Part III- Hodge Theory Lecture Notes

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1 Part III- Hodge Theory Lecture Notes Anne-Sophie KALOGHIROS These lecture notes will aim at presenting and explaining some special structures that exist on the cohomology of Kähler manifolds and to discuss some of the properties and consequences of these structures from the point of view of complex algebraic geometry. We will concentrate on nonsingular projective varieties, a special case of compact Kähler manifolds. Overview A complex manifold is a topological space that is locally modelled on open polydiscs of C n and equipped with holomorphic transition functions. The natural isomorphism C n R 2n endows with the structure of a differentiable manifold of dimension dim R = 2 dim C. These two structures on the underlying topological space turn out to behave quite differenttly in general; this reflects the fact that holomorphic function theory is in some sense much more rigid than differentiable function theory. If is a complex manifold, the tangent bundle of the associated differentiable manifold T,R can be complexified T,C = T,R C and equipped with a Hermitian metric h = g iω. This metric is inherited from the identification of T x,,c with a complex vector space at each point x, and it varies smoothly with x. The complex manifold is Kähler when the metric ω is closed for the exterior differential; this condition ensures compatibility between the complex structure of and its differentiable structure. Nonsingular projective varieties are Kähler, and in fact Kähler manifolds can be thought of as a differential geomeric generalisation of these. However, this analogy should be taken with a pinch of salt; for instance, a Kähler manifold does not in general have any complex submanifold. Note that it is in general difficult to decide whether a given complex manifold is Kähler, or even to construct non projective Kähler manifolds. Let be a complex manifold and T,R its tangent space (when viewed as a differentiable manifold). At every point x, T x,,r is equipped with a complex structure J x, i.e. an endomorphism of T x,,r such that J 2 x = id, 1

2 which can be extended to the complexified tangent space T x,,r C = T x,,c, and induce a decomposition: T x,,r C = T 1,0 x, T 0,1 x, where T 1,0 x, = {u T x,,c J x u = iu} and T 0,1 x, = {u T x,,c J x u = iu}. This direct sum decomposition holds at the bundle level, and dualises to a similar decomposition on the cotangent bundle, the vector bundle of differential 1-forms Ω 1,C = (T,C), and by extension, on the bundle of differtial k-forms Ω k, namely: Ω k,c = Ωk C = Ω p.q, p+q=k where we define Ω p,q = p Ω 1,0 q Ω 0,1. This decomposition satisfies the Hodge symmetry, i.e. Ω p,q = Ωq,p, where complex conjugation acts naturally on Ω k,c = Ωk,R C. Let C (, Ω k,c ) denote the space of complex differential forms of degree of degree k on C -sections of Ω k,c and denote d: C (, Ω k,c ) C (, Ω k+1,c ) the exterior differential, with d d = 0. Recall that the kth de Rham cohomology group of are then defined as: H k (, C) = ker(d: C (, Ω k,c ) C (, Ω k+1,c ) im(d: C (, Ω k 1,C ) C (, Ω k,c ). Theorem 0.1 (Hodge Decomposition). Let be a compact Kähler manifold. If H p,q () H k (, C) is the set of De Rham cohomology classes that can be represented by a closed form α of type (p, q) at every point x, then we have a decomposition: H k (, C) = H p,q (), p+q=k and the summands satisfy Hodge symmetry H p,q () = H q,p (). Remark 0.2. The Hodge decomposition theorem states that on a Kähler manifold, the decomposition of degree k differential forms into forms of type (p, q) with p + q = k descends to the De Rham cohomology. Remark 0.3. Note that in the statement of the Hodge Decomposition, we have written the = sign between various cohomology rather than ; this is because the isomorphisms in question will be shown to be canonical. 2

3 Remark 0.4. Even though the Kähler condition is crucial in order to prove Theorem 0.1, we will see that the Decomposition itself does not depend on the choice of Kähler metric, it only depends on the complex structure. The principle behind the proof of the Hodge Decomposition Theorem is that each De Rham cohomology class has a unique representative that is a harmonic form for an elliptic differential operator, the Laplacian d. The Kähler hypothesis is crucial in the proof of this identification and also provides a decomposition of harmonic forms into forms of type (p, q). Another consequence of this principle of representation of cohomology classes by harmonic forms is the Lefschetz Decomposition. As has been mentioned above, if is a complex manifold of dimension n, can be endowed with a Hermitian form that arises from its differentiable structure h = g iω. When is Kähler, the form ω is a representative of a (cohomology) class [ω] in H 2 (, R). The exterior product with the class [ω] defines an operator L: H k (, R) H k+2 (, R). Theorem 0.5 (Hard Lefschetz Theorem). Let be a compact Kähler manifold. For every k n = dim C, the map L n k : H k (, R) H 2n k (, R) is an isomorphism. In particular, L: H k (, R) H k+2 (, R) is injective for k < n. The primitive cohomology is then H k (, R) prim = ker(l n k+1 : H k (, R) H 2n k+2 (, R)). Theorem 0.6 (Lefschetz Decomposition). Let be a compact Kähler manifold. The natural map i: H k 2r (, R) prim H k (, R) k 2r 0 is an isomorphism for k n (α r ) Σ r L r α r The existence of these decompositions on the De Rham cohomology groups of a compact Kähler manifold has important consequences. For instance, define the Betti numbers by b k () = dim C H k (, C) and the Hodge numbers by h p,q () = dim C H p,q (). By Theorem 0.5, the odd (resp even) 3

4 Betti numbers b 2k 1 () (resp. b 2k ()) increase with k for 2k 1 n (resp. 2k n). Theorem 0.1 shows that b k () = Σ p+q=k h p,q (), and by Theorems 0.1 and 0.6, we have: h p,q () = h q,p () = h n p,n q () = h n q,n p (), so that, in particular, the odd Betti numbers are even. The Hodge Decomposition has further reaching consequences when it is combined with the integral structure on the cohomology H k (, Z). It would be nonsensical to consider the De Rham complex (i.e. degree k differential forms) to compute H k (, Z)- we will use the methods of sheaf cohomology to compute H k (, Z). Using the language of sheaves, we will identify the summand H p,q () in Theorem 0.1 as the Dolbeault cohomology groups H q (, Ω p ) the q-th cohomology groups of with values in the sheaf of holomorphic differential forms of degree p (this identification is possible because of the Kähler hypothesis). We will formalize these results on the cohomology of compact Kähler manifolds by introducing the notion of Hodge Structures. An integral Hodge structure of weight k is an abelian group of finite type H Z and a Hodge decomposition H C = H Z C = H p,q, p+q=k with H p,q = H q,p. We have seen that a Hodge structure exists on the degree k cohomology of a Kähler manifold. Here is an example of application of these Hodge Structures. If is a nonsingular projective variety, (H 1 (, Z), H 1 (, C) = H 1,0 () H 0,1 ()) is a weight 1 Hodge structure. To this Hodge structure, we may associate a complex torus T = C k /Γ = H 0,1 () /H 1 (, Z), where k = b 1 () and Γ is a lattice of rank 2k. This complex torus is the Picard variety of and we will see that it parametrises the holomorphic line bundles L on which have trivial Chern class c 1 (L). As I mentioned in Remark 0.4, the Hodge structure only depends on the complex structure and not on the differentiable structure, that is on the choice of a Kähler metric. We can ask how these Hodge Structures vary with the complex structure. These questions amount to studying varying decompositions on a fixed vector space. Indeed, the De Rham cohomology groups are invariant under diffeomorphism (differentiable isomorphism), 4

