Complex Manifolds. Lectures by Julius Ross. Notes by Tony Feng. Lent 2014

Transcription

1 Complex Manifolds Lectures by Julius Ross Notes by Tony Feng Lent 2014

2 Preface These are lecture notes for a course taught in Cambridge during Lent 2014 by Julius Ross, on complex manifolds. There are likely to be errors, which are the fault of the scribe. If you find any, please let me know at tonyfeng009@gmail.com.

3 Contents Preface i 1 Complex Manifolds Several complex variables Complex manifolds Almost Complex Structures Linear algebra preliminaries Almost complex structures Differential Forms Differential forms on Complex Manifolds Dolbeault Cohomology The Mittag-Leffler Problem The Poincaré Lemma Holomorphic Functions on C Holomorphic functions of several complex variables The Poincaré Lemma Sheaves and Cohomology Sheaves Cech Cohomology Properties of Cech cohomology More on Several Complex Variables Hartog s Theorem Holomorphic Vector Bundles Basic definitions Bundle Constructions Holomorphic Line Bundles ii

4 Contents iii 7.3 Line bundles on Projective Space Ample Line Bundles Kähler Manifolds Kähler metrics The Kähler Identities Hodge Theory The Hodge Decomposition Lefschetz Theorems Lefschetz, I Lefschetz, II Hermitian Vector Bundles Hermitian metrics The Chern connection Curvature Kodaira s Theorems Example Sheet Example Sheet Example Sheet 3 87

5 Chapter 1 Complex Manifolds 1.1 Several complex variables Identify C = R 2 in the standard way, x + iy (x, y). Similarly, C n = R 2n by (x 1 + y 1 i,..., x n + y n i) (x 1, y 1,..., x n, y n ). Definition 1.1. A smooth function f : C n C is holomorphic if it is holomorphic in each variable. Remark 1.2. There is a more natural definition multivariable holomorphic function (you can probably guess what it is), but it is equivalent to this one, and this is more useful for applications. Writing f = u + iv, the theory of single-variable complex functions implies that being holomorphic is equivalent to satisfying the Cauchy- Riemann equations: for all j, u = v x j y j u = v. y j x j Formally define the differential operators = 1 ( i ) z j 2 x j y j = 1 ( + i ) z j 2 x j y j Then the Cauchy-Riemann equations are equivalent to f z j = 0 j = 1,..., n. 1

6 Complex Manifolds 2 Again, this is probably familiar from single-variable complex analysis. Proposition 1.3 (Maximum Principle). Let U C n be open and connected. Let f be holomorphic on a region D such that D U. Then max D f(z) = max f(z) D and f achieves this maximum at an interior point if and only if f is constant. Proof. For n = 1, this is the usual maximum principle. Apply the singlevariable principle repeatedly to deduce it in higher dimensions. Proposition 1.4 (Identity Principle). Let U C n be open and connected and f : U C be a holomorphic function. If f vanishes on an open subset of U, then f 0 on U. Proof. This follows from the single-variable identity principle, applied repeatedly. Lemma 1.5. Let U, V be open sets in C n. A smooth map f : U V is holomorphic if and only if df is C-linear. Proof. Picking frames Also, x k and y k, we have that df = J =. u j x k v j x k..... u j y k... v j y k Now, df is C-linear if and only if Jdf = (df)j, and we see that. Jdf =... v j x k... u j x k.. y k... u j y k... v j. and (df)j = u j y k v j j k.. x k... x k... u j v j The equality of these two matrices is precisely the content of the Cauchy- Riemann equations..

7 Complex Manifolds 3 Theorem 1.6 (Inverse function theorem). Let U, V be open sets in C n and f : U V a holomorphic function. Suppose that z 0 U is such that det J C (f)(z 0 ) 0. Then there exists an open subset U containing z 0 such that f U : U f(u ) is a biholomorphism. Proof. By the real inverse function theorem, there is a local smooth inverse f 1. Furthermore, df 1 f(x) = (df) 1 x, which is C-linear because df is, so f 1 is holomorphic by Lemma 1.5. Theorem 1.7 (Implicit function theorem). Let U C n and f : U C m a holomorphic function such that ( f z i (x)) m i=1 has full rank. Then there exists a holomorphic function g defined in a neighborhood of U of x such that (z m+1,..., z n ) (g(z m+1,..., z n ), z m+1,..., z n ) is a biholomorphism onto U f 1 (0). Proof. This is deduced from the (holomorphic) inverse function theorem in the usual way. Let w = (z 1,..., z m ) and z = (z m+1,..., z n ). Then we define the map C n C n by (w, z) (f(w, z), z). By the hypothesis, this is a diffeomorphism at x, so there exists a local holomorphic inverse. In particular, the set (0, z) maps via this inverse to (0, z) (g(z), z). 1.2 Complex manifolds Complex manifolds lie at the intersection of several different mathematical areas: several several complex variables, differential geometry, and algebraic geometry. This makes their theory very rich and naturally interesting. Let X be a smooth manifold of real dimension 2n. We now develop the basic theory of complex manifolds, which is completely analogous to smooth manifolds except that one demands all functions to be holomorphic instead of smooth. Definition 1.8. A holomorphic atlas for X is a collection of charts (U α, ϕ α ) where ϕ α : U α V α R }{{} 2n = C n open such that

8 Complex Manifolds 4 1. X = α U α, and 2. the transition functions are holomorphic. ϕ αβ = ϕ α ϕ 1 β Definition 1.9. Two holomorphic atlases (U α, ϕ α ) and (Ũα, ϕ α ) are said to be equivalent if ϕ α ϕ 1 β is holomorphic for all α, β. Definition A complex manifold X is a smooth manifold with a choice of equivalence class of holomorphic atlases. This is also called a complex structure on X. Note that if X has dimension 2n as a real manifold, then it has dimension n as a complex manifold. Example We gives some examples of complex manifolds. C n is a complex manifold. M n (C) = C n2 is a complex manifold. Hence GL n (C) is a complex manifold, as it is an open subset of a complex manifold. Complex projective space CP n. As a set, this consists of one-dimensional subspaces of C n+1. A point can be written as [z 0,..., z n ]. A holomorphic atlas is given by taking U i = {z i 0}, and ϕ i : U i C n [z 0,..., z n ] ( z 0 z i,..., ẑi z i,..., z n z i ). The transition function on U i U j is ϕ i ϕ 1 j (w 0,..., ŵ j,..., w n ) = ( w 0 w i,..., ŵj w i,..., w n w i ). In contrast to our previous examples, CP n is compact. Definition A function f : X C is holomorphic if for all holomorphic charts (U, ϕ), the composition f ϕ 1 : ϕ(u) C is holomorphic. Definition A morphism F : X Y between complex manifolds is holomorphic if for all holomorphic charts (U, ϕ) on X and (V, ψ) on Y, ψ F ϕ 1 is holomorphic.

