Topics in Complex Geometry and Holomorphic Foliations. César Camacho, Hossein Movasati

Topics in Complex Geometry and Holomorphic Foliations César Camacho, Hossein Movasati

Contents 1 Introduction 5 1.1 The organization of the text........................... 6 1.2 Notations..................................... 7 2 Preliminaries 8 2.1 Local rings.................................... 8 2.2 Analytic varieties................................. 9 2.3 Embedding dimension.............................. 11 2.4 Algebraic varieties................................ 13 2.5 Analytic sheaves................................. 13 2.6 Restriction of sheaves.............................. 15 2.7 Homomorphism between sheaves........................ 16 2.8 Complementary notes.............................. 17 3 Cech cohomology 18 3.1 Cech cohomology................................. 18 3.2 Definitions..................................... 19 3.3 How to compute Cech cohomologies...................... 21 3.4 Cohomology of manifolds............................ 21 3.5 Line bundles................................... 22 3.6 Short exact sequences.............................. 22 3.7 Chern classes................................... 23 3.8 Serre Duality................................... 24 3.9 Complementary notes.............................. 25 4 Stein varieties 27 4.1 Stein varieties................................... 27 4.2 Coherent sheaves................................. 28 4.3 Some properties of Stein varieties........................ 29 4.4 Stein covering................................... 29 5 Strongly convex/plurisubharmonic functions 30 5.1 Tangent space................................... 30 5.2 Strongly convex functions............................ 31 5.3 Some properties.................................. 32 5.4 Strongly pseudoconvex domains......................... 33 5.5 Plurisubharmonic functions........................... 34 5.6 Complementary notes.............................. 37 2

6 Cohomological properties of pseudoconvex domains 39 6.1 A theorem of Grauert.............................. 39 6.2 Exceptional varieties............................... 42 6.3 Complementary notes.............................. 43 7 Positive and negative line bundles 44 7.1 Positive forms................................... 44 7.2 Positive and negative bundles in the sense of Grauert............. 45 7.3 Positive line bundle in the sense of Kodaira.................. 45 7.4 The equivalence of two definitions for line bundles.............. 46 7.5 Complementary notes.............................. 46 8 Kodaira vanishing theorem 47 8.1 Neighborhoods.................................. 47 8.2 Neighborhood of zero sections.......................... 48 8.3 The case of a Riemann surface......................... 49 8.4 Birkhoff-Grothendieck theorem......................... 50 9 Grauert vanishing theorem 51 9.1 Some lemmas................................... 51 9.2 Main vanishing theorem............................. 53 9.3 Restriction of line bundles............................ 53 10 Blow-up 56 10.1 Blow up...................................... 56 10.2 Blow-up of a singularity............................. 56 10.3 An embedding theorem............................. 57 10.4 Blow up along a submanifold.......................... 58 11 Foliated neighborhoods 60 11.1 Holomorphic foliations.............................. 60 11.2 Construction of functions............................ 61 11.3 Equivalence of transverse foliations....................... 62 11.4 Construction of line fields............................ 63 11.5 Construction of holomorphic foliations..................... 64 11.6 Grauert s theorem................................ 65 11.7 Arbitrary codimension.............................. 65 11.8 Proof of Theorem 11.5, codimension greater than one............. 66 11.9 Proof of Theorem 11.7.............................. 67 11.10Rational curves.................................. 67 11.11Complementary notes.............................. 68 A Remmert reduction 69 A.1 Proper mapping and direct image theorems.................. 69 A.2 Equivalence relations in varieties........................ 69 A.3 Cartan s theorem................................. 71 A.4 Stein factorization................................ 72 A.5 Remmert reduction................................ 72 A.6 Complementary notes.............................. 73 3

B Formal and finite neighborhoods 74 B.1 Formal and finite neighborhoods........................ 75 B.2 Some propositions................................ 76 B.3 Geometric interpretation............................. 78 B.4 Obstructions to formal isomorphism...................... 78 B.5 Calculating the obstruction........................... 80 B.6 Breaking Aut(ν)................................. 82 B.7 The case of a Riemann surface......................... 82 B.8 Complementary notes.............................. 83 4

Chapter 1 Introduction There are many theorems and statements in Algebraic Geometry which are of complex nature and are not covered in a classical Algebraic Geometry course. The present book pretends to partially fill this gap and discuss all the necessary materials for proving one of these theorems. This is namely the following theorem of Grauert and its generalizations to higher dimensions: Theorem. (H. Grauert [Gr62]) Let A be a Riemann surface of genus g embedded in a complex surface X and with negative and less than 4 4g self-intersection. Then a topological neighborhood of A is biholomorphic to the neighborhood of the zero section of the normal bundle of A in X. See Figure 1.1. A typical example of the above theorem is a rational curve A = P 1 obtained by a blow-up of the origin in C 2. If π is the blow-down map and ψ : C 2 R +, ψ(x, y) = x 2 + y 2, then A has strongly pseudo-convex neighborhoods {x X ψ π(x) < ɛ}. This picture, is also true in a more general context. We have many reasons to choose such a theorem and its generalizations to higher dimensional varieties A and X, as a basic theme of our book. First, it uses the usual topology of algebraic varieties over complex numbers. The same statement is trivially false with Zariski topology. Therefore, it cannot be stated in a purely algebraic context. The usage of formal neighborhood both in analytic or algebraic context would not imply the Grauert theorem. Second, in higher dimension the complex manifold A must be necessarily algebraic and so we need the machinary of algebraic geometry applied to A. This means that we will apply the machinery of algebraic and complex geometry in the same time. Third, in the way we learn its proof we learn many other classical tools and theorems. This includes, Cech cohomology, positive (ample) and negative bundles, Kodaira vanishing theorem and embedding and so on. Fourth, we learn how the machinery of holomorphic foliations can be used to prove statements in complex Algebraic Geometry. Because of this, the book is also is an introduction to the theory of holomorphic foliations on complex manifolds. We will deal with only foliation whose leaves are analytic varieties and so they will not have any dynamics. We don t prove all the theorems which appear in this text. Instead, we want that the reader feel himself comfortable with all such theorems, by explaining them in special cases and using examples. We provide the necessary references for missing proofs so that the curious reader iterates back to the original proofs. The basic idea is to prepare the reader for consulting the already published books like [GuI90, GuII90, GuIII90, GrRe79, GPR94] whenever he finds it necessary. The mathematics is growing fast and sometimes 5

