Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis (Encyclopaedia of Mathematical Sciences)

H. Grauert . Th. Peternell - R. Remmert (Eds.) /I n/

Several Complex Variables VII Sheaf-Theoretical Methods in Complex Analysis

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Encyclopaedia of Mathematical Sciences Volume 74

Editor-in-Chief:

RX Gamkrelidze

Contents Introduction

Chapter

I. Local Theory of Complex R. Remmert 7

Spaces

Chapter II. Differential Calculus, Holomorphic Maps and Linear Structures on Complex Spaces Th. Petemell and R. Remmert 97 Chapter

Chapter

Chapter

IV. Seminormal Complex Spaces G. Dethloff and H. Grauert 183

V. Pseudoconvexity,

Chapter

III. Cohomology Th. Peternell 145

the Levi Problem and Vanishing Th. Petemell 221

VI. Theory of q-Convexity H. Grauert 259 /

Chapter

and q-Concavity

VII. Modifications Th. Peternell 285

Theorems

Chapter VIII. Cycle Spaces F. Campana and Th. Petemell 319 Chapter

IX. Extension of Analytic Objects H. Grauert and R. Remmert 351 Author Index 361 Subject Index 363

Introduction Of making many books there is no end; and much study is a weariness of the flesh. Eccl. 12.12. 1. In the beginning Riemann created the surfaces. The periods of integrals of abelian differentials on a compact surface of genus g immediately attach a gdimensional complex torus to X. If g 2 2, the moduli space of X depends on 3g - 3 complex parameters. Thus problems in one complex variable lead, from the very beginning, to studies in several complex variables. Complex tori and moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex coordinates zi, . . . , z,; holomorphic functions are, locally, those functions which are holomorphic in these coordinates. In the second half of the 19’h century, classical algebraic geometry was born in Italy. The objects are sets of common zeros of polynomials. Such sets are of finite dimension, but may have singularities forming a closed subset of lower dimension; outside of the singular locus these zero sets are complex manifolds. Even if one wants to study complex manifolds only, singularities do occur immediately: The fibers of holomorphic maps X + Y between complex manifolds are analytic sets in X, i.e. closed subsets which are, locally, zero sets of finitely many holomorphic functions. Analytic sets are complex manifolds outside of their singular locus only: A simple example of a fiber with singularity is the fiber through 0 E (I? of the function f(z,, z2) := zlzz resp. z: - z:. Important classical examples of complex manifolds with singularities are quotients of complex manifolds, e.g. quotients of (c* by finite subgroups of SL,((lZ). For the group G := {(i

3, (-i

-i)}

the orbit space OZ*/G is

isomorphic to the affine surface F in (c3 given by z$ - z1z2, the orbit projection (lZ* + F is a 2-sheeted covering with ramification at 0 E (c* only. Hence F = c*/G is not a topological manifold around 0 E F. This is true whenever the origin of (c* is the only fixed point of the acting group. All these remarks show that complex manifolds cannot be studied successfully without studying more general objects. They are called reduced complex spaces and were introduced by H. Cartan and J-P. Serre. This was the state of the art in the late fifties. But soon reduced spaces turned out to be not general enough for many reasons. We consider a simple example. Take an analytic set A in a domain U of (c” and a holomorphic function f in U such that f JA has certain properties. Can one find a holomorphic function f in a neighborhood

2

Introduction

V c U of A such that f/A = f/A and that the properties of flA are conserved by f*? This is sometimes possible by the following step-wise construction: Let A be the zero set of one holomorphic function g which vanishes of first order. Then we try to construct a convergent sequence f, of holomorphic functions on V, A c V c U, such that f,+i = fvmodg’+‘, where f0 := J. The limit function f* may have the requested properties. This procedure suggests to form all residue rings of local holomorphic functions on U modulo the ideals generated by g’+l. This family leads to a so-called sheaf of rings over U which is zero outside of A. We denote the restriction of this sheaf to A by oAV. Sections in (?4, are called again holomorphic functions on A. For v = 0 these sections are just the ordinary holomorphic functions on the reduced complex space A, i.e. caO = 6&. For v > 0 the sections can be considered as power series segments in g with coefficients holomorphic on A in the ordinary sense. This is expressed geometrically by saying that A, with the new holomorphic functions, is a complex space which is infinitesimally thicker than A. We call (A, 1!9~“)the v-th infinitesimal neighborhood of A. The sheaf oAV has, at all points of A, nilpotent germs #O. This phenomenon cannot occur for reduced complex spaces. Infinitesimal neighborhoods are the simplest examples of not reduced complex spaces. The topic of this book is the theory of complex spaces with nilpotent elements. As indicated we need sheaves already for the definition. Sheaf theory provides the indispensable language to translate into geometric terms the basic notions of Commutative Algebra and to globalize them. 2. Sheaves conquered and revolutionized Complex Analysis in the early fifties. Most important are analytic sheaves, i.e. sheaves 9’ of modules over the structure sheaf 0, of germs of holomorphic functions on a complex space X. Every stalk Y,, x E X, is a module over the local algebra 0x,x, the elements of Yx are the germs sx of sections s in Y around x. An analytic sheaf 9’ is called locally finite, if every point of X has a neighborhood U with finitely many sections si, . . . , sP E Y(U) which generate all stalks Yx, x E U. This condition gives local ties between stalks. For the calculus of analytic sheaves it is important to know when kernels of sheaf homomorphisms are again locally finite. This is not true in general but it certainly holds for locally relationally finite sheaves, i.e. sheaves 9’ having the property that for every finite system of sections si, . . . , sP E Y(U) the kernel of the attached sheaf homomorphism 0; + Y;, (fi, . . . , f,)~ cfisi, is locally finite. Locally finite and locally relationally finite analytic sheaves are called coherent. Such sheaves are, around every point x E X, determined by the stalk Yx; this is, in a weak sense, a substitute for the principle of analytic continuation. Trivial examples of coherent analytic sheaves are all sheaves Y on (c”, where 9” = 0 for x # 0 and yb is a finite dimensional Gvectorspace (skyscraper sheaves). It is a non-trivial theorem of Oka that all structure sheaves Ogn are coherent. Now a rigorous definition of a complex space is easily obtained: A Hausdorff space X, equipped with a “structure sheaf” c?, of local c-algebras, is called

Introduction

3

a complex space, if (X, 0,) is, locally, always isomorphic to a “model space” (A, 0’) of the following kind: A is an analytic set in a domain U of (c”, n E IN, and there is a locally finite analytic sheaf of ideals in the sheaf O,, such that 9 = 0, on U\A and O,., = (Q/9)1,4. In the early fifties complex spaces were defined by Behnke and Stein in the spirit of Riemann: Their model spaces are analytically branched finite coverings of domains U in Cc”.In this approach the structure sheaf Ox is given by those continuous functions which are holomorphic outside of the branching locus in the local coordinates coming from U. It is known that Behnke-Stein spaces are normal complex spaces. A complex space X, even if a manifold, may not have a countable topology. If the topology is countable the space admits a triangulation with its singular locus as subcomplex. Hence the topological dimension is well defined at every point: it is always even, half of it is called the complex dimension. Furthermore all complex spaces are locally retractible by deformation to a point, in particular all local homotopy groups vanish and universal coverings always exist. Sheaf theory is a powerful tool to pass from local to global properties. The appropriate language is provided by cohomology. This theory assigns to every sheaf 9’ of abelian groups on an arbitrary topological space X so called cohomology groups Hq(X, Y), q E IN, which are abelian. There are many cohomology theories, for our purposes it suffices to use Tech-theory. For analytic sheaves all cohomology groups are (C-vector spaces. These spaces are used to obtain important results which, at first glance, have no connection with cohomology. E.g. vanishing of first cohomology groups implies, via the long exact cohomology sequence, the existence of global geometric objects. For Stein spaces, which are generalizations of domains of holomorphy over c”, all higher cohomology groups with coefficients in coherent sheaves vanish (Theorem B), this immediately yields the existence of global meromorphic functions with prescribed poles (Mittag-Leffler, Cousin I). If X is compact all cohomology groups with coefficients in coherent sheaves are finite dimensional (C-vector spaces (Theo&me de Finitude). 3. Stein spaces are the most important non compact complex spaces. Historically they were defined by postulating a wealth of holomorphic functions and are characterized by Theorem B. They can also be characterized by differential geometric properties of convexity, more precisely by the Levi-form of exhaustion functions. Stein spaces are exactly the l-complete complex spaces, i.e. all eigenvalues of the Levi-form are positive. Natural generalizations are the qcomplete and q-convex spaces, where at most q-l eigenvalues of the Levi-form may be negative or zero. The counterpart of q-convexity is q-concavity. For all such spaces finiteness and vanishing theorems hold for cohomology groups in certain ranges, such theorem generalize as well the finiteness theorems for compact spaces as the Theorem B for Stein spaces. Most important examples of convex/concave spaces are complements of analytic sets in compact complex spaces. If A, is a d-dimensional connected complex submanifold of the

4

Introduction

n-dimensional projective space lPn then the complement lP”\Ad is (n - d)-convex and (d + I)-concave. The notion of convexity is also basic in the theory of holomorphic vector bundles on compact spaces. A vector bundle is called q-negative, if its zero section has arbitrarily small relatively compact q-convex neighborhoods. If q = 1 the bundle is just called negative; duals of negative bundles are called positive or ample. The Andreotti-Grauert Finiteness Theorem can be used to obtain Vanishing Theorems for cohomology groups with coefficients in negative or positive vector bundles. As a consequence compact spaces carrying ample vector bundles are projective-algebraic. For normal compact spaces the notion of a Hodge metric can be defined. Spaces with such a metric always have negative line bundles, hence normal Hodge spaces are projective-algebraic. For a complex torus a Hodge metric exists if and only if Riemann’s period relations are fulfilled. Serre duality holds for q-convex complex manifolds if the cohomology groups under consideration have finite dimension. For compact spaces, i.e. Oconvex spaces, duality is true in every dimension. For concave spaces the field of meromorphic function is always algebraic. For details on all these results see Chapters V and VI. 4. Whenever there is given a complex space X and an equivalence relation R on X the quotient space X/R is a well defined ringed space. It is natural to ask for conditions on R such that X/R is a complex space. To be more precise let X be normal and of dimension n and assume that R decomposes X into analytic sets of generic dimension d. Then R is called an analytic decomposition of X if its graph is an analytic set in the product space X x X. Under certain additional conditions the quotient X/R is an (n - d)-dimensional normal complex space and the projection X + X/R is holomorphic. In important cases the analytic graph is a decomposition of X only outside of a nowhere dense analytic “polar” set. Then the limit fibers of generic fibers are all still pure d-dimensional and we get a “fibration” 4 in X whose fibers may intersect. We call C$a meromorphic decomposition resp. a meromorphic equivalence relation of X. A simple regularity condition guarantees that, by replacing polar points by the fiber points through them, one obtains a proper modification r? of X such that 4 lifts to a true holomorphic fibration 6 of r? with d-dimensional fibers. The quotient Q := r?/J is called the quotient of X by the meromorphic equivalence relation 4, this space Q is always normal. Simple examples are obtained by holomorphic actions of complex Lie groups; e.g. if (c* acts homothetically on (c”, the family 4 consists of all complex lines through 0 and we have Q = IE’“‘,-r. The theory of analytic decompositions is set-theoretic and not ideal-theoretic. An ideal-theoretic approach seems to be possible only for “proper” decompositions; then the theorem of coherence of image sheaves can be applied. The theory of decomposition is discussed in Chapter VI, @2-4, and in Chapter V, 91.

Introduction

5

5. There is a kind of surgery of complex space called proper modifications. Roughly speaking one replaces a nowhere dense closed complex subspace A of X by another complex space B in such a way that Y := (X\A) u B becomes a complex space with a proper holomorphic map n : Y + X which map Y\B biholomorphically onto X\A. If X, Y are normal, the fields of meromorphic functions are isomorphic, we say that X and Y are bimeromorphically equivalent. Bimeromorphic geometry, i.e. the theory of complex spaces modulo bimeromorphic equivalence, is the topic of Chapter VII. Classical is the blowing up of points: e.g. replace 0 E (l2” by the projective space lPml of all line directions at 0. This procedure can be generalized: Every closed complex subspace A can be blown up in a natural way along A (monoidal transformation). The most important applications of such modifications are the elimination of indeterminancies of meromorphic maps and the desingularization of reduced complex spaces (Hironaka). General proper modifications are not too far away from blow-ups: they always are dominated by a locally finite sequence of blow-ups (Hironaka’s Chow lemma). 6. For every compact complex space X the set of all closed complex subspaces carries a natural complex structure. This complex space is called the Douady space of X, its analogon in algebraic geometry is the Hilbert scheme. In contrast the Barlet space or cycle space of X (supposed now to be reduced) parametrizes all finite linear combinations (cycles) cnvZ,,, n, E IN, where Z, is an irreducible analytic set in X; here the corresponding algebraic object is the Chow scheme. Douady and Barlet spaces are discussed in Chapter VII. They play an important role in the theory of compact complex spaces, e.g. the existence of the Douady space implies easily that the holomorphic automorphisms of every compact space form a complex transformation group. The global structure of cycle spaces is best understood for spaces which are bimeromorphically equivalent to compact Klhler manifolds; then the components of the cycle spaces are compact. It is also remarkable that convexity properties of X are reflected in its cycle space: If X is q-complete then the space of (q - l)-dimensional cycles is a Stein space. The problem of extending analytic sets into analytic sets of at most the same dimension was initiated in the years 1934-1953, later on growth conditions were used. In the sixties coherent sheaves were first extended into isolated points and then into q-concave smooth boundary points. In order to obtain sufficient conditions for extendability gap sheaves were invented. An important application is the Hartogs continuation theorem for meromorphic maps, cf. Chapter IX. It is our pleasure to thank Prof. J. Peetre for reading carefully the original manuscript and for his extensive linguistic advice which improved the text considerably. We also express our sincere thanks to Springer Verlag for its patience.

Chapter I

Local Theory of Complex Spaces R. Remmert

Contents Introduction

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

10

$1. Local Weierstrass Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Division Theorem and Preparation Theorem . . . . . . . 2. Structure of the Algebra of Convergent Power Series . 3. The Category of Analytic Algebras . . . . . . . . . . . . . . . 4. Finite and Quasi-Finite Modules and Homomorphisms 5. Closedness of Submodules . . . . . . . . . . . . . . . . . . . . . . . . 6. A Generalized Division Theorem . . . . . . . . . . . . . . . . . 7. Finite Extensions of Analytic Algebras . . . . . . . . . . . . 8. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

11 11 12 13 14 16 17 18 19

. $2. Presheaves, Sheaves and (C-ringed Spaces 1. Presheaves and Sheaves . . . . . . . . . . . . . . . 2. Etale Spaces and Sheaves . . . . . . . . . . . . . 3. Sheaves of Modules over a Sheaf of Rings 4. Image Sheaves and Inverse Image Sheaves 5. The Category of (L-ringed Spaces . . . . . . .

. . . . .

. . . . . .

. . . . . .

20 20 21 22 23 24

.. .. . .. .. ..

.. .. . .. .. ..

26 26 27 28 29 30 30

. . . . . .

. . . .

.. .. .. .. ..

.

0 3. The Concept of Complex Space . . . . . . . . . . . . . . . . 1. Complex Model Spaces . . . . . . . . . . . . . . . . . . . 2. Complex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Holomorphic Maps . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . . .. . . .. . . .. . 4. A Gluing Device 5. Analyticity of Image and Inverse Image Sheaves 6. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . .

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

$4. General Theory of Complex Spaces ..................... 1. Closed Complex Subspaces .......................... 2. Factorization of Holomorphic Maps .................. .......................... 3. Anti-Equivalence Principle 4. Embeddings and Embedding Dimension. Jacobi Criterion 5. Analvtic and Analvticallv Constructible Sets . . . . . . . 2

~2

. . . . .

.. .

.

31 31 32 33 34 35

8

R. Remmert

5 5. Direct Products, Kernels and Fiber Products 1. Direct Products .......................... 2. Kernels ................................. 3. Fiber Products. Graph Lemma .............

31 31 38 39

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

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

§ 7. Coherence Theorems ................................ ........................... 1. Weierstrass Projections ................................. 2. TheoremofOka 3. The Sheaf of Meromorphic Functions and the Sheaf of ................................... Normalization 4. Locally Free Sheaves ............................. 5. Coherence of Torsion Modules ..................... 6. Weierstrass Spaces and Weierstrass Algebras .........

. .. . .

9 6. Calculus of Coherent Sheaves ................. 1. Finite Sheaves. Relationally Finite Sheaves .... 2. Coherent Sheaves ......................... 3. Yoga of Coherent Sheaves .................. ....................... 4. Extension Principle

9 8. Finite Mapping Theorem, Riickert Nullstellensatz Spectra ................................................ .............................. 1. Finite Mapping Theorem ................................ 2. Riickert Nullstellensatz 3. Applications ......................................... ............................. 4. Open and Finite Mappings ...................................... 5. Analytic Spectra

. . . .

..

44 45 45

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

46 48 49 50

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

51 51 52 53 54 55

and Analytic

56 56 59

4 9. Coherence of the Ideal Sheaf of an Analytic Set .............. ............................... 1. Theorem of Oka-Cartan ................................ 2. The Reduction Functor 3. Active Germs, Thinness and Torsion Modules for Arbitrary Complex Spaces ...................................... $10. Dimension Theory ......................... ......... 1. Analytic and Algebraic Dimension .......................... 2. Active Lemma .. 3. Invariance of Dimension. Open Mappings 4. Convenient Coordinates. Purity of Dimension 5. Smooth Points and Singular Locus ........ ............................. 0 11. Miscellanea 1. Homological Codimension. Syzygy Theorem 2. Analyticity of the Sets S,JsP) ............. 3. The Defect Sets 0,(,4p) .................. 4. Cohen-Macaulay Spaces ................ 5. Noether Property ......................

40 40 41 42 44

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

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

60 .. .. .. .. .. ..

.. ,.

61 61 62 63 64 65 66 66 68 69 70 72

1. Local

Theory

of Complex

Spaces

9

9:12. Analytic Coverings ........................................ 1. Coverings and Integral Dependence ........................ 2. Examples of Coverings ................................... 3. Weierstrass Coverings ................................... 4. Local Embedding Lemma ................................ 5. Existence Theorem for Coverings. Riemann’s Extension Theorem ..............................................

79

9 13. Normal Complex Spaces ................................... 1. General Remarks ....................................... 2. Criteria for Normality ................................... 3. TheoremofCartan ...................................... 4. Determinantal Spaces. Segre Cores ........................ 5. Divisor Class Groups and Factoriality .....................

80 80 81 82 83 84

9 14. Normalization ............................................ 1. Theorem of Cartan-Oka ................................. 2. Normalization of Reduced Spaces ......................... 3. Irreducible Spaces. Global Decomposition .................. 4. Historical Notes ........................................

85 86 87 89 90

0 15. Semi-Normalization ....................................... 1. Function-theoretic Characterization of 6 ................... 2. Semi-Normalization ..................................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 91 94

74 74 76 77 78

10

R. Remmert

Introduction Da es darauf ankommt, Begriffe auf Begriffe zu haufen, so wird es gut sein, so viele Begriffe als miiglich in ein Zeichen zusammenzuhaufen. Denn hat man dann ein fiir alle Ma1 den Sinn des Begriffes ergriindet, so wird der sinnliche Anblick des Zeichens das ganze Rasonnement ersetzen, das man fri.iher bei jeder Gelegenheit wieder von vorn anfangen musste.(Variation of a sentenceof C.G.I. Jacobi) A fundamental tenet of contemporary Complex Analysis is that geometric properties of complex spacesand algebraic properties of their structure sheaves are living in happy symbiosis. This introductory chapter is a rambling through basic notions and results of Local Complex Analysis based on local function theory, local algebra and sheaves. There are many advantages to develop the theory in a general context. However, as in algebraic geometry, one has to burden oneself with a considerable load of technical luggage. Sheaves are a powerful and versatile tool, they provide the natural way of keeping track of continuous variations of local algebraic data on topological spaces.The revolutionary slogan of the fifties “il faut faisceautiser” is a truism long since. In focus are coherent analytic sheaves.We discussfour fundamental results: -

Coherence of Structure Sheaves in Q7, Finite Mapping Theorem in 4 8, Coherence of Ideal Sheaves in Q9, Coherence of Normalization Sheaves in $14

All local function theory originates from the Weierstrass Preparation and Division Theorems. These theorems, which prepare us so well, form the cornerstones of 9:1. In sections 2 to 6 we introduce and discussbasic notions. Dimension theory is developed in $10, while 0 11 is devoted to homological codimension, Cohen-Macaulay spaces,Noether property and analytic spectra. From Riemann’s point of view pure dimensional reduced complex spaces look locally like analytically branched coverings of domains in Cc”.Such coverings are finite holomorphic maps which are locally biholomorphic almost everywhere. In 5 12 we study such coverings. Section 13 to 15 are dealing with normal spacesand (semi-)normalizations. Local Theory of Complex spaces is by now a well understood and rather elegant topic which has been polished by many mathematicians. One may wonder whether K. Oka still would write: Le cas de plusieurs variables nous apparait commeun pays montagneux, trds escarpt.

1. Local

Theory

of Complex

11

Spaces

5 1. Local Weierstrass Theory All local function theory originates from the famous Weierstrass Preparation Theorem. This theorem expresses the fundamental fact that the zero set of a holomorphic function displays, locally in suitable coordinates, an “algebraic” and thus “finite” character. Thanks to this theorem one can obtain many results by induction on the number of complex variables, this procedure is sometimes called the “one variable at a time” approach. Let zi, . . . . z, denote complex coordinates in c”. We denote by 0, = C:(Z 1, . . . . z,,} the c-algebra of convergent power series around 0 E Cc”; the elements of 0, are called germs of holomorphic functions at 0. We write 0; for (C{zi,..., z.-i}. In what follows z, will be a priviledged variable, and we often write w for z,. We consider 0; and the polynomial ring cO;[w] as subalgebras of 0,. The degree of r E O&[w] in w is denoted by deg r. 1. Division Theorem and Preparation Theorem. For polynomial rings there is the powerful (Euclidean) division algorithm: If g = go + gi w + ... + gr,wb E cOb[w] is such that gb(0) # 0, then to every polynomial f E ob[w] there exist uniquely determined polynomials q, r E 0b[w] such that f = qg + I and deg r < b. This division algorithm can be generalized to convergent power series in w. We say that an element g E 0, has order b E IN in w if

g = f&W”,

gv E 6$),

go(o) = ” ’ = &l(o)

= 0, gb(O) # 0.

0

Weierstrass Division Theorem 1.1. If g E B. has order b in w then for every germ f E 0, there exists a germ q E 0, and a polynomial r E Ob[w] such that f = qg + r

and

degr -K b.

(*)

The elements q, r are uniquely determined by f.

For a simple proof using an appropriate Banach-algebra in 8, see [Gas], 40-41. This proof works for all ground fields k with a complete valuation. - The decomposition (*) gives rise to the so-called Weierstrass map @O-+6%bT

fH(r0,...,rb-1)9

induced by g; here r,, . . . , rbel are the coefficients of r. We put on record: Proposition 1.2. If g E Lo, has order b in w, the Weierstrass @A-module epimorphism with kernel 0,g.

map Lo, + 06” is an

A Weierstrass polynomial o (in w) over 0; is a polynomial w := Wb + a, wb-l + ... + ab E d$,[w], Such polynomials

U,(o) = ... = U,(o) = 0,

have the property

(1.3) If q E 0, and qo E 0&[w]

then q E Ob[w].

b 2 1.

R. Remmert

12

Proof. Obviously f := qo is the Weierstrass decomposition off with respect to O-E CQ. Now f E Q,[w] also has a decomposition f = 40 + Y in the polynomial ring Q,[w] with respect to o E Ob[w]. Uniqueness yield q = q E Sb[w]. 0

The Division Theorem easily implies the Weierstrass Preparation Theorem 1.4. If g E 0, has order b 2 1 in w, then there exists a uniquely determined Weierstrass polynomial co E Ob[w] of degree b and a unit e E 0, such that g = ew. If g E cOb[w] then e E Ob[w]. Proof. Write wb = qg + r with deg r < b. Then g(0, w) = wb&(w)with .6(O)# 0, hence r(0, w) = 0 and q(0, w) = l/&(w). Thus q is a unit in 0,. Now g = 60 with e := l/q and o := wb - r is the required equation. 0

A most important

corollary is

Proposition 1.5. Let g E (?I0have finite order, and let g = ew (according to the Preparation Theorem). Then the injection Ub[w] + 0, induces a C-algebra isomorphism Ob[w]/&,[w]w + cO,/O,g. In particular w is prime in Ob[w] if and only if it is prime in 0,.

It can be said without exaggeration that all of the coherence theorems in complex analysis trace their roots to the maps described in (1.2) and (1.5). 0 For applications of the Weierstrass theorems one needs germs of finite order in w, i.e. germs such that g(0, w) f 0. This can be arranged by a change of coordinates: For given non-zero germs gl, . . , g1 E 0, there always exist coordinates(z;,...,zA-,,w)inC’withz~=z,+c,w,c,~C, 1 ~v
1.6. The ring O0 is noetherian and factorial.

Proof (induction on n). Noether property: Let n > 0. It suffices to show that all residue rings cO,/cO,g, g E 0,, g # 0, are noetherian. We may assume that g has finite order in w. By (1.2), we have an &-module isomorphism O,/O,g r Cobb. Since, by assumption, 0;” is noetherian, O,,/cO,g is a noetherian &$-module and therefore a noetherian ring. Factoriality: Let n > 0 and let g E 0, be a non unit. By (1.4) we may write g = eo. Since Sb is factorial by induction hypothesis, the polynomial ring Sb[w] is factorial by Gauss’ lemma. Therefore the Weierstrass polynomial w is a product of manic prime polynomials or, . . . , o, E 0; [w]. Then all wj are Weierstrass polynomials. Now (1.5) tells us that all wj are prime elements in 0,. Hence g = em,.... . o1 is a factorization of g into prime factors. 0

I. Local

Theory

of Complex

Spaces

13

Hensel’s Lemma 1.7. Let o = o(.z, w) = wb + a, wbP1 + ... + ub E Ob[w]. Let ~(0, w) = (w - c~)~’ . . . .(w - c,)~~ with different roots cl, . . . , c, E Cc. Then there exist unique manic polynomials ol, . . . , w, E Ob[w] of degree b,, . . . , b, such that cc) =

O,‘...‘W,

and

~~(0, w) = (w - c~)~J, 1 < j I

t.

Sketch of proof (induction on t). Let t > 1. Applying the Preparation Theorem to w E c”&[w - c,] we obtain an equation o = o,e with ol, e E Ob[w - c,], deg cr)i = b,, where oi is a Weierstrass polynomial in w - cl. Now e is a manic polynomial in w of degree b, + ... + b,. Since e(0, w) = n (w - cj)bj, by induction ionic

hypothesis

we get e = I+. . . . . o,, where oj E 0; [w] is

of degree bj such that ~~(0, w) = (w - cj)bj.

Hensel’s Lemma makes algebraically

q

precise a geometrically

clear fact:

If the zero set N of a manic polynomial in w meets the w-axis in t different points pl, . . . , pt, then N is the union of the zero sets N,, . . . , N, of t manic polynomials in w such that pj E Nj. 3. The Category of Analytic Algebras. A local (C-algebra A is called a (local) analytic algebra, if it is isomorphic to a residue class algebra 0,/a, where a # Co, is an ideal in 0,. An analytic algebra is called regulur if it is isomorphic to 9,. Every analytic algebra A is - as (T - vector space - a direct sum A = (IZ @ m,, where mA is the maximal ideal of all non units of A. By applying (1.6), (1.7) and Krull’s Intersection Lemma* we obtain: Proposition more:

1.8. Every analytic algebra A is noetherian and henselian;further-

rni = (0).

fi 1

(C-algebra homomorphisms A -+ B between analytic algebras are called analytic, they are eo ipso local, i.e. map m, into mg. Clearly analytic algebras together with analytic homomorphismsform a category. Using (1.8) we seethat each analytic homomorphismA + B is determined by its values on a system of generators of m,. Analytic homomorphisms are obtained by “substitutions”. A simple argument of convergence shows that for every finite set fi, . . ., fk E m, there exists an analytic homomorphism I++:C{z,, . . . , z,} -+ A such that $(z,) = f,, 1 I K I k. The following “lifting device” is very useful:

* Krull’s Intersection Lemma. Let R be a (commutative) let M be a finite R-module. Then fi (N + m’M) I

= N

local noetherian

for any submodule

ring with maximal

N of M.

ideal m,

I. Local

Theory

of Complex

15

Spaces

m,Kb, + ... + ntKKb4 by assumption, the module M is finite over the subring R := (c + m,Z? of Z?. Dedekind’s Lemma* produces a germ g = wd + c1 wdel + ... + cd E R [w] such that gM = 0. Since ~~(0, . . . , 0, w) E Cc for all j, the germ g E Z? is w-general, say of order b 2 1. Then, by (1.2), there is a K-moduleisomorphism Kb r l?/l?g. Now let ,u: l? + M be an Z?-epimorphism. Due to gM = 0, we get an K-epimorphism (1?/Kg)q + M and hence an K-epimorphism Kbq -+ M.

0

An analytic homomorphism cp: A + B of analytic algebras is called finite if (via cp)the A-module B is finite. Noether’s Finiteness Lemma 1.12. To every analytic algebra A there exists a finite analytic monomorphism cp: C{z,, . . , zd} + A. Proof. Write K, := (G (zl, . . . , z”} and let d be the smallest number such that there exists a finite analytic homomorphism cp: K, + A. If there were a germ g # 0 in K, with cp(g) = 0, consider the induced finite homomorphism (p : K,/K,g + A. By (1.2) there exists a finite homomorphism $: K,-, + K,/K,g. Then (p o $: K,-, + A is finite contrary to the choice of d. Hence cpis injective. 0

The number d occurring in Noether’s Lemma is the dimension of A. Geometrically the Lemma roughly means that every complex space X of dimension d is, locally, a “branched analytic covering” of ad, cf. (12.12). 0 An analytic homomorphism cp: A + B is called quasi-finite, if B is a quasifinite A-module, i.e. dim. B/Bq(m,) < co. This is equivalent to saying that Bq(m,) is an “ideal of definition” in B, i.e. rn; c Bq(m,) for large t. Finite homomorphisms are quasi-finite. Conversely we have Theorem 1.13. Every quasi-finite

analytic homomorphism

A + B is finite.

Proof (cf. [AS], p. 91). By lifting according to (1.9) we can arrange that A = C{zl, . . . . z,}, B = A{w,, . . . . w,}. Then the assertion is clear by (1.11). 0

It is possible to start Weierstrass Theory with the above theorem: it easily yields the Division and the Preparation Theorem. This line of approach is taken in [ENS60/61], Exp. 18. Another direct proof of (1.13), which includes estimates, is given in [Bo67]. The geometry behind (1.13) is that a holomorphic map f: X + Y between complex spaces is already finite at a point x E X if x is an isolated point of the f-fiber over f(x), cf. (8.8). Here is a simple consequence of (1.13): * Dedekind-Lemma. Let R be a subring of a commutative ring S with 1 E S. Let M be an S-module which is finite over R and let s E S. Then there is a manic polynomial g E R[s] such that gM = 0. Proof. Let xl, . . . . x, generate A4 over R. Then sxj = crijxj det (~6, - rij) E R [s] we get gxi = 0 for all i by Cramer’s rule. We attribute this Lemma math. Werke III, p. 93.

to Dedekind,

who

introduced

with

this gadget

rij E R. Putting

in the proof,

g := 0

cf. his Ges.

R. Remmert

16

(1.14) Let cp: A -+ B be analytic, assume mB = Bq(m,). Proof. B is a finite A-module. Since dim c B/Bq(m,) erates this A-module by (1.10). Hence q(A) = B.

Then cp is surjective. = 1, the unit 1 E B gen0

Using (1.14) we easily get a Criterion for Isomorphy 1.15. The map cp: A + B is an isomorphism induced (C-linear maps ‘pj: A/mi -+ B/Bq(m,)i, j 2 1, are bijective.

if all

Proof. Since A/m,,, = (c, surjectivity of cpl means m, = Bq(m,). Thus cp is surjective. Furthermore Ker ‘pj = 0 means Ker cp c m;. Hence Ker cp = 0 by (1.8). 0 In geometric language we have just proved that a holomorphic map f: X + Y already is biholomorphic at a point x, if f induces an isomorphism between all infinitesimal neighborhoods of x E X and f(x) E Y, cf. Chapter II, 3 4.2. For every analytic algebra A the cotangent space m,/mj is a finite dimensional (C-vector space. We call emb A := dim. m,/mi the embedding dimension of A (this notation will become clear in Q4.4). Proposition 1.16. The number emb A is the minimal number of generators of the ideal mA and the smallest integer n 2 0 such that there exists an analytic epimorphism cp: C(z,, . . . , z,,> + A. Proof. The first assertion follows from the Nakayama-Lemma. - If cp is onto, then cp(zi), . . . , cp(z,) generate m,, hence emb A I n. Now let k := emb A and let fi, . . . , fk generate m,. Choose a homomorphism cp: (l{zl, . . . , zk} + A with cp(z,) = f,. This map cp is onto by (1.14). 5. Closedness of Submodules. For every analytic algebra A all (C-vectorspaces A/me, e = 1, 2, . . . , have finite dimension and hence carry a natural topology. The weak topology on A is the coarsest topology on A such that all ((C-linear) residue class maps E,.. A --* A/m’ are continuous. The weak topology on A is a Hausdorff topology satisfying the first axiom of countability. The algebra A provided with this topology is a topological C-algebra* (cf. [AS], p. 31 and 81/82). We equip every A-module Aq, 1 I q < co, with the product topology. Lemma 1.17. Every A-submodule

N of A4 is closed in Aq.

Proof. Take a sequence fj E N with limit e 2 1. Now s,(N) is a closed Gsubvectorspace

cc(f) E E,(N),

i.e.

* Note that the Krull-topology (=m-adic me, e 2 1, form a basis of neighborhoods

f E A. Then lim s,(h) = se(f) for all of A/m’. Therefore

f E 0 (N + m’Aq) = N

(Krull).

0

topology) which is characterized by the fact that the sets of 0 E A, is genuinely finer than the weak topology.

I. Local

Theory

of Complex

17

Spaces

In the case A = 0, a sequence f;. = 1 ayj,,,V,z;l . . . z,‘n E 8, converges in the weak topology to f = c a”, “,z;’ . . . zln E &, if and only if tix+rna!!,), “, = a,,, “, for all n-tuples (v,, . . , v,,) E IN”. The Cauchy-inequalitiel t?n- coefficients of Taylor series tell us that uniform convergence in neighborhoods U of 0 E (c” implies convergence in the weak topology of A. Hence we have (1.18) Let 4 be a sequence of q-tuples of functions holomorphic in an open neighborhood U of 0 E C”. Assume that the fj converge uniformly in U towards a q-tuple f, and assume furthermore that all germs fj,, E 08 belong to an 0, - submodule M of 0:. Then f. E M. This immediately

yields:

Let X be a complex manifold and let 9’ be an analytic subsheaf of a sheaf OR, of global sections in Y is a closed subspace of the space 0$(X) with respect to locally uniform convergence on X.

1 I q < co. Then the space Y(X)

The Closedness Lemma was proved by H. Cartan, [C44], p. 610. 6. A Generalized Division Theorem. A more sophisticated version of the Division Theorem (1.1) is needed to obtain in 9 7 the coherence of structure sheaves. Let 0 E (I? and consider a manic polynomial o E O,[w] of degree b 2 1. Let c r, . . . , c, be the distinct roots of ~(0, w). We set xj := (0, cj) E Q’+l and denote by 6J+j the ring of germs of holomorphic functions at Xj. Every polynomial p E O,[w] determines a germ pxj E Oxj at each point xj, 1 I j I t. Generalized Division Theorem 1.19. For any choice of t germs fj E Uxj there exist t germs qj E CIxj and a polynomial r E 0, [w] of degree
4j"xj

+ rxj

forj=

l,...,t.

The germs qj and the polynomial r are uniquely determined by Proof ([CAS], 53-54). By Hensel’s Lemma w,,...,o,~LO~[w]suchthat

and

coXj = olxj’...‘w,,j

~~(0,

W)

=

fi, . . . , ft and o.

there are manic polynomials (W

-

Cj)*j,

1<j I

t.

The germ ojxj E O,[w - cj] is a Weierstrass polynomial of degree bj. Since ei := n wj is not zero in xi, we have - by (1.1) - equations j#i = q:Ojx, + I where qj E oxjxj,rj E O,[w - cj] and deg rj < bj.

Existence.

fie;f

rj,.,

Setting eij := n

ok, we define

k#i,j

qj:=

q;-

c r,,,eijxj i#j

E oxj

and

r := Crjej E O,[w]

with deg r < b.

Then fj = qjax, + rxj, 1 I j I t, since oj~j = ejxjo,, and eijxjojxj = eix,.

18

R. Remmert

Uniqueness. Assume 0 = qioxj + s,, with qJ E O,,, s E O,[w], 1 I j I t. It suffices to show that s E 0, [w] o. Let s # 0. Then pj := s/(ol . . . . . oj) # 0 and s =

PlOl,

pjml = pjoj

for 2 I j I t, s = pto.

(*)

Now sXj = - qJwxj implies ptx, = -4; and pjxj = - qJ(wj+I . . . . . CO,),! E cOxj, 1 5 j < t. Since Oj~, E Q,[w - cj] is a Weierstrass polynomial the equations (*) and (1.3) successively yield that pl, . . . , pt E cO,[w]. Hence s = p,w E O,,[w]w. 0 We may express (1.19) more elegantly.

We define two maps into the ring

direct sum of the t analytic algebras OXj/OX,x,w,,as follows:

7~:GCwl+ @~xi/~xj~xJ,

P+-+~(P~,

mod oxj),

(so,...)S*-l)H11. *g spwp 4:0; -+@coxjJoxjwxj’ ( > Then ( 1.19) precisely says: (1.20) The map $ is an 0, - module isomorphism. The map x induces a Calgebra isomorphismOO[w]/OO[w]o s Ox,/Ox,o,, @ ... @ Ox,/Ox~x,o,l. We shall apply. (1.20) in the next paragraph and in 9 7.1; see also 4 7.6. 7. Finite Extensions of Analytic Algebras. Analytic algebras admit finite Ccalgebra extensions which are not analytic algebras, e.g. direct sums of rings. Using (1.20) we show that indeed all finite extensions are of this type. Proposition 1.21. If A -+ B is a finite C-algebra homomorphismof an analytic algebra into a C-algebra, then B is a unique direct sum A, 0 .. ’ 0 A, of analytic algebras, 1 5 t < co. If B is an integral domain, then B is an analytic algebra. Proof. The unicity of the summands is clear. In order to construct them we may assume that A = 0,. Let b,, . . . , bk generate B over A. The extended map ACw 1, . . . . %I +B, g-g@,, . . . . bk) is surjective. Every b, annihilates a manic polynomial gk E A[w,], hence we have an epimorphism $: A [w,, . . . , wk]/a + B whereaisgeneratedbyg,,...,g,.IfweshowthatA[w,,...,w,]/a=A1O...O A, with analytic algebras Aj then Ker I/ is a sum of ideals aj c Aj and B is the ring-direct sum of the analytic algebras Aj/aj. We proceed by induction on k. The induction step is routine, the case k = 1 just being (1.20). q

We emphasize an important application. For every reduced analytic algebra A the integral closure A^of A in its quotient ring Q(A) consists of all elements of Q(A) which are integral over A, i.e. annihilate a manic polynomial p E A[w]. The set A^is a C-algebra extension of A; we call A^the normalization ofA. If A^ = A we say that A is normal. Factorial algebras are normal. Proposition 1.22. If the analytic algebra A is an integral domain then A^is the maximal ring extension of A in Q(A) which is a finite A-module. Furthermore A^is an analytic algebra.

I. Local Theory of Complex Spaces

19

Proof. By (1.12) there exists a finite monomorphism K, + A. Then Q(A) is a finite (separable) field extension of Q(K,,). Since K, is normal and noetherian, a well-known theorem of Dedekind-Noether says (E. Noether: Abstrakter Aufbau der ldealtheorie in algebraischen Zahl- und FunktionenkBrpern, Coll. Pap. p. 503) that A^is the maximal ring extension of A in Q(A) which is a finite A-module. Since A^has no zero divisor, A^ is an analytic algebra by (1.21). 0 Every analytic algebra A has (as a noetherian ring) finitely many minimal prime ideals p 1, . . . , pt. If A is reduced we have r\ nj = 0 and the residue maps A + Aj := A/pj extend uniquely to epimorphisms Q(A) + Q(Aj) of the quotient rings; these maps give rise to an isomorphismQ(A) 3 Q(A,) @ ... @ Q(A,) which sends A^onto Alp 1 @ . . . 0 A/p,. By applying (1.22) we obtain: Proposition 1.23. Let A be a reduced analytic algebra with minimal prime idealsp,,..., pt. Then thAnormalizxtion A of A is isomorphic to the direct sumof the t normal algebras Alp,, . . . , A/p,. The normalization A is the maximal ring extension of A in Q(A) which is a finite A-module.

The results of this paragraph are the “stalk versions” of later theorems concerning the analytic spectrum of coherent analytic algebras and the normalization of structure sheaves, cf. 6 8.5 and 9 14.2. 8. Historical Notes. The Preparation Theorem was known to Weierstrass since 1860, though only published in 1886, cf. [W], p. 105. Already in 1893 the theorem is discussed in E. Picard’s Trait6 d’Analyse, vol. 2; this discussion is reproduced 1929 in W.F. Osgood’s Lehrbuch der Funktionentheorie [LF]. In the course of time several proofs were given, among others by F. Hartogs (1908), A. Brill (1910, already 1891 for two variables), W. Wirtinger (1927), H. Spath (1929), H. Cartan (1944), H. Grauert and R. Remmert (1962), C. L. Siegel (1968). For details consult Cartan’s article Sur le thdoremede preparation de Weierstrass in [C66], 875-888, see also [AS], 35-36, and [Si68]. The Division Theorem does not go back to Weierstrass. Till 1968 it was folklore that it was stated and proved, for the first time, in 1929 by H. Splth, cf. [Sp29]. Function theorists had forgotten that, already in 1887, L. Stickelberger in [Sti1887] gave a widely algebraic proof of the Preparation Theorem and deduced from it, by a simple and elegant argument, the Division Theorem. Splth was aware of Stickelberger’s work, lot. cit. p. 95, after him it fell into oblivion; only in 1968 Siegel called attention to it again; cf. [Si68], 5-6, and also [AS], p. 39ff. Algebraization of local function theory started in 1905 with E. Lasker (world champion of chess from 1894 till 1924). In [Las051 he showed that all rings 8, are noetherian and factorial. However, Lasker’s paper remained unknown; even W. Riickert, a student of W. Krull, does not refer to it in his now classical paper [Rii33] finished in 1931 but published only in 1933.* Riickert’s landmark paper * Walter Riickert, 1906-1984. His father was a minister in the government of Baden till 1933. For political reasons, Riickert could not start a career at a university. From 1964-1970 he was President of the Oberschulamt Nordbaden.

20

R. Remmert

is written in the spirit of the upcoming age of modern algebra; his proofs that 0, is noetherian and factorial are the proofs of today. He proudly emphasizes the power of algebraic methods in local function theory: “In dieser Arbeit wird gezeigt, da13 eine sachgemlBe Behandlung nur formale Methoden, also keine funktionentheoretischen Hilfsmittel benotigt” (lot. cit. p. 260). The importance of Ruckert’s work was not recognized at the time; it was more than twenty years later that complex analysts slowly became algebraically minded. Hensel’s lemma is not dealt with by Ri.ickert. It occurs explicitly in local analytic geometry only in [ENS61], Exp. 19, p. 9; its geometric content, however, has been used ever since the days of Weierstrass. There is also a Division Theorem and a Preparation Theorem for differentiable functions. This was conjectured by R. Thorn and proved, in 1962, by B. Malgrange; cf. Seminaire Cartan 1962/63, exp. 11, 12, 13 and 22.

0 2. Presheaves, Sheaves and C-ringed

Spaces

Sheaf Theory is a general tool for handling questions which involve local solutions and global patching. Here we briefly discuss basic facts; standard references are Godement’s [TF], Hirzebruch’s [TMAG] and Serre’s [FAC]. Furthermore we introduce (P-ringed spaces as precursors of complex spaces. 1. Presheavesand Sheaves. Let X denote a topological space and let f be the category of open subsets of X with inclusion maps as morphisms. Suppose that to every set U E t there is associated an abelian group S(U) and that for every pair U, V E t with V c U we have a group homomorphism (restriction map) py”: S(U) + S(V), s H SI I’, such that p{ = idso, and & o p,? = pi whenever W E t and W c I/ Then the family S = {S(U), py”} is called a presheuf (of abelian groups on X): Thus a presheaf is just a couuriunt functor of the category t into the category of abelian groups. - If S’ = {S’(U), p;“} is another presheaf, a presheuf map @: S’ + S is a family { ab”} of group homomorphisms @,: S’(U) + S(U) such that QV 0 pv” = py” o aa whenever V c U. Presheuves on X together with presheaf mapsform a category.

0

A presheaf (S(U), p:} is called a sheaf (of groups) if the following condition is satisfied (Serre’s conditions, [FAC], 200/201): (F) Given an open set U and an open covering (U=} of U and elements s, E S(U,) satisfying s,I U, n U, = sglU, n U, for all ~1,j?, there exists exactly one element s E S(U) such that s[ U, = s, for all a. The definition of a (pre)sheaf is not restricted of the category of abelian groups. In particular, we shall consider (pre)sheaves of sets or of rings (with or without a unit element) etc. Historical Note. (Pre)sheaves occured lirst 1946 in a short paper by J. Leray: L’unneuu d’homologie d’une reprksentution, C.R. Acad. Sci. 222, 1366-68. An

I. Local

Theory

of Complex

Spaces

21

excellent presentation of general sheaf theory was already given in [ENS50/51], Exp. XIV. Sheaves became folklore in 1955/56 with [FAC] and [TMAG]. For a detailed history we refer to Ch. Houzel: Les debuts de la theorie des faisceaux, in the Grundlehren volume Sheaves on Manifolds by M. Kashiwara and P. Schapira, Springer 1990. 2. Etale Spaces and Sheaves. A pair (9,~) consisting of a topological space Y and a local homeomorphism n: Y + X is called an etale space ouer X. The fiber Y: := rc-r(x) is called the stalk of Y at x E X. A continuous map s: U + 9, x-s, with n o s = id is called a section (in Y over U); the value s, is called the germ of s at x. All points in Y? are germs of sections. Often we write Y for (9, n). The etale spaces over X are the objects of a category, the etale morphisms (9, x) +, (Y, 7~‘)are all continuous maps f: Y + Y’ with n = rc’ o f. If Y is an etale space and U is open in X, then the restriction Y; := 71-l (U) c Y is an etale space over U, clearly Y;r = 97 Every etale space 9’ determines a sheaf of sets: Denote by Y(U) the set of all sections s: U + Y and by & the restriction map s)--* sJV. Then (Y(U), pv”> is a presheaf and even a sheaf of sets on X, the so called sheaf of sections in 9. Every presheaf S of sets determines an etale space Y and a presheaf map S + (Y(U), pf}. Take for Y the union of all direct limits 9” = 1% S(U), where U

runs though all open neighborhoods of x E X. There are natural maps p,“: S(U) + 9,. Each s E S(U) induces the map 3: U + Sq x H pts. The family {S(U): U it} g en era t es a topology in Y such that rc: Y +X, n(Yx) := x is a local homeomorphism. Hence (Y, rc) is an etale space. All maps S are sections in Y, therefore the canonical maps S(U) + Y(U), s H S define a presheaf map S + {s”(U), &}. This map is an isomorphism if and only if S is a sheaf. Let us denote by r(Y) the sheaf of sections of an etale space Y and by i’(S) the etale space determined by a sheaf S. We have a canonical sheaf isomorphism S + rf(S). Etale maps Y’ + Y resp. sheaf maps S’ + S induce sheaf maps r(Y) + T(Y) resp. etale maps F(S) -+ f(S) thus giving rise to covariant functors

I? {etale spaces} -+

{sheaves},

I! {sheaves) -+

{etale spaces}.

These functors are inverse to each other (i.e. there are natural isomorphisms of rf and fr’ with the identity). Hence the categories of sheaves of sets and etale spaces over X are canonically equivalent. For two etale spaces (9, rr), (Y’, rr’) over X the subspace of Y x Y’ given by Y xx 9” := {(p, p’) E Y x 9”: rr(p) = n(p’)}

= u (9Tx x 9:) x

(fiber product)

together with the map Y xx Y’ + X, (p, p’)~ z(p) is an etale space. The corresponding sheaf is called the direct sum of Y and Y’. For a sheaf of groups (rings) the stalks of the associated etale space carry group (ring) structures “depending continuously” on the points of X. This is made precise as follows: We call 9’ an etale space of abelian groups, zf each stalk

22

R. Remmert

Yx is an (additive) abelian group such that - the map 9’ xx 9’ + 9, given stalkwise by (sx, t,) H s, - t,, is continuous, - the map X -+ 9, x H O,, is continuous (zero-section). Etale spacesof rings are defined mutatis mutandis: all stalks Y? are commutative rings with unit element l,, addition and multiplication define two continuous maps Y xx Y + 9, both maps x H 0, and x H 1, are continuous. Note that there may be stalks with 1, = 0,. The etale spaces of groups resp. rings over X form a category, morphisms Y + 9’ are those etale morphisms which induce, for each x E X, a homomorphism 9” + 9: of groups resp. rings. The functors r and p give the following Equivalence Principle 2.1. The functor I- from the category of isomorphy classesof sheaves of groups (rings) over X to the category of etale spaces of groups (rings) over X is a covariant equivalence. In all what follows we do not distinguish between an etale space and its sheaf of sections, and we denote both objects by the same letter. Sometimes etale spacesare more convenient, sometimes sheaves are easier to handle. - In the older literature etale spaces were called sheaves whereas sheaves were called canonical presheaves. Sheavescan be constructed by “gluing”, see0 3.4. 3. Sheavesof Modules over a Sheaf of Rings. Let & denote a sheaf of rings on X. A sheaf of abelian groups Y on X is called an &‘-module or d-sheaf, if each stalk Yx is an ,r8,-module such that the map lc4 xx Y + 9, defined stalkwise by (a,, s,) H axsx, is continuous. The category of d-modules is well defined, .d-morphisms being those sheaf maps cp:9” + 9, which induce s$,-linear stalk maps cpx:9; + 9!!, x E X. For every d-module Y on X the support of Y is the set supp Y := {x E x: 9, # O), this set may be, even for complex-analytic sheaves,rather complicated. Constructions of module theory carry over to d-modules. Thus we have &-submodules, sumsof .zI-submodules,finite intersections of &-submodules of Y etc. Especially d-submodules 9 of ~4 are called ideals in &. The product JJ. Y with stalks -0,. Yx is a submodule of 9’. The radical rad 9 with stalks rad ,aX:= {a, E 5$,: ai E 9, for suitable t 2 l} is an ideal in x2; the nilradical Jlr := rad 0, 0 := zero ideal in G?,will play an important role later. For a submodule Y’ of 9’ the set Y/Y’ = u .YJ9?? provided with the quotient topology is an d-module called the quotient sheaf. We have canonical exact sequences0 --* Y(U) + Y(U) + YplY( U). The last map is not surjective in general; this fact is a point of departure for cohomology theory, cf. Chap III. If cp:Y + Y is an d-morphism, the kernel A% cp:= U Ker cpxresp. the image 9~ cp:= u Im cp, is a submodule of Y resp. 5 The cokernel of cp is the

I. Local

quotient sheaf S&h

Theory

of Complex

Spaces

23

cp := ~-/&JR cp.An &-sequence Y’ % Y f 9” is called JZIcp: Y -+ 9 induces two d-exact

exact, if 3~ cp = .%a $. Every &-morphism

sequences

The direct sum 9’ xx y of d-modules with stalks (9 xx F), = Y: 0 ,Kx is an d-module; we write Y 0 5 The definition extends to any finite number of d-modules, in particular we have d-sheaves yp, 1 I p < GO. The tensor product Y Od 9 of &-modules is defined by the family y(U)); there are canonical isomorphisms (9 O.., y),K * Y: O.&x TX, {WJ) CL(“) x E X. Every d-exact sequence Y’ -+ 9 + 9” + 0 gives rise to an exact sequence Y’ & F -+ 9’ C& 9 -+ 9”’ @,, 5 -+ 0 (right-exactness). The sheaf ApSq p E N, is the sheaf @‘9/M, where the stalk JV~ of the sheaf ,&’ is the dxmodule generated by all elements s1 0 . . .@ sp, Si E yI, such that s, = s, for at least one couple p, v, ~1# v. The sheaf X”m,&!?‘, F) of germs of &-morphisms from Y to 9 is the family of all dU{Hom+,(%, %), 6’1 where Hom~~,d Y u , yU) is the d(U)-module morphisms 9” + yU and the restrictions r; are canonically at hand, clearly this is an d-sheaf. Note that F) 2 F

.x5wc,(dfl,

(canonical isomorphism).

Each germ s, E &%K+,(S@, r), defines, by passing to a representative s of s,, an &x-homomorphism 9X + 9I. The induced &x-homomorphism px from %‘uKQ(~, 9), to Hom.,=(yx, ,KJ is, in general, not bijectiue; see however 0 6.4b). 4. Image Sheaves and Inverse Image Sheaves. Let f: X + Y be continuous and Y a sheaf of abelian groups on X. If f(X) n I/ = 0, we put Y(f-‘( V)) := 0. The family

{9(f-l(V)),

pt;},

W c V open in Y, pg := canonical restriction,

is a sheaf of abelian groups on Y. This sheaf is denoted by f,(Y) and called the with 9(f-r(V)). Clearly:

f-image sheaf of 9? We identify f*(Y)(Y)

SUPP f*(Y) = f(Supp 9). Every germ t E f*(Y)fcx, is a germ of a section in Y along the f-fiber through x, hence there is a natural group homomorphism & f,(Y), -+ Y: for y = f(x). A sheaf map cp: 9” + Y induces, via sections, a sheaf map f,(q): f*(y’) -+ f,(Y). Thus f, is a couariant functor of the category of sheaves of abelian groups on X to the category of sheaves of abelian groups on Y (image functor). If g: Y -+ 2 is another continuous map, then g,(j,(9)) = (g of),(Y) and g,(f,(cp)) = (g of),(q) for any d-map cp: 9” + 9. Furthermore the functor f, is left exact: every exact sequence 0 --) Y’ + Y --) Y” --) 0 is transformed to an exact sequence 0 + f*(9’) -+ f*(Y) + f,(y”). The functor f, is not right exact: epimorphisms $: Y + 9” usually are not turned into epimorphisms f,(G):

24

R. Remmert

f,(9) -f,(y”); this precisely is the reason why one has to introduce “higher image sheaves” later. It is of utmost importance for Local Analytic Geometry that the functor f, is exact for finite maps. A continuous map f: X -P Y is called finite if f is closed (i.e. maps closed sets onto closed sets) and if each f-fiber is a finite set. Finite maps between Hausdorff spaces look locally as follows: Let U be a neighborhood off -l(y) = {x1, . . . , x,}. Then there exists an open neighborhood V of y E Y and pairwise disjoint neighborhoods U,, . . . , U, of x1, . . ..x.suCh that h Uj=f-‘(V)

c u and all induced maps4.: Vj + V are finite.

1

Using this description

one can prove easily:

Exactness Lemma 2.2. Let f: X + Y be a finite map. Then for every sheaf 9’ of abelian groups on X there are canonical isomorphisms

The functor f, is exact.

For every sheaf (r, rc) of abelian groups on Y the etale space (fiber product) := X xy 9 = {(x, t) E X x y: f(x) = n(t)} is a sheaf of abelian groups on X with stalks (f-IF), = 9& this sheaf is called the inuerse image sheaf of

f -‘y

9 with respect to f. 5. The Category of (C-ringed Spaces. A topological space X together with a sheaf d = ~4~ of rings on X is called a ringed space.We write (X, &‘x) or just X and call &’ the structure sheaf of X. Sometimes we write 1x1 for the underlying topological space. We denote by R := X x Cc the constant sheaf of fields (l2 on X. If .J$ is a R-module, we call JZIa sheaf of (C-algebras.In important cases R is a submodule of &’ and 1, E R, = (c is the unit element of dx. For such sheaves we identify (c. 1, c &‘x with (c. If, moreover, every stalk &x is a local ring with (unique) maximal ideal m(&.J such that &x = (l2 @ m(&J as (C-vectorspace, we call d a sheaf of local (II-algebras, clearly Supp d = [Xl. A ringed space (X, a) with a structure sheaf of local (C-algebras is called a (C-ringed space. Examples. 1) The sheaf of continuous functions. For any open set U in X the set w(U) of all continuous functions in U is a (C-algebra. For V c U we have the restriction r$‘: %?(V)+ w(V), f H f 1I’. Clearly %?x:= {%?(U), rf } is a sheaf of local (C-algebras called the sheaf of continuousfunctions on X. The space (X, %x) is the prototype of a c-ringed space. 2) The sheaf of holomorphic functions. Let D be a domain in (c”. The set Co(U) of all holomorphic functions in U c D is a (G-algebra, we have restrictions r/: O(U) + 0(V). Obviously &, := {O(U), r,“} is a sheaf of local (C-algebras called the sheaf of holomorphic functions on D. The (C-ringed spaces (D, 0,) are the point of departure of all Complex Analysis. The classical Identitiitssatz says that

I. Local Theory of Complex Spaces

25

6JDis a Hausdorff sheaf (i.e. the etale space is Hausdorff). The sheaf Q, is a subsheaf of local C-algebras of the sheaf gD. Note that wr, is not a Hausdorff sheaf if n > 0. 3) One point C-ringed spaces are all pairs (x, -01) where x is a point and ~2 a local C-algebra. If d is artinian, these spaces are complex spaces, e.g. (0, C(z)/z”0(z)) with 0 E C is the n-fold point. 0 In order to motivate the subtle notion of a morphism between C-ringed spaces we first consider a contmuous map f: X + Y of topological spaces. We have lifting-homomorphisms fV: G&(V) + ‘ik;L(f -l (V)) commuting with restrictions and ergo a @-algebra-homomorphismf: %?r-+ f,(V&). This example motivates the general definition: A morphism(f, T): (X, dx) + (Y, ~4~) of @-ringed spacesconsists of a continuous map f: X -+ Y together with a C-algebra lifting homomorphismf: zfy + f,(s8,), i.e. a family {f,: dy(V) + &x(f-‘(I/)), V open in Y} of @-algebra homomorphisms commuting with restrictions in &‘y and dx. (Note that f*(dx) may not be a sheaf of local C-algebras). We often write f: X + Y for such morphisms. Examples. 1) The pair (f, f): (X, %&) + (Y, %r) is a morphism. 2) Let X, Y be domains in C”, Cm. Take functions fi, . . . , f, E c”(X) such that f: x + @“, z H (h(z), . . ., f,(z)) sends X to Y. The lifting fV: G&(V) + 5$&-‘(V)) induces (chain rule) a homomorphism &: O,(V) + Co,(f-l(V)) and hence a C-algebra homomorphism 5 0, + f*(Q). Clearly (f, f): (X, ox) + (Y, 0,) is a morphism of C-ringed spaces (determined by fi, . . . , 1,). 3) Take point spaces (p, .zZ), (q, 9). Any local homomorphism fi 9? + & gives rise to a morphism (p, &) + (q, 99). Alre_ady the double point (0, @0 EC), E’ = 0, has continuously many morphisms(id, f), f(u + bs) := a + cbe, c E @, into itself. 4) If (X, dx) is a C-ringed space and if U is open in X, then (U, J&‘“) is a C-ringed space. The inclusion z: U + X induces a C-algebra homomorphism ? dx -+ z,(&‘“) (which is the identity on U, the zero map outside of U, and may behave badly on au). The space (U, &‘u) together with the inclusion morphism (z,9 is called an open C-ringed subspuceof (X, dx). cl

Example 3) shows that - contrary to the first guess - the map f does by no means determine the sheaf map 5 However this is true if dx, ZZ’~are subsheaves of %?x,%$, cf. 6 3.3. Every morphism (f, [): (X, JB,) + (Y, &‘r) induces stalk maps x: s&‘~,~+ j&@‘~),. Since restricting germs along the fiber through x to germs at x is a C-algebra homomorphism f&~‘,)/,,, + ~x,x, the composition of these maps is a local @-algebra homomorphism f,: MY,/

--* dx.,,

x E x.

These stalk maps determine the map f, since germs along a fiber are determined by their germs at all points of the fiber. Two morphisms (f, f): (X, JzZ~)+ (Y, ZZ’~),

26

R. Remmert

(y, g): (Y, dr) -+ (Z, dz) are composed (h, 6): (X, cdx) + (Z, dz)

as follows

with h := g o f and i := g,(f)

c-ringed spaces form a category. A morphism topological map and f a sheaf isomorphism.

o Q.

(f, f) is an isomorphism,

if f is a 0

If (f, /) is given, the sheaf f*(9) is for any &x-module Y an f,(s4,)-sheaf and hence, by means off: &y -+ f,(dx), an &r-module. Furthermore dx-morphisms cp: 9 + 9 give rise to &r-morphisms f,(q): f*(9) + f,(r), so that f, is a covariant functor of the category of zdx-modules into the category of &r-modules.

6 3. The Concept of Complex

Space

Complex spaces are (C-ringed HAUSDORFF spaces which locally are isomorphic to complex model spaces, i.e. (C-ringed subspaces of domains D in (c” defined by finitely many holomorphic functions in D. The finiteness condition will guarantee (later) that structure sheaves of complex spaces are coherent. 1. Complex Model Spaces. Take finitely many holomorphic functions fi , . . . , fk in a domain D of (c” and form their ideal sheaf 9 := 9D := O,fi + ... + OJjk c 0,. The quotient sheaf cO,/YD is a sheaf of rings on D. Put x := Supp(co,/Y&) = ( x E 0:-a,,,

# Loo,,}, 0, := (co,/&)lX.

Clearly X is the set N(4) := N(f,, . . . ,fk) = (x E D :fi (x) = . . = fk(x) = 0) of commonzeros of fi, . . . , fk in D and, moreover, closed in D. Every stalk Ox,, = O,,,/JJ~ is an analytic algebra, hence 0, is a sheaf of local C-algebras on X. The (C-ringed space (X, 0,) is called the (complex) model space, defined (in D) by Y. We write V(fl,. . . ,fk) or simply V(9) for this space. Note that 1V(9)I = N(9). Examples. 1) Clearly V(0) = (D, O,), while V( 1) is the empty space. 2) Neil’s parabola in (c2 (with coordinates w, z) is the model space V(w2 - z3). The origin is a cusp of this space. 3) The model spaceof coordinate axes in (c2 is defined by wz. The topological space N(wz) consists of two complex lines through 0 E c2. 4) The m-fold point is the model space V(zm), m E IN\ {0}, in a:. Here N(z”) is the origin p of (c and 0p = O,r/Ogz” g Cc” has m generators 1, E,. . . , sm-i with sm= 0 (Artin-algebra). In case m > 1 there live nilpotent germs # 0 on p. The space V(z2) is called a double point. 5) The cone Y in a23is given by w2 - zr z2. The space Y\ (0) is a manifold of real dimension 4, but the origin has no neighborhood homeomorphic to a ball in IR4. q All analytic algebras occur as stalks of model spaces,more precisely: (3.1) To every analytic algebra A there exists a complex model space X and a point x E X such that O,,, is isomorphic to A.

9015328 Theory of Complex

I. Local

Spaces

27

Proof. Suppose that A = OgW,x/a. Since O,rn,x is noetherian, there exists a neighborhood D of x E (JZ:”and functions fr, . . . , fk E O(D), whose germs fr,, . . . , fkx generate a. Then X := V(f,, . . . ,fk) is a desired model space.

2. Complex Sphces. A (C-ringed space (X, ox) is called a complex space, if X is a Hausdorff space and if every point of X has an open neighborhood U such that the open (C-ringed subspace (Cr, 0,) of (X, 0x) is isomorphic to a complex model space. Thus, locally, complex spaces are determined by finitely many holomorphic functions in domains of number spaces. Complex model spaces are complex spaces. In a complex space (X, 0,) every open set U c X defines an open complex subspuce (U, 0,). Complex spaces form a (full) subcategory of the category of c-ringed spaces. Morphisms (isomorphisms) are called holomorphic (biholomorphic) maps; Ox-modules on a complex space (X, 0,) are called unalytic sheaves on X. Every section s E O(U) gives rise to a (C-valued function [s]: U + c by putting Csl(x) := cx, if s, = c, + t, E (c 0 m(0J. By arguing with model spaces one easily shows, that [s] is continuous. A direct verification yields: (3.2) For every complex space (X, Co,) the maps O,(U) a (C-algebra homomorphism 0, -+ %Tx.

+ (6Tx(U), s H [s], define

This evaluation homomorphism may have a kernel # 0. E.g. ifs E O,(U) has a nilpotent germ s, at x E U, then [s] E %Tx[U] vanishes in a neighborhood of x. This means that the nilradical JV of 0, is contained in the kernel of the map (fix -+ %7x.Thus a section s in the structure sheaf is more than just the continuous function [Is]. There may live sections # 0 invisible to the geometric eye (the simplest example is given by the double point). Nevertheless sections in 0x are called holomorphic functions in X. Using the Riickert Nullstellensatz we shall see in 9 8.3 that the nilradical of 0, is exactly the kernel of 0, -+ Wx; hence for reduced spaces sections in 0x can be identified with their functions. 0 Algebraic properties of the analytic algebras ox,, are used to introduce geometrical notions. a) Smooth points are those points x E X for which O,,, is regular, i.e. isomor-

phic to a (C-algebra c {zr , . . . , z,,}. Complex spaces with smooth points only are called complex manifolds; standard examples are domains in (I? and Riemann surfaces. - Non smooth points are called singular points; the origin of Neil’s parabola and of the cone w2 - zlz2 = 0 in (c3 is singular. b) Irreducible points are those points x E X for which O,,, is an integral domain. Spaces with irreducible points only are called locally irreducible. Neil’s parabola is locally irreducible. - Non irreducible points are called reducible. The origin of the space of coordinate axes wz in 4Z2 is reducible. c) Factorial points are those points x E X for which Ox,, is a factorial ring. Smooth points are factorial points, but the origin of V(w2 - z3) is not factorial. d) Reduced points are those points x E X for which O,,x is a reduced ring. Irreducible points are reduced points. Spaces with reduced points only are

R. Remmert

28

called reduced complex spaces. The space of coordinate axes in (c2 is reduced (although it has a reducible point). The double point is the simplest example of a non-reduced complex space. e) Normal points are reduced points x E X for which O,,, is a normal ring. Factorial points are normal points; normal points are irreducible points. Spaces with normal points only are called normal spaces. The cone Y in (c3 is a normal space; the origin of Neil’s parabola is a non normal point. The sets of such points are, in general, open in X; in point of fact, we shall see: The locus of singular resp. non normal resp. non reduced points is analytic in X. The locus of non irreducible resp. non factorial points is not necessarily open. 3. Holomorphic Maps. If (f, f): (X, 0,) + (Y, 0,) is holomorphic, every section s E O,(V) has a “lifting” f(s) E f*(cO,)(V) = O,(f-i(V)). For the associated continuous function [s] E %?r(V) this is the usual lifting:

[f(s)](x)

= [s] (f(x))

for all x E f-‘(V).

We see, in particular:

If 0, is a subsheaf of %&, then the sheaf map f: 8, + f,(cO,) is uniquely determined by the underlying continuous map f: X + Y, more precisely: f-is “the lofting map” defined by O,(V) -+ Co,(f -‘( V)), s H [s] 0 f. This explains, why in the old days of Complex Analysis - when Co,, Co,always were subsheaves of %&, G& - no one ever looked explicitly at the lifting homomorphism 1 When non-reduced spaces came into being one was forced to proceed with more care. - The following general fact is easily established (and true in the category of (C-ringed spaces): Swp(%4Ker

f) = S~PP f,(G)

= f(X).

Even in classical theory it is worthwhile to consider the sheaf map i in particular its kernel. We discuss the Osgood map f: C2 + C3,

(u, u) H (x, y, z) := (u, uu, uue”).

Denoting by H the plane in c3 given by x = 0 and by M the zero set of z yeYix E 0 (@\H) in (C3\Zf, we have f(C2) = (0) x M and f((C2) = H u M (all points of H are essential singularities of M). Now Ker f is easily computed: In C3 \H the ideal Ker f is generated by z - yeYIX and hence is finite; however (Ker &, = 0 for all points p E H. We see that Supp(Ker f) = M; furthermore Ker f contains no finite BCs-ideal # 0.

A holomorphic function s on X determines a holomorphic map (f, f): X + C such that f(x) = [s] (x) and fz = s. There is a better result. Every holomorphic map (f, f): X + Cc” determines n sections fzi, . . . , fzfi, in f,(c,)((C”) = Ox(X). We denote by Hol(X, (C”) the set of all holomorphic maps (f, f ): X + cc”. Lemma 3.3. The map Hol(X,

C”) + C&(X)“, (f, f) H (A zl,. . . ,fzz.), is bijectiue.

I. Local

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of Complex

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29

Sketch of proof. Znjectiuity: Let (f, f’), (g, g) E Hol(X, (I?‘) with fzfi, = Qz,. Then z,(f(x)) = [fzz,](x) = &z,](x) = z,(g(x)), i.e. f = g. Now 2 and gx are analytic homomorphisms O,rn,f(xJ + 0x.x coinciding on the generators z, ). This implies f’, = 0,. Hence (f, f”) = (g, g). W(x)) of m(%,f,x, Surjectivity: Take fi, . . . , fn E O(X). If X is a model space in D c (c” such that fi, ‘..2 fn are induced by FI, . . . , F, E O(D), then the restriction of the map D+C:p++(F,(p) ,..., F,,(p))toX isamap(f,f): X + cC”withfz, = F,,lX =f,. If X.is arbitrary we patch together these local maps (using their uniqueness). 0

The Lemma immediately

yields:

(3.4) Let (f, f) and (g, g) be h&morphic

maps X + Y and x E X a point such

that f(x) = g(x) and 2 = g,. Then (f, f) = (g, 4) in a neighborhood of x.

Proof. The question being local, we may assume that Y is a model space in (c”. We may even assume that Y = cc” (since two holomorphic maps X + Y are the same if the maps X + Y -+ D c cc” coincide). By assumption the functions fzV, gz, have the same germs at x, i.e. there is a neighborhood U of x such that fzy = Gzy in U. Hence (f, f) = (g, 8) in U by the Lemma. 4. A Gluing Device. Complex spaces are often obtained by “gluing”. A general topological construction is at the bottom of this procedure. Let there be given a topological “base” space Y and an open covering {K>, i E I, of Y. A family {Xi, fi} of topological spaces Xi together with continuous maps A: Xi + v is called topologically glued over {vi> if for every pair i, j E I there is given a topological (possibly empty) map gij from X, := fj-‘(v n 5) c Xj onto Xji = A-‘(r/I n 5) c Xi which fulllils the gluing conditions

h = fi 0 gij

on Xij,

gik = gij 0 gjk

for all i, j, k E I.

GL)

For such families we have the Gluing Lemma 3.5. There exist, up to isomorphism, a unique topological space X, a continuous map f: X + Y and u homeomorphism gi: f-‘(K) * Xi such that j = J o gi

on f -‘(VJ

and gij = gi o g,:’

If all the spaces Xi, Y are huusdorff

on f-‘(v

n Vj), i, j E 1.

then the space X is huusdorff.

Sketch of proof. Take the disjoint union u Xi and call points p E Xi, q E Xj equiuulent if p = gij(q). Due to (GL) we obtain an equivalence relation and hence a topological quotient space X. The continuous projection rc: u Xi + X is open and induces a homeomorphism ni: Xi + 71(Xi). Clearly 7ZjlX = 7ri 0 gij. On 7r(Xi) we define f by h o 7~~‘; due to (GL) this gives a continuous map f: X + Y We have f-‘(v) = rc(Xi) and f-‘(v n 5) = .rr(Xij) = rc(X,,). Now put gi := 7t;‘. iJ

This lemma has many applications. If all spaces Xi are sheaves over F and if all maps gij are sheaf isomorphisms, then X is a sheaf over Y (recollement des faisceaux, [FAC], 314-315). We are interested in holomorphic gluing. If all spaces Xi, Y are complex spaces and if all maps fi resp. gij are holomorphic resp.

30

R. Remmert

biholomorphic, we call (Xi, fi, gij} a holomorphically glued family of complex spaces ouer (v >. For such families the lemma can be stated as follows: Proposition 3.6. There exist, up to isomorphism, a unique complex space X, a holomorphic map f: X -+ Y and a biholomorphic map gi: f -‘( r/I) + Xi such that

f =Aogi This proposition

onf-‘(vi)

andgij=giOgJrl

is crucial in the construction

analytic spectra, in particular

onf-‘(KnVJ.

of complex product spaces and

normalizations.

5. Analyticity of Image and Inverse Image Sheaves. For every holomorphic map f: X + Y and every 8,-module Y the sheaf f*(Y) is an f,(Q)-sheaf and hence, by means of f: 0, + f,(O,), an O,-module. We call f*(Y) the analytic image sheaf of Y. Clearly f, is a left exact covariant functor of the category of O,-modules into the category of &-modules. For every Or-module 9 the sheaf f -‘y is an f -‘Or-sheaf. Since 0, is an f -‘O,-module (the map f’: 0, --) f,(Q) induces a morphism f -lcO, + Ox), the sheaf f *(q

:= f -15

@pcy

Lo,

is an C”,-module. We call f*(y) the analytic inverse image sheaf of r. It is easily shown that f* is a right exact covariant functor of the category of &-modules into the category of (!&-modules. The functor f* is the left adjoint off,, i.e. for any Ox-module 9 and any Q-module y there is a canonical isomorphism In particular

we have canonical sheaf maps r + f,( f *y) and f *( f*9’)

+ 9’.

Right exactness together with f *(OF) % 0$ and the coherence of 0, immediately yields: (3.7) For every Lo,-coherent sheaf 5 the sheaf f *(y) is Q-coherent. 6. Historical Notes. It all started in 1851 when B. Riemann, in his dissertation, conceived the “Idee der Riemannschen Flache”. The higher dimensional analoga of Riemann surfaces however became widely known only 100 years later. Today it is hard to believe that complex manifolds did not come up before 1950. Though first introduced 1932 by C. Caratheodory in his address at the International Congress in Zurich, his rather clumsy definition was not adopted at that time. In 1944 complex manifolds reappear in 0. Teichmiiller’s work on “Veriinderliche Riemannsche Fliichen” (Collected Papers, p. 714). But only through the work of topologists, in particular of H. Hopf in 1948 and 1951, complex manifolds became a fundamental notion of complex analysis. From that time on the development was breathtaking. Already in 1953 the Theorems A and B for Stein manifolds and the Finiteness Theorem for compact complex manifolds were well established by H. Cartan and J-P. Serre.

1. Local

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31

From the very beginning it was clear that the notion of a complex manifold had to be generalized in order to include singularities. In 1932 H. Behnke and P. Thullen in their Ergebnisbericht Theorie der Funktionen mehrer komplexer Vertinderlicher only allowed locally schlicht domains over Cc. In 1951 K. Oka complains that “almost nothing is known about domains (over a?‘) with interior ramification”, [OSl], p. 109. At that time two suggestions were made for a category of complex model spaces larger than the category of open sets in (I?: H. Behnke and K. Stein considered in CBS511 triangulated finite analytically ramified coverings over domains of Cc”, while H. Cartan ([ENS51/52], Exp. 13, p. 3) used special analytic sets in domains of (I?‘. Both these definitions lead using today’s language and the main result of [GR58a] - to the notion of a normal complex space. In 1954 Cartan called these spaces “espaces analytiques gentraux ([ENS53/54], Exp. 6, p. 8). But they were not general enough: In his Gaga paper [Se561 Serre allowed all analytic sets in domains of (c” as local models. In this way he obtained all reduced complex spaces. However this was not yet journey’s end: The libre of the map (E + (E:, z H z* over 0 E a2 is the double point. In 1960, H. Grauert finally arrived at the notion of a not necessarily reduced complex space, [G60], p. 9/10. The way to this most general concept of complex space was paved in Algebraic Geometry by A. Grothendieck: he wrote in [ENS60/61], Exp. 9, p. 3: ‘Vest la le trait le plus saillant qui destingue la geomttrie analytique avec elements nilpotents de l’ancienne geometric, oti les faisceaux structuraux etaient toujours supposes realises comme des faisceaux d’applications dans Cc”.

0 4. General Theory of Complex

Spaces

We discuss notions and facts which are more or less formal but are needed to provide a sound basis for Analytic Geometry. All these results hold for any completely valued field k (instead of c). By X = (X, Q), Y = (Y, 0,) we always denote complex spaces. 1. Closed Complex Subspaces. z := N(9)

:= { x E X:&

For every ideal 9 in 0x the zero set

# Ox} = (x E x:9x

c rn(c!J~,,)> = Supp(B,/$)

of 9 in X is closed in X (its complement is the open set {x E X: 1, E ,aX}. We put 0z := (0,/4)1Z and denote by I resp. L the inclusion JZI 4 1x1 resp. the residue class map Ox + ox/9 = zJ0z). Clearly V(9) := (Z, Oz) is a (G-ringed space and (I, i): (Z, 0z) + (X, ox) is a morphism of c-ringed spaces. In general V(9) is not a complex space. An ideal 9 c Ox is called finite (or: of finite type) if every point of X has a neighborhood U with holomorphic functions h,, . . . , h, E O,(U) generating &, i.e. $u,, = WL + . . . + h,,8,,, for all x E U (cf. also $6.1).

32

R. Remmert

Proposition 4.1. If .Y is finite, then V(Y) is a complex space and (I, 9 is holomorphic. Proof. The problem

V(fl, D so

O,(X) Then,

being local we assume that X is a model space fk) in a domain of (I?‘. Fix z E 2 c D. Since 9 is finite, we may choose ...? small that there are functions gi, . . . , g1 E O(D) whose residue classes in generate JJ~. Now (Z, 4) = V(ft, . . ., fk, gr, . . ., gr) is a model space. by definition, (I, Q is holomorphic. 0

We call V(9) the closed complex subspaceof X defined by 9. Model

spaces

Vfl, . . .?a in a domain D c (c” are closed complex subspaces of D. For every p E X, the ideal 3 c Ox given by 9p := m(Ox,,), 9x := Ox,x for x # p, is finite. Hence (p, Cc)is a closed complex subspace of X. For closed complex subspaces V(S), V(Y) of (X, Ox) the union V(Y) u V(Y) := V($ n 9’) resp. intersection V(Y) A V(Y) := V(Y + 9’) is a closed complex subspace of X, since the ideals

9 n 9’ and 9 + 3’ are finite. If Z = V(9) is a closed complex subspace of X, all spaces Z,, := V&P’), n 2 1, are such spaces. Clearly lZ,,l = IZI and furthermore Z, is a closed complex subspace of Z,,, . We have a sequence Z 4 Z, 4 Z, 4 . . . 4 X of holomorphic inclusions. The space Z, is called the n-th infinitesimal neighborhood of Z in X. With these spaces we shall construct in Chapter II, $4.2 the completion of Z along X. Warning. A closed subset Z of a complex space X may be the underlying topological space of a complex space (Z, Co,) such that there exists a holomorphic map (z, 9: (Z, Co,) -+ (X, fix) with injection I, but (Z, Co,)may not be a closed complex subspace. The map 7 may not be surjective; examples are plentifully provided by semi-normalizations, cf. 0 15. Finite ideals are of utmost importance in Complex Analysis, we exhibit two very useful properties: (4.2) Let J be an arbitrary O-ideal and let x E X. If J’ is any finite O-ideal with J: c J,, then J; c Ju for a suitable neighborhood U of X. There always exists a neighborhood W of x and a finite Cow-ideal9 such that Yx = J, and 9, = Jw. Proof. a) Let fi, . . . , fk E O(X) generate J’. Since fix, . . . , fkx E J,, there is a neighborhood U of x such that fi 1U, . . . , fkl U E J(U). This implies J; c Jo. b) Since (9x is noetherian, there is a neighborhood W of x with sections gm E U(W) whose germs at x generate J,. Put & := Owgl + *.. + O,g,. gt,..., 2. Factorization of Holomorphic Maps. Let f: X + Y be holomorphic and let 1: Z -+ Y be a closed complex subspace of Y. An equation f = I 0 g, where g: X + Z is holomorphic, is called a factorization off through Z. Then, if Z = V(Y), every stalk mapfx: O,,, -+ Co,,, (with y := f(x)) has the factorization

0-L Y,Y

OY.,l~Y

= G,, j,

0x.x

with Ker By = iy (Ker f+,).

I. Local

Theory

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33

A necessary condition for the existence of g is that [f(X)1 c (Z(, however this is not sufficient. Zf Z = V(9), then f: X + Y factors through Z if and only L$ 9 c Ker $ The factorization off through Z is unique. Holomorphic maps admit “maximal” factorizations along their fibres. It is easy to show: To each point x E X there exists a neighborhood V of y := f(x) such that the induced map f: f-‘(V) + V has a factorization I o g with an injection 4,: c?$-~--, 0 x,x.

In general, Ker jcontains no finite ideals # 0 and f factors through Y only; this may even happen locally, as the Osgood map a2 + Q? at 0 E (c2 shows. The optimal case is when Ker f”is a finite &,-ideal. Then f(X) := V(Ker f’) is called the complex image space (of X under f ). If f (X) exists, the set If( = f(lXl) is analytic in X and f(X) is, for any factorization X -+ Z 4 Y of f, a closed complex subspace of Z. Using coherent sheaves we can state (cf. theorem 8.2): (4.3) Zf f,(Ox) is Or-coherent, then the complex image spacef(X) exists. Proof. The ideal Ker f is coherent by sheaf yoga.

cl

Note that analyticity of f(lXl) d oes not guarantee the existence off(X). Let X’ = V(J), Y’ = V(9) be closed complex subspaces of X, Y with inclusions i, j. A holomorphic map f ‘: X’ + Y’ is called a restriction off: X + Y to X’, Y’ if f 0 i = j 0 f ‘, i.e. if f 0 i: X’ + Y factors through Y’. There is at most one restriction off

to X’, Y’. A restriction exists if and only if p(S) c f,(J).

3. Anti-Equivalence Principle. A germ X, of a complex space is a complex space X together with a marked point x E X. Morphisms X, -+ Y, of germs are the germsof holomorphic mapsf of neighborhoods U of x into Y with f(x) = y. The germs of complex spaces form a category. To every germ X, we associate its analytic algebra ox,,. To every morphism X, --* Y, there belongs, via a representing holomorphic map f: U + V, an analytic homomorphism Oy,y + Ox,,. We obtain evidently a contravariant functor X, + 8x,x. We claim: Theorem 4.4 (Anti-Equivalence Principle [ENS60/61], Exp. 13). The functor from the category of isomorphy-classes of germs of complex spaceto xx @x.x the category of isomorphy classes of local analytic algebras is a contrauariant isomorphism.

One has to verify three statements: a) To every analytic algebra A there is a germ X, such that Ox,, E A. b) To every (C-homomorphismcp:O,,, + Ox,, there is a morphismf,: X, + Yy

such that jrx = cp. c) Zf f,, gx are morphismsX, + Y,, then fl, = g, implies f, = gx.

Here a) is clear and c) is obvious by (3.4). It remains to prove b). First assume Y = (I?, y = 0. Choose U = V(x) with sections s, E Ox(U) such that cp(z,,) = s,,, 1 I v < n. By (3.2) there is a morphism f: U + a?,:”with fz, = s,. Clearly f(x) =

34

R. Remmert

y. Furthermore fx and cp coincide on the generators z,, of m(O,,,,,). Hence fx = cp. - We now deal with the general case. We may assume Y = N(4) c D c (c”. Denote by I+!I:O,rm,y+ cO,,xthe composition of Qcn,y+ 0,,,,j9y = Coy,,, with cp.By what we already proved, there is a morphism h: U + (c” with h(x) = y, I& = I/I. Then (Ker &)y = 9,, hence there exists a neighborhood I/ c D of y with h(U) c I/ such that 9” c (Ker 6)“. Hence the map h: U + V has a factorization f 0 I through Y Clearly fx = cp. 0 The Anti-Equivalence Principle is a key for translating local analytic geometry into local algebra and vice versa. We give a typical example: The set of all smooth points of X is open in X. Proof. By definition, a point x E X is smooth if 0,,x is isomorphic to O,rn,O, with a suitable n. But then, by the Principle, a neighborhood of x is biholomorphic to a neighborhood of 0 E Q? and ergo has smooth points only. 4. Embeddings and Embedding Dimension. Jacobi Criterion. A holomorphic map f: X -+ Y is called a (closed) embedding if f admits a factorization f = I o j through a closed complex subspace2 of Y such thatj: X -+ Z is biholomorphic. This holds if and only if f inducesa topological map 1XI -+ If(X)1 and if the complex image spacef(X) exists and f: 0, + f*(cO,) is surjectiue. From this one easily derives: Let f: X -+ Y, g: Y + Z be holomorphic such that g is an embedding. Then g o f: X + Z is an embedding if and only if f is an embedding.In particular, if X is a closedcomplex subspaceof Y and if Y is a closedcomplex subspaceof Z then X is a closed complex subspaceof Z. 0 f: X + Y is called an embeddingat x (into Y) if there are neighborhoods U, V of x, y := f(x) with f(U) c V such that the induced map f: U -+ V is an embedding. (4.5) f is an embeddingat x if and only if fx: Oy,fCxJ+ Co,,, is surjective. Proof. The necessity is evident. Let, conversely, jx be surjective. Then there is a neighborhood I/ of y := f(x) and a factorization z 0 g of f: f-‘(V) + V through a complex subspace Z of V such that gx: Oz,y + Ox,, is injective. Since Ii = 8x o 29(x)is onto, ox is bijective. Then, by the Anti-Equivalence Principle, g: X + Z is biholomorphic at x and ergo f an embedding at x. cl Embeddings at x E X into (c” exist, if a neighborhood U of x is biholomorphic to a complex model space D c Cc”.The smallest number n with this property is called the embeddingdimensionof X at x and denoted by emb,X. Using (4.5) we show that this number is the embedding dimension of the algebra Ox,, introduced in 9 1.4. (4.6) emb,X = emb O,,, for all x E X.

1. Local

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35

Proof. By (4.5) emb,X is the smallest number n such that there exists an epimorphism cO,r,,0+ O,,,. By (1.16) this is just the number emb 8x,,. q

There is a simple classical device for computing

the embedding

dimension.

Jacobi Criterion 4.1. Let X = V(J) be a complex model space in D c (c”, and let J be generated by fi, . . . , fk E O(D). Then

emb,X where rank,(f,,

+ rank&-,,

. . , fk) = n for all x E X,

. . . , fk) is the rank of the jacobian matrix of fl, . . . , fk at x.

The proof makes use of the Implicit

Function Theorem, cf. [CAS], p. 114. q

Corollary 4.8. The equation emb,X

= n holds if and only if J, c m(Oe,,x)2.

This result immediately yields (4.6) again: the canonical epimorphism m(O,,,,) to m(0,,,) induces an isomorphism of the cotangent spaces.

from q

Given any pair d, e E IN with 2 I d < e one can use (4.8) to construct ddimensional normal complex spaces having points x such that emb,X = e. 5. Analytic and Analytically Constructible Sets. A subset A of X is called analytic at x E X if there exists a neighborhood U of X and finitely many

functions fi , . . . , fk E @x(U) such that A n U = N( fi , . . . , fk). If A is analytic at all points of X we call A an analytic set in X; such sets are always closed in X. An analytic set A in X is called thin at x E X if A is nowhere dense in a neighborhood of x. If A is thin everywhere we call A thin in X. In connected complex manifolds X all analytic sets A # X are thin and do not disconnect X locally. Finite intersections and finite unions of analytic sets in X are again analytic sets in X. A much deeper result is that the intersection of any number of analytic sets is an analytic set. This is trivial in Algebraic Geometry; in Analytic Geometry however the neighborhoods of x occurring in the definition above may shrink to {x} if infinitely many sets are involved, cf. (11.19). q The zero set of every finite ideal I c fYxis analytic analytic set A in X the family 3A := {f E B(U):N(f) an ideal in Ox. We call 9A the ideal of A. Every stalk rad 9A,x. The ideal $A reproduces A, namely N($A) in ,YAaround a E A generating YA,o we easily see:

in X. Conversely for every 3 A n U, U c X open} is 9A,, is reduced, i.e. ,a,,, = = A. By choosing sections

(4.9) To every point a of an analytic set A in X there exists an open neighborhood U of a and a finite ideal 3 in 8, such that A n U is the underlying spaceof V(9). The space V(Y) is reduced at a. The fundamental theorem of Oka-Cartan will tell us later that we always have 3 = YAno for small U; this means that 3A itself is a finite ideal. q

R. Remmert

36

An analytic set A in X is called irreducible at a E A if ,a,,, is a prime ideal in Ua, otherwise A is called reducible at a. An equation AnU

= AIu...uAS,

U = open neighborhood

of a in X,

is called an irredundant decomposition of A at a if A,, . . . , A, are analytic sets in U which are irreducible at a such that Aj n V $ A, n V

for all j # k

Since, by the Lasker-Noether tive noetherian

and all neighborhoods

I/ c U of a.

theorem, every reduced ideal a in a commuta-

ring can be written uniquely

in the form a = h Pj, Pj prime 1

ideal, pj $ pk forj # k, we easily derive: Local Decomposition

Lemma for Analytic

Sets 4.10. Let &,,

= fj pj be the

Lasker-Noether decomposition of the reduced ideal Caa,, c Ox,, into piime ideals. Then there exists an irredundant decomposition AnU=A,u...uA,

suchthatYAj,,=Pj,

lljls.

(*)

u Ai is an arbitrary irredundant decomposition of A at a, IfAnU’=A;u... then r = s and there exists a neighborhood W c U n U’ of a such that (after suitable reordering): A(i n W = Aj n W, 1 I j I s.

Thesets AI,..., A, in (*) are called prime components of A around a, while the set germs AI,, . . . , A,, are called prime germs of A,. There is an important generalization of the notion of an analytic set. We consider the family of all subsets of X generated by the family of analytic sets in X by taking complements and locally finite unions. Every such set is called analytically constructible. A set M c X is of this type if and only if M = U (A,\BJ, where iA,), {BoI) are locally finite families of analytic sets in X (such that B, $ A,). The closure of every analytically constructible set is an analytic set in X; for more details see [ICAG], 248-253. There do exist interesting analytically constructible sets. The set of all reducible points of X is analytically constructible, but in general not closed in X, e.g. for Whitney’s umbrella V(x2 - y2z) c (c3, it consists of all points (0, 0, z), z # 0, cf. 0 12.2b). Another analytically constructible not necessarily closed set is the set of all factorial points of X, cf. 0 13.5. Historical Note. Analytic sets were already around in the nineteenth century, long before complex spaces were born. Analytic sets occur implicitly in Chapter 2 of Osgood’s book [LF]. They were called “Gebilde” by Weierstrass. If one wants to believe Osgood (loc.cit. p. 132), a Local Decomposition Lemma was already known to Weierstrass; cf. Math. Werke III, p. 79. The first modern presentation of this result along with an ideal theoretic interpretation was given by Riickert; cf. [Rii33]. A first systematic treatment of analytic sets, which looks clumsy from our present day’s view, is in [RS53].

1. Local Theory of Complex Spaces

37

;55. Direct Products, Kernels and Fiber Products For any two topological spaces X, Y there exists the direct product space X x Y with continuous projections p: X x Y + X, q: X x Y + Y. For continuous maps f, g of X into a space S there exists the kernel K := (x E X :f(x) = g(x)} off and g; K is closed in X if S is a Hausdorff space. If f: X + S, g: Y -+ S are continuous, the fiber product X xs Y := {(x, y) E X x Y: f(x)‘=g(y)}ofXand YooerSisthekerneloffop:X x Y+Sandgoq:X x Y + S. Our aim is to give meaning to these devices in the category of complex spaces. By C we denote an arbitrary category. 1. Direct Products. A direct product of X, Y E C is an object T E C together with morphisms p: T + X, q: T + Y, called projections, such that the following universal property holds: For any pair f: Z + X, g: Z + Y of C-morphisms, there exists exactly one C-morphism h: Z + T such that the diagram is commutative:

Clearly if T and T’ with C-morphismsp, q and p’, q’ are direct products of X and Y then the (unique) C-morphism h: T’ + T with p’ = p 0 h and q’ = q 0 h is an C-isomorphism.If a direct product of X, Y exists, it will be denoted by X x Y Proposition 5.1. In the category of complex spaces,for every pair (X, Ox), (Y, Co,)there exists a complex direct product (X x Y, Ox x u). The underlying topological space (X x YI is the topological direct product 1x1 x 1YI.

The proof is given in three steps. First we obtain, by applying (3.3): (a?, Lo,,) x (a?, 0,“) = (a?+., ogm+“). Using this one shows that model spaces admit direct products: Let u Ir . . . , uk resp. ul, . . . , O, be holomorphic functions in open sets U c (c” resp. V c: Cc”. Denote by p resp. q the projection U x V-P U resp. U x V + V. ThenthemodelspaceV(u,op,...,u,op,v, oq,...,utoq)inU x Visacomplex direct product of the model spacesV(u,, . . ., u,J in U and V(u,, . . ., u,) in V.

Now the general case is handled by applying two times the Gluing Lemma 3.5 in the following way: Zf {F} is an open covering of Y such that all complex direct products X x 5 exist, then the complex direct product X x Y exists. q For holomorphic maps f: X + X’, g: Y + Y’ there exists a unique holomorphic product map f x g: X x Y + X’ x Y’. If f and g are closed embeddings so is f x g.

R. Remmert

38

Basic properties of X, Y are inherited by X x Y. For instance a point (x, y) E resp. reduced resp. normal if and only if both points x E X, y E Y have the corresponding property; for proofs see [AS], p. 179-204; see also 4 9.2 and $14.2 of this Chapter. Let us finally mention that the embedding dimension and the dimension itself behave as expected: X x Y is smooth resp. irreducible

emhw, X x Y = emb,X + emb,Y, dim,,,,,X x Y = dim,X For proofs see [AS], p. 193 and (10.7) of this Chapter.

+ dim,Y.

2. Kernels. Let f, g: X -+ S be C-morphisms. A C-morphism i: K +X f 0 i = g o i is called a kernel off and g in C if the following holds:

with

To any C-morphism k: Z -+ X with f 0 k = g 0 k there exists exactly one Cmorphism h: Z + K such that k = i 0 h.

The existence of kernels is not clear, but the uniqueness is trivial: Zf i: K + X and j: L + X are kernels off, g in C, then the (unique) C-morphism j’: L + K with j = i 0 j’ is a C-isomorphism.

Denoting

by Mor(X,

S) the set of C-morphisms

(5.2) The morphism i: K + X is a kernel off

X + S, we have:

and g tf and only if the map

Mor(Z,K)+{kEMor(Z,X):fok=gok},

hHioh

is well defined and bijective for every object Z E C.

For morphisms (f, f”), (g, 8): (X, SB,) + (S, &‘s) of (C-ringed spaces a kernel is obtained as follows: Take the topological kernel K off, g, denote for x E K by 9X the ideal in dx,, generated by the set (fx - ~,JG&‘~,~(~)c m(s9,,,), and consider the sheaf of ideals 9 c dx determined by the dx( U)-ideals y(U) := {h E &,JU):

h, E yx

for all x E U A K}.

(0)

Clearly N(9) = K. Pass to the closed c-ringed subspace I’(3) = (K, zIK) with s$K = &x/$(K and the injection (z, r): (K, dK) + (X, dx). Using (5.2) it is a matter of routine to show that the morphism (I, ?): (K, ~4~) --,(X, &x) is a (C-ringed kernel of (f, f), (g, 4). This construction of V(y) = (K, dK) directly implies a localization property: If U c X, W c S are open sets such f( Vi v g(U) c W, then V(Nt,) c U is a c-ringed kernel off 1U: U + W, g[ U: U + W. Now the case of complex spaces can be handled easily: Proposition 5.3. The c-ringed kernel (I, r): V(9) + X of two holomorphic maps f, g: X + S is a complex kernel. The space V(9) = (K, 0,) with Ott = Ox/91 K is a closed complex subspace of X. Proof. It suffices to show that V(x) is a complex space. In view of the ization property we may assume that S is a model space in a domain D with injection j: S -+ D. The maps f, g and j of, j 0 g clearly have the c-ringed kernel, hence one only has to consider the case of holomorphic

localof (c” same maps

I. Local

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of Complex

Spaces

39

f, y: X + V. Now f, g and f - g, 0 have the same ringed kernel. Therefore we only have to verify that for every holomorphic map h: X + (c” the (C-ringed kernel V(9) + X of h and X + 0 E a? is the closed complex subspace h-‘(O) of X. In this situation the ideal .a defined by (o), evidently, has the following properties: N(9)

= Ih-‘(O)l,

y1 = 0x,,.i,(m(9,,,0)

for all points x E h-‘(O).

Hence .a is the analytic inverse image of the ideal sheaf defining the origin of 0. Thus I’(,$) = h-‘(O). Examples. (1) The m-fold point is the kernel off, g: (c -+ (JZ:,where f(z) := z”‘, g(z) := 0. - (2) The fixed point space Fix f of f: X -+ X is the kernel off and id,. 3. Fiber Products. Graph Lemma. Every object S E C gives rise to the category Cs of C-objects over S: The objects of Cs are all C-morphisms X + S, while a C,-morphism between objects X + S, Y + S is any C-morphism X -+ Y inducing a commutative diagram (Figure). X-Y \/

S

The definition of a direct product applies to C,: The direct product of X + S and Y + S is denoted by X xs Y. It is common to write just X, Y instead of X --t S, Y -+ S. One calls X xs Y a fiber product of X, Y ouer S. If X xs Y exists, the universal property implies uniqueness up to isomorphisms. The existence of fiber products in C, can be proved if direct products and kernels exist in C,. Thus for complex spaces we obtain: Proposition 5.4. Let f: X --) S, g: Y + S be complex spaces over a complex space S. Let the closed complex subspace K of X x Y be a complex kernel of the holomorphic maps f 0 p, g 0 q. Then K together with the holomorphic map fopIK=gOq1K:K-tSisacomplexfiberproductXx,YofX,YoverS. Examples. (1) The direct product X x Y is the fiber product X xP Y over P = (P, C). (2) The inverse image f-‘(Z) of a closed complex subspace Z 4 Y under a holomorphic map f: X + Y is isomorphic to X x ,, Z. (3) The intersection A n A’ of closed complex subspaces A, A’ 4 X is the complex fiber product A xx A’ (follows from (2)). (4) The graph Xr 4 X x Y of a holomorphic map f: X + Y is defined as the fiber product of f: X + Y and id,: Y + Y, the map X, + Y is the restriction q/Xr: X, -+ Y. The graph of id, is called the diagonal of X x X. Graphs

are very important;

we exhibit their main properties

as

Graph Lemma 5.5. The graph X, of a holomorphic map f: X + Y is a closed complex subspace of X x Y; the space lXfl = {(x, f(x)): x E 1x1) c (Xl x (Y( is

40

R. Remmert

the “topological” graph of J The map pIXr: X, + X is biholomorphic, map f admits the factorization x Proof.

LXf

3

and the

Y, where g := (plXJ)-l.

The map (id,, f): X + X x Y determines a map g: X + Xs such that

the diagram commutes. Hence (pi X,-) 0 g = id, and f = (qlXs) 0 g. Taking into account the additional arrow ----+, we also have g o (plXJ) = id on X,. Thus plX, is biholomorphic and g = (pIX,)-‘. 0 By passing to the graph one can replace f by the induced projection which often can be handled more easily.

0 6. Calculus of Coherent

qlXf

Sheaves

Coherent sheaves play a fundamental role in Local and Global Complex Analysis. Coherence is, in a vague sense, a principle of analytic continuation from a point to a neighborhood. If Y is coherent at x E X, the stalk 9?x determines all stalks near by. In this section we develop the calculus of coherent sheaves. We consider the category of &-modules 9, F, . . . on a ringed space (X, ~4. An &‘-map $: J&’ + Y is determined by its p values si := $(ei) E 9’(X) on the canonical basis e,, . . . , eP of d”(X). Conversely, every sequence si, . . . , sP of sections in Y(X) defines an &-map P

II/: dp + 9, (a,,, . . . , apx)t-+C

aixsix, x E X,

where si = t,Qei).

1

The sections sl, . . . , sp are said to generate Yx resp. Sq if $(&“), $(dP) = Y.

= 9!! resp.

1. Finite Sheaves. Relationally Finite Sheaves. An &‘-sheaf Y is called finite if every point of X has a neighborhood U such that Y; is generated by finitely many sections of Y(U), i.e. if 9” is a quotient sheaf of Soup,with a suitable p. All sheaves &‘“, 1 I n < co, are finite; all quotient sheaves of finite sheaves are finite. If Y is finite, every stalk 9?! is a finite &‘x-module. However, finiteness means more: The UC-ideal 9 with stalks & := 0 and SZ := 0= for z # 0 is not finite (at 0). Note that in the definition of complex model spaces the ideal sheaves are finite.

I. Local

Theory

of Complex

If .4p is any d-sheaf and M a finite submodule hood U of x and a finite subsheaf Y’ of 9” such finite subsheaves of an d-sheaf is finite. If Y is then J. .4p is finite. ~ All important properties of Proposition

41

Spaces

of Yx, there exists a neighborthat 9; = M. Any finite sum of finite and J c d a finite ideal, finite sheaves are contained in

6.1. The support of a finite d-sheaf

is closed in X.

Proof. Let Yx = 0. Choose U = U(x) with sections sl, . . . , sP E Y(U) erati,ng 9,. Since six = 0, we have si = 0 for small U, 1 I i I p. Then x\supp Y,

genU c 0

Note that for the ideal 4 c O,r discussed above we have Supp 4 = (E:\ {O}. We collect some corollaries. 1) Let .Y be finite and 9” 5 9’ z 9”’ an d-sequence. Then the sets {x E X: Im cpx = 9Yx} and {x E X: $, = 0) are open in X. 2) If 9 is finite and if sl, . . . , sp E 9’(V) generate the stalk .Z& x E V, then x has a neighborhood U c V such that s1 ( U, . . . , s,,( U generate 9”. 3) Let YI, 9’; be submodules of 9’. Zf YI is finite, the set {x E X: Y;, c 9”x} is open in X. If C4”2is finite too, the set {x E X: Y,44,= Y;,} is open in X. Proofs. ad 1). Apply the propqsition to the finite sheaf Y’/$m cp resp. Y/xet ad 2). Apply (1) to the map cp: &,7 + Y; defined by sl, . . . , s,. ad 3). The first set is the complement of the closed set Supp Y1/Y1 A 9”.

II/. 0

An &‘-sheaf Y is called relationally finite if for every open set U in X and every &“-homomorphism &‘$ + Y the kernel is finite on U, i.e. if for every set sl, . . . , sp E Y(U) the sheaf of relations ~)GP(S1) . . . ) s,):=

IJ (a,,,..., xeu i

apx) E JzZ~: f aixsix = 0 1

I

is finite on U. All subsheaves of relationally finite sheaves are relationally finite. However quotient sheaves of relationally finite sheaves may not be relationally finite: the 0,-sheaf 0, is relationally finite by Oka’s Theorem, but 0,/2, where f0 = 0 and $= = I”; for z # 0, is not, since the kernel 9 of the residue map fails to be finite at 0. Exercise. the quotient

Zf Y’ is a finite submodule of a relationally module 9/Y’ is relationally finite.

finite module 9, then

2. Coherent Sheaves. An d-module Y is called coherent, more precisely: d-coherent, if Y is finite and relationally finite. The property of a sheaf to be coherent (resp. finite resp. relationally finite) is local in character, we call Y coherent at x E X if 9” is coherent for a neighborhood U of x. Coherence permits to pass from point-properties to local properties when finiteness alone fails. Here is a first example how coherence works: (6.2) Let 57, F be coherent. Then every s8,-map cpx: 9X + Fx extends into a neighborhood U of x to an &v-map cp: Y; + Fv.

R. Remmert

42

Proof. For small U there are morphisms $: &i -+ 5&, x: d& + YU such that 3/(&;) = 9” and xx = cpx o II/,. Then we obtain Ker II/, c Ker xx and hence xe2: $ c X&2 x for a suitable U, since x&r $ is finite. Now x induces the desired map from $, = &;/Ker $ to &. Cl The next evident observation

is applied very often.

(6.3) Every finite subsheafof

a coherent sheaf is coherent.

In particular: If Y’, 9”’ are coherent subsheaves of a coherent sheaf 9 and if f is a finite ideal, then the sheaves Y’ + 9”’ and I. Y are coherent. A most useful device in the calculus of coherent sheaves is the Three Lemma 6.4. ([Fat], p. 321). Let 0 + Y’ + Y -+ 9”’ + 0 be an exact sequence of &-sheaves. Then Y, 9, 9” are all coherent if any two of them are coherent. We have two immediate

corollaries:

(6.5) The direct sum of finitely

many coherent sheaves is coherent.

(6.6) Let cp: 9 + Jo be an &-homomorphism between coherent sheaves. Then the sheaves 352 cp, An cp, Vu.4~ cp are coherent. Here is another example of how to pass from points to neighborhoods. (6.7) Let Y’ s Y $ Y” be a sequence of coherent sheaves. Then the set of points x E X such that Y; + Yx + Y?” is exact is open in X. In particular coherent sheaves are locally isomorphic if they are stalkwise isomorphic. Proof. Since &.PZ cp and Z.PZ $ are coherent by (6.6), the set {x E X: Im cpx = Ker tj,} is open in X by properly 3) in (6.1). - The last assertion now follows from (6.2). 0 Every coherent sheaf is, locally, the cokernel have a converse:

of a morphism

JP + dp. We

(6.8) Let & be coherent. Then Y is coherent, if locally there always exists an exact d-sequence SP + dp + Y + 0. Finally we state a “principle

of changing the base sheaf”.

(6.9) Let & be coherent and let 2 be a finite ideal. Then an &/y-sheaf Y on X is d/y-coherent if and only if Y is &-coherent. In particular, SJ$ is a coherent sheaf of rings. 3. Yoga of Coherent Sheaves. In order sheaves, we discuss some examples.

to become familiar

with

coherent

a) Tensor product. If 5“ is, locally, a cokernel of a morphism zJq + dp, then Y OJ Y is, locally, a cokernel of a morphism zJq @.d Y -+ dp @.d .Y. Since d4” & Y is isomorphic to Y’, we see using (6.6):

I. Local

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43

Spaces

(6.10) If Y and ~7 are coherent, then Y @ d F is coherent. b) The Sheaf Y~+z~,(Y,

F).

Referring

to Q2.3 we show:

(6.11) Zf Y is coherent, every stalk map px: Z&ZJ~‘, F), -+ Horn ,dx(Yx, TX) is bijective. If Y and 9 are coherent, then %HzJY, F) is coherent. Proof. If II/: Y + 5 is d-linear around x and if $, = 0 then $ is zero around x by property 1) in (6.1). Hence px is injective. The surjectivity of px is just (6.2). - Now let F be coherent too. The problem is local. Let d4 --+ J&‘~ + Y + 0 be exact on X. The induced d-sequence 0 --) %‘~Gvz~(Y, F) + Z’~~.~(~p, F) -+ &?GP~z,(&~, 9) may be, a fortiori, not exact (contrary to what is true for the usual Horn-functor). However, for every point x E X, we have a commutative diagram 0 -

2ruwz,(Y,

F),

-

~um&P,

0 -

Px I Hom,,x(% C)

-

I Hom.dXWi’,

9-),

-

*@md(dq,

K)

-

I Hom.dx@‘,4, Z)

n,

with an exact sequence below. We know already that px is bijective. The other two vertical maps are bijective since ZO~.~(JZI”, F) z F. Hence we have an d-exact sequence 0 + &?0*n,(Y, F) -+ .Yp 5 Tq. So J~?u~vzJ~, 9) = xe* 5 is coherent by (6.6). 0 (6.11’) If LZI and Y are coherent, then the dual sheaf Y* := H0m,~(9’, zd) and the bidual sheaf Y** are coherent and, furthermore, (9’*), = (Y,)* for all x E X. c) Transporter Ideal and Annihilator. over a ring R, the ideal An M := (a E R: aM = 0}

If P, Q are submodules

of a module M

resp. Q: P := (a E R: aP c Q>

is called the annihilator of M resp. the transporter ideal of P into Q. Clearly Q: P = An((P + Q)/Q) and An M = 0: M, furthermore An M is the kernel of the homomorphism CI: R + Hom,(M, M), aw(st--+ as). - We generalize these notions to sheaves. We have the canonical &-homomorphism ~1:d + %umJY, 9’) attaching to each section a E J&‘(U) the d(U)-homothety s-as of 9’(U). We call &oln 9’ := Ker u the annihilator of the sheaf Y; for subsheaves 9,2 of Y, we call 9: 9 = dn((9 + 9)/.2?) the transporter sheaf of 9 into 22. (6.12) Let JZI and Y be coherent and let 9, Z? be coherent submodules of 9’. Then the sheaves &n Y and 22: 9’ are coherent ideals; we have (&n Y), = An Yx, (2!:9), = Z?,:L?~ and Supp Y = N(dn 9). Proof. By (6.11) and (6.6) the ideal d+~ Y is coherent. Since 9 + 9 is coherent by (6.3), the ideal 9:Y is coherent. The assertions about the stalks follow from (6.11); the last assertion is obvious. 0 The equation Supp Y = N(J$+z 9) is used again and again in Complex Analysis to show that important sets are analytic.