5 they are even topological invariants. In particular, when the complex structure varies and the differentiable structure is fixed, these groups will not change, but the decomposition into direct summands will. We will show how to construct a period domain D that parametrises deformations of Hodge Structures when the complex structure varies (that is when H Z = Γ 0 and H C = Γ 0 C are fixed, but the decomposition on H C varies). In fact, we will see that these questions are related to the study of families of compact complex manifolds. If B is a family of compact complex manifolds over a contractible base B, then the fibres t, t B are all diffeomorphic to the central fibre 0. We will see that in some cases, there is a universal family of deformations B of the central fibre 0. The period map P : B D associates to t B the Hodge structure on H k ( t, C) H k ( 0, C). A fundamental result due to Griffiths is: Theorem 0.7. The period map is holomorphic. In fact, in nice cases, this period map is even an embedding or a submersion, and understanding how the Hodge Structure varies locally gives much information about the deformations of the manifold itself. 1 Complex manifolds In these notes, I will assume some familiarity with basic notions of differential and algebraic geometry. Let U C n be an open subset and f : U C a complex valued function. The function f is differentiable if after some identification C n R 2n and C R 2, the induced function f : R 2n R 2 is differentiable. This is independent of the choice of identification. In these notes, I always use the term differentiable for C, but most of the statements will hold under weaker assumptions. 1.1 Local Theory: holomorphic functions of several variables The reader who is not familiar with these notions should read [Voi02, Ch.1] or [Huy05, Section I.1]. Fix a standard system of coordinates (z 1,, z n ) on U C n, and let x j = Rz j and y j = Iz j be the canonical linear coordinates of R 2n. Definition 1.1. Let U C n be an open set and f : U C a differentiable function. The function f is holomorphic at ω = (ω 1,, ω n ) U if for all j = 1,, n, the function z j f(ω 1,, ω j 1, z j, ω j+1,, ω n ) 5

6 is holomorphic at ω j, that is if f z j (ω) := 1 2 ( f x j + i f y j )(ω) = 0. The function f is holomorphic on U if f is holomorphic at ω for all ω U. When f : U C m is a differentiable function, f is holomorphic if each f i : U C is holomorphic, where f = (f 1,, f m ). Lemma 1.2. If f : U C is holomorphic and does not vanish on U, 1/f is holomorphic on U. If f, g are holomorphic maps U C, fg, f + g and g f (when it is defined) are holomorphic on U. Proof. Exercise 1.3. Show that f is holomorphic at ω U precisely when the R-linear application df ω : C n C is C-linear. Let u = Rf : U R and v = If : U R be the real and imaginary parts of f. Show that f is holomorphic on U if and only if for all j = 1,, n, u and v satisfy the Cauchy-Riemann Equations: u x j = v y j and u y j = v x j. Definition 1.4. Let ω = (ω 1,, ω n ) U be a point and R = (R 1,, R n ) (R +) n. The polydisc around ω with multiradius R is: D(ω, R) = {(z 1,, z n ) C n z j ω j < R j, j = 1,, n}. Theorem 1.5. [Voi02, 1.17] Let U C n be an open subset and f : U C be a differentiable map. The following are equivalent: 1. f is holomorphic at ω for all ω U, 2. For all ω U there is a polydisc D(ω, R) U such that f admits a power series expansion f(z + ω) = Σ I α I z I for multi-indices I = (i 1,, i n ) N n that converges absolutely for ω + z D. 3. If D = D(ω, R) is a polydisc contained in U, for all z D, f(z) = ( 1 dζ 1 2iπ )n f(ζ) dζ n, ζ j ω j =R j ζ 1 ω 1 ζ n ω n where the integral is taken over a product of circles, with the orientation that is the product of the natural orientations. 6

7 Definition 1.6. Let U C n be an open set. A holomorphic map f : U C n is locally biholomorphic at ω U if there is a neighbourhood V of ω with V U such that f V is bijective onto f(v ) and f 1 V is holomorphic. Definition 1.7. Let U C n be an open subset, and f : U C m be a holomorphic map. The (complex) Jacobian of f at ω U is the matrix J f (ω) = ( f k z j (ω) ) 1 k m,1 j n. The point ω U is regular if J f (ω) is surjective, f(ω) is a regular value if every point z {f 1 (f(ω))} is regular. Exercise 1.8. Let U C n be an open subset and f : U C n a holomorphic map. Show that f is locally biholomorphic at ω U if and only if det J f (ω) 0. Finally, the following result can be extracted from [Huy05, 1.10,1.11], and shows that a holomorphic map whose Jacobian matrix has locally constant rank locally has a canonical representation. Theorem 1.9. Let U C n be an open subset and f : U C m a holomorphic map. Let ω U be a point such that rk J f (z) = k for all z in a neighbourhood of ω. There are open neighbourhoods V of ω U and W of f(ω) in C m, and biholomorphic maps ϕ: D n V and ψ : W D m such that the composition ψ f ϕ: D n D m is given by (z 1,, z n ) (z 1,, z k, 0,, 0). 1.2 Complex manifolds: definitions and first examples Definition A complex manifold of dimension n is a connected Hausdorff topological space equipped with a complex atlas {(U i ), φ i }, where (U i ) i I is a countable covering by open subsets, and each φ i : U i V i C n is a homeomorphism from U i onto an open subset of C n, and for all i, j I, the transition functions φ j φ 1 i : φ i (U i U j ) φ j (U i U j ) are biholomorphic. Two complex atlases are equivalent if their union defines a complex atlas. There is a maximal atlas equivalent to any given complex atlas on, we say that such a maximal atlas is a complex structure on. A complex manifold is compact if its underlying topological space is compact. 7

