Complex Manifolds

Home , Complex manifold, Dolbeault cohomology, Holomorphic function

Michael Groechenig March 23, 2016

Contents

1 Motivation 2 1.1 Real and complex diﬀerentiability ...... 2 1.2 A proof of the Cauchy formula using “real techniques” ...... 3 1.3 Holomorphic functions on complex manifolds ...... 4 1.4 Complex projective space ...... 5 1.5 Complex projective manifolds ...... 7 1.6 Complex tori ...... 8

2 Holomorphic functions of several variables 11 2.1 The basics ...... 11 2.2 The Inverse Function Theorem ...... 13 2.3 The Implicit Function Theorem ...... 14 2.4 ∂-derivatives ...... 14 2.5 Hartog’s Extension Theorems ...... 15 2.6 Complex vector ﬁelds ...... 16 2.7 Complex diﬀerential forms ...... 17

3 Complex manifolds 20 3.1 The deﬁnition ...... 20 3.2 Holomorphic functions, complex vector ﬁelds, and forms ...... 21 3.3 Smooth complex projective varieties ...... 23 3.4 Holomorphic maps between manifolds ...... 26

4 Vector bundles 28 4.1 The basics of holomorphic vector bundles ...... 28 4.2 Line bundles ...... 31 4.3 Line bundles on projective space ...... 32 4.4 Line bundles and maps to projective space ...... 35

5 Sheaves 38 5.1 Presheaves ...... 39 5.2 Sheaves ...... 41 5.3 Locally free sheaves and vector bundles ...... 42 5.4 Exact sequences of sheaves ...... 45

1 6 Sheaf cohomology 47 6.1 Overview: why do we need sheaf cohomology? ...... 48 6.2 The exponential sequence revisited ...... 48 6.3 Sheaf cohomology as cohomology ...... 51 6.4 Construction of sheaf cohomology ...... 54

7 GAGA 63 7.1 Projective algebraic geometry in a nutshell ...... 63 7.2 Coherent sheaves ...... 69 7.3 A short exact sequence for twisting sheaves ...... 74 7.4 The proof of the ﬁrst GAGA theorem (modulo basic cohomology computations) . . 76 7.5 Holomorphic sections of coherent sheaves ...... 79

8 Computing Cohomology: Dolbeault and Cechˇ 80 8.1 The Dolbeault resolution ...... 80 8.2 Partitions of unity and sheaf cohomology ...... 84 8.3 Cechˇ cohomology ...... 86 8.4 Cohomology of complex projective space ...... 90 8.5 Line bundles on complex manifolds ...... 92

1 Motivation 1.1 Real and complex differentiability This course is concerned with the theory of complex manifolds. On the surface, their definition is n identical to the one of smooth manifolds. It is only necessary to replace open subsets of R by open n subsets of C , and smooth functions by holomorphic functions. This might lead you to think that the notion of complex manifolds has been created as a mere intellectual exercise, which cannot hold any surprises for those already versed in the classical theory. It is the main goal of this course to show by how far this hypothesis fails. The complex and real worlds are fundamentally different. The complex analogues of “real” definitions often yields more restrictive, and rigid mathematical objects. To illustrate this principle we take a look at the notion of differentiability. Let U ⊂ R be some open subset, and f : U / R a function. We say that f is differentiable, if for every x ∈ U the limit f(y) − f(x) f 0(x) = lim (1) y→x y − x exists. It is well-known that the thereby defined derivative f 0(x) can be quite pathological, in particularly it doesn’t even have to be continuous (for instance the continuous extension of the 2 1 function x 7→ x sin( x ) for x 6= 0 has a non-continuous derivative). Assume now that U ⊂ C is an open subset of the complex numbers, and f : U / C a complex- valued function. The limit in (1) makes sense as it stands, for complex variables, and we say that f is complex differentiable, or holomorphic, if and only if the aforementioned limit exists for every x ∈ U. In contrast to the example of a real-valued, only once differentiable function given above, we have that the derivative of a complex-differentiable function is always continuous, in fact the following is true.

2 Theorem 1.1. Every holomorphic function f is infinitely many times complex differentiable. This theorem is easily deduced from an even stronger result. Theorem 1.2. Every holomorphic function f can be locally expressed by a converging power series. P n That is, for every z0 ∈ U there exists > 0, and (an)n∈N ∈ C, such that f(z) = n≥0 an(z − z0) for |z − z0| < . These two theorems demonstrate the philosophical principle mentioned earlier: “The complex analogues of “real” definitions often yields more restrictive, and rigid mathematical objects.” While analytic functions (that is, those locally expressible by converging power series) are examples of differentiable functions in both the real and complex world, they are revealed to be the only complex example thereof, while being considered special amongst real differentiable functions. Pretty much everything that is known about holomorphic functions is a consequence of an identity of Cauchy. For a holomorphic function f as above, z0 ∈ U, and > 0, such that U = {z| |z − z0| < } ⊂ U, we have Z f(z)dz = 0. (2) |z−z0|=

1.2 A proof of the Cauchy formula using “real techniques” Despite the diﬀerences between real and complex analysis, it is possible to use the real-valued theory to say something about the complex theory. In this paragraph we will deduce Cauchy’s formula (2) from the so-called Stokes Theorem, which should be familiar from last term’s manifolds lecture. Theorem 1.3 (Stokes). Let M be a bounded compact n-dimensional manifold with boundary ∂M. For every smooth (n − 1)-form ω we have the identity Z Z dω = ω. M ∂M

We will consider the disc D(z0) = {z ∈ C | |z − z0| ≤ } as a bounded manifold, such that 1 the boundary S is oriented in a counterclockwise manner. We will then attach to a holomorphic function f(z) a complex 1-form ω = f(z)dz, and apply Stokes Theorem in order to deduce Cauchy’s formula (2). Before getting there, we need some preparatory work. 2 We use the standard bijection C =∼ R , obtained by decomposing a complex number z = x + iy into real and imaginary part, z 7→ (x, y). In other words, we view C as a smooth manifold of dimension 2, and similarly this allows us to view D(z0) as a smooth manifold of dimension 2 with boundary. Similarly we may decompose f into real and imaginary parts, f(x + iy) = u(x, y) + iv(x, y), yielding real-valued functions u and v. Holomorphicity of f can be read oﬀ the functions u and v.

Theorem 1.4 (Cauchy-Riemann). The function f : U / C is holomorphic, if and only if u and v ∂u ∂v ∂v ∂u are differentiable and satisfy the Cauchy–Riemann differential equations ∂x = ∂y and ∂x = − ∂y . This theorem follows easily from the following observation in linear algebra. We can embed the field of complex numbers C into the ring of (2 × 2)-matrices, by assigning to a + ib the matrix a −b . b a

3 The Cauchy–Riemann Equations can now be rephrased as stipulating that the Jacobi matrix

∂u ∂u ! ∂x ∂y ∂v ∂v ∂x ∂y is of the shape above, that is, corresponds to multiplication of (x + iy) by a complex number. On the manifold with boundary D(z0) we now introduce the complex 1-form dz = dx+idy. It is a formal sum of a real and imaginary part, both of which are smooth 1-forms in the usual sense. We deﬁne a second complex 1-form by formal complex multiplication ω = f(z)dz = (u+iv)(dx+idy) = (udx − vdy) + i(udy + vdx).

Lemma 1.5. A complex-valued function f, such that real part u and imaginary part v are differentiable, satisfies the Cauchy–Riemann equation, if and only if ω = f(z)dz satisfies dω = 0. Proof. We compute dω by independently differentiating real and imaginary part: ∂u ∂v ∂u ∂v d[(udx − vdy) + i(udy + vdx)] = dy ∧ dx − dx ∧ dy + i dx ∧ dy + i dy ∧ dx. ∂y ∂x ∂x ∂y Using that dx ∧ dy = −dy ∧ dx, we obtain the following result: ∂u ∂v ∂u ∂v dω = [( + ) + i( − )]dx ∧ dy. ∂y ∂x ∂x ∂y The equation dω = 0 is equivalent to vanishing of both the real and imaginary part, which corresponds to the Cauchy–Riemann Equations.

Now we apply the Stokes Theorem to the manifold with boundary D(z0), and the complex 1-form ω = f(z)dz. For f a holomorphic function, we obtain Z Z f(z)dz = dω = 0. ∂ D(z0) D(z0) 1.3 Holomorphic functions on complex manifolds In this paragraph we will give a sloppy definition of complex manifolds, which will be revised at a later point. We will ignore some technical points, like that the topological space underlying a complex manifold should be Hausdorff and second-countable. Moreover, we pretend that we already know what a holomorphic function in several variables is. Moreover, we will ignore differences between atlases and maximal atlases, etc.

Deﬁnition 1.6. Let X be a topological space, together with an open covering {Ui}i∈I , and home- ' 0 0 n omorphisms φi : Ui / Ui , where Ui ⊂ C is an open subset. If for every pair (i, j) the induced bijection −1 φj ◦φi φi(Ui ∩ Uj) / φj(Ui ∩ Uj) is holomorphic, we say that X is endowed with the structure of a complex manifold of complex dimension n.

4 S The pairs (Ui, φi) are called charts. The covering X = i∈I Ui introduces a system of locally deﬁned complex coordinates. A complex manifold of complex dimension n = 1 is also called a Riemann surface.

Definition 1.7. Let X be a continuous manifold, and f : X → C be a continuous map. We say that f is holomorphic, if for every chart (U, φ) as above, the composition f ◦ φ−1 : U 0 / C is a holomorphic function on U 0. To emphasise the difference between complex and real manifolds, we will show that a compact complex manifold has only very few holomorphic functions. We will deduce this from a lemma about holomorphic functions, which states that for a non-constant holomorphic function g defined n on an open subset U 0 ⊂ C , the image g(U 0) ⊂ C is an open subset. Lemma 1.8. If X is a compact, connected, complex manifold, then every holomorphic function f : X / C is constant.

Proof. Let us assume that f is not constant. For every chart (Ui, φi) we obtain a holomorphic S 0 function gi = f ◦ φi. We see that f(X) = i∈I g(Ui ) ⊂ C is an open subset. On the other hand, since X is compact, the image f(X) ⊂ C is compact as well, thus in particularly closed. The only open and closed subsets of C are ∅ and C. The second case is not possible, because C is not compact.

n Corollary 1.9. A compact complex manifold X cannot be embedded into any C , unless X is 0-dimensional.

n n Proof. Let us assume that we have an injective holomorphic map h: X,→ C . Let pi : C / C n the projection to the i-th coordinate, that is, the map sending (z0, . . . , zn) ∈ C to zi. The maps pi ◦ h are holomorphic, and therefore constant on every connected component of X. This implies that h can only be injective, if X is a disjoint union of points. So far this is probably the most striking diﬀerence with the theory of real manifolds. Every n smooth manifold can be embedded into R , for n large enough!