9 Almost Complex Structures 5 We say that X and Y are isomorphic (or biholomorphic) complex manifolds if there exists a holomorphic function F : X Y with holomorphic inverse. An important theme is that there are fewer holomorphic functions than smooth functions. In the smooth theory, functions can be constructed locally and patched together using bump functions (partitions of unity). In the analytic theory, everything is much more rigid: for instance, a function is determined globally by its local behavior (the Identity Theorem). Holomorphic functions on complex manifolds are much more like regular functions on algebraic varieties. Proposition Let X be compact and connected. Then any holomorphic function on X is constant. Proof. Since X is connected, it suffices to show that f is locally constant. Suppose otherwise. Then f has a local maximum at some x X. Composing with a chart, we get a function f ϕ 1 : ϕ(u) C assuming a local maximum in the interior. By the maximum principle, f ϕ 1 is constant. Corollary A compact complex manifold cannot embed in C N. Proof. Indeed, otherwise one could obtain nonconstant holomorphic functions by pulling back holomorphic functions on C N (e.g. the coordinate functions). This contrasts with the Whitney embedded theorem for smooth manifolds, which says that any real manifold embeds in a finite-dimensional Euclidean space. Proposition Let X be a connected complex manifold. If f : X C is holomorphic vanishes on an open subset of X, then f is identically zero. Proof. Let ϕ be a chart around such an accumulation point, and apply the identity theorem for complex functions. In particular, this prohibits a holomorphic analogue of partitions of unity. This means that it is much hard to pass from local to global, as we are accustomed to do in the theory of smooth manifolds. Definition Let Y X be a smooth submanifold of dimension 2k. We say that Y is a closed complex submanifold if there exist holomorphic charts (U α, ϕ α ) for X where Y U α and ϕ α : U α Y ϕ α (U α ) C k with the canonical inclusion of C k C n as (z 1,..., z k ) (z 1,..., z k, 0,..., 0). Definition We say that X is projective if it is isomorphic to a closed complex submanifold of P N for some N.

10 Chapter 2 Almost Complex Structures We might ask, to what extent can the complex structure of a manifold be captured by linear data? In particular, if X is a complex manifold, we should get some extra structure on T X, the tangent bundle of X. This leads to the notion of almost-complex structures. A model case is X = R 2n = C n, with real coordinates x 1, y 1,..., x n, y n. Consider the linear map R 2n R 2n (x 1, y 1,..., x n, y n ) ( y 1, x 1,..., y n, x n ). Under the identification with C n, this corresponds to multiplication by i. Similarly on T X = X R 2n, we have frames x j, y j and we can define an endomorphism of the tangent bundle J : T X T X by ( ) J = ( ) J =. x j y j y j x j Note that again, J 2 is 1. We want to investigate what extra structure we get from having this kind of endomorphism of the tangent bundle. 2.1 Linear algebra preliminaries Let V be a real vector space. Definition 2.1. A linear map J : V V such that J 2 = 1 is called a complex structure on V. On R 2n, the endomorphism (x i, y i,..., x n, y n ) (y 1, x 1,..., y n, x n ) is called the standard complex structure. Given a complex structure J, the relation J 2 = 1 implies that its eigenvalues are ±i. As V is real, this means that J has no eigenspaces in V. But if we consider the complexification V C = V R C, and any real 6

11 Almost Complex Structures 7 linear map T : V V extends to a complex linear map T : V C V C. In particular, J extends to an endomorphism of V C satisfying J 2 = 1, and it now has two eigenspaces. We let V 1,0 and V 0,1 denote the eigenspaces with eigenvalues i, i. Lemma 2.2. With the notation above, 1. V C = V 1,0 V 0,1. 2. V 1,0 = V 0,1. Proof. 1. This is standard linear algebra. It is worth noting that one can explicitly write this decomposition as v = 1 (v ijv)) + 1 (v + ijv). } 2 {{}} 2 {{} V (1,0 V (0,1) 2. Straightforward computation. Note that it is obvious from the above explicit representation. Let J : V V be the dual map, defined by J α(v) = α(j(v)) for all α V. This is necessarily an almost-complex structure on V. Therefore, its complexification decompses VC into a direct sum of +i and I eigenspaces. Lemma 2.3. (V C ) = Hom(V C, C) = Hom R (V, C), and we have the decomposition (V C ) = (V ) 1,0 (V ) 0,1. where and similarly for (V ) 0,1. (V ) 1,0 = {α V C : α(jv) = iα(v)} 2.2 Almost complex structures This discusion extends to real vector bundles. Given a real vector bundle V on X it makes sense to consider V C, obtained by complexifying fiber-byfiber. Definition 2.4. Let V X be a real vector bundle. A bundle morphism J : V V satisfying J 2 = 1 is called an almost complex structure on V.

12 Almost Complex Structures 8 where Given such a bundle, we have a decomposition V C = V 1,0 V 0,1 J V 1,0(v) = iv J V 0,1(v) = iv. This decomposition exists fiber by fiber, of course, but the point is that it varies smoothly in X. That is clear from the fact that V 1,0 can be viewed as ker J i id. Definition 2.5. Let X be a real manifold. An almost complex structure on X is an almost complex structure on T X. We now again assume that X is a complex manifold. Recall that there was a standard almost complex structure on T R 2n, which we denote by J st. Now, given a holomorphic chart ϕ : U R 2n, we get a bundle map J : T U T U by J = Dϕ 1 J st Dϕ. Theorem 2.6. The J defined above is independent of the choice of holomorphic chart, and gives a well-defined almost complex structure on X. Proof. Suppose that ϕ, ψ are two different charts defined in neighborhoods of some point. We must check that or equivalently, Dϕ 1 J st Dϕ = Dψ 1 J st Dψ D((ϕ ψ 1 ) 1 ) J st D(ϕ ψ 1 ) = J st. Now, ϕ ψ 1 is a biholomorphism between subsets of C n, so we are done since if f holomorphic, then df commutes with J st. By the preceding discussion, we get a splitting (T X) C = T 1,0 X T 0,1 X. Definition 2.7. T 1,0 X is the holomorphic tangent bundle of X. We have a similar splitting (T X) C = (T X) 1,0 (T X) 0,1. In terms of local coordinates, suppose that ϕ : U R 2n is a holomorphic chart. We say that x j + iy j are holomorphic coordinates. By definition, ( ) J = ( ) J = x j y j y j x j and J(dx j ) = dy j J(dy j ) = dx j.

13 Differential Forms 9 Definition 2.8. We define Then you can check that dz j := dx j + idy j dz j = dx j idy j. J(dz j ) = J(dx j ) + ij(dy j ) = dy j + idx j = i(dx j + idy j ) = idz j so dz 1,..., dz n give a frame for (T X) 1,0. frame for (T X) 0,1. Similarly, dz 1,..., dz n give a Definition 2.9. We define z j = 1 2 ( x j i y j ) = 1 ( + i ) z j 2 x j y j These are local frames for T X 1,0 and T X 0,1, respectively.