FIGURE Figure 1.1: A biholomorphism of two neighborhoods it is necessary to learn the art of reading, understanding and using many theorems, without going into details of their proof. 1.1 The organization of the text The book is organized in the following way: In Chapter 2 we present all the necessary background in complex analysis and several complex variables in order to read and follow the book. In this chapter we explain concepts like germ of varieties, structural sheaf, Stein varieties, coherent sheaves and so on, and announce many theorems related to all these concepts. Whenever it is possible, we explain theorems by examples and particular cases and give the idea of the proof. Chapter 3 is devoted to the basics of sheaf cohomology and in particular the construction of Cech cohomology. It is aimed to prepare the reader as fast as possible in order to feel himself comfortable with the notation H i (X, S). We also explain line bundles and Chern classes in the framework of Cech cohomology. In Chapter 4 we review basic facts and theorems on Stein varieties. Chapter 5 is devoted to pseudoconvex domains. For some technical reasons, we have preferred to work with C 2 convex functions instead of C 2 plurisubharmonic functions. A convex function carries just the convexity information of its level varieties and is easy to handle. In Chapter 6 we prove that the cohomologies of strongly pseudoconvex domains with coefficients in a coherent sheaf are finite dimensionals. The same is also true for compact analytic varieties instead of strongly pseudoconvex domains. Using Remmert reduction theorem in Appendix A this leads us to the notion of exceptional varieties. In Chapter 7 we apply the machinery of pseudoconvex domains to the neighborhoods of zero sections of line bundles and we get the notion of negative and positive vector bundles. We prove that Grauert positivity of line bundles is the same as Kodaira s positivity and we give a proof of Kodaira vanishing theorem. Chapter 9 is dedicated to one of the most important cohomological properties of strongly pseudoconvex domains. This is namely Theorem 9.2 which states that if a coherent sheaf vanishes enough along the exceptional divisor of a strongly pseudoconvex domain then its cohomologies are zero except the 0-th cohomology. In Chapter 10 we review the notion of blow-up and we see that an exceptional variety can be obtained by a usual blow-up of a singularity. We also explain the notion of a 6

blow-up along a divisor. This plays a fundamental role in the study of neighborhoods of varieties with codimension stricktly bigger than one. In Chpater 11 we start the study of holomorphic foliations in neighborhoods of varieties. We first consider the most simple foliations which are those transversal to the variety. This will be used in order to give a geometric proof of Grauert s Theorem. We also study foliations with tangencies. In Appendix A we discuss Remmert reduction theorem. Using this one can show that strongly pseudoconvex domains are the point modification of Stein varieties. This leads to the notion of exceptional or negatively embedded varieties. In Appendix B we introduce the notion of a finite neighborhood and the extension problems in this context. The content of this Appendix are part of the machinary which Grauert used to prove his theorem and we do not need them for our proof. However, it can be used as a nice collection of exercises to the concepts introduced in Chapter 2 and Chapter 3. 1.2 Notations The letter A is mainly used to denote an analytic variety, complex manifold or an algebraic variety. It is usually embedded in another variety X. In this case we denote by (X, A) a neighborhood of A in X. This notation means that, if necessary, we can take such a neighborhood smaller. In other words, we think of (X, A) as a neighborhood of A in X, however, when (X, A) appears in a statement or property, such a neighborhood can be smaller. For instance a function f : (X, A) C is defined in a neighborhood U of Ain X. It satisfies a property P if there is another neighborhood U U of A in X such that f U satisfies P. We use the same letter i for the complex number 1 and for indexing; being clear in the text the distinction between them. For a C-vector space V we denote by V the dual vector space, that is, the vector space of all linear maps V C. For topological spaces A and B we write A B to denote that A is relatively compact in B, i.e. the closure of A in B is compact in B. 7

Chapter 2 Preliminaries In this chapter we review the notion of analytic and algebraic varieties, coherent sheaves and basic facts about Stein varieties. The main references for contents of this chapter are [GuI90, GuII90, GuII90, GrRe79]. 2.1 Local rings For a topological space X and a point x X we denote by (X, x) a neighborhood of x in X. This means that in our statements and arguments we fix a neighborhood of x in X but we can take it smaller if it is necessary. A C-algebra is a commutative ring containing the field C as a subring, with 1 C as the identity element of the ring. A homomorphism between two C-algebras is a ring homomorphism that induces the identity mapping on the subfield C. An example of C-algebra we use in this text is: O C n(u), the space of holomorphic functions in an open subset U of C n. O C n,x, the ring of germs of holomorphic functions in a neighborhood of x in C n ; O C n,x = {f O C n(u) for some open set U in C n, x U}/ where f 1 O C n(u 1 ), f 2 O C n(u 2 ), x U 1 U 2, f 1 f 2 f 1 = f 2 in some open set U 3 U 1 U 2, x U 3. The maximal ideal of O C n,x is M C n,x := {f O C n,x f(x) = 0}. We have O C n,x/m C n,x = C and so O C n,x is a local ring. One can consider O C n,x as the ring of convergent power series i a i (y x) i, where i = (i 1, i 2,..., i n ) runs through (N {0}) n and (y x) i = (y 1 x 1 ) i 1 (y 2 x 2 ) i2 (y n x n ) in. A germ of an analytic subvariety (X, x) of (C n, x) is given by f 1 = 0, f 2 = 0,..., f r = 0, where f 1, f 2,..., f r M C n,x. The defining ideal of the germ (X, x) is defined to be I X,x := {f O C n,x f X = 0}; O X,x := O C n,x/i X,x, the germs of holomorphic functions in a neighborhood of x on X. One can view O X,x as a set of functions f (X, x) C which extend to holomorphic functions in (C n, x); 8