R. Remmert

44

4. Extension Principle. Let Y denote a closed subspace of X and let I: Y + X be the inclusion. For every sheaf y of groups in Y the image sheaf z.+r is a sheaf of groups on X characterized by r,r/ Y = r and I,~(X\ Y = 0; we call r,r the trivial extension of y to X. If 5? is a sheaf of rings on Y and 9 an @-module, then I,&? is a sheaf of rings on X and I,F is a +%module. One easily verifies:

(6.14) A g-sheaf

9 on Y is B-coherent

if and only if I,F

is z,B-coherent

on X.

This is the Extension Principle in its simplest form. We need a refinement for (C-ringed spaces (X, dx). Every ideal J c dx gives rise to the (C-ringed space (Y, &y) where Y := N(J) and &,, := (zZx/J)I Y, cf. 4.1. Clearly s&‘~/J is the trivial extension of dy. Thus the trivial extension z,r of every sB,-module y is an &,/J-module. Hence (6.14) and (6.9) yield: (6.15) Let ~4~ be coherent and J c &x a finite ideal. Then an dy-module dy-coherent if and only if the trivial extension I,$ is dx-coherent. Since structure sheaves of complex Theorem, cf. 0 7.2, we conclude:

F is

spaces are always coherent by Oka’s

Extension Principle for Coherent Analytic Sheaves 6.16. Let (Y, 0,) be a closed complex subspace of a complex space (X, 0,). Then an analytic sheaf F on Y is &-coherent if and only if the trivial extension r,F of 9 to X is Ox-coherent.

This principle is applied again and again in Complex Analysis to reduce questions of coherence to domains in (c”: The coherence of an ox-sheaf Y is local. Hence one assumes X to be a model space in a domain D of Cc”and proves that the trivial extension of Y to D is O,-coherent.

5 7. Coherence Theorems The calculus developed in section 6 might be a theory of the almost empty set as long as no convincing examples of coherent structure sheaves are given. It was the Japanese mathematician K. Oka (1901-1978) who, in 1948, proved (using another terminology) that the structure sheaf of every complex space is coherent. We outline a simple proof using yoga of coherent sheaves (formal part) and a Coherence Lemma for Weierstrass projections (analytic part). In subsection 3 we define for every reduced space X the sheaf 4 of meromoron X. Locally these “functions” are quotients of holomorphic functions. Due to the existence of points of indeterminancy (like 0 E (c2 for z1/z2) and singular points in the space we use sheaf language from the very beginning. Using Oka’s theorem we collect basic properties of meromorphic functions. The coherence of 0 implies the coherence of all sums On, 1 I n < co, and more generally - the coherence of all locally free sheaves; such sheaves are discussed in subsection 4. phic functions

I. Local

Theory

of Complex

Spaces

45

1. Weierstrass Projections. Most easily to handle are complex model spaces defined by one manic polynomial

o(z, w) = w* + q(Z)W*-l D := a domain

+ .‘. + a*(z) E O(D)[w], in Cc”, 1 I b < 00.

(*I

The attached space (W, 0,) in D x Ccis called a Weierstrass (model) space. The projection D x (c + D induces the Weierstrass projection $: (W, Co,) -+ (D, Co,) which enjoys excellent topological properties: (7.1) Every Weierstrass

projection

if finite and open.

The proof uses Hensel’s Lemma and “continuity

of roots”, cf. [CAS], 53-54. q

Every Weierstrass projection II/: W + D induces an OD-homomorphism $: 0: + $,(O,): For U c D, s = (so, . . . , sb-i) E 0:(V), the polynomial 1 sswBml E 0i(U)[w] c 0(U x Cc) induces a section in (O/oO)(U x Cc) and therefore, by restriction to W, a section J E (0/00)(U x Cc)\W = 0,($-‘(U)) = $*(0,)(U). The map OL( U) + $,(I!&) (U), s H J is an O,( U)-module homomorphism. These maps are compatible with restrictions and hence give an Co,-homomorphism $: 0; -+ J/,(0,). Every stalk $,(I!&), is the ring-direct sumOof the analytic algebras 0 Dx c.xj/ODx c,xjcooxj,Xj E $-l(z), while the stalk map rc/=:c%,, -+ $,(O,), is just the map considered in Q1.6. Hence (1.20) yields: 7.2. The homomorphism

$1 0; -+ @JO,)

is an On-module isomorphism.

Now the road is well paved to obtain the crucial Coherence Lemma 7.3. Let $1 (W, c!&) + (D, 0,) be a Weierstrass projection, and assume that the sheaf Lo, is coherent. Then the sheaf Ow is coherent too, and for every coherent O,-sheaf Y the image sheaf e,(Y) is @,-coherent. Proof. By (7.2) all sheaves $,(cO$) g 0;“’ are @,-coherent. Using this it is easily seen by sheaf yoga, that 0, is relationally finite and hence coherent. - In order to show that $*(9’) is coherent we may assume that there is an O,-exact sequence OP, + 04, + Yw + 0. The functor $, is exact by (2.2), therefore we get an oD-exact sequence Okp + Skq + *,(9& -+ 0. The coherence of Lo, implies the coherence of ICI,(Y). 2. Theorem of Oka. 7.4. The structure coherent.

sheaf C!& of every complex space X is

The problem being local we may assume that X is a model space in a domain 6.16). The coherence proof for (c” makes use of the following

D of (c”. If 0,” is coherent, so is 0, (Extension Principle

Formal Criterion for Coherence 7.5. Let S! be a Hausdorff sheaf of rings on a topological space X such that all stalks ~4~ are integral domains. Then ~4 is coherent if the following condition is fulfilled:

46

R. Remmert

For any open set U c X and any section s E d(U), on U is coherent at every point x E U where s, # 0.

the sheaf of rings sB,/s&v

The proof is a nice application of sheaf yoga, cf. [CAS], p. 58/59. Now it is fun to prove Oka’s theorem for (I?’ by induction on n. The case n = 0 is clear. It suffices to verify the condition of the formal criterion. Let U be open in (c” and s E O(U) and x E U such that s, # 0. We may assume x = 0 and s(x). = 0. Choose coordinates (z, w) in (I?’ = (G”-’ x (c such that ~(0, w) # 0. Then, by the Preparation Theorem, there is a neighborhood D of 0 E Cc”-’ and a manic polynomial o = o(z, w) E O(D) [w] such that s,&‘~ = w,O,. We consider in D x (lZ the Weierstrass model space (W, 0,) defined by w and its Weierstrass projection (W, 0,) -+ (D, 0,). Since 0, is coherent by induction hypothesis, the sheaf 0, is coherent by (7.3). Then, by the Extension Principle (6.16), its trivial extension 1~0~ = 0, x cfoOg x ,r is a coherent sheaf of rings. Since 0, x c/00,~ g and O,/scO, coincide around x, the sheaf cO,fsO, is coherent at x. 0 Oka’s Theorem is at the bottom of all theory of coherent analytic sheaves. Its importance for Complex Analysis cannot be put into evidence in just a few lines. Here is a first application: (7.6) The support of every Ox-coherent

sheaf Y is an analytic set in X.

Proof. Since 0x is coherent, the annihilator ideal dn Y of Y is coherent and Supp Y = N(&n Y), cf. (6.12). Clearly N(An Y) is analytic in X. Cl

It is no exaggeration to claim that Oka’s theorem became a landmark in the development of function theory of several complex variables. By shealifying one suddenly was able to obtain results one had not dared to dream of in 1950. Historical Note. The problem of coherence was posed 1944 by Cartan, [C44], p. 572 and 603. In 1948, Oka proved the theorem for (c” by using the Weierstrass Division Theorem, cf. [OSO], p. 87, and Cartan’s comments to this paper. In algebraic geometry, Serre proved in 1955, cf. [FAC], that the structure sheaves of algebraic varieties are coherent. This result, however, is much easier to obtain than Oka’s theorem. 3. The Sheaf of Meromorphic Functions and the Sheaf of Normalization. A complex space X is called reduced if all points of X are reduced, i.e. if no stalk 0x has nilpotent elements #O. All locally irreducible spaces are reduced. In this subsection X always denotes a reduced space. The set dc, of all non zero divisors in ~9~is multiplicative, i.e. 1, E &‘c, and g,., g: E dc, implies g,g: E &c,, x E X. Germs in dc, are also called active. By looking at the homothety 0 + 0, f H gf, we obtain (using Oka’s theorem and sheaf yoga): (7.7) For g E O(X) the set {x E X: gx 4 &c,} is the support of the coherent o-sheaf &‘n g0 and hence analytic in X. In particular the set {x E X: gx E &c,} is open in X.

1. Local

Theory

of Complex

Spaces

It follows directly from (7.7) that the set dc := (,J dc, XEX

is open in 0, i.e. a subsheuf of sets of Co.If U is open in X, the set &c(U) is a multiplicative system of non zero divisors in O(U), hence the quotient ring is well defined. We obtain a presheaf WJ) := {f/s: f E WJ), 9 E ~W)) {M(U), r:}; the corresponding sheaf ~4’ is called the sheaf of meromorphic functions in X. Sections h E J.&(U) are called meromorphic functions in U. In general such functions are not quotients of global holomorphic functions, i.e. M(U) $ J&‘(U). Thus the presheaf may not be a sheaf. & is an O-algebra containing 0. Since dc is open in 0, every stalk JZx is the quotient ring of 13~. In general, the sheaf JZ?is not O-coherent. However Oka’s Theorem easily yields (7.8) Every finite O-subsheafof JH is coherent.

For every function h E A(X) the sheaves h8 n 0 and hO/(h0 n 0) are coherent. Hence the annihilator ideal D := An(hU/(hB n 0)) with stalks 3, = {vx E 0,: v,h, E O”,) is coherent. We call ha resp D the sheaf of numerators resp. denominators of h. The zero set resp. polar set of h is the analytic set N(h) = N(hD)

resp. P(h) = N(D)

the set N(h) n P(h) is the set of indeterminacy of h. Clearly these sets are analytic in X. Since every stalk 3, contains active germs we can say more. We have a general useful Proposition 7.9. a) The zero set N(g) of a function g E O(X) is thin at x (i.e. nowheredensein a neighborhood of x) zf and only if g, E &‘c,. - b) An analytic set A is thin at x if and only $ the ideal $A,, contains active germs. Proof. a) Clearly thinness entails activeness. Conversely, if gx E &‘cx then g is active everywhere in a neighborhood U of x by (7.7). If z E N(g) n U were an interior point we would have gZ = 0.

b) Follows from a) by looking at a local decomposition Now (7.9) immediately

yields for every meromorphic

of X at x.

0

function h E A(X):

(7.10) The set P(h) is thin in X and the smallest subset of X such that h is holomorphic in X\P(h). - The set N(h) is thin in X if and only if h is a unit in A(X); in this casewe have N(h) = P(h-‘), P(h) = N(h-‘). Meromorphic functions with non empty polar sets may be continuous where. E.g. on Neil’s parabola X in (I? given by wz - z3 the function active everywhere, hence h := (w/z)lX is meromorphic on X. We have P(h) = 0 E X; clearly h is continuous on X and hZ = z(X E O(X). In

everyz/X is N(h) =

general

R. Remmert

48

meromorphic functions cannot be written locally as a quotient f/g of holomorphic functions such that N(h) = N(f) and P(h) = N(g), the reason is that the rings cOxmay not be factorial. Cl A most important subsheaf of M is the sheaf 8 of normalization. For every point x E X the normalization 8x of 0, is the integral closure of cOxin Ax, cf. $1.7. It is trivial to show: (7.11) The set 6 := U fix c A, is open in &, more precisely: If h E A(U) XPX

and h, E &x for x E U, then there exists a neighborhood functions a,, . . . , a,~O(V)suchthat(h~V)“+a,(h~V)“-‘+.+.+a,=O.

V c U of x and bounded

Wesee that 8 c &! is an O-algebra containing 0. We call this subsheaf 8 of (sheaf) of 0. We shall see in 0 14.1 that d is coherent; we shall need 6 in 9 12.4 to formulate Riemann’s Extension Theorem. We know from 0 1.7: ~2’ the normalization

(7.12) Let pl, . . . . p, be the minimal prime ideals of 0,. There is a canonical Lox-isomorphism J%‘~; @ (quotient

field of O.Jpj) which maps 6x onto &a 1

4. Locally Free Sheaves. On every not necessarily reduced complex space X all free sheaves OXp,1 I p < co, are coherent by Oka’s theorem and by (6.5). By gluing, we can construct many other coherent sheaves, namely locally free ones (vector bundles). An @,-sheaf 9’ is called locally free at x E X of rank p 2 1 if there is a neighborhood U of x such that Y; r 0;. Such sheaves are coherent. Using Oka’s theorem we get a converse: (7.13) If a coherent sheaf Y is free at x E X, i.e. if the stalk 9, is isomorphic to cO,P,then Y is locally free at x of rank p. In particular the set of all points where Y is free is open in X. For a closer study we introduce, for every Ox-coherent sheaf 9, the rank function. All (C-vectorspaces 9X := YJtnxspX, x E X, are of finite dimension. The assignment Y’ + y? is, for fixed x, a covariant functor which is right-exact (since y: = 9: Ocx C and 0 is right exact). The integer rk 9?? := dirn,yx E IN is called the rank of 9’ at x; clearly rk 0: = p. By (1.10) the rank is the number of elements of every minimal system of generators of 9,. If r := rk L& there exists a surjection Or + Y in a neighborhood of x, hence the function rk 9$ x E X, is upper semi-continuous on X. The following device for computing rk 9” is useful: (7.14) Let oq 5 Or + Y + 0 be exact, and let + be given (with respect to bases of 04 and Or) by the p x q - matrix with entries fij E O(X). Denote by $x the p x q -matrix with entries [hj] (x) E C. Then rk 9X = p - rank 6, for all x E X. Proof

We have an induced C-exact sequence Cq 2 Cp + px + 0.

0

The rank function of a locally free sheaf is locally constant. The converse is not true: take the double point (x, 0) and Y := red 0. However (7.14) yields:

I. Local

Theory

of Complex

Spaces

49

Proposition 7.15. Let X be reduced and let 9’ be an Ox-coherent sheaf such that rk Yx is locally constant on X. Then 9’ is a locally free sheaf on X. Proof. Fix x E X and put p := rk 9,. We may assume that there is an exact sequence Co45 0* s Y --) 0 on X. By (7.14) we have rank I$~ = 0 for all x E X. Thus the matrix [fij](x) representing $, is zero everywhere on X. Since X is reduced, this implies l;j = 0 for all i, j, i.e. $ = 0. Thus cp: Co*+ 9’ is an isomorphism.

The general behaviour of the rank function is described by the following Proposition 7.16. Let Y be Ox-coherent. rk Yx > k}, k E JN, is analytic in X.

Then every set Bk(9’) := {x E X:

Proof. We may start with an exact sequence Co4z 0* -+ Y -+ 0 on X. Then B,(Y) = {x E X: rank $, < p - k) by (7.14), and this set consists of those points of X where all subdeterminants of order p - k of the matrix (fij) are nilpotent. 0

Note that B,(Y) = Supp 9 - The (bad) set S(Y) of all points of X where a coherent sheaf Y is not free is called the singular locus of 9’. Proposition analytic in X.

7.17. The singular

locus S(Y) of every C&-coherent

sheaf Y is

If X is reduced, this set is thin in X.

Proof. a) We may assume that there is an epimorphism cp: @ + Y on X. This map cp induces a sheaf map 4: %~EJz(S~ Up) -+ ZUPZ(~‘, 9). Now Y is free at x if and only if there exists an Ox-homomorphism 0: Yx + 0: such that cpx0 CJ= id (exercise). Clearly 0 exists if and only if @, is surjective. Hence S(Y) = Supp(Coker 4) is analytic in X by (7.6). b) Let X be reduced. Assume that S(Y) contains an open set V # 4. For m := min{rk 9”: x E V} the set M := V\B,(Y) is not empty, and open in X by (7.16). Since, by choice of m, the rank function is constant on M, the sheaf 9’ is, by (7.15), locally free on M in contradiction to M c S(9). 0

In general, the singular locus S(Y) is different from all sets B,(Y). However if X is reduced and irreducible, then S(Y) = B,(Y) holds for m := inf{rk Yx: XEX}. 5. Coherence of Torsion Modules. In this subsection X denotes again a reduced space. For every O-module 9’ we define T(Y) := u

T(YJ, w h ere T(Yx) := {sx E Yx:: gxs, = 0 for a suitable gx E dc,}.

XEX

Since &c is open in 0 with multiplicative stalks &cx, we obtain an Co-submodule of 9’; evidently T(9’Sx = T(9??). We call T(Y) the torsion sheaf of 9’: the assignment Y - T(Y) is a covariant functor. We call 9’ torsion free at x if T(9)x = 0. The sheaf Y/T(Y) is torsion free everywhere. Subsheaves of locally free sheaves are torsion free.

R. Remmert

50

Proposition 7.18. Let Y be coherent. Then T(9) is the kernel of the natural &homomorphism IJ: 9 + Y** of Y into its bidual (attaching to each section s E 9(U) the map 9*(U) + d(U), A H A(s)). In particular, T(9) is coherent. Proof. By (6.11’) the bidual sheaf Y** is coherent and we have (P’**)X = y’** for all stalks. Thus Ker o, = {sX E yX: the map yX* + OX”,,i H Q.J is zero}. Now it is well-known that for every finite module M over a reduced noetherian ring R the torsion module T(M) is the kernel of the canonical map M + M** attaching to each element m E M the linear form M* + R, AH A(m). Hence Ker o, = T(yx) for all x, i.e. T(9) = Ker 0. q 6. Weierstrass Spaces and Weierstrass Algebras. There is a canonical and important generalization of the notion of a Weierstrass space introduced in subsection 1. Let Y be a complex manifold, let ol, . . . , ok E O(Y) [ T] be manic polynomials, and let w 1, . . . , wk be linear coordinates in (C“. The closed complex subspace W of Y x UZkdefined by w,(y; w,), . . . , o,(y; wk) is called the Weierstrass space belonging to ol, . . . . ok. The projection Y x (Ek -+ Y induces the Weierstrass projection *: W 4 Y; it is easily seen (cf. 7.1):

(7.19) The map +: W + Y is finite. The @,-sheaf $,(O,) is an O,-algebra. There is a simple description of this algebra. We denote by 8, [w 1, . . . , wk] the &,-algebra of polynomials in wl, . . . , wk and call the residue-algebra

; /

~:=~YlT~1,...,~~l/~~~,...,~~~~Yc~~,...,~~l the Weierstrass algebra belonging to ol,

isomorphic

. . . , ok. Clearly the (!&-module

-ty is

to Oi, with b := fi deg 0,. We now consider the map 1 q&%,

. . . . wkl

+

PH {F,: x E V(Y)),

h&hi’),>

Y E Y;

here { px: x E $-l(y)} denotes the family of holomorphic germs px E 0, x Ck,x induced by the polynomial germ p on the $-fiber over y and F, E O,,, denotes the germ induced by px. Clearly these stalk maps define an Or-algebrahomomorphism O,[w,, . . . . wk] + $,(O,); its kernel is the ideal sheaf generated by WI, . . . . Ok. Division Theorem (Final Form) 7.20. The map cO,[w,, . . . , wk] + $,(0,) induces an @.-algebra isomorphism -ly- G $,(O,). In particular, the O,-module $,(Lo,)

is isomorphic to 0; with b := fi deg w,. 1

For k = 1 this is, in essence, the Generalized Division Theorem 1.19. The general case k > 1 is handled by induction on k; for details we refer to [CAS], 56-57 or [HF], 83-85. The Division Theorem 7.20 will be used in $8.5 to obtain analytic spectra. Weierstrass projections II/: W + Y play an important role in the theory of analytic coverings, cf. $12.3.

1 j I

I. Local Theory of Complex Spaces

6 8. Finite Mapping Theorem, Analytic Spectra

Riickert Nullstellensatz

51

and

A culmination of Local Weierstrass Theory is the Finite Mapping Theorem which says that coherence is preserved by finite holomorphic maps. This theorem is proved in subsection 1, it is just the tip of the iceberg: Grauert’s Direct Image Theorem guarantees the coherence of all image sheaves of a coherent sheaf for proper holomorphic maps. The celebrated Riickert Nullstellensatz establishes a fundamental and deep connection between coherent ideals and their analytic sets, it corresponds to the Hilbert Nullstellensatz in algebraic geometry. We discuss two versions of the Nullstellensatz and give convincing applications bridging algebraic and geometric properties. To every finite holomorphic map f: X -+ Y belongs the coherent &,-algebra f’.(O,). In subsection 5 we show that, conversely, every coherent &-algebra is the “analytic spectrum” of a finite map; this yields an Anti-Equivalence Principle between the category of finite complex spaces over Y and the category of coherent &,-algebras. By X, Y we always denote arbitrary complex spaces. 1. Finite Mapping Theorem. Weierstrass projections are finite maps and preserve coherence, cf. (7.3). We first generalize this result as follows: Proposition 8.1. Let f: X --) Y be a holomorphic map, and supposethat x E X is isolated in f -'(f(x)). Then f induces a finite holomorphic map h: U + V of a neighborhood U of x into a neighborhood V off(x). the image sheaf h,(Y) is &-coherent.

For any Q-coherent sheaf Y

Sketch of Proof (for details see [CAS], 61-64). Using the Graph Lemma 5.5 we may assume that X is a model space in a domain D x E of (I? x Q? and that f is the map X -+ D induced by the projection D x E + D such that x = f -‘( f (x)). We proceed by induction on k, the step k + k + 1 being routine. Let k = 1 and 9 c oDxE be the ideal defining X. Furthermore, let x = 0 E D x C and w be a coordinate in (c. Since X intersects the w-axis only in 0, there is a Weierstrass polynomial w(z, w) in &. We can arrange that o E oD[w] n #(D x E). The Weierstrass projection II/: W + D defined by o is finite, and for suitably chosen D x E the induced map W n (D x E) --* D is finite too. Now X n (D x E) is a closed complex subspace of W, hence the map f: X n (D x E) + D, being the restriction of JI, is finite. The trivial extension 9” of Y to W is O,-coherent by the Extension Principle 6.16. Since f,(Y) z $,(Y’) the coherence of f*(Y) follows from (7.3). 0

The Proposition

and the Exactness Lemma 2.2 now give

Finite Mapping Theorem 8.2. Let f: X + Y be a finite holomorphic map. Then, for any Ox-coherent sheaf 9, the image sheaf f,(Y) is &-coherent.

52

R. Remmert

Proof. Take y E Y and let f-‘(y) = {xi, . . . , x,}. By (8.1) there is a neighborhood Uj of xj and 5 of y such that the induced map fj: Uj + I$ is finite and fi*(y[ Uj) is coherent over l$l I j I t. We may arrange that y = I’ for all j and that f&Y’), z @&(94p Uj), cf. (2.2). Hence f,(y) is @,-coherent. 0 A direct consequence is that for every finite holomorphic complex image space exists (cf. Q4.2).

map f: X + Y the

Note. The Finite Mapping Theorem was proved in [GR58b], Satz (“eigentlich, nirgends entartet” means “finite”); it was deduced at that a semi-global direct image theorem for coherent sheaves proved in A finite mapping theorem for ideals was used already in [OSl]; cf. 1, p. 115, also [ENS53/54], Exp. XI, p. 1.

Historical

27, p. 289 time from [GR58a]. Theorem

The first assertion in (8.1) is a most useful criterion for finiteness; it was shown already in [G55], proof of Satz 1, p. 238. We shall use this criterion in § 10.4; it is the first step towards Theorem 12 that every pure d-dimensional reduced complex space is locally an open analytic covering of a domain in Cd. 2. Riickert Nullstellensatz 8.3. Let 9’ be Lo-coherent, and let f E 0(X) uanish on Supp Y. Then, for each point x E X, there exists a neighborhood V of x and a positive integer t such that ftYt = 0.

The problem being local one easily reduces the assertion, by applying Graph Lemma 5.5 to f: X + (c, to the following special case:

the

Let 9’ be O,-coherent over a neighborhood D of 0 E C” such that a coordinate function w vanishes on Supp 9. Then (w’9’), = 0 for suitable t 2 1.

The proof is by induction on n. Essential use is made of Proposition details see [CAS], p. 66. For Y := 0 the Nullstellensatz yields: (8.3’) If f E O(X) vanishes everywhere

8.1; for

on X then all germs f, are nilpotent.

This corollary is in fact equivalent to the Nullstellensatz: The annihilator ideal 3 of Y is coherent by (6.12) and f induces a holomorphic function p on the closed complex subspace Y of X defined by A Since f(Y) = 0, all germs f”,, x E Y = Supp y, are nilpotent. Now f’,” = 0 means f: E 9x, i.e. (f ‘Y), = 0. We now turn to the ideal theoretic version of the Nullstellensatz. For every ideal 9 in 0 the radical rad 9 of 3 is the ideal sheaf with stalks rad 9X := {f E cOx:f m E & for large m E N}. Since $x is finitely generated this implies (rad &)” c $x for large k. For every analytic set A in X we denote as before by JJA the ideal sheaf of all germs of holomorphic functions on X vanishing on A. Riickert Nullstellensatz (Classical with zero set A. Then 3A = rad 3.

Version) 8.4. Let 9 c 0 be a coherent ideal

Proof. It suffices to show that $A c rad j. Take f E $rA(U). Then f vanishes on Supp 9, where Y := Co,/& is Q-coherent. Hence there is a neighborhood I-J V t U of x and a t E N such that f ‘,4pV= 0, i.e. f’ E Y(V), i.e. f E (rad y)(V).

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Observe that no use was made of possible coherence of the sheaf -aa. Historical Note. Since prime ideals are their own radical, Riickert’s Theorem says in particular: 9N(IJ,x = 91xfor all prime ideals ,aX in ox. It is this statement which occurs in essence in Riickert’s paper [Rii33]; on p. 278 he says: “Ferner gehiirt jedes Element von ,a,, das auf U verschwindet, zu p.” This directly implies the Nullstellensatz (by using the Lasker-Noether decomposition of radical ideals), hence it is justified to ascribe the theorem to Riickert. - There are other prodfs of the Nullstellensatz. A nice one using functional-analytic arguments is given in [La71]. 3. Applications. The nilradical 4’” of Cois the radical of the zero ideal in 0, i.e. the sheaf of all nilpotent germs in Lo.This definition is algebraic. The Nullstellensatz permits a function theoretic characterization. (8.5) The nilradical

is the kernel of the evaluation map 0, -+ %&.

Proof. We have Ker(Co, + 9$) = 9. by definition

and ,aX = JV by Riickert. 0

Reduced complex spaces are characterized by JV = 0. Hence (8.5) implies: (8.6) A complex space (X, Ox) is reduced if and only if Ox + ‘ik;i is injective.

The next application

is crucial for dimension

theory.

(8.7) A point x E X is isolated in X if and only if c?, is artinian. Proof. (Recall that 0, is artinian if and only if m(0,) = xx). The ideal 9 := jfx, is coherent, and rn(ox) = (rad .a),. Since N(9) = {x}, the point x is isolated in X if and only if X = N(9) around x. This holds, due to Riickert, if and only if rad 9 = Jf around x. 0

The next application concerns finite maps. We call f: X + Y finite at x E X if there are neighborhoods U, V of x, f(x) with f(U) c V, such that f induces a finite map U + V. With the help of (8.7) we can express this differently. (8.8) For a holomorphic mapf: X + Y the following statementsare equivalent: i) x E X is an isolated point of the fiber f -'(f(x)), ii) f is finite at x, iii) lx: 0, f(x) + Ox,, is finite, iv) fX: Oy,/(,t + Co,,, is quasi-finite. Proof. i) j ii): Clear due to (8.1) - ii) * iii): Assume f: X -+ Y is finite and that f-'(f(x))={x}.S' mce f,(Q) is @,-coherent, the Oy,f(x) -module f*(OX)/cX,~ 1!9~,,is finite. - iii) + ii). Trivial. - iv) =z-i). F := f -'(f (x)) is a closed complex subspace of X. Since O,,, = o,,,/o,,,f,(m(Oy,J(x)) is artinian, the point x is isolated in F by (8.7). Cl

Note that we just gave an “analytic”

proof of Theorem

1.13.

R. Remmert

54

Finally, we prove two statements about meromorphic which is intuitively clear. (8.9) Let h E M(X),

functions, the first of

and assume that the set Supp(h0) is thin in X. Then h = 0.

Proof. All stalks of the ideal sheaf 9 of Supp(h0) contain active germs. The assertion being local we may assume that there is a function g E y(X) n &c(X). Then, by Ruckert, we have for each x E X an equation gih, = 0, t 2 1. Since g: E dc, we get h, = 0. Cl The next fact has important applications:

(8.10) Let h E A(X). Then to every point x E X there exists an integer t 2 1 such that h,JL(,,,,, c cOx. Proof. We have h,B, c 0,. Since N(D) = P(h) the Nullstellensatz t 2 1 such that J&+

gives a

c a,.

4. Open and Finite Mappings. A map f: X -+ Y is called open at x E X if every neighborhood of x is mapped onto a neighborhood of f(x) in Y. We call f open if f is open at all points of X, i.e. if f maps open sets in X onto open sets in Y. Nullstellensatz and Finite Mapping Theorem yield a simple necessary resp. sufficient algebraic criterion for openness of a holomorphic map f: X + Y. Criterion 8.11. Zf f is open at x, then Ker TX c Ny,fC,j. Zf f is finite and if Y is irreducible at f(x) and if f*(Ox)fcx, is torsion free, then f is open at x. Proof. a) Let I/ be a neighborhood of y := f(x) and g E Co,(V) such that fl,(g,) = 0. Then f(g) E 0,(f-i(V)) vanishes in a neighborhood U of x. Hence g vanishes on the neighborhood f(U) of y. Therefore g,, E Jvr,), by Ruckert. 0

b) Consider any finite map fu,“: U + V induced by f, where U = U(x) and V = V(y). The sheaf Y := f,(Q) is @,-coherent (by 8.2) and Supp Y = f(U) is the zero set of the coherent ideal &n Y c Lo,. If f is not open at x, we can choose U, I/ such that there is a section h in &n Y around y := f(x) with h, # 0. By Ruckert we have h:yy = 0 for suitable t 2 1. Since Y is irreducible at y, we conclude that 9y has Or,,-torsion. Since 0 # 9, c f*(O,)y, we see that f*(O,)y is not torsion free. 0 Using the coherence of torsion sheaves on reduced spaces we now get: Open Mapping Lemma 8.12. Let f: X + Y be a finite holomorphic map, and assume that Y is locally irreducible. Let X be irreducible at x and let f be open at x. Then f is open at all points near x. Proof. We may assume that x = f -‘( f(x)). By the Finite Mapping Theorem the sheaf f,(O,) is Or-coherent. By (7.18) the torsion sheaf S(f,(S,)) is also or-coherent, thus its support is closed in Y. Therefore, in view of (8.1 l), it suffices to show that f,(O,) is torsion free at y := f(x). By (8.11) the map fl,: O,,, + O,,, is injective. Since x = f-‘(y), we have f,(O,), = O,,,. Therefore all torsion elements in f*(O,)y are zero divisors in Ox,,. Since Lo,,, is an integral domain, it follows that f,(O,) is torsion free at y. 0

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If X is not irreducible at x, the Lemma is false, e.g. for a line and a plane through 0 E (c3 with projection onto the plane. The Lemma will be used several times, e.g. for proving that complex spaces are pure dimensional at irreducible points, cf. 10.17. 5. Analytic Spectra. Let Y be a fixed complex space. A finite complex space (X, f) over Y is a complex space X together with a finite holomorphic map f: X + Y. For every such space the image sheaf f,(S,) is a coherent O,-algebra by Theorem 8.2. If JZZ’is any coherent &-algebra, a finite space (X, f) over Y is called an analytic spectrum of &’ provided the Or-algebras d and f*(Q) are isomorphic. In this case the maximal ideals of each stalk &,, = @ ox,, are in xs/-‘(y)

to the points of X over y; this motivates the terminology. The Division Theorem (7.20) gives the

one to one correspondence

Key Fact 8.13. Every @,-coherent residue algebra JZI of a Weierstrass W has an analytic spectrum (X, f).

algebra

Proof. We use the notations of 9 7.6. By (7.20) there is a finite space (W, II/) over Y such that -W = $,(0,). Write d = $,(0,)/Z’ with an @,-coherent ideal %? c 11/J&,). There is a unique coherent ideal 9 in 6Jw such that +,(x) 7 xx. We claim that X := V(9) c W together with the finite map f: X 4 W + Y is an analytic spectrum of &‘. First we have f.(ax) 1: $,(0,/9). Since Ic/, is an exact functor by (2.2), we get an &-exact sequence 0 + t+?,(9) + I,+*(&,,) --+$,(&,/$) -+ 0. From this we conclude J&’= w/x = 1,+,(0,)/$,(9)0 ll/*v%v/a = f*(G). Using (8.13) we can show Local Existence Lemma 8.14. Let d be a coherent @,-algebra. Then to every point y E Y there exists an open neighborhood V of y such that the @,-algebra ~4~ has an analytic spectrum.

Proof (cf. also the proof of (1.21)). Choose V so small that dV is generated by sections sl, . . . , sk E S/(V). Since the algebra J;syis a finite Coy-module, all germs sKyare integral over Co,.Hence we may assume that every section s, annihilates a manic polynomial o, E Co(V)[ T]. Denote by V the Weierstrass algebra belonging to ol, . . . , ok. Since sl, . . . , sk generate dV, we have an O,-epimorphism w --) dV. Now the Lemma follows from (8.13). 0 The globalization of (8.14) is obtained by a formal technique. All finite spaces (X, f) over Y form a category, the morphisms of two finite spaces (X, f), (X’, f’) over Y are defined by Hol,(X,

X’) := (h: X -+ X’ holomorphic

and

f = f' o h}.

A canonical contravariant fun&or of this category into the category of all coherent &,-algebras & with morphism sets Horn, ,,(&, a’) := {all O,-algebrahomomorphisms

&’ + ZZ!’>

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56

is defined by (X, f) -

f,(0x) and

@:HOW, X’) -, Hom,r(fJ%T), f+h%)), ~H(G(~‘-~W))

5 Wf-‘W)),

where the last map on the right is induced by the lifting homomorphism 6: O,, + h,(Co,). Now it is a matter of routine to check that 0 is boectiue. This implies that the analytic spectrum of & is unique up to a natural isomorphism, if it exists. This enables us to apply to the local solutions given by (8.14) the Gluing Device of 0 3.4, since the gluing isomorphisms are uniquely determined and clearly satisfy the gluing condition. In this way we obtain Theorem 8.15. Every coherent C&-algebra has, up to isomorphism, a unique analytic spectrum.

It is customary to denote the analytic spectrum of d by Specan &. Clearly d - Specan .M is an inverse functor to (X, f) ,-+ f,(O,). Hence we may state: Anti-Equivalence Principle 8.16. ([ENS60/61], Exp. 19). The functor ~4 -+ Specan d from the category of isomorphy classesof coherent &-algebras to the category of isomorphy classesof finite complex spacesover Y is a contraoariant isomorphism.

Analytic spectra will be used in 5 14 to construct normalizations. alization of the notion of analytic spectrum see Chapter II, Q3.3.

For a gener-

9 9. Coherence of the Ideal Sheaf of an Analytic Set Analytic sets were the forerunners of complex spaces. They became somewhat outdated at that moment when Oka and Cartan proved that the ideal sheaf of an analytic set is coherent. This Theorem is at the center of this section. It implies, among others, that analytic sets are always, in a canonical way, reduced closed complex subspaces. Hence, for a purist, there is no need to consider analytic sets at all. - By X we denote an arbitrary complex space. The ideal sheaf of an analytic set A in X is denoted by 9,. 1. Theorem of Oka-Cartan 9.1. For every analytic set A in X the ideal 9A is coherent.

This theorem is as basic as the coherence theorem for structure sheaves. Since 9 N(Yj = rad 3 for any coherent ideal (Riickert), we have: Corollary 9.2. For every coherent ideal 9 c 0 the radical rad 9 is coherent.In particular, the nilradical JV = rad 0 of 0 is coherent.

The Theorem is, in fact, equiualent to the coherence of nilradicals: Choose x E A, a neighborhood U of x and a coherent ideal Y c 0, such that N(9) = A n U. Consider the exact Q-sequence 0 + 3 + rad 9 = YAnv + rad Y/Y +

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0. It suffices to show that rad Y/S is Q-coherent. Now the 0,/3-sheaf rad 4/4 is the trivial extension of the nilradical JV of the complex space (A n U, (Q,/.Y)lA n U). The @Ana-coherence of .Af implies the O,-coherence of rad .a/$ by the Extension Principle 6.16. 0 A direct consequence of (9.2) is (9.3) For every compact set K in X there exists an integer k 2 1 such that .A$ A 0.

In the following we give a “self-contained” proof of (9.1) using only Riickert’s theorem and the Implicit Function Theorem. By sheaf yoga and the Extension Principle it is easy to reduce (9.1) to spaces X which are balls D in C’. It suffices to show the coherence of $A at each point a E A. We may assume that A # D and that A is irreducible at a. Indeed: if A,, . . . , A, are the prime components of A at a we have $A = n YAj around a and coherence of all ideals sAj at a implies the coherence of $A at a. Now the proof of coherence of $A c 0, at irreducible points of A rests heavily on the following Lemma 9.4. Let A # D be analytic in D and irreducible at 0 E A. Then we can shrink D to a ball around 0 such that there exists a finite ideal SJ c On, a function A E U(D) and an integer r, 1 I r I n, with the following properties: a) N(3) = A, 9, = $A,,. b) For every f, E C!Jx,x E D, we have f, E & if A:f, E & for suitable t 2 1. c) For every point x E A\N(A) there exist local coordinates wl, . . . , w, at x such that the germs wlx, . . . , w,, E 0X generate 3;.

This Lemma gives YA = 9 and hence coherence of 9A at 0 by an elegant argument: It suffices to show that $A c 9. For x E A\N(A) the set A = N(9) is around x the zero set of wi, . . . , w,. Hence if f = 1 avI ,,,“” w;l.. . . . w,“nvanishes on A around x all coefficients with vi = . . . = v, = 0 vanish, i.e. f, E Yx. Now let x E N(A) and f, E YA,x.Choose a representative f E 3A(V) in a neighborhood V c U of x, and consider the (!&-coherent sheaf

9 := (fG +&M

= VMJQ.

Since Y and -OA coincide on D\N(A), we have Supp Yc N(A) and hence 0 for suitable t 2 1 by Riickert. It follows Aif, E &, i.e. f, E &. 0

A:Yx=

It remains to prove the Lemma. If 0 is a singular point of A, this is not evident at all. In [CAS], p. 86, we applied the Local Description Lemma (cf. also 0 12.4 of this chapter). Here we reproduce a direct rather intriguing argument of [La78]: The problem being local we may start with an ideal 9 = ODfi + .** + O,f,, f, E Co(D), such that a) is true. We choose coordinates zt , . . . , z, in (I? and denote by r I n the maximal integer such that 1spurn = r lSVS?l

for points x E A in every neighborhood

of 0.

58

R. Remmert

We have r 2 1 since A # D. Let (8fP/azV)r5,,,,, have rank r on A arbitrarily close to 0. We now put wr := fr, . . . , w, := f,, w,+r := z,+~, . . . , w, := z, and E Co(D). 1
Then 0 is in the closure of B := A\N(A), which implies A, $ -OO.Now the homothety f H Af induces an Co,-endomorphism 6: 0,/Y -+ 0,/4. Since -00 = YA,, is a prime ideal, 6, is injective due to A,, 4 9e. Since COD/Y is coherent the map 6 is injective for suitable D. Then all powers 6’ are injective too. This proves b). The functions wr, . . . , w, are complex coordinates at each point of D\N(A). Hence for every f E O(D) all partial derivatives af/aw, exist in D\N(A) and we have af/aw, = h,/A in D\N(A) with h, E O(D) [apply Cramer’s rule to ($&

. . . . ET”“’

= it4($,

. . . . $)“““‘,

where M := ($)I,i,j5n].

Hence we have the equations

af = h,,/A on D\N(A) aw, 2

with functions

h,, E O(D), 1 I p I m, 1 I v I n.

(*)

Now for p, v > r all functions af,/dw, vanish in a neighborhood of 0 everywhere on I3 (otherwise the rank of the jacobian of fi, . . ., f, would excede r at points of A arbitrarily close to 0). Then (*) shows that for such p, v all functions Ah,, vanish on A in a neighborhood of 0. Hence A,h,,,, E YAA,O= & and ergo hp+ E x0 for all p, v > r, since A, $9, and Y0 is a prime ideal. It follows that h,, E Y(D) for all p, v > r if D is chosen smaller. We conclude: E ,aX for all x 4 N(A) and all p > r, v > r. This implies that all partial derivatives of f,+r , . . ., f, with respect to w,+r, . . . , w, of any order are in Y(D\N(A)) and hence vanish on B. By looking at the Taylor expansion off, at x E B with respect to wr, . . . , w, we seethat fr+l, . . . , f, are power seriesin w1, . . . , w, only. Hence ,aX= OxwIx + . . + oxw,, for all points x E B. This proves c). q Another proof of Theorem 9.1 can be based on Serre’s conditions (R,) and (S,); for details we refer to [ENS60/61], Exp. 21, p. 3-9. For later use we stressa remarkable by-product of (9.1) and (9.4): (9.5) To every irreducible point 0 of X there exists an open neighborhood U of 0 and a thin set N in U, such that all points of U\N are smooth points of X. Proof. Let X be a model space in D c (c” and $ c UD the defining ideal. To A := 1x1 and x = 0 E A we choose 9, A according to (9.4). Then 9 c YA, f c $A and $a = $&, = $,. Since $, is coherent by (9.1), we have $, = 9*,” = ,& for a neighborhood U of 0. Now all stalks 0,,x = OD,x/,$x, x E A n U\N(A) are

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regular by (9.4). If N := A A N(d) n U were a neighborhood of some point p E U then A, would be in ,a,,, = -ap contrary to (9.4). Hence N is thin in U and all points x E U \ N are smooth. 0 Historical Note. The Theorem was probably known to Oka in 1948; cf. the last lines of [OSO], p. 106. The first proof was published by Cartan; cf. [CSO], p. 631. In 1951, in his 8th paper, Oka published his proof. Having not at his disposal the language of sheaves, he states the theorem as follows, [OSI], p. 113; Tout ideal gdometrique de domaines indetermines possede les pseudobases locales.

Oka’s arguments are rather breathtaking; here and elsewhere he refers to the subtle Local Weierstrass Theory just by saying “grace a WeierstraD”, cf. p. 114. For further details the reader may consult Cartan’s comments in Oka’s Collected Papers; 106- 108 and 132- 134. 2. The Reduction Functor. There is no elementary way to provide a given analytic set globally with the structure of a complex space. As an important direct consequence of (9.1) we obtain: Proposition 9.6. Every analytic set A in X is a closed complex subspace of X with structure sheaf OA = Ox/j*1 A and injection (1, i): (A, OA) + (X, Co,), where i is the epimorphism 0, -+ z,(O*) = O,lYa. The space (A, a,.,) is reduced. ~ In particular the ringed space red X := (X, red 0) with red 0 := Of..V is a closed complex subspace of X. The injection red X -+ X consists of id, and the epimorphism 0 --f o/Jv”.

The space red X is called the reduction of X, while the injection red X + X is likewise denoted by “red” resp. “red,” and is called the reduction map. Every holomorphic map f: X + Y canonically induces a holomorphic map red f: red X + red Y; obviously red, o red f = f 0 red,. Clearly red is a covariant functor. Furthermore: (9.7) The set of non reduced points of X is the analytic set Supp .M in X; the reduction map red,: red X + X is biholomorphic outside of this set.

Note that all points of X may be non reduced; more general we have: (9.8) For every analytic set A in X there exists a complex structure sheaf 0; on X with red 0; = red Ox such that A is the non reduced locus of (X, 0;).

Consider X as an analytic subset of X x (IZ and denote by $A, & the coherent ideals of A, X in 0, x c. Put 0; := 0, x c/,aa. &. q Obviously X and red X have the same analytic sets. For any two complex spaces X, Y the reduction and product maps induce a holomorphic map red(X x Y) --, (red X) x (red Y) which even is biholomorphic. A point (x, y) E X x Y is reduced if and only if x E X and y E Y are reduced points. Note that for reduced spaces X, X’, Y the fiber product X xy X’ need not be reduced.

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3. Active Germs, Thinness and Torsion Modules for Arbitrary Complex Spaces. In a reduced space X the zero set N(g) of g E O(X) is thin at x E X if and

only if g, is active, i.e. no zero divisor in 0, cf. (7.9). In general this is not true: The “line” L := V(wz, w’) c (I? is not reduced at 0; the function g := zl L has an isolated zero in 0, but go is a zero divisor in OL,O. There is an easy way to remedy matters. If X is an arbitrary complex space, we call a germ gx E cOxactive if red gx is not a zero divisor in red 0,; i.e. if f,g, E Jv; implies f, E Mx. We have a simple criterion for activity: (9.9) A germ gx E m(UJ is active if and only if gx lies in no minimal prime ideal Of OX.

P~*oof. Letp,,...,

pt be the minimal

prime ideals of ox. Then Jv; = h pj. For

gx $ u pj and f,g, E JI/‘, we have f, E () pj = Xx. If gx E pl, choose al E pj\Pl for j 2 2. Then gJa, . . . . a,) E Nx but a, . . . . . a, 4 JI$. 0 By using the continuous epimorphism (9.10) The set dc x := red-‘(&c,,,x)

red: 0, -+ COredXwe see is open in O,, every stalk &‘c~,~ is multi-

plicative.

Now (7.7) and (7.9) carry over almost verbatim: Proposition

9.11. For

every

g E O(X)

we have (x E X: gx $ &c,,,)

=

Supp((J1’: gO)/Jlr). An analytic set A is thin at x if and only if YA,, contains active germs.

Due to (9.10) we have for every U-module T(Y) := u

Y a torsion-module

T(Y’), where T(Y,.) := { s, E 5f$ gxs, = 0 for a suitable gx E dc,,,}.

XEX

We cannot carry over proposition

7.18 verbatim, nevertheless we have

Theorem 9.12. For every complex space X and every coherent sheaf Y on X the torsion sheaf T(Y) is coherent. The support of T(Y) is thin in X. Zf the support of Y is thin then Y = T(Y). Proof. All O-coherent sheaves Yj = Nj-‘Y/J”jY, j 2 1, are red Co-coherent since .Afq = 0. Therefore all red O-torsion sheaves T(Yj) are red Co-coherent by (7.18). Now T(q) is, by definition, also the Co-torsion sheaf of 3, hence all sheavesT(Yj) are O-coherent. Now fix p E X. Since the problem is local we may assume JV = 0 for suitable t 2 1; cf. (9.3). Clearly there is an active germ ap E dcP such that a,T(Y& = 0 for j = 1, . . . . t. Since T(Yj) is coherent we may choose X so small that there is an active function a E &c(X) such that aT(Yj) = Oforj= l,..., t. Then a direct calculation shows that aT(xj-‘Y) c T(JlrjY), 1 I j I t. We conclude a’T(Y) c T(JVY) = 0. Thus T(Y) is the kernel of the homothety Y + Y defined by a’. Hence T(Y) is coherent and Supp T(Y) c N(a) is thin. If Supp Y is thin, we have Y = T(Y) by (9.11). cl

The Theorem just proved is in [G60], 61-62.

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There are three possibilities to introduce the notion of dimension of a complex space X at a point x E X: - as the dimension of the topological space X at x (Urysohn-Menger) ~ as the smallest integer d such that d holomorphic functions have x as an isolated zero (Chevalley) ~ as the largest integer d such that there is a “flag” AZ $ Ai $ .. . $ A$ of prime germs of analytic sets at x (Krull). All three notions coincide; the interplay between them provides a satisfactory dimension theory for complex spaces. The Active Lemma is the key for our approach, it allows proofs by induction. The topological notion will not be pursued in what follows. 1. Analytic and Algebraic Dimension. We write dim, X I k if there are k functions fi, . . . , fk holomorphic in a neighborhood U of x E X such that N(fl, ..., fk) = {x}. We write dim, X = d, if dim, X I d, but not dim, X < d 1, and call this number d E IN the (analytic) dimension of X at x. The global dimension of X is defined by dim X := sup{dim, X: x E X> E IN u { NI}. The following statements are obvious:

(a) dim,X = 0 if and only zif x is an isolated point of X. (b) dim, X I d if X is a model space in a domain of Cd. (c) If f: X + Y is finite and holomorphic, then dim, X < dimf,,, Y, x E X. The next statement is extremely useful: (10.1) dim, X < d if and only if there exists a finite holomorphic map f: U --) D of an open neighborhood U of x into a domain D in (Cd. In particular, the function dim, X, x E X, is upper-semicontinuous: dim, X I dim, X for all points x near by p. Proof. Assume dim, X I d. By definition there is a neighborhood I/ of 0 E cd and a holomorphic map f: V + (III’ such that f -‘( f(x)) = {x}. By a suitable restriction f 1U: U + D is finite by (8.1). cl

Since the reduction map red X + X is finite we have dim, red X s dim, X. A simple reasoning gives the converse; hence: (10.2) A complex space and its reduction are of same dimension everywhere. For any analytic set A in X we define dim, A to be the dimension of the attached reduced closed complex subspace of X. By (10.2) we may as well use any complex subspace structure of A around x. We write dim ~9~= n if n is the minimal number of germs gl, . . . , gn E m, such that the ring OJ(gI, . . . , g,)CIx is artinian. We call n E IN the (algebraic) dimension of X at x (Chevalley-dimension).

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Dimension

Formula

10.3. dim, X = dim 8,,, fir all x E X.

Proof. Let d := dim, X, II := dim 0,. There are functions fi, . . . , fd, gr, . . . , g,, E O(U) around x such that N(f,, . . . . fd) = {x} and O,/(glx, . . . . g,,)cOx is artinian. Now (8.7), i.e. the Nullstellensatz, tells us that O,/(fI,, . . . , fdx)Ox is artinian and that x is isolated in N(g,, . . . , g,,). This means d 2 n and rr 2 d. 2. Active Lemma. For every g E 0(X) we clearly have dim, N(g) 2 dim, X - 1. We want information about when equality holds. Using the notion of activeness introduced in $9.3 we claim:

10.4. Zf g E O(X) is active at x E N(g), then dim, N(g) =

Active Lemma

dim,X

- 1.

The proof uses (10.1) and exploits that g* is integral over COg,f(xJ,see [CAS], p. 100, or [HF], p. 185 for details. 0 This lemma is most useful for induction application is

proofs. An immediate

dim, (Ed = d for all d E IN, z E Cd.

convincing (10.5)

Proof. Take z = 0. The case d = 0 is trivial. Let d 2 1. Let z1 be a coordinate function. Since z1 is active at 0 and since N(z,) = (Cd-‘, the Active Lemma and the induction hypothesis yield: d - 1 = dim,, (Ed-l = dim, (Cd - 1. 0

All applications of the Active Lemma require active germs appropriate for the occasion. Since m(@J is the stalk at x of the ideal of (x} c X, we get directly from (9.11): (10.6) If dim, X > 0 then there always are actiue germs in m(0.J. An improved criterion for activity is given in [CAS], p. 99.

0

We collect further results. Product Formula

10.7. dim+,,

(X x Y) = dim, X + dim,, Y.

Proof. Let d := dim, X 2 1. Choose f E O(U) in a neighborhood U of x, with = 0 such that A := N(f) is thin in U. Lift f to f^ E 0( U x Y). Then N( f^) = A x Y is thin in U x Y. Hence f resp. f is active at x resp. (x, y), ergo dim, A = d - 1 and dim(x,yj A x Y = dim(x,yj (X x Y) - 1. Since dim(,,yj A x Y = dim, A + dim, Y by induction hypothesis we are done. III

f(x)

An important Intersection

corollary of the Product Formula Inequality

is the

10.8. Zf A, B are analytic sets in X, then

dim, (A n B) 2 dim, A + dim, B - emb, X for every point x E A n B. Another consequence of the Active Lemma is

I. Local

Ritt’s

Lemma

Theory

10.9. An analytic

of Complex

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63

set A in X is thin in X if and only if

dim, A < dim, X for all x E X. If dim, A = dim, X at an irreducible point x of X then A is a neighborhood

of x in X.

There is a simple device for computing

the dimension

of analytic sets.

(10.10) Let A be an analytic set in a domain D of C:“. Then dim, A I d if and only tf there exists a linear complex subspace E in Cc” through x of (linear) dimension >n - d such that x is isolated in A n E. In 1953, when the theory of analytic sets was put on firm grounds, (10.10) was used as definition of dimension, c.f. [RS53]. Cl We now discuss Krull’s characterization of the dimension algebra 8,. The point of departure is a simple

of an analytic

Remark 10.11. Let p, p’ be prime ideals in c?, such that p $ p’. Then dim 0Jp >

dim 0Jp’. Proof. The kernel of the epimorphism 0Jp --) &,/p’ is #O. All germs #O in 0Jp are active. Hence the assertion follows from the Active Lemma. q

The remark implies directly that for every chain p. $ pi $ . . . z pk of prime ideals in c!J~we have k I dim ox. More is true: Theorem 10.12 (Krull). The dimension of 0, is the largest integer d such that there exists a chain p0 $ p1 $ ... $ pd of prime ideals in Co,. Proof. It s&ices to construct such a chain with d := dim 0,. For d = 0 we put no := m(0J. Let d 2 1. Then m(0.J contains an active germ a by (10.6). Since dim OxfaO, = d - 1 there is by induction hypothesis a chain in C!I,..0~a which provides a chain pi $ . . . $ pd of prime ideals in cOx.Since pi contains the active germ a, this ideal is no minimal prime by (9.9) and hence there is a prime ideal p. $ pl. 3. Invariance

er the dimension.

of Dimension.

Open Mappings.

Finite holomorphic

maps low-

There is a better result.

Theorem 10.13. (Invariance of Dimension). Let f: X + Y be finite and holomorphic. Then the image set f(X) is analytic in Y and its dimension is given by

dim, f(X)

= max (dim, X: x E f-‘(y)}

for all y E f(X).

Zf, in addition, f is open at x E X, then dim, X = dim,(,) Y.

The routine proof proceeds by induction

on dim, X using the Active Lemma. 0

Using (10.9) and (10.13) we easily obtain Lemma 10.14. Let f: X -+ Y be finite at x and let Y be irreducible Assume that dim, X = dimf,,, Y. Then f is open at x.

at f(x).

64

R. Remmert

Proof. We may arrange that f-‘(f(x)) = x. The finiteness at x gives arbitrarily small neighborhoods U of x such that f(U) is analytic at f(x). By (10.13) we have dimf(,, f(U) = dim, X. Hence f(U) is a neighborhood of f(x) by (10.9). 0 We call X pure dimensional at p E X if the function dim, X is constant around p. Then (10.14) immediately ensues: 10.15. Let X, Y be pure d-dimensional and let Y be locally irreducible. Then every holomorphic map f: X + Y with discrete fibers only is open. In particular: Zf f is injective, then f(X) is open in Y and f induces a homeomorphism X + f(X). Note that holomorphic

homeomorphisms

need not be biholomorphic.

4. Convenient Coordinates. Purity of Dimension. Let X be a reduced closed complex subspace of a domain D in (c” and p E X. Our concern is to represent X around p as a “kind of covering”. We put d := dim, X. We can choose, in many ways according to (lO.lO), linear coordinates zi, . . . , zd, wi, . . . , wk in Q? with p = (0,O) such that p is an isolated point of the intersection of X with the subspace (Ck of c:” given by z1 = * *. = zd = 0. Such coordinates are called convenient for X at p; their power is displayed by Proposition 10.16. Let zl, . . ., there exist neighborhoods Y and such that the projection Y x V + open at p and satisfies ~~-l(z(p)) then 7cis open.

z,,, wl, . . ., wk be convenient for X at p. Then V of 0 E (Cd and 0 E (lZk with U := Y x V t D Y induces a finite map rt: X n U + Y which is = p. - If moreover X n U is pure dimensional

Proof. Clear by (8.1) and (10.14).

Cl

This Proposition is a forerunner of the Local Embedding Lemma 12.8. We stress an important consequence of (10.16). If X is irreducible at p, then rcis open in a neighborhood of p by the Open Mapping Lemma 8.12. Hence dim, X = dim+, Y = d for all points x E X near p by (10.13). This shows: Theorem 10.17. (Criterion of Purity). at all irreducible points of its reduction.

Every complex space is pure dimensional

Now we can obtain more insight into the local decomposition of an arbitrary complex space X, cf. $4.5. For given x E X the zero ideal (0), c Loxadmits a primary

decomposition (0), = h qj with associated prime ideals pj = rad qj. Let 1

PI,

..a,

p,, s I t, be all minimal prime ideals of Co,. Then Jv; = h pi, and the

coherence of the nilradical

JV implies the existence of an open nelfghborhood U of x and of coherent ideals ‘pi, Qi c 0, such that Q, = qix, ‘$$ = rad Q, (0)” = 0 Qi, Jv; = 0 ‘pi. Clearly u = x, u * *. v x,,

where Xi := N(‘Pi).

I. Local

Theory

of Complex

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65

is an irredundant decomposition of 1x1 at x into prime components. By applying Theorem 10.17 and Ritt’s Lemma we conclude: (10.18) For an appropriate sional in U and every set Xi n

choice of U all analytic

is thin in Xi, 1 < i < s.

u Xi (

i#i

sets Xi are pure dimen-

>

Every set {x E X: dim, X 2 k}, k E IN, is analytic in X. 5. Smooth Points and Singular Locus. At smooth points complex spaces look like domains in number spaces. For the investigation of the set of non smooth points we call in the embedding function emb, X. [Recall that emb, X is the smallest integer n such that a neighborhood of x is biholomorphic to a model space in a?]. Clearly emb, X 2 dim, X; furthermore the function emb, X is upper semi-continuous. This and Jacobi’s Criterion 4.7 easily yield:

(10.19) Every set X(k) := { x E X: emb, X L k}, k E N, is analytic in X. Using the function emb, X, we immediately

get a Criterion for Smoothness.

(10.20) A point x E X is smooth if and only if emb, X = dim, X. Proof. We only have to prove sufficiency. Put A := Co,,, and d := emb A. Choose an epimorphism cp: Ocd,e -+ A. If Ker cp # 0 the Active Lemma would give the contradiction dim A = dim Oga,O/Ker q < d. Hence cp is an isomorphism. 0

The set S(X) of all singular (= non smooth) points of X is called the singular locus of X. This set is closed in X, furthermore: (10.21) S(X) = X’ u S(red X), where X’ := the set of not reduced points of X. In order to investigate S(X) further on, we first deal with special cases. (10.22) Zf X is pure dimensional, then S(X) = {x E X: emb, X > dim, X} is analytic in X. Zf X is irreducible at x, then S(X) is nowhere dense around x. Proof. The first assertion is clear by (10.20) and (10.19), while the second assertion is clear by (9.5). 0

For a pure d-dimensional model space X = V(fl, . . . , f,) in D c (c” Jacobi’s Criterion 4.7 and (10.22) yield an effective device to determine S(X): S(x)={xtX:rank(g)
(10.23)

we now handle the general case.

Proposition 10.24. The singular locus S(X) is an analytic reduced, the singular locus is thin in X.

set in X. Zf X is

66

R. Remmert

Proof. In view of (10.21) we may assume that X = red X. The problem is local. Fix x E S(X). Choose a neighborhood U of x and a decomposition X, u . . . u X, of U into prime components such that Xj n X, is thin in Xj, j # k. Then it is easily seen: S(X) n U = T u u S(Xj),

where T := jyk (Xj A X,).

(*)

Clearly, T is thin in U. The set Xi is irreducible at x and hence pure dimensional around x. Thus (10.22) implies that S(X,) is thin in U for small U. q The singular locus can now be viewed as a closed reduced complex subspace of X. The associated ideal 9& carries additional information about S(X); it can be shown (cf. [CAS], p. 118):

If X is reduced then no ideal 9&s(xJ,x,x E S(X), is a principal ideal in Co,,,.

$11. Miscellanea The first three subsections are devoted to the general theory of homological codimension;subsection 4 deals with Cohen-Macaulay spaces.In the fifth subsection we discuss the Noether property for coherent sheaves, in addition we touch upon gap sheaves. 1. Homological Codimension. Syzygy Theorem. Let A be an analytic algebra ideal m and let M # 0 be a finite A-module. A finite sequence a,), ai E m, is called an M-sequence if ai is not a zero divisor in M/ E:‘h;..; ... + ai- M), 1 I i I q. Such a sequence is called maximal if there is no M-sequence (a,, . . . , a4, a). It is well known that every M-sequence can be extended to a maximal M-sequence, and that two maximal M-sequences have the same length (= number of elements). The length of a maximal M-sequence is called the homological codimension of M and is denoted by cdh M. In case M = 0 we put cdh M := co. Quite often cdh M is also called the depth or the profondeur of M. We collect basic facts; for proofs see [AS].

with maximal

(a) If a E m is no zero divisor of M, then cdh M/aM = cdh M - 1. M = 0 then there exists an element s # 0 in M such that m’s = 0. The number cdh M is independent

If cdh

of the algebra A in the following sense:

(b) Let A + B be finite, e.g. B = Ala. Then, for every finite B-module M, the B-module and the A-module M have the samecodimension. In case K = K, = (c{z,, . . . , z.} there always exist convenient M-sequences: (c) To every finite K,-module M with k := cdh M c co there exist generators w1,..., w,ofmsuchthat(w, ,..., w,J is an M-sequence (hence cdh M I n).

Using these facts, we obtain

I. Local

Theory

of Complex

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67

Syzygy Lemma 11.1. Let M be a finite K,-module with k := cdh M < co. 1) M is free if and only if k = n (in particular cdh K, = n). 2) Ifk
z,N -+ (K f$ z,K)~ + M/$

z,M + 0

is exact, 1 5 m I n. For m = n we have 0 + NImN -+ (K/mK)P $ MImM + 0. Since K/mK = Cc, it follows from the choice of p, that @ is bijective. Hence N/mN = 0, i.e. N = 0 (Nakayama) and M “= KP. ad 2). Assume N c KP. We have N # 0 by 1). Thus z1 is not a zero divisor of N, i.e. cdh N 2 1. We proceed by induction on k. Let k = 0. It suffices to show that every f E m is a zero divisor of N/z, N. Since f is a zero divisor of M, there exists a g E KP\N such that fg E N. Then f(z,g) E z1 N, but zig 4 z1 N. If 1 I k < n, we have - as in the proof 1) - the K-exact sequence where

O+fl--+KP+fi-+O,

fi:=NfzrN,

i?:=K/z,K,

fiii:=MIz,M. (*I

We view (*) as a sequence over l? = C(z,, . . . , z,). By (b) the codimensions the K-resp. E-module F coincide, and the same is true for M. Furthermore cdh E = cdh N - 1 and

Now the induction

of

cdh ii?l = k - 1 by (a).

hypothesis yields cdh fi = k.

Cl

A free resolution of finite type of an A-module M is an A-exact sequence . . . -+Fj 24-1 -+ . . . + Fl - ‘PI F, zM’-1-0, (F) where all 4 are finite and free A-modules. Every finite A-module admits (many) such resolutions. It is convenient to decompose (F) as follows: 0 -+ Ker qj + Fj + Ker ‘pj-r + 0, j E IN, where Ker cp-i = M.

F’)

The Syzygy Lemma can now be restated as Syzygy Theorem 11.2. Let (F) be a free resolution of finite type of an K,module M with cdh M = k -C 00. Then j = n - k is the smallest number such that Ker ‘pjml is free. Proof. We know 0 I k I n. By (11.1.1)) we have k = n if and only if Ker q-1 is free. Let k < n. Then cdh Ker ‘pjwl = k + j for 0 I j I n - k by (11.1.2)). Hence by (11.1.1)) the first free kernel is obtained for j = n - k. cl Remark. If Ker (P,,, is free, all kernels Ker qjpi, j 2 m, are free. This follows immediately from (F’), since in an A-exact sequence 0 + N z F : L + 0 the module N is free if F and L are free.

R. Remmert

68

The Syzygy Theorem says in particular Every finite K,-module

(classical version):

M # 0 has a free resolution of finite type

0 + F, + F,-, + . . . + F, + F, + M + 0. For M # 0 the resolution (F) is called of length p E IN u {a} if 4 = 0 for j > p but FP # 0. The homological dimension of M # 0 is the smallest m E IN u { CCI} such that M admits a resolution (F) of length m. We write dh M = m and dh M := ---co for M = 0. Clearly dh M = 0 if and only if M is free and M # 0. Using the homological dimension the Syzygy Theorem can be rephrased as follows: (11.3) For every finite K,-module

M # 0 we have dh M + cdh M = n.

For arbitrary analytic algebras there is no Syzygy Theorem; e.g. if A is artinian have a free resolution + A + A +~~~-*A+m+Ooflengthco,i.e.dhm=co,butcdhm=O.As a matter of fact it can be shown: lf dh M < co for every finite

By standard (11.4) resolution of x and C&-exact

A-module

sheaf techniques

with m # 0, we

M # 0, then A is regular.

we now obtain easily:

Let Y be Ox-coherent, and assume that the stalk 9”, x E X, admits a free of finite type of length up. Then there exists an open neighborhood U a free resolution of finite type of the Ou-sheaf Y; of length I p, i.e. an sequence 0 + @Urn,-+ cO;/m,-’+ . ~-+O;‘+Oum,+Y~+O.

By the Syzygy Theorem we can be apply (11.4) to smooth points. Syzygy Theorem for Coherent Sheaves 11.5. Let Y be a coherent sheaf on an n-dimensional complex manifold X. Then Y admits, in a neighborhood of every point of X, a free resolution of finite type of length
global resolu-

2. Analyticity of the Sets S,(Y). To every coherent sheaf 9’ on an arbitrary complex space X we attach the sets S,(Y) := (x E X: cdh Y? < k}, k E IN. We show Theorem 11.6. The set Sk(Y) is analytic in X of dimension I k. In particular, the function X + IN u (oo}, x H cdh 977is lower semi-continuous. Proof. Since cdh Y, is “independent” of 0x by (11.1(b)), we may assume that X is a domain in (c” and that there exists, according to (11.4), a free resolution + ... -go -Y-+0 of finite type . ..+g3g-on X. By (11.2) the set Sk(Y) is the singular locus hf the’c:herent sheaf Ker (P”-~ and hence analytic by (7.17). We prove dim Sk(Y) I k by induction on k. Let k = 0 and x E S,(Y). The subsheaf 9” of Y generated by all local sections in Y vanishing on X\x is coherent, the quotient .Y := Y/9” coincides with Y on X\x (gap sheaf). We claim x $ S,(Y). Otherwise there is a section t in Y around x such that t, # 0,

I. Local

Theory

but m,t, = 0; cf. (11.1(a)). The last borhood V of x. For small enough t = s mod Y’(V). Then s, $ Sp? but tradicts the definition of 9’. Hence is a neighborhood U of x such that conclude U n S,(Y) = x.

of Complex

69

Spaces

equation implies tl(V\x) = 0 for a neighV there exists a section s E Y(V) such that s, E 9;, i.e. s,, = 0 for y E V\x. This conx $ S,,(Y). Since S,(Y) is closed in X there U n S,(Y) = 0. Since Y = Y on U\x, we

Now assume k > 0. Suppose that dim Sk(Y) > k at some point p. We can choose p & So(Y). Then there is a neighborhood W of p and a function f E Co(W) such that f E mP is no zero divisor for Sp,. Then, for small enough W, the sheaf map 9, -+ Yw, s, H~,s,, is injectioe. For all points x E WA N(f) this means cdh (&,/fSP,), = cdh Y??- 1 (cf. 11.1(a)) and thus Sk(Yty) n N(f) c Sk-,(~~/f~w). Hence dim, S,(yl,) n N(f) I dim, Sk-,(spW/~&,). Here the number on the right is 5 k - 1 by induction hypothesis, while the number on the left is > k - 1 by the Active Lemma. Contradiction. 0 Since dh Yx + cdh Y: = dim, X for smooth points by (11.3), we have Corollary 11.7. Zf X is a complex manifold, every set S,*(Y) := {x E X: dh Yx 2 k}, k E IN, is analytic in X of codimension 2 k. Zf n := dim, X everywhere we have for k = 0, 1, . . . . n.

S$(Y) = S&Y),

In particular, the function X + IN v { -co}, x H dh 9” is upper semi-continuous.

Theorem 11.6 is in Scheja’s paper [Sche64], Satz 5. The sets Sk(Y) are fundamental for studying gap sheaves; see [Tea], 13 1- 146. 3. The Defect Sets D,,,(9).

homological

dimension

There is a fundamental and dimension.

inequality

between co-

Lemma 11.8. Let A be an analytic algebra, M # 0 a finite A-module and p c A a prime ideal associated with M (i.e. p = AnAa for an a E M). Then cdh M I dim A/p I dim A/An M < dim A.

The

proof

proceeds

by constructing

to an M-sequence

(a,, . . . , a,),

k := cdh M, recursively a chain p0 $ pr $ +*. $ pk, where p0 := p and p, is associated to M/(a, M + . . . + a,M). Then (10.12) gives cdh M I dim A/p. cl

We now define, for every coherent sheaf 9 on X, the defect function def,Y := 0 if 9X = 0 and Since dim, Supp 9’ = dim(&J&n

def,Y = dim, Supp 9 - cdh Yx

Yx), we have def,Y 2 0 by (11.8). The set

Q,,(Y) := {x E X: def,Y 2 m}, m E IN, is called the m-th defect set of 9, clearly De(Y) = X

and

if 9” # 0.

D,,,(Y) c Supp Y for m 2 1.

70

R. Remmert

Theorem 11.9. Every set D,,,(Y) is analytic in X. We have

codim(D,(Y),

Supp 9) 2 m for m 2 l.*

Proof. Assume m 2 1. We can suppose X = Supp Y. The problem being local we may also assume that d := dim X is finite. Then D,,,(Y) is empty for m > d. For 1 I m I d one verifies directly that

RW)

=

t.)

A,,

where A,, := {x E X: dim, X 2 p} n S&Y).

mspsd

Every set A,, is analytic in X by (10.18) and (11.6), hence D,(Y) Applying again (11.6) we obtain for all x E X:

is analytic in X.

dim, A, 5 dim, S,,-,,,(Y) I p - m I dim, X - m, m I u I d. It follows dim, Dm(9’) I dim, X - m, i.e. codim, D,,,(Y) 2 m. Historical

0

Note. The analyticity

Exp. 21, proposition

of the sets D,,,(Y) was proved in [ENS60/61], 3; the inequality in (11.9) is in [Sche64], Satz 7.

4. Cohen-Macaulay Spaces. An analytic algebra A is called algebra if cdh A = dim A. From (11.8) we obtain directly

a Cohen-

Macaulay

(11.10) If A is Cohen-Macaulay then dim A/p = dim A for all prime ideals p associated with A, in particular all these ideals p are minimal. We collect basic facts about Cohen-Macaulay algebras. 0) Every O-dimensional algebra is Cohen-Macaulay. 1) Every reduced algebra A of dimension 2 1 has cohomological dimension 2 1; hence every reduced l-dimensional algebra is Cohen-Macaulay. For A := (c{w, z}/(w”, wz) we have dim A = 1, cdh A = 0. 2) Every normal algebra A of dimension 22 has cohomological dimension 22, hence every normal 2-dimensional algebra is Cohen-Macaulay. The reduced algebra A := flI(w, x, y, z)/( w, x). (y, z) is pure 2-dimensional, but cdh A = 1. 3) Every regular algebra (c{z,, . . . , zn} is Cohen-Macaulay. 4) For every k 2 3 there exist k-dimensional normal algebras which are not Cohen-Macaulay. There is a simple device to obtain Cohen-Macaulay algebras from given ones. Proposition

11.11. Let A be Cohen-Macaulay

and let a,, . . . , ak E mA such that

dim A/(a,, . . . , ak) = dim A - k. Then A/(a,, . . . , ak) is Cohen-Macaulay. Proof. It s&ices to handle the case k = 1. Then a, lies in no prime ideal p associated with A since (11.10) would give dim A > dim A/(a,) 2 dim A/p = dim A. Hence a, is not a zero divisor of A and therefore cdh A/a,) = cdh A - 1. 0 *For analytic sets A, B in X such that A c B we write A in B, i.e. for min (dim, B - dim, A) E IN. XEA

codim(A,

B) for the

complexdimension

of

I. Local

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71

We now turn to complex spaces. We call x E X a Cohen-Macaulay point if Ux is a Cohen-Macaulay algebra; we call X a Cohen-Macaulay space if all points of X are Cohen-Macaulay. From (11.9) and (11.10) we derive: Theorem 11.12. For every complex space X the set of non Cohen-Macaulay points is thin in X. Every Cohen-Macaulay space is locally pure dimensional.

A partial geometric translation

of the facts listed above reads as follows:

Reduced l-dimensional spaces, normal 2-dimensional manifolds are Cohen-Macaulay spaces.

spaces and all complex

There is a classical way to construct Cohen-Macaulay spaces. Let M be a pure n-dimensional complex manifold and X a closed complex subspace of M with defining ideal 3 c 0,. Every stalk has at least codim,(X, M) := n dim, X generators. The space X is called a complete intersection (respectively a set-theoretic complete intersection) at x if ,aX has codim, (X, M) generators (respectively if there is an ideal .I, c O,,, with codim, (X, M) generators such that rad J, = rad ,a;). Since the regular algebra O,,, is Cohen-Macaulay we obtain at once from (11.11) Proposition 11.13. If X c M is a complete intersection Cohen-Macaulay point of X.

at x E X, then x is a

X need not be a complete intersection at Cohen-Macaulay points. Consider the holomorphic homeomorphism (c + x c c3, t H (x, y, z) := (P, t4, tS) of cc onto the locally irreducible space curve X := 1/(x3 - yz, y* - xz, z* - x*y). X is Cohen-Macaulay, but X is not a complete intersection at 0 E X, however X is a set-theoretic complete intersection at x; for details see E. Kunz: Introduction to Commutative Algebra and Algebraic Geometry, Birkhauser 1985, 137-140. We conclude this subsection with a quite helpful criterion for reducedness. Proposition 11.14. A Cohen-Macaulay space X is reduced if the set of reduced points of X is dense in X (i.e. if S(X) is thin in X). Proof.

We show that Jv;, = (0), for given a E X. Let (0), = h qi be a primary 1

decomposition.

Then J0 = b pi, pi := rad qi. It suffices to show: pi c qi for all

i. The problem being local d may start with coherent ideals ‘pi, Qi c 8x such that Qi, = qi, ‘pi = rad Qi, (0)x = fi Qi, sx = n vi. Since all prime ideals pi are minimal by (11.10) we can assume that Xi := iV(‘$$) is a prime component of X at U. Then (0), = Qi, and Jx = ‘pix for x 4 u Xi. NOW {X E X: Mx = (O),) is

dense in X by assumption

i#i

and Xi\ u Xi is open in X and dense in Xi by (10.18). i#i

Hence {x E Xi: $Jix = 12,) is dense in Xi. Ergo the analytic subset S := Supp(yi/Qi) of X, which is contained in Xi = iV(‘$,), is thin in Xi. Consequently x S,a$ ‘pia = pi and thus, by the Nullstellensatz (8.3), we get a germ g. E oO\pi such that g,(‘$$/Qi), = 0. This means g0pi c qi. Since no power g,” is in qi and since qi is primary we conclude pi c qi. 0

72

R. Remmert

This proposition from [Boh86].

is, in essence, in Schumacher’s paper [Schu76], the proof is

5. Noether Property. In a noetherian module every ascending chain L, c L2 = ..’ of submodules is stationary: there exists an index m such that L, = L, for all n 2 m. Coherent analytic sheaves enjoy as well this Noether Property 11.15. Let X be a complex space and Y a coherent Oxmodule. Then every ascending chain YI c Yz c . . of coherent submodules9, of Y is locally stationary (=stationary on all compact sets). Warning. For every x E X there is an m such that 5& = y,, for all n 2 m. Hence there are neighborhoods U, of x, such that ynl U,, = 9,1 U,, n 2 m. How-

ever n U, may be just x. n2m Proof (by induction on d := dim X). The case d < 0 is trivial. Suppose d 2 0. Take x E X. By (10.1) there exists a finite holomorphic map rc: U(x) -+ D c Cd. By using the coherence of all sheaves rr*(YI U) one is reduced to the case, where X is a connected domain D in (Cd (note that n,(q) = rr,(yj) implies pi = ~j). Now sheaf yoga shows that one only has to deal with Y = 0,. Then all yn are coherent ideals. Let 9, # 0 for a certain k. The closed complex subspace Y of D defined by yk is different from D, whence dim Y < d. Since all ideals x 1 9, are 0,-coherent, the induction hypothesis yields the assertion. 0 A family {M,} of subsets of a set M such that chain in {M,} is stationary has maximal elements. {M,} is increasingly filtered (i.e. if to every pair i, Mj c Mk), then u M, is the only maximal element. implies:

every (countable) If - in addition j there is a k such Hence the Noether

ascending the family that Mi u Property

Maximality Property 11.16. Let Y be ox-coherent and let {x} be an increasingly filtered family of coherent submodulesof 9. Then u Y: is a coherent Loxmodule.

We have an important Corollary 11.17. Let Y be Ox-coherent and let (q} be any family of coherent submodulesof 97 Then the sumc Y: is a coherent ox-module. In particular, every set {s,} of sections s, E 9’(X) generates a coherent module 1 Oxs,,

Proof. The family {Y;} of all finite sumsof sheaves Y: is increasingly filtered. Since all Y; are coherent, their union U yJ = 18 is coherent. 0 Historical Note. The Noether Property was first stated and proved in [G60], cf. Satz 8 on p. 23, by applying Cartan’s version of the Theorem of privileged neighborhoods (Basissatz of Cartan-Riickert), cf. also [CAG], 34-36. The proof given above is due to Serre, [Se66], 278-79.

We give some applications.

Since ,a;, c 9B for A 3 B, we get directly:

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Descending Chain Condition for Analytic Sets 11.18. Euery descending chain A, 3AA, 2 . . . of analytic sets X is locally stationary. Intersection Property for Analytic Sets 11.19. For any family {A,} of analytic sets A, in X the intersection n A, is an analytic set in X.

Property (11.19) was already stated in [H49], p. 549; Hermes uses a “Satz von der wesentlichen Umgebung.” q A simple consequence of the Noether property and of theorem B for Stein spaces is Proposition 11.20. Let K c C” be a compact Stein space such that every point of 3K has a neighborhood U in K which is real-analytically isomorphic to a polyhedron in JR”. Then the ring O(K) is noetherian.

For a short proof see [La77]. Proposition 11.20 is a special case of a theorem of Frisch which says that for every compact semi-analytic Stein space K in an arbitrary complex space X the ring O(K) is noetherian, cf. [F67], p. 123, and Chapter II, theorem 4.8. 0 Our last application

deals with “gap sheaves”.

Proposition 11.21. Let 9 be a coherent submodule of a coherent module 9’. (1) To every analytic set A in X there exists a unique maximal coherent subsheaf $[A] of Y such that Y[A] 3 9 and Supp $[A]/$ c A. (2) To every integer q E TN there exists a unique maximal coherent subsheaf 9q of 9’ such that Yq 3 9 and dim Supp jQ/S I q. Proof. The family FI resp. F, of all coherent submodules 8 of Y such that Y: 1 9, and Supp Sq/$ c A resp. dime Supp Sq/9 < q is increasingly filtered, since Y1, YZ E F,, implies Sp, + Y; E F, (due to Supp(Y1 + YZ)/$ c Supp Y1/ 9 u Supp spZ/Y). Hence FI as well as Fz has a unique maximal element. 0

The sheaf $[A] is called the gap sheaf of 3 in 9 with respect to A; the sheaf Yq is called the qth gap sheaf of 3 in Y. These sheaves can also be obtained quite differently: (1’) The family (s E Y(U):

slU\A $[A]

E Y(U\A))

is the sheaf Y[A].

Moreover:

= fi (9: 9;). k=O

(2’) The sheaf Yq is the family

{SELqU):S~U\BE~(U\B)f Moreover

Yq = 9[A],

or some analytic set B in U with dim B I q}.

where A := S&Y/9)

= (x E X: cdh,(Y/S)

I q}.

Gap sheaves were introduced in 1962 by W. Thimm [Th62]. They play a key role in the theory of coherent extension of coherent sheaves, cf. Chapter 9.

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0 12. Analytic Coverings Since the days of Riemann analytic coverings have played a fundamental role in Complex Analysis. Analytic coverings are “unbranched” outside of thin sets. Here we collect basic properties of such coverings and give revealing examples. One concern is to show that around pure d-dimensional points reduced spaces are always analytic coverings of balls in (Cd. A key result is the Theorem of Primitive Element. The reasoning is algebraic and analytic; essential use is made of the classical Riemann Extension Theorem. Let Y be a complex manifold and T a thin set in Y and let f be a holomorphic function on Y\T. Assume that f is bounded near a E T or that T is thin of order 2 at a E T. Then f has a unique holomorphic extension to a E Y.

Here “analytically thin of order k 2 1 at a E T” means that for all points x E T near a we have dim, T I dim, T - k. 1. Coverings and Integral Dependence. A finite surjective holomorphic map 7~:X + Y is called an (analytic) covering (of Y) if there exists a thin set T in Y such that n-‘(T) is thin in X and the induced map X\n-‘(T) + Y\T is locally biholomorphic. Note that n-‘(T) is eo ipso thin if 7~is open.

The set T is called a critical locus of 71.The set B all points in X where z is not locally biholomorphic, is called the branch locus of 7~.Clearly B c x-‘(T); however, it is not trivial that B is analytic in X. We call 71unbranched, if B is empty. We denote by b(y) the (finite) number of different points in the fiber K’(y). Since z is finite, the function b(y) is locally constant on Y\T. If it is constant there, say b := b(y), we call 7ca b-sheetedanalytic covering. Analytic coverings may have non empty branch loci though they are unbranched in the sense of topology (i.e. locally topological): The holomorphic homeomorphism 5: (c + Y, u H (u3, u’) onto Neil’s parabola V(x2 - y3) in (cz is a one-sheetedanalytic covering with branch locus (0) c Cc.Further examples of coverings are given in the next subsection. 0 Analytic coverings 7~:X + Y preserve thinness: If M c X resp. N c Y is thin then n(M) c Y resp. n-l(N) is thin. The lifting map it: 0, + 7~,.(0,) is an O,sheaf monomorphism; since thinness is preserved it is easily seen: (12.1) The monomorphism ii: 0, + x*(0,) morphism it: 4, + it*.

extends uniquely to an C&-mono-

For the rest of this subsection 7~:X + Y always is a b-sheeted analytic covering of a connected complex manifold Y with critical locus T c Y. By A we denote a thin set in X. Then n(A) is thin in Y of sameorder. We assume A c Cl(T) and z-‘(n(A)) = A. We regard O(Y) resp. Lo(Y\T) as a subalgebra of O(X) resp O(X\C’(T)) via 5.

I. Local Theory of Complex Spaces

For every function f E O(X\A)

the “characteristic

of(w) := Of(w, y) := x~~l~y~(w -f(x))

75

polynomial”

= Wb- a,(y)wb-’ + “’ + (- l)bab(y),

y E Y\T, has coefficients as E O(Y\T) [Every point y E Y\T has a neighborhood V c Y\ T, such that C1( V) consists of domains U,, . . . , U, which are “schlicht” over V. Then fin-i(V) determines “branches” fa E 6(V) such that fj Us = fs 0 nl U,. The coefficients atrJV are the elementary symmetric functions off,, . . . . fb, i.e. upI V E O(V)]. A straightforward application of Riemann’s Extension Theorem yields the fundamental Theorem of Integral Dependence 12.2. Every holomorphic function f E 8(X\A) has a characteristic polynomial or E O(Y\n(A)) [w] such that or(f) = 0. If f is bounded near A or if A is thin of order 2, then or extends (uniquely) to

a polynomial

or E O(Y) [w].

This theorem has an important Corollary 12.3. Every meromorphic function h E M(X) has a characteristic polynomial oh(w) = wb - uIwb-l + . . -( - l)bub E A(Y) [w]. We have co,,(h)= 0. Proof. The set A := n-‘(n(P(h)) is thin in X. Take the characteristic polynomial wb + I(1)8u,wb-8 E CI(Y\z(A))[w] of hlX\A E Co(X\A). We claim that all coefficients aP extend meromorphicully to Y. Since x(A) is the support of the 8,-sheaf y := 7t.,JhC!&/hO,n 0x), the Nullstellensatz gives to every point y E n(A) a function g f 0 holomorphic in a neighborhood V of y such that gyV = 0. Then g.(hlU) E Co(U)on U := rc-‘(V). Now wb + c (- 1)8(gausj(V\7t(A))) wb-” is the characteristic polynomial of (gh)l U \ A with respect to the induced covering U\A + V\z(A). By (12.2) each gBuaE O(V\z(A)) extends holomorphically to V, i.e. aa E &Z(V). 0 Historical Note. The classical reference for the Theorem(s) of Integral Dependence is Riemann’s dissertation, cf. his Werke, 3-45, especially p. 39. In his paper “Theorie der Abel’schen Functionen” (Werke, 88-142) Riemann states the theorem explicitly and performs the above construction, p. 108/109.

A function e E Co(X) is called a primitive element over U(Y) if the discriminunt d E O(Y) of its characteristic polynomial o, E O(Y) [w] is not the zero function; this is equivalent to saying that there exists a point y E Y such that e has b different values on the fiber C’(y). In general there are no primitive elements. Locally however primitive elements always exist. Theorem of Primitive Element 12.4. Let e E O(X) be a primitiue element with A E O(Y). Then to every function f E O(X\A) there corresponds a unique polynomial Sz, E O(Y\z(A))[w] of degree I b - 1 such that Af = Q,(e). If f is boundednear A or if A is thin of order 2 then Q, extends to a polynomial in O(Y) [w] and f extends (uniquely) to a meromorphic function f on X which is a section in the sheaf 6 of normalization.

discriminunt

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The construction of s2, is classical; we refer the reader to [CAS], 140-142. The theorem will be used in 9 12.5 to generalize Riemann’s Extension Theorem. 2. Examples of Coverings. Quite often the singular locus S(Y) of Y occurs as a critical locus, then X\rr-‘(S(Y)) is a manifold too. The Implicit Function Theorem yields a simple device for constructing coverings: A finite holomorphic surjection rc: X + Y of a pure dimensional space X onto a pure d-dimensional closed complex subspace of a complex manifold 2 is an analytic covering if everywhere on X\S(X) the jacobian of the map X 5 Y G 2 has rank d. Covering maps can behave rather erratic in the branch locus. This already occurs for one-sheeted coverings, as the following examples a)-c) show. a) The map 5: Cc -+ Y, u H (1 - u2, u( 1 - u2)) is a one-sheeted covering of the curve Y := I/(x3 - x2 + y2) in (E2 with (0) c Y as critical locus and { + 1) as branch locus. There are two prime components of Y at 0 E Y with “normal crossing”. In old days Y was called Newton’s parabola nodata. b) The surface Y := I/(xk - ykz), k 2 2, in (c3 has as singular locus the line in (lZ3 given by x = y = 0. The map 5: a2 + Y, (u, O)H(UU, u, u”) is a one-sheeted covering with critical locus S(Y), the branch locus being the line in a2 given by u = 0. The fiber t-‘(O) has one point, all fibers t-‘(y), y E S(Y)\O, have k points. The space Y is irreducible at 0 and has k prime components at all other points of S(Y). Sometimes Y is called Whitney’s umbrella; cf. [CAV], p. 57, 113. c) The cone Y = I/(x3 - z(x2 - y’)) in (c3 is the homogenized parabola nodata. The singular locus S(Y) consists of two lines S,, S, in (c3 given by x = y = 0 and x = z = 0. The (homogenized) map

5: c2 + y, (u, U)H(U(U2 - u2), u(u2 - u2), u3) is a one-sheeted covering with critical locus S(Y), and the branch locus consists of three lies in (c3 given by u(u - u)(u + u) = 0. The fiber t-‘(O) has one point, all fibers over S,\O have six and all fibers over S,\O have three points. Sometimes Y is called Cartan’s umbrella, c.f. CEUVRE II, p. 716. d) The map (c” + (lZ2n, (u, u2, . . . , u,,)H(u~, u2, . . . , u,, u3, u2u, . . . , u,u) induces a one-sheeted covering t: (c” + Y of an n-dimensional reduced closed complex subspace Y of a2” with S(Y) = (0) c Y as critical locus and (0) c Cc” as branch locus, n 2 1 (generalized Neil’s parabola). The space Y has remarkable analytic properties at its isolated topologically smooth singularity 0.

f: Y + Cc

CI) For all n 2 1 there exists a continuous meromorphic function which is holomorphic in Y\O but not in 0; we haue f. E ay,,,. B) We haue cdh O,,, = 1 for all n 2 1.

Proof. c() Since 5 is topological, the inverse l-l: Y -+ (c” is continuous holomorphic in Y\O. Now 5-l can be written in the form

C-l:Y+C”,

y=(z,

)...) Z,,Wl)...)

W,)H(f(Y),Zz

)...)

Z”)

with f(y) := u = (wl/zl)l Y. Since f2 - zl = 0, this function fullfills a).

and

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fl) The algebra A := Or,, is isomorphic to (c{u’, u2, . . . , u,, u3, LQU, . . . , u,u}. We claim that u2 is a maximal A-sequence. It suffices to show that mA = rad q for a primary ideal q occurring in a primary decomposition of Au’. Since u3 +! Au’, there is a q with u3 $ q. Clearly (u,u)‘, . . . , (u,u)~, u2, (u”)’ are in q. Since u,u3 = (u,u)u’ E Au2, u3 # q, the primary property of q implies ui E q for large t. Hence all generators u2, u2, . . . , u,, u3, u2u, . . . , u,u of m, are in rad q. 0 e) The map

n: (c2 --+ Y, (u, u)H(u*,

v*, uv), b 2 2, onto

the cone Y :=

V(zb - xy) in (lZ3 is a b-sheeted covering with critical locus 0 E Y and branch locus 0 E (c2. Since GZ2\0 + Y\O is a universal covering, the cone Y is not a

topological manifold around 0 E Y, cf. also !j 13.3. The maps in a)-c) are normalizations in the sense of 4 14.2, they are not open. This phenomenon can only occur if Y is not locally irreducible, cf. (10.15). 3. Weierstrass Coverings. Let Y denote a connected complex manifold and let w,(y, wK) E o(Y)[w,] be a manic polynomial, 1 I K < k. We consider the Weierstrass space W in Y x (Ck belonging to oi, . . . , ok and the induced map $: W + Y, cf. 5 7.6. Proposition 12.5. The map $: W + Y is finite and open. The space W is a (global) complete intersection and hence a Cohen-Macaulay space.

Proof. Finiteness and openness of + is shown verbatim as in the case k = 1, cf. (7.1). Then (10.13) yields dim O,,, = dim Y for all x E W. Now, by definition, W c Y x a? is a complete intersection and ergo, by (11.13), a Cohen-Macaulay space. cl In general Weierstrass spaces are not reduced. We show Theorem 12.6. Assume that the discriminant A, E O(Y) of CD, is not the zero function, 1 I K I k. Then W is reduced and $: W + Y is an open analytic covering with critical locus T := N(A, . A,. . . . . Ak). Proof. For every point p E Y\ T the polynomial coK(p, wK) E cC[w,] splits into b, := deg ok distinct linear factors. By Hensel’s Lemma, there is a neighborhood V c Y\ T of p such that 0,) V x (Ck splits as follows:

*. % = n (WK - cJY)), v=l

c,, E @W), C,,(Y) + C,,(Y) forp Z v and Y E 1/.

Hence every point x E W\n-l(T) has a product neighborhood U := V x I/’ c Y x (Ek such that the ideal defining W in U is generated by k linear functions w, - c,,~(Y) E O(U), 1 I K I k. Th us W n U is a dim Y-dimensional complex manifold and $ maps W n U biholomorphically onto V (e.g. by the Graph Lemma 5.5). Now T is thin in Y. Therefore $-l(T) is thin in W by (12.5). Since all points of W\I,-‘(T) are smooth, the Cohen-Macaulay space W is reduced by (11.14) and $: W -+ Y is an open analytic covering. 0 The covering +: W + Y obtained by (12.6) is called the Weierstrass covering belonging to the (reduced) polynomials oi, . . . , ok.

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Every manic polynomial o E O(Y) [ T] has a unique factorization 4:’ * . . . . qf” into manic prime polynomials over the quotient field of O(Y). Since Co(Y) is normal (by Riemann’s Extension Theorem) all qi are in LO(Y)[T]. The manic polynomial redo := ql-..:qt E o(Y)[Y] is called the reduction of o; evidently o is reduced, i.e. w = red w, if and only if the discriminant of w is not the zero element of O(Y). Therefore Theorem 12.6 directly gives Corollary 12.7. If W is the Weierstrass space in Y x Ck belonging to ol, . . . , wk, then the reduction red W of W is the Weierstrass space in Y x Ck belonging to red or, . . . . red ok.

Weierstrass spaces W are rather easy to handle due to the projection II/: W + Y. In the next subsection we embed complex spaces locally into Weierstrass spaces. 4. Local Embedding Lemma. Let Y, V be connected open neighborhoods of 0 E (Cd, 0 E (Ck and let X be a reduced closed complex subspace of U := Y x V through 0 = (0,O) E U such that the projection Y x V + Y induces a finite map x:X--, Ywithn-’ (0) = 0. We “embed” n around 0 as follows: Local Embedding Lemma 12.8. For an appropriate choice of there exists a reduced Weierstrass covering +: W+ Y with 0 E Wand 0 such that the following is true: 1) The space X is a closed complex subspace of W and nlX is the of II/ with the injection I: X 4 W. 2) If n is open in a neighborhood of 0 the map 7cJX is an analytic

U = Y x V @‘($(O)) = product Ic/ 0 I covering.

Proof. Since r~.+(0x~~)~(~) = Ox,-, is a finite O,,,-module, every germ w,,~ := (w, IX)0 E ~x.0 annihilates (by Dedekind’s Lemma, cf. p. 15) a manic polynomial pK E Oy,OIT]. We may assume that pK is a Weierstrass polynomial. Since X is reduced at 0 we may assume, furthermore, that pK is reduced in the factorial ring O,,,[T]. Then the discriminant A,,, E 0,,0 of pK is not the zero germ. Now for suitably small Y there are holomorphic representatives A, E O,(Y), aK(y, T) E 1 I K I k. Clearly the Weierstrass covering $: W + WY) CT1 for 4, and pK3

Y belonging to wr , . . . , ok fulfills 0 E w and e-l ($(O)) = 0. ad 1). By construction all polynomials w,(y, wK) vanish on X around 0. Since $ is finite and t+Q-‘($(O)) = 0 we can choose U = Y x I’ in such a way that X c W c U as sets. Since X and W are reduced spaces it follows that X is a closed complex subspace of W and that 7cJX = $0 z. ad 2). We choose U in such a way that 7~:X + Y is open. Let T be a critical locus of the Weierstrass covering I(/. It suffices to show z: X 4 W is open at every point x E X\n-‘(T). Choose a neighborhood U, c W\+-‘(T) of x E X such that Ic/ induces an isomorphism IJ~~: U, + $(Ul). Then z(Xn U, = I++;’ o (nlX A U,) and both maps here are open. 0 In general there will occur besides X “parasitic branches” in the Weierstrass covering, i.e. X $ W. In fact this always happens if X is not a complete intersec-

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tion at 0, e.g. for the space curve X c (c3 which is the image of (c + a’, t w (t3, t4, t’), cf. Q11.4. Here the map X + (c, (x, y, z) H x is a 3-sheeted open covering, and an attached Weierstrass covering +: W + (c is given by the polynomial y3 - x4, z3 - x5 and is 9-sheeted. 0 Historical Note. The Local Embedding Lemma was known in principle to Weierstrass. In several situations Oka just refers to it by saying “thanks to Weierstrass”, e.g. in [OSl], p. 114. In the “presheaf” days, in 1953, this Lemma - in a relined form - was called the “Einbettungssatz”; at that time it was the point of departure for the theory of analytic sets, cf. [RS53], p. 267. For the rest of this subsection X denotes an arbitrary reduced complex space. A direct consequence of (12.8) is Theorem 12.9. Every open and finite holomorphic plex manifold is an analytic covering.

map 71:X + Y onto a com-

Proof. The assertion is local. So let Y be a domain in cd with 0 E cd and n-‘(O) = 0. By passing to the graph of 71we may assume that X is a subspace of a product Y x I/ c cd x (CL and that 7c is induced by the projection onto Y. Now apply (12.8). q

Corollary 12.10. (Umkehrsatz). Let X, Y be connected complex manfolds of samedimension and let f: X + Y be a holomorphic injection. Then f(X) is a

domain in Y and f: X + f(X) is biholomorphic. Proof. First f is finite f(X) is finite and thus a holomorphic outside of holomorphic everywhere

at all points and hence open by (10.14). Then f: X one-sheeted analytic covering. Ergo f-’ : f(X) + X a thin set. Since f-l is continuous the map f-l by Riemann’s Extension Theorem.

+ is is

q

There is no “elementary proof” of the Umkehrsatz. The Implicit Function Theorem easily gives a thin set A in X such that f is biholomorphic around all points of X\A. However it is not trivial that f(A) is analytic in f(X). This is taken care of above by the finiteness of J For a direct proof see R. Narasimhan’s nice booklet: Several Complex Variables, Chicago Lectures in Mathematics 1971,86-88. The theorem can be. generalized. By rather routine arguments it is seen: Theorem 12.11. Every open and finite holomorphic map X + Y onto an arbitrary reduced complex spaceis an analytic covering. 5. Existence Theorem for Coverings. Riemann’s Extension Theorem. The first assumption made in the Local Embedding Lemma can always be fulfilled by choosing convenient coordinates, cf. (10.16); then the openness is eo ipso fulfilled if the space is pure dimensional. Thus, by passing to model spaces, we immediately obtain: Existence Theorem for Open Analytic Coverings 12.12. To every point x of a pure d-dimensionalreduced complex spaceX there exists an open neighborhood U of x and an open analytic covering U + Y of a ball in cd.

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This theorem reveals en passant that S(X) is nowhere dense in X. Combining this Existence Theorem with Theorem 12.4 we obtain Riemann Extension Theorem for Locally Pure Dimensional Complex Spaces 12.13. Let X be a reduced locally pure dimensional complex space. Let A be a thin subset of X and let f E O(X\A) be a holomorphic function on X\A. Assume that f is bounded near A or that A is thin of order 2. Then f has a unique meromorphic extensionf to X, and the element f belongs to b(X). Zf X is a normal space, then f is holomorphic in X. Proof. The problem is local. So apply 12.12 and let X be a b-sheeted analytic covering of a ball Yin (cd. We may assume that there exists a primitive element. Then (12.4) gives the assertion.

0 13. Normal Complex Spaces In this section X always denotes a reduced complex space. The space is called normal if all points x E X are normal, i.e. if 6, = 0,. Every complex manifold is a normal space. Neil’s parabola is not normal. 1. General Remarks.

Our first result is rather clear:

(13.1) Every normal space X is locally irreducible and pure dimensional. Proof. If X were reducible at x, there would exist germs h, E dx\O, such that hZ = h,(!). The second assertion now follows from the Criterion 10.17 of Purity. 0

The next statement is fundamental,

for a proof we refer to [CAS], p. 128.

(13.2) The singular locus S(X) of every normal space X has codimension I 2. Thus normal spaces of dimension 1 are Riemann surfaces and normal spaces of dimension 2 have at most isolated singularities: Simple examples are algebraic surfaces in (c3 defined by a polynomial z$ - zlzz, b 2 2, where the origin is the only normal singularity, cf. next paragraphs. 0 In 1961 Mumford showed that a 2-dimensional normal space with singular points can never be a topological manifold [Mu61]. The first examples of singular normal spaces of dimension 2 3 which are topological manifolds were given in 1966 by Brieskorn; he showed [Br66]: (13.3) Let n 2 4 be even. Then the hypersurface X in C” defined by the polynomial z: + z: + .. . + z,‘-~ + zi is a topological manifold though 0 E X is an isolated normal singularity. There are - already in dimension 2 - many isolated normal singularities which are not isomorphic, e.g. the origins of the surfaces zf; = zlzz (for details see paragraph 3). There is only one factorial singularity in dimension 2; more precisely ([Mu611 p. 245, [Br68]):

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(13.4) Let X be singular at x and assume dim, X = 2. Then x is a factorial point if and only if the space germ X, is isomorphic to the space germ at 0 of Klein’s icosaeder surface z: + zz + z: in C3. 0 A classical theorem of Chow and Kodaira (Proc. Nat. Acad. Sci. USA 38, 319-325 (1952)) says that compact 2-dimensional complex manifolds with two independent meromorphic functions are projective. For spaces this is no longer true; Grauert and Hironaka showed independently, cf. [G62], p. 332: (13.5) There exist connected compact 2-dimensional normal complex spaces having exactly one singular point which have two analytically independent meromorphic functions but which are not projective. Such spaces can be obtained from smooth rational surfaces by blowing down a smooth non rational curve; the following device is due to Hironaka: Let X be obtained by blowing up ten points in general position on a smooth cubic C in lP2. The proper transform of C on X has self intersection - 1 and thus can be blown down, [G62]; the result is the desired normal surface. Theorem 12.13 immediately yields: Riemann Extension Theorem for Normal Complex Spaces 13.6. Let X be a normal space, let A be a thin set in X and let f be holomorphic on X\A. Assume that f is locally bounded near A or that A is thin of order 2. Then f has a unique

holomorphic extension to X. Since S(X) is thin of order 2 in X by (13.2), we conclude: (13.7) Zf X is normal, then every holomorphic function on the complex mani$old X\S(X)

extends uniquely to a holomorphic function

on X.

Another consequence of Riemann’s Theorem is the Criterion of Connectedness 13.8. If X is a connected normal space then for each thin set A in X the space X\A is connected. Proof. Let X\A = Y u Y’ be a decomposition into open sets. Put f(x) := 0 for x E Y and f(x) := 1 for x E Y’. Then f E O(X\A) is bounded near A: Its holomorphic extension f has values 0 and 1 only, hence Y = @ or Y’ = @. 0

This criterion is important to characterize irreducible further consequence of the Extension Theorem is:

spaces, cf. (14.15). A

(13.9) If h is a meromorphic function on a normal space, then the polar locus of h is empty or of codimension 1 everywhere. 2. Criteria

for Normality.

The point of our departure is

Lemma 13.10. The following

statements are equivalent:

i) X is a normal space. ii) The singular locus S(X) of X is thin of order 2 in X and for every open set U in X the restriction

map O(U) + cO(U\S(X))

is surjectioe.

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iii) Zf U is open in X every bounded holomorphic uniquely to a holomorphic function on X.

function

in U\S(X)

extends

Proof. i) + ii). Clear by (13.2) and (13.6). - i’1)+ iii) Trivial. - iii) = i). Take a germ h, E 6x,. Choose U around x and a section h E 6(U) through h,. Then P(h) c S(X) and hi U\S(X) is bounded for suitable U, cf. (7.11). Now h has an extension i E O(U). Clearly h, = h, E Ox. Hence 8X c &.

In important cases condition ii) can be exploited. due to W. Thimm, [Th59], Satz 9a:

The following

theorem is

Theorem 13.11. Let X be a reduced Cohen-Macaulay space (e.g. a complete intersection at all its points). Then for every thin set A of order 2 in X the restriction map O(X) -+ O(X\A) is surjective. If, moreover, the singular locus of X is thin of order 2, then the space X is normal.

For complete intersections this result was obtained, independently and by a different method, by Abhyankar [A60], p. 186. For further details we refer to [Sche61] and [K62]. Corollary 13.12. Let X be a reduced hypersurface in a domain that the singular locus S(X) is of dimension
of

(c”. Assume Then X is a

This is Oka’s famous Lemma 1, [OSl], 118-l 19, which he proved by using Cartan’s theorem on three annulli. The corollary is extremely useful; it implies trivially that all surfaces in (lZ3 given by zi - zlzz, b 2 2, are normal. Using the analytic sets S,(O,) = {x E X: cdh 0, I k} Markoe obtained in [Ma741 the following criterion which is the most general version of the results of Abhyankar, Oka, and Thimm: Proposition

13.13. A reduced space X is normal if and only if

dim[S(X)

n S,+,(O,)]

I k

for

all k E Z, k 2 - 1.

There are other criteria for normality. In 9 14 we shall discuss a criterion (14.3) which easily yields the coherence of the sheaf 6,. Using the technique of localization with respect to prime ideals Serre introduced his conditions (R,) and (S,,) and thus derived another simple criterion for normality; for details see [ENS60/61], Exp. 21, p. 3. 3. Theorem of Cartan. A group G of (holomorphic) automorphisms of X acts properly discontinuously on X if for every compact set K c X the set {g E G: gK n K # @> is finite. Then all isotropy groups G, are finite, the orbit space X/G, provided with the quotient topology, is hausdorff, and the orbit projection rr: X + X/G is continuous and open. The family {s E q(U): s o rr E 0,(x-‘(U))} is a sheaf O,,, of local (E-algebras on X/G, and the map n: (X, Ox)+ (X/G, Ox,,) is a morphism of (C-ringed spaces. In [C57] Cartan showed:

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Theorem 13.14. Let G c Aut X act properly discontinuously on the reduced space X. Then (X/G, OX/o) is a reduced complex space and the projection rc: X + X/G is a holomorphic map. If X is normal, then X/G is normal.

Using this device we can easily construct interesting normal spaces. For every complex manifold X and every subgroup G of Aut X of finite order b 2 2 we obtain a normal space X/G; the singular locus is contained in M := n((x E X: G, # (id)) and 7~:X\n-‘(M)+X/G\M is a b-sheeted covering. Beautiful examples are given by the fiue types of finite subgroups of SL(2, (IJ acting linearly on (c. We indicate the procedure for the cyclic group G generated by the automorphism U? + (c’, (u, u)H(~u, q-iv), where rl = exp(2rri/b), b 2 2. The G-orbits are the fibers of the map f: (E2 + (I?, (u, u)H(u*, u*, uu). Since fW') = {(z,, z2, z3) E (I?: zi = z,z2} we see that the algebraic surface Fb in Q? defined by zi - z1z2 is isomorphic to the orbit space @‘/G and hence a normal space. Since n: c2\0 -+ Fb\O is a b-sheeted universal covering the surface Fb is not a manifold around 0. The local fundamental group of Fb at 0 is cyclic of order b, hence Fb and F, are never isomorphic around 0 if b # c. Similarly one can deal with the binary dihedral groups and the binary tetrahedral, octahedral and icosahedral groups; for details we refer to the booklet Singularittiten, Birkhiuser 1991, by D. Battig and II. Knorrer. 4. Determinantal Spaces. Segre Cones. Let m, n, r be integers such that 2 I m I n and 0 < r < m; let Z = (z,,) be an m x n-matrix of mn complex variables. The closed subspace O;,n of cm” defined by the ideal generated by all (r + l)minors of Z is called a determinantal space. Proposition 13.15. The space DL,. is a normal Cohen Macaulay space of (pure) dimension r(m + n - r) with singular locus 0. We have emb, Ok,. = mn.

The assertions about the dimension, the singular locus and the embedding dimension can be obtained rather easily. The proof that D&,, is Cohen Macaulay is more involved; we refer the reader to [BVSS]. Once this has been shown, the space is reduced by (11.14) and the normality of D;,, follows from theorem 13.11, since 2r + 1 I m + n. 0 The number

of (r + 1)-minors of the matrix

Z is

(rrI)(rY

1)’ These

minors form a minimal system of generators of the ideal defining &,, at 0, hence DL,, is a complete intersection at 0 if and only if r + 1 = m = n. The minimal number of germs needed to describe Ok,. set-theoretically around 0 is mn (r + 1)2 + 1; hence Oh,, is set-theoretically a complete intersection at 0 if and only if r + 1 = m = n; for details see [BVSS] and [BruSchw90]. Let us have a closer look at the space S := Di,,. Putting z, := zlv and w, := zlv this space S is defined by the ideal $ which is generated by the l quadratic 0 polynomials wjzk - wkzj, 1 I j < k I n. We call S c (c’” the Segre cone with vertex s := 0 E S. It is easy to see:

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(13.16) All points of S\s are smooth. The cone S c (lJ2nis of dimension n + 1 everywhere. The vertex s is a singularity: emb, S = 2n. The spacered S is irreducible at s. Proof. Take x E S\s, say zi(x) # 0. Then, around x, the ideal 9 is generated by the n - 1 functions w, - (wi/zr)z,, 2 5 v I n. Thus x is a smooth point and dim, S = n + 1. Now dim, S = n + 1 since s is not isolated in S. Since -a0 c m(0,,,,,)2 we have emb, S = 2n by 4.8. If red S would have two prime components S,, S, at s, both would have dimension n + 1 and the Intersection Inequality 10.8 would yield dim, (S, n S,) 2 2, which contradicts S, n S, c S(S) = s. 0 It needs some labor to prove directly that S is reduced and hence normal at s. ~ We list some remarkable geometric properties of the Segre cone. Since

s=

5=...=f! i Wl

wnI ’

the set S is the union of all n-dimensional planes E, := {(z, w) E (G”‘: z = aw},

a E lP,,

with E, = {(z, w) E c”‘: w = O}.

Every plane E, is of codimension 1 in S, but E, n E, = {s} for all a # b (in complex manifolds such intersections are always of codimension at most 2). The map h: S\s + lP, attaching to x E E, the value a is a meromorphic function on S with s as only point of indeterminancy (in manifolds sets of indeterminancy are always of codimension 2 everywhere). Finally we note:

If fi> '..3 fk are holomorphic in a neighborhood U N(f,, . . . , fk) = E, n U, then k 2 n.

of

s E S and if

This is clear since on every n-plane Et,, b # a, these functions have s as only common zero (in a manifold every hypersurface is, locally, the zero set of one holomorphic function). 5. Divisor Class Groups and Factoriality. For every normal point x E X the divisor classgroup Cl(0J is the factor group of the group @ Zp of all divisors hfp=l

in x by the subgroup of all principal divisors div f, f E Ax\(O). The group Cl(0J measuresthe deviation of the family of pure 1-codimensional analytic sets at x from the family of zero sets of germs in ox. In general the group Cl(0J is not finitely generated, e.g. we have Cl(0,) = UZ for X := T/(x2 + y3 + z7) c (c3, cf. [Sto84], 194-195. Applying Grauert’s Coherence Theorem to a proper resolution of singularities of X it can be shown, cf. [Sto69], p. 97: (13.17) If X is normal, the set of all points x E X, where Cl(0,) is not finitely generated, is analytic in X.