14 Chapter 3 Differential Forms 3.1 Differential forms on Complex Manifolds Recall that if X is a complex manifold, we have local holomorphic coordinates z j = x j + iy j. We showed last time that dz 1,..., dz n is a local frame for (T X) (1,0), dz 1,..., dz n is a local frame for (T X) (0,1), z j is a local frame for (T X) (1,0), z j is a local frame for (T X) (0,1). We already showed that if f : X R 2 = C is smooth, then f is holomorphic if and only if the Cauchy-Riemann equations hold, which is equivalent to f z j = 0 for all j. Let u : X R be a smooth function. We have a map Abusing notation, we also let du x : T x X T u(x) R = R. du x : T x X C C be the complexified map. If f : X C is smooth, say f = u + iv, then df = du + idv, which is a smooth section of T X C. Exercise 3.1. In a local frame, we have an equality of the expression df = j f j dx j + j f j y j = j 10 f z j dz j } {{ } f + f dz j. z j j }{{} f

15 Differential Forms 11 This shows that we have chosen our definitions well (or at least, we have not chosen them to be ridiculous). Remark 3.2. The condition that f be holomorphic is equivalent to f = 0. Recall that T X C = (T X) (1,0) (T X) (0,1). We now study what happens upon taking exterior powers. We will get summands of the following form. Definition 3.3. Let p,q (T X) = p (T X) (1,0) q (T X) (0,1). A smooth section of p,q (T X) is called a (p, q)-form, and (p, q) is called the bi-degree. Locally, it loos like fα,β (z)dz α dz β where α = p and β = q, i.e. a smooth function times a form with p holomorphic coordinates and q antiholomorphic coordinates. We then define A k C (U) to be the space of smooth sections of Λk (T X C) = p,q (T X). In particular, A 0 C (U) is the set of smooth complex-valued fucntions on U. We will suppress the subscript C when the context is clear. over U, and A p,q C Lemma There is a natural identification k (T X C) = p,q (T X) p+q=k and in particular, A k C(U) = p+q=k A p,q C (U). 2. If α A p,q (U) and β A p,q (U) then α β A p+p,q+q (U). Proof. At the level of fibers, this is all formal multilinear algebra. If V = A B, with corresponding representations v j = a j + b j, then v 1... v k = (a 1 + b 1 )... (a k + b k ) p+q=k p V1 q V2. At the level of vector bundles, we can establish it using local frames, or by recognizing the subspaces as kernels of the relevant operators, which are wedge powers of J and the identity.

16 Differential Forms 12 What does this discussion look like in local coordinates? Choose local holomorphic coordinates z 1,..., z n, write A frame for p,q T X is dz I = dz i1... dz ip I = (i 1,..., i p ). dz I dz J I = p, J = q. Example 3.5. Elements of A (1,0) (C) include z dz and z dz. In particular, we allow non-holomorphic functions to multiply the holomorphic differentials. 3.2 Dolbeault Cohomology Definition 3.6. Denote by d : A k (U) A k+1 (U) the C-linear extension of the usual exterior derivative. In addition, we can define the operators : A p,q (U) A p+1,q (U) which is the composition of d with projection to A (p+1,q). : A p,q (U) A p,q+1 (U) which is the composition of d with projection to A (p,q+1). On A (0,0), these agree with the differential operators and that we already defined. This is very concrete in locally coordinates. Locally, if α = fdz I dz J A p,q (U), then dα = f dz j dz I dz J + f dz j dz I dz J. z j j z j j }{{}}{{} α α The following lemma is apparent from this example. Lemma 3.7. On A p,q, we have 1. d = = 0, 2 = 0, and =. 3. If α A p,q (U), then (α β) = α β + ( 1) p+q α β

17 Differential Forms 13 and (α β) = α β + ( 1) p+q α β. Proof. The first part follows from the above local calculation. The second follows from the fact that d 2 = 0. The third follows from the corresponding identity for d. Remark 3.8. The operators and commute with restriction because d does. This will show that they induce maps of sheaves. Definition 3.9. The (p, q) Dolbeault cohomology group is H p,q (X) = ker : Ap,q A p,q+1 (X) im : A p,q 1 A p,q (X). Remark One could make the obvious analogous definition for. However, this gives no extra information, since and are essentially the same (up the conjugation). This is more natural because the kernel of corresponds to holomorphic forms. Example The convention is that negative degree groups are zero, so H 0,0 (X) is the space of holomorphic functions on X. Lemma 3.12 (Functoriality). Let F : X Y be holomorphic. Then the map F : A k (Y ) A k (X) respect the bigrading and commutes with, hence induces a map f : H p,q (Y ) Hp,q(X). 3.3 The Mittag-Leffler Problem Let S be a Riemann surface (a one-dimensional complex manifold). principal part at a point x S is a Laurent series A P = n a k z k k=1 where z is a holomorphic coordinate centered at x. The Mittag-Leffler problem is: given distinct points x 1,..., x n S and principal parts P 1,..., P n at these points, does there exist a meromorphic function on S, which is holomorphic away from the x i, and has precisely these principal parts at the x i? This is, in some sense, answered by Dolbeault cohomology. Clearly, locally the answer is yes, i.e. there exist neighborhoods U j around x j and meromorphic functions f j on U j with principal part P j. Consider a partition of unity (ρ j ) subordinate to the U j, such that ρ j = 1 near x j

18 The Poincaré Lemma 14 and supp ρ j U j. Then j ρ jf j is a function which is smooth except at the x j, with the correct principal parts. The smooth form g = j (ρ jf j ) satisfies g = 0 and is identically zero in a neighborhood of x α. (There is actually something subtle here - f j is not a smooth function on U j. But since f j is locally meromorphic, f j 0 on the punctured neighborhood of x j.) So g represents a cocycle in Dolbeault cohomology. If it were zero in cohomology, then g = h and the function f = ρ j f j h would be a solution to the Mittag-Leffler problem. So the Dolbeault cohomology groups are obstructions to Mittag-Leffler problems.