1.4 Complex projective space In the preceding paragraph we saw that a compact complex manifold cannot be holomorphically n embedded into C . This means the theory of smooth manifolds favourite strategy to produce n examples, an application of the Regular Value Theorem to produce smooth submanifolds of R , will not directly apply to the theory of complex manifolds. However, at this stage we haven’t even seen a single example of a compact complex manifold.

2 2 Example 1.10. Consider the 2-sphere S as a topological space, and let x ∈ S be an arbitrary point. We denote by N = (0, 0, 1) the “north pole”, and by S = −N the “south pole”. In the 2 manifolds course, stereographic projection was used to produce a homeomorphism between S \x 2 2 with R . If we further use the standard identiﬁcation C =∼ R , we obtain a homeomorphism (or 2 ' 2 chart) φ1 : S \N / C. We also denote S by C, and refer to it as the Riemann sphere, and N as ∞.

5 We will now construct a second chart, endowing C with the structure of a complex manifold. 2 Since the underlying topological space is homeomorphic to S , it is our ﬁrst example of a compact complex manifold.

1 Lemma 1.11. Consider the map z 7→ z , which is well-defined on C \0. It extends to a homeomorphism of i: C → C, by sending 0 to ∞, and ∞ to 0. Proof. We begin by proving that the map i is continuous. This is certainly the case for every 2 point belonging to C \0 =∼ S \{N,S}, so it remains to verify continuity at 0 and ∞. We use that 0 has a neighbourhood basis given by -neighbourhoods U = {z ∈ C | |z| < }, while ∞ has a neighbourhood basis by VR = {z ∈ C | |z| > R} ∪ {∞}. So to show continuity at 0, we need to show that for every R > 0, there exists an > 0, such that i(U) ⊂ VR. This is automatically satisfied for 1 1 1 z = 0, so we may focus on z ∈ U \ 0. Since |i(z)| = | z | = |z| , we have that for = R the property i(U) = VR is satisfied. Continuity at ∞ is verified similarly. We have i2 = id , since this property is satisfied in the dense subset \0, and the functions i2 C C and id are continuous. Therefore, we see that i is its own continuous inverse, in particular it is a C homeomorphism.

This lemma now directly implies that the Riemann sphere C is a complex manifold of dimension 1, that is, it is a Riemann surface. As charts we use U0 = C ⊂ C, with the identity map φ: U = id C / C, and V = C \ 0 with the map ψ = i|V : V → C. Since V = i(U), and ψ = φ ◦ i, we obtain from Lemma 1.11 that ψ is a homeomorphism. The change of coordinates map is given by

U; ∩ V φ−1 ψ

1 # z7→ z C \0 / C \0 which is a bijective holomorphic map, with holomorphic inverse (that is, a biholomorphic map). This implies that C is a complex manifold. 1 The Riemann sphere C is also often denoted by P . It belongs to a family of complex manifolds n P , known as complex projective spaces. n n+1 Deﬁnition 1.12. For n ≥ 0, we denote by P the set of equivalence classes in C \0, with respect × to the equivalence relation (z0, . . . , zn) ∼ (w0, . . . , wn), if and only if there exists λ ∈ C = C \0, n+1 n n such that (z0, . . . , zn) = (λw0, . . . , λwn). We have a canonical map C \0 / P . We endow P n with the quotient topology, that is, consider a subset U ⊂ P as open, if and only if the preimage n+1 p−1(U) ⊂ C \0 is open.

The equivalence class [(z0, . . . , zn)] will be denoted by (z0 : . . . , zn). As a next step, we construct n charts on P , and verify that they define the structure of a complex manifold. n For 0 ≤ i ≤ n we let Ui ⊂ P be the subset consisting of all (z0 : ... : zn), such that zi 6= 0. The −1 n+1 preimage p (Ui) is {(z0, . . . , zn) ∈ C |zi 6= 0} is an open subset, hence Ui is by definition open S n n+1 in the quotient topology. Moreover, we know that Ui = P , since for every (z0, . . . , zn) ∈ C \0, at least one coordinate zi must be non-zero. Hence, we have defined an open covering (Ui)i=0,...,n n of P .

6 −1 n 1 We define a continuous map φi : p (Ui) / , by sending (z0, . . . , zn) to (z0,..., zi, . . . , zn) e C zi b (omitting the i-th component). Since the value of φei is the same for every representative of a given n equivalence class (z0 : ... : zn) ∈ Ui, we obtain a well-defined continuous map φi : Ui / C . n −1 We have a continuous map ψei : C / p (Ui), sending (w1, . . . , wn) to (z0, . . . , zn), where zj = wj+1 for 0 ≤ j ≤ i − 1, and zi = 1, and zj = wj for j ≥ i + 1. The composition p ◦ ψei defines n n a continuous map ψi : C → Ui. Since φi ◦ ψi = idC , and ψi ◦ φi = idUi , we have that φi is a homeomorphism. The only fact that remains to be verified, is that the change of coordinates maps are holomorphic. We will do this for i < j, and the charts (Ui, φi), (Uj, φj). The case j > i follows by relabelling coordinates. Let’s assume that (z0 : ... : zn) belongs to Ui ∩ Uj, that is, we have zi 6= 0 and zj 6= 0. n The image φi(Ui ∩Uj) is equal to {(w1, . . . , wn) ∈ C |wj 6= 0}. The resulting change of coordinates map is now given by the map

n n φi(Ui ∩ Uj) = {(w1, . . . , wn) ∈ C |wj 6= 0} / C ,

w1 wi 1 wi+1 wj−1 wj+1 wn sending (w1, . . . , wn) to φj(ψi(w1, . . . , wn)) = ( ,..., , , ,..., , ,..., ). Since wj wj wj wj wj wj wj the components of this map are rational functions, they are in particular holomorphic. Therefore, we have established the following result.

n Proposition 1.13. The topological space P , together with the charts deﬁned above, is a complex manifold.

n One can also show that the topological space underlying the complex manifold P is compact. The easiest way to see this is by noting that every equivalence class [z0 : ... : zn] has a repre- 2 2 sentative (z0, . . . , zn) satisfying |z0| + ··· + |zn| = 1 (simply rescale appropriately). The subset n+1 2 2 2n+1 {(z0, . . . , zn) ∈ C | |z0| + ··· + |zn| = 1} is by deﬁnition equivalent to S , that is, deﬁnes n 2n+1 n a compact topological space. Since P = p(S ), we see that P is the image of a compact topological space, that is, compact itself.

1.5 Complex projective manifolds

Let G = G(z0, . . . , zn) be a polynomial in n + 1 variables. We represent it as a ﬁnite sum P a zi0 ··· zin , and refer to max(i + ··· + i |a 6= 0) as the degree deg G of i0,...,in≥0 i1...in 0 n 0 n i0...in G.

Deﬁnition 1.14. The polynomial G is called homogeneous of degree d if for every λ ∈ C, and n+1 d every z = (z0, . . . , zn) ∈ C we have G(λz) = λ G(z). This deﬁnition can be reformulated in more explicit terms, as the following lemma shows. The proof is left as an exercise.

Lemma 1.15. A polynomial G = P a zi0 ··· zin is homogeneous of degree d if and i0,...,in≥0 i1...in 0 n only if ai1...in = 0 for i0 + . . . in 6= d. In particular we see that for a non-zero homogeneous polynomial of degree d, we have d = deg G.

Note that homogeneous polynomials are quite a general class of polynomials. In fact, if g(z0, . . . , zn−1) is an arbitrary polynomial of degree d, then there exists a unique homogeneous polynomial G(z0, . . . , zn) 2 of degree d, such that g(z0, . . . , zn−1) = G(z0, . . . , zn−1, 1). Take for example g(z0, z1) = z0 z1 +

7 2 2 2 2 5z0 − 3z1 . It is easy to see that G(z0, z1, z2) = z0 z1 + 5z0z2 − 3z1 z2 satisfies the requirement. We say that G is the homogenisation of g. The reason to introduce homogeneous polynomials is that they can be used to define subsets n of P . For instance, if G is a non-zero homogeneous polynomial of degree d, then the subset {(z0 : ... : zn)|G(z0, . . . , zn) = 0} is well-defined. Indeed, if (w0 : ... : wn) = (z0 : ... : zn), then × there exists λ ∈ C = C \0, such that (w0, . . . , wn) = λ(z0, . . . , zn). Thus, we have G(w0, . . . , wn) = d λ G(z0, . . . , zn); and G(z0, . . . , zn) is zero, if and only if G(w0, . . . , wn) is zero.

Deﬁnition 1.16. Let G1,...,Gk be non-zero homogeneous polynomials of degrees di. The subset n {(z0 : ... : zn) ∈ P |Gi(z0, . . . , zn) = 0 ∀i = 1, . . . , k} is called a complex projective subvariety of n P , and will be denoted by V (G0,...,Gk)

It is not true that V (G0,...,Gk) is always a complex manifold. However, we will see that this is true generically. The criterion for it to be a manifold is reminiscent of the Regular Value Theorem in the theory of smooth manifolds and will be proven using an analogue of this result for holomorphic functions.

Proposition 1.17. Let G1,...,Gk be non-zero homogeneous polynomials of degrees di, and as- ∂Gi n+1 sume that the Jacobi matrix ( )i,j is regular for every value of (z0, . . . , zn) ∈ \0, such that ∂zj C Gi(z0, . . . , zn) = 0 for all i = 1, . . . , k. Then there exists a natural structure of a complex manifold on V (G0,...,Gk). We can now state one of the first big theorems. We will learn more about its proof at a later point. Theorem 1.18 (Chow). Let X be a compact complex manifold, such that there exists a holomorphic n map f : X / P . Then, there exist non-zero homogeneous polynomials G1,...,Gk, such that f(X) = V (G0,...,Gn). n Chow’s theorem implies in particular that every compact complex submanifold of P can be described in terms of purely algebraic data, such as a finite number of homogeneous polynomials. This is a surprising result, because the class of holomorphic functions is in general much bigger than just algebraic ones. At a later point we will deduce this result from a theorem of Serre, which is referred to as GAGA (abbreviating the title of Serre’s article Géométriealgébriqueet géométrie analytique). Stating Serre’s result properly requires the language of sheaves, and cohomology. For now let us just explain its meaning by the cryptic remark that every holomorphic object n (submanifolds, vector bundles, etc.), which we can define on P is in fact algebraic, that is, defined in terms of polynomials and rational functions.