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In many interesting cases - for example for isolated singularities - the group Cl(&) carries, in a natural way, the structure of a complex Lie-group. Sometimes CI(0J is finitely generated, e.g. we have CI(0,) = Z for the origin of all determinantal spaces, cf. [Sto69], p. 99, for the Segre cone and [BVSS], p. 96 for the general case. An important tool for studying and computing the groups C1(&.) is a comparison method about the behavior of these groups when passing from the complexalgebraic to the complex-analytic situation and to formal completions. 0 A normal point x E X is factorial if and only if CI(ox) = 0. The following criterion goes back to Grothendieck; cf. Cohomologie locale des faisceaux coherents et Theoremes de Lefschetz locaux et globaux (SGA2), North-Holland Publ. Comp. 1968, furthermore [Sto69] and [Bi77]: (13.18) Let the normal space X be a complete intersection at all its points and assume that 0, is factorial for all points outside of an analytic set in X of codimension 2 4. Then all stalks ox, x E X, are factorial. The set of all non factorial points of a normal space is not necessarily closed in X. For example the 3-dimensional space X := I/(x2 + y3 + z3 + z2w) c (E4 is normal and has the line given by x = y = z = 0 as singular locus; the origin 0 E X is a factorial point but Cl(cOJ = 2/32 for all points p E S(X)\(O); for details see [BF84], p. 37. In 1976 Bingener showed: (13.19) For every normal space X the set of all (not) factorial cally constructible. For a proof and a generalization

points is analyti-

of this statement, see [BF84].

9 14. Normalization For every reduced complex space X = (X, 0) the sheaf 6 of normalization of sheaf 0 is a subalgebra of the sheaf J%’ of meromorphic functions, cf. 4 7.3. The stalk 8x,, x E X, is the integral closure of 0x in the quotient ring Ax; we recall the fundamental property (1.23): the structure

C?~is the maximal ring extension of 0, in J#~ which is a finite O,-module. The study of the sheaf d leads to deep insights into the geometric structure of the underlying space. We discuss a Criterion of Normality which immediately implies the openness of the set of normal points. Using this fact we easily obtain the Co-coherence of d (Theorem of Cartan-Oka). In 9 12.2 we obtained normal complex spaces from reduced ones by “manipulating” at their singular loci. Such “normalizations” exist for all reduced spaces X, they are nothing else than the analytic spectrum of the coherent @,-algebra 6, (Theorem 14.9). In view of (13.2) passing to normalization resolves all singular loci of codimension 1. In this section X, Y, Z always denote reduced spaces.

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1. Theorem of Cartan-Oka. Our aim is to show that the sheaf 8 is coherent. First we exhibit a rather unexpected connection between & and ideals in 0. Lemma 14.1. Let Y c 0 be a reduced, finite ideal with a thin zero set in X. Then there exists a canonical O-monomorphism o: Z’MZ(& 9) + A? such that

(Im (T), = {h, E fix,: h,& Proof.

c Ox},

0 c Im 0 c d.

(*)

Since N(9) is thin, every stalk Yx contains active germs.

1) Construction of a,: Write a := Yx. Every germ h E jae, with ha c a induces an O,-homomorphism a + a, f t+ hf. Conversely, every c1E Hom(a, a) is of this form: just choose an active germ g E a and put h := cc(g)/g. In this way we obtain an O,-monomorphism a,: Hom(a, a) + Ax, CIH h; hence Hom(a, a) 2 Im o, = {h E Jld,: ha c a}.

(0)

Im a, is a ring extension of 0x in Jll,. Since Hom(a, a) is a finite Lo,-module, we have Im a, c 6x and therefore Ox c Im o, c {h E 6x,: ha c O,}. We claim that Im o, = {h E &x: ha c ox}. Let h E &x and ha c ox. Since h” = with c, E cOx,n 2 1 suitable, we conclude n-l (hf )” = “& (c,f “-‘)(hf)’ E a for every f E a.

co + c,h + ... + c,-Ihn-l

Thus ha c rad a = a, hence h E Im CJ,by ( 0 ). 2) Sheafifying: Since Hom(& YJ = %‘uM(~, 9), by the coherence of 9, the maps a, define a map Q: J&P@ $) + JV. Clearly r~ is continuous and hence an O-monomorphism. 0 We shall apply the Lemma to the ideal sheaf 9 := .Yss(x) of the singular locus of X. This sheaf 3’ is tied to & in the following way: (14.2) To every germ h, E 6X there exists an integer t 2 1 such that h,-YJ c Ox. Proof. Choose a neighborhood U of x and a section h E 6(U) representing h,. Since P(h) c S(X) we have JPchi3 9” and hence h,9’; c Ox by (8.10). 0

Now we get a surprisingly

simple

Criterion for Normality 14.3. Let o: ~uw@&, 9&) + A denote the canonical O-monomorphism. Then a point x E X is normal if and only if Im a, = ox. Proof. If fix = 0, then Im o, = Co, by (14.1). Now_let Im o, = ox. According to (14.1) it suffices to prove h,Px c Co, for all h, E cOx.By (14.2) there exists a minimal t 2 1 such that h,Pi c ox. Assuming t > 1 we find a germ gx E .5$-’ with h,g, $ cOx.Since h,g, E & and (h,g,)5Zx c O,, we derive from (14.1) the contradiction h,g, E Im a, = ox. Hence t = 1. 0

A straightforward

consequence is the renowned

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Theorem 14.4. The non-normal locus N(X) of a reduced space X is an analytic set in X. In particular, the set of normal points of X is open in X. Proof The Criterion of Normality yields N(X) = Supp(Im a/0). Since Im 0 is coherent, the sheaf Im g/O is coherent. Hence its support is analytic. Cl

Now - using the local existence of analytic spectra (Lemma openness of normal points - it is fun to show Theorem of Cartan-Oka &-coherent.

8.14) and the

14.5. For every reduced space X the sheaf 8, is

Proof. Let p E X. Choose sections si, . . . , sk E 8(U) in a neighborhood U of p which generate &P as @,-module. We denote by J&’ the $-algebra generated

by ~1, . . . . sk. Clearly Co, c &’ and 58, = &jP. Since si, . . . , sk are integral over 0(U) for small U, the (?&-sheaf & c 4” is finite for small U and hence coherent by (7.8). Furthermore, we have & c 0,. The coherence of 8, at p will follow if we show 8, c d for suitable U. For U small enough there exists by (8.14) a finite holomorphic map c: V + U such that & g [JO,). Then J&‘~ r @ Co,,, for all x E U. Hence &‘x is integrally m=x closed in J%J~if and only if each stalk of 0, over x is normal. Since &‘, = fiPp,all stalks over p are normal. Now (14.4) yields that all stalks Co”,=,z E c-‘(U), are normal for U small enough. Hence all algebras s$x, x E U are integrally closed. Since J&‘~ EJ &J~it follows that (?x c JG’~for all x E U, i.e. 6, c d. 0 There is no direct argument to see that the normal points form an open set in X. Of course, now this follows trivially, since N(X) is the support of the coherent sheaf 6/O. Another immediate consequence of (14.5) is the local existence of “universal denominators” u for 6, over 0,; more precisely: (14.6) For every point p E X there exists an open neighborhood U and a function u E O(U) such that every germ ux, x E U, is active and u&v c 0,. The nonnormal points of U are contained in the zero set of u. Proof.

Since Supp &/lo is thin in X, all stalks of&n

&/lo contain active germs

by (7.9). 2. Normalization of Reduced Spaces. Recall that a finite surjective holomorphic map q: Y -+ X is called a one-sheeted (analytic) covering if there exists a (critical) thin set A in X such that n-‘(A) is thin in Y and v]: Y\q-i(A) + X\A is biholomorphic, cf. Q12.1. Such a covering is called a normalization of X if Y is a normal space. Normalizations “dominate” all one sheeted coverings: Proposition 14.7. If n: Y --, X, [: Z -+ X are one sheeted coverings and tf Y is normal, there is a unique holomorphic map f: Y + Z such that n = [ of. If Z is normal too the map f is biholomorphic.

critical set A c X for r7 and [. Then f := [-' o q: is biholomorphic. Fix y E q-‘(A). It is easily seen that

Proof. There is a common

Y\?-‘(A)

+ Z\c-‘(A)

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88

there exist neighborhoods I/ of y and W of i-‘(y) such that f maps V\q-‘(A) into Wand, moreover, W is isomorphic to a model space in a bounded domain of (c”. This map is given by n bounded holomorphic functions. Since V is normal, there exists a holomorphic extension to I/ by (12.13). This yields a holomorphic map f: Y + Z. Since q-‘(A) is thin in Y the map f is unique and q = [ 0 f on Y. -If Z is normal we may extend f-‘: Z\[-‘(A) + Y\q-‘(A) as well. Thereforef is biholomorphic now. cl Clearly the examples a)-d) in 0 12.2 are normalizations. Locally all normal spaces are normalizations of hypersurfaces in number spaces; more precisely: Proposition 14.8. To every normal point x of a d-dimensional complex space X there exist a neighborhood U of x, a ball B in Cd and a Weierstrass space W in B x C defined by one manic polynomial w E O(B) [w] such that U is isomorphic to the normalization of W. Proof. Realize X around x as an analytic covering of a ball in ad and choose for o the characteristic polynomial of a primitive element of such a covering, for details see [CAS], p. 162. cl In general normalizations are obtained as follows: Theorem 14.9. The analytic spectrum 5: X -+ X of 6x is a normalization

of

X.

The map 5 is finite and 6x,, g t,(c”a), = @ Ow,, for all x, cf. $8.5. w=x Since 6 x,x is integrally closed, all stalks O%,=,z E c-‘(x), are normal, i.e. the space r? is normal. The non-normal locus N of X is thin in X. Since OX,N = OX,N the map 5: X\<-‘(N) + X\N is biholomorphic. It remains to show that t-‘(N) is thin in 2. Let p E N and choose u E Lo,(U) in a neighborhood U of x as in (14.6). Then u vanishes on N n U, hence u := u o 5 vanishes on t-‘(N n U). It suffices to show that the zero set of u is thin or equivalently that every germ uz, z E t-‘(U), is active in Og,,. Now u, = @ u,, if u, is viewed as an element of m)=x K!?,,, 2 @ Oa,,. Since u, is active in 6 x,x by (14.6), all germs u, are active. 0 ‘%)=x Remark. In view of (14.7) the existence of 5: 8 + X follows immediately from the Gluing Lemma 3.5 and the Local Existence Lemma 8.14. Hence the general technique for constructing analytic spectra is not needed to obtain normalizations. In what follows 5: X + X always denotes a normalization of X. Proof.

of

(14.10) The number of points of the fiber c-‘(x), prime germs of the space germ X,. Proof.

By (7.12) the algebra 6, x z 5,(cOa)x = @ A

x E X, equals to the number

W)=x

0~~~ is also the ring-direct

sum of the analytic algebras (?~‘~,,/p, where p runs through all minimal prime ideals of d,,,. In both direct sums we have the same number of summands. Since there are as many minimal prime ideals as prime germs of X,, we obtain (14.10). 0

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If we denote by X’ the set of all prime germs of X and by 5’: X’ -+ X the surjection sending prime germs to their corresponding points, it follows from (14.10) that there is a bijection T: J? --, X’ such that 5 = C;’0 5. It is possible to define - in an intrinsique way without looking at 2 - a topology on X’ such that there exists exactly one homeomorphism z: 8 -+ X’ satisfying 5 = r’ o z; for details cf. [CAS], 162-163. 0 From (14.10) we obtain by applying (10.15) to <: (14.11) 1f X is locally irreducible, the map 5: 2 + X is a homeomorphism. For every one-sheeted covering ‘I: Y-P X the lifting map q: 0, + q.JO,) is injective and extends to an O,-monomorphism @: 4, + q&My). By applying Riickert’s Nullstellensatz it can be shown, [CAS], p. 154- 155: (14.12) For every one-sheeted covering r~: Y + X, in particular for every normaliza;ion, the sheaf map q: Ax + ~.+(..&r) is an Ox-algebra isomorphism which maps Ox onto q*(O,). The normalization of a direct product is the direct product of the normalizations of the factors; in particular X x Y is normal if and only if X and Y are normal. Let us emphasize however that “passing to normalization” is not a fun&or. If 5: 2 -+ X, q: p + Y are normalizations and f: Y + X is holomorphic there may exist no holomorphic map f^: ? -+ 2 such that < of* = f o q. The familiar example is Whitney’s umbrella: Using earlier notations the map f: C2 + X; z-(0,0, z) into the singular locus of X cannot be lifted (the lifting condition is ub = z). In general a lifting off: Y + X is possible (and unique) if the f-inverse of the nonnormal locus of X is thin in Y. 3. Irreducible Spaces. Global Decomposition. Let 5: 2 + X be a normalization. The space X is called irreducible if there are no proper analytic sets A # 0, B # (21in X such that X = A u B. Irreducible spaces may have reducible points. Proposition 14.13. Let X be irreducible. Then the spaces X\S(X) and 2 are connected. Furthermore, every proper analytic set in X is thin in X. Moreover, Jt(X) is a field and 8(X) is an integral domain. Conversely, each of these properties implies that X is irreducible. For a proof cf. [CAS], p. 168/169. - Here are some further properties: (14.14) Let X be irreducible. Then X is pure dimensional. If A is an analytic set in X such that dim, A = dim X for at least one point x, then A = X. Two meromorphic functions on X are equal if they coincide on an open set #@. Zf X --) Y is holomorphic and surjective the (reduced) space Y is irreducible. Every connected reduced locally irreducible space is irreducible.

In (4.5) and (10.4) we have “decomposed” analytic sets, locally, into prime components. Using normalizations we can establish a global analogue. We call a family {X,} of irreducible analytic subsets of a reduced space X a decomposi-

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tion of X into irreducible components if {X,} is a locally finite covering of X such

that Xi q? Xj for all i, j, i # j. Global Decomposition Theorem 14.15. There exists a unique decomposition of X into irreducible components. These components are just the topological closures of the connected components of X\S(X), i.e. the <-images of the connected components of 8.

For a proof we refer to [CAS], p. 172/73. - Using irreducible easily obtain a general

components

we

Global Maximum Principle 14.16. Zf f E O(X) has a local maximum at p E X, then f is constant on all irreducible components of X passing through p. 4. Historical Notes. The concepts of normality and of normalization, which are purely algebraic in character, were introduced long ago by R. Dedekind in Algebraic Number Theory (see his Vorworte to Dirichlets Vorlesungen iiber Zahlentheorie and his famous Supplement XI ober die Theorie der ganzen algebraischen Zahlen, all in Ges. Math. Werke III). In Algebraic Geometry 0. Zariski recognized the fertility of these notions (see his papers The Fundamental Ideas of Abstract Algebraic Geometry, Coll. Pap. III, 363-375, in particular p. 368, and Sur la normalite analytique des uarietts normales, Coll. Pap. III, 162- 165). In Analytic Geometry the power of these notions were seen and stressed by K. Oka and H. Cartan. Already in the days of Weierstrass it was intuitively clear that - locally - all hypersurfaces in (c” admit normalizations, cf. [LF], Theorem on p. 135/36. Oka expresses this in the three words “grace a Weierstrab”. In his paper [OSl] he makes large strives towards an existence proof in his endeavours to understand better the local structure of “domaines interieurement ramifies”: Property (H) on p. 118 is his definition of normality; the (W)-functions on p. 119/20 are universal denominators, while normalization sheaves already occur in a disguised form. Oka’s approach is analytic and rather breath-taking, cf. Cartan’s comments in [OSl], p. 132. Oka’s ideas and results were put in a rigorous setting by Cartan in his seminar [EN53/54], Exp. VI-XI. In Expose X we find the algebraic definition of normalization sheaves, the theorems 14.4 and 14.5 are stated there explicitly on p. 3 and p. 4, while the existence theorem 14.9 occurs only implicitly. The proofs of theorem 14.4 and 14.5 given above were known to the authors around 1962. Another proof of theorem 14.5 using Serre’s condition (R,) and (S,,) can be found in [EN60/61], Exp. 21. In 1961 N. Kuhlmann proved the existence of normalizations, cf. [Ku61]; another proof is in Abhyankar’s book [LAG], p. 464. The proof via analytic spectra is in [ENS60/61], Exp. 21, p. 11. Sometimes it is claimed that the Global Decomposition Theorem was known, in principle, to Weierstrass for subspaces of domains in (c”. In modern setting the theorem is in [C44], p. 611/12. Readers interested in more history concerning global decomposition may consult [He49].

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0 15. Semi-Normalization The definition of the sheaf 6 is algebraic. Due to Riemann’s Extension Theorem (12.13) for bounded holomorphic functions the sheaf 6 turns out to be equal to the analytically defined sheaf of germs of weakly holomorphic functions on X. An interesting subsheaf of 6, is the sheaf 6, = 6, n SX of germs of continuous weakly holomorphic functions on X. This sheaf is coherent too and its analytic spectrum is a semi-normalization of X. All spaces X, x 2 are assumed to be reduced. 1. Function-theoretic Characterization of 8. A weakly holomorphic function on an open set U c X is a holomorphic function f: U \ A + (c which is defined outside of a thin set A in U and which is bounded near A. In view of Riemann’s Extension Theorem for manifolds one always can assume that A = S(U). The set O’(U) of all such functions is an O( U)-algebra; clearly we obtain an O-algebra 0’ 3 0. Theorem 15.1. The sheaf 0’ is the normalization sheaf 6 of 8. Proof. Fix x E X. The inclusion 6X c 0; is an immediate consequence of (7.11). The other inclusion 6’: c 6X is clear by (12.13) if X is pure dimensional. The general case is reduced to this one as follows: Let U = U, u .. . u U, be a decomposition of a neighborhood U of x into prime components such that S(U) n Uj is thin in Uj, 1 I j I t. Then the “restriction map”

o;(u)+~o~j(uj,, 1

sHS~UIO~~~@S(Ut

is bijectiue and sends 6,(U) c O;(U) into @ 6nj(Uj). Since U can be chosen arbitrarily small we get a bijection cp: 0: + @ Obj,, mapping &x c 0; into @ ~9”~~~.Now all spaces Uj are pure dimensional around x. Thus we know Shj,, = c%“~,,.Since cpis bijective we conclude 0: = &x. 0 2. Semi-Normalization. A weakly holomorphic function s E &x(U) is called continuous weakly holomorphic if the holomorphic function sI U\S(X) admits a (unique) continuous extension to U. We denote the attached sheaf by 8, and call this Ox-algebra the semi-normalization (sheaf) of cx. Neglecting formal

trouble we may write fix = &x n Vx. The sheaves 8x, Ox are uniquely determined by Ox,A whenever A is a thin set in X containing S(X). In general the inclusions are genuine (e.g. for the space of coordinate axes in C2). A point x E X is called semi-normal(or maximal) if 6x,, = Ox,,. Spaces with semi-normal points only are called semi-normal (or maximal). Normal points are semi-normal. Whitney’s

umbrella is semi-normal,

but not normal. Neil’s parabola is not semi-normal.

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The definition of semi-normality yields right away a “weak Riemann theorem of removable singularities for holomorphic maps”: (15.2) Every continuous map X -+ Z of a semi-normal space into a reduced space which is holomorphic outside of a thin set in X is holomorphic. Clearly the natural to see that and a loan

2 := (X, 6,) is a ringed subspace of (X, %‘x). Since Ox t 6x we have morphism I: (X, fix) + (X, Ox) with 111= id. There is no simple way 2 is a complex space. A detour via a normalization 5: X + X of X from the theory of analytic equivalence relations are necessary. Since

a function f E @a( t-‘( U)) is of the form f = f 0 c$? with f E 6.J U) if and only if f is constant on all c-fibers, the ringed space (X, 6x, is isomorphic, in a natural way, to a ringed quotient space (X/R, Oa,a) of (X,82): The equivalence relation R has the l-fibers as equivalence classes and the sheaf #flR is given by the R-invariant holomorphic functions in R-saturated open sets of 2. We now bor-

row the following result from the theory of analytic equivalence relations*: ( 0 ) Let R be a finite analytic equivalence relation on a reduced complex space (X, Ox). Assume that each point x E X has an open R-saturated neighborhood U such that there exists an R-invariant holomorphic map U + Y with discrete fibers into some complex space Y. Then the ringed quotient space (X/R, OXIR) is a reduced complex space.

Taking this for granted we easily obtain TheoremJ5.3. The ringed space r? := (X, fix) is a semi-normal complex space. The map t: X + X is holomorphicLand there is a unique holomorphic map p: 8 + X such that { = I 0 p. The sheaf 0, is Co,-coherent. The set Supp 6’,/Ox c N(X) of non semi-normal points of X is thin, Proof. We identify (X, 6x) with (X/R, OXIR). The equivalence relation R on X is analytic (its graph is the 4 x c-inverse of the diagonal X x X). Since 5 is finite, the relation R is finite. Furthermore, the condition of ( 0 ) is fulfilled for 5: X + X. Hence (X, I!?~) is a complex space. Clearly I is holomorphic. Since I is finite, the Ox-sheaf 1,(8x) = 6x is Ox-coherent. The unique existence of p is trivial. From z(S(r?)) c S(X) and 6x-s(xJ = Ox-,(,, we get g(U) = I!?(U) for all open sets U in X. Hence (X, 8x) is semi-normal. 0

Holomorphic homeomorphisms characteristic for semi-normality:

need not be biholomorphic.

*An equivalence relation R on a reduced space xRx’} is an analytic set in X x X. The relation R set is compact and if every equivalence class is analytic equivalence relations are encouraged to

This property is

X is called analytic if its graph ((x, x’) E X x X’: is called finite if the R-saturation of every compact finite. - Readers who want to learn more about study Chapter IV, $54-7.

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(15.4) A reduced space X is semi-normal if and only if every holomorphic homeomorphism f: Z + X of a reduced space Z onto X is biholomorphic. If the condition holds, the map I: r? + X has to be biholomorphic i.e. X = 2 is semi-normal. Conversely, let X be semi-normal. The set A := S(Z) u f-‘@(X)) is th’ m in Z. Hence f(A) is thin in X. Since dim, Z = dimJ(,) X for all z E Z, and since Z\A, X\f(A) are complex manifolds, the restriction f: Z\A + X\f(A) is biholomorphic by (12.10). We may assume that IZJ = 1x1 and IfI = and hence Ez = 8,. Since ax = 6, c Co,c fiz we obtain id. Then oziA = OX,,,, 0, = 8,. 0 Proof.

We see that structure sheaves of semi-normal spaces are maximal in the sense that they cannot be enlarged to a reduced complex structure sheaf; this motivates the old terminology. From (15.4) and (14.11) we derive directly: (15.5) Every locally irreducible semi-normal space is normal. Another corollary of (15.4) is a converse of the Graph Lemma 5.5. (15.6) A continuous map f: X + Y of a semi-normal space X into a reduced space Y is holomorphic if its topological graph lXfl is analytic in X x Y. Proof. Denote, as in (5.5), by p, q the holomorphic projections of XJ- to X, Y. Since f is continuous p is a homeomorphism. Hence p is biholomorphic and f = q o p-l is holomorphic. 0

A holomorphic map I: r? + X is called a semi-normalization tion) of X if 2 is semi-normal and if I is a homeomorphism. (15.4) we derive at once:

(or a maximalizaFrom (15.3) and

Theorem 15.7. Every reduced space X has a semi-normalization I: r? + X (this is the analytic spectrum of 8,). Zf ‘I: Y -+ X is another semi-normalization, there exists a unique biholomorphic map f: Y G z such that 9 = 1 0 f.

Passing to semi-normalization “maximalizes” the structure sheaf without changing the underlying topological space. In contrast to the situation of normalization we now have a functor: (15.8) Zf I: 2 + X, I~: zI + X, are semi-normalizations, every holomorphic map f: X -+ X, admits a unique holomorphic lifting 5 2 + r?, such that zI 0 f = f 0 1. Proof. We have to put IfI := Ifl. By (15.6) it suffices to show that (Xi\ = lXfl is analytic in 2 x 2,. This is obvious since I x zl: 2 x g1 -+ X x X, is holomorphic and IX,-1 is analytic in X x X,. q Finally, we note that the semi-normalization of a direct product is the direct product of the semi-normalizations of the factors; in particular, X x Y is seminormal if and only if X and Y are semi-normal.

R. Remmert

References* Monographs [LAG1 CAM1 [BVSS]

CEW [CA’4 PI CAN ICASI [TMAG]

CHFI [ICAG]

PI WAC1 iTEA1 [GS71]

WV1

Abhyankar, S.: Local Analytic Geometry. Acad. Press 1964,Zbl.205,504. B&ii&, C. and Stanasila, 0.: Algebraic Methods in the Global Theory of Complex Spaces. Wiley 1976,2b,1.334.32001,2b1.284.32006. Bruns, W. and Vetter, U.: Determinantal Rings. Lect. Notes Math. 1327, Springer 1988, Zbl.673.13006. Cartan, H.: Seminaire H. Cartan, EC. Norm. Super. 1950/51, 1953/54, 1960/61, Zbl.l16,244. Fischer, G.: Complex Analytic Geometry. Lect. Notes Math. 538, Springer 1976, Zb1.343.32002. Godement, R.: Topologie algebrique et theorie des faisceaux, Hermann 1963, 3. ed, 1973,Zbl.80,162. Grauert, H. and Remmert, R.: Analytische Stellenalgebren. Springer 1971, Zbl.231.32001. Grauert, H. and Remmert, R.: Coherent Analytic Sheaves. Springer 1984, Zbl.537.32001. Hirzebruch, F.: Neue Topologische Methoden in der Algebraischen Geometrie. Erg. Math., Springer 1956; translated and extended 1966 by R.L.E. Schwarzenberger, Topological Methods in Algebraic Geometry, last edition 1978,Zbl.138,420. Kaup, L. and Kaup, B.: Holomorphic Functions of Several Variables. de Gruyter 1983,Zbl.528.32001. Lojasiewicz, St.: Introduction to Complex Analytic Geometry. Birkhauser 1991, Zbl.747.32001. Osgood, W.F.: Lehrbuch der Funktionentheorie 11.1, 2. Auflage, Teubner 1928, Jbuch 54,326. Serre, J-P.: Faisceaux algtbriques coherents, Ann. Math., II. Ser. 61, 197-278 (1955) Zb1.67,162; (Euvres I, 310-391. Siu, Y-T.: Techniques of extension of analytic objects, Lect. Notes Pure Appl. Math., Marcel Dekker Inc. 1974, Zbl.294.32007. Siu, Y-T. and Trautmann, G.: Gap - Sheaves and Extension of Coherent Analytic Subsheaves. Lect. Notes Math. 172, Springer 197l,Zbl.208,104. Whitney, H.: Complex Analytic Varieties. Addison - Wesley 1972, Zbl.265.32008.

Articles I.4601 CAP703 [BF84] [BSSl] [Bi77]

Abhyankar, S.: Concepts of order and rank on a complex space, and a condition for normality. Math. Ann. 141, 171-192 (1960) Zbl.107,150. Abhyankar, S. and Van Der Put, A.: Homomorphisms of analytic local rings. J. Reine Angew. Math. 242, 26-60 (1970) Zbl.193,5. Bingener, J. and Flenner, H.: Variation of the divisor class group. J. Reine Ang. Math. 351, 20-41 (1984) Zbl.542.14003. Behnke, H. and Stein, K.: Modilikation komplexer Mannigfaltigkeiten und Riemannsche Gebiete. Math. Ann. 124, 1-16 (1951) Zbl.43,303. Bingener, J.: Uber die Divisorenklassengruppen lokaler Ringe. Math. Ann. 229, 173-179 (1977) Zbl.338.13016.

* For the convenience of the reader, compiled using the MATH database, have, as far as possible, been included

references to reviews in Zentralblatt and Jahrbuch iiber die Fortschritte in this References.

fiir Mathematik der Mathematik

(Zbl.), (Jbuch)

I. Local [Boh86] [Bo67] [Br66] [Br68] [BruSchw90] K481 K501 rc571

P361 F671 P551 P3601

[G62] [GR58a] [GR58b] W471 W611 K621 [La711 [La771 [La781 [Las051 [Ma741 [Mu611

co501

Theory

of Complex

Spaces

95

Bohnhorst, G.: Einfache holomorphe Abbildungen. Math. Ann. 275,513-520 (1986) 2131.585.32029. Bosch, S.: Endliche analytische Homomorphismen. Nachr. Akad. Wiss. Gottingen, Math. - Phys. Kl., 41-49 (1967) Zb1.154,37. Brieskorn, E.: Examples of singular normal complex spaces which are topological manifolds. Proc. Nat. Acad. Sci. USA, 55, 1395-1397 (1966) Zbl.144,450. Brieskorn, E.: Rationale Singularitaten komplexer Fllchen. Invent. Math. 4, 336358 (1968) Zbl.219.14003. Bruns, W. and Schwiinzl, R.: The number of equations defining a determinantal variety. Bull. Lond. Math. Sot. 22,439-445 (1990) Zb1.725.14039. Cartan, H.: Idtaux de fonctions analytiques de n variables complexes. Ann. Sci. EC. Norm. Sup., III, Ser. 61, 149-197 (1948) Zbl.35,171; (Euvres II, 565-613. Cartan, H.: Ideaux et modules de fonctions analytiques de variables complexes. Bull. Sot. Math. Fr. 78,29-64 (1950) Zbl.38,237; (Euvres II, 618-653. Cartan, H.: Quotient d’un espace analytique par un groupc d’automorphismes. Algebraic Geometry and Topology, A symposium in honor of S. Lefschetz, Princeton Math. Ser. 12,90-102 (1957). Zbl.84,72; (Euvres II, 687-699. Cartan, H.: Sur le theoreme de preparation de Weierstrass. Arbeitsgemeinschaft Forsch. Nordrhein-Westf. 33, 155-168 (1966). Zbl.144,78; (Euvres II, 875-888. Frisch, J.: Points de platitude dun morphisme d’espaces analytiques complexes. Invent. Math. 4, 118-138 (1967) Zbl.167,68. Grauert, H.: Charakterisierung der holomorph vollstlndigen komplexen Raume. Math. Ann. 129, 233-259 (1955) Zbl.64,326. Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulraume komplexer Strukturen. Publ. Math. Inst. Hautes Etud. Sci., N”5, 5-64 (1960) Zbl.100,80. Grauert, H.: Uber Moditikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331-368 (1962) Zb1.173,330. Grauert, H. and Remmert, R.: Bilder und Urbilder analytischer Garben. Ann. Math., II, Ser. 68, 393-443 (1958) Zb1.89,60. Grauert, H. and Remmert, R.: Komplexe Raume. Math. Ann. 136, 245-318 (1958) Zbl.87,290. Hermes, H.: Analytische Mannigfaltigkeiten in Riemannschen Bereichen. Math. Ann. 120, 539-562 (1949) Zbl.32,67. Kuhlmann, N.: Die Normalisierung komplexer Raume. Math. Ann. 144, 1 lo-125 (1961) Zbl.96,278. Kuhlmann, N.: Uber die normalen Punkte eines komplexen Raumes. Math. Ann. 146, 397-412 (1962) Zbl.115,68. Langmann, K.: Ein funktionalanalytischer Beweis des Hilbertschen Nullstellensatzes. Math. Ann. 192,47-50 (1971) Zbl.203445. Langmann, K.: Zum Satz von Frisch. Math. Ann. 229, 141-142 (1977) Zbl.339.32008. Langmann, K.: Zur KohLrenz der Idealgarbe. Arch. Math. 31, 565-567 (1978) Zbl.407.32012. Lasker, E.: Zur Theorie der Moduln und Ideale. Math. Ann. 60, 20-116 (1905) Jbuch 36,292. Markow, A.: A characterization of normal analytic spaces by the homological codimension of the structure sheaf. Pac. J. Math. 52,485-489 (1974) Zbl.268.32006. Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. Inst. Hautes Etud. Sci., N” 9, 5-22 (1961) Zb1.108,168. Oka, K.: Sur les fonctions analytiques de plusieurs variables VII: Sur quelques notions arithmetiques. Bull. Sot. Math. Fr. 78, l-27 (1950) Zb1.36,52; COB. Pap. 80-108.

96

IO511 [RS53] [Rii33] [Sche61] [Sche64] [Schu76] [Se561 [Se661 [Si68] ~~~291 [Sti1887] [Sto69] [Sto84] [Th59] [Th62]

WI

R. Remmert Oka, K.: Sur les fonctions analytiques de plusieurs variables VIII: Lemme fondamental. J. Math. Sot. Japan 3, 204-214. Zbl.43,304, 259-278, Zb1.45,41 (1951); Coll. Pap. 1099132. Remmert, R. and Stein, K.: Uber die wesentlichen Singularitaten analytischer Mengen. Math. Ann. 126, 263-306 (1953) Zbl.51,63. Riickert, W.: Zum Eliminationsproblem der Potenzreihenideale. Math. Ann. 107, 259-281 (1933) Zb1.5,98. Scheja, G.: Eine Anwendung Riemannscher Hebbarkeitssatze fib analytische Cohomologieklassen. Arch. Math. 12, 341-348 (1961) Zbl.l16,290. Scheja, G.: Fortsetzungssatze der komplex-analytischen Cohomologie und ihre algebraische Charakterisierung. Math. Ann. 157, 75-94 (1964) Zbl.136,207. Schumacher, G.: Ein topologisches Reduziertheitskriterium fiir holomorphe Abbildungen. Math. Ann. 220,97-103 (1976) Zbl.305.32013. Serre, J-P.: Geometric algtbrique et gtometrie analytique. Ann. Inst. Fourier 6, l-42 (1956) Zbl.75,304, (Euvres I, 402-443. Serre, J-P.: Prolongement de faisceaux analytiques cohtrents. Ann. Inst. Fourier 16, 363-374 (1966) Zb1.144,80; (Euvres II, 277-288. Siegel, C.L.: Zu den Beweisen des Vorbereitungssatzes von WeierstraB. Abh. Zahlentheor. Anal., 297-306 (1968) Zbl.189,365 Ges. Abh. IV, l-8. Spath, H.: Der Weierstragsche Vorbereitungssatz. J. Reine Angew, Math. 161, 95100 (1929) Jbuch 55,206. Stickelberger, L.: Uber einen Satz des Herrn Noether. Math. Ann. 30, 401-409 (1887) Jbuch 19,399. Starch, U.: Uber die Divisorenklassengruppen normaler komplex-analytischer Algebren. Math. Ann. 183, 93-104 (1969) Zbl.174.334. Starch, U.: Die Picard-Zahlen der Singularitlten t;’ + t;’ + t3 + t? = 0. J. Reine Angew. Math. 350, 188-202 (1984) Zbl.527.14009. Thimm, W.: Untersuchungen iiber das Spurproblem von holomorphen Funktionen auf analytischen Mengen. Math. Ann. 139, 95-114 (1959) Zbl.196,344. Thimm, W.: Liickengarben von koharenten analytischen Modulgarben. Math. Ann. 148, 372-394 (1962) Zbl.111,82. Weierstrass, K.: Einige auf die Theorie der analytischen Funktionen mehrerer Veranderlichen sich beziehende Satze. Math. Werke II, (1895) 135-142, Jbuch 26,41.

Chapter II

Differential Calculus, Holomorphic Maps and Linear Structures on Complex Spaces Th. Peternell and R. Remmert Contents Introduction

. .. . . . .. . . . .. , ,. . . . .. , . .. . . .. . . . .. . . .. . . .. . . .. . .. .

9 1. Differential Calculus of Complex Spaces . . . . . . . . . . . . . . . 1. Sheaves of Germs of Differential Forms on Manifolds . . . .. . . . .. . .. . . . .. . . .. . . .. . . . . . . . . . . . . 2. The Cotangent Sheaf Qi and the Tangent Sheaf& . . .. . . .. . . . .. . . .. . .. . . . .. . .. . . 3. Conormal Sheaves and Standard Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Criteria for Smoothness, Submersions . . . . . 5. Singularities of Holomorphic Maps . . . . . . . 6. Immersions . . . .. . . .. . . . .. . .. . . . .. . .. . .

99

. 100 . 100 . 101 . . . .

104 105 106 108

$2. Flatness ................................................... 1. Flat Modules ............................................ 2. Flatness in Complex Analysis .............................. 3. Criteria for Flatness ...................................... 4. Smooth and Etale Maps ..................................

109 109 111 113 114

Q3. Vector Bundles, Linear Spaces and Analytic Spectra ............. 1. Holomorphic Vector Bundles .............................. 2. Holomorphic Linear Spaces ............................... 3. Analytic Spectra ......................................... 4. Homogeneous Spectra. Projective Bundles ...................

115 115 118 121 124

0 4. Formal Completions ........................................ 1. Inverse Limits ........................................... 2. Formal Completion of a Complex Space along a Closed Complex Subspace ....................................... ................... 3. Coherent Sheaves on Formal Completions

127 127 129 130

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and R. Remmert

4 5. Cohen-Macaulay Spaces and Dualizing Sheaves ................. 1. Cohen-Macaulay Spaces .................................. ............................... 2. Ext-groups and &‘zt-sheaves 3. Dualizing Sheaves ........................................ 4. Gorenstein Spaces .......................................

132 132 135 138 140

References

143

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

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Introduction In this chapter we study complex spaces in greater detail as far as we can go without using the cohomology machinery (with a few exceptions in 95). The main themes are: differential calculus, structure of holomorphic maps of complex spaces, formal completions and, finally, special classes of complex spaces (Cohen-Macaulay) and dualizing sheaves. Since these subjects are quite different in nature, we describe the sections separately. 9 1 is - so to speak - the first order study of complex spaces. As everybody knows, tangent bundles, differential forms, differentials of differentiable maps are important concepts in the theory of real differentiable manifolds. All these notions have their translations in the theory of complex spaces; of course due to possible singularities, things become more subtle. Thinking functorially, it is often an advantage to take a co-point of view: we deal with cotangent sheaves rather than tangent sheaves, with conormal sheaves rather than normal sheaves. Cotangent sheaves and their relative analoga carry a lot of information on the underlying complex space: they somehow measure the singularities, they influence the geometry of subspaces etc. The most “regular” objects are of course manifolds; their relative analoga are submersions and smooth maps which form important classes of holomorphic maps. The important principle of “generic smootheness” says that a holomorphic map from a manifold to an irreducible reduced space is generically smooth. Another important class of holomorphic maps are the flat maps. They form the subject of 0 2. Philosophically, flatness should be thought of in the following way. Suppose we are given a holomorphic map f: X + Y. For y E Y consider the fiber f-‘(y). As y varies we obtain a “family” of complex spaces. But if f is arbitrary the fibers need not behave well: the dimension may jump etc. To say that f is flat means that these ugly phenomena cannot occur: the “family” really behaves like a family. This will become very clear in chap. III, (f-‘(Y)),,* when the cohomological behavior of flat maps is studied. In $3 we study linear structure on complex spaces: vector bundles, linear spaces, analytic and homogeneous spectra. Vector bundles are the geometric pictures of locally free sheaves; in the same spirit linear spaces correspond to coherent sheaves. The procedure of attaching linear spaces to coherent sheaves can be generalized: this leads to the notion of analytic spectra. Analytic spectra have an universal property; particular examples are normal cones; normalizations and also Stein factorization can be described by an analytic spectrum. A vector bundle - which is a particular an-bundle - can be converted into a lP,,-,-bundle by taking hyperplanes in the fibers. This procedure might be called “projectivization” and can be carried out also for linear spaces or coherent sheaves. It is a special case of the notion of homogeneous spectra. The importance of this construction will turn out in chap. V and chap. VII in connection with positive sheaves, vanishing theorems and modifications.

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In 5 4 the formal completion of a complex space along a subspace is studied. While the conormal sheaf informs us merely about the first order structure of the embedding, the formal completion carries much more information, and sometimes even tells us everything about the embedding (see the “formal principle” in chap. VII). Let us finally describe the content of 4 5. If X is complex manifold of dimension n, we denote by sZ$ the (locally free) sheaf of holomorphic l-forms and by ox = Q; = JQi the sheaf of n-forms. ox is called the dualizing (or canonical) sheaf of X. Of course it carries less information on X than Sz: but has the big advantage to be of rank 1. Therefore ox plays a very important role in the classification theory of compact complex manifolds. In particular, wx is indispensable for Serre duality which relates certain cohomology groups. In its simplest form Serre duality on a compact Riemann surface X states that the number of holomorphic l-forms is the genus of X. Serre duality will be discussed in Chap. III since it deals with cohomology. Concerning singular spaces, ox (to be defined suitably!) is often not so well behaved (and so does Serre duality), so one looks for special classes of singular spaces such as Cohen-Macaulay spaces and Gorenstein spaces where the behaviour is better. These spaces are the main objects of investigation in $5.

5 1. Differential

Calculus of Complex Spaces

In this section we introduce some fundamental concepts of the differential calculus of complex spaces: sheaves of germs of holomorphic differential forms, tangent sheaves, conormal sheaves etc. We characterise smooth points via the cotangent sheaf and study the singularities of holomorphic maps. Basic references are [Fi76] and [SC60]. It will be helpful to compare the results with those in algebraic geometry, see [Ha771 and [EGA]. 1. Sheaves of Germs of Differential Forms on Manifolds. Let X be a complex manifold of dimension n. Viewing X as a real manifold of dimension 2n we denote by & the sheaf of germs of complex valued r-forms on X. In local coordinatesx,,...,~,, every r-form o can be written as follows:

OJ= i,
coordinates

z, := x, + ix”+“,

1 < v I n, we may also write

gil...i,jl...j, dzi,A. . . A dzip o=cp+q=r il <...
A

dz,,

A

. . . A

dFj,.

j,

Thus every r-form is a sum of (p, q)-forms where p + q = r. Now the notion of a of the chosen holomorphic chart. Hence the sheaf dpsq

(p, q)-form is independent

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(p, q)-forms on X is well defined and we have

For every open set U c X we have the differential defined in local coordinates by

operator d: d’-+

sS+l

do := c dfi ,,,, i, dxi, A ... A dxiv = .fl i
A dxi, A ... A dxi/

If o E &pvq(U), then do E zJpfq+r can be written uniquely as do = cp + t,b, where cp E J&‘+‘~~(U),

$ E &‘p*q+‘(U).

Putting do := cp,20 := II/ we get differential operators a: dp.4 --) d;gP+l.q, 3: dp.4 + dn4+1, d = a + 3. In holomorphic

coordinates we have (using multi-index

notation)

e.g.

a(C gIJ dz, A dz,) = C aglJ A dz, A dz,.

A (p, 0)-form o is called a holomorphic p-form if & = 0. Locally such forms have the form o = c gi,

i, dzil

A

...

A

dzip

with gi,

i, holomorphic.

Clearly all germs of holomorphic p-forms on X form an ox-sheaf Q$. This sheaf n . The differential d induces a differential d: Q$ + sZi+‘, 0P in particular d: 0x + sZ$. Every holomorphic map f: X -+ Y of complex manifolds induces pull back maps

is locally free of rank

f *:

Q{(V) +

sZg(f-l(V),

V open in Y.

f* can be viewed as an Ox-morphism f *(Qe) + Q$. For details we refer to text books on complex analysis, e.g. [GrRe77]. 2. The Cotangent Sheaf a; and the Tangent Sheaf 5x. Most important in differential calculus is the sheaf of germs of holomorphic l-forms. In order to define this sheaf for arbitrary complex spaces, we first consider a model space I/ in a domain D c (c” with ideal 9 c UD. The map 9 + Szk, f H df, sends 9’ into 962; and hence induces, by passing to residue classes, a morphism a: X/S2 + S2;,/$52,.

(1.1)

We put Q? := Coker a; this is a coherent sheaf on V. The case of an arbitrary complex space is handled by gluing. Let {Vi} be an open covering of X such that there exists a biholomorphic map fi: Vi 3 Vi onto a model space k$ Then the sheaves Or?I and hence sZ/ = !2& (via jJ are well

Th. Peternell

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and R. Remmert

defined. The isomorphisms L-’ o A.: Ui n Uj + Ui n Uj give rise, by differentiating and passing to residue classes, to isomorphisms Oij: Q/ 1Ui n Uj + Q; 1Ui A Uj such that OijOj, = Oi,. Hence we have an Ox-sheaf atlas and thus an Ox-sheaf Qi on X such that a$\ Ui z Qui (Gluing Lemma I. 3.5). This sheaf is coherent on X and called the sheaf of germsof holomorphic l-forms on X or the cotangent sheaf of X. It is a matter of routine to verify that, up to isomorphisms, the sheaf Szi is independent of the choice of the local models and of the choice of the covering. For every point x E X the embedding (x} + X induces an exact sequence

(This follows immediately x c UY).

from the definition

of Szt via a local embedding

Proposition 1.2. The map a is bijective. Proof. Clearly !2txi = 0, hence u is surjectiue. The injectivity valent to the surjectivity of the dual map

of a is equi-

The first space here is isomorphic to the space of all derivations D: 0x,x -P C, i.e. (C-linear maps obeying the rules D( 1) = 0 and D( fg) = f (x)Dg + g(x)Df. Since D(mf) = 0, every D induces, by restriction to m,, a homomorphism m,/mf + C:; clearly a*(D) is just this map. In order to show surjectivity of tl*, let v, E Hom,(m,/m$ Cc). For arbitrary f E Ox,, we have g := f - f(0) E m,; clearly -+ (c, f H cp(g mod rnz) is a derivation and a*(D) = 4p. 0 D: 0x.x For the rank rk.J&, i.e. dimg Qi,,/m,Q$,, rk,Q: = emb, X (=dim,

we now obtain m,/mz),

x E X.

This shows again that at every smooth point x E X the sheaf Szi is locally free and rk,!Si = dim, X. Since x E X is a singular point if and only if emb, X > dim, X we derive furthermore (cf. also Chapter I, $10.5): The singular locus S(X) of every pure dimensional complex space X is an analytic set in X, namely the set {x E X: rk,Qi > dim X}. Remarks 1.3. The map c1of (1.1) is crucial for computing Q$. In general, this map is rank decreasing at points, where the defining equations of 9 become more singular. In case of a principal ideal 9 = Of, the map c1has rank 0 exactly at those points where all partial derivatives off vanish. For example, if X is Neil’s parabola in (c2 defined by f := z: - zz, the morphism c( has rank 1 at all smooth points of X, however a(x) = 0 for x = 0. The map ccis easily computed when X is a plane in 47 given by z1 = . . . = zk = 0. Functions vanishing on X near 0 are given by the series f(z)=

~fil,,,i~(~k+l,...,~,)~f'.....~~,

where i, + ... + i, > 0.

II, Differential

Calculus,

Holomorphic

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and Linear

The map CIassociates to the residue class [f] i +,C,i,=, L

fil...ik(Zk+l~

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103

at 0 the linear part ...)

zn)

of the power series: Forming df and passing to the residue class just means taking the linear part off in the directions z1 , . . . , zk. q We now define the tangent sheaf Fx of a complex space by & := ~&m,~(sZ$, 0,) = the dual sheaf of L$. Sections s E r(9x) are called holomorphic uector fields on X. The finite dimensional (C-vector space Fx,x/mxFx,x is called the tangent space at x and denoted by T,X. For a complex manifold X the tangent sheaf yx is locally free with rk,Fj = dim, X. The sheaf & can be viewed as the sheaf of germs of holomorphic sections in the holomorphic tangent bundle T, which assigns to every point x E X the vector space T,,, of tangent vectors at x. From a linguistic point of view it seems strange to introduce first cotangent sheaves and then tangent sheaves as their duals. And in fact, when dealing with manifolds and their geometry, tangent sheaves are the more natural objects. In the singular context however functorial properties are very important and the cotangent sheaves do behave much better. Cl Next we introduce cotangent sheaves and tangent sheaves with respect to a given holomorphic map f: X + Y of complex spaces. If X and Y are manifolds we already associated to f a morphism df: fW;

+l2$

of Ox-modules (cf. subsection 1, where df was denoted by f *). This morphism is now easily constructed in the singular case as well. In fact, the problem being local in X and Y, we may assume that X and Y are model spaces in domains U c (Em, V c a”, and that f can be lifted to a holomorphic map fi U + UZ:“. Then, for every local l-form w on Cc” we put

df(Col) := C&41, Definition

where [ ] denotes corresponding

1.4. Zf f: X + Y is a holomorphic

residue classes.

map, the Ox-sheaf

G,Y := Coker(df: f *Cl: + 0:) is called the sheaf of germs of relative l-forms

with respect to f.

This sheaf is coherent; its importance will become evident when we will study singularities of holomorphic maps. The dual sheaf 5 XIY:=

Jf-%x(Qi,Y,

Q)

is called the relative tangent sheaf of jI By the very definition of Q$ we have an exact sequence

Th. Peternell

104

and R. Remmert

Observe that df is, in general, not injectiue (the rank of df may decrease; take e.g. f: (c2 -+ (c with f(zl, z2) := z: - 23). As an elementary fact we state Proposition 1.6. Let f: X + Y be holomorphic and let Xy be the “full” fiber over a point y E Y (i.e. f-‘(y) is equipped with the structure sheaf coming from Im( f *(m,) + 0,). Then sZi,,lX, 2: Qiy. 3. Conormal Sheaves and Standard Exact Sequences. If Y is a closed complex subspace of X with defining ideal 9 c 0, the coherent &,-sheaf

.N$

:= (Y/Y’)1 Y

resp. JV$ := ~uw~~,(JV&,

0,)

(1.7)

is called the conormal sheaf resp. the normal sheaf of Y in X. We easily get a global version of the maps TVconsidered in (1.1). Proposition 1.8. There exists an exact sequenceof l&-sheaves

(here 52:1Y denotes analytic restriction, i.e. Q$l Y = (C$/ZJQ$I Y). The map c( is injective, if Y is a submanifold of a complex manifold X.

There is, of course, also a relative version. Proposition 1.9. If f: X + Z is holomorphic, then there exists an exact sequence

Note that in case of submanifolds (1 .S), namely an exact sequence

of a manifold there is a dualized version of

Geometrically the normal vectors of Y in X and the tangent vectors of Y span the tangent spaces of X. There is another important situation when the map c( is injective. Call a closed complex subspace Y of a complex manifold X locally a completion intersection if the ideal 9 can be generated, locally, by codim( Y, X) holomorphic functions. In this case $a/$’ is locally free of rank codim( Y, X), cf. Q5. If now, in addition, the space Y is reduced, then the sequence

is exact: In fact, generically Y is a submanifold, hence ccis injective generically. Thus Ker u E 0 everywhere, since 9/x2 is locally free. Note however that c1 need not to have constant rank, since Q,? is not locally free at singular points. There is also a relative version of the exact sequence (1.5).

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Proposition 1.10. Zf f: X + Y and g: Y --) Z are holomorphic, exact sequence f*c&

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105

then there is an

-+ ix:,, -+ cl:,, + 0.

Clearly (1.5) is obtained if g is a constant map onto a point.

0

Finally let us mention an exact sequence for conormal sheaves of embeddings. Proposition 1.11. Let Z be a complex space and let X c Y c Z be closed complex subspaces. Then there exists an exact sequence N~~Ix+Jv&+Jv$7+0. 4. Criteria

for Smoothness,

locally free. It is an important

Submersions. For manifolds X the sheaf L2.j is fact that the converse is also true.

Theorem 1.12. A point x of a complex space X is smooth if and only if Qi is (locally) free at x.

The proof is based on the following Fact 1.13. Assume that there exist local vector fields vl, . . . , vk E TX,, which are linearly independent modulo mx9x,x. Then there is an open neighbourhood U of x in X and a domain D c Ck and a complex space Y such that U is isomorphic to the product space D x Y.

A proof can be found in [Fi76], p. 92-93. Now the proof of (1.12) is easy: If Szi is free at x, so is & and we have k := rk,& = rkJ2.j local vector fields fulfilling (1.13). Since k = emb,X by (1.2), the embedding dimension of D x Y, where D c (Ck, is k. Hence Y is smooth at x and U is isomorphic to D. cl We wish to emphasize that in (1.12) it is not required that rk,S2$ = dim, X. For reduced spaces this is obvious; it should also be mentioned that for reduced spaces (1.12) is really an easy consequence of the implicit function theorem. As a corollary of (1.12) we note once more: The singular locus S(X) of X is the singular locus S(Q$) of the coherent sheaf !G?iand hence an analytic set in X.

Next we introduce the notion of a submersion. A holomorphic map f: X -+ Y of pure dimensional complex spaces X, Y is called a submersion if Q-& is locally free of rank dim X - dim Y. Instead of assuming X and Y to be pure dimensional we could also require that for all irreducible components 2, y of X, Y with f(r?) c F the numbers dim x - dim P coincide; then L& has to be locally free of fixed rank. Theorem 1.14. For a holomorphic map f: X -+ Y of connected complex manifolds the following assertionsare equivalent:

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Th. Peternell

and R. Remmert

i) f is a submersion. ii) Every f-fiber (with its canonical structure) is smooth dim X - dim Y. iii) f has maximal (Jacobian) rank dim X - dim Y everywhere.

of dimension

Proof. Consider the exact sequence f *G?t 2 04 + Q&Y + 0. If i) holds, then df is injective and of maximal rank, so iii) holds. On the other hand, i) and ii) are obvious consequences of iii): first L2iIy = Coker df is locally free of rank dim X - dim Y, and furthermore $2$,,lX, ‘v Oty, so X, is smooth with the

correct dimension.

Cl

Remark 1.15. If f is a submersion of arbitrary spaces X, Y, then one can find, for every point x E X, an open neighbourhood U of x, an open neighborhood V of f(x) with f(U) c V, an open set W in (Ek and a complex space Z such that U 2: W x Z and such that the following diagram commutes (cf. [Fi76], p. 100 ff.):

u2wxz f

w-j I v&

I Z

0

In 9 2 we shall introduce the notion of flatness. Flat submersions will be called “smooth maps”. An example of a submersion which is not smooth is the normalization of a curve with a normal crossing singularity. 5. Singularities

the singularities

of Holomorphic

of a holomorphic

Maps. We want to measure the behavior of map f: X + Y. For every k E N we call

Singk(f) := {x E Xjrk,f

> k}

the k-th singular set off. The set Degk(f) := {x E Xldim,

X&) > k}

is called the k-th degeneracy set off. Here, as usual, Xffx) denotes the fiber over f(x) with full fiber structure. Obviously Singk(f) = {x E Xlrk,S2,$,, > k}, so that Sing”(f) is an analytic set in X. To say that f is smooth is equivalent to saying that Singk(f) is empty for k = dim X - dim Y. The analyticity of Degk( f) is not so obvious and, in fact, a result of Cartan [Car331 and Remmert [Rem57]. Theorem

1.16. For

every holomorphic

map

f: X + Y all degeneracy sets

Degk( f ), k E N, are analytic in X. For a simple proof, see CFi76], p. 137. In particular, (1.16) implies that the function X + IN, x H dim, Xfcx) is upper semi-continuous in the Zariski topology.

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f is called equidimensional if all f-libres have the same dimension For irreducible spaces this means: Deg”(f)

= @

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107

at all points.

for k = dim X - dim Y.

In fact, if Degk(f) = 0, then all fibers have dimension 5 k. By (1.17) below there are no fibers of dimension
dim, X I dimf(,, Y + dim, Xfcxj

(1.17)

The proof of (1.17) uses the following fact. If k = dim, Xfcxj, then there is a neighborhood U of x and a commutative diagram u 9

YxCk

of holomorphic maps such that x is an isolated point of g-‘(g(x)); this description is immediate since the definition of k permits to map X,-.(xj around x to (Ck with x isolated in its fiber. In general, (1.17) is not an equation: for the map f: (c2 + UZ2 given by (z,, z2)+-+(z1, z1z2) the fiber X,-,,, is the line (zl = O}. Hence (1.17) gives the strict inequality 2 I 2 + 1; note that this map f is not open. Theorem 1.18 [Rem57]. Let f: X + Y be holomorphic locally irreducible. Then the equation

and assume that Y is

dim, X = dirnftxJ Y + dim, Xfcxj holds for all points x E X if and only if the map

f is open.

For a proof we refer again to [Fi76].

cl

Next we consider the f-images of the sets Deg”(f) and Singk(f), i.e. we look at those points y E Y where the fiber is singular to some extent. Theorem 1.19 [Rem57]. Let f: X + Y be a proper surjective map, assume that X is reduced and irreducible. Then the set

holomorphic

{ y E Y 1dim, X,, > dim X - dim Y for some x E X,,} is analytic in Y and of codimension at least 2.

In other words, fibers of dimension bigger than the generic fiber dimension occur only over an analytic subset of codimension 22. The proof goes by induction over dim X. One considers the analytic set A := {x E X(dim,

Xrcx, > dim X - dim Y>.

Th. Peternell

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and R. Remmert

By Remmert’s proper mapping theorem (see Chapter III), f(A) is analytic in Y. The induction proceeds by passing from X to components A’ of A and studying fl A’. For details consult [Fi76], p. 140. q Using (1.18) one easily concludes: Corollary 1.20. Under the assumption of (1.19) there is an analytic set B in Y of codimension at least 2 such that f: X\f -l(B) + Y\B is open provided Y is locally irreducible.

Concerning

the existence of smooth fibers we first state

Proposition 1.21. Let f: X + Y be a holomorphic surjection of reduced irreducible complex spaces. Then there exists a Zariski open set U c X such that f 1U is a submersion. Proof. Since X is reduced and irreducible there is a Zariski open set U in X such that the coherent sheaf L&&Yis locally free on U. So we have only to check that rklj$,, = dim X - dim Y at one point. We may assume, if necessary passing to a smaller U, that X and Y are smooth. But then the assertion just comes from local function theory (implicit function theorem). cl

A priori we may have f(U)

= Y. In general we have:

Theorem 1.22. Let X be smooth, let Y be irreducible, and assume that f: X + Y is holomorphic. Then the set N := (y E YlX, is not smooth} has Lebesgue measure 0 in Y, and the map f: X\f -l(N) + Y\N is a submersion. If, moreover, f is proper, then N is analytic in Y. Proof. We may assume that Y is smooth too. Now consider X, Y as differentiable real manifolds and f as a real differentiable map. Then outside of some set N of measure 0 in Y, f has real rank dim, X - dim. Y, by Sard’s theorem. N is just the set of critical values, and f-‘(y), y E N, is not a differentiable manifold. Now clearly f has complex rank dim, X - dim. Y in X\ f -l(N), while in f -i(N) the rank decreases. This proves the first assertion. Since

N = f({x

E XIX,,,,

is not smooth at x}) = f(Sing Q.&r),

the second assertion follows from the proper mapping

theorem.

0 In summary, (1.22) states that for a general point y E Y the fiber X, is smooth, or that there is a neighborhood U of y such that f (f -l(U) is a submersion. 6. Immersions.

A holomorphic

- an embedding,

if there is a induces a biholomorphic map f: X - an immersion at x, if there respectively f(x) E Y with f(U) c

map f: X + Y is called closed complex subspace Z of Y such that f + Z, are open neighborhoods U resp. V of x E X V such that f 1U: U + V is an embedding.

f is called an immersion if it is an immersion

The following proposition

at all points of X. is easily verified:

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Proposition 1.23. A holomorphic map f: X + Y is an immersion at x E X if and only tf the induced map f: Or,/(X) + O,,, is surjective.

The connection with differential calculus is given by Proposition

1.24. The following

statements about a holomorphic map

and a point x E X are equivalent: i) f is an immersion at x, ii) (df )x: f *Qj,, + Qi,, is surjective, iii) the jacobian T,f: yx,, + f *5r,x iv) Qilr,, E 0.

f: X + Y

is injective,

Proof. Clearly ii), iii) and iv) are equivalent, moreover, i) implies ii). If (df ), is surjective, then, passing to minimal embeddings into number spaces, one may assume that X and Y are domains in (c” and (cm, respectively. Now i) is just the implicit function theorem. 0

Noteworthy embedding.

is also the following

criterion

for an immersion

to be an

Proposition 1.25. Let f: X -+ Y be an immersion. Assume moreover that f is injective and a closed map. Then f is an embedding.

Of course immersions need not be injective: just consider the normalization of curves in (lZ2 with ordinary double points.

0 2. Flatness The topics of this section are flat holomorphic maps and flat coherent analytic sheaves. Since flatness is an algebraic concept, we first review the relevant facts about flat modules over rings. The real importance of flatness in complex analysis will turn out in Chapter III in connection with the semicontinuity theorem etc., and in Chapter VIII in connection with the theory of cycle spaces. 1. Flat Modules. By R we always denote a commutative ring with unit element; standard references for this subsection are [Bou61 ff], [Fi76], [Dou68], [BaSt76], [SC60/61]. An R-module M is called R-flat or flat over R or just flat if and only if for every injective homomorphism N, + N2 of R-modules the induced homomorphism NI OR M + N, QR M is injective. Proposition 2.1. An R-module equivalent conditions are satisfied: (1) For every finitely generated (2) For every exact sequence quence .. . + Nkwl OR M + Nk OR

M is flat over R if and only if the following ideal I c R the map I @ M + M is injective. ... + Nkml + Nk --) Nk+l + *a. the induced seM + Nk+l OR M + *** is exact.

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In short: M is R-flat if and only if the functor QR M is left exact (this functor is always right exact). Next we collect basic properties of flat modules. Proposition 2.2. (1) Zf M is R-flat and R + R’ is a ring homomorphism,then M OR R’ is R-flat (base extension). (2) If R’ is a flat R-algebra and M a flat R’-module, then M is a flat R-module. (3) Assume that R is a local noetherian ring. Then a finitely generated Rmodule M is R-flat if and only if M is free. (4) Let 0 -+ M, + M, + M, + 0 be an exact sequenceof R-modules. Zf M,, M, are flat, so is M,. If M,, M, are flat, so is M, .

Now we are going to introduce the so-called “Tor-modules”, which give a measure for non-flatness of an R-module M. We pick a projective resolution ... + Ai + Aipl -+ . ..+A.+A,,+M+O

of M. [Recall that an R-module A is called projective if for every exact sequence B, + B, -+ B, of R-modules the induced sequence Hom,(A, B,) + Hom,(A, B,) + Hom,(A, B3) is exact. Most prominent examples of projective modules are free modules, and in fact in our situation we can choose all Ai free.] We take another R-module N and consider the induced map di: Ai OR N -+ Aiel OR N. Clearly Im di+l c Ker di, so that the R-modules To&(M,

N) = Tor’(M,

N) := Ker d,/Im di+l

are well defined. Basic properties of these Tor-modules

are collected in

Proposition 2.3. Let M, N be R-modules. (1) Tor’(M, N) doesnot dependon the choice of the projective resolution of M. (2) Tor’(M, N) N M 0 N. (3) Tor’(M, N) N Tor’(N, M). (4) Zf 0 + N, + N, + N3 -+ 0 is exact, then there is an exact sequence

. ..Tor’(N.,

M) + Tor’(N,,

M) + Tor’(N,,

M) + Tori-l(N,,

M) --* ...

. ..~Tor’(N.,M)~N,O,M~N,O,M~N,O,M-rO. Corollary 2.4. M is R-flat if and only if one of the following equivalent conditions is fulfilled: (1) Tor’(M, N) = 0 for all R-modules N (2) Tor’(M, N) = 0 for all R-modules N and all i 2 1.

Of course condition of greatest importance.

(1) is not easily verified. Therefore the following result is

Theorem 2.5. Let R, R’ be local noetherian rings and f: R’ + R a local homorphism. Let m be the maximal ideal of R and I c R’ an arbitrary ideal. Let M be a finitely generated R-module. Then the following conditions are equivalent: (1) M is flat ouer R’. (2) M/lkM is flat over R’Jlk for every k 2 1. (3) MIIM is flat over R’fI and Tork. (R/Z, M) = 0. (4) Tor’(R/m, M) = 0.

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2. Flatness in Complex Analysis. Using the language of subsection 1 we introduce the notion of a flat map and of a flat sheaf. A holomorphic map f: X + Y is called flat at x E X if Ox,, is a flat O,,/(,,-module. [Here 8x,x is considered as Qr,+i- module via the canonical map 11,: Oy,I(,) + c?,,,]. The map f is called flat if f is flat everywhere. Let 9’ be a (coherent) sheaf and let f be as above. Y is called f-flat at x E X if Y: is a flat Oy,l(xJ- module. Y is called f-flat if 9’ is f-flat everywhere. Clearly f is flat if and only if Ox is f-flat. For examples of flat resp. non-flat maps we refer to subsection 3. Here we collect some basic functorial properties. Proposition 2.6. (1) (base change). Let f: X -+ Y, g: 2 -+ Y be holomorphic and Y an f-flat Ox-module. Consider the fiber product X xr Z with projections pr, respectively pr, to X resp. Z. Then pry(Y) is pr,-flat. In particular, iff is flat then pr, is flat. (2) Let f: X -+ Y, g: Y + Z be holomorphic maps. Let Y be an f-flat Lo,module and assume g to be flat. Then Y is (g 0 f)-flat. (3) A coherent Ox-module Y is flat over X (i.e.: id-flat) if and only Y is locally free. Proof. Of course (2) and (3) are immediate consequences of 2.2(2) and 2.2(3). Note however that (1) is not just a formal consequence of 2.3( 1) since the delinition of X x y Z requires a completion of tensor products (see Chapter I). We sketch a proof of (1) for 9’ = 0, following [Fi76]; for the general case we refer to [SC60/61]. Locally g can be decomposed into a finite map and a submersion. So it is sufficient to show (1) for these types of holomorphic maps (use (2)).

(a) The finite case: Let r? = X xy Z. Take p E r?. Then (*I G,, = *z,,m @“y,/(,r,(,)) %w,w This is a consequence of finiteness (see [Fi76, p. 1531). Hence the claim follows from 2.2( 1). (b) case where g is a submersion: Now (*) need not hold. From the local description of X xy Z in terms of local rings we see that the following has to be proved: Assume R’, R to be analytic rings and let u: R’ + R be a flat morphism (in our situation R = Ox,x, R’ = O,,f(x, and u = fx). Let n E lN. Then the induced homomorphism R’(Z 1, .a*, ~,)+R(zw..,z,)

is flat. Now take I c R’(zl, . . . . z,} to be the ideal generated by zi, . . . , z,. Since all induced maps R’ + R’ {z 1, . . . , z,}/lk, k 2 1 are “finite” (obviously), we can apply (1) to conclude that R(z,, . . . . zn}/lkR{zI, . . . . z,} is flat over R’{z,, . . . . z,}/Zk for all k 2 1. Now apply (2.5)! Corollary Y is flat.

2.7. Let X, Y be complex spaces. Then the projection pr,: X x Y +

Th. Peternell

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and R. Remmert

Proof. Apply 2.6(l) with f the constant map X + {0}, g the constant map Y + (0) and observe that X x (01 Y z X x Y and f is flat. 0 The set where a coherent module is not flat is analytic. This is the content of a theorem of Frisch: Theorem 2.8. Let f: X + Y be a holomorphic map of complex spaces and let Y be a coherent @,-module. Then the “non-flat locus” A = (x E X19’

is not a flat O,,+,-module}

is an analytic subset of X. If Y is reduced and if X has countable topology, then f(A) is nowhere dense. moreover, f is proper, then f(A) is thin in Y.

If,

For a proof see [Fr67], [Dou68] or [BaSt76]. If f is the identity we find again that the set where 9 is not locally free is analytic. Assume now for simplicity X and Y to be reduced and f surjective. By 0 1 f is a submersion outside a proper analytic set B of X. So outside B, f is locally a projection of a product. If we also enlarge B in such a way that Y is locally free outside B, then we see from (2.7) that Y is f-flat outside B. Hence in this situation A is at least contained in some analytic set and Y is generically f-flat. 0 If Y is generically f-flat one may ask whether Y can be made f-flat everywhere by performing “blow-up’s” (see Chapter VII), i.e. by changing X and Y only in thin sets. This is indeed the case and is a remarkable deep theorem of Hironaka [Hir75], which can be viewed as a kind of “desingularization” of the coherent sheaf Y (Chapter VII.7). Theorem 2.9. Let f: X -+ Y be a proper holomorphic map where X is countable at infinity and Y is reduced. Let Y be coherent on X. Then there is a proper holomorphic map 71: Y’ -+ Y, which is an isomorphism almost everywhere and in fact a locally finite sequence of blow-ups, such that the following holds: Consider the diagram Xx,Y’xI

x

f’ I

I

Y'

and let F c n’*(Y)

I

"-Y

be the coherent subsheaf of n’*(Y)

given by

Fx = {f, E 7t’*(P’),I there is a non-zero divisor g E OYP,f,(xj with g Then n’*(Y)/F

'f, = O}.

is f ‘-flat.

This procedure to flatten a sheaf is often referred to as “flatification”. Theorem 2.9 has an important consequence: The Chow Lemma, which will be discussed in Chapter VII. Of course there is no hope to make Y flat without dividing by the torsion sheaf y. For example let X = Y = (c2 and f the identity

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map. Consider the full ideal sheaf Y = m(,) of (0) in (I?. Then the set where 9’ is not f-flat is just the set where Y is not free, i.e. (0). Now blow up (0) in (c2 (see Chapter VII) and call the map 7~:&:’ + (c’. Then Z*(Y) has torsion along x-l (0), but dividing by the torsion it will become locally free. 3. Criteria for Flatness. Up to now the only flat maps we encountered are projections and locally trivial maps. In this subsection we investigate in more details under what conditions holomorphic maps are flat. We begin with finite maps. For such a map f: X -+ Y we put

v(y) :=

c dim. xEf-‘(Y)

Q,,x,

where X, = f-‘(y) is the fiber over y equipped with the natural structure given by Im(f*(m,) -+ ox). Since X, is O-dimensional, the stalk Loxy,, is a finite dimensional c-vector space. Proposition 2.10 (Douady). Let f: X -+ Y be a finite holomorphic map. Then f is flat if and only iff*(O,) is locally f ree. If, moreover, Y is reduced, then f is flat if and only if the function v(y) is locally constant. Proof. The equivalence

.f*(%), = ,($,

of the first two assertions follows from 2.2(3), since The last assertion follows since on a reduced space a co@Lx.

herent sheafxis locally free if and only if its rank is locally constant (1.7.15).

0

Using 2.10 it is easy to construct non-flat finite maps, see subsection 4. For non-finite maps we first state Proposition

2.11. Zff: X + Y is flat then

dim, X = dimf(,) Y + dim, Xs(,.)

for all x E X.

This is easily shown by induction on dim Y and taking local hypersurface sections in Y, cf. [Fi76], p. 156 for details. Cl (2.11) implies e.g. that non-finite modifications (see Chapter III for this notion) are not flat. In other words: a flat holomorphic map which is locally biholomorphic almost everywhere cannot have fibers of positive dimension. An easy consequence of (2.11) is Corollary

2.12. Every flat holomorphic

map is open.

In fact, if Y is locally irreducible (of course we may assume Y to be reduced) this follows from 2.11 by 1.18. In the general case normalize Y and apply base change. 0 We will see in 0 4 that an open map is not necessarily flat. On the other hand we have Theorem 2.13 (L. Kaup, Kerner). Let f: X + Y be a holomorphic complex manifolds. If f is open, then f is flat.

map of

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and R. Remmert

So if X and Y are pure dimensional (say connected) then f: X + Y is flat if and only if every fiber if of pure dimension dim X - dim Y. This is one of the most useful criteria for flatness. Some words concerning the proof (cf. [Kau68], [Ker68], [Fi76]). Since (2.13) is local in Y, we may assume Y = c”. Now proceed by induction on n. Write f = (fi, . . . , f,) and put X’ = {f,, = 0} c X. Since X’ is not necessarily smooth, we must prove a slightly more general statement for a fixed point x E X: if O,,, is a Cohen-Macaulay module (see [GrRe71] and $5), then f is flat at x. Indeed, in our induction 0x,,, is again Cohen-Macaulay (we assume of course f(x) = 0). Since it is easy to see that dim, X’ = n - 1 and (fr, . . . , f,-i): X’ + (En-l is open at x (use the Active Lemma of 1.10.4), the induction argument works. 4. Smooth and Etale Maps. A holomorphic

map

f: X -+ Y is called smooth, if

f is a submersion and a flat map. Example 2.14. In 9 1 we already remarked that a submersion need not be smooth: e.g. the normalization of a curve with a normal crossing singularity (double point) is a submersion but not flat as follows immediately from (2.10). More generally the normalization of a singular curve is never flat. By contrast any submersion of manifolds is automatically flat. Proposition 2.15. Let X, Y be complex f: X + Y is flat.

manifolds.

Then any submersion

In fact, f is “equidimensional”, hence open (1.18), hence flat (2.13). Of course a flat map of complex manifolds is in general not a submersion: there may be singular fibers. Now consider a finite map of complex manifolds. Since the dimension formula holds, f is necessarily flat. But f need not be a submersion. For instance take a d-sheeted covering f: JP, + IP,, d > 1. If f were a submersion, then But IP, f*&%J = G,, so f would be unramified, i.e. locally biholomorphic. being simply connected, this is not possible. This leads us to the following Definition 2.16. A holomorphic spaces is called &tale if and only if (1) Qi,, = 0, (2) f is flat.

map f: X + Y between

arbitrary

complex

In view of(l), f has discrete fibers only. If f is finite and flat, f is often called an “unramified covering”. Remark 2.17. Etale maps are related to the fundamental group xi(X); e.g. if x1(X) has a finite quotient of degree d, then X admits a finite &ale cover 2 + X of degree d (i.e. the general fiber consists of d elements and conversely). The universal cover of a complex manifold is always &ale.

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Example 2.18 (Douady). Let X c C4 be the union of the planes (z3 = z4 = 0) and (z3 - zi = 0, z4 - z2 = 0). The complex structure of X is given by the product ideal of (z,, z4) and (z3 - zi, z4 - z2). Let f: X + Cz be the restriction of the projection C4 + C2 onto (z,, z,)-space. Then v(s) = 2 for s # 0, and one computes easily v(0) = 3, since the fiber X0 is given by the ideal

(z,,

z-29

z:,

z3z4,

a.

Hence f is not Jut. Since f is finite and Y = C2 locally irreducible, f is open by (1.18). So we have an example of an open non-flat holomorphic map. Example 2.19 (Douady). If f: X + Y is a flat holomorphic non-reduced, then red f: red X + Y is not necessarily flat.

map and X is

This is shown at the hand of the following example, see [Fi76], p. 151 for details. Let X c (c3 be given by the equations 42: + 2725 = 0, z: + zlz3 + z2 = 0. Let Y c (c2 be given by 42: + 27~: = 0; Y is reduced. Let f be the restriction of g: C3 + C2, (Zl) z2, z3) H (zi, z2). Since vJ is constant, f is flat. But v,,~~ is not locally constant, so red f is not flat.

5 3. Vector Bundles, Linear Spaces and Analytic Spectra In this section we describe several methods for constructing linear structures. We discuss vector bundles and linear spaces as geometric pictures of locally free respectively coherent sheaves. Furthermore, we consider the functor “Proj” and, in particular, projectivizations of coherent sheaves and vector bundles. 1. Holomorphic Vector Bundles. A holomorphic vector bundle of rank r over a complex space X is a complex space E with a holomorphic map 7~:E + X (called “projection”) such that there exists an open covering {Ui} of X and biholomorphic maps hi: n-i(Ui) + C:’ x Ui with the following property: For every pair (i, j) the biholomorphic map Qij := hi o h,:‘: Ui n Uj x C’ -+ Ui n Uj x C’

is of the form Bij(x, v) = (x, gij(x)v). The maps hi are called local trivializations of E, while the maps gij are called transition functions. Every fiber E, = Z-‘(X) is a C-vector space isomorphic to a?. A vector bundle homomorphism between two vector bundles E $ X, E’ 5 X is a holomorphic map f: E + E’ such that f(E,) c E! and f IE,: E, + EL is linear, x E X. If f is biholomorphic the bundles are called isomorphic. Of course there are analogous notions of topological complex (or real) vector bundles over topological spaces and of differentiable complex (or real) vector bundles over differentiable manifolds.

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The transition

functions

and R. Remmert

satisfy the so called cocycle relations gijgjk

= gikt

&Iii = id;

(3.1)

gij should be viewed as a section over Ui n Uj of the sheaf CL,(O) of germs of invertible holomorphic matrices, in other words: gij: U + CL(r) is holomorphic. We can reconstruct the vector bundle from the transition functions. More precisely, let ( Ui)ipl be an open covering of X, and let gij E GL,(O,(Ui n q)) satisfy the cocycle relations (3.1). Then there is a vector bundle E of rank I with these transition functions: The disjoint union ,!?= u Ui x (c’ is (in a natural i.51

way) a complex space. For (x, u) E Ui x (c*, (x’, u’) E Uj x (c’ introduce the following equivalence relation: (x, u) - (x’, ?I’)0 x = x’,

u’ = gij(x)u.

Now put E = @/-. There is a natural projection rr: E + X making E to a holomorphic vector bundle of rank I as one easily verifies. 0 A holomorphic vector bundle E of rank r is called trivial if E is isomorphic to the trivial bundle X x Cc’. With this notation we can say that every vector bundle is locally trivial. If E is a holomorphic vector bundle of rank r over X and if U is open in X, every holomorphic map s: U + E, .*= x-‘(U) such that s(x) E E, for all x E U is called a (holomorphism)section in E over U. The sheaf of germs of holomorphic sections in E over X is denoted by O,(E) or just O(E). Since E is locally trivial, the sheaf O(E) is locally free of rank r. If s E O(E)(X) is a section in E over X, and if hi are trivializations of E with corresponding transition functions gij, then the maps si := hi o SIUi fulfill the relations si = gijsj.

(3.2)

Conversely, if functions si E Or(Ui) are given such that (3.2) holds then there exists a unique section s E O(E)(X) whose local expression in the trivialization hi is just si. A subbundle of E of rank p is a closed complex subspace F of E with the following two properties: Every set F, := E, n F is a complex linear subspaceof E, of dimension p. There exists an open covering { Ui} of X with local trivializations hi: Euj + Uj x (c’ and furthermore biholomorphic maps h;: Fui := (xIF)-l(Ui)

+ Vi x Cp

such that the following diagram commutes (with 4 as inclusion) Eui hi

Ui x C’

J

Vi x Cp

Fu, hj

J

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Let f: E + E’ be a homomorphism of vector bundles. If the rank function r&f,: E, -+ E:) is constant, then Ker f := u Ker f, is easily seen to be a subbundle of E. Of course this bundle exists only if the rank is constant. Similarly one can introduce the bundles Im f and Coker J The map f induces a sheaf homomorphism f”: O(E) --) Lo(E’). To say f has constant rank means that Ker f” and Im f’ are locally free. Warning. f may be injective without f being injective. For example take trivial l-bundles E, E’ on (c and consider a map f: E = C x C + E’ = Cc x C, (x, V)H(X, g(x)u) with g E 0((c), g # 0, g(0) = 0. This map is not injective, but f: Oc -+ Oc is just multiplication by g, hence injective. Gi,ven holomorphic vector bundles E, F over X we can construct - essentially pointwise - new holomorphic vector bundles as E @ F, E @ F, S”‘(E), NE, Hom(E, F), E* etc. (linear algebra of vector bundles). Of course these constructions commutive with taking sections: O(E @ F) N O(E) @ O(F),

O(E @ F) 2: O(E) C&, O(F) etc.

If f: Y --* X is a holomorphic map of spaces and E a holomorphic bundle over X, we define the pull-back f *E by

(f*E), = Em. If gij are transition functions for E, then f *(gij) are transition Note that 0( f *E) N f*(O(E)).

0 vector

functions for E.

There is - in general - no push forward f,(F) of a bundle F on Y. Of course the image sheaf f,(O(F)) exists. This indicates that the category of vector bundles is, for many purposes, not flexible enough; sheaves behave much better. Proposition 3.3. The map E + O(E) induces a bijection of the family of isomorphismclassesof holomorphic vector bundles over X of rank r onto the family of isomorphismclassesof locally free C&-sheavesof rank r.

Proof. It suffices to construct, for a given locally free sheaf 8 of rank Y, a vector bundle E such that O(E) 5: 8. Choose an open covering { Ui} of X with “trivializations” gi: &lUi 1 cObi.Then gij := gi 0 gy’ E GL(r, O(Vi n U,)), and it is easy to check that gij o gjk = gik on Ui n Uj n U,. Thus there exists a holomorphic vector bundle E on X with gij as transition functions. Now O(E) N B is obvious. cl For every complex manifold X the holomorphic tangent bundle Tx of X is “the” holomorphic vector bundle whose associated sheaf is the tangent sheaf 9x of X. Analogously, the holomorphic cotangent bundle T,* is the bundle for which B(T,) = S-2;.

If Y is a complex submanifold of X (or more generally locally a complex intersection) we define the normal bundle Nylx of Y in X by the sheaf equation WW = 4,x, where Jl/;,, denotes the normal sheaf of Y in X. We obtain an exact sequence 0 -+ Tr + TX1Y + Nvlx + 0 (Tx( Y := restriction).

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and R. Remmert

Geometrically, the tangent space T,X of X at x is spanned by the tangent T,Y of Y at x and the normal space (Nrlx),., consisting of those tangent vectors of T,X which are normal to Y Of course the tangent bundle can be constructed directly, without using the tangent sheaf. Namely, take an open covering of X by holomorphic charts and define the transition functions (of Tx) to be the Jacobian of the change of charts. Then let Tx be the vector bundle defined by these transition functions. Holomorphic vector bundles of rank 1 are of utmost importance. They are also called line bundles. The “canonical” line bundle K, of a manifold X is defined by K, := A” T,*,

where n := dim X.

The associated sheaf o x := cO(K,) = Qi (= sheaf of holomorphic n-forms) is called the dualizing or canonical sheaf of X. This sheaf plays an important role in cohomology theory (duality, vanishing theorems etc. see Chapters III and V). There is another important line bundle which is ubiquitous in algebraic and analytic geometry: the Hopf bundle (or its dual) on the complex projective space lF’“. We interpret IPn as the space of lines in cC”+l through 0 and define the Hopf bundle H as the space of all lines H, where H, represents x E lP”. It is easily seen that H is a holomorphic line bundle on II’“. Its sheaf of germs of holomorphic sections is usually denoted by OPn( - l), while the dual sheaf is denoted by Up,(l). In general Opn(k) is a short writing for Op,(l)@‘“, k > 0, respectively oIp n(- l)@lkl, k < 0. One shows easily that, for k > 0, the sections in OP,(k)(lPJ can be identified with the homogeneous polynomials of degree k in n + 1 variables, whereas Up,(k) has no global sections for k < 0. For details and more material around these topics see e.g. [We801 and [GH78]. The theory of line bundles is much simpler than the theory of bundles of higher rank. The reason is that GL(r, (c) is abelian if and only if r = 1: line bundles behave “in abelian way”, bundles of higher rank do not. For instance the set of all isomorphism classes of line bundles - called Pit X - is an abelian group and can be computed by cohomology theory as H’(X, Co*), see Chapter III. Line bundles are closely related to hypersurfaces in complex manifolds, whereas vector bundles of rank r have some relations to submanifolds (respectively locally complete intersections) of codimension r, in particular for r = 2. Needless to say, submanifolds of codimension 2 are more complicated than hypersurfaces. 2. Holomorphic

Linear Spaces. We have seen that there is a correspondence

vector bundles c* locally free sheaves Locally free sheaves are coherent. So the question arises whether “something like a vector bundle” can be associated to any coherent sheaf. The answer is “yes” but this new general correspondence does not extend the above one. The new objects are called (holomorphic) linear spaces.

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Detinition 3.4. A linear space L over a complex space X is a complex space L with a holomorphic map 71:L --+X (projection) and with holomorphic maps +: L xx L + L (addition), .: (C x X) xx L -+ L (multiplication by scalars), 0: X + L (existence of a zero section) which commute with 7c and satisfy the usual module axioms.

We usually identify ((c x X)x x L with (lZ x L. In categorial language linear spaces over X are unitary (c x X-modules in the category of complex spaces over X (see [Fi67], [Fi76], [0066]). The notion of linear space was introduced by Grauert [Gra62] and Grothendieck [SC60/61, exp. 121. All fibers L, = K’(X) of a linear space L $ X are reduced CO0661 and L, = (c”, n de,pending on x E X. Definition 3.5. A holomorphic map f: L + L’ between linear spaces L $ X, L’ s X is called a homomorphism of linear spaces tf IC’ 0 f = 71and tf the following diagrams commute L x,L

+I L-

3

L’xxL’

/

(CXL~CXL,

I+ ’ .I /

L’

L-

I, L’

In particular, if L c L’ is a closed complex subspace and if the inclusion is a homomorphism, then L’ is called a linear subspace of X. We want to associate a coherent sheaf 3 to every linear space L. We first note that for every open set U of X and every pair L, L’ of linear spaces over X the space Hom(L,, En), of homomorphisms of the restriction L, to the restriction Lh is an O(U)-module in a natural way: if v: L, + Lb is a homomorphism then we put f. v := ( .Lt) o [(id,, f) 0 7c,v]

for every f E O(U).

Now we define Y(U) := Hom(L,, X x U) and clearly obtain a sheaf Y of Ox-modules. We call 2’ the sheaf of germs of linear forms on L. The correspondence L -+ 9 defines a contravariant functor from the category of linear spaces over X to the category of Ox-modules (note that, in contrast, E + O(E) is a covariant functor from the category of vector bundles over X to the category of locally free sheaves). The main result of this section is the following theorem of Fischer ([Fi67], also [Fi76]). Theorem 3.6. The correspondence L + Y, associating to every linear space over X its sheaf of linear forms, defines a duality between the category of linear spaces over X and the category of coherent C&-modules. Proof. 1) First 9 is always coherent (this is the hardest part of the theorem). This coherence relies on the following two facts: a) (Prill, [Pr68]). Locally L is a subspace of a space X x (CN, N suitable. This yields an epimorphism 0: + 9” + 0, i.e. 9’ is finite.

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and R. Remmert

b) (Fischer, [Fi67]). Every linear subspace L c X x a?” is locally the kernel of a homomorphism U x UZN + U x (I?‘. This implies that 9 is relationallyfinite.

2) To a given coherent sheaf Y on X we associate a linear space V(Y). First we construct V(Y) locally. Let 06 5 0; 5 Y; + 0 be an exact sequence over U c X. The map a is given by a holomorphic matrix, hence a can be viewed as a homomorphism a: U x (cp + U x (c4. Let a*: U x (cq + U x (cp be the dual of a and put V(Y;) := Ker a*. The subspace V/(9”) of U x (cq is given by the equations 2

uijzi

Now V(9) is obtained [Fi76].

=

0,

1 <j I p, where

u = (Uij).

by gluing the pieces V(Y;). For details we refer to Cl

If &’ is a locally free sheaf, we have the associated vector bundle E whose sheaf of local sections is just 8’ and furthermore the linear space V(b). The constructions easily yield:

Definition 3.1. Let Y be a closed complex subspuce of X defined by the ideal 9 c 8,. Then the normal “bundle” or normal linear space of Y in X is the linear space N YIX := V(Y/P).

By the previous remark we get back the old definition Y is a submanifold.

if X is a manifold

and 0

We continue the discussion of the general theory of linear space with Proposition 3.8. Let L be a linear space on a reduced space X and assume that the function x H dim L, is constant. Then L is a vector bundle. Proof. Let Y be a coherent sheaf with V(9) 2: L. Since dim L, is the rank of 9” (= dim yx/rn,SQ the sheaf Y is locally free by 1.7.15. Hence L is a vector bundle. 0

The procedure of associating a linear space to a coherent sheaf can be reformulated in categorial language ([SC60/61], Exp 12). For this purpose fix a complex space and a coherent sheaf Y on X. Let C, denote the category of complex spaces over X (a complex space (Y, rc)over X is just a complex space Y with a holomorphic map rc: Y -+ X; a morphism f: (Y’, d) + (Y, ?t) is a holomorphic map f: Y’ + Y such that n o f = 7~‘). Let S be the category of sets. For a given coherent Ox-sheaf 9 we define a functor F,: Cx + S as follows: For

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map g: Y + X we put

F.AY) := Hom,y(s*(.4P), 0,). For a morphism

J: Y’ -+ Y of complex spaces over X we define

Fy(f) := {f*: H omt, (g*(Y), 0,) + Homey, ((f 0 d*(Y),

COY)>.

Then we have Theorem 3.9. (Grothendieck [SC60/61], Exp. 12). The functor FY is representable and, in fact, represented by the linear space V(Y) associated to Y.

In the proof V(9) is constructed as a complex space only; but it is clear how V(Y) becomes a linear space. Of course (3.9) includes the existence of morphisms between linear spaces: a sheaf homomorphism CL:Y + Y’ of coherent sheaves gives rise to a homomorphism V(u): V(9”) + V(y) of linear spaces (a priori V(a) is a holomorphic map only). Proposition 3.10. closed immersion.

Zf CC:Y+

9”

is surjectioe

then V(a): V(9’)

--f V(9)

is a

For a proof see [SC60/61], Exp. 12. In particular, if 0% -+ Y is an epimorphism, then V(9) is a closed complex subspace of X x cN via V(E). This yields again a) in the proof of (3.6). 0 Another remarkable Zff:

fact is the following:

Y + X is holomorphic

then V(f*(y))

N V(Y)

xx

Y

If Y is a point x E X and f is the inclusion, we obtain again that ,4Px/m,9!! is isomorphic to V(9’)p)x. 3. Analytic Spectra. In this section we describe a general procedure associating to every finitely presented Q-algebra d a complex space Specan ,rQ, the analytic spectrum of d. The importance of this construction will become clear by various examples. The general reference is [SC60/61], Exp. 19. A sheaf of ox-algebras d on a complex space X (or an Q-algebra for short) is called finitely presented or of finite presentation if every point x E X admits an open neighbourhood U such that

~“l.OUCtl,...,tnl/(fi,...,fm),

wheref,E(~~Ctl,...,t,l)(U).

Examples. (1) Every Ox-algebra which is coherent as Ox-module, presented. (2) For every coherent Co,-module Y the symmetric algebra

is finitely

w3 = kToskv) is finitely presented, [SC60/61].

q

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and R. Remmert

The analytic spectrum Specan(&) of a finitely constructed as follows: Locally we may write d = O,[t

presented

d

is

1,...,t,ll(fi,..,,f,).

Then we let Specan(&) be the complex subspace of X function fi, . . . , f,. Afterwards, glue the local pieces. Now we will see that Specan(&‘) represents a functor. the category of complex spaces over X and S the category finitely presented @,-algebra G? we define a finctor F,: C, KAY)

Ox-algebra

:= Howy

where g: Y + X is the structure homomorphisms”.

(g*W),

morphism

x cc” defined by the Let Cx denote again of sets. For a given + S by

h), and

Horn

means

“algebraic

Proposition and Definition 3.11. ([SC60/61], Exp. 19). The jiinctor F,, is represented by the analytic spectrum Specan(&‘) of LZI. There is a structure morphism g: Specan(&) -+ X and a homomorphism p: g*(d) + Or, Y = Specan(d), i.e.: p E F,(Specan(d)). Sketch

of proof. The

assertion

is local.

So we

may

assume

~4 =

QCtt, . . . . t.ll(ft, . . . . &). First let & = O,[t 1, . . . , t.]. Then F..,, is represented by X x (I?: If g: Y -+ X is a complex space over Y, then g*& = Qu[t,, . . . , t,]; every homomorphism 1: g*& + 0, of Or-algebras is determined by the elements h, := A(t,) E O,(Y), hence f: Y+X

x Cc”, YH(g(y),hl(y),...,h,(y))

is holomorphic and pr, o f = g. But this just means that F++ is represented by x x cc”. In the general case JZ? = Oy[tt, . . . , tn]/(ft, . . . , f,) the space Specan(d) is the closed complex subspace of X x a?’ defined by the equations x E x, z E cc”.

fi(x,z)=o,...,f,(x,z)=o, An easy verification

yields that this subspace represents

Fd.

0

Examples. (1) If 9 is coherent, consider its symmetric algebra S(Y). Then Specan S(P) is just the linear space attached to Y. This obviously is (3.9). (2) Let Y be a closed complex subspace of X defined by the ideal 9. Then mTo YrnlY m+l is of finite presentation, the associated spectrum Specan

(

@ $m/$m+l ItI>

>

is called the normal cone qf Y in X. If Y is, locally, a complete intersection, then Y(X/S2) z FlF+l and the normal cone coincides with the normal bundle Nylx = Specan S(S/3’). Cl We collect basic properties

of Specan.

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Proposition 3.12. (1) If cp: & -+ 98 is a homomorphism presentation, then there is a canonical holomorphic map

on Complex

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123

of finite

Specan cp: Specan B + Specan &. & -+ Specan d is a contravariant then Specan(cp) is a closed immersion.

(In other words: (2)

functor).

If cp is surjective,

If JZZand 99 are finitely presented, so is SS?Oc, 98, and Specan(& Oc, B) N (Specan &) xx (Specan 28).

(3) Let & be a finite presented Ox-algebra and let g: Y + X be a holomorphic map. Then Specan(g*&) N (Specan ZZ’) xx Y (base change). Remarks 3.13. Let us consider the simplest case when X is just a reduced point (0). Then an ox-algebra d of finite presentation is nothing else but an analytic algebra A. The fact that Specan represents the functor FA can be expressed as follows: there is a bijection Hol({O}, Specan A) -+ Hom(A, (c). The space Hom(A, (c) of (C-homomorphisms A -+ (I2 is usually called the spectrum of A; this motivates the notation “Specan”. If Specan &’ 3 X and x E X are given, there is a bijection from the set {points y with g(y) = x) to the set {maximal ideals n c J&‘~with n 1 rn,dx and &Jn = Cc}.For details see [SC60/61], Exp. 19. q

We now consider an important

special case.

Theorem 3.14. Let X be a complex space and ~4 a coherent Ox-algebra. (1) The structure morphism Specan & 3 X is finite. (2) The points of every fiber g-‘(x), x E X, correspond bijectively to the maximal ideals n of &, with &Jn N Cc. (3) There is a canonical isomorphism (namely g,(p), see (3.11))

d z g*(%xcan sd). (4) Anti-Equivalence

Principle:

Consider the two functors

FI : {complex spaces finite over X} + (coherent Co,-algebras}, F,: (coherent Ox-algebras} given by Y-+g.+(&$) inverse to FI.

respectively

+ {spaces finite over X},

&bSpecan

&‘. Then F2 is a quasi0

We see that for a finite map g: Y + X we have Y 2: Specan g,(cO,) so that Y can be reconstructed from the image sheaf g,(Lo,). There is no space here to go into details of the proof and we refer to [SC60/61], Exp. 19. Let us just mention that the Weierstrass Preparation Theorem is essentially used and that furthermore the Continuity of Roots is needed. The use of these tools is not too surprising since it is easy to derive from (3.14) the coherence of direct images under finite maps.

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We conclude this subsection with two applications. The first deals with normalizations (see also Chapter I Q14). We consider a reduced space X and denote by 8x its sheaf of germs of weakly holomorphic functions (if U is open in X then 8.J U) consists of all functions f E 0x( U\S(X))_which are locally bounded_on U). The space X is normal if and only if 0, = OxLIt can be shown that 0, is a coherent Ox-algebra. So the space r? := Specan 0x is well defined and we have a holomorphic map v: 2 + X such that 0, z v,(Of). It can be shown that r? is normal (see e.g. [Fi76], p. 115). 0 Next we discuss the so-called Stein factorization [St561 and Cartan [Car60].

which goes back to Stein

Theorem 3.15. Let f: X + Y be a proper holomorphic map of (arbitrary) complex spaces. Then there exists, up to isomorphism, a complex space Y’ and holomorphic maps g: X + Y’, h: Y’ + Y such that f = h 0 g, the map g has connected fibers, 0,. N g,(O,), while the map h is finite. Sketch of proof. Set-theoretically, Y’ is just the quotient space of all connected components of all f-fibers and g and h are the obvious homomorphisms. By construction g has connected fibers and h is finite. So the essential point is to show that there is a complex structure on Y’ such that g and h are holomorphic. This means in essence that (Y’, g,(8,)) is a complex space. Now the assumption of properness off guarantees - by Grauert’s Direct Image Theorem (Chapter 111.4.1) - that f,(O,) is O,-coherent. Hence the complex space Specan f*(Ox) is well-defined, and it is easily seen ([BaSt76], 111.2.12) that (Y’, g,(cO,)) N Swan

f,(G).

4. Homogeneous Spectra. Projective Bundles. In subsection 2 we associated to every coherent sheaf Y over a complex space X its linear space V(9). In a similar manner we may associate to Y its projective bundle K’(Y). We define a functor

F,: (complex spaces over X} + {sets} by associating to f: Y + X the set {locally free quotient sheaves of rank 1 off *(sP)}. Proposition 3.16 (Grothendieck [SC60/61], Exp. 12). The functor F, i:s, for every Ox-coherent sheaf Y’, representable. F, is represented by a coherent sheaf OPCy)( 1) on a complex space n: lP(9) + X, the so-called projective fiber space (or bundle) associated to Y. There is a canonical epimorphism n*(Y) + Co,,,,(l). Sketch of proof. Locally we may start with an exact sequence 0;: 0; + Y; + 0, where u is given by an (n, m)-matrix (xii) with entries aij E Lo(U). Now let lP(9’“) c U x lPml be the closed complex subspace defined by the homogeneous system of m linear equations given by the functions clij. Then glue local pieces. The sheaf 0,(,&l) is just p*(O,“_,(l)), where p is the projection to IPn-i and O,-,(l) denotes the dual Hopf bundle. 0

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Remark. Instead F, we may as well Grk(9’). For details several sheaves 9i, Basic properties

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of taking quotients of rank 1 in the definition of the functor take rank k-quotients. Then we obtain “Grassmann-bundle” see [SC60/61], Exp. 12. In the same spirit we may start from . . . , y, and generalize Gr,Jy). of projective bundles are collected in

Proposition 3.17. Let X be a complex space and let 9, T be coherent sheaves on X. (1) Every sheaf map ~1:Y -+ T determines a holomorphic map IF’(m): lP(T) + Jw?. (2) There is a canonical map lP(9’) x lP(T) -+ lP(Y @ F). (3) If g: Y - X is holomorphic, then lP(g*Y) 2: IF’(Y) xx Y. For proofs see [SC60/61],

Exp. 12.

0

Remarks. The canonical map in (2) is called Segre embedding. In fact, if X is just a simple point, then this map is just the classical Segre embedding pm x Jpn+ ~?n+,. If Y is just a point x E X, then (3) says: lP(YI{x})

= IP(9x.m,9x)

In other words: n-‘(x) eral, r depends on x.

2: lP(Y) xx {x} = n-i(x),

is a projective

n being the projection.

space lP, with r = dim(9x/m,9x).

In gen-

Examples 3.18. (1) Let 9 be a locally free ox-sheaf or rank r. Then lP(9) is locally free of the form U x lPr-i, i.e. a lPr-,-bundle. The sheaf Co,,,,(l) induces on all fibers the sheaf O,~Jl). If F denotes the vector bundle associated to 9, then we set II’(F) := IF’(F). This procedure of passing from F respectively 9 to a “projectivized” object generalizes the procedure of passing from (c” to lP-i. In our setting lP,,-, arises by taking hyperplanes in a” rather than lines to be points of lP-i. This corresponds to the fact that we have taken quotients of 9. In the literature lP(9) is often defined by taking lines, so one has to be careful. The relation between the two definitions is obvious: if we denote this other projectivization by P(F), then P(F) = lP(F*). (2) We use the notations of (1). The procedure of passing from 9 to lP(F) is an important method of reducing problems for vector bundles to problems for line bundles; cf. Chapter V. For relating sheaves on P(F) with sheaves on X the following facts are often useful: For every coherent sheaf Y on X we have (with the projection 7~:lP(Y) + X) a) .rr&*Y 0 QP,,,, 0 Opt&l)) = 9 0 @ilO 9, b) n,(x*Y 0 O,,,,(m)) ‘v 9’ 0 Sm(9), m E IN, in particular: ~,(Gt&)) For proofs see e.g. [ShSo85].

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(3) Again we keep the notations of (1). Then there is an epimorphism n*w A f%,,(l) -+ 0. The kernel of II is given by the so-called relative Eulersequence 0 + Q:,,,,,(l)

-+ n*w

L G(,)(l)

+ 0

where R’ P(f),X(l) := Qb,,,,, 0 Co,,,,(l). For X := (0) and 9 = (l?l quence reads 0 + sZi,( 1) -+ 0;:’ -+ Opn( 1) + 0, or dualized: O~OP”(-l)~~~~~~~~“(-l)~O.

(Sl) the seCM

The map ,U is given by the so-called Euler vector fields .ciL (locally) which are sections in yr,( - 1) (i.e. vector fields vanishing on a hypkrplane) generating rp,( - 1) everywhere. Hence Ker ,Uis of rank 1. By Chern class considerations or by computing cohomology (see Chapter III) it is easy to see that Ker p = OPn( - 1). The sequence (S,) is called the Euler-sequence on lP,,; clearly (S,) is a relative version of (S,). 0 Next we generalize the procedure of attaching coherent sheaf. Let

a projective fiber space to a

be a graded Ox-algebra. Analogously to Specan ~4 we want to define a homogeneous spectrum Projan d. We assume &’ to be of finite presentation, so locally we have an isomorphism & N Q[t,, . . . , t,]/9 with a finitely generated homogeneous ideal 9 in CO,[t,, . . . , t,,]. Let Projan

U) c U x IPn

be the subspace defined by 9. The (global) space Projan & is obtained by gluing together these local pieces. In this way we obtain Proposition 3.19. To every graded Ox-algebra ~4 of finite presentation there is associated a homogeneous spectrum Projan d G X. The space Projan J@’carries, in a canonical way, a line bundle O(1) such that n,(O(m)) = SZZ’,,,.

Every epimorphism d + .G@of graded @,-algebras induces a closed immersion Projan 9? + Projan s$. Examples 3.20. (1) Let Y be a coherent sheaf on X and S(Y) = mFo S”‘(9)

be its symmetric algebra. Then we can form Projan S(Y); clearly Projan S(9) N IP(9). (2) Let Y be a closed complex subspace of X defined by the ideal 9. Then

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The attached space

Projan

@ F/9m+1 ( mt0 > plays an important role in the theory of blow-ups, see Chapter VII. If Y is a submanifold of X or, locally, a complete intersection, the canonical map syY/9’) -+ Y/Y+l is an isomorphism. Hence Projan

@ 9m/$m+1 N lP(4’/92), ( ?llkO > but in general Projan (@F/9m+1) is a proper subspace of lP(9/x2). (3) Let E be a holomorphic vector bundle over a complex space X and O(E) its sheaf of sections. The total space of E can be compactified by a hyperplane at cc at every fiber z-‘(x) of the projection rc: E + X. This compactilication of E can formally be described as follows: The epimorphism 0 @ O(E) + 0 yields an embedding of lP(0) into the space lP(0 @ O(E)) such that IP(0) is just the hyperplane at co (and hence isomorphic to X), so that P(0 @ O(E))\lP(B)

N E.

3 4. Formal Completions Let Y be a closed complex subspace of a complex space X with 9 c Ox as defining ideal. If we replace 9 by 9”“, v 2 1, then P’+l defines the v-th infinitesimal neighbourhood of Y in X. This complex structure on Y is “thicker” than the original one but certainly will not reflect completely the embedding of Y in X. In order to get more information one has to take the limit over all v E N. Then we obtain the so-called formal completion of X along Y. This object will be looked at more closely in this section. 1. Inverse Limits. Let C be a category. A sequence ... + A, 2 A,-, + ... + A, 2 A I of objects and morphisms of C is called an inverse system, and we write (A,, cr,) or simply (A,). An inverse limit A = I@ A, of (A,) is an object A of C together with morphisms p,,: A + A, such that ~1,o p,, = P~-~, satisfying the fol-

lowing universal property: Given any object B of C, together with the morphisms +,,I B + A, such that a, 0 +. = 1+5~-~,there exists a unique morphism Ic/: B + A such that +,, = p,, 0 + for alln 2 1.

The inverse limit is unique, but it may not exist. A morphism (A,, a,)+(B,,, B,,)of. inverse systems is a sequence of morphisms u,: A, + B,, such that u,-i o CI, = 8. o u,. If I@ A, and l&n B,, exist, every morphism (u,) uniquely defines a morphism u: A + B such that u, 0 p,, = q,, o u, where q,,: B -+ B,, is the morphism belonging to B. 0

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In the category of abelian groups, local (C-algebras, . . . inverse limits always exist: Take for A the group, . . . of all “connected” sequences { (ai, a,, . . . ): cr,a, = a,-,} in the product nAY and for p. the restriction of the n-th projection 0 rp”-,A”. We collect some properties of inverse limits (see [At MC 19691, [EGA O,,,], [Ha77]). Proposition 4.1. Let 0 + (A,,) -+ (I?,,) + (C,) + 0 be an exact sequence of inverse systems of abelian groups. (1) The induced sequence 0 + 19 A,, -+ lf-” B, 3 I@ C, is exact, but p may be not surjective. (2) If (A,,) satisfies the so-culled Mittug-Leffler condition, then p is surjective.

The Mittag-LeffIer

condition

can be stated as follows:

If LT,,,,:4, -+ A,, m > n, denotes the canonical map, then for every fixed n the decreasing family {a,,(A,) c A,,, m > n} of subgroups of A,, is stationary.

This condition is certainly satisfied if all maps ~1,: A, + A,-, are onto. The Mittag-Leffler condition is important in cohomology theory, cf. chapter III. We now turn to the category of analytic sheaves on a complex space (X, 0). Proposition 4.2. For every inverse system (9,) of O-sheaves on X the inverse limit Sp := I@ Yn(U) exists. For every open set U in X we have 9’(U)

= l$r Y”(U)

in the category of 0(U)-modules.

For every point x E X there is a canonical Lox-map jx: 9, + lirn Y&, in the cutegory of O,-modules, induced by restrictions .Y( U) + Yx, Yn( U) + Ynx, x E U. Proof. First note that (Y”(U)) is an inverse system of O(U)-modules. In this category l$n Y”(U) exists by what was said above. For V c U we have a canonical restriction morphism (Y”(U) + Y”(V)) and hence a unique morphism r;: lim Yn( U) + lim Y”(V). Clearly Y := { lim Y,(U), r;} is an analytic presheaf. One v&ities direct6 that this presheaf is a &eaf. Now let Y be an O-sheaf with compatible maps tin: Y + sc7,.From the uinversal property of inverse of limits of O(U)-modules we obtain, for each U, unique maps F(U) + Y(U). These give the unique A-homorphism $: 9 + Y which ensures that 9 = l@r Sp,. Since elements of Y(U) respectively 19 Y”, are “connected sequences” in n Yn( U) resp. n Y&, the restrictions Y”(U) + y7nxdefine a map’Y( U) + l@r 5& x E U. Since germs of Y? are induced by sections around x, we obtain the map j, we are looking for. Remark 4.3. In general, the map j, is neither injective nor surjective. However it is easily seen, by using Stein theory (Theorem A) for small Stein-neighborhoods of points of X:

Zf all sheaves Yn are coherent, then every map 9” + l@r Yn, is surjective.

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Remark 4.4. Let (yn, CI,) be an inverse system of Ox-modules, and let $n: y -+ 9, be a morphism of an O,-module y into y”, n E N. Then, by the universal property of inverse limits, there is a canonical morphism Ic/: 9 + 19 9”. Clearly we have Ker Ic/ c n Ker $,,. 2. Formal Completion of a Complex Space along a Closed Complex Subspace. Let X be a given complex space and Y a closed complex subspace

defined by the ideal 3 c 0,. The closed complex subspace of X defined by the ideal sheaf P’+l, v E N, is called the v-th infinitesimal neighborhood of Y in X. All these neighborhoods have Y as underlying topological space, since supp(O,/9”+‘) = supp(0,/9). Clearly we have induced morphisms Ox/$‘” + 0,/4”, v 2 1, and thus an inverse system (0x/Sy), v 2 1. Definition 4.5. The formal completion of X along Y is the ringed space (Y, Oi), where L9i denotes the sheaf of rings Oi = 1Lrn0,/F (which exists by proposition

4.2). It is easily seen that Oi is a sheaf of local (C-algebras, hence (Y,fi) is a C-ringed space. The formal completion is sometimes also denoted by (X, Coi) or by (t 0t) (though these notations are, unfortunately, also used for normalizations). Local sections in 0i are called local formal (holomorphic) functions. We collect some basis properties. (1) The formal completion depends only on the analytic set Y and not on the choice of the ideal sheaf 9.

In fact, if J is another coherent ideal with Y = N(J), then - by the Nullstellenintegers m, n such J” c 9 and x” c J. Hence the inverse limits lp 0x/9v and lt_mUx/Jv coincide. 0 satz - there are, locally,

(2) The residue maps x,: 0, + O/F induce an injection x: 0, + 02. In fact, we have Ker 2cc n Ker x, = n xv by Remark 4.4, and 0 9’ = 0 by Krull’s Y Y Intersection Lemmg. Examples. a) The formal completion of X along X is the space (X, Co,). b) Let Y be a point x E X defined by the maximal ideal m, of Ox,,. Then the sheaf of germs of formal functions of the formal completion of X along x is the formal completion of the local ring O,,,. In case X = (I?’ and x = origin the sheaf 19i,~ is isomorphic to the (C-algebra UZ[[zl, . . . , z,]] of formal power series in n variables. If, more generally, Y c (c” is defined by z~+~ = ..* = z, = 0, then the germs of 0&0 are just those formal power series in zl, . . ., z, which are convergent in zl, . . . , zk. cl Remark 4.6. There is a notion of formal complex space generalizing the concept of formal completion. Roughly speaking, a formal complex space is, locally, the completion along a subspace. Formal complex spaces will be considered in Chapter VII in connection with the theory of (formal) modifications,

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3. Coherent Sheaveson Formal Completions. As before X denotes a complex space and Y a closed complex subspace defined by the ideal 9 c 0,. To every Ox-sheaf Y we associate the Og-sheaf

9 := l&n Y/$vY, which exists by Proposition

4.2. We call 3 the restriction of Y to the formal This Opsheaf is well defined up to isomorphism: choosing another coherent idea1 J c 0, with 1/(J) = Y yields an ok-sheaf isomorphic to 9. Obviously we have 6, = Oi. Observe that for every open set U in X there is a canonical isomorphism

completion 2, we also write 9 2 912.