19 Chapter 4 The Poincaré Lemma Our goal is to show that the Dolbeault groups H p,q (X) vanish when X is a product of discs in C n. This is essentially an analytical problem, like Poincareé s Lemma for de Rham cohomology. 4.1 Holomorphic Functions on C Definition 4.1. A disc in C is a set B R = { z < R} for some R [0, ]. Suppose that B is a bounded disc, u is a smooth complex-valued function on B. Then Stokes formula gives B u(z) dz = If u is holomorphic, then u z B du dz = B u z u dz dz = 2i B z dx dy. = 0 and we recover Cauchy s formula. Theorem 4.2 (Cauchy Integral formula). If u is smooth on B, then u(w) = 1 ( ) u(z) 2πi z w dz + u 1 B z z w dzdz. B If u is holomorphic, we recover the usual Cauchy integral formula. Proof. Apply Stokes theorem B \ B ɛ (w): B\B ɛ(w) u (z) z z w dzdz = B u(z) z w dz B ɛ(w) u(z) z w dz. We compute the second term by explicit parametrization, and let ɛ tend to 0: u(w + ɛe iθ ) idθ 2πiu(w). 15

20 The Poincaré Lemma 16 We now give a converse to this formula, which you can think of as the -lemma in one variable. Theorem 4.3. Let B be a bounded disc in C such that B B U, for some open set U. Let g C (U). Then the integral u(w) = 1 g(z) 2πi z w dzdz is well-defined, smooth, and satisfies u (w) = g(w) on B. w This says that if α = g dz A 0,1 (B) where g has compact support, then there exists u such that u = α. Remark 4.4. We cannot just differentiate under the integral sign here, which would give g(z) w z w = 0, since the integrand is not continuous on the domain of integration. As above, we get around this by analyzing the contribution in a small neighborhood of w. Proof. Let ρ be a smooth bump function such that ρ = 1 on a neighborhood V of B such that supp ρ U. Then ρg is smooth. 1 g(z) 2πi z w dzdz = 1 ρ(z)g(z) 2πi z w dzdz + 1 (1 ρ(z))g(z) dzdz. 2πi z w B B The second integrand has supported away from w, so differentiating by w kills it. Therefore, we reduce to the case where g is smooth and compactly supported in C. We then reparametrize 1 2πi C g(z) z w dzdz = 1 2πi = 1 2πi B C 2π 0 B g(z + w) dzdz z 0 g(w + re iθ ) 2irdrdθ re iθ. Written in this way, we see that the integral is well-defined and the integrand is smooth on all of C, so we can differentiate under the integral sign and trace our steps backwards to deduce that 1 2πi C g(z) z w dzdz = 1 2πi C g(z) z 1 z w dzdz. Then Cauchy s integral formula says that this is g(w).

21 The Poincaré Lemma Holomorphic functions of several complex variables Definition 4.5. A polydisc in C n is a set of the form B = B R1... B Rn, where the R i [0, ]. For a polydisc B, we define 0 B = j B Rj. Theorem 4.6. If B is a bounded polydisc, u is a smooth function on B and holomorphic in B, then u(w) = 1 u(z) (2πi) n (z 1 w 1 )... (z n w n ) dz 1... dz n. B Proof. This is essentially by applying the Cauchy-formula several times. Notation: If α = (α 1,..., α n ), with α j N, then we use the usual multi-index notation z α = z α zn αn α z = α1 α z α... αn 1 1 zn αn and α! = α 1!... α n!. Corollary 4.7. We have the Taylor series expansion u(w) = α a α w α, where Proof. We write 1 z 1 w 1 = α a α = 1 α! z u 0. α 1 z 1 (1 w 1 /z 1 ) = 1 z 1 α 1 0 w α 1 1 z α 1 1 for w 1 z 1. So Then 1 (z 1 w 1 )... (z n w n ) = 1 z 1... z n u(w) = α ( 1 (2πi) n 0 B α w α z α. ) u(z) z 1... z n z dz α 1... dz n w α. Again, differentiating the Cauchy integral formula gives a α = α u z α 0.

22 The Poincaré Lemma The Poincaré Lemma Theorem 4.8. Let B be a polydisc. Assume B B U for some open U C n. Suppose that α A p,q (U) with α = 0. Then there exists β A p,q 1 (B) with α = β on B. Proof. We induct on the highest index dz n that appears in this sum. We can write α = β dz n + α where α involves only dz 1,..., dz n 1. If β = g I dz I. Write gi (w 1,..., w n 1, z n ) dz n u I (w 1,..., w n ) =. z n w n Then Setting we obtain u I w n (w 1,..., w n ) = g I (w 1,..., w n ). δ = ( 1) q 1 I u I dz I δ = I = I u I z n dz I dz n + ( 1) q 1 i n g I dz I dz n + ( 1) q 1 i n u I z i dz i dz I u I z i dz i dz I Then we can modify by the coboundary δ: the cocycle α δ is closed and doesn t involve dz n. We claim that it doesn t involve dz i for i > n, since α is closed, g I z i dz i dz I dz n = for any i > n. This reduces to the case of fdz 1... dz q, and then the same argument shows that this is a coboundary. Lemma 4.9. Let B be a bounded polydisc, B B B U where B is a larger bounded polydisc. Then for any multi-index α there exists a constant C α such that for all holomorphic functions u on U, α u C 0 (B) C α B u(z) dz dz. Proof. This follows from the multivariate Cauchy integral formula, as in the one-variable case. Corollary Let u n be holomorphic on U, say u n u uniformly on compact sets. Then u is holomorphic.

23 The Poincaré Lemma 19 Proof. By the previous lemma, u n z u m z u converges uniformly on compact sets. Also, n = 0. So u is C 1 and z satisfies the Cauchy-Riemann equations, hence is holomorphic. Theorem If B is a (possibly unbounded) polydisc in C n, then H p,q (B) = 0 for q > 0. In other words, if α A p,q (B) with α = 0, then there exists β A p,q 1 (B) with β = α. Proof. The case p = 0, q = 1 is the most difficult. The others reduce easily to this one, so let s study it first. Say B = B r1... B rn. We approximate this region by an increasing sequence of bounded polydiscs. Letting ɛ i (m) increase to r i as i, and B m = B ɛ1... B ɛn. Then B 1 B 2... is an ascending chain of open polydiscs whose union is B. Let α be a cocycle; we want to write it as a coboundary. We already know that we can do this after restricting to a smaller open set, so we filter our space as an increasing union of open sets, and patch together solutions. Specifically, we claim that we can find a sequence β m A 0,1 (B m ) such that 1. β m = α on B m, and 2. β m+1 β m C 0 (B m 1 ) < 2 m. Assuming this for now, we obtain that β m form a Cauchy sequence, hence converge uniformly to some smooth function γ on B. Moreover, (β m+1 β m ) = 0 on B m, so β m β m0 γ β m0, and each β m+1 β m0 is holomorphic on B m0. So γ B m0 is a uniform limit of holomorphic functions, hence is holomorphic on B m0. In particular, γ is smooth on B m0, but since this is independent of m 0 we get that γ is smooth on B and γ = α on B. Now it suffices to prove the claim. Suppose that β 1,..., β m have been defined, and we want to construct β m+1. By the -Poincaré Lemma applied to B m+1 B m+2, there exists a smooth β A 0,0 (B m+2 ) such that β = α on B m+1. After multiplying by some smooth bump function ρ which is identically 1 on B m+1 and with support contained in B m+2, we can assume that β is smooth on B. In order to get the norm bound, we modify β by a holomorphic function on B; since β β m is holomorphic on B m, it suffices

24 Sheaves and Cohomology 20 to take a sufficiently high degree Taylor polynomial approximation to their difference. The case for higher p, q is completed in Example Sheet 1, Question 8. In fact, the same proof shows the following slightly stronger result. If ɛ = {z C: z < ɛ} and ɛ = ɛ {0}. Theorem If q 1 and p 0, then H p,q ( ɛ 1... ɛr ɛk+1... ɛl ) = 0.