1.6 Complex tori In the last paragraph we saw how systems of homogeneous polynomials can be used to produce examples of compact complex manifolds. There is another, even easier strategy to produce a lot of examples. n n Deﬁnition 1.19. A lattice in V = R is a subgroup Γ ⊂ (R , +), such that V/Γ is a compact topological space, and Γ ⊂ V is a discrete topological space. For example, if V = R is a 1-dimensional vector space, then Z ⊂ R is a lattice. Indeed, the 1 complex exponential map induces a homeomorphism R / Z ⊂ S , and the circle is known to be a compact topological space. There is a more general class of example.

8 Example 1.20. Let V be a ﬁnite-dimensional vector space with a basis ω1, . . . , ωn. Then the subgroup Γ(ω1, . . . , ωn) = Z ω1 + ··· + Z ωn is a lattice. It is clear that the subgroup deﬁned above is discrete, in the lemma below we show that the quotient is indeed compact.

Lemma 1.21. Let V be a ﬁnite-dimensional real vector space, and Γ = Γ(ω1, . . . , ωn) ⊂ V a lattice as in the example above. Then, the topological space V/Γ is a torus, that is, homeomorphic to a 1 ﬁnite product of copies of S (and in particular it is compact). 1 Proof. The complex exponential map induces a homeomorphism R/Z =∼ S . We choose a basis ∼ n ω1, . . . , ωn, such that Γ = Γ(ω1, . . . , ωn). Choose the unique isomorphism of vector spaces V = R , ∼ n n ∼ transforming (ωi) to the standard basis (ei). This induces a homeomorphism V/Γ = R /Z = 1 (S )n.

We will show first that every lattice Γ can be written as Γ(ω1, . . . , ωn) = {λ1ω1 + ··· + n n λnωn|(λ1, . . . , λn) ∈ Z },where (ω1, . . . , ωn) is a basis of V = R viewed as a real vector space of dimension n. Afterwards, we will verify that V/Γ is a complex manifold, if V is a complex vector space. ∼ n Lemma 1.22. Let Γ ⊂ V = R be a lattice, then there exists a real basis ω1, . . . , ωn, such that Γ = Γ(ω1, . . . , ωn). In the proof of this lemma we will use a theorem from algebra, which follows from the classifi- cation of finitely generated abelian groups.

Theorem 1.23. Let A be a finitely generated torsion-free abelian group, then A is free. That is, n there exists an isomorphism A =∼ Z . Proof of Lemma 1.22. Consider the linear span W = R ·Γ ⊂ V , that is, the smallest real subspace, containing Γ. We claim that W = V . This follows from the fact that we have a surjective continuous map V/Γ / V/W , sending [v]Γ to [v]W . The right hand side V/W is also a finite-dimensional vector space. Therefore, not compact, unless V/W = 0. Since images of compact spaces under continuous maps are compact, we must have V/W = 0, which implies W = V . n Linear algebra tells us, that the set Γ contains tuples (v1, . . . , vn), forming a real basis of the n vector space V . We denote this subset of Γ by BΓ. We know that Γ ⊃ Γ(ω1, . . . , ωn). Consider now the quotient Γ/Γ(ω1, . . . , ωn) ⊂ V/Γ(ω1, . . . , ωn), −1 which is closed with respect to the quotient topology, since its preimage p (Γ/Γ(ω1, . . . , ωn) = Γ in V is closed as a discrete subspace of V . Closed subsets of compact topological spaces are themselves compact, this implies that Γ/Γ(ω1, . . . , ωn) is compact. On the other hand Γ is a discrete topological space, that is, every subset is open. This implies that every subset of Γ/Γ(ω1, . . . , ωn is open with respect to the quotient topology. Therefore, Γ/Γ(ω1, . . . , ωn is both compact and discrete, that is, it must be a finite set. Choosing representatives for elements of this finite sets, we obtain that Γ is a finitely generated abelian group. We also know that Γ ⊂ V is torsion free. By Theorem 1.23, torsion free finitely generated abelian groups are free. Hence there exists (ω1, . . . , ωn) ∈ BΓ, such that Γ = Γ(ω1, . . . , ωn). Now let V be a finite-dimensional complex vector space of dimension n. We can also treat V as a real vector space of dimension 2n. If Γ ⊂ V is a lattice in V as a real vector space, then V/Γ has

9 the structure of a complex manifold. In order to define this structure, we need to exhibit complex charts, and verify that the change of coordinates maps are holomorphic. In order to define the charts, we choose > 0, such that v ∈ Γ and |v| < 2 implies v = 0. This is possible, because Γ ⊂ V is discrete, that is, 0 ∈ Γ must possess a neighbourhood, which doesn’t contain another element of Γ. Let p: V → V/Γ be the canonical projection. If we restrict p to U(v) = {w ∈ V | |w − v| < }, we obtain an injective map. Indeed if p(w1) = p(w2), then w1 − w2 ∈ Γ. But this is impossible for w1, w2 ∈ U(v), since we also have |w1 − w2| < 2 by the triangle inequality. Moreover, for every open subset V ⊂ U(v), the image p(V ) is open in the quotient topology, −1 S since p (p(V )) = w∈Γ(w + V ) is a union of open subsets. We have therefore defined a bijective continuous map ψ,v : U(v) / p(U(v)), mapping open sets to open sets. Hence ψ,v is a homeomorphism with continuous inverse denoted by φ,v. We define charts by (p(U(v)), φ,v), choosing once and for all. It only remains to verify that the change of coordinates maps are holomorphic. Let v, w ∈ V be two element of V , the composition

−1 φv φw φv(U(v) ∩ U(w)) / U(v) ∩ U(w) / φw(U(v) ∩ U(w)) is equivalent to the translation z 7→ z + c, for some c ∈ V . Hence, it is holomorphic. Proposition 1.24. Let V and W be two ﬁnite-dimensional complex vector spaces, and Γ ⊂ V and Γ0 ⊂ W be lattices. Then every holomorphic map f : V/Γ / W/Γ0, is induced by a complex linear map φ: V / W, such that φ(Γ) ⊂ Γ0, such that f([v]) = [Av + b] for some b ∈ W and all v ∈ V .

In the rest of the subsection we provide a sketch of the proof. A sketch because we haven’t yet given a complete deﬁnition of the notions of complex manifolds, and holomorphic maps between them.

n m m×n Proof. We choose complex isomorphisms V =∼ C and W =∼ C . We denote by C the complex n m vector space of linear maps C → C . At first we will show that we have a well-defined holomorphic m×n map Df : V/Γ → C . This map is defined as follows: for [v] ∈ V/Γ, we choose a representative v ∈ V , and also a representative w for f([v]). Choosing and δ appropriately, we may assume that for every v ∈ V −1 we have f(U(v)) ⊂ Uδ(w). We define Df(v) to be the Jacobi matrix of φw ◦ f ◦ φv . This is 0 0 −1 well-defined, since for another choice of representatives v and w , the map φw0 ◦f ◦φv0 differs from the above one by pre- and postcomposition of translations, which have zero derivative. Therefore, the chain rule implies that the Jacobi matrices are independent of these choices. By definition of −1 the function Df we have for the chart (p(U(v)), φv) that Df ◦ φv is the Jacobi matrix of the holomorphic map U(v) → Uδ(w). In particular, it is a holomorphic function. Using that V/Γ is compact and connected, it follows from Lemma 1.8 that Df is in fact a −1 constant function. We denote the corresponding matrix by A. This implies that φw ◦ f ◦ φv is equivalent to z 7→ Az + bv,w, where bv,w is a constant depending on the charts labelled by v respectively w. 0 0 If U(v) ∩ U(v ) 6= ∅, then bvw − bv0w ∈ Γ . This follows from [Az + bvw] = [Az + bv0w] for 0 0 0 z ∈ U(v) ∩ U(v ). Similarly, if Uδ(w) ∩ Uδ(w ) 6= ∅, we must have bvw − bvw0 ∈ Γ . Since for 0 0 every v, v ∈ V , there exists a sequence of points v0, . . . , vk ∈ V , such that v0 = v, vk = v , 0 and U(vi) ∩ U(vi+1) 6= ∅, we see that the element [b] = [bv,w] ∈ W/Γ is well-defined, that is, independent of the choice of v and w.

10 This implies that f(z) = [Az + b] for any z ∈ V . Since z ∈ Γ implies [v] = 0, we must have f(z) = [Az + b] = f(0) = [b]. Hence, f(z) ∈ Γ0. Corollary 1.25. If two complex tori V/Γ and W/Γ are biholomorphic complex manifolds, then there exists a complex linear isomorphism φ: V ' / W , such that φ(Γ) = Γ0.

It is an interesting question when a complex torus V/Γ can be realised as a complex submanifold n of a projective space P . For dimC V = 1 this turns out to be always the case. Those complex tori are also known as elliptic curves. For dimC V ≥ 2 this is in general not possible.

2 Holomorphic functions of several variables 2.1 The basics In the study of holomorphic functions in several variables we will often be inspired by methods of the 1 variable case. The most important example being Cauchy’s formula below. However, there are also some genuinely new phenomena, as Hartog’s Extension Theorem 2.14.

Proposition 2.1 (Cauchy). Let U ⊂ C be an open subset, and f : U / C a holomorphic function. If D(z0) := {z ∈ C | |z − z0| < } ⊂ U is a closed disc fully contained in U, then for every z ∈ U(z0), we have 1 Z f(w) f(z) = dw. 2πi w − z ∂ D(z0) Proof. We will use the same strategy as in the proof of another identity of Cauchy in 1.2. That is, we apply the Stokes Theorem to an appropriate manifold with boundary. We denote by A,δ the annulus D(z0) \ Uδ(z), for δ small enough. It is a manifold with boundary being a union of two circles. As in 1.2, we deﬁne a complex 1-form f(z) dz by decomposing into real and imaginary parts: z−z0 f = u+iv, and dz = dx+idv. This 1-form is well-deﬁned on A,δ, because the singularity at z = z0 f(z) f(z) lies outside of A,δ. As in 1.2 we have that d( dz) = 0, since is holomorphic. Therefore z−z0 z−z0 the Stokes Theorem implies

Z f(w) Z f(w) Z f(w) dw − dw = dw = 0. w − z w − z w − z ∂ D ∂ Dδ ∂A,δ

This also implies that the value of R f(w) dw does not depend on δ. We will therefore study ∂ Dδ (z) w−z what happens when δ → 0. To compute the integral we parametrise ∂ D(z0) by the path γ : [0, 2π] / C, which sends t 7→ z + δeit. By substitution we obtain

Z f(w) Z 2π f(z + δeit) Z 2π dw = δieitdt = f(z + δeit)idt w − z δeit ∂ D 0 0 which tends to 2πif(z), as required.

