/12

7 1

0}.

+ dz 2

2 1 1 + z + z 1 1 1 a 1 dz + a2 dz 2 1 1 1 2 1 + a1 z + a 2z 2 2 1 a 1 dz + a2 dz 2 2 2 2 1 1 + a 1 z + a2 z and

2

T(F («:);log 5 where D = uj=OHj.

1

v ,...._. (1/i (v), ... ,1/J

D)~

Then, as in (3.6),

~is

Consider two equations: d (1/Jl ,1/12,1/13 ,1/14)

1

2

1

0,

2

()(z,z,v,v) a (1/Jl ,1/12,1/13 ,1/15) 1 2 1 2 a(z,z,v,v)

0 •

5

(v))

E

5 a:,

a proper morphism.

248 By

(3. 7) we have

(v 2 - v 1)(•1 v z 1 + v 2 z 2 )a 1i a i2 - {v 2 + (v 2 - v 1 )z 1 }v1 a 1i

(3. 8)

+ {v

1

+ (v

1

We want to choose

2 2 2 i -v )z }v a 2 = 0, i = 1,2 •

a~ so that

'I'

I (T (JP 2 (CC);

log D) j((JP 2 (CC)-D) -0)

is an isomorphism onto its image and the degenerating locus of 'I' is contained in the union of the zero section and T(JP 2 (cc); log D) !D.

We may assume that 1 2 v = 1 and v 'I 1

Put zl

(v

zl

(v

2

2

l)z

1 ('f 2 0 on lP (CC)- D) , 2 2

l)v z

Substituting these to (3.8), we have (a 1i a 2i - a 1i )Z1 + (a 1i a i2 - a12·) z2 + (a i2 - a 1i )v 2

(3.9)

i

If v 2

=

0 '

1,2 0, then z 2

0, so that i

i

a 1 (a 2 - 1) = 0 This contradicts the assumption that H.,j=O, ... ,S, are in J

general position.

If the determinant 6 of coefficients of (3.9) in the variables z1 ,z 2 ,v 2 , is not zero, then 'I' satisfies

our requirements.

By calculation we have

1 1 1 a1a2 - al

1 1 1 ala2 - a2

2 2 2 ala2 - al

2 2 2 ala2 - a2

1 1 1 a1a2 - al

1 1 1 ala2 - a2

2 2 2 a1a2 - al

2 2 2 ala2 - a2

2

0

Therefore i f

then T(lP 2 (CC); log D) is quasi-negative over JP 2 (CC)- D,

'I

0

•

249

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H. Cartan, Ideaux de fonctions analytiques de n variables, Ann. Ecole Normale, (3) 61(1944), 149-197.

[ Dl]

P. Deligne, Theorie de Hod9e II, Inst. Hautes Etudes Sci. Publ. 40 (1971), 5-57.

[ Fl]

H. Fu~imoto, Remark& to the uniqueness problem of meromorphic maps into WN(~). IV, N~goya Math. J. 83(1981), 153-181.

[ Gl]

H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann.l46 (1962), 331-368.

[G-Gl] M. Green and P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979, pp. 41-74, Springer-Verlag, New YorkHeidelberg-Berlin, 1980. [ Il]

s. Iitaka, Lo9arithmic forms of algebraic varieties, J. Pac. Sci. Univ. Tokyo Sect. IA 23(1976), 525-544.

[ I2]

S. Iitaka, On logarithmic Kodaira dimension of algebraic varieties, Complex Analysis and Algebraic Geometry, pp. 175-189, Iwanami, Tokyo, 1977.

[Nl]

J. Noguchi, Holomorphic curves in algebraic varieties, Hiroshima Math. J. 7(1977), 833-853.

[ N2]

J. Noguchi, Supplement to "Holomorphic curves in algebraic varieties", Hiroshima Math. J. 10(1980), 229-231.

[ N3]

J. Noguchi, Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Na9oya Math. J. 83(1981), 213-233.

[N4]

J. Noguchi, On the value distribution of meromorphic mappings of covering spaces over ~m into algebraic varieties, to appear in J. Hath. Soc. ,Japan.

[N-Sl] J. Noguchi and T. Sunada, Finiteness of the family of rational and meromorphic mappings into algebraic varieties, Amer. J. Math 104(1982), 887-900. [01]

T. Ochiai, On holomorphic curves in algebraic varieties with ample irregularity, Invet. Hath. 43(1977), 83-96.

[Sl]

K. Saito, Theory o£ logarithmic differential forms and lo_gari thmic vector fields, ,J. Fac. Sci. Univ. Tokyo, Sec. IA 27 (1980), 265-291.

[Stl]

W. Stoll, Die heiden Hauptsatze der Wertverteilungstheorie bei Funktionen mehrerer komplexer Veranderlichen (I), Acta Math. 90 (1953), 1-115; ibid (II), Acta Math. 92(1954), 55-169.

[Vl]

A.L. Vitter, The lemma of logarithmic derivatives in several complex variables, Duke Math. J. 44(1977), 89104.

Remarks on the Nakano Vanishing Theorem Bernard Shiffman Johns Hopkins University Baltimore, Maryland 21218, U.S.A.

In this note we give a numerical version of k-ampleness for line bundles
This vanishing

theorem yields a Lefschetz-type theorem
We begin

by reviewing the Nakai-Moishezon-Kleiman criterion for ampleness on which our numerical condition is based. Let

L

be a holomorphic line bundle on a compact

n-dimensional complex manifold of a meromorphic section of where the

mi£Z

can regard

D

and the

X, and let

L.

D

We can write

be the divisor D = }miDi,

Di are complex hypersurfaces.

We

as a current

coefficients> of type

<1,1)

on

X

by the formula

(1)

for

c~

(n-1,n-1)-forms

«

on

x.

It is well known that

represents the Chern class (with real coefficients> of c1.

One way to see this is as follows:

hermitian metric section of

L

where {ha}

8

on

with divisor

AJJ

(2)

s

II II

aa

L, and let D.

A JJ 2

is the curvature form of and

L.

{sa}

with respect to a local covering

llsll2=ha1sat2 •

Consider a be a meromorphic

We note that

log llsll =

represent the metric

s

8 + D ,

To verify <2>, we let represent the section

{Ua}• so that

The identity <2> then follows from the

curvature formula

-aa

D

L,

8 = log ha and the Poincare-Lelong formula

252

,/-1 ,.;;

1 og

1111

= o·1v

sa •

A

and the left side 2 n 8 is a representative of (2) is exact, it follows that D is also a representative

Since of

lsal

of

c1

as claimed.

It is the observation <2> which led to the CarlsenGriffiths [2] generalization of the First and Second Main Theorems of Stoll [13] to maps into algebraic varieties. Suppose

L

is an ample line bundle on a compact complex

X, and let

d-dimensional algebraic subvariety of is ample,

X

X,

be a Since

<1~d~).

there is a hermitian metric on

at all points of

Y L

A

with

8

L

>

0

and hence

(3)

Since

Fi 2n

8

c1 (L),

is a representative of

any hermitian metric on

L

(3) is valid for

and is equivalent to

(4)

where

Y

is regarded in (4) as a real

We shall restate (4) in algebraic terms. divisor on

X.

We let

[DJ

with the divisor

D;

[DJ

whose divisor is

D.

Hence

Suppose

D is a

denote the line bundle associated has a global meromorphic section D

is a representative of

c1([DJ>, and the intersection number od.y

=

d,y)

<See [7] or [11, pp. 135-139].)

Thus

• (4)

is equivalent to

the inequality

od·v > o

(4.)

where

D

is the divisor of a meromorphic section of

L.

The

Nakai-Moishezon-Kleiman criterion [7J, [9J, [10] (see also If

[4, pp. 29-33]) gives the converse:

L

is a line bundle

on a projective manifold such that (4) or (4') holds for all d-dimensional subvarieties

Y

of

X

for

1~~.

then

L

is

ample. The Nakano Vanishing Theorem [1] states that for an ample

253 line bundle

L

en a projective manifold

=

HQ<X,OP(L))

(5)

where

QP(L)

0

X,

fer p+q>n,

denotes the sheaf of germs of L-valued

hclcmcrphic p-fcrms.

We shall also write

HP • q <X, U

(6)

=

Hq ( X, aP
which by Hedge theory can be identified with the space of harmonic
X

compact Kahler).

Recall

(5) is equivalent to

(7)

Hedge theory also provides a proof of the First Lefschetz Theorem <with rational coefficients).

This observation,

originally due to Kcdaira and Spencer [8] is embodied in the following lemma: Lemma 1.

Let

D

be a smooth hypersurface in a compact Kahler

manifold

X, and let

HP•Q<X,[-DJ)

be fixed.

k~

=

If

Hp,q-1<0,[-Dllo>

0

fer p+q=k ,

then

Fer a proof of Lemma 1, see [11, Th. 3.44].

[-Dllo

D.>

is the dual to the normal bundle of

Lemma 1 and

the Nakano Vanishing Theorem (7) immediately imply the theorem of Lefschetz:

~allowing

First Lefschetz Theorem.

Let

projective manifold and let of

L.

X D

be an n-dimensicnal

be a smooth hyperplane section

Then the restriction maps Hj<X;Q) ~ Hj
are bijective fer

j~-2

and injective fer

j=n-1.

We recall next the concept of k-ampleness due to Scmmese [12]: that

Let L

is

L

be a hclcmcrphic line bundle en

X.

We say

k-ample if there exists a positive integer

and global hclcmcrphic sections

s1, ••• ,sm

of

LN

N

such that

254

at least one of the

Sj

does not vanish at each point of

X

and the map (8)

has at most k-dimensional fibres. A differential-

Note that 0-ample corresponds to ample.

geometric analogue of k-ampleness is k-positivity. holomorphic line bundle

L

carries a hermitian metric whose curvature form semi-positive and has at least each point.

A

is defined to be k-positive if n-k

8

L

is

positive eigenvalues at

The following summarizes two generalizations of

the Nakano Vanishing Theorem: Theorem 1. a>

L

If either is a k-ample line bundle on a projective manifold

x, or b>

L is a k-positive line bundle on a compact Kahler manifold

X,

then HP,q<X,L> = 0

for

HP•q<x,L-1) = 0

p+q)n+k,

for

p+q
Part (a) of Theorem 1 is due to Sommese [12] and part (b) is due to Girbau [3].

(See also [11, Ch.

IIIJ.)

We shall

generalize Theorem 1(a) to nu•erically k-a•ple line bundles, which we define below. First, we recall the following definition: is an algebraic variety. family of subvarieties of

We say that {As}sES X

if

S

YcXXS

Ynn-1<s>=Asx{s}

n:Xxs~x

projection.

Let

We say that

algebraic family in

X

is an algebraic such that is the

Definition 1. X.

sES, where

X

is an algebraic variety

and there is an algebraic subvariety for all

Suppose

D

D

be a divisor on a projective manifold

is numerically k-ample if there exists an {As}sES

of subvarieties of dimension

~k

such that for every irreducible algebraic subvariety

255 vex

such that dim V = d

0-dimensional for all

1

~

and

VnAs

sES, we have

holomorphic line bundle

L

is empty or

nd-v

>

0.

We say that a

is numerically k-ample if any

divisor of a meromorphic section of

L

is numerically

k-ample, or equivalently if above. We have the following observation: Proposition 1.

Let

L

projective manifold.

be a holomorphic line bundle on a If

L

is k-ample, then

L

is

numerically k-ample. Proof:

Suppose

manifold

L is a k-ample line bundle on a projective

X, and let

L=t*H, where

H

tL:x~pm-1

be given as in .

is the hyperplane section bundle on

We consider the algebraic family of fibres SEI(X).

Let

vex

Then ~pm-1.

As=t-1(s)

be given as in Definition 1.

for

Choose

sEt; since

o,

dim Vnt-1 <s> it follows that dim t

dim V

d.

Thus d, [V]l = d, and hence

L

t*['f])

>

0 ,

is numerically k-ample.

The following result generalizes Theorem 1(a): Theorem 2: p~ojective

Let

L be a numerically k-ample line bundle on a

manifold

X.

Then

HP•q<X,L> = 0 HP,q<x,L-1> = 0

for for

p+q>n+k, p+q
Theorem 2 yields the following generalization of the First Lefschetz Theorem: Theorem 3:

If

D

is a numerically k-ample smooth

hypersurface in a projective manifold Hj<X,D;Ql = 0

for

X, then J
256 We first prove that Theorem 2 implies Theorem 3: X

be as in Theorem 3 and let

J
Let

L=[DJ.

o

p+q=j.

Let

D,

By Theorem

2,

HP•q<x,L-1> = One easily checks that bundle on

LID

D. Thus Hp,q- 1

and thus for p+q=J
for

is a numerically k-ample line 0

for

p+q-l
The conclusion of Theorem 3 then

follows from Lemma 1. Note that the conclusion of Theorem 3 is also valid for k-positive line bundles on compact Kahler manifolds.

This

follows from the Birbau vanishing theorem 1(b) and the above In order to prove Theorem 2, we need the following

proof.

slicing technique of Sommese <see [12J, [11, Lemma 2.

Let

manifold

X

Let

p,q~.

D

<3.24)]):

be a smooth hypersurface in a compact Kahler

and let

E

be a holomorphic vector bundle on

If

a)

HP•q<X, [DJ8E) = 0 ,

b)

HP-1,q-1

c)

Hp,q-1
0

HDJ8E> ID>

0 ,

then HP,q<X,E> = 0 • Proof:

Consider the short exact sequences of sheaves

<10>

o ~ ng-1eEID Ad~ ID ~ nge<[DJ8E> ID ~ o ,

where

s

E

r<X,[Dl>

with

Div s =D.

By ,

(c), and the

long exact sequence associated with <10>, it follows that (11)

Hq-1
<~8[DJ8E> I D) = 0 •

The desired vanishing then follows from ,

<11), and the

long exact cohomology sequence associated with (9).

X.

257 In order to prove Theorem 2 by induction on

k, we need

the following result of Hironaka: Lemma 3

(Hironaka [5Jl:

projective variety

Let

L

be an ample line bundle on a

X, and let

of subvarieties of X.

{As}sES be an algebraic family

Then there exists a positive integer

such that the generic element of

N

rcx,LN) does not vanish

identically on any positive-dimensional irreducible component of

As

for all

SES.

Far a proof of Lemma 3, see [5, Sec. 2J or [11, pp.62-63J. We now prove Theorem 2: and let k-ample.

{As}sES

Let

L, X

be as in-Theorem 2,

be as in the definition of numerically

We use induction on

k.

If

k=O, then

L

is ample

by the Nakai-Moishezon-Kleiman criterion, and the conclusion fallows by the Nakano Vanishing Theorem. that

k~1

and the theorem is valid for numerically

-ample line bundles.

X

such that

E9L

Choose an ample line buntl-le

is ample.

hypersurface

D

EN

D.

an

N

and a global

whose divisor is a smooth

such that dim DnAs

for all sES.

E

By Lemma 3 and Bertini's

Theorem, we can find a positive integer holomorphic section of

bundle on

Thus we can assume

Then Let

LID

~

k -

1

is a numerically
p+q>n+k.

Then

( 12)

0

'

(13) by the Nakano Vanishing Theorem.

Furthermore, by our

inductive hypothesis, (14) since p-1+q-1>n-1+k-1.

The first conclusion of Theorem 2

follows from <12>,<13>,<14>, and Lemma 3.

The second

conclusion follows by Serre duality. The following special case of Theorem 2 is a Nakano-type

258

version of the Kawamata-Viehweg vanishing theorem. Corollary 1.

Let

D

be a divisor on a projective manifold

X, such that

od·v > o for all pure d-dimensional subvarieties

Proof: X

0

for

p+q>n+k,

HP•Q<X,[-DJ>

0

for

p+q
Let

Hl•···•Hn-k

such that the intersection

subvarieties

Y

A.

is

be hyperplane sections

A=Hln ••• nHn-k

has

It suffices to show that the set of

such that

dim Y

>

n-k

and

dim AnY

~

0

is

Suppose

>

dim Y = d n-d.

D

We consider the algebraic family consisting of

k.

the single subvariety

Choose

X, fer

By Theorem 2, it suffices to show that

dimension

empty.

of

HP•Q<X, [DJ)

numerically k-ample. in

Y

Then

l~:!>n-k.

Hn-k+l•••••Hd

such that

n -

k.

H1n ••• nHd

has dimension

Then the intersection number Y·H1·····Hd

<where

[HJ

=

d, Y>

> 0

is the hyperplane section bundle). YnA

~

Therefore

YnHln ••• nHd ¢ 0

which implies that dim YnA

~

d+k-n

>

0,

which completes the proof. Corollary 1 is an analogue of the following vanishing theorem of Kawamata [6] and Viehweg [14J: Theorem 4
Let

D

be a numerically

effective divisor- on a pr-ojective manifold 0 n-k.Hk > 0 wher-e

H

is a hyper-plane section of Hq <X, [-DJ)

Recall that

D

algebraic curves

=

0

for

' x.

Then

q < n

is numerically effective if ccx.

such that

X

.

k

D·C~O

for- all

Note that the hypotheses of Theor-em 4

are weaker than the hypotheses of Cor-ollary 1.

259 We end this report by giving a variation of Theorem 2: Theorem 5.

Let

L

be a holomorphic line bundle on a

projective manifold family that

{As>sES ,C>

such that

X.

Suppose there exists an algebraic

of subvarieties of dimension

>0

C~As

in

X

such

for all irreducible algebraic curves

for all

exists a positive integer base point set of

~k

LN

ccx

sES.

Suppose further that there

N

such that the dimension of the

is ~k.

Then the conclusion of Theorem

2 holds. Proof:

We note that if

(3.38>,

(7.5)J.)

k=O

then

L

is ample.

<See [11,

The theorem then follows by induction

exactly as in the proof of Theorem 2. Theorem 5 is the numerical analogue of Theorem <3.37> in [11].

REFERENCES 1.

Akizuki, Y., and Nakano, S., Note on Kodaira-Spencer·s proof of Lefschetz theorems, Proc. Jap. Acad. 30 <1954>, 266-272.

2.

Carlson, J., and Griffiths, P.A., A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Hath. 95 (1972>, 557-584.

3.

Girbau, J., Sur le th~or~me deLe Potier d'annulation de la cohomologie, C. R. Acad. Sci. Paris 283 (1976>, Serie A., 355-358.

4.

Hartshorne, R., A•ple subvarieties of algebraic varieties, Lecture Notes in Math. 156, Springer-Verlag, New York, 1970.

5.