9(U

n V) N lp

(y/$‘y)(U);

furthermore there is a canonical sheaf map Y + 9. Every Co,-morphism cp: Y -+ r determines an @i-morphism I$: 9 + @, thus we have a functor from the category of @,-modules to the category of Of-modules. In order to study this functor we need a device for computing “formal” stalks (g),. The following procedure heavily uses Stein theory. A compact set K in X is called a Stein compact if K admits a fundamental system of (open) Stein neighbourhoods. Now Theorems A and B for Stein spaces (cf. Chapter III) easily yield (for details we refer to [BaSt76], VI; 0 2): Lemma 4.7. Let Y be coherent over X, and let K be a semi-analytic Stein compact in X. Then the O(K)-module l@(Y/Y’Y)(K) is canonically isomorphic to the completion 9@C) = l& 9(K)/Y(K)“9’(K) (which is also an Co(K)-module). For every point x E X there is a canonical isomorphism

(L?), N 1% Y(K), the limit being taken over all semi-analytic Stein compacts containing X as an interior point.

This lemma shows that “taking sections” commutes with “passing to completion” on semi-analytic Stein compacts, furthermore it gives a device for computing forma1 stalks. The lemma would be rather useless without the following Theorem 4.8. Let K be a Stein compact in X. Then O(K) is noetherian tf and only tf for every set A analytic in a neighborhood of K the set A n K has only finitely many connected components. In particular, for every semi-analytic Stein compact K in X the ring O(K) is noetherian.

This theorem is due to Frisch [Fr67], Grothendieck [SGA2] [Siu69]. See also [BaSt76], V, Q3. Frisch and Grothendieck considered where K is semi-analytic. Note that by a Theorem of Lojasiewicz compact semi-analytic sets have only finitely many components. The Theorem 4.8 again uses the main theorems of Stein theory.

and Siu the case, [Loj64], proof of 0

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We now state the main properties of the functor Y-2’. Theorem 4.9. a) The sheaf 02 is a coherent sheaf of_ rings. b) If Y is OX-coherent, then Y is @i-coherent and Y N Y Qc, 02. c) The functor

y-++g

is exact on coherent sheaves.

d) Let yn be a coherent sheaf on (Y, (!IX/P’), II E lN, and let q,,,: ym + gn be an epimorphism, m 2 n, such that Ker a,,,, = P’yn. Then the inverse limit l@Yn is 0i-coherent. Conversely, for every OR-coherent sheaf 9 we have 9 = lLrnYn, where 9, := p/F@‘.

We want to give some indications of the proof, for details see [BaSt76]. For a point x the theorem belongs to commutative algebra. Consider a commutative noetherian ring R, an ideal I in R and an R-module M. Denote by fi := lim M/I’M the I-adic completion of M. Then the following is well known (proofs &n be found in [Bou61] or [AtMc69]): Proposition 4.10. a) The ring R^ is noetherian. b) If M is a finite R-module then the canonical isomorphism. c) The factor M-h?

map M OR l? -+ d

is an

is exact on finite modules.

d) Let (M”) be an inverse system such every M, is a finite R/P-modules. Assume that all induced maps a,,: M, + M, are surjective, and that Ker c1,,, = ImM,. Then fi := lirn M,, is a finite R-module and M,, N h?/I”ii?. We now sketch a proof of theorem 4.9. ad c) Let 0 + B + G + Y? + 0 be an exact sequence of coherent @,-sheaves. For any semi-analytic Stein compact K in X one obtains, by Stein theory (see again Chapter III), an exact sequence 0 + R(K) + G(K) + YE(K) + 0 of finite B(K)-modules. Since O(K) is noetherian by (4.8), we obtain an exact sequence 0 + Fg) + G@) + 3yi\K) + 0 by applying (4. lob)). Now the assertion follows from the second statement of (4.7). ad a) Let U c X be open, x E U and K c U a semi-analytic Stein compact -‘p + 6, be a morphism; it must be shown the with x as an interior point. Lzt a: Co, Ker a is finite at x. Since O(K) is noetherian by (4.8) and (4.10), there exists an exact sequence

This sequence gives rise to a sheaf map 6$, -+ @& and hence to a sequence (p. AP 5 0,o o,o + 0. Now it is a matter of routine to show that this last sequence is exact, for details we refer to [BaSt76], VI, Q2. ad b) Assume that Lo4+ Op + Y + 0 is exact. Then gq -+ &p + 9 -+ 0 is exact by c) and hence 9 is coherent by a). ad d) This is easily deduced from (4.10, d).

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Finally, we mention some further functorial properties. First of all completions behave well under algebraic operations, e.g. we have

the first isomorphism was already used above in the proof of b). Furthermore, if f: X + X’ is a holomorphic map such that f(Y) c Y’ for closed complex subspaces Y, Y’ of X, X’, then f*(y)

-N f^*(s’)

for every coherent sheaf 9” on X’.

(Here * on the left hand side denotes completion with respect of Y, while * on the right hand side denotes completion with respect to Y’ and f* denotes the induced map (2, Oi) -+ (x“, 0~). 0 Formal completions book, e.g. in

will turn out to be a helpful tool in later chapters of this

- Chapter III: formal cohomology, comparision theorem, base change etc. - Chapters V and VII: theory of modifications, l-convex spaces, formal principal etc. As already mentioned, the theory of formal completions can be developed in more abstract terms by introducing the notion of a formal complex space. Such spaces will be discussed in Chapter VII. As a general reference see [Bin78].

$5. Cohen-Macaulay

Spaces and Dualizing Sheaves

In this section we first treat complex spaces X which are Cohen-Macaulay, i.e. every local ring O,,, is Cohen-Macaulay. After giving various criteria for X being Cohen-Macaulay, we will discuss in particular the remarkable fact, that a Cohen-Macaulay space X is normal if and only if the singular set Sing(X) has codimension at least 2. In order to be able to define dualizing sheaves on complex spaces, we deal in section 2 with Ext-groups and &‘&-sheaves. Section 3 gives the construction of the dualizing sheaf wx and some properties. In sect. 4 we discuss dualizing sheaves on normal spaces and introduce Q-Gorenstein spaces. The main application of dualizing sheaves, namely Serre duality, will be discussed in Chap. III which deals with cohomology. Part of the material presented here can also be found in Chap. I, 9 11. 1. Cohen-Macaulay spaces. In this section we study complex spaces which are Cohen-Macaulay. We start with the algebraic background, general references are: [Bou61], [Ser65], [Mat70], [GrRe71] if you like German, or [EGA].

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(5.1) Let R be a local ring with maximal R-module.

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ideal m, M be a finitely generated

In (1.10) the notions of M-sequences, homological codimension (profondeur) codh M and homological dimension dh M were introduced, which notions we will use freely. A module M is called a Cohen-Macaulay module if codh M = dim M. (we have always codh M I dim M). Recall that dim M is the smallest number k such that there are fi, . . . , fk with ring if it is CohendimRim Ml(f, y . . . , fk)M < co. R itself is a Cohen-Macaulay Macaulay as R-module. 0 We collect some fundamental facts about Cohen-Macaulay-modules course, we are only interested in local rings 0x,, of complex spaces):

(of

Proposition 5.2. (1) Every regular noetherian local ring is Cohen-Macaulay. (2) If R is a regular noetherian local ring, and if fi, . . . , f, E m is a regular (or R-)-sequence, then R/( fi, . . . , f,) 1s ' again Cohen Macaulay. It should also be noted that in a Cohen-Macaulay ring R a set ( fi, . . . , f,} c m is regular (R-sequence) if and only if dim R/( fi, . . . , f,) = dim R - r. More generally let M be a Cohen-Macaulay module over a noetherian local ring R, and fi, . . . . f, an M-sequence, then M/(f,, . . . , f,)M is Cohen-Macaulay over R. (3) Every two-dimensional normal noetherian local ring is Cohen-Macaulay (in dimension 2 3 this is no longer true, see e.g. [Lin67]). Definition 5.3. (1) A complex space X is called Cohen-Macaulay (sometimes also the terminus “perfect” is used) if every local ring Co,,,, x E X, is a CohenMacaulay ring. More generally, X is said to be Cohen-Macaulay at x if ox,, is Cohen-Macaulay. (2) Let X be a complex manifold, and let Y c X be a closed complex subspace. Y is said to be a local complete intersection if its ideal sheaf $ is locally generated by codim (Y, X) elements. The global translation of (5.2) now reads: Proposition 5.4. (1) Every complex manifold is Cohen-Macaulay. (2) Every local complete intersection Y c X is Cohen-Macaulay. (3) Every two-dimensional normal complex space X is Cohen-Macaulay.

It should be noted that a Cohen-Macaulay space may well have nilpotent structure: take e.g. f E 0((F) and let X c (c” be defined by f k, k 2 2; then X is even a global complete intersection, hence Cohen-Macaulay, but is nowhere reduced. Note also that X is Cohen-Macaulay at x if and only if there is a nonzerodivisor f E m, such that O,,, is Cohen-Macaulay where Y = {f = O}; in particular if dim X = 3 and X is normal, then X is Cohen-Macaulay at x iff and only if there is f E m, as above such that Y = {f = 0} is normal (near x). Cl It is convenient to make the following

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Definition 5.5. Let X be a complex m E IN, let

space, 9 a coherent sheaf on X. For

S,,,(9) = {x E Xlcodh

3KxI m}.

The following result is due to Scheja [Sja64] Theorem 5.6. S,,,(F) is a closed analytic set with

dim S,,,(B) I m. The proof uses Ext-sheaves, more details will be given in the next section. Corollary 5.7. For every complex space X the set {x E Xl19~,, is not CohenMacaulay} is analytic of codimension 2 1. In other words, every complex space is Cohen-Macaulay almost everywhere.

On Cohen-Macaulay spaces normality is a very simple property. To be precise, we need the following theorem due to Scheja and Trautmann ([Sja64], [Tra67] see also [SiTr71], [BaSt76], compare also chap. I, 6 10). For a treatment of cohomology theory see chap. III. Theorem 5.8. Let X be a complex space,A c X a closed analytic set and 9 a coherent sheaf on X. Fix a nonnegative integer k. Then the following statements are equivalent (1) inf codh Fx 2 k + 1

x Ex (2) dim A A S,+,+,(R) I p for all ~1

(3) for every open set U c X, the restrictions H’( U, 9) + H’( U\A, 9) are bijective for i < k and injective for i = k. Corollary 5.9. Let X be Cohen-Macaulay. Let A c X be analytic of codim A = c. Then the restriction map o(X) + O(X\A) is bijective for c 2 2 and injective for c = 1.

In fact, apply (5.8) to 9 = 0, and k = 1 resp. k = 0. Corollary 5.10. Let X be a Cohen-Macaulay space. Then X is normal sff codim

Sing(X) 2 2. X is reduced if and only if X is generically reduced. Proof. Observe that, X being generically reduced, X must be reduced everywhere in view of (5.9). Assume codim Sing(X) 2 2 and take x E X. In order to prove normality of ox,, we take f,/g, E Q(0,,,) which is integral over Ox,,. We have to show that f,/g, E Ox,,. By assumption there are h,, . . . , h, E O,(U), with U an open neighborhood (so small that f, g E O(U)), such that (f~+hI+ff-l+...+h,=O.

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Hence i is locally bounded near A and thus f extends by the Riemann 9 sion Theorem to U\Sing(X), hence to U by (5.9). So k E ox,*. x

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Exten0

2. Ext-groups and b&-sheaves. In this section we introduce groups Ext’(P, ~3) and sheaves &&(F-, 9) for every @,-modules 9,s. This generalizes or “shealilies” the well-known construction of Extk(A4, N) for modules M, N over a commutative ring R. As a general reference, see e.g. [Go58]. Let us fix a ring space (X, 0,). Definition

5.11. An O-module 3F is called injectiue if the functor Y + HomCx (9, F)

is exact, i.e. if the following holds. Given any exact sequence o+q

+sz +?33 -+o,

the induced sequence is again exact. [Equivalently:

if

is exact, then Hom(F&, 2F) + Hom(CC&,, 9) + Hom($,

9)

is again exact]. It is now easily verified: Proposition 5.12. Every Ox-module 9 4 9 into an injective Co-module 9.

As a consequence every Q-module we will call I.(B)).

9 admits

an injective

homomorphism

Q admits an injective resolution

(which

which is the same as to say that a) every 4 is injective b) the sequence is exact. We now assume familiarity with the notion of complexes and cohomology of complexes as developed in (111.1). We fix an injective resolution I.(%) for every Q-module 9. To I.(%) we can apply the functor Hom(s, .), where 9 is another &$-module. Hence we obtain a complex Hom(F,

I.(a)).

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We introduce the Ext-groups as the cohomology Definition

3) = H’(Hom(S, I.(Y)).

5.13. Extix(F,

If we take the functor %‘MQ.(~, lead to the following. Definition

5.14. &‘zd(B,

of this complex:

9) instead of HomGx(9,

9) = H’(.YFowz(F,

Y), then we are

I.(9)).

The set Ext’(9, 3) carry naturally the structure of abelian groups, while the c?z:~\~ (9,59) are sheaves of Ox-modules. Remark 5.15. The construction of Ext and &zt only is a special case of a very general procedure. Start with an “abelian” category 59, e.g. the category of Ox-modules on a ringed space or of coherent Co,-modules on a complex space. “Abelian” includes the existence of Hom(A, B) for objects A and B. Now an object A is called injective if Hom(. , A) is left exact. Assume furthermore that any object A admits an injective resolution I.(A). Let F: %?-+ %” be a covariant left exact functor into another category. Then we define the right derived functor

@F(A) = H’(F(I.(A)). So in our situation, Ext and &CC& come up as right derived functors of Hom(9, .) respectively &MZ(~-) .). Another important example is the cohomology H’(X, 9), which is the right derived functor of the functor 9 + 9(X) (in fact, we have constructed cohomology already here, since the functor 9 + P(X) is nothing but Hom(O,, .)). In general, however, this approach is often too abstract for explicit computations. Therefore special nice resolutions play an important role in the computation of Ext or cohomology. For an excellent presentation of this categorial material see - besides [Go581 - also [Ha77]. 0 We now collect basic properties of Ext and 8~~2 which immediately follow from 111.1. Proposition 5.16. Let 9, (1) Ext”(9,9) = Hom(9, 6zd0(9, 3) = #mt(S, (2) ~zt’(O,, 3) = ojor i > (3) If 0 + Pi -+ 9 + F2 +

Y be Ox-modules. Then we have: 9) Y) 0. 0 is exact, then there is an exact sequence

0 -+ Hom(&, -Ext’(gi,

3) + Hom(9, %)+Ext’(S,

Y) + Hom(gi,,

9)

B)+...;

and analogously for &‘zt. (4) Let 3 be a locally free C&-module. Then (a) Ext’(F 0 9, Y) N Ext’(9, 9’* @ ‘3) (b) &z&(9 0 2, Y) N &zti(9, JZ* @ 9) N &i&‘(9-,

‘3) 0 2*

or easily

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(5) If U c X is an open subset, then &dp-,

3)lU = h$“(9p42,

Sp2).

From (2) and (5) we conclude that EzL’(F,

$9) = 0,

i > 0,

provided F is locally free. This corresponds to the fact that Hom(P, if 9 is locally free. In the complex case we have in addition

.) is exact

Proposition 5.17. Let X be a complex space, and let F, 29 be coherent Lo,modules. Then all c.F?zJ~~(9, 9) are coherent. Moreover: &z&

(9, %), ‘v l?~t~,,, (3Fx, %J

for every x E X.

The coherence of J?zt’(B, 3) is seen as follows. By coherence of 9, Y and O,, every x E X admits an open neighborhood %! and an exact sequence o+~+(yy+~~~+o

~+Fp?+O.

(*I

Since &&,(O#, 31%) = 0 for r > 0, we obtain easily (after splitting short exact sequences and applying &‘~t( *, 9)) an exact sequence Afum, Jo;-‘,

31%) + icw?zc,(x~~,

3p2) + &zdi,&q@,

(*) into

SpY) + 0,

hence EzC’, *(F Ia’, B 1%) is coherent.

0

(5.18) We now indicate a proof of (5.6). It is easily seen that we may assume X to be a manifold (by local embedding). Let n = dim X. We make use of the following basic algebraic fact: Lemma 5.19. Let R be regular noetherian local ring of dimensionn, and let M be an R-module of finite type, M # 0. Then:

codh M > q if and only if Extk(M, Moreover dim Extk(M,

R) = 0 for i z n - q.

R) I n - i.

For a proof see e.g. [BaSt76, 1.1.15, 11.1261.

q

Now we deduce from (5.19) S,(9)

=

(J Supp(C%tP(~, 0)) p>“-ffl (5.20) We are going to explain the word “Ext” which stands for “extension”. Of course, everything that follows can be formulated also for modules over rings, but we use the context which is suitable for us. So let (X, 0,) be again a ringed space. An exact sequence of C&-modules 0-+~~+9+2F~+o is called an extension of Fz by Pr. Suppose there is another extension o+F~+s+-+--+o.

(El

138

Th. Peternell and R. Remmert

Then these two extensions are called isomorphic, diagram O----*F~-----*

if there is a commuta’

:

9-&-O

There is a bijection @: (extensions of .& by 9i}/isomorphism

--* Ext’(PZ,

5i)

given in the following way. From (E) we have a “connecting” S: Hom(FZ,

t

homomorphism &) + Extl(.&,

(apply Hom(gZ, a) to (E)). Now Hom(9& namely id,. Then let

F1)

gZ) has a distinguished

element, 1

~([0-~~-~-~*~0])=6(id,~). For details see [GH78], [HiSt70]. So if Exti(&, 9i) # 0, one can always construct new coherent sheaves fron fll and &. This is an important method for constructing locally free sheave? (= vector bundles) of rank 2 2. See e.g. [GH78], [OSSSO]. iA 3. Dualizing Sheaves. The aim of this section is to construct dualizing sheaves on complex spaces. For complex manifolds the definition is easy: We just set

ox = a;,

n = dim X.

? 2 Cl Ii tc

But in the singular case Fiji is not an appropriate candidate: It is in some 2 :’ ‘I sense too singular. ‘n As general references for this section we mention [AK70], [Lip84], [RR70$ [BaSt76]. For our purpose it is important to state Lemma 5.21. Let X be a complex manifold, and let Y c X be a closedcomplex ge subspaceof pure codimensionr. Then SP b~t!~~(Co,, ox) = 0, i -c r. ca

Proof. Since the problem is local, we may assume X to be an open ball in Cc”. Let $ = &~ti~(Or, wx). Since g is coherent, it is sufficient (by Theorem B, . Chap. 111.3) to show g(X) = 0. Now the main point is the existence of an set isomorphism 9X-V = Ext’,,(%

ox),

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139

rich results from the spectral sequence (see 111.519) EQ4 = HP(X, &+(O*, nverging ves

to Ext;:q(O,,

ox))

wx). Together with Theorem

B the spectral sequence

HP(X, &:t4,,(COy, ox)) = 0 for p > 0 and all q. -“his yields easily the isomorphism 3n spectral sequences. Definition

we have been looking for. See III.5 for details q 5.22. Let X and Y be as in (5.21). Then we define coy = &zt;x(oy,

ox).

By definition, or is a coherent sheaf on Y A priori or might depend on the embedding Y G X. However we have the following result. Lemma 5.23. coy is independent of the embedding.

:,

Proof. It is sufficient to consider the following situation. Let X,, X, be complex manifolds of dimensions ni, such that X, is a submanifold of X,. Let Y c X, be a closed subspace of pure codimension r. Then: azt;x, (Oy, ox,) ‘v &t;~yyOy,

t !> J

‘ET

re

ox*).

This follows easily from the spectral sequence (see 111.5). 21

eq = pax, (@Xl?~4J~Y, %*)) converging to G?zC~$$(ox,, c+J, combined with (5.21). Corollary 5.24. coy is well defined for every pure-dimensional It is called the dualizing sheaf of Y.

Cl complex space Y

Proof. Take local embeddings in (cm, define or locally by (5.22) and use (5.23) to prove independence of the local embeddings. 0

3 The name “dualizing sheaf” comes from the Serre duality theorem, (see ‘II.4 and Chap. VI), which is particularly useful on Cohen-Macaulay spaces or manifolds. ,% We give now some properties of dualizing sheaves. The first fact is obvious. 1; :’

ex

Lemma 5.25. Let X be a reduced pure-dimensional complex space. Then ox is generically locally free of rank 1, namely on X\Sing(X). Proposition 5.26. Let X be a complex space, and let Y c X be a closed subspace defined locally by regular sequences, ( fi, . . . , f,) c 3,(%!). Then there is a canonical isomorphism (often called “local fundamental isomorphism”)

C”. B, an

In particular, if X is a complex manifold section of codimension r, then my = 4

and Y c X a local complete inter-

Y 0 detW’&),

(5.26a)

140

Th. Peternell

where NY,, = &‘omC1(3/J2, “det” means taking fl. This algebraic geometry. If Y is a easily by taking determinants

and R. Remmert

Co,) is the normal sheaf (bundle) of Y in X and formula is usually called ‘adjunction formula” in submanifold of the manifold, (5.26.a) follows also of the exact sequence

-0. 0-+-4$X +l2$IY+sz: For a proof (in the algebraic context), see e.g. [AK70]; 3/32 is locally free of rank r by assumption.

q it is important

that

Remark 5.27. (1) If X c C” is a hypersurface, i.e. defined by one equation, then by (5.26) ox is locally free of rank 1, whether or not X is smooth. On the other hand A”-’ Qi is locally free of rank 1 only if X is smooth. So in general ox is different from fi Szi, n = dim X. (2) The sheaf ox is very useful for classifying compact manifolds. For example, consider smooth hypersurfaces X c P,, of degree d. By (5.26.a) we have

ox = (w,IX)

0 G&f)

= G,(d - n - 11,

using the notation UP”(l) = dual of the Hopf bundle (see § 1). Using the notations of positivity (chap. V.), we have: (1) wx is negative if and only if d < n - 1 (2) wx = 0, if and only if d = n - 1 (3) ox is positive if and only if d > II - 1. Manifolds of class (1) are called “Fan0 manifolds”; they are of “Kodaira dimension K(X) = -co”, i.e. H”(w3) = 0 for all p > 0; this class includes projective spaces, quadrics, etc. class (2) has K(X) = 0; if n = 2, X is an elliptic curve; for n = 3, X is a so-called K3-surface. Manifolds of class (3) are called of general type; compare e.g. [Ue75], [Ha77]. A final remark. Dualizing sheaves play - as already mentioned - a two-fold role. First they are important - at least in the normal case - in classification theory. Second they are indispensable for duality theory. However if X is arbitrary, the “dualizing” sheaf ox is not adequate; instead one needs a more general object: the dualizing complex. For this concept, we refer to [RR70], [Ha66], [BaSt76], [Weh85]. 4. Gorenstein Spaces. In this section we consider mostly normal comp!ex spaces. If X is normal, ox is generically locally free of rank 1 and has possibly higher rank only on an at least 2-codimensional set, namely Sing(X), the set of singular points of X. Definition 5.28. Let X be a complex space. A coherent Ox-module 9 is called reflexive if the natural map 9 + 9 ** is an isomorphism. Here as usual ‘9* = ZU~,, (9,0x) by definition.

Note that locally free sheaves are always reflexive. Let us mention two basic facts about reflexive sheaves. First codim (Sing(S), X) 2 3 for any reflexive sheaf S on a complex manifold, second every reflexive sheaf of rank 1 on a

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141

Spaces

manifold is automatically locally free (see [OSSSO]. On a singular space, a reflexive sheaf of rank 1 is in general not locally free. In algebraic geometry this reflects the difference between Weil and Cartier divisors. A remarkable property of reflexive sheaves is the following Riemann Extension Theorem. Proposition 5.29. (Serre [Ser66]) Let X be normal, let 9 be a reflexive sheaf on X and let A c X be analytic of codimension at least 2. Then the restriction map F(X)

+ F(X\A)

is bijective.

0

Lemma 5.30. Let X be a reduced complex space, A c X analytic of codimension 2 2. Then the restriction map WAX) + ox(X\A) is an isomorphism.

A proof can be found in [GrRi70] p. 278. The main point is to “localize” the problem: treat first the case of X being embedded in a ball G c (cm and use here a free resolution O+Rm+

. . . + 90 --+ c?, + 0.

Then apply %?MPz(., oc) and investigate the resulting long exact sequence. Corollary

5.31. Let X be normal and consider the inclusion i: X\Sing(X)

Then ox p iJ~xwngwJ For the proof just observe that Sing(X) normal X.

+ X.

is at least 2-codimensional

Corollary 5.32. Let X be normal. Then wx is reflexive. In particular (& sZ$)** if X is of pure dimension n.

for cl ox 1:

Proof. Since o%* is reflexive, we have i.+(O,\sine(x)) = cot* by (5.29). So (5.31) gives ox N II@*. 0

Having (5.32) in mind it is interesting to know when ox is locally free (X normal). If X is a local complete intersection of codimension r in a complex manifold Y defined by ideal J, then we have by (5.26.a): ox = %4x 0 A’(3/3”)? so ox is locally free of rank 1, even if X is not normal. Lemma and Definition 5.33. Let X be normal and let i: X\Sing(X) the inclusion. (1) For r E IN let o&l = i*(o$&,,&. Then 05;~ is a reflexive sheaf. (2) X is said to be r-Gorenstein if c&l is locally free. If X is r-Gorenstein r, then we say also that X is Q-Gorenstein. (3) X is Gorenstein if it is Cohen-Macaulay and 1-Gorenstein.

+ X be for some

142

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and R. Remmert

There is a point to be cautious: 1-Gorenstein Gorenstein. For an example see [I&7].

spaces are not necessarily

Example 5.34. Here we give an example of a normal complex space X which is not QGorenstein. Let E be the vector bundle associated to the locally free sheaf S,,( - 2) @ O,,( - 2) on lP,. We identify lP, with the zero section in the total space Y of E. We will see in chap. V that there exists a normal complex space X and holomorphic map cp: Y + X such that cp(lP,) is a point x,, E X and cpI Y\lP, + X\{x,} is biholomorphic. The reason is that the normal bundle N P,,y 2: E is negative. Let us verify that X is not 1-Gorenstein. So assume the contrary. Then ox is locally free on all of X. Since

we have Hence oyllP, N 0. On the other hand, the adjunction

formula (5.26.a)

(-1 2: oyP, 0 det Npliy gives oy IlP, 2: O(2), contradiction. The argument for c# is the same. Note that X is nevertheless Cohen-Macauly (“rational always Cohen-Macaulay”, see e.g. [Rei87]).

singularities

are 0

Example 5.35. If we substitute O( - 2) 0 U( - 2) in (5.34) by 0( - 1) 0 0( - 1) then it is easily seen that X will be Gorenstein. Compare also [Lau81]. Remark 5.36. (1) Q-Gorenstein spaces (with additional properties on the singularities) play an important role in the classification theory of algebraic varieties, in particular for finding minimal models etc. See e.g. [KMM87]. Of course, the notion of Gorenstein spaces is also very important in the theory of singularities. (2) In algebraic geometry the notion of a Weil divisor on a normal complex space plays an important role. A Weil divisor is a formal finite sum Eni& where n, E Z and x are irreducible reduced analytic sets of codimension 1 in X. Since X is not required to be smooth, the ideal sheaf Sri is not necessarily locally free, hence r; is not a divisor (= Cartier divisor) in the usual sense. But we can still associate to yi a reflexive sheaf @x(x) = (syi)*. Thus we get a one-to-one correspondence (modulo “isomorphisms”)

Weil divisors ++ reflexive sheaves of rank 1. See [Ha771 for details (in the algebraic category). In particular, ox corresponds to a Weil divisor K,, which is called a canonical divisor of X (unique up to “linear equivalence”). Then we can say: X is r-Gorenstein if and only if the Weil divisor rK, is in fact a Cartier divisor.

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Maps and Linear Structures on Complex Spaces

143

References* [AK701 [AtMc69] [BaSt76] [Bou61] [Bin781 [Car331 [Car601 [Dou68]

LEGAl [Fi67] [Fi76] [Fr67] [Gra62] [GH78] [Go583 [GrRe71] [GrRe77] [GrRi70] [Ha661 [Ha771 [HiSt71] [Hir75] [Is871

Altman, A.: Kleiman, S.: Introduction to Grothendieck duality theory. Lect. Notes Math., Springer 146, 197O,Zbl.215,372. Atiyah, M.F.; McDonald, LG.: Introduction to Commutative Algebra. AddisonWesley 1969,Zbl.175,36. Banica, C.; Stanasila, 0.: Algebraic Methods in the Global Theory of Complex Spaces. Wiley 1976,Zbl.284.32006. Bourbaki, N.: Algibre commutative. Elements de Mathematique 27, 28, 30, 31. Hermann, 1961-1964,Zb1.108,40; Zbl.l19,36; Zbl.205343. Bingener, J.: Formale komplexe Raume. Manuscr. Math. 24, 253-293 (1978) Zb1.381.32015. Cartan, H.: Determination des points exceptionnels dun systeme de p fonctions analytiques de n variables complexes. Bull. Sci. Math., II, Ser. 57, 334-344 (1933) Zbl.7,354. Cartan, H.: Quotients of complex analytic spaces. In: Contrib. Funct. Theor., Int. Colloq. Bombay 1960, 1-15 (1960) Zbl.122,87. Douady, A.: Flatness and privilege. Enseign. Math. II, Ser. 14, 47-74 (1968) Zbl.183,351. Grothendieck, A.: Elements de geometric algebrique. Publ. Math., Inst. Hautes Etud. Sci. 4, 8. II, 17, 20, 24, 28, 32 (1960-1967) Zb1.118,362; Zbl.122,161; Zb1.136,159; Zb1.135.397; Zbl.144,199; Zb1.153,223. Fischer, G.: Lineare Faserriiume und koharente Modulgarben iiber komplexen Raumen. Arch. Math. 18,609-617 (1967) Zbl.177,344. Fischer, G.: Complex analytic geometry. Lect. Notes Math. 538, Springer 1976, Zbl.343,32002. Frisch, J.: Points de platitude dun morphisme d’espaces analytiques complexes. Invent. Math. 4, 118-138 (1967) Zbl.167,68. Grauert, H.: Uber Modilikationen und exzeptionelle analytische Mengen. Math. Ann. 246, 331-368 (1962) Zb1.173,330. Grifliths, Ph.; Harris, J.: Principles of Algebraic Geometry. Wiley 1978,Zbl.408,14001. Godement, R.: Topologie algebrique et thiorie des faisceaux. Hermann 1958, Zbl.80,162. Grauert, H.: Remmert, R.: Analytische Stellenalgebren. Grundlehren math. Wiss. 176, Springer 1971,2b1.231.32001. Grauert, H.: Remmert, R.: Theorie der Steinschen Raume. Grundlehren math. Wiss. 227, Springer 1977,Zb1.379.32001. Grauert, H.; Riemenschneider, 0.: Verschwindungssiitze fiir analytische Kohomologiegruppen auf komplexen Rliumen. Invent. Math. 11,263-292 (1970) Zb1.202,76. Hartshorne, R.: Residues and duality. Lect. Notes Math. 20, Springer 1966, Zb1.212,261. Hartshorne, R.: Algebraic Geometry. Graduate Texts Math. 52, Springer 1977, Zb1.367.14001. Hilton, P.J.; Stammbach, U.: A Course in Homological Algebra. Graduate Texts Math. 4, Springer 197l,Zbl.238.18006. Hironaka, H.: Flattening theorem in complex-analytic geometry. Am. J. Math. 97, 503-547 (1975) Zbl.307.32011. Ishii, S.: Uj-Gorenstein singularities of dimension three. Adv. Stud. Pure Math. 8, 165198 (1987) Zb1.628.14002.

*For the convenience of the reader, references to reviews in Zentralblatt fiir Mathematik (Zbl.), compiled using the MATH database, have, as far as possible, been included in this bibliography.

144 [Kau68] [Ker68] [KMM87] [Lau81] [Lip841 [Loj64] [Mat701 [0066] [0066] [OSSPO] [Pr68] [RR701 [Rem571 [SC60/61] [ShSo85] [SGAZ] [Ser56] [Siu69] [SiTr71] [Sja64] [St561 [Tra67] [Ue75] [We801 [Weh85]

Th. Peternell

and R. Remmert

Kaup, B.: Ein Kriterium fur Platte holomorphe Abbildungen. Bayer. Akad. Wiss., Math.-Naturw. Kl., S.B. 1968, Abt. II., 101-105 (1969) Zbl.207,380. Kerner, H.: Zur Theorie der Deformationen komplexer Raume. Math. Z. 103, 3899 398 (1968) Zbl. 157,404. Kawamata, Y.; Matsuda, K.; Matsuki, K.: Introduction to the minimal model problem. Adv. Stud. Pure Math. 10, 283-360 (1987) Zbl.672.14006. Laufer, H.: On CIP, as an exceptional set. In: Recent developments in several complex variables. Ann. Math. Stud. 100, 261-275 (1981) Zb1.523.32007. Lipman, J.: Dualizing sheaves, differentials and residues of algebraic varieties. Asterisque I 17, 1984,Zbl.562.14003. Lojasiewicz, S.: Triangulation of semi-analytic sets. Ann. SC. Norm. Super. Pisa 28, 4499474 (1964) Zbl.128,171. Matsumura, H.: Commutative Algebra. Benjamin, New York 197O,Zbl.211,65. Oort, F.; Commutative group schemes. Lect. Notes Math. 15, Springer 1966, Zbl.216,56. Oort, F.: Algebraic group schemes in characteristic zero are reduced. Invent. Math. 2, 79-80 (1966) Zb1.173,490. Okonek, C.; Schneider, M.; Spindler, H.: Vector Bundles on Complex Projective Spaces. Prog. Math. 3, Birkhauser 198O,Zbl.438,32016. Prill, D.: Uber lineare Faserraume und schwach negative holomorphe Gcradenbiindel. Math. Z. 105, 313-326 (1968) Zbl.164,94. Ramis, J.P.; Ruget, G.: Complexe dualisants et theoreme de dualite en geometric analytique complex. Publ. Math., IHES 38, 77-91 (1970) Zb1.205,250. Remmert, R.: Holomorphe und meromorphe Abbildungen komplexer RPume. Math. Ann. 133, 328-370 (1957) Zb1.79,102. Stminare H. Cartan 1960/61: Familles des espaces complexes et fondements de la geometric analytique. Paris, EC. Norm Super., 1962, Zbl.124,241. Shiffman, B.: Sommese, A.J.: Vanishing Theorems on Complex Manifolds. Prog. Math. 56, Birkhauser 1985,Zb1.578.32055. Grothendieck, A., et al.: Seminaire de geometric algibrique 2. Cohomologie local des faisceaux coherents. North Holland 1968,Zbl.197,472. Serre, J.P.: Algebre locale. Multipliciti. Lect. Notes Math. II, Springer 1965, Zb1.142,286. Siu, Y.T.: Noetherianness of rings of holomorphic functions on Stein compact subsets. Proc. Am. Math. Sot. 21,483-489 (1969) Zbl.175,374. Siu, Y.T.; Trautmann, G.: Gap-sheaves and extension of coherent analytic subsheaves. Lect. Notes Math. 172, Springer 197l,Zb1.208,104. Scheja, G.: Fortsetzungssatze der komplex-analytischen Cohomologie und ihre algebraische Charakterisierung. Math. Ann. 157,75-94 (1964) Zb1.136,207. Stein, K.: Analytische Zerlegungen komplexer Riiume. Math. Ann. 132, 63-93 (1956) Zb1.74,63. Trautmann, G.: Ein Kontinuitatssatz fur die Fortsetzung koharenter analytischer Garben. Arch. Math. 18, 188-196 (1967) Zbl.158,329. Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Lect. Notes Math. 439, Springer 1975,Zbl.299.14007. Wells, R.0 Differential Analysis on Complex Manifolds. 2nd ed. Springer 1980, Zbl.435.32004, Zbl.262.32005. Wehler, J.: Der relative Dualitatssatz fib Cohen-Macaulayrlume. Schriftenr. Math. Inst. Univ. Miinster, 2, Ser. 35, 1985,Zb1.625.32010.

Chapter III

Cohomology Th. Peternell

Contents Introduction

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

.................... 5 1. Flabby Cohomology ............ 1. Cohomology of Complexes 2. Flabby Sheaves ...................... ................. 3. Flabby Cohomology 4. Fine Resolutions and the de Rham Lemma

. . ..

..

146

. . . . .

. .. . .. ..

147 148 149 150 152

. . . . .

......................................... Q2. Tech Cohomology ........................................ 1. Tech Complexes ...................................... 2. Tech Cohomology 3. Leray’s Lemma ......................................... 4. Dolbeault Lemma and Dolbeault Cohomology

.. .. .. ..

153 153 154 155 157

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

158 158 159 161

5 3. Stein spaces .............................................. 1. Stein spaces: Definition and Examples ...................... ...................................... 2. TheoremsAandB 3. The Cousin Problems .................................... $4. Cohomology of 1. Direct Image 2. Comparison, 3. Riemann-Roth 4. Serre Duality

Compact Spaces ............................. Theorem ................................... Base Change and Semi-Continuity Theorems ................................. Theorem ......................... and Further Results

....

........................................ $5. Spectral Sequences ............................ 1. Definition, Double Complexes 2. The Frolicher Spectral Sequence .......................... 3. The Leray Spectral Sequence ............................. 4. Some more Spectral Sequences ............................ References

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

162 162 163 167 170 173 173 175 177 179 180

Th. Peternell

146

Introduction Cohomology (with values in a sheaf) - attaching to every sheaf 2 of abelian groups “cohomology groups” Hq(X, Yip), q 2 0 - was invented in the late 1940s but implicitly it has been present since the 19th century. We want to explain this and demonstrate the necessity for a cohomology theory at the hand of several classical or “basic” problems. 1. Let X be a compact Riemann surface. A basic problem of the last century was to construct non-constant meromorphic functions on X. More specifically let x 1, ..., xk E X be points, and associate to xj some integer nj E IN. Does there exist a meromorphic function f with poles at xj of order at most nj and holomorphic outside the xj’s? If 0,(x njxj) = 2 d enotes the locally free sheaf of rank 1 of local meromorphic functions with the prescribed pole orders as above, then the problem is to show the non-vanishing HO(X, 9) # 0. In general H”(X, d;p) is nothing else than P(X). X being compact, both H”(X, .Y) and H’(X, 2) are finite-dimensional and the famous Riemann-Roth theorem says: dim H”(X,

2) - dim H’(X,

2) = 1 - g + 1 ni,

with g being the genus of X. So if cni > g - 1 we can conclude H”(X, 2) # 0 and obtain our meromorphic function. 2. The classical first Cousin problem asks the following: Let X be say a complex manifold, (ai) a covering by open sets. Let hi E &(+Yi) be a meromorphic function on ei. We ask for a global meromorphic function h E &Z(X) with hlai - hi E .,H(+Yi). We will see later (3.9) that the hi give rise to a section s E H”(X, A/07) and that h exists, if and only if 6(s) = 0, where 6: HO(X, Ji!/O) -+ H’(X,

0)

is the so-called connecting homomorphism. In particular, h exists always if H’(X, 0) which in turn is true for instance for Stein manifolds. 3. We are going to explain “connecting homomorphisms”. Let O+%-+-++~-+O be an exact sequence of sheaves of abelian groups on a topological Taking global sections we obtain an exact sequence 0 + HO(X, %) + HO(X, 3) 5 HO(X, YF),

space X. (S)

but c1is in general not surjective. Let for example X = (c*, and consider the exponential sequence 0 + Z + 0 + Co* + 1, f t+exp(2rrif). Then the non-surjectivity

of CImeans that on Cc* there is no global logarithm.

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147

Now the problem is to measure the non-surjectivity connecting homomorphism 6: HO(X, H) + H’(X,

of (Y.This is done by the

9)

which allows us to extend (S): 0 + HO(X, 9) + z-P(X, 9) + HO(X, X) + H’(X, And “proceeding

9)

further”:

Coming back to the Cousin problem or the above counterexample one can state that cohomology groups in complex analysis often describe how to go from the local to the global, how to patch local things together to obtain a global object respectively what obstruction occur. 4. We would like to mention one other basic problem which is immediately related to cohomology. Let X be a complex space, Y c X a subspace with ideal sheaf J,. The problem is to extend holomorphic functions from Y to X. For this purpose write down the exact sequence O+Jy+Ox-,O,-+O

and “take cohomology”: 0 -+ H”(X,

Jy) + H”(X,

Q) : H’(Y,

0,) A H’(X,

Jy)

Hence if H’(X, Jy) = 0, c( is surjective. This happens for instance if X is a socalled Stein space. In this chapter we describe mainly cohomology of coherent sheaves on complex spaces. We introduce cohomology in two different ways: as flabby cohomology and as Tech cohomology. From a general point of view, flabby cohomology might be more satisfactory but for many purposes in complex analysis, Tech cohomology is better adapted. In the next chapters we describe the cohomology of Stein spaces and of compact spaces including base change and semi-continuity theorems as well as the Riemann-Roth theorem. The last section deals with an important tool to compute cohomology: spectral sequences. We tried to avoid a too abstract presentation of cohomology theories, in particular derived categories etc. If one is familiar with the standard cohomology theories, one might find it easy to understand also the general abstract point of view. Another remark: we have not tried to trace the historical origins too carefully, especially not in the first sections. Instead, we included only “standard references”.

0 1. Flabby Cohomology In this 0 we first introduce in a very general way cohomology of complexes. As a special case, we will construct for any sheaf a “flabby resolution”, which gives rise to a complex by taking global sections and leads us to the notion of (flabby) cohomology of a sheaf. General references are [God641 and [Ser55].

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1. Cohomology Definition

We let R be a commutative

of complexes.

ring.

1.1. a) A sequence K’-+K’+

do

d’

. . . +

K4

!$,

K4+’

+

of R-modules

K’ and R-homomorphisms is called a complex of R-modules if dq+’ o dq = 0 for all q 2 0. We write for short K’ = (K‘J, dq) for this complex. Each element tl E Kq is called a cochain, dq is called a coboundary map. b) Let K’ = (Kq, dq), K” = (Klq, dlq) be two complexes of R-modules. A morphism of complexes cp’: K’ + K” is a collection (cp’) of homomorphisms of Rmodules (pq: Kq + K’q such that

Definition 1.2. Let K’ = (Kq, dq) be a complex of R-modules. a) We define Zq(K’) = Ker dq, the group of q-cocycles, and Bq(K’) = Im dQ-‘,

the group of q-coboundaries, B’(K’) = 0. b) Since d q+ldq = 0 , we have Bq(K’) c Zq(K’). tient and define Hq(K’)

Hence we can form the quo-

Zq(K ‘) = ~ Bq(K’)

to be the q-th cohomology module of K’. c) If cp’: K’ -+ K” is a morphism then (pq(Zq(K’)) (pq(Bq(K’))

c Zq(K”),

and

c Bq(K”),

so that cp’ induces homomorphisms Hq((p’): Hq(K’)

+ Hq(K”).

If cp’: K’ --* K”, II/‘: K” + K”’ are two morphisms, cp’ 0 I,+‘, and one has

it is obvious how to define

W((p’ 0 II/‘) = W((p’) 0 Hqp). In other words, Hq is covariant functor of the category of complexes of Rmodules to the category of R-modules. Definition

1.3. A sequence

K. 5 K’. 2 KU. of complexes of R-modules

is called exact if for all q K4 s K’4 2 K”q

is exact, i.e. Im ‘pq = Ker eq. An exact sequence O-,K’+K”+K”‘+O

III.

149

Cohomology

will also be referred to as a “short” exact sequence. Here 0 is the zero complex (P, dq) with Kq = 0 for all q. The following result is of vital importance

for the sequel

Proposition 1.4. Let 0 + K’ % K” % K”’ + 0 be a short exact sequence of complexes of R-modules. Then there exists a “long” exact cohomology sequence Ho(q’) Ho(K’.) + . . . + H4(K’) k!?!, fjq(K”) + 0 + H’(K’) Ha(ll! Hq(K”.)

2 Hq+‘(K’)

with canonical “connecting”

morphisms

--) . . .

dq: H‘J(K”‘)

--) W+‘(K’).

Moreover, given a commutative diagram of exact sequences of complexes O+K’+K”-,K”‘+O 1

1

1

0 + L’ + L” + L”’ + 0 one has a commutative diagram of long exact cohomology sequences . . . -+ Hq(K’)

+ H’J(K”)

-+ Hq(K”‘)

+ Hq+‘(K’)

1

1

1

1

. . . + fp(L’)

+ fp(L”)

+ fp(L”‘)

--* fp+‘(L’)

-+ . . .

-+ . . .

2. Flabby Sheaves Definition 1.5. Let X be a topological

space, and 9’ a sheaf of say groups on X. The sheaf Y is called flabby if the restriction

is surjective for every open set U c X. Construction

and Definition 1.6. Let Y be a sheaf. Then we are going to sheaf F(9) containing 9’.

associate to 9’ aflabby

For this purpose we let rc: S + X be the espace &ale associated to 9’ and we put S, = rt-i(x). Then for U c X open define

In other words, we take all not necessarily continuous sections of 9. It is obvious that F(Y) is a flabby sheaf. Every morphism cp: Y + Y’ of sheaves determines a morphism 9((p): 9(9q

+ F(Y).

Th. Peternell

150

If o-+y+Y’+Y”+o is exact, then in general 0 + Y(X) + Y’(X) -+ Y’(X) is exact, however Y(X) + Y”(X) is not necessarily surjective. This is one of the reasons for introducing cohomology. But if Y is flabby we nevertheless have Proposition 1.7. Let (X, z&‘) be a ringed space (e.g. a complex space, a topological space. . . ) and let o-iy-+~-iY”+o be an exact sequence of &-modules. 0 + Y(X)

If Y is flabby, + Y(X)

the sequence

-+ Y”(X)

+ 0

is exact. Definition

1.8. If Y is an &-modul,

any exact sequence of d-modules

0 + Y + yb + y; -+ ... is called a resolution of 3 The resolution flabby.

is called flabby

if all Yq, q 2 0, are

It follows from (1.7) that Corollary

1.9. A flabby

resolution 0 + Y + yb + y; -+ *..

induces a complex O-,~(X)~~~(X)~~~(X)-r~~~.

Next we construct for a given Y in a canonical way a flabby resolution. Let Ye = F--(Y) be the flabby @-(Ysp) and let cpe:yb -+ Y; be the sheaf constructed in (1.6). We set Y1 = 9 ~ ( Y > canonical map. Proceeding inductively we get a canonical flabby resolution Construction

1.10. Let Y be an &-module.

0 + y -+ yb 2 9; 1. . . . 3. Flabby Cobomology. Let (X, ~2) be a ringed space, Y and &-module. use the canonical flabby resolution of Y to define flabby cohomology.

We

Definition 1.11. Let 0 + 9 + yb + Y1 + . .. be the canonical flabby resolution of .Y. By (1.9) this resolution gives rise to a complex

o~~(x)~yb(x)~~t(x)~....

III.

Hence letting R = d(X)

151

Cohomology

and Kq = Yq(X), we obtain a complex K’ and define H4(X, 9) = Hq(K’),

q 2 0,

the flabby cohomology modules of 9. We collect basic properties of flabby cohomology

(1.4):

1.12. Let 9, 9”, Y” be d-modules. (A) Hq(X, Y 0 9’) ‘y Hq(X, 9) 0 Hq(X, 9’) (B) Any exact sequence Proposition

O+Y+y’+~4p”+O gives rise to a long exact cohomology sequence 0 + HO(X, 9) + zP(X,

9’) + HO(X, Y”) 3. H’(X,

Y) + . . * .

(C) Zf 9’ is flabby then Hq(X, 9’) = 0 for q > 1. (D) Any commutative diagram of exact sequences o-Y-y’-y-0

implies a commutative diagram

. .. +

Hq(X,Y)

-

Hq(X,9")

. .. -

HW) I H4(X,y)

-

Hq(X,y')

Remark.

H”(X,

I

-

Hq(X,9"')

-

Hq(X,F')

I

-

Hq+'(X, 9') -

...

-

Hq+'(X, y) -

I

"'

3) is nothing else but Z’(X).

A natural question arising now asks what happens if we take another flabby resolution of Y for defining cohomology, or, more generally, another resolution for which the sequence arising when we take global sections remains exact. In order to get an answer we state the following elementary proposition (it is done in a rather informal way but we hope that it will be clear to the reader what is meant). Proposition 1.13. The cohomology theory is uniquely determined by the properties (B), (C), (D) of (1.12), and by H’(X, 9’) = 9’(X), Ho(~) = ‘px. Definition

1.14. A resolution

is acyclic if Hq(X, YP) = 0

for all p 2 0, q 2 1.

Th. Peternell

152

Given an acyclic resolution as above, we obtain a complex K” by setting K4 = Yq(X). By applying (1.13) we obtain Proposition

1.15. We have Hq(X, Y) N Hq(K”).

So any acyclic resolution and, in particular, any flabby resolution acyclic by (1.1 1,C)) can be used to compute Hq(X, 9’). 4. Fine resolutions

(which is

and the de Rham Lemma

Definition 1.16. Let X be a topological fine (sometimes the term “soft” is used) if

rest:

space, 9’ a sheaf on X. Y is called

Y(X) + Y(A)

is surjective for every closed A. Here Y(A) is a short-hand for (91,4)(A). The most important

examples are:

(1) the sheaf %?of continuous functions on a metric space (2) the sheaf d of differentiable functions on a differentiable generally all b-modules.

manifold

and more

The proof of (2) relies on the partition of unity. It is also easy to see that flabby sheaves on a metric space are fine. Fine sheaves are not necessarily flabby (take d of example 2!). Nevertheless one has the Proposition

1.17. Let (X, JS?)be a paracompact

ringed space.

(1) Let 0 + Y’ + 9’ -+ 9”’ + 0 be an exact sequence of &-modules

with 9’ fine. Then 0 + Y(X) + Y(X) -+ Y”(X) + 0 is exact. (2) Let Y be a sheaf of &-modules. Let 0 --* 9’ + Yb + 9, --, .. . be a fine resolution of 9, i.e. all $ are fine. Then 0 + Y(X) + go(X) + Sp(X) + * .. is a complex. If Y is also fine, the sequence is exact.

It has the following important

consequence

Proposition 1.18. Let (X, SCJ’)be ringed with X metric, and let 9’ be a fine sheaf of &-modules on X. Then H4(X, 9’) = 0, q > 0.

In other words, fine resolutions are acyclic, and so fine resolutions can be used to compute cohomology. We will demonstrate this at the hand of an important example. Example 1.19. Let X be a differentiable (real) manifold, and let 64 the sheaf of differential q-forms on X. Let lR be the sheaf of locally constant real-valued functions on X. Then the sequence

0+IR~*d@~&.+...

(S)

is exact. The exactness (which is a local statement !) is nothing but the classical lemma of Poincare:

III.

153

Cohomology

Let U be, say, a ball in lR” (or a convex domain), and cp a r-form with dq = 0 then there is an (r - l)-form + such that cp = dll/. By the remark following (1.16) it is seen that (S) is a fine resolution particular it is acyclic and hence (1.15) yields:

of lR. In

H”(X, IR) s Ker(d: a(X) + S’(X)) Hq(X w) ~ KerW EqW -, gq+‘(X)) 7 Im(d: &H(X) + &7(X))

twith b-’ = O).

These statements are called the lemma (or theorem) of de Rham, while the right hand sides are often referred to as de Rham cohomology groups. The main point of de Rham’s theorem is that it relates topological invariants of X (namely the groups Hq(X, lR)) to differential invariants.

0 2. tech Cohomology For many cohomological problems in complex analysis flabby cohomology is not very well adapted, in particular when open coverings come into the play. Also for explicit computations flabby cohomology is often not very suitable. So we are now going to introduce the most popular cohomology theory in complex analysis, Tech cohomology. Again we refer to [God58], [Set%], [GrRe84]. 1. tech Complexes. We fix a ringed space (X, &), an d-module covering 22 = ( Ui)i EI by open sets Ui c X. Definition

9’ and a

2.1. For q 2 0 we put

cq@!,%Yq =

n

9(uio,...,i4)7

(io,....iq)E1q+’

where UiO,,,,,i, = Ui, n ... n Ui9. Cq(@, Y) is an &(X)-module, a =

its elements are called q-cochains ~1.We shall write (a(io,

. . . 9 iq))(io,...,i,)E,4+l.

We now define a coboundary map 6 = 6,: cqp&, 9) + cQ+l(%Y,9) by

q+l W4(io, . . . , iq+l

) =

~~o~-~)Y~~~o,...~5~..~~~,+~~l~io,....i,,,~

Here !,, means omission of i,. The following fact is easily verified.

Th. Peternell

154

Proposition

2.2 S,,, 0 8, = 0

Thus (Cq(%, Y’), S,), is a complex, the so-called tech complex C’(SY, 9’) of 9’ with respect to %. Definition 2.3. Hq(@, LY’)~G~Hq(C’(%, of Y with respect to 4? Remarks

9)) is the q-th tech cohomology module

2.4. (1) Let cp: Y + Y’ be a morphism.

Then cp induces maps

CyiY, cp): eye, Y) + cya, 9’) and H4(@, cp):Hy!%, 9) + Hq(e, Y’). (2) We put Ker S, = Zq(%!, Y), the module of q-cocycles, and Im S,-, = F(%, Y), the module of coboundaries. 2. tech Cohomology. The cohomology modules Hq(%, Y) depend on the open covering @. In this section we are getting rid of %. SO let ?3! = (Ui)ier, Y = (k$)j,r, YY = (Wk)keK be open coverings of X. Definition 2.5. (1) We say that V is finer than @, in signs V < %!, if there exists a map z: J + I such that for all j E J: y c Urcjj. (2) If V < a, and T: J + I is the corresponding map, then z induces a map

C(2): cy42, Y) + cyv-, 9) setting CqWW(io,

. . . . iq)

=

Mid,

.-.,

4iq))lFo,...,i

4.

Now one verifies Lemma 2.6. (1)

If V < 49, and z is the defining map, then dqcJ(z) = cq+‘(z)dq.

Hence 7 induces morphisms Hq(.r): Hq(@, 9’) + Hq(V-, 9’). (2)

Zf 5’ is another map defining Y < 9!!, then Hq(.r) = Hq(r’).

Thus Hq(~) depends only on the open coverings.

Now we can form the inductive define:

limit

of the system (Hq(‘B, 9’), Hq(~)). We

III. Cohomology

155

Definition 2.7. fiq(X, 9’) = 15 Hq(%, 9) is the q-th tech cohomology module of Y.

Given an exact sequence O+Y+Y’+~+O, we do not have a long exact sequence for H”(%, .). For Hq(X, .) this however holds. We state the result. Proposition 2.8. Given an exact sequenceon the paracompact spaceX o-+~P~+y’-*o

of d-modules,

there is a long exact cohomology

. . . -+ tjqx,

Y) + Ijyx,

Remarks 2.9. Sometimes cochains, putting either

sequence

Lq -+ 2+(X, 9”) : tiq+‘(x,

one works with a slightly

cq(@~

y,

=

n i. < .

Y) + . . .

different notion

of q-

yP(“io,....i,) < i,

or letting cq.(@9

be the subset of “alternating”

yP)

n (i0,...,i,)

y(“io,.

, i,)

cochains (tlio,, , i4)’ defined by c(i.,o,,

for permutations

c

,i.,,,

=

w

n%o..

, i4

z of (0, . . . , q}, with ci.10,,,,,is=O

ifi,=i,

(forsomevfp).

Using one of these definitions for the cochains in building ends up with the same tech cohomology theory.

up the theory, one still

3. Leray’s Lemma. In this section we are going to deal with the following two basic questions (notation as above):

(A) When is a Tech cohomology

module Hq(&, Y) already the inductive limit

Hq(X, Y)?

(B) Are tech cohomology

and flabby cohomology

the same?

Definition 2.10. An open covering 3! = (Ui) of X is called acyclic or a Leray covering with respect to Y if

Hq(Uio,...,iP, 9) = 0 for all q 2 1 and all p 2 0.

Th. Peternell

156

Theorem 2.11. (Leray’s Lemma) Let X be paracompact locally finite covering of X with respect to 9 Then:

and 92 an acyclic

H4(sY, Y) ‘v H4(X, Y) (the right hand side being flabby

cohomology for the moment)

It follows from the paracompactness of X that such a covering % exists, and the proof proceeds in the following steps: (1) We construct a special resolution of 9’ corresponding to the covering %. Namely we set

YP=

n

i,(W”io,...,ip)y

(io,....i,)~lp+l

where i: Ui,,,,,,i -+ X is the inclusion; so i,(Y 1Ui,, ,i,) is nothing but the trivial extension of 9+~Ui,,,,,,,i by 0. Since Yp(U) can be viewed as a module of pcochains, we obtain a &solution ()+y~yO$y’$.. (the exactness has to be proved, of course). (2) One has the following isomorphism for flabby cohomology: H’(Xt

9’)

2:

n (i0,...,i,)

Hq(X3

(here locally finiteness of 9Y is important). Hq(X~

&.(WQ,

,...,

i,))

2:

i*(~l~i,,...,i,))

Since Hq(Uio

,...,

ip3

W”io

,...,

iph

we obtain: Hq(XY

yp)

2:

n (i0....,i,)

Hq(Uio

,...,

ip7

yI

uio ,.._, i,h

(*I

(3) Now assume @ to be acyclic with respect to 9’. Then (*) tells us that the resolution of (1) is acyclic. Hence H4(X, 9) N H4(K’), where K’ is the complex (Yi(X), d’). But K’ = C(%, 9’) by the very definition Yp and the differentials. So our claim follows.

of

The assumption of paracompactness can be omitted in (2.11) but this is of little practical use. Since by Leray’s lemma all modules kZq(@,9’) vanish if Y is flabby, we obtain as Corollary

2.12. Let X be paracompact, Eiq(X, 9) = 0

and let Y be a flabby sheaf. Then for all q 2 1.

Now we have seen that the Tech cohomology (1.12). Therefore we obtain

fiq fulfills all requirements

of

Theorem 2.13. Let X be paracompact ringed space, and let Y be a sheaf of &-modules. Then Hq(X, 9) 2: fiq(X, 9’); in other words 6ech and flabby cohomology are the same (and we will not distinguish them in what follows).

III.

Cohomology

1.57

4. Dolbeault Lemma and Dolbeault Cohomology. This section is the holomorphic analogue of de Rham’s lemma (sect. 1.4). Let X be a complex manifold, and let @‘denote the sheaf of holomorphic p-forms. We denote by &Fq the sheaf of complex-valued Cm+, q)-forms. So locally o E ap*“(U) is of the form o = Cfi ,,..., ip,j ,,,,., j, dzi, A ... A dziP A d~jl A ... A d”j,,

with the coefficients fi ,,,,,, i,,j, ,,,,, j, being complex-valued P-functions. In the multi-index notation one has w = c fiJ dz, A dZ,. One obtains maps 2: &Pq --) &p,q+l by setting &J = 1 8fiJ A dz,

A

(locally),

d5,

where

The main result of this section is Theorem 2.14 (Dolbeault).

The sequence

0 *

is exact, i.e. a resolution complex.

The mathematical

QP

i+

gp.0

4

&P.

14

. . .

of Qp. The complex (JFpg’(X), 3) is called the Dolbeault

content of 2.14 is the

Lemma of Dolbeault 2.15. Let U(r) = U c C” be a polycylinder radius r and center 0, that is U={z~C”~~zY~
r=(rI

,...,

with poly-

rJ.

Let s < r, q 2 1 and let CIJE cF?~,~(U) with &o = 0. Then there exists an element cpE kFpvq(U(s)) with w = &.

(In fact, this result is also for true s = r, but for our purposes this is not necessary). A proof of (2.15) (relying on Cauchy theory) can be found e.g. in [GrRe77] (even in a more genera1 form). Since all sheaves &p*q are fine (see 1.16), our resolution (2.14) is acyclic by (1.18). Therefore we deduce: Theorem 2.16. We have Hq(X in particular,

(here

we set ep*-‘(X)

= 0).

7

Qp) N 10 E ~P*qwIa~ &yP+rl(X)

= 01 ’

158

Th. Peternell

Theorem (2.16) is sometimes also called Dolbeault’s theorem Lemma (2.15) can now rephrased (in its strong form s = r) as follows:

(lemma).

Hq(U, Qp) = 0 for all q 2 1 and p 2 0. Let us point out that the cohomology modules Hq(X, 9’) with a coherent sheaf Y (very often locally free) contains important geometric information and it is desirable to have criteria for when these modules vanish or at least are linitedimensional (C-vector spaces.

§ 3. Stein Spaces In the 50’s sheaf cohomology complex analysis:

was the main tool for the great breakthrough

in

Theorems A and B of Stein theory, solution of Cousin problems, Levi problem etc. In this section we want mainly to describe the cohomological aspects of Stein theory. As principal reference as well as a source for further references we quote the book [GrRe77]. The “convexity” aspects of Stein theory (pseudo-convex spaces, q-completeness, Levi problem) will be discussed in chap V and VI. 1. Stein Spaces: Definition

and Examples.

Stein manifolds

have been intro-

duced - not named -by K. Stein [StSl]. Definition

3.1. Let X be a complex space with countable

topology.

is called a Stein space if the following two conditions hold. (A) Every compact analytic set A c X is finite. (B) For each compact set K c X there exists an open neighborhood in X such that Z? n U is compact, where I? = {x E XjIf(x)l

I sup (If(z)1

for allfE

Then X

U of K

Lo(X)}

ZEK

is the holomorphically

convex hull of K in X.

Remark 3.2. It can be shown (see [GrRe77, Gr55]) that for manifolds these conditions are equivalent to the following three conditions: (C) X is holomorphically separable: for x1 # x2 E X there exists f E Lo(X) such that f(xi) # f(xZ). (D) (Uniformization) For any x,, E X there are functions fi, . . . , f, E 8(X) (m = dim X) such that in local coordinates

III. Cohomology

159

(E) X is holomorphically convex, i.e. (B) holds for any K c X with U = X, or equivalently: for every discrete sequence (xi) in X there is a function f E O(X) such that the set { If( i E IN} is unbounded. Examples 3.3. (1) a? is Stein.

(2) An open set U of a Stein space X is Stein if and only if it is holomorphitally convex. (3) Any closed subspace of a Stein space is again Stein. (4) If X and Y are Stein, so is the product X x Y. (5) X is Stein if and only if its reduction is. A reduced space is Stein iff its normalization is Stein. (6) If X is Stein, and h E Co(X), h # 0, then X\(h = O> is Stein (7) If X is a normal Stein space, and A c X is an analytic subset of codim A 2 2, then X\A is not Stein: Every f E O(X\A) can be extended to all of X by Riemann’s extension theorem, hence for any sequence (xi) converging to x E A, the set (If(xi)l Ii E JN} is bounded. In particular, different:

(c”\(O} is not Stein for n 2 2. For n = 1 things are completely

(8) Every non-compact Riemann surface (which is connected by definition) is Stein. This is a non-trivial theorem of Behnke-Stein [Best491 and the main point is precisely to construct one non-constant, holomorphic function (or to prove cohomology vanishing H’(X, Y) = 0 for locally free sheaves of rank 1 on X). More examples of Stein spaces can be found in [GrRe77]. The fact that a domain G c (lZ:”is Stein if and only if its a domain of holomorphy is especially noteworthy. 2. Theorems A and B. The main theorem

Theorem B of Cartan-Serre Theorem 3.4 (Theorem

on Stein spaces is the so-called

[SC52]. B). Let X be a Stein space. Then zP(X, 9) = 0

for all q > 0 and all coherent sheaves9 on X.

For its (complicated) application is

proof we refer e.g. to chap VI and [GrRe77].

A first

Theorem 3.5 (Theorem A). Let X be a Stein space, and let 9 be a coherent sheaf. Then the global sections of 9 generate every stalk 9?Zas Ox,,-module.

The proof is a nice example of how cohomology theory works. Fix x E X and let m, c O,,, be the maximal ideal. We identify m, with the “trivial” extension of m, on X, i.e. the ideal sheaf of {x} in X. Then we obtain the exact sequence O+m,+Ox+Ox/m,+O.

160

Th. Peternell

Here 0,/m, is nothing but the sheaf of holomorphic functions on the reduced space {x}, so it equals (c on x and 0 outside x. Now tensor the exact sequence by 9 and let m,P be the image of the natural map

Then we obtain an exact sequence

By Nakayama’s

lemma, our claim is equivalent

to the surjectivity

H’(X,

N 9 0 6$/m,,

F) 1: H’(X,

9 @ 0,/m,)

of

where the map ICjust attaches to each section s of 9 its value s(x). By the long exact cohomology sequence, K is surjective if H’(X, m,R) = 0. But this is guaranteed by Theorem B. By similar methods we can prove that holomorphic functions on Stein spaces separate points etc. It is not difficult to see that in fact Theorems A and B are equivalent. Moreover, one has the following converse of Theorem B (see [GrRe77]): Theorem 3.6. Let X be a complex space with countable topology. Assume that H’(X, J) = 0 for every coherent ideal sheaf J (it would have been sufficient to assume this for those sheaves with supp(o/J) discrete). Then X is Stein.

Since Hq(X, Qp) = 0 for 4 > 0 on a Stein manifold

X we obtain from (2.16):

Theorem 3.7. Let X be a Stein manifold, p 2 0, q > 0. Let o be a &closed (p, q)-form. Then o = a(p for some (p, q - l)-form cp.

It is an interesting open problem whether on a complex manifolds X the vanishings Hq(X, Qp) = 0 for all p and all q > 0 already force X to be a Stein manifold. Compare [Pe9 11. From the Poincare lemma for holomorphic forms we have a resolution O+(C+O~~a’&-22+... If X is Stein, this resolution is acyclic, hence Proposition

3.8. Let X be a Stein manifbld. Then H’(X,

Q N Ker(dILD(X)),

Hq(X c) N Kdd: fJq(W -+ Qq+l(X)) 3 Im(d: aqml(X) + Qq(X)) ’ In particular,

Hq(X, (c) = 0 for q > dim X on a Stein mani$old X.

For more information [Na67], [Ha83].

on the topology

of Stein spaces, we refer to [Gr58],

III.

Cohomology

161

3. The Cousin Problems. The Cousin problems influenced in a very significant way the development of the theory of several complex variables in the first half of this century. They are the analogues of the classical theorems of MittagLeffler and WeierstraD in one variable. 3.9. The first Cousin Problem. Let X be a complex space, and let (Ui) be a covering by open sets Ui. For any index i let hi E A?(Ui) be a meromorphic function on Ui. The problem is now to find h E .4(X) such that hJ Ui - hi E O(Ui). To bring cohomology into the picture we look at the exact sequence

040+~~~/040. The collection (hi) determines an element s E H’(X, A/O) and to solve the problem means to find h in H’(X, A!) with q(h) = s. This h exists if and only if d(s) = 0 where 6: HO(X, .M/O) 4 H’(X, is the connecting homomorphism. 0. Hence:

co)

If X is Stein, H’(X,

Lo) = 0 and hence 6(s) =

Theorem 3.10. The first Cousin problem on a Stein spacecan be solved for all (Vi, hi). (In fact, it can be solved for all X with H’(X, 0) = 0 or, even more general, for all complex spacesX for which the map H’(X, 0) + H1(X, 4) is injective). 3.11. The second Cousin Problem. Given again a complex space X and a covering (Ui) by open sets, we now let hi E A*(Ui) (.A!* being the sheaf of units in A) and ask for h E A*(X) such that

The data (Ui, hi) determine an element D E H’(X, 9), where 9 = A*/O* “sheaf of divisors” on X. If we consider the sequence 0-+0*4A!*&240

is the

9

then to solve the problem for (Vi, hi) means to find h E A’*(X) a similar way as above we obtain:

with $(h) = D. In

Theorem 3.12. The secondCousin problem is solvable for (Ui, hi) if and only if 6(D) = 0, 6: H’(X, 9) + H’(X, O*) being the canonical map. In particular, it is solvable for all (Ui, hi) if and only if H’(X, O*) + H’(X, A*) is injective (this condition is fulfilled if H’(X, O*) = 0).

We want to investigate the group H’(X, Co*) of holomorphic line bundles modulo isomorphy more closely. To this end let us look at the exponential sequence 04Z+O+O*+l.

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Th. Peternell

We obtain:

H’(X, 6) + H’(X, o*) + HZ(X, iz) + H2(X, Co) This yields Corollary 3.13. Let X be a complex space with H’(X, 0) = H*(X, Z) = 0. Then the second Cousin problem is always solvable. In particular, this is the case for all Stein spaces with H*(X, Z) = 0.

Z) = 0 holds e.g. for any non-compact Weierstral3 theorem on these spaces).

(H’(X,

We have already mentioned rank 1 the class

given by the transition H’(X,

Riemann

surface implying

the

that by associating to a locally free sheaf Y of

functions with respect to some open covering, we have

O*) N {holomorphic

line bundles}/isomorphy.

The map H’(X, O*) + H*(X, Z) induced by the exponential sequence associates to 2 its first Chern class cl(y). Hence on a Stein space 2 is determined by cl(~), which is a topological invariant of the line bundle corresponding to 2’. The group H*(X, Z) can also be interpreted as group of topological line bundles (modulo isomorphy), because of the “topological” exponential sequence

and H’(X, %?)= H*(X, %‘) = 0. Therefore a holomorphic line bundle L on a Stein space X is (holomorphically) trivial if and only if it is topologically trivial. This is a very special case of the so-called Oka-Grauert principle. For more information on this topic, see [Lei90].

54. Cohomology

of Compact Spaces

In this section we discuss the cohomology of compact complex spaces: Iiniteness theorems, cohomology of families, base change, semi-continuity etc. 1. Direct Image Theorem. Given a continuous f: X + spaces and a sheaf Y - say of abelian groups - we denote by of abelian groups associated to the presheaf U + Hq(f-‘(U), usually writes f,(Y) for R”f,(P’). One of the most important theory of complex spaces is

Y of topological R4f,(Y) the sheaf 9’). If 4 = 0, one theorems in the

Grauert’s Direct Image Theorem 4.1. Let f: X + Y be a proper holomorphic map of complex spaces, and let Y be a coherent sheaf on X. Then Rqf,(Y) is coherent for every q > 0.

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Cohomology

163

For a simplified version of the original proof ([Gr60]) we refer to [FoKn71] and [GrRe84]. In the special case where Y is a point, we obtain Corollary 4.2. (Cartan-Serre, [CaSe53]). Let X be a compact complex space, and let 9’ be a coherent sheaf on X. Then the C-vector spaces H4(X, 9’) are finite dimensional.

Of course (4.1) respectively (4.2) are false if f is not proper respectively X is not compact. A classical case of (4.2) is when X is a compact manifold and Y a locally free sheaf. Then finite-dimensionality can be proved via the theory of elliptic operators. (One applies this theory to the Laplace operator acting on vector-valued differential forms with respect to hermitian metrics on X and on the vector bundle associated to 9 Compare e.g. the book [We80]. The approach goes back to Hodge and Kodaira. In the case of Riemann surfaces, finite dimensionality has been already known to 19th century (Riemann).) Another important consequence of (4.1) is Remmert’s mapping theorem ([Re58]): Corollary 4.3. (Remmert). Let f: X + Y be a proper holomorphic map of complex spaces and A c X a closed analytic subset. Then f(A) is analytic in Y. Proof. Equip A with the reduced structure. Then 0A is a coherent Q-module (since the full ideal sheaf IA is coherent). Consequently, f,(OA) is a coherent &-module. Since supp( f,(OJ) = f(A), f(A) is analytic. Another rather easy consequence is the Stein factorization. Theorem 4.4. Let f: X + Y be a proper holomorphic Then there exists a diagram

with a complex space Z, such that h is finite !3*(&) = Pz. 2. Comparison,

and g has connected fibers,

Base Change and Semi-Continuity

tion we refer in general to [BaSt76]. X + Y, and a coherent sheaf Y on equipped with the “analytic preimage reduced point y, the ideal sheaf&i,, off

Rqf,(9’)t

is just the inverse limit

12 Wd~P)ylmyk~qf*(~40)y).

and

For this secholomorphic map f: consider the fiber X, my is the ideal of the of the canonical map

Theorems.

We fix a proper X. For y E Y we structure”. So if -l(y) is the image

f *(my) + %

The formal completion

map of complex spaces.

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Th. Peternell

On the other hand, the formal cohomology ffqGfy, 9) is nothing but the inverse limit l&n H4(Xy, sp/@Y). Formal cohomology can also be defined abstractly for any coherent sheaf on a completion X (or more generally on formal complex spaces). Compare [Bi78]. So via the canonical maps Rqf*(mylmykRqf*(my -+ HQ(Xy, w$,% which are induced by the exact sequence o+dl,“Lf+c+-,sp/ti;~+o, we obtain a canonical map (py”:Rqf*(Y); Now Grauert’s isomorphism:

comparison

theorem

-+ H’I(Jizy, 9). ([Gr60],

[BaSt76])

states that (py”is an

Theorem 4.5. (1) (p; is an isomorphismfor all q and all y (recall the properness assumption!). (2) There is a function h: IN -+ IN such that W~qf,&3’),

--, Rqf,Wy)

= m,h”“Wqf,W)J

(2) is the essential content of the theorem, while (1) is a rather straight forward consequence of (2). For the proof the Mittag-LelTler condition is important (see chap.II.4). This condition implies that cohomology commutes with inverse limits [BaSt76, V.1.91. As a particular case of (4.5) we mention the isomorphism

Roughly speaking the comparison theorem says that formal completion commutes with taking direct images. Next we discuss the base change theorem. We assume now that Y is f-flat. Let g: Y’ -+ Y be any holomorphic map. Then there is a canonical map

wheref’:X xr Y’+X’andg’:X xr Y’ + X are the projections. In general tiq is not an isomorphism. The base change theorem says under which conditions I,+~is an isomorphism. Theorem 4.6. Assume that Y is f-flat (and f: X -+ Y proper as usual). Fix q E IN. Then the following conditions are equivalent. (1) For any basechange g: Y’ + Y the canonical map Ic/,(defined above) is an isomorphism.

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Cohomology

(2) The canonical restriction R’f,(yiy

+ Ff,

V’I~,W,

is onto for all y E Y. (3) The canonical restriction Rqf*(L+ii;9)

-+ R“f*(y&9’)

is onto for all k 2 1 and all y E Y. (4) The functor 9 H R4f*(9

0 f *(9))

(from the category of coherent sheaves on Y to the category of coherent sheaves on Y) is right exact (left exact).

An important corollary (of a more general version) of the base change theorem is Grauert’s theorem ([Gr60]) Theorem 4.7. Assume that Y is f-flat. (a) (semi-continuity): For any q E IN the function y-dim

HqWy, Wf-'(y))

is upper semi-continuous. (b) Zf base change holds for Y and q and q - 1 then

y H dim Hq(Xy, 91X,,) is locally constant. The converse holds if Y is reduced. (c) The function

is locally constant.

(d) Zf y-dim Rqf*(Y)

HQ(X,,, YIX,)

is locally

constant and Y is reduced, then

is locally free of rank dim Hq(X,, YIX,,),

W&9,lqJ%J~“),

moreover

= HqWy, WXJ

Remarks. (1) In the theorem, X,, is always understood as the complex subspace of X given by the ideal sheaf fi,,. t-3 x(3 = C (- l)qhq(X, 9) 1s . as usual the (holomorphic) Euler characteris4 tic of 9. Corollary 4.8. Assume that Y is f-fat and that for fixed q E IN one of the equivalent conditions of the base change theorem is fulfilled. Then the following two assertions are equivalent. (1) Ry*(P’) = 0 for all p 2 q (2) HP(Xy, 9’) = 0 for all p 2 q, and all y.

We wish now to explain by some examples how these results work.

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Th. Peternell

Example 4.9. Let Y be a normal complex surface (germ) with just one singularity ye. The point y, is called a rational singularity if for one (and hence for all !) desingularisations 7~:X + Y one has R’n,(O,) = 0. From the comparison theorem we deduce

Let E be the reduced space n-‘(ye). We claim that H’(E, infinitesimal neighborhood E, we have

fmEJ

0,) = 0. In fact, for any

= 0,

as there are epimorphismus fw%P)

+ m&z-,I

(note that H’(E, Ip-‘/Zp) = 0, since dim E = 1, where I denotes the ideal sheaf of E). Thus H’(oE) = 0 and it follows that E is a “tree” of smooth rational curves (This explains the name “rational singularity”). Example 4.10. (1) Let X be a smooth complex surface, Y a Riemann surface, and f: X + Y a smooth proper surjective holomorphic map. Assume furthermore that f has connected fibers. So X can be viewed as a family of compact Riemann surfaces parametrized by Y. Now it is clear that every fiber X,,, y E Y, is reduced, hence dim H’(X,, 0x,) = 1. From (4.7) we conclude that dim H’(X,,, Oxy) is locally constant, hence constant, which just says that the genus of X,, does not vary. This follows of course also from differential topology: f is C” - locally trivial. For the theory of surfaces it is important that dim H’(X,,,

Loxy) is constant

even if f is not necessarily smooth and has disconnected libres, see e.g. [BPV84], so if e.g. the general fiber X, is Ip,, then every fiber X,, has to fulfill H’(Xy,

q)

= 0,

implying that a (reduced) singular X, can only be a tree of lP,‘s. (2) We would like to have a closer look at the case when Y is compact and X a lP, - bundle over Y. That is, X is a ruled surface. If dp is a line bundle on X, let d = deg(YIX,). It is an easy topological fact that d is independent of y. Hence dim H’(X,,,

91X,,) = d + 1 if d 2 0

and Zf’(X,,, So by (4.7)

91X,,) = 0

if d < 0.

f*(Z) is locally free of rank d + 1 whenever d 2 0. In particular, f*v%) = 0,

Now H’(X,,

TpIX,,) = 0

ford 2 1,

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Cohomology

hence R’f,(T)

= 0.

Since the higher groups Hq(X,,, P’[X,,) vanish anyhow, we have also R4f,(Lf) = 0 for 4 > 0. We will see in the next section (via Leray’s spectral sequence) that this implies that zP(X, 9) N zP(Y, f*(L-Eq), d 2 - 1. In particular H2(X, 9) = 0 (for d 2 - 1). If 9 = 0, we obtain: H2(X, 0,) = 0, i.e. dim H’(X,

H’(X, w 0,) is the genus of Y.

= H’(Y, &I,

3. Riemann-Roth Theorem. Although the Riemann-Roth theorem is quite different in nature from all the other material presented in this paragraph, it is one of the most basic methods to compute cohomology on compact manifolds, so it should be mentioned here. 4.11. Chern classes. Let X be a complex manifold of rank r. We can associate with d the Chern classes c,(S) E H2’(X,

lR),

and d a locally free sheaf

0 I i 5 r.

(In fact one can define Chern classes of complex vector bundles on differentiable manifolds). For a construction of c,(S) using connections see e.g. [We80]. We list a few of their properties: (a) ci(f*(b)) = f*(c,(S)), where f*: H”(X, lR) + H2’(r?, lR) is the pull-back map and f*S the pull-back of 8, induced by a holomorphic map f: 2 + X. (b) Ci(b*) = (- l)‘ci(~). (c) cl(P) = deg 9 for a locally free sheaf of rank 1 on a compact Riemann surface. (d) c,(d) = 1. We define the Chern polynomial c,(d) by c,(cq = c,(B) + c,(b)t + .** + c,(B)t’. (e) If 0 -+ 9 + d -+ 3 + 0 is an exact sequence of locally free sheaves, then W)

= c,(T). CA%

(the dot denotes the intersection product in H*(X, 4.12. Definition.

ronX.

Let X be a compact manifold,

IR)).

6 a locally free sheaf of rank

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Th. Peternell

(1) The exponential Chern character is ch(B) = i

e”‘,

i=l

where we write formally c,(a) = ir (1 + ait), i=l

eai being defined as 1 + ai + $ + . . . in H*(X,

lR).

(2) The Todd class of d is defined by

This formula is interpreted power series expansion

in the following sense (ai as in (1)): If we consider the

X

1 - ePx

-

1 +;x+&x2-&x4+-.,

then td(&‘) = n

1+ ; + $ - &

+ . .. >

(since dim X is finite, td(d) is clearly a finite expression). Remark. One can show (with ci = Ci(a)) that

(1) ch(&‘) = r + cr + (fc: - C2) + i(C; - k, C2 + 34 + ’ **, (2) td(d) = 1 + ;cl

+ :,(c:

+ c2) + :,c,c,

+ ... .

4.13. Theorem of Riemann-Roth. Let X be a compact manifold of dimension n and 6 a locally free sheaf on X. Then the holomorphic Euler characteristic

x(X, 8’) = t (- l)i dim H’(X, 8) can be computed as follows: i=l x(X,

4

=

(ch(4.

td(%)hn,

where & is the tangent bundle (sheaf) and ( degree 2n, i.e. in H2”(X, IR).

)2n means taking the part of

For line bundles on compact Riemann surfaces the theorem is due to Riemann and Roth, but this was all there was for almost one century. In 1953 Hirzebruch [Hir56] proved (4.13) in the case of projective manifolds. The general case is a consequence of the Atiyah-Singer index theorem [AtSi63]. There are generalizations for coherent sheaves on projective manifolds ([BoSe59]) and

III. Cohomology

169

compact manifolds ([ToTo76]). Moreover, Grothendieck proved RiemannRoth in a relative algebraic situation, i.e. for maps; singular algebraic versions are due to Baum, Fulton and Mac Pherson, and to Verdier (see [Fu184]). Examples 4.14. (1) For surfaces one has x(0,) = &c:(X) + c*(X)), whereas for 3-folds, the formula reads x(0,) = &ci(X)c,(X). Since one is very often able to compute x(0,), Riemann-Roth formulae give important informations about the Chern classes of X. For instance, if X is a Fano 3-fold, meaning that - Kx = A3 TX is ample, (see V.4) then Hq(X, Co,) = 0 for q > 0 (Kodaira vanishing theorem), hence ~(0,) = 1 and cl(X)c,(X) = 24 by Riemann-Roth. On the other hand, if cl(X) = 0 for a 3-fold X, we see that x(0,) = 0. (2) Now let X be a compact surface, L? a locally free sheaf of rank 1 on X. Then Riemann-Roth reads:

If c,(L?)’ > 0, we conclude that either dim H’(X, like $. Since H2(X, LP) 2: HO(X, Jr’

zZ’~),or dim H2(X, Yfl) grows 0 Kx)*

by the so-called Serre duality, with K, = (A2 Y.)*, we have produced sections either of Lp or of L-” @ K,. This argument is very important in surface theory. (3) Often useful is the following remarkable theorem of Hopf: if X is a compact complex manifold of dimension n, then c,(X) = K&X), where x,,,(X) = f. ( -

l)ibi(X)

is the

topological

Euler

characteristic

and

bi = hi(X) =

dim H’(X, lR) are the Betti numbers of X. This has a holomorphic counterpart: if E is a holomorphic vector bundle on X of rank n = dim X admitting a section s whose zero set {s = 0} is finite, then c,(E) = #(s = 0}, counted with multiplicities. We refer to [GH78]. (4) We demonstrate the power of “Chern class theory” by indicating a proof of the famous theorem that every compact complex surface X homeomorphic to IP2 is in fact lP2. For details we refer to CBPV843. By Hopfs theorem 4X)

= c*(P2) = 3.

By the so-called index theorem, the index of the topological H2(X, W) is computed by t(X) = 3(&X)

intersection form on

- 2$(X)).

It follows in general that c: is a topological invariant of compact surfaces. In our situation we conclude that c:(X) = c:(IP2) = 9. Since c:(X) > 0, X is projectivealgebraic (the argument in (2) shows already the existence two algebraically independent meromorphic functions). From Hodge decomposition on X (see 9 5) we get Hq(X, Co,) = 0 for q = 1,2.

170

Th.

Hence the exponential

Peternell

sequence gives Pit(X)

2: H2(X, Z) 2: Z,

where Pit(X) is the group of holomorphic line bundles modulo g E HZ@‘,, Z) be the generator with g = ~i(O~~(l)). Then

isomorphy.

Let

Cl(X) = +3g. It follows that either ox’ = ,4’Tx, or ox is ample (for the notion of ampleness see V.4). Assume first that OX’ is ample (this is the case which really occurs). Let O,( 1) be the ample generator of Pit(X). Riemann-Roth gives x(0,( 1)) = 3. Since H’(X, O,(l)) 2: H’(X, 0X( - 1) @ wx) = 0 (Serre duality), it follows dim H’(cO,(l))

2 3,

and even equality holds by applying the Kodaira vanishing theorem (V.6) to H’(X, O,(l)). Now it is easy to see that the map f: X + lP, defined by HO(Ox( 1)) is biholomorphic. It remains to show that wx cannot be ample. This was unknown for a long time. Up to now the only known way to exclude this case is to apply Yau’s theorem on the existence of a Kahler-Einstein metric on X [Yau78]. This metric together with the equality c:(X) = 3c,(X) implies that the universal cover of X is the unit ball in C2, in particular X is not simply connected, contradiction. 4. Serre Duality and Further Results. In this section we shortly review other important results on the cohomology of compact complex spaces. One of the most important and most basic results is the Hodge decomposition. We shall discuss this in 6 5 in connection with the Frolicher spectral sequence. For further results in this direction, see [We80], [GH78]. Another fundamental result is Serre duality. Theorem 4.15 (Serre). Let X be a n-dimensional compact complex manifold, 8 a coherent sheaf on X and ox = A”04 the dualizing sheaf of X. Then

Hq(X, 8) 2: Ext”C;q(&, wx) (more precisely there are functiorial

maps

Ext”-q(&, ox) + Hq(X, a)* which are all isomorphisms).

In fact, one can construct a natural pairing ExCq(&

wx) x Hq(X, b) + C

which in case q = n is just the composition Hom(&

of the canonical maps

ox) x H”(X, d) + H”(X, ox) $ Cc,

where t is the so-called trace map. In most applications,

in particular

when d is

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locally free, just the equality of dimensions

171

is used, e.g.:

dim Hq(X, b) = dim Hndq(X, 6* 0 wx). If X is non-compact, one has to use cohomology with compact support; if X is singular, ox has to be replaced by the so-called “dualizing complex” (see [RR70], [BaSt76]), except when X is Cohen-Macaulay (e.g. a locally complete intersection): Theorem 4.15a. Let X be a compact Cohen-Macaulay n. Then the conclusion of (4.15) still holds.

For the definition

of dualizing

space of pure dimension

sheaves on Cohen-Macaulay

spaces see chap.

II. Finally, there are relative versions of Serre duality. General references are: [RR70,74], [BaSt76], [Weh85], [Ser55-21. It should be mentioned that the algebraic case is due to Grothendieck [SGA2], and to Hartshorne [Ha66]. See also [Lip84], [Kun75,77,78]. Serre type duality on compact Riemann surfaces was already known in the 19th century. In the most basic form it states H’(X, 0) N H’(X, @), i.e. the genus of X is the number of independent holomorphic l-forms. Up to now we here discussed mainly finiteness theorems for cohomology. Vanishing theorems are likewise of great importance: under which conditions can one conclude that Hq(X, P’) = 0 for a coherent sheaf Y and a certain q? This will be a topic of chap. V. In order to work with cohomology one must know basic cohomology groups on certain “model” manifolds. Let us mention here the “Bott formula” ([Bo57]) in the special case of projective space. Theorem 4.16 (Bott). One has (k+;-p)(kpl)

for;;>

Olpln,

fork=O,

O
dim Hq(P,,, sZp @ Lo(k)) =

I

i-F;p)(;*,‘)

for;;;,-tSpSn,

0

otherwise

For the cohomology of Grassmannians and other homogenous rational manifolds we refer to [Sn86]. Very often one is not only interested in the dimension of a single cohomology group Hq(X, 2) for say a line bundle 9 but also in the asymptotical behavior of dim Hq(X, P’),

p --) 00

Th. Peternell

172

The most important special case leads to the definition sion rc(X) of a compact complex manifold X.

of the Kodaira

dimen-

Theorem-Definition 4.17. Let X be a compact maniJold of dimension n and set ox = Ilnf2:. (1) Assume that dim H’(X, coy) 2 2 for some m E IN. Then there exists an integer k E IN with 1 I k I n and constants a, b such that amk I dim H’(X,

o$‘) I bmk

for sufficiently large m. In other words, dim H’(X, (2) We define: K(X) = -cc

if H’(X,

wx”) = 0

K(X) = 0

$ dim H’(X,

coy) grows like mk. for all m E IN

coy) I 1 for all m E IN

with equality for some m.

K(X) = k > 0

if dim H’(X,

coy) grows like mk.

The Kodaira dimension K(X) is a very important invariant in the theory of compact complex manifolds. For instance we have @I’“) = -co, more generally K(X) = -cc if X is covered by a family of rational curves. Manifolds with K(X) < dim X have in a certain sense a special structure, therefore manifolds with K(X) = dim X are called of “general type”. For more information and for the proof of (4.17(l)) we refer to [Ue75]. For recent results - in particular for algebraic manifolds - see [Mo86]. Other important results on asymptotic estimates of cohomology groups here in connection with curvature - are Demailly’s Morse inequalities [Dem85]. We close this section with some remarks on q-convex and q-complete manifolds. For Stein spaces we have Theorem B discussed in Q3, while on compact spaces holds the finiteness of cohomology. Is there something in between? The answer is given by the notions of q-convexity and q-completeness, for which we refer to chap. VI. Here we only mention Theorem 4.18 (Andreotti-Grauert [AG62]). Let X be a complex space, Y a coherent sheaf on X. (1) Zf X is q-convex, then the vector spaces Hk(X, 9’) are finite-dimensional for k 2 q. (2) Zf X is q-complete, then Hk(X, 9’) = 0 for k 2 q.

O-convex spaces are nothing but the compact spaces, while l-complete spaces are exactly the Stein spaces. Hence the theorem of Andreotti-Grauert generalizes both Theorem B and the Cartan-Serre finiteness theorem. l-convex spaces are particularly interesting; they are proper modifications of Stein spaces. There is also the notion of q-concavity, see again chap. VI.

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Cohomology

0 5. Spectral Sequences A very important tool for computing cohomology are the so-called spectral sequences. This section is an introduction to the theory of spectral sequences with focus on examples rather than on general abstract theory. As a general source for spectral sequences we recommend Godement [God581 and CartanEilenberg [CaEi56]. 1. Definition, Definition

Double Complexes

5.1. A spectral

sequence is a sequence (E,, d,),Zo of bigraded

groups E, =

@

E,P”

P,Y20

and “differentials”

with d,? = 0 and the following property: Epq _ Ker(d,: E,Pq-+ EF+r*q--*+l) r+1

-

Im(d,:

Elp-r,q+‘-l

+

E$)

’

(in order to avoid technical diffkulties we set EFq = 0 for p < 0 or q < 0). This last property means that E,+l is the cohomology of E,: E r+l = H*(E,).

Sometimes it is necessary to allow E,Pqto be non zero for negative p or q, too. Definition 5.2. (1) We say that a spectral sequence (E,, d,) converges if there exists r, 2 0 such E, = ErO for all r 2 r,. Then we write E, = ErO and (E,) * E,. (2) We say that (E,) degenerates at E,O if E,Pq= El”,’ for all r 2 r, and all p, q.

In order to construct complexes.

spectral sequences we have to consider filtered

Definition 5.3. (1) Let K’ = (Kq, dq) be a complex of, say, (abelian) groups (see (1.1)). Let be given subcomplexes F’K’ c K’ such that

K’=FOK’~F’K’~...~F’K’={o}. Then (FPK’), is called a filtered complex. (2) Given a filtered complex (FPK’) we define the associated graded complex GrK’ by GrPK’ = FPK’/Fpi’K’,

the definition

of the differentials being obvious.

Th. Peternell

174

(3) We introduce a f&ration

on the cohomology

FPW(K’) Furthermore,

of K’ by setting

= FPZyK’yFPByK’).

we have a graduation on cohomology GrPHq(K’)

by

= FPHq(K’)/FP+‘Hq(K-).

Proposition 5.4. Let (FPK’) be a filtered complex. Then there exists a spectral sequence (E,) such that (1) E!q = FPKP+qIFP+‘KP+q, (2) ET4 = HP+q(GrPK’), (3) (E,) converges to E, with Eg = GrP(HP+q(K’)).

For a proof see [God58,1.4]. Proposition

As a special case of (5.4) we have

5.5. In the situation of (5.3) we have an exact sequence -+ E;*’ 1 EZ,*O-+ H2(K’).

0 + E;,’ + H’(K)

Now we turn to double complexes and associated spectral sequences. These will provide the first explicit examples of spectral sequences. Definition

5.6. A double complex (of groups) is a bigraded group K’.’

=

@

KP.4

p,qto

together with “differentials” d:

KP.4

+

KP+‘.‘l

6:

KP.4

m,

KP.4+’

such that d2 = 0, h2 = 0 and d8 + 6d = 0. The associated “single” complex (K’) is defined by K’ =

@ Kp.4

p+q=r

with differentials d = d + 6. Note that d6 + ad = 0 yields J2 = 0. Proposition 5.7. Given a double complex (K’s’, d, 6) there are two spectral sequences (E,), (‘E,) with E;vq = Hq((KP.‘, 6)), ‘E$*q = Hp( {H%(K’*‘),

‘ETvq = HP((K’*q, d)), d}),

‘Epq = H;( (HP(K’*‘),

(1) S}),

(2)

[where HP({H{(K’*‘), d}) just means the cohomology of the complex Lp = Hq((KP*‘, 6)) with fixed q and the differential being d (operating on cohomology).] (~9, W,) * H*W’), where K’ is the associated single complex.

(3)

III. Cohomology

175

In fact, the associated single complex (K’, d) carries two filtrations: FPK’ = @ KSs4, s+q=r S2P ‘FPK’ = @ KP.‘. p+t=r rtq

Now apply (5.4) to these filtered complexes ! 2. The FrSIicher Spectral Sequence. We are now going to construct the lirst explicit example of a spectral sequence. Let X be a complex manifold. Let b’ and &‘P*qdenote the sheaf of complex valued P-forms of degree r respectively bidegree (p, q). We define a double complex by setting K”*q = c$‘~*~(X), d = ~3, 6 = 3.

The associated single complex is K’ = CT(X)

with differential d = d, the ordinary exterior derivative. Then FPK’ is the set of all cp E b’(X) such that if we decompose cp into types, cp = 1~~~ with ptq E &‘vq(X), then (P~,~# 0 for t 2 p. By symmetry one usually considers only the first of both spectral sequences. Theorem 5.8. On every complex manifold X there is a spectral sequence(E,) with ET*q2 Hq(X, Qp) converging to H*(X, Cc). We call (E,) the Friilicher spectral sequence.

Theorem 5.7 follows by applying serving that

(5.6) to our special double complex ob-

Hq(X Q”) N {cp E ~p7q(X)I% = 01 by (2 16) 3 &r’(X)

and Hr(X c:) N b E ~‘lX)ldv 3 d&‘-‘(X)

= 01 by (1.18).

Strictly speaking, EfTqis exactly the right hand of (1) and @ Egq is the right p+q=r

hand side of (2). In the following we will identify these spaces. In case X is compact we define the numbers hP*q = hPsq(X)= dim Hq(X, Qp). These numbers hp,q are called the Hodge numbers of X. We let b,(X) = dim H’(X, (IY)be the r-th Betti numbers of X. From the existence of the spectral sequence we deduce

Th. Peternell

176

Corollary 5.9. Let X be a compact complex manifold. Then the so-called Friilither relations hold: (1) 1 hJ’*q 2 b,, p+q=r (2) I,,, = C (- l)‘b, = 1 (- 1)P*4hP*4.

I P.4 Up to now we have not said anything about degeneration of the Frolicher spectral sequences. If (E,) degenerates at E,, this information is almost useless. On the other hand we have:

Theorem 5.10. Let X be a compact Kiihler manifold. Then the FrBlicher tral sequence degenerates at (E,).

spec-

A proof of (5.10) can be found e.g. in [We80]. For the notion of a Kahler manifold we refer to chap. V. In one word, X is Kahler if there is a d-closed (1, I)-form w on X which is positive, i.e. in local coordinates

where the matrix (ctjk(z)) positive definite for all z. Most prominent submanifolds of P,, or C”. As a formal consequence we obtain Corollary Then

5.11 (Hodge decomposition).

H’(X, Remark

examples are

Let X be a compact Kiihler

C) 2: @ W(X, p+q=r

manifold.

QP).

5.12. (1) Usually one considers also the Hodge symmetry hpTq = hq.p (all p, q)

as part of the Hodge decomposition. On the other hand, Hodge symmetry is not related to the spectral sequence. (2) Degeneration of (E,) at E, holds on compact manifolds bimeromorphitally equivalent to a Kahler manifold. For a proof see [Ue75]. Explanation 5.13. Let us understand what degeneration of the Frolicher spectral sequence (E,) means and how (5.10) is proved. Fix p and q. In order to show Ef*q N E$gqwe must prove that for all p d, : ET.4 + &+‘*‘I

is zero.

This amounts to showing that, given cpE &‘p*q(X) with & = 0, one has also [acp] = 0, i.e. dq = &J for some (p, q - l)-form y. Now the Kahler condition comes into the game. Fix a Klhler metric on X and let d, = dd* + d*d, AJ = %* + a*% be the Laplacians of d and 3. Then the fundamental Klhler identity says:

III.

Cohomology

177

Of course, for a general hermitian metric there is no relation between da and d,. The identity is however easy for the euclidean metric, and one can base a proof of the general case on the fact that the KHhler metrics are just those hermitian metrics which are locally euclidean of second order. The form cp can be chosen to be da - harmonic, i.e. damp = 0. So by the Kiihler identity, d,cp = 0, too. But then dq = 0, hence even acp = 0. For details we refer again to [WeBO], [GH78] or [DV74]. Once one has E, = E,, it clear that E, = E, and (5.10) holds. The isomorphism of (5.9) can be also easily understood in terms of harmonic forms. Let c1E H’(X, C) and let 40 E b’(X) be a harmonic representative. Let cp = c (P,,~ be the decomposition into types. From da = *A, one concludes easily that every (PP,~is As - harmonic: A~(P~,~ = 0. Now the image of c( is just the class Remarks 5.14. (1) For a compact manifold which is not Ktihler or bimeromorphic to a Kahler manifold it may still happen that the Frolicher spectral sequence degenerates at (E,). For instance, this is true for every compact surface (see [BPV84]). (2) On the other hand, there are compact manifolds X with E, # E,. A famous example is the so-called Iwasawa manifold (see e.g. [GH78, p. 4441). 3. The Leray Spectral Sequence. The Leray spectral sequence is of purely

topological

nature but nevertheless of great importance

in complex analysis.

Theorem 5.15. Let X, Y be topological spaces, f: X -+ Y a continuous map and Y a sheaf of abelian groups on X. Then there exists a spectral sequence (E,) - called the Leray spectral sequence associated to f and 9’ - such that (a) E$vq 2: HP( x F&(9’)), (b) 6%) * H*(X, 9) (so H’(X, Y) = @ Egq). p+q=r

The proof is obtained by constructing canonical flabby resolution (1.9)

a double complex. We start with the

0 + Y + yb -+ 91 + *. . Taking direct images we note that all f,(q) a double complex by setting

are again flabby. Now we construct

Kp*q = ZI’(Y, SB,,),

where

0 -

f*(Yq) II doI?

d lq -

d al -

...

178

Th. Peternell

is the canonical flabby resolution of f,(yq). This double complex (the morphisms being obvious) leads to the spectral sequence. For details, see [God58]. An easy consequence of (5.15) is Corollary 5.16. Let X, Y be topological spaces,f: X + Y a continuous map and Y a sheaf of abelian groups. Assume Rqf*(Y) = 0 for all q > 0. Then: Hq(X, Y) ?: Hq(Y, f,(Y))

for all q.

Proof. Let (E,) be the spectral sequence associated to f and 9 assumption: E$gq = 0 for all q 2 1 and all p.

By our

(*I

We have to show that E$*’ 2: Ego for all p. This follows at once from (*) and the fact that E, is the cohomology of E,-, . Example 5.17. (1) Let X be a lP,-bundle over the complex space Y. First note that - denoting by rr: X --) Y the projection - rc*(cOx) = 0,. Next we have Rqrr*(O*) = 0, q > 0. This follows from (4.8). Hence it follows by (5.14): Hq(X, 0,) N Hq(Y, 0,) for all q 2 0. (2) We would like to have a closer look at projective bundles. Let d be a locally free sheaf of rank r on the complex space Y. Let

x = lP(b) be the associated lP_,-bundle (see chap. 2). The space X carries a distinguished line 0x( 1) = O,,,,( 1) which is 0( 1) restricted to the fibers and which has the basic property: n*(G(l)) 7~:X -+ Y denoting the projection.

= 4

Furthermore:

~n,tGA4)

= W-3

for P > 0,

~*(~x(P))

= 0

for p < 0,

H4(X, %(PL) 0 n*(W)

= ffqtr, SW? 0 9)

(4 (P > 0)

04

for every locally free sheaf 9 on Y. (a) (which is obvious in case /J < 0) is just the relative version of the fact that the sections of flPn(p) can be viewed as homogeneous polynomials of degree p in (n + 1) variables which in turn can be identified with P(C”+l). (b) follows via Leray’s spectral sequence from the projection formula

111. Cohomology

together with the “obvious”

179

vanishing

for q > 0, p > 0 (see 4.10(2)). (3) (see (4.9)) Let f: X + Y be a desingularisation of a normal rational surface singularity, so that R1f,(Ox) = 0. Since clearly Rqf,(O,) = 0 for q 2 2, we obtain fP(X,

co,) = W( y, cl,),

q 2 0.

4. Some More Spectral Sequences. Here we gather some more spectral sequences which are often useful. Theorem 5.18. Let f: X + Y, g: Y + Z be continuous maps of topological spaces. Let Y be a sheaf of abelian groups on X. Then there is a spectral sequence (E,) with E;” = R”s,UW.V’)) converging to R’(g 0 f),(Y). For a proof, see [HiSt71]. spectral sequence.

This spectral sequence is called the Grothendieck

Theorem 5.19. Let (X, J&‘) be a ringed space, and let ~$9 be d-modules. there exists a spectral sequence (E,) with

conuerging to Ext,.,(& 9). (This relates dxt-sheaves

Then

to Ext-groups).

As an application we obtain easily the following fact: If X is a projective manifold with an ample line bundle 2 (cf. chap. VI), 9, 9 being coherent sheaves on X, then for n 2 n,: H”(X,

EzCtq(F, 3 @ 2’“))

N Extq(y,

9 @ P”)),

In fact, it is sufficient to show Ezq = 0 for p > 0. But Eqq = HP(X, &zd”(P, 3 @ 9”)) z HP(X, &‘zc!~(~, 9) @ 2”)) = 0 for n 2 no and p > 0, since 2 is ample and &ztq(F, 9) is coherent. (this last vanishing is the socalled coarse Kodaira vanishing theorem, seechap. VI). If X is a topological space, A c X a locally closed set and 9 a sheaf of abelian groups on X, one can define local cohomology groups

Here Hj(X, 9) is nothing but the space of those sections s E H’(X, 9) whose support are in A. We have the remarkable exact sequence 0 + Hi(X, 9) + HO(X, 9) + HO(X\A, 9) -+ Hi(X, 9) + H’(X, LF) -b.. .

180

Th. Peternell

For details see [SGA2], [BaSt76]. We define the sheaves of local cohomology s’?.; by taking the sheaf associated to the presheaf u H Hi( u, 9). Theorem 5.20. There exists a spectral sequence (E,) with Eqq = HP(X, &j(F)) converging to HAp+q(X, 9). (See [BaSt76, chap. 21). Note that for A = @ we get back our ordinary cohomology. The sheaves s;(P) are important for extension theorems. In fact the spectral sequence yields Corollary Then

5.21. Assume that HP(X, 22(F)) H;(X,

9) N H’(X,

= 0 for q < k (k fixed)

Z/(S))

and p 2 1.

(p I k + 1).

Thus it follows Corollary 5.22. The following (1) #i(9) = 0 for i I q. (2) The restrictions

assertions are equivalent.

H’(U,

9) + H’(U\A,

9)

are isomorphic for i < q, injective for i = q. Hence for instance the second Riemann extension theorem for holomorphic functions on a normal complex can be stated in the following way (up to injectivity on the Hi-level): A$Ox)

= 0

for i I 1,

for any analytic set A c X of codim A 2 2. For the general extension theory it is important to know under which conditions on a complex space X the cohomology sheaves #i(9) of a given coherent sheaf vanish or are coherent. We refer to Siu-Trautmann [SiTr71].

References* [AC621 [AtSi63] [Best491

Andreotti, A.; Grauert, H.: Theoremes de tinitude pour la cohomologie des espaces complexes. Bull. Sot. Math. Fr. 90, 193-259 (1962) Zb1.106,55. Atiyah, M.F.; Singer, I.M.: The index of elliptic operators on compact manifolds. Bull. Am. Math. Sot. 69,422-433 (1963) Zb1.118,312. Behnke, H.; Stein, K.: Entwicklung analytischer Funktionen auf Riemannschen Flathen. Math. Ann. 120,430-461 (1949) Zbl.38,235.

*For the convenience of the reader, compiled using the MATH database,

references to reviews in Zentralblatt have, as far as possible, been included

fiir Mathematik (Zbl.), in this bibliography.

III. [BaSt76] [Bo57] [Bose591 [BPV84] [CaEi56] [Case531 [Dem85] [DV74] [FoKn71] [Fu184] [GH78] [God581 [Gr55] [Gr58] [Gr60] [GrRe77] [GrRe84] [Ha831 [Ha661 [Hir56] [HiSt71] [Kun75] [Kun77] [Kun78] [Lei90] [Lip841 [Na67]

Cohomology

181

Banica, C.; Stanasila, 0.: Algebraic Methods in the Global Theory of Complex Spaces. Wiley 1976. Zb1.284.32006. Bott, R.: Homogeneous vector bundles. Ann. Math., II. Ser. 66, 203-248 (1957) Zbl.94,357. Borel, A.; Serre, J-P.: Le theortme de Riemann-Roth. Bull. Sot. Math. Fr. 86, 97-136 (1959) Zbl.91,330. Barth, W.; Peters, C.; van de Ven, A.: Compact Complex Surfaces. Erg. Math. 4, Springer 1984,Zbl.718.14023. Cartan, H.; Eilenberg, S.: Homological Algebra. Princeton Univ. Press 1956, Zbl.75,243. Cartan, H.; Serre, J.P.: Un theoreme de linitude concernant les varietts analytiques compactes. C.R. Acad. Sci. Paris 237, 128-130 (1953) Demailly, J.P.: Champs magnetiques et inegalites de Morse pour la d”-cohomologie. Ann. Inst. Fourier 35, No. 4, 185-229 (1985) Zbl.565.58017. Douady, A.; Verdier, J.P. (ed.): Differents aspects de la positivite. Asterisque 17. Paris 1974. Forster, 0.; Knorr, K.: Ein Beweis des Grauertschen Bildgarbensatzes nach Ideen von B. Malgrange. Manuscr. Math. 5, 19-44 (1971) Zbl.242.32008. F&on, W.: Intersection theory. Erg. d. Math., 3 Folge, Bd 2. Springer 1984. Zb1.541.14005. Griftiths, Ph.; Harris, J.: Principles of Algebraic Geometry. Wiley 1978, Zb1.408.14001. Godement, R.: Topologie algtbrique et theorie des faisceaux. Herman, Paris 1958, Zbl.80,162. Grauert, H.: Charakterisierung der holomorph-vollsttindigen Raume. Math. Ann. 129, 233-259 (1955) Zbl.64,326. Grauert, H.: On Levi’s problem and the embedding of real-analytic manifolds. Ann. Math., II. Ser. 68,460-472 (1958) Zb1.108,78. Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulraume komplexer Strukturen. Publ. Math., Inst. Hauter Etud. Sci. 5, 5-64 (1960) Zbl.lOO,SO. Grauert, H.; Remmert, R.: Theorie der Steinschen Raume. Grundl. 227, Springer Math. Wiss. 1977, Zb1.379.32001. Grauert, H.; Remmert, R.: Coherent Analytic Sheaves. Grundl. 265, Springer 1984, Zbl.537.32001. Hamm, H.A.: Zum Homotopietyp Steinscher Riiume. J. Reine Augew. Math. 338, 121-135 (1983) Zbl.491.32010. Hartshorne, R.: Residues and duality. Lect. Notes Math. 20, Springer 1966, Zb1.212,261. Hirzebruch, F.: Neue Topologische Methoden in der Algebraischen Geometrie. Grundl. Math. Wiss. 131, Springer 1956,Zbl.70,163. Hilton, P J.; Stammbach, U.: A Course in Homological Algebra. Graduate Texts Math. 4, Springer 1971, Zbl. 238.18006. Kunz, E.: Holomorphe Differentialformen auf algebraischen Varietaten mit Singularitlten I. Manuscr. Math. 15,91-108 (1975) Zbl.299.14013. Kunz, E.: Residuen von Differentialformen auf Cohen-Macaulay-Varietaten. Math. Z. 152, 165-189 (1977) Zb1.342.14022. Kunz, E.: Differentialformen auf algebraischen Varietaten mit Singularitlten II. Abh. Math. Semin. Univ. Hamb. 47,42-70 (1978) Zbl.379.14005. Leiterer, J.: Holomorphic vector bundles and the Oka-Grauert principle. In: Encycl. Math. Sci. IO, 63-103, Springer 1990,Zb1.639.00015. Lipman, J.: Dualizing sheaves, differentials and residues of algebraic varieties. Astirisque 2 2 7, 1984, Zbl.562.14008. Narasimham, R.: On the homology groups of Stein spaces. Invent. Math. 2, 377-385 (1967) Zbl.l48,322.