11 This formula implies directly that holomorphic functions are analytic. Let’s take a look at the strategy, because it will also serve us well in the higher variable case. Recall that we have for |z − z0| < |w − z0|

1 1 1 1 X z − z0 = = = ( ))k. z−z0 w − z (w − z0) − (z − z0) (w − z0)(1 − ) (w − z0) w − z0 w−z0 k≥n

Therefore, we deduce from Cauchy’s formula that

Z Z k Z 1 f(w) 1 X (z − z0) X 1 f(w) k f(z) = dw = f(w) k ) = [ k+1 dw](z−z0) , 2πi w − z 2πi (w − z0) + 1 2πi (w − z) ∂ D(z0) ∂ D(z0) k≥0 k≥0 ∂ D(z0) which converges uniformly for |z−z0| ≤ 0 < 1. |w−z0| The most direct way to deﬁne what it means for a function in several variables to be holomorphic is the following.

n Definition 2.2. Let U ⊂ C be an open subset, and f : U / C a continuous function. We say that f is holomorphic, if for each (z1, . . . , zn) ∈ U, and each i = 1, . . . , n, the function f(z1, . . . , zi−1, w, zi+1, . . . , zn) is holomorphic in the variable w, defined in a small neighbourhood of w = zi. As a first step, we will derive a multivariable version of Cauchy’s integral formula.

n Proposition 2.3. Let f : U / C be a holomorphic function, deﬁned on an open subset U ⊂ C .

For a point z = (z1, . . . , zn) ∈ U we choose 1, . . . , n > 0, such that the polydisc D1,...,n (z) =

Dn (z1) × · · · × Dn (zn) is a subset of U. Then, for each w = (w1, . . . , wn) ∈ D1,...,n (z) we have Z Z 0 0 1 f(w) f(w1, . . . , wn) = n ··· 0 0 dw1 ··· dwn. (2πi) (w1 − w ) ··· (wn − w ) ∂ D1 (z1) ∂ Dn (zn) 1 n Proof. We prove the formula by induction on n, the anchor case being n = 1, for which it is the classical integral formula of Proposition 2.1. By induction we assume that the statement of the proposition has already been proven for holomorphic functions in n − 1 variables. The deﬁnition of holomorphic functions in several variables tells us that for a ﬁxed tuple 0 0 0 0 w1, . . . , wn−1, the function w1 7→ f(w1, w2, . . . , wn) is holomorphic in w1. Therefore, applying the classical Cauchy formula, we obtain

Z 0 0 0 0 1 f(w1, w2, . . . , wn) 0 f(w1, . . . , wn) = 0 dw1. 2πi (w1 − w ) ∂ D1 (z1) 1

0 0 Inserting the induction hypothesis for the holomorphic function in n−1 variables (w2, . . . , wn) 7→ 0 0 f(w1, w2, . . . , wn), we obtain the asserted identity. n Corollary 2.4. A holomorphic function f : U / C, for U ⊂ C an open subset, is analytic. That is, for a small enough open neighbourhood V of each point (z1, . . . , zn) ∈ U, the there exists a power sum P a (w − z )m1 ··· (w − z )mn , convergent for w ∈ V , such that f(w) = m1,...,mn≥0 m 1 1 n n P a (w − z )m1 ··· (w − z )mn for each point w of V . m1,...,mn≥0 m 1 1 n n

12 The strategy of the proof is the same as in the 1-dimensional case, that is, uses the geometric series in each variable. We avoid proliferation of indices by leaving the details to the reader. n 2n Given an open subset U ⊂ C , we may view it as an open subset of R . With respect to this transition, the complex coordinates (z1, . . . , zn) get replaced by 2n real coordinates (x1, yn, . . . , xn, yn), satisfying the relation zi = xi + iyi. n n Let A: C / C be a complex linear function, given by an (n × n)-matrix also denoted by A = (aij)i,j=1,...,n. The corresponding real linear function will be denoted by Ae. It is represented by a (2n × 2n)-matrix Ae = (ãk,j)k,j=1,...,2n. n / 2n Remark 2.5. The map A 7→ Ae defines an injective morphism of rings EndC(C ) EndR(R ). This implies that if B = Ae is invertible, then also its inverse B−1 is induced by a complex matrix −1 2n (namely A ). A real (2n × 2n)-matrix C = (ck,l)k,j=1,...,2n ∈ EndR(R ) lies in the image, if and only if for every i, j = 1, . . . , n, the “linear Cauchy–Riemann equations” are satisfies, that is c2i,2j = c2i+1,2j+1, and c2i,2j+1 = −c2i+1,2j.

A (real) differentiable function f = u + iv : U / C is holomorphic, if the partial derivatives with respect to the real variables xi, yi satisfy for each i = 1, . . . , n the Cauchy–Riemann equations ∂u = ∂v , and ∂u = − ∂v (see 1.4 for more details on the Cauchy–Riemann equations). ∂xi ∂yi ∂yi ∂xi k m Given a partially differentiable function f = (f1, . . . , fk): V / R , for V ⊂ R an open n subset, we denote the Jacobi matrix by DRf. Likewise, if U ⊂ C is an open subset, and f = i=1,...,k / k ∂fi (f1, . . . , fk): U C we have a Jacobi matrix of complex partial derivatives Df = ∂z . j j=1,...,n n 2n With respect to the standard identification C =∼ R we have the following remark. k Remark 2.6. A function f : U / C is holomorphic, if and only if it is differentiable as a function 2n defined on an open subset of R , and for every z ∈ U the real Jacobi matrix DRf(z) is of the shape n Ae for A ∈ EndC(C ). We then have A = Df(z).

2.2 The Inverse Function Theorem We’ll use the insight above to deduce a holomorphic version of the Implicit Function Theorem.

n Theorem 2.7 (Inverse Function Theorem). Let U ⊂ C be an open subset, and f = (f1, . . . , fn): U → n C a holomorphic function. Assume that w ∈ U is a point, such that the complex Jacobi matrix ( ∂fi ) is invertible at z = w. Then, there exist open neighbourhoods V of w, and W of f(w), as well ∂zj as a holomorphic function g : W / V , such that g ◦ f = idV , and f ◦ g = idW .

n 2n Proof. We identify C with R , and apply the Inverse Function Theorem for smooth functions, to deduce the existence of neighbourhoods V and W as above, and a smooth function g : W / V with the required property. It only remains to show that g is actually holomorphic. The chain rule implies that the Jacobi-matrix of g, with respect to (x1, y1, . . . , xn, yn) is equivalent to the inverse of the Jacobi-matrix of f with respect to (x1, y1, . . . , xn, yn). Since the Jacobi- n n matrix of f corresponds to a complex matrix, by means of the isomorphism C =∼ R , this must also be true for the Jacobi-matrix of g (by Remark 2.5). Therefore we deduce from Remark 2.6 that g is holomorphic.

13 2.3 The Implicit Function Theorem The Implicit Function Theorem gives us a suﬃcient condition, when the solution set of a system of holomorphic equations, can be parametrised by holomorphic functions, near a given solution. We will use it in the future to construct several examples of complex manifolds, for instance smooth complex projective manifolds, which arise as simultaneous zero sets to a system of homogeneous polynomials.

n k Theorem 2.8 (Implicit Function Theorem). Let U ⊂ C be an open subset, and f = (f1, . . . , fk): U / C ∂fi a holomorphic function. If f(0) = 0, and the determinant | (0)|0≤i,j≤k 6= 0 is non-zero. Then ∂zj k n−k there exists a holomorphic function g = (g1, . . . , gk): V / C , for some open subset V ⊂ C , such that for some neighbourhood W of 0 in U we have f(z) = 0 is equivalent to zi = gi(zk+1, . . . , zn) for 0 ≤ i ≤ k, and z ∈ W .

n Proof. Consider the function F : U / C , given by (z1, . . . , zn) 7→ (f1, . . . , fk, zk+1, . . . , zn). Since ∂Fi all components are holomorphic functions, F is holomorphic as well. Moreover, we have | (0)|0≤i,j≤n = ∂zj ∂fi | (0)|0≤i,j≤k 6= 0, since the Jacobi matrix of F has a block upper triangular shape, with invertible ∂zj diagonal blocks ∂fi ! ∂z (0) ∗ j i,j=1,...,k . 0 1 We can therefore apply the Inverse Function Theorem 2.7 to deduce the existence of a locally n deﬁned holomorphic inverse G: V 0 /U, where V 0 ⊂ C is an open neighbourhood of F (0) = 0 (by 0 0 assumption). We denote the components of G by (g1, . . . , gn). We let g1, . . . , gn be the holomorphic n−k functions on V = V 0 ∩ {0}k × C obtained by restriction. It follows from the deﬁnition of F , and the fact that G is an inverse, that gk+1, . . . , gn are just the functions (zk+1, . . . , zn) 7→ zi for i = k + 1, . . . , n The relation F ◦G = id implies that for (zk+1, . . . , zn) ∈ V we have fi(g1(zk+1, . . . , gn), . . . , gk(zk+1, . . . , gn)) = 0. Similarly, for W = G(V 0), and z ∈ W , such that f(z) = 0, we obtain from G◦F = id the relation (z1, . . . , zn) = (g1(zk+1, . . . , zn), . . . , gk(zk+1, . . . , zn), zk+1, . . . , zn). This concludes the proof. Although the statement of the Implicit Function Theorem is already quite general, it is important to remember the existence of a biholomorphic map G as constructed in the proof.

n k Porism 2.9. Let U ⊂ C be an open subset, and f = (f1, . . . , fk): U / C a holomorphic function n as in the statement of Theorem 2.8. Then there exist open neighbhourhoods W, W 0 ⊂ C of 0, such that W ⊂ U, and a biholomorphic map G: W 0 / W , such that (f 1, . . . , f k) ◦ G(z1, . . . , zn) = (z1, . . . , zk).