Hironaka, H., Smoothing of algebraic cycles of small

6.

Kawamata, Y., A generalization of Kodaira-Ramanujam·s

7.

Kleiman, S.L., Toward a numerical theory of

dimensions, A•er. J. Hath. 90 <1968>, 1-54. vanishing theorem, Hath. Ann. 261 (1982>, 43-46. ampleness, Ann. of Hath. 84 <1966>, 293-344.

260 B.

Kodaira, K., and Spencer, D., On a theorem of Lefschetz and the lemma of Enriques-Severi-Zariski, Proc. Hat.

Acad. Sci. U.S.A. 39 <1953>, 1273-1278. 9.

Moishezon, B.G., A criterion for projectivity of complete algebraic abstract varieties, A•er. Hath. Soc.

Translations 63 (1967>, 1-50. 10.

Nakai, Y., A criterion of an ample sheaf on a projective scheme, A•er. J. Hath. 85 (1963>, 14-26.

11.

Shiffman, B., and Sommese, A.J., Vanishing theorems on

co•plex •anifolds, Progress in Math. 56,

Birkh~user,

Boston, 1985. 12.

Sommese, A.J., Submanifolds of abelian varieties, Hath.

Ann. 233 (1978>, 229-250.

f3.

Stoll, W., Die beiden Hauptsatze der Wertverteilungstheorie bei Funktionen merherer komplexen Varanderlichen ,

(II~,

Acta Hath. 90 (1953), 1-115, and 92 (1954>,

55-169. 14.

Viehweg, E., Vanishing theorems, J. Reine AngeN. Hath. 335 (1982)' 1-8.

Curvature of the Weii-Petersson Metric in the Moduli Space of Compact Kihlet-Einstein Manifolds of Negative First Chern Class Yum-Tong Siu Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138 USA

For compact Riemann surfaces of genus at least two, using Petersson's Hermitian pairing for automorphic forms, Weil introduced a Hermitian metric for the Teichmuller space, now known as the Weii-Petersson metric.

Ahlfors [1,2) showed that the Weil-

Petersson metric is Kahler and that its Ricci and holomorphic section curvatures are negative.

By using a different method of curvature computation, Royden [8) later showed

that the holomorphic sectional curvature of the Weii-Petersson metric is bounded away from zero and conJ"ectured

the best bound to be -

~ , where g is the genus.

.. 11,g-IJ

Recently Wolpert (12] and also Royden proved Royden's conjecture on the bound of the holomorphic sectional curvature and obtained in addition the negativity of the Riemannian sectional curvature.

Wolpert's method used some SL(2,l) invariant first-order differential

operators obtained by Maass [7].

Royden's computation is based on the fact that the

Fbincar~ metric on a compact Riemann surface of genus at least two is Einstein.

Since Yau's proof [13) of the existence of a Kahler-Einstein metric on

a Kahler

manifold of negative first Chern class, the question naturally arose about the negativity of the curvature of the Weil- Petersson metric induced by the Kahler-Einstein metric on the moduli space of such a

manifold.

length of a tangent vector in the moduli space

In this case the square of the

with the Weii-Petersson metric is

simply the L 2 norm, computed from the Kahler- Einstein

metric, of the harmonic

Research partially supported by a grant from the National Science Foundation

262 (0, 1 }-form with coefficients in the tangent bundle representing it.

Koiso [6J showed that

this Weii-Petersson metric is always Kahler, In this paper we introduce a method of computing the curvature of the Weii-Petersson metric in the hi~Jher-dimensional case,

We compute the curvature of the Weii-Petersson

metric for any local complex submanifold in the moduli space even when the moduli space itself is singular. The method involves canonically lifting a vector field on the moduli space to a vector field on the total space of the family of deformations. canonical

This method of

lifting greatly simplifies the computation of the full curvature tensor and makes

it possible to express the full curvature tensor in simple explicit terms, dimensional case the final explicit form

In the one-

of the full curv11ture tensor agrees with that

obtained by Wolpert [12] which so far is the most" manageable form of the full curvature tensor, To a certain extent this seems to indicate that our final explicit form of the full curvature tensor in the general higher-dimensional case may be the most manageable form. From this explicit expression of the full curvature tensor we conclude that, for a compact Kahler-Eirostein manifold the

E;,n

Mu

with negative first Chern class and with tangent bundle TMo,

holomorphic bisectional curvature of the Weii-Petersson metric in the direction of £

H 1 (Mo,TM 0 ) is negative if the elment in H 2 (M 0 ,A 2 TM 0 } defined by E;" n in the

natural way vanishes, where

A2T

Mo

is the exterior product of two copies of TM 0

•

In

particular, if H'(M o.A 2 TM 0 } vanishes, then the holomorphic bisectional curvature of the Weii-Petersson metric for any local complex submanifold of the moduli space of M o is negative.

It is not clear whether the sufficient condition of the vanishing of H2 (M 0 ,A 2 M 0 }

(or the vanishing of the exterior product of two copies of substantially weakened,

H1 (Mo,TM 0 ) ) can be

because the formula for the curv11ture explicitly contains an

obstruction term from the exterior product of two copies of

H1 (M 0 ,TM 0 ) .

Though we

have no counter-example, from the explicit expression for the curvature it seems rather impossible to absorb the obstruction term into the other terms,

The vanishing of

263

~{Mo,A 1 TM 1 ) is

very restrictive.

For example no two-dimensional compact Kahler surface

with negative first Chern class can satisfy it, because for such a surface the nonvanishing of the second plurigenus follows from the theorem of Riemann-Roch and the Kodaira vanishing theorem. The canonical lifting of a vector field on the moduli space to a vector field on the total space of the family of deformations is defined by the property that the tangentbundle-valued {0, 1 )-form computed at each fiber from the lifting is harmonic with respect to the Kahler-Einstein metric on that fiber.

The use of the canonical lifting makes it

unnecessary to use the computationally complicated process of w:>ing to the harmonic projection. Moreover, by using the canonical lifting, one gets the vanishing of all variations of the volume form and a very simple expression for the variations of the Kahler-Einstein metric. Having a canonical lifting is the same as having a canonical smooth trivialization of the total space of the family of deformations with one real parameter, or equivalently a canonical way of regarding the family of deformations as obtained by deforming the complex structure on a fixed underlying smooth manifold. Our concept of canonical lifting is motivated by Hitchin's method of deforming the complex structure on a K3 surface with a Kahler-Einstein metric [3].

The smooth trivialization given by his deformation of the

complex structure corresponds precisely to the canonical lifting though in his case the Ricci curvature tensor is zero instead of negative.

In the case of

K3 surfaces there is a

possibility of explicitly constructing a Kahler-Einstein metric and obtaining an alternative direct proof of the Kahler property of every K3 surface by trying to find a canonical smooth trivialization in a given holomorphic deformation. We will, however, not pursue this possibility in this paper. Royden [9] told me that he could prove the negativity of the holomorphic sectional curvature and the nonpositivity of the bisectional curvature under the assumption of the

264 vanishing of the Lie bracket on H1 (Mo,TM 0 ) , by using his proof for the case of a compact hyperbolic Riemann surface and adding the following two new ingredients.

The first

ingredient is that on a compact Kahler-Einstein manifold of negative Ricci curvature the tensor of covariant rank two obtained by lowering the contravariant index of a harmonic (0, 1 )-form is always symmetric.

(This ingredient was also used by Koiso in his proof that

the Weil-Petersson metric is Kahler.)

The second ingredient

is to use, in the place of the

Bers coordinates in the one-dimensional case, the holomorphic local coordinates of the moduli space constructed by Kodaira-Nirenberg-Spencer [5] in their proof of the existence of deformations of complex structures. on H1 (Mo,TM 0 )

The assumption of the vanishing of the Lie bracket

is needed for the second ingredient.

Recently Schumacher [10]

considered the Weil-Petersson metric for polarized

compact Kahler manifolds with zero first Chern class and showed explicitly that in the case of polarized tori and polarized symplectic manifolds respectively it agrees with the Maass metric on the Siegel upper half space and the Bergman metric on a symmetric space of the third type and is therefore Kahler-Einstein with negative curvature. I would like to thank Koji Cho, Alan Fekete, and Antonella Nannicini for having very carefully checked my lengthy curvature computations and spotted some misplaced signs, missing complex conjugations, and missing curvature terms in an earlier version.

Table of Contents §1. Canonical Lifting of Vector Fields §2. Lie Derivatives §3. Kahler Condition §4. First Part of the Curvature Computation §5. Second Part of the Curvature Computation

265 §1. Canonical Lifting of Vector Fields.

Let I be the unit interval (-1,1) in lit and

(1.1)

(0., w~, I

I

t),

11- 1 (t).

For t e: I let Mt =

complex manifolds.

We cover M by coordinate charts

Let n be the complex dimension of M 0 • a

wi = fij Let U. = M 0 I

n 0.I

I

I

~I M 0 x {0} is the identity on M 0 ,

We have coordinate transformations

1 n (wj, •••,wj' t).

a

w~ I U..

and z~

M+ I be a smooth family of compact

11:

Take a diffeomorphism ~:

I

M 0 x I + M so that

In local coordinates ~is given by a 1

a

n

wi = Fi (zi , ... zi, t), where F~ is C"' in z. and t. I

field

~

I

The diffeomorphism ~ corresponds to a lifting of the vector

on I to the vector field

a v =- + at

a~

L

a

I

-+

at

awa

a on

M.

The

T Mo -valued

(0,1)-form

a on ui with q,i =a(

(1,0)-component of v ~ M 0 •

I

a~

L

I

at

a

representing

at).

a~

aw~ I

the

infinitesimal

deformation

is

In other words, q, ~ual to the a of the

Note that, while vis globally defined, the (1,0)-component of

v is not globally defined because its projection onto a vector tangential to M 0 depends on the local coordinate chart. another lifting

v,

So

is not a-exact in general.

However, if we have chosen

then the (1,0)-component of v-v' is globally defined on M 0 and

the ~obtained by using

vdiffers from

The infinitesimal deformation

by the a of the global vector field v-v'.

can also be described by the Cech cohomology class

266

is the limit as t in

0.

I

+

0 of ..!t times the discrepancy of going from M0 to M along w~ = constant t I

and going along w~ = constant in I

0.. I

In this paper we will not use the description of

the infinitesimal deformation by Cech cohomology.

Now assume that M 0 carries a Hermitian metric.

(1.3)

+ there is a unique harmonic representative 1(1. The difference +-1(1 is given

represented by by the

a of a global (1,0) vector field u on M

vector field

von M

0

0

a

a

1

dz~I

u.

(J

von M

O•

Write u =E u~ I

the canonical lifting of

it•

a
on

u..

Define a new

I

It is characterized by the fact that the a~

its (1,0)-component (with respect to the charts w~, t,

-

_a_

by

v=v- '\. L

We call

In the cohomology class

-

I

0.)I is the harmonic representative of -

the infinitesimal deformation. When the first Chern class of M0 is negative, the canonical lifting is unique, because on M0 there exists no nonzero holomorphic vector field. One way to see the nonexistence of nonzero holomorphic vector field ta is to use Yau's result on the existence of a KahlerEinstein metric gaB on M0 , because

and gait a t 6 is a strictly subharmonic function on the compact manifold M0 •

Here, as in

the rest of the paper the summation convention is used and the following commonly used notations in Kahler geometry especially those in [4) are used without special explanations.

267 a =_a_ II azll'

_a_ a;"'

a-= \1

Rai= allaiof the determiniant of (glli) V =covariant differentiation in the direction of _a_ II azll V-= covariant differentiation in the direction of a II azll (g 11 \l) = the inverse matrix of (glli) · h t he su bs cnpt · The Coordl· nates z 1 , ••• ,z n mean some coord"mate c hart z 1 i, ...,zin w1t suppressed.

§2. Lie Derivatives

(2.1)

We continue with the notations of §1 and assume that each Mt has negative first

Chern class.

We endow each Mt with a

Kahler-Einstein metric glla
Raa
Let v be the unique

~ with respect to the

Kiihler-Einstein metric of M • t

b

Let Ta(t) be a tensor of rank two on Mt' where a,b and other lower case Latin 1,2, ••• ,n,1,2, ••• ;il (in contrast to lower

letters in this paper denote indices with the range case Greek letters of

T~

t~,B,

etc. with the range 1,2, ••• ,n). We now compute the Lie derivative

with respect to the vector field v and evaluate it at t = 0.

trivialization

~: M0 x I+

M given by

v

a =at

+

w~I = f.'(z~ , •••,l,t) so that I I I

L (I

That is, the smooth trivialization

~

Choose a smooth

(I

aFi

_a_+

at aw~ I

'L at ar:- a

(1.

a

aw.

I

corresponds to the vector field v. We use here and in

the rest of the paper the following notation. The dot • in the superscript position denotes

268 the differentiation of a scalar function or a component of a tensor with respect to t when z 1 , •••,zn are kept constant. The operator L denotes the Lie derivative with respect to the v

vector field v. For example, when t is set to be 0,

b Here L T means the derivative with respect to t of the pullback of Ta(t) by the v

diffeomorphism tt: M0

+

Mt induced by t.

d. The term (Tc) means the derivative with

n __n da respect to t with fixed z 1, ... ,z of the component T- of T for dw ~--. c awe

The factor

a

a

azc

- - - azc

(~)·means the derivative with respect tot with fixed z 1, •••,zn of the function~. Let J(t) denote the almost complex structure of Mt. B a

J_ = -.r='icS

B B B (where cS is the Kronecker delta), la a a

= 0,

B and J = 0, we obtain at t a

=0

fron

the above formula for L T v

B

(L ]) = 0 v a

B

(L ])_ = 21"=i'Lv

v a

B

(L J)_ v a

B

a

B = 2.r=1~a

=0

B B Here ~- = a_v denotes the harmonic T1 •0 -valued (0,1)-form representing the Mo a a

infinitesimal deformation at t = o.

(2.2)

We apply the same method to compute the Lie derivative of the Kahler-

269 Einstein metrics gab and get at t = 0

y

y

(L~>all = grll•a + gar•ll

(l g) -= (g -)• + g -a vY + g -a-:-~~Y v all all yll a ay ll The other components of Lvg can be obtained by taking conjugates, because both g

and v are real.

Here we have used

is unsatisfactory.

+aY to denote

,.r..a

The expression for (l g) a v a.,

We are going to use the Einstein condition Raa= e gaB to prove

that (lvg) aB = 0. We have at t = 0

= (-(a

a

i·)

a

A

- (a

a

i·) a-) A

a- log det (g -) ll yo

+a (-(a-v\1) a - a v\1 a-) log det (g -6) + a a-:-(log det (g 7 ))• (I ll )I ll )I Y (I ll Yu At the point under consideration we use normal coordinates with dgaB = 0 and a-a-;:g 0 = O. Then y o a.,

(R -)·=-(a vA)R - - aav\1 R -+a a-;;-
270

A

where 41-- = g -41- and BA ~A B

+ay

= cjl-ay-· Using LvR = e L 2, we get

v-

At this point we need the fact that

+(ia is symmetric

in a and B. One way

to see it is to apply the Bochner-Kodaira formula to the harmonic TM 0 -valued

a

(0,1 )-form 41- and get

B

o=

-

a

-= - 111

A~

B

(see e.g. [11, (1.3.4)]).

Here

0

a

v-v •-~AB =

at R

a

a a a -8 • - + R •t a B

aa- +IT and RaBy6= aaa"Biy6 in normal

coordinates. lowering the index a and using the Einstein condition, we get A~

O

t

= -g v-;v A41 a'if- Ra(i" B41at + e 41a'if"

Since R - t -8 is symmetric in a and B, it follows that aa

o = -l~v-v (41--- c~~--) + e(c~~--- c~~--). ~ A aB Ba aB Ba . Multiplying both sides by g
>o that

•ae- •sa must vanish identically

on M 0 • From the symmetry of

-

follows that g Y6 vY

-

•ae in a and B and from the harmonicity of · -aBi t

•acs and g Y 6 v~ ay both vanish identically on M 6 aa a-;;-< -;-) 8 g Y (L v g) yu

=e

0•

Hence

(L g) B v

a

and contracting it with gail we conclude that g y"6(L g) -;-must vanish identically V

yu

otherwise it is a nonzero subharmonic function on M 0 • The vanishing of g Y6 (Lvg\6 in turn implies that (Lvg)ai vanishes identically. Our final result is that

271 (Lvg)aB= 2 +aB (Lvg)ai= 0 • From g bgbc = 6c (where 6c is the Kronecker delta) and these two equations a a a direct computation yields (Lvg)

ai

= 0

(L g)aB=- 2 +aB v (where +aB is obtained by raising the indices in (2.3)

•as>·

We now compute the Lie derivative of the volume form of the Kahler-Einstein

metric. Let 1 a b a B w=Adw =g-dw Adw 2 wbdw a aB with wab skew-symmetric in a and b. Then wai= gai waB = -gBa

w =0 aB

was= o. Using the suymmetry of +a B in a and B and using (Ly8) aB = 0, we conclude that Lvw vanishes identically on M0 • The volume form is, up to a constant, the exterior product of n copies of w. Hence the Lie derivative with respect to v of the volume form of the Kahler-Einstein metric is zero.

§3. Kahler Condition.

(3.1) Though the Kahler property of the Weii-Petersson metric was proved already by Kosio [6), to compute the curvature of the Weii-Petersson metric we have to

first differentiate the Weii-Petersson metric once as an intermediate step. differentiation immediately yields the Kahler property.

This

We are going to do this

272 intermediate step of first-order differentiation of the Weii-Petersson metric here in this section. Now assume that ll is an open neighborhood of 0 in coordinates t = (t 1 , ... ,tN).

~ with holomorphic

We now modify the situation in §1 as follows.

replace the unit open interval I by ll so that

11:

We

M + ll is a holomorphic family of

compact Kahler-Einstein manifolds with negative Ricci curvature.