182 [Pe91] [Re57] [RR701 [RR741 [X52] [Ser55] [Ser55-21 [SGAZ] [SiTr71] [Sn86] [St511

[ToTo [Ue75] [We801 [Weh85]

Th. Peternell Peternell, Th.: Hodge-Kohomologie und Steinsche Mannigfaltigkeiten. In: Complex Analysis, Wuppetal, Ed. K. Diederich. Vieweg 1991. Remmert, R.: Holomorphe und meromorphe Abbildungen komplexer Raume. Math. Ann. 133, 328-373 (1957) Zbl.79,102. Ramis, J.P.; Ruget, G.: Complexe dualisant et theoremes de dualite en geometric analytique complexe. Publ. Math. Inst. Hautes Etud. Sci. 38, 77-91 (1970) Zbl. 206,250. Ramis, J.P.; Ruget, G.: Residus et dualite. Invent. Math. 26, 89-131 (1974) Zbl.304.32007. Seminaire Cartan. Theorie des fonctions de plusieurs variables. Paris 1951/52. Serre, J.-P.: Faisceaux algebriques coherents. Ann. Math., II. Ser. 61, 197-278 (1955) Zb1.67,162. Serre, J-P.: Un theoreme de dualitt. Comment. Math. Helv. 29,9-26 (1955) Zbl.67,161. Grothendieck, A.: Stminaire de geometric algebrique 2. Cohomologie locale des faisceaux cohtrents. North Holland 1968,Zbl.197,472. Siu, Y.T.; Trautmann, G.: Gap-sheaves and extensions of coherent analytic subsheaves. Lect. Notes Math. 172, Springer 1971, Zbl.208,104. Snow, D.: Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces. Math. Ann. 276, 159-176 (1986) Zbl.596.32016 Stein, K.: Analytische Funktionen mehrerer komplexer Vefnderlichen zu vorgegebenen Periodizitatsmoduln und das zweite Cousinsche Problem. Math. Ann. 123,201222 (195 1) Zbl.42,87. Toledo, D.; Tong, Y.L.L.: A parametrix for 2 and Riemann-Roth in tech theory. Topology 15, 273-301 (1976) Zbl.355.58014. Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Lect. Notes Math. 439, Springer 1975,Zbl.299.14007. Wells, R.O.: Differential analysis on complex manifolds. 2nd ed. Springer 1980, Zb1.435.32004; Zb1.262.32005. Wehler, J.: Der relative Dualitltssatz fiir Cohen-Macaulay-Raume. Schriftenr. Math. Inst. Univ. Milnster, 2. Ser. 35, Zbl.625.32010.

Chapter IV

Seminormal Complex Spaces G. Dethloff

and H. Grauert

Contents Introduction

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

Q 1. Analytically Branched Coverings . .. . . . .. . . . .. . .. . . . .. . .. . .. . . . .. . . .. . 1. Basic Definitions and Elementary Properties 2. Holomorphic Structure and Further Elementary Properties . . . .. . . .. . . . ,. . . .. . . .. . . . .. , .. . . . 3. The Main Theorem

.

.

.

.

.

.

9

.

.

.

.

.

185 185 186 188 189

$2. Proof of the Main Theorem on Analytically Branched Coverings .. .. . . .. . . .. . . . .. . .. . . . .. . .. .. . .. . . . .. . . 1. Some L*-Methods 2. Proof of the Main Theorem using an L2-Theorem .. . .. . . . .. . .

190 190 192

. .. . . . .. . . . .. . .. . . . .. . . 9 3. Some Related Results and Applications . . . . .. . . .. . . . . .. . . . .. . . 1. An Inverse of the Main Theorem 2. Analytically Branched Coverings over Normal Complex Spaces 3. Extension of Analytically Branched Coverings . . . .. . .. . . . . . .

193 193 194 196

.. . . .. . .. . .. . Q4. Analytic Decompositions 1. Analytic Equivalence Relations on Complex 2. Holomorphic Maps . .. . . . .. . .. . . .. . . 3. Restrictions . . . .. . . . . . . . .. . . .. . 4. Finer Equivalence Relations . . .. . . . .. . .

. .. . .. . Spaces . . . . .. . . .. . .. . .. . . .. . .. . .. . .

.. .. ..

197 198 198 199 199

.. .. $5. Spreadable and Semiproper Equivalence Relations .. 1. Spreadable Analytic Equivalence Relations ....... .. 2. Semiproper Equivalence Relations ..............

.. .. ..

199 199 200

6 6. Normal Equivalence ....................................... 1. Maps of Complex Spaces ................................. .................... 2. Normal Analytic Equivalence Relations

201 201 201

9 7. The Main Theorem ........................................ 1. Indication of the Theorem ................................ 2. IdeaoftheProof ........................................ 3. Simple Equivalence Relations ............................. 4. A Geometric Construction of Simple Equivalence

203 203 204 204 206

Relations

...

G. Dethloff

184

5. ExamplesofRandR” 6. Analytic Dependence

and H. Grauert

206 207

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

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

......................... 9:8. Meromorphic Equivalence Relations ........................................ 1. Meromorphicity 2. The Fibration Given by R ................................ 3. Regular Meromorphic Equivalence Relations ................ 9:9. Meromorphic Dependence of Maps .......................... ............................. 1. Proper Equivalence Relations 2. The Notation of a Simple Meromorphic Equivalence ............................... 3. Meromorphic Dependence 4. Meromorphic Bases (m-Bases) .............................

Relation

........................................... 5 10. Non Regularity .................. 1. A Simple Non Regular Algebraic Relation 2. A Non Regular Relation which cannot be Enlarged to a ........................................... RegularOne 3. Reduction to a Moishezon Space .......................... 0 11. Applications . . . . . .. . . . .. . . 1. Complex Lie Groups . .. . . 2. One Dimensional Jets . . . . 3. The Non Hausdorff Case . . 4. Cases where X is Not Normal Historical Note . .. . .. .. . .. . References

.. .. .. .. ..

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

207 207 208 208 210 210 . 211 211 212 212 212 213 215 216 216 216 218 218 218

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

IV. Seminormal

Complex

Spaces

185

Introduction Section 1 deals with the notion of analytically branched coverings. The main theorem states that analytically branched coverings are normal complex spaces. Its proof makes use of L*-methods as developed by Hormander, and is sketched in 4 2. In 8 3 some applications are given, some of which are used in 5 7. Then we begin to develop the theory of analytic decomposition. The definitions are given in $4. In order that the quotient space be a complex space two conditions are needed: The equivalence relation has to be spreadable and in addition semiproper. These definitions are contained in 0 5, while $6 deals with the rather simple case of normal equivalence. We consider the quotients of pure dimensional seminormal complex spaces requiring that all fibers have the same dimension in all the points, and one further condition. Then the quotient space is again a seminormal complex space while the quotient map is open. In subsections 7.1 and 7.2 the main result concerning semiproper spreadable analytic decompositions of seminormal complex spaces is given, to the effect that the quotient is a seminormal complex space. The remaining subsections of 4 7 deal with the notion of analytic dependence of holomorphic maps. In 5 8 we come to the notion of meromorphic equivalence relation in normal complex spaces. Here some proper modifications are employed. An analytic equivalence relation always leads to a meromorphic equivalence relation. But the quotient space by the corresponding meromorphic equivalence relation may be totally different. To get results we must restrict ourselves to meromorphic equivalence relation that are regular, however it turnes out that this condition is nearly always fulfilled. The sections 0 9 to $11 bring some applications. First (in Q9) the notion of meromorphic dependence of holomorphic and meromorphic maps is considered. Next we give some examples of non regular meromorphic equivalence relations and finally (in 5 10) the reduction of a compact normal complex space X to a Moishezon space which is a biregular invariant of X. In 4 11 the action of complex algebraic Lie groups on normal complex spaces is considered. It leads to the definition of the non parametrized jet. These objects are better than the ordinary ones. We give also an example of a quotient space which is not a Hausdorff space and discuss some cases where X is not normal.

9 1. Analytically

Branched Coverings

In order to treat semi normal complex spaces we must first prove a result on analytically branched coverings. Roughly speaking, an analytically branched covering X is a proper covering over a domain G in the n dimensional complex number space C” with only finitely many sheets. It may be branched, but only over a nowhere dense analytic subset of this domain (cf. subsection 1.1).

G. Dethlofl

186

and H. Grauert

Historically Behnke and Stein used analytically branched coverings as local model spaces in order to generalize the notion of complex manifolds, but it could be shown already in 1958 that the objects they had defined were precisely the normal complex spaces (cf. subsection 1.3 and 3.1). Even in our days analytically branched coverings are still important: First they tell us a lot about the local structure of normal complex spaces. Second they yield us some device of extending certain normal complex spaces (cf. subsection 3.1 and 3.3). 1. Basic Definitions and Elementary Properties. The aim of this subsection is to define the concept of analytically branched coverings. Furthermore we list some properties of such analytically branched coverings, most of which are elementary. We do this mainly in order to make clear how such objects look like. In the next subsection we then will supply them with a natural holomorphic structure. A general reference for this and the next subsection is [GR58]. First we recall two basic notions: Let X and Y be topological spaces. A map f: X + Y is called finite if it is continuous and closed and if every fiber f-‘(y), y E Y, consists of finitely many points only . Let now A be a closed and nowhere dense subset of X. We say A does not separate locally X if for every point P E A and for every connected open neighborhood U of P in X there exists a neighborhood V of P contained in U such that V\A is connected. Definition 1.1. Let X be a locally compact space, G a connected open domain in C” and rc: X + G a finite and surjective mapping. Assume further that there exists an analytic subset A # G of G such that: (1) 71-‘(A) does not separate locally the set X. (2) rr: (X\rc-l(A)) -+ (G\A) is locally topological. Then 7~:X + G is called an analytically

branched covering with critical locus A.

We list some elementary properties: We start with two properties of rc (assertions 1) and 2)), then we turn to X (assertion 3)) and finally we take a look at our analytically branched covering from the perspective of G (assertions 4) and 5)): 1) rr: X -+ G is proper, while n: (X\Cl(A)) + (G\A) is proper and unbranched. 2) n: X - G is an open mapping. 3) X has a countable basis of topology. Moreover, every point P E X possesses a countable basis of neighborhoods U,,, v E IN in such a way that every (U,,, n, n( U,)) is again an analytically branched covering. 4) There exists a positive integer b such that #(n-‘(P))

I b

for P E G with equality if P $ A.

Hence rc: X + G is said to have b sheets.

IV.

Seminormal

Complex

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187

Before giving a more precise statement it is convenient to introduce some more terminology. Let 7~:X + G be as in Definition 1.1. A point P E X is called of order k if it has a basis of neighborhoods such that every neighborhood in this basis is an analytically branched covering with k sheets (cf. 3) and 4) above). We denote this number k by o(P). The point P is called a schlicht point if o(P) = 1, otherwise it is called a branching point. Now we have: 5) For every point P E G there exists a neighborhood U of P in G such that the following is true: n-‘(U) decomposes into connected components Vi, . . . , V, such that each v contains exactly one inverse image point Qi of P. Each set r/Tgives again rise to an analytically branched covering 7~: q + V with o(Qi) sheets. Moreover, x: v + U is topological if and only if Qi is a schlicht point. Hence we especially have the formula xl=, o(Qi) = b, which strengthens 4): We have #(z-‘(P)) = b exactly if there are only schlicht points lying over the point P E G, and that is the case exactly if there exists a neighborhood U of P in G such that the inverse image of U consists of exactly b connected components, which all are mapped topologically onto U by rc. Next we wish to obtain some information about the behaviour of rc: X + G near branching points. 6) The critical locus A is not uniquely determined (e.g. we can take any nowhere dense analytic subset B of G and get a new critical locus A u B). 7) The critical locus A can be chosen to be empty or pure one codimensional. Assertion 7) is not so easy to prove as the other ones. We continue with an example of an analytically branched covering, which will turn out to be very important, since “most” of the branching of a general analytically branched covering “looks like” this example (see assertion 8)). To make clear what the phrase “looks like” shall mean, we first need another definition: Let 7rr: X, + G and rr2: X, + G be analytically branched coverings over the same base space G. They are called equivalent if there exists a topological map t: X, -+ X, with the property n1 = rc2 o t. Now we give the example of an analytically branched covering which was promised: Let G = {lzl < l} c Cl?’ and X, = ((w, z) E C x G: wb - z1 = 0}, b E IN. Let rc: X, + G be the canonical projection. Then 71:X, -+ G is an analytically branched covering, which we denote by w,. If b 2 2, it has the (minimal) critical locus A = (z E G: z1 = 0} and we have o(P) = b for all points P lying over A. Returning to general analytically branched coverings we may assume, with respect to property 7), that the critical locus A is pure one codimensional in G. Then we have: 8) If P E A is a smooth point and Q E n-‘(P), then there exist (possibly after a homothety) neighborhoods U(Q) c X, V(P) c G such that 71: U + V is an analytically branched covering equivalent to wr, Y = o(Q).

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From 7) and 8) we can now finally, by using the decomposition into irreducible components, conclude:

of analytic

sets

9) The projection of all branching points of rc: X -+ G yields itself an empty or pure one codimensional analytic set in G (which then, of course, is the minimal critical locus). 2. Holomorphic Structure and Further Elementary Properties. So far we have discussed the topological structure of an analytically branched covering rr: X + G in some detail. We are now going to introduce a holomorphic structure on the covering space X canonically induced by the projection rc onto the base space G. The holomorphic functions on X are defined to be the continuous functions on X which are holomorphic in the schlicht points in the sense of domains over C”, or, to be more precise: Definition 1.2. Let rc: X + G and rc’: X’ + G’ be analytically branched coverings and let Q c X be an open subset. (1) A continuous function f: Q + C is called h&morphic if for every schlicht point P E Q there exists an open neighborhood U(P) c 52 such that rc: U(P) + n(U(P)) is topological and the function f 0 (rcl”)-’ is holomorphic in n(V(P)). The set of such functions is denoted by Co’(Q). The sheaf given by this presheaf is written 0;. (2) A subset M c Q is called an analytic set in Q if for every point P E Q there exists a neighborhood U(P) c Sz and functions fi, . . . , f, E O’(U(P)) such that M n U(P) = /‘$‘& {f;: = O}. (3) A continuous mapping +: 52 -+ X’ is called a holomorphic map if for every f E O’(U), where 52’ is an open subset of X’, we have f o $ E S’($-l(Q)). If $: X + X’ is bijective and both $ and $-’ are holomorphic, II/ is called biholomorphic. Now we can continue our list of elementary properties. Namely, with an assertion relating the global holomorphic functions on X to those on G and with a version of a Riemann Extension Theorem on analytically branched coverings: 10) A continuous function holomorphic functions (f(X))’

f: X + Cc is holomorphic if and only if there exist a,, . . . , a,: G + (c such that +

i$l

44x)).

(f(X))*+

E

Cl

on

X.

Moreover we always can achieve r I b. 11) Let M be a nowhere dense analytic subset of X. Let f E U(X\M) be locally bounded around every point P E M. Then f can be extended to a function

f E O'(X). Let us finally come back to the question what the branching of an analytically branched covering looks like: From assertion 8) one knows how rr: X + G

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looks like, up to the branching over the (at least two codimensional) analytic set S(A) of singularities of A. Especially, one now can say that over every point of G\S(A) the covering space X, supplied with its holomorphic structure, is at least uniformizable as a manifold point. A simple example shows that this need not be true any longer over points of S(A). The branching can become more complicated there: Let G = {IzI < l} c (c’, X = {(w, z) E C x G: wz - zlzZ = 0} and let rr: X -+ G be the natural projection. Then n: X + G is an analytically branched covering with (minimal) critical locus A = (zi z2 = O}. Every point lying over A is a branching point of order 2, and for every P E A, P # 0 this analytically branched covering is locally equivalent to 9KZ. Above the origin, however, it is more complicated. There X is no longer uniformizable, but only a normal complex space. 3. The Main Theorem. In this subsection we state two important theorems on analytically branched coverings. The main assertion of the first theorem says that the covering space of an analytically branched covering is a normal complex space. In order to prove it we use a second theorem which yields the local existence of holomorphic functions separating the sheets. We start by recalling the definition of an analytic covering (cf. [GR84]): A finite surjective map n: X + Y between reduced complex spaces is called an analytic cooering of Y if there exists a nowhere dense analytic subset T of Y with the following properties: a) The set 7t-l( T) is a nowhere b) The induced map rc: (X\rc-l(

dense analytic subset of X. T)) -+ (Y\T) is locally biholomorphic.

Then we have our main theorem: Theorem 1.3. Every analytically branched covering is an analytic covering over a connected open domain in C” the covering space of which is a normal complex space. The converse is also true. In order to understand this theorem, the following remark might be helpful: What we mean here is that the covering space X, together with the sheaf 0; (defined in subsection 1.2), is isomorphic to a normal complex space in the category of C-ringed spaces (cf. chapter I), or, equivalently, that (X, 0;) is a normal complex space. How can such a theorem be proved? First the property of the (C-ringed space (X, 0Iy) (derived from an analytically branched covering 7~:X + G) to be a normal complex space is a local property. Hence it suffices to show that for every P E G there exists a neighborhood U such that (V = rc-i(V), 0;) is a normal complex space. In order to prove this we will show the following theorem: Theorem 1.4. Let rc: X -+ G be a b-sheeted analytically branched covering. Then for every P E G there exists a neighborhood U(P) and on V := z-‘(U(P)) a function f E O’(V) which separates the sheets, i.e. there exists a point Q E U out-

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side the singular locus such that f takes pair-wise different values in the b points lying over Q. This theorem was first proved in [GR58]. Another proof was given by Siu in [Si69]. The proof which we want to give here is completely different. It is based on a special L2-method due to Hormander, in which we obtain the holomorphic function separating the sheets as a solution of a differential equation with growth conditions. This proof, the main idea of which is also due to Siu, will be sketched in the next section. It might be of interest to know how to pass from a holomorphic function separating the sheets to a normal complex structure. So let us explain the idea how one passes from Theorem 1.4 to Theorem 1.3: Let 7~:X -+ G have b sheets. We may assume that G is chosen so small that there exists a function f E Co’(X) separating the sheets. Then there exists a manic polynomial w(w, z) E O’(G)[w] of degree b the coefficients of which are holomorphic functions on G such that w(f(x), n(x)) = 0 on X (cf. property 10) of subsection 1.2). Let D c G be the analytic subset of G where the discriminant of o vanishes. Further define M := {( w, z) E (c x G: w(w, z) = 0} and @: X -+ M; x --f (f(x), n(x)). The restrictions of X and M to the points which lie over G\D are both smooth and the holomorphic map @ maps them biholomorphically onto one another. If (y: N -+ M is the normalisation (cf. chapter 1) one can show from the topological properties of the maps @ and !P that the biholomorphic of X and N to those points lying over map Y’-’ o @ between the restrictions G\D can be extended to a topological map t: X -+ N. Since in (X, 0;) and in (N, ON) the (first) Riemann Extension Theorem holds, these spaces are biholomorphically equivalent under t. The converse of Theorem 1.3 is true, as in a normal complex space a nowhere dense analytic subset does not separate locally.

5 2. Proof of the Main Theorem on Analytically Branched Coverings 1. Some L2-methods. Roughly speaking the philosophy of L2-methods in complex analysis goes as follows: If one tries to solve a problem involving objects with holomorphic or, at least, C” coefficients, one passes to the corresponding objects which have only square integrable coefficients with respect to a suitable chosen metric. Now one can apply Hilbert space techniques. At the end one tries to get a solution of the original problem, or at least information about it, from the solution of the corresponding L2-problem. Using L2-methods farereaching results have been obtained, concerning e.g. the existence of holomorphic functions with special properties, the approximation of holomorphic functions with holomorphic functions defined on larger domains, the computation of cohomology groups, and concerning many other problems.

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Our aim in this subsection, however, is only to state a very special theorem from L2-theory, which we use for the proof of our main theorem, and to give the main ideas of its proof. (It is also for this reason that the literature at the end of this chapter is, what L*-methods are concerned, far from being complete). Hence this subsection can at most serve as a first introduction how L*-methods work. An already classical but nevertheless standard reference for those readers who want to learn more about L2-methods is the paper [Ho651 of Hormander. The theorem from L*-theory which we need is the following: Theorem 2.1. Let n: X + G be an analytically branched covering with a bounded and pseudoconvex base space G and empty critical locus. Let 4: X + IR u {--MI} be plurisubharmonic. Let further g E C”(X)(,,,, with ag = 0 and lx lg12e-~ dV=: c, < cc (where dV denotes integration with respect to Lebesgue measure lifted from G by z). Then there exists a function u E Cm(X) with au = g and a constant k depending only on the diameter of G such that lul*e-” dV I k.c,. (1) sX For simplicity we only deal with the case X = G, the details of which also can be found in [Hii73]. The general case, the proof of which goes along the same lines, can be found in [NS77] and, with more details, in [De90]. The basic ideas of the proof are as follows: Step 1: Let f$i, . . . . & be real valued C” functions on G and define L*(G, 4i)tp.q) to be the set of all (p, q) forms with coefftcients which are square integrable over G with respect to the Lebesgue measure and the weight function e-41. Further assume that g E L*(G, d2)(,,ir. Then we have the sequence L2(G 41 ho) J+ L*(G,

42ho,1)

5

L*(G

hko.2~

(2)

where T = % and S = 2, taken in the sense of distribution theory, are densely defined and closed linear operators between the Hilbert spaces L*(G, ~ii)(o,i-l) with inner products denoted by (G, .)i, i = 1, 2, 3. What we have is g E Ker(S), and what we want to show in the first step of this proof sketch is g E Im(T), since then we have an L*-solution of the equation I% = g, while u E L*(G, #l)(O,Oj yields the additional growth condition. Let T* be the adjoint operator of T and denote by D,, Ds, D,* the sets where the corresponding operators are defined. The main difficulty is to show that the functions di can be chosen in such a way that the inequality

l<s,f>zI I c,(dIIT*flI,,

fob

(3)

holds with a positive constant c,(g). It is also this point where the pseudoconvexity of G is needed. This inequality shows that T*(D,.)

+ c:;

T*f + (a fh

is a bounded antilinear operator. Hence the Hahn-Banach Theorem Riesz Representation Theorem yield a function u E L*(G; dl)(,,Or with au = g and jG(u(2e-41 dV I c,(g).

(4) and the T**u =

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Step 2: This is needed since, on the one hand, we want the solution u of au = g to be bounded with the same weight function 4 as the function g, and, on the other hand, we have to take care of the fact that 4 is only plurisubharmonic and not C” smooth. What we can do is approximating the weight functions in a suitable way, and, for every such function, applying the procedure given in step 1. Then, since we have uniform bounds, we can pass successively to convergent subsequences. This yields us an Lz-solution of the equation & = f with the “correct” weight functions. Step 3: We show by means of Sobolev Theory that, by using that g was C” smooth, our L2-solution u is, up to changes on a set of measure zero, automatically C” smooth. Hence in all we have got a function u E Cm(G) with 3u = g and the desired growth condition, i.e. we have proved Theorem 2.1. 2. Proof of the Main Theorem Using an L2-Theorem. What still needs to be done is to prove Theorem 1.4 using the L2-result Theorem 2.1. We are going to prove somewhat more, namely the following Proposition 2.2. Let 7~:X -+ G be an analytically branched covering with critical locus A. Furthermore, assume that G is bounded and pseudoconvex. Let z,, E (G\A) and n-‘(z,) = {x1, . . . , xb}. Then there exists a holomorphic function f E O’(X) with pairwise different f(xi), i = 1, . . . , b.

Again we sketch only its proof here. Details can be found in [De90]. Parts of it can be located already in [NS77]. The proof consists of two parts: In part 1 we are going to construct a function h E 0’(Y) with pairwise different h(x,), i = 1, . . . , b, and the growth condition IhI2 dV < co,

(5)

s Y

where Y := X\~C-‘(A) and dV denotes integration with respect to Lebesgue measure lifted from G to Y. We find first a function p E Cm(Y) with pairwise different p(xi) which is holomorphic in a neighborhood of each xi and has compact support in Y. The existence of such a function is evident since 7~: Y + (G\A) is unbranched. Our construction is complete if we can find a function u E C”(Y) with the following properties: on Y

(6)

i = 1, . . .> b

(7)

& = Jp u(xi) = 0,

1~1’ dJ’<

CO,

(8)

s Y

since then we can define h := p - u. The properties (7) and (8) can be enforced by the modified growth condition sY

Iu12e-rpon dV < CO, q(z) := 2n loglzl:

G + lR u (-co}.

(9)

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But Theorem 2.1 yields us precisely such a function u with the properties (6) and (9), which finishes the first part of the proof. Notice that here the additional growth condition on the solution of the a-equation yields the zeroes of its solutions over the point zO. In part 2 of the proof of Proposition 2.2 we first show that after having multiplied the function h E O’(Y) constructed above with the square of an appropriate chosen function t E Co’(X) which vanished identically on z-l(A) and takes t.he value 1 in xi, i = 1, . . . , b, we can extend it to a function f E S’(X), which then, of course, still separates the sheets. Since G is a Stein domain, such a function t exists. We first prove that there exists a holomorphic function h’ E Co’(X\n-‘(S(A)) with h’l, = t.h, where S(A) denotes the (at least two codimensional) set of singular points of A. The proof uses the special structure of the branching over smooth points of the critical locus A (cf. property 8) in subsection 1.1) to reduce the extension problem to the well known fact that a holomorphic function defined on a domain B\{z, = 0}, (B c a”) which is square integrable there can be extended to B. Notice that here the growth condition on h is needed a second time to get extendability into the branching OfX. Finally, we show that there exists a holomorphic function fe O’(X) with fix, = t. h’. This can be done for all schlicht points lying over S(A) by the ordinary second Riemann Extension Theorem. We then use property 10) of subsection 1.2 to show, again by the ordinary Second Riemann Extension Theorem, that h’ is bounded near branching points lying over S(A). But then the function t. h’ can be extended continuously to these points. Since this function f~ O’(X) separates the sheets, Proposition 2.2 and hence our main theorem is proved.

5 3. Some Related Results and Applications 1. An Inverse of the Main Theorem. The assertion of our main theorem was that every analytically branched covering is an analytic covering over a domain in C”, the covering space of which is a normal complex space, and conversely. It is now a natural question if every normal complex space yields (at least locally) an analytically branched covering in this way. This indeed is true: Theorem 3.1. Let (X, Ox) be a normal complex space and P E X a point. Then there exist a neighborhood U(P) c X, an open domain V c C”, and a map rc: U(P) -+ V such that the latter is an analytically branched covering. Especially, we have O,I, = Ob, where 0; again is the holomorphic structure sheaf induced through the analytically branched covering.

This theorem was proved already in [GR58]. Nevertheless the proof to be given here makes use of a result from [GR84], because it might be more familar to the reader: Since a normal complex space is locally pure dimensional, it can

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locally be realized as an analytic covering n: U + I/ over a domain V’ c C:” with critical locus A (for the definition cf. subsection 1.3). Now the assertion follows from the main theorem. As this “proof” already has shown, one has generalized in modern theory Theorem 3.1 representing much more general complex spaces locally as analytic coverings over domains in C”. Those theorems are valuable tools in order to get information about the local structure of such complex spaces. For more information about this see [GR84]. As an application of Theorem 3.1 and the main theorem we are in a position to prove all our theorems and most of the elementary propositions for analytically branched coverings (stated in subsection 1.1 and 1.2) over normal complex spaces, which we will define in the next subsection. The following further application should also be given even it is more of historical than of theoretical interest: In 1951 Behnke and Stein used these analytically branched coverings as local model spaces to define the concept of a complex space in order to generalize the concept of a complex manifold [BSSl]. This kind of complex space, which was called complex a-space in [GR58], is defined as follows: A Hausdorff space R is called a complex cr-space if there exists an open covering R,, I E I, with the following properties: 1) For every I E I there exists an analytically branched covering 71,: : Z, + B, and a topological map $,: R, -+ Z,. 2) If R,, n RI2 # 0, the map II/,, 0 +,;‘: II/,,(R,, n R,J + $,,(R,, n RI11 is biholomorphic. For every I E I, the tripe1 (R,, $,, rc,: Z, + B,) is called an cr-chart. A continuous function f: Q + C, defined on an open subset 52 c R, is called holomorphic if f o II/,-‘: $,(Q) + C is holomorphic for every N-chart (R,, $,, rc,: Z, + II,) with Q n R, # a. From this point of view, our main theorem and Theorem 3.1 just say that the complex cl-spacesare exactly the normal complex spaces.In 1958 the aim that Grauert and Remmert had in mind when proving these two theorems in [GR58] was exactly this application! 2. Analytically Branched Coverings over Normal Complex Spaces. We generalize the notion of analytically branched coverings over domains in (c” to those over normal complex spaces: Definition 3.2. Let X be a locally compact space and G a connected normal complex space and consider a finite and surjective mapping 7~:X + G. Assume further that there exists an analytic subset A # G of G such that: (1) Cl(A) does not separate locally the set X. (2) rr: (X\n-‘(A)) + (G\A) is locally topological. Then rr: X -+ G is called an analytically branched covering with critical locus A.

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Notice that the only change which has been made in comparison to the corresponding Definition 1.1 is that we have replaced the words “open domain in (c”” by “normal complex space “! All the other definitions of subsection 1.1 and 1.2 (e.g. schlicht point, branching point, holomorphic function, analytic set) are verbally the same and hence need not be repeated here. We can also give the following different characterization of analytically branched coverings over normal complex spaces, which will become the key for generalizing most of our results on analytically branched coverings over domains in C’ to such coverings over normal complex spaces: Proposition 3.3. Let X be a topological space, G be a connected normal complex space and 7~:X ---f G a continuous map. a) 7~:X + G is an analytically branched covering if and only if for all points g E G there exists an open neighborhood U(g) c G and an analytically branched covering 71’: U(g) --f V over a domain V c C” such that the composed map 7t 0 If: 7c-l(U(g))

+ D

is again an analytically branched covering. b) The holomorphic structures induced on Y by the coverings rc: Y + U(P) and R’ o z: Y + V coincide.

The first part of a) is an immediate consequence of Theorem 3.1. The difficult point in the second part of a) is how to define a global critical locus in G for K: X + G. We can proceed as follows: It is easily seen that under our assumptions 7~:(X\Cl(S(G)) + (G\S(G)), where S(G) denotes the singular locus of the normal complex space G, is an analytically branched covering over a manifold. From our assertion 9) (cf. subsection 1.2) it follows that this analytically branched covering has an empty or pure one codimensional (minimal) critical locus A. Since S(G) is at least two codimensional in G, the set A u S(G) again is analytic by the extension theorem for analytic sets (cf. [GR84]), and it can now again easily be shown that this analytic set can serve as critical locus. Part b) is proved using the Riemann Extension Theorem. It is an immediate consequence of this proposition that our main theorem also holds for analytically branched coverings over normal complex spaces. Furthermore it can now relatively easily be shown that the elementary properties l)-6) and 10)-l 1) stated for analytically branched coverings over domains in C” in subsection 1.1 and 1.2 also hold for analytically branched coverings over normal complex spaces (the first part of property 3) is, of course, only true if the base space itself has a countable base of topology). From property 3) and the fact that on analytic subsets of a C” there exist holomorphic functions separating a finite number of given points it follows that Theorem 1.4 also still holds. The assertions about the structure of the branching, however, are deeper: What remains true is the fact that the projection of the branching points of K: X + G is an analytic set, and hence again the minimal critical set. It is not difficult to see this, since, as a consequence of the main theorem, we have locally finitely many holomorphic functions on X separating the points of X. The

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discriminant sets of these functions (built from the coefficients a, in assertion 10) in subsection 1.2) are analytic in G, and their intersection is just the projection of the set of all branching points of x: X + G. It is, however, not any longer true that this set is empty or pure one codimensional, as can be shown with the following example: Take the two dimensional cone in C3 given by the equation z: - z1z2 = 0 in C3. As stated in 1.3.1 it is a normal complex space G with only one singular point P (the origin). Now the map 7~:C2 + G c C3; (tl, t2) + (tf, t:, t, t2) is an analytically branched covering of G with 2 sheets. The projection of the set of all branching points is exactly the point P, which is two codimensional in G. We wish to close this subsection with the following two remarks: If one wants to define the concept of “analytically branched coverings” for more general than normal complex spaces, one has to make stronger assumptions on the holomorphic structure on the branching, as this is done e.g. in the definition of analytic coverings (cf. subsection 1.3). Then one looses, furthermore, the possibility to prove extension theorems of the kind that we will give in the next subsection (Theorem 3.4 is false for analytic coverings with more general than normal complex spaces). The fact that all our theorems and most of the elementary properties of analytically branched coverings over normal complex spaces are easy consequences of the corresponding properties for the analytically branched coverings over domains in C’ is the reason why we dealt only with the latter during the first two sections. 3. Extension of Analytically Branched Coverings. We show the following extension theorem, which yields us the astonishing fact that an analytically branched covering can be extended over any nowhere dense analytic subset, if we only can extend some critical locus over it: Theorem 3.4. Let N be a normal complex space, and let B c N be a nowhere dense analytic subset. Let 71: Y + (N\B) be an analytically branched covering with a critical locus A c (N\B). Assume that A u B c N is analytic. Then 71: Y + (N\B) can be extended to an analytically branched covering z’: X -+ N, which is uniquely determined up to equivalence of analytically branched coverings.

It is mainly this theorem which makes analytically branched coverings so important. We will give two applications which both illustrate this. The first application is the proof of the Main Theorem on holomorphic equivalence relations given in section 7 of this chapter. The second one, which deals with the compactification of analytically branched coverings over Zariski open subsets of normal projective varieties, will be given below as Theorem 3.5. It also might be worth while to stress again that Theorem 3.4 in particular implies that an analytically branched covering is uniquely determined by its behavior outside the critical locus, where it is an unbranched covering. It is of some interest to say a word about the proof of Theorem 3.4, which is purely topological: Since we can use A u B as the new critical locus, we only

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have to show that our analytically branched covering can be uniquely extended as a topological branched covering in the sense of Stein CSt.561. But this is shown in [St56], even in a much more general topological situation. Theorem 3.5. Let N c IP,, be a normal projective variety, and let B c N be a nowhere dense complex analytic subset. Let 7~: Y --) (N\B) be an analytically branched covering, Assume further that there exists a critical locus A such that A u B is analytic (which e.g. is always the case if A is a subvariety of N). Then there exists a compact analytically branched covering n: X + N which extends z: Y + (N\B). Moreover, X is again a normal variety (in some II’,,,). Let us give an idea of the proof: The first assertion is clear from Theorem 3.4. In order to prove the second assertion, let Fk = 0; @ Zk, where 2’ is the sheaf obtained by lifting the standard positive line bundle from lP,. Then the direct image sheaf n,(F’) is a coherent sheaf over IF’,,,,which, by Serre’s theorem (cf. [GR58a]), satisfies Theorem B for all sufficiently large k. But then Fk also satisfies Theorem B. Hence we can find finitely many sections in 9’ mapping X biholomorphically onto an analytic subset of some II’,,,, which, by Chow’s theorem, is a variety. Corollary 3.6. Let N c C” be a normal affine variety, and let 7t: Y + N be an analytically branched covering with an affine critical locus A. Then Y is again an affine variety. We can prove this by applying Theorem 3.5, where we use as the set B the hyperplane at infinity of (c”. We have only to take care that there exists a hyperplane H in lP,,, with Y = X\H (where we now have used the notations of the theorem). To this end let s be a section of our line bundle 2 such that the projection of the zero set of s to IPn is the hyperplane B. If we now add the section 1 @ sk to the finitely many sections of Fk yielding the projective imbedding, we can define H to be the hyperplane in IP,,,+i where the component onto which this section maps is zero.

0 4. Analytic Decompositions* Since old times some times it is necessary to consider functions f as functions of the values of another function g. Such functions f and g are analytically dependent: If they are holomorphic their Jacobian has rank 1, only. In year 1953 K. Stein suggested a student K. Koch to develop a general theory (see [St53]). Koch considered a domain G in the n-dimensional complex number space (c” together with a non constant complex holomorphic function g. He showed that the ring R = R, of holomorphic functions f on G which are analytically depen* For this and the following sections proofs (see [Ka93] and [Sb93])

we have to thank

B. Kaup

and B. Siebert

for completing

some

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dent on g can be viewed as the ring of holomorphic functions on a Riemann surface. Some years later K. Stein built a more general theory of analytic decomposition. Other authors followed. We shall consider here the modern version of this, that is the theory of analytic equivalence relations. We shall throughout assume that our complex spaces have countable topology. 1. Analytic Equivalence Relations on Complex Spaces. These will be equivalence relations whose graph is an analytic set. Of course, in complex analysis also more general equivalence relations appear (whose graph is not closed). But these will not be of interest here. Let us always denote by X a reduced complex space. The Cartesian product X x X is again a reduced complex space. We take an analytic subset R c X x X with the following two properties: 1) R contains the diagonal D = {(x, x)1x E X} c X x X; 2) R is invariant under reflexion (xi, x2) + (x,, xi). For any two points in X we write then x1 N x2 if and only if the pair (xi, x2) belongs to R, and we call R an analytic equivalence relation if and only if ‘v satisfies the transitive law. Then N is an ordinary equivalence relation in X, and we have the quotient set Q = X/R and a quotient map q: X + Q which is surjective. We denote the projection X x X + X onto the first respectively the second component by pi respectively p2. If x E X is a point, we denote by X, the (reduced) analytic set pl(p;‘(x) n R), which we call the fiber through x. We assumefrom now on that R is an equivalence relation. If x’ E X, then X,, = X, and {X,: x E X} is a decomposition of X. We equip Q with the quotient topology: a set I/ c Q is open if and only if the inverse image U = q-l(V) is open in X. Then q: X -+ Q is continuous. Moreover, there is a natural structure sheaf on Q. A complex function g on an open subset I’ c Q is called a holomorphic function if and only if the lifted function f = g o q is holomorphic on the inverse image U. It follows that f is constant on the fibers over 1/ The inverse image f-‘(w) c U of any open W c C is then fiber saturated and open. Hence, g is continuous. We have the local holomorphic functions over Q. These define a sheaf 0, of local C-algebras over U. The local cross-sections over an open V c Q are special continuous complex functions over V They coincide with the holomorphic functions there, and are brought back by q to holomorphic functions over U. Therefore q is called a holomorphic map. We also write Co,= q*R(l!Jx)and call it the fiber-constant direct image of the structure sheaf on X. 2. Holomorphic ‘Maps. Assume that Y is another (reduced) complex space and that F: X + Y is a holomorphic map. The libered product X xr X is a complex subspace of X x X. The zero set of its ideal sheaf in X x X lies over the diagonal D c Y x Y and over D its stalks are generated by the local holomorphic functions f(xl) - f(xi) with f(x) = g 0 F(x), where g is any local holomorphic function in Y. In general, the O-setis not reduced. We denote by R = R,

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its reduction, i.e. we consider X xF X as an analytic set in X x X. Then R is the union of the analytic setsF-‘(y) x F-‘(y) with y E Y and the fibers with respect to R are the (set theoretic) fibers of F. It follows that R in an analytic equivalence relation. We call it the analytic equivalence relation defined by F. 3. Restrictions. Assume now that Z is a complex spaceand that H: Z + X is a holomorphic map. The Cartesian product H x H: Z x Z + X x X, which maps (a, b) -+ (H(a), H(b)), is again a holomorphic map. If R is an analytic equivalence relation in X, then the inverse image in Z x Z is an analytic equivalence relation in Z. We call it the lifting of R to Z. In the case where Z is an open subset or an analytic subset of X, we consider it as the restriction RIZ of R to Z. The fibers in Z are then the intersections of the fibers in X with the set Z. 4. Finer Equivalence Relations. If R’ and R are analytic equivalence relations on X, the relation R’ is called finer than the relation R if the set R’ is contained in R. In this casewe write R’ I R. Then the fibers with respect to R’ are always contained in the fibers to R. Assume now that R, with p E M is a family of complex equivalence relations on X. We denote by R the intersection of all these R,. Then R is an analytic subsetof all R, and is an analytic equivalence relation in X with R I R, for all p. This possibility enables us to construct the finest analytic equivalence relation with certain given properties.

9 5. Spreadable and Semiproper Equivalence Relations 1. Spreadable Analytic Equivalence Relations. Let us first consider the following complex manifold X c IP, x (c2 over (c2 with X = {(w’: w, z’, z): w’z = wz’}, where (w’: w) denote homogeneous coordinates in the complex projective space lP,. The map F: X + 42’ is the holomorphic projection (w’: w, z’, z) + (z’, z). Then the fiber over 0 E (c2 is II’, and over all other points a single point. Moreover, F: X - IP, + C2 - (0) is b’h 1 o1omorphic. So every fiber is irreducible. The fiber dimension is not constant. The quotient space Q is (c’, hence a complex space. Is this always true? A complex space X has many holomorphic functions in each of its points. So, if Q is a complex space there are many fiber constant holomorphic functions around each of the fiber in X. Consider the following simple example: Put X = lP, and denote by L c X a projective line and take for R the union D u (L x L) c X x X, where D stands for the diagonal. Then R again is an analytic equivalence relation in X and one libre in X is L. But there are arbitrary small neighborhoods of L which are pseudoconcave (seechap. VI). Hence every holomorphic function in a neighborhood of L has to be constant. To avoid this unpleasant situation we define:

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Definition 5.1. An analytic equivalence relation is said to be a spreadable analytic equivalence relation if for every point y E Q there are an open neighborhood V(y) and a holomorphic map F: U = q-‘(V) -+ (l2:” such that F is con-

stant on the fibres and the fibers of F have locally the same dimension fibers to R.

as the

This means that on the fibers, locally, the fiber to R and to R, are the same. If F: X + Y is a holomorphic map of X into a complex space, then Y can locally be embedded into open subsets I/ of a (Em and we can consider F as a holomorphic map of U into I/. So R, always is spreadable (and, moreover, the fibration is locally separable by fiber-constant holomorphic functions). Then also every liner analytic equivalence relation, whose fibration is locally the same on the libres, is spreadable. On the other hand, if the quotient space Q is complex, the analytic equivalence relation R has to be spreadable. The condition spreadable is much weaker than the property that Q be a complex space: If Q is a complex space, then the fiber-constant holomorphic functions in U separate the fibers, but spreadable does not imply this property. It is difficult to further weaken the condition. It is impossible to use a condition of pseudoconuexity alone. But it may be that this can be done by using formal power series. To get an example we take a smooth family 2 of compact Riemann surfaces of genus 2 1 over the unit disc A c (IJ and embed it in a 3 dimensional complex manifold X, such that the normal bundle of Z with respect to X on every fiber Z,, t E A, of Z is strictly negative. We have the analytic equivalence relation R = (D x D) u (Z x d Z) c X x X. If the embedding of Z is twisted enough along Z in the direction of A, there will be no smooth surface S which contains just one libre Z, since there is an obstruction in the cohomology H’(Z,, Lot), t E A, where 6Jt denotes the structure sheaf of Z,. If R is spreadable along a generic fiber such a surface S does exist. But all the fibres to R in X have a strongly pseudoconvex neighborhood. So pseudoconvexity alone cannot give a sufficient condition for the spreadability. Two analytic equivalence relations R’ and R are called equivalent if and only if the connected components of the fibres are the same. Assume that R is spreadable and that R’ is liner than R. Then also R’ is spreadable. 2. Semiproper Equivalence Relations. Consider the following example: Put X and Y equal to (c* and denote by F the holomorphic map X -+ Y given by w’ = z’, w = z’. z. The image does not contain points (0, w) with w # 0. We have F(X) = Y - { (0, w)} u { (0, O)}. The equivalence relation R, is spreadable. However, the quotient space Q is F(X) as a set. But, its topology in 0 = (0,O) is different from the relative topology. An open neighborhood of 0 means an open neighborhood of 0 x (c in X. Therefore, 0 does not have a countable base of neighborhoods. Moreover, Q is not locally compact in 0. Hence, Q is not a complex space. We need a further condition:

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Definition 5.2. An analytic equivalence relation R is called semiproper if every point y E Q has an open neighborhood V(y) such that there is a compact set K in X with q(K) =) V.

In [Gr83]

is proved:

Proposition 5.3. If R is a semiproper analytic equivalence relation in X, then Q is a locally compact Hausdorff space. Necessary and sufficient for “semiproper” is that Q is a Hausdorff space and that every point in Q has a countable base of neighborhoods.

For proof see p. 140/141 and p. 147/148.

@6. Normal

Equivalence

1. Maps of Complex Spaces. If R and X are complex spaces and F: R + X is a holomorphic map, then each irreducible component R’ of R is mapped into an irreducible component X’ of X. We call F locally dense if 1) on every R’ there are in arbitrary small neighborhoods of the points some smooth points r in which the rank of the Jacobian of F is equal to the dimension of X’ (of course, we take the Jacobi rank after composition with a local embedding of X around F(r) in a smooth domain), 2) if r E R is a point and U(r) an arbitrary small neighborhod, then F(U(r)) contains points of the difference of any irreducible component X’ of X through F(r) and the rest of X. In points r E R’ with Jacobi rank equal to dim X’ the map F: R’ --f X’ is open. Moreover, if F is locally dense and A c X is a nowhere dense analytic subset, then the inverse image F-‘(A) is empty or nowhere dense in R. We have the following: Proposition 6.1. Assume that R and X are pure dimensional complex spaces and that p: R -+ X is a locally dense holomorphic map such that the fiber dimension is everywhere the same. Then the map p is open.

The proof follows like this: Since a local embedding of R in smooth domains is possible there are locally branched cross sections S in R over neighborhoods in X. The map p: S + X of analytically branched coverings always is open. If R and X are pure dimensional and X is locally irreducible (for instance, normal) and R is an analytic equivalence relation in X, with constant fiber dimension dim R - dim X then the condition of the proposition is satisfied. Hence, the projections pP: R -+ X are open. A complex space X is called semi complex function which is holoa nowhere dense analytic set is holomorphic. Since the

2. Normal Analytic Equivalence Relations. normal if and only if every local continuous

morphic

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Riemann theorem is valid for normal complex spaces it follows that every normal complex space also is semi normal. Assume from now on always at least that X is semi normal and that X and R are pure dimensional. (See [BR90]). Definition 6.2. An analytic equivalence relation R in X is called nowhere degenerated if the projections pP: R -+ X are locally dense and the dimension of the fibers in the points of R is constant.

We define a multiplicity for the irreducible components of the fibers of a nowhere degenerated R + X in the following way: for generic fibers we define the multiplicity of the irreducible components of the libres to be 1. For special fibers it is obtained using the homology class of the maximal cycle, i.e. it is the number of sheets if in generic points we represent neighboring fibers as branched coverings of the special fiber. The following definition is important: Definition 6.3. Assume that R is an analytic equivalence relation in a semi normal (pure dimensional) complex space X with the following two properties: a) R is nowhere degenerate; b) For every point x’ E X there is counted with multiplicities (in the above sense) a unique limit of tibres lim,,,. X, from generic libres X, c X which equals the libre X,. in X. Then R is called a normal equivalence relation*. The codimension of R is the codimension of the fibers X, in X.

It follows at once that the projections pP: R + X are open. Since a normal complex space is locally irreducible and hence lim,,,. is independent of the direction, it follows that a nowhere degenerate analytic equivalence relation in such a space is a normal equivalence relation, always. We have the following theorem (see [Gr85, p. 1421): Theorem 6.4. Zf X is a semi normal complex space and R is a normal equiualence relation in X, then R is semi proper and spreadable.

For a brief indication of the proof we take for any point x E X a point r E R over x (by the projection p2) and in a neighborhood of r a complex subspace S = p;‘(s’) n p;‘(f), where S’ is a local complete intersection in X such that the dimension of S is everywhere the codimension of the libres in X and this local complete intersection S’ intersects the Iibre through x only in x. The intersection number of S’ with the Iibres p,p;‘(x’) for x’ near to x is always the same. By passing to the arithmetic mean of values of holomorphic functions on S’ with respect to the intersection points of S’ with the libres we obtain, using the fact that X is semi normal, fiber constant holomorphic functions along the fibers which separate the fibers. Hence, R is spreadable. If we denote by U the set * In [G&5] the definition of the normal equivalence relation on p. 115 was too weak. Indeed on p. 146 is written: “6. In the case of the Main Theorem . . . well defined multiplicity . . .“. We have to use that multiplicity. But that means, that we used our stronger delinition of “normal equivalence relation” there: We need the same multiplicities from all directions.

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R n p;l(S’), then p1 (U) c X is open and tibre saturated and hence, qpl (U) is an open neighborhood of q(x). We have q(U) = q(Y). From this it follows that R is semi proper. If X is a semi normal complex space, R an analytic equivalence relation in X such that the quotient Q is a complex space and the quotient map q: X + Q is locally dense, then Q is semi normal also. If, moreover, X is normal, and instead of q the maps p,: R + X are assumed to be locally dense, then also q is locally dense and no identification between lower dimensional analytic subsets of the components of X takes place. Hence, the Riemann theorem for Q is valid. So Q has to be normal also. The condition in Definition 6.3a to the effect that the pP are open is essential. Consider for instance (c2. Equip (T with the reduced complex substructure of the Neil parabola. We denote by S the l-dimensional complex space obtained in this way. The local ring in 0 E S is smaller than the full ring of convergent power series. Then also the Cartesian product X’ = S x (c is a complex substructure of a’. We make the following shift: z’ = w’, z = w’ + w. Another complex substructure X‘ of (c2 is obtained. But we have the same complex planes E = 0 x c2 in X’ and in X-. We put X = X’ v X’ by identifying X’ and X- identically transversally on E. Then X is a reduced complex space. We have topologically X’ = X‘ = (c’. If we identify the components X’ and X’ identifically we have a semi proper analytic equivalence relation in X. All the fibers are O-dimensional. We also have E c Q. Since local holomorphic functions on X’ and Xhave vanishing derivatives on E in different transversal directions to E with respect to the common complex structure of (c2, it follows that the inverse image of every local holomorphic function on Q has to be constant on E. Hence, Q is not a complex space, since there are not enough local holomorphic functions. - We embed X’ and also X’ as an analytic set in (E3 and do the same analytic equivalence. So we obtain a semiproper analytic equivalence relation in a normal complex space which is the disjoint union of two copies of (c3. The fibers have everywhere dimension 0. But now there is an irreducible component of the equivalence graph whose p,-image is nowhere dense. The equivalence relation is not spreadable. The quotient space again is not a complex space. In [Gr85, p. 1421 such a space is called a sutured complex space.

0 7. The Main Theorem 1. Indication of the Theorem. If X is a reduced complex space and R is a semiproper spreadable equivalence relation in X then, in general the quotient space Q will not be a complex space. We consider the following example: We take two copies of the complex plane (c and identify the infinitesimal neighborhoods (0, O/z’+’ . 0) identically with each other. We obtain for p = 0, 1, 2, . . . reduced complex spaces X,, which lie over Cc by a natural holomorphic map. We take the disjoint union X of all these X,. There is a natural holomorphic projection F: X -+ c:. The equivalence relation R < R,, which identi-

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lies all the X, identically over C, is analytic, it is semiproper and spreadable by the projection F. The quotient space Q is topologically isomorphic to every X, and contains a point 0 which is the image of all the O-points. However, the inverse image of a holomorphic function f in a neighborhood of 0 E Q has to be holomorphic in all points 0 E X,,. This means that the restrictions f' and f" of f to the two sheets of Q over C coincide in 0 of arbitrary high order, if f' and f' are considered as functions on C. So we get that f on both sheets is the same. We cannot separate points by local holomorphic functions and Q is not a complex space. This means, we cannot expect a good theorem if X is a general reduced complex space. Therefore, we shall always assume that the complex space X is semi normal. Theorem 7.1. If R is a semi proper spreadable analytic equivalence relation in a semi normal complex space X, then the quotient space Q = X/R is a semi normal complex space. 2. Idea of the Proof. We can pass over to the normalization X” of X and can lift R to an analytic equivalence relation R” on X” such that by this points of X” over the same point of X are again identified. R” is spreadable and semi proper, again, and gives the same quotient space. So we can assume without loss of generality that our space X is normal. We have to prove that the structure sheave gives the structure of a complex space to the topological space Q. Now Q is a Hausdorff space. Therefore, in proving the theorem we may assume: a) there is a closed set K c X such that the quotient map q: K -+ Q is proper; b) there is a holomorphic map F: X + (cm with R I R, such that the fibers to R and R, have the same dimension in all the points. The map F admits the following factoring: F:XsQ$C”. Following [Gr83, p. 1431 we may assume that there is a subdomain I/ c Cm such that _F: Q + I/ is finite and open. Denoting by E the degeneration set of R we see that the normalization Q” of Q over I” = V’ - F(E n K) is an analytically branched covering. The set F(E n K) can be considered as an analytic subset of V of codimension at least 2. The branching locus of Q” has codimension 1 and by the theorem of Remmert-Stein [RS53] it extends to an analytic set in V. It follows that also the analytically branched covering can be extended to an analytically branched covering over V (see9 1 to 9 3). But by the main theorem of 0 1 every analytically branched covering is a normal complex space.Now Q is obtained from this by an analytic gluing. Of course, the exact proof is more complicated. It is contained in [Gr83]. 3. Simple Equivalence Relations. Assume now again that X is any reduced complex space.Analytic dependence of holomorphic maps of X does generalize to analytic dependenceon a given analytic equivalence relation R. In order to

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describe the nature of this dependence we associate to R a canonical equivalent liner analytic equivalence relation R” which we call the simple equivalence relation corresponding to R. This relation R” will be the finest analytic equivalence relation such that the fibres to R and to R” are locally the same. We do the construction of the graph R” in X x X. To this end it will be necessary to define the degeneration set E, of order n, n = 0, 1, 2, . . , to the fibration given by R. We put E, = {x E X: dim, X, 2 n}. By [G&3, 1 Proposition l] all these E, are analytic subsets of X. This follows directly from an old theorem of Remmert. We thus obtain a properly decreasing sequence. Let us now take an irreducible component X’ of X. We write R’ = RIX’ and n = dim X’. Furthermore, we denote by d’ the minimal dimension of an irreducible component of a fibre in X’ and by R”” the irreducible component of R’ which contains the (irreducible) diagonal D’ c X’ x X’. This component R”” is uniquely determined, since R’ is an equivalence relation. R”” is mapped by p1 and p2 onto X’. To prove the uniqueness of R”” we take a point (a, b) E R’ such that X’ is smooth in a and b and such that R’ is smooth in (a, b) and such that pl, pz are regular, there. Then R’ is smooth in (a, a), (b, b) E R’, while pi, p2 are regular there. So only one irreducible component of R’ can pass through (a, a) and (b, b), hence through D’. In a neighborhood of (a, a) the sets R”” and R’ are equal. So the transitive law holds for points (c, d) E R”” of a neighborhood of (a, a). For general points of R”” this need not be true. We denote by R* c R’ the finest analytic equivalence relation in X’ containing R”“. We put X* = {x E X’: dim, X, > d’} and take the union R, of all R”” to all irreducible components X’ of X and, moreover, the union X, of all X* to them. We repeat now for X, what we did for X and obtain R, and X, and go so on. The sequence R, is locally finite. Therefore, the union R- of the R, is an analytic set in X x X which is contained in R. However, R- will in general not be an analytic equivalence relation. But, there is a unique smallest analytic equivalence relation RA 2 R- in X. This is the desired simple one. There is a simple example that R” # R* in general. We take in C2 = ((z’, z)} the diagonal A = {z’ = z}, take two copies X’, X’ of C2 and glue X’, Xtogether along the two A identically and transversally, so that they have transversal crossing in the reduced complex space X obtained by this procedure. In C2 we have an analytic equivalence relation by the holomorphic map (z’, z) + z: C2 + C. This equivalence relation gives an analytic equivalence relation R in X taking the union of the fibers. The set R- lies in (X’ x X’) u (X” x X’), but R A does not. The simple analytic equivalence relation R* is liner than R. If 2 c X is an open subset of X or an analytic subset in X, then in general (RIZ)” will be different from R” IZ, but we always have (RIZ)” I R” IZ. The two libres X x RA c X,,, have the same dimension in X; moreover, locally on every libre we have X,,. n = X,,. and globally every libre X,,.,. is a union of connected components of X,. R. Hence, R A is spreadable if R is.

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If F: X -+ Y is a holomorphic map, Stein [St631 called the quotient the complex base of F.

space

Q = X/(R,)”

4. A Geometric Construction of Simple Equivalence Relations. Let us assume now that the complex space X is irreducible and has dimension d and, furthermore, that R is an analytic equivalence relation in X. It can be seen rather easily that in general points of R” the projections p,: R” -+ X are open. Now R” can be constructed, as was already done by Stein’s student Koch in a special case. To this end we denote by X” the set of non-singular points of X and by R” the biggest open subset of R with the following properties: a) R” c X” x X” and is smooth: b) the projections pP: R” + X” are open and have everywhere maximum Jacobi rank: they are smooth. Especially we call two points x’ and x in X equivalent if x’ = x. Furthermore we require that: if there are equivalent points y’ and y in X” with (y’, y) E R” together with a path 4(t), t E [0, l] in R with +5(O)= (y’, y) and d(l) = (x’, x) such that d(t) E R” if t # 1, then also x’ and x are equivalent. More precisely, we take the finest equivalence relation having this property. Of course, transitivity is used everywhere. Since R”” contains points of the diagonal and R”” n R” is connected all pairs in R”” are equivalent. - To get all points of R” we have to use the transitive law of equivalence relation finitely many times. We consider the special case of a holomorphic map X + Y and take R = R,. The path 4(t) then consists of two paths qSl(t) and d2(t) in X over the same path in Y. The statement c)(t) E R” means that dl(t) and d*(t) run through smooth fibers of F in X”. So the simple equivalence relation to R can be constructed in a very simple way in this case. We call an analytic equivalence relation R itself simple if RA = R. 5. Examples of R and R”. We denote by G the Riemann surface of z112over the complex plane C = (z} and by X the Cartesian product of G with (c. We have the natural projection F: (z”~, w) + z mapping X onto C. The fiber X, is C but all other fibres consist of two disjoint copies of C, and are thus disconnected. The analytic equivalence relation R = R, c X x X consists of 4 irreducible components. Hence, R is not simple. The simple relation R” has the fiber C x C over every point of G and the quotient Q = X/R^ is G. We consider the following domain in C2:

G = {(w)~IzI

< 1 or 1 < IwI < 2

or 3 < /WI},

and denote by F the projection (w, z) + z: G + (c. If IzI < 1, the fibers are C. However, if IzI > 1, the fibres consist of two disjoint disc rings. We can use the geometric construction of the simple equivalence relation R” to R = R,. It follows that R = R”. So in the case of a simple equivalence relation the fiber is in general not connected. However, clearly the following is true: if every fiber of R = R, is irreducible then R is simple.

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6. Analytic Dependence. Assume that X is a (reduced) complex space and that R is an analytic equivalence relation on X. If Y is another complex space and F: X -+ Y a holomorphic map, then F is called analytically dependent on R if F is constant on the fibers to R”. If the quotient space Q = X/R” exists, it follows immediately that there is a unique holomorphic map _F: Q 4 Y with F = _F o q. This factorization is called the Stein jktorization. The quotient space Q is uniquely determined by the property that the Stein factorization exists for all holomorphic maps F which are analytically dependent on R. - Stein called Q a h&morphic base for R. The construction of holomorphic bases was the starting point for his theory of analytic decomposition. If G is any holomorphic map of X, then F is called analytically dependent on G if F is analytically dependent on R,.

3 8. Meromorphic

Equivalence Relations

1. Meromorphicity. We denote by X an n-dimensional connected normal complex space. If R is an analytic equivalence relation in X, the quotient space may not exist as a complex space. But we can construct out of R, after having passed over to a proper modification of X, an analytic equivalence relation whose fibers everywhere have the same dimension. Then in many more cases we will have a complex quotient space. This will be most suitable in the study of meromorphic maps. In Stein [St63 + 641 the quotient space was called an mbase.

To achieve the highest generality, we require only that the analytic set R c X x X has the following 2 properties: a) R contains the diagonal D c X x X; b) R is mapped by reflexion (x’, x) + (x, x’): X x X ~1 X x X onto itself. We define: Definition 8.1. R is a meromorphic equivalence relation in X of codimensionc if and only if the following three properties are satisfied: a) there is a nowhere dense analytic set P c X (a polar set) such that the intersection R n (X x P) is nowhere dense in R; b) RJ(X - P) = R” := R n ((X - P) x (X - P)) is a normal analytic equivalence relation of codimension c in X - P; c) all irreducible components of R enter in R”.

We define the fibers in X again as the sets X, = pl(R n piI(x which are analytic subsets of X. In X - P they have the dimension n - c everywhere. The degeneration set E in R consists of those points in R where the fibre dimension is bigger than n - c. It is an analytic subset in (X x P) u (P x X). Outside P the graph R is a spreadable and semi proper equivalence relation. The quotient of X - P is a normal complex space of dimension c.

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2. The Fibration Given by R. In the whole space X the graph R does not give an equivalence relation in general. But we can extend our fibration from X - P

to X in sucha waythat all thefibershavedimensionM- c everywhere.Wepass to the normalization tt: R* -+ R of R and denote by E* the inverse image of E in R* and by E’ c R* the inverse image of p,(E) under the map tl = pz o z: R* + R +X. The last set is not an analytic set, in general, but it is an at most countable union of local nowhere dense analytic subsets, like p,(E) is in X. We put R’ = R* - E* - cl-‘(P) and obtain a “Zariki” open subset of R*. If xh E R* is a point, there are many holomorphic maps II/: A -+ R* of the unit disc A c Cc with the following 3 properties: a) rj(O) = x^; b) I,-‘(E-) is an at most countable set; c) $(A - (0)) c R’. We take the set theoretic fibred product R* xx A, that is we lift the fibration given by CIin R* to A and take the union 2 of all irreducible components which do not lie over a single point of A, completely. All fibers of Z over A i.e. Z, := Z n (R: x t) with x = CIo $(t), t E A have pure dimension n - c. If t is generic, we have Z, = R,*, i.e. Z, is the fiber given by c1in R*. So we call all the Z, fiber in R*. If we take their (pl 0 x)-image in X, we get a family of libres in X. The set of all the fibers obtained in this way is called a fibration in X. We denote it by 4. Some of the libres of 4 may cross. All S E 4 are contained in a pip;‘(x), x = u o II/(t). 3. Regular Meromorphic Equivalence Relations. The set {S} is a fibration 4 in X in pure dimensional analytic subsets. This is considered to be the fibration given by the meromorphic equivalence relation R. We shall construct the quotient space by 4 or, as we say, by R. We follow [G&5, p. 1171. However, in general it does not exist as a complex space. For this we need a further assumption. Some peperations are necessary. First we have to consider families of fibers. Assume that X’ c X is an open subset and that Z is a purely (n - c + l)-dimensional complex space over the unit disc A such that the holomorphic map Z + A is nowhere degenerate and no irreducible component of Z lies over one point of A only. Moreover, we assume that there exists a holomorphic map H: Z + X’ such that every fibre is mapped topologically and holomorphically onto an intersection S n X’, where S E 4. Then we call Z + X’ a bunch of fibres in X’. We use the definition in [Gr87, p. 1821: Definition 8.2. The meromorphic equivalence relation R in X is called regular if for every compact subset K CC X there is an open subset B CC X with K c B such that the map 4 + 4 n B is bunch-injective with respect to K: there is no open subset X’ with B c X’ CC X together with a bunch in X’ of infinitely many fibers S n X’ with S n K # 0 such that the intersection with B consists of one fiber only.

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If S n X’ belongs to a bunch of infinitely many fibers such that the intersection with B consists of one ftbre only, then every irreducible component S’ of S which is moved in the bunch is in the (pi o x)-image of the degeneration setE*. In [Gr85] the following theorem is proved: Main Theorem 8.3. If R is a regular meromorphic equiudence relation of codimensionc in X, there is a unique proper modification 7~~:X‘ -+ X together with an open holomorphic map q: X- -+ Q of X‘ onto a c-dimensional normal complex space Q such that: a) the map n’ mapsthe fibers S’ in X- topologically holomorphically onto the s = 7c’(S’) E 4; b) the map of the S’ to S is a certain kind of normalization Q + 4. Proof idea. We can also use [Hk73]. We put Q = X’/R and call Q the (generalized) quotient space of X by R. We obtain Q as the quotient space by an normal analytic equivalence relation from the proper modification X- which is a weakly normal complex space. But first we have to construct X’ as a normal complex space. This has to be done as in [Gr85] by gluing normal blown up cones. But the analytic equivalence relation employed there has to be made somewhat liner in such a way that the result is again a normal complex space. Otherwise the proof does not go beyond relative compact open subsetsK” of X (at least not in the general case). Briefly speaking: We replace those points x of K”, through which runs an eventually higher dimensional generalized bunch of infinitely many fibers S E 4 n K”, by elements which are nothing else but these fibers. If we pass over to a bigger K”, it may happen that we have to replace all the points of a single S by an generalized infinite bunch of fiber intersections with the larger K”. This means a blowing up in all the points of the set S. Of course, we cannot do this infinitely many times. The procedure breaks off when K” increasesif and only if R is regular. This proves also that a complex quotient space does not exist if R is not regular. Finally, we exhaust X by an increasing family of setsK” and glue together the (normal) X’ belonging to the K” and then glue together those points that are mapped onto the same point of the corresponding fiber S (but only so far that the quotient spaceQ becomesa normal complex space).Then X” will no longer be normal: it is only semi normal. We shall give a simple example showing that X’ is not normal in general, and another example of the samekind to the effect that the map rc-: S- + S need not be biholomorphic. First we take for X the well known normal complex space {w’ = zlzZ} over C2 = {(z,, z,)} and as fibers S the inverse images over the lines through 0 E C2. Then X’ is obtained by blowing up 0 E Cz and lifting X to the result. The O-point of X is replaced by a lP,, but all generic fibres S- through the points of lF’, consist of 2 intersecting lines. Hence X’ is not normal in lP,.

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To get the other example we take in (c2 the fibration by curves {t(x - tx’)’ = (x + t~‘)~} with t E lP,. Now the generic fibres are Neil parabolas in 0. The fibers cross in 0 and in other points of arbitrary small neighborhoods of 0. But there is a two sheeted analytic covering X over (lZ2 with one point 0” over 0 and a fibration with Neil parabolas in 0” which all pass through 0” and have no other point in common, so that the fibration of (c2 is just the image. Every fiber in X is mapped biholomorphically onto a fiber in (c’. By passing to X- the point 0” E X is blown up just simply by blowing up 0 E (lZ2. This transforms the Neil parabolas of X into smooth curves, i.e. their normalizations. So rc- is the map from the normalization which is not biholomorphic. See for more details [Sb92].

tj 9. Meromorphic

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In 3 9 to # 11 we shall consider some examples in which the main theorems for analytic and meromorphic equivalence relations can be applied. 1. Proper Equivalence Relations. We call an analytic equivalence relation R in a complex space X proper if all fibers to R are compact and around every fibre S c X there is a relatively compact open neighborhood U such that every fiber with some points very near to S is completely contained in U. We prove in this case first that R is semiproper. We denote by M the closure of the set theoretic union M’ of all fibers passing through points of X - U. Then V = X - M is an open neighborhood of S which is contained in U. Every fiber through a point of V is contained in U. If such a fibre had a point x in common with M, then there would be fibers in M’ passing through points arbitrary close to x. Then these would be contained in U, which is a contradiction to the construction of M’. So I/ is a relatively compact open neighborhood of S which is fiber saturated. Hence, W = q(V) is an open neighborhood of the point q(S) and the compact closure of V is mapped onto W. So R is semiproper and Q is a Hausdorff space. Moreover, the map q is proper over Wand therefore proper everywhere. This also implies that the projections pP: R + X are proper. If on the other hand one pp is proper, then trivially also R is a proper analytic equivalence relation. The same is true if Q is a Hausdorff space and q is a proper map. Clearly, an analytic equivalence relation is proper if all the fibers are compact and connected. Let us assume now that X is normal and that R is a simple proper equivalence relation in X. We may assume that X is connected. In this case R is irreducible and contains the diagonal D of X x X. We denote by R” the set of those smooth points (x’, x) of R lying by p,, over smooth points of X and such that moreover the maps pr are regular in (x’, x), i.e. have Jabobi rank equal to the dimension of X. Then R” is a connected complex manifold. So also the set of

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connected components of the fibres in R” is connected. There are some tibres in R” which meet the diagonal D. This shows by the identity theorem that every connected component of any general libre of R” meets the diagonal of X x X. Hence, there is only one connected component and since every point on an arbitrary fibre in R is a limit point of generic libres and X is locally irreducible, it follows that all the fibres in R and in X are connected. If, more generally, R is a proper analytic equivalence relation in our (normal) complex space X, then also the attached simple equivalence relation R” is proper. Locally the fibres to R and R” are the same. So the fibre to R” through a point x E X consists of the connected component of the fibres to R which contains x. 2. The Notation of a Simple Meromorphic Equivalence Relation. Assume that X is a normal (connected) complex space and that R is a regular meromorphic equivalence relation in X. Then there is a polar set: a nowhere dense analytic set P c X such that R/X - P is a normal equivalence relation of codimension c. We take the simple analytic equivalence relation R” c (X - P) x (X - P) to RIX - P. This again is a normal equivalence relation in X - P. The analytic set R” can be extended analytically by the theorem of [RS53] to the full spaceX x X. So we obtain a regular liner meromorphic equivalence relation in X which we call the simple meromorphic equivalence relation R^ to R. Its delinition is independent of the choiceof the polar set P. 3. Meromorphic Dependence. Following Remmert we define meromorphic maps F: X -+ Y of a reduced complex spacesX into a (reduced) complex space Y. Such a meromorphic map is nothing else but an analytic subset A c X x Y which is mapped properly onto X by the projection p: X x Y -+ X of X x Y onto the first component such that outside a nowhere dense analytic set P c X this map is biholomorphic. Moreover p-‘(P) has to be nowhere densein A. The set A is uniquely determined and is also called the graph of the meromorphic map F. If M c X is a subset and q: X x Y --) Y denotes the product projection the set q(p-l(M) n A) is called the image of M. Since a holomorphic map always has a graph A, it also is a meromorphic map. On the other hand, off P every meromorphic map is a holomorphic map. We can form the intersection of all possible P. Therefore we may and shall assumethat P is minimal. It can be proved that this property is of local nature. Hence, this P is uniquely determined and is called the polar set of F. If H: Y + Z is another meromorphic map with polar set P’ c Y such that F-‘(P’) = p(q-‘(P’) n A) is nowhere densein X, then the composition H o F is well defined and is a meromorphic map X + Z. If R is a meromorphic equivalence relation in the connected normal complex spaceX, and F: X + Y is a meromorphic map, then F is called meromorphically dependenton R if F is constant on the generic libres to the corresponding simple meromorphic equivalence relation R”.

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On the other hand every meromorphic map F: X + Y defines a meromorphic equivalence relation R = R, in X. We take the graph A of F and the

analytic equivalencerelation R’ c A x A to the holomorphic map A 4 Y. The graph R is then simply the meromorphic equivalence relation to the analytic image set of R’ under the proper holomorphic projection p x p: A x A + X x X. The libres over X - P are the libres of FIX - P. Since we can pass to intersections of meromorphic equivalence relations, F is meromorphically dependent on a meromorphic equivalence relation R if and only if F is locally constant on the general libres to R. Another meromorphic map H: X + Z is called meromorphically dependent on F if and only if H is meromorphically dependent on R,. This is the case if and only if in general points of X the differential dH is spanned by the differential dF. 4. Meromorphic Bases(m-Bases). Assume that R is a meromorphic equivalence relation in a connected normal complex space X. We assume moreover that R is regular. Then also the corresponding simple meromorphic equivalence relation R” is regular. By the main theorem we have the proper modification X’, the meromorphic quotient space Q = X/R” and the holomorphic map q: X’ -+ Q, where every libre has the same dimension. Take another reduced complex space Y and a meromorphic map F: X + Y which is meromorphically dependent on R. Then F is locally constant in general points on general fibres to RA. Le us lift F to a meromorphic map Fe: X- + Y and take X- to be normal (by loosing some superfluous structure). Then F- is constant on the generic fibers of q. Also the graph A’ of F- in X’ x Y is constant on generic fibers. Since every fiber of q is the limit of general fibers, this is true for every fiber of q. This means that A- is the lifting of a graph A’ c Q x Y The graph A’ gives a meromorphic map F’: Q + Y with F- = F’ 0 q. So we have a Stein factorization of F. It can be seenrather easily that in our case Q is uniquely determined up to bimeromorphic equivalence by the property that there is a meromorphic map q: X + Q and that for Q a Stein factorization exists for any meromorphic map X -+ Y. We call q: X + Q an m-baseto R. It is determined up to bimeromorphic equivalence.

$10. Non Regularity 1. A Simple Non Regular Algebraic Relation. We construct a simple example of an algebraic meromorphic equivalence relation which is not regular. Let us denote by C3 the 3-dimensional complex number space with variables w, z, t. We have in (c3 a meromorphic equivalence relation with l-dimensional fibres whose general fiber is given by the equations (aw = l/t, bz = l/t: t E Cc*} c (c3. Here a, b are complex numbers both #O. For a + co or b + cc limit fibers exist. We obtain a good libration coming fom a meromorphic equivalence relation. Only the set L = {w = z = 0} is attached with all complex lines running

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through 0 in the plane E = {t = 0} and therefore with a bunch of limit libres. This means that we do not have an analytic equivalence relation in (lZ3. In order to get X’, the set L will be blown up. Hence Q is obtained from c2 by blowing up the point 0. Of course, this meromorphic equivalence relation still was regular. In order to get a non regular relation our construction has to depend on another parameter s E UZ. We take the complex space a4 in the variables w, z, t, s. For fixed s we perform always the translation w” = w, z” = z, t” = t + l/s and apply after it our construction. We get a meromorphic equivalence relation in (lZ4 again with l-dimensional fibers. Ifs # 0, then there is always a blowing up by the lines in E = E, = {t” = 0, s = const.) through the O-point. For s = 0 our (limit) fibers in (E3 are parallel lines to the t-axis. Hence, no blowing up will occur and the set E, is empty. By passing to the quotient for s # 0 the O-point of (lZ* has to be blown up, for s = 0 nothing has to happen. So the quotient does not make sense as complex space. This is why our meromorphic equivalence relation is not regular. Of course, we obtain a good quotient Q when we pass to the closure lP, of a?. But then the points of Q do not correspond to libres in (lZ3, any longer. The quotient is obtained by taking the field of fibre constant functions in (c4, not by a geometric construction out of c4. The definition is ring theoretic, i.e. categorical. We work geometrically in this chapter of the book. A non geometric definition is something which has to be avoided. Clearly we have: Proposition 10.1. Assume that X is a connected normal complex space which is compact. Then every meromorphic equivalence relation in X is regular. 2. A Non Regular Relation which cannot be Enlarged to a Regular One. We construct a non regular meromorphic equivalence relation in a 3-dimensional complex space X such that X cannot be enlarged so that it becomes regular. We follow [Gr87, p. 1771. Assume that X, = (c3 and that n,: (z”, z’, z) -+ (z’, z) is the holomorphic projection X, -+ (c2. We obtain in X, a smooth analytic equivalence relation R, which fibers X, into l-dimensional complex lines. We apply in 0; = 0 E X, the ordinary monoidal transform (the Hopf a-procedure). We obtain a complex manifold X, and by lifting of z0 a holomorphic projection 7~~:X, -+ (c*. This maps the 2-dimensional complex projective space lP”l, which was implanted in 04, into O,, = 0 E c2. The l-dimensional analytic fibration in X, - lP”’ belonging to x1 extends to a l-dimensional meromorphic equivalence relation R, in Xi. This fibers IpA’ into a bunch of lines all passing through a fixed point TI E IF”“. This TI is just the intersection of lP”’ with the closure A of the set (Xi - lP”l) n {z’ o 7~~= z o z1 = O}. Outside of TI the fibration given by R, is smooth and analytic. The main theorem for meromorphic equivalence implies: Proposition 10.2. The proper modification X; of X, is obtained by the ordinary monoidal transform along A and the quotient space Q1 = X,/R, is the omodification of (c* in 0,.

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The quotient space Q. is just C2 itself. We repeat our construction in the point 0; which in Xi is the intersection of lP^’ with the closure of the set (Xi - Iphi) n {z^ 0 7ci = z’ 0 rri = O}. We obtain a o-modification X2 of Xi in 0; and a holomorphic map rc2: X2 + C2, which gives a meromorphic equivalence relation R, in X2. The point 0: is in lF’“’ (T,} c Xi and by the holomorphic map X; + Qi over a point 0, E lP’ c Qi, where lP’ denotes the l-dimensional complex projective space which was implanted in 0, E Q. = C2. The quotient Q2 = X,/R, is obtained from Qi by a a-procedure in 0,. We now apply our construction to the 2-dimensional complex projective space lP; which was implanted in 0; during the construction of X2. We then obtain the point 0; E lP”’ outside the strict transform of lP”’ and the new bunch point T, E lPA2. The situation in X2 is near to 0; the same as in Xi near to 0;. Similarly, the image point 0, E lP2 c Qz is outside the strict transform of IP’. We repeat and go on so. We obtain an infinite sequence consisting of 3dimensional complex manifolds X,, which o-modifications of X,-i in points Ok-, E lPAp-l c xpml, meromorphic equivalence relations R, in X, and the 2dimensional quotient manifolds Q, = X,/R,. We have the image points 0, E Q, of 0; where Q, is a-modification of Q,-i in O,-,. The meromorphic equivalence relations R, come from holomorphic maps rcP:X, + C2. We have a bunch point T, E IP”’ 1 X,. We use the notation T, also for the image of T, in X, for L 2 ~1.The libration R, is smooth outside the union of the T, with 3cI p. The bunch points T, are always in the strict transform of IpAP-‘, while the point 0: is always outside of this set (for p 2 2). The last statement is also true for the 0, E Q,. The domain X,-, - (O,^_,} can be considered as an open subset of X,. We define X as the union of X, - (0; } and obtain a 3-dimensional connected complex manifold and a holomorphic map rc: X --* C2. This X is a (non proper) modification of all X,. The modification maps commute with n and the rep. From rt we get a meromorphic equivalence relation R on X. The modification X + X, = C3 takes place in 0 E C3 only. The inverse image is the union of the PAp - {Ok} for p = 1, 2, 3, . . . . Over 0 E Cz the relation R defines a uniquely determined libre F of dimension 1 in the following way: The modification image of F in P”” is the line which connects T, with Ok. In X, this is E while F is nothing else but the minimal set in X with these properties. If now Q = X/R were a complex space,then F would be a point y E Q. There is always a natural holomorphic map Q -+ Q,. We pass to a desingularization (in the sense of Hironaka) to obtain a complex manifold Q” and a natural holomorphic map r” : Q” + C2 which is composition of Q” -+ Q with z: Q -+ C2. This map r factorizes over the sequence of a-modifications ... -+ Q, + Qp-1 --t.. . -+ Q. = (c’. Assume that y” E Q” is a point over y E Q. This point always is mapped into the point 0, E Q,. Since the functional determinant of Q, + Qpml is 0 in 0, and the functional determinant of composed maps is a product of determinants, the functional determinant in y” has to vanish of

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infinite order. Since also Q” as well as X is connected the functional determinant of zA has to vanish identically and so z” has to be constant. This is a contradiction. Hence our assumption is false and X/R is not a complex space. If we could enlarge X to a normal complex space X’ such that the meromorphic equivalence relation could be extended to X’, then the libre F would be part of a l-dimensional libre F’. So the point y” would still be there, and we could repeat the conclusion. Therefore even this is not possible, as it is always in the algebraic case. Clearly, our meromorphic equivalence relation R in our complex manifold X is not regular. The points F n (X, - 0:) have to be blown up infinitely many times.

3. Reduction to a Moishezon Space. We define the Moishezon reduction. Assume that X is a compact connected normal complex space of dimension n. We denote by M = M(X) the field of meromorphic functions on X. By definition a meromorphic function f is meromorphically dependent on meromorphic functions fi, . . , f, if the meromorphic map f: X -+ IP, = (I2 u (co} is meromorphitally dependent on the meromorphic map F: X --+ IPc which is defined by the functions fi, . . . , f,. In generic points x E X the functions f, fi, . . . , f, are holoindependent if morphic. It is easy to see that f, fr, . . ., f, are meromorphically and only if there is such a point x such that the Jacobian off, fi, . . . , f, has rank c + 1. Hence, there are at most n meromorphically independent meromorphic functions on X. Assume that c < n is the maximal number. Let fi, . . . , f, be meromorphically independent. These functions are holomorphic nearly everywhere and define a normal analytic equivalence relation outside a nowhere dense analytic set P c X. The equivalence relation extends from (X - P) x (X - P) to a meromorphic equivalence relation R in X. If we take another meromorphic function f E M, then f is locally constant on the libres of the libration 4 belonging to R. We pass to the simple meromorphic equivalence relation R” corresponding to R. Then f is constant on the libres belonging to R”. From this it follows: R A is independent of the choice of fi, . . . , f,. Since X is compact the meromorphic equivalence relation R” is regular. So the quotient space Q of X exists and it has dimension c. If f is in M(X), then f can be considered as a meromorphic function on Q. So we have c = dim Q meromorphically independent meromorphic functions on Q. The field M(X) = M(Q) is an algebraic function field and c is the degree of transcendency tr(Q). A compact normal complex space Q with tr(Q) = dim Q is called a Moishezon space. It has an etale algebebraic structure in the sense of M. Artin (see [Ar70]). We have the semi normal complex space X-, which is a proper modification of X, and the quotient map 9: X- -+ Q. These objects are uniquely determined by the function field M(X). They are biregular inuariants of X. We call Q the Moishezon reduction of X. We have:

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Theorem 10.3. Every connected compact normal complex space has a well defined Moishezon reduction which is a compact normal algebraic space in the sense of M. Artin.

It can be seen rather easily that the Moishezon reduction of a compact complex torus is always an abelian variety. The Moishezon reduction of a Hopf manifold is always a complex projective space. We also obtain: Theorem 10.4. Assume that X is a compact normal algebraic space (in the sense of M. Artin) and that R is a meromorphic equivalence relation in X. Then Q = X/R is algebraic again.

9 11. Applications 1. Complex Lie Groups. We consider the quotients of complex spaces by complex Lie groups. Assume that X is a normal complex space such that every connected component of X is n-dimensional, and let L be a complex Lie group acting holomorphically on X. We assume that there is no smaller union of connected components of X on which L acts. We denote the dimension of the generic orbit by d and put c = n - d. We define R, as the graph of the orbits: R, = {(x’, x) E X x Xix’ E L 0 x}.

In general, the closure R of R, is not an analytic set in X x X. But it contains the diagonal and is invariant under reflexion X x X N X x X. We can prove (see [G&5], $4): Theorem 11.1. If R is an analytic set of dimension n + d then R is an meromorphic equivalence relation in X.

If X is an algebraic space and L is an algebraic group acting algebraically on X, the assumption of the theorem is always satisfied. However, also in this case the meromorphic equivalence relation R may not be regular. In the case of Subsection 1 of 4 10 the fibration of C3 and hence that of C4 may be obtained by an action of the algebraic Lie group C*. Now we define: Definition

dimension

11.2. L acts analytically n + d and the meromorphic

closed on X if R is an analytic set of equivalence relation R is regular.

In this case we have a normal complex space Q which can be considered as the (geometric) quotient X/L. The meromorphic functions on Q are just the L-invariant meromorphic functions on X. Of course, if X is compact, then a meromorphic equivalence relation R is always regular. 2. One Dimensional Jets. We define the space of jets of order m = 0, 1,2, . . . in the origin 0 E C”. We denote by 0, the O-dimensional complex subspace of

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the complex plane UZwith the complex variable t, consisting of the point 0 and the Artin ring A,,, = Co/t”‘+’ . &! Here 0 is the ring of convergent power series in 0. The holomorphic functions on 0, can be given uniquely determined in the form z = a, + a,t + ... + u,t”’ with a,, E (IZ. Hence, 0, also is called an inlinitesimal neighborhood of 0 of oder m. The infinitesimal neighborhood of order 0 is just the point 0. On 0, there operates the complex Lie group L of automorphisms z= a,t + ... + a,tm with a, # 0. The group L has a normal subgroup L’ = (z = t + a,P + ... + amtm}. The quotient group L/L’ is the multiplicative group (c*. By a p-e-jet we understand a holomorphic map F: 0, + (I?’ with F(0) = 0 E cc”. If j z Pm is the space of these pre-jets, then L operates on j via transformation of the variable t. The space of projective jets in 0 will be just the quotient space PJ := j/L. To prove its existence we have to study the relation R corresponding to L. The group L is algebraic and operates algebraically on j. So we have just to prove that it is regular. We have to study its fibers. The dimension d of generic fibre is equal to m. A pre-jet II/ always is given in the unique form: z1 =allt+~~~+a,,tm z, = anIt + ... + anmtm

We define the order o = o(ll/) as the minimal number 3c- 1 with uxP # 0 for some ,D.If $ = 0 then we put o = m. So x is a number between 0 and m. We take for R the closure of the space of orbits. It follows that the libre with respect to R through + has dimension m + (n - 1). o($). Clearly, R is invariant for multiplication t aa. t with a E (c*. We put a 0 II/ = Il/(at) and [$I = max((a,,(). If $ # 0 we can multiply so that 111/lbecomes 1. The set of these II/ is a compact space K. To prove regularity we can choose for the relative compact open subset B (in $6.3) any open neighborhood of K in j* = j - {0}, i.e. one which does not contain $ = 0. So R* = RI j* is regular. We call the quotient space Q = j*/R* the projective jet space PJ. A pre-jet $ is called regular if o($) = 0. There is a natural map of the quotient manifold by L’ from the manifold of regular prejets into PJ. This extends to a meromorphic (or even holomorphic) map of j*/L’ onto PJ. The graph is a proper modification of j*/L’. We add a O-element 0 to the graph over the origin 0 E j/L’ and obtain the complex space J = J,,, of the (so called non parametrized) jets of order m in 0 E (I?‘. The space J is normal, as it can be seen. The multiplication II/ + a. $ with a E (c, II/ E j gives a multiplication in J with obvious properties. So the meromorphic equivalence relations are needed to define the notion of good jets. The pre-jets are not good since their structure is too weak if they are not regular. In the case m = 1 we obtain just the ordinary tangent vectors.

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3. The Non Hausdorff Case. We take just two copies (c, and a:2 of the complex plane (lZ. We have the holomorphic identity map r: (IZr u (G, + c:. We identify now over the open unit disc d c Cc.The equivalence relation R in (c, x (lZ2 is not a closed analytic set. So R is not an analytic equivalence relation. The quotient space Q exists with a good complex structure in a neighborhood of every point, but it is not a Hausdorff space.

4. Cases where X is Not Normal. There are some cases of semi proper and holomorphic analytic equivalence relations where the quotient space of a complex space X exists as a complex space even if X is not normal. In these cases the assumption “spreadable” is replaced by a much stronger one: Locally two different fibres can always be separated by fibre constant holomorphic functions. One case is that of Remmert reduction of any holomorphically convex space.

For the proof the direct image theorem is used here (see chapter V.l). The structure sheaf of the space X may also contain nilpotent elements. Assume now that X is an arbitrary complex space and that L is a discrete group acting properly discontinuously on X. That means, the group L consists of biholomorphic transformations X = X such that for every open B cc X every point x E X has an open neighborhood U with I(U) n B # 0 for finitely many 1E L only. The set R is the union of the graphs ((x’, x)1x’ = I(x)) c X x X. In this case R will in general have nilpotent elements in its structure rings. Since L operates properly discontinuously this union is locally finite. Hence, R is a complex subspace. It satisfies the axioms of an analytic equivalence relation. It is also semi proper and is spreadable in a stronger sense. It follows immediately that the quotient space X/R = X/L is a complex space. Take now an arbitrary complex space Y Then Y has a covering with open subsets X,, for p = 1, 2, 3, . . . such that every X, is a complex subspace of a domain of a complex number space. We denote by X the disjoint union of the sets X,,, which is again a complex space. But the identification of X, and X, over the intersections X, n X, gives an analytic equivalence relation R (with nilpotent elements) in X, which is semi proper and spreadable in a stronger sense. We obtain Y for quotient space. Historical Note. In Chapter IV we have considered quotient spaces Q of complex spaces X which are produced by an analytic libration in X which we call an analytic equivalence relation. In general, X should contain nilpotent elements in its structure rings. This should be admitted also for the graph of the equivalence relation R c X x X. Experience shows that then we can only derive results by the direct image theorem on coherent analytic sheaves. That implies that we have to suppose that R is proper. An application is the Remmert reduction theorem which is considered in chapter V. But in the chapter IV we only are interested in the set theoretic case. This means we shall assume that X and R are reduced complex spaces. We can then replace the assumption “proper” by the weaker notion “semi proper”. However, in general, even if we require, as it is trivially necessary, that there are locally

IV.

Seminormal

Complex

Spaces

219

along the libres many libre constant holomorphic functions in X, a complex quotient will not exist. We can develop the theory only if X is a normal or a semi normal complex space. The direct image theorem is then replaced by the local existence of holomorphic functions f on any analytically branched covering such that the functions f give the prescribed branching. This basic theorem was first proved in [GRS]. Here, a new proof is given using methods of real analysis. The whole theory of set theoretic analytic equivalence relations was opened up by K. Stein already in the early fiftieth. Other authors who went into this area, lateron, are K. Koch, H. Holmann, B. Kaup, Wiegmann and some further people. It is also possible to treat the more general meromorphic equivalence relations. For this it is no longer necessary to require the local existence of libre constant holomorphic functions. The quotient Q exists under a very weak condition, namely the assumption “regular”. This result has therefore very many applications. It finishes in the complex analytic case an ad hoc theory given by Mumford (see [Mu65], where algebraic methods are employed; see also [Ne78], which is a survey article and is more geometric). For the proofs one needs the first considered theory of analytic equivalence relations.

References* [Ar70] [BR90] [BSSl] [Ca55]

[De901 [Gr83] [Gr86]

[Gr87] [GR58] [GR58a]

Artin, M.: Algebraization of formal moduli II. Existence of modifications. Ann. Math., II, Ser. 91, 88-135 (1970) Zb1.185,247 Bonhorst, G.; Reiffen, H.J.: Uber offene analytische Aquivalenzrelationen auf komplexen Rlumen. Osnabriick 1990. Behnke, H.; Stein, K.: Moditikation komplexer Mannigfaltigkeiten und Riemannscher Gebiete. Math. Ann. 124, 1-16 (1951) Zb1.43,303. Cartan, H.: Quotient d’un espace analytique par un groupe d’automorphismes. Algebraic geometry and topology. Princeton Univ. Press, Math. Ser. 12, 90-102 (1955) Zbl.84,72. Dethloff, G.: A new proof of a theorem of Grauert and Remmert by Lz-methodes. Math. Ann. 286, 129-142 (1990) Zb1.681.32007. Grauert, H.: Set theoretic complex equivalence relations. Math. Ann. 265, 1377148 (1983) Zbl.504.32007. Grauert, H.: On meromorphic equivalence relations. In: Contributions to several complex variables Hon. Stall, Proc. Conf. Notre Dame/Indiana 1984, Aspects Math. E9, 115- 147 (1986) Zbl.592.32008. Grauert, H.: Meromorphe Aquivalenzrelationen. Anwendungen, Beispiele, Erganzungen. Math. Ann. 278, 175-183 (1987) Zbl.651.32008. Grauert, H.; Remmert, R.: Komplexe Raume. Math. Ann. 136, 245-318 (1958) Zbl.87,290. Grauert, H.; Remmert, R.: Bilder und Urbilder analytischer Garben. Ann. Math., II, Ser. 68, 393-443 (1958) Zb1.89,60.

* For the convenience of the reader, compiled using the MATH database,

references to reviews in Zentralblatt have, as far as possible, been included

fur Mathematik (Zbl.), in this bibliography.

220 [GR84] [Hk74] [Ho651 [Ho733 [Ka67] [Ka69] [Ka75] [Ka93] [Ku641 [KK83] [Mu651 [Ne78] INS773 [RS53] [Sb93] [Si69] [Si74] [St531 [St561 [St63 +64]

G. Dethloff

and H. Grauert

Grauert, H.; Remmert, R.: Coherent Analytic Sheaves. Springer 1984,Zbl.537.32001. Hironaka, H.; Lejeune-Jalabert, M.; Teissier, B.: Platiticateur lolocal en geometrie analytique et aplatissement local. Asterisque 8,441-463 (1974) Zb3.287.14007. Hormander, L.: Lz-estimates and existence theorems for the a-operator. Acta Math. 113, 89-152 (1965) Zbl.l58,110. Hormander, L.: An Introduction to Complex Analysis in Several Variables. NorthHolland 1973, Zbl.271.32001; Zb1.138,62. Kaup, B.: Aquivalenzrelationen auf allgemeinen komplexen RIumen. Diss. Fribourg 1967, Schiftenr. Math. Inst. Miinster 39 (1968) Zb1.182,416. Kaup, B.: iiber offene analytische Aquivalenzrelationen. Math. Ann. 183,6-16 (1969) Zb1.172,105. Kaup, B.: Zur Konstruktion komplexer Basen. Manuscr. Math. 15, 385-408 (1975) Zbl.341.32017. Kaup, B.: Grauerts Satz iiber Quotienten semi-normaler komplexer Rime nach semieigentlichen Aquivalenz-Relationen. Math. Gottingensis 7/93. Kuhlmann, N.: ijber holomorphe Abbildungen komplexer Rlume. Arch. Math. 15, 81-90 (1964) Zbl.122,87. Kaup, L.; Kaup, B.: Holomorphic Functions of Several Variables. Gruyter 1983, Zbi.528.32001. Mumford, D.: Geometric Invariant Theory. Erg. Math. 34, Springer 1965,Zbl.147,393. Newstead, P.E.: Lectures on Introduction to Moduli Problems and Orbit Spaces. Tata Inst. Bombay, Springer 1978,Zb1.411.14003. Norguet, F.; Siu, Y.T.: Holomorphic convexity of spaces of analytic cycles. Bull. Sot. Math. Fr. 105, 191-223 (1977) Zb1.382.32010. Remmert, R.; Stein, K.: ijber die wesentlichen Singularitaten analytischer Mengen. Math. Ann. 126,263-306 (1953) Zbl.51,63. Siebert, B.: Fibre cycles of holomorphic maps. I. Local flattening. Math. Annalen 1993. II. Fibre cycle space and canonical flattening. To appear. Siu, Y.T.: Extension of locally free analytic sheaves. Math. Ann. 179, 285-294 (1969) Zb1.165,99. Siu, Y.T.: Techniques of extension of analytic objects. Lect. Notes Pure Appl. Math. 8, M. Dekker (1974) Zbl.294.32007. Stein, K.: Analytische Projektion komplexer Mannigfaltigkeiten. Centre Belg. Rech. Math., Colloque Bruxelles 1953,97-107,Zb1.52,86. Stein, K.: Analytische Zerlegungen komplexer Raume. Math. Ann. 132, 63-93 (1956) Zbl.74,63. Stein, K.: Maximale holomorphe und meromorphe Abbildungen, I, II. Am. J. Math. S&298-315 und 86,823-868 (1963, 1964) Zbl. 144,339.

Chapter V

Pseudoconvexity, the Levi Problem and Vanishing Theorems Th. Peternell

Contents Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

.............. Q1. Plurisubharmonic Functions and Pseudoconvexity ................. 1. The Notion of Plurisubharmonic Functions .................. 2. Properties of Plurisubharmonic Functions 3. Pseudoconvex Domains .................................. Q2. l-convex Spaces . . . . . . . . . . . . . . . . . . . . . . 1. Remmert Reduction . .. . .. .. . .. . . . .. . . 2. The Levi Problem for l-convex Spaces . . . 3. Maximal Compact Analytic Sets . . . . . . . . 4. Positive Sheaves and the Normal Bundle . 5. The Cohomology of l-convex Spaces . . . .

.. .. .. .. ..

......................... $3. The Levi Problem ............... 1. The Classical Levi Problem 2. Counterexamples ........................ 3. Characterizing Stein Spaces ............... 4. The Local Stein Problem ................. $4. Positive Sheaves and Vanishing Theorems . . . . . . 1. The Projective Bundle . . . . . . . . . . . . . . . . . . . . 2. The Vanishing Theorem for Positive Sheaves . 3. The Embedding Theorem . . . . . . . . . . . . . . . . . . 4. Characterization of Positivity by Cohomology Vanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Functorial Properties of Positive Sheaves . . . . 6. Differential-Geometric Positivity Notions . . .. 7. Hodge Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Relative Positive Sheaves . . . . . . . . . . . . . . . . . .

.. .. .. .. ..

. . . . .

. . . . .

224 224 226 228 . .. . .. . .. .

.

236 236 237 238 240

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

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

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

228 229 229 231 232 235

. ..

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

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

241 242 242 243 244 245 246 249 249

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. .. . . . .................. 9 5. More Vanishing Theorems . .. . ............ 1. Demailly’s Vanishing Theorem . . . . . . . . .. ............... 2. The Notion of k-ampleness and Direct Images 3. Grauert-Riemenschneider Vanishing Theorem . . of Dualizing Sheaves ..................... References

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

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Introduction Convexity was introduced in complex analysis by E.E. Levi around 1910 when he discovered that the (smooth) boundary of a domain of holomorphy in (c” is not arbitrary but satisfies a certain condition of pseudoconvexity. The question whether conversely such a pseudoconvex domain is a domain of holomorphy became famous as the so-called “Levi problem” and influenced the development of complex analysis over several decades. The Levi problem was first solved in two variables by Oka (1942) and then in general by Oka, Norguet and Bremermann in the early fifties. A main tool for its solution was provided by the notion of plurisubharmonic (sometimes called pseudoconvex) functions. This generalizes the notion of subharmonic functions in one variable. Plurisubharmonic functions are today indispensable for complex analysts. After the solution of the classical Levi problem for pseudoconvex domains in (c”, the problem was modified: one asked for holomorphic convexity of strongly pseudoconvex domains in arbitrary complex spaces. Grauert provided in 1958 a solution in the smooth case, while the general solution was given by Narasimhan. Going further, one now asked for “weakly” pseudoconvex domains. Grauert showed by an example that in this context the Levi problem has a negative answer. This of course raises the question whether there are special classes of spaces for which the “pseudoconvex Levi problem” has a positive solution, or whether additional assumptions on the boundary will help. We will deal with these questions in 5 3. The most interesting problem which grew out of these questions is certainly the local Stein problem (6 3.4), which is still unsolved. In summary, one can say that the Levi problem asks whether certain geometric conditions force a complex space to be Stein. The Levi problem for strongly pseudoconvex domains has far-reaching consequences in the global theory of complex spaces. In Q2 we will treat “l-convex” spaces, which spaces by definition can be exhausted by strongly pseudoconvex domains. Using the holomorphic convexity of these domains and the Remmert reduction, Grauert showed that l-convex spaces are point modifications of Stein spaces: they can be constructed from Stein spaces by substituting some points by compact spaces. Moreover, l-convex spaces can be characterized by cohomology (generalizing Theorem B for Stein spaces). The compact subspace of a l-convex space which is contracted by the Remmert reduction is called “exceptional”. The Remmert reduction of a l-convex space is a so-called modilication; the theory of modifications in general will be treated in Chap. VII. A natural problem arising is how to characterize exceptional analytic sets or subspaces A. This can be done via the conormal sheaf of A: if, say, A and the ambient space X are smooth and A is exceptional in its normal bundle, then Grauert proved that A is exceptional in X, too. Vector bundles on compact spaces whose zero sections are exceptional are called negative and their duals, consequently, positive. They will be studied in $4. It turns out that positive vector bundles (especially of rank 1) and, more

224

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generally, positive sheaves greatly influence the geometry of X. In particular, they force X to be “algebraic”. We should note that the general theory for q-convex spaces (q 2 1) is the content of Chap. VI, but since the theory for q 2 2 differs significantly from the l-convex case, it is certainly justified to treat the case q = 1 separately. Vanishing theorems and pseudoconvexity (in the total space of vector bundles) are closely related. This will be clear in 0 4, where we sketch a proof of the “coarse Kodaira vanishing theorem” using pseudoconvexity. But it lies in the nature of this approach that precise vanishing theorems cannot be obtained. These require differential geometric methods. In $4 we therefore also discuss various differential geometric notions of positivity of bundles, and their basic vanishing theorems. New developments are finally given in 0 5, such as GrauertRiemenschneider type theorems, k-ampleness and vanishing theorems for tensor powers of positive vector bundles.

5 1. Plurisubharmonic

Functions

and Pseudoconvexity

Plurisubharmonic functions constitute a very important and indispensable tool for the investigation of complex spaces. They were introduced by Oka [Oka42] and Lelong [Le145]. A systematic treatment is given in Richberg [Ric68] and in Lelong’s book [Le168]. 1. The Notion of Plurisuhharmonic

Functions

Definition 1.1. Let G c Cc” be a domain. A function cp: G + [-co, co) is called plurisubharmonic if (1) cp is upper semicontinuous and cp # -co, (2) for every z0 E G and a E Cc”, a # 0, and for every map r: (IZ + (c”, r(z) = z0 + az, the function cp 0 r is on every connected component of t-i(G) (which are domains in (c) either -cc or subharmonic. Subharmonic functions are defined as follows. Given a domain G c lR” and an upper semi-continuous function cp: G -+ [-co, co), cp is called subharmonic if for x,, E G and all r > 0 with B(x,, r) = {xl Ilx - x011 I r} c G:

&)dx, s-Bt-%.r) where a is the euclidian volume of B(x,, r) and integration is taken with respect to the Lebesgue measure. Let f: G + IR be a C*-function. The Levi form off in z E G is given by the hermitian form

V. Pseudoconvexity,

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225

Theorems

Now we are going to define the Levi form of upper semi-continuous functions, which will be a hermitian form of distributions. Namely, given z E G, a Cmfunction g with compact support in G and a = (ai, . . , a,) E (I?‘, we set

where integration The following Lelong [Le168]:

is - of course - taken with respect to the Lebesgue measure A,,. characterization of plurisubharmonic functions is due to

Theorem 1.2. An upper semi-continuous function cp: G -+ [-co, subharmonic if and only its Levi form is positive semi-definite, i.e.

CUdlkd(4

CO) is pluri-

20

for every P-function g with compact support and every a E 6Y. In particular, q is C”, then cp is plurisubhurmonic if for every point z E G:

if

UcpW(4= i,j=l i &(zJuizj 2 O. We now define strictly

plurisubharmonic

functions.

Definition 1.3. A strictly (or strongly) plurisubhurmonic function is an upper semi-continuous function cp: G + [-co, co) such that for any P-function f: G -+ lR with compact support there is some a > 0 such that for all t E lR with ItI I a the function q + tf is again plurisubharmonic. In other words, a plurisubharmonic function is strictly plurisubharmonic if sufficiently small perturbations are still plurisubharmonic. It is easy to verify that a C2-function cp is strictly plurisubharmonic if the Levi form is positive definite at every point: 1 &uiZj

> 0

for all a E Cc”\(O).

J

So far we have defined plurisubharmonicity only for functions on domains in (c”. We note that the condition of (semi-)positivity of the Levi form is invariant under biholomorphic maps. Therefore we can define plurisubharmonic function on arbitrary complex manifolds. Concerning arbitrary complex spaces we make the Definition 1.4. Let X be a complex space. A (strictly) plurisubharmonic function on X is a function cp: X + [-co, co) having the following property. For every x E X there is an open neighborhood U with a biholomorphic map h: U -+ V onto a closed complex subspace I/ of some domain G c 47” and a (strictly) plurisubharmonic function 4: G + [-co, co) such that

226

Th.

Peternell

If 4 can be chosen continuous (or differentiable), (differentiable) strictly plurisubharmonic function.

then cp is called a continuous

Some remarks should be made at this point. First, if X is smooth, we get back the previous notion. Second, the definition of plurisubharmonicity does not depend on the choice of local charts (see [Nar62] for a proof). Third, it is obvious that a continuous plurisubharmonic function is continuous but it is not clear a priori that conversely a continuous function which (strictly) plurisubharmonic is continuous (strictly) plurisubharmonic. This is a theorem by Richberg [Ric68]. If f: X + Y is a holomorphic map of complex space, then it is clear that cp 0 f is plurisubharmonic for any plurisubharmonic function cp on Y. In particular, this is true for all maps f: A + Y, where A c CGis the unit disc. Conversely, Fornaess and Narasimhan proved in [FoNs80]: Theorem 1.5. Let X be a complex space, and let cp: X + [ -00, 00) be an upper semi-continuous function. Assume that cp 0 f is (pluri)subharmonic for all holomorphic maps f: A + X (or --CO). Then cp is plurisubharmonic. In other words, our definition given in [GrRe56].

of plurisubharmonicity

is the same as that one

Example 1.6. (1) The most basic example of a strictly plurisubharmonic function on Cc” is given by z + 1z12= c ziZj. (2) If f is holomorphic (not identically 0 on connected components) on a complex space X, then log1 f 1is plurisubharmonic. (3) loglz12 is plurisubharmonic on Cc”. (4) If G c (c” is a pseudoconvex (see below) relatively compact domain whose distance function to the boundary is denoted d, then -log d is plurisubharmonic. We close by stating functions:

some trivial

properties

of (strictly)

(1) both notions are local, (2) both classes form cones, (3) a limit of a sequence of plurisubharmonic functions pact convergence in again plurisubharmonic. Two properties

plurisubharmonic

in the topology

of com-

which are easily proved are:

(4) the mean value property, (5) the maximum principle: every plurisubharmonic compact complex space is constant.

function

on a connected

2. Properties of Plurisubharmonic Functions. Plurisubharmonic functions can be approximated by smooth ones. The first result in this direction is due to Lelong, see [Le168].

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221

Theorem 1.7. Let G c (I? be a domain and cp a plurisubharmonic function on G. Then cp can be approximated by a decreasing sequence of C2-diffferentiable plurisubharmonic functions, where convergence takes place in the topology of compact convergence.

The proof uses convolution and the mean value property of plurisubharmanic functions. The global situation for strictly plurisubharmonic functions has been treated by Richberg [Ric68] in the following two theorems. Theorem 1.8. Let X be a complex space, and let X, c X be open subspaces, m E N, with X, cc X,,,,, and

Let (E,) be a decreasing sequence of positive real numbers converging to 0. Let cp be a continuous strictly plurisubharmonic function on X. Then there exists a differentiable strictly plurisubharmonic function (in the sense of (1.4)) II/ such that 0 I $(x) - q(x) I E, for all x E X\X,,, and all m E IN. Theorem 1.9. Let X be a (paracompact) complex manifold and let cp be a continuous strictly plurisubharmonic function on X. Let G c X be a domain. Then there is a continuous strictly plurisubharmonic function $ on X such that (1) $,o is dtfferentiably strictly plurisubharmonic, 6’) $lX\G = dX\G

(3) 0 I * - cp I E. The proof of (1.8) and (1.9) makes use of (a semi-local version of) the following extension theorem of Richberg, which is an important technical tool also in other contexts. Theorem 1.10. Let X be a paracompact complex space, and let Y c closed complex subspace. Furthermore, let cp be a continuous (differentiable) ly plurisubharmonic function. Then there is an open neighborhood U of and a continuous (differentiable) strictly plurisubharmonic function 4 on @lY= $9.

X be a strictY in X U with

This theorem was conjectured (and proved in a special case) by Grauert [Gra62]. The last property of plurisubharmonic functions that we want to discuss is the “Riemann removable singularity theorem” for such functions, proved by Grauert-Remmert [GrRe56]. Theorem 1.11. Let X be a complex manifold, and let A c X be a nowhere dense analytic set. Let cp: X\A + [-co, 00) be a plurisubharmonic function. Under one of the following conditions, cp can be extended uniquely to a plurisubharmonic function on all of X: (1) cp is bounded from above, (2) codim A 2 2.

Th. Peternell

228

3. Pseudoconvex Domains. We introduce main in a complex space. The significance connection with the Levi problem.

the notion of a pseudoconvex doof this will be clear in @2, 3 in

Definition 1.12. Let X be a complex space, and G c X a domain. We say that G has (strictly) pseudoconvex boundary at x0 E aG if there is a neighborhood U of x0 in X and a (strictly) plurisubharmonic function (D: U + [-co, 00) such that (a) GnU={x~U~cp(x)
< 4

are relatively compact in X. For examples of pseudoconvex domains we refer to Q3. There we will seethat e.g. for a domain G c Cc” the notions of pseudoconvexity and holomorphic convexity are the same. Some basic examples, which are easily checked to be pseudoconvex by hand, are convex domains in C”, in particular the unit ball.

$2. l-convex Spaces Strongly pseudoconvex domains and l-convex spacesare the themes of this section. l-convex spacesare exhausted by strongly pseudoconvex domains, and by solving the Levi problem on these domains we get a lot of information on l-convex spaces.Finally, one is lead to the characterization of l-convex spaces as “point modifications” of Stein spaces.The analytic sets contracted by these modifications are called exceptional and their study is begun in this section, too.

V. Pseudoconvexity,

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1. Remmert Reduction. Holomorphically “fiber spaces” over Stein spaces:

and Vanishing

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229

convex spaces can be viewed as

Theorem 2.1. ([Rem56, Car60]) Let X be a holomorphically conuex space. Then there exists a Stein space Y and a proper surjective holomorphic map cp: X + Y with the following properties: (1) cp has a connected fibers, (2) cp*(%) = @Yr (3) the canonical map Lo,(Y) + 0,(X) is an isomorphism, (4) (unioersal property) if 0: X + Z is a holomorphic map into a Stein space 2, then there exists a uniquely determined holomorphic map T: Y + Z such that the diagram

Geometrically, cp collapses all positive-dimensional compact analytic sets of X; up to Stein factorization, Y is the quotient of X by the equivalence relation Xl

-x2

The construction theorem.

if and only if f(xl)

= f(x2)

for all f 6 O,(X).

of the complex structure on Y requires Grauert’s direct image

2. The Levi Problem for l-convex Spaces. Generally speaking, the Levi problem asks whether a space, which is pseudo-convex in some sense, is also holomorphically convex. This problem goes back to Levi (1911) [Levi l] and will play a prominent rale in this chapter. G CC X will always denote an open relatively compact set in the complex space X. We recall from sect. 1 that G cc X is called strongly pseudoconvex if for any boundary point x,, E 8G we can find an open neighborhood U in X and a strictly plurisubharmonic function cp: U + IR such that U n G = {x E Glq(x) < O}. The solution of the Levi problem was given in this context in 1962 in full generality by R. Narasimhan - the smooth case has been previously treated by [GraBI: Theorem 2.2 [Nar62]. Let X be a compact space, and let G CC X be a strongly pseudoconvex domain. Then G is holomorphically convex.