2.4 ∂-derivatives There is a formal device, which simplifies many arguments concerning multivariable holomorphic n functions. We fix an open subset U ⊂ C , and a continuous function f : U / C. Using the n 2n standard identification C =∼ R , which on the level of coordinates works by decomposing each complex component zi into real and imaginary part, zi = xi + iyi, we may ask that f is partially differentiable with respect to the 2n real coordinates xi and yi.

14 Definition 2.10. For f as above, we define a differential operator ∂f 1 ∂f ∂f = ( + i ). ∂zi 2 ∂xi ∂yi We also define ∂f 1 ∂f ∂f = ( − i ). ∂zi 2 ∂xi ∂yi Using this notation we can repackage the Cauchy–Riemann equations. Lemma 2.11. A function f = u + iv as above, satisfies the Cauchy–Riemann equations for the ∂f coordinates (xi, yi) if and only if = 0. ∂zi Proof. Let us compute ∂f , with respect to the decomposition f = u + iv. We have ∂zi 1 ∂f ∂f 1 ∂u ∂v ∂u ∂v ( + ) = ( − + i + i ). 2 ∂xi ∂yi 2 ∂xi ∂yi ∂yi ∂xi And by considering real and imaginary part separately, we see that the Cauchy–Riemann equations ∂f with respect to (xi, yi) are satisfied, if and only if = 0. ∂zi This differential operator allows us to check that certain functions, defined by integration, are holomorphic.

2 Lemma 2.12. For a smooth function f : U / C, with U ⊂ C and open subset, and γ : [0, 1] / C a continuous path, we have ∂ Z Z ∂ f(z, w)dz = f(z, w)dz, ∂w γ γ ∂w R whenever well-deﬁned. In particular, if f is holomorphic in w, then so is w 7→ γ f(z, w)dz, whenever well-deﬁned.

2.5 Hartog’s Extension Theorems The result of this paragraph highlights an interesting diﬀerence between holomorphic functions in 1 and several variables. If f is a holomorphic function on an open subset of 2-dimensional complex space, deﬁned away from a point, then it extends to a holomorphic function on that point. This means that the set of singularities of a holomorphic function in 2-variables is 1-dimensional or empty. Moreover, the proof given below, also gives us an explicit method to construct a holomorphic extension. In the proof we will use the following lemma, the proof of which is left to the reader.

Lemma 2.13 (Principle of analytic continuation). Let f : U / C be a holomorphic function, and n U ⊂ C a connected open subset. If there exists an open subset ∅= 6 V ⊂ U, such that f|V = 0, then f = 0.

2 We denote by U1,2 = {(z, w) ∈ C | |z| < 1, |w| < 2} the polydisk and by D1,2 its closure.

15 / Theorem 2.14 (Hartog). Let i > δi > 0 for i = 1, 2, and f : U1,2 \Dδ1,δ2 C be a holomorphic function, then f extends to a holomorphic function on U1,2 .

Proof. Choose r between 2 and δ2, and define F : U to be Z 1 f(z1, w2)dw2 F (z1, z2) = , 2πi |w2|=r w2 − z2 which is well-defined for all (z1, z2) ∈ U(0). Moreover, F agrees with f, whenever |zi| > δi, by Cauchy’s integral formula. A 3-variable version of Lemma 2.12 tells us that ∂ applied to the ∂zi above integral is zero, because we may exchange the order of this differential operator with the integral.

2.6 Complex vector fields n Let’s fix an open subset U ⊂ C of a complex space. Our goal in this paragraph is to define complex vector fields on U. A careful study of this local situation will allow us later to give a definition of complex vector fields on complex manifolds. From now on we modify our notation slightly, turning some subscripts into superscripts, to be consistent with Ricci index notation. Definition 2.15. A complex vector field V on U is a first order differential operator of the shape n X ∂ Vi , (3) ∂zi i=1 1 n n where (V ,..., V ): U / C is a holomorphic function on U. n Equivalently, we could have defined a complex vector field V as a holomorphic function U / C . ∂ Since the differential operators ∂zi are linearly independent (which can be seen by using the relation ∂zi i ∂zj = δj), no information is lost by forming the linear combination as in (3). The advantage of thinking of V in terms of the differential operator is that it’ll become clear how biholomorphic changes of coordinates affect complex vector fields. It is instructive to think this through in some detail: consider a biholomorphic map φ: V / U, with inverse ψ : U / V . k A holomorphic function g : U / C induces a holomorphic function g ◦ φ on V by composition. So, given a vector field V on U, represented by an n-tuple of holomorphic functions (V1,..., Vn), we could be tempted to simply consider the composition (V1 ◦φ, . . . , Vn ◦φ). This however would be inconsistent with the viewpoint of a complex vector field V as a differential operator. Indeed, if U 0 ⊂ U is an open subset, and f : U / C a holomorphic function, then we obtain a holomorphic function h = f ◦ φ: ψ(U 0) / C. Equivalently, we can write f = h ◦ ψ, and use the chain rule to compute n n n X ∂(h ◦ ψ) X X ∂h ∂ψj V f(z ) = V(h ◦ ψ)(z ) = Vj (z ) = Vj (ψ(z )) (z ). 0 0 ∂zj 0 ∂wi 0 ∂zj 0 j=0 i=0 j=0 This defines a holomorphic function on U 0, but by composing with φ, we see that we have defined a first order differential operator

n  n  X X ∂ψi ∂h h 7→ (Vj ◦φ) ◦ φ .  ∂zj  ∂wi i=0 j=0

16 Thinking of the complex coordinates of V , denoted by (w1, . . . , wn) in biholomorphic dependence of the complex coordinates (z1, . . . , zn) in U, we can rewrite this as the suggestive identity

n n n X ∂ X X ∂wi ∂ Vi = (Vj ◦φ) . ∂zi ∂zj ∂wi i=1 i=1 j=1

This inspires the following deﬁnition.

n Definition 2.16. For open subset U, V ⊂ C , a biholomorphic map ψ : U / V , and a complex Pn i ∂ vector field V = i=1 V ∂zi on U, we define

 1  V ◦φ n  n  X X ∂ψi ∂ ψ V = Dψ  .  = (Vj ◦φ) . ∗  .   ∂zj  ∂wi Vn ◦φ i=0 j=0

Similarly to our treatment of complex vector ﬁelds we can study complex analogues of n-forms introduced in the smooth manifolds lecture.

2.7 Complex differential forms n As before we fix an open subset U ⊂ C . A complex m-form ω on U is an object, which assigns 0 to m complex vector fields V1,..., Vm, defined on an open subset U ⊂ U, a holomorphic function 0 ω(V1,..., Vm) on U . Several properties need to be satisfied. For instance we want this construction to be compatible with restriction, that is, for an open subset U 00 ⊂ U 0 we want

ω(V1 |U 00 ,..., Vm |U 00 ) = ω(V1,..., Vm)|U 00 . (4)

If W is another complex vector ﬁeld on U 0, and f a holomorphic function on U 0, then we have that the following multilinearity condition is satisﬁed for each index i = 1, . . . , m

ω(V1,..., Vi−1, Vi +f W, Vi+1,..., Vn) = ω(V1,..., Vi−1, Vi, Vi+1,..., Vn)+fω(V1,..., Vi−1, W, Vi+1,..., Vn). (5) Moreover, for every pair of indices 1 ≤ i < j ≤ m an m-form satisﬁes

ω(V1,..., Vi−1, Vj, Vi+1,..., Vj−1, Vi, Vj+1,..., Vm) = −ω(V1,..., Vm). (6)

For every ordered m-tuple 1 ≤ i1 < ··· < im ≤ n we denote by ωi1...im the holomorphic function on U given by ∂ ∂ ω( ,..., ). ∂zi1 ∂zim Since every complex vector field can be expressed as a linear combination (3) of the coordinate ∂ vector fields ∂zi , the multilinearity (5) and anti-symmetry conditions (6) of m-forms guarantee that those holomorphic functions determine ω completely. We write dzi1 ∧ · · · ∧ dzim for the holomorphic m-form which satisfies for

1 ≤ j1 < ··· < jm ≤ n

17 the condition ( ∂ ∂ 1, if (i1, . . . , im) = (j1, . . . , jn) (dzi1 ∧ · · · ∧ dzim )( ,..., ) = ∂zj1 ∂zjm 0, else.

It is then clear that we have

X i1 im ω = ωi1...im dz ∧ · · · ∧ dz (7)

1≤i1<···

n Deﬁnition 2.17. The complex vector space of complex m-forms on U ⊂ C will be denoted by Ωm(U). By convention we agree that a complex 0-form is a holomorphic function. We will denote the complex vector space of holomorphic functions on U by O(U). Exercise 2.18. For an inclusion of open subsets U ⊂ V deﬁne a restriction map Ωm(V ) /Ωm(U).

The collection of complex m-forms for varying m interact with each other by means of the exterior, or wedge product ∧, and the exterior differential d. Definition 2.19. The wedge product ∧:Ωm(U) × Ωk(U) / Ωm+k(U) is defined to be the unique collection of maps satisfying for ω, ω0 ∈ Ωm(U), η ∈ Ωk(U), and f ∈ O(U) (a) ω ∧ η = (−1)mkη ∧ ω

(b) (ω + ω0) ∧ η = ω ∧ η + ω0 ∧ η (c) (fω) ∧ η = f(ω ∧ η).

i1 im i1 im (d) for 1 ≤ i1 < . . . im ≤ n, (dz ) ∧ · · · ∧ (dz ) agrees with the m-form dz ∧ · · · ∧ dz deﬁned above.

0 (e) For U ⊂ U an open subset we have (ω ∧ η)|U 0 = (ω|U 0 ) ∧ (η|U 0 ). The exterior differential is a linear map Ωm / Ωm+1, uniquely characterised by a few axioms. Definition 2.20. The exterior differential is the unique collection of maps Ωm(U) / Ωm+1(U), which satisfies for ω, ω0 ∈ Ωm(U), η ∈ Ωk(U), and f ∈ O(U)

(a) d(ω + ω0) = dω + dω0

Pn ∂f i (b) d(fω) = i=1 ∂zi dz ∧ ω + fdω (c) d(ω ∧ η) = dω ∧ η + (−1)mω ∧ dη.

0 (e) For an open subset U ⊂ U we have (dω)|U 0 = d(ω|U 0 ). Diﬀerentiating a complex m-form twice leads to a vanishing (m + 2)-form.

n Lemma 2.21. For a complex m-form ω ∈ Ωm(U) deﬁned on an open subset U ⊂ C we have d2ω = 0.