We keep, with

obvious modifications, the notations of §1. Let t' k and t' k be respectively the real part and imaginary part of tk. liftings of

-fir,k ata' k•

Let

Let v'k and v' k denote respectively the canonical

'(F),'( a:• )denote respectively the tangent-bundlek k

valued harmonic (0,1 )-forms representing the infinitesimal deformation in the direction of

a a at'• at"'' k

Define

k

Note that in general the Lie bracket [vk',vk'] is not zero so that corresponding to vk', vk' there is no smooth trivialization of the family M over a two-realdimensional plane inn with t'k, t' k as variables. Let

Since

it follows that

273

(L

a

a

a

J) =(L J) =(L Jl-=0. vk a vk a vk a

The tensor L- J is the complex conjugate of L J. vk vk complex conjugate of

The tensor

a

~ (at.)

is the

k

~(a: ). k

(3.2)

Let dV denote the volume form of the Kahler-Einstein metric gai"

We

N

define the Weii-Petersson metric }_ hil dti & d~ on n by i,j=1

In order for the metric to be positive definite we assume from now on the element of H 1 (Mt,T M/ defined by any nonzero vector of ll of type (1,0) at t is nonzero. Because of (2.3) we have

+ J

(L-L J)P(L J)Qbg::gabdV vk v. a v. pq I M0 I

(L J)P (L v. a v. M0 I I

J)~ (L-vkg) pq-gab dV

274 The last two terms vanish, because (lvg) 11i, (lvg) 118 , (lvJ):,

(lvJ)~

vanish for all

canonical liftings v of real vector fields on n and because of type considerations. We have also used the fact that the Lie derivative of the volume form is zero. Since d11(vk,vi] = at every point.

la~·a~il

=

o,

it follows that (vk,vi] is tangential to M0

Hence the (1,0)-component of (vk,vi] is globally well-defined on

M 0 and direct computation of L 1-

vk,vi

l = L- L

vk vi

- L L- yields

vi vk

B r-: B (l[]J)_ = 2,.. -1a-[vk,v.] vk,vi 11 11 1

(cf. (2,1J).

B

B Hence (l[]J)_ is a-exact. vk,vi II

Since (L J )_ is harmonic, it follows

vi

II

that

J)q a"bdv- 0 a v. b gpqg - • JM (L (vk' v.J J)P(l J 0

I

Hence

J)P(l J)qbg-gabdV=J (l L-J)P(l J)qbg-gabdV JM (l-L vk v. a v. pq v. vk a v. pq J M J I

0

0

I

We claim that this last integral vanishes. Since the only nonzero components of B L J are of the form (l ])....., because of type considerations it suffices to show that v. v. Cl l B (l L- J)_ = O. Now

v. vk

l

II

1

B B. B a b B L- Jl- = ((L- J)_) + (L- J) a-v +(L- J)_abv , vi vk 11 vk 11 vk a 11 vk 11 _ B but the nonzero components of L- J must be of the type (L- J) , vk vk 11 B (l L- J)-= 0 and vi vk II (L

J

Thus

275 Since M0 is no different from other M , the formula continues to hold when 0 is t replaced by t and M0 is replaced by Mt. The Kahler property of the Weii-Petersson matric follows from the vanishing of

which is a consequence of the a-exactness of (L[ ~

(L

~

])_ and the harmonicity of

vk,v. 1 a I

Jl-

vi a

§4. First Part of the Curvature Computation.

(4.1) In this section we compute the curvature tensor of the Weii-Petersson metric and reduce it to such a form that in the one-dimensional case it agrees with the expression obtained in Wolpert [12).

Further and more complicated reduction will

be done in the next section on the second part of the curvature computation. Because of (2.3) the curvature tensor R<~-fliTof the Weii-Petersson metric at 0 is given by

=I+ II+ Ill+ IV,

where

276 II=J

(l J)p(l-L

v. a

M0

III=J M0

L J)qbgabg-dV pq J

v. vk v.

I

..

(l J)p(l L J)qb(l g)abg-dV v. a vk v. v. pq J

I

..

Our goal is to show that the holomorphic bisectional curvature of the Weil-Petersson metric is negative in certain directions.

For this purpose the term I is good. The

term II is the most difficult one to handle. We first look at term Ill.

(4,2) From our computations of L g in (2,2) and of L J in (3.1 ), we have

v

=21-=-1 ~(...!...>::ati

(L

a11

v.

I

L J)ll

vk vi fl

(-2~(~aat laa). R.

By applying L L to both sides of !bleb= -6c and using the fact that the only vk vi a a fl

fl

r-:-

fl

nonzero components of L 1 are of the type (l 1)_and the fact that 1 =., -1 6 vi vi a a a fl

fl

y

and 1- = -1'=1 6 , we conclude that (l L I) = 0. Hence the term .Ill vanishes, a a vk vi a In exactly the same way we show that the term IV vanishes.

(4,3) We now deal with the term II. Since L- L

L

= L -

vR. vk vi

L

+ L

[vR.,vk] vi

=L L + L L [vR.,vk] vi vk [vR.,vi] =L + L L [ [v R.'vk],vi]

L- L

vk vR. vi + L

vi [v R.'vk]

L L-

vk vi vR.

+ L

L-

vk [v R.'vi]

+ L

L L-

vk vi v .t'

277

it follows that II =A+ B + C + D, where

_J

A-

M0

J

B=

I

)

"

q ab

p

(L J) (L L[]J)bg v. a v. v ,,vk

MD

D=

q ab g -dV (L J) p (L[ 1] ]J)bg v. a v ,,vk ,v. pq

J M0

J

I

..

g -dV pq

(L J)p(L L L- J)qb gab g -dV. v. a vk v. v. pq J

I

..

We immediately have the vanishing of A, because [(vJ.,vk], vi] is tangential to Mo II >. (l[ 1l no and (L J)_ are respectively a-exact and harmonic vJ.,vk ,v. 1 P v. a 10 J I -valued (0,1 )-forms on M 0 • The second easiest term to handle is D. In (3.2)

and

\i

a

o

we computed (L L- J)_ to be zero. Hence vi vk a (L

L L-J)~=(l L-J)~"_a_)~-(l L-J)~cH-a-)~ vk vi v J. a vi v J. " atk a vi v J. a at k II

Applying L L- to both sides of Jblbc = -6c and using the computations for L- J vi v J. a a v J. and L J from (3.1 ), we obtain vi

y

a

a

a y

(L L- J) =2M+(-) +(-)_ v. v. a ar a at. a J .. J. J

r

a a

a y

(L L- J)-=-2r-1+(-)-+(-) vi v J. a ati a at;, a Thus

and

278

(4.4) The terms B and C are similar and can be obtained by switching i and k.

So

we will handle only the term B. Since

JM (Lv iJ)ap t

q

(L(v v ]J)b g L' k

abgp-q dV =

J M t

).

A

IJ

-;a gA"ildV

(LV J)_(L(v v ]])_ g i a L' k B -

)J

vanishes due to the harmonicity of (L ])_and the a-exactness of (L 1J)6, it vi a vL,vk 1 foiiONs by applying

a~

to it that

)

We write B = B 1 + B 2 + 8 3 , where

B3

=-)r M0

A ii aB (L-L J)_ (L[Ill- g ~IJ dV. v.,vk B " v. v. a

J

1

"

Since from the computations of (L L- J) Y and (L L- J)~ in (4.3) vi vL a vi v L a

279

it follows that 8 1 =

o.

likewise 8 1 =

o.

We now handle the term 8 2 • This is the

most difficult step in the first part of the computation of the curvature tensor of the Weii-Petersson metric.

It will be achieved by computing a" of the T~·~-valued

A.

(0,1)-form (L-L J)~ vi vi a

(4.5)

II II For notational sill'l>licity we let T- = (L J)- and v = v.. a vi a 1

trivialization wa

=wa(z,t)

for the family

Choose a smooth

a

M + g such that ar.

11:

= -vi

at M 0 and

J

a

We use the dot • in the superscript position to denote at: of a j

scalar function or a component of a tensor.

As before, a a' aa mean respectively

a

-a· --==a· az az Since on each Mt the g

a

ai

Y T_) Y II

--(~

awa

Applying

T~:0 -valued

(0,1)-form

T~

is harmonic, we have

=0 on Mt for every t, that IS.

a~ and setting t = 0, we get on Mo i

Using ). ti(a a)• 0 = (L ~ )ai= (8 ail. -8 a"0(-a 0 w 11 - g tw

280

y

Ye

y--.

t

ye

(LT)_=(T_) +T_(a wal -T_(a w ), v a a a a a t we get

+g

aa a

a

((-~

t

ut

--a • Y

•

(a w l - g- (a w l )T-l y

ay

a

0

Multiplying out and using normal coordinates at the point under consideration and

a/· •

the fact that (--) t=O awa by

applying

= -(a aw ). )•t=O a

respectively

al •

and ( 0 ) t=O aw

=-cac;w ). )•t=O (obtamed 0

and

to

at.

I

o=

). - az). a a wa + - - a--w ) we obtam 0

az

awa

g

aa

- g

ai

ai

,

y

a T_ + g

uY a

aca

t

a

uY "

a

w l

ut a

-- • Y

g- a
ai

a •

t

~

y a T_a

uy a

g

ai

). •

(a w l a

~

y

a, T_

a

uy "

A • ~ a7"T-Y ai t • Y ai t • Y g ~ a ca w l T_- g lh (a w l a T-

(a w l a

t

a •~

-g

- g

awa

t

ai

y

a

-- •

ut

Y

g-
y

ai

~a

uY a

a

a

Y

(l T)v

a

Y -- • ai Y -- • ail Y y • - g ~ L.a (a wal + g &7 (a T_)(a w ) a a a uy a a a uy a a t

&7 (a T_)(a wal

uy

+ g

ai

y • a a (a t w l

t

&7 T_a

uy

=

o.

The following pairs of terms on the left-hand side cancel out after a change in the dummy indices:

the first and the tenth terms, the second and the third terms, the

281

fifth and the thirteenth terms. .

(0,1)-form on M00

Moreover, the sixth, the eighth, and the twelfth

. . Y 10 a T_ = 0 due to the harmon1C1ty of T_ as a TM ' -valued a 13 13 o The eleventh term vanishes because

terms vamsh because g

al3

Y

-. a a due to the a -closedness of +(at,l(i"" Thus J g

ail

~ya

u

a

Y al3 ~ • Y ai --a • Y (LT)_=g (a w )~LT_+g g-a (aw )T~ v 13 a uY A 13 cry a 6 13

A For notational sifTl>licity we let S_ = a

A

a A +(at)_ which i a

A is equal to a (w )• at t = 0. a

A

Let S = S_. Since a a

--. --a. a a·(a a) =
13 13 Since T_ is harmonic and in particular a-closed, it follows that a-T_ is a Y a symmetric in a and y. From (2.2) we know that T- 0 =gil T~ is symmetric in a (4.6)

a..,

and 13.

..,y a

Hence in normal coordinates a;Tail is symmetric in a, 13, y.

13 (L-T)--= g- (L-T)~ Then in normal coordinates vi ay yl3 vi a g aia (L-T)--=g aiS~a-T--+g al3 (a-s A)T-av.l3y ayllA yalll. J

Let

282

"fa

= a-{S T_), Y a A I! I! I! Since (L L-J)-=0 from (3.2), it follows that (L-T)-=(L[- )J)- is a-exact. v. v. a v. a v.,v. a I J f! f! So there exists H such that (l-T)_ =a_ Hs.

c Let Uab = gacT b'

J

J

I

vi a a Since the Lie derivative of a symmetric tensor is again y

symmetric, (L-yY>ab is symmetric in a and b.

Since (L-g)::- = 0 and T_ = 0, it ~ ay I! follows that (Lv.niii = (L-v.U>aa and (Lv_T)0 ,r is symmetric in a and f!, Let J

a

J

J

J

Ha =gaSH • Then V~Ha= (Lv.T>-a!f is symmetric in a and f!, Hence J

g all a (l-T)-- = g all V V-Ha v. flY a I! Y J ai =g V V-H- a Y I! =ga 11 v-v H-- gai!R -HY a I! ay I! =V-:-(V Ha)- e Hy a Y and

Aa a a-{S T_- V H ) = -e H-. y a A a y It follows that H- is a-exact and we can write H- = a--f, where f is a complexy

-

valued smooth function on M0 •

y

y

Let D= -gai!V V- be the operator on smooth a I!

functions, Then

Aa

A a

-

a-{S T_ + (0 + e)f) = a A

y

o

-

and S T_ + (0 +e)f is holomorphic on M 0 and therefore must be equal to a constant a A ConM 0 • Letf=f -~.Then e

on M 0 and f

a = - (0 + e) -1 (SAa T_), A

It follows that A

V,H =-Of= 0(0 +e) "

We are now ready to compute B2 •

-1

A

a

(S T_), a A

283

Bz =-

=

=-

JM

JM

H 0

>.

-

(L-L ll- (L- L j)!:8 gaS g,-dV v.t vk "\1 v. v. a o J I

>.

( V (L -

a

);a

JMo (S a L)) >.

L

v J. vk

-

11 8

J)- g

t~B g-

a s

>.11

) dV

a a

(21-1t(at) t(a-tl-) dV J. a k 8

284 (4.7)

Our final conclusion in the first part of the computation of the curvature

tensor of the Weii-Petersson metric is

R(WP)-.-:-= Jf R.k 1 J

-4

JM

(l M0

l J)p(l l J)q gabg dV v. v. a v v. b pq .. I k J

a6aYaaaa tyt<-at )..A(a"f >at
a

i

j

0

k

R. 13

Except in the one-dimensional case, this formula does not give negative holomorphic sectional curvature right away. This can be seen as follows. Let i = j = k = t. At any point under consideration delta) and

a

t(at~-

i

p

we can assume that gaa=aaa (the Kronecker

a

=a>. ;(at)--. The integrands in the last three integrals can p

i

pp

be expressed at that point as follows.

285

The sum of the last three integrals becomes

a

which is negative unless {•(-T.::-} _

ati aa a-1, ••• ,n

has only one nonzero component at

every point. One way to get negative holomorphic sectional curvature is to use the integral

to help cancel the undesirable contribution from the last three integrals.

In the

case of Reimann surfaces and the Teichmuller space this integral vanishes. One can see it as follows.

We choose normal coordinates t1, ••• ,tN at the point t = 0 under

consideration so that a.h.-k = 0 at that point. (l

Vt

J)~ •••,(L a

I

VN

J

In the case of the Teichmuller space

J)~ form a basis over ([ in the space of all harmonic TM 1 •0 -valued

(0,1)-forms on M0 •

a

o When M0 is a Riemann surface every TM: •0 -valued (0,1)-form is B

a-closed.

a- -closed.

We will verify below that (L L J)_ is always vi vi a B (l L J)_ is harmonic, but vi vi a [

0=(3-:-hk.-:-)(0)=; ij

A

II

(L J)_(L L J)_g v. a vk ~ B I M0 J

Thus, being perpendicular to a basis, (l

B

aB

Hence

g:l.-dV 1.1

L J)_ is zero. So, for the case of the vk vi a Teichmuller space the curvature tensor of the Weii-Petersson metric is simply

286 (WP)

R

r

11 1· = 4e

J M0

((0 +e)

-1

a i a Y a i a a <+y+<-at )..,..))(+a+-)dV 15 j i k I. a

This expression agrees with that obtained by Wolpert (12). Now we return to the general case.

We are going to verify that (L L J >! v. v. I

I

is always 3"-closed.

In the computation in (4.5) if we use v = v. instead ofv., we I

I

get in normal coordinates aa Y aa --,: • Y ai --a • Y g. g-;- a (L T)_= g (a w > 87 a,r_+ g It:" a T_ uy a v a a uy 1\ a ay a 6 a

= g

ai -a--A Y aS --a-a Y +<ar.>- ~ ai"L + g It:" ->L j a y a ay j a a

a b which vanishes because the only nonzero components of + (at:) a are of the form I

a i

-

+ ( ar-.) a. Here when we use v = v. instead of v. the eleventh term g I

Lj

I

ai - - . a (a wa) a

a

on the left-hand side of the equation in (4.5) vanishes not because of the

-..

a a a -closedness of • (at>- but because j a

a a a and+<ar.>cJ=O. Thus,a"(L T)=O. Thatis,(L L J)_isa"-closed. I vi vi vi a §5. Second Part of the Curvature Computation.

(5.1)

We continue with the notations of §4 and set out to transform the first

287 integral in the expression for R(WP)t.kiTin (4.7). We will do it first for the special case i

= j = k = I.

and then get the general case by polarization. As a first step in

the transformation we compute

a of Lv. Lv.J. y I

Since T

position denote

is a-closed,

I

a-T_ is a II

syiTVTletric

in

a

and

11.

That is,

a~ of a scalar function or a component of a tensor. Then i

is symmetric in a and 11. Using y (L T)_ vi II we conclude that at t

y

y-1: y wa )• - TJ (w )•, a II II -.:

= (T_)• + T_(a II

=0

Ae Y --;~,• Y Y y--. -.: Y• -(a-w) a,T--(a w) a,T_+a-((L T>--T- +TJ (w)) a " II a " II a vi II a II II -.: is symmetric in a and 11. Hence

1-=1" ). y -- • y y y -- • - 2-TJ,T_-(a wA)a,T_+a-:-(L T)_-(a-T-Ha wa> a " II a " II a v. II a a II y --- • 1: • 1: 1-=1" y -L.(a a wa> +a-L..(a wy) +TJ (---T....:.) a all all 1: 111: 2 a

is symmetric in a and 11.

Since the sum of the second and the fourth term is

symmetric in a and II and also both the fifth term and the sixth term are symmetric in a and II, by taking the skew-symmetrization we obtain 1: y 1: y y y 3-(L T)-- ao(L T)- = r-f(TJ L- TJ L). a vi II " vi a II 1: a a 1: II (Note that one can also obtain this result by differentiating with respect to aat. the I

288

integrability condition at(t} = i[t(t},t(t}] and using t• = - 1-T, where t(t} is the

2r-'i

T~; 0 -valued (0,1}-form on M0 such that a local function f is holomorphic on Mt if as a function on M0 through the smooth trivialization f satisfies

(a- t(t}"a,_}f = 0

t = 0 due to the on M0 .} We now use normal coordinates. Since a T_ a- -closedness t

a

of T, it follows that y y a-(L T}-- a-:-(L T}- = a vi a a vi a

M

ty ty a (LT-- LL}. t a a a a

Let a----= T--T--- T--T-aa,ty at ay at ay and let aaa__ be obtained from a.,--- by raising the first two indices. Then the ty pa;y last equation can be rewritten as

when (i}

01

>. means applying a to the first index of (L T}-;;- (which is g, :-:{l T}-}, i.e. v.

regarding ( l

vi

T}2:_ as a TM 1 ,O -valued (0,1}-form, a

o

PY

~pplying a and

"Y

v.

a

1

lowering the index

y, (ii} 0 2 • means applying

a'

to aaa'Ty as a A2 TM 01 •0 -valued (0,2}-form and then

lowering the indices a and a, or in other words

a* is applied to the second set of

indices of 6----. aa,ty

(5.2}

Let X denote the space of all tensors :!-;;--if satisfying the following three aP,Y

symmetry relations: (i} :!----=-:!----(skew-symmetry in the first two indices} aa,yll aa,yll (ii} :!0 -:.= :!-F0 (symmetry in the two sets of double indices} a.,,yu yu,ap (iii}

:!---- + :!---- + :!---- = 0 (vanishing of the sum from the cyclic aa,yll ay,lla all,ay

permutation of the last three indices}.