25 Chapter 5 Sheaves and Cohomology 5.1 Sheaves Let X be a topological space. Definition 5.1. A presheaf F of groups consists of a group F(U) for all open subsets U X and restriction homomorphisms r V U : F(U) F(V ) for each inclusion of open subset V U. Remark 5.2. In other words, a presheaf is a contravariant functor from the poset of open subsets of X to Grp. You should think of F(U) as being some class (e.g. continuous, smooth, holomorphic, etc.) functions on U, and the restriction maps as being restriction of functions to a smaller domain. In keeping with this intuition, for s F(U) we write s V for r V U (s). Example 5.3. C 0 (U) is the group of continuous functions on U, and the restriction maps are restriction of functions. Remark 5.4. One can similarly define a presheaf of sets, rings, vector spaces, etc. If O is a pre-sheaf of rings, then one can define pre-sheaves of O- modules by asking each F(U) to be an O(U)-module, in a compatible way. Definition 5.5. A presheaf F is a sheaf if 1. For all s F(U), if U = U i is an open cover and s Ui = 0 for all i, then s = If U = U i is an open cover and s i F(U i ) is such that s i Ui U j = s j Ui U j for all i, j then there exists s F(U) such that S Ui = s i for all i. Example. C 0 (U) is a sheaf. More generally, the presheaf functions satisfying some nice purely local property will be a sheaf, e.g.: 21

26 Sheaves and Cohomology 22 Z(U) = locally constant functions f : U Z, R(U) = locally constant functions f : U R. If X is a smooth manifold, we have sheaves C (U) = smooth functions on U, A p (U) = smooth p-forms on U, For E a vector bundle on X, C (E)(U) = smooth sections of E U. If X is a complex manifold, then we have further sheaves O(U) = holomorphic functions on U, O (U) = nowhere vanishing holomorphic functions on U, Ω p (U) = holomorphic p-forms on U. Definition 5.6. Let U C n. A meromorphic function on U is a function f : U \ S C where S U is nowhere dense, such that U has an open cover U = U i and there exists g i, f i O(U i ) with f Ui \S = f i Ui \S. g i Ui \S Exercise 5.7. Define a sheaf M of meromorphic functions on X, and show that it is a sheaf. If X is a complex manifold, we say that f is meromorphic on X if there exists a nowhere-dense S X such that f X S is meromorphic on some open cover by holomorphic charts. Definition 5.8. A morphism α : F G between (pre-)sheaves on X consists of homomorphisms α U : F(U) G(U) for all open subset U X, such that F(U) α U G(U) r V U r V U F(V ) αv G(V ) Remark 5.9. If we view sheaves as functor, then this is just the notion of natural transformations, i.e. morphisms in the functor category. Definition We say that is exact if for all U, the sequence 0 F α G β H 0 0 F(U) α U G(U) β U H(U)

27 Sheaves and Cohomology 23 is exact and if s H(U) and x U, there exists an open neighborhood V U of x and t G(V ) such that β V (t) = s V. These are the notions of exactness in the functor category. In particular, this is not the same as the short exact sequence of abelian groups being exact at the level of each open set U. Definition 5.11 (Kernels). Let α : F G be a morphism of sheaves. We define the kernel sheaf by (ker α)(u) = ker(α U : F G(U)). Cokernels are trickier, since the naïve definition does not necessarily satisfy the sheaf axiom. Definition 5.12 (Cokernels). If 0 F α G H 0 is a short exact sequence of sheaves, we define coker α = H. This does not imply that G(U) H(U) is surjective. In general, there is a sheaf which is the cokernel of any sheaf morphism F G (in the category theoretic sense), but it is not defined as the object-wise cokernel, as this is a pre-sheaf but not necessarily a sheaf. One has to take the sheafification of this naïve definition. Example If X is a complex manifold 0 2πiZ ι O X exp O X 0 is a short exact sequence of sheaves. Given any non-vanishing holomorphic function, we can locally take its logarithm (but not globally). We will call this the fundamental short exact sequence. Definition A complex of sheaves is exact if is short exact for all n. α F 1 α 1 2 F ker α n F n ker α n+1 0 If X is a complex manifold, we have sheaves A p,q. The maps give rise to a complex of sheaves... q 2 A p,q 1 q 1 A p,q q A p,q+1 q+1...

28 Sheaves and Cohomology 24 The -Poincaré Lemma says that this is exact. Indeed, consider the sequence 0 ker q A p,q ker q+1 0. The -Poincaré Lemma said that we can locally find a lift of any form in ker q+1 (in particular, over any polydisc). Recall that we also defined the sheaf Ω p (U) = {σ A p,q (U): σ = 0}, the holomorphic (p, 0 forms on U. In our new language, So Ω p = ker : A p,0 A p,1. 0 Ω p ι A p,0 A p,1... is a resolution of Ω p. We will later see that the sheafs A p,q are fine, hence acyclic, and can therefore be used to compute the cohomology of Ω p. 5.2 Cech Cohomology We consider a toy example to illustrate the idea. Let X be a topological space with open cover X = U V. Suppose that F is a sheaf on X. Say we have sections s U F(U) and s V F(V ) - when do they come from a global section? By the sheaf axiom, this is the case if and only if s U U V = s V U V. Define a map δ : F(U) F(V ) F(U V ) by sending (s U, s V ) s U U V s V U V. Then the gluing condition is precisely δ(s U, s V ) = 0, so F(X) = ker δ. So we have identified an algebraic obstruction to patching local sections into a global one. Let X be a topological space and F a sheaf on X. Let U = {U α } be an open cover of X, indexed by a subset of N (though this isn t really necessary). We introduce the notation U α0...α p = U α0... U αp. Define C 0 (U, F) = α F(U α ) and C 1 (U, F) = α<β F(U αβ ). More generally,

29 Sheaves and Cohomology 25 Definition C p (U, F) = α 0 <...<α p F(U α0 α 1...α p ). By convention, for any multiindex α 0,..., α p we set σ α0,...,α i,α i+1,...,α p = σ α0,...,α i+1,α i,...,α p. If α = (α 0,..., α p ), we say that σ = (σ α ) C p (U, F) is a p-cochain. Definition Define the boundary map δ : C p (U, F) C p+1 (U, F) by p+1 (δσ) α0,...,α p+1 = ( 1) j σ α0,..., α j,...,α p+1 Uα0,...,α p+1. j=0 Lemma The composition δ δ : C p (U, F) C p+2 (U, F) is the zero map. Proof. If σ C p (U, F), the δ δ(σ) α1,...,α p+2 is [( 1) i ( 1) j 1 + ( 1) i ( 1) j ]σ α1,..., α i,... α j,...,α p+2 i,j where the two opposite signs depend on whether i or j is deleted first. Example Let U = {U, V, W } be an open cover of X. Then a σ C 1 (U, F) looks like σ = (σ UV, σ UW, σ V W ). Then δσ = σ UV σ UW + σ V W. Definition We say that σ C p (U, F) is a cocycle if δσ = 0, a coboundary if σ = δτ Define Ȟ q (U, F) = ker δ : Cq (U, F) C q+1 (U, F) im δ : C q 1 (U, F) C q (U, F). In other words, this is the group of cocycles mod coboundaries, or the cohomology of the chain complex described earlier. Obviously, this depends on the open cover and is not our final definition of Cech cohomology.