8 Recall that a smooth manifold M of dimension m is a topological space M and a maximal differentiable atlas {(W i ) i I, ψ} i I, where (U i ) i I is a countable covering by open subsets, and each ψ i : W i R m is a homeomorphism from W i onto an open subset of R m, and for all i, j I, the transition functions ψ j ψi 1 : ψ i (W i W j ) ψ j (W i W j ) are diffeomorphic. In particular, every complex manifold of dimension n has a natural structure of smooth manifold of dimension 2n. Remark Even though the definitions of complex and smooth manifolds are very similar, they have crucial differences. For instance, while a differentiable manifold M m can always be covered by open subsets that are diffeomorphic to R m, a complex manifold n cannot in general be covered by open subsets that are biholomorphic to C n. For instance, take to be the unit disc D C, then Liouville s theorem shows that there is no non-constant holomorphic map C. Examples 1. Let U C n be an open subset. Then U is a complex manifold, with complex atlas {U, id}. More generally, if U is a connected open subset of a complex manifold, then it has a complex structure induced by that of. 2. Let V be a complex vector space of dimension n + 1. Let P(V ) denote the set of lines through {0} V, i.e. P(V ) = {l V l is a subspace of dimension 1} = Gr(1, V ), then P(V ) is a complex manifold of dimension n (when V = C n+1, P(V ) = CP n ). For every point v V \ {0}, denote [v] = C v V the corresponding point of P(V ), conversely, for every point l P(V ), there is an element v V \ {0} that is unique up to multiplication by a constant λ C such that [v] = l. We have a surjective map: π : V \ {0} P(V ), which endows P(V ) with the quotient topology defined by π and the standard topology on V. We endow P(V ) with a standard complex structure as follows. Choose a C-linear isomorphism V C n+1, and 8

9 for every point v = (v 0,, v n ), denote [v] = [v 0 : :v n ] the homogeneous coordinates of [v] P(V ). For each i = 0,, n define the open set U i = {[v] = [v 0 : :v n ] P(V ) v i 0} P(V ) and the homeomorphism φ i : U i C n The transition functions are [v 0 : :v n ] ( v 0 v i,, v i v i,, v n v i ). φ j φ 1 i : {(z 1,, z n ) C n z j 0} {(u 1,, u n ) C n u i 0} these are biholomorphic. (z 1,, z n ) ( z 1 z j,, ẑj z j,, z i 1 z j, 1 z j,, z n z j ), Remark As a differentiable manifold, CP n S 2n+1 /S 1 (see example sheet 1), in particular CP n is compact. 3. Let Λ C n be a lattice of rank 2n. Denote π : C n C n /Λ the quotient map and = C n /Λ the quotient. Then is a complex manifold. Endow with the quotient topology of C n. If U C n is a small open subset such that U (U + (Λ 0)) =, then U π(u) is bijective. Covering by such open subsets gives a complex atlas, whose transition functions are just translations by elements in Λ. More generally, let be a complex manifold and Γ Aut a subgroup of the group of automorphisms of that acts properly discontinuously on, i.e for any two compact subsets K 1, K 2, γ(k 1 ) K 2 for at most finitely γ Γ. Assume further that Γ acts without fixed point i.e. γ x x for all x and 1 γ Γ then is a complex manifold and /Γ is a locally biholomorphic map. 4. (Affine hypersurfaces) Let f : C n C be a holomorphic function such that 0 is a regular value. Consider = f 1 (0) = {z C n f(z) = 0}. By Theorem 1.9, there is an open cover = i U i, open subsets V i C n 1 and holomorphic maps V i C n inducing bijective maps φ i : U i V i. The transition maps φ j φ 1 i are biholomorphic, and is a complex manifold of dimension n 1. 9

10 5. (Projective Hypersurfaces) Let f : C n+1 C n be a homogeneous polynomial, and assume that 0 C is a regular value for the induced holomorphic map f : C n+1 \{0} C. Then, the affine hypersurface f 1 (0) is a complex manifold. The projective hypersurface = {[z 0 : :z n ] CP n f(z 0,, z n ) = 0} CP n is a complex manifold of dimension n 1. The open subsets U i form an open cover of, where U i are the standard charts of CP n defined above. Using the isomorphism U i C n above, U i is identified with fi 1 (0), for f i (z 1,, z n ) = f(z 1,, z i 1, 1, z i+1, z n ) and by the previous example, we can find a complex atlas. Definition Let be a complex manifold of dimension n, equipped with a complex atlas {U i, φ i : U i C n } i I and Y a complex manifold of dimension m equipped with a complex atlas {W j, ψ j : W j C m } j J. A holomorphic map f : Y is a continuous map such that for all (i, j) I J, ψ j f φ 1 i : C n C m is holomorphic. If Y = C, we say that f is a holomorphic function on. Two complex manifolds and Y are biholomorphic if there exists a holomorphic homeomorphism f : Y. Definition Let be a complex manifold, define the structure sheaf O as the (pre)sheaf (see Definition 1.50, and the discussion there): O (U) = Γ(U, O ) = {f : U C f is holomorphic}, where U is an open subset of. The presheaf O is a sheaf of rings (the restriction morphisms are ring morphisms). If x, define O,x = lim x U O (U), where the limit is taken over all open subsets that contain x. Remark Let be a complex manifold and (U, φ: U C n ) be a holomorphic chart. By definition, if x U is such that φ(x) = 0 C n, there is a natural identification O C n,0 O,x. Remark When is a differentiable manifold, we define the sheaf of differentiable functions A = C () on as the (pre)sheaf U A (U) = {f : U R f is differentiable}. 10