M X,x := {f O X,x f(x) = 0}, the maximal ideal of O X,x ; M k X,x, the sub C-algebra of M X,x generated by Π k i=1 g i, g i M X,x. We collect all the necessary statements on C-algebras which we need in the following proposition: Proposition 2.1. The following statements are true: 1. M m C n,0 is exactly the set of holomorphic functions with the leading term (in the Taylor series) of degree greater than or to equal m; 2. O C n,0 is a Noetherian ring, i.e. every ideal in O C n,0 has a finite basis; 3. k=1 Mk C n,0 = {0}; Proof. We first prove the nontrivial part of the statement 1., i.e. if f O C n,0 with the leading term of degree m then f M m C n,0. The proof is by induction on n. The case n = 1 is trivial. By a linear change of coordinates we can assume that f is regular in the variable x 1, i.e. f(x 1, 0,..., 0) is not identically zero. Let us write f = f(x 1, 0,..., 0) = x l 1h(x 1 ), h(0) 0. By Weierstrass preparation theorem (see [GuII90] Theorem A4) we can write f = u.(x l 1 + a 1 x l 1 1 + + a l 1 x 1 + a l ), where a 1, a 2,..., a l are holomorphic functions in x 2, x 3,..., x n with a i (0) = 0, i = 1, 2,..., l and u is a holomorphic function in x 1, x 2,..., x n with u(0) 0. Since f and f u have the same leading term up to multiplication by a constant, it is enough to prove that x l 1 + a 1x1 l 1 + + a l 1 x 1 + a l M m C n,0. By our hypothesis l m and the degree of the leading term of a i is bigger than or equal m (l i). Now our assertion follows from the hypothesis of induction for n 1. The statement 2. can be found in [GuII90] Theorem A8. The statement 3. is a direct consequence of the first part (or the second part in the general context of local rings). Remark 2.1. For a holomorphic function f defined in some open subset U of C n we will denote by f x its classes in O C n,x, x U. Many times we use again f instead of f x and simply say f in O C n,x. We will also use this simplification later for sheaves on analytic varieties. 2.2 Analytic varieties In order to define an arbitrary variety we have to define the affine varieties and then try to glue them. Definition 2.1. Let D be an open domain in some C n. A closed subset X of D is called an analytic subvariety of D if for any point a X there is an open neighborhood U of a in D and holomorphic functions f 1, f 2,..., f r in D such that X U := {x U f 1 (x) = f 2 (x) = = f r (x) = 0}. 9

U 2 3 V={x y =0} C 2 U U V=D C V={xy=0} C 2 Figure 2.1: Analytic variety We also call X an (analytic) affine variety. We look at X as a topological space equipped with a sheaf O X of C-algebras (2.1) O X (U) := {f : U C f O X,x }, U open subset of X which is called the structural sheaf of X. The definition (2.1) means that O X (U) contains all complex valued functions f : X C such that for all x X we can find a holomorphic finction f : (D, x) C such that f restricted to X D is equal to f. Let X and Y be two affine varieties. A continuous map τ : X Y is called holomorphic if the pull-back of functions, given by τ (f) = f τ, defines a map τ from O Y,τ(x) into O X,x, which is a morphism of C-algebras for all x X. The map τ is called a biholomorphism if there is a holomorphic map τ : Y X such that τ τ and τ τ are identity maps respectively on X and Y. Definition 2.2. Let X be a second-countable Hausdorff topological space and C X be the sheaf of complex valued continuous functions on X. We say that X with a sheaf of C- algebras O X C X is an analytic variety if every point of X has an open neighborhood U such that (U, O U ) is isomorphic to a (V, O V ), for some affine variety V, i.e. there is a homeomorphism ψ : U V such that ψ : O V O U, ψ (f) = f ψ is an isomorphism of sheaves of C-algebras. In Figure 2.1 we have roughly depicted a variety X with three points p i, i = 1, 2, 3 on it. Let X be a variety. For every point x X there exist an open set U around x, V a closed analytic subset of an open domain D in some C n and a homeomorphism ψ : U V which induces an isomorphism between O V and O U. A rough picture of this definition is depicted in Figure 2.1 This is called a chart around x and we denote it simply by ψ : U V D C n, a chart around x. Given two such charts ψ α : U α V α D α C nα and ψ β : U β V β D β C n β around x, the first is called a subchart of the second if there is an embedding em : (D α, ψ α (x)) (D β, ψ β (x)) such that ψ β = em ψ α. They are called equivalent if one is a subchart of the other and n α = n β. In this case the map em is a biholomorphism. This is an equivalence relation. Note that n, the dimension of D, differs chart by chart. For this reason it is better to define a variety using the language of C-algebras rather than the formal definition by charts and transition functions, for instance see [GuII90], Definition B16. 10

Definition 2.3. Let X be an analytic variety and Y a subset of X. We say that Y is an (analytic) subvariety of X if for every y Y there exists an open set U, y U and f 1, f 2,..., f r O X (U) such that U Y = {x U f 1 (x) = = f r (x) = 0}. It is easy to see that Y is an analytic variety in such a way that the inclusion Y X is analytic. The following proposition is Theorem B14 of [GuII90]. We give its proof because it is instructive. Proposition 2.2. Let (X, 0) (C n, 0) and (Y, 0) (C m, 0) be the germs of two affine varieties. Every holomorphic map τ : (X, 0) (Y, 0) is induced by a holomorphic map from (C n, 0) to (C m, 0). Proof. We have a morphism τ : O Y,0 O X,0 of C-algebras. Since it sends the units to units, it sends the maximal ideal M Y,0 into the maximal ideal M X,0 and so τ (M k Y,0 ) M k X,0, k = 1, 2,.... Let us denote the coordinate functions of (Cm, 0) by y 1, y 2,..., y m ( M Y,0 ) and define f i := τ (y i ). The map f : (C n, 0) (C m, 0) defined by f = (f 1, f 2,..., f m ) is the desired map. We consider the diagram (2.2) O C m,0 j O Y,0 f O C n,0 ı τ O X,0 where ı, j are the canonical maps. We observe that the maps ı f, τ j : O C m,0 O X,0 coincide on polynomials in y i s. For an arbitrary k N, every g O C m,0 can be written as g 1 + g 2, where g 1 is a polynomial in y i s and g 2 M k C m,0 (here we have used Proposition 2.1,1). Therefore (ı f τ j)(g) M k X,0 for all k = 1, 2,.... Now Proposition 2.1, 3 implies that ı f = τ j. I Y,0 is a subset of the kernel of τ j and so of the kernel of ı f. This implies that whenever a g O C m,0 is zero on (Y, 0) then it is zero on f(x, 0) and so f(x, 0) (Y, 0). Since τ, f : X Y induce the same map τ = f, the proof is finished. 2.3 Embedding dimension For a germ of an analytic variety (X, x) we set Tx X := M X,x /M 2 X,x, the cotangent space of X at x; T x X := the dual of T x X. T x X is called the tangent space of X at x. A holomorphic map f : (X, x) (Y, y) induces the map T x f : T f(x) Y T x X It would be instructive to check that the definition of the tangent space in the case where X is smooth coincides with the usual definition of tangent space with differential of transition 11