The proof is based on the following: (say X smooth) Proposition 2.3. Hq(G, 0) is a finite-dimensional vector spacefor all q > 0.

The idea of the proof of 2.3 is sketched below. We take a covering (Ui) of c such that Ui n G is Stein for all i. Next we build up new coverings (Ut, . . . , (Vi”) of c such that

230

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Th. Peternell

Uik=

{X E

G n uik[4Di,k(x) < O), with certain strictly plurisubharmonic

func-

tions (P~,~. Changing (P~,~slightly, we can obtain a strongly pseudo-convex domain G, such that (a) G\U’ = G,\U’. (b) 8G n aG, n U’ = 0 (“bumping

technique”).

Now we repeat the same process with G, and the covering (Uf). Inductively we obtain domains G, and coverings (Vkm)with I/km = G, n Ur. It is then an easy exercise in Tech cohomology (see [Gra58, p. 464/465]) to show that the restriction map Hq(GN, Co)+ Hq(G, 0) is surjective. Then a standard argument involving the following theorem of L. Schwartz (which is used again and again in cohomology theory) finishes the proof: let V, W be Frtchet spaces, let CI: V -+ W be continuous, surjective, let p: I/ + W be compact. Then dim(W/Im(cr

linear and + /I)) < co.

Using 2.3 the idea of the proof of 2.2 can be described as follows. Given x0 E 8G one has to construct h E B,(G) with lim Ih( = co. Choose x-+x0 an open neighborhood U of x0 such that there exists f E Co,(U) with {f = O} n c = {x0} Choose a strongly pseudoconvex set c with G cc G having positive distance from { f = 0} at all points of a U. Let Vi c V, be small neighborhoods of x0. We choose a function g;: G\ vi + (c to be identically 0, and define g:’ E M(V,) by g:’ = l/f’. Then the pair (g:, g:) forms a “Cousin I-distribution”, that is, they define a cohomology class g, E H’(E, 0) in an obvious way. By 2.3 we can determine - if r is sufficiently large - a,, . . . , a, E Ccwith a, # 0 and C Lligi = 0. Hence there is h E A(c) with h - c aif-’ is the function we are looking for. A strongly pseudoconvex X carrying a P-exhaustion harmonic outside a compact Hence a l-convex space ( f < c} for c >> 0. It follows from (2.3): Corollary

E O,(Vz) and h E O,(c\v,).

Now h(G

or l-convex space is by definition a complex space function f: X --* [O, co) which is strictly plurisubset. is exhausted by strongly pseudoconvex domains

2.4. Every l-convex complex space is holomorphically

convex.

V. Pseudoconvexity,

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231

Theorems

Note that (2.4) holds even without any smoothness assumption on the exhaustion function (see [CoMi85] and theorem 3.9 below). This property of l-convex spaces being holomorphically convex makes the theory of these spaces completely different from the theory of general q-convex spaces: a q-convex space will in general not carry any non-constant holomorphic tion. But in the l-convex case we can expect new insights by looking Remmert reduction. This is the starting point of the theory.

funcat the

One last remark. Of course, proposition 2.3 is nothing but a very special case of the Andreotti-Grauert finiteness theorem (Chap. VI). But since the proof of 2.3 is rather simple and also quite instructive, and since moreover 2.3 is crucial for all what follows, we have included a sketch of proof. 3. Maximal Compact Analytic Sets. In order to study the Remmert tion of l-convex spaces, we need the following

reduc-

Definition 2.5. Let X be a complex space, and let A c X be a compact analytic set of positive dimension at every point. A is called maximal compact analytic set of X if every compact analytic set B c X of positive dimension at every point is contained in A. Proposition 2.6. [Gra62] Let G CC X be a strongly pseudo-convex domain. Then there exists a compact set K c G such that every nowhere discrete compact analytic set A is contained in K. The same holds (by exhaustion) for any l-convex space. Here we use the following terminology: an analytic set is called nowhere discrete if it has positive dimension at every point. The proof is based on the fact that, given G cc X, there exists an open neighborhood of all of aG (not only locally !) and a strictly plurisubharmonic function cp: U + lR with U n G = {x E Ulq(x) The other basic ingredient is the maximum tions. As a consequence one obtains

< O}.

principle

Theorem 2.7. [Gra62] Let G CC X be strongly its Remmert reduction. Then the degeneracy set A = {x E Gldim cp-‘q(x)

for plurisubharmonic

pseudoconvex

func-

and cp: G -+ Y

> 0}

is the maximal compact analytic set of X. So the picture is as follows: cp contracts phic map outside A. It is convenient to make the following

A to a finite set and is a biholomor-

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Definition 2.8. Let X be a complex space and A c X a compact nowhere discrete and nowhere dense analytic set. A is called exceptional if there is a complex space Y and a proper surjective holomorphic map p: X + Y such that (1) q(A) is finite, (2) cp: X\A + Y\(p(A) is biholomorphic, (3) cp*(G)

= 0,.

We say that cpcollapses or blows down A. If V is a Stein neighborhood of q(A) then cp-‘(V) is a l-convex space with maximal compact analytic set A and (pIq-l(1/) is the Remmert reduction. In particular, the uniqueness of the Remmert reduction shows that the blow-down of A is unique, too. Of course, (2.7) has the following analogue in the general l-convex case: Corollary 2.9. Let X be a l-convex space with Remmert reduction cp. Then the degeneracy set A of cp consists of compact connected components Ai such that: (1) Ai is the maximal compact analytic set of small neighborhoods of Ai, (2) q(A) is a discrete set. So the l-convex spaces arise from Stein spaces by “point modification”. general theory of modifications will be the subject of Chap. VII. It is now easy to characterize exceptional sets:

The

Theorem 2.10. [Gra62] Let X be a complex space, and let A c X be a compact analytic nowhere dense set. Then A is exceptional if and only if there is a strongly pseudo-convex open neighborhood U CC X whose maximal compact analytic set is just A. In Chap. VII we will construct many examples of exceptional sets, the easiest of them given by the so-called a-modifications, where a point in an n-dimensional manifold is substituted by an (n - 1)-dimensional projective space lPnml. For the theory of relatively exceptional sets we refer to the paper [KS713 of Knorr and Schneider. The next step in the theory is to obtain good criteria for an analytic set to be exceptional. Here the normal “bundle” of A comes into the game: we do not want to make assumptions on a whole neighborhood of A in X but only on its “linear approximation”. In general, the normal bundle will not be a vector bundle, merely a linear space. Often it is better to consider the conormal sheaf instead. The notion which is needed to characterize exceptional sets - positivity for conormal sheaves, or negativity for normal spaces - is introduced in the next section. 4. Positive Sheaves and the Normal Bundle. We recall from chap. 2 the notion of a linear space and, in particular, the linear space associated to a coherent sheaf. Let X be a complex space and let 9 be a coherent sheaf on X. By V(9) we denote the linear space associated to 9. One should bear in mind that if 9 is locally free and E the vector bundle whose sheaf of holomorphic sections is 9, then V(P) N E*.

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Definition 2.11. a) A linear space (or a vector bundle) V on a compact complex space is called negative if the zerosection in V is exceptional. b) A coherent sheaf 9 on X is called positive if the linear space V(9) associated to 9 is negative.

Positive sheaves and their cohomological properties will be discussed in detail in $4. Here we are only interested in their connection with exceptional sets. Theorem 2.12. [Gra62] Let X be a complex space, and let A c X be a compact nowhere dense complex subspace defined by the ideal sheaf I. If l/I2 is positive, then A is exceptional in X.

The deeper reason why this theorem holds is the so-called deformation to the normal cone, which we state for simplicity only in the case of a locally complete intersection, to the effect that the normal cone is the same as the normal bundle. This case will be sufficient for our purposes. We refer to [Kos83] or [Fu184] (in the algebraic category). Theorem 2.13. Let Y c X be a locally complete intersection. commutative diagram 1

Then there is a

with the following properties. 1) prZ is the projection, i an embedding and q apat map; 2) q[ Y x (0) is the embedding of Y as zero-section in its normal bundle NyIx; 3) when restricted to C\(O) the diagram reduces (up to isomorphism) to Yx C\(O)

-

x x ~\{Ol

Let us indicate how 2.13 implies 2.12. The first difficulty is to come into the situation of a locally complete intersection. We let cp: 2 -+ X be the blow-up (see chap. 7) of the subspace A in X. Then A^= q-‘(A) - the analytic preimage of A - is a hypersurface, in particular a locally complete intersection. It is a general fact that the positivity of Z/Z2 implies the positivity of J/J2, J being the ideal sheaf defining A^ c 2. Namely, A^is a subspace of lP(J/J2) and J/J2 = 0 p(I,12J(l)Ii (for details see again chap. 5). By 4.1 below, OpC1,lZ,(l) is a positive line bundle. Hence the conormal bundle J/J2 of A^ is a positive line bundle. If we know that A^ is exceptional, then obviously A is exceptional, too. Hence we may a priori assume A to be a hypersurface. Now we apply theorem 2.13: we choose a strongly pseudoconvex neighborhood U of A in its normal bundle N. Then U deformes to a strongly pseudoconvex neighborhood of A in X, using the results of Richberg of 5 1.

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Grauert proved (2.12) by a projection method. Next, we ask whether the converse of theorem 2.12 holds. It turns out, however, that this is not true. Example 2.14. (Laufer [Lausl]) We construct a 3-dimensional manifold X with exceptional set A 2: lP, such that the normal bundle Nalx 2: Olp, 0 O,,( -2). If you prefer an example where the exceptional set is a hypersurface, blow up A in X! We let A be a projective line with coordinates (w,, wz). Let U, and U, be afflne 3-spaces (I? with coordinates (w,, zl, z2) respectively (w,, y,, yz). Let k E IN. By the gluing

21 = w;y,

+ W,YL

zz = Yz, Wl = l/w, we obtain a complex manifold

X with a submanifold

A given by

{zl = z2 = O} = {y, = y, = O}. Define f = (fl, f2, f3, JJ: X -+ c4 by fllU1

= Yzv fllU2

=

Y2,

filU,

=

Zl,

=

4Yl

f3lUl

=

WlZ2,

f4i”1

= w:zl

filU2

w,y:,

+

f3lU2 = W2Yl + Y2, -

w2z:,

f,i”2

=

Yl.

Then f is a proper holomorphic map, f-‘(O) = A, and flX\A *f(X)\(O) is an isomorphism. Moreover f(X) is isomorphic to (z E (c41z: + z$ + z: + z:” = 0}, in particular normal. Consequently, A is exceptional in X with blow-down J It is easy to compute the normal bundle NAIx: N N 0(-1)00(-l), .4X i 0 0 @t-2),

k= 1 k22’

So for k 2 2 the normal bundle of A is not negative. On the positive side we have ([Anc82, Pet82]). Theorem 2.15. Let A be an exceptional set in a complex space X. Then there exists a coherent ideal sheaf I with sup~(0~/I) = A such that the conormal sheaf l/I2 is positive.

Hence the conormal A with an appropriate

sheaf of red A is not necessary positive, but if we equip complex structure, then its conormal sheaf will be posi-

tive. The proof of 2.15 rests on an important theorem for modifications: Hironaka’s Chow lemma (Chap. VII). Roughly speaking, the Chow lemma says that a blow-down can be dominated by a monoidal transformation. These are much easier to deal with than arbitrary modifications.

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It follows also from (2.15) that an exceptional set A is not arbitrary: Since it carries a positive torsion free coherent sheaf with support A, A must be Moishezon, i.e. has “many” meromorphic functions (see Chap. VII). This result is due to Ancona and Van Tan (see [AT82]). 5. The Cohomology of l-convex Spaces. Let X be a l-convex space with Remmert reduction cp: X + Y whose degeneracy set is A. We will assume A to be compact, hence A is exceptional in X. For our purposes it is also no loss of generality to assume that A be connected. So q(A) consists of one point y. Let X denote completion of X along A (Chap. II). Proposition 2.16. Let 9 be a coherent sheaf on X and q 2 1. a) The restriction map Hq(X, 9) + Hq(X, &) is bijectiue. b) Let I c tJ!Zxbe a coherent ideal sheaf with supp(&/I) = A. Then there exists k E IN (depending on 9 and I) such that the canonical map W(X,

Zk9) -+ lP(X,

9)

vanishes. (Ik+F is the image of lk @ 9 + 9).

Part (a) is essentially the “comparison

theorem” (chap. 3):

fP(2, &) N l&p*(9); where Rq 0, the maps

are isomorphisms. Now, Y being Stein, we have Hq(X, 9) 1: Hq( Y, Rqq,(9)) canonically, hence (a) follows. For (b) see [Kos83, Gra62]. Corollary

2.16. In the situation of 2.16 the canonical map zP(X, 9) + W(X,

is injective for q > 0, k suitable and sufficiently

S/IklF) large

If 9 is locally free, then SJlk.% = 9 0 QJlk. So FJlk9 is nothing but the restriction 9 1A,-, to the (k - l)-th infinitesimal neighborhood of A in X. In summary, the cohomology of a l-convex space is concentrated on the (completion of the) exceptional set. Stein spaces can be characterized by Theorem B. For l-convex spaces we have the following analogue: Theorem 2.17. ([Nar62], [Kos83], [CoMi85]) Let X be a complex space, and A c X a compact nowhere dense analytic subset. Then X is l-convex with exceptional set A tf and only if the following two conditions are satisfied for all coherent sheaves B on X and all q 2 1:

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(1) dim Hq(X, 9) < co; (2) the restriction

maps IP(X,

9) + W(rZ,

3)

are isomorphisms.

In order to be able to conclude that some complex space is l-convex, it is even sufficient to have dim H’(X, 9) < co for any coherent ideal sheaf whose support is of dimension 0.

0 3. The Levi Problem This section gives a more detailed treatment of various versions of the Levi problem. This problem was over several decades one of the outstanding problems in complex analysis and has influenced its development significantly. Even today there are interesting unsolved problems connected with it. As a general reference to the Levi problem we recommend [Siu78]. 1. The Classical

Levi Problem.

Our starting point is the following theorem

of E.E. Levi [Levll]. Theorem 3.1. Every domain of holomorphy is pseudoconvex.

G c C:” (with smooth boundary aG)

“Domain of holomorphy” means that there is f E O(G) which cannot be extended to a larger domain. For a modern proof of (3.1) including a rigorous definition of “domain of holomorphy”, see [GrFr74]. In one variable every domain is a domain of holomorphy. In sharp contrast to this, to be a domain of holomorphy in (c” is a remarkable property for n 2 2. A famous example in (c* runs as follows. Let A, = {z E Cl IzI < r}. Set G = (A, x AI,*) u (A1\11,2) x A,). Then G is not a domain of holomorphy. In fact, the restriction map

o(A, x A,) + O(G) is onto, as easily seen by Cauchy’s formula. The converse of (3.1) is the “classical” Levi problem, solved by Oka [Oka42] for n = 2, and by Oka [Oka53], Bremermann [Bre54] and Norguet [Nor541 in general. Theorem 3.2. Every pseudoconvex

domain in C” is a domain of holomorphy.

Oka proved even a more general theorem for unbranched Riemann domains over (I?. The connection to the Levi problem of sect. 2 is Cartan-Thullen’s famous ([CaTh32]). Theorem 3.3. A domain in C:” is a domain of holomorphy holomorphically convex.

if and only if it is

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A proof can be found in [GrFr74]. Thus via (3.3), theorem (3.2) is in the case of relatively compact domains which are strongly pseudoconvex a special case of (2.2). 2. Counterexamples. By (2.2) every relatively compact strongly pseudoconvex domain in a complex space is holomorphically convex; on the other hand, (3.2) states that every pseudoconvex domain in Cc” is holomorphically convex. So the natural question arises: does (3.2) hold for domains in an arbitrary manifold X? Grauert showed by the following example that this is false. Example 3.4. Let Tc IR’” be the lattice generated by ek= (0, . . . , 0, LO, . . . , 0), 1 I k I 2n, the canonical generators of IR’“. Let

lc: IR2” + lu2”/T be the projection. Now choose an IR-linear isomorphism ~1:QZ:”--* R2” such that there is a vector z E (c” with (a) a(c.z) c {x1 = 0}, (b) rccr((c.z) dense in rr({xr = O}). Let X = C/Ci(r) be the torus defined by the lattice C’(r). Denoting by IC:a:” + X be the canonical map, we define G = n(a-l({lxll

< )})).

Clearly G is pseudoconvex but nowhere strictly pseudoconvex. We claim that O(G) = C. Let SE O(G) and consider the holomorphic map g = (fln(C.z)) 0 7~:(c -+ (c. By Liouville’s theorem g is constant, hence f is constant on n(cc-l((x, = 0}), which is of real codimension 1. Thus f is constant and G cannot be holomorphically convex. Of course, this example gives the motivation for just slightly modifying the original question: Assume additionally that aG is strictly pseudoconvex at some point. However we have the following Example 3.5. ([Gra63]) There is a complex manifold X with a pseudoconvex domain G cc X whose boundary is smooth and strictly pseudoconvex at some point, but G is not holomorphically convex.

In order to construct X and G, let us start with a projective algebraic manifold Y (i.e. a submanifold of lP,,) whose first Betti number b,(Y) = dim H’(X,

C)

does not vanish. The last assumption implies that there exists a holomorphic line bundle T on Y which is topologically trivial but no tensor power Tk is trivial. In Kahler theory it is proved that T carries an Hermitian metric of zero curvature. Now let X = lP(Or @ L*) where L is a negative line bundle on Y (for the existence of such an L see sect. 4). So X arises from L by adding a point at

Th. Peternell

238 infinity

in every line L,, y E Y. The bundle L being negative, its zero section Y,

(which can identified holomorphic map

with Y) can be blown down to a point. Thus we obtain a cp:X-+X’

to a compact complex space X’. Moreover

X’ is easily seen to be again projec-

tive algebraic (see [Gra62]). Now take a negative line bundle F’ on X’ and choose a metric of negative curvature on F’ (see again sect. 4). The zero section

of F’ has a neighborhood

U’ with strongly pseudoconvex boundary given by

U’ = (4 Ilull < 11, the length /lull of u E Fi given by our metric. This is an easy computation. F = q*(F). Then we can lift the metric and

Put

u = (4 II4 < 1) gives a neighborhood of the zero-section in F whose smooth boundary is pseudoconvex everywhere and strongly pseudoconvex almost everywhere (but not over the exceptional set of cp). Let 71:X + Y be the projection and consider the line bundle H = F @ n*(T),

which bundle H inherits a natural metric by taking tensor products of the given metrics. Let G = {u E HI Ilull < l}. Then G is pseudoconvex everywhere and strongly pseudoconvex almost everywhere, since the curvature of H is semi-positive and positive almost everywhere. Now let X, be the zero section of X = lP (0 0 L*). Then HIX, N T if we identify X, and Y. We wish to see that G is not holomorphically convex. It is sufficient to show that every f E O(G) is constant on p-‘(X,), where p: H +X is the projection, because then we can for no discrete sequence in p-‘(X0) find f E Co(G) which is unbounded on this sequence. So let f~ Co(G). Then flp-‘(X,) gives rise to sections s E H’(X,, H*k), k E IN, via power series expansion along the fibers of H (for more details on this procedure see sect. 4). Since H*klXo = T*k, k 2 1, all these sections have to vanish by our assumption on T. Thus flX, must be constant. In (3.5) an interesting method to construct pseudoconvex domains appeared: take a holomorphic line bundle (or vector bundle) on a compact manifold with semi-positive curvature. Let G = {u E Ll Ilull < l}, the norm being computed from our metric. Then G is pseudoconvex (but not strictly pseudoconvex at x E aG, unless the curvature is positive at x). 3. Characterizing Stein Spaces. In his famous paper [Oka53] Oka established the existence of nice exhaustion functions on domains in Cc”.Specifically, let G c C” be a domain of holomorphy with smooth boundary. Let d be the

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euclidean distance function to aG. Then, assuming G is bounded, the function -log d is plurisubharmonic and also an exhaustion function for G: (cp
(cEIR).

If G is not bounded, -log d is still plurisubharmonic, tion function. But the function -log

but clearly not an exhaus-

d + 1z12

exhausts G, and is even strictly plurisubharmonic. Oka showed that, conversely, every G supporting a strictly plurisubharmonic exhaustion function is a domain of holomorphy. Grauert [Gra58], using the technique described in (2.3), generalized this considerably: Theorem 3.6. Let X be a complex manifold. If X carries a smooth (C’) strictly plurisubharmonic exhaustion function, then X is a Stein manifold.

The result was in turn generalized by Narasimhan

[Nar62]:

Theorem 3.7. Let X be a complex space. Zf X carries a continuous strictly plurisubharmonic exhaustion function, then X is a Stein space. Conversely, every Stein space admits a real-analytic strictly plurisubharmonic exhaustion function.

An important theorem.

step in the proofs of these theorems is the following Runge type

Theorem 3.8. Let cp be a strictly plurisubharmonic continuous function complex space X. Let c E IR and Y = {‘p < c}. Then the restriction map

on the

has dense image in the topology of compact convergence.

One says also that (X, Y) is a “Runge pair”. Another similar characterization of Stein spaces was obtained by NorguetSiu [NoSi77]. A complex space X is called K-complete or holomorphically spreadable if for any x E X there is a holomorphic map f: X + (lZp(p depending on x) such that x is an isolated point off-'(f(x)). Then we have: Theorem 3.9. Let X be a K-complete complex space. Assume that X admits a continuous exhaustion function which is plurisubharmonic on the regular part of

X. Then X is a Stein space.

What happens if we do not require continuity of the exhaustion function cpin (3.6), but only semi-continuity? In this case we should allow cpto assume also the value -cc and it is clear that we cannot expect X to be a Stein space any longer: the set { cp = -m} may contain compact subspaces as easy examples show. In fact, Coltoiu [Co1851 and Coltoiu-Michalache [CoMi85] showed

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Theorem 3.10. Let X be a complex space. Assume that X admits a strictly plurisubharmonic exhaustion function cp: X -+ [-a, co). Then X is l-convex. Conversely, every l-convex space X carries a strictly plurisubharmonic exhaustion function which is --oo on the exceptional set and real-analytic outside.

Another “discontinuous Narasimhan [FoNs80,6.1].

version” of the Levi problem

is due to Fornaess-

4. The Local Stein Problem. Let us go back to Oka’s results described in the previous section. They tell us that a domain G c C” is holomorphically convex or Stein if every point x E 8G admits a neighborhood U c (c” such that U n G is Stein. It is now natural to ask whether this remains true for general X instead of a?. We say that a domain G c X, where X is a complex space, is locally Stein if every x E aG has a neighborhood U in X with U n G Stein. The local Stein problem amounts to the following:

Is every locally Stein domain in a Stein space itself Stein? So far only partial solutions are known. Docquier-Grauert the problem in the smooth case:

[DoGr60]

solved

Theorem 3.11 The local Stein problem has a positive solution for X smooth.

This is a direct generalization of (3.2) with Stein manifolds instead of C”. For special singular spaces Andreotti-Narasimhan [AnNs64] showed Theorem 3.12. Let X be a Stein space with singular set S. Let G c X be a domain which is locally Stein. Assume, in addition, that S n aG admits a neighborhood U such that U n G is Stein. Then G is Stein. In particular, the additional assumption is always satisfied if S n c?G is discrete (or empty).

The general philosophy is, of course, to construct strictly plurisubharmonic exhaustion functions. This is the more difficult the more singular X is. In the smooth or almost smooth case, as in (3.11), (3.12), one can use (via embeddings) the negative logarithm of the euclidian distance. Of course, one might be tempted to admit a larger class of spaces than just Stein spaces in the local Stein problem. But one has to be very careful: Grauert constructed a domain G in a (Zdimensional) torus which is pseudoconvex, in particular locally Stein, but not Stein (example 3.4). For positive results on domains in homogeneous manifolds, see Hirschowitz [Hir74], [Hir75]. He proved e.g. that in a rational homogeneous manifold X every domain G admitting a continuous plurisubharmonic exhaustion function is holomorphically convex and, if X is not a product, then G is even Stein. This is not in conflict with Grauert’s example because there domains in tori (tori are obviously not rational) are considered. Hirschowitz makes heavily use of -log d, where d is a distance computed from the holomorphic vector fields. Particularly noteworthy is the following result of Michel [Mic76] building up on Hirschowitz’s work.

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Theorem 3.13. Let G c X be an open locally Stein set in a compact homogeneous mantfold X. Assume that there exists a smooth point x E aG where aG is strictly pseudoconvex. Then G is Stein.

Again this does not contradict to Grauert’s example where the boundary is strictly pseudoconvex at no point. On the other hand Elencwajg [Eln75] proved the following generalization of Docquier-Grauert’s theorem. Theorem 3.14. Let X be a complex manifold and G CC X be a locally Stein domain. Assume the existence of a continuous strongly plurisubharmonic function in a neighborhood of c. Then G is Stein.

This result was recently generalized to the singular case by ColtoiuMichalache [CoMi89]. They also point out that a pseudoconvex domain of a K-complete space (for instance a Stein space) is Stein. Let us now return to the local Stein problem in Stein spaces. Using Runge techniques Fornaess and Narasimhan [FoN&O] showed Theorem 3.15. Let X be a reduced Stein space, G CC X a domain. Suppose that (1) G is locally Stem,

(2) H’(G, 0) = 0, (3) whenever f E O(X) is non-constant on every irreducible component of X of positive dimension, then G n {f = 0} is Stein. Then G is Stein.

Moreover

Fornaess and Narasimhan

obtained the following

Corollary 3.16. Let X be a Stein space, and let G CC X be locally Stein. Hq(G, 0) = 0 for q > 0, then G is Stein.

If

Without any further assumption on G, the same authors proved that a locally Stein domain in a normal Stein space is a domain of holomorphy. We close our discussion of the Levi problem and pseudoconvexity by mentioning some important fields of research in the neighborhood. (1) Pseudoconvex domains in (c” have been studied intensively in the last two decades - independently from the Levi problem - and form still a very active area of research. See [DiLisl] and [BeNs90] for surveys. (2) Characterizations of Stein manifolds by curvature conditions, complete Klhler metrics, etc. See the survey article [Die86]. (3) The theory of q-convex spaces, q-convex functions, etc. See the next chapter.

$4. Positive Sheaves and Vanishing Theorems In 4 2 we introduced the notion of a positive coherent sheaf. Here we want to study them in greater detail. Their most striking property is a vanishing property for cohomology. This has important geometric consequences. Furthermore,

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we study other positivity notions (coming from differential tions, and vanishing properties (Kodaira, Le Potier).

geometry),

their rela-

1. The Projective Bundle. Let X be a complex space and d a coherent sheaf on X. Let rc: II’(d) + X be the associated projective fiber space (11.3.4). lP(b) carries a natural line bundle 0 pea,(l) which restricted to a fiber rc-‘(x) 2: lP,+.i is just the usual bundle Co(l). We recall (111.5.17) that

Furthermore, phism

one has for any coherent sheaf 9’ on X the fundamental Hqw(a,

n*(y)

0 Gy,,(cL))

isomor-

= H4(X, 9 0 W8)).

The following

is of great importance

Proposition positive.

4.1. Let X be compact. Then d is positive if and only if ~9,~~,(1) is

The proof proceeds by establishing L\zero

an isomorphism

section -+ V\zero

where L and V are the linear spaces associated Corollary

section, to Co,,,,(l) and 8 respectively.

4.2. LoPn(1) is positive.

(Apply 4.1 to X = {0} and d = C’+i.) 4.1 gives a method to reduce the study of positive sheaves to that one of line bundles, and positive line bundles are often easier to handle. 2. The Vanishing Theorem for Positive Sheaves. Let X always be compact and I a positive sheaf on X; S”‘(a) denotes the m-th symmetric power of 8. Theorem 4.3. (Grauert [Gra62]) Let Y be a positive coherent sheaf on X. Then there is a number m, such that for all m 2 m, and all q > 0: W(X,

Y @ Srn(&)) = 0.

Sketch of Proof. (1) We reduce the problem to the case of a line bundle 8: Take E’(b) 5 X; then O,,,, (1 ) is positive by 4.1. If the vanishing holds for Qiro,(l) then we use the fundamental isomorphism of section 1 to conclude. (2) We may now assume that 8 is locally free of rank 1 and let E be the line bundle whose sheaf of sections is 8. In fact, all the following considerations are valid for arbitrary vector bundles, too. The essential point of the proof is the existence of a canonical injective map 6 where

W(X,

Y 0 Yk/Yk+l)

-+ l$l Hq( U, rr*(q),

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a) d is the ideal sheaf of the zero-section - which we identify with X - in E, b) the limit is taken over all neighborhoods U of X in E, c) rc: E -+ X is the projection. This map is given by expanding cohomology classes into power series. Now choose a strongly pseudoconvex neighborhood U of X in E. Then the above injection yields an injection Q H4(X, Y 0 xk/Yk+l)

-+ Hq( u, 7t*(,a)).

The latter group being a finite-dimensional vector space by the AndreottiGrauert finiteness theorem (VI.3), we conclude by remarking that 9k/9k+1 2: &Ok In the next section we shall see that this vanishing theorem (4.3) characterizes a positive sheaf. 3. The Embedding Theorem. We now prove that compact complex spaces carrying a positive vector bundle must be algebraic. Theorem 4.4. Let X be a compact complex space and d a locally free sheaf of rank r on X. Assume that 6 is positive. Then there exists a number k, such that for k 2 k, the sections of S”(8) define an embedding qi: X 4

Gr(r, N)

into the Grassmann manifold of r-planes particular, X is projective.

in (CN, with N = dim H’(X,

Sk&). In

The map 4 is given by choosing a basis si, . . . , sN of H’(X, Sk&) and associating to x E X in a local trivialization the r-plane in (CN spanned by the vectors sl tx), . . . , sN(x). Sketch of Proof. Every linear subspace T/ c H’(X, Sk&Y)defines a meromorphic map 4: X- Gr(r, m), m = dim H’(X, Sk&). So one has to prove that for V = H’(X, Sk&) and k >> 0 the map 4 is well defined everywhere and in fact an embedding. The first part amounts to prove the existence of k such that: (1) for all x E X and all u E E, there is s E H’(X, Sk&?)with s(x) = v. The embedding property translates into the following two statements: (2) for all x, y E X, x # y, there is s E H’(X, Sk&) such that s(x) # s(y) and (3) for all x E X and v E SkE, @ m,/mz = m,Sk6?Jm~Sk&~ there is s E H’(X, Sk&) with s(x) = 0 such that s,/m~Sk&~ = u. Here m, denotes the maximal ideal at the point x.

Via the exact sequences O~m,OSkd~Skb-,Sk~OLo,/m,~O

and 0 + mxY @ Sk& + Sk& + Sk& 0 COx/m,, -9 0, (l), (2) and (3) follow from the cohomology

vanishing:

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Th. Peternell

(4) there is k, E N such that for k 2 k, and all x, y E X: H’(X,

m, Q Skb) = 0,

H’(X,

mxy Q Sk&) = 0.

Here by abuse of language, m,, m,.. are the full ideal sheavesof {x}, {x, y}, and forx=y,m,,=mf. Now for fixed x respectively (x, y) the cohomology vanishing (4) holds by theorem (4.3). Since (4) is an open condition (in x resp. (x, y)) we conclude by compactness that k, can by chosen independently of x respectively (x, y). Remark. If in theorem (4.4) & is merely a positive coherent sheaf of rank > 0, then we can construct only an almost everywhere defined map to a Grassmannian and only conclude that X is Moisezon, i.e. bimeromorphically equivalent to a projective variety. This will be discussed in (VII.6).

Since cOPn(1) is ample, we can reformulate

(4.4):

Corollary 4.5. Let X be a compact complex space. X is projective if and only if X carries a positive line bundle. Remarks 4.6. (1) In algebraic geometry line bundles with the embedding property (4.4) are called ample. A vector bundle d is called ample if and only if O,,,,( 1) is ample. (2) Assume the existence of a line bundle B such that the sections of bk define an embedding i: X 4 lPN, i.e. d is ample. Then

i*(LOPN(l)) = bk. Hence, bk is clearly positive (4.8) and so d is positive, too (4.9(l)). So ampleness and positivity are the same for line and hence vector bundles. In other words: the embedding theorem (4.4) characterizes positive line bundles. (3) If in (2) 8 is merely a vector bundle defining the embedding (2), then Sk& N i*(3), FJ being the dual of (the sheaf of holomorphic sections of) the tautological vector bundle V on Gr(m, n). I/ is defined by the rule: I’, := the 67” in Cc”given by the point x. Now V is not a positive vector bundle, and this is the reason why bundles of rank > 1 cannot by characterized by (4.4). (4) The following is easily derived from all what we know up to this point: A coherent sheaf 6 is positive if and only if for every coherent sheaf F there is a number n, such that for n 2 n, there is a canonical epimorphism oN+sQSn(cq+O for some N E IN. 4. Characterization of Positivity by Cohomology Vanishing. In this section

we discuss the converse of theorem (4.3):

V. Pseudoconvexity,

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245

Theorem 4.7. Let X be a compact complex space and 8 a coherent sheaf on X. Assume that for all coherent sheaves 9’ on X there is some k, E IN such that for all k 2 k, H’(X,

Y @ Sk(&)) = 0.

Then d is positive. Proof. By passing to E’(d), if necessary, we may assume that d is locally free of rank 1. The proof of (4.4) shows that for some k the sections of bk define an embedding i: X 4 lPN such that bk = i*(O,“(l)). Hence gk and, consequently, d is positive (4.9( 1)). Remark. If in (4.7) 6 is locally free of rank 1 it is sufficient to have vanishing for sheaves Y of the form m,, mxy, m,2 in order to conclude the positivity of 8. This is clear from the construction of the embedding i: X 4 lPN. 5. Functional

Properties

of Positive

almost obvious from the definition Proposition

Sheaves. The following

proposition

is

of positivity:

4.8. Let X be a compact complex space and d a coherent sheaf on

X.

(1) Let Y c X be a compact subspace. Zf 8 is positive then bl Y is positive. (2) 6 is positive if and only if dlred X is positive. Less obvious is (cp. [Gra62],

[Ha66]):

Proposition 4.9. Let X be a compact complex spaceand F a coherent sheaf on X.

(1) & is positiue if and only if Sk(&) is positive for some k E IN. (2) Let Y be a compact complex space and f: Y + X a finite map. Then d is positive if and only tf f *(8) is positive. (3) Zf 9 is a coherent sheaf on X together with an epimorphism 6’ + 9, then the positivity of &’ implies the positivity of 9. (4) Let $,9 be locally free sheaveson X together with an exact sequence

If 99and 9 are positive, then 8 is positive.

Sketch of Proof. (For details see [Gra62],

[Ha66].) (1) can be reduced to the case where 6 is locally free of rank 1. Let E be the associated line bundle. The holomorphic map E* + E*k, u + u @ ... @ u maps l-convex C*-invariant neighborhoods of the zero-section of E* those of E*k and vice versa. (2) is easy, going back to the definition, in one direction. The other is not so obvious. (3) The epimorphism d -+ 9 yields an embedding i: lP(P) + E’(I) such that ~pc,,U) = i*(G&)).

246

Th. Peternell

(4) First observe that V 0 9 is positive. Then using the decomposition s”(Y 0 9) = one obtains the vanishing

@ P(3) p+q=n

0 PyB)

for a given coherent sheaf Y:

H’(X,

5 @ P(Y)

0 P(5))

= 0,

i > 0, p + 4 2 n,, with n, depending on Y. Now use the filtration Sp(%) @ Sq(9), p + 4 = n, and deduce the cohomology vanishing H’(X,

9 0 Y(&))

= 0,

of Y’(a) by

n 2 no.

So 6 is positive. Important is also Grauert’s positivity criterion [Gra62]: (5) a line bundle L on the compact space X is positive if for every irreducible compact analytic set A c X there is k E N and s E H’(LI A), s # 0, having a zero. This is a Nakai-Moishezon type result (cpIShSo851). Using representation

theory of Gl(r) one can show (see [Ha66]):

Proposition 4.10. Let X be a compact complex space and d a locally free sheaf on X. Then all tensor powers and all exterior powers of d are positive, too. Concerning

direct images Ancona has shown

the following

(see [AT82])

Theorem 4.11. Let X, Y be compact complex spaces, f: X + Y be a (surjectiue) holomorphic map and 9 a positive coherent sheaf on X. Then there is some no E IN such that f*S”“o(F) is positiue for all n E IN. 6. Differential-Geometric Positivity Notions. Here we want to restrict ourselves to vector bundles on manifolds; for generalizations to linear space and complex spacesseee.g. [GR70-11. Let E be a vector bundle of rank r on a complex manifold X of dimension n equipped with an hermitian metric h. Let D be the uniquely determined connection of E (often called “Chern connection”) which is compatible with both the hermitian metric and the holomorphic structure of E. Let c(E) = D2 be the associated curvature. If we choose local coordinates (z,, . . . , z,,) of X and (e,, . . . , e,) of E, then

c(E) =

1

cijnp dzi A dzj 0 e; 0 e,.

lci,jsn 1
Observe that ciji,, = Cjiprl,so it(E) can be viewed as hermitian form on T, 0 E, T, the holomorphic tangent bundle of X. For the details seee.g. [We80], [ShSo85]. Definition 4.12. (1) E is called positive in the senseof Griffhs if there is an hermitian metric on E such that the curvature c(E) of the associated Chern

V. Pseudoconvexity,

the Levi

Problem

connection satisfies the following condition J-1c(E)x(5

0

v)

=

1 i,j,

and Vanishing

Theorems

247

(in local coordinates): cijlp(x)Si~jvlvp

>

O

((3

LP

for all x E X, all 0 # 5 = 1 ti ,:“, E T,X, and all v = 1 vlel E E,. (2) E is called positive in the’ sense of Nakano if the condition replaced by J-l

C

cijAp(x)uiAuju

’

O

(G) in (1) is (NJ

i,j,l,p

for all x E X and all v = (vii) E C”‘\{O>. In practice, it turns out that Nakano positivity is a too strong condition to be of great use. The relations between the different positivity notions are collected in Theorem 4.13. (1) For line bundles, the notions of positivity, Griffiths positivity and Nakano positivity coincide. (2) For vector bundles, we have: Nakano positivity * Griffiths positivity * positivity. Proof. (1) It is clear that Nakano positive bundles are Grifliths positive and that the converse holds for line bundles. (2) Let L be a Grifliths positive line bundle. Using the Kodaira vanishing theorem (4.14) and Kodaira’s blow-up method one obtains a number n, such that for all x, y E X and all n 2 n,: wyx,

Y 0 L”) = 0,

9’ = m,, mxY, mz. So L is positive. For details see [We80]. It is also possible to construct directly from the metric on L a strongly pseudo-convex neighborhood of the zero-section of L*, see [Gra62]. (3) Now assume that L is positive. It is a standard fact that L is Griffiths positive if and only if there is a positive (1, 1)-form o such that c,(L) = [o] in H’(X, IR), see e.g. [GH78, p. 1481. L being positive, there is an embedding i: X 4 IPN and some k E IN such that Lk = i*(OpN(l)). Since OpN(l) is positive in the sense of Griffiths (use the Fubini-Study-metric), Lk is Griffiths positive and c1(Lk) = kc,(L)

is represented by a positive (1, 1)-form o, say. Then F represents

c,(L) and by our above remark, L is Griffiths positive. For an complex-analytic

proof see [Gra62]. (4) Finally, if E is a Grifliths positive vector bundle, the tautological line bundle 0,(,,(l) is Griffiths positive, too. A metric on cOPo,(l) of positive curvature can be explicitly constructed from an analogous metric on E. So 0,&l) is positive by (2) and hence E is positive. The differential-geometric positivity allows us to prove “precise” vanishing theorems rather than “coarse” vanishing theorems (4.3).

Th. Peternell

248

Theorem 4.14. Let X be a projective manifold of dimension n. vector bundle on X of rank r. If E is Griffiths positive then

(1) Let E be a holomorphic

Hq(X, E @ Qf;) = 0 for p + q 2 n + r. (Le Potier vanishing theorem). The case r = 1 is the classical Kodaira-Nakano vanishing theorem. (2) Let E be a holomorphic vector bundle of rank r. Zf E is Nakano positive, then H4(X, E @ Q;) = 0 for q > 0. (Nakano

vanishing theorem).

For a proof see [Sh-So85], [D-V74]. Using the equivalence of positivity and Grifliths positivity for line bundle and passing to IF’(E) one deduces from (4.14): Corollary dles.

4.15. The Le Potier vanishing theorem holds for positive vector bun-

At this point we should mention the following very important generalization of the Kodaira vanishing theorem due to Kawamata and Viehweg, the most important vanishing theorem in algebraic geometry. Theorem bundle on X. (a) (c,(L). (b) c,(L)” Then

4.16. Let X be a projective manifold of dimension n and L a line Assume C) 2 0 for all irreducible curves C c X. > 0. Hq(X,L@SZjG)=O

forq>O.

Here c,(L) denotes the first Chern class of X.

For a proof and applications

see [ShSo85],

[KMM87].

Example 4.17. The tangent bundle Ton lPn is Griffiths positive - this follows from the Euler sequence

0 + 0, + Q.(l),+’

+ T + 0.

But it is not Nakano positive; otherwise we would have H’(IP,,, T @ Kp”) = 0, for all i > 0 but by Serre duality: H”-‘(lF’w,

T @ KpJ 1: H’(IP”,

Szh,) N c:.

For more informations on the differential geometry of positive vector bundles, for further vanishing theorems and applications we refer to [GH78], [D-V74], [We80], [Sh-So851 and [Dem88]. Remark

GrifIiths

4.18. It is not known whether the notions of positivity positivity coincide for vector bundles of rank > 1 or not.

and of

V. Pseudoconvexity,

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249

7. Hodge Metrics. Let X be a (compact) complex manifold. A Kahler form w on X is said to be a Hedge firm (and hence the underlying Kahler metric a Hodge metric) if its de Rham class [o] E H2(X, IR) is actually in the image Im(H2(X,

2) -+ H2(X, IR))

via the natural inclusion Z -+ IR. A complex manifold admitting a Hodge metric is said to be a Hodge manifold. If L is a positive line bundle on X with positive curvature form o, then io is a Hodge metric, since &. [io]

Hence every projective manifold theorem of Kodaira [Ko54]:

= c,(L) E H’(X,

Z).

is a Hodge manifold.

The converse is a famous

Theorem 4.19. Every compact Hodge manifold is projective.

In fact, given a Hodge metric o, then [io] E H2(X, Z) n H’(X, a’), so there is a line bundle L with c,(L) = [iw] (see [We80]). L is positive (see the proof of 4.13), hence X is projective. Using Hodge decomposition ([We801 and chap. V) it is easy to deduce from 4.19 another theorem of Kodaira: Theorem 4.20. Any compact KBhler manifold X with H2(X, tive.

0) = 0 is projec-

Remark 4.21. The notions of a Kahler metric can be carried over to singular spaces, so one can speak of Kiihler spaces and also of Hodge spaces. For normal spaces X, Grauert [Gra62] has generalized theorem 4.19:

A normal compact Hodge space is projective. For further information [Var89].

on Kghler

spaces we refer to [Bin83],

CMoi7.51,

8. Relative Positive Sheaves. All results discussed in section 4 up to 7 have relative versions. Let us shortly describe the most important ones. We fix a proper morphism Z: X --* S of complex spaces. Let d be a coherent sheaf on X. d is called positive relative JTor n-positive if the following holds: For every coherent sheaf B on X and every compact set K c S there is some n, E IN such that for all n 2 n, the canonical map

7c*7c*(P @ S”(8)) --) s 0 S”(fY!q is an epimorphism over 7r-l (K). Note that for S a reduced point we obtain the old notion of positivity (4.6(4)). Relative positivity has been introduced first by Grothendieck. Relative positive sheaves can be characterized by cohomology vanishing:

250

Th. Peternell

Theorem 4.21. Let d be a coherent sheaf on X. 6 is rc-positive if and only if the following statement holds. For every coherent sheaf 9 on X and every compact set K c S there is some n, E IN such that for all n 2 no and all q > 0: Rqz,(9

@ S”(&‘))lK

= 0.

For a proof see [KS711 - at least for the case Q locally free. For the general case one has to pass to II’(d). Compare also [AT82]. Remark 4.22. In [KS711 Knorr-Schneider introduce the notion of relative exceptional sets. We do not want to go in the details here but mention that using this notion - one can reformulate the original definition of positivity using strongly pseudo-convex neighborhoods of zero-sections. Theorem 4.23. Let 6’ be a x-positive projective morphism.

locally free sheaf on X. Then rc is a

A proper morphism rc: X + S of complex spaces is called projective if for any relatively compact open set U c S there is an embedding x-‘(U) 4 II’,, x U such that the following diagram commutes: 7c-‘&q

t

IPn x u

n

Pr2

\J

U

In particular all fibers rc-‘(x) are projective. Theorem 4.23 is a generalization of (4.5). For a proof we refer to [KS71]. is essential to prove that the canonical map

It

7Tn*7c*(LP)-+ LP is surjective over U for any n-positive line bundle 9 on X.

3 5. More Vanishing Theorems In this section we discuss briefly several vanishing theorems which are useful in complex analytic geometry. A general remark should be made There is an abundance of vanishing theorems in the literature. We have neither able nor willing to collect all of them here. For a manifold X of dimension n we let K, be the canonical divisor which is nothing but (the line bundle associated to) 0;.

often here: been of X

1. Demailly’s Vanishing Theorem. Let E be a positive ( = ample) vector bundle on a compact manifold X of dimension n. In geometry it is often necessary to consider the symmetric powers Sk(E) or the tensor powers Ek. Since rk Sk(E)

V. Pseudoconvexity,

the Levi

Problem

and Vanishing

251

Theorems

grows very fast, the Le Potier vanishing theorem is practically useless in the commutation of the cohomology of Sk(E). In this situation Demailly [Dem88] proved Theorem 5.1. Let X be a compact n-dimensional manifold, E a vector bundle of rank r, and L a line bundle on X. Assume that E is ample and L semi-ample (in the sense that some tensor power Lk is generated by global sections) or that E is semi-ample and L ample. Then Hq(X, P(E) 0 (det E)h 0 L 0 K,) = Ofor all q 2 1, where P(E) is the bundle associated to E by the irreducible power representation of Cl(E) of highest weight a E Z’ and h E { 1,. . . , r - l} such that a, 2 a, 2 . . ’ 2 ah > ah+l = . . . = a, = 0. For a precise definition of P(E), see [Dem88]. In particular P(E) for a = (k, 0, . . . , 0) and P(E) = Ak(E) for a = (1, . . . , 1, 0, . . . , 0) with bers “1”. For P(E) = Sk(E) we obtain Hq(X, Sk(E) 0 det E 0 LO K,) which result is due to Griffrths

= 0,

= S“(E) k num-

q 2 1,

[Gri69].

Theorem 5.2. Let X, E and L be as in (5.1). Then Hq(X, Ek @ (det E)’ @ L @ $2:) = 0 forallp+q>n+

l,k>

l,l>n-p+r-

1.

The proof of (5.1) and (5.2) uses representation theory of Cl(E), in particular the decomposition into irreducible representations and Bott’s theory of homogeneous line bundles on flag manifolds ([Bot57]). An example in [PPS87] shows that in (5.1) the factor det E cannot be omitted. More specifically, it is proved that given integers r, m 2 2 there is a projective manifold X of dimension n = 2r and an ample vector bundle E of rank r such that H”-‘(X,

Sk(E) @ K,)

# 0

for 2 I k 5 m.

For H”-’ such nonvanishing is not possible if E is generated by sections; this is proved in [PPS87]. This reference gives also geometric applications for vanishings of Sk(E). The necessity to have the factor det E in the vanishing (5.1), (5.2) is plausible in view of the following result of Demailly-Skoda [DeSk80]: if E is positive in the sense of Griffiths, sense of Nakano.

then E @ det E is positive in the

This implies already that Hq(X,E@detE@K,)=O for q 2 1 by Nakano’s vanishing theorem (4.14(2)). For some other vanishing theorems covering also in the case of Ak(E) see [Man91].

252

Th. Peternell

2. The Notion

of k-ampleness.

Sommese) is very useful for particular

The notion purposes.

of k-ampleness

(due to A.J.

5.3. Let X be a compact manifold. (1) Let L a line bundle on X. L is said to be k-ample (0 I k I dim X - 1) if there is some m E IN such that L” is generated by global sections and such that the induced map Definition

cp: x + lP(HO(X, L”))

has at most k-dimensional fibers. (2) A vector bundle E is called k-positive or k-ample if 0,(,,(l) is k-ample on W-9. By (4.4), positivity (= ampleness) coincides with O-ampleness. The following result due to Sommese generalizes the Kodaira-Nakano vanishing theorem respectively the Le Potier vanishing theorem. Theorem 5.4. Let E be a k-ample vector bundle on the projective manifold X. Then Hq(X, E @ 9”) = 0 for p + q 2 n + k + rk(E).

The proof is done by reducing to the Kodaira-Nakano vanishing and by passing to IP(E), see [ShSo85]. For applications see [Som78] or [PPS87]. There is also a differential-geometric notion of k-positivity, due to Girbau; see [ShSo85] for details. 3. Grauert-Riemenschneider Vanishing Theorem and Direct Images of Dualizing Sheaves. A vector bundle E on a compact manifold X is called almost positive if there is a metric on E whose curvature is semi-positive everywhere (in

the sense of Grifliths) and positive at some point. Geometrically, we deduce the existence of neighborhoods of the section in E* having pseudoconvex boundary which is strongly pseudoconvex at some point. Then the GrauertRiemenschneider vanishing theorem ([GR70]) states Theorem 5.5. Let X be a projective (or Moishezon) almost positive vector bundle on X. Then W(X,E@K,)=O,

manifold, and let E be an

q>o.

Again, this is a direct generalization of the Kodaira-Nakano vanishing theorem. The proof in [GR70] uses again harmonic theory. The assumption “X Moishezon” is automatically fulfilled: this is the content of the so-called “Grauert-Riemenschneider conjecture” solved by Siu and Demailly; see Chap. VII.6. There is one point to be cautious of: (5.5) does not hold in general for Hpvq, i.e. W(X,

E @ QpX) # 0

(p + q 2 n + rk(E)),

unless p = n.

V. Pseudoconvexity,

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Problem

and Vanishing

Theorems

We consider the following example due to Ramanujan [Ram72]. the blow-up of Ip, in one point and let E = 0*(0,~(1)). Then H’(X,

253

Let c: X -+ ll’,

E 0 52:) # 0,

as is easily verified by computation. The Grauert-Riemenschneider vanishing (which in the case of line bundles is slightly weaker than the Kawamata-Viehweg vanishing (4.16)) holds also for torsion - free coherent sheaves and on reduced singular spaces, the “canonical sheaf” being chosen suitably, see [GR70]. Note that if a line bundle L has a metric with semi-positive curvature (such line bundles L are called semi-positive) then (c,(L). C) 2 0 for every curve C c X. If however a line bundle L on a projective manifold X satisfies (c,(L). C) 2 0 for all curves C c X then it does not follow the existence of a metric on L with semi-positive curvature. See [DPS91] for an example. Line bundles with this last property are called “nef” (numerically eventually free, see [KMM87]). A local consequence of (5.5), which is easily proved using the Leray sepectral sequence, is Corollary 5.6. Let X be a projective manifold and f: X + Y a generically finite proper holomorphic map to a projective variety Y. Then Pf*(K,)

This was generalized by Takegoshi

= 0,

q > 0.

[Tak85]

to

Theorem 5.7. Let X be a complex manifold, X + Y a proper generically finite map. Then Pf*(K,)

= 0,

Y be a complex space and f:

q > 0.

In [Pet851 an analogue of the Kodaira vanishing theorem on l-convex spaces was proved, a particular case had been previously proved in [GR70-21: Theorem 5.8. Let X be an irreducible reduced l-convex space, and let E be a semi-positive vector bundle (in the sense of Grtffiths). Then Hq(X, E @ K,)

= 0,

q > 0.

Semi-positivity means that the corresponding metric exists on all of X but curvature is computed only on the regular part. The canonical sheaf K, has to be defined in a suitable way; for normal X it is just the sheaf associated to the presheaf u

H

0

E Q~\singdU\SiW(X)

i

where n = dim X. As a consequence one obtains

254

Th. Peternell

Corollary 5.9. Let f: X + Y the blow-down irreducible and reduced). Then Rqf*(K,)

= 0,

of the exceptional

set A c X (X

q > 0.

(5.9) can also be deduced from (5.7) using desingularization spaces. (5.7) has the following important generalization case q = 0 being due to Ohsawa [Ohs84]).

due to Kollar

of complex [Ko186] (the

Theorem 5.10. Let f: X + Y be a holomorphic map of projective varieties, where X is assumed to be smooth. Then (1) Rqf,(K,) is torsion free for every q; (2) HP( Y, L 0 Rqf,(K,)) = 0 for q 2 0, p > 0 and every ample line bundle L on Y.

If f is generically finite, then the support of Rqf,(K,), q > 0, is a nowhere sense analytic set in Y. On the other hand, Rqf,(K,) is torsion free by (5.10), so it must vanish and we get back (5.6). Theorem (5.10) has a lot of important applications in the classification theory of algebraic varieties, see [Ko187] for comments and references. Finally, let us mention that there are many other vanishing theorems on non-compact manifolds X (of course not Stein). They are in part related to convexity properties of X (e.g. on weakly pseudoconvex spaces), in part to Kahler geometry (e.g. on complete Kahler manifolds) or deal with L*cohomology. Some references: [AnVe65], [Nak74], [Ohs83], [KoKo90], [TaOh81]. For more algebraic aspects see [EV86], [EV93], especially for the Hodge-theoretic approach to vanishing theorems.

References* [Anc82] [AnNs64] [AnVe65] [AT821 [BeNs90] [Bin831

Ancona, V.: Faisceaux amples sur les espaces analytiques. Trans. Am. Math. Sot. 274, 899100 (1982) Zb1.503.32014. Andreotti, A.; Narasimhan, R.: Oka’s Heftungslemma and the Levi problem for complex spaces. Trans. Am. Math. Sot. I II, 3455366 (1964) Zbl.134,60. Andreotti, A.; Vesentini, E.: Carleman estimates for the Laplace-Beltrami equation on complex manifolds. Publ. Math., Inst. Hautes Etud. Sci. 25,313-362 (1965) Zbl.138,66. Ancona, V.; Tomassini, G.: Modifications analytiques. Lect. Notes Math. 943, Springer 1982, Zbl.498.32006. Bell, S.; Narasimhan, R.: Proper holomorphic mappings of complex spaces. In: Encycl. Math. Sci. 69, l-38, Springer 199O,Zb1.733.32021. Bingener, J.: Deformations of Klhler spaces I. Math. Z. 182, 505-535 (1983) Zbl.584.32042.

*For the convenience of the reader, references to reviews compiled using the Math database, have, as far as possible,

in Zentralblatt been included

fur Mathematik in this References.

(Zbl.),

V. Pseudoconvexity, [Bre54]

[Bot57] [Car601 [CaTh32] [CoMi85] [CoMi89] [Co1851 [Dem88] [DPS91] [Desk801

[Die861

[DiFo77] [DiLiBl] [DoGr60] [DV74] [Elw75] [EV86] [EV93] [For791 [Ful84] [Grass] [Gra62] [Gra63] [GrFr74] [GrRe56] [GR70-I]

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and Vanishing

Theorems

255

Bremermann, H.: iiber die Aquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum von n kompleken Veriinderlichen. Math. Ann. 228, 63-91 (1954) Zb1.56,78. Bott, R.: Homogeneous vector bundles. Ann. Math., II. Ser. 66, 203-248 (1957) Zb1.94,357. Cartan, H.: Quotients of complex analytic spaces. In: Contrib. Funct. Theor., Int. Collog. Bombay 1960, l-15 (1960) Zbl.122,87. Cartan, H.; Thullen, P.: Zur Theorie der Sigularitlten der Funktionen mehrerer komplexer Verlnderlichen. Math. Ann. 106, 617-647 (1932) Zbl.4,220. Coltoiu, M.; Michalache, N.: Strongly plurisubharmonic exhaustion functions on lconvex spaces. Math. Ann. 270,63-68 (1985) Zb1.533.32009. Coltoiu, M.; Michelache, N.: Pseudoconvex domains on complex spaces with singularities, Compos. Math. 72,241-247 (1989) Zbl.692,32011. Coltoiu, M.: A note on Levi’s problem with discontinuous functions. Enseign. Math., II, Ser. 31, 2999304 (1985) Zb1.588.32021. Demailly, J.P.: Vanishing theorems for tensor powers of on ample vector bundle. Invent. Math. 91, 203-220 (1988) Zbl.647.14005. Demailly, J.P.; Peternell, Th., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. Preprint (1991). J. Alg. Geometry 1993/94. In press Demailly, J.P.; Skoda, H.: Relations entre les notions de positivites de P.A. Grifliths et de S. Nakano pour les Iibres vectoriels. In: Lect. Notes Math. 822, 304309, Springer 1980, Zb1.454.55011. Diederich, K.: Complete Kahler domains. A survey of some recent results. In: Contributions to several complex variables, Hon. W. Stoll, Proc. Conf. Notre Dame/Indiana 1984, Aspects Math. E9,69-87 (1986) Zbl.594.32017. Diederich, K.; Fornaess, J.E.: Pseudoconvex domains: bounded strictly plurisubharmanic exhaustion functions. Invent. Math. 39, 129-141 (1977) Zb1.353.32025. Diederich, K.; Lieb, I.: Konvexitat in der komplexen Analysis. DMV Seminar, B. 2. Birkhauser 1981,Zbl.473.32015. Docquier, F.; Grauert, H.: Levisches Problem und Rungescher Satz fur Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 240,94-123 (1960) Zbl.95,280. Douady, A.; Verdier, J.L. (eds.): Diffcrents aspects de la positivite. Asterisque 17 (1974). Elencwajg, G.: Pseudoconvexitt local dans les varietes Kahleriennes. Ann. Inst. Fourier 25, 295-314 (1975) Zb1.278.32015. Esnault, H.; Viehweg, E.: Logarithmic de Rham complexes and vanishing theorems. Inv. Math. 86, 161-194 (1986). Esnault, H.; Viehweg, E.: Lectures on Vanishing Theorems. DMV-Seminar, Birkhauser (1993). Fornaess, J.E.: The Levi problem in Stein spaces. Math. Stand. 45, 55-69 (1979) Zb1.436.32012. Fulton, W.: Intersection Theory. Erg. Math., 3. Folge, B. 2. Springer 1984. Grauert, H.: On Levi’s problem and the imbcdding of real analytic manifolds. Ann. Math., II. Ser. 68,460-472 (1958) Zbl.108,78. Grauert, H.: iiber Modilikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331-368 (1962) Zbl.173,330. Grauert, H.: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Z. 81, 377391 (1963) Zbl.151,97. Grauert, H; Fritzsche, K.: Einfiihrung in die Funktionentheorie mehrerer Veranderlither. Springer 1974,Zbl.285.32001. Grauert, H.; Remmert, R.: Plurisubharmonische Funktionen in komplexen Raumen. Math. Z. 65, 175-194 (1956) Zbl.70,304. Grauert, H.; Riemenschneider, 0.: Verschwindungssltze fur analytische Kohomologiegruppen auf komplexen Raumen. Invent. Math. 11, 263-292 (1970) Zbl.202,76.

256

Th. Peternell

[GR70-21

[Gri69]

[GH78] [Ha661 [Hir74] [Hir75] [KMM87] [Ko54] [KoKo90] [Kol86] [Kol87] [Kos83] [KS711 [LauSl] [Le145] [Le168] [Levl

l]

[Man911 [Mic76] [Moi75] [Nar61] [Nar62] [Nk55] [Nor541 [NoSi77] [Ohs831

Grauert, H.; Riemenschneider, 0.: Kahlersche Mannigfaltigkeiten mit hyper-qkonvexem Rand. Problems in Analysis, Symp. Hon. S. Bochner, Princeton 1969, 6179 (1970) Zb1.211,103. Griffiths, Ph.A.: Hermitian differential geometry, Chern classes and positive vector bundles. In: Global analysis, Pap. Hon. K. Kodaira Princeton Univ. Press, 185-251 (1969) Zbl.201,240. Grifliths, Ph.A., Harris, J.: Principles of Algebraic Geometry, Wiley 1978, Zbl.408.14001. Hartshorne, R.: Ample vector bundles. Publ. Math., Inst. Hautes Etud. Sci. 29, 63-94 (1966) Zb1.173,490. Hirschowitz, A.: Pseudoconvexite au-dessus d’espace plus on moins homogtnes. Invent. Math. 26, 303-322 (1974) Zb1.275.32009. Hirschowitz, A.: Le problime de Levi pour les espaces homogenes. Bull. Sot. Math. Fr. 103, 191-201 (1975) Zbl.316.32004. Kawamata, Y.; Matsuda, K.; Matsuki, K.: Introduction to the minimal model problem. Adv. Stud. Pure Math. 10, 283-360 (1987) Zbl.672.14006. Kodaira, K.: On Kahler varieties of restricted type. Ann. Math., II. Ser. 60, 28-48 (1954) Zbl.57,141. Kosarew, I.; Kosarew, S.: Kodaira vanishing theorems on non-complete algebraic manifolds. Math. Z. 2(X,223-231 (1990) Zbl.734.14005. Kollar, J.: Higher direct images of dualizing sheaves I. Ann. Math., II. Ser. 123, 1 l-42 (1986) Zbl.598.14015. Kollar, J.: Vanishing theorems for cohomology groups. Proc. Symp. Pure Math. 46, 233-243 (1987) Zbl.658,14012. Kosarew, S.: Konvergenz formaler komplexer RIume mit konvexem oder konkavem Normalenbiindel. J. Reine Angew. Math. 340,6-25 (1983) Zbl.534.32002. Knorr, K.; Schneider, M.: Relativ-exzeptionelle analytische Mengen. Math. Ann. 193, 238-254 (1971) Zbl.222.32008. Laufer, H.: On CIP, as an exceptional set. In: Recent developments in complex analysis, Proc. Conf. Princeton 1979, Ann. Math. Stud. 100,261-275 (1981) Zb1.523.32007. Lelong, P.: Les fonctions plurisousharmoniques. Ann. EC. Norm. Super., III. Ser. 62, 301-338 (1945) Zbl.61,232. Lelong, P.: Fonctions plurisousharmoniques et formes differentielles positives. Gordon and Breach 1968, Zbl.195,116. Levi, E.E.: Studii sui punti singolari essenziali delle funzioni analitiche di due o piu variabli complesse. Ann. Mat. Pura Appl. 17, 61-87 (1911). Manivel, L.: Un thiroreme d’annulation pour les puissances exterieures dun libre ample. In: J. Reine Angew. Math. 422, 91-116 (1991) Zbl.728.14011. Michel, D.: Sur les ouverts pseudoconvexes des espaces homogtnes. C. R. Acad. Sci., Paris, Ser. A 283, 779-782 (1976) Zbl.355.32019. Moishezon, B.G.: Singular klhlerian spaces. Proc. Int. Conf. Manifolds, relat. Top. Topol., Tokyo 1973,343-351 (1975) Zbl.344.32018. Narasimhan, R.: The Levi problem for complex spaces I. Math. Ann. 142, 355-365 (1961) Zbl.106,286. Narasimhan, R.: The Levi problem for complex spaces II. Math. Ann. 146, 195-216 (1962) Zbl.131,308. Nakano, S.: On complex analytic vector bundles. J. Math. Sot. Japan 7, 1-12 (1955) Zbl.68,344. Norguet, F.: Sur les domains d’holomorphie des fonctions uniformes de plusieurs variables complexes. Bull. Sot. Math. Fr. 82, 137-159 (1954) Zbl.56,77. Norguet, F.; Siu, Y.T.: Holomorphic convexity of spaces of analytic cycles. Bull. Sot. Math. Fr. 105, 191-223 (1977) Zb1.382.32010. Ohsawa, T.: Cohomology vanishing theorems on weakly l-complete manifolds. Publ. Res. Inst. Math. Sci. 19, 1181-1201 (1983) Zbl.537.32014.

V. Pseudoconvexity, [Ohs841 [Oka42] [Oka53] [Pet821 [Pet851 [Pot751 [PPS87] [Ram721 [Rem561 [Ric68] [Sch74] [ShSo85] [Siu78] [Som78] [Tak85] [TaOh81] [Var89] [We801

the Levi Problem

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Theorems

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Ohsawa, T.: Vanishing theorems on complete Kahler manifolds. Publ. Res. Inst. Math. Sci. 20, 21-38 (1984) Zbl.568.32018. Oka, K.: Sur les fonctions analytiques de plusieurs variables VI. Domaines pseudoconvexes. Tohoku Math. J. 49, 15-52 (1942) Zbl.60,240. Oka, K.: Sur les fonctions analytiques des plusieurs variables IX. Domaines finis sans point critique interieur. Japan J. Math. 23,97-155 (1953) Zb1.53,243. Peternell, Th.: Uber exzeptionelle Mengen. Manuscr. Math. 37, 19-26 (1982) Zb1.546.32004, Erratum et Addendum, ibid. 42,259-263 (1983) Zbl.546.32005. Peternell, Th.: Der Kodairasche Verschwindungssatz auf streng pseudokonvexen RPumen I, II. Math. Ann. 270, 87-96, ibid. 603-631 (1985) Zbl.538.32022; Zbl.538.32023. Le Potier, J.: Annulation de la cohomologue a valeurs dans un librt vectoriel holomorphe positif de rang quelconque. Math. Ann. 218, 35-53 (1975) Zbl.313.32037. Peternell, Th.; Le Potier, J.; Schneider, M.: Vanishing theorems, linear and quadratic normality. Invent. Math. 87, 573-586 (1987) Zbl.618.14023. Ramanujam, C.P.: Remarks on the Kodaira vanishing theorem. J. Indian Math. Sot., New Ser. 36,41-51 (1972) Zbl.276.32018. Remmert, R.: Sur les espaces analytiques holomorphiquement ¶bles et holomorphiquement convexes. C.R. Acad. Sci., Paris 243, 118-121 (1956) Zbl.70,304. Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 17.5, 257-286 (1968). Schneider, M.: Ein einfacher Beweis des Verschwindungssatzes fur positive holomorphe Vektorblndel. Manuscr. Math. II, 95-101 (1974) Zb1.275.32014. Shiffman, B.; Sommese, A.J.: Vanishing Theorems on Complex Manifolds. Prog. Math. 56, Birkhauser 1985, ZbI.578.32055. Siu, Y.T.: Pseudoconvexity and the problem of Levi. Bull. Am. Math. Sot. 84,481-512 (1978) Zbl.423.32008. Sommese, A.J.: Submanifolds of abelian varieties. Math. Ann. 233, 229-256 (1978) Zbl.381.14007. Takegoshi, K.: Relative vanishing theorems in analytic spaces. Duke Math. J. 52, 273-279 (1985) Zbl.577.32030. Takegoshi, K.; Ohsawa, T.: A vanishing theorem for HP(X, Q4(B)) on weakly l-complete manifolds. Publ. Res. Inst. Math. Sci. 27, 723-733 (1981) Zb1.482.32008. Varouchas, J.: KIhler spaces and proper open morphisms. Math. Ann. 283, 13-52 (1989) ZbI.632.53059. Wells, R.O.: Differential Analysis on Complex Manifolds. 2nd ed. Springer 1980, Zbl.435.32004; Zb1.262.32005.

Chapter VI

Theory of q-Convexity and q-Concavity H. Grauert Contents Introduction

261

. . . . . . . . . . .._.....................................

$1. Domains in (c” ........................... ................ 1. The Dolbeault Complex 2. Families of Domains of Holomorphy ....................... 3. Pseudoconvexity ....................... 4. Pseudoconcavity

.........

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

......

9 3. Finiteness Theorems ...................... 1. Finiteness ............................. 2. Some Further Results ................... 3. Projective Spaces ...................... ............................. 0 4. Applications 1. Complex Spaces with Holes ............. 2. Two Dimensional Complex Manifolds 3. Vanishing Theorems .................... .................. 4. Hulls for Cohomology

..

264 264 265

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266 267

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0 2. Complex Spaces .......................... 1. The Syzygy Theorem ................... 2. q-Convex and q-Concave Complex Spaces . 3. The Frtchet Topology in the Space of Chech Cocycles .............................. ............... 4. Extension of Cohomology

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4 5. Serre’s Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . .. . . . .. . . .. . . .. . . . .. . . .. . . .. . . . .. . . . . 1. Resolutions 2. Compact Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . .. . . .. . . .. .. . .. . . . .. . .. . . . .. . . . 3. Applications 8 6. Algebraic Function Fields ................ 1. Pseudoconcave Complex Spaces ......... .................. 2. The Schwarz Lemma

261 261 262 263 264

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269 269 269 270 271 271 273 273 274 275 275 276 277 279 279 280

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Functions 3. Analytically Dependent Meromorphic 4. Modular Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References

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

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Introduction Section 1 deals with the cohomology with coefficients in the sheaf of local holomorphic functions 0 near to q-convex and q-concave boundary points of domains in the complex number space (I?‘. Section 2 carries over the results to complex spaces and to arbitrary coherent analytic sheaves, proves extension theorems and introduces the Frechet topology in the set of local cross sections. In 0 3 the finiteness of the dimension of cohomology vector spaces is proved in certain cases. Section 4 gives some applications: When can a hole in a complex space be filled. When do hulls for cohomology classes exist and when does the cohomology vanish? Section 5 applies the theory to prove the old (and simple) form of Serre’s duality theorem for q-convex spaces and, finally, in 5 6 we establish the fact that the fundamental domain for the Siegel modular groups of degree n > 1 is qconcave. Hence, the field of modular functions is algebraic.

5 1. Domains in C:” 1. The Dolbeault Complex. For general context see chapter III, § 2.4. In this chapter we assume that D c Cc” is a domain, i.e. an open subset. If K c D is a subset, we put K” = K; = {z = (z~,..., z,) E D: If(z)1 I supIf for all f holomorphic in D} and call K” the holomorphic hull of K (with respect to D). Clearly, K” is always a closed subset of D. The domain D is a domain of holomorphy if and only if it is holomorphically convex: this means that for every compact subset K c D the hull K” is also compact. It is well known that a domain D c (I? is a domain of holomorphy if and only if D is a Stein manifold. If D’ c D are two domains of holomorphy then (D’, D) is called a Runge pair if and only if for every holomorphic function f in D’ there exists a sequence gr of holomorphic functions in D which converges in D’ locally uniformly against J. If D is a domain, we denote by H’(D, 0) for p = 0, 1, . . . the (flabby) p-th cohomology group (= complex vector space) of D with coefficients in the sheaf 0 of germs of local holomorphic functions in D. If D is a domain of holomorphy, then we have H”(D, 0) = 0 for all p = 1, 2, . . . . This is a special case of the famous theoreme B for Stein manifolds and more general for Stein spaces. We denote by A’,’ the (E-vector space of complex forms CIof type (i, j) over D:

We assume that the coefficients a xI ,,,nj(z) are complex P-functions in D, i.e. they are infinitely often continuously differentiable there. We have the deriuatives a = ai,j: AiTj --, A’+‘,j and 2 = ai,j: Ai,’ + Ai*jfl with:

aa = Z(Ux,...xi,*,...&kc

A dzx,-dzxi d%,-.d%j

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and Ja = qa x,... x,,~,... n,),&

A dz,l...dz,idz,,...dz,j.

We have 8 = 0, 8 = 0, 8 = --ad. The total derivative d = 3 + a is real and maps the space of l-dimensional complex exterior form A’ = xi+j=lAi,j linearly into A’+’ . We have d2 = 0. So these forms form a complex in the same way as A’*’ with respect to a and Ai*’ with respect to 2 (for fixed j, resp. i). We call Et A’ the total complex and xi A ‘ai the Dolbeault complex. The total complex belongs to a resolution of the sheaf of locally constant complex functions on D, while the Dolbeault complex corresponds to a resolution of the structure sheaf cc! We define the Dolbeault groups H’*‘(D) to be the complex vector spaces ker ai,j/im 8i,j-1. It is clear that H’*‘(D) = 0 for i or j > n and that ZP’(O) = Q’(D), where a’(o) denotes the vector space of holomorphic i-forms on D. There is a spectral sequence leading from the Dolbeault complex to the complex of cross sections in the flabby resolution of 6! Since this spectral sequence is trivial in both directions, we have a natural isomorphism H’*‘(D) 2: Hj(D, 0). If D is a domain of holomorphy, this means that H’*‘(D) = 0 for j > 0. By the Banach theorem on surjective continuous linear maps of Frtchet spaces we get a proposition of the following kind: Assume that a E A’,‘, j 2 1 is small (in the Schwartz-Frtchet 8cr = 0. Then there is a small B E Ao7j-’ with a/? = a.

topology) with

We consider the unit cube I = {t r, . . . , t,): 1t, 1I 1 } < lR” and P-forms in D depending on t. The type (i, j) of such forms is defined with respect to z only. The space of forms of type (i, j) is denoted by Ai*’ = A’*j(D x I). The derivative will be applied to the variable z E D, only. We now assume that D is a domain of holomorphy and that D” 3 D is a larger domain of holomorphy such that (D, D”) is a Runge pair. We get: Proposition 1.1. Assume that j 2 1 and that czE L4O-j with & = 0. Then there is a /? E AO*j-’ such that aj? = a. Zf f E A',' is a function with 8f = 0, then f can be approximated (in the Schwartz topology with respect to z and t) arbitrary well by functions g E Ao3’(DA x I) with ag = 0.

By using a partition of unity in open subsets of lR” it follows directly that this result remains true if I is replaced by an open subset of lR”. 2. Families of Domains of Holomorphy. The proposition can be generalized to non trivial families of domains of holomorphy. Assume that G c 47 x lR” is a domain such that every fiber G, of the projection rc: G + lR” is empty or a domain of holomorphy. Assume moreover that this family of domains of holomorphy is regular: If I c lR” denotes the open set rc(G) then, there exists a domain of holomorphy D c (c” such that a) n-‘(I) c D x I, b) for t E I the pair (G,, D) is a Runge pair with G, cc D.

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We denote by 17‘,j = 17’si(G) the cohomology of the parameter Dolbeault comof type plex .Ejja ‘,j. Here A’,’ denotes the complex vector space of P-forms (0, j) in G. We can use an exhaustion of G by locally finite unions of Cartesian products of domains of holomorphy with open subsets I’ c lR” and Runge approximations from D x I’. Then we get: Lemma 1.2. Assume that G c Cc” x JR” is a regular family of domains of holomorphy. Then ZT”,j = 0 if j > 0. The functions f E IT’*’ can be approximated by functions g E ZT’*‘(D x I) (in the Schwartz topology).

We take an integer q with 1 I q I n and put m = 2 .(q - 1) and use for the number n in the lemma the value n - m/2. Using an induction on the number of differentials dzj in lR” of forms c1we prove the following Theorem 1.3. Assume that G c C:” is a regular family of domains of holomorphy of dimensionn - m/2 over lR2’4-“. Then Ho,’ = 0 for j 2 q. The cocycles Z’+-’ c A”,q-l can be approximated by cocycles Z”*4-‘(D x I) c AO*q-‘(D x I). 3. Pseudoconvexity. In the following we have to use the notion of strictly q-pseudoconvex function. Assume that G c (c” is a domain, z E G an arbitrary point and p a C” real function in G. Definition 1.4. The function p is called strictly q-pseudoconvex in z if and only if the Levi form UP) = C p,,,-,(z) dz, dz, x.1

has at least n - q + 1 positive eigenvalues (with q = 1, . . . , n + 1). The function p is called strictly q-pseudoconvex in G if and only if p is strictly q-pseudoconvex everywhere in G. Thus q = 1 is the strongest property. Every function p is strictly (n + l)pseudoconvex. The strictly 1-pseudoconvex functions are just the strongly plurisubharmonic functions. We call a domain G c (c” q-convex if there exists a strictly q-pseudoconvex function p in G such that p(z) converges to a fixed value b I co with p(w) < b for w E G as z tends to the ideal boundary of G. Using the Levi theorem in chapter V it follows that every l-convex domain is a domain of holomorphy. We obtain: Proposition 1.5. Assume that p is a strictly q-pseudoconvex function in a domain G c C, that z’ E G is a point and that D = {z E G: p(z) < p(z’)} CC G. Then after an unitary rotation about z’ there is a polydisc Q = {z E C:“: IzP - z/I < E,, p= l,..., n} c G with E, > 0 centered at the point z’ such that for the projection 71:(z,, . . .) z,) + (z,, . . .) z~-~) the family n: Q n D + R2’(q-1) is a regular family of domainsof holomorphy.

The property that every pair (Q, n D,, Q,), t E R2(q-1),is a Runge pair follows from [DG60, p. 96, theorem (5)]. This follows after enlarging Q somewhat. By our last theorem we obtain:

H. Grauert

264

Theorem 1.6. Assumethat p is a strictly q-pseudoconvex function in a domain G t C”, that z’ E G, and that D = {z E G: p(z) < p(z’)}. Then there are arbitrary small Stein neighborhoods(= domainsof holomorphy) Q of z’ with Hj(D n Q, 0) = 0 for j 2 q, and that the cocycles of Z o*q-l(D n Q) can be approximated by those of z O*“-‘(Q). 4. Pseudoconcavity. We consider in Cc”for 1 5 q I n the following domains: A, = (z E C”: +
withp=q,...,n

and A,=(z~C’:~z,~<)for;l=l,...,

q-l,lzll
forI=q

and put A = Up=0,q,...,nAp9 P = i z E C”: Iz~J < l}. The following to J. Frenkel (Strasbourg).

,..., n}, lemma is due

Lemma 1.7. Hj(A, 0) = 0 for all j # 0, n - q, and every holomorphic function

f E H’(A, 0) extends analytically to H’(P, 0). From this it can be derived: Theorem 1.8. Assume that G c cc” is a domain and that p is a strictly qpseudoconvex function with 1 I q < n and let z’ E G be a point. Put D = {z E G: p(z) > ~(2’)). Then there are arbitrary small Stein neighborhoodsQ(L) c G such that a) the restriction map H”(Q, 0) + H”(D n Q, 6) is bijective, b) Hj(QnD,@=OforO<j
We call such a domain D q-concave in the points of aD n G.

$2. Complex Spaces 1. The Syzygy Theorem. For general theory see chapter I, 5 1.7. In this chapter denote by 0’ the ring of convergent power series in a point z’ E C” and by % a finite module over 0’. Since 0’ is a Noetherian ring we can find a free resolution %“I’ of %’ of the form: . . . +~+5&... + % 0 -+ %“I’ + 0.

Here the 6 are finite free modules over 0’. We put %-1 = %’ and %-Z = 0. By the classical syzygy theorem it follows that there is a least integer e 2 0 such that the kernel of the map yewI: %e-l + %e-Z is free. This number is independent of % and is called the homological dimensionof %‘. We always have e 5 n. If % is a coherent analytic sheaf over a domain G c C” and z’ E G is a point we, can perform the free resolution over a full neighborhood U(z’) c G. Then the stalk of the kernel sheaf of the sheaf homomorphism ye-I: %e-1 -+ %e-z is free over z’. It follows that the kernel sheaf is locally free over a full neighborhood W(z’) c U. Hence we have for z E W the inequality e(z) I e = e(z’), i.e. the homological dimension is upper semicontinuous.

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In this chapter we shall always assume that the complex spaces have a countable topology; otherwise they may be arbitrary. If X is such a complex space and F is a coherent sheaf over X and if x’ E X is a point there is a neighborhood U(x’) c X and a biholomorphic embedding I,/?:U + G of U into a domain G c C”. The sheaf 91 U becomes a coherent sheaf 9;’ over G. We put z’ = $(x’) and define the homological codimension cdh,.(F) of 9 in x’ as the number n e(z’) 2 0. It is seen rather easily that cdh,.(F) is independent of the embedding $ and that the inequality cdh,.(P) I dim,. X is always valid. We define cdh,(X) = cdh,.(0), where 0 is the structure sheaf of X. One has the following properties: a) cdh,.(F) is lower semicontinuous. b) If X is a reduced complex space and x E X is a point with dim, X 2 1, then cdh,X 2 1 also. c) If X is a normal complex space and x E X is a point with dim, X 2 2, then cdh,X 2 2 also. d) We call a complex space X a complete intersection at a point x’ E X if there exist a biholomorphic embedding $: U(x’) --* G c C” and holomorphic functions fi, . . . , f,-i in G with i = dim,, X which generate the ideal sheaf of the complex space $(U) at each of the points z E G. In this case we have dim, X = i for all x E U and the relation cdh,X = dim, X, there. The assumption is always satisfied if x’ E X is a smooth point of X. On the other hand, if cdh,X = dim, X for all x E X, then the space X need not be a local complete intersection, i.e. a complex space which is a complete intersection at all of its points. However, such complex spaces have locally constant dimension. They are called Cohen-MacCauley spaces. We put cdh(F) = min cdh,(F) and call cdh(X) the homological codimension of X. 2. q-Convex and q-Concave Complex Spaces. We consider an arbitrary complex space X. If p is a continuous real function in X and x’ E X is a point, then p is called strictly q-pseudoconvex in x’ if there is a neighborhood U(x’) and a biholomorphic embedding Ic/: U + G of U in a domain G c C” and a strictly q-pseudoconvex function p’ in G with pl U = p’ 0 I,$. Of course, the existence of p’ does not depend on the embedding $. Assume that B c X is a non empty open subset of X and that x’ E 8B = B B is a boundary point. We define Definition 2.1. B is q-convex in x’ if there exists a neighborhood U(x’) c X and a strictly q-pseudoconvex function p in U such that B n U = {x E Ulp(x) < p(x’)}. We call B q-concave in x’ if U and p can be so chosen in such a way that B A U = {x E Ulp(x)

> p(x’)).

We say that B is q-convex (respectively q-concave) if B CC X and 8B is q-convex (respectively q-concave) in all of its points. This notion carries over to complex spaces without boundary:

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Definition 2.2. A complex space X is q-conuex (q-concaue) if there is a compact subset K c X and a continuous function p in X with the following properties: a) p is strictly q-pseudoconvex in X - K, b) there is a number b E R u {co} (respectively E R u -co) such that limx,,, p(x) = b, where 13x denotes the ideal boundary of X, c) p(x) < b (respectively p(x) > b) for x E X.

For such a function p we get: The domains B, = {x E X/p(x) < c} with c < b (respectively = {p(x) > c} with c > b) are contained as relatively compact sets in X and are for c > cO = sup(p(K)) (resp. c < c0 = inf(p(K))) q- convex (respectively q-concave) domains in X. Their ideal boundary contains that of X.

Here X has an exhaustion. We call X q-complete if X is q-convex and K can be chosen as empty. If X is compact, then X is called O-convex. This is the strongest property, while q-convex implies (q + 1)-convex. Because of the maximum principle for q-convex functions (if q I dim, X) an analogue of q-completeness does not exist for the concavity. Of course, for q > 1 the stronger (q - 1)-concavity implies the weaker q-concavity. 3. The Frkhet Topology in the Space of Chech Cocycles. We denote by 9 a coherent sheaf over our complex space X and by 9(X) the module of cross sections in 9 over X. We introduce a Frechet topology in 9(X). If x’ E X is a point then, there are neighborhoods U(x’) cc V(x’), a biholomorphic embedding I,+: V + G c (c” of I/ in a domain of holomorphy G, a smaller domain of holomorphy B cc G such that U = e-‘(B), and a sheaf epimorphism p: Cop-+ $,(R[ V) over G, where 0” denotes the direct sum of 0 with itself p times. Then, ifs E 9(X) is a cross section the image section $,(sl U) is the image of a bounded p-tuple f E Lop(B) of holomorphic functions: We just use the results of Stein theory for domains of holomorphy. We put llsll c = min/ sup 11f (B)II. Now we take a (fixed) open covering U of X with such domains U. The llsllU are seminorms in 9(X) and introduce there a Frechet topology. The open mapping theorem of Banach on surjective continuous linear maps of Frtchet spaces states: Assume that a’ and 9 are Frechet spaces and that T: S’ + 9 is a continuous epimorphism. Then z is an open map.

Since two of our coverings U always have a common liner, it follows that our Frechet topology in F(X) is independent of the covering U and the maps $. So S(X) has a unique Frechet structure. If U c X is an open subset, then U is again a complex space. So the module of cross sections 9(U) is a Frechet space. If U is a locally finite open covering of X and Cj(U, 9) is the complex vector space of (countable) j-dimensional Chech cochains with coefficients in 9 respectively to U, then Cj(U, 9) is a

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countable direct product of spaces of cross sections over open sets and therefore Cj(U, 9) has a unique Frechet structure. The set of cocycles is equipped with the relative topology. We introduce in the set Hj(X, 9) of (flabby) cohomology classes a topology, which however, in general, is not Hausdorff. First, we take a locally finite Stein covering U of X. There is a trivial spectral sequence which connects Hj(X, 9) with the Chech cohomology group Hj(U, 9). In this way we can identify H’(X, 9) and Hj(U, 9). We obtain a topology in Hj(U, 9) since there is the epimorphism from the space of cocycles o: Zj(U, 9) + Hj(U, 9). A neighborhood of a point is just the image of a neighborhood in Zj. By using translations in Zj(U, 5) it follows that w is continuous. So we have also a topology in H’(X, 9). Since for two different Stein coverings U’ and U there is a trivial spectral sequence identifying Hj(U’, LP) with Hj(U, 9) and the canonical maps from the total cocycles to the cocycles with respect to u’ and U are surjectice and continuous, we get the same topologies in Hj(X, 9) regardless of whether we use U’ or U. This follows again from the Banach theorem. So our topology is unique. There are simple examples where it is not Hausdorff. We also get: a) Assume that U is a locally finite open covering of X and x E Zj(U, 9) is a small cocycle (in the Frtchet topology of Zj(U, 9)) then x represents a small cohomology class x E Hj(X, 9). b) Assume that-X = G c (c” is a domain and that c( is a &closed P-form of type (0, j) in G which is small in the Schwartz Frechet topology, then CIrepresents a small cohomology class g E Hj(X, 9). It should be remarked that restrictions are always continuous. 4. Extension of Cohomology. Assume that 9 is a coherent sheaf on the complex space X, that B cc X is a q-convex domain and that x’ E 8B is a boundary point. There is a neighborhood U(x’), a biholomorphic embedding Ic/: U + G of U into a domain of holomorphy of (c”, a strictly q-pseudoconvex function p’ in G with U n G = {p(x) < 0} for p = p’ 0 $ and a free resolution

0 -+ co’e + (p,-l + . , . + (po --+ $*(~I with cdh,(F)

U) + 0

= n - e. By using theorem 1.6 we get:

Theorem 2.3. There are arbitrary small Stein neighborhoods Q(x’) c U such that H’(B n Q, 9) = 0 for all j 2 q and such that the Chech cocycles of Z4-‘(B n Q, 9) can be approximated by those of Q.

The cocycles are given with respect to arbitrary locally of B n Q respectively Q. The map Zqel(Q, 9) -+ Zq-l(Q n a chosen map of sets of indices, as usual. In the same way we treat the q-concave case. Assume cave, that x’ E aB and that 9 is a coherent sheaf over X.

finite Stein coverings B, 9) corresponds to

that B c X is q-conThen we have:

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Theorem 2.4. There are arbitrary small Stein neighborhoods Q(x) such that a) tf q < cdh,,(P), then the restriction map H”(Q, P) + H”(B n Q, 9) is bijective, b) H’(B n Q, 9) = 0 for 0 < j < cdh,.(s) - q.

We have the canonical flabby resolution

of F over X. For instance, w. just consists of the germs of not necessary continuous local cross sections in 9. The j-th cohomology over B n Q is represented by a cross section s in 3 which maps to 0. If j 2 q in the q-convex case and 0 < j < cdh(P) - q in the q-concave case, we can alter s just over B n Q without changing its cohomology class such away that we get SIB n Q = 0, Now we can enlarge B by adding a small bump in Q which smoothes in C” manner to B near to 8Q, such that the new B is again q-convex (q-concave). We denote this new B (simply) by B”. Then s also is a cross section over B” and our cohomology class is extended to B”. We see that this extension can be established simply by extending the original s to B^. In the q-concave case follows that the extension is uniquely determined, since in the case j > 0 the cohomology of dimension j - 1 of B n Q can also be extended to Q (also if j - 1 = 0). The same is true in the case of q-convexity if j > q. In the case j = 0 of q-concavity with q < cdh(%) the cohomology classes are just cross sections in 9 and their extension is unique. If B is q-convex and j = q, then the (j - l)dimensional Chech cocycles in B n Q can be approximated by cocycles in Q. So two different extensions from B to B” differ only by the cohomology class of an arbitrary small cocycle in B” which is supported in B n Q. To prove this we have to use a partition of unity to the covering of B” with the two elements 4 Q. Assume now that X is q-convex. Then applying our bumping methods in various points x’ E f3B,, c 2 co we obtain an extension of Hj(B,, 9) to a Hj(Bc+v F) for j 2 q for a number E > 0 with c + E < b. This E cannot become arbitrary small as long c I b - 6 with 6 > 0. For j > q this extension is unique, for j = q the difference of two different extensions is arbitrarily small. The extension can always be established just by the extension of a cross section in I$$ over B, to such a cross section over B,,,. If the given cross section is small, the extension is also small. Thus we have proved: Theorem 2.5. If B is q-convex and j 2 q, then the restriction map Hj(X, 9) + Hj(B,, F), c > co is surjective. If j > q then, this map is injective, while in the case j = q the di#fernce of two extensions is arbitrarily small.

A similar result holds in the q-concave case: Theorem 2.6. If 0 I j < cdh(9) H’(B,, 9) to H’(X, 9) if c < co.

- q, then there is a unique extension of

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0 3. Finiteness Theorems 1. Finiteness. Assume that 1 < q I n and that X is a q-convex (respectively q-concave) complex space. Then there is a compact set K c X and a continuous function p which is strictly q-pseudoconvex in X - K and a real number b I co (respectively a real number b 2 -co) such that p(x) < b (respectively p(x) > b) forx~X-Kandlim,,,,p(x)=b.WeputB,=Ku{p(x) c} for c0 > c > b). Then all the B, are q-convex (resp. q-concave) and there is a positive number E such that for any coherent sheaf 9 the cohomology classes of Hj(B,, 9) extend to Hj(B,+,, 9), if j 2 q (resp. extend to Hj(B,-,), if j < cdh(9) - q). We take locally finite Stein coverings U of B, and U” of B,,, (respectively B,-,) such that U is completely finer than U”: i.e. there is a map r: U + U^ such that for all U E U the Stein space U CC r(U) E U”. The vector spaces I/ = Zj(U, 9) and I/” = Cl-‘(U, 9) x Zj(U”, 9) are Frechet spaces. The restriction map u: VA + V, (d, e) -+ e[U is completely continuous or compact as one also says and the map U: VA + V, (d, e) --* 6d + elU is surjective. Here 6 denotes the Chech coboundary. By a theorem of L. Schwartz (cf. [Sc53]) then follows:

a) The image of the map (u - u)(V”)

= Bj(U, %) is closed (here Bj is the space of coboundaries). b) The quotient oector space V/(u - u)(V”) = Hj(U, %) has finite complex dimension. Frtchet

In the q-concave case with j = 0 we have to modify the proof somewhat. We just have to put the Cj-’ to 0 in I/“. In all cases we find that Hj(B,, 9) has finite dimension. In the q-convex case we get, since Bj(U”, 9) is closed and the difference of two extensions of a cohomology classes to #@I”, 9) is arbitrary small, that the extension is unique: i.e. the restriction Hj(B,+,, 9) + Hj(B,, 9) is an isomorphism for j 2 q. In the q-concave case we have proved already that Hj(B,-,, %) -+ Hj(B,, 9) is an isomorphism if j < cdh(9) - q. Hence by extension to the full space X we get: Theorem 3.1. Assume that X is a q-convex (q-concaue) complex space, that c > c,, (respectively c < co), that % is a coherent sheaf on X, and that j 2 q (resp. j < cdh(6) - q). Then a) The cohomology groups Hj(X, %) have finite dimension. b) The restrictions Hj(X, %) + Hj(B,, %) are isomorphisms of vector spaces. 2. Some Further Results. First, we consider here the case of a O-convex complex space X. This means that X is compact. In this case the methods for proving the finiteness of cohomology are without exception. So we have: Theorem 3.2. If X is a compact complex space, and % is a coherent sheaf on X then, all cohomology groups Hj(X, %) with j = 0, 1, 2, . . . haue finite complex

dimension.

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The vanishing of cohomology holds for q-complete complex spaces. In this case the compact set K is empty and we can also for B, take the empty set. Therefore we have: Theorem 3.3. Zf X is a q-complete complex space, then we have the vanishing Hj(X, %) = 0 for all j 2 q.

If no irreducible component of the reduction of X is compact, then it is possible to prove that X is n-complete with n = dim X (see for proof [Oh84]. This works for non reduced complex spaces, also). So we get: Theorem 3.4. If no irreducible component of a n-dimensionalcomplex spaceX is compact, then H”(X, %) = 0 is true.

This important result implies e.g. that the first cohomology with coefficients in an arbitrary coherent sheaf 9 of a connected non compact Riemann surface always vanishes and this implies the existence of many non constant holomorphic and meromorphic functions. The Mittag-LelIIer and Weierstrass problems can always be solved. Moreover, all non compact Riemann surfaces are Stein manifolds. Take now a complex space X and an analytic subset A c X which has in all of its points at least the codimension q. We denote by A” the manifold of smooth points of A, which as a set is dense in A. Of course, X may be non smooth in some points of A”. In every point x’ E A” we can find a neighborhood U(x’) c X - (A - A”) and a strictly q-pseudoconvex function p in U which vanishes precisely on A. So X - A is q-concave in x’. Assume now that F is a coherent sheaf in X and that j is an integer with j < cdh(9) - q. Then by theorem 2.6 every cohomology class x E Hj(X - A, 9) can be extended uniquely to Hj(X - (A - A”), 9). Of course, we have to go into the proof of theorem 2.6 again and have to see that the bumping is such that we reach the points x’ of A” immediately. By this we get an extension to any relatively compact open subset A’ c A”, but then also to A” itself. We then apply the same procedure for A - A” and go on so. Finally we get: Theorem 3.5. Assumethat A c X is an analytic set with codim, A 2 q for all points x E A, that % is a coherent sheaf on X and that j < cdh(9) - q. Then the restriction map Hj(X, %) + Hj(X - A, %) is an isomorphism.

A theorem of this type proved already in [Sj61]. 3. Projective Spaces. An interesting example of q-convexity is given by complex submanifolds Y of the n-dimensional complex projective space lP,,. There are infinitesimal projective transformations 4 which move Y in lR,,. By passing to the quotient of the tangent bundle of lPn by the tangent bundle of Y the vectors of ~,8give holomorphic cross sections s in the normal bundle N of Y in lP,,. So we can construct very many of such cross sections. If y E Y is an arbitrary point then the restriction of s to the first infinitesimal neighborhood of y in Y can be prescribed. From this it can be derived that N is positive (in the sense of Griffith).

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Proposition 3.6. Zf Y is a complex submanifold of IP,,, then the normal bundle N(Y) is a positive vector bundle.

Assume now that Y is a complex subspace of lf’,, which is locally a complete intersection of codimension q. But instead of lP,, we take, more generally, an arbitrary purely n-dimensional complex manifold X. So Y has codimension q everywhere and for any point y E Y there is a neighborhood U(y) c X and holomorphic functions fi, . . . , f, in U such that the ideal sheaf of Y in U is spanned by these functions. The homological codimension of Y is n - q = dim Y everywhere and the direct image of the structure sheaf by a finite local map of an open W c Y onto a domain G c C”-q is always locally free. The normal bundle of Y with respect to the nilpotent structure of Y is a vector bundle N of rank q on Y. It is well known that in this case with nilpotent structure it can happen even if X z lP,, that N is negative. We therefore assume that N is positive (in the sense of Griffith using differentiable functions on Y in the sense of Spallek, see [Sa65]). Now we get from [Fr76] and [Fr77] that X - Y is q-convex. Theorem 3.7. Assume that Y c IP” is a complex submanifold of codimension q or that Y is a local complete intersection with positive normal bundle in a complex manifold X. Then IF’” - Y, respectively X - Y is q-convex.

In the case of this theorem we get: If 9 is a coherent sheaf on X then the cohomology groups Hj(X, 9) have finite dimension for j 2 q. The assumption: Y is a complex submanifold of IP, is essential. If Y is an analytic set of codimension
- {0}, R’n,(CoP4-r))

with a subspace of H2(lP4 - Y c!JJ~~-~).

The sheaf R%,(O.. .) is not coherent but an infinite direct product of the structure sheaf of C2 - (0). So the dimension of H2(P4 - Y, 8) is infinite. There are examples of q-codimensional submanifolds Y such that for a coherent sheaf 9 in lPn - Y the cohomology Hj(lP” - Y, 9) for j 2 q does not vanish. Hence, lE’” - Y is not q-complete (see [Ba70] and [BF81]). On the other hand if Y is a pure q-codimensional analytic subset which is a set theoretical complete intersection then lF’” - Y is q-complete. If Y is such an intersection in a neighborhood U(Y), then lP” - Y is q-convex. It would be conceivable that stronger results are true but nothing is known in this direction.

3 4. Applications 1. Complex Spaces with Holes. Assume that X is a complex space and that U c X is an open subset with a strictly 1-pseudoconvex function p which has

the following properties:

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a) there are real numbers a, b with -co < b < a 5 co such that b < p(x) < u for x E U and such that for a’ and b’ with b < b’ < a’ < a the subdomain U’ = {x E U/b’ < p(x) < a’} is always contained as a relatively compact subset in U, b) if lim p(x) = b then, x converges to 3X (we call this limit set the inner boundary of X), c) forallxEXisdim,X 2 1. We say that such a complex space is a complex space with a hole. We call the complex structure of a complex space X” maximal if after a local embedding in a smooth domain, the ideal sheaf of X” cannot be enlarged in isolated points. If cdh X” 2 2, then its complex structure is maximal. - We indicate the proof of the following Theorem 4.1. Assume that X is a complex space with a hole and assume that the homologicul codimension satisfies cdh(X) 2 3. Then the hole can be filled: There is a maximal complex space X” which contains X us a subdomain such that for every b’ with b < b’ < a the union X’ = (X” - X) u U’ with U’ = {x E VP(X) < b’) 1s . oPen and relatively compact in X^ and every irreducible component of X’ enters in U’. This complex space X^ is uniquely determined up to an obvious isomorphy. Proof. (See [Ro65]). We take a, b’ with b’ < b^ < a and denote by U” the open set {x E U: p(x) < b”j and by 0 the structure sheaf of X. The boundary of UA consists of two parts corresponding to b and b”. We call them 3’ and a”. The first one is l-concave, the other one l-convex. We have q = 1 and j = 1 < cdh(X) - q and j 2 1 and the so called mixed case. In this case we can prove the finiteness of cohomology as well (see also [Si74]). So we find that the cohomology H’( U “, 0) is finite and the restriction H’( U “, 0) -+ H’( U’, 0) is an isomorphism. Moreover, if b” is sufficiently small for every point x’ E U n au, there is adivisorD c U” - U’ which is the O-set of a local holomorphic function h such that D n aU’ is just {x’>. The cohomology of the Cousin-I distribution given by the local meromorphic function l/h vanishes. So we have a meromorphic function f in U^ which is holomorphic in U^ - D and has on D the principal part l/h. If x, E U’ is a sequence of points which converges to x’ then f (x,) converges to co (it is not necessary that X has homological codimension 3 at least on U A - U’). This gives even more freedom in constructing f since the infinitesimal behaviour in x’ can be prescribed. Now we take many, but only finitely many of such functions f and construct a holomorphic map F: U’ -+ (I?’ such that we have the following properties: a) there is a spherical shell S = H - H’, where H 3 H’ are two concentric balls about the origin 0 in a:” such that U- = F-‘(S) is relatively compact in U’ while F is a biholomorphic embedding of U- into S; b) we have sup p(F-‘(8H’)) < inf p(F-‘(dH)). We denote by 9 the coherent ideal sheaf of the complex subspace Y = F(U-) in S. The homological dimension of 9 is at least 3. By [Bc76], p. 358, there is an extension 2 of 4 to a coherent ideal sheaf in H. If ,$ is maximal, then f is uniquely determined (it may happen in isolated points only that f is not maxi-

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mal). Clearly 2 is the ideal of a complex subspace Y A in H which does not have O-dimensional irreducible components. In S the spaces Y and Y” coincide. So by identifying X - F-‘(H’) with Y” over U’ we get a complex space X”. The identification map extends uniquely to U- u F-‘(H’) (the structure there is always maximal since cdh(X) 2 3). In the same way follows: If X”” is another hole repair of this kind by another complex space Y A“, then X” and X” A are isomorphic. 0 2. Two Dimensional Complex Manifolds. There are examples which state that in the 2-dimensional case a hole repair is not possible in general. Let us denote by X a line bundle over IP, with Chern class c1 =: -c < - 1. We denote by 9 the sheaf of local holomorphic cross sections in _X. We have dim. H’(IP,, 9) = c - 1 # 0. We take a non zero cohomology class given by a cocycle y = {Y,~} E Z’(U, F), where U is an open covering of Ip,. We twist _X by y employing over U,, the identification _Xl U,, ‘v _Xl U,, by z = z’ + yxA for the points of the fibres. We obtain a new fibred complex manifold Y over lP,. The libres are again (c, but the structure group is now the full affine group. This libre space Y has a holomorphic cross section s if and only if the cocycle y is cohomologuous to 0. There is a neighborhood v/cc _X of the zero cross section 0 with smooth boundary which is strongly pseudoconvex. The fundamental group of _X - v is Z/(c - 1)Z. If y is small enough, we obtain such a subdomain I/ also in Y just by a small pertubation. This V does not contain any non trivial compact analytic subset. Otherwise there would be an irreducible l-dimensional analytic set in 1/ which is a possibly branched multivalued holomorphic cross section in Y. By passing over to the barycenter we would get a holomorphic cross section and y would be cohomologuous to 0. Hence, V is a Stein manifold. The fundamental group of W = Y - v is also Z/(c - 1)Z # 0. So there is an unramilied covering X of W with c - 1 sheets. The 2-dimensional complex manifold X has a hole. Assume now that the hole can be filled in. Then we get a compact (normal) complex space X”. The covering map can be extended from X to X^. So X” is an analytic covering of Y. Since I/ is simply connected the covering X” has to have a branching locus A which is a multivalued cross section in Y. Passing to the barycenter we get an ordinary holomorphic cross section in Y which is a contradiction. So we have proved: Theorem 4.2. There are 2-dimensional cannot be repaired.

complex manifolds with holes which

3. Vanishing Theorems. Some vanishing theorems were proved in chapter V already. There are some such theorems which come from q-convexity. Assume that X is a q-convex complex space and that V is a (holomorphic) vector bundle over X. We call V (weakly) negative if there is a tube W around the zero cross section 0 in I/ with a strongly pseudoconvex (l-convex) differentiable boundary. We assume that the projection w -+ X is proper. By smooth-

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ing we prove that W is q-convex (take always q 2 1). So the cohomology with coefficients in any coherent sheaf on V for dimensions j 2 q is finite. The dual I/* of any negative vector bundle is said to be positive: the linear forms on a negative vector bundle form a positive vector bundle. The same is true for a symmetric power S’V* for i = 1, 2, . . . . These S’V* are given by the homogeneous polynomials of degree i on the fibers of V. We take a coherent sheaf 9 on X and lift 9 to a coherent sheaf J? on I’. This sheaf on V is the (topological) direct product of F and all the tensor products .F @ s’V*. Since the cohomology of &? is finite for dimensions j 2 q, we obtain Theorem 4.3. Assume that X is a q-convex complex space, that 9 is a coherent sheaf on X and that V is a positive vector bundle on X. Then there is a integer i, such that for all i 2 i, the cohomology Hj(X, 9 @ S’V) vanishes for j 2 q. Another interesting case is that X is a compact complex space (O-convex) and that V is a (holomorphic) vector bundle over X. We call V q-negative if there is a q-convex tube W cc V around the zero cross section. The dual of V is called q-positive. A vector bundle is l-positive if and only if it is positive. We find again Theorem 4.4. Assume that X is a compact complex space, that 9 is a coherent sheaf on X and that V is a q-positive vector bundle on X. Then there is a number i, such that for i 2 i, the cohomology H’(X, 9 0 S’V) with j 2 q vanishes. If X is a complex manifold and 9 is a locally free sheaf on X, we have analoguous theorems for q-negative vector bundles by the Serre duality. If X has singularities, a vanishing theorem will not be true in general. - The Serre duality theorem in its most simple form (for complex manifolds) will be treated in the next section. 4. Hulls for Cohomology. dratic form

(See [Grsl]).

Consider

a positive

definite qua-

where Re stands for the real part. We put D = (z E Cl?‘: Q(z) < 11, G = {(w, z): z ED, lg(lwl) where t(z) denotes a nowhere i

c,z,Z,

< t(z)}

and

n= m+ 1

negative function with 0 < ci I c2 < ..’ < c,.

a=1

Then D is an Euclidian ellipsoid. Hence it is elementary convex and then (strongly) pseudoconvex. It is clear that G is a Hartogs domain over D, which is 1-pseudoconcave in each of its boundary points (w, z) with z E D. We put t,(z) = t(z) - cq *(Q(z) - 1)

for q = 1, . . . , m.

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We then have t(z) =: to(z) I t1(z) I ... I t,(z) and, with G, := {(w, 4: z E D, Mlwl)

< t,(z)),

the inclusions The boundary of G, is weakly (n - q)-convex (a notion which is defined using local weakly (n - q)-pseudoconvex functions p). The domain G,,, is a domain of holomorphy. We prove: Theorem 4.5. Every cohomology class y E H’(G, 0) has a unique extension to G,,,-,. There is a cohomology class @ E H’(G,,-,, 0) which is singular in each of the boundary points of G,,-, over D, i.e. it cannot be extended locally into a full neighborhood of the point.

We call G,,-, the hull of G for the l-dimensional cocycles. In the situation of general domains G the analytic extension is not unique as it is in the case 1 = 0 of holomorphic functions. So a hull like the hull of holomorphy for holomorphic functions does not exist for cohomology then.

0 5. Serre’s Duality

Theorem

1. Resolutions. Assume that X is a n-dimensional complex manifold and that S is a locally free sheaf on X. As usual we denote by 0 the structure sheaf of X. We have the Dolbeault resolution of 0, i.e. the resolution of 0 by the germs of C” exterior forms of type (0, j) with j = 0, 1, . . . , n. We can tensor this with 9 and passing to the global cross sections over X we get the complex of (0, j)-forms on x:

A’ = A’(9q: A”q$q

-+ AO”(&F) -+ . . . + AOqq

+ 0.

The derivative in this complex again is denoted by a. By the Dolbeault theorem we have: Hj(X,

9) 2: Hj(A’)

for j = 0, . . . , n.

Now Dolbeault’s lemma is of essential importance: Assume that P c C” is a polydisc. Then the Dolbeault complex A’ over P is exact. We need also that 9 is trivial over P. That means that 9 1: 0’ where C!Y

denotes the r-fold direct sum of 0. This follows from [Gr58] (see better [Ca58]). For the duality we need moreover the notion of currents. A current is briefly speaking an exterior form on X whose coefficients are distribution cross section in 97

276

H. Grauert

More general currents can be defined even on non oriented differentiable manifolds. Then a current is in local coordinates like an exterior form with distribution coefficients. But if we change the local orientation by a coordinate transformation the current multiplies by - 1 and then transforms like an exterior form. The name current is derived from this property. It behaves like a stream.