11 Proposition Let be a compact connected complex manifold, then Γ(, O ) = C, i.e. every global holomorphic function is constant. Proof. Let f : C be a holomorphic function. Since is compact and f is continuous, f admits a maximum at a point x. If (U, φ) is a holomorphic chart with x U, f φ 1 is locally constant by the maximum principle, and hence constant because is connected. Remark Using Hartog s Theorem (cf Example Sheet 1), if is a complex manifold of dimension at least 2, Γ( \ {x}, O ) = Γ(, O ) for all x, so that if is compact and connected Γ( \ {x}, O ) = C. Remark In a sense, complex manifolds define a much more rigid structure than smooth manifolds; this reflects the properties of holomorphic functions vs. differentiable functions. We may give an equivalent definition of complex (resp. differentiable) manifolds that is closer to the spirit of algebraic geometry: a complex (resp. differentiable) manifold is a ringed space (, O ), where is a topological space and O is the structure sheaf, whose sections we define to be the holomorphic (resp. differentiable) ones. Definition Let and Y be complex manifolds. A holomorphic map f : Y is a submersion (resp. an immersion) if for all x, there is a neighbourhood U(x) of x such that rk J f (z) = dim Y (resp rk J f (z) = dim ) for all z U(x). The map f is an embedding if it is an immersion and if f is an homeomorphism from onto f(). Remark The rank of the Jacobian matrix does not depend on the choice of coordinate charts. Definition Let be a complex manifold of dimension n and Y be a closed subset. The subset Y is a closed submanifold of of codimension k if for all x Y, there is an open neighbourhood U of x and a holomorphic submersion f : U D k such that U Y = f 1 (0). Example Let and Y be complex manifolds of dimension n and m, f : Y be a holomorphic map and y Y such that rk J f (z) = m for all z f 1 (y); f 1 (y) is a submanifold of dimension n m. If f : Y is an embedding, then f() is a submanifold of Y. Definition A projective manifold is a submanifold CP N such that there exist homogeneous polynomials f 1,, f k C[ o,, N ] of degrees d 1,, d k with = {x CP N f 1 (x) = = f k (x) = 0}. (1) 11

12 Remark Note that if the Jacobian matrix J = ( f j z l )1 j k,0, l N has rank k everywhere, is a submanifold of CP N of codimension k. More generally, (1) defines a submanifold of CP N if the Jacobian matrix has maximal rank. If the Jacobian matrix does not have maximal rank everywhere, (1) defines a projective algebraic variety and the points where the Jacobian has less than maximal rank are the singular points of the variety. Definition A projective manifold CP n of dimension m that is defined by m n homogeneous polynomials such that the Jacobian has rank n m in every point is a complete intersection. Exercise Show that C = {[x 0 : :x 3 ] P 3 x 0 x 3 x 1 x 2 = x 2 1 x 0 x 2 = x 2 2 x 1 x 3 = 0} is a submanifold of dimension 1. Is it a complete intersection? Exercise Let f = (f 1,, f n ): C n C n be a holomorphic map, and (z 1,, z n ) the standard coordinates on C n. Define x k = Rz k, y k = Iz k and u j = Rf j, v j = If j. The (real) Jacobian of f at a C n is J R (f)(a) = ( (u 1, v 1,, u n, v n )) (a). (x 1, y 1,, x n, y n ) Show that det J R (f)(a) = det J(f)(a) 2, and deduce that any complex manifold is orientable. 1.3 Vector bundles We will want to distinguish between two different notions: complex vector bundles are differentiable vector bundles that have values in C, i.e. they have differentiable transition functions, holomorphic vector bundles have holomorphic transition functions. Definition Let be a differentiable manifold. A complex vector bundle of rank r over C is a differentiable manifold E endowed with a surjective map π : E such that: 1. For all x, the fibre E x = π 1 (x) C r has the structure of a C-vector space of dimension r, 12

13 2. There is an open cover = i U i and local trivialisations (U i, h i ), where h i : π 1 (U i ) U i C r is a diffeomorphism with π π 1 (U i ) = p 1 h i and p 2 h i : E x C r a C-vector space isomorphism for all x U i. The manifold E is the total space of the vector bundle and is its base space. Given two local trivialisations (U i, h i ) and (U j, h j ) of E, h i h 1 j : (U i U j ) C r (U i U j ) C r induces a differentiable map g i,j : U i U j GL(r, C), where g i,j (x) = h x i (h x j ) 1 is a C-linear automorphism. The g i,j are the transition functions of E. Exercise Check that the transition functions satisfy the cocycle conditions: g i,j g j,k g k,i = Id on U i U j U k, and g i,i = Id on U i. Remark A complex vector bundle E of rank r is determined uniquely by the differentiable cocycle {U i, g i,j : U i U j GL(r, C)}. Let Ẽ = iu i C r and define (x, v) (y, w) if x = y U i U j, and w = g i,j (x) v, then E = Ẽ/. Example Let be a differentiable manifold of dimension m; if {U i, φ i } is a differentiable atlas of, the real tangent bundle T,R is the vector bundle associated to the cocycle {U i, g i,j = J R (φ i φ 1 j ) φ j }, where J R is the (real) Jacobian. Via the inclusion GL(m, R) GL(m, C), the transition functions g i,j : U i U j GL(m, C) define a complex vector bundle T,C = T,R C, the complexified tangent bundle of. Definition Let be a complex manifold and π : E a complex vector bundle associated to a cocycle {U i, g i,j : U i U j GL(r, C)}. The vector bundle E is holomorphic if E is a complex manifold and if the transition functions g i,j are holomorphic. Remark A natural question is to ask in how many (non-isomorphic) ways a given complex vector bundle can be seen as a holomorphic vector bundle. This is in general a non-trivial question; In some cases, no holomorphic structure exists, but there can also be several different holomorphic structures. This is already the case for line bundles: on a complex torus C n /Γ, the trivial complex bundle of rank 1 admits many holomorphic structures. 13

14 The first example of holomorphic vector bundle is of course that of the trivial vector bundle E = C. Example (The holomorphic tangent bundle) Let be a complex manifold of dimension n endowed with a complex atlas {U i, φ i : U i V i C n }. Then, T is the holomorphic bundle of rank n associated to the cocycle {U i, g i,j = J(φ i φ 1 j ) φ j : (U i U j ) GL(r, C)}, where J is the complex (holomorphic) Jacobian. In other words, T is the holomorphic vector bundle which is trivial over each U i, and whose transition functions correspond to the Jacobian of the change of (holomorphic) coordinates from those defined by the chart φ i to those defined by the chart φ j. Remark Recall from the previous section the definition of the two sheaves of algebras A and O ; the stalks of these sheaves at a point x, A,x and O,x, are the C-algebras of germs of differentiable and holomorphic functions respectively. A derivation of the C-algebra A,x (resp. O,x ) is a C-linear map D : A,x C (resp. O,x C) that satisfies the Leibniz rule, i.e. for any f, g A,x (resp. f, g O,x ), D(fg) = D(f)g(x) + f(x)d(g). The complexified tangent space T,x,C is the space of derivations of A,x, while the holomorphic tangent space T is the space of derivations of O,x. Exercise Let O C n,0 be the C-algebra of germs of holomorphic functions at {0} C n. Let z 1,, z n be standard coordinates on C n and z i := z i 0 be defined by z i : f O C n,0 f z i C. Show that the 0 z i are complex derivations of O C n,0 and form a basis of T C n,0 over C. Deduce a basis of T,x for any complex manifold. Example The tautological vector bundle U(r, V ) Gr(r, V ) over the Grassmannian is defined as the vector bundle with total space U r (V ) = {([U], x) Gr(r, V ) V x U} Gr(r, V ) V, and π : U r (V ) Gr(r, V ) the projection onto the first factor. Using the complex atlas of Gr(r, V ) determined in Example Sheet 1, show that U r (V ) is holomorphic. Examples As in the differentiable situation, any canonical construction in linear algebra gives rise to a geometric version for holomorphic vector bundles. Assume that E and F are holomorphic vector bundles over a complex manifold, we can construct in this way: E F, the direct sum, 14