U U variety X U a subchart of U Figure 2.2: Embedding dimension maps of X. In the singular case the bundle of tangent spaces {T x X, x X} has a natural structure of an analytic variety (see [GuII90] J) and so we can define in a natural way the notion of a vector field in a variety. Let X be a variety and x X. Using some chart around x we can identify the germ of the singularity (X, x) as an analytic subspace of C n, for some n. The smallest integer n with this property is called the embedding dimension of X at x and is denoted by emb x X. In Figure 2.1 we have roughly depicted a germ of variety (X, x) whose embedding dimension is 2 and not 3. The following proposition can be also found in [GrRe84] p. 115. Proposition 2.3. We have emb x X = dim C Tx X. More precisely, for a point x X if x 1, x 2,..., x m M X,x form a basis for Tx X then ψ : (X, x) C m given by ψ = (x 1, x 2,..., x m ) is a chart map around x whose associated affine space is of dimension dim C Tx X. Every two charts with the dimension of the affine spaces equal to dim C Tx X are equivalent and every chart has a subchart whose associated affine space is of dimension m = dim C Tx X. Proof. Since our statement is local, we can assume that X (C n, 0) and x = 0. Let λ : I X,0 M C n,0/m 2 C n,0 be the canonical map. Its coimage (M C n,0/m 2 C n,0)/im(λ) is isomorphic to M X,0 /M 2 X,0. Therefore if r := dimim(λ), m := dim CM X,0 /M 2 X,0 then r + m = n. Let f 1, f 2,..., f r I X,0 such that their image by λ form a C-basis for Im(λ). This means that the linear part of the map f = (f 1, f 2,..., f r ) has the maximum rank r. Therefore f is a regular map and N = {x (C n, 0) f(x) = 0} is a smooth complex submanifold of (C n, 0) and dim C N = m. But we have also X N. We have proved that each chart has a subchart whose associated affine space is of dimension m = dimt0 X. Let us be given two charts for (X, 0) whose associated affine spaces are of dimension T0 X. This means that (X, 0) is embedded in two different ways in (Cm, 0), say X 1, X 2. By Proposition 2.2 the map induced by the identity ı : (X 1, 0) (X 2, 0) can be extended to a holomorphic map f : (C m, 0) (C m, 0). Using the argument of the previous paragraph and the dimension condition we have (2.3) T 0 C m = T 0 X i, i = 1, 2 But we know that ı : O X2,0 O X1,0 is an isomorphism of C-algebras and so it induces an isomorphism M X2,0/M 2 X 2,0 M X 1,0/M 2 X 1,0. The equality (2.3) and the inverse mapping theorem imply that f is a biholomorphism. 12

Let x 1, x 2,..., x m M X,0 form a basis for T x X. The map ψ : (X, x) C m given by ψ = (x 1, x 2,..., x m ) is a holomorphic map. Take an arbitrary embedding of (X, 0) in (C m, 0). According to Proposition 2.2 ψ is obtained by restriction of a holomorphic map f : (C m, 0) (C m, 0). Since T 0 X = T 0 Cm, the map T 0 f : T 0 Cm T 0 Cm is an isomorphism and so f is a biholomorphism. This proves that ψ is an embedding. Proposition 2.4. For a holomorphic map f : (X, x) (Y, y) if T x f is surjective then f is an embedding. Proof. Let σ be the canonical map M Y,y T y Y and n = dim C T x X. Choose g 1, g 2,..., g n M Y,y such that their image by T x f σ form a basis of T x X. The map g = (g 1, g 2,..., g n ) : (Y, 0) (C n, 0) has the following property: g f is an embedding of X in (C n, 0), for this see the first part of Proposition 2.3. We identify X with its image by g f in (C n, 0). The set X 1 := f(x) is an analytic variety because it coincide with g 1 (X). The inverse of f : X X 1 is given by g. 2.4 Algebraic varieties All the terminology of the previous sections can be introduced in algebraic context. For simplicity we consider the field of complex numbers but the whole discussion is valid for an algebraically closed field k. The reader is referred to [Ha77] for further details on algebraic varieties. We define an affine variety V in C n to be the zero set of polynomials in C[x 1, x 2,..., x n ] and we denote by O V (V ) the restriction of polynomials in C[x 1, x 2,..., x n ] to V. In V we consider the Zariski topology, i.e. the the complement of an open set is given by zeros of the elements in O V (V ). A function f : U C in an open subset U of V is regular if there are f 1, f 2 O V (V ) such that f 2 does not have zeros in U and f = f 1 f 2. We denote by O V (U) the C-algebra of regular functions in U. We have defined the algebraic structural sheaf O V of V. An algebraic variety (X, O X ) is a topological space equipped with a sheaf of C-algebras such that every point of X has a neighborhood U such that (U, O X U ) is isomorphic to some affine variety with its canonical structural sheaf. The first example of algebraic varieties are projective varieties P n. A closed subvariety of a projective variety is called a projective variety. An algebraic variety X is an analytic variety in a canonical way. One usually denotes by X an the underlying analytic variety. We may keep in mid the following classical theorem Theorem 2.1. (Chow s theorem) Any closed analytic suvariety X of P n is an undrlaying variety of an algebraic variety Y P n, that is, Y an = X. For a proof of the above theorem see [GrHa78]. This is a part of a bigger theorem called Serre s GAGA principle, see [Se56] and also Grothendieck s articles [Gro58], [Gro69]. 2.5 Analytic sheaves For an analytic variety X we have used the structural sheaf O X. It is the collection of all holomorphic functions defined on some open subset of X. We have also used the stalk of O X over a point x X. Some other examples of sheaves are the following: 13