18 Proof. Since ω can be written as P ω dzi1 ∧ · · · ∧ dzim it suﬃces to check this 1≤i1<···

n n n X ∂f X X ∂2f d2 = d dzi = dzj ∧ dzi. ∂zi ∂zj∂zi i=1 j=1 i=1

Since dzi ∧ dzj = −dzj ∧ dzi, and dzi ∧ dzi = 0, the sum on the right hand side consists of n(n − 1) terms, which can be paired to cancel each other out.

This discussion shows that as far as computations are concerned, a complex m-form is just a n collection of m holomorphic functions. However, their definition as alternating multi-linear forms on holomorphic vector fields, and equation (7) are more than mere bookkeeping devices. They help us to pin down the behaviour of m-forms with respect to biholomorphic coordinate changes. The key difference to the theory of complex vector fields is that complex m-forms can be pulled back along an arbitrary holomorphic map, without imposing the existence of a holomorphic inverse.

n n Deﬁnition 2.22. For open subsets U ⊂ C 1 , V ⊂ C 2 , a holomorphic map ψ : U / V , and a complex m-form ω on V , we deﬁne

∗ X ∗ 1 ∗ m X 1 m ψ ω = (ωi1<···

1≤i1<···

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ One sees easily that ψ (ω1+ω2) = ψ ω1+ψ ω2, ψ (fω) = (f◦ψ)f ω, and ψ (ω∧η) = ψ ω∧ψ η. Moreover, pullback is compatible with exterior diﬀerentiation ψ∗dω = d(ψ∗ω).

n n Lemma 2.23. Let U ⊂ C 1 , V ⊂ C 2 be open subsets, and ω ∈ Ωm(V ). For a holomorphic function φ: U / V we have dφ∗ω = φ∗dω. Proof. Since φ∗ commutes with sums and wedge products, and d is compatible with wedge products by means of a sort of Leibniz rule (see (c) of Definition 2.20), it suffices to prove the identity for complex 1-forms (since every general complex m-form can be written as a finite sum of m- forms obtained by wedge products of complex 1-forms). Henceforth, we assume m = 1, and write Pn2 i ∗ Pn2 i ∗ Pn2 i ω = i=1 ωidz . We have φ ω = i=1(ωi ◦ φ)dφ . Therefore, dφ ω = i=1(d(ωi ◦ φ) ∧ dφ + 2 i ∗ Pn2 Pn1 j 2 (ωi ◦ φ) ∧ d φ ) = φ ( i=1 j=1 dωi ∧ dz ). In our computation we used the identity d = 0 from Lemma 2.21.

ni Lemma 2.24. For open subsets Ui ⊂ C for i = 1, 2, 3, and holomorphic maps φ: U1 / U2, ψ : U2 / U3, we have an equality of linear maps

∗ ∗ ∗ m m (ψ ◦ φ) = φ ψ :Ω (U3) / Ω (U1).

Proof. Since pullback commutes with d and ∧, it suﬃces to prove this assertion for m = 0, that is, ∗ ∗ holomorphic functions f ∈ O(U3). We have (ψ ◦ φ) f = f ◦ (ψ ◦ φ) = (f ◦ ψ) ◦ φ = φ (f ◦ ψ) = φ∗ψ∗f.

n Remark 2.25. For m > n and U ⊂ C , ever complex m-form ω on U vanishes. As we have seen in (7) we have ω = P ω dzi1 ∧ · · · ∧ dzim . But for m > n there is no ordered 1≤i1<···

19 3 Complex manifolds

In this section we introduce the principal object of interest of this course: a complex manifold. A (real) manifold was deﬁned to be a topological space X, satisfying mild technical conditions, endowed with a so-called atlas. An atlas is a collection of open subsets U ⊂ X, and homeomorphisms ' n φ: U −→ U 0, where U 0 ⊂ R is an open subset of Euclidean space. A pair (U, φ) is also called a chart. Given two charts (U, φ) and (V, ψ), we may deﬁne the so-called change-of-coordinates map. It’s the horizontal arrow in the commutative diagram below.

U ∩ V ψ φ

−1 y φ|U∩V ◦ψ |ψ(U∩V ) % ψ(U ∩ V ) / φ(U ∩ V )

−1 In the definition of a (real) manifold one demands that the transition map φ|U∩V ◦ ψ |ψ(U∩V ) is smooth, that is infinitely many times differentiable. The definition of a complex manifold is similar. We demand that each chart (U, φ) defines a n homeomorphism between U and an open subset U 0 ⊂ C . The transition map above is required to be holomorphic.

3.1 The definition Let’s repeat the definition of a complex manifolds with the necessary technical details. To prepare the ground we need to introduce the notion of an atlas. Definition 3.1. Let X be a connected topological space. (a) A (complex n-dimensional) chart is a pair (U, φ), where U ⊂ X is an open subset, and ' n φ: U / U 0, where U 0 ⊂ C is an open subset of a complex space. (b) Two charts (U, φ), (V, ψ) are called compatible, if the change-of-coordinates map

−1 φ|U∩V ◦ ψ |ψ(U∩V ) : ψ(U ∩ V ) → φ(U ∩ V ) is holomorphic. (c) A (complex) atlas U is a set of charts covering X, such that every pair of charts in U is compatible. (d) An atlas is called maximal, if it is not strictly contained in another atlas (that is, it contains every chart (U, φ), which is compatible with all charts in U). Now we are ready to give the definition of a complex manifold. Definition 3.2. A complex manifold is a second countable, Hausdorff topological space X, together with a maximal complex atlas U for every connected component of X. In practice the definition above is impractical. It is next to impossible to “write down” a maximal atlas, trivial cases excluded. In concrete examples, we define the structure of a complex manifold on a topological space X by defining an atlas, and refining it to a maximal atlas.

20 Exercise 3.3. Show that every atlas is contained in a maximal atlas. Do you need the axiom of choice?

n Let’s take another look at the examples covered in Section 1. We have deﬁned atlases for P and V/Γ, where Γ ⊂ V is a lattice in a complex vector space. To complete the proof that these are complex manifolds it remains to verify that the topological spaces are Hausdorﬀ and second countable. S Lemma 3.4. If X = Ui is a countable covering by open subsets, such that each Ui is second i∈N countable, then X is second countable as well.

Proof. Recall that Ui is second countable, if and only if there exists a sequence of open subsets {V } , such that every open subset W ⊂ U can be expressed as the union S W = W . ij j∈N i Vij ⊂W Since × is countable, we have produced a countable collection of open subsets {V } 2 of N N ij (i,j)∈N X with an analogous property: let W ⊂ X be an arbitrary open subset, then we have that [ [ [ [ Vij = Vij = (W ∩ Ui) = W. Vij ⊂W i∈N Vij ⊂W ∩Ui i∈N This shows that X is also second countable.

n As we know from our construction of the charts, P admits a finite covering my topological n n spaces homeomorphic to C , that is, P is second countable by the Lemma above. Similarly, we have remarked that the spaces V/Γ are compact and covered by topological spaces homeomorphic n to open subsets of C . Compactness tells us that a finite number of these is already a covering, and the Lemma above implies that complex tori are second countable too. Since V/Γ is homeomorphic 1 n to (S )2n, we see that complex tori are also Hausdorff spaces. It remains to conclude that P is a complex manifold.

n Exercise 3.5. The topological space P is Hausdorﬀ.

3.2 Holomorphic functions, complex vector fields, and forms The definition of a complex manifold has been set up in a way that it is meaningful to define holomorphic functions on them. From now on we fix a complex manifold X, and denote the maximal atlas, inducing the complex structure on X, by U.

k Definition 3.6. A continuous function f : X / C is called holomorphic, if and only if for every chart (U, φ) ∈ U we have that f ◦ φ−1 is holomorphic. In fact we could have stated the definition above in a slightly different way. We could also say that f is holomorphic, if and only if for every x ∈ X, there exists a chart (U, φ), containing x, such that f ◦ φ−1 is holomorphic at φ(x). However, if (U1, φ1) and (U2, φ2) are two charts containing x, it follows from the definition of −1 −1 a complex manifold that f ◦ φ1 is holomorphic at φ1(x) if and only if f ◦ φ2 is holomorphic at −1 −1 −1 −1 φ2(x). Indeed, we have f ◦ φ1 = (f ◦ φ2 ) ◦ (φ2 ◦ φ1 ), and the change of coordinates map φ2 ◦ φ1 is a biholomorphic map (where it’s defined). This implies the assertion. Similarly we can define complex vector fields on a manifold. The definition is inspired by the one above.

21 Definition 3.7. A complex vector field on X is a rule which assigns to every chart (U, φ) ∈ U a 0 n complex vector field V(U,φ) on U ⊂ C , such that for (U1, φ1), (U2, φ1) ∈ U we have

(φ21)∗ V(U1,φ1) |φ1(U1∩U2) = V(U2,φ2) |φ2(U1 ∩ U2),

−1 where φ21 : φ1(U1 ∩ U2) / φ2(U1 ∩ U2) is the biholomorphic change of coordinates map φ2 ◦ φ1 .

Let’s unravel this definition a little bit. The expression (φ21)∗ refers to the push-forward operation of complex vector fields that we introduced in Definition 2.16. More concretely, we demand that V1 ◦φ−1 (U1,φ1) 21 n  n  . X X ∂φi ∂ V | = Dφ  .  | = ( 21 Vj ◦φ−1) . (U2,φ2) φ2(U1∩U2) 21  .  φ2(U1∩U2)  ∂zj 21  ∂wi n i=0 j=0 V −1 (U1,φ1)◦φ21 Unlike in the definition of a holomorphic function, this definition produces a global object from scratch, by assembling local data (the complex vector fields V(U,φ)) in a consistent way. When we will have introduced the notion of sheaves, we will see that this definition fits very well into a sheaf-theoretic framework. Definition 3.8. A complex m-form on X is a rule which assigns to a chart (U, φ) an m-form ω(U,φ), such that for every pair of charts (U1, φ1), (U2, φ2), and the change of coordinates map φ21 : φ1(U1 ∩ U2) / φ2(U1 ∩ U2) we have

∗ (φ21) ω(U2,φ2)|φ2(U1∩U2) = ω(U1,φ1)|φ1(U1∩U2).

