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...

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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


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

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 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,

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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.

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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 (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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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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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

Th. Peternell

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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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


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

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

III.

Cohomology

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].

148

Th. Peternell

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~ 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.

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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.

III.

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.

164

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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165

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,

III.

167

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

168

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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Cohomology

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 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


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

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

Spaces

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).

188

G. Dethloff

and H. Grauert

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

IV.

Seminormal

Complex

Spaces

189

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-

190

G. DethloiT

and H. Grauert

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.

IV.

Seminormal

Complex

191

Spaces

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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and H. Grauert

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’
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,

242

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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),

V. Pseudoconvexity,

the Levi Problem

and Vanishing

Theorems

243

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,

the Levi Problem

and Vanishing

Theorems

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&)).

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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 . 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:

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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,

the Levi

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

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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.

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[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

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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

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

266 267

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

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

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

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

....

. . . . .

.. .. .. .. ..

. . . . .

. . . . .

. . . . .

.. .. .. .. ..

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

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

269 269 269 270 271 271 273 273 274 275 275 276 277 279 279 280

H. Grauert

260

Functions 3. Analytically Dependent Meromorphic 4. Modular Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References

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

VI. Theory of q-Convexity and q-Concavity

261

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

262

H. Grauert

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:

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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”: + 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) 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 . 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).

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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

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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,


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.

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[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.

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[Kau65] [Ker60] [I(0621

[KoS60] [Kr75]

[Le90] [Lel57] [Lel68] [Lie781 [Ma571 [Moi67]

[NS77] [Pot1691 [St861 [SC60/61] cu751 IV841 IY781

[Yo77]

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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


..

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

. . . . .

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

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.

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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.

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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

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

Several Complex Variables VI: Complex Manifolds (Encyclopaedia of Mathematical Sciences)

Analysis of several complex variables

Analysis of Several Complex Variables