For a holomorphic vector bundle π : E X over a complex manifold X we can talk about the sheaf of holomorphic sections of E, i.e to each open set U X we can associate the O X (U)-module Ẽ(U) := {s : U E s is holomorphic and π s = id}. For a complex manifold X we have the sheaf of holomorphic vector fields which is the sheaf of holomorphic sections of the tangent bundle of M. For a complex manifold X we can talk about the sheaf Ω i of holomorphic differential i-forms in X. This can be interpreted as the sheaf of holomorphic sections of the vector bundle T X T X, i times. In a local chart U X with coordinate functions z 1, z 2,..., z n : U C one can write: Ω i (U) = { f j1,j 2,...,j i dz j1 dz j2 dz ji f j1,j 2,...,j i O X (U)}. j 1,j 2,...,j i Let Y be a subvariety of an analytic variety X. The sheaf M Y defined in the following way: of ideals of Y is M Y (U) = {f O X (U) f Y = 0}, U X open. For a natural number n we can define the sheaf O n X = O X O X O X, n times. Its elements in an open set U are the n-tuples (f 1, f 2,..., f n ), f i O X (U), i = 1, 2,..., n. It is natural to extend the notion of sheaf S of holomorphic forms or vector fields to the case of analytic varieties. First of all note that in all these examples we can multiply the elements of O X (U) with the elements of S(U), i.e. S is an analytic sheaf. There are many other sheaves that are not analytic. For instance For a ring R and topological space X the sheaf of constants in X with coefficients in R associates to each open set in X the ring R. One usually denotes again by R the sheaf of constants. The sheaf of real valued continuous functions on a topological space X. The sheaf O X of invertible holomorphic functions on an analytic variety X: O X(U) := {f O X (U) x U, f(x) 0}, U open in X. In all these examples we note that: First, we have associated to open subsets of X some algebraic structure. Second we have the restriction maps to smaller open subsets. This leads us to the following abstract notion of a sheaf: Definition 2.4. A sheaf S of abelian groups on a topological space X is a collection of abelian groups S(U), U open subset of X For U 1 U 2 open sets in X, it is equipped with morphisms r U2,U 1 abelian groups such that they satisfy: : S(U 2 ) S(U 1 ) of 14

1. For U an open set of X, r U,U is the identity map; 2. For open sets U 1 U 2 U 3 X we have r U2,U 1 r U3,U 2 = r U3,U 1 ; 3. If for U = i I U j, U and U j s open subsets of X, and f j S(U) we have r Ui,U j U i (f i ) = r Uj,U j U i (f j ) then there is a unique element f S(U) such that r U,Ui (f) = f i. r U2,U 1 is called the restriction map from U 2 to U 1. The stalk S x of S at a point x X is defined in the following way: S x = {f S(U), for some open set U in X, x U}/ where f 1 S(U 1 ), f 2 S(U 2 ), x U 1 U 2, f 1 f 2 f 1 = f 2 in some open set U 3 U 1 U 2, x U 3. Note that the uniqueness condition 3 in the above definition implies that if the restriction of g S(U) to each U j is zero (of the corresponding abelian group) then g must be zero. Remark 2.2. We will usually use f U1 instead of r U2,U 2 (f); being clear that f is an element of S(U 2 ) for some open set U 2 which contains U 1. By definition S( ) = and r U, =. Throughout the text, for a given sheaf S over a topological space X, when we write x S we mean that x is a section of S in some open neighborhood in X or it is an element in a stalk of S over X, being clear from the text which we mean. We now define an analytic sheaf. Definition 2.5. An analytic sheaf S on an analytic variety X is a collection of O X (U)- modules S(U) for all open sets of X such that 1. The groups (S(U), +) form a sheaf of abelian groups; 2. The restriction maps are morphism of O X -modules. 2.6 Restriction of sheaves Let us be given an analytic sheaf S on X and let Y be a subvariety of X. In this section we introduce the notion of restriction of elements S to Y. We can define the sheaf of abelian groups S Y : S Y (U) = {f S(V ), for some open set V in X, V Y = U}/ where f 1 S(V 1 ), f 2 S(V 2 ), V 1 Y = V 2 Y = U, f 1 f 2 f 1 = f 2 in some open set V 3 V 1 V 2, V 3 Y = U. 15

The obtained sheaf on Y is not analytic. Now consider the sheaf M Y S (2.4) (M Y S)(U) := {f S(U) x U Y, f 1 S x, f 2 M Y,x, f = f 1 f 2 in S x } For all examples of analytic sheaves that we have seen the restriction of elements of M Y S to Y is zero. Therefore, we may define the quotient sheaf S/M Y S as a candidate for the restriction of S to Y. This sheaf is zero outside Y, i.e S(U) = {0} for open sets U in X which do not intersect Y. We finally get the following definition Definition 2.6. The structural restriction of of an analytic sheaf S on X to its subvariety Y is defined to be res(s) = S Y = (S/M Y S) Y. It is left to the reader to check that S Y is an analytic sheaf in a canonical way. Remark 2.3. Every definition and construction with sheaves has a local nature. Sometimes such a definition coincides with the trivial definition or construction. For instance for two sheaves S 1 S 2 on X we may define (S 2 /S 1 )(U) = S 2 (U)/, f 1 f 2 x U, (f 1 f 2 ) = 0 in S 2,x /S 1,x It turns out that this coincides with the trivial definition (S 2 /S 1 )(U) = S 2 (U)/S 1 (U). This is not always the case. For instance, consider the the definition of M Y S in (2.4). 2.7 Homomorphism between sheaves Definition 2.7. A homomorphism f between two sheaves S and S of abelian groups on a topological space is X is a collection of group homomorphisms f U : S(U) S (U), U X open which is compatible with the restriction maps, i.e. for U 1 U 2 open sets in X, we have f U1 r U2,U 1 = r U2,U 1 f U2. A homomorphism f is an isomorphism if all the f U s are isomorphisms. In the case where X, S and S are analytic, we say that f is analytic if each f U is a morphism of O X (U)- modules. Definition 2.8. Let f : S S be a homomorphism of sheaves of abelian groups. The kernel Kernel(f) and image Image(f) of f are defined in the following way: Kernel(f)(U) := {a S(U) x U, a x kernel(f x : S x S x)} Image(f)(U) := {a S (U) x U, a x Image(f x : S x S x)} f is called injective (resp.surjective) if Kernel(f) = 0 (resp. Image(f) = S ). In this case we write S f S 0 ( resp. 0 S f S ) It is easy to see that Kernel(f) and Image(f) are sheaves of abelian groups, respectively analytic sheaves if X, S and S are analytic. According to the properties of restriction maps, the definition of Kernel(f) does not change if we define Kernel(f)(U) = kernelf U. But this is not the case for Image(f). This is another example for the situation discussed in Remark 2.3. 16