We remind the reader of the operation ψ∗ω defined in Definition 2.22. Formally it is characterised by the fact that ψ∗dω = dψ∗ω, ψ∗(ω ∧ η) = ψ∗ω ∧ ψ∗η, and for 0-forms (that is, holomorphic functions) f we have ψ∗f = f ◦ ψ. Maybe you’re worried that a maximal atlas is too big as a collection of charts (U, φ) to work with complex m-forms or vector fields. The next lemma addresses this question, and as we’ll see in its proof, holomorphic compatibility implies that we don’t have to work with all possible charts to define those objects. S Lemma 3.9. Let i∈I Ui = X be an open covering of X, such that (Ui, φi) ∈ U are charts. Assume m 0 that for every i ∈ I we have a complex m-form ωi ∈ Ω (Ui ) (respectively a complex vector field Vi 0 on Ui ), such that for i, j ∈ I the compatibility condition

∗ (φji) ω2|φj (Ui∩Uj ) = ωi|φi(Ui∩Uj ), respectively

(φji)∗ V1 |φi(Ui∩Uj ) = V2 |φj(Ui ∩ Uj) is satisﬁed. Then, there exists a unique complex m-form ω on X (respectively a complex vector ﬁeld

V), such that ω(Ui,φi) = ωi for all i ∈ I (respectively V(Ui,φi) = Vi). Proof. We will only discuss the case of m-forms, leaving the analogous situation of complex vector fields to the reader. Given an arbitrary chart (U, φ), we need to define a complex m-form ω(U,φ). 0 0 n To do so, we observe that Vi = φ(Ui ∩ U) ⊂ U defines an open covering of U ⊂ C , and that the

22 0 restrictions ωi|Vi still define a compatible collection of locally defined m-forms for U . Therefore, in m 0 0 n order to define ω(U,φ) ∈ Ω (U ), we may temporarily assume that X = U ⊂ C is an open subset of X. n Using the temporary assumption that X ⊂ C , we define for every 1 ≤ j1 < ··· < jm ≤ n, and every i ∈ I, a holomorphic function ωj1...jm;i ∈ O(Vi). The functions are defined by means of the decomposition of the complex m-form on Vi

∗ X j1 jm φi ωi = ωj1...jm;idz ∧ · · · ∧ dz .

1≤j1<···

∗ 0 The compatibility condition (φji) ω2|φj (Ui∩Uj ) = ωi|φi(Ui∩Uj ), implies that for i, i ∈ I we have

0 ωj1...jm;i|Vi∩Vi0 = ωj1...jm;i .

Or, in other words, the family of holomorphic functions ωj1,...,jm;i agree on overlaps of the open n covering {Vi}i∈I of X ⊂ C . Therefore, there exist holomorphic functions ωj1...jm on X, such that P j1 jm ωj1...jm |Vi = ωj1...jm;i. We let ω = (ωU,φ) be 1≤j <···

−1 φ21 = φ2 ◦ φ1 : φ1(U1) / φ2(U2). We have to show that ∗ φ21ω(U2,φ)|φ1(U1∩U2) = ω(U1,φ1)|φ2(U1∩U2). (8)

It suﬃces to establish (8) for an open neighbourhood U of every x ∈ U1 ∩ U2. Choose i ∈ I, such that x ∈ Ui, and consider the chart (U3, φ3) = (Ui ∩ U1 ∩ U2, φi|Ui∩U1∩U2 ). We now obtain a commutative triangle of change of coordinates maps

φ φ (U ∩ U ∩ U ) 31 / φ (U ∩ U ∩ U ) 1 1 2 3 3 5 1 2 3

φ21 ) φ32 φ2(U1 ∩ U2 ∩ U3).

∗ By deﬁnition of ωU1,φ1 we have ω(U1,φ1)|φ1(U1∩U2∩U3) = φ31ω3, where ω3 = ωi|φ1(U1∩U2∩U3). And ∗ ∗ ∗ ∗ similarly, ω(U2,φ2)|φ2(U1∩U2∩U3) = φ21ω3. This implies that φ21ω2 = ω1, since φ31 = (φ32 ◦ φ21) = ∗ ∗ φ21 ◦ φ23 by Lemma 2.24.

3.3 Smooth complex projective varieties We have already seen in Corollary 1.9 that a compact complex manifold cannot be embedded into n C . Nonetheless, the notion of complex submanifolds plays as much of a role in the theory of complex manifolds to construct examples, as it does in the theory of smooth manifolds. The formal deﬁnition of complex submanifolds is as follows.

23 Deﬁnition 3.10. An m-dimensional complex submanifold of an n-dimensional complex manifold X is a subset Y ⊂ X, such that for every y ∈ Y there exists a chart (U, φ) of the complex manifold X, satisfying the condition

' 0 n−m m 1 n 0 φ|Y ∩U : Y ∩ U / U ∩ {0} × C = {(z , . . . , z ) ∈ U |zn−m+1 = ··· = zn = 0}. Every complex submanifold Y ⊂ X is of course a complex manifold itself. The pairs (Y ∩ U, φ|Y ∩U ) for complex charts (U, φ) of X as in the definition above, give rise to charts for Y . We m only need to consider φ|Y ∩U as a homeomorphism from Y ∩ U to an open subset of C . This gives us a covering of compatible charts for Y , and we simply pass to the maximal atlas containing these charts. Since X is a complex manifold, the underlying topological space of X is Hausdorff and second countable. This implies that those topological properties are also held by the underlying space of Y . Summarising this discussion we see that a complex submanifold carries indeed the structure of a complex manifold. We fix k homogeneous polynomials Gi = Gi(z0, . . . , zn) of degree di in n+1 variables. Moreover, n+1 we assume that w = (w0, . . . , wn) ∈ C \0 is a vector, such that Gi(z0, . . . , zn) = 0 for every i = 1, . . . , k, the condition i=,1,...,k ∂Gi j (w) is a surjective linear map (9) ∂z j=0,...,m

n is satisﬁed. Recall that we denote by V (G1,...,Gk) ⊂ P the subset given by tuples (z0 : ··· : n zn) ∈ P , such that Gi(z0, . . . , zn) = 0 for all i = 1, . . . , k.

Proposition 3.11. If the homogeneous polynomials G1,...,Gk satisfy condition (9) for every w ∈ V (G1,...,Gk), then V (G1,...,Gk) is a complex submanifold We will deduce this proposition as a special case of a holomorphic version of the Regular Value Theorem. At ﬁrst we need to explain what it means for a map between complex manifolds to be holomorphic. If X and Y are complex manifolds, and f : X / Y is a continuous map we are going to deﬁne next what it means for f to be holomorphic. For a chart (V, ψ) of Y , we consider a chart (U, φ), such that f(U) ⊂ V . This is possible because f is continuous. With respect to these choice we obtain a commutative diagram f U / V

φ ψ U 0 / V 0. ψ◦f◦φ−1

f Deﬁnition 3.12. A continuous map X /Y between two complex manifolds is called holomorphic, if and only if ψ ◦ f ◦ φ−1 is holomorphic for every pair of charts (U, φ) of X and (V, ψ) of Y , such that f(U) ⊂ V . Since the transition function between overlapping charts is a biholomorphic function, the deﬁ- nition above has a lot of redundancy built in, which makes it feasible to verify in explicit situations that a continuous function is holomorphic.

24 In practice it is suﬃcient to choose charts {(Uiφi)}i∈I for X, and {(Vj, ψj)}j∈J for Y , such that S S i∈I Ui = X, j∈J Vj = Y , and the image f(Ui) of every Ui is contained in some Vj(i). One is then −1 only required to show that for every i ∈ I that the resulting map ψj(i) ◦ f ◦ φi is holomorphic.

f Theorem 3.13 (Regular Value Theorem). Let X / Y be a holomorphic map between complex manifolds, and y ∈ Y a so-called regular value, that is, for every x ∈ f −1(y), and every chart (U, φ) −1 i ∂(f◦φ ) −1 containing x, we have that ( ∂zj )i,j is a surjective linear map. Then, f (y) ⊂ X is a complex submanifold.

k Proof. Without loss of generality one can assume that Y = C is a complex space. This is possible because y ∈ Y belongs to a chart (V, ψ), which is by means of ψ biholmorphically equivalent to an f| k f−1(V ) open subset V 0 ⊂ C . The preimage f −1(y) ⊂ X only depends on f −1(V ) /V . Henceforth, k we replace X by the open subset f −1(V ) and Y by C . Let (U, φ) be a complex chart containing x ∈ f −1(x). We have a holomorphic function f ◦ φ−1 n on U 0 ⊂ C . We use the Implicit Function Theorem 2.8 to construct a smaller open neighbourhood W ⊂ U of x, belonging to a chart (W, ψ), such that we have a homeomorphism ψ : f −1(y)∩W with n−k n {0}k × C ∩W 0 for W 0 an open subset of C . n Using the chart φ we identify U with U 0 ⊂ C . To simplify the notation we will refrain from distinguishing U and U 0. By translating the open subset we may assume without loss of generality n that x = 0 ∈ U and f(0) = 0. We now have an open neighbourhood of 0 ∈ U ⊂ C and a i=1,...,k k ∂f i holomorphic function f : U / C , such that the Jacobi matrix k (0) is a surjective linear ∂z j=1,...,n i=1,...,k ∂f i map. This implies that there exist 0 ≤ i1 < ··· < ik ≤ n, such that i (0) is surjective. ∂z j j=1,...,k By permuting coordinates we may assume ij = j for every j = 1, . . . , k. By Porism 2.9 we see that n there exist open neighbhourhoods W , W 0 of 0 in C , and a biholomorphic map h: W / W 0, such that (f ◦ φ)(z1, . . . , zn) = (z1, . . . , zk). Thus, by restriction we have a homeomorphism

−1 ' 1 n 0 h: f (0) ∩ W / {(z , . . . , z ) ∈ W |z1 = ··· = zk = 0}.

This allows one to conclude that f −1(y) ⊂ X is a complex submanifold.

We cannot directly apply Theorem 3.13 to the system of homogeneous equations G1,...,Gk. n+1 Despite having holomorphic functions Gi : C \ 0 / C, we cannot directly define a holomorphic 1 n 0 n function Gi :(z : ··· : z ) 7→ Gi(z , . . . , z ), because choosing a different representative (e.g. 0 n di λ(z , . . . , z )) modifies the value of Gi by λ . Proof of Proposition 3.11. Let G be a homogeneous polynomial of degree d. By expressing G as a linear combination of degree d monomials (z0)d0 ··· (zn)dn , we see that

n X ∂G zi = d · G(z0, . . . , zn). (10) ∂zi i=0

0 n n+1 In particular if (z , ··· , z ) ∈ C \0 is a non-trivial zero of G, that is G(z0, . . . , zn) = 0, then Pn i ∂G 0 n i=0 z ∂zi = d · G(z , . . . , z ) = 0.