289 Simple direct verification shows that 9- 0 -"i" belongs to X. a.,,yu For s = 1,2, the operator D applied to a covariant tensor with two sets of s . skew-symmetric indices of antiholomorphic type (e.g, an element of X) means the operator

a applied

to the sth set of skew-symmetric indices,

Let

o• s denote the

adjoint operator of Ds which is the same as applying a" to the sth pair of indices. Let 0 = s

o·sDs + Dso·s and let Hs denote the projection operator onto the kernel

of 0 • Let G denote the Green's operator which is zero on Ker 0 aad equals the

s

s

s

inverse of (the identity operator minus H ) on the orthogonal complement of Ker 0 •

s

s

For !! in X we have the following properties

(d)

(e) [] 1 !! belongs to X

(gl if

o,:: = o, then

ID 2 •,0 1 1:: = e

o• .::, where !'D. •,0

1

1 means the commutator

Properties (a) and (c) follow from simple straightforward computations. To prove property (b), by definition we have

A

where

a

\1

means that the index

a

\1

is omitted and

290

(j)

2 = ~ (-1) v+1 gcn R - "--- ~ aa v -ta,a ••••av···az v=O 2

@

= ~ (-1 )v+1 R- t_P !!--- ~ _ L a a tP,a ••• .a ••• az v=O v v

2 G)= }_ (-1 )v+1

l:

2

v=O

11=0

R- ' - P " - - -

f3

V

a

-

D

-ta,a •••• (p) ••• a ••• az II V

II

II¢V

the subscript

(p)

II

The term

meaning that the subscript i

II

is replaced by

p.

(j)

vanishes because R - = e g and because of the aav aav symmetry property (iii) of E. The term '2' vanishes because R- t_P is \!:1 a a -

-

symmetric in t and p whereas !!--.,

/" 0-

a

'tP,Po•••P\)••• 2

is skew-symmetric in

v

Y" and p.

The term G) is the sum of six terms which can be grouped in three pairs so that the two terms in each pair cancel out because of the symmetry of R-8- 6 in a and a y y and of the skew-symmetry of E-;;--i" in y and 6. Property (d) is obtained from a.,,yu property (b) because of the symmetry property (ii) of !!. To prove properties (e), (f) and (g) we compute explicitly 0 1!! and obtain (01!!)----=- ga'v V-!!----- gat[v-v ]!!---aa,AII a t aa,A11 a' a ta,AII

=- ga'v V-!!----- 2e E---o ' afl,AII afl,AII

+ R-t_P !!---- + R-t_p !!---a II ta,pA a A ta,PII where R "Q = e g "Q is used. The skew-symmetry of (0 1!! }-;;-,-in a and a and the a., a., a .. ,"ll symmetry of (0 1 !! }- 0 , - in (a,a) and (A,II) are clear from this expression. To get a.,,,.ll

291 the symmetry property (iii) for 0 1 E, we cyclically permute I!,A,Il and take the sum from the last four terms of the above expression and get - R-'t_P "----- R-'t_P "----+ R-'t_P "---+ R-'t_P "---a A -'tll,Pil I! II -'ta,pA a II -'t8,PA I! A -'ta,Pil

where the two terms in each of the following six pairs cancel out: the first and the eleventh terms, the second and the twelfth terms, the third and the fifth terms, the fourth and the sixth terms, the seventh and the ninth terms, the eighth and the tenth terms. Property (f) follows from property (e) because applying 0 2 to E is equivalent to switching the two pairs of indices, applying 0 1 , and then switcllng the two pairs of indices again. perty (d), we have

D2.0 1 E = oto 1•D 1 E and 0 1D 2•E = D 1D1•o2•E.

Direct

computations from definitions yield - ·D -(D D . E)---= g pA g a't ( V V-V 2 1 1

p a a

all,ll

E----- V V-V E---;:j 't8,A11 p 8 a 'ta,Ail

-- . E)---=g pA g a't (-V-V V E----+ V-V V E---j. (D D ·D 2 1 1 a 8, 11 a p a 't 8, All 8 P a 't a, All

Hence - • 0 ]::)---= ([D g pA [V-V ](g a't V ::---::i 2 ' 1 a8,1.1 I!' p a 'ta,A11

- (the expression obtained by switching a and I!) A p = [R-I! a

(g

a'[

V

-

a

::----) 'tP,Ail

p

A p a't A a-:t + R-, (g V E---::i + R-- (g V E----)] I! " a 'ta,Pil 8 II a 'ta,Ap

- [the expression obtained by switching a and 8]

292

- (the expression obtained by switching a and a), is

symmetric

in

the symmetry of R-A_p

a

II

a

and

a

and

in A,p and the

skew-symmetry of ::---,in A,p, From symmetry property (iii) of:: we obtain ta,p,.

-.

([0 2

Let

X0

'D/01- 0 1Dz•

be

_

,Od ::)as,-;=- e g the set of

= e oz• on

.

x,

0"[

all

-.

V o"as,'Til= e(Dz Elas,-;·

::

in

X

such that D 1 :: = O.

by property (gl, it follows that

.

Smce 0 1 = 0 2 on X, 1t follows that (0 1 +e)

'D/01

=

Since

<0 1 +el'D/

-1- •. -. 0 2 0 2 = 0 2 on X 0 • Moreover,

we have G 1 =G 2 on X and hence D 1G 2 = D 1 G 1 = G 1 D 1 = 0 on X 0 • Thus G 2 maps . -1-. -. X 0 to Itself and (0 1 +e) 0 2 0 2 G 2 = 0 2 G 2 on X 0 •

Since lJ 2 G 2 :: =::- H 2 :: and D 2.H 2 :: = 0 for :: in X, it follows that

(5.3)

We now return to our tensor 6----= T--T--- T--T-- arising from the a a, t y

formula D 1 (L

vi

-- • T) = /-10 2 e.

From the

at a y

at a y

a harmonicity of T_we conclude from straa

ightforward computations that e belongs to X 0 , LetTdenotel T-H 1 (L T). v. v. I I

SinceD(L T)=MD 2 •eandD 1 •L T=O 1 v. v. I I

(from the end of (4.7)), it follows that DIT =MD/ e and H 1T = 0 it follows that

ol•i = o.

From

293 We use the notation (•,•)M 0 to denote the global inner product, for example,

Thus (l T,L T)M = (HL T,HL T)M + (T,T)M • vi vi o vi vi o o We now use

i = o,*c 1{1-=lD •• a)

and

Dl T=MD •• a

to transform cT,T)Mo.

= (Ct(I-=1D .. a),D 1 T)Mo

= (c,o;a,o:a>Mo = ((C 1

-

{0 1 +e)

-1- •

-

•

D 2 a,D 2 a)M 0 + ((01+e)

-1- •

-

•

D 2 a,D. a)Mo

From property (g) we obtain (T,T)M 0

= ((C

1 -

(0 1+e)

-1- * - * - • - • )D 2 a,D 2 a)Mo + (D 2 Cza,D 2 a)Mo -1 - • - • )D 2 a,D 2 e)Mo + (iJ 2 C 2 a,a)M 0

= ((C 1

-

(l] 1 +e)

= ((C 1

-

. -1 - • - • (LJ 1 +e) )D 2 a,D 2 a)Mo + (e,a)Mo- (H 2 a,H 2 6)Mo'

where to get the second last expression we have used

D1 a =

02a

= 0 which follows from

0 and the symmetry property (ii) of a. We now compute (6,6)M 0 •

(a,a)M o

= 41

The reason for the factor

1

-i

I J Mo

-a-f--aa-

A \l A 11 (LT-- LLJ ( L L - LL) dV a 13 a >. 11 >. 11

a

is that for

;,

A= - 1 - 1 E A- - (dz p.q. ap .. ap,a, ••• aq one has

a

" ••• "dz P) ~ (dz

a, " •••lldz il Q),

294 and in the formulae we have been using for correspond to A-

-

-

-

al•••a.p 8 1••• 8 q

with p

0 1 ,0 1 *,'5 2 ,02 •,

the components

eA a 8 , II

= q = 2. Thus, when we choose local coor-

dinates so that

-.. a a T--=2.1-11!1(-}:-=U-16 1!1(-}::aS at. aS aS at. aa I

I

at the point under consideration, we have

(6,6} M 0 = 8

2 - 8 j l. 1•<..!..>-=-1-· jMo <~a ~~~~<.!.->=-1•> at. aa at. aa Mo a I

(WP) Thus R .-,- . ..,-at t II II

I

= 0 becomes

+ ((G 1

-

(0 1 +e)

-1- •

-.

)Da 6,0 2 6)Mo

+ (HL L J,HL L J)M - (H 1 6,H 1 6)M 0 vi vi vi vi o

Here we drop the subscript 1 for H in HL L J because we regard l l J as a v. v. v. v. 10 I I I I TMo ' -valued (0,1 )-form on M 0 and there is no need to distinguish between two different kinds of covariant indices.

We will drop subscripts in similar circum-

stances later without further explicit mention, We now further simplify the second and the third terms on the right-hand side of the above expression for ((G1- (Ul +e)

-1 -

R(WP>.-,-.~ II II

•

- • )D2 6,0 2 6)M

o

=

((G 1

-

·-

(U1 +e)

-1 -

)D 1 l

vi

T,D 1 l

vi

T)M

o

(with L l J regarded as TM 1 •0 -valued (0,1)-form) vi vi o

295

(because L L

vi vi

(because HL L

v. v. I

J has

Let

v. v. I

II II

£•

J as

with HL L

v. v. I

J)

I

R(WP~..,... . ..,- to I Ill

get the expression for

We Introduce the following notations to make the expression simpler.

<+(a~ ), +(a~ I

I

= 0 becomes

We now polarize the expression for

R(WP)iJk

from the end of (4.7))

the same inner product with L L

I

The final expression for R(WP>. ..,-.-:-at t

(5.4)

J is a" -closed

I

)> denote the function on M 0 which is the pointwise inner product

296 10 a a of the TM 0 ' -valued (0,1)-fonns +
a

Let +

1\

a

+ denote the

A2TM:· 0 -valued (0,2)-form with components

a a >.11 1 a >. a 11 <+< a\» ,.. •< at/aB" = 2 I +< at/a" at/ifa

>.

a 11

a

>.

a >.

a

11

+< at/a" at/a

a 11

+ •< a\ >a• i- •< atk >i• a1•

Then at t = 0

+4e ((01 +

+e

e)-1<4>(a~k),4>(a~t·<+(a:R.)>)Mo

(OJ 1 + e)-\

L

J,L L

J)M

vi vk vi v R. o a a a a - (H(+(ati)A ''a\)),H(+(at/,.. +»Mo'

Here we have used the fact that L[ ]J is a a-exact and L l J is 3"-closed _ vj'v R. vi vk and the fact that OJ 1 + e) 1 is self-adjoint. Let '1': H 1 (M 0,TM 0 ) x H1 (M 0 ,TM 0 )

+

H2 (M 0,A 2 TM 0 ) be defined by taking the

skew-symmmetric part of the tensor product.

T~:0 -valued

That is, if t, n are a-closed

(0,1)-fonns on M0 defining the classes [tl.(n] in H1 (M 0 ,TM 0 ), then

'I'([ t],[ n)) is defined by the a-closed A2 TM1: 0 -valued (0,2)-fonn t

1\

n with compo-

nents

(t

A

A\1

n)_aB

=-12

>. II

>. II

>. II

>. II

a B

B a

a B

B a

( f;_n_- f;_n_ + n_f;_- n_E;__),

(Straightforward verification shows that E;

1\

n is always a-closed.)

We can now state our result on the negativity of the bisectional curvature of the Weil-Petersson metric,

(5.5) Theorem. let

11:

M + 1'1 be a holomorphic family of compact complex mani-

folds parameterized by an open neighborhood 1'1 ~

0!!!

ct,

Assume ·that each fiber

297 M

t

= 11- 1 (t)

carries a Kahler-Einstein metric whose Ricci curvature is equal to the

metric times a fixed negative constant.

Assume that the element of H 1 (M ,TM ) t

defined by any nonzero tangent vector of 0 ~ (0, 1) !! t is nonzero.

t

Let

X 1,X 2 be two tangent vectors of type (0,1) ~ 0!! 0 and 111 1 ,111 2 be the elements of

H 2 (M 0 ,A 2 TM ) defined by taking the skew-symmetrization of the product of 111 1 and D

--

1112 is zero, then the holomorphic bisectional curvature of the Weii-Petersson metric in the directions of X 1 ,X 2 is negative. To prove this theorem, without loss of generality X1

= a~, i

X2

= aat,

where t 1 , . . .,tN are the coordinates of

we can assume that

~.

Then the vanishing

j

of '1'(111 1 ,111 2 ) implies that

H(cJI(a~.>" cJI(a~.>> = 0

and

J

I

( WP) . -1 a a a a R .-,-.-,-= 8e ((] 1 +e) (cJI(at),cJI(at)>,)M I I J J i i j j D + e (OJ!+ e)-1Lv.Lv.J'Lv.Lv.J)Mo•

J

I

Since the operator {]

1

the right-hand side is positive.

J

First we show the following:

valued smooth function f on M 0 , Suppose the contrary.

I

+ e) - 1 is positive, it suffices to prove that the first term on

Let g =

0

{] 1

1

+ e) - 1 f is also a nonnegative-valued function.

+ e) - 1 f. Then the infimum -A of g is a negative

number and it is achieved at some point P of M 0 (note that that g is real-valued). From OJg)(P• -eA

= eg(P)

for any nonnegative-

S, 1 we ~(4]

1

0

is a real operator so

conclude that

+ e)g)(P)

= f(P),

contradicting the fact that f(P) ~ 0. We now apply it to the function f

= . 1

Since a Kahler-Ein-

I

stein metric is real-analytic, it follows from the harmonicity of cJI(

a~.)

that f is

I

real-analytic and QJ 1 + e)

_,f

is also real-analytic.

From the nonnegativity of the

298 two non-identically-zero real-analytic ftnctions

4J 1

+ e)

-1

f and (t(

a a at.>,+< at.>>, J

we conclude that((]

1

+ e)- 1 f,

,+<-/->>) tj tj M

J

is positive. 0

REFERENCES

1.

L. Ahlfors, Some remarks on Teichmiiller's space of Riemann surfaces, Ann. of Math. 74 (1961 ), 171-191.

2.

L. Ahlfors, Curvature properties of Teichmiiller space, ). Analyse Math. 9

( 1961 ), 161-176. 3.

N. Hitchin, Compact four-dimensional Einstein manifolds, ). Diff. Geom. 9

(1974), 435-441. 4.

K·. Kodaira and). Morrow, Complex Manifolds, New York: Holt, Reinhardt and Winston, 1971.

5.

K. Kodaira, L. Nirenberg, and D.C. Spencer, On the existence of deformation of complex analytic structures, Ann. of Math. 68 (1958), 450-459.

6.

N. Koiso, Einstein metrics and comp!ex structures, Invent. Math. 73 (1983),

71-106. 7.

H. Maass, lber eine neue Art von nichtanalytischen automorphen Funktionen, Math. Ann. 121 (1949), 141-183.

8.

H. Royden, Intrinsic metrics on Teichmiiller spaces, Proceedings International Congress Math. 2 (1974 ), 217-221.

9.

10.

H. Royden, Oral communication, detailed paper in preparation. G. Schumacher, On the geometry of moduli spaces. Preprint 1984.

11.

Y.-T. Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, ). Diff. Geom. 17 (1982), 55-138.

12.

S. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves. Preprint 1984.

13.

s.-T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation, 1. Comm. Pure Applied Math. 31 (1978),

339-411.

Extension Problems and Positive Cunents in Complex Analysis H. SKODA Universit~ de Paris VI 4 Place Jussieu 75230 PARIS CEDEX 05

SUMMARY

This paper is a survey of recent developments in the theory of the extension of analytic sets and closed, positive currents.

INTRODUCTION

Since the fundamental works of P. Lelong ([ 17] and

and [19])

Y.T. Siu [ 22], it is well known that positive closed

currents are a vecy

good generalization of the notion of

analytic set because the analytic sets are exactly the sets of density of positive closed currents and because many properties of the analytic set

X are in fact properties of the associated current

of integration [ X] and also because the Lelong-Poincare equation of currents function F

~ aa Log If I = [ X]

am

giVE5 a simple fundamental relation

the analytic hypersurface X

= f-1 (0)

bet~

in the case of codilnen-

sioo one [ 18] . It is natural to ask if the basic theorems al::x:Jut the analytic sets are really theorems of analytic geanetry involving the hard structure of analytic set or if these theorems are in fact

theorems of carplex analysis only involving the soft structure of positive closed current.

the

300 Since the Ramert-Stein extension theorem for currents of Y. T. Si u

f 22]

and R. Harvey [ 12] it became clear that the extension of

closed, positive current was a natural generalization of the extension of analytic set. In this paper, we shall make a sw:vey of recent theorems of extension of analytic objects, essentialy of closed, positive currents.Our main reference will be the very interesting paper of N. Sibony [ 21].

1. EJcr'ENSICN OF ANALYTIC SEI'S AND OF CLOSED POSITIVE CURRENTS.

Perhaps the first IOOtivation of these problems of extension of closed, positive currents across an exceptional set was to obtain a better understanding of the following classical result of E. Bishop (1964,[4]) :

Theorem 1 : Let A be a subvariety of the c:::arplex hermitian manifold

n and X be a subvariety of n \ A, of pure dimension p,

such that for all cx:npact set

K of

n :

(whare vol 2 p means the euclidean volune of real d:imension

X

is a subvariety of

2p)

then

n (of ~ dimension p) .

Of course the main interest of the theorem is to give a necessary and

sufficient condition of extension of dim

X~

X across A in the case

dim A which is not covered by the Remnert Stein theorem. The

special case where dim X W. Stoll [ 26]

= dim

A, is due tow. Stoll [ 25]. In 1964,

gave the following beautiful characterization of

301

algebraic sets which is closely connected with the Bishop's theorem.