Example 2.1. Let X be an analytic variety and Y be a subvariety of X. We have the following exact sequence i r 0 M Y O X Õ Y 0. where i is the inclusion and r is the restriction map. Here ÕY is the trivial extension of the sheaf O Y to X, i.e. Õ Y (U) = {0} if U does not intersects Y and = O Y (U Y ) if U intersects Y. We usually omit the tilde and simply write O Y to denote ÕY. Example 2.2. For an analytic variety X we have the following exact sequence: 0 Z i O X e O X 0 where i is the inclusion map and e(f) = e 2πif. Example 2.3. Let S be an analytic sheaf on X and [f ij ] i=1,2...,n,j=1,2,...,m be a matrix with entries in O X (X). Then we have an analytic sheaf homomorphism f : S m S n, s [f i,j ]s tr, s = (s 1, s 2,..., s m ) S m 2.8 Complementary notes The complementary material to section 2.1 can be Weierstrass preparation and division theorems, Theorem 4A,5A of [GuII90]. One can also include sections A,B of [GuIII90] for more details on the notion of a sheaf. Section I,J of [GuII90] are devoted to the tangent space of an analytic variety and can be included in section 2.1. Particularly Theorem 16I claims that the both notions of tangent space there and here are the same. This will be useful for section B.4 if one wants to follow the arguments in a general case of an embedded A in a variety X. One can also include the notion of linear spaces over varieties from the survey [GPR94] chapter 2 section 3. Section M of [GuIII90] covers various equivalent definitions of Stein spaces and fill the proof of the equivalent definitions of a Stein variety stated in the beginning of section 4.1. 17

Chapter 3 Cech cohomology In this Chapter we explain the construction of Cech cohomology of sheaves. The categorical approach to sheaf cohomology and the way that it is used in mathematics, show that in most of occasions we need only to know a bunch of axioms for the sheaf cohomology and not its concrete construction. However, in some other occasions, mainly when we want to formulate some obstructions, we obtain elements in some Cech cohomologies and so just axioms of sheaf cohomology would not work. Therefore, we introduce some properties of sheaf chomology which can be taken as axioms and we also explain its explicit construction using Cech cohomology. We assume that the reader is familiar with sheaves of abelian groups on topological spaces, see for instance Chapter 2. The reader who is interested in a more elaborated version of this section may consult other books like [2], Section 10, [7] Section 4.3. 3.1 Cech cohomology A sheaf S of abelian groups on a topological space X is a collection of abelian groups S(U), U X open with restriction maps which satisfy certain properties, for instance see Chapter 2. In particular S(X) is called the set of global sections of S and the following equivalent notations S(X) = Γ(X, S) = H 0 (X, S) is used. It is not difficult to see that for an exact sequence of sheaves of abelian groups (3.1) 0 S 1 S 2 S 3 0. we have 0 S 1 (X) S 2 (X) S 3 (X) and the last map is not necessarily surjective. In this section we want to construct abelian groups H i (X, S), i = 0, 1, 2..., H 0 (X, S) = S(X) such that we have the long exact sequence 0 H 0 (X, S 1 ) H 0 (X, S 2 ) H 0 (X, S 3 ) H 1 (X, S 1 ) H 1 (X, S 2 ) H 1 (X, S 3 ) H 2 (X, S 1 ) 18

that is in each step the image and kernel of two consecutive maps are equal. Let us explain the basic idea behind H 1 (X, S 1 ). The elements of H 1 (X, S 1 ) are considered as obstructions to the surjectivity of H 0 (X, S 2 ) H 0 (X, S 3 ). This map is not surjective, however, we can look at an element f S 3 (X) locally and use the surjectivity of S 2 S 3. We fix a covering U = {U i, i I} of X such that the exact sequences corresponding to global section of (3.1) over U i hold, that is, 0 S 1 (U i ) S 2 (U i ) S 3 (U i ) 0. We take f i S 2 (U i ), i I such that f i is mapped to f under S 2 (U i ) S 3 (U i ). This means that the elements f j f i, which are defined in the intersections U i U j s, are mapped to zero and so there are elements f ij S 1 (U i U j ) such that f ij is mapped to f j f i. It is easy to check that different choices of f i s lead us to elements f ij + f j f i, where f i S 1 (U i ). This lead us to define H 1 (U, S 1 ) to be the set of (f ij, i, j I).. 3.2 Definitions Let X be a topological space, S a sheaf of abelian groups on X and U = {U i, i I} a covering of X by open sets. In this paragraph we want to define the Cech cohomology of the covering U with coefficients in the sheaf S. Let U p denotes the set of p-tuples σ = (U i0,..., U ip ), i 0,..., i p I and for σ U p define σ = p j=0 U i j. A p-cochain f = (f σ ) σ U p is an element in C p (U, S) := σ U p H 0 ( σ, S) Definition 3.1. Let π be the permutation group of the set {0, 1, 2,..., p}. It acts on U p in a canonical way and we say that f C p (U, S) is skew-symmetric if f πσ = sign(π)f σ for all σ U p. The set of skew-symmetric cochains form an abelian subgroup C s (U, S). For σ U p and j = 0, 1,..., p denote by σ j the element in U p 1 obtained by removing the j-th entry of σ. We have σ σ j and so the restriction maps from H 0 ( σ j, S) to H 0 ( σ, S) is well-defined. We define the boundary mapping We have to check that p+1 δ : Cs p (U, S) Cs p+1 (U, S), (δf) σ = ( 1) j f σj σ Proposition 3.1. The above map is well-defined, that is, if f C p (U, S) is skewsymmetric then δf is also skew-symmetric. Proof. For simplicity, and without loss of generality we can assume that π is permutation of 0 and 1. j=0 19