30 Sheaves and Cohomology 26 Example Let X = P 1 and F = O. We pick the open cover U = [z : 1] and V = [1 : w]. Both are isomorphic to C, and their intersection is C. So C 0 (U, O) = O(U) O(V ) and C 1 (U, O) = O(U V ). The map to C 1 (U, O) is δ(f + g)(z) = f(z) g(1/z). The kernel consists of {(f, g)} where f = g is constant. The image of δ consists of all holomorphic functions on C, since all holomorphic funtions on C have Laurent series. So Ȟ0 (U, O) = C and Ȟi (U, O) = 0 for i > 0. Definition Given two covers U and V, we say that V refines U, and write V U, if there exists α : N N such that for all β, V β U α(β). If V U, we have natural restriction maps ρ V,U : C p (U, F) C p (V, F) such that ρ V,U (σ β1,...,β p ) = σ α(β1 ),...,α(β p). This commutes with δ, so it descends to a map on cohomology: ρ V,U : Ȟ(U, F) Ȟ(V, F). Note that the definition of ρ depends on α. However, you can check that on the level of cohomology, it is actually independent of α. Definition The Cech cohomology of a sheaf F on X is Ȟ p (X, F) = lim U Ȟ p (U, F) For X a complex manifold and F = O, this holds if each intersection is a polydisc. So the cover we used for P 1 in Example 5.20 was in fact acyclic. Example Trivially from the definition, Ȟ 0 (U, F) = F(X) for any open cover of X. Therefore, Ȟ 0 (X, F) = F(X). Example Ȟ q (X, A r,s ) = 0 for all q 1. The point is that the sheaf A r,s is a module over the smooth functions on X, making it much flabbier than holomorphic sheaves like O. To see this, let σ Ȟq (X, A r,s ) be represented by some σ C p (U, A r,s ). We have δσ = 0, so let ρ α be a partition of unity subordinate to {U α } (assuming that the cover is locally finite). Define τ α0,...,α q 1 = ρ β σ β,α0,...,α q 1. β }{{} extend by zero Here, since ρ β is supported in U β, multiplying by it allows us to extend to the complement of U β, so that σ β,α0,...,...,α q 1 is defined on U α0,...,α q 1. As a guiding example, consider U = {U, V, W }. Then by hypothesis, 0 = δσ = σ UV σ UW + σ V W.

31 Sheaves and Cohomology 27 We have τ U = w ρ W σ W U. Then (δτ) UV = τ V τ U = ρ U σ UV + ρ W σ W V ρ V σ V U ρ W σ W U = ρ U σ UV + ρ V σ UV + ρ W (σ W V σ W U ) = (ρ U + ρ V + ρ W )σ UV. Now we tackle the general case. From the definition, we have (δτ) α0,...,α p = = p ( 1) j τ α0,..., α j,...,α p j=0 p j=0( 1) j β ρ β σ βα0,..., α j,...,α p. Now, since σ is a cocycle we have 0 = (δσ) βα0...α p p = σ α0...α p + ( 1) j+1 σ βα0,... α j,...,α p. j=0 Substituting this above, we find that p (δτ) α0,...,α p = β ρ β j=0 σ βα0,..., α j,...,α p = β ρ β σ α0...α p = σ α0...α p. This kind of sheaf, with partitions of unity, is said to be fine. 5.3 Properties of Cech cohomology Let f : F G be a morhpism of sheaves. This induces f : C p (U, F) C p (U, G) for any open cover U. Furthermore, these maps commutes with δ, and hence induce a map on cohomology: Theorem Suppose that f : Ȟp (X, F) Ȟp (X, G). 0 E g F f G 0.

32 Sheaves and Cohomology 28 is a short exact sequece of sheaves on a manifold X. Then there exist natural maps δ : Ȟ p (X, G) Ȟp+1 (X, E ) such that there is a long exact sequence in cohomology,... Ȟp 1 (X, G) δ Ȟp (X, E) f Ȟp (X, F) g Ȟp (X, G)... Proof. First we define δ. Suppose that σ Ĥp (X, G) represented by some σ C p (U, G). The short exactness implies that there is a refinement V U such that ρ VU σ = f(τ), for some τ C p (V, F). Consider δτ: f(δτ) = δf(τ) = δρ VU σ = ρ VU δσ = 0. By exactness, there exists some u C p+1 (V, E) such that g(µ) = δτ. Now g(δµ) = δg(µ) = δ 2 τ = 0. But injectivity of g implies that δµ = 0, so we can define δ [σ] = [µ] Ȟp+1 (X, E). We will prove this under the assumption that there exists arbitrarily fine covers U such that 0 C p (U, E) C p (U, F) C p (U, G) 0 are exact for all p. This is an exact sequence of chain complexes, so by the Snake Lemma we obtain boundary homomorphisms δ : Ȟ p (U, G) Ȟ p+1 (U, E) such that... Ȟp 1 (U, G) δ Ȟp (U, E) f Ȟp (U, F) g Ȟp (U, G)... is exact. Moreover, these commute with the restriction maps ρ U,V to finish off the proof. Theorem Let X be a complex manifold. Then there exist natural identifications H p,q (X) = Ȟq (X, Ω p ). Recall that Ω q denoted the sheaf of holomorphic q-forms. Remark We have shown that 0 Ω p A p,0 A p,1... is an acyclic resolution for the sheaf Ω p. The derived functor cohomology then defines H (X, Ω p ) to be the cohomology of the complex 0 A p,0 A p,1... which is a high-level explanation of the theorem. In nice situations, the derived functor cohomology will agree with Cech cohomology. The proof we give here basically reworks the fact that in derived functor cohomology, one can take an acyclic resolution (instead of an injective resolution).