Theoran 2 : A subvariety

of!,

X in

of pure d:imension p , is alge-

braic if and only if there exists a constant

c

>

0

such that for

r »0 :

all

vo1 2 (X n B(O,r)) ~ er 2 P p

(where B(O,r)

is the euclidean ball of ce1ter

The proof of

0, radius

r).

W. Stoll was independent of Bishop's theorem. He

used the value distribution theo:ry of holanorphic maps. cne can now

of!

give the following short proof : ('there

lPn_ 1

is isanorphic to

is the hyperplane to the infinite), the assumption

about the growth of the volune of

X in

finite volume for the Fubini metric on theoran (with X in that

lPn

X

Q

= lP n

, A = lP

n-1

is a subvariety of and

lPn\ lP n- 1

)

of!

means that

X has

lPn , therefore the Bishop's

claims that the closure

X

of

lPn. Then Cb::>w' s theorem proves

X are algebraic.

The Bishop's theorem has a nice generalization to the case of closed,

positive currents.

Theoran 3 : Let

A be a closed canplete pluripolar subset of the

oorrplex hennitian manifold on

Q \

A and if T

trivial extension

en

fl.

:f

fl. If

T

is a closed, positive current

has locally finite mass near A, then the of

T

to

Q

is a ~, positive current

302 (A

is cx:mplete pluripolar in

where

u E PSH(n), i.e.

u

n

neans that

A

is plurisubharm:>nic in

and J. Polking proved the theorem in 1974 [ 15] bidillensioo

(p,p)

n

and T

E

n).

when

T

n, u (z)

when

= - ""}

R. Harvey is of

and A is an analytic subset of d:imansion

I proved the theorem in 1981 [ 24] of

= {z

p.

A is an analytic subset

is an arbitrary positive, closed current. My proof

was qUite different fran that of

R. Harvey and J. Polking. I

directly worked with the current

T

potential associated to

T

instead of the plurisubhanronic

by R. Harvey and J. Polking. In 1982,

H. El Mir [ 8] [ 9] generalized my method to the case \\here

A is only

closed, pluripolar. Nevertheless, these two proofs used technical estilnates of the distributioo of the mass of

T

in rome neighborhood

of A. Quite recently, N. Sibony [ 21] found a very nice fonnalization of these proofs. I shall now explain the Sibony' s proof. For the sake of sinplicity,

suppose, T

\\1e

(it is the decisive case) and T

is of biclimension (1 , 1)

has finite mass in

if necessazy, the problan is local relatively to

n

n \ A (\\le shrink A) • The two

following lannas give the basic infonnation about the distribution of the mass of the current

Lemna 1 :

T

near

(Essentially a variation of a Chern-I.evine-Nirenberg estimate)

(Q1e only needs to suppose that subset

u

E

of

€."" (n)

A.

A is closed). let

n. There exists a oonstante

n PSH (n)

such that

u

is

sane neighborhood of A (depending on ! K

TA

i

K be a oanpact

C(K,n) such that for all

;;;. 0, is bounded and vanishes in u), we have :

aau.;;; C(K,Q).IJTII

(n \

A).IJull ..

303

11'111

where

is the mass-measure of the current T.

Proof :

I.et be
E

=1

such that :
~(m a test-fnnction with ccmpa.ct support in an K and

0 .;;;

T A i a1i u

neighborhood of A, the current positive (because

u E PSH (fl) and T

=0

u

fl

is sane

is well defined and is

is positive) . Therefore, we have

the inequality !

T A i

!

aau .;;;

K

Because

fl

is closed, an integration by parts proves that

T

!

=!

fl

ida (
T) u

fl

=

! u T A i a a

Then we have obviously :

! T A i aau.;;;

K

J\lT A i fl

aa;, I .;;; Jli a'liuJICO

11'111

(fl \

A)

llull

Lemna 2 :

With the same hypothesis as in lama 1, we have ! T A i au A au.;;; C(K,fl) K n ta .;;; u (z) .;;; b}

for all such that

of A.

a

(b-a) 11'11J (fl \ A)

and b, 0
and for all

IJu\Jco

u E

t: (fl)

n PSH(!l)

is positive bounded and vanishes in snne neighborhood

304

Proof :

For fixed

fnnction of class

a

and

b let

h : lR

+

lR

be a oonvex increasin]'

c"", vanishing in sare neighborhood of o, such that h" (t) ;;;. _1_ 11. [

b-a

! h" (t) .;;; 2 lR

,

a,

bl

,

h(t) .;;; 2t ,

the graph of h" beeing the follc:Ming

r

1

b-aJ I

a

0

( 'nl a,b] h o u,

b

is the characteristic function of [ a,b]). Then

~(' (n) n PSH {r!) and ho u

E

=0

in sare neighborhood of

A. We

can apply the lemna 1 to h o u : !

T A ida (h o u) .;;; C(K,n)

11'111

(!"l \

A)

llh o uJI.,

K

But

we have : i aa(hou) =h' ou

iaau+h" ou

i auA au

Beoause each tenn in this last equality is positive, we obtain

! h" o u T A iauA au

.;;; C (K, n) 11'111 (n \ A) 2llull

K

As we have

lity :

h" o u

;;;.

~a

1l. a<;;u<;;b , we obtain the annonnced inequa-

305

K

11'!11 (Q

1 T A i
n

{a .;;;; u (z) .;;;; b}

\

A)

lluil

Lemna 3 :

Let

n

A

be a closed canplete pluripolar subset of

if necessru:y, there exists a sequence

0 .;;;;

uv .;;;; 2 , uv

u..,

E PSH(Q)

vanishes is soma neighl:x:lrhood of

and such that pointwise

uv

lim \)

...

= 11 n \

n. Shrmking

n

f6""

such that

A (depending oo

v)

A.

00

Proof : A= {z E

pose

n,

v < 0

v(z) = -oo} an

n.

for

Let be

sane

v

E PSH(n).

Shrinking

w\) := exp(Y) , then w\) = 0 \)

n, we oo

can sup-

A,

wv ->-11 n\A pointwise and wv .;;;; 1. Let now

x be a oonvex increasing function of

T

whose graph is the

following :

1 0

1

Then 1

4

of

Wv E PSH (Q)

and now

vanishes in same neighl:x:lrhood of and

anly need 1o ~la.rlze x o w\)

o

X

x

o

wv ..,. ll

n\

A

X o Wv

Q

pointwise. We

in order to archiele the oonstructian

u. \)

The following lemna is the Cauchy-Schwarz inequality.

306 Lerma 4

If

T

is positive of bid.inension (1,1) and if

and

ljJ

E~1 , 0 (11)

then we have :

I

1

~>I,.;;;

A

() 2

1

(

T,i ljJ A

~>) 2

ProOf of theorem 3 : 9 E

I.et

&"" (lR)

be a rut-off function \ofu)se graph is the following : 1

0

I.et

m

u \) be the sequence

9 o uv

=0

1

1

4

2

PSH(fl)

n e(n)

m sane neighborhood of lim \) -+

A

and

9 o uv = -i1 11 \ A ,

m

By the defmition of the trivial extension

:f

given by the lentna 6,

=

1' ,

we have

lim9ou\) T

and therefore a:f=lim m the \\'eak sense. that

have

:f

:f

is obviously positive. The diffirulty is to prove

is closed. I.et

-

'\,

=

9'ou\) TAau, \)

lim \) -+

m

~,O (n)

be a fonn of bidegree (1,0). We

307 and we need to prove that the l.imi t is

0. But the Lemna 4 gives the

following inequality : I
1 0 u i au A au\) >I "' (
1{1

(
a 1 o uv

But

b

=2

9 1 o uv

and +

0

i

1{1

is bounded and his support is a subset of

{%.;;; uv .;;; !}. 1

uv

The application of I..amla 2

K

= supp

=%

proves that the secxnd tenn is bounded. Because

1{1

pointwise, 9 1 o uv

third tenn goes to

to uv with a

is bo1.mded and T has finite mass, the v

0 when

+ "". We

have proved :

Remark :

Of course the theorem 3 implies the Bishop 1 s theorem because if T

= [X],

it is easy to see that

~

= [ X1

where

X

the structure theorem of R. Harvey and J. King [ 14]

elenentar structure theorem of

is analytic, using or the Jrore

J.King [ 16].

N. Sibony also proved a Jrore generalresult than theorem 3. In this result, the structure of the exceptionnal set

bidirnension of the current

A deperrls on the

T :

Definition 1 Let be

p

E

N • The subset

pluripolar if for all

a

E

n

A

of

n is said to be

there exists a neighborhood

local system of cxx:n:dinates in

V (z 1 , z 2 , ••• ,zn)

p ccnplete V of

a, a

and a finite nunber

308 of projection

II. :

J

J

1 .;;; j .;;; N

such that for alloost all

1 .;;; j .;;; N

i p

is carplete pluripolar in the carplex linear

J

(2)

and where

a?,

Bp

-

dz 1 A dz 1 A••• A dzp A dzp.

on subspaces of dimension J not only projections on the coordinates subspaces of dimension 1t11e

p

of

p

(z 1 .•• zn)

need enough projections

+

(z . , z . , .•• , z . ) 12 1p 11

We have the following

s:L~~ple

II.

for a given coordinates systan on

2) If A is a closed subset of A is

Let

a!?

of Hausdorff 2p-neasure 1r- 1

if

1r

(a)

cr-finite,

is countable for

is a projection of

a?

in

o.

If

T

a!?.

([21) )

A be a closed

positive current on of T

a?

pluripolar catl>lete set.

p-pluripolar catl>lete, because

almost all a in

p-pluripolar subset of

o \ A of bi.dilrension (p+l ; p+l)

is a closed,

and i f the mass

is locally finite in a neighborhood of A, then the trivial

extension

1.C

of

T

a?.

exanples

1) 0-pluripolar carplete sets are the usual

Theorem 4 :

a!? ,

is the volun fonn on

-

Notice that

then

Z0

0 ) and such that the differential fonns II~ Bp J J the space of cx:nplex fonns of bidegree (p,p) where

generate

=

,

n-:- 1 (Z

manifold V n

Bp

+ ctP

v n n-:- 1 (zo)

II . {V) , A n

in

a?

is a closed positive current on

o.

309 Proof :

The problem is local. We suppose that

t~ 1

is open in

of

V

on

of

t~ 2

and

locally flat current en if

= v = •\

{y}

A2 • The currents

x

v. If

is a fonn of degree 1

Jd:fA.pA/w= V

is the injection of

{y}

cii'

is the slice of

and is a locally flat current en

We have [10 ] :

<:f,

11, y>

=

i',

=

<:f,

A2

x

(p,p) on A1 V, we

(cf .[10 ])

in

V and where y

11- 1 (y) . 11, y>

for alm::>st all

11- 1 (y) n A

proves that

Therefore we have for all

cii'

are

y.

is the trivial extension of the current

y>

(1,1) , closed and because 11- 1 (y) , the theorem 3

d:f

which is defined for alllost all

11- 1 (y) \ A. Because

en

and

(y) n A

"' y> A i *y .p]w, [ J
J A1

iy

:f

with carpact support in

have the fl.mdamental foiiiiUla of the slicing of

where

-1

y E A1 , 11

is a fonn of bidegree

w

where

x A2

is the projection

fil-P. If 11

A1 , we also suppose that for alllost all

is pluripolar carplete in

and

is open in

n

is of bidimension

is pluripolar carplete in

~' y> is closed.

=

* =0

11 w

Then we have

elf

A / w = 0

for several projections

11

and then

cii' = 0. 2)

Pluripositive currents. In [ 24],

we observe that a positive current T

such that

ddG:r

310

is also positive, has a well defined I.elong's number. Recently, gaves a generalizatioo of the preceedings results to

N. Si.bony [ 21]

pluripositive currents.

Definition 2 : A current positive if

T oo

n of bidimensioo

(p,p) is said to be pluri-

T is no:rnal (i.e. the coefficients of ddc T is positive and if

neasures) , if

T and dT

are

is positive or negative.

T

Exal!ples : 1) T

=u

where

u E PSH (Q) is positive or negative.

2) T '"' u0 ddcu1 A••• A ddc~ where where

u0

is;;;.

0

or

and B. A. Taylor [ 3]

<

uj

E

PSH:(O) n L"' (O,loc) and

0 (ooe needs the definition of

for the product

E. Bedford

ddcu 1 A••• A ddc~) •

3) If T is a closed, positive current and if u E PSH (0) positive, then one can prove that

uT

is

is pluripositive (cf. [ 21] ) .

We hope, pluripositive currents will be a good generalization of plurisubhal:Ironic ftmctions. N. Si.bony proved in [ 21] the following result.

Theoran 5 : Let

A be a closed, p-pluripolar canplete set in

n and let T

be a pluripositive current in

n \ A, of bidimension (pH,p+l). We

suFPQse that

have locally finite mass in sane

T, dT and ddc.r

neighborhood of A. Then we have :

311 i)

d:JI = ffi.

ii)

"' = ddc:f + s dd"r is a closed current, carried by A. :M)reover,

where

S

if

is positive, negative

T

if

T

S

is positive

is negative.

The proof uses integration by parts and a Cl'em-Levine-Nirenberg inequality (as in Lemnas 1 and 2) :

J

T A du A

dcu ..;;

C (K, n,T)

(b-a) !lull

K n{a<;u<;b}

where

is positive, of bid:imension (1,1), pluripositive in

T

and where

u

E

PSH (Q) n

C"' (n)

n\ A

is positive and vanishes in sane

neighborhood of A (K cc n) •

corollary 6 : Let

A be a closed p -pluripolar canplete set in n and

pluripositive current , of bidimension (p+l,PH)

i) d(~ A T)

= -:dA

dT

ii) ddc ('!d.AT)

;;. 0

if

Proof :

Let be

is positive

T' : =Tin\ A , then YJe have

11A T

YJe

T

=T

- ".!1

Q \

A

apply theorem 5 to the current

T

=T

T'.

-

:J'•

in

n. Then

T

a

YJe

have

312

3) Extension across

c. R.

sul:manifold

N. Sibony also proves in [ 21]

extension theorems across c.R.

sul:lnanifold. The main interest of these results in the fact that they den 't need an asSI.Ullption of finite mass.

Theorem 7 : let

be a totaly real sul:manifold of class

A

c2

of

ll (i.e. the

real tangent space to A does not ocntain any canplex line) and let T be a positive current on 1) If

T

is closed in

of biclimensicn

ll \ A

ll \ A

(p ,p) ,

and if p ;> 1, then the mass of

T

is

locally finite in a neighborhood of A. 2) If

T

~

is pluripositif and p

2 , then the mass of

T

is also

locally finite in a neighborhood of A.

Prcof :

We on-ly prove the

with

1)

p

=1

(the important ca.se) , It is a

local result, therefore we SUP.[X)se that the euclidean ball B(0,1) is ocntained in

n and that in B we have A= p- 1 (0)

positive function of class c 2

in

ll

where

p

is a

such that

ddc p ;> CB ,

in

B(0, 1) ,

cfl.

We choose any function

where

C

is a oonstant

h

h< 0

h E

too (B)

=o

in

>

and

B is the Kahler fonn en

such that

B(O,t> ,

in the set

h .;;; 0

0

{z ; ~ <; Jzl

everywhere.

"'jl

313

o>

If

is small enough, we have :

0

cH0 (p+Ohl ;.;. ~ a

o.

B(0,1). We fix such a

in

We define

p

e:

P

e:

E

PSH(B(0,1)) II ~ 0 (B(0,1))

= sup(p+oh-e:,O). dd0 u

Then the current

g

Let

and

For

E

lzl .s;;

i

, we have

and

g

=1

on

B(0,1).

B(O,i)

lzl :>t·

for

Iz I .s;; i

is defined, positive and closed in

AT

e:

a positive function such that

&""(B(0,1))

=0

g

by setting

p e:

p :> e: , pe:

= sup (p-e:, 0)

(because

=

diF pe:

and

p-e:

h

= 0) •

= dd0

Therefore for

p. Because of the

choice of g, we have < dd0 p

A T, g> ;.;. C

e:

f

T A

1 ) n {e: B(0,4

a < p}

By Stokes follllUla we have :

< T, P

e:

c'Jd'=g

>

:>

c

1 T A

a

1 ) n {e: < p} B(0, 4

When

pe: > 0, we have

p+c5h - e: > 0, that is p > e:-c5h > -c5h •

But the suwort of 0

<

e:

is contained in

i .s;; Iz I .s;; i}

{

and

on this canpact set. Therefore, there exists a constante

that p

ddcg

-c5h :> c 0

>

0

on the support of

ddcg, we have : p

>

-c5h ;> c

0

>

0

h

is

c0

ddcg. Then on tre support of

such

314

This imply that :

Let

T

K

=

{z ;

~

.s;; lzl .s;;

has finite mass en

t

e:, we have the estimate

p (z) ;> c 0 }. For all

1

B (0 ,4)

\

A.

p =0 £

Theorem 7' :

Let

A

be a

(i.e. dim__ -·u: HTz

C.R. subnanifold of

= dinL(T n u; z

iTz )

positive current of bidimension 1) If

T

is closed and

p ;> k+l,

=k (p,p) T

n

of c.R. d.imensicn

for all

z

E A)

and let

k

T

be a

on n \ A. has locally finite mass in a

neighborhood of A. 2) If

T

is pluripositive and p ;> k+2, T

a neighborhood of A.

has locally finite mass in

315 Sketch of the proof

We cmose a systan of coordinates in

fil

such that

0

E

A and

such that :

= zn = 0}. We set

V1

Z

1

and V"

z~ E V 1

:

(z 1 , ••• ,zk) of

0

in

z"

of-

=

(zk+l'''''zn). There exists neighborhood

and fil-k

respectively such that for all

the set

A(z 01 )

=

{z

A·, z 1·

E

= Z0

1 }

is a totally real sul::Jnailifold of {z 01 } x V" • We awly theorem 7

to the slice '

of T (where

11

:

fil +

af

is the canonical projection). Ole obtains a unifom estilrate of the mass of the slice of T

0

and therefore the finiteness of the mass

using classical estimates of slicing theocy

The first proof of theorem 7 1

in [ 8]

[ 10] •

assertion 1 given by H. Elmire

was false. The first correct proof was given by N. Sibany in

[ 21] • After H. Elmire gave a new proof in [ 9] • His proof is the

as Sibcny 1 s proof when

k=O

but is different when

k ;;;. 1

same

and don 1 t

use slicing theocy.