From now on we identify σ with i 0 i 1 i p and write a p-cochain as f = (f i0 i 1 i p, i j I). For simplicity we also write where î j means that i j is removed. Proposition 3.2. We have Proof. Let f C p s (U, S). We have p+1 (δf) i0 i 1 i p+1 := ( 1) j f i0 i 1 i j 1 î j i j+1 i p+1 (δ 2 f) i0 i 1 i p+2 = = p+2 j=0 δ δ = 0. ( 1) j (δf) i0 i 1 î j i p+2 j=0 p+2 p+2 j=0 k=1, k j ( 1) j ( 1) kf i0 i 1 î k i p+2 = 0 where k = k if k < j and k = k 1 if k > j. See [2] Proposition 8.3. Now 0 C 0 s (U, S) δ C 1 s (U, S) δ C 2 s (U, S) δ C 3 s (U, S) δ can be viewed as cochain complexes, i.e. the image of a map in the complex is inside the kernel of the next map. Definition 3.2. The Cech cohomology of the covering U with coefficients in the sheaf S is the cohomology groups H p (U, S) := Kernel(Cp s (U, S) δ Cs p+1 (U, S)). Image(Cs p 1 (U, S) δ Cs p (U, S)) The above definition depends on the covering and we wish to obtain cohomologies H p (X, S) which depends only on X and S. We recall that the set of all coverings U of X is directed: Definition 3.3. For two coverings U i = {U i,j, j I i }, i = 1, 2 we write U 1 U 2 and say that U 1 is a refinement of U 2, if there is a map from φ : I 1 I 2 such that U 1,i U 2,φ(i) for all i I 1. For two covering U 1 and U 2 there is another covering U 3 such that U 3 U 1 and U 3 U 2. It is not difficult to show that for U 1 U 2 we have a well-defined map H p (U 2, S) H p (U 1, S). which is obtained by restriction from U 2,φ(i) to U 1,i. For details see [2], Lemma 10.4.1 and Lemma 10.4.2. Definition 3.4. The Cech cohomology of X with coefficients in S is defined in the following way: H p (X, S) := dir lim U H p (U, S). We may view H p (X, S) as the union of all H p (U, S) for all coverings U, quotiented by the following equivalence relation. Two elements α H p (U 1, S) and β H p (U 2, S) are equivalent if there is a covering U 3 U 1 and U 3 U 2 such that α and β are mapped to the same element in H p (U 3, S). 20

3.3 How to compute Cech cohomologies Definition 3.5. The covering U is called acyclic with respect to S if U is locally finite, i.e. each point of X has an open neighborhood which intersects a finite number of open sets in U, and H p (U i1 U ik, S) = 0 for all U i1,..., U ik U and p 1. Theorem 3.1. (Leray lemma) Let U be an acyclic covering of a variety X. There is a natural isomorphism H µ (U, S) = H µ (X, S). See [7] Theorem 4.41 and Theorem 4.44. A full proof can be found in the book of Godement 1958. See also [GrHa78] page 40. For a sheaf of abelian groups S over a topological space X, we will mainly use H 1 (X, S). Recall that for an acyclic covering U of X an element of H 1 (X, S) is represented by f ij S(U i U j ), i, j I f ij + f jk + f ki = 0, f ij = f ji, i, j, k I It is zero in H 1 (X, S) if and only if there are f i S(U i ), i I such that f ij = f j f i. Remark 3.1. For sheaves of abelian groups S i, i = 1, 2,..., k over a variety X we have: H p (X, i S i ) = i H p (X, S i ), p = 0, 1,.... 3.4 Cohomology of manifolds The first natural sheaves are constant sheaves. For an abelian group G, the sheaf of constants on X with coeficients in G is a sheaf such that for any open set it associates G and the restriction maps are the identity. We also denote by G the corresponding sheaf. Our main examples are (k, +), k = Z, Q, R, C. For a smooth manifold X, the cohomologies H i (X, G) are isomorphic in a natural way to singular cohomologies and de Rham cohomologies, see respectively [7] Theorem 4.47 and [2] Proposition 10.6. We will need the following topological statements. Proposition 3.3. Let X be a topological space which is contractible to a point. Then H p (X, G) = 0 for all p > 0. This statement follows from another statement which says that two homotopic maps induce the same map in cohomologies. Proposition 3.4. Let X be a manifold of dimension n. Then X has a covering U = {U i, i I} such that 1. all U i s and their intersections are contractible to points. 2. The intersection of any n + 2 open sets U i is empty. Using both propositions we get an acyclic covering of a manifold and we prove. Proposition 3.5. Let M be an orientable manifold of dimension m. 1. We have H i (M, Z) = 0 for i > m. 21

2. If M is not compact then the top cohomology H m (M, Z) is zero. 3. If M is compact then we have a canonical isomorphism H m (M, Z) = Z given by the orientation of M. If M is a complex manifold of dimension n, then it has a canonical orientation given by the orientation of C and so we can apply the above proposition in this case. Note that M is of real dimension m = 2n. 3.5 Line bundles Let X be an analytic variety and OX be the sheaf of invertible holomorphic functions in X. Elements of the cohomology H 1 (X, OX ) are called line bundles in X. It corresponds to the geometric notion of a line bundle as follow. Let us take the covering U i, i I and assume that g H 1 (X, O) is given by g ij O (U i U j ). We can glue the data U i C, i I according to the biholomorphisms f ij : (U i U j ) C (U i U j ) C (3.2) f ij ((x, v)) = (x, g ij v) and obtain an analytic variety L with a canonical projection π : L X whose fibers L x := π 1 (x) are isomorphic to C. The fact that δ(g) = 0, ensures us that the gluing process is consistent. An equivalent way of saying the equality (3.2) is the following: we take local holomorphic without zero sections of L, namely f i : U i L, i I, and we have f j = g ij f i, i, j I A meromorphic section of a line bundle L = {g ij } is a collection { f i, i I} of meromorphic functions f i in U i, such that f j = gij 1 f i, i, j I. In particular, f i and f j have the same zero and polar set in their common domain of definition U i U j. See Figure 3.1. Note that f i f i = f j f j in U i U j and so we get a global section of L in a geometric sense. Exercise 3.1. For an analytic sheaf S and a line bundle L = {g ij } on X show that a cocyle in H 1 (X, S OX L) is represented by {s ij, s ij O X (U i U j )} such that 3.6 Short exact sequences s ij + s jk g jk + s ki g ji = 0, i, j, k I. In this section we return back to one of the main motivations of the sheaf cohomology, namely, for an exact sequence of sheaves of abelian groups 0 S 1 S 2 S 3 0. 22