15 E F, the tensor product, i E, the ith exterior product, E, the dual bundle Hom(E, C) (i.e. fibre wise C-linear maps E C), det E the determinant line bundle. In all cases, a common local trivialisation of E and F will yield transition functions for these constructions. For example, if E is associated to the cocycle {U i, e i,j } and F to the cocycle {U i, f i,j }, E F is ( the vector ) bundle ei,j 0 of rank rk E + rk F associated to the cocycle {U i, g i,j = }; E 0 f i,j is the vector bundle of rank rk E associated to the cocycle {U i, g i,j = t e i,j } (Check that these are cocycles...). Definition Let be a complex manifold and T its holomorphic tangent bundle. The cotangent bundle of is the dual of the tangent bundle Ω = T, the bundle of holomorphic k-forms is Ωp = p Ω, the canonical bundle is K = det Ω, and its dual is the anticanonical bundle K = det T. Remark The definitions of T, Ω and K are independent of the choice of complex structure on, they are invariant of the manifold. π Definition Let E 1 π 1 and 2 E2 be two complex (resp. holomorphic) vector bundles of rank r over C. Then E 1 and E 2 are isomorphic if there is a diffeomorphism (resp. biholomorphism) φ: E 1 E 2 such that π 1 = π 2 φ. Let be a complex manifold and {U i, g i,j : U i U j C } be a collection of holomorphic functions that satisfy the cocycle condition. This defines a holomorphic line bundle L. The holomorphic line bundle L is trivial (i.e. isomorphic to C) precisely when, possibly after refining the cover {U i } i I, there exist holomorphic functions s i : U i C such that g i,j = s i s j on U i U j. Definition Let f : Y be a holomorphic map between complex manifolds and E a holomorphic vector bundle over. If {U i, g i,j } is a cocycle for E, define f E as the holomorphic vector bundle over Y associated to the cocycle {f 1 (U i ), g i,j }. For all y Y, there is a canonical isomorphism (f E) y E f(y). In particular, if Y is a submanifold of, the restriction of E to Y is E Y = i E, where i is the inclusion map i: Y. 15

16 Definition Let π : E be a complex (resp. holomorphic) vector bundle over a differentiable (resp. complex) manifold. A global section of E is a differentiable (resp holomorphic) map s: E such that π s = Id. Remark While a complex vector bundle always has many sections (because we may use local bump functions), this is not necessarily the case for a holomorphic vector bundle. Using fibrewise addition, the set of sections C (, E) (resp. Γ(, O (E))) of a complex (resp. holomorphic) vector bundle has a natural C-vector space structure. If {U i, g i,j } is a cocycle for E, over an open set U i, C (U i, E) (resp. Γ(U i, E)) has dimension rank r. A basis s i1,, s ir of C (U i, E) (resp Γ(U i, E)) is a local frame (resp. local holomorphic frame) of E. The data of an open set U i and a local frame s i1,, s ir is equivalent to a local trivialisation of E. If C (, E), Γ(, E) have dimension r, then a basis s 1,, s r is a global (holomorphic frame) of E, and E is trivial, i.e. isomorphic to C r. Example Exercise 1.37 shows that for any complex chart {U, ϕ} of a complex manifold, if z 1,, z n are local holomorphic coordinates at u U, z 1 u,, z n u is a local holomorphic frame of the holomorphic tangent bundle T. Example Recall the identification CP n = P n = Gr(1, n + 1) and define the tautological line bundle O P n( 1) P n C n+1 P n as the tautological line bundle defined on Gr(1, n + 1) in Example Define O P n(1) = O P n( 1) and O P n(k) = O P n(1) k for k Z. Set O P n(0) for the trivial line bundle. If U i = {[l 0 : :l n ] P n l i 0}, define s i : U i O P n( 1) by: [l 0 : :l n ] ([l]; ( l 0 l i,, l n l i )). The section s i does not vanish anywhere on U i ; the associated local trivialisation is: h i : π 1 (U i ) U i C ([l], x) ([l], x i ) where x i is the unique complex number such that x = x i s i ([l]). The transition functions g i,j : (U i U j ) (U i U j ) C are g i,j : [l] ([l], l j l i ) (this is well defined on U i U j ). The sections of O P n(k) for k N are homogeneous 16

17 polynomials of degree k in the variables l 0,, l n. The transition functions g i,j associated to the line bundle O P n(k) are of the form gi,j k = s i s j for s i, s j homogeneous polynomials of degree k. Exercise Describe the global sections Γ(P n, O P n(k)) for k Z. Definition Let E be a complex (resp holomorphic) vector bundle of rank R over a complex manifold. A submanifold F E is a subbundle of rank m if: 1. For all x, F E x is a subvector space of dimension m, 2. π F : F has the structure of a complex (resp holomorphic) vector bundle induced by that of E. In other words, F is a subbundle of E if E and F are represented by cocycles {U i, e i,j } and {U i, f i,j } such that: ( ) fi,j e i,j = 0 g i,j Examples. 1. The tautological line bundle O P n( 1) is a subbundle of the trivial bundle P n C n Let Y be a submanifold of, T Y is a subbundle of the restricted tangent bundle T Y. Definition Let φ: E F be a vector bundle homomorphism. There are well defined holomorphic vector bundles ker φ and Coker φ. If E and ker φ are represented by cocycles {U i, e i,j } and {U i, k i,j } with ( ) ki,j e i,j =, 0 g i,j Coker φ is associated to the cocycle {U i, g i,j }. Sheaf theory was introduced as a unified way of dealing with problems concerned with the passage from local data to global data; as such it is clear that sheaves are useful to the study of (topological, differentiable, complex) manifolds and of (topological, complex, holomorphic) vector bundles over these. Recall the definitions of sheaves: 17