33 Sheaves and Cohomology 29 Proof. Let Z p,q = ker : A p,q A p,q+1. By the -Lemma, we have short exact sequences of sheaves and 0 Z p,q 1 0 Ω p A p,0 Z p,1 A p,q 1 Z p,q 0 0 for all q. Of course, the content of the -Lemma is the surjectivity. The long exact sequence for the first short exact sequence reads... Ȟr (A p,q ) Ȟr (Z p,1 ) Ĥr+1 (Ω p ) Ȟr+1 (A p,0 )... but since A p,q is fine, the middle map is an isomorphism. That shows Ȟ r (Z p,1 0 ) = Ĥr+1 (Ω p ). if r 1, and the long exact sequence for the second implies similarly that Ȟ r+1 (Z p,q ) = Ȟr (Z p,q+1 ). if q 1. This says that we can increase r by dropping q, and if we do this dropping q to 1, and then using the first row of the long exact sequence in the last step, we find: Ĥ q (X, Ω p ) = Ȟq 1 (X, Z p,1 0 ) =. = Ȟ 1 (X, Z p,q 1 ) H 0 (Z p,q ) = im : H 0 (A p,q 1 ) H 0 (Z p,q = H p,q (X). ) Theorem Let X be a complex manifold. Suppose that U is an open cover of X such that Then Ȟq (X, O) = Ȟq (U, O). Ȟ q (U α1... U αs, O) = 0 for q 1. Proof. We have Ȟr (U α1... U αs, O) = H 0,r (U α 1... U αs ) = 0 by the Dolbeault theorem. This implies that for r 1, we have an exact sequence 0 Z 0,r 1 (U α1... U αs ) A 0,r 1 (U α1... U αs ) Z 0,r (U α 1... U αs ) 0

34 More on Several Complex Variables 30 As this is true for all multi-intersections, the exactness passes to the level of cochains. 0 C p (U, Z 0,r 1 ) C p (U, A 0,r 1 ) C p (U, Z 0,r ) 0 is exact. This gives a long exact sequence in cohomology, with the middle terms vanishing, giving Ȟ p (U, Z 0,r ) = Ȟp+1 (U, Z 0,r 1 ). We then repeat the proof of the above theorem for this open cover, which give an isomorphism to Dolbeault cohomology. As before, Ȟ p (U, O) = Ȟp (U, Z 0,0 ) =... = Ȟ1 (U, Z 0,p 1 ). Ȟ 1 (U, Z 0,p 1 ) = Ȟ 0 (U, Z 0,p ) im = H 0,p (X) = Hp (X, O). Remark The same proof works for Ω p. Actually, the following is true: if Ȟ p (U α, O) = 0 for all p 1 (no higher intersections), then Ȟ1 (U, O) = Ȟ 1 (X, O). Corollary If dim X = n, then Ȟq (X, O) = H 0,q (X) = 0 if q > n. Example Ȟ q (C n, O) = H 0,q (C n ) = 0 for all q 1 (this is the - Lemma). In addition, Ȟ q (C k (C ) l, O) = 0 if k + l 2, by an extension of the -Lemma. It is a fact that if X is contractible, then Ȟq (X, Z) = 0 for all q 1. Indeed, the Cech cohomology with coeficients in a constant sheaf is isomorphic to the singular cohomology with the corresponding coefficients. Recall the exact sequence 0 Z O O 0. So then if X is contractible, its higher Z-cohomoogy vanishes, and we get that Ȟ q (X, O) = Ȟq (X, O ) for q 1. Example H p (P n, Ω q ) = { C p = q n 0 otherwise. The idea of the proof is to cover P n by the standard charts and compute everything explicitly.

35 Chapter 6 More on Several Complex Variables 6.1 Hartog s Theorem There are two useful result on extending holomorphic functions of several complex manifolds, both of which have been called Hartog s theorem in the literature. The intuition is that the singularities of a multivariable analytic function must look like an analytic hypersurface. This tells us a couple of things: if the singular locus has codimension at least 2, then it should be removable. Also, if the singular locus is compact, then in higher dimensions we also expect the function to extend. As a simple example, consider 1 z 1 : we get a pole along z 1 = 0, which defines a noncompact hypersurface. Contrast with this with the one-dimensional case. There are no nonempty (complex) codimension 2 subsets of C. On the other hand, we know that there are holomorphic functions in a punctured disk that do not extend, e.g. a pole or essential singularity. Theorem 6.1 (Hartog s Theorem 1). Let B ɛ = B ɛ1... B ɛn be a polydisc in C n for n 2. Suppose that 0 < ɛ < ɛ for all i so that B ɛ B ɛ. Let f : B ɛ B ɛ C be holomorphic. Then there exists a unique extension f : B ɛ C. Proof. Without loss of generality, let ɛ i = 1. For δ > 0, let V i = {z B ɛ : 1 δ i < z i < 1} and V = V i. So for sufficiently small δ, V B ɛ B ɛ, hence f is holomorphic on V. We want to formalize our intuition that f cannot have a polar part, which would cut out a non-compact singular locus. 31

36 More on Several Complex Variables 32 For fixed w = (z 2,..., z n ), z i < 1, consider the complex function of one variable g w (z 1 ) = f(z 1,..., z n ) defined on A = {1 δ < z 1 < 1}. Therefore, this has a Laurent expansion g w (z 1 ) = n= a n (w)z n 1. We claim that the a n (w) are holomorphic in w. One way to see this is to use Cauchy s integral formula a n (w) = 1 g w (ξ) dξ 2πi ξn+1 ξ =1 δ 2 and note that g w is holomorphic on this region. If we choose w so that 1 δ < z 2 < 1, then we see that g w is holomorphic on the entire disc z 1 < 1. This implies that a n (w) = 0 for n < 0, for all such w, i.e. 1 δ < z 2 < 1. So by the identity theorem, a n (w) = 0 for all z 2 < 1. This implies that f(z) = n=0 a n(w)z1 n is holomorphic in z 1 over the entire polydisc (it converges absolutely since it does when z 2 = 1, and the a n are holomorphic so the maximum modulus occurs on the boundary). Theorem 6.2 (Hartog s Theorem 2). Let U be an open subset of C n for n > 2, and f a holomorphic function on U U {z 1 = z 2 = 0}. Then f extends to a holomorphic function on all of U. Proof. Let D = {(z 1,..., z n ): z i r i } U be a polydisc and D = D D {z 1 = z 2 = 0}. We show that for each z 1,..., z n ) D, we have a Cauchy formula f(z 1,..., z n ) = 1 (2πi) n D f(ζ 1,..., ζ n ) dz 1... dz n (ζ 1 z 1 )... (ζ n z n ). Since the right hand side is holomorphic in D, it defines a holomorphic extension. If D ɛ = {(ζ 1,..., ζ n ): z 1, z 2 ɛ, ζ i r i for i > 1} is a small neighborhood of (z 1,..., z n ) contained in D, then Cauchy s formula implies f(z 1,..., z n ) = 1 (2πi) n f(ζ 1,..., ζ n ) dz 1... dz n D ɛ (ζ 1 z 1 )... (ζ n z n ). Therefore, we just have to show that D ɛ can be homotoped to D through D. This is where the codimension 2 hypothesis comes in.