To get an exact definition of currents of type (i, j) we need test forms. We take the dual P* of 9 and denote by A;-i*n-j = Ai-i*“-j(B*) the exterior forms y of type (n - i, n - j) in X with coefficients in 9* and with compact support. The space At-i-n-j with the Schwartz topology is a Frechet space and its (topological) dual, which we denote by T’,j(9), is the space of currents of type (i, j). If x is an exterior form of type (i, j), on X we have the integral Ix x A y for all test forms y. The map A;-isn-j + Cc is linear and continuous. Hence 1 is a current. We have A’~j(F) c T’*‘(F). The derivative 3: T’,‘(9) + T’*j+‘(Y) is defined as follows: (J(x))(y) = -(- l)“j.x(&). It generalizes the derivative for exterior forms. It is possible to prove that the sequence of the sheaves of germs of currents of type (0, j) is a resolution of the sheaf 97 It is essential that in the space of P-exterior forms of type (i, j) with j < n on a shell of polydiscs the set of coboundaries is closed in the Frtchet-Schwartz topology and that in the case n = 1 the set of those holomorphic functions in such a shell which can be extended analytically to the full polydisc is closed. Then in a polydisc the coboundaries B in the space of test forms A, is closed and by the Hahn-Banach theorem a continuous functional on B can be extended to a continuous functional on A,. For the proof we make the following consideration: For j < n - 1 every cocycle on a shell of polydiscs is a coboundary. So in this case there is no problem. We have only to consider the (n - 1)-dimensional cohomology. We need the following situation: Assume that A c (c” is the unit polydisc. For a real number t with 0 < t < 1 denote by U, the polydisc {z = (zl, . . . , z,,) E A: t < lzll < l}. Then U = {Un} is a Stein covering of a shell of polydiscs. We put U = U, n ... n U,. Then every holomorphic function over U is an (n - l)-dimensional cocycle with coefficients in (??It has a Laurent series L and it is a coboundary (in the shell) or in the case n = 1 can be extended analytically to the full polydisc if and only if all terms in L with all indices negative vanish. From this follows immediately the desired fact that the coboundaries are closed. We have the complex of currents:

and the isomorphy Hj(X, 9) N Hj(T’). We can also take the cohomology with compact support: H&(X, 9) and the currents T’,*‘(9) with compact support. Then we get an isomorphism H&X, 9) N Hj(T’,). The same is true for the resolution by exterior forms. 2. Compact Support. We denote by x the canonical sheaf on X, i.e. the sheaf of holomorphic (n, 0)-forms. Then the germs of C” exterior forms of type (n, j) with coefficients in 9 and of currents to 9 of type (n, j) give two resolu-

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277

tions of the sheaf 9 0 3?. We take a continuous linear form x on A”,j(F). Then x is defined on forms with compact support as well. So x is a current of type (0, n - j) with coefficients in >r*, But this x has compact support by the following argument: If y is a test form of type (n, j) with support very close to the ideal boundary of X, then x(y) has to be 0. Otherwise x could not be continuous. We have 8, = 0 if and only if all X(&C) = 0 for all test forms c( of type (n, j - 1). In this case x defines a linear map -x: Hj(An7’) -+ (c, provided x has compact support. We assume now that X is q-convex with q = 0, 1, 2, . . . , n and that j r q. Since the cohomology is finite and the space of coboundaries is closed, any linear map x: Hj(X, B @ K) + C comes from a continuous linear map from the vector spa& of cocycles of type (n, j) into C:. By Hahn-Banach it can be extended to a continuous linear form x: A”*j --, C. This is a &closed current of type (0, n - j) with compact support. If x’ is obtained in the same way from 1, then we put $(&) = (x - x’)(a) for all u E A”*‘. Thus we get a continuous line& map II/ from the space of coboundaries in A”*j+l into C such that $ can be continued to a current of type (0, n - j - 1) with compact support (since j + 1 2 q and the space of coboundaries is closed). This continuation will be also denoted by I,$. We have a(+) = x - x’. So we have a unique map of the dual T: (H’(X, 9 Q Jr))* + zYz;;-‘(X, F*). On the other hand, if x is a current of type (0, n - j) with coefficients in R* and compact support, then x defines a continuous linear form A”,’ + C. If the current x = a($), where II/ again has compact support, then X(U) = 0 for every cocycle CL.So x is 0. This means that z is injective. If we have a cohomology class of H;f-j(X, F5) then this can be given by a cocycle c(E A”*“-j(9*) with compact support. Then CI defines a x: A”*j(9) + Cc. So r is an isomorphism. We have proved: Serre’s Duality Theorem 5.1. Assume that X is a q-convex complex manifold and that 9 is a locally free analytic sheaf on X then for each j 2 q there is a natural isomorphism 5: (Hj(X, S Q ,X))* N H;;-j(X, 9*). It is given by a pairing Hj(X, 9 Q ,X) x H:-‘(X, S*) --t C, which is nothing else than the cup product. Zf the cohomology classes are given by differential forms, then the pairing is the integral Jx x A *. 3. Applications. First assume that X is a compact complex manifold of any dimension n. If 9 is the sheaf of local cross sections in a vector bundle V on X, we have H’(X, 9) = H”(X, S* 0 3). If 9 is positive there may be many holomorphic cross sections in I/. Then also the nth-cohomology of X with coeflicients in 9* 0 Z? will be large. This is the case on the n-dimensional complex projective space Pn for the sheaf D 0 Z where S* = Q is the sheaf of covariant vectors. If V is a q-negative vector bundle on X then the dual I/* is q-positive. So the cohomology H’(X, x @ S’V*) vanishes for j 2 q if i is sufficiently large. From

278

H. Grauert

the Serre duality follows the vanishing of Hj(X, S’V) for j I n - q and large i. There are examples displaying that such a vanishing theorem for negative vector bundles is valid in general on smooth manifolds only. If X is a compact complex manifold and I/ is a negative line bundle then by the vanishing theorem of Kodaira all cohomology groups Hj(X, V) are zero for j < n. It can be seen that the vector bundle V = Sz of covariant vectors over X = lPnis negative. But from Kahler theory it follows that even H’(X, V) has dimension 1 for arbitrary large n. So the Kodaira oanishing theorem, which is called a vanishing theorem of the strong kind, since it does not employ the S’, is not true for vector bundles of higher rank. For vector bundles we have a stronger negativity (in the senseof Nakano). For this the strong vanishing is valid. For proof we have to use Kahler theory, and in our context all this cannot be done and even the Kodaira vanishing theorem for line bundles cannot be proved (one needs elliptic or p-adic theory; for the last case seethe methods of Deligne and Zllusie). If X is a (connected) compact n-dimensional complex manifold, there are only the constant holomorphic functions on X. So dim. H’(X, 0) = 1. By Serre’s theorem we get dim. H”(X, 2) = 1. Assume now that D c X is a diuisor. So D is the union of 1-codimensional irreducible analytic setswith integral multiplicity. The divisor -D is the same union but with the negative of the previous multiplicities. By (D) we denote the sheaf of local meromorphic functions belonging to D. That are the meromorphic functions which only have poles on D of order equal to the multiplicity belonging to D (negative order means zeros!). This sheaf is the sheaf of local cross sections in a line bundle, which is also denoted by (D), and (-D) is the dual of (D). We call a divisor D holomorphic if all the multiplicities are positive. Since then H’(X, (-D)) is zero, we establish the vanishing of H”(X, (D) 0 .X). A special caseis when X is a compact Riemann surface. Here H’(X, (D) @ 2’) is zero for any holomorphic divisor D. We have an isomorphism H’(X, 0) = H’(X, Q) with Sz = &‘Y The elements of H’(X, Q) are the holomorphic l-forms on X and were called abelian differentials of the first kind in the classical literature. We consider the topological cohomology group H’(X, C) of X with coeflicients in the constant sheaf C. An element of this group can be given by a exterior a:“-form $ = I,+‘*’ + $‘T’ with dl(/ = 0. We have a$l~o = 0 and &Go9’= 0. We put g = dim. H’(X, 0) and call this number the genus of X. The elements of H’(X, 0) are represented by forms $“T’ with & = 0. If x E Z’(X, Q) = Q(X), then the conjugate 2 is a form $‘*l with &Go3’ = 0. If x # 0 the cohomology of X in H’(X, Co)is different from 0. Otherwise we could find a function f on X with 3f = x. But then we would have Af = 48f = - 4& = 0. So f would be harmonic and thus constant in view of the maximum principle and 8f = x could not be valid. Because of the Serre duality there are forms xl, . . . , xs E Q(X) such that 2, span H’(X, 0). Now it follows: If II/ is a closed l-form on X then there Xl,..., is a x E 0(X) such that Go*’ - X = c?f and IJ~- d(f) E Z’(X, Q) + Z’(X, a).

VI.

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219

Every cohomology class of H’(X, Cc)is represented uniquely by an element of the direct sum H’(X, Q) + H’(X, 0). The two summands are isomorphic. So dim. H’(X, a:) = 2g and the genus g is a topological invariant. We have established Theorem 5.2. If X is a compact Riemann surface of genus g, then there are g abelian differentials of the first kind on X.

It is now easy to prove the theorem of Riemann Roth. Denote by B the sheaf of local holomorphic cross sections in a holomorphic line bundle on X. Then the Euler-Poincare characteristic is defined by x(F) = dim. H’(X, 9) dim. H’(X, 9) = dim. H’(X, 9) - dim. H’(X, 9* 0 a). If D is a divisor on X, the number of points (with multiplicity) of D is called the degree of D and is denoted by d = IDI. It is an integer. In the case of a holomorphic divisor D we have the exact cohomology sequence 0 + H”(X, 9) + H”(X, 9 @ (D)) + H”(X, 9 @ (D)l D) = Cd

Like in [Hi561 it follows from this fact that ~(9 0 (D)) = ~(9) + 1Dj. For holomorphic divisors D and D’ this means the equation: x(D) = x(D - D’) + ID’/. For arbitrary devisors we have )D, + D, I = ID, ) + ID, I. An arbitrary divisor D is a difference of two holomorphic divisors. Since x(O) = 1 - g we get the equation x(D) = 1 - g + 1DI. Since any (line bundle) sheaf B can be written as (D) for some divisor D, we finally obtain ~(9) = 1 - g + ID I. The divisor D corresponding to 9 is not uniquely determined but its degree is. We call ID 1 the Chern number c(9). So we have the: Riemann-Roth Theorem 5.3. Assumethat .?Fis the sheaf of local holomorphic cross sections of a holomorphic line bundle on a compact Riemann surface X of genusg. Then the Euler-Poincare characteristic ~(9) is given by 1 - g + c(9).

5 6. Algebraic Function

Fields

1. Pseudoconcave Complex Spaces. In this section 6 we use the papers [AG61] and [An63]. We assume always that X is a n-dimensional connected normal complex space. If U c X is an open subset and K c U is non empty, we have the holomorphic hull K” = (x E U( )f(x)1 I sup If(K)1 for all holomorphic functions f in U}. The set K” is closed in U and contains K. We call X pseudoconcave if there is a relatively compact subdomain B cc X such that for every boundary point x E 8B there exist arbitrarily small neighborhoods U(x) cc X such that x is an interior point of (U n B)“. If U is a relatively compact neighborhood which is covered by a set of l-dimensional analytic subsets A with 8A c B then U has the desired property. We get:

280

H. Grauert

Proposition 6.1. A complex space X is pseudoconcave if for some subdomain B CC X every point x’ E aB has the following property: There is a neighborhood U(x’) and a strictly (n - I)-pseudoconvex function p in U such that U n B = {x E u: p(x) > O}.

The assumption of the proposition means that LJB is (n - 1)-concave in x’ in the sense of the definition 2.1. But in general our pseudoconcavity is much weaker. 2. The Schwarz Lemma. There are at most n analytically (= meromorphitally) independent meromorphic functions on X. Let m I n be the maximal number. We choose m such functions: fi, . . . , f,. Then fi, . . . , f, are also algebraically independent. If x’ E X is a point, we can find a neighborhood W(x’) which can be represented as a b-sheeted analytic covering Z: W -+ G over a ball G c C” around the point 0 E C” with x(x’) = 0. First we perform an arbitrary small biholomorphic transformation G N G such that thereafter 0 is no longer in the branching locus of 7~and all the functions fi, . . . , f, are holomorphic in the inverse image S = 7(-‘(O) and give a smooth fibration of a neighborhood of S in m-codimensional analytic sets. We denote by U the inverse image of a concentric, somewhat smaller ball G’ CC G. If h is a holomorphic function in U (which in this case can be, more generally, an arbitrary subdomain) we put 11 h 11Li = sup Ih( U)l. If V(x’) is a neighborhood of S with V CC U, then there exists a number q with 0 < q < 1 such that for every holomorphic h in U vanishing of order k in the b points of S, the inequality

Ilhll, I d‘llhllu is valid. This is the Schwarz lemma. If f is a meromorphic function on W, then there is a nowhere identically vanishing holomorphic function d in W (denominator) such that h = f. d is holomorphic. We then put llflld = Ilflld,v = Ilhll,. If d’ is another denominator in an open subset W’ c W, we make W’ somewhat smaller. Then there is a number A4 2 1 with IIf II,,, I M. llfll,, over W’. However, M depends on d and d’ but is independent on J The Schwarz lemma gives Ilf I[,,” I qk Ilf II,,” provided if f (that means h) vanishes in x’ of order k. We assume always that f is analytically dependent on fi, . . . , f,. That f vanishes of order k in S puts some conditions on J The number of linearly independent such functions f, which satisfy the conditions, is: b.(k+z-l)=(b/m!)km+-.. 3. Analytically Dependent Meromorphic Functions. We take open coveringsof~:2D={W,:~=l,..., ~(,},U={U,:~=l,..., ~*}and’I)=(I$:~= 13 . . . . p*} with V, CC U,, CC W, and W,, UP having the properties of the last

VI. Theory

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subsection. We do this in such a way that U, is always contained in holomorphic hull with respect to W, of W,’ n B, i.e. in the set (W,’ n B)“, where is a relatively compact subdomain of W,. We take for b and q the maximum the b respectively q belonging to p. We denote with 11 11iI the maximum of norms with respect to the elements For the functions fi, . . . , f, in 1, . . . , CL*and a common M 2 1. If have another number M’ 2 1. We of the following formal expressions: f= withx-0

,...,

i-land2,+...+I,=O

281

the W,’

of the of U. We do the same for the covering ‘$3. W, we have common denominators d,, u = g is another meromorphic function on X, we consider the vectorspace ~i,j of formal sums

gx~fp~..:f,“,..., j-l.Wehave

For the functions out of Fo,j the j-th powers di are good for denominators. Therefore the j-th power Mj will serve for the estimates. Similarily for the f E Fi, j the constant (M’)’ . Mj will do. If a function f E Fi, j vanishes in the S,, of order k, then we have the inequality llfll n I qk. I/f 11II. By looking at the intersections V, n W, n U” for fixed p and passing to W,’ n B and then to U, by the holomorphic hull, we get II f II U I (Ml)‘. Mj. Ilf II \n and hence IIf I( n I qk. (Ml)‘. Mj. 11f 11n. Here U- is a small open neighborhood of B. This means that f has to vanish if the constant q’(M’)‘Mj is < 1. That is the case if k = - i. bW’)lb(d - j. WWldq) - l/lg(d. The number of linearly independent functions f E Fi, j which do not vanish in all the S, of order k is at most (b/m!)p*(-lg(M)/lg(q))“. j”’ + ... . So if i is so chosen that i > bp,( - lg(M)/lg(q))” then for big j there are more expressions in Fi,j than there are functions. A non trivial linear combination of expressions has to give the zero function. This means that all g are algebraically dependent f, with a degree pi. In other words we have off,,..., Theorem 6.2. Assume that X is a connected pseudoconcave normal complex space of dimension n and that m I n is the maximal number of analytically independent meromorphic functions on X. Then the field K(X) of meromorphic functions on X is an algebraic extension of degree li of the rational function field in m indeterminants over Cc. 4. Modular Groups. We shall show here that the quotient of the Siegel upper half plane by the Siegel modular group in pseudoconcave. This is also true for many other quotients by discontinuous groups. For the Hilbert modular group it was proved in [Sp63]. The Siegel upper half plane of degree n is the set H = H, c Cn’(n+l)iz which is the connected open set of symmetric square matrices Z = X + iY of dimension n with Y being positive definite. One can show that H is a homogeneous domain

H. Grauert

282

with respect to the biholomorphic transformations H N H given by Z + (AZ + B) o (CZ + D))‘, where A, B, C, D are real n x n matrices such that the following two equations are satisfied:

The Siegel modular group r consists of all such matrices with integral entries. It acts properly discontinuously on H. The quotient H/T is a normal complex space which, however, is not compact. It has a cusp at infinity co, but has finite volume (with respect to the Bergmann metric). Using this cusp we can define what it means that a point Z E H is far out in H (with respect to H/T). We call a transformation y E r a transformation in co if all far out points are moved to far out points. It can be proved that for such a transformation the determinant det(CZ + D) does not depend on the last line and the last row of Z. We define k(Z) = - Ig(det(Y)). For y E r we have Q(Z))

= k(Z) - 2lg(ldet(CZ

+ D)l),

We put p(Z) = min,, ,- k(y(Z)). Then p is continuous on X = H/T converges to --oo for x + ax. The Levi form of the function k is: L(k)

:=

;

,z;2;z, r,n

It is positive definite everywhere in H. The p is not differentiable. But since the Levi follows: We can consider X as to be n(n ber is smaller than n(n + 1)/2 = dim. X. found

dzi,

n d&,

and p(x)

n

.J.n

same is true for p in X. The function form is positive definite a concavity 1)/2 + 1 concave. If n > 1 this numThus X is pseudoconcave. We have

Theorem 6.3. The field of Siegel modular functions field if n > 1.

is an algebraic function

Thus for n > 1 for algebraicity was proved in [AG61].

is needed. This theorem

no further

condition

Historical Note. This chapter deals with the extension of complex analytic cohomology classes to larger domains. In the case of O-dimensional cohomology, especially in the case of holomorphic functions, this is a classical problem which came up in the research of Hartogs on the simultaneous continuation of holomorphic functions in the beginning of this century. In the thirties Thullen and Car-tan considered the construction of the hull of holomorphy of domains G c (c” (see [CT321 and [BT33], where the whole theory is given). This hull is the smallest domain G” containing G such that every in G holomorphic function f can be analytically continued to G”. It is uniquely determined. In the theory of several complex variables the analytic cohomology classes are the obstructions against the construction of holomorphic functions. It is

VI. Theory

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283

necessary to extend them as far as possible. The theorems 2.5 and 2.6 give such a possibility. Some applications are obtained: finite dimension of the vector spaces of cohomology groups; vanishing of cohomology; repairing of holes of complex spaces. In a very special case also the definition of unique hulls for l-dimensional cohomology is possible. But in general the extension of cohomology classes in not unique. An important application is the famous duality theorem of J.P. Serre. We state it here only in the simplest case for complex manifolds and locally free sheaves. The theory of “currents” is used. Another application is to the (very general) pseudoconcave complex spaces X. The field of meromorphic functions on such spaces is algebraic. In particular the fundamental domains for the Siegel modular forms are pseudoconcave. Of course, this also is true if X is compact. For such compact spaces X the theory was started in special cases by W. Thimm [Th54] and then continued by R. Remmert [Re56] using the projection theorem for analytic sets. We follow in this book the ideas of J.P. Serre and C.L. Siegel.

References* [An631 [AC611 [AC621 [Ba70] [Bc76] [BF81] [BT33] [Ca58] [CT321

[DC601 [Fr76] [Fr77]

Andreotti, A.: Theoremes de dtpendance algibriques sur les espaces complexes pseudoconcaves. Bull. Sot. Math. Fr. 91, l-38 (1963) Zbl.113,64. Andreotti, A.; Grauert, H.: Algebraische Korper von automorphen Funktionen. Nachr. Akad. Wiss. Giittingen, II. Math.-Phys. Kl. 1961, 39-48 (1961) Zbl.96,280. Andreotti, A.; Grauert, H.: Theortmes de tinitude pour la cohomologie des espaces complexes. Bull. Sot. Math. Fr. 90, 193-259 (1962) Zbl.10655. Barth, W.: Transplanting cohomology classes in complex projective space. Am. J. Math. 92, 951-967 (1970) Zbl.206,500. Banica, C.; Stanasila, 0.: Algebraic Methods in the Global Theory of Complex Spaces. Wiley 1976, Zbl.284.32006. Buchner, M.; Fritzsche, K.; Sakai, T.: Geometry and cohomology of certain domains of the complex projective space. J. Reine Angew. Math. 323, l-52 (1981) Zbl.447.32003. Behnke, H.; Thullen, P.: Theorie der Funktionen mehrerer komplexer Verlnderlichen. Erweiterte Auflage, herausgegeben von R. Remmert, Springer 1970, Zbl.8,365, Zb1.204,395. Cartan, H.: Espaces librts analytiques. Symp. Int. Topol. Algebr. Mexico 1956, 97-121 (see H. Cartan, Collected Works, Vol. II, Springer 1979) (1958) Zbl.121,305. Cartan, H.; Thullen, P.: Zur Theorie des Singularitlten der Funktionen mehrere komplexe Vednderliches. Regularitatsund Konvergenzbereiche. Math. Ann. 106, 617-647 (1932) Zbl.4,220. Docquier, F.; Grauert, H.: Levisches Problem und Rungescher Satz fiir Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 140, 94-123 (1960) Zb1.95,280. Fritzsche, K.: q-convexe Restmengen in kompakten komplexen Mannigfaltigkeiten. Math. Ann. 221,251-273 (1976) Zbl.327.32007. Fritzsche, K.: Pseudoconvexity properties of complements of analytic subvarieties. Math. Ann. 230, 107- 122 (1977) Zb1.346.32025.

* For the convenience of the reader, compiled using the MATH database,

references to reviews in Zentralblatt have, as far as possible, been included

fur Mathematik (Zbl.), in this References.

284 [Gr58] [Gr81] [Hi561 [Oh841 [Re56] CR0653 [Sa65] [Se551

[SC531 [Si74]

CWll CSp631 [Th54]

H. Grauert Grauert, H.: Analytische Faserungen iiber holomorph-vollstlndigen Raumen. Math. Ann. 235, 263-273 (1958) Zb1.81,74. Grauert, H.: Kontinuitatssatz und Hiillen bei speziellen Hartogsschen Kiirpern. Abh. Math. Semin. Univ. Hamb. 52, 179-186 (1981) Zbl. 493.32015. Hirzebruch, F.: Neue Topologische Methoden in der Algebraischen Geometrie. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer 1956,Zbl.70,163. Ohsawa, T.: Completeness of noncompact analytic spaces. Publ. Res. Inst. Math. Sci. 20, 683-692 (1984) Zbl.568.32008. Remmert, R.: Meromorphe Funktionen in kompakten komplexen Raumen. Math. Ann. 132, 277-288 (1956) Zbl.72,80. Rossi, H.: Attaching analytic spaces to an analytic space along a pseudoconcave boundary. Proc. Conf. Complex Analysis, Minneapolis 1964, 242-256 (1965) Zb1.143,303. Spallek, K.: Differenzierbare und holomorphe Funktionen auf analytischen Mengen. Math. Ann. 161, 143-162 (1965) Zbl.166,338. Serre, J-P.: Un thiortme de dualitt. Comment. Math. Helv. 29,9-26 (1955) Zbl.67,161. Schwartz, L.: Homomorphismes et application completement continues. C.R. Acad. Sci., Paris 236, 2472-2473 (1953) Zbl.50,333. Siu, Y.T.: The mixed case of the direct image theorem and its applications. Complex Anal., C.I.M.E., Bressanone 1973,281-463 (1974) Zbl.338.32012. - - Techniques of extension of analytic objects. Lect. Notes Pure and Appl. Math. 8, M. Dekker (1974) Zbl.294.32007. Scheja, J.: Riemannsche Hebbarkeitssltze fur Cohomologieklassen. Math. Ann. 144, 345-360 (1961) Zb1.112,380. Spilker, J.: Algebraische Korper von automorphen Funktionen. Math. Ann. 149, 341360 (1963) Zb1.124,293. Thimm, W.: Uber meromorphe Abbildungen von komplexen Mannigfaltigkeiten. Math. Ann. 128, l-48 (1954) Zbl.56,306.

Chapter VII

Modifications Th. Peternell

Contents Introduction 9:1. Definition 92. Blow-ups

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

286 287

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

290

0 3. Criteria for Blowing Down ................................... 1. Criteria for Blowing Down by a Monoidal Transformation 2. Fujiki’s Contraction Theorem ..............................

293 293 294

.....

$4. The Formal Principle and Extension of Analytic Objects ......... 1. The Problem ............................................ 2. The Formal Principle - Problem (A) ........................ 3. Extension of Analytic Objects - Problem (B) .................

297 297 297 299

$5. Formal Modifications ....................................... 1. Formal Complex Spaces .................................. 2. Formal Modifications .................................... 3. Existence Theorems ......................................

300 300 301 302

0 6. Moishezon Spaces .......................................... 1. Algebraic Dimension ..................................... 2. Basic Properties of Moishezon Spaces ....................... 3. Positive Sheaves and Moishezon Spaces ..................... 4. AlgebraicSpaces ......................................... 5. Examples ............................................... 6. Projectivity Criteria ......................................

303 303 304 306 307 309 310

3 7. Desingularization .......................................... 1. Statement of the Problems ................................. 2. Desingularisation in the Algebraic Case - Hironaka’s 3. Embedded Resolutions ................................... 4. The Complex-Analytic Case ...............................

.

311 311 312 313 3 15

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

315

References

Theorems

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Th. Peternell

Introduction This chapter is devoted to the study of the bimeromorphic geometry of complex spaces. Bimeromorphic geometry is the study of complex spaces up to bimeromorphic equivalence. Roughly speaking, two complex spaces X, Y are bimeromorphically equivalent if they are isomorphic outside thin analytic sets. If X and Y are irreducible, this means that their fields of meromorphic functions are isomorphic. For two given spaces X and Y which are bimeromorphically equivalent, one can find another complex space Z and a diagram Z a A x-y

B

Y with y the bimeromorphic equivalence described above and ~1,/l holomorphic everywhere defined maps which are isomorphisms almost everywhere. The maps tl and /3 are called modifications. So the study of bimeromorphic geometry is the study of modifications of complex spaces. To modify a complex space means to take out an analytic set and to substitute it by some other analytic set; so this procedure is a kind of surgery. In chap. V we have already met several modifications: the Remmert reductions of l-convex spaces. In our terminology l-convex spaces are just the modifications of Stein spaces in a discrete set D: D is taken out of the Stein space X and a higher dimensional set A (the exceptional set) is put in instead. Of course A cannot be arbitrary, as there are restrictions due to the local geometry of X. In dimension 1 there is not enough space for interesting bimeromorphic geometry. In dimension 2 2 things change completely; there is a rich bimeromorphic geometry. The most basic example of a modification is the blow-up of a point x E X in a complex manifold of dimension n. The point x can be replaced by a projective space lP-i which can be viewed as the space of all tangent directions in x. In particular, this blow-up separates all curves in X meeting transversally in x. It is also possible to blow-up higher dimensional subspaces. These blow-ups are treated in sect. 2 and are the most important examples of modifications. One reason is that by applying repeatedly blow-ups one can smooth a reduced complex space (without changing its bimeromorphic nature); this is called “desingularisation” and is discussed in Sect. 7. Another reason is the so-called Chow lemma (Hironaka) to the effect that every modification can be dominated by a blow-up. Given a complex subspace A c X, it is important to know when it can be “blown down” to a lower-dimensional complex space. This problem is treated in sect. 3. The next two sections deal with formal geometry, we refer for any explanations to the appropriate places. Compact complex spaces which are bimeromorphically equivalent to projective varieties (i.e. subvarieties of projective spaces II’,,) are called Moishezon spaces. They need not be projective

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but are not far from being projective. Examples and the basic theory is presented in sect. 6. The final section gives a short treatment of the theory of desingularisation.

0 1. Definition In this section we introduce the notion of modification and discuss some first elementary properties. Generally speaking, a modification changes a complex space X only along a “small” analytic subset A but leaves it unchanged outside A. The precise definition is given below. Definition 1.1. A proper surjective holomorphic map @: X + Y of complex spaces X and Y is called a (proper) modification if there are closed analytic sets A c X and B c Y such that (1) B = @(A) (2) @(X\A: X\A -+ Y\B is biholomorphic (3) A and B are “analytically rare”. (4) A and B are minimal with the properties (l)-(3). A is called the exceptional set of 0. One also says that X is blown down along A; B is often called the center of the modification. Sometimes we write @: w, A) --+ (r, B).

We have to explain what analytically

rare means:

Definition 1.2. An analytic set A in a complex space X is called analytically rare if for every open set U c X the restriction map

is injective. It is obvious that for X reduced the following statements are equivalent: (1) A is analytically rare in X, (2) A has codimension at least 1 at every point x E A, (3) No irreducible component of X is contained in A. If X is not reduced, (2) or (3) does not imply (1). Modifications

do not affect meromorphic

functions. More precisely:

Proposition 1.3. Let @: X + Y be a modification. (1) The canonical map 8: 0, + @,(6&) is injective, (2) The canonical map 8: JZ%!~ + @.+(Jllx) is an isomorphism of sheaves of meromorphic functions. In particular A(Y) 21 A(X), i.e. CDinduces an isomorphism of function fields. Qi is also culled bimeromorphic.

For a detailed proof see [Fi76]. ((1) is obvious). Let us consider a simplified but typical case. Assume that X and Y are irreducible and reduced, let A c X be the exceptional set and B = @(A). Assume that codim, B 2 2. In order to check the surjectivity of 8, let U c Y be open and f E JH~(@-‘(U)). Then f can

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be considered as meromorphic function on U\B, and by codim, B 2 2 and the Riemann extension theorem for meromorphic functions, f can be extended to g E &#J). Then clearly f = 6*(g). Remark 1.4. Let f: X + Y be a modification with Y reduced. Then X is necessarily reduced, too: Assume that x E A (the exceptional set off) and f, E Ox,, such that fp = 0 for some m. Represent f, by f E O,(U). Then f 1U\ A = 0 and by the injectivity of O,(U) -+ 0x(U\A) we conclude f, = 0. However, if Y is normal, it is in general not true that X is normal. For example let 2 be the total space of a negative line bundle E over lP, x I’,. Fix x,,, x1 E lP, and let Co = Ip, x {x0>, C, = {x1} x lP,. Furthermore, fix an isomorphism i: C,, 2: Cr. Let X be the reduced space arising by identifying ,cO and C, c 2 via i. Clearly X is not normal. We obtain a modification g: X + X. Since IP, x lP, is exceptional in 2 (V.2.4), its g-image A is exceptional, too. Let f: X -+ Y be the blow-down. Then f o g is nothing than the blow-down of lP, x IP,, in particular Y is normal. For another example see (2.4). Corollary 1.5. Let CD:X + Y be a finite modification. biholomorphic.

Zf Y is normal, @ is

This is known in algebraic geometry as (a special case of) Zariski’s main theorem. Important examples of modifications are normalizations and blow-ups (monoidal transformations), which are introduced in the next section. A first insight into the structure of modifications is given by the so-called purity-of-branch theorem of Grauert-Remmert [GR55]. Theorem 1.6. Let X be a normal complex space, Y a complex manifold and

f: X -+ Y a surjective holomorphic map. Let A = {x E XI f is not biholomorphic at x} and assumethat f is generically finite. Then codim, A = 1 for all x E A. In particular, tf f is a modification, then the exceptional set A for f is of codimension 1 everywhere. (Zn the modification caseit is sufficient to assumeX to be reduced).

We sketch the proof following [Ker64]. We assume that codim A 2 2 and have to show that A = 0, i.e. f is &ale (locally biholomorphic). The problem being local with respect to Y we may assume that Y is an open subset of Cc”. Now take x0 E A and choose an open Stein neighborhood U of x,, in X. Put U, = U \ A. Since f 1U,: U, + Y c (c” is locally biholomorphic, U, is a domain over (c”, and hence we can construct the hull of holomorphy consisting of a Stein space oc, and a finite map f: o,, + Cc” into Cc”. For the construction of hulls of holomorphy, see e.g. [GF74]. Then f is locally biholomorphic and has the following property: there is a holomorphic map g: U, + G,, such that h

Moreover

=flu3. cO(U,) N 0( Deb,.Since 0( U,,) N Lo(U) we obtain O(U) N O( cO) and, as

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U and & are Stein spaces, this last isomorphism is induced by a biholomorphic map @: &, + U (this is a theorem on Stein algebras, see [GR77]). Moreover, f o 0 = f and hence fl U is locally biholomorphic, since f enjoys this property. Remark. We have encountered with modifications already in Chap. V. There they occurred as Remmert reduction of l-convex spaces. Sometimes they are called “point modifications”: a compact connected analytic set is substituted (blown down) to a point. Or, conversely, a point is “blown up”. Next we introduce the concept of meromorphic maps of complex spaces as given by Remmert in [Rem57]. Definition 1.7. Let X and Y be reduced complex spaces. A meromorphic map f: X + Y associates to every point x E X a subset r(x) c Y such that the following conditions hold: (1) The graph Gf = {(x, y)ly E r(x)> c X x Y is a connected complex subspace of X x Y with dim G, = dim X. (2) There exists a dense subset X, c X such that z(x) consists of exactly one point for every x E X. Remark 1.8. Here we collect some basic facts on meromorphic maps (see [Rem57]). (1) Let f: X -+ Y be meromorphic. Then there exists an analytic set N c X, the set of indeterminacies off, such that flX\N is in fact holomorphic. The set N has codimension at least 2 if X is normal. (2) Examples of meromorphic maps: a) If f: X + Y is a modification, then f-‘: Y + X is a meromorphic map, b) If f E A(X) is a meromorphic function on X, then it can viewed as a meromorphic map X + lP,. c) Let 9 be a locally free sheaf of rank 1 on X and let se, . . . , sN E H’(X, 9). Then these sections define a meromorphic map X + lP( V), where V is the vector space generated by the si, and the set of indeterminacies is just the common zero locus of the si. See [GH78], [We801 for details and also Chap. V. (3) If f: X -+ Y is meromorphic, then f induces a pull-back map f*: 4?(Y) + J%‘(X). In particular, if f is an isomorphism almost everywhere and X, Y are, say, irreducible and reduced, then f* is an isomorphism.

The study of meromorphic maps can essentially be reduced to the study of holomorphic maps by virtue of the following result. Theorem 1.9 (Elimination of indeterminacies [Rem57]). Let f: X + Y be meromorphic. Then there exist modifications (r: 8 + X and a holomorphic map z: 8 + Y such that the following diagram commutes

Th. Peternell

290

8 2. Blow-ups The most important modifications are certainly blow-ups. In some sense an arbitrary modification is not very far from being a sequence of blow-ups. Blowups were introduced (in a special case) around 1950 by H. Hopf but certainly they have been around implicitly much earlier. First let us give the definition of blow-up in a rather algebraic way. Later we will see explicit descriptions in terms of local coordinates. We fix a complex space X and a closed subspace Y c X defined by the ideal sheaf J. Put

Here “Proj” Q-algebras

denotes the analytic homogeneous

which is of finite presentation tion map

spectrum

of the graded sheaf of

(11.3.4). The space r? comes along with a projec0:2+X.

Both r? and/or 0 are called the blow-up of Y in X or of X along Y or the monoidal transformation with center Y. The analytic preimage F = a-‘(Y) defined by Im(a*(J)

i* set of 0. Since F = Y xx 2 N Proj (Y(FJ$

is called the exceptional have F-

Proj(

+ 02) We

F J”/J”‘+‘).

Remarks 2.1. (0) The blow-up of an analytically rare subspace is a modification (cp. [Fi76]). Moreover it is a projective map (see 2.7). (1) If J is invertible, 0 is an isomorphism. (2) J being invertible outside Y, a~Z\i!Z\~+X\Y is an isomorphism. (3) Assume that Y is locally a complete intersection. Then in particular Jm/Jm+’ N Sm(J/J2) and hence Proj(@ So p= lP(N&),

Jm/Jm+l) N Proj(@

with N&

Sm(J/J2))

denoting the conormal

= lP(J/J2).

bundle of Y in X.

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291

If Y in X is locally defined by regular sequences (fO, . . . , f,) (see e.g. [GR71]), with 1; E O(U), then o-‘(U) is given in U x lPr by the equations fitj - j&,

0 5 i, j I I,

where to, . , . , t, are homogeneous coordinates in E’,. This follows from the local description of Proj(@ J”). If Y is a submanifold in X, then Y is locally defined by a regular sequence and so our remarks apply to this important special situation. The blow-up of a subspace has a universal property: Proposition 2.2. Let X be a complex space and Y c X a closed subspace defined by the ideal J c 0,. Let 6: 2 + X be a proper map of a complex space 2 to X such that (1) the analytic preimage P = a-‘(Y) is a hypersurface, i.e. the ideal J = Im(a*(J) -+ 0%) is invertible; (2) for all morphisms g: 2 + X of a complex space Z to X having property (1) (i.e. g-‘(Y) is a hypersurface), there exists a unique morphism h: Z + 2 such that g=aoh. Then o is - up to isomorphism - the blow-up of Y in X.

Because of (2), it is sufficient to show that the blow-up of Y in X satisfies (1) and (2). Now (1) is clear, and (2) is first reduced by uniqueness to the local case and then proved directly in coordinates. A local description of blow-ups 2.3. Let U c (cm be an open domain and A c U a closed subspace defined by the coherent ideal sheaf J c 0,. Assume that J is generated by fo, . . . . f,eLO(U) with fi#O. Let Gc U x Ip, be the closure of

((x,y)~U

x lP~lx~Aandy=(f,(x):...:f,(x)}.

In other words, we let G the graph of the meromorphic map (fo, . . . , f,): U + IP”. Then the map T + U, induced by projection U x lP” + U, is the blow-up of A in U. This is a simple verification of the conditions of (2.2). Example 2.4. Let Y be the subspace of a? (with coordinates denoted by zl, z~) given by z: = z: = 0. Let 0: 2 + (cz be the blow-up of (c* along Y. Then X c (c* x lP, is given by the equation t,z: - t,z: = 0.

Clearly 2 is singular along (0) x Ip,, this is in fact the non-normal locus of 2. The normalization is up to isomorphism the space {(z, t} E (c* x IP, 1t,z, t,z,}, i.e. just the blow-up of the simple point 0 E (E*. Definition 2.5. Let f: (X, A) + (Y, B) be a modification and Z c Y be a closed subspace, Z Q B. Let 2 c X be the smallest complex subspace of X containing f -‘(Z\B) (with structure). 2 is called the strict transform of Z.

292

Th. Peternell

Now assume that f is furthermore the blow-up of A c Y. Then f/Z: Z + 2 is the blow-up of the analytic intersection A n 2 in Z. This follows immediately from (2.2). Proposition 2.6. Let X be a reduced complex space, Y c X a closed complex subspace,and (T:2 + X the blow-up of X along Y. Then r? is also reduced.

This is a special case of (1.4) since rr is a modification, see also [HR64]. we are going to describe the conormal sheaves of blow-up%.

Now

Conormal sheaves2.7. Let f: 2 + X be the blow-up of the subspace A c X defined by the ideal sheaf J. Let A” be the analytic preimage defined by 2 = f*(J). 0% . Then we have

J/J2 N og(i)lX, where f%(l) is the natural line bundle on r? = Proj(@ J”). Observe that (1) and if A is locally a complete intersection, o,i41)IA N @iii(l) = Groj(@Jm/Jm+l) J/P In particular, if X is a manifold 2 N lP-r and j/j2 2: O,“-,(l). As a conclusion we can state:

N CO~~~,~~~(~). of dimension

n and A a simple point,

J/J2 is positive if and only if j/j’

then

is positive.

(*I More generally, j/j’ is positive in the new directions arising from the blow-up (i.e. j//J”’ is f-ample, to make a precise statement). So f is projective (V.4.23). In fact, the canonical epimorphisms Sm(J/J2) + Jm/Jm+l

give an embedding Proj(@ Jm/Jm+l) c lF’(J/J’). By (IV.4), J/J2 is positive if and only if 0,(,,J2J( 1) is positive. Since j/?/J”’ = Or.r,,j(oJ,,,,J,,,+l, = 0,1J,J2)( l)[ A”, the claim (*) follows. A very important problem on the structure of a general modification is to decide how far it is away from being a blow-up. An answer is given by Hironaka’s Chow Lemma [Hir75]. Theorem 2.8. Let f: X + Y be a modification, with Y reduced. Then there exists a modification g: Y’ + Y which is a locally finite sequenceof blow-ups and a holomorphic map h: Y’ --) X such that the diagram Y’

commutes.

A -

x

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293

In other words, every modification is dominated by a (locally finite with respect to the base) sequence of blow-up’s, in particular is a Moishezon morphism (VIII.3.5). A local version of the Chow lemma is Corollary 2.9. Let f: X + Y be a modification of reduced complex spaces. Let V c Y be an open relatively compact set. Then there exist a blow-up g: U + V with center D c V and a blow-up h: U + f-‘(V) with center f-'(D), the analytic preimage of D, such that the diagram

u -

h

f-'(v)

V commutes.

The Chow lemma turns out to be an important tool in the study of general modifications; it can often be used to reduce a general problem to a problem on blow-ups. The Chow lemma itself is a corollary of Hironaka’s flattening theorem (11.2.9).

4 3. Criteria for Blowing Down This section will be devoted to the following problem: Given a complex space X and a subspace A c X, under which conditions can A be blown down? In Chap. V we studied the case when A is blown down to a point. Here we consider the following more general problem: Assume that there is a surjective holomorphic map 4: A -+ B onto another complex space B. Find conditions under which there exists a modification $: X + Y such that B c Y, $[A = 4 and $IX\A is biholomorphic. 1. Criteria

for Blowing

Down by a Monoidal

theorem is proved by Nakano

Transformation.

and Fujiki in [Nak71],

The following

[FN71].

Theorem 3.1. Let X be a complex manifold, and let A c X be a closed subspace of the form A = P(9), where 9 is a locally free sheaf on a complex space B. Let p: A + B be the projection. Let Ja be the ideal sheaf on A in X. Assume furthermore:

(1) codim(A, X) = 1, (2) J,/J,’ 2: 0,(,,(l) locally with respect to B. Then there exists a monoidal transformation that $JA = p. Remarks. (1) MoiSezon [Moi67] tion that X is a MoiSezon space.

$: X + Y with center B c Y such

proved (3.1) under the additional

assump-

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Th. Peternell

(2) If B in (3.1) is a point, then A is a projective space lRn-i. If X is smooth of dimension n and if J,/J,’ = 0 P,_,(1), then the modification 4 is the blow-up in a simple point. This had previously beenproved by Kodaira [Kod54]. If n = 2 we get back the classical criterion of Enriques. If X in (3.1) is projective then in general Y will not be projective, see Example 6.22 below. Instead we have the following criterion of GrifIiths [Gri66]. Theorem 3.2. Let X be a projective manifold, and let A c X be a submanifold of X. Let B be a projective mantfold and 9 a locally free sheaf on B. Assume that A = lP(9) and that

J,/J,’ = &y,,(s) for some s > 0. If F-* is positive, then there exists a modification that $1 A = p, p being the projection, and Y is projective. Note that + is a monoidal 1 ands= 1.

transformation

$: X + Y such

if and only if A is of codimension

2. Fujiki’s Contraction Theorem. In this section we discuss important contraction criterion due to Fujiki [Fuj74].

the following

Theorem 3.3. Let X be a complex space, and let A c X be an effective Cartier divisor, and B another complex space. Assume that (1) the conormal bundle N4x = J,/J,’ is f-positive (V.4.8) and that (2) R’f,(N,*“) = 0 for all p > 0. Then there exists a modification II/: X + Y with 1+9 1A = f. Moreover $ has the additional property

where the coherent sheaf 9 is defined by the sequence

We indicate how to prove divisior A c X. Assume that Remmert reduction onto the into (c” with coordinates (zi, Define II/: A + IR by

theorem (3.3). Fix a complex space X and a Cartier A is holomorphically convex. Let p: A -+ A’ be the Stein space A’, and let j: A’ + (CN some embedding . . . , zN).

Then II/ is a plurisubharmonic exhaustion function of A and A is weakly lcomplete (by definition). Let A, = I+-‘(( - co, c)). The main point in the proof of (3.3) is its local version. Proposition 3.4. Let X be a complex space, and let A c X be a holomorphically convex Cartier divisor. Assume that:

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Moditications

295

(1) the conormal bundle N& is Griffhs-positive (see V.4.6), and that (2) H’(A, N2p) = 0 for all p > 0. Then for every c E IR there is a neighborhood UC of A, in X with A, = A n UC and a proper holomorphic map h: UC-+ C” which is isomorphic on U,\A,. Theorem (3.3) follows from (3.4) by taking small Stein open sets U’ c B, applying (3.4) to X = f-‘(U’) and then patching the local pieces together. Let us now pass to the proof of (3.4)! Let J!, = O,(A). Step 1: Construction of a neighborhood I/ on A, in X such that L* 1V is Grifhths-positive. Using the vanishing theorem (3.6) below we can construct an embedding 4, of the v-th infinitesimal neighborhood A,,, onto a locally closed subspace of some projective space lP, by sections in L*m. For technical reasons one has to prove this for Ad,“, with d > c. We shall write A, = A,,,. If v is chosen that H’(A,

L*“)

= 0,

p 2 v,

we can extend the sections of L*m to any A,, .D 2 v. Fix p > v sufficiently large. Then our sections in H’(A,, L?r) can be lifted to A, as 5P’-sections of L*“’ on some open neighborhood U of A. So the embedding 4, is extended to a ‘P-map 4: U + lPr. After possibly shrinking U, 4 is a 9?‘-embedding and there is a %?“-isomorphism 4*v%,(l))

= G”

inducing d:(&~,.(l))

= L*mlA,.

Now pull back the Fubini-Study-metric on Otr(l) shows that this metric has positive curvature.

and a local computation

Step 2: Find a neighborhood W of A, which is weakly l-complete with W c 1/ The main part here is to find a neighborhood W, of A,(d E IR) and a plurisubharmonic function on W, which is strictly plurisubharmonic outside A,. For this use the embedding q5of Step 1. Step 3: L* being positive on the weakly l-complete space W, we may fix a relatively compact open set K in W and find do, . . . , 4, E H”( W, L*m) inducing an embedding

such that 2*(0(l)) N L*“‘. This is in analogy to the compact case (Chap. V). The sections viewed as elements fj E H”( W, Jr) which in turn define a holomorphic f: K + ccr+l. Since 1 is an embedding, we have f-‘(O) = K n A. Fix a weakly l-complete neighborhood W, of A.

dj can be map

296

Th. Peternell

Using the vanishing theorem (3.6) and assumption (2) of (3.4) any s E H’(A, 0,J can be lifted to S E HO(W,, oWd), such that SIIF: n A = sll( n A. In particular, the map j o p: A -+ (CN as defined before can be lifted to F: K + CN.

In summary we obtain a holomorphic

map

g = (A, f, F): K + IP, x C’+’

x CN.

We put h = pr o g, where pr is the projection onto cC’+l x (CN. Right away from the definition of 2 and f it is clear that Y = g(K) is the blow-up cr of (l?+i x (EN with center D = (0) x cN. Moreover prl Y = 0. It follows easily that hlK\(A n K) is biholomorphic. Now we take an appropriate shrinking U, of K,

wheres1,s2>Oandf=(fl

,..., f,+l),F=(Fl De, = {z E c’+lI

,..., F,).Let lZil < El},

DE, = {t E (c”I C ltj12 < EZ}.

Then hl U,: U, + DE, x DE2is a proper holomorphic map and the composition of h with some embedding DE, x DE,+ a? gives the map we have been looking for. Remarks 3.5. (1) The algebraic analogue of (3.3) (in the category of algebraic spaces) has been proved by M. Artin [Art70]. (2) There is the following generalization of (3.3) for A c X of codimension > 1. Assume the situation of (3.3) except for the assumption that A is a Cartier divisor in X. Assume that (a) the normal cone C,,, can be blow down along f: there is a modification g: C,,, + Z to some complex Z such that B c Z and gJA = f (identify A with the zero-section of C,,,). [The normal cone C,,, is by definition Spec(@ J”/J’“)]. (b) R’f*(P/P+‘) = 0 for all p > 0. Then the conclusion is the same as in (3.3). (3) For further results see [Cor73]. We have still to discuss the vanishing theorem used in the proof of (3.4). Let X be a complex space (possibly non-reduced). We define ad hoc a line bundle L on X to be positive if there is a metric h on L and a cover (U,) by open sets U, c X with LI U, trivial such that for the local representatives h, of h on U,, the functions -log h, are strictly plurisubharmonic on U, (in the sense of V.l). If the functions -log h, are only plurisubharmonic, L is said to be semi-positive. It is easily seen that L is (semi-)positive if and only if L/red X is. Then we have: Theorem 3.6. Let X be a weakly l-complete complex space (i.e. X carries a plurisubharmonic exhaustion function +). Let Y be a coherent sheaf on X, L a positive line bundle on X, and F a semi-positive line bundle on X. Let c E IR

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Modifications

291

and XC = {tj < c}. Then there is an index no E JN such that for all n 2 no and all i2 1: H’(X,, Y @ L” @ F) = 0. A proof can be found in [Fuj74]. Of course (3.6) is very similar to the coarse Kodaira vanishing theorem (V.4.4).

04. The Formal Principle

and Extension of Analytic Objects

This section treats the so-called formal principle, which for the first time occurred in [Gra62], and considers also the problem of extending analytic objects from “high” infinitesimal neighborhoods of a subspace A to a full neighborhood. This question was first treated by Grifliths [Gri66]. In the context of exceptional subspaces A, these questions are related to some of Artin’s approximation theorems (4.3,4.4). 1. The Problem. Let X be a complex space and A c X a closed subspace. We denote by A,, the p-th infinitesimal neighborhood of A in X, so that A,, = (A, cOx/Jfl+‘), where J is the ideal sheaf of A. Furthermore let A^ denote the formal completion of A in X. The Formal Principle asks whether it is possible to extend some object on A^ or on A,, ,Usufftciently large, to some neighborhood of A in X. More exactly, we formulate the following problems: (A) Given another complex space Y with closed subspace B c Y, a formal isomorphism g: Al--+ g, and n E IN, are there neighborhoods U of A in X and T/ of B in Y and a biholomorphic map f: U + V such that f 1A,, = g[ A,? If (A) holds for any B, we say that the formal principle holds for (X, A).

(B) Given an analytic object on A^ (a vector bundle, a coherent sheaf, a cohomology class etc.) and p E N, is there an analytic object in a neighborhood of A inducing the original one on A,? Is the extension unique? We will only deal with the case “A compact” and our main interest will be exceptional sets A. As a general survey we recommend [Kos86]; there one can find results on the Stein case, too. 2. The Formal Principle-Problem (A). Let X be a complex space and A a compact subspace. In “general” the formal principle will not hold: Counterexample 4.1. In [Am761 Arnol’d constructed a smooth surface X containing an elliptic curve A whose normal bundle NAlx is topologically but not analytically trivial and enjoying the following property. Let Y be the total space NAlx and B the zero-section. Then A and B are formally isomorphic (A N 8) but there is no convergent isomorphism. On the positive side one has

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Theorem 4.2. ([Kos81], [Anc80]) If A is exceptional in X, the formal principle holds for (X, A). Moreover, if 4: (X, A) -+ (Y, B) is a modification (with A, B compact), then the formal principle holds for (X, A) if and only zf it holds for ( r, B). In case X and A are smooth, 4.2 has been proved by Grauert [Gra62] and if X\ A smooth, by Hironaka and Rossi [HR64]. The proof of (4.2) is based on the Chow lemma, existence theorems for formal modifications (to be discussed in Sect. 5) and Artin’s approximation theorem [Art68]. This approximation theorem is of local nature; it can be applied in the theory of l-convex spacesbecause by blowing down the exceptional set we are in a local situation. Before stating Artin’s results, let us fix some notations. Consider variables y,). Let Cc[[xl] be the ring of formal power series X=(X1,...,Xn),Y=(Yl,..., in xi, . . , x,, tlZ{x} that one of convergent power series.Let m c C[ [x]] be the maximal ideal. Theorem (Artin) 4.3. Let fi, . . . , fk E C{x, y}, and let f = (fi, . . . , fk). Suppose that gl, . . . . Zj, E C[[x]] are formal solutions of the equation

f(x, Y) = 0, i.e. f(x, S1(x), . . . >9*(x)) = 0. Assume that gi has no constant term. Then for given c E IN there are gl, , . . , g,,, E (c{x} with gi - Si E mc, 1 I i I m,

solving f(x, y) = 0. An often useful version is Theorem 4.4 (Artin). Let f be as in (4.3). Let I c C(x} be an ideal. Let gl, . . . , g,,, E l&n (c{x}/Z’ be formal solutions of Y f(x, Y) = 0.

Assume that the gi have no constant terms. Let c E IN. Then there are gl, . . ., gm E C(x} solving f(x9 Y) = 0 such that gi - Si E p. A very general criterion for the formal principle is due to Kosarew [Kos88]. As an application he is able to prove that the formal principle holds for (If’“,, A), with A a local complete intersection. Commichau-Grauert [CG81] proved the formal principle for (X, A), with X and A smooth, and with a certain positivity assumption for N,,,. It is however unknown whether the formal principle holds for embeddings with positive normal bundle. See also [Gri66].

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Another type of theorems-assuming the existence of enough deformations of A - for the validity the formal principle is due to Hirschowitz [Hir79] and SteinbiD [Ste86], see also VIII.l. 3. Extension

of Analytic

where A is exceptional.

Objects - Problem B. First let us consider the case

Then we have

Theorem 4.5 ([PeWI). Let X be a complex space and let A be an exceptional subspace. Let & be a locally free sheaf on the formal completion A^ of A. Then there is a locally free sheaf 9 on a neighborhood U of A in X such that F”I(A^ 2: 8. Moreover, 9 is uniquely determined on the germ of the embedding A + X.

In fact, it is sufficient to have a locally free sheaf 6 on a sufficiently high infinitesimal neighborhood A,, in order to be able to extend d uniquely to a small neighborhood of A in X. The number p depends only on dl A. Main ingredients of the proof of (4.5) are the Chow lemma and the approximation theorem (4.4). A special case has been proved earlier by Griffiths [Gri66]. In the case of positive normal bundles Grifliths established: Theorem 4.6. Let X be a complex mani$old of dimension n and let A c X be a compact submantfold of codimension 1. Assume that the normal bundle Nalx is positive and n 2 4. Then every locally free sheaf d on A^ can be extend uniquely to a whole neighborhood of A in X.

If A has codimension > 1, Grifhths introduces a notion of “suflicient positivity” for NAlx such that - under this stronger assumption - (4.6) remains valid. The proof of (4.6) is based on extension results for cohomology classes. In order to describe these we need a notation. Definition 4.7. Let X be a complex manifold and E a holomorphic hermitian vector bundle with Chern connection D (see V.4.7). Let c(E) be the associated curvature.

If for all x E X and v E E, the quadratic form ic(E)A. 0 v) has exactly s positive and t negative eigenvalues, we say that E has signature 6, Q

In particular, dim X.

E is Grifliths

positive if and only if it has signature (n, 0), n =

Theorem 4.8 [Gri66]. Let X be a complex manifold and A c X a compact submanifold of dimension n whose normal bundle has signature (s, t), s + t = n. Let 8 be a locally free sheaf on a neighborhood of A in X. Let a E Hq(A, &IA). Then there exists a number p0 with the following property. If p 2 ,uLoand ap E Hq(A,, 61 AJ with a,(A = cc,then there is an uniquely determined a E Hq(X, bl A) (here &‘I A is the set - theoretically restriction, so 6 is a germ of cohomology classes near A in X) with &IA, = ~1~.

Th. Peternell

300

The proof of (4.8) in turn is based on Theorem 4.9. Let X be a complex manifold, and A c X a compact submanifold of dimension n. Assume that the normal bundle has signature (s, t), s + t = n. Let & be a locally free sheaf on X. Then H4(X,JP.&lA)=0

forOIq<s-l,q>n-t.

Here J’. bl A is the set-theoretical

restriction

§ 5. Formal

to A.

Modifications

In this section we deal with the so-called existence theorems for modilications. Roughly speaking, these theorems state that a formal modification (to be defined below) is the formal restriction of a “convergent” modification. This modification is uniquely determined. So if one wants to blow down some subspace, it is sufficient to do it formally. 1. Formal Complex Spaces. We c-ringed space (L!Z, O,r) of (C-algebras For every x E 57 there is an open with a closed subspace A defined by

define a formal complex space 9 to be a with the following property: neighborhood U and a complex space X the ideal J such that

Recall that 0, = l@ O,/Jk is the sheaf of formal functions along A. So a formal complex space is locally isomorphic to the completion of a complex space along a subspace. A morphism of formal complex spaces CAP): (fc 0,) + WY 0,) is given as follows: If locally (3, O,f) = (A, Co.& (%, 0,) = (B, ok), with A c X, B c Y, then there exists a holomorphic map (f, f): (X, 8,) + (Y, CO,) such that )f=f,

+jT

If (3, Co,.) is a formal complex space, we define a kind of “Cartan ideal sheaf” as follows. Denote by m, c 8,-,, the maximal ideal. For U c 57 open put

z(U) = {fE ~.AWL~m,,x~

u>.

An ideal sheaf J c cO,f is called a defining ideal if and only if for all x E % there is an open neighborhood U and k E IN such that ZklU c JIU c ZIU. Finally, a morphism f: !Z + g of formal complex spaces is called adic if for any defining ideal J c Co*, the ideal f*(J) . OI is a defining ideal on 95.

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301

Defining ideals are best understood if SY is the formal completion A^ of a complex space X along a subspace A c X. In this case, a defining ideal of % = A^is just 1lSY, where I is a coherent ideal sheaf on X with supp(cO,,,) = A. Moreover, if f: (X, A) + (Y, B) is a modification, then the formal completion f: A -+ 6 is adic. For a general theory of formal complex spaces we refer to [Bin78]. 2. Formal Modifications. Let (/,p): (!Z, 0,) --+(g, 0,) be an adic morphism of formal complex spaces. First we are going to construct the so-called Jacobi and Cramer (or Fitting) ideal sheaves associated to/: These were first considered by Artin [Art701 in the algebraic context; for the analytic case see [AT82]. Construction

5.1. Let x E X, y =f’(x). let us look at the map A: OK, -+ o.f,r

It gives rise to an isomorphism 0 9-J -0 - !dx a?,bd~ for some ideal B (see [AT82, p. lo]). Choose generators fi, . . . , f, of B. Then we define J(B) to be the ideal generated by the N x N-minors of the matrix 36 .

a7j C-h J(B) is called the Jacobi ideal of B. Here Ti, , . . , TN are variables in (EN. Now let t aij&, 1 I j I m, be a generating system of relations for fi, . . , f,. j=l

We let C(B) the ideal generated by (q - N) x (q - N) minors of (aij); C(B) is called the Cramer (or Fitting) ideal of B. Coming back to our map fi let J(A) be the image of J(B) in LOS,,, and let C(&) have an analogous meaning. One checks easjly that J(pY) and C(FY) are independent of all choices. Finally, we patch all J(fPY), C(/,) to obtain coherent ideal sheaves J(p), C(p), the Jacobi resp. Cramer ideal of $ Definition 5.2. An adic morphism of formal complex spaces tp: X -+ ?Y is called a formal modification if the following conditions are satisfied. (1) f is proper and surjective. (2) For all x E 3 there is a defining ideal J c Or such that I, c J(f)x n C(f 1,. (3) If x is the ideal defining the diagonal A N X in the fibre product % x )y X and if 9’ is a defining ideal of the formal complex space % x ?yX, then, given x E Z x J SY,there is some k E N and a neighborhood U of x such that

Lzk.XIU

= (0).

(4) Any adic formal morphism

(03~:cCzll) + y

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Th. Peternell

is induced by a formal morphism

The definition of formal modifications category, [Art70]).

is due to M. Artin

(in the algebraic

It is not difficult to show that formal completions of modifications are formal modification. More specifically: if f: (X, A) + (Y, B) is a modification, A^ the completion of X along A, g the completion of Y along B, and f^: i-, fi the restriction off, then fis a formal modification (see e.g. [AT82]). The motivation for definition 5.2 is the following characterization of modifications (see [AT82]): Proposition 5.3. Let f: X + Y be a holomorphic map of complex spaces. Let A c X and B c Y be closed subspaces. Let I be the ideal sheaf of B and assume that f*(l). GJx is the ideal sheaf J of A in X. Then f is a modification if and only if the following conditions are satisfied. (1) f is proper. (2) For all x E X there is somek E IN such that Ji c J(f), n C(f), (3) For all w E X x y X there is somem E IN such that 9”‘. &” = 0 in a neighborhood of w; here x denotes the ideal of the diagonal in X x ,, X and 9 the ideal of A x,AinX x,X. (4) For any y E Y and any local adic homomorphismCY:8, y -+ C[[t]] (the ring of formal power series in the variable t), there exists x Ef -‘l(y) and a local adic homomorphisma: &x.x --* cC[ [t]] inducing ~1. Remark 5.4. It is not very difficult to seethat condition (2) in (5.3) is equivalent to saying that f IX\A is locally biholomorphic, that (3) there just means that f lX\A is injective and that (4) is nothing but surjectivity off IX\A. Proposition (5.3) is the motivation for definition (5.2): it translates the fact that f is a modification into purely algebraic terms which make sensealso in the formal category. 3. Existence Theorems. In this section we state the main convergence (or “existence”) theorems on formal modifications. Theorem 5.5. Let X be a complex space,let A c X be a closedsubspaceand A^ the formal completion of X along A. Let g be a formal complex spaceand assume that there is a formal modtficationp: A^-+ CV.Then there exist a complex space Y, a closed subspaceB and a modification f: (X, A) -+ (Y, B) such that (1) fi Lxq (2) f* = /(up to formal isomorphism). The map f is unique up to isomorphism. Theorem 5.6. Let Y be a complex space, and let B c Y be a closed subspace. Denote by fi the completion of Y along B. Let X be a formal complex space and /: X + l? a formal modification, Then there exist a complex space X, closed subspaceA c X and a modtfication f: (X, A) + (Y, B) such that

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Modifications

303

(1) q&J?“, (2) f = /(up to formal isomorphism). Here f is unique up to isomorphism.

Theorem (5.5) is often referred to as “existence of contraction”, while (5.6) is referred to as “existence of dilatations”. Both theorems were proved first in the category of algebraic spaces by M. Artin [Art70]. In the “smooth” analytic case they are due to Krasnov [Kra73]; in the general case with an additional assumption they have been proved by Ancona-Tomassini [AT79], in full generality, by Bingener [Bin81].

0 6. Moishezon

Spaces

Introduction. Let X be an irreducible reduced compact complex space. To study X from a bimeromorphic point of view means to study its field of meromorphic functions d(X). In fact, if X’ is a modification of X, then the “function fields” d(X) and J&X’) are isomorphic and vice versa. The size of A(X) is measured by the transcendence degree over Cc, called the algebraic dimension a(X) of X, so a(X) measures how algebraic X is. The algebraic dimension being bounded by dim X, we draw particular attention to the spaces for which a(X) = dim X, the so-called Moishezon spaces. This section is devoted to the study of their basic properties. 1. Algebraic Dimension. We always let - unless otherwise stated - X be an irreducible reduced compact complex space. Let &Z(X) be its field of meromorphic functions. 6.1. a(X) = tr deg, A(X), the transcendence Cc,is called the algebraic dimension of X. Definition

degree of A(X)

over

A well known theorem due to Siegel and Thimm in the smooth case and to Remmert [Rem561 in the general situation bounds a(X) by the dimension: Theorem 6.2. a(X) I dim X.

For a proof compare also [Fi76]. Definition 6.3. X is said to be a Moishezon space if a(X) = dim X. A general reduced compact complex space is called Moishezon if all its irreducible components are. We call also a non-reduced compact complex space Moishezon if its reduction is Moishezon.

The name “Moishezon space” was introduced by Artin [Art701 because of Moishezon’s intensive and fundamental study of these spaces [Moi67]. Before studying Moishezon spaces we consider shortly those spaces X with a(X) < dim X. Example 6.4. The algebraic dimension a(X) can take every integer value in [0, dim X]. We give first the following two basic examples.

304

Th. Peternell

1 0 &2 Jz 0 1 Jz Fi. > Then it is easy to show that every r-invariant meromorphic function on (c2 is constant, hence the torus X has no meromorphic function, i.e. a(X) = 0. (b) We consider the “original” Hopf surface X as an example of a surface with a(X) = 1. Let S3 c QZ2be the 3-sphere (a) Let X = 6Z2/r, with the lattice r =

s3 = {(z,, Then (c’\(O)

is diffeomorphic

~2)11~112

+

b212

= 1).

to S3 x lR via (z19 z2, t)~(e’z,,

efz2).

Let Z act on (c*\(O) by m.(z,, z2)H(emz,,

emz2).

Then (c* \ (0)/Z is a compact complex surface X which is diffeomorphic to S3 x S’. Since b,(X) = 1, we conclude a(X) # 2 (otherwise X would be algebraic, 6.11, hence b, even), on the other hand X is an elliptic fiber space over lP,, hence a(X) # 0. Hence for surfaces a(X) takes all possible values. In general, given n E IN, n 2 2 and m E IN u (0) one can always construct a torus T with dim T = n, a(T) = m by carefully choosing the lattice. The study of compact manifolds X with 0 < u(X) < dim X can - to some extent - be reduced to that the study of projective varieties via the so-called algebraic reduction: Definition-Theorem 6.5. Let X be a compact manifold. A surjective holomorphic map cp:r? -+ Y is culled an algebraic reduction of X if the following conditions hold. (a) r? is smooth and there exists a proper modification 2 + X (b) Y is a projective muniJold with dim Y = a(X) (c) cp*: A(Y) + .4!(x) is an isomorphism.(hence d(Y) low). Every compact manifold X has an algebraic reduction.

For more informations and Chap. VIII.

1: 4’(X), see (6.7) be-

on algebraic reductions we refer to [Ue75],

[Ue83]

2. Basic Properties of Moishezon Spaces. The simplest examples of Moishezon spaces are projective varieties: Example 6.6. Every reduced projective complex space is a Moishezon space. This is a basic fact in algebraic geometry since every meromorphic function on X is rational by Chow’s theorem (see [GH78]). Alternatively, use the fact that

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Modifications

305

every n-dimensional compact projective variety can be realized as finite cover of P,,; then take n algebraically independent meromorphic functions on lP” (e.g. sections in O,“(l)) and pull them back to X. The following is of basic importance: Proposition 6.7. Let X, Y be irreducible compact complex spaces and 4: X + Y a proper modification. Then the induced homomorphism fj*: A(Y)

+ J?qX),

obtained by lifting meromorphic functions,

is an isomorphism (of fields).

For a proof see [Fi76]. A meromorphic map 4: X + Y is called bimeromorphic if there are proper closed analytic sets A c X, B c Y such that dIX\A -+ Y\B is biholomorphic. We also say that X and Y are bimeromorphically equivalent. From (6.7) we obtain by elimination of indeterminacies (see Sect. 7): Corollary 6.8. Every bimeromorphic map 4: X + Y induces an isomorphism d*: A(Y) -+ A(X). Conversely, given irreducible reduced Moishezon spaces X and Y, with an isomorphism a: A(Y) then there is a bimeromorphic

+ .&if(X),

map 4: X -+ Y such that d* = M.

The second part can be deduced easily from the analogous statement projective varieties (see e.g. [Ha77, 1.4.41) applying one part of (6.9). By using algebraic reduction we also see

on

Theorem 6.9. Let X be an irreducible reduced compact complex space. Then X is Moishezon tf and only zf X is bimeromorphically equivalent to a projective variety.

By using a strong form of elimination (9 7) we have more precisely

of indeterminacies

([Hir64],

[Moi67])

Corollary 6.10. Let X be an irreducible reduced Moishezon space. Then there exists a blow-up 7~:8 + X such that X is projective. If X is smooth, we can achieve this also by a finite sequence of blow-ups with smooth centers. Corollary

6.11. Every smooth Moishezon

surface is projective.

Indeed, one shows easily the following: if X is a compact manifold, x E X, then X is projective if and only if the blow-up of X in x is projective (see [GH78]). Now we discuss functorial properties of Moishezon spaces. In fact, we will see that these spaces behave “more functorial” than projective varieties do, showing the importance of the category of Moishezon spaces.

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Th. Peternell

Proposition 6.12. ([Moi67]) (1) Let X be a Moishezon space, Y a reduced compact complex space, and f: X -+ Y a holomorphic surjective map. Then Y is a Moishezon space. (2) Let X be Moishezon, and Y c X a compact subspace, then Y is again Moishezon.

Note that we may assume X, Y to be irreducible (reduced) and further that (2) is an immediate consequence of (1) via (6.10). Claim (1) can be easily deduced from the following Proposition 6.13. ([Ue74]) Let X, Y be irreducible reduced compact complex spaces, and 4: X + Y a surjective holomorphic map with connected fibers. Then a(X) I a(Y) + dim 4, where dim CPdenotes the dimension of a general smooth fiber of q5. 3. Positive Sheaves and Moishezon Spaces. In Section V.4 we saw that a compact complex space is projective if and only if it carries a positive line bundle. It is natural to ask for a similar characterization for Moishezon spaces. There are two ways of weakening the notion of a positive line bundle:

a) one takes a positive coherent sheaf which is not locally free, b) one substitutes positivity by “almost positivity” in a differential sense.

geometric

According to a) we have Theorem 6.14. An irreducible reduced compact complex space is a Moishezon space if and only if it carries a positive torsion-free coherent sheaf 9 with

supp(3)

= x.

In fact, if X is Moishezon, let by (6.9) rc: X + X be a modification with X being projective. Take a positive line bundle LZ on 2. Then rr&‘P) is a torsion free positive sheaf on X with some p >> 0 by Ancona’s theorem (V.4.11). In the other direction, given a positive sheaf ~3, then E’(P) carries a positive line bundle, namely O(1). Thus lP(6p) is projective, whence X is Moishezon by (6.12). Coming to b) we define a line bundle L to be almost positive if it carries a hermitian metric whose associated canonical connection has semi-positive curvature everywhere which is positive at some point. Then we have the following generalization of Kodaira’s embedding theorem (see Chap. V), conjectured by Grauert and Riemenschneider [Gr-Ri70]. Theorem 6.15. (Siu-Demailly [Si84], [Dem85]) Let X be a compact manifold carrying an almost positive line bundle. Then X is a Moishezon space.

Siu’s and Demailly’s methods culminate in the creation of many sections in exactly they prove dim H”(X, LP) - p” for p + co (n = dim X). In other words, the Iitaka dimension fullilles rc(X, L) = n. Then one has easily L”. More

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307

Proposition 6.16. Let X be an irreducible compact complex space of dimension n and L a line bundle on X with K(X, L) = n. Then X is Moishezon.

In fact, sections of L” produce a meromorphic generically finite map of X to a subvariety of lPN. It should be noted that there is a notion of almost positive coherent sheaves which allows us to restate (6.15) as follows: An irreducible compact complex space X is Moishezon if and only if X carries an almost positive torsion free coherent sheaf 9 with supp 9 = X. For details see [GrRi70], [Ri71]. Remark 6.17. Let X be a compact manifold, and L an almost positive line bundle on X. Then X is easily seen to be Moishezon provided Kodaira vanishing holds in the category of compact manifolds and almost positive line bundles: zP(X,LOK,)=O,

q>o.

(*I

Observe that if X is a priori Moishezon, this is just the Grauert-Riemenschneider vanishing theorem [GrRi70]. But this is not how Siu and Demailly proved (3.2); they showed first the asymptotic estimate dim Hq(X, L”) I C. $-i

for positive 4.

So it would be interesting to have a direct proof of (*). Demailly’s

proof is based on his famous Morse inequalities: dimHq(X,Lk~E)~~~*~q~(-l)q(~4~+

j& (- 1)q-j dim Hj(X, Lk @ E) < rs . jx,.,,(-l,‘(&%>”

o(k”), + o(k”),

where X is a compact n-dimensional manifold, E a holomorphic vector bundle of rank r and L a line bundle. Furthermore, a hermitian metric on L is given with curvature z (see Chap. V) and we set X(q) = (x E XII’&(x)

has exactly q negative and n - q positive eigenvalues},

and X(14)

= X(O)u...uX(q).

4. Algebraic Spaces. The famous GAGA theorems of Serre [Ser56] state that any projective algebraic complex space can be viewed as a projective scheme of finite over (IJ. To be more precise, there is a functor

an: (category of scheme of finite type/(C) -+ (complex spaces) associating to every scheme X of finite type over (c a complex space X,,. In fact, locally (in the Zariski topology) X is given by polynomial equations fi, . . . , fk in some affine space IAN and now considering fi, . . . , fk as holomorphic functions

308

Th. Peternell

one obtains by the some local data a complex functor an induces an equivalence of categories

space. GAGA

states that the

(projective schemes of finite type/C) + (projective algebraic complex space). Now consider the category of projective complex spaces as a subcategory of Moishezon spaces. Then we ask: What are the “algebraic objects” corresponding to the Moishezon spaces (if there are any)? In the next section we shall see that there are Moishezon spaces X which are not of the form Y,, with Y a complete scheme. In other words, the category (Y,,l Y complete scheme of finite type / C) is a category strictly contained between the projective one and the Moishezon category. So in order to represent Moishezon spaces by algebraic objects one has to enlarge the category of schemes. This was done by M. Artin [Art701 and Knutson [Knu71] who invented the “algebraic spaces”. We do not want to give the precise definition of algebraic spaces here, but instead refer to Knutson’s basic Lecture Notes. Intuitively, instead of using affine spaces as local pieces of schemes, points in an algebraic space have only &tale neighborhoods and these “&tale pieces” then have to be glued. In particular, schemes are algebraic spaces in a natural way. Now Artin proves Theorem 6.18. There is a “natural” functor

an: (algebraic space of finite type / C) - (complex spaces) extending the functor an on the category (schemesof finite type / C). This functor inducesan equivalence of categories (complex algebraic schemesof finite type / Cc)--P(Moishezon spaces).

In other words, every Moishezon spaces “is” in an unique way an algebraic space. We conclude that for the needs of birational geometry the category of schemes is often not adequate: there are too few objects. But the category of algebraic schemes is: you cannot leave this category by modifications. Some comments to the proof. First one shows that given a (possibly nonreduced) Moishezon space X, there is a diagram

with cp and I,+ modifications and X” projective. Since X” is algebraic by the classical GAGA theorem, it is now sufficient to prove the following statement: (*) given a modification f: X + Y with degeneracy sets A c X, B c Y, then if one of the spaces X and Y is algebraizable and if the formal completion p: A^ + fi is algebraizable, then f is algebraizable.

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Modifications

309

In fact, in our situation {( = $J, I,&) will be algebraizable by using induction and the fact that subspaces of Moishezon spaces are again Moishezon. The proof of (*) relies essentially on the algebraic version of (5.5) and (5.6) and the uniqueness statements of (5.5) and (5.6). A useful consequence of (6.18) is Corollary 6.19. Let X be a Moishez_on space. Then every x E X admits a neighborhood U, an affine complex space U (that is, fi is given in some space CN by polynomical equations) and an ttale map f: 0 + U. 5. Examples. We have seen in Sect. 2 that every smooth Moishezon surface is projective. Thus in order to get examples of non-projective Moishezon spaces we either have to look for singular surfaces or for smooth 3-folds. In the sequel we shall consider only non-projective Moishezon 3-folds. The first example is due to Hironaka [Hir60] and is already classical. Example 6.20. Let Y be any smooth projective 3-fold, and let C c Y be an irreducible curve which is smooth except for one double point y, with a normal crossing. Let U be an open neighborhood of y, in Y and rrl: U, + U the blowup of one of the two irreducible components of C n U, rc2: U, + U, the blow-up of the strict transform of the second one. Let II/: Z -+ Y\{ y,} be just the blow-up of C\{y,}. We can glue II/ and rcl o rc2 to obtain a compact manifold X and a modification p:X-+Y. In particular, X is Moishezon. Now X is not projective: p-l(y) is a smooth rational curve for y E C, y # y,, and p-l(y,) = C, u C, with two smooth rational curves Ci. If C, comes from 7c1,then it is easily shown that C, is homologous to 0, so X cannot be projective. By substituting the singular curve C by two smooth curves meeting in exactly two points transversely a similar construction gives a Moishezon 3-fold X containing two smooth rational curves C,, C, with C, + C, homologous to 0. This X “is” even a complete scheme but not projective. The next example is due to Fujita. Example 6.21. Let Y c IP, be a smooth quadric, so Y N IP, x IP,. Let b 2 3 be an integer and C c Y a smooth curve of type (3, b). Let 7~:X + IP, be the blow-up of C c IF’,. Let Y c X be the strict transform of Y. Let F be a fiber of the first ruling of Y N Y, i.e. the projection p1 to the first factor. Then one computes easily: (CF)

= -1,

i.e. the restriction of the normal bundle to F is nothing but OF( - 1). So by Sect. 3, X can be blown down along the fibers of pr; we obtain a Moishezon manifold X with a blow-up map G: X -+ X. Now it is easy to verify that Pit(X) N Z. As X is Moishezon, there is exactly one generator 9 of Pit(X) such that dim H’(X,

yp) - p3

as p + co.

Th. Peternell

310

But if C c X denotes the curve which is the o-image of P then one computes (cI(~)~c)=

b - 3 5 0.

Hence X cannot be projective, since it does carry an ample line bundle. Example 6.22. In [Hir62] Hironaka constructed a family of compact manifolds K), s /, such that X, is projective for t # 0 and X, is Moishezon but not projective. For details seeHironaka’s paper. For more examples and informations seee.g. [Wer87], [Pet93]. 6. Projectivity Criteria. Given a Moishezon manifold X we ask for criteria guaranteeing projectivity of X. The first basic result is Moishezon’s Theorem 6.23. Let X be a Moishezon manifold. projective.

If X is Kiihler, then X is

For a proof see[Moi67], for a shortened version [Pet86,93]. Observe that a compact Kahler manifold cannot carry a curve C homologous to 0 becausefor any Kahler form w one has

sw>0. C

In all known examples of non-projective Moishezon 3-folds there appear effective curves c n,C, (ni > 0) homologous to 0. So we ask conversely whether these are the only obstructions to projectivity. This is unknown up to now. In order to formulate a partial result, we introduce the finite-dimensional space N,(X) = (1 niCilni E Z, Ci c X irreducible curve}/ =, where z denotes numerical equivalence, i.e. C, = C,

ifandonlyifD.C,

= D.CZ

for every divisor D on X. Let NE(X) be the cone in N,(X) generated by the classesof irreducible curves and NE(X) its closure. Then: Theorem 6.24. Let X be a Moishezon 3-fold. Assume that there is no irreducible curve C c X homologousto 0 and that NE(X)n

-NE(X)=

(0).

Then X is projective. For an analytic proof see [Pet86], for proofs using “Mori theory” see [Kol90], [Pet93]. If m(X) n -m(X) = (0) and X is a scheme, then (5.2) is classical and rather easy to verify: this condition enforces rather directly the existence of an ample line bundle on X, see [Har70]. The problem stated above is whether our condition can be weakend to NE(X) n -NE(X) = (0) plus the non-existence of irreducible curves homologous to 0 (note that a com-

VII.

Moditications

311

plete scheme cannot carry an irreducible curve homologous to 0 - this follows easily from the existence of afline neighborhoods). For more informations, partial results and problems in this context see [Pets& 931. We construct now 2-dimensional normal Moishezon spaces which are not projective. For this recall that a normal surface singularity y E Y is called rational if for one (and hence for all) desingularization (see $7) f: X + Y we have R’f,(Co,) = 0. Otherwise y is called irrational. First note Proposition 6.25. Let X be a normal singularities.

Moishezon

surface

with

only

rational

Then X is projective.

This result is due to Brenton [Br77]. Example 6.26. We now construct normal non-projective Moishezon surfaces. Necessarily such an example has to have an irrational singularity. (1) Let C be a curve of genus at least 2. Choose a rank 2 vector bundle E on C as a non-split extension O-+L+E+Q+O

with a line bundle L of positive degree d, such that for no covering c + C the pulled back sequencesplits. There is a unique section C,, of r? = lP(E) + C with Ci = -d. Hence there is a holomorphic map f: r? -+ X to a normal Moishezon surface blowing down C,. Assume that X is projective. Then we find an irreducible curve not passing through the singular point f(C,). Thus there exists an irreducible curve B c r? with B n C, = 0. B can be considered as multisection of r? + C, hence after passing to the covering B + C, the bundle E has to split, contradiction. Compare [Gra62]. (2) By blowing down elliptic curves, such an easy example is not possible. Instead, consider a cubic C c lP, and let xi, . . . , xl0 be general points on C. Let f: X + lPZ be the blow-up of these point. Then the strict transform C of C in X is elliptic with C2 = - 1. It can be shown that the blow-down of C is not projective.

9 7. Desingularization One of the main applications of blow-ups is the bimeromorphic smoothing of complex spacesto be explained in this section. Due to the complexity of the material we can give here only a very rough exposition without going into any details of proofs. 1. Statement of the Problems Definition 7.1. Let X be a reduced complex space. A desingularization of singularities of X is a proper modification f: r? -+ X such that (a) 2 is smooth, (b) the center off is the singular locus of X.

resolution

or

Th. Peternell

312

The original problem of desingularization asks whether any reduced complex space can be desingularized. Often one wants f to be a (locally finite) sequence of blow-ups with smooth centers. Another important version of the problem is the so-called embedded desingularization (or resolution of singularities): Definition 7.2. Let X be a complex manifold and Y c X a reduced closed complex space of X which is nowhere dense. An embeddeddesingularisation of X is a proper modification f: r? -+ X such that (a) 2 is again smooth, (b) f-‘(Y) is a hypersurface in X and has only normal crossings.

The last term has to be explained: Definition 7.3. Let X be a complex manifold, and Y c X a reduced hypersurface. We say that Y has only normal crossingsif every point y E Y has an open neighborhood % in X with coordinates zi, . . . , z, (n = dim X) such that every irreducible component of Y n ??/is of the form {zk = 0) for some k. The problem of “embedded resolution of singularities” is now to prove for a given reduced closed subspace Y c X the existence of an embedded desingularization. Desingularization problems are of course only interesting for complex spaces of dimension at least 2; for reduced curves desingularization is nothing but normalisation. For algebraic surfaces (over C) the problem (7.1) was first solved by Walker [Wa135], and then also (in characteristic 0 in general) by Zariski [Zar39], [Zar43]. For complex algebraic 3-folds Zariski [Zar44] obtained a slightly weaker result than stated in (7.1). Finally, problem (7.1) for general for complex algebraic spaces (or for schemes in characteristic 0) was solved by Hironaka [Hir64]. This paper settles also the problem of embedded resolution (7.3) in the algebraic case. We have to specify what we mean by an algebraic space in this context: Definition 7.4. A complex space X is called algebraic if there is a scheme S? of finite type over C whose associated complex space xa^,,is just X (up to isomorphism).

For the definition of C&b.see Section 6.4. In particular, a complete algebraic complex space is Moishezon. As for the general complex analytic case, some special cases were proved already by Hironaka [Hir64]; later (7.1) was proved by Aroca-HironakaVicente in [AHV77]. A new constructive and much simpler proof was obtained by Bierstone-Milman [BM91], [BM92]. 2. Desingularization in the Algebraic Case- Hironaka’s Theorems. The main

results of [Hir64]

can be given in the following two theorems.

Theorem 7.5. Every irreducible reduced algebraic complex spaceadmits a desingularization which can be chosenas a finite sequenceof blow-ups with smooth centers.

VII.

Modifications

313

Theorem 7.6. Let X be an algebraic manifold, and let Y c X be a closed reduced algebraic subspace. Then Y c X admits an embedded desingularization which can be chosen as a finite sequence of blow-ups with smooth centers.

The proofs of (7.5) and (7.6) are very complicated and depend on an inductive process. Roughly speaking (for (7.1)) one has first to find a suitable subspace in the singular locus of X, then to explain why the singularities of the blow-up become better or at least not worse and then to prove that the process stops after finitely many steps. In the proof of (7.1) the notion of normal flatness plays an important role. Definition 7.7. Let X be a complex space, and let Y c X be a complex subspace defined by the ideal sheaf I. X is called normally flat along Y at the point x E Y if Z!JI:+’ is a locally free Or,, -module for all v E IN; X is normally flat along Y if X is normally flat along Y at every point x E Y.

If Y c X is locally a complete intersection, then X is normally flat along Y. Note that if X is normally flat along Y, then the multiplicity of O,,, is constant along Y. This gives a hint why normal flatness is important in desingularization theory: the multiplicity is a measure for the complexity of the singularity. In particular x is a smooth point of X if and only if the multiplicity of 0,,x is 1. Also the following fact is remarkable and sheds some light on the general philosophy for desingularization described above: Let X be irreducible and reduced and Y c X a smooth closed subspace such that X has constant multiplicity along Y. Let 7~:X -+ X be the blow-up of Y. Then multiplicity for all Z?E X. Actually Hironaka

(@a,;) I multiplicity

(0,,,&

proves in fact more than Theorem 7.5:

Theorem 7.8. Let X be an algebraic complex space. Then there exists a finite sequencexi: Xi+I + Xi (0 I i I r) of blow-ups with centers x c Xi such that (1) x0 = x, (2) x is smooth (for all i), (3) Xi is normally flat along yi (for all i), (4) if y E x then either y is a singular point of the reduction red Xi or Xi is not normally flat along red Xi, (5) red X, is smooth and X, is normally flat along red X,.

One can view (7.8) as a desingularization theorem for non-reduced spaces. Of course one cannot make the nilpotent elements vanish but they can be forced to behave nicely. 3. Embedded Resolutions. Now we discuss Theorem

7.6.

(7.9) The easiest case is dim X = 2 and dim Y = 1. Let y,, . . . , yP be the singular points of Y. Let 7~~:X, + X be the blow-up of y, and E, be the exceptional curve rc;i(yl) and denote by Y, the strict transform of Y in Xi. Then

314

Th. Peternell

(E, . Y,) is nothing but the multiplicity my,(Y) of Y in y,. Moreover, it is easy to see that for the singular points y;, . . . , yi of Yr one has:

Now blow up successively the points y;, . . . , yi. Proceeding in this way it is easy that after finitely many steps the strict transform Y, will be smooth “over yr”. Now take y, and proceed in the same manner. Thus after finitely many blowups the strict transform Y, c X, is globally smooth. Now the full preimage of Y in X, consists of smooth curves, however not necessarily with transverse intersections. But this can be clearly achieved by some more blow-ups of the “critical” points of intersection. 0 Theorem 7.6 turns out to be a special case of a far more general theorem, called “simplification of coherent ideal sheaves”, which will be explained next. Definition 7.10. Let X be a complex manifold and I a coherent sheaf of ideals. (1) Let x E X. Then we define v(Z,) to be the maximal integer ,U such that I, c rn: where m, is the maximal ideal in cO,,x. (2) Let Y c X be a complex submanifold and Q: r? + X the blow-up of X along Y. The weak transform of I by (r is the coherent ideal sheaf J c 02 given by the following property: Let p = v(Z,) for generic x E X. Let Zo-Lu,j be the full ideal sheaf of a-‘(D). Then

Im( f *(I) -+ 0,) = Zgml,,, . J. The number v(Z,) should be considered as a measure how singular the subspace defined by Z is at the point x. Now the generalization

of (7.6) reads

Theorem 7.11. Let X be an algebraic mantfold, and let Z c 0, be a non-zero coherent ideal sheaf. Let u = max v(J,). Let E, c X be a reduced hypersurface XPX with only normal crossings. Then there exists a finite sequence of blow-ups 7ti: Xi+l + Xi, 0 5 i I r, with X, = X such that the following holds: (1) The center x of 7ciis smooth and connected; (2) Let Ii be the weak transform of I,-, by IC-~ and I, = I. Then v(Z~,~) 2 u for all y E xi; (3) Let Ei be defined inductively by Ei = red(n,:\(Ei-,)

u ~c~I\(I’-~))

for i 2 1.

Then Ei has only normal crossing with x::; (4) E, has only normal crossings and v(Z,,,) < u for all y E X,.

An important indeterminacies:

(but not totally

obvious) consequence is the elimination

of

VII. Modifications

315

Theorem 7.12. Let us assumefor simplicity that X and Y are compact algebraic complex spaces,and let f: X + Y be a meromorphic map. Then there exists a finite sequencex: X -+ X of blow-ups with smooth centers in the indeterminacy set of f respectively the map induced by f such that there is a commutative diagram

with a holomorphic map p.

For details and more precise results see [Hir64]. 4. The Complex - Analytic Case. Hironaka proved already in [Hir64] a “semi-local” version of the desingularization of complex spaces: assume X to be a reduced complex subspace of Y, x Y, with Y, Stein and Y, projective-algebraic. Let pr, be the projection onto Y,. Let y E Y,. Then there exists an open neighborhood C&of y in Y, such that X, = pr;‘(%) can be desingularized in the sense of (7.1). In the same spirit he proved an embedded resolution theorem. In the general case one has the following theorem due to by Aroca-HironakaVincente [AHV77] and Bierstone-Milman [BM91,92] Theorem 7.13. (1) Every reduced complex space can be desingularized. (2) Every reduced closed complex subspaceof a complex manifold admits an embeddeddesingularization.

For more detailed statements see [AHV77] and [BM91]. The method of Bierstone and Milman gives canonical way of resolution and is much simpler than the methods of [AHV77].

References* [AHV77] [Anc80] [Am761 [Art681 [Art701

Aroca, J.M.; Hironaka, H.; Vicente, J.L.: Desingularisation theorems. Mem. Math. Inst. Jorge Juan No. 30. Madrid 1977,Zbl.366.32009. Ancona, V.: Sur l’equivalence des voisinages des espaces analytiques contractibles. Ann. Univ. Ferrara, Nuova Ser., Sez. VII 26, 165-172 (1980) Zbl.459.32008. Arnold, V.I.: Bifurcations of invariant manifolds of differential equations and normal forms in neighborhoods of elliptic curves. Funkts. Anal. Prilozh. 10, No. 4, 1-12 (1976). English transl.: Funct. Anal. Appl. 10, 249-259 (1977), Zbl.346.58003. Artin, M.: On the solution of analytic equations. Invent. Math. 5, 277-291 (1968) Zbl.17253. Artin, M.: Algebraization of formal moduli II: Existence of modifications. Ann. Math., II, Ser. 91, 88-135 (1970) Zbl.185,247.

*For the convenience of the reader, references to reviews in Zentralblatt fiir Mathematik (Zbl.), compiled using the MATH database, have, as far as possible, been included in this References.

316 [AT791 [AT821 [Bin781 [Bin811 [BM91] [BM92] [Br77] [CG81]

[Cho49] [Cor73] [Dem85] [Fi76] [FN72] [Fuj75] [GF74] [GH78] [Gra62] [Gri66] [CR553 [CR713 [GR77] [GrRi70] [Ha771 [Hir60] [Hir62] [Hir64] [Hir75] [Hir81]

Th. Peternell Ancona, V.; Tomassini, G.: Thtoremes d’existence pour les modifications analytiques. Invent. Math. 51,271l286 (1979) Zbl.385.32017. Ancona, V.; Tomassini, G.: Modilications analytiques. Lect. Notes Math. 943, Springer 1982, Zbl.498,32006. Bingener, J.: Formale komplexe Rlume. Manuscr. Math. 24, 253-293 (1978) ZbI.381.32015. Bingener, J.: On the existence of analytic contractions. Invent. Math. 64, 24-67 (1981) ZbI.509.32004. Bierstone, E.; Milman, P.: A simple constructive proof of canonical resolution of singularities. Prog. Math. 94, 11-30 (1991). Bierstone, E.; Milman, P.: Canonical desingularisation in characteristic zero: a simple constructive proof. To appear Brenton, L.: Some algebraicity criteria for singular surfaces. Invent. Math. 41, 129-147 (1977) 2131.337.32010. Commichau, M.; Grauert, H.: Das formale Prinzip fur kompakte komplexe Untermannigfaltigkeiten mit 1-positivem Normalenbtindel. Ann. Math. Stud 100, 101-126 (198 1) Zb1.485.32005. Chow, W.L.: On compact complex analytic varieties. Am. J. Math. 71, 893-914 (1949) ZbI.41.483. Cornalba, M.: Two theorems on modifications of analytic spaces. Invent. Math. 20, 227-247 (1973) Zbl.264.32006. Demailly, J.P.: Champs magnetiques et inequalites de Morse pour la d” - cohomologie. Ann. Inst. Fourier 35, No. 4, 185-229 (1985) Zbl.565.58017. Fischer, G.: Complex analytic geometry. Lect. Notes Math. 538. Springer 1976, ZbI.343.32002. Fujiki, A.; Nakano, S.: Supplement to “On the inverse of monoidal transformations”. Publ. Res. Inst. Math. Sci. 7, 637-644 (1978/2) Zbl.234.32012. Fujiki, A.: On the blowing-down of analytic spaces. Publ. Res. Inst. Math. Sci. 10, 473-507 (1975) Zb1.316.32009. Grauert, H.; Fritzsche, K.: Einfiihrung in die Funktionentheorie Mehrerer Veranderlither. Springer 1974, Zb1.285.32001. Griffith, P.A.; Harris, J.: Principles of Algebraic Geometry. Wiley 1978,ZbI.408.14001. Grauert, H.: ijber Modilikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331-368 (1962) Zbl.173,330. Grifliths, P.A.: The extension problem in complex analysis II. Am. J. Math. 88,366-446 (1966) Zbl.147,75. Grauert, H.; Remmert, R.: Zur Theorie der Modilikationen I. Math. Ann. 239, 274-296 (1955) Zb1.64,81. Grauert, H.; Remmert, R.: Analytische Stellenalgebren. Grundlehren math. Wiss. 176. Springer 1971, Zbl.231.32001. Grauert, H.; Remmert, R.: Theorie der Steinschen Raume. Grundlehren math. Wiss. 227. Springer 1977, Zb1.379.32001. Grauert, H.: Riemenschneider, 0.: Verschwindungsdtze fur analytische Kohomologiegruppen auf komplexen Rlumen. Invent. Math. II, 263-292 (1970) Zbl.202,76. Hartshorne, R.: Algebraic Geometry. Springer 1977, Zb1.367.14001. Hironaka, H.: On the theory of birational blowing-up. Thesis, Harvard (1960). Hironaka, H.: An example of a non-Klhlerian complex-analytic deformation of Kahlerian complex structures. Ann. Math., II. Ser. 75, 190-208 (1962) Zb1.107,160. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero., I, II. Ann. Math., II Ser. 109-326 (1964) Zbl.122,386. Hironaka, H.: Flattening theorem in complex analytic geometry. Am. J. Math. 97, 5033547 (1975) Zb1.307.32011. Hirschowitz, A.: On the convergence of formal equivalence between embeddings. Ann. Math., II. Ser. 113, Sol-514 (1981) Zbl.421.32029.

VII.

CHRW [Ker64] [Knu71] [Kod54] [Ko190] [Kos8

l]

[Kos86]

[Kos88] [Kra73]

[Moi67]

[Nak71] [Pet811 [Pet861 [Pet931 [Rem571 [Ri71] [Ser56] [Si84]

CSp831 [Ue75] [Ue83] [Wal35] [We801 [Wer87] [Zar39] [Zar43]

Modifications

317

Hironaka, H.; Rossi, H.: On the equivalence of imbeddings of exceptional complex spaces. Math. Ann., II. Ser. 156, 313-333 (1964) Zbl.l36,208. Kerner, H.: Bemerkung zu einem Satz von H. Grauert und R. Remmert. Math. Ann. 157, 206-209 (1964) Zb1.138,67. Knutson, D.: Algebraic spaces. Lect. Notes Math. 203. Springer 1971,2b1.221.14001. Kodaira, K.: On Kahler varieties of restricted type. Ann. Math., II. Ser. 60,28-48 (1954) Zb1.57.141. Kollar, J.: Flips, flops and minimal models. Surv. Differ. Geom., Suppl. J. Differ. Geom. I, 113-199 (1991). Kosarew, S.: Das formale Prinzip und Modifikationen komplexer Rlume. Math. Ann. 256,249-254 (198 1) Zb3.468.32004. Kosarew, S.: On some new results on the formal principle for embeddings. Algebraic Geometry, Proc. Conf. Berlin 1985. Teubner Texte Math. 92, 217-227 (1986) Zb1.631.32008. Kosarew, S.: Ein allgemeines Kriterium fiir das formale Prinzip. J. Reine Angew. Math. 388, 18-39 (1988) Zb1.653.14002. Krasnov, V.A.: Formal modifications. Existence theorems for modilications of complex manifolds. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 848-882 (1973). English transl.: Math. USSR, Izv. 7, 847-881 (1974) Zb3.285.32009. Moishezon, B.G.: On n-dimensional compact varieties with n algebraically independent meromorphic functions. Izv. Akad. Nauk SSSR, Ser. Mat. 30, 133-174,345-386,621656 (1966) English transl.: Am. Math. Sot., Transl., II. Ser. 63, 51-177 (1967) Zbl.161,178. Nakano, S.: On the inverse of monoidal transformations. Publ. Res. Inst. Math. Sci. 6, 483-502 (1971) Zb1.234.32017. Peternell, Th.: Vektorraumbiindel in der Nahe von kompakten komplexen Unterraumen. Math. Ann. 257, 111-134 (1981) Zbl.452.32013. Peternell, Th.: Algebraicity criteria for compact complex manifolds. Math. Ann. 275, 653-672 (1986) Zb1.606.32018. Peternell, Th.: Moishezon manifolds and rigidity theorems. Preprint 1993. Remmert, R.: Holomorphe und meromorphe Abbildungen komplexer Raume. Math. Ann. 133, 328-370 (1957) Zbl.79,102. Riemenschneider, 0.: Characterizing Moishezon spaces by almost positive coherent analytic sheaves. Math. Z. 123, 263-284 (1971) Zb1.214,485. Serre, J-P.: Geomttrie algtbrique et gtometrie analytique. Ann. Inst. Fourier 6, l-42 (1956) Zbl.75,304. Siu, Y.T.: A vanishing theorem for semipositive line bundles over non-Kahler manifolds. J. Differ. Geom. 19,431-452 (1984) Zbl.577.32031. Spivakovsky, M.: A solution to Hironaka’s polyhedra game. Arithmetic and geometry, Vol. II, Prog. Math. 36,419-432 (1983) Zb1.531.14009. Ueno, K.: Classification theory of compact complex spaces. Lect. Notes Math. 439. Springer 1975,Zbl.299.14007. Ueno, K.: Introduction to the theory of compact complex spaces in class C. Adv. Stud. Pure Math. I, 2199230 (1983) Zb1.541.32010. Walker, R.J.: Reduction of singularities of an algebraic surface. Ann. Math., II, Ser. 36, 336-365 (1935) Zbl.l1,368. Wells, R.O.: Differential Analysis on Complex Manifolds. 2nd ed. Springer 1980, Zb1.435.32004; Zb1.262.32005. Werner, J.: Kleine Aullosungen spezieller 3-dimensionaler Varietlten. Thesis, Bonn 1987, Bonn. Math. Schr. 186. Zbl.657.14021. Zariski, 0.: The reduction of singularities of an algebraic surface. Ann. Math., II. Ser. 40, 639-689 (1939) Zb3.21,253. Zariski, 0.: Reduction of singularities of algebraic three dimensional varieties. Ann. Math., II. Ser. 45,472-542 (1944).

Chapter VIII

Cycle Spaces F. Campana and Th. Peternell Contents Introduction

320

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

320 320

01. TheDouady Space ......................................... ................................... 1. The Existence Theorem 2. Application: Spaces of Holomorphic Maps and Automorphism Groups ................................................. 3. Properties of the Douady Space ............................ ....................................... 4. Some Applications

322 323 324

Q2. The Space of Cycles (Barlet Space, Chow Scheme) ............... 1. Construction of the Barlet Space (Chow Scheme) .............. 2. Compact Subsets of Cycle Spaces ........................... ........................... 3. Meromorphic Families of Cycles

326 326 329 330

5 3. Cycle Spaces and the Structure of Compact Complex Spaces ............................... 1. Compact Families of Cycles ................................. 2. An Algebraicity Theorem .................................. 3. Algebraic Connectedness 4. Manifolds of Class V ..................................... 5. The Albanese Variety ..................................... 6. Automorphism Groups ................................... 7. Structure of Compact Manifolds in V ....................... $4. Convexity of Cycle Spaces ................................... 1. Convexity of Cycle Spaces of q-complete Spaces .............. ............................... 2. The Method of Norguet-Siu 3. q-convexity and the Cycle Space ............................ References

......

332 333 334 334 336 331 339 340 342 342 344 346

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

320

F. Campana

and Th. Peternell

Introduction Given a complex space X, the Douady space 9(X) parametrizes all puredimensional compact complex subspaces of X. 9(X) carries a natural complex structure and moreover there is an universal family over it (see $1). When X is projective, 9(X) is just the Hilbert scheme of X, as constructed by A. Grothendieck. Clearly the construction of the Douady space for a general complex space is harder than in the algebraic situation. The cycle space %‘(X) or Barlet space of a reduced complex space X in contrast parametrizes linear combinations (with positive integer coefficients) of irreducible compact analytic sets, all of the same dimension (these are called cycles). The construction of g(X) is somehow easier than that of 9(X) and is sketched in $2. The algebraic counterpart (of V?(X)) is the Chow scheme. Almost all applications of 9(X) and U(X) arise from situations where compact families of subspaces are studied. Therefore the following theorem of Lieberman and Fujiki is important: if X is a compact Kahler manifold, then the connected components of%?(X) and 9(X) are compact. This depends on a theorem of Bishop saying that limits of analytic sets with bounded volume are again analytic. In case X is projective, the components of %‘(X) and 9(X) are again projective (Grothendieck). 0 3 gives several applications of the cycle space: to the structure of compact manifolds in general, and to the structure of manifolds in class % (that is, manifolds bimeromorphic to Kahler manifolds) in particular. $4 finally deals with convexity properties of %‘(X) for q-complete and q-convex complex spaces X (Andreotti-Norguet, Norguet-Siu, Barlet).

0 1. The Douady

Space

The aim of this section is to introduce the Douady space and give some applications. The construction of the Douady space being rather complicated, we will not go into the details of its construction and instead refer to Douady’s very clear original paper [Dou66]. Theorem. We understand every complex space to be of The existence of Douady spaces can be formulated as follows.

1. The Existence

finite dimension.

Theorem 1.1. Let X be a complex space, and let d be a coherent sheaf on X. Then there exist a complex space $3 = 9(B) and a coherent sheaf 92 on 9 x X with the following properties: (a) 9 is a quotient of pi-f(B), where pr, denotes the projection, (b) 9%’is flat ouer 9 and prr Isupp(W) is proper, (c) (universal property) If S is a complex space (of finite dimension) and if9 is a coherent quotient of pr,(B) on S x X such that 9 is flat ouer S and pr2(supp(F)) is compact, then

VIII.

Cycle Spaces

there exists a uniquely determined holomorphic

321

map

such that 9 2: (f x id,)*(B).

In other words, $3 = $3(&) parametrizes pact support. Taking & = 0, we obtain:

coherent quotients

of 6 with com-

Corollary 1.2. Let X be a complex space. Then there exists a complex space 9 = 9(X) and a subspace Y c 92 x X (“the universal family”) such that: (a) Y is j7at over 9 and pr2 1Y is proper, (b) if S is a complex space, Z c S x X a subspace having the properties stated in (a), then there exists an unique map f: S + 9 such that Z N S x 9 Y.

In fact, applying (1.1) with d = 0, we just let Y = supp(W) be equipped with the structure defined by Ker(0, x x + ~3). The complex space $3(X) parametrizes compact subspaces of X and is called the Douady space of X. We now reformulate theorem (1.1) in the language of categories. Let X be a complex space and 8 a coherent sheaf on X. Let %?be the category of complex spaces and Y the category of sets. We define a contravariant functor by setting F(T) = {coherent quotients

9 of prf(&‘) on T x X which are flat over T and which have compact support over T}

and letting F(q) be the pull-back map for every holomorphic Then theorem 1.1 can be reformulated as follows. Theorem 1.3. The jiunctor F is representable

map cp: T + S.

(by a complex space g(8)).

Theorem 1.1 has been conjectured by A. Grothendieck. He proved it in the case when X is projective [Gro61], even in a relative version (X projective over S). The analytic relative version is due to Pourcin [Pou69]. Another proof is due to Bingener [Bin80]. Just one word about Douady’s method. First he shows that the functor F can be represented by an infinite dimensional space, a socalled Banach-analytic space. Then he shows that this Banach-analytic space is of finite dimension at each of its points. Or, equivalently, its Zariski tangent space is locally compact. This is a variant of Ascoli’s theorem, and the properness of pr, Isupp(%‘) is used there in an essential way. The analogue of the Douady space in algebraic geometry is the “Hilbert scheme”; it parametrizes complete subschemes of a given scheme X. For X projective the existence of Hilbert schemes was proved by Grothendieck [Gro61], as mentioned above, while the general case was settled by [Art69]. Artin showed that the “Hilbert functor” is represented by an algebraic space, a more general notion than the notion of schemes (cf. VII.6). Therefore the Hilbert “scheme”

322

F. Campana

often is not a scheme but only over C then by GAGA complete complex space X,, associated X and Hilb, are “the same”, which

and Th. Peternell

an algebraic space. If subschemes of X and are “the same”, hence is the same as to say

Wilb,),,

X is a projective scheme compact subspaces of the the Douady space 9(X,,) that

= WLJ.

The same thing still holds true for X an algebraic space of finite type over C (cf.VII.6). We end this section by formulating Pourcin’s relative version of the Douady space. Theorem 1.4. Let X and S be complex spaces, and let f: X + S be a holomorphic map. Let & be a coherent sheaf on X. Then there exists a complex space 9&(S) with a holomorphic map to S, and a coherent quotient 9 of the pull-back of 8 to 5&(&F) xs X such that (a) 92 is pri-flat and pr, lsupp 9 is proper, (b) (universal property) for any complex space Z + S and any coherent quotient 9 of the pull-back of d to Z xs X enjoying the properties of (a), there is a unique holomorphic map f: Z + &(a) such that (f xs id,)*(B) N 9. Here pr,: &.(8) xs X + @(a) is the projection. 2. Application: Spaces of Holomorphic Maps and Automorphism Groups. Let X be a compact complex space and Y an arbitrary complex space. We consider the set Hol(X, Y) of holomorphic functions f: X -+ Y. The problem is to introduce a complex structure on Hol(X, Y). For this purpose we make the following identification. Hol(X,

Y) = {s E 9(X x Y)(the compact subspace R(s) c X x Y given by s is the graph of a holomorphic map f: X + Y}.

Here 9(X x Y) denotes as usual the Douady space of X x Y, coming along with a universal subspace R c 9(X x Y) x X x Y. Then Douady proved in [Dou66] Theorem 1.5. (1) Hol(X, Y) is an open subset of 9(X x Y), hence inherits a complex structure. (2) R n (Hol(X, Y) x X x Y) is the graph of a “unioersal” holomorphic map @: Hol(X x Y) x X -+ Y. (3) (universal property). Zf S is any complex space and f: S x X -+ Y a holomorphic map, then there is a uniquely determined holomorphic map g: S -+ Hol(X, Y) such that f=

(4) The topology underlying ogy of compact convergence. As a corollary

one obtains

@ o(g

x

id,).

the complex structure

on Hol(X,

Y) is the topol-

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Corollary 1.6. Let X be a compact complex space. Then the group Aut(X) of automorphisms is an open set in Hol(X, X) and, in particular, a complex Lie group. This has also been proved by W. Kaup [Kau65] and in the reduced case previously by Kerner [Ker60]. As we see, the Douady space (and also the cycle space of Barlet to be explained later) plays an important rBle in the study of the automorphism groups of compact Klhler manifolds. For more information on this topic, consult Fujiki [Fu78-21 and Liebermann [Lie78]. 3. Properties of the Douady Space. First let us mention that the construction of the Douady space gives also a description of the (Zariski-)tangent space to every point: Proposition 1.7. Let X be a complex space and d a coherent sheaf on X. Let s E g(8) correspond to a quotient 9 of 8. Then there is a canonical isomorphism for the tangent space at s: T&3(&) N Horn,@,

9)

with 9 = Ker(b + 9).

In particular, if & = 0,: W(X)

= Homcx(Jz, 0,) z Hom,Z(JzlJ~2, 09.

where s corresponds to the compact subspaceZ defined by the ideal sheaf J,. For instance, if X is a complex manifold and Y c X a compact submanifold, then the Zariski tangent space to the space of deformations of Y in X (at the “point Y”) is H”( Y, NyIx), Nylx denoting the normal bundle of Y in X. However in general g(X) might not be smooth at Z, so we have only an estimate dim, Q(X) I dim Hom,z(Jz/Ji,

Q).

For an example, let X be a complex manifold of dimension 3 and C c X an exceptional smooth rational curve with normal bundle NC,* = W)O

q-3)

(seeV.2.4). Then C corresponds to an isolated point of &S(X) since it does not deform in any positive-dimensional family. On the other hand 7;,-9(X)

2: H”(Nc,x) N c2.

So g(X) is non-reduced at the point [C]. For more informations on embedded deformations see [Gri66], [Pa190]. The global structure of the Douady space is described by the following theorem. Theorem 1.8. (Fujiki [Fu78-11, [Fu82], [Fu84]) Let X be a compact complex space whosereduction is bimeromorphically equivalent to a KBhler manifold. Let Do be an irreducible component of 9(X). Then:

F. Campana

324

(1) D, is compact. (2) Q,,red is again bimeromorphically

and Th. Peternell

equivalent to a Kiihler manifold.

The analogous statements hold also for g(a), if & is a coherent sheaf on X. Moreover there exists a relative version (see Fujiki’s papers). The first part of (1.8) is proved via Barlet’s cycle space (see sect.2,3), where compactness holds for every component (proved also by Campana [Ca80]). In algebraic geometry the analogous statements hold for every component, due to Grothendieck [Gro61] in the projective case respectively Artin [Art69,70] in the Moishezon case ~ of course then the Douady space (Hilbert scheme) is algebraic. For the significance of the category of compact complex spaces bimeromorphically equivalent to Kahler manifolds, often called spaces of class %?,see sect.3. Without any assumption on X, theorem 1.8 is in general false. This is illustrated by the following example from [Ue81]. Example 1.9. Let X be the classical Hopf surface: let i E (L:, 121< 1, and let cp: (c2\{O} + C2\{O} b e g iven by q(z) = AZ, G the cyclic group generated by cp and let X = (cC2\{O})/G. X being diffeomorphically S’ x S3, we have b,(X) = 0 and hence X cannot be bimeromorphically equivalent to a Kahler manifold (otherwise one would have a positive, closed (1, I)-current on X coming from a Kahler form of a Kahler manifold bimeromorphic to X which defines a nonzero element in H2(X, W)). Now it is well known

and easy to prove that il 0 Aut(X) is just Gl(2, (L) o I operating on I( > to check that the stabilizer H of the set given (0, 1) is an infinite discrete group and also containing the graphs of all maps II/ E Aut,(X) so is not compact. Fujiki also proved ([Fu79])

the identity component

Aut,(X)

of

X in an obvious manner. It is easy by the residue classes of (1,0) and that the component of 9(X x X) actually reduces to Aut,(X) and

Theorem 1.10. The Douady space of a complex space has only countably many irreducible components. 4. Some Applications. In this section we describe some “typical” applications of the Douady space in order to demonstrate the importance of this concept. The first application deals with the formal principle (for more informations on this topic see chap. VII, sect. 4). Let X be a complex space, and consider A a closed complex subspace. Let (X, A) be the germ of X along A. We say that (X, A) fulfills the formal principle if the following holds: every germ (Y B) which is formally isomorphic to X.

isomorphic

to (X, B) is in fact analytically

We will use the following notations. Let D = g(X) be the Douady space of X and Y c D x X the universal subspace. We denote by E c D the set of all subspaces Z c X meeting A and let YE = E xg D be the induced family.