Theorem 7" (N. Sibany) Let

A be a C.R. sul:manifold of

T be a positive current in 1) I f T

n of C.R. dimension k

n \ A of bidirnension (p,p)

is closed and p = k+l, then

1'

is nonnal

and

316

2) If

T

sicn to 3) If

is closed and

(this was also proved by H. Elrni.re).

rl

T

p ;> k+2, T has a unique closed positive exten-

d~ has a locally

is pluripositive and p ;> k+2, then

flat extensicn to

Q

dd~

Notice that

and cne has :

has bidimension

(p-1,p-1)

and that

p-1 ;> k+l ,

~

therefore

has finite mass and

difT

dd~

is a no:r:mal current using

assertion 1) • Easy counterexanples prove that the results are shiup.

The fundamental tool of the Sibcny' s proof is still a Chern-LevineNirenberg estimate. T A dp Ad~

f

<:

C(K,rl) (b-a)

£11'111 (S'i\p.)-+jldd~l

(rl \F)]

{a <: I pI <: b} n K

where

p

is a

c2

n, I pI <: 1 , F := p- 1 (0) , and T

function in

positive, pluripositive current in that the mass of

T

F

of bidimensicn (1,1), such

are finite. NoW' the important trick is

ddc.r

and

Q \

is

the fact that the inequality is valid for all

a

and

b, 0 < a < b < 1.

A s.imilar inequality is true for currents of bidimension (2,2). The main idea of the proof is similar to that of the theoran 3.

N. Sibcny also awlied these last results to the following theorem which extends results of H. Alexander [ 1] , J. Becker [ 2] and

s.

Shiffman [ 20] •

Theorem 8 : let

A be the set

(z 1 , ••• ,zn) 11

(Q) = rl

+

Jif x

(z1 , ••• ,zn) •

and if

T

If

{ 0} Q

c

af1

and

11

the mapping :

is an open set in

afl

is a positive closed current of

such that

bidimensicn

317

(1,1) of

in

A, tha1. there exists a positive, closed, current

fl \

bidiroensien (1,1)

en

such that :

fl

S ;;. T ,

on

fl \

S

A.

Proof : Because

T

is positive of bidimension (1, 1) , -11*T

We ccnsider the closed, positive currents en

fl \

is positive.

A :

S := T -11* T

Because

neighborOOod of defined on fl

\ole

have :

is a totally real sul:manifold, S

A

As

11 o 11 = Id ,

has locally finite mass in a

A (thoorem 7). The trivial extensioo

and is locally flat. Therefore

fl

~

~

is \>~ell

is locally flat oo

and verify

1T

*

1T

*

But

~ is carried by

a classical result of

~

= -~

~=-~

A

and

H. Federer

1T

*

We have

The current

1T

*

11* lA =I d. Because

[10 ]

:inplies that :

~ = ~

~=~=~

~

~ is locally flat,

=0

~ is positive and closed and

318

on

n \ A.

4) The canplex M:m.ge-J\Illp€re operator and the extension of mercm:>rphic

~· Let

class

M be a closed subset of

c"" in n \

111

E

PSH(n)

a function of

or in L"" (n \ M, loc) , we want to prove that in

sane cases the operator N. Sil:xmy [ 21]

and u

n

(
has locally finite mass in

n \ M.

introduces the following definition.

Definition 3 : cne says, the closed subset M of if for all

(C)

rhood

w of

exists 'P M n

E

x

n c

of!

verify the condition

there exists a strictly pseudoconvex neighbo............... such that : ;;; c n and x fJ M n aw i.e. there

x

E M

PSH (n) n

f:" (n)

such that 'P (x) > 1

rut 'P < 0

on

aw.

ExClll"£les : 1)

M

is a closed subset of a totally real sul:rnanifold

r.

2)

M

is a cc:npact subset of the pseudoconvex open set

n. In this

case, aw n M = cp. 3)

M is a strictly pseudoconvexe real hypersurface in

4)

n = B2

the euclidean ball in

M = {z E B2 ; z 2

=

(Thullen's situation).

0 ,

a: 2

Iz 1 I ;;;. r}

n.

and where

r

is

>

0.

319

Theorem 9 : Let for all

(N. Sibany} (C) • Tha1

M be a subset of n 'Which verifies the condition u E PSH(n}n

The case where

&""en\

M}

and all canpact set

Kc

n, we

have :

M is a canpact subset of the ball is due to P. Griffiths

[ 11] • The proof is .:in tie s:me vein as the proof of theorem 7, using ingenious integration by parts. Using the slicing theory, it is

easy to deduce the following corollary.

Corollary 10 : · If M is a C.R. sutmanifold of for all

The

u

final

e&"'cn \

M} n PSH(n}

n

of

k, then

C.R. dimension

and for all canpact

n,

Kc

we have

aim of the theory is the following result in the same vein

asP. Griffiths [11] and

Y.T. Sju [23].

Theorem 11 :(N. Sibony} Let

M be a closed subset of the open set

n

a canpact, kahler manifold. Tha1 a mercm:>rphic map extends to a merarorphic map fran following assumptions about

n

to

and

N be

f : n \ M+ N

N 'When we make one of the

n

of C.R. dimension

exists a (n-2) pluripolar canplete closed subset M.

~

M

1) M is a C.R. sul::manifold of

tains

in

A

k.;;; n-2
n

and there

which con-

320 M is oettpact and there exists a closed canplete pluripolar set A

2)

which contains M.

Sketch of the proof : For the sake of s.inplicity, be the kahler fonn of

"We

=l

N. T :

suwose

is a current on

w

locally finite mass. The restriction of finite mass. Because A

1'

Q

to

Q

u

PSH (Q) n

E

Q \

has also locally

A

c"" in

f:"" (Q \

proved that

Q \

M)

T

Restricting

A.

such that :

aau = 1'

= tP =

b.n- 2 x 8 2

is the product of bvo poly-

~- 2 and a: 2 and that for all z 0 = (z~, ... , z~_ 2 )

discs in

T has

.inplies that

to

which is of class

Q

if necessary, there exists

can suwose that

T

w

M of bid.irnen-

Q \

is (n-2) -pluripolar, theorem 4

i We

is holarorphic: Let

k <: n-2 , theorem 7'

sicn (n-l,n-1). Because

has an extension

f

M(z) := M n {z 1 o

E

b.n- 2 ,

0 = z01 , ••• ,zn-2 = zn2}

is a totally real submanifold. Let be

r

= {(z,f (z)) z0

For ellery

to Then

({z0 "We

}

E

; z

E

b.n- 2 , let

lln \ M}

the graph of

f.

r (z 0 ) be the graph of the restriction of

f

x 8 2 ) \ M(z 0 ).

have

vol (r (z 0 )) = ! (f3-t1:'1dcu) 2 = ! (eA (3 + 2(3 A ddcu + (ddcu) 2 > 8 2 \ M(z 0 ) 8 2 xM(z 0 ) 9

bcnmded by a constante

C(z 0 ). The details of the

proves that

C (z0 )

(with n

=

Applicaticn of theorem

2)

proves that the last malllber is

is ba.mded by a constante

c

proo~

of this theorEm

for all

z

o

E

n-2 8 .

321

Fubini 1 s t:.heorEm allow to estimate the volutre of

=

vol (r)

J

vol(r (z 0 )) dA. (z 0 )

An-2 Because

r

(!1 x N) \

is

< cJ

dA. (z 0 )

r

<

+ "'

An-2 n of

is an analytic subvariety of dimension

of locally finite volume and because A x N

(M x N)

(n-2) -pluripolar catplete, theorem

subvariety of

5) Extention

S1 x N, i.e.

fran an

oem

f

4 proves that

r

is a

has a merarorphic extension to

set to

11.

rfl.

J.P. Danailly in [ 6] considers the following problem : let

be

T

a positive,closed, current of bidl.mension (p,p) on an open set S1 of

rfl.

When has T a closed, positive extension to

of density Ec = {z e S1 , v (T,z) ;;;. c} > 0

and

? Of course, the sets

of the current T (where c

is the Lelong's number of T

v(T,z)

rfl

is

at the point z) are

analytic because of the Siu 1 s theorem and are obstructions to the extension of T

to

v (T, z)

of

T

density to Let

Therefore, one needs to suppose that the is snall enough on

S1

in order to extend T

rfl. a =_!_ T A

pl

where B (z, r) and where

aP

be the trace measure of T, and

is the euclidean open ball of center

z e nr

with

S1r := {r; e

a:l1 ,

d(r;,

Cn)

a (z, r) = a (B (z, r) ) ,

z, radius

r

> r} .

Theorem 12 : J.P. Demailly ( 6] Let

current on

S1 cc

be a Runge danain and

T be a closed, positive

S1 whose coharology class vanishes. We suppose, the

322 following conditicns about the mass For all

>

E

I

sup

z

r

(2)

o

E K

0

0 ~-~)

dr < + oo

T

are verified

K c QE ,

and all carq;>act set

0

E

(1)

o (z,r) of

if

is of bid.egree (1,1),

T

r 1

sup

z

E K

2 o(z,r)

dr <

+

00

rn

T

is of bidegree (q ,q)

1 < q < n

o> n

Then for all real nunber

closed and positive en

if

afl

is of class coo outside

>

0, there exists a current

such that

T

=a

an

a

and such that

Q0

e

Q

n

It is easy to extend the result to Stein manifolds. The main difficulty of the proof is to wild a good potential V such that modulo coo

(n0),

using an appropriate kemel when

The construction is easy When

oorrlitian (1)

T

i il a V = T

1 < q < n.

is of bidegree (1, 1) because the

means that the function

V is locally bounded.

J.P. Demailly build ingenious counterexarrples proving that the condition (1) is sharp and the reascnable sufficient condition for 1 < q < n

is the following :

sup

z

E K

He also gives a nore constructive proof of the existence of totally real

Coo

sul:manifold of

rfl

of real dimensioo

n-1

which are

pluripolar catplete (the existence of such manifolds is due to Diederich-Fomaess [ 7] ) .

323 6) Q?en problens 1o

-

There is still one open interesting question about extension of

positive, closed, current.

tP

I.et

= flk x fln-k

afl.

be a polydisc in

z" 1T

fln

:

I..et

flk

+

T

, z

+

z1 .

ne a positive, closed surrent

lil

f1

k

x f1

n-k

\

(fl

k

x

{0}) of

bid:ilnensian (k, k) . We ccnsider the set E

=

{Z 1

E

/).k, II II <+co}

where

E

::<

has

0

is the slice of the current T

by

U!besgue neasure or more generally that

pluripolar in

t:.k • Has

T

TT. Suppose that

E

is not

a closed positive extension across

t:.k x {0} to

t:.n (R.Harvey and J.Polking 1 s problem) ? It is true when

k = n-1 (R. Harvey and Polking [ 15] ) . For

I don 1 t know the answer. When en an analytic subset of

X of

T

= [ X]

k

<

n-1 ,

is the current of integration

fln \ (llk

x { 0})

,

X

is an analytic subset

fln (V.1. Stoll [ 25]) .

2° - I f

T

is positive and pluripositive current an

is still

defined (cf. [ 24] ) . For

c > 0 , it is easy to see that the set :

= {x

E

is closed and of Haudorff };Dla.rin general.

n ?

Q

v(T,x)

~

at the point

x

v (T,x)

c

T

I have proved

that the I..elang number

E

of

n,

C}

2p-measure locally finite. Is

it is also easy to see that

E

c

Ec pluri-

is not analytic in

324 3'!... The tangent cane

~

If

T

afl

r > 0.

is a closed, positive current an a neighborhood of

0, the

the mapping :

problE!ll is to prove the existence of

-+

lim r -+ 0

exists, then

sense. If the limit

T0

and is a current an

afl.

If

= [X] ,

T

H. Federer [ 10]

is an algebraic conic set an

4o

-

z-+-

i*

T0

in the TNeak

T

.!* r To = To

is conic

has proved that the limit exists and

fil.

The Demailly' s oonjecture Let

TNe

z r

afl ,

for

We denote by

X

denote by

be a projective manifold. Following J.P. Demad.lly [ 5] ,

Jp (X)

the set of current [ Z]

is an irreducible analytic set of dimension

E

3J'p,p (X) p

in

where

X. S P

Z

~(X)

is the set of strongly positive, closed current of bidimension (p,p) an

X

which are "rational". T is "rational" if the

ool'XJnology class

is in the JR -vectorspace generated by the

cl (T)

classes of bidegree (k,k)

in

H2k(X,:?l) (p+k=n).

J.P. Danailly asked in [ 5] if one has the equality

S P

where

~

/'\ (X)

= Jp (X)

A means the closed convex hull. It is not very difficult

to see (cf. [ 5] ) that this conjecture implies the Hodge conjecture every ratiannal coharology class of bidegree (k,k) is the coharology class of a holarmphic cycle [ X]

of dimension

p. But probably this

canjecture is much stronger than the Hodge oonjecture because rwghly speaking this conjecture means that each can be represented as an integral cycle

(Z:>..) :>. E A where

ll

! A Z:>.. dll (:>..)

T

E

S P ~(X)

of a family of

is a probability neasure an

A.

325

Hodge conjecture only says that

T = [ Z]

where

Z

+ dS

is a holano:rphic cycle and S a current.

J.P. Demailly [ 5] proved his conjecture for

p

= n-1.

326 BI BL I 0 GRAP HI E '{l]H.ALEXANDER.- Continuing

!-dimensional analytic s.ets. Math. Annalen, 191,

1970, p. 14 [2] J.BECKER. -

Continuing analytic sets across lRn • Math. Annalen, 195, 1972, 103-106.

[ 3] E.BEDFOF.D and B.A. TAYLOR. -

A new capacity for plurisubharmonic functions.

Acta Math., 149, 1982, 1-39. [4] E.BISHOP. -

Conditions for the analyticity of certain sets. Michigan Math. Jour., II, 1964, p. 289-304.

[ 5] J.-P.DEMAILLY. -

Courants positifs extremaux et conjecture de Hodge. Inventiones

Mathematicae, 69, 1982, 347-374. [ 6] J.-P.DEMAILLY. -

Propagation des singularites des courants, positifs, fermes.

Arkiv fi.ir

~1athematik,

[7] K.DIEDERICH and J.E.FORNAESS.-

1983. Smooth but not complex-analytic pluripolar sets.

Manuscripta Mathematica, 37, 1982, p. 121-125. [8] H.EL MIR. -

Sur le prolongement des courants,positifs,fermes. These d'Etat soutenue

[ 9] H.EL MIR. -

a

l'Universite de Paris VI en Novembre 1982.

Sur le prolongement des courants,positifs,fermes. Acta Mathematica, 153, 1984, p. 1-45.

[10] H.FEDERER. -

Geometric Measure Theory. Springer Berlin, Heidelberg, New-York,

1969. [ tr] P.GRIFFITHS. -

Two theorems on extension of holomorphic mappings. Inventiones

Uath., 14, 1971, 27-62. [12 ]R.HARVEY. -

Removable singularities for positive currents. Amer. Jour. of Math., 96, 1974, 67-68.

( 13] R.HARVEY. -

Holomorphic chains and their boundaries. Proceeding of Symposia in Pure Mathematics. Several complex variables, Williamstown,l975,

327 [14] R.HARVEY and J.R.KING.-

On the structure of positive currents. Invent. Math.

15, 1972, p. 47-52. (15] R.HARVEY and J.POLKING. -

Extending analytic objects. Comm. pure and applied

Math., 28, 1975, 701-727. [16] J.KING. -

The currents defined by analytic varieties. Acta Math. 127, 1971, p. 185-220.

[17] P.LELONG. -

Integration sur un ensemble analytique complexe. Bull. Soc. Math. France, 85, 1957, p. 239-262.

(18] P.LELONG. -

Fonctions entieres (n variables) et fonctions plurisousharmoniques d'ordre fini dans

~n . Jour. Anal. Math. Jerusalem, 12, 1964,

p. 365-407. [19] P.LELONG •. -

Fonctions plurisousharmoniques et formes differentielles positives. Gordon and Breach, Dunod, Paris, Londres, New-York, 1968.

:(20] B.SHIFFMAN. -On the continuation of analytic sets. Math. Ann. 185, 1970, 1-12. [21] N.SIBONY. -

Quelques problemes de prolongement de courants en analyse complexe. Preprint Universite d'Orsay, 1984,

~22] Y.T.SIU. -

a

paraitre au Duke Math. Jour.,l984.

Analyticity of Sets associated to Lelong Numbers and the extension of closed positive currents. Inventiones Math., 27, 1974, p. 53-156.

~23]

H.SIU

Extension of meromorphic maps into Kahler manifolds. Annals of Mathematics, 102, 1975, 421-462.

124] H.SKODA. -

Prolongement des courants, positifs, fermes, de masse finie, Inventiones Mathematicae, 66, 1982, p. 361-376.

f25] W.STOLL. -

Uber die Fortsetzbarkeit analytischer Mengen endlichen Oberflacheninhaltes. Archiv der Mathematik, 9, 1958, 167-175.

i[26] W.STOLL. -

The growth of the area of a transcendental analytic set I Math. Annalen, t. 156, 1964, p. 47-48 et 144-170.

and

II.

On the Uniformization of Parabolic Manifolds Pit-Mann Wong Department of Mathematics University of Notre Dame Introduction The concept of an open parabolic Riemann surface is classical.

It is well-known that there are many different but

equivalent characterizations, each of which can be generalized in some way to complex manifolds of higher dimension.

However,

as is to be expected, these generalized concepts are in general not equivalent.

In this article parabolicity is defined via

the existence of an (unbounded) exhaustion satisfying the homogeneous complex Mange-Ampere equation which, in the case of Riemann surfaces is simply the Laplace equation.

There are

several reasons for choosing this as the definition of parabolic manifolds.

First of all it includes affine algebraic

manifolds and with this definition, the classical value distr~ bution theory over ~n can be extended to parabolic manifolds in such a way that the defect relation is still intrinsic (cf. section 1 below). quite intrinsic

Secondly, this definition is indeed

as it is possible to obtain (under appropri-

ate assumptions on the regularity of the exhaustion function) uniformization theorems for parabolic manifolds.

Thirdly, by

allowing bounded exhaustions (so that the manifolds under consideration are no longer parabolic but rather, Kobayashi hyperbolic), it is discovered that the Mange-Ampere condition is intrinsically related to the condition of certain complex curves being qeodesics(or extremal disc) of the Kobayashi metric (cf. sections 4 and 5). This article is a

(partial) survey of known results.

However, some of the results and proofs are new.

The new gre-

dient here is the observation that global arguments which work very well in the unbounded case can be replaced by local arguments, which work also in the bounded case, using the fact that a solution of the complex homogeneous Monqe-Ampere equation is real analytic when restricted to the leaves of the associated foliation.