37 Holomorphic Vector Bundles 33 We choose a homotopy through regions of the form D(t) = {(ζ 1,..., ζ n ): ζ i tz i α i (t) i = 1, 2; z i = r i i > 2}. We require α i (0) = r i so that D(0) = D, and α i (1) = ɛ i so D(1) = D ɛ. In order for D(t) to stay in D, we must have α i (t) < r i tz i. For D(t) to stay in D, it also has to avoid (z 1,..., z n ) and {z 0 = z 1 = 0}. The first requirement is equivalent to (1 t) z i = α i (t) for any t. Since r i tz i > (1 t) z i at t = 0 and t = 1, we can choose α so that (1 t) z i < α(t) < r i tz i. To avoid {z 0 = z 1 = 0}, we cannot simultaneously have α 1 (t) = t z 1 and α 2 (t) = t z 2. That is fine, since we have two degrees of freedom in choosing α 1 (t), α 2 (t) and we impose only one condition. (This comes down to choosing a curve (t, α 1 (t), α(t)) in a cube that avoids the curve (t, t z 1, t z 2 )). Corollary 6.3. If X is a complex manifold and f a holomorphic function on X Z, where Z is a complex submanifold of codimension at least 2, then f extends to a holomorphic function on all of X.

38 Chapter 7 Holomorphic Vector Bundles 7.1 Basic definitions Let X be a differentiable manifold. Recall that a complex vector bundle E of rank r on X is a smooth manifold E X with a smooth projection map π : E X such that each fiber E x := π 1 (x) has the structure of a C-vector space of dimension r, which is locally free: X has an open cover Uα with trivialization ϕ α : π 1 (U α ) = U α C r. If r = 1, we call E a complex line bundle. Definition 7.1. If X is a complex manifold, we say that E is a holomorphic vector bundle if E is a complex manifold, π is holomorphic, and the ϕ α are holomorphic. Note that holomorphic vector bundles have stricter demands than complex vector bundles. Recall that a vector bundle is determined by its transition functions ϕ αβ : ϕ α ϕ 1 β : (U α U β ) C r (U α U β ) C r. Note that ϕ αβ can be considered as a map U α U β GL n (C). The bundle E is holomorphic if and only if the ϕ αβ can be taken to be holomorphic. From the definition, the ϕ αβ satisfy the cocycle conditions ϕ αβ ϕ βγ = ϕ αγ ϕ αβ = ϕ αγ ϕ γα for all α, β, γ. Let GL r (C) denote the sheaf GL r (C)(U) = {g : U GL r (C) holomorphic}. (This notation is sligtly confusing, since it would be also used to denote the locally constant sheaf.) If E is holomorphic, then the data {ϕ αβ } forms a cochain in C 1 (U, GL r (C)). The cocycle condition says that δ : C 1 (U, GL r (C)) C 2 (U, GL r (C)) has δ{ϕ αβ } = 0, i.e. this cochain is actually a cocycle (hence 34

39 Holomorphic Vector Bundles 35 the name), so data of the transition functions descends to an element of Ȟ 1 (X, GL r (C)). Note that we have technically only defined cohomology for sheaves of abelian groups, but we will soon specialize to r = 1. Definition 7.2. Let π E : E X and π F : F X be holomorphic vector bundles. A morphism f : E F is a holomorphic map such that π F f = π E and such that the induced map f x : E x F x is C-linear and rank(f x ) is locally constant in x. Remark 7.3. The constancy of rank is required in order to ensure that we can take kernels and cokernels of morphisms. Be warned that this isn t how vector bundle morphisms are usually defined in differential geometry. Definition 7.4. We write O(E) for the sheaf of local sections of E: O(E)(U) = {s : U E : π E s = id}. Example 7.5. Locally (for small enough opens) we have since the vector bundle is trivial. O(E)(U) O(U) r So there is an essentially surjective embedding from the category of holomorphic vector bundles to that of locally free sheaves, but this is not an equivalence of categories, since we imposed the constant rank condition Bundle Constructions If E and F are holomorphic vector bundles, we can form holomorphic vector bundles E F, E F, i E, S k E, E, etc. This can be easily checked using transition functions: if E has transition functions ϕ αβ and F has transition functions ψ αβ, then E F has transition functions ϕ αβ ψ αβ, etc. 7.2 Holomorphic Line Bundles Let O denote the trivial line bundle X C (we are sort of abusing notation here; O evokes the sheaf of sections of X C). Observe that if L 1, L 2 are holomorphic line bundles, then so is L 1 L 2. Also, L 1 L 1 = O. Definition 7.6. The Picard group Pic(X) is the group of holomorphic line bundles on X up to isomorphism, with the operation of tensor product. Given a holomorphic line bundle L and an open cover U = {U α } with trivializations ϕ α : L Uα U α C a biholomorphism, the collection of

40 Holomorphic Vector Bundles 36 transition functions for L {g αβ = ϕ α ϕ 1 β } C1 (U, GL 1 (C)) = C 1 (U, O ) satisfy the cocycle condition, and hence descend to an element [g L ] Ȟ 1 (X, O ). Proposition 7.7. The map Γ: L [g L ] induces an isomorphism of groups Pic(X) = Ȟ1 (X, O ). Proof. First we must show that this is actually well-defined. If L = M are holomorphic line bundles, choose an open cover U trivializing both L and M, so we have ϕ α : L Uα U α C and σ α : M Uα U α C This gives cocycles gαβ L = ϕ αϕ 1 β and gαβ M = σ ασ 1 β. We have an isomorphism f : L M commuting with projections to X, giving f α : L Uα M Uα. Define h α = σ α f α ϕ 1 α : U α C U α C. This is multiplication by some (non-vanishing) constant on each fiber, so can be regarded as a section of O (U α ). Moreover, (δh) αβ = h α h 1 β = σ α f α ϕ 1 α (σ β f β ϕ 1 β ) 1 = (g L αβ) 1 g M αβ (f α f 1 β ) }{{} =1 since f comes from a global section Clearly Γ is surjective, since any set of transition functions satisfying the cocycle condition can be patched to form a holomorphic line bundle L with those transition functions. We have to establish that it is injective (or said differently, this inverse is well-defined). Suppose [L], [M] Pic(X) such that [g L ] = [g M ]. That means that there exists h = {h α } C 0 (U, O ) such that h α h 1 β = (gαβ) L 1 (gαβ). M We can use the h α to define an isomorphism of L and M. Let f α : L Uα M Uα be f α = σα 1 h α ϕ α. We have to verify that these glue properly, but of course that is precisely encapsulated by the above equation (details left as exercise). That Γ is a homomorphism is evident from the definitions. Recall the exact sequence of sheaves 0 2πiZ i O exp O 0. This gives an exact sequence on cohomology,... Ȟ1 (X, O) Ȟ1 (X, O ) δ Ȟ2 (X, Z)...