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V. SteinbiB [St861 proved

Theorem 1.11. Let X, A be reduced. Assume that for every x E A there is a compact subspace Z c A with x E Z having the following two properties: (1) There is an open neighborhood of Z in E containing a dense subset M such that for all Z’ E M, Z’ is reduced with every irreducible component of Z’ meeting A. (2) The projection map of germs of complex spaces P: (Y,,

CZI x {x>, + (X3 x)

is surjective. Then the formal principle holds for (X, A). Intuitively, (1) means that sufficiently many small deformations of Z meeting A are automatically reduced and, in fact, every irreducible component of the small deformations meet A. This is a kind of technical assumption. The more important is the second assumption: small deformations of Z still meeting A fill up a neighborhood of x in X. In particular, many compact subspaces of A can be moved out of A, a condition which is never satisfied if A is exceptional in X (see Chap. V). So the second condition can be viewed as an assumption in the direction of (semi-)positivity of the normal bundle. In any case, we see that the neighborhood structure of a subvariety can sometimes be well described via the Douady space. (1.12) Let X be a smooth compact complex surface. The r-th symmetric power S’(X) is by definition the quotient X x . . . x X/S, of the r-th product of X by the symmetric group S,. The set S’(X) carries a natural complex structure such that the quotient map f: X x . . . x X + s’(X) is finite. But s’(X) will have singularities along the image of the “diagonal” (see below). Now let Xt*] c 9(X) be the closed complex subspace of O-dimensional compact subspaces Z c X of length r, i.e. 1 dim. oz., = r. Then there is a canonical map 71:Xt’] -+ S*(X) ZSZ

obtained

by associating

to the subspace

Z the O-cycle

i

rizi, where

Z =

c=1

{zi, . . . , z,} as set and dim Oz,,i = ri. The space Xt’] can be considered ral” desingularisation of s’(X) by virtue of Proposition is the set

1.13. (Fogarty D = {fh,

Moreover

[Fo68])

as “natu-

XC’] is smooth. The singular locus of s’(X)

..-, x,)/xi = xi

for some i # j}.

the map II is bimeromorphic.

In algebraic geometry Xt’] is denoted by Hilb’(X) and called the Hilbert scheme of points of length r. This object reflects a lot of geometry of X and has been studied intensively. See the survey article [Ia87] for details. Beauville has proved in [Be831

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Theorem 1.14. Let X be a K3-surface nected). Then X[‘l is Kiihler and symplectic on X such that A’o has no zeroes).

(i.e. Qi N 0, and X is simply con(i.e. there is a holomorphic 2-form w

This is remarkable since there are not so many known examples of higher dimensional symplectic manifolds (especially non-algebraic). Compare (3.42).

0 2. The Space of Cycles (Barlet Space, Chow Scheme) In contrast to the Douady space, the Barlet space or cycle space or Chow scheme parametrizes cycles with multiplicities. Its construction and properties (which are analogous to the properties of Douady spaces) form the topic of this section. 1. Construction

of the Barlet Space (Chow Scheme)

Definition 2.1. Let X be a complex space and n E W an integer. An n-cycle of X is a finite linear combination Z = c n,Z, where the Z,‘s are irreducible anais1

lytic compact subsets of X dimension n which are pairwise distinct. The support of Z, denoted IZI, is the union of all Zls. The set of all n-cycles of X is denoted by %JX), and the set of all cycles of X is the union of all q”(X) for n E N, denoted by %(X). We call q(X) the Barlet spaceor Chow schemeof X. In [Ba75] a natural structure of a complex space is introduced give a short introduction to this construction. First, we will define analytic families of n-cycles of X.

on g(X). We will

Definition 2.2. A scale E = (U, B, f) of X is an open subset X, of X with an embedding f of X, as a closed analytic subset of an open neighborhood in (lY”+p of 0 x B, where U and B are polydiscs of (c” and (cp respectively. The scale E = (U, B, f) is said to be adapted to the n-cycle Z if f(lZ() does not meet v x aB. Remark 2.3. In this case f(X n X,) appears as a ramified covering of degree k = deg,(X), possibly zero, of a neighborhood of !? in Cc”, account being taken

of the multiplicities

of the local branches of Z n X,.

Such ramified coverings are now parametrized to Symk(B) defined as follows:

by holomorphic

maps from U

Construction 2.4. Let Symk(ap) be the quotient

of the k-th symmetric

variety ((EP)k under the action group acting by permutations of the factors; this is an k-l

affine variety admitting

a natural embedding

in VP,, = ,g sj(c’), where sj(ap)

is the jth component of the symmetric algebra of a?‘. In particular, parametrizes the k-tuples of points (with multiplicities) of (cp.

Symk((CP)

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Let Symk(B) be the (open) image of Bk by the quotient from ((Cp)k to Symk(ap). We are now in position to define the notion of an analytic family of n-cycles 0fX: Definition 2.5. Let S be a complex space and (Zs)sEs be a family of n-cycles of X parametrized by S. Then this family is said to be analytic if for each sOE S there exists an open neighborhood W of lZsol in X such that lZ,l c W for each s sufficiently near to sO, and if for every scale E = (U, B, F) of X which is adapted to ZsO, there exists an open neighborhood SE of s,, in S such that: i) E is adapted to Z, for each s in SE. ii) deg,(Zs) = deg,(Z,J = k, for every s in SE. iii) The map gE: SE x U -+ SymkE(B) is holomorphic, where g&, .) is, for each s of SE, the holomorphic map associated to the ramified covering f(Z, n X,) of u.

2.6. This definition enables us to construct a contravariant functor Fi from the category of complex reduced spaces to the category of sets, associating to each S the set of analytic families of n-cycles of X parametrized by S. Barlet’s theorem states that this functor is representable by a reduced complex space which is finite dimensional at every point. Before giving some indications on the construction of the complex structure of %7,,(X), let us state a result which shows the geometric signification of the preceding definition: Theorem 2.7. Let (Zs)scs be an analytic family of n-cycles of X parametrized by S, and let IG,l c S x X be defined by: (s, x) E IG,l if and only if x E lZ,l. Then IG,l is a closed analytic subset of S x X. The restriction of the first projection ps of S x X to IG,l is proper, surjective, and its fibers have pure dimension n. For an irreducible component IGz 1of I G,l, there exists a positive integer no such that for s generic in So = p,( I G,” I) all irreducible components of IZ,l contained in 1Gp I have multiplicity no. The closed analytic cycle G, = 1 n”G,O is called the graph of the analytic family (Z,), Es parametrized

by S.

Let us now give some indications on the proof of the representability of Fi, which can be viewed as a generalization of the classical Cartan-Serre’s proof of the finiteness of the dimension of S = H”(X, F) for X a compact complex manifold and F a holomorphic vector bundle over X, the reduced image of an analytic section s E S being a cycle Z, of F. In this case the family (Zs)sss is nothing but the connected component of V(F) containing the zero-section Z, of F. First step: Parametrization of local pieces of cycles. Let (Ei)i,r be a finite set of scales of X adapted to a given n-cycle Z, on X such that Ei = (Ui, Bi, fi) and (f;-‘(U, x Bi))i,l covers Z,. For each i E I, let pi = yEi,ki with ki = deg,,(X,) be defined as follows:

Let H(ui, Vpi,k,) be the Banach space of continuous functions on vi, analytic on Ui, with values in Vpi,k, where pi = dim(&).

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and Th. Peternell

Then Bi is the Banach-analytic subset of H(U,, VPi) consisting of those functions which take their values in Symf(Bi) and for which the associated ramified covering of degree ki of vi contained in vi x Bi is contained in f;(Z,,). Then fi parametrizes local pieces of cycles sufficiently close to {X0 n f,-‘(U, X Bi)}. Second step: Embedding of a neighborhood of Z, in G&(X) as a Banachanalytic subset of n yi. For each (i, j) E I, let (EJoeAij be a finite number of iel

scales E, = (U,, I?,, f,) on Pj = Uj x Bj such that where XEa = (1) XEa is relatively compact in X, = fi-‘(Pj) n f;-‘(Pi), 6’ (L~‘U’a))~ and (2) each component of X, n Z, meets at least one XEl, and for all a in A, one has: fi = f, o fi on X, . Such scales exist if (i, 3) is ordered in such a way that pi 2 pi. The neighborhood W of Z,, in g,,(X) consisting of all cycles Z for which all scales E are adapted to Z with the same degrees for Z,, is now realized as the closed Banach-analytic subset of g = n fi defined as the reciprocical image of 0 by the natural product is1

of differences of restriction mappings res: B + r&

.G,, H(uaT,, Symka(&) ‘J

(where k, = degEa(Zo n X,)).

The key point is the analyticity of the restriction mappings resE,e.: ~e,~ -+ gE,,k, for scales where E and E’ are adapted to Z, and k = deg,(Z,), k’ = deg,(Z,,) if XE, cc x,. Third step: Restriction of the scales. We choose now scales (E$,, and ((Eb)aeAij)(t.j)eI 2 in such a way that X,; CC X,, for all i and XEh CC XEG for all a. We then get a commutative diagram (where W’ is defined as Win the preceeding step, but for all scales E replaced by I?)

What is now needed for the conclusion (finite dimensionality of W’, as well as the representability of Fi near Z,) is: i) res is holomorphic and induced by a compact linear mapping of the ambient Banach spaces. ii) res restricted to CI(W’) is a holomorphic isomorphism to c(‘(W’). The main difficulty in the construction of V,,(X) is that i) and ii) (as well as the analyticity of restrictions at the end of step 2) turn out to be false in general. However, these statements are almost true, namely are true for liftings to the weak normalisation of W’. This is due to the fact that holomorphic maps 0: S x U + Sym“(B) (for S an analytic set, U and B polydiscs of Cc”and Cp) do not

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necessarily remain analytic if S is not weakly normal, after one passes to the associated map 0’: S x U + Symk(B) by the ramified covering defined by 0. The maps 0 which remain holomorphic after such operations are called isotropic. Now it turns out that, taking the notations of step two above, there exists for every i a Banach-analytic set Bi; and an analytic homeomorphism h := gi -+ pi such that an analytic map 5: S + gi corresponds to an isotropic morphism if and only if it lifts to an analytic map c: S + Bi. Moreover, the restriction properties of 9;s needed to realize the above program are true for the (~J’s. This allows the construction of e,,(X). 0 However, in most applications of the Chow scheme, only weakly normal parameter spaces are needed. The proof of the representability of F$ in this case is much easier, as sketched above. Proposition 2.8. (1) If X’ is a closed analytic subset of X, the natural inclusion of %‘,,(X’) in g,,(X) is a closed holomorphic embedding. (2) If X is a projective variety, then 9?“(X) with the above complex structure, is isomorphic to the complex space defined by the Chow scheme of X. In particular, F,,(X) is a countable union of projective varieties. (3) Zf A is a closed analytic subset of X, then 9$(X), := {YE C,(Z)lA n 1YI # a} is a closed analytic subset of Vd(X). Example 2.9. Let X be a complex manifold and let Y be a d-dimensional compact complex submanifold of X. Let Nrlx be the holomorphic normal bundle to Y in X. From [Ko62] we get: if H’(Y, A$,,) = 0, then %?JX) is smooth at {Y}, and the natural map 8: TIy)%$(X) + H”(Y, NrIx) is an isomorphism. In particular: dim +$(X){,) = h o( Y N, Y). Recall the description of 19:if t is a tangent vector to %?JX) at {Y} and y E Y, we let 2 E (T (lyl,y,G) be such that p,(t) = t, where G c %$(X) x X is the graph of the universal family parametrized by %$(X), with p: G + gd(X) and q: G +X being the natural projections. Then: O(t) = v o q*(f), where v: TyX + A& is the natural projection over y.

We conclude the subsection by mentioning space.

the relative version of the cycle

Theorem 2.10. Let f: X + S be an holomorphic map between complex spaces. Let %(X/S) be the subset of V(X) consisting of cycles Z whose support IZI is mapped to a single point of S by f. Let f,: %(X/S) + Z be the map which sends Z as above to f(lZl) E S. Then %(X/S) is a closed analytic subset of U(X) and f* is holomorphic. 2. Compact Subsets of Cycle Spaces Definition 2.11. Let X be a complex manifold and Z c X be a p-dimensional connected compact complex submanifold. Let h be a hermitian metric on X, and set v,,(Z) := Jz Im(h)“P. It is known classically [Le157] that these notions still make sense when X and Z are reduced analytic spaces. We extend the function v,,: g,,(X) + lR by linearity to w(X), and call v,,(Z) the h-volume of Z. It

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and Th. Peternell

is a continuous function on G??(X)(for the quasi-projective general [Ba78]).

case see [AN66],

in

Example 2.12. When X = lF’N and h is the Fubini-Study metric on X, u,,(Z) is nothing but the degree of the variety Z. When X is Kahler and h is a Kahler metric, Stoke’s formula and a triangulation of analytic sets show that uh is constant on the connected components of U(X). Theorem 2.13 )[Li77]). Let S be a subset of s(X). Then S is relatively compact in q(X) if and only if i) there is a compact subset K of X containing lZ,l for any s E S, and ii) v,,(ZJ is untformly bounded on S for some (hence any) hermitian metric h on X. Remark 2.14. 1) If X is compact, condition i) is fulfilled by any set S. 2) If X is a compact Klhler manifold and h is a Kahler metric on X, u,, is constant on the connected components of U(X). Hence we obtain the fundamental Corollary 2.15 ([Li77]). Let X be a compact Kiihler manifold. Then the connected components of q,,(X) are compact.

Observe that, when X is projective, the connected components projective, hence compact. Remark 2.16. It is in general false that the components if X is merely compact.

of V(X) are

of S’(X) are compact

Idea of the proof of 2.13: The theorem is actually of local nature and can be reduced to a theorem of Bishop ([Bi64]), asserting the following. If (Z,,),,>e is a sequence of pure p-dimensional closed analytic subsets of a domain U of C”, if u(Z,) is bounded, where u is the euclidian volume and if finally (Z,),,, converges (in the Hausdorff metric on closed subsets of U) to some nonempty closed subset A of U, then A is analytic of pure dimension p. For the proof of Bishop’s theorem, i.e. the analyticity of A at a E A, one chooses a linear projection from Cc” to 0, which is simultaneously finite on all (Z,Jnzo (this choice uses the assumption on u(Z,) already). One is thus reduced to the case where all Z, are finite ramified covers of I/ c Cp of degree d. The consideration of symmetric functions reduces to the case d = 1, which is nothing else than Ascoli’s theorem. 0 Remark 2.18. Note that if X is compact of pure dimension n, the irreducible components of %“-i(X) (i.e.: components of effective Weil divisors of X) are compact. This is because all but finitely many of the irreducible divisors of X are given by meromorphic functions on X (see [Kr75], [FiFo79] for this last assertion). In particular, if X is a compact surface, the irreducible components of %?(X) are compact. Corollary 2.19. Assume that X is countable at infinity. at infinity, too.

Then g(X) is countable

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Using [Hi741

331

or, more elementarily,

[Ba79], one easily shows: Proposition 2.20. Let X and S be irreducible complex spaces. There exists a natural identification between: i) meromorphic maps p: S + %?JX), and ii) S-proper pure (d + p)-dimensional cycles G of S x X (see definition). We call G the graph of the meromorphic family determined by (p, S) and ,u the map defined by G.

When p is holomorphic or when S is normal and the fibers of p: G + S of pure dimension, the correspondance is the standard one. In general, we just take the meromorphic extension over the complement of a suitable Zariski dense open subset of S. Definition 2.21. (1) Recall ([Ue75]) that if X is compact irreducible, then there exists a surjective meromorphic map r x: X + A(X) to a projective variety A(X) which dominates all such maps (i.e.: if r’: X + A’ is another map, there exists c(: A(X) + A’ such that: CI0 r, = r’). Of course, r,: X + A(X) is unique, up to bimeromorphic equivalence, and its generic fiber is irreducible. It is called “the” algebraic reduction of X, and a(X) = dim A is called the algebraic dimension of X. It is defined algebraically as the transcendance degree over (c of the field of meromorphic functions on X. Compare VII.6 (2) If X is compact reducible, a(X) is the maximum of the algebraic dimensions of its irreducible components. It is an integer between 0 and n := dim X. (3) When a(X) = dim X, X is said to be a Moishezon space: it is then bimeromorphic to A. More precisely, a deep theorem of Moishezon ([Moi67]), which can be deduced from Hironaka’s flattening theorem, asserts that X becomes projective after finitely many blow-ups with smooth centers. It is easy to show that the irreducible components of g(X) are compact Moishezon provided X is Moishezon. For more information of Moishezon spaces see VII.6.

A canonical algebraic reduction can be constructed ([Ca81-11) (or without using it: [Gr85]).

by using cycle spaces

Remark 2.22. The link between algebraic dimension and cycle space is the following: for X a compact reduced irreducible complex space z(X) is the maximal number of effective prime divisors which meet transversally at a generic point of X. In particular: a(X) 2 1 if and only if X is covered by irreducible divisors. A strengthening is due to Krasnov ([Kr75] and [Ko62] for the case of surfaces): if a(X) = 0, then X contains only finitely many (at most dim X + h’(X, Qi) if X is smooth) effective prime divisors. For a generalization of this, see [FiFo79]. In the special case when X is a complex torus, we conclude that a(X) = 0 if and only if X does not contain any effective divisors. (Use translations of X).

Let us mention some basic properties of algebraic dimension:

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and Th. Peternell

Proposition 2.23. Let f: X + Y be a surjective meromorphic map between irreducible compact complex spaces X and Y. Let a(f) := inf (a(X,,)) and set a*(X) := YSY

dim(X) - a(X). Finally let dim(f)

:= dim X - dim Y.

Then:

i) a(Y) 5 a(X) I a(Y) + a(f) I a(Y) + dim(f). ii) Let Z c X be an irreducible compact analytic subset of X. Then a*(Z) I a*(X).

The proof of these results is easy (see [Ca81-l] [Ue75] for other proofs).

for more details, and [Moi67],

Corollary 2.24. Let f: X + Y and let Z be an irreducible compact analytic subset of X. Assume that X is Moishezon. Then so are Y and Z. Remark 2.25. Even when the fibers off: X + Y are projective and Y is projective, it does not follow that X is Moishezon. For example: compact surfaces of algebraic dimension one are elliptic fibrations over a curve (Kodaira).

A relative algebraic reduction exists also in certain cases: Theorem 2.26 [Ca81-11. Let f: X + Y be a fiber space (i.e.: X and Y are irreducible, f is surjective and the general fiber is irreducible) of reduced compact complex spaces. Assume that all irreducible components of U(X) are compact. Then there exists an algebraic reduction off, i.e. a commutative diagram f X-Y

such that for general y E Y, the map h,: X,, + Z, is an algebraic reduction X, = f-‘(y). Here “general” means that y belongs to a countable intersection Zariski open dense subsets of Y.

of of

In the special case where dim X = 3, theorem 2.26 was proved by Kawai [Ka69] without any assumption on q(X).

0 3. Cycle Spaces and the Structure of Compact Complex Spaces In this section we study compact complex spaces via cycle spaces. As an example we will see that an analytic set of an irreducible subvariety S of the cycle space w(X), parametrizing cycles containing a fixed point, is Moishezon regardless, whether X is Moishezon or not. This theorem has many interesting applications. We study also Albanese reductions of compact KIhler manifolds of class %Zand their classification theory (via algebraic reductions etc.), automorphism

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groups and almost-homogeneous manifolds (with emphasize on constructing them in a systematic way from projective and simple manifolds). 1. Compact

Families of Cycles

(3.1) We begin by fixing some notation. Let X and S be irreducible reduced complex spaces and (Zs)sss an analytic family of cycles parametrized by S. Let G c S x X be the graph of this family with projections p: G + S, q: G + X. Recal.1 that p is proper. When no confusion can arise (e.g. when S is an analytic subset of %7(X)) we call (Zs)seS just the family S. We say that the family S is compact if S is compact. Moreover we say that S is a covering family if (a) Z, is irreducible for generic s E S (b) q is surjective. Recall that V*(X) is just the open subset of V?(X) parametrizing irreducible cycles and let S* = V*(X) n S. Moreover, S is called prime if S* # 0. 0 Remark 3.2. (1) If S is prime, then G (the graph of the family S) is irreducible and reduced. (2) We let S, = p(q-l(X)) be the subfamily of cycles of S passing through x E X. Then S, is a closed analytic subset of S (by Remmert’s Projektionssatz) (3) A family S respectively (Zs)s.s is a covering family if and only if X = u lZ,l. If in addition S is compact, then S is covering if and only if q is SGS open at some point. q (3.3) Assume that S is a compact covering family for a compact analytic set A c X and let $4 = PW’(4) be the closed analytic subset of S parametrizing all cycles in S which meet A. We also define: S(A) = q(p-‘(S,)). This is the subset of X consisting of all x E X which can be joined to some a E A by a cycle Z, with s E S (observe that lZ,l is connected since S is prime). Since S is compact and q is proper, S(A) is a compact analytic set in X. Define now inductively: Sy4) = S(rl(‘4)) (and So(A) = A). Then every Sm(A) is a compact analytic subset of X and Y(A) 3 r-l(A). on X by

Put Y(A)

= u Y(A).

w e introduce an equivalence relation

I?220

x ws y

if and only if y E Sm({x}).

In other words: x wS y if and only if x and y can be joined by a connected chain of cycles in S. Then we have Theorem 3.4. [Ca81-21 Let X be a normal complex spacewith S c %7(X)as in (3.3). Then there is a surjective meromorphic map cp:X -+ Y and Zariski open subsetsX* c X, Y* c Y such that:

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(a) ‘plX* is holomorphic, proper and open; cp(X*) = Y*, X* = qp-‘(Y*); (b) for every x E X*, cp-‘q(x) is precisely the equiualence class of {x} with respect to wS.

In other words: the quotient X/-s 2. An Algebraicity

exists almost everywhere as a complex space.

Theorem.

Definition 3.5. (1) A holomorphic map f: X + Y is said to be projective if and only if X carries an f-ample line bundle (see V.4.1). (2) f is said to be Moishezon if and only if f is bimeromorphically equivalent to a projective map, i.e. if there is commutative diagram

x-x / \/

h

.i

Y

where h is bimeromorphic (see chap. VII) and f’is projective. If f is Moishezon, every fiber f-‘(y) is a Moishezon space (VII.6) but the converse is not true. Theorem 3.6. Let X be a complex space, and let S c s(X) be a prime family (i.e. S* # a). Let G be its graph. Then the holomorphic map q: G +X is Moishezon. In particular, S, = p(q-l(x)) is Moishezon for every x E X.

More generally, if A c X is Moishezon, then S, = p(q-l(A)) is again Moishezon. And if X is Moishezon itself, then every irreducible component of g(X) is Moishezon. A proof of (3.6) can be found in [Ca80] or in [Fu82] (in the context of Douady spaces). 3. Algebraic Connectedness. Definition 3.7. A normal irreducible braically connected if

compact complex space X is called alge-

(a) all irreducible components of %Tl(X) are compact and (b) any two general points of X can be joined by a connected compact complex curve. Condition (a) is e.g. satisfied if X is a compact Klhler connected spaces can be characterized as follows: Theorem 3.8. [CaM-21 A compact irreducible ically connected if and only tf X is Moishezon.

manifold.

Algebraically

complex space X is algebra-

Loosely speaking, X is already Moishezon if there are enough curves in X. Of course 3.7(b) is obvious for Moishezon spaces (take a projective model and consider hyperplane sections). We give the idea for the other direction in the case where there is a compact covering family (CJseS of X such that any two

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points of X can be joined by a cycle Cs, + ... + C,*, si E S. In this case we have Sm( {x}) = X. Now (3.8) follows by virtue of (an inductive application of) Lemma 3.9. If A c X is a Moishezon analytic subset, then S(A) is Moishezon, too. Indeed, S, = p(q-‘(A)) is Moishezon by (3.6). Since p: G + S has one-dimensional fibers, p-‘(S,) will be Moishezon by (3.10) below, hence S(A) = q(p-‘(S,)) is Moishezon. q Lemma 3.10. Let f: X + Y be a holomorphic surjective map of compact irreducible spaces. Assume that (a) f has a holomorphic section 0: Y + X and (b) dim X = dim Y + 1. Then f is Moishezon. In particular, if Y is Moishezon, then X is Moishezon. It should be noted that (3.10) is false without the assumption (a). For example, it is easy to construct compact complex surfaces X of algebraic dimension 1 (VI1.6), e.g. Hopf surfaces or tori. These surfaces admit a holomorphic surjective map f: X + Y to a compact Riemann surface Y, but there is no compact curve C c X with f(C) = Y. In fact, if such a C would exist, one can take an ample line bundle 9 on Y and by setting A? = Q(aC)

@f*(ZZb)

with suitable a, b, one has c,(Z)* > 0, which [BPV84] for details. This phenomenon can be generalized:

forces X to be algebraic.

See

Theorem 3.11. [CaSl-21 Let X be an irreducible compact complex space such that all components of%?(X) are compact. Let f: X + Y be surjectiue and holomorphic. Let A c X be a closed analytic set with f(A) = Y. Assume that f IA is Moishezon and that the fibers off are Moishezon. Then f is Moishezon. This is a consequence of (3.8).

0

Another application of (3.8) is the construction of an “algebraic coreduction”: Theorem 3.12. [Ca81-21 Let X be an irreducible compact complex spacesuch that all componentsof %7(X)are compact (e.g. X a KBhler manifold). Then there exists a meromorphic map r: X + Y such that (a) the conditions of 3.4(a) are fulfilled (b) for general x E X (i.e. x E X*), the fiber X, = r-‘(y), with y = r(x), is irreducible and Moishezon and is moreouer the biggest connected analytic set M, which is Moishezon and contains x. The map r: X -+ Y is called the algebraic coreduction of X. It is unique up to bimeromorphic equivalence. In particular, we note that for general x E X, there is a biggest connected Moishezon subvariety containing x. For details of the proof see[Ca81-21.

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of Class ‘Z.

Definition 3.13. Let Y be a class of compact reduced complex spaces which is stable under the following operations: (a) taking products, (b) taking images of holomorphic maps, (c) taking preimages of Moishezon morphisms (in particular modifications); i.e. if f: X --+ Y is Moishezon and surjective and if Y E 9 then X E x Moreover, we require for X E Sp that all components of U(X) are compact. Then we say that 9’ is geometrically stable. Theorem 3.14. Assume that 9’ is geometrically components of g(X) are again in 9

stable. Let X E 9’. Then all

(3.14) is due to [Ca80] (and Fujiki [Fu82] for the Douady space instead of g(X)). It is a direct consequence of (3.6). Remark 3.15. The class of projective varieties is not geometrically VII.6). The smallest class Y which is geometrically stable containing tive varieties is the class of Moishezon spaces.

stable (see all projec-

In a similar way the smallest class of reduced compact complex spaces which is geometrically stable and which contains all compact Kahler manifolds is the so-called class % introduced by Fujiki [Fu78-11: Definition 3.16. A reduced compact complex space X is said to belong to the class %?if there is a compact Kahler manifold r? and a surjective holomorphic mapf:X+X.

We note (cf. VII.6) that (reduced) Moishezon

spaces are in %?.

Proposition 3.17. The class V is geometrically stable.

This is a consequence of Hironaka’s flattening theorem (11.2.9,VII.7). See [Fu78-l] for details. Varouchas [Va84] has given a very simple characterisation of the class %: Theorem 3.18. (Varouchas) A reduced compact complex spaceX belongsto V if and only if X is bimeromorphically equivalent to a compact Kiihler manifold. Remarks 3.19. (1) One might be tempted to consider the class Y of those spaces X for which all components of g(X) are compact. But unfortunately Y is not geometrically stable: if X is a homogeneous Hopf surface, then X x X is no longer in Y (see 1.9). (2) Small deformations of a compact Klhler manifold are again Kahler. This is a stability theorem of Kodaira-Spencer [KoS60]. For manifolds in %?this is no longer true as shown in [Ca91-21, [LeB90].

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Spaces

5. The Albanese Variety

(3.20) Let X be a connected compact complex manifold. Blanchard [BlSS] attached to X a holomorphic map CC:X + A(X), A(X) being a complex torus, the Albanese torus of X. The map c(is called the Albanese map or reduction of X. It is determined by the following universal property: if t: X + T is a holomorphic map to a complex torus T, then z factors through ~1:there is an alline holomorphic map p: A(X) + T such that T = p o ~1.In particular CIis unique up to translations. Let us sketch the construction fF(X) let

of 01. Fix a point x0 E X. For x E X and o E

I(xo, Y) =sYw where y is any path from x0 to x. Let H be the smallest connected complex Lie sub group in Q’(X)* containing all linear maps OJI-+ ~&I, Y), where y ranges over all loops based at x0. Then put A(X) = Ql(X)*/H, CIbe the canonical map X + A(X).

and let

Remarks 3.21. (1) In general a is neither surjective (take X to be a compact Riemann surface of genus g 2 2) nor has c1connected fibers (see [Ue75] for an example). (2) c1is a bimeromorphic invariant: if f: r? + X is a modification of compact complex manifolds, then CI~0 f = cc%. This is because f respects holomorphic l-forms. Definition

3.22. alb(X) = dim a(X) is called the Albanese dimension of X.

Since the induced map c(*: H’(A(X), Q&,,) -+ H’(X, Q$) is always surjective [BlSS], one has the inequality alb(X) I dim H’(X, 0;). (3.23) If X is a compact Klhler A(X) = H’(X,

(note that then every holomorphic

manifold,

then

Qi)*/(H,(X,

@/torsion)

l-form is d-closed), hence

dim A(X) = dim H’(X,

0;) = dim H’(X,

0,).

The same holds more generally for compact manifolds in GC Since b,(X) = dim H’(X, C) = 2 dim H’(X, Q$) by Hodge decomposition, dim A(X) is a topological invariant of manifolds in V?.On the other hand, Blanchard has given examples of compact manifolds X (necessarily not in class W) such that dim A(X) < dim H’(X,

ai).

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Example 3.24. ([Ue75]) Let X be a compact manifold denoting the algebraic dimension, cf. VII.6).

with a(X) = 0 (a(X)

Then CLis surjective and has connected fibers. Proof. Since a(a(X)) = 0, cc(X) must be a subtorus of A(X). By the universal property of u, we get u(X) = A(X). Thus a(A(X)) = 0. Now consider the Stein factorisation X 5 Y J+ A(X) of u. Since a(A(X)) = 0, A(X) cannot carry any hypersurface (by homogeneity!), so the finite map y must be unramified. Hence Y is torus, too. Again by the universal property, y is isomorphic, hence CLis connected. 0 We would like to mention the following remarkable additivity property the Albanese dimension, due to Fujiki [Fu83-1] and Blanchard [BlSS].

of

Theorem 3.25. Let X be a compact Kiihler mani$old, and let f: X + Y be a surjective holomorphic map with connected fibers. Assume that f is smooth over the Zariski open set Y* c Y and that A(X,,) is independent of y E Y*, where X, = f-‘(y). Then there exists a generically finite map g: f -+ Y, unramified over Y*, with the following property: if 2 denotes a desingularization of the unique component of X xy p which is mapped onto X by the projection, then b,(X)

= b,(X) + dim A(X,,).

Loosely speaking, the above formula holds after base change. We discuss now a relative version of the Albanese reduction. 3.26. Let X be a compact manifold, and let f: X + Y be a holosurjective map with connected fibers (such an f will be called fiber space). An Albanese reduction off is a commutative diagram of meromorphic maps Definition

morphic

XAT

such that: (a) a and r are holomorphic over a Zariski dense open subset Y* c Y, (b) for y E Y*, a,,: X,, + T, is the Albanese map of X,,. Theorem 3.27. [Ca85-l] Let X be a compact manifold in %, and let f: X + Y be a fiber space. Then an Albanese reduction of f exists and is unique up to bimeromorphic equivalence.

This relative Albanese reduction is an important tool in the bimeromorphic classification of manifolds in % The proof rests essentially on the following fact. If Z is a compact manifold with Albanese map a: Z + A and if a,,: Z” = z x *.. x Z + A is defined by a,(zl, . . . . z,) = a(zl) + ... + a(~,,), where + is the addition in A for any chosen origin, then a, is surjective and all fibers have the same dimension. The remaining part is cycle space theory.

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Groups

Theorem 3.28. Let X be a connected compact Ktihler mantfold. Let G be the connected component of the group Am(X) of automorphisms containing id,. Define a holomorphic morphism of complex Lie groups

J: G + A*(X) = AutO(/l(X)) as follows.

For g E G, J(g) is the translation J(g).u(x)

= C&J(X))

of A(X) such that for every x E X.

Let L = Ker J. Then J(G) is a compact subtorus of A*(X) and L carries natural way the structure of a complex linear algebraic group.

in a

(3.28) is due to Lieberman [Li78]. It was carried over to class % by [Fu78]. Let us give the general idea of the proof: the first assertion is easy; let us show the second. Let H be the unit component of L and let 0, be the Lie algebra of H; it is the subalgebra of H’(X, T,) consisting of holomorphic vector fields V on X for which the contraction map: i(V): H’(X, Qi) + H’(X, 0;) = C is zero. More generally, one has: Proposition 3.29. ([Li78]) The vector field V belongs to 0, if and only if i(V): HO(X, Q$) -+ HO(X, Q-‘) is zero for every p 2 1.

Let now x be in X; for any n, let H act diagonally on X” = X x ... x X in the natural way. Observe that H has a natural compactification H in %(X x X) because X- is Kahler, and that the action of H on X extends to a meromorphic map CC,,:H x X” + X”. By choosing n sufficiently large and 2 = (x1, . . . , x,) generic in X”, we can assume that the map uf: H + X” defined by @g(h) = a,(h, 2) is injective. Let Y be the closure of H. Then 2 = a%(H) in X”. By applying Hironaka’s equivariant resolution of singularities, one can assume that Y is smooth and that Lie(H) c O,(Y). But now H acts on L with a dense orbit. Hence H”( Y, 52;) = 0. Thus Y is projective and one can choose a very ample line bundle 9 on Y which acts equivariantly. One is thus reduced to a classical situation (see [Li77] for details). Definition 3.30. A compact connected manifold X is said to be almost homogeneous if Auto(X) acts with an open orbit on X.

From the preceding result one gets the following, already due to [B074] (see [BoR62] for the homogeneous case). Recall that F is said to be unirational if there exists a surjective meromorphic map @: lP,,(C) + F for some n. Theorem 3.31. Let X be an almost homogeneous compact Kiihler manifold. Then ~1:X -+ Alb(X) is a smooth surjective flat fiber bundle with fiber F an almost homogeneous unirational manifold. (Here “flat” means the following: if rr: A -+ A(X) is the universal couer and X, = X x,(z) A, then X, N A x F). Remark 3.32. The fibers of cxin (3.31) are the closures of the orbits of L. If L is abelian, F is rational. This is expected to be true in general.

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For other structure theorems about the Albanese map, not to be discussed here, we refer to [DPS93-1,2]. 7. Structure of Compact Manifolds in W. Here we wish to discuss some structure theorems on manifolds in %?.All these are bimeromorphic in nature. Only the non-algebraic case will be considered, i.e. a(X) < dim X. Definition 3.33. A complex space X E % is said to belong to class A iff a(Y) > 0 for every compact irreducible subspace Y c X of positive dimension. Theorem 3.34. [CaSO] Let X E W. Then there exists an A-reduction of X; this is a surjective meromorphic map a,: X + Y, unique up to bimeromorphic equivalence, such that Y belongs to A and such that any other map b: X -+ Z with these properties factors through a,.

Observe that the general fiber of a, has algebraic dimension 0. The map a, is obtained by taking successively algebraic reductions. Of course, one has a(X) = a(Y). If some fiber space f: X -+ Y has this last property and the fibers are Moishezon, then one has: Proposition 3.35. [Ca85-l] Let f: X + Y be a fiber space with X a compact mantfold in %‘. Assume that all fibers off are Moishezon and a(X) = a(Y). Then the general fiber off is almost homogeneous.

Combining [Fu83-l] Corollary

(3.34) and (3.35) one can prove the following result due to Fujiki 3.36. Let X be a compact manifold in A with algebraic

reduction

f: X + Y. Let a: X + T and z: T + Y be the Albanese reduction off. Then there is a Zariski dense open subset Y* c Y such that a is surjective, connected almost homogeneous rational fibers over Y*.

smooth, with

By (3.34) the study of the structure X E % is to some extent reduced to the study of a) X E A; here we have the structure theorem (3.36); b) a(X) = 0. So we study now X E %Zwith a(X) = 0. Definition 3.37. ([F&3-1]): (1) A Kummer manijiold is a compact manifold which is bimeromorphically of the type T/G with T a torus and G a finite group. (2) Let X be a compact manifold. A Kummer reduction of X is a meromorphic fiber /I: X + B with B a Kummer manifold such that every /?: X -+ B’ of the same type factors uniquely through p. The number k(X) = dim B is called the Kummer dimension of X (if a Kummer reduction exists). Proposition 3.38. ([F&3-1]) Let X be any compact manifold with a(X) = 0. Then a Kummer reduction of X exists (unique up to bimeromorphic equivalence).

The proof relies on (3.24).

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It can be proved ([Fu83-11) that k(X) = sup{q(T) = h’(Q)lr? is a finite covering of X}. The most interesting class of manifolds X with a(X) = 0 is of course that with k(X) = 0; otherwise one can study the Kummer reduction. Definition 3.39. Let X E %?,dim X 2 2. Then X is called simple if there is no u Z,. family (Z,), ET of compact cycles with 0 < dim 2, < dim X such that X = fET X is called semi-simple if there is a compact space Y a product Z, x . ..Z, of simple manifolds and generically finite maps Y + X, Y -+ Z, x . . . x Z,. Remark 3.40. If X is simple then obviously a(X) = 0 and either k(X) = 0 or k(X) = dim X (i.e. X is Kummer).

Concerning

the structure of manifolds

with a(X) = k(X) = 0 one has

Proposition 3.41. [Fu83-21 Let X E 59, with a(X) = k(X) = 0. Then there exists a non-constant surjective map z: X -+ Y with connected fibers and with Y semi-simple such that every map with these properties factors through 0. We say 0 is the semi-simple reduction of X.

The proof is essentially the same as for the existence of algebraic reduction. This result reduces to some extent the structure of manifolds X of class %?with a(X) = 0 to the classification of those which are simple. Remark 3.42. The manifolds X of class %?which are simple with k(X) = 0 seem to be very scarce. Let X be such a manifold and let n = dim X. If n = 2, then X is a K3 surface. No example is known for n = 3. For n = 4, the first example was found by Fujiki ([Fu83-21). We describe it shortly. Let S be a K 3 surface and let i be the involution of S x S which exchanges the factors. Let E be the blow-up of S x S along its diagonal, i: z + z being the lifting of i to C. Then X, = (C/$is smooth, its Kuranishi space at {X,} is smooth of dimension 21 (one more than the dimension of that one of S) and the general deformation of X, is simple and non-Kummer. The proof relies heavily on the fact that X,, is symplectic (i.e.: carries a holomorphic 2-form o of maximal rank everywhere), so that in particular KxO is trivial. This construction has been generalized in two ways in [Fu83-l] and [Be831 and gives other examples of simple manifolds. Note that presently all known examples of simple manifolds are bimeromorphic to some symplectic manifold (in the above sense, but maybe with some mild singularities). 0

The structure of the general fibers of semi-simple reduction is still unknown. This structure would be very clear if it could be shown that, given a fiber space f: X + Y with X E %, a(Y) = 0 and the general fiber X,, being simple, then the smooth fibers off are all isomorphic. In one case (X,, symplectic) this is known to be true ([Ca89]). We conclude this subsection with describing the bimeromorphic structure of non-algebraic threefolds in G??as given by Fujiki [Fu83-11. Note that by Kodaira’s classification (see e.g. [BPV84]) a non-algebraic minimal surface in V (which is the same as to say the surface is Kahler) is either an elliptic libration

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over a compact Riemann surface or a complex torus or a (non-algebraic) surface (i.e. Szi N 0, and n,(X) = 0). Theorem 3.43. (Fujiki) a(X) I 2. Then

Let X be a smooth compact

K 3-

3-fold in 5~7.Assume

(1) a(X) = 2. The general fiber of the algebraic reduction is an elliptic curue (so X is called an elliptic 3-fold) (2) a(X) = 1. Then there are two cases. A. The algebraic reduction f: X -+ Y is holomorphic and the general smooth fiber is a complex torus or a flat IF’,-bundle over an elliptic curve. B. X is bimeromorphic to C x S/G, with C a compact Riemann surface, S a torus or a K3-surface, and G a finite group operating on C, on S, and on C x S diagonally. (3) a(X) = 0. Then either A. X is Kummer B. k(X) = 0, 2 and X is a fiber space over a normal compact surface S with a(S) = 0, the general fiber being II’,. C. k(X) = 0 and X is simple (no such example is known).

For the proof and many more results on compact manifolds in % we refer to [Fu83-11.

$4. Convexity of Cycle Spaces If X is a l-convex complex space in the sense of chap. V, then X is a modilication of a Stein space; in particular, X is holomorphically convex. If X is qconvex (in the sense of Chap. VI), then it is no longer true that X is holomorphitally convex: if Z is a compact manifold and Y is a q-codimensional submanifold whose normal bundle NY,* is positive in the sense of Grilliths (Chap. V; take e.g. Z = lP”., then this condition is automatic), then X = Z\Y is q-convex but if q 2 2, it is never holomorphically convex (O(X) N (c by the Riemann extension theorem). So we consider instead of X itself the space of (q - 1)-cycles and it turns out that +&,(X) is Stein if X is q-complete. Unfortunately it is not true that 59-,(X) is holomorphically convex if X is q-convex. An additional assumption is needed. The results presented in this section are due to D. Barlet [Ba78] and NorguetSiu ENS771 and are based on earlier papers of Andreotti-Norguet [AN66], [AN67]. Observe that there is a shift in the notation of q-convexity in [Ba78] and [NS77]: q-convex there is (q + 1)-convex in the usual Andreotti-Grauertsense. 1. Convexity

of Cycle Spaces of q-complete Spaces

(4.1) All complex space X will be assumed finite dimensional, we will also work only with reduced spaces (since we investigate cycle spaces). Recall the

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343

notion of (strictly) plurisubharmonic functions from chap. V. In a similar way, (p, q)-forms can be defined on X (by local embeddings). Given a (p, p)-form cp we define the map F,: VP(X)

by

+ IR

FJU =sI-cp. Even if r is a singular cycle this definition makes senseby a classical result of Lelong [Le157]. For the definition of q-convexity and q-completeness see Chap.VI. 0 The main aim of this subsection is to discussthe following theorem of Barlet [Ba78]. Theorem 4.2. Let X be a reduced

q-complete

complex

space. Then %?-,(X)

is

Stein.

The first to prove convexity properties of cycles spaceswere Andreotti and Norguet [AN66], [AN67]. They considered only quasi-projective manifolds, because the cycle space was not known to exist in general at that time. In the algebraic case, however, its existence follows from the existence of Chow varieties (see [AN66/67]). Recall that a manifold (or complex space) is quasi-projective if it is a Zariski-open set of a closed complex subspaceof lP,,.Then NorguetSiu in [NS77] proved Theorem 4.3. Let X be quasi-projective holomorphically moreover

convex

(with

the complex W(X,

for all coherent

q-convex structure

9)

sheaves 9 on X, then Wq-I(X)

manifold. Then %?q-l(X) defined by [AN66/67]).

is

If

= 0 is Stein.

It should be remarked that the complex structure considered on the cycle space is semi-normal, i.e. every continuous function which is holomorphic on the regular part is holomorphic. The first assertion of (4.3) will be discussedin the next section. Notice that the cohomological condition is satisfied if X is q-complete. This is the vanishing part of Andreotti-Grauert’s theorem [AG62], see chap. VI. At present it is still unknown whether this cohomology is enough to force X to be q-complete. In this sense(4.3) is not included in (4.2). We are now discussing the main ingredients in the proof of (4.2) which are interesting for their own sake. The tool to recognize %‘-i(X) to be Stein is the following result of Narasimhan (seeV.3.7). Theorem 4.4. Let X be a (reduced) strictly

pluri-subharmonic

continuous

complex space and let f: X + [0, co [ be a exhaustion function. Then X is Stein.

So the problem is to construct such an exhaustion function on 9?-,(X) when X is q-complete. We will in the sequel call a C2-function (say) f to be q-convex

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if the Levi form L(f) has at least n - q + 1 positive eigenvalues (see chap. V,VI). If h is an hermitian metric on X, then we denote by oh = o the associated (1, I)-form (see also chap. V.6). Proposition 4.5. Let X be reduced and q-complete with the q-convex exhaustion function f. Let h be a hermitian metric on X of class C2. Then there is another hermitian metric h’ of class C2 and a strictly increasing convex function c: [0, a[ + [0, a[ such that (a) h’ 2 h, (b) i8(c 0 f. o&1) > 0 in the sense of distributions (currents) (cf.V.1).

For the easy proof of (4.5) see [Ba78]. [Ba78, theoreme 33:

Using cycle space theory Barlet shows

Proposition 4.6. Let X be reduced, cp a real (p, p)-form on X. Assume that i&p 2 0 everywhere and ia& > 0 on an open non-empty set U of X. Then the map Frp: Q?,,(X) + IR, defined by F,(T) = s’p, is continuous and strictly plurisubharmonic at every point I-E %?JX) with the property that every component I-, of rmeets U.

By (4.5) we can now apply (4.6) to our q-complete space X and the proof is finished by virtue of (4.4). Continuity and plurisubharmonicity in (4.6) are guaranteed by Lemma 4.7. ([Ba78]) Let X be reduced and cp a (p, p)-form coefficients. Consider the function F,: ‘ix,(X) -+ IR,

F,(T)

=

s I-

with continuous

cp.

Then F, is continuous and plurisubharmonic.

Continuity follows easily from the description of cycle spaces; plurisubharmonicity is seen as follows. Let A c (l2 be the unit disc; then it is sufficient to show that F,,, o g is subharmonic on A for every holomorphic map g: A + %YJX). Let G = {(t, x)lt E A, x E X such that x E Ig(t)l) be the graph of the family of cycles corresponding to g. Generically one may assume that Ig(t)l = g(t) and letting 4 be the pull-back of cp to G, one obtains i&@ 2 0. Let rc: G + A be the projection, then the current n,(e) is just F, 0 g; on the other hand, we have z+(e) 2 0 since ii%@ 2 G (see [Le168] for details on positive currents). 0 Properness of F in (4.6) is proved in [Ba78,1.B]. 2. The Method of Norguet-Siu. In this section we discuss the proof of (4.3). The idea to prove holomorphic convexity of gqel(X) is to mimic the Remmert reduction of holomorphically convex spaces.

(4.8) Fix again a C2-exhaustion function f of X which is q-convex outside a compact set K (with K = 0 if X is assumed even to be q-complete). We may

VIII.

Cycle Spaces

345

assume K = (f > A,} with a suitable 2,. Define cp: gq-,(X) + IR

by

cpm = ,“,“,p,,f(x). Then it can be shown that cp is a continuous exhaustion function and that cp is plurisubharmonic on (‘p > A,,}. Let 2, > 2, and $ = max(cp, A,). Then II/ is clearly a plurisubharmonic exhaustion function (of every component of %‘-i(X)). The Steinness criteria which is used in [NS77] is slightly weaker than (4.4) and reads: Theorem 4.9. Let X be a reduced holomorphically spreadable complex space (i.e. for every x E X the set {y E Xlf(x) = f(y) for all f E O(X)} is discrete; compare 111.3). Let cp be a continuous plurisubharmonic exhaustion function of X. Then X is Stein.

An important Proposition

step is now [NS77, prop.2.21: 4.10. Let X be a quasi-projective

manifold.

(a) Zf X is q-convex, then for a given r~ gqe,_,(X) all components of L(T) = {r’

E %q-I(X)lf(r)

= f(r’)

for all f E O(%?-l(X)}

are compact

(b) If Hq(X, 9) = 0 for all coherent sheaves 9 on X, then %Yq-,(X) is holomorphically

separable.

Sketch of proof of (b). Let as in (4.8) cp be an exhaustion

plurisubharmonic Define a function

function, which is on { cp > A,,}. The essential point of the proof is the following.

bv 9

iT, (

44

= >

C islo

44

where i E I, if and only if 4 n {‘p > &} # a. Say 1 E Z,,. Now the claim is: if g(T) # g(r’) then f(r)

# f(r’)

for some f E c?@?~-,(X)).

The function f is constructed as follows. By a “separation find a (q - 1, q - 1)-form o on X with

argument”

one can

Now define f(CniZJ

= Cni

W. sG

Finally, the compactness of L(T) comes from the conclusion that r’ E L(T) must be a translate of g(T) by a linear combination 1 nit’ with 141 c is1

Cf 5 &I.

cl

346

F. Campana

and Th. Peternell

Although Q?-i(X) is not holomorphically separable in general, (4.10) allows us to reduce X to a holomorphically separable space in the spirit of the Remmert reduction (V.2): Proposition ponents of

4.11. Let X be reduced and suppose that for every x E X all com-

are compact. Then there exist a holomorphically spreadable complex space Y and a proper holomorphic surjective map 7~:X + Y such that n,(Co,) = 0,.

The space Y is obtained as the quotient Xl- with x - y if and only if x and y belong to the same component of L, for some z E X. For details see [NS77]. 0 (4.12) The second part of (4.3) follows now from (4.9) and (4.10). For the proof of the first part Norguet and Siu argue as follows. By (4.10), (4.11) we find a holomorphically separable space Y and a proper holomorphic map 7~: ‘t;b-r(X) + Y as in (4.11). Now consider the continuous plurisubharmonic exhaustion $ from (4.8). By properness of rr and the maximum principle for plurisubharmonic functions, $ descends to a continuous function +’ on Y. It is easy to see that $’ is plurisubharmonic, hence Y is Stein by (4.9). Thus gq,-i(X) is holomorphically convex. 0 3. q-convexity and the Cycle Space. As seen in (3.4), G??-,(X) is holomorphically convex if X is a q-convex quasi-projective manifold. It is natural to ask whether this holds for every q-convex complex space. But, unfortunately, this is not the case. Example 4.13. (Barlet) Let p: Y + (c4 be the blow-up of the origin 0 E (c4. Identify the exceptional divisor E = p-‘(O) with lP,. Choose a line L c E and a smooth conic C c E. Fix an abstract isomorphism i: C + L (both curves being rational). Introduce the following equivalence relation: x - y if and only if x E C,yELandi(x)=y.LetX=Y/-. Then X carries the structure of a reduced complex space, and p descends to a holomorphic map @:X + (E4 showing that X is l-convex, hence 2-convex. Now we claim that vi(X) is not holomorphically convex. Using the map K*: %‘i (Y) + %i (X) it is clear that the irreducible components of %i (X) are compact, since they are images of those of %‘r(Y). The point is, however, that C and 2L are in the same connected component of %‘i( Y) 2: %?i(lP,), hence by taking rc* we conclude that K*(L) and 1c,(2L) are in the same connected component. The same is true for K,(L) and q(nL), n E IN. Hence %i (X) is not compact but connected. In summary %?i(X) cannot be holomorphically convex. q

We should note that Y can be compactified by the blow-up of lP4 along 0, so that also X has a natural compactilication 1. But x is merely Moishezon and not projective as follows from (4.3). On the positive side Barlet proves in [Ba78]:

VIII.

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Spaces

347

Theorem 4.14. Let X be a reduced complex space. Let f: X + Cl, co] a proper C*-function which q-convex on f -‘(Cl, c0[). Assume that there is an open neighborhood u%Iof the “exceptional” compact set f -‘( 1) carrying a hermitian metric h (of class C’) such that iaJ(o,4-‘) 2 0 (in the senseof currents). Then ~F?,-~(X)is holomorphically convex. We remark that the condition i8w~-1) is satisfied provided h is a Klhler metric. The existence of a Ktihler neighborhood of f -‘( 1) is satisfied if X is a quasi-projective manifold (restrict the Fubini-Study metric to X; see Chap. V). Hence (4.3) is contained in (4.14).

References* [AC621 [AN661 [AN671

[Art691 [Art701 [Ba75]

[Ba78] [Ba79] [B074] [Be831 [Bi64] [BinSO] [BlSS] [BoR62] [Bo74]

Andreotti, A; Grauert, H.: Theoremes de linitude pour la cohomologie des espaces complexes. Bull. Sot. Math. Fr. 90, 193-259 (1962) Zbl.106,55. Andreotti, A.; Norguet, F.: Problime de Levi et convexite holomorphe pour les classes de cohomologie. Ann. SC. Norm., Super. Pisa, Cl. Sci., III, Ser. 20, 197-241. Andreotti, A.; Norguet, F.: La convexitt holomorphe dans l’espace analytique des cycles dune variete algebrique. Ann. SC. Norm., Super. Pisa, Cl. Sci., III. Ser. 22, 31-82 (1967) Zbl.176,40. Artin, M.: Algebraization of formal moduli I. In: Global Analysis, Papers in Honour of K. Kodaira, 21-71 (1969) Zbl.205,504. Artin, M.: Algebraization of formal moduli II. Ann. Math., II. Ser. 92, 88-135 (1970) Zbl. 185,247. Barlet, D.: Espace analytique reduit des cycles analytiques complexes compacts dun espace analytique complexe de dimension finite. Lect. Notes Math. 482, 1-158. Springer (1975) Zbl.331.32008. Barlet, D.: Convexite de l’espace des cycles. Bull. Sot. Math. Fr. 373-397 (1978) Zbl.395.32009. Barlet, D.: Majoration du volume des libres gentriques et forme geometrique du theoreme d’aplatissement. C.R. Acad. Sci., Paris, Ser. A 288, 29-31 (1979) Zbl.457.32015. Barth, W.; Oeljeklaus, E.: Uber die Albanese - Abbildung einer fast-homogenen Kahler-Mannigfaltigkeit. Math. Ann. 212,47-62 (1974) Zbl.276.32022. Beauville, A.: Varietb Klhltriennes dont la premiere classe de Chern est nulle. J. Differ. Geom. 28, 755-782 (1983) Zbl.537.53056. Bishop, E.: Condition for the analyticity of certain sets. Mich. Math. J. I I, 289-304 (1964) Zbl.143,303. Bingener, J.: Darstellbarkeitskriterien fur analytische Funktoren. Ann. Sci. EC. Norm. Super., IV, Ser. 13, 317-347 (1980) Zbl.454.32017. Blanchard, A.: Sur les varittes analytiques complexes. Ann. Sci. EC. Norm. Super, 73, 157-202 (1958). Borel, A.; Remmert, R.: Uber kompakte homogene Kahler-Mannigfaltigkeiten. Math. Ann. 145,429-439 (1962) Zbl.111,180. Bogomolov, F.A.: Kahler manifolds with trivial canonical class. Izv. Akad. Nauk SSSR, Ser. Mat. 38, 11-21 (1974), English transl.: Math. USSR, Izv. 8, 9-20 (1975) Zbl.292.32020.

* For the convenience of the reader, compiled using the MATH database,

references to reviews in Zentralblatt have, as far as possible, been included

fur Mathematik (Zbl.), in this References.

348 [Ca80] [Ca81-l] [Ca81-21 [Ca83] [Ca85] [Ca89] [Ca9 l] [Ca92] [De721 [Dou66] [DPS93-l] [DPS93-2]] [Fi-Fo79] [Fo68] [Fu78-l] [Fu82-21 [Fu79] [Fu82] [Fu83-l] [Fu83-21 [Fu83-31 [Fu84] [Gr86]

[Hi751 [Hi811 [Ia87]

F. Campana

and Th. Peternell

Campana, F.: Algtbricite et compacite darts l’espace des cycles dun espace analytique complex. Math. Ann. 251, 7-18 (1980) Zb1.445.32021. Campana, F.: Reduction algebrique dun morphisme faiblement Kahlerien propre et applications. Math. Ann. 256, 157-189 (1981) Zbl.461.32010. Campana, F.: Coreduction algebrique d’un espace analytique faiblement Kahlerien compact. Invent. Math. 63, 187-223 (1981) Zbl.436.32024. Campana, F.: Densite des varietes Hamiltoniennes primitives projectives C.R. Acad. Sci., Paris, Ser. I 297, 413-416 (1983) Zbl.537.32004. Campana, F.: Reduction d’Albanese dun morphisme propre et faiblement Kahltrien I et II. Compos. Math. 54, 373-416 (1985) Zbl.609.32008. Campana, F.: Geometric algebraicity of moduli spaces of compact KIhler sympletic manifolds. J. Reine Angew. Math. 397,202-207 (1989) Zbl.666.32021. Campana, F.: The class ?? is not stable by small deformations. Math. Ann. 290, 19-30 (1991) Zbl.722.32014. Campana, F.: An application of twistor theory to the non-hyperbolicity of certain compact sympletic Kahler manifolds. J. Reine Angew. Math. 425, l-7 (1992). Deligne, P.: Theorie de Hodge II. Pub]. Math. Inst. Hautes Etud. Sci. 40, 5-57 (1972) Zbl.219,91. Douady, A.: Le probltme des modules pour les sous espaces analytiques compacts dun espace analytique donnt. Ann. Inst. Fourier 16, l-95 (1966) Zbl.146,311. Demailly, J.P.; Peternell, Th.; Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Alg. Geom. 1993, in press. Demailly, J.P.; Peternell, Th.; Schneider, M.: KIhler manifolds with numerically effective Ricci class. Comp. Math. 1993, in press. Fischer, G.; Forster, 0.: Ein Endlichkeitssatz fur Hyperflachen auf kompakten komplexen Rlumen. J. Reine Angew. Math. 306,88-93 (1979) Zbl.395.32004. Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math. 90, 51 I-521 (1968) Zb1.176,184. Fujiki, A.: Closedness of the Douady spaces of compact Kahler spaces. Publ. Res. Inst. Math. Sci. 14, l-52 (1978) Zb1.409.32016. Fujiki, A.: On automorphism groups of compact Kahler manifolds. Invent. Math. 44, 225-258 (1978) Zbl.367.32004. Fujiki, A.: Countability of the Douady space of a complex space. Japan. J. Math. 5, 431-447 (1979) Zbl.437.32005. Fujiki, A.: On the Douady space of a compact complex space in the Category C. Nagoya Math. J. 85, 189-211 (1982) Zbl.445.32017. Fujiki, A.: On the structure of compact complex manifolds in +? Adv. Stud. Pure Math. 1, 231-302 (1983) Zbl.513.32027. Fujiki, A.: On primitively symplectic compact Kiihler V-manifolds of dimension four. Prog. Math. 39, 71-250 (1983) Zb1.549.32018. Fujiki, A.: Semisimple reductions of compact complex varieties. Inst. E. Cartan, Univ. Nancy 8, 79-133 (1983) Zb1.562.32014. Fujiki, A.: On the Douady space of a compact complex space in the category g, II. Pub]. Res. Inst. Math. Sci. 20,461-489 (1984) Zbl.586.32011. Grauert, H.: On meromorphic equivalence relations. In: Contributions to Several Complex Variables, Hon. W. Stall, Proc. Conf. Notre Dame/Indiana 1984, Aspects Math. E9, 115-147 Zbl.592.32008. Hironaka, H.: Flattening theorem in complex analytic geometry. Am. J. Math. 97, 503-547 (1975) Zbl.307.32011. Hirschowitz, A.: On the convergence of formal equivalence between embeddings. Ann. Math., II. Ser. 113, 501-514 (1981) Zbl.421.32029. Iarrobino, A.: Hilbert scheme of points: overview of last ten years. Proc. Symp. Pure. Math. 46, 297-320 (1987) Zbl.646.14002.

VIII. [Ka69]

[Kau65] [Ker60] [I(0621

[KoS60] [Kr75]

[Le90] [Lel57] [Lel68] [Lie781 [Ma571 [Moi67]

[NS77] [Pot1691 [St861 [SC60/61] cu751 IV841 IY781

[Yo77]

Cycle

Spaces

349

Kawai, S.: On compact complex analytic manifolds of complex dimension 3, I and II. J. Math. Sot. Japan 17, 438-442 (1965) and 21, 604-616 (1969) Zbl.136.72 and Zbl.192,441. Kaup, W.: Inlinitesimale Transformationsgruppen komplexer Rlume. Math. Ann. 160, 72292 (1965) Zbl.146,311. Kerner, H.: Uber die Automorphismengruppen kompakter komplexer RPume. Arch. Math. II, 282-288 (1960) Zbl.112,312. Kodaira, K.: A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds. Ann. Math., II. Ser. 75, 1466162 (1962) Zbl. I 12,384. Kodaira, K.; Spencer, D.C.: On deformations of complex analytic structures III. Ann. Math., II, Ser. 71, 43-76 (1960) Zbl.128,169. Krasnov, V.A.: Compact complex manifolds without meromorphic functions. Mat. Zamethi 17, 119-122 (1975). English transl.: Math. Notes 17, 69-71 (1975) Zbl.321. 32017. Lebrun, C.: Asymptotically scalar flat self-dual metrics. Preprint 1990. Lelong, P.: Integration sur un ensemble analytique complex. Bull. Sot. Math. Fr. 8.5, 239-262 (1957) Zbl.79,309. Lelong, P.: Fonctions plurisousharmoniques et formes differentielles positives. Gordon and Breach 1968,Zbl.195,116. Lieberman, D.: Compactness of the Chow Scheme. Lect. Notes Math. 670, 140-186. Springer (1978) 2131.391.32018. Matsushima, Y.: Sur la structure du groupe des homtomorphismes analytiques dune certaine variete Kahlerienne. Nagoya Math. J. II, 145-150 (1957) Zbl.91,348. Moishezon, B.C.: On n-dimensional compact varieties with n algebraically independent meromorphic functions. Izv. Akad Nauk SSSR, Ser. Mat. 30, 1333174, 345-386, 621-656 (1966). English transl.: Am. Math. Sot., Transl., II. Ser. 63, 51-177 (1967) Zb1.161,178. Norguet, F.; Siu, Y.T.: Holomorphic convexity of spaces of analytic cycles. Bull. Sot. Math. Fr. 105, 191-223 (1977) Zbl.382.32010. Pourcin, G.: Theoreme de Douady au-dessus de S. Ann. Sci. EC. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 23,451-459 (1969) Zbl.186,140. SteinbiB, V.: Das formale Prinzip fur reduzierte komplexe Raume mit einer schwachen Positivitatseigenschaft. Math. Ann. 274,485-502 (1986) Zbl.572.32004. Grothendieck, A.: Technique de construction en geometric analytique, in Semin. Cartan 13 (1960/61), No. 7717 (1962) Zbl.142,335. Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Lect. Notes Math. 439. Springer (1975) Zbl. 299.14007. Varouchas, J.: Stabilite de la classe des varietes Klhleriennes par certain morphismes propres. Invent. Math. 77, 117-127 (1984) Zbl.529.53049. Yau, ST.: On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation I. Commun. Pure Appl. Math. 31, 339-411 (1978) Zbl.362.53049. Yoshihara, H.: On hyperelliptic manifolds. Mem. Jap. Lang. Sch. 4, 104-l 15 (1977), (In Japanese).

Chapter IX

Extension of Analytic Objects H. Grauert

and R. Remmert

Contents Introduction

..

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

. . . . .

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

352 352 353 353 353

0 2. Extension of Analytic Sets ............... 1. The Remmert-Stein Theory ............ ............ 2. The Stoll-Bishop Theorem

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

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

354 354 354

............................... 03. Corners 1. Extension into Corners ............... 2. High Quotient Gap Sheaves ........... 3. TheCaseofX=IP” ..................

. . . .

. . . .

. . . .

. . . .

.. .. .. ..

. . . .

. . . .

355 355 356 356

. § 4. Extension of Coherent Sheaves ........... . 1. Sheaf Extension ...................... ... . 2. Extension into q-Concave Boundaries 3. Quotient Sheaves in Cartesian Products . . . Historical Notes ........................

. . . . .

. . . . .

. . . . .

. . . .

. . . . .

. . . . .

. . . . .

356 356 357 357 358

8 1. Continuation into q-Concave Boundaries 1. GapSheaves ........................ 2. The Siu-Trautmann Theorem .......... 3. A Counter Example .................. 4. The Case q = dim A ..................

References

.. .. .. .. ..

. . . . .

352

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

359

352

H. Grauert

and R. Remmert

Introduction Section 1 deals with the extension of coherent quotient sheaves into q-concave boundaries. The old notion of gap sheaf is essential here. The end of 9 1 and the beginning of 42 is devoted to the theorems of Remmert and Stein for the extension of analytic sets into other analytic sets. It is of importance that the boundedness of the area gives a necessary and sufficient condition (Stall-Bishop theorem). If the boundary is q-concave but has corners stronger conditions are necessary. Of especial interest is the extension into the full complex projective space (see 3 3). The last section gives the extension of absolute (i.e. not quotient) sheaves into q-concave boundaries. This also gives the “Kontinuitltssatz” for meromorphic maps; in the case of meromorphic functions the theorem first was proved by Hellmut Kneser.

5 1. Continuation

into q-Concave Boundaries

1. Gap Sheaves. Such sheaves were already considered in Chapter I,1 1.5; the notion was introduced by Thimm [Th62]. Let X be a complex space and let 9 be a coherent submodule (analytic sub sheaf) of a coherent analytic sheaf Y on X. For every integer q there exists a (unique) maximal coherent subsheaf Yq of 9’ such that 9, 3 9 and dim supp(~$) I q, cf. I.1 1.21 (2). Clearly 9 c J$, c ... c .a, c 97 Definition 1.1. The sheaf Yq is called the qth gap sheaf of 9 in 9 The coherent sheaf 9?)4:= 91Yq is called the qlh quotient gap sheaf to 9. The sheaf W := Ypl9 is called q-complete if 9q = 9 and thus 9%?4 = W. It is clear that q-completeness implies p-completeness for p I q. The qth gap sheaf of -9, is 9q again. Following [Si73] the sheaf J$ can also be defined as follows, cf. I.1 1.21 (2’): Attach to every open set U c X the LO(U)-module: {s E Y(U): SI U - A E Y(U - A) for some analytic set A in U with dim A I q}; this gives an analytic presheaf with Yq as corresponding sheaf. A simple example is given by the analytic sets A c X. In this case the sheaf 9’ is the structure sheaf Co of X and 9 = -OA is the ideal sheaf of A. The set A = u A, decomposes into irreducible components A,. We denote by B = B4 c A the union of those A, whose dimension is bigger than q. Then 9, is the ideal sheaf of B and W, = O/$$ is the structure sheaf of the (reduced) complex space B. The structure sheaf of an analytic set A is q-complete if and only if the local dimension of A is > q everywhere. We call then A itself q-complete. Assume that Y is a coherent sheaf on a purely n-dimensional complex space X and that A is a nowhere dense analytic subset of X. Then we have dim A I n - 1. We put 9 = YA. Y and obtain a coherent subsheaf of 9 The support of 9 is then contained in A since we have 91X - A = YIX - A. But now ,~$!~-i is the zero sheaf 0.

IX.

Extension

of Analytic

Objects

353

2. The Siu-Trautmann Theorem. This theorem was proved in [ST71], p. 148. Assume that X is an arbitrary complex space and that Y is a coherent analytic sheaf on X. Assume moreover that Y c X is an open subset which is q-concave, q= 1,2,.... Theorem 1.2. If 0 E dY is a boundary point of Y then all quotient gap sheaves B,rA with q” 2 q have a unique q-complete extension into a neighborhood of the point 0. All these extensions are q h -complete. If A c Y is an analytic set whose local dimension is > q everywhere, then we take for 9 the structure sheaf of X and 9? is the structure sheaf of A. The q-th quotient gap sheaf is B again. So A has a unique analytic extension into 0 E a Y. Every space Y is q-concave if Y = X - D, where D c X is an analytic set whose local dimension everywhere is smaller than q. Hence, A has a unique analytic extension into D. This extension is the closure 2 of A in X. Here we need for the local dimensions the inequalities dim, A 2 dim. D + 2. This is a very strong assumption. There is a theorem by Remmert and Stein (see [RS53]) which is much weaker in its assumptions: Theorem 1.3. Assume that D is an analytic set in the complex space X and that A is in analytic set in Y = X - D such that always for the local dimension dim. A 2 dim. D + 1. Then the closure of A in X is the unique analytic extension of A to X with local dimension >dim, D everywhere. 3. A Counter Example. We take for X a neighborhood of 0 in the complex number space (c” with n > 2 and for Y the n-dimensional manifold X - (0). Cleary, Y has in 0 the strongest concavity, it is l-concave. We take an analytic curve A c X with 0 E A as only singularity. We consider the first infinitesimal neighborhood (A, UJ(&)‘) of A. The normal bundle N of A is defined in this infinitesimal neighborhood. In N there exists a flag space along A - {0} with linear flags of dimension 1 such that the holomorphic map of A - 0 into the projective bundle lPN which assignsthe flags to the points of A - 0 becomes transcendentally singular in 0. The flag space gives a coherent ideal sheaf 9 c 0x of local holomorphic functions which vanish on the flags. Its zero set is A. If the quotient sheaf 9J?= 0x19 could be extended into 0, it would be possible also to extend the flag space into 0. However, this is not possible, because of transcendency. Hence the Remmert-Stein theorem is valid for analytic sets only. 4. The Case q = dim A. Let E = {z E (cllzl 2 l} denote the closure of the outside of the unit disc in the complex z-plane. There is a differentiable complex function f on E which is holomorphic in the interior E”, but is singular in every boundary point of E. We denote by G the graph {(z, w)l w = f(z), z E E} and by G’ its restriction to aE. We take for X - Y a closed ball around 0 E X = Cc2 which contains G’ such that the intersection of G’ and aY is not empty. Then the one dimensional analytic set A = Y n G cannot be extended analytically into any boundary point of aY n G’ though Y is l-concave there.

354

H. Grauert

and R. Remmert

5 2. Extension of Analytic Sets 1. The Remmert-Stein Theory. Assume that X is an arbitrary complex space and that D c X and A c X - D are pure dimensional analytic subsets. We have seen that if dim A > dim D the closure 2 of A is the smallest analytic extension of A to X. This theorem has many applications in complex analysis, for instance to prove the proper mapping theorem for analytic sets. In the case dim A = dim D the theorem is no longer true. We need a further assumption (see also [RS53]). We decompose D into irreducible components D = u D,. We assume that the closure of A does not contain any D, and get: Theorem 2.1. Under the assumptions analytic extension of A to X.

made the closure 1 of A is the smallest

For hypersurfaces in domains of CC:”this theorem had already been proved by P. Thullen [Thu35] in 1935. There are only a few applications of this generalization of the fundamental Remmert-Stein theorem. One application is a generalized version of the famous Rado theorem [Ra24] for holomorphic functions: Denote by G a domain in C” and by f a multivalued holomorphic function in G. These functions are defined as follows: Assume that z E G is a point and that w E C. Then there is a neighborhood U(z) c G and a neighborhood V(w) such that the graph of f in V x U is the zero set of a manic polynomial

4w, z) E qu) [WI. Theorem 2.2. Assume that G c B are domains of Cc” and that f is a multivalued holomorphic function over G such that the graph off tends to 0 for z -+ i?G n B. Then f has a unique analytic extension to B as a multivalued holomorphic function. 2. The Stall-Bishop Theorem. Assume that X is a reduced complex space.By a Hermitian metric p on X we understand a differential metric on X (which gives the length of tangent vectors) and has the following local property: If x E X is an arbitrary point there exist a neighborhood U = U(x) and a biholomorphic embedding of U into a domain G c (CNand an ordinary C”-Hermitian

metric in G whoserestriction to U is pi U. In the same way we define the differential forms on X. We associate with p a corresponding real form o = i. p of type (1, 1) by replacing the symmetric products dz . dZ in p by the antisymmetric Grassmann products multiplied by i = (- 1)1’2.Let us denote by X” the set of smooth points in X. Then plX” and 01x” are ordinary Hermitian metrics and differential forms. The open set X” is dense in X. If Xi is an irreducible component of X it has a fixed (complex) dimension n. We define the volume V(Xi) of Xi to be the integral over XT with respect to o” and call the sum of these V(Xi) the volume of X. It can be infinite or else a finite number > 0. In [Bi64] Bishop proved among more general results the following

IX.

Extension

of Analytic

Objects

355

Theorem 2.3. Assume that X is a reduced complex space with an Hermitian metric p, that B c X is a nowhere dense analytic subset and that A c X - B is an analytic set without isolated points. Assume that for every point x E B there is a neighborhood U such that U n A has finite volume. Then the closure 2 is an analytic set in X.

If X is the n-dimensional complex projective space II’,, we have the FubiniStudy metric on X. We take for B c X a hyperplane and get X - B = Cc”. If A c Ic” is an analytic set without isolated points, then A is algebraic if and only if A has finite volume, since then it can be analytically extended to IF’,,. This special case was proved by W. Stall [St63].

93.

c orners

1. Extension into Corners. Assume that Y c X is open. A boundary point x E 8Y is called q-concave with corners if there is a neighborhood U = U(x) c X together with q-convex functions c1i, . . . , elkin U such that U n Y is the union of the sets {c+ > ui(x)}. So U n Y is the union of ordinary q-concave subdomains. Hence, it may have corners. But in these points the concavity should be very strong and a continuation theorem of the Siu-Trautmann type should be valid. However, this is not the case. We consider the following example. We put X = IP4 with inhomogeneous coordinates zl, z2, wl, w2. The planes {zi = z2 = 0} and {wi = w2 = 0} in (c4 intersect in the O-point 0 E (c4, only. Each plane has arbitrary small 2-concave neighborhoods U and U’ whose intersection is a small neighborhood of 0. Their boundaries are smooth and intersect transversally. We put Y = U u U’. The intersection A := U n B with B = (zi - w1 = 1) c (c4 is analytic in Y if Y is sufficiently small. It is clear that A has dimension 3. Of course, B also intersects the plane (wi = w, = 0} in a set disjoint from A. If everything is done well there are increasing families of domains U, and U; which exhaust (E4 such that the domains have similar properties as U and U’. We take the increasing family I: = U, u Vi. If the continuation theorem of Trautmann and Siu would hold for every t we could extend A to every large ball in (c4, hence to (E4 and then to lP4. But the extension would be always equal B, which would enter Y yet a second time. We define Y” as the first x such that the continuation in the sense of Trautmann and Siu is not possible into all boundary points. We denote such a point by y” and get y^ E B. For every k = 1, 2, 3, . . . there is a meromorphic function f in (c4 which has {zi - w1 = l} as polar set of order k and is holomorphic elsewhere. The restriction flA is a Cousin-I distribution in Y and hence an element II/ E H’(Y, 0). One can extend Ic/ to I: as long as A can be extended. So an extension into Y” is possible but not an extension into a neighborhood W of y”. Since k can be arbitrary the first cohomology with coefficients in the structure sheaf of every intersection W n Y h is infinite.

356

H. Grauert

and R. Remmert

2. High Quotient Gap Sheaves. We assume again that X is a complex space and that Y c X is a q-concave subdomain with corners. We take a coherent analytic sheaf Y on X, a coherent quotient sheaf 5%of Y on Y and put 9” = a4* with q” 2 2q - 1. We then call aA a high quotient gap sheaf (with respect to q). In [GrSl] it was shown: Theorem 3.1. The quotient sheaf ~8~ can be extended into every boundary point of Y as a coherent quotient sheaf of 9. The extension is done by continuing gA across the corners of any two qconcave hypersurfaces and by proving that the sheaves obtained are independent of this pair. If B c X = lPn is an algebraic subset everywhere of codimension ~q, then - using the Fubini-Study metric on X - we obtain arbitrarily small neighborhoods Y around B which are q-concave with corners (see [Ba70]). There is an increasing continuous family of q-convex subdomains with corners which extends Y to X. Hence, 9” with q” 2 2q - 1 extends as a quotient sheaf of Y to X and is an algebraic sheaf by a GAGA theorem. 3. The Case of X = lP,,. If B c X is an analytic set of pure dimension n - 1, we have q = 1. Then also every cohomology class $ E H’( Y, P’), where Y is a coherent sheaf on X, and Y is a suitable open neighborhood of B, can be extended to X if i < cdh(Y) - q. Here cdh(Y) denotes the homological codimension of y, cf. Chapter I. 11.1-2. If B is a smooth (n - q)-dimensional manifold and q > 1 it follows that this cohomology is finite. But in general an extension to X is not possible. Barth [Ba69] proved however that it can be done for all i I cdh(Y) - 2q. Finally, Ogus [Og76] derived the same result if B is locally a complete intersection. The continuation of quotient sheaves $5”’ is much simpler. Fakings [Fa80] showed: Theorem 3.2. Assume that B is irreducible and of dimension n - q. Then for q” 2 q already the gap sheaf &?4Ahas a unique coherent extension to X. Of course, it is essential that B is irreducible. The example given in Subsection 1 of this section is also a counterexample to this theorem if B is reducible.

04. Extension of Coherent

Sheaves

1. Sheaf Extension. Let X be a complex space and Y a coherent analytic sheaf on X. For every integer i the ith absolute gap sheaf P$, of Y is the analytic sheaf associated to the family of 0( U)-modules Y;,,(U) = lim ind y(U

- A),

A

where the inductive limit is taken for all analytic sets in U of dimension si. In general, Y;,, is not coherent; a necessary and sufficient condition for this is given

IX. Extension

of Analytic

Objects

357

in [Si73] (see especially: The mixed case . . . , p. 137). If the codimension of A n C in C, where C is the support of an arbitrary local section in 9, is always at least two everywhere, then the sheaf Y;,, is coherent. We always assume that this condition is fullfilled. If the canonical homomorphism Y + 9&, is bijective, the sheaf is called absolutely q-complete. Theorem 4.1. Let B be analytic in X of dimension
The proof is rather involved. In [Si73] (especially: The mixed case . . .) Y.T. Siu uses, for proving the theorem on p. 417 of coherence of direct images, essentially the mixed case of a holomorphic projection rr: X + D, where the boundary of X over D consists of two disjoint parts whose intersection with the fibers is 3-concave in the first case and l-convex in the second case. Then the first direct image R’rc,(Y) is used. It can be seen rather easily that even for the extension of locally free sheaves the inequality q < cdh(X) is not enough. We take for X the complex number space c2 = {(z, z’)} and put Y = X - 0 or Y = (c2 - a closed ball. Then Y is l-concave. We take for Y the locally free sheaf of local meromorphic functions in X which have poles on the z-axis at most of order k. Since there are holomorphic functions on the z-axis which have no analytic continuation into 8Y it is clear that Y cannot be extended beyond aY n {(z, z’): z’ = O}. 2. Extension into q-Concave Boundaries. Let Y c X be an open subset whose boundary in X is q-concave. We take a boundary point y’ E aY. After an embedding of a neighborhood of y’ in some n-dimensional complex manifold Z there is an (n - q)-dimensional complex manifold A through y’ which does not enter X - Y but has contact of second order with dY. By moving A transversally we get locally a holomorphic libration of Y with l-concave, l-convex fibers. We consider a coherent absolutely q-complete sheaf Y on Y. Around y’ there is a Hartogs figure in the sense of [Si74] (especially: Techniquesof.. . , Theorem 7.4, p. 243). By applying the theorem to this Hartogs figure we get the unique extension of 9’ in y’. Finally, we arrive at the following generalization of (4.1): Theorem 4.2. Assumethat 9’ is coherent and absolutely q-complete in Y. Then Y has a unique extension to a coherent absolutely q-complete sheaf into y’.

There are simple necessary.

examples which show that the condition

9&, = 9’ is

3. Quotient Sheavesin Cartesian Products. We discuss results of Ivashkovich (see [Iv86]). Assume that Y is a q-concave subdomain of X with a boundary point y E 8Y and that K is a compact Kahler manifold. We have the projection p: K x X + X. We take a coherent sheaf 9’ on K x Y and a coherent subsheaf 3 such that for every irreducible component of the support A of the quotient

358

H. Grauert

and R. Remmert

sheaf .B = ,40/y the induced map p: A + X is generically B? is absolutely q-complete.

finite. We assume that

Theorem 4.3. There is a unique extension .%YAof .!8 into y such that a) 5fA is q-complete. b) every irreducible component of supp(W^) is a continuation of an irreducible component of supp(B). Of course, the projection p may have degenerations on A. There is a simple application for meromorphic maps I++:Y + K. Assume that X is a n-dimensional normal complex space and that q -C n. We take for W the sheaf of holomorphic functions on the graph A of II/ in K x Y. Then B = W,-, and the other conditions are satisfied. So, by the theorem 4.3 the set A can be extended analytically to K x 8Y. This means that the meromorphic map has an analytic continuation into f3Y. If K is a projective algebraic manifold we have a positive line bundle F on K. If the positivity is high enough we get the direct image of 0,., on Y as a quotient sheaf of the O-direct image of F. Hence the extension of (?4 in theorem 4.3 can be obtained by extension of this quotient sheaf. So the extension theorem for meromorphic maps follows from 0 1.2. Historical Notes. In 1851 Riemann proved his famous theorem about removable isolated singularities of holomorphic functions of one complex variable (art. 12 of his dissertation). In 1906 Hartogs extended all holomorphic functions of several complex variables from the (connected) boundary of a bounded domain holomorphically into every interior point, cf. [Ha06], p. 231 and 239; later on, in 1932, H. Kneser extended meromorphic functions meromorphically, cf. [Kn32]. In the thirties and the forties the Kontinuitiitssatz played an important role. First results for the extension problem for analytic sets were proved by P. Thullen (cf. [Th35]). He studied the continuation of analytic sets of codimension 1 into analytic sets of lower or same dimension. His theorem was generalized by Remmert and K. Stein [RS53]. A special case of this generalization is very useful, e.g. it gives immediately Chow’s theorem to the effect that all analytic sets in projective space are algebraic; furthermore, it can to a certain extend replace the direct image theorem for proper images of coherent sheaves. Rothstein CR0493 was the first to obtain results about the continuation of analytic sets from a domain G into a larger domain B, where, in general, B - G contains interior points. He proved for instance: If G is an euclidian shell B - K between two balls B and K and if A is an analytic set in B - K whose dimension is 22 everywhere, then A can be analytically extended to the full ball B. These results are more or lessincluded in the results of Siu and Trautmann (see[ST711 and [Si74]). The appropriate terminology is that of extension of q-complete quotient sheavesas quotient sheaves. Already Rothstein considered q-concave boundaries with sharp corners by tranversal intersection. One should expect that this caseis even better for exten-

IX.

Extension

of Analytic

Objects

359

sion. But this is not the case; we need stronger gap sheaves (high gap sheaves). It is only fair to say that in the work of Rothstein many gems are still hidden, cf. also CD1781 and [D190]. In the early sixties Trautmann [Tr67] began to study the continuation of absolute coherent sheaves on arbitrary complex spaces X. Here the homological codimension enters. Quite recently Zvashkovich has developed an extension theory for (the graphs) of meromorphic mappings into compact Kahler manifolds (see [Iv86]). In his plenary talk 1950 at the International Congress in Cambridge, Mass., H. Cartan said: SW ce probEme [of the extension of analytic objects] nous n’avons aujourd’hui que des rbultats fragmentaires. Meanwhile there is a plethora of deep results. Time is ripe for a detailed well commented on Ergebnisbericht.

References* [Ba99]

Barth,

W.: Transplanting cohomology classes in complex projective space. Am. J. Math. (1970) Zbl.206,500. Barth, W.: Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projektiven Raum. Math. Annalen 287, 105-162 (1970) Zbl.184,313. Bishop, E.: Condition for the analyticity of certain sets. Mich. Math. J. II, 289-304 (1964) Zbl.143,303. Dloussky, G.: Anylycite s&par&e et prolongements analytiques (d’apres le dernier manuscript de W. Rothstein). Lect. Notes Math. 683, 173-202, Springer (1978) Zbl. 409.32010. Dloussky, G.: Analycitt stparte et prolongement analytique. Math. Ann. 286, 1533168 (1990) Zbl.701.32003. Faltings, G.: A contribution to the theory of formal meromorphic functions. Nagoya Math. J. 77,99-106 (1980) Zbl.401.14001. Grauert, H.: Kantenkohomologie. Compos. Math. 44, 79-101 (1981) Zbl.512,32011. Hartogs, F.: Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veranderlichen. Sitz. Ber. Math. Phys. Kl. Akad. Munch. 36, 2233242 (1906) Jbuch 37,443. Ivashkovich, S.M.: Extension of locally biholomorphic mappings into complex projective space. Izv. Akad. Nauk SSSR, Ser. Mat. 47, 197-206 (1983). English transl.: Math. USSR, Izv. 22, 181-189 (1984) Zbl,523.32009 (except theorem 2). - Extension of locally biholomorphic mappings into a product ofcomplex manifolds. Izv. Akad. Nauk SSSR, Ser. Mat. 49,884-890 (1985). English transl.: Math. USSR Izv. 27, 193-199. (1986) Zbl. 584.32025 Hartogs phenomenon for holomorphically convex Klhler manifolds. Izv. Akad. Nauk SSSR, Ser. Mat. 50,866-873 (1986). English transl.: Math. USSR, Izv. 29, 225-232. (1987) Zbl. 618.32011 - The Hartogs-type extension theorem for meromorphic mappings into compact Klhler manifolds. Invent. Math. 109,47-54 (1992). - Spherical shells as obstructions for the extension of holomorphic mappings. Preprint Bochum. See also: Mat. Zamethi 49, No. 2, 141-142 (1991). English transl.: Math. Notes 49, No. 2,215-216 (1991) Zb1.727.32007. Kneser, H.: Ein Satz iiber die Meromorphiebereiche analytischer Funktionen von mehreren Verlnderlichen. Math. Ann. 106,648-655 (1932) Zbl.4,358. 92, 951-967

[Ba70] [B&l] CD1783 CD1901 [Fa80] [Gr81] [Ha061

[Iv861

[Kn32]

* For the convenience of the reader, compiled using the MATH database, have, as far as possible, been included

references to reviews in Zentralblatt and Jahrbuch iiber die Fortschritte in this References.

fur Mathematik der Mathematik

(Zbl.), (Jbuch)

360 Cog761 [Ra24] IRS533 CR0491

[Si73]

[Si74]

[ST711 [St631

[Th62] [Thu35] [Tr67]

H. Grauert

and R. Remmert

Ogus, A.: On the formal neighborhood of a subvariety of projective space. Am. J. Math. 97, 1085-1107 (1976) Zb1.331.14002. Rado, T.: ijber eine nicht fortsetzbare Riemannsche Mannigfaltigkeit. Math. Z. 20, l-6 (1924) Jbuch 50,255. Remmert, R.; Stein, K.: ijber die wesentlichen Singularitaten analytischer Mengen. Math. Ann. 126,263-306 (1953) Zbl.51,63. Rothstein, W.: Die Fortsetzung vier- und hoherdimensionaler analytischer FlPchen des R,, (n 2 3). Cousinsche Verteilungen 2. Art. Math. Ann. 121, 340-355 (1950) Zbl.37,183. - iiber die Fortsetzung analytischer Fhichen. Math. Ann. 222.424-434 (1951) Zb1.44,308. - Zur Theorie der Singularitaten analytischer Funktionen und Flachen. Math. Ann. 126, 221-238 (1953) Zb1.51,63. - Zur Theorie der analytischen Mannigfaltigkeiten im Raume von n komplexen Veranderlichen. Math. Ann. 129,96-138 (1955) Zb1.64,80. - Zur Theorie der analytischen Mannigfaltigkeiten im Raume von n komplexen Verlnderlichen. Die Fortsetzung analytischer Mengen vom Rande eines Gebietes her ins Innere; bzw. Die Fortsetzung analytischer Mengen in Gebieten mit analytischen Schlitzen. Math. Ann. 233, 271-280; bzw. 400-409 (1957) Zb1.77,289 bzw.84,72. - Zur Theorie der analytischen Mengen. Math. Ann. 174, 8-32 (1967) Zbl.172,378. - Das Maximumsprinzip und die Singularitaten analytischer Mengen. Invent. Math. 6, 163-184 (1968) Zbl.164,382. Siu, Y.T.: The mixed case of the direct image theorem and its applications. Complex Anal., C.I.M.E., Bressanone 1973,281-463 (1974). Zbl.338.32012. -A pseudoconcave generalization of Grauert’s direct image theorem I and II. Ann. SC. Norm. Super. Pisa, Cl. Sci., III. Ser. 24, 278-330 and 439-489 (1970) Zbl.195,368 and Zb1.206,91. - The l-convex generalization of Grauert’s direct image theorem. Math. Ann. 190, 203-214 (1971) Zbl.203,82. A pseudoconvex-pseudoconcave generalization of Grauert’s direct image theorem. Ann. SC. Norm. Super. Pisa., Cl. Sci., III. Ser. 26, 649-664 (1972) 2171.248.32014. Siu, Y.T.: Techniques of extension of analytic objects. Lect. Notes Pure Appl. Math. 8, Dekker, New York 1974,Zb1.294.32007. - Absolute gap sheaves and extensions of coherent analytic sheaves. Trans. Am. Math. Sot. 141, 361-376 (1969) Zbl.184,110. - Extending coherent analytic sheaves. Ann. Math., II. Ser. 90, 1088143 (1969) Zbl.181,363. - An Osgood type extension theorem for coherent analytic sheaves. Lect. Notes Math. 185, 189-241, Springer (1971) Zb1.205,94. - A Hartogs type extension theorem for coherent analytic sheaves. Ann. Math., II. Ser. 93, 166-188 (1971) Zb1.208,104 - A Thullen type extension theorem for positive holomorphic vector bundles. Bull. Am. Math. Sot. 78, 775-776 (1972) Zbl.253.32016. Siu, Y.T.; Trautmann, G.: Gap sheaves and extension of coherent analytic subsheaves. Lect. Notes Math. 172, Springer (1971) Zbl.208,104. Stall, W.: The growth of the area of a transcendental analytic set of dimension 1, Math. Z. 81, 76-98 (1963) Zb1.109,308; Part I and II, Math. Ann. 156, 47-78 and 144-170 (1964) Zbl.126,95. Thimm, W.: Liickengarben von koharenten analytischen Modulgarben. Math. Ann. 148, 372-394 (1962) Zb1.111,82. Thullen, P.: iiber die wesenthchen Singularitlten analytischer Funktionen und F&hen im Raume von n komplexen Verlnderlichen. Math. Ann. 111, 137-157 (1935) Zb1.11,124. Trautmann, G.: Ein Kontinuitatssatz fur die Fortsetzung kohlrenter analytischer Garben. Arch. Math. 18, 188-196 (1967) Zb1.158,329.

Author Index Abhyankar, S. 90 Ancona, V. 234,235,246,298,303,306 Andreotti, A. 172,240,243, 343 Arnold, V. 297 Aroca, J. 312, 315 Artin, M. 215, 296,298, 301, 303, 308, 321, 325 Atiyah, M.F. 168 Barlet, D. 326, 327, 329, 330, 343, 344 Barth, W. 339 Beauville, A. 325, 341 Behnke, H. 31,159,186, 194 Bierstone, E. 312, 315 Bingener, J. 85, 132, 301, 303, 321 Bishop, E. 331,352 Blanchard, A. 337,338 Borel, A. 168,339 Bott, R. 171,251 Bourbaki, N. 109 Bremermann, A. 236 Brenton, L . 311 Brieskorn, E. 85 Brill, A. 80 Campana, F. 324,332-336,338,339 Caratheodory, C. 30 Cartan, H. 19, 31,46, 59, 72, 90, 124, 159, 163,229,236,282,359 Chow, W.L. 112, 358 Coltoiu, M. 235,239 Commichau, M. 298 Dedekind, R. 15 Deligne, P. 278 251, 253, 306, 307 Demailly, J.P. Diederich, K. 240 Docquier, F. 240 Dolbeault, P. 157 Douady, A. 113, 115,320,321,322 Elencwajg, G. 240 Enriques, F. 294 Faltings,

G.

356

Fischer, G. 114,119,33 1 Fogarty, J. 325 Fornaess, J.E. 240 Forster, 0. 163, 331 Frenkel, J. 264 Frisch, J. 112, 130 Frolicher, A. 175 Fujiki, A. 293, 294,323-325, 338-342 Fujita, T. 309

334,336,

Girbau, J. 251 Godement, R. 173 Grauert, H. 19, 31, 84, 119, 124, 162, 164, 165, 172, 194,227,229,237, 239, 240,242,251, 288,298,306,343 Gritliths, Ph. A. 246,270,271,294,299 Grothendieck, A. 85,119, 121, 130, 171,249, 321,325 Hartogs, F. 19, 358 Hartshorne, R. 171,246 Hilbert, D. 281 Hironaka, H. 112,234,292,293,298,305, 309,310,312,336,339 Hirschowitz, A. 240, 293, 325 Hirzebruch, F. 168 Hodge, W.V.D. 175 Holmann, H. 219 Hopf, H. 169,290 Hormander, L. 185,190,191 Houzel, Ch. 21 Illusie, L. 278 Ishii, S. 141 Ivashkovich, S.M.

357,359

Kiihler, E. 278 Kaup, B. 219 Kaup, L. 113 Kaup, W. 323 Kawamata, Y. 248 Kerner, H. 113,288,323 Kneser, H. 352, 358 Knorr, K. 163,232,249

362 Knutson, Koch, K. Kodaira, 341 Kollar, J. Kosarew, Krasnov, Krull, W. Kuhlmann,

Author D.

308 197,206,219 K. 170,247-249,278,294,

329,336,

254, 310 S. 235,298 V. 303, 330,331 19 N. 90

Lasker, E. 19 Laufer, H. 234 LeBrun, C. 336 Lelong, P. 225, 329, 343 Leray, J. 20, 156, 177,271 Levi, E.E. 223,229,236 Lieberman, D. 330,339 Lindel, H. 133 Lojasiewicz, S. 130 Malgrange, B. 20 Michalache, N. 235, 239 Michel, D. 240 Milman, P. 312, 315 Moishezon, B.G. 215,293,303,305,310,332 Mumford, D. SO, 219 Nakano, S. Nakayama Narasimhan, Noether, E. Norgeut, F.

247,248,278,293 14 R. 229,235,239-241,343 19 236,239,343,345

Oeljeklaus, E. 339 Ohsawa, R. 254 Oka, K. 19, 46, 56, 79, 90, 162, 236 Osgood, W.F. 14, 19,36 Peternell, T. 234, 251, 253, 299, 310 Picard, E. 19 Poincare, H. 152 le Potier, J. 248, 251 Pourcin, G. 321, 322 Prill, D. 119 Ramanujam, C.P. 252 Remmert, R. 19, 107, 108, 163, 194,205,211, 218,283,288,289,303,333,339,353,358

Index de Rham, G. 153 Richberg, R. 227,228 Riemann, B. 30, 74, 168,279, 358 Riemenschneider, 0. 25 1,306 Roth, G. 168,279 Rossi, H. 298 Rothstein, W. 358 Riickert, W. 19, 36 Scheja, G. 134 Schneider, M. 232,249,251,253 Schwartz, L. 69 Serre, J.P. 30,46, 90, 141, 159, 163, 168, 169, 278,283,307 Siegel, C.L. 19,281,283, 303 Singer, I. 168 Siu, Y.T. 130, 180, 190,239, 306, 343, 345, 355,358 Skoda, H. 251 Sommese, A.J. 251 Splth, H. 19 Spallek, K. 271 Spencer, D. 336 Stein, K. 30, 124, 158, 159, 186, 194, 197, 198, 206,207,219,352,353,358 SteinbiB, V. 299, 325 Stickelberger, L. 19 Stoll, W. 352, 355 Takegoshi, K. 253 Thimm, W. 73,283,303,352 Thullen, P. 31, 236, 282, 354, 358 Tomassini, G. 303 Trautmann, G. 134,180,355,358,359 Ueno,

K.

306,324,332,338

Van Tan, V. 235 Varouchas, J. 336 Vicente, J. 312, 315 Viehweg, E. 248 Walker, R. 312 WeierstraD, K. 19, 36, 79 Wiegmann, K. 219 Wirtinger, W. 19 Zariski,

0.

90,3 12

Subject Index active germ 46,60 active lemma 62 adic morphism 300 adjunction formula I40 Albanese dimension 337 Albanese map 337 Albanese reduction (of a map) 338 Albanese torus 337 algebra, analytic 13, 18 artinian 14 Cohen-Macaulay 70 noetherian 13 normal 18 reduced 14 sheaf of C 24 Weierstrass 50 algebraic coreduction 335 algebraic dimension 61, 301, 331 algebraic group 2 16 algebraic reduction 304, 331 algebraic reduction, relative 332 algebraic space 308, 321 algebraically connected 334 almost homogeneous 339 ample bundle 244 analytic covering 189 analytic decomposition 198 analytic equivalence relation 198 defined by F 199 equivalent 200 liner 199 normal 202 nowhere degenerated 202 proper 210 restriction RIZ 199 semiproper 201 simple 205, 206 spreadable 200 analytic family (of cycles) 327 analytic spectrum 121 analytic algebra 18 covering 74 dimension 61 equivalence relation 92

homomorphism 13, 15 set 35 sheaf 27 spectrum 55 analytically branched covering 186 branching point 187, 195 critical locus 186, 188, 195 equivalence 187 holomorphic function 188, 195 holomorphic map 188 order of point 187 schlicht point 187, 195 sheet 186 analytically constructible set 36 analytically dependent 197,204,207 analytically rare 187 anti-equivalence principle 22, 33, 56 anti-equivalence principle of complex spaces and local analytic algebras 33 finite complex spaces over Y and coherent &-algebras 56 approximation theorem (of Artin) 298 automorphism group (of a complex space) 323,339 Barlet construction 327 Barlet space 326 base change 164 Betti number 169 bimeromorphically equivalent blow-up 290 branch locus 74

305

canonical line bundle 118 canonical sheaf 118 Cartan’s umbrella 76 Cartan-Oka, theorem 87 Cartier divisor 142 Tech cohomology 154 Tech complex I54 center (of a modification) 287,290 Chern character 168 Chern class 167 Chem polynomial 167

364 Chow scheme 326 coboundary 148, 154 coboundary map 153 cochain 153 cocycle 148,154 codimension, complex 70 codimension homological 66 Cohen-Macaulay algebra 70 point 71 space 71 Cohen Macaulay module 133 Cohen Macaulay ring 133 Cohen Macaulay space 133, 171 Cohen-Macauley spaces 265 coherence lemma 45 of ideal sheaf 56 inverse image sheaf 30 normalization sheaf 87 structure sheaf 45 torsion sheaf 50 coherent sheaf 41 cohomology (module) 148 cohomology, flabby 15 1 cohomology group 261 comparison theorem 164 complete intersection 71,265 completion 32 complex (of R-modules) 148 complex (holomorphic) base 206, 207 complex exterior form 262 total derivation 262 complex forms of type (i, j) 261 derivatives 261 Complex Lie group 216 acts analytically closed 216 geometric quotient 216 complex cc-space 194 a-chart 194 holomorphic function 194 complex space holomorphically spreadable 239 K-complete 239 normal 134 1-convex 230 projective algebraic 394, 307, 310 strongly psuedo-convex 230 Chow-lemma 234,292,298 complex space, algebraic 312,313 complex space of class C 324,336 complex structure 272 hole repair 273 maximal, with a hole 272

Subject

Index complex space 27 codimension 70 image space 33,52 manifold 27 model space 26 pure dimensional 64 reduction 59 cone, in C3 26,28 cone, Segre 83 connection 246 conormal sheaf 104,233,292 cotangent bundle 117 cotangent sheaf 102 continuously, weakly, holomorphic function 91 Cousin problems 161 covering acyclic 155 covering, analytic 74 b-sheeted 74 unbranched 74 Weierstrass 77 covering family (of cycles) 333 Cramer ideal 301 criterion, for coherence 45 connectedness 81 isomorphy of an analytic map 16 Jacobi 35 normality Slf, 86 openness of a holomorphic map 54 purity of dimension 64 smoothness 65 critical locus 74 current 344 current complex 275,276 curvature 246 cycle 326 cycle space 326 de Rham cohomology 153 decomposition into irreducible components 89f Dedekind Lemma 15 defect, function 69 defect, set 69 defining ideal 300 degeneracy set (of a map) 106 derivation 102 descending chain condition for analytic set 73 desingularization 311, 312, 315 determinantal space 83 differential 101 differential form, on complex spaces 354

Subject dimension formula 62 of an analytic algebra 15 a complex space 61 embedding 16,34 homological 68 invariance 63 direct image theorem 162, 229 division algorithm 11 division generalized 17 division theorem 11, 15 divisor 84 divisor class group 84 divisor principal 84 Dolbeault complex 262 Dolbeault groups 262 Dolbeault resolution 275 Dolbeault theorem 275 domain of holomorphy 236,261 domain, locally Stein 240 Douady dimension 323 Douady space 320,321 double complex 174 double point 26 dualizing sheaf 118, 17 1 elimination of indeterminacies 289 elliptic curve 140 embedded desingularisation 312, 314, 315 embedding 34, 108 embedding dimension 16, 34 espace &tale 149 eta16 map 114 Euler characteristic 165 Euler sequence 126 exact sequence 148 exceptional analytic set 231, 233 exceptional set (of a modification) 287, 290 exhaustion function 228, 294 Ext-group 136 Ext-sheaf 136 extension 137 extension, theorem Riemann 74 for locally pure dimensional complex spaces 80 normal complex spaces 81 factorization of holomorphic map 32 fiber product 21,39 space 39 Fano manifold 140, 169 finite analytic homomorphism 15, 18 extensions of analytic algebras 18

Index

365

holomorphic map 51, 54 ideal sheaf 31 sheaf 40 relational w sheaf 41 finite map 186 finite mapping theorem 51 finiteness lemma 1.5 finiteness theorem of Andreotti-Grauert 243 Fitting ideal 301 flat map 111 flat module 109 flat sheaf 111, 164 flatification 112 flattening theorem (of Hironaka) 292, 336 formal cohomology 164 formal completion 129, I63 formal complex space 129,300 forma1 modification 301, 302 formal principle 297, 324 Frechet topology 266 free resolution of finite type 67 sheaf 48 locally 48 Fubini-Study-metric 355 function, plurisubharmonic 224, 294 function, strict pluri-subharmonic 225, 239, 343 function subharmonic 224 functor, right derived 136 GAGA 307 gap sheaf 69,73,352 absolute 356 quotient gap sheaf 352 high quotient 356 genus 278 global decomposition theorem gluing lemma 29 Gorenstein space 141 graph lemma 39 Grassmann manifold 243 GriIIiths positivity 246,295,298

90

Hartogs Iigur, in the sense of [Si74] Hermitian metric on complex spaces Hilbert scheme 321 Hodge decomposition 175 Hodge manifold 249 Hodge metric 249 Hodge number 175 holomorphic base 207 holomorphic function, multi-valued holomorphic hull 261,279 holomorphic locally dense 201

357 354

354

366

Subject

holomorphic map 27, 198 continuously weakly 91 factorisation of 32 finite 51, 54 weakly 91 holomorphically convex 159,236,261 holomorphically convex hull 158 holomorphically separable 345 holomorphically spreadable 345 homogeneous spectrum 126,290 homological codimension 66, 133, 265, 356 homological dimension 68 Hopf bundle 118 Hopf Surface 304,324 hull of holomorphy 282,288 ideal finite type 31 sheaf 22,26 immersion 108 indeterminacy, points of set of w 47 index theorem 169 inIinitesimal neighborhood invariance of dimension inverse limit 127 inverse system 127 irreducible analytic set locally w space 27 point 27, 36 space 89 irredundant decomposition Jacobi criterion 35 Jacobi ideal 301 jet 216 jet, non parametrized jet, projective 217

44

63, 129, 297 63

36

217

K3 surface 140,326 Kiihler manifold 175, 310 Klein’s icosaeder 81 Kodaira dimension 140, 172 Kodaira vanishing theorem 170 Kontinuitltssatz 358 Krull dimension 63 Krull intersection lemma 13 Kummer dimension 340 Kummer manifold 340 Kummer reduction 340 Lemma of de Rham Dolbeault

153 157

Index Leray 156 Nakayama 160 Poincare 152 Leray covering 155 Levi form 224,344 Levi problem 229,236 line bundle 118, 162 line bundle, nef 253 linear space 119 linear space, positive 233 negative 233 local cohomology 179 local complete intersection 104, 133 local decomposition lemma (of analytic sets) 36 embedding lemma 78 existence lemma (of analytic spectra) local Stein problem 240 locally free sheaf 48 irreducible space 27 m-base 207,212 manifold of class C 336 manifold, of general type 140 manifold, quasi-projective 343 manifold, simple 341 map, bimeromorphic 287,305 map, equidimensional 107 map, flat 111 map, holomorphic 27 factorization 32 quasi finite 15 reduction 59 map, meromorphic 288 map, projective 250, 334 maximal space 91 maximality property 72 maximalization (of a complex space) 93 meromorphic equivalence relation 207 bunch of tibres 208 tibration in X 208 polar set 207 regular 208 simple 211 meromorphic function 47 meromorphic map 211 graph 211 image 211 polar set 211 meromorphically dependent 211 0nF 212 Mittag-Lelller Condition 127, 164 modification 287

55

Subject module, injective 135 module, projective 110 Moishezon maps 293,334 Moishezon space 215,303,331,346 monoidal transformation 290 morphism of a formal complex space morphism of C-ringed space 25 Morse inequality 307 multiplicity 313 multiplicity for components 202 Nakayama lemma 14 negative (weakly) 273 neighbourhood, inlinitesimal 32 Neil’s parabola 26 Newton’s parabola modata 76 nilradical 22, 53 Noether finiteness lemma 15 normal algebra 18 point 28 semi point/space 91 space 28 normal bundle 117, 119 normal cone 122,295 normal crossings 3 12 normal flatness 3 13 normal sheaf 104 normalization 124 normalization of an algebra 18 a complex space 87 sheaf 48,91 semi 91 Nullstellensatz (Riickert) 52 ideal theoretic 52 Oka-Grauert principle Oka, theorem 45 Open Mapping Lemma Osgood map 28

162 54

p-form, holomorphic 101, 157 point Cohen Macaulay 71 double 26 factorial 27 irreducible 27 maximal 91 normal 28 of indeterminacy 44 reduced 27, 59 reducible 27 semi-normal 91

300

Index singular 27 smooth 27, 34, 65 positive 274 preparation theorem 12, 15 prime components (of an analytic principal divisor 84 Projan 126 projective bundle 124,242 projective jet space 217 projective space 169, 171 pseudoconcavity 279 pseudo-convex domain 236 q-complete 172, 343 q-complete sheaf 352 absolutely 357 q-concave 263,265,266 q-concave with comer 355 q-convex space 172,263,265,266,343 O-convex 266 q-complete 266 q-negative 274 q-positive 274 q-pseudoconvex function (strictly) quasi finite homomorphism 15 module 14

367

set)

36

263, 265

radical (ideal) 22 (ideal sheaf) 52 rational singularity 166, 311 reduction functor 59 map 59 of a complex space 59 relationally finite sheaf 41 relative l-form 103 relative tangent sheaf 103 Remmert reduction 218, 229, 346 resolution 150 resolution, acyclic 151 resolution, flabby 150 resolution, injective 135 resolution of singularities 311,312, 315 resolution, projective 110 Riemann Extension Theorem 74, 141 for locally pure dimensional complex spaces 80 normal complex spaces 80 Riemann surface 278 ringed space 24 C-ringed space 24 Ritt’s lemma 63

368 Rtickert Nullstellensatz ideal theoretic 52 Runge pair 239,261

Subject 52

Schwarz lemma 280 Segre cone 83 Segre embedding 125 Semi normal 201 semi normalization sheaf 91 map 93 separates locally 186 Serre duality 170 set of indeterminacy 47 analytic 35 analytically constructible 36 sheaf 20 sheaf, locally free 167 sheaf analytic 27 coherent 41, 159, 162 fine 152 finite 40 flat 111, 164 flabby 149 free 48 gap 73 locally free 48 normalization 48 of holomorphic functions 24 of local C-algebras 24 of meromorphic functions 47 reflexive 140 relationally finite 41 semi normalization 91 singular locus 49,65 smooth point 105 space Cohen Macauly 70 complex 27 C-ringed 25 determinantal 83 irreducible 89 maximal 28 normal 28 semi-normal 91 Weierstrass (model) 45, 50 Specan 122 spectra1 sequence 173 spectral sequence, convergence of 173 degeneration of 173 of Frolicher 175 of Grothendick 179 of Leray 177,253

Index spectrum, analytic 55 Stein compact set 130 Stein factorization 124, 163,207,212 Stein space 73, 158 submersion 105 SYZYkTY lemma (for finite modules) 67 theorem (for modules) 67 theorem (for coherent sheaves) 68 tangent bundle 103, 117 holomorphic 118, 168 tangent sheaf 103,168 tangent space 103, 118 theorem of Riemann-Roth 168 A 159 B 159 theorem of Cartan 82 Cartan-Oka 87 Chow and Kodaira 81 integral dependence 75 Oka 45 Oka-Cartan 56 Weierstrass division 11 generalized Weierstrag division 17 Weierstrass preparation 12, 17 Syzygy 67 thin 35 of order k 74 Todd class 168 topology in flabby cohomology Tor-module 110 torsion sheaf 60 coherence 50 total complex 262 transition function 115 Umkehrsatz unirational

classes

79 339

vanishing theorem of Grauert-Riemenschneider of Kawamata-Viehweg of Kodaira 247 of Le Potier 248 of Nakano 248 vector bundle, holomorphic almost positive GriRiths positive 246 k-ample 252

307 248

115, 167

267

Subject Nakano positive 246 vector field, holomorphic 103,339 volume (of cycle) 329

weak transform 314 weakly holomorphic function continuously 91

91

Index Weierstrass algebra 50 coverings 77 decomposition 11 division theorem 11, 17 generalized 17 preparation theorem 12, 17 space 45,50 Weil divisor 142 Whitney’s umbrella 76

369