330

The definition of parabolic

manifold described above, as

well as many results in this area are due to Professor Wilhelm Stoll.

We dedicate this paper in his honor, on the occasion

of his sixtieth birthday. The author would like to thank the National Science Foundation and the Sloan Foundation for their support. §1

Value Distribution on Parabolic Manifolds

The fundamental paper of R. Nevanlinna in 1925 on the theory of meromorphic functions was, in the words of H. Weyl, one of the few great mathematical events of our century (cf. Weyl [34], p. 8).

This theory was brought to a new heisht by

L. Ahlfor's work [1], a magnificent demonstration of the interplay between topology, geometry and analysisron holomorphic curves in 1941. Ahlfor's theory was extended to parabolic Riemann surfaces by H. Weyl and J. Weyl in the famous monograph [34] published in 1943.

Modern treatment of

this theory can also be found in Chern [8] and Wu [38]. Extension of the theory to higher dimension was first achieved by Stoll in a series of papers ([26], [27]) in the late forties and early fifties.

Among other things the following

defect relation was essentially obtained by Stoll. Theorem 1.1 .

~Pn be a linearly noh-degener-

ate meromorphic map and let a be a set of hyperplanes in eral positions in ~Pn.

gen~

The sum of defects satisfy the follow-

ins estimate,

For an alternate proof of the above theorem see Vitter [33].

In the sixties and early seventies, value distribution

theory attracted the attention of many mathematicians, notably the works of Carlsen-Griffiths [7], Griffiths-Kinq [12] extend the theory to algebraic manifolds in the equi-dimensional case. However in this article we shall restrict ourselves to the ideas of Stoll in extending the defect relation to a larger class of manifolds other than ~m. This class of manifolds should be large enough so as to include all affine algebraic

331

manifolds and on the other hand, manifolds in this class should not admit any non-constant bounded holomorphic functions (otherwise we cannot expect to have a nice simple defect relation) . Such manifolds can be produced bv the followin0 procedure (cf. Stoll [26].).

F'or simplicitv we consider only the case Let ~·!"\ be a complex manifold with Kahler

of Kahler manifolds.

metric g and w the associated Kahler form.

Choose a sequence

of relatively compact domains with smooth boundaries {G.}, G.

J

cc

J

Solve the Dirichlet problem for the

G.+l and uG. = M. J

J

~

Laplace-Beltrami operator

on Gj-G 0 ,

u.

0

on G.-G

u.

{~

on M-G.

J

J

]

0

(1. 2)

J

on G

(1.1)

0

The harmonic condition (1.1) is equivalent to ddcu.

wm-1

/1

J

i(a- 3).

(1. 3)

0

Define constants

J

c. =J

dcu.

ClG.

11

wm-l

(1.4)

J

J

These constants are strictly positive (by Hopf's lemma) and satisfies limit

c.

J

>

Ck if k

~

j

(by maximum principle).

C (M) =lim C. > 0 oo

j+oo

J -

Thus the (1.5)

exists and is called the capacity of M at infinity. Remark.

In many applications we often encounter the situation

where instead of a Kahler form, we have only a closed nonneaative form

w.

We can simply replace (1.1) by (1.3), which

still makes sense, then the concept of capacity can still be defined, provided that the Dirichlet problem is solvable. Proposition 1.1 .

A connected Kahler manifold M with zero

capacity at infinity does not admit any non-constant bounded holomorphic function. Proof.

Let w be the fundamental Kahler form and suppose there

332 is a non-constant bounded holomorphic function f, say lfl < 1. Then by (1.2) we have,

0 < A

/1

From (1.1),

m-1

w

(1.2) and Stoke's theorem, the last term above is

equal to log(l+ I f 12 )d c U.IIW m-1 < - J log(l+ I f !2 )d c U.IIW m-1 ~c.log 2 J -
J

aG 0 -

G.

]

diction since c. --+ 0 as j --+ J

~elated

For

QED.

oo,

results on capacity see Sibony-Wonq [23]

and [24], also Alexander [2]. The concept of zero capacity introduced above depends on the metric, thus it is desirable to have a more intrinsic notion of "parabolicity".

The following definition defining

parabolicity via the homegenous complex Mange-Ampere equation is due to Stoll (cf. [29]). Definition 1.1 .

A continuous non-negative function T on a

complex manifold Mm is parabolic if (i) (ii)

T is C00 on M* = M- {T=O}; T is plurisubharmonic on M and is strictly plurisubharmonic at some point of M*;

(iii)the function u =log T satisfies the complex homogeneous Monge-Ampere equation on M*, i.e. det (uaS) -

(1. 6)

where u - = a 2 u;az~az 8 .

aB

A complex manifold M is parabolic if there exists a continuous exhaustion T: M--+

[Q,oo)

which is parabolic on the

complement of some compact set K. Examples. (1) Open parabolic Riemann surfaces are parabolic in the above sense; (2) Euclidean space ~m with T(z) = lzl 2

333

is parabolic;

(3) an affine alaebraic manifold H can be realized as a finite branched cover 7T : M ___,. a:m over a:m, then M 2 with '(z) = 17T cz > I is parabolic; (4) Let L be a nec:rative holomorphic line bundle over a projective manifold then there exists a hermitian metric h with negative chern form. is equivalent to the parabolicity of h.

This

Thus (L,h) is para-

bolic; (5) Product M1 x M2 of parabolic manifolds (M 1 ,, 1 ) and (M 2 ,, 2 ) is parabolic with exhaustion ' l + c 2 ; (6) A manifold obtained by blowing up a finite number of points of a parabolic manifold is parabolic. On a parabolic manifold (M,T) take M r

{T

0

For

the function

,

on M

0

u

so that the set

0

r } contains the exceptional compact set K.

<

0

any r > r 0

r

log r log r

r

-

-

log '"[ loa ro

M r

-

on M r on M r

1

-

M r

0

0

solves the Dirichlet problem (1.2) and (1.3) with w= ddcloa T. Capacity can be defined as before.

It is easily verified that

since T is unbounded, the capacity at infinity

C 00

(M) vanishes.

We obtain from proposition 1.1 the following: Corollary 1.1 .

Let M be a parabolic manifold then C (M) 00

=

0

and so it does not admit any non-trivial bounded holomorphic function. The following defect relation generalize theorem 1.1 to the case of parabolic manifolds (cf. Wong [35], Stoll [32]). Let (M,T) be a parabolic manifold and let Theorem 1.2 f: M ___,.
L of(a)

~

n+l +

n(n+l) Rf + 3n(n+l)Yf • 2

O.E!l

The term Rf' which vanishes in the case of a:m, reflects the topology of the manifold M. Denote by n ddclocr det(Ta.S)

334 then Rf is defined as lim sup S(r)/Tf{r) where r

S(r)

=

Jrro

-+

dt t2m-l JT
A

(ddcT)m-l

In the classical theory over Riemann surfaces, the term S(r) is replaced by the Euler characteristic via the Gauss-

=

Bonnet theorem, namely if X(t)

Euler characteristic of the

open set {T < t} then Rf = lim sup

{-j~

X(t) d:} I Tf(r) 0

If M is affine algebraic then it can be shown that S(r)

=

O(log r), so that Rf

0 if f is transcendental and

=

Rf = deg M/deq f if f is rational (cf. Wong [35]). The appearance of the term Yf is due mainly to technicality.

The linearly non-degenerate condition on the map f

can be expressed as a condition on the derivatives, namely f

A

f'

A

A

t

f(n)

0.

Due to the global nature of the

theorem, it is necessary to define these derivaties by a global (meromorphic) vector field Z in such a way that f

A

Zf

A

•••

A

Znf t

0

The term f occurs in the estimate

of the growth of Z, for which one does not have good control in general. A natural way of constructing a vector field with "good" growth condition is to take the (complex) gradient vector field Z ofT, namely Z

=

TaBT- (lj(lzol 6

where (TaB) is the inverse metrix of (TaB) and where we have used the summation convention. The Mange-Ampere condition (1.6) is equivalent to IJzJJ 2 = T where II II denotes the norm of the metric (TaB). not holomorphic.

The problem here is that Z is in general

However, if Z is holomorphic, the term Yf

vanishes, this is the case for parabolic Riemann surfaces and for affine algebraic manifolds. Corollary 1.2

Let M be as in theorem 1.2 and assume in

addition that the complex qradient vec.tor field Z of T is holomorphic, then the term Yf in the defect relation vanishes.

335

Corollary 1.3 . to

If M is affine algebraic, theorem 1.2 reduces

Iof{a) ~ n + 1

(i)

(ii)

i f f is transcendental

l,of(a) < n + 1 + n(~+l)

deg M if f is rational . deq f '

Critertion for the holomorphicity of the vector field Z will be given in the next section. §2

Geometry of the Complex Homoqeneous Monge-Amp~re Equation Let M be a coTiplex manifold of dimension m with a posi-

tive strictly plurisubharmonic function (not necessarily an > 0 of class c 2 on H, with the property,

exhaustion) T

(2.1) Let u

=

log T then (2.1) together with the assumption that T

is strictly plurisubharmonic imply that the hermitian form (uaS) is positive semi-definite and is of constant rank m-1. Thus it has exactly one zero eigenvalue everywhere. By a direct calculation it is easily seen that (2.1) is equivalent to the identity TaST T- = T a S The complex qradient vector field Z is given by a

Z = T

a/
then (2. 2) is simply the identity

(2.2)

aS TS

(2. 3)