18 Definition Let be a topological space. A presheaf F of abelian groups (vector spaces, rings, algebras, etc..) on consists of an abelian group (vector space, ring, algebra, etc..) Γ(U, F) = F(U) for every open set U, and a group homomorphism (linear map, ring homomorphism..) r U,V : F(U) F(V ) associated to each pair of nested open sets V U, satisfying the compatibility conditions: 1. r U,U = Id for any U, 2. r U,W = r U,V r V,W for any W V U. The presheaf F is a sheaf if it satisfies the further two conditions. Denote U = i I U i an open cover. 3. If s, t F(U) are such that r U,Ui (s) = r U,Ui (t) for all i I, then s = t, 4. If s i F(U i ) is a collection of objects such that for all i, j I, r Ui,U i U j (s i ) = r Uj,U i U j (s j ), there is an element s F(U) such that r U,Ui (s) = s i for all i I. The presheaves U C (U, E) and U Γ(U, E) are sheaves of abelian groups on, in both cases (complex and holomorphic vector bundles), we denote E the sheaf of sections of E. If E is a complex (resp holomorphic) vector bundle, E is a sheaf of A -modules (resp O -modules). Remark If R is a sheaf of rings over, F is a sheaf of R-modules over if for every open U, F(U) has an R(U)-module structure compatible with its group structure. The restriction maps F(U) F(V ) are morphisms of R(U) modules, where F(V ) is equipped with an R(U)- module structure via the restriction R(U) R(V ). Definition Let be a connected topological space. A sheaf F of R-modules over is locally free of rank r if F is locally isomorphic to R r as a sheaf of R-modules. Proposition Let be a complex manifold. The map E E is a bijection between the set of holomorphic vector bundles E of rank r and locally free sheaves of rank r over. Proof. We have seen that if E is a holomorphic vector bundle, E is a sheaf of O -modules. For any local trivialisation, E Ui U i C r and hence E Ui O r U i and E is locally free of rank r. Conversely, let = U i be an open covering over which E Ui O r U i is an isomorphism of sheaves. On 18

19 each U i U j, O r U i U j E Ui U j O r U i U j is an isomorphism of sheaves that corresponds to an invertible r r matrix M Ui,U j with holomorphic entries, i.e. to a holomorphic map g i,j : U i U j GL(r, C). The sheaf axioms for E ensure that the maps g i,j are cocycles. Define E as the vector bundle associated to the cocycle {U i, g i,j } Remark Beware that when working with this bijection between the category of locally free sheaves and that of holomorphic vector bundles, morphisms of vector bundles are required to have constant rank, but morphisms of sheaves are not. This bijection in fact defines an equivalence of categories between the category of holomorphic vector bundles over and the category of free sheaves of O -modules (this is one way to fix the rank of morphisms..). Remark An analogous statement holds for complex vector bundles and locally free sheaves of A -modules. 1.4 The complexified tangent bundle We now start to examine systematically the relationship between the complex and differentiable structures on a complex manifold of dimension n. Let V be an R-vector space of dimension m, an almost complex structure on V is an endomorphism I End(V ) such that I 2 = Id. An almost complex structure endows V with a structure of C-vector space via: (a + ib) v = a v + b I(v), for v V, a and b R. The dimension m has to be even, m = 2n. The C-vector space is denoted V C = V C. Define a complex conjugation on V C induced by I by: v α = v α, where v V and α C. Conversely, a C-vector space W of dimension n is an R-vector space of dimension 2n, endowed with an almost complex structure I given by multiplication by i. If z 1,, z n are coordinates on C n, denote x i = Rz i and y i = Iz i the associated coordinates of R 2n. The standard complex structure on C n is the endomorphism ( of) R 2n given by the matrix whose only nonzero entries are n 0 1 blocks on the diagonal. 1 0 Let (V, I) be an almost complex structure on a R-vector space V of dimension 2n; I extends to an endomorphism of V C by I(v λ) = I(v) 19

20 λ, and still satisfies J 2 = Id. The C-vector space endomorpshism I is diagonalisable, and this gives a direct sum decomposition: V C = V 1,0 V 0,1, where V 1,0 is the eigenspace associated to the eigenvalue i, and V 1,0 to i. Define a conjugation on V C by v α = v α, we have V 0,1 = V 1,0. Definition Let be a differentiable manifold of (real) dimension m. An almost complex structure on is a differentiable vector bundle isomorphism J : T,R T,R such that J 2 = Id. Remark In general, there does not exist an almost complex structure on a differentiable manifold, even when its dimension is even. In fact, any even dimensional vector space admits a linear complex structure. As a consequence, for an even dimensional differentiable manifold, we may always define a linear transformation I p : T,P,R T,P,R such that IP 2 = Id. The existence of an almost complex structure on is equivalent to determining whether this local construction can be patched up to a vector bundle diffeomorphism (it is then uniquely determined by its action on each fibre). This becomes a question of reduction of the structure group of the tangent bundle from GL 2n (R) to GL n (C) and is a purely algebraic topological question.the sphere S 4 is an example of a differentiable manifold which admits no almost complex structure. Proposition Let be a complex manifold, then induces an almost complex structure on its underlying differentiable manifold. Proof. If {U i, φ i : U i C n } is a complex atlas of, define a differentiable atlas on the underlying real manifold by {U i, (u i, v i ): U i R 2n }, where u i = Rφ i and v i = Iφ i. The real tangent bundle T,R is trivial over U i, and { u i1, v i1,, u in, v in } is a local frame. The holomorphic tangent bundle T is also trivial over U i and a local holomorphic frame is { φ i1,, The identification T,R U U R 2n U C n T U φ in }. endows T,R with a (complex) vector bundle diffeomorphism I such that I 2 = Id (I is induced by the standard complex structure on C n ). This definition is independent of the choice of chart in the complex atlas. Indeed, the transition functions φ i,j = φ i φ 1 j are holomorphic, so that the real Jacobian of ψ i,j, the corresponding differentiable transition functions ψ i,j, 20