FIGURE Figure 3.1: Line bundle we have a long exact sequence (3.3) H i (X, S 1 ) H i (X, S 2 ) H i (X, S 3 ) H i+1 (X, S 1 ) that is in each step the image and kernel of two consecutive maps are equal. All the maps in the above sequence are canonical except those from i-dimensional cohomology to (i + 1)-dimensional cohomology. In this section we explain how this map is constructed in Cech cohomology. Let us take a covering U i, i I and f = {f j, j I i+1 } H i (X, S 3 ). By taking the covering smaller, if necessary, we can assume that f is in the image of the map S 2 S 3, that is, there is g = {g j, j I i+1 } such that each g j is mapped to f j under S 2 S 3. Now we have δf = 0 and so δg is mapped to zero under S 2 S 3. We conclude that there is h = {h j, j I i+2 } such that h is mapped to δg under S 1 S 2. We have the map H i (X, S 3 ) H i+1 (X, S 1 ), f h which is well-defined and gives us the long exact sequence (3.3). 3.7 Chern classes Consider the short exact sequence 0 Z O O 0 over a complex manifold M. The map O O is given by the exponential map f e 2πif. We write the corresponding long exact sequence H 1 (M, O ) c H 2 (M, Z) and call c the Chern class map. For a line bundle L H 1 (M, O ) on M, c(l) is called the Chern class of L. If L = (g ij, i, j I) then c(l) = {c ijk } i,j,k I, c ijk = ln g ij + ln g ik + ln g ki. 2πi 23

Now let us consider the case in which M is a compact Riemann surface. Using Theorem 3.5, part 3, we have an isomorphism H 2 (M, Z) = Z given by the orientation of M. In this case we usually compose the Chern class map c with this and consider c as a map H 1 (M, O ) Z. Proposition 3.6. Let M be a Riemann surface and L be a line bundle over A. Take a meromorphic section s of L. We have c(l) = mulltiplicity(s, x) x pol(s) zero(s) If one is not familiar with the isomorphism H 2 (M, Z) = Z then he may take the above proposition as a definition of Chern class in the case of Riemann surfaces. Note that for two meromorphic section s 1 and s 2, the quotient s 1 s 2 is a meromorphic function on A and its corresponding sum of multilicities is zero. Therefore, this new definition does not depend on the meromorphic section s. Exercise 3.2. For a Riemann surface M and its canonical bundle Ω 1 and Tangent bundle T show that c(ω 1 ) = 2g 2, c(t ) = 2g 2 Remark 3.2. By definition if c(l) < 0 then L does not have any global holomorphic section. In othe words, if L has a global holomorphic section s with at least one zero then c(l) 0 and if it has no zero then L is the trivial line bundle (the isomorphisim is given by A C L, (x, t) ts(x). ) 3.8 Serre Duality Using Serre duality we can compute some Cech cohomologies easily. Let M be a complex manifold, T M be its tangent bundle and T M be its cotangent bundle. The sections of Ω 1 = Ω 1 M := T M are called differential forms. The sheaf of differential p-forms is the sheaf of sections of the vector bundle Ω p = Ω 1 Ω 1 Ω 1 }{{} p times Theorem 3.2. (Serre Duality) Let M be a complex manifold of complex dimension n and V a holomorphic vector bundle over M. Then there exists a natural C-isomorphism H q (M, Ω p V ) = (H n q (M, Ω n p V )) For a proof of this theorem the reader is referred to [Ra65]. It is useful to consider the following special cases: 1. V is the trivial line bundle H q (M, Ω p ) = (H n q (M, Ω n p )) 24

The numbers are called Hodge numbers and so 2. q = n and p = 0. h q,p := dim C H q (M, Ω p ) h q,p = h n q,n p. H n (M, V ) = (H 0 (M, Ω n V )) Note line bundle Ω n is called the canonical bundle of M. In particular, if rankv = dim M and the zero section of W := Ω n V has a negative self intersection then W does not have a global holomorphic section and so 3.9 Complementary notes H n (M, (Ω n ) W ) = 0. There is a nice description of Chern classes in de Rham cohomology which we have described it below. Proposition 3.7. The Chern class of a line bundle L A over a complex manifold in the de Rham cohomology H 2 (A, C) is represented by a real (1, 1)-form ω. More precisely, in a covering U i, i I of M by open sets we can write the Chern class of L in the complex de Rham cohomology in the form q i, where q i is a pure imaginary C function in U i and e πiq i = h ij e πiq j and L is given by {h ij } H 1 (A, O ). The converse of the above statement is known as Lefschetz (1, 1)-theorem: Every element of H 2 (A, Z) which is represented by a real (1, 1)-form in the de Rham cohomology H 2 (A, C) is a Chern class of some line bundle over A. Proof. In Cech cohomology the Chern class of L is obtained by δ{f ij } H 2 (A, Z), where f ij := 1 2πi log h ij (write the long exact sequence associated to 0 Z O e2πi. O 0 and recall the construction of the coboundary map δ : H 1 (A, O ) H 2 (A, Z)). Now let us look at the diagram which produces an isomorphism between Cech cohomology and de Rham cohomology (see [BT82] Chapter 2): (3.4) 0 Ω 2 (A) Π i Ω 2 (U i ) Π ij Ω 2 (U ij ) Π ijk Ω 2 (U ijk ) 0 Ω 1 (A) Π i Ω 1 (U i ) Π ij Ω 1 (U ij ) Π ijk Ω 1 (U ijk ) 0 Ω 0 (A) Π i Ω 0 (U i ) Π ij Ω 0 (U ij ) Π ijk Ω 0 (U ijk ) C 0 (U, C) C 1 (U, C) C 2 (U, C) 0 0 0 where the right arrow maps are δ s and the up arrow maps are d s. We start with {f ij } Π ij Ω 0 (U ij ). We have δ{f ij } C 2 (U, C). The equality dδ{f ij } = 0 implies δ(d{f ij }) = 0 and so there is a collection {ω i } of 1-forms such that (3.5) δ{ω i } = {f ij } 25