= T

~~~~

norm defined by the Kahler netric i33T.

II II

= T when

is the

An immediate conse-

quence of (2.2) is that the vector field Z is the unique zero eigenvector of the hermitian form

=

0

(2.4)

Since aau is a closed form, it follows that Z in integrable and r1 is foliate by the (complex) inteqral curves of M.

Con-

dition (2.5) implies that u is harmonic (first observed by Bedford- Kalka [3]) along each leaf of this foliation (henceforth will be referred to as the Mange-Ampere foliation) . The followinq easy consequence is quite useful. Lemma 2.1 . The function T of class c 2 satisfying (2.1) is real analytic when restricted to· a leaf of the t!onge-Ampere

336 foliation. From now on we assume that Tis of class examine the hioher order derivatives of T.

c4 .

We shall

We shall use the

'as to raise or lower indices, for instance as a av ' 's' 'sy = ' 'svy etc. The Christoffel symbols

metric

'

a

=

are

given by, (2.5) and the components of the curvature by, (2. 6) R --

aSy:\

where,

r

=

-T

-

(2.7)

-

aSy,:\

denotes covariant differentiation w.r.t.

a;az-:\ .

We begin by differentiating (2.2), l

a

=

l

T- + T)l\! T T\! ll va

,a

+

l

a

from which we qet the followino identity involvinq Christoffel symbols (2. 8)

Note that T = T follows from the Kahler condition. a,)l )l,a Curvature informations are obtained by differentiatinq (2 • 8)

1

(2. 9)

which implies the following theorem (cf. Wong [36] for a different derivation). Theorem 2.1 .

(i)

Ricci tensor,

(ii)

J~zJ~ + S(Z,i)

=

0 where S denotes the

the hol6morphic sectional curvature K(Z)

for the plane section spanned by Z, vanishes identically. To prove (i)

we contract (2.9) with the metric,

aS]l aS ll T T , i3' a , ll + T T Ta , ll B

0 •

The first term above is IJ8z 1!2 and the second term is the Ricci tensor, as can be easily deduced via the Ricci commutation formula,

337 \)

R -

(2.10)

'1!.:r

0.\!f.lp

where the first equality follows from

T

afl

0 •

= T -

(), i3 I fl

(),I fJ

For the proof of (ii) we contract (2.S) with z, TO.Tf.l - T + TO.Tf.IT - = 0 ,8 a.,f.l a.,f.l8

where the first term vanishes by (2.8) and so we have (2.11)

which implies (ii). Remarks.

(i)

Let H = a:m or more generally an affine algebraic

manifold with the usual exhaustion (cf. example in section 1) and metric, is Ricci flat thus Z is holomorphic by part (i) of Theorem 2.1 . (ii)

If m

1 then the vanishing of the holomorphic sec-

tional curvature is equivalent to the vanishing of Ricci, thus Z is holomorphic for parabolic Riemann surfaces.

Actually,

this can be seen more easily as follows, a.

\)

T

by (2.8).

T

I

-

\)

= T

\)

T

a.8

8,v = 0

(2.12)

T- -

The above identity means that Z is holomorphic in

the direction of each leaf and so is holomorphic if m

=

1.

Note that even though T is only assumed to be of class 4 c , it is however real analytic in the direction of Z (cf. lemma 2.1) and thus so is ll"dz 11 2 . Differentiating in the direction of Z, we get lemma 2.2

If T is of class

c5

then

(i)

v _

R-

-ba.S.

+ R Proof. Ta(llazll 2

'v

f.! aa.B bn " a.b a y-8 -T T R- - T T + 2F"' - T T R - T T aa.8 nf.lb aa.b f.IY8

By theorem 2.1 (i) ,a

bBa. TTT-

-Ta(R -Ta.T"S) a.8 ,a

we have, a a. 8 T Ra.B,a T T

R

~T

a.8

a. T 8

(2.13)

338 where we have also used the identities ,a

,a

= oa and a

Using the following commutation formula

,a,~,a=O

(cf. Appendix).

T]JVa I Sa = Tava I SJJ (we need T of class c 5 here) and we have

R = T]JVR - aS,a )JVaS,a

-T

]JV

T

-

-

).!Va,Sa

Applyinq (2.11) to the above, we have ,aR _ ,a,S = ,a,JJV R _ _ ,a,S aS,a avaS,JJ T

]JV

aS )J aaS -2R -T T - R - - T T T aS avaS 'JJ Substituting the expression above in (2.13) qives (i). Now for ,b,a ( Jlaz ll2l

(ii) we differentiate the identity in (i), ,a

b- =

- R~- ,b,S,a - + Tb(R)J _,a,aTS ) bbaS ,v aaS '\1 '

II azll2

where the last term above is equal to ]Jv iJ JJv a a iJ s RavaS T T T T ,]Jb T T RavaS,b + T

s

By (2.10) we have T

,).lb

s

v and

R-- T V]Jb

=

by (2.11)

aab aab T T T RavaS,b = ' ' T Ravab,B ( TaTaTb Rav_N_b ) , 0 -2T a - Ta Tb R - - - Ta Ta R - ~ ~ 'S avab avaS

Combining all of these we have, TbTa(ll'azll

2l

,ab

=

113zll 2 -

~

--ba!r T

b T BT a

,V

s

R\1- T a T a T S ,).! aaS

aba JJ aas vb -2Ravab T T T ,S T 'JJ + RaaB T T RV\.lb T T which is (ii) by the following identity, Ta _ Trf ,S ,\1

(TrfTa 10 ) ~

_

,)J

,B,a _ ,S\1

where we have used (2.10) and (2.12).

TST yRa _ Y\.lS

339 The function 11az11 2 is real analytic when restricted to a leaf of the foliation, thus if it isn't identically zero then we can define a pseudo-hermitian metric on each leaf,

w = m ~ Jlazll 2 i dz" dz Corollary 2.1 .

If

2 a ll ..,ziJ

(2.14)

t 0 on a leaf of the Mange-Ampere

,

foliation, then Ric w > w. Proof.

For convenience define tensors A and B by

A~ B then lemma 2.2

=

R~ a a B~ aaB T T ' a

=

R~ a 8 aaB T T

(i) and (ii) can be expressed as, Ta(Jiazll 2 >,a =

11az11 2 +

TbTacllazll 2 >,ab"= Jlazll 2 - - + IIAII 2 + 2 IIBII 2 From these we get, Tb"Ta(logllazJI 2 >

,a -b

( II az II 2 II A II

~I I 2 +2 II az II 2 II B II 2 ) Iii az 114 > 2lJBII 2 ;11azll 2 .

a

Observe that llaz II 2

-R -T T

8

Ta\!B

aB

a\! = - trace of B and

since (trace of B) 2 < m liB II 2 , we conclude that b a T T (log II azll 2> > ~ II az II 2 ,ab -m proving Corollary 2.1. A well-known theorem of Ahlfor's asserts that there is no pseudo-hermitian metric on

~

satisfying the conclusion of

Corollary 2.1, hence we have Theorem 2.2 .

If the universal cover of each leaf of the Mange-Ampere foliation is ~ then lla z II 2 0, that is the Monqe-

=

Ampere foliation is holomorphic. The above theorem is due to Burns [6], our computation is perhaps more systematic. §3

Uniformization Theorems

Let M be a complex manifold with a parabolic exhaustion T on M.

Namely T: M--+ [O,c) is proper and continuous (3.1)

and satisfies properties (i), on M* = M- {T=O}.

(ii) and (iii) of definition 1.1

The exhaustion is unbounded or bounded

340 or <

depending on c

oo

lize so that c

l.

longer parabolic.

in the later case, we always norma-

oo

In the bounded case, the manifold is no

(cf. Corollary 1.1).

The following Uniformization theorem is due to Stoll [30] (see also Burns [6], Wong [36] for alternative proofs). Theorem 3.1 .

m

Let M

be a _conn!i:!.s;;ted complex mani£ol d with a

parabolic exhaustion T. 00

C

~surr.te in add~ti2__I!_"t_~~_!_ T is of_--...£1AS_§

and is strictly plurisubharmonic everz_~~.!.-~-1:!... M.

Then_ M

with t%e Kahler metric iClaT is b~ho_!omorp!:!_~~~l.~LJ:.§.Qmet!.:ic to (i) ~ with Euclidean metric if T is unbounded, or (ii) the

IDrn

unit ball

in ~m with Euclidean metric if T is bounded.

We present here the basic ideas of the proof of this theorem.

For a strictly plurisubharmonic function, it is

well-known (cf. Harvey and Wells [13] ) that the Taylor series at a minim4m point (i.e., where T=O) assumes the followina form, T(z)

m

L { (l+A.)z.2

2 + (1-A.)y.} + hiaher order terM :i =1 J 1 J 1 . m and z. = x. + iv. is a local coordinate

where A;

€

system.

Since lao T satisfies the homoqeneous Monae-Ampere

1

J

- J

equation, this implies that A.= 0 for all j. J

Thus we have

T(z) =Liz. 1 2 + hiqher order term. J

Consequently the set {T=O} is discrete and by (2.3) we know that the vector field Z is nonvanishing on M* so that M can be homotopically deformed onto the set {T=O} alonq the integral curves of

z.

Thus {T=O} consists of exactly one

point {o} because M is connected. If T is an unbounded exhaustion then it is unbounded and harmonic when restricted to each leaf of the foliation (cf. lemma 2.1), thus the universal cover is

~.

Mange-Ampere foliation is holomorphic.

From this and theorem

By theorem 2.2 the

2.1, it is easily seen (cf. Wona [36]) that the exponential map from the point o is a holemorphic

isometry from the com-

plex tangent space T M onto M. In fact, one has 2 ~ m T(exp(z)) = lzl for-z E: T M = ~. 0

341

If T is bounded we shall show that the foliation is holomorphic under the additional assumDtion that T is real analytic, the general case will be treated in the next section.

II az II 2

We shall prove inductively that ~,

ite order at

vanishes to infin-

so that it vanishes identically if T is real

analytic. From theorem 2.1, it vanishes at o.

foll~ws that ll~zll

Thus we have Tv

,~

By differentiatinq (2.11) w.r.t.

2 = o(l)

since Z

= 0 at o for all v,~.

a;az 'Y

and

a;az-A

respectively,

we get ( 3. 2)

Ta~ T R - - + 2T a ,,T ~v T R - - + Ta~v T T R - - aS~

A

"

aB~v

aB~v

- 0

,A -

(3. 3)

The last two terms in (3.2) obviously vanish up to third order at

~,

hence so is the first term. Thus - T

aBR--T~Tv aB~v

is of order 3 at 0. Similarly from (3.3) we see that TaT~ R - - = o(3)

(3.4)

aB~A

which implies that the second term in (3.2) is of order o(4). By contractinq (3.2) with T'Y and using (2.11) and (2.12), we qet TyTaT~TV R - aB~v

Differentiating (3.5) w.r.t.

=

,y

a;azn

(3.5)

0

and using the commutation

formula we obtain 3TaT~TV R - aB~v

from which

,n

+ TyTaT~TV

,n

R -aB~v

,y

+ TyTaT~TVR--

aS~v,

= 0

yn (3.6)

we conclude that the first term (= third term in

(3.2)) is o(4). Thus the first term in (3.2), hence also IIJ"zll 2 , is of order o(4). This of course implies that Tv = o(2). ,l-J

Now by (3.4) we conclude that the second term in (3.2) and (3.6) are actually o(S). By contracting (3.6) with Tn and

342 using (3.5) and (2.12), we obtain TnTyTaT~Tv R - -

= 0 .

aS~v,yn

Differentiating the above w.r.t. a;azP we obtain 4TyTaT~Tv R - -

aS~v,yp

+ TnTyTaT~Tv

,p

R--

aS~v,yn

+ TnTyTaT~TVR--

aS~v,ynp

0

from which we see that the first term (=third term in (3.6)) is o(S).

Thus the first term in (3.6)

(=third term in (3.2))

is also of order o(S). This forces the first term in (3.2), hence also lld"Z 11 2 , vanishes at ~ up to fifth order. The process described above can of course be repeated as often as desired, hence ll3zll 2 vanishes at o up to infinite order as claimed. Let Lm be a negative holomorphic line bundle over a prom-1 jective manifold N , then there exists a fiber metric h satisfying all the hypothesis of theorem 3.1 with the exception that it is only strictly plurisubharmonic outside of the zero section= {h=O}.

The following theorem is due to Burns

[ 6 J.

Theorem 3.2 .

Let (M,T) be a connected complex manifold with

unbounded exhaustion T satisfying all hypothesis of theorem 3.1 except that T is strictly plurisubharmonic only on M*. there exists a negative

hol~morph~c

Then

line bundle (L,h) such

that the Remmert quotients M and L are biholomorphic . The Remmert quotients M and L are obtained by blowing down {T=O} and {h=O} respectively. remark about the above theorem.

We shall only make one

Namely, since the exhaustion

is unbounded, the leaves of the foliation are uniformized by ~.

hence the foliation is holomorphic. We shall discuss the counterpart of Theorem 3.2 for

bounded exhaustion in the next section.

343

§4

Bounded

~onqe-~mpere

exhaustions

We beqin with an existence theorem of Lempert [14]. Theorem 4.1 .

Let D be a bounded strictly

[m with Coo boundary. a

u~e

80.

Then for each Point P s D, there exists

exhaustion T: D---+ [0,1]

and satisfies (i),

w~th

T-i(o) = p, T

(1)=30

(ii) and (iii) of definition (1.1) up to

In addition, log T(z) =log lz-pl

Remark D -

convex domain in

2

+ 0(1) near p

(4 .1)

The exhaustion T is actually strictly convex on

{p}.

Lempert also qave a geometric description ofT: Theorem 4.2 .

Let D and T be as in theorem 4.1, then there

exists a homeomorphism ~:D---+ ~ = unit ball in [m such that ~(p) =

Cil

o

and~ is a coo-diffeomorphism of

JB m - { 0}, (ii)

o-

{p} onto

~ map each leaf of the Mange-Ampere foliation

biholomorphically onto a disc throuah the oriqin, T(z) = ICzll 2 .

(iii)

Furthermore there is an explicit relationship between T and the Kobayashi metric. Theorem 4.3

Let D and T be as in theorem 4.1 then

o0 = loq (1 + IT/1-/T) where o0 (z) =Kobayashi distance from P to z. Furthermore, the leaves of the foliation are extremal discs (or geodesic) of the Kobayashi metric. Recall that a holomorphic map f: 6

--+

D from the unit

disc with f(o) = p and f' (o) =vis said to be extremal for the Kobayashi metric if for any holomorphic map q: 6 ---+0 with q(o) = p, a' (o) = Av and A

>

0 then A < 1.

We shall qive some motivations for the construction of T in theorem 4.1. of D. on

Let ¢ be a strictly convex definina function

Suppose T exists as claimed then since u = loa T = 0

ao we have u = h¢ with h

>

0 and so we have ua = h¢a on 80.

Recall the identity (2.4), -=T 6 u-=O (4.2) a, S aS which means that u = T /T is holomorPhic when restricted to a a a leaf (except at the ooint p = T-l(o)) of the foliation. With Tsu

344 these in mind we recall the followincr definition of Lempert [14].

A map f: K ~ 0 is said to be stationary if f is of class c 1 1 2 on K, (ii) fl~= ~--+Dis a proper

Definition (i)

holomorphic map and (iii) there exists a positive function h of class c 1 / 2 defined for ~E8~ such that for all a, the function

fa(~)

=

~h(~)

¢a(f(~)))

extends holomorphically

to~.

The existence of stationary maps will provide us with a foliation and also the u 's. Since 0 is a bounded domain, it a is Kobayashi hyperbolic and extremal discs exist. The existence of stationary maps are then quaranteed if one can show that they are extremal discs.

To see this let q:

~

--+ D be a

holomorphic map with q(o) = f(o) and cr' (o) = Af' (o) ,A > 0. By property (iii) of a stationary maP, the function a

Rei:(fa(~)-qa(~))h(~)¢a(f(~)) =Rei: a

a

(~~-q (~)~h(~)¢a(f(~))

f

a

extends holomorphically to all

~

By strict convexity the

E ~.

left-handed side above is > 0 (with eauality iff f

= a).

Thus we have , at the oriqin 8fa Re L(l-A)~ (o) fa(o) ~ o with eauality iff f

=

a. In particular, takina a to be a con8fa stant map, we aet Re L(l-) ~ (o) fa (o) ~ 0, and so in

general we have A·< 1. Thus a stationary map is extremal. It can be shown easily that a stationary map is a biholomorphic map of T onto its image. tion T

o

We conclude that the exhaus-

f is strictly subharmonic on

is harmonic on~ and soT

f(~) = 1~1

o

~

and u

o

f = loa T

2 by theorem 3.1

o

f

(actu-

ally for m=l, it is not difficult to see this). In other words T is obtained by first constructina the stationary maps (= extremal discs) f and then pullina back the absolute value function on

~

via f.

By examinina the situation of the strictly convex domain carefully, one realizes that the special relationship between T and the Kobayashi metric remains valid if the Monqe-Ampere exhaustion T satisfies,

345 T T

-1

{~}

(o) consists of exactly one point

(4. 3)

is Coo everywhere after blowing up the point {o}

and u = log

T

(4. 4)

satisfies (4.1) on a neighborhood of o.

Let (r!,T) be a complex manifold with a bounded Theorem 4.4 Monae-Arnpere exhaustion satisfyina (4.1), (4.3) and (4.4) then the leaves of the Honae-Amoere foliation of .M* extends across ~

and are extremal discs of the Kobayashi metric of M.

Remark.

The exhaustion

T

restricted to a leaf is then the ~

oull-back of the absolute value function on disc throuqh o.

In particular

T

via an extremal

is real analytic on each leaf

throuah o. We also remark that the conditions on

T

do not characterize strictly convex domains.

in theorem 4.1 However, if we

know in addition that the Monqe-Ampere foliation is holomorphic then we have, Theorem 4.5 .

Let (M,T) be as in theorem 4.4 and assume in

addition that the Monae-Amoere foliation is holomorphic then M is biholomorphic to a bounded strictly pseudo-convex circular domain in ~m. A domain D in ~m is circular if z complex numbers

A with !AI

~

1.

E

D then

AZ

E

D for all

For details of theorems 4.4

and 4 . 5 see [ 18 J , [ 19 J , [ 2 0 J , and [ 3 7 J • Returnina now to the proof of theorem 3.1 in the previous section where we had established the holomorphicity of the Honqe-Ampere foliation in the case of bounded exhaustion under the additional assumotion that under the C

00

T

is real analytic.

However,

assumption, what we had shown in §3, clearly

implv that the hypothesis of thereom 4.4 is satisfied, thus the leaves of the foliation extend across o as extremal discs and T is real analytic (hence so is !1-;J Z II 2 ) when restricted to each leaf. Since 1!8zl! 2 vanishes up to infinite order at~, it vanishes identically on each leaf.

Thus Z is holomorphic,

completing the proof of theorem 3.1. Notice that in theorem 4.5 (unlike theorem 3.1) the exhaustion T is not assumed to be of class C at the origin, 00

346

therefore holomorPhicity of the foliation has to be assumed. A bounded circular domain D in ~m can always be defined by D = {z E ~mjT(z) < 1} where T(z) = ea jzj 2 and the function q is constant alona each complex line through the oriain, i.e., n-1 ~P • Thus T is not smooth at the oriain

q is a function on

unless D is biholomorphic to the ball. §5

Intrinsic metrics in the bounded case

From the results of the Previous section, we know that the Honcre-Ampere exhaustion is intrinsically related to the Kobayashi metric, we shall further exploit this relationship in this section. Let T be a non-negative

~~onqe-Ampere

sarily an exhaustion) which is bounded. erality assumed that supT

=

1.

function (not necesWithout loss of aen-

In section 2, we have studied

the geometry of the Kahler metric h = Tasdz a dz B•

Consider now

also the hermitian metric g = (1-T)

-2

h

(5 .1)

The followina is a consequence of theorem 2.1, Theorem 5.1 .

tenor of cr and h.

Let K

a

ture of the metric a. (i)

Sq = Sh

(ii)

K

cr

(Z,Z)

and Sh respectively the Ricci

be the holomorPhic sectional curva-

Then on M* we have 2m (1-T)q = Sh

l!azl! 2

-

s0

Denote by

2mT/l-T < -

2mh l-T

1

in particular

o,

-1.

By a direct computation we have

~-Sy]..!V where ively.

~

=

(l-T)- 2 R - - + 2(1-T)- 4 T -[-T- + T 2 u -] By]..!V

Sy

]..!V

]..!V

- - and R - - are the curvature of a and h respect-

SyJ.lV

BYJ.lV

From this we qet

4

(1-T) -

- - 1.

Specialize to the case where M is a strictly convex domain with sMooth boundary in ~m or, more crenerally where M

347

satisfies the hypothesis of theorem 4.4, then the leaves of the foliation extend through the fixed point and are in fact extremal discs of the Kobayashi metric. The metric g = iaaT/(l-T) 2 pull back to the unit discs in ~ via the extremal maps is the Poincare metric

on~.

Thus it can be ~e

thouaht of as a generalization of the Poincare metric. have the

followin~

theorem concernina isometries of this

metric. Theorem 5.1 .

Let (M,T) and (M,T) be complex

fyina the hypothesis of theorem 4.4.

satis-

Let <j>: H-+ M be an

isometry of the correspondina metrics a and defined by (5.1).

manif~lds

q

respectively

If ¢ preserves the correspondina Monoe-

Ampere foliations then ¢ is biholomorphic or antibiholomorphic. The above theorem was proved in [16] for strictly convex domains with the exhaustion obtained in theorem 4.1.

The

?roof in the more general situation above is analoaous.

Basi-

cally the assumptions guaranteed that ¢ maps leaves onto leaves and is biholomorphic (or anti-biholomorphic) on each leaf.

Thus X¢

=

0 if X is tanoent to a leaf.

direction at the point

~

Since for each

there is a leaf through

direction, we have a¢ ·= o at ~·

~

in that

By an argument similar to the

aroument at the end of section 4 we see that 8¢ actually vanishes to infinite order at o; But lla¢11 2 is real analytic on each leaf because both¢ and the metrics are.

Hence IIY¢11 2

vanishes identically. For manifolds (M,T) and (M,T) in theorem 5.1, let Mr and Mr be respectively the Kobayashi balls of radius r from o and o. (cf.

The following is an easy conseauence of theorem 5.1. [16]).

Corollary 5.1

With the notations as above, then every biho-

lomorphic map ¢: Mr

~

Mr is the restriction of a biholomor-

phic map from M onto M. The corollary above was first observed by Bland, Duchamp and Kalka [5] where M and M are strictly convex domains.

This

theorem can be thouaht of as a "weak form" of uniaue analytic continuation for the complex homoaeneous Monae-Arnpere eauation.

348 For applications of this result we refer to [5] and [16]. Consider now a bounded strictly convex domain D with smooth boundary in ~m.

Denote by oD and CD respectively the

Kobayashi and Caratheodory distances from a fixed point p c D. Let T be the unique Honc:re-Arnpere exhaustion centered at p.

X

Take an extremal disc f: throuqh p. f(8D)

c

D of

~

the Kobayashi metric

Construct a complex vector bundle E over

aD by assigning to each point z 0 c f(8D) the complex m

tanoent space to D at z 0

,

i.e., E

zo

{zJ Iu (z ) (za-za) a=la o o

=

O}

where u

(log T)a. Since an extremal disc is a stationary a map, the vector bundle E extends to a holomorphic vector bundle TI:E

~

f(~)

over f

(~).

By the convexity of 8D, the

domain D lies entirely on one side of each tanoent space to the boundary, thus D

c

E. The restriction TijD: D ~ f(~) is

holomorphic and surjective onto the extremal disc. bounded holomorphic function on

f(~)

Thus every

extends to a holomorphic

function on D with the same bound, i.e.,

f(~)

is also an

extremal disc for the Caratheodory metric, therefore CD

= oD.

More generally for any convex domain (bounded or unbounded and not necessarily with smooth boundary), bv takinq an exhaustion with strictly convex domains with smooth boundaries, a limit argument shows that the same is true. Theorem 5.2 .

Let D be a convex doMain in ~m then the Cara-

theodory metric and the Kobayashi metric of D are identically. The above theorem was proved by Lempert [15] and independently by Royden-T•Tonq [21].

For the more aeneral situation of

manifolds satisfyino the hypothesis of theorem 4.4, one expects to have a bounded A > 0 so that it is not clear how A deoends on

~~.

CD~

AoD on D, however,

A clear understandina of

this constant will be extremely helpful on the problem of existence of bounded holomorphic functions.

349 Appendix ~e

collect here, for the convenience of the reader, some

commutation formulas that were used in the computation in §2. (I)

For 1-tensors ¢.l, k..,.. J

{II)

s Rik] ¢s

- ¢ l, . ..,..k J

For 2-tensors ¢i],k'f

¢i],k"f

s RiH ¢SJ

-

t Rjtk ¢it

s Rik"f ¢SJ

-

s Rik] ¢s"f

¢k-.'0 J,lx, ¢ l. ] ..,.., k" )(,

s s Rkj"f ¢si + Rik"f ¢ sj (III)

For 3-tensors

Rs. _ lpq

,~, '¥

s]k -

Rs ikq

,~, '¥

SJP

¢ - +rs ¢ - - r~lp ¢ s]kq + r~lp r~ JO stk io sika

+

rl~k

r!Jq ¢ stp

¢i]p,qk

¢i]k,qp

+ R!]qp ¢itk

- r~]q

- r~lp ¢s]kq + r sik ¢s]pq

-

t r~ Rjqk ¢itp + r~]q lp ¢stk

s rik ¢stp

For the special case in §2 where is the metric, we have

T

is a function and

T

-

pa

350

-

where

T~lp

RI]qp

T

~-

SJkq

+ TS T ~- + ik SJpq

f~Jq

TS T ip Stk

Titk

s £ s = -TJSqpTik + Tsp TJ£q Tik

t s £ s Rjqk Titp = TJSqk Tip- Tsk TJ£q Tip t s Tjq Tip Tstk t s s £ -Tjq Tik Tstp= -TJ£qTikTsp• from which we get the following formula,

351

References [1]

Ahlfors, L., The theory of meromorphic curves, Acta Soc. Fenn. Nova Ser. A i (4), (1941).

[2]

Alexander, H., Projective Capacity, Recent Development in Several Complex Variables, edited by J. Fornaess, Ann. Math Studies 100 (1981).

[3]

Bedford, E., and Kalka, M., Foliations and complex Mange-Ampere equations, Cornrn. Pure and Appl. Math, 30, 543-571 (1977).

[4]

Bedford, E. and Taylor, B.A., Variational properties of the complex Mange-Ampere equation II, intrinsic norms, Am . J . Hath 1 0 1 , 1131-116 6 ( 19 77 ) .

[5]

Bland, J., Duchamp, T. and Kalka,~~., On thfi automorphism group of strictly convex domains in ~ , preprint, (1984).

[6]

Burns, D., Curvature of ~onae-Amoere foliations and bolic·manifolds, Ann. of Math. 115, 349-373 (1982).

[7]

Carlson, J., and Griffiths, P., A defect relation for equidimensional holomorphic mappings between alqebraic varieties, Ann. of Math. ~' 557-584. (1972).

[8]

Chern,

[9]

Chern, S.S., Holomorphic curves in the plane, Diff. Geometry in honor of K. Yano, Kinokuniya, Tokyo, 7394 (1972).

[10]

Foote, R., Curvature estimates for Mange-Ampere foliations, Univ. of Michigan Thesis (1983).

[11]

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