21 J R (ψ i,j ) commutes with the matrix of I, the standard complex structure induced by C n. Recall that T,R is given by {U i, J R (ψ i,j ψj 1 ) ψ j }, and by the Cauchy Riemann ( equations, ) J R (ψ i,j ) ψ j is an n n matrix of 2 2 a b blocks of the form. b a In fact, the same argument shows: Lemma Let f : U C n V C m be a holomorphic map and x U; then df(t 0,1,x ) T 0,1 1,0 f(x) and df(t,x ) T 1,0 f(x). Let f : U C n V C m be a holomorphic map; we also denote by f the induced differentiable map R 2n R 2m. The real Jacobian of f is: ) J R (f) = ( uj x i v j x i u j y i v j y i where the holomorphic coordinates on U are z i = x i + iy i, with x i, y i R and f j = u j + iv j, where u j, v j : U R. Recall that J R (f) viewed as a matrix with coefficients in C defines the transition functions of T,R C. When f is holomorphic, after an appropriate change of basis of T x R 2n C and of T f(x) R 2m C J R (f) = ( J(f) 0 0 J(f) where J(f) is the complex Jacobian of f. Proposition Let be a complex manifold. The subbundle T 0,1 T,R is naturally diffeomorphic to the holomorphic tangent bundle. Proof. Let be a complex manifold and {U i, φ i : U i C n } a complex atlas. The complexified tangent bundle T,C is represented by the cocycle {U i, J R (φ i φ 1 j ) φ j }. When φ i φ 1 j is holomorphic, by what precedes, the subbundle T 0,1 is associated to the cocycle {U i, J(φ i φ 1 j ) φ j }. It follows that T 0,1 and T are naturally isomorphic as complex vector bundles. Remark This shows that every complex manifold has a natural orientation. Remark When is a complex manifold, this shows that T 0,1 naturally has a holomorphic structure; however, we view it as a complex vector bundle. In other words, sections of T 1,0 are always differentiable sections. 21 ),

22 We now go back to the general case; let (, I) be an almost complex manifold. There is a direct sum decomposition T,C = T 1,0 T 0,1, where the summands are ker(i i Id) and ker(i + i Id). This induces a dual direct sum decomposition on the (complexified) cotangent complex: Ω,C = Ω 1,0 Ω0,1, and extends to a direct sum decomposition Ω k,c = Ω p,q, p+q=k where Ω p,q = p Ω 1,0 q C Ω 0,1. Further, Ω1,0 = Ω0,1. These decompositions are obvious at the vector space level (i.e. over a point x ), and they are defined on the bundle via local trivialisations. Definition The sheaves A k and Ap,q are the sheaves of sections of the complex vector bundles Ω k,c and Ωp,q. Consider d: Ak Ak+1 the C-linear extension of the exterior differential. If Π k : A Ak and Π p,q : A k Ap,q are the natural projection maps, define the operators and as = Π p+1,q d: A p,q A p+1,q, and = Π p,q+1 d: A p,q A p,q+1. Lemma The Leibniz rule holds for and, i.e. if for some open set U, α A k (U) and β Ak (U), (α β) = α β + ( 1) k α β (α β) = α β + ( 1) k α β. Proof. Follows from the Leibniz rule for d. Definition-Lemma Let (, I) be an almost complex manifold, I is integrable if the following equivalent conditions hold: 1. d = +, 2. Π 0,2 d = 0 on A 0,1. If is a complex manifold, the almost complex structure induced by I is integrable. 22

23 Proof. We check that the conditions are indeed equivalent. If d = +, since d 2 = 0, Π 0,2 d = 0. Conversely, assume that Π 0,2 d = 0 on A 1.0 and let α A p,q. We want to prove that dα Ap+1,q A p,q+1. Fix a common trivialising subset U for Ω 1,0 U and Ω0,1 U and associated local frames (ω i ) 1 i n and (ω i ) 1 i n. Note that for any i, ω i A 1,0 (U). A section α A p,q (U) is of the form α = ΣI,JfI,JωI ωj, where I = p and J = q, and where for each I, J, f I,J is a differentiable function on U, ω I = ω i1 ω ip, for I = {i 1,, i p } and ω J = ω j 1 ω j p, for J = {j 1,, j p }. For each I, J, df I,J A 1,0 (U) A2,0 A 1,1 (U) A0,1 (U) and dω i (U) by assumption, so that df I,J ω I ω J, and f IJ dω I ω J are sections of A p+1,q (U) A p,q+1 (U). For any i, dω i = dω i, so that dω i A 2,0 (U) A1,1 (U) = A0,2 (U) A1,1 (U) and the result follows. When is a complex manifold, using trivialisations associated to any holomorphic chart, one sees that I is integrable. Lemma Let (, I) be an integrable almost complex manifold. The operators and satisfy 2 = 2 = + = 0. Proof. This follows from d = + and d 2 = 0. Remark In fact, one can show that if 2 = 0, I is integrable. The notion of integrability is very important because of the following (hard) theorem, for which a proof in the real analytic case is given in [Voi02]. Theorem 1.68 (Newlander-Nirenberg Theorem). Any integrable almost complex structure is induced by a complex structure. Remark It would be natural to ask how much of complex geometric methods can be applied to the setting of non-integrable almost complex manifolds. The proof of the Newlander-Nirenberg theorem relies on finding holomorphic coordinates near each point. If (, I) is an almost complex manifold, a differentiable function f : C is I-holomorphic if Idf = idf, that is if du = Idv and dv = Idu, where u = Rf and v = If. When has dimension 2n, this is a set of 2 equations in 2n variables, and there may be few holomorphic functions. When I is integrable, the derivatives of holomorphic functions span (T 1,0,R,x ) for all x, and this yields a complex structure on. When dim = 2, an almost complex structure 23

24 is always integrable. On a general almost complex manifold however, there are in general few holomorphic functions f : C. In particular, there are few submanifolds Y defined by holomorphic injections of dimension 2 dim Y dim. The complex submanifolds Y of dimension 2 are I-holomorphic curves, and are in general well behaved, they are studied in Symplectic Geometry. Proposition Let, Y be complex manifolds and f : Y a holomorphic map. The pull back of differential forms respects the decomposition of k-forms into forms of type (p, q), i.e. f A p,q Y Ap,q for all p, q, and f is C-linear and compatible with and. Proof. Since f is differentiable, the pull back map f : A k Y Ak satisfies f d Y = d f. The pullback is C-linear, and as in Lemma 1.59, f respects the decomposition of k-forms into forms of type (p, q) for p + q = k because f is holomorphic. The compatibility with and follows from the compatibility with d. 1.5 Sheaf cohomology and Dolbeault Cohomology groups Let be a complex manifold, we have seen that for integers p, q we may define vector bundles Ω p,q = p Ω 1,0 q Ω 0,1. Let A p,q denote the sheaf of (C ) sections of the bundle Ω p,q. The exterior diffential d = + induces differential operators A p,q A p+1,q and A p,q A p,q+1 with 2 = 2 = 0. These operators define cohomological complexes, and we now examine the cohomology of the Dolbeault complex, associated to. Denote Z p,q (U) = {α Ap,q (U) : α = 0} and Bp,q(U) = { β; β A p,q 1 (U)}. The Dolbeault cohomology groups of are: H p,q We first examine the case q = 0. Lemma For all p 0: () = Zp,q ()/Bp,q(). H p,0 () = Γ(, Ωp ) := H0 (, Ω p ). 24

25 Proof. First note that since q = 0, H p,0 () = Zp,0 () = ker( : Ap,0 () A p,1 ()). Locally, if z 1,, z n is a set of holomorphic coordinates on, dz 1,, dz n is a local frame for Ω 1,0 and dz 1,, dz n is a frame for Ω 0,1. The forms dz I dz J where I, J run through multiindices with I = p and J = q form a frame for Ω p,q. If α Ap,0 (), locally, α = Σ I α I dz I, where α I is a differentiable function, and: α = Σ n j=1σ I α I z j dz j dz I. The {dz j dz I } I,j form a local frame of Ω p,1, so that if α = 0, α I z j = 0 for all I, j. Each α I is a holomorphic function, and by the canonical identification between the holomorphic cotangent bundle Ω and Ω 1,0, this shows that α is a holomorphic section of Ω p. We will extend this statement to the higher cohomology groups of the holomorphic vector bundle Ω p. Proposition The Dolbeault complex {A p,, } is an exact complex of sheaves and is a resolution of Ω p, i.e. the sequence of sheaves is exact. 0 Ω p Ap,0 A p,1 A p,n 0 Let be a topological space and E,F and G be sheaves of abelian groups/ modules, etc.. over (see definition 1.50). A morphism of sheaves α: F G is a collection of homomorphisms α(u): F(U) G(U) for each open set U that are compatible with the restriction maps r F and r G, i.e. for every pair of open sets V U, r G UV α(u) = α(v ) rf UV. If α is a morphism of sheaves, ker α is the sheaf U ker α(u) and Coker α and im α are the sheaves associated to the presheaves U Coker α(u) and U im α(u) respectively. Remark These two last presheaves are not sheaves in general.. E.g., if exp(2iπ ): O O, the map {z z} O U is not in α(o U) for U = C {0}, but {z z} α(o V ) for any contractible V U. Given a morphism of sheaves α: F G, a section s Coker α is an open cover = i I U i and a collection of sections s i G(U i ) such that s i Ui U j s j Ui U j α(u i U j )(F(U i U j )). Two such sections {U i, s i } i I 25

26 and {V j, t j } are identified if for any P U i V j, there is a neighbourhood V of P such that s i V t j V α(v )(F(V )). A sequence of sheaf morphisms 0 E α F β G 0 is exact if E = ker α and G = Coker β. A sequence of sheaves αn 1 α F n α n+1 n Fn+1 is exact if for each n N, ker α n+1 = im α n, i.e. if for each n N, is exact. 0 ker α n F n ker α n+1 0 Remark Let 0 E α F β G 0 be a short exact sequence of sheaves. Let U be an open set, 0 E(U) F(U) G(U) is exact (i.e. α(u) is injective, and im α(u) = ker β(u)) but β(u) is not in general surjective. As we will see, the failure of surjectivity is measured by the cohomology of the sheaves E, F, G. Note that the sequence of stalks 0 E P α FP β GP 0 is exact for each P. Proof of Lemma 1.71 shows that Ω p = ker( : Ap,0 Ap,1 ) so that: 0 Ω p Ap,0 is exact. We want to show that for each p, im ( : A p,q 1 A p,q This follows from Lemma Ap,1 ) = ker ( : A p,q ) Ap,q+1. Lemma 1.75 ( -Poincaré lemma). Let U C n be an open neighbourhood of D(0, R), the closure of a bounded polydisc and let α A p,q (U) a -closed form. There is a polydisc D D and β A p,q 1 (D ) such that α D = β. Proof. We first reduce to the case p = 0. Let α be a section of A p,q (U). Locally, we may write α = Σ I,J α I,J dz I dz J = Σ J α I dz I, where the first sum runs over multi-indices I and J of length p and q respectively, and where α I = Σα I,J dz J A 0,q (U). Write: α = Σ n l=1 Σ I,J α IJ α J dz l dz I dz J = Σ j,l dz l dz I. z l z l 26

27 Since dz l dz I dz J is a local frame for Ω p,q+1, α = 0 precisely when α α J = Σ J l z l dz l = 0 for all J. Similarly, using local frames, wwe see that α = β, precisely when α I = β I for all I. We now assume that α A 0,q (U) is such that α = 0. Locally, write α = Σ J =q f J dz J and define: k = min{l : no dz i appears in α for i > l}. We may write α = α 1 dz k + α 2, where no dz i for i k appears in α 1 A 0,q 1 (U) or α 2 A 0,q (U). Define operators i by: = Σ n i=1 dz i = Σ n z i=1 i. i If α = 0, we immediately see that i (α 1 ) = i (α 2 ) = 0 for all i k + 1, so that all f J are holomorphic in the variables z k+1,, z n. By the -Poincaré lemma in 1-variable (see Example Sheet 1), for some 0 < ε k < R k, if g J (z) = 1 2iπ D(0,ε k ) f J (z 1,, z k 1, ω, z k+1,, z n ) dω dω, ω z k then, on D = D(0, R ) for R = (R 1,, R k 1, ε k, R k+1,, R n ), g J z k = f J and g J is C and holomorphic in the variables z k+1,, z n. Set γ = ( 1) q Σ k J g J dz J {k}, then α+ γ is -closed and does not involve any monomial dz l for l k. We conclude by induction on k. We have proved that the Dolbeault complex {A p,, } is an exact complex of sheaves, the proof of Lemma 1.71 shows that this complex is a resolution of the sheaf Ω p. We now show that the cohomology of the Dolbeault complex computes the cohomology of the sheaf Ω p. We first recall the setup of Čech cohomology. Let be a (paracompact) topological space, F a sheaf on and U = {U i } i I a locally finite open cover of. Define the Čech complex (C (U, F), δ) of F associated to the cover U as: C p (U, F) = Π J =p+1 F(U Jp ), where J runs through multi-indices of length p + 1, and U J = U i0 U ip for J = {i 0,, i p } I, and the boundary map δ : C p (U, F) C p+1 (U, F), sends σ to the p + 1-chain whose component on F(U i0 U ip+1 ) is: δ(σ) i0,,i p+1 = Σ p+1 j=0 ( 1)j (σ i0,,î j,,i p+1 ) Ui0 U ip. 27