25 n n n n n+1 Recall the charts (Ui, φ) of P , and the inverses ψi : C / P , presented by a map ψei : C / C \0. We will apply the Regular Value Theorem 3.13 to (G1,...,Gk)◦ψei, to see that V (G1,...,Gk)∩Ui is n+1 a submanifold for every i = 0, . . . , n. This will imply that V (G1,...,Gk) ⊂ P is a submanifold. Without loss of generality we may assume i = 0, because the general case may be reduced to this one by relabelling the coordinates z0, . . . , zn. n n 1 n 1 n The map ψe0 : C / P is given by (w , . . . , w ) 7→ (1, w : ··· , w ). Hence we have

1 n 1 n 1 n (G1,...,Gk) ◦ ψei :(w , . . . , w ) 7→ (G1(1, w , . . . , w ),...,Gk(1, w , . . . , w )).

The chain rule implies

i=1,...,k j=0,...,n ! i=1,...,k i ! ∂(Gi ◦ ψe0) ∂Gi ∂(ψe0) = · . ∂wj ∂zj ∂wj j=0,...,n j=0,...,n `=1,...,n

This implies that the matrix on the left hand side is obtained from the Jacobi matrix A(z0, . . . , zn) = i=1,...,k ∂Gi 0 ∂zj j=0,...,n by setting the variable z = 0 and omitting the 0-th column. Omitting the 0-th column n corresponds to considering a complex linear subspace C with the basis (e1, . . . , en) rather than n+1 n (e0, . . . , en) for C . It remains to show that the resulting linear map A|C is still surjective. n+1 To see this we consider the basis of C  1  1 ( w  , e , . . . , e ). .  1 n .wn

Pn i ∂G 1 n As we have seen above, we have i=0 z ∂zi |z0=1,zi=wi = d · G(1, w , . . . , w ) = 0, and therefore the n matrix A vanishes on the ﬁrst vector of this basis. Since A is surjective, we see that A|C has to be surjective as well, which is what we wanted to check.

3.4 Holomorphic maps between manifolds It is time to study more examples of holomorphic maps between manifolds.

n+1 n Example 3.14. The canonical map p: C \0 / P , sending (z0, . . . , zn) to (z0 : ... : zn) is holomorphic.

n Complex projective space P is covered by the n + 1 standard charts (Ui, φi). The preimage −1 0 n i p (Ui) is by deﬁnition equal to the open subset {(z , . . . , z )|z 6= 0}. We choose the family of −1 n+1 n −1 charts {(p (Ui), id)}i=0,...,n for C \0, and {(Ui, φi)}i=0,...,n for P . The map φi ◦ p ◦ id is 0 n z0 zi−1 zi+1 zn given by (z , . . . , z ) 7→ ( zi ,..., zi , zi ,..., zi ), which is indeed holomorphic for zi 6= 0.

f Deﬁnition 3.15. Let X / Y be a holomorphic map between complex manifolds. For ω ∈ Ωm(Y ) we deﬁne f ∗ω to be the complex m-form, which assigns to every chart (U, φ) of X with image ∗ −1 ∗ f(U) ⊂ V for a chart (V, ψ) of Y , the form (f ω)(U,φ) = (ψ ◦ f ◦ φ ) ω(V,ψ). In principle it is possible that X has a chart (U, φ), such that f(U) 6⊂ V for every complex n+1 n chart (V, ψ) of Y . Take for instance the holomorphic map p: C \0 / P . The left hand side

26 n+1 n+1 is already an open subset of C , that is, we have a tautological chart (U, φ) = (C \0, id). But n the image p(U) = P is too big to be contained in a single chart. ∗ This illustrates that in Definition 3.15 we have a priori defined (f ω)(U,φ) only for a subset of the atlas U. However, since f is continuous, the collection of these charts covers X. As we have checked in Lemma 3.9, this is sufficient to define a differential form. n f n In this section we will study an interesting example of a holomorphic map: a blow-up Bl0(C ) / C . n −1 The complex manifold Bl0(C ) has the property that f (C \0) / C \0 is a biholomorphic map, n−1 and f −1(0) is equivalent to a projective space P . This sudden chance in dimension of the fibre is the reason why this construction is referred to as blow-up. n−1 Definition 3.16. Remember that as a set P is the set of 1-dimensional complex subspaces (or n n n n−1 lines) ` ⊂ C . As a topological space, we define Bl0(C ) to be the subset of C × P , as defined by the incidence relation z ∈ `:

n n n−1 Bl0(C ) = {(z, `) ∈ C × P |z ∈ `}. n n We let f : Bl0(C ) / C be the map induced by the projection (z, `) 7→ z. n n n−1 Since Bl0(C ) is defined as a topological subspace of C × P it is clear that it is Hausdorff and second countable. For the same reason we see that f is a continuous map. n n−1 n−1 There’s another continuous map, g : Bl0(C ) / P , mapping (z, `) to ` ∈ P . By definition, n n−1 we have the interesting relation g` = ` ⊂ C for every point of ` ∈ P . n−1 We’ve already introduced standard charts (Ui, φi) for P , and will use a similar construction n −1 n to define charts for Bl0(C ). Since g is continuous, g (Ui) ⊂ Bl0(C ) is open. Changing back to 0 n−1 0 −1 n our usual notation ` = (z : ··· : z ) we have a continuous map φi : g (Ui) / C z0 zi−1 zi+1 zn (u, (z0 : ··· : zn−1)) 7→ (ui, ,..., , ,..., ) zi zi zi zi n −1 with a continuous inverse C / g (Ui) w0(w1, ··· , wi, 1, wi+1, . . . , wn−1), (w1 : ··· , wi : 1 : wi+1 : . . . , wn−1) .

n Exercise 3.17. Verify that coordinate changes are holomorphic, and hence conclude that Bl0(C ) is a complex manifold. n n n n−1 Using these charts we can check that f : Bl0(C ) / C and g : Bl0(C ) / P are holomor- 0 −1 0 n 0 1 i i+1 n−1 phic maps. We have f ◦ (φi) (w , . . . , w ) = w (w , ··· , w , 1, w , . . . , w ), which is certainly 0 −1 0 n 1 i i+1 n−1 holomorphic. Similarly, φi ◦g ◦(φi) (w , . . . , w ) = w , ··· , w , 1, w , . . . , w is a holomorphic map. As we have observed above, the ﬁbres of g are naturally 1-dimensional complex vector spaces. Moreover, we have produced biholomorphic maps

−1 ' p (Ui) / Ui × C

g p1 # | Ui which are ﬁbrewise linear. As we will see in the next section, this is an example of a line bundle (that is, a complex vector bundle of rank 1).

27 4 Vector bundles 4.1 The basics of holomorphic vector bundles We fix a complex manifold X and will study so-called holomorphic vector bundles on X. To simplify the notation we will often omit holomorphic and only speak of vector bundles. Definition 4.1. A (rank n) vector bundle E over X is a holomorphic map of complex manifolds E π / X, such that (a) every fibre π−1(x) has the structure of a finite-dimensional complex vector space,

S −1 αi n (b) there exists an open covering i∈I Ui = X and biholomorphic maps π (Ui) / Ui × C , n (c) such that p1 ◦ αi = π, where p1 : Ui × C / Ui is the canonical projection to the second component. That is, the following diagram commutes

−1 n π (Ui) / Ui × C

# { Ui,

−1 n (d) αi is ﬁbrewise linear, that is, ∀x ∈ X the induced map π (x) × C is a linear isomorphism of complex vector spaces.

In heuristic terms we can think of a vector bundle E π / X as a family of (complex) vector spaces over X. To each point x ∈ X we associate a vector space π−1(X). This family is not supposed to behave pathologically, hence we stipulate that X can be covered by open subsets over n which the vector bundle is trivial, that is, of the shape Ui × C . Vector bundles are locally trivial, but often allow to encode interesting global phenomena. Let E π / X be a vector bundle. The complex manifold E is called the total space, while X is referred to as the base. The holomorphic map π is often called the structure map. This terminology is useful, but distinguishing between the total space of a vector bundle and the vector bundle itself (which actually is a triple (E, X, π)) is too cumbersome in practice. We will therefore often refer to a vector bundle E on X, and suppress the structure map π from our terminology. Deﬁnition 4.2. A (rank n) cocycle datum on a complex manifold X is a pair {Ui}i∈I , (αij)ij∈I2 , S n×n where X = i∈I Ui is an open covering for X, and αij : Ui∩Uj / GLn(C) ⊂ C is a holomorphic function, such that for every triple of indices (i, j, k) ∈ I3 the so-called cocycle identity

αik = αij · αjk is satisﬁed, where · denotes pointwise matrix multiplication.

2 Note that taking (i, j, k) = (i, i, i) we obtain αii = αii, that is, since αii ∈ GLn(C), we must −1 have αii = 1. Similarly, taking (i, j, k) = (i, j, i), we obtain αii = 1 = αijαji, that is, αji = (αij) (taking the inverse matrix).

28 S If E is a vector bundle of rank n on X, then there exists an open covering i∈I Ui = X, and −1 ' n fibrewise linear biholomorphic maps αi : π (Ui) /Ui ×C . Over the intersections Uij = Ui ∩Uj −1 n n we therefore obtain a fibrewise linear map αi ◦αj : Uij ×C /Uij ×C , defined as the composition of the top row of the following commutative diagram

α−1 n j −1 αi n Uij × C / π (Uij) / Uij × C

π p1 p1 & x Uij.

−1 By commutativity of this diagram, and fibrewise linearity, we see that αi ◦ αj is of the shape x 7→ (x, αij(x)), where αij : U / GLn(C) is a holomorphic function taking values in the open n×n subset GLn(C) ⊂ C . This process defines a cocycle datum. To see this, it suffices to check that for every triple of indices (i, j, k), the relation αik = αij · αjk is satisfied on Uijk = Ui ∩ Uj ∩ Uk. −1 This follows from the fact that x 7→ (x, αij) is by definition equal to αi ◦ αj . Hence, we have −1 −1 −1 αi ◦ αj ◦ αj ◦ αk = αi ◦ αk on Uijk, which implies the cocycle relation αik = αij · αjk. Lemma 4.3. Every rank n cocycle datum on a complex manifold can be realised by a vector bundle E π / X. Proof. As a first step we have to construct a topological space E and a continuous map π : E /X. Next, we have to construct an atlas on E, and show that π is holomorphic, and construct bundle charts to show it is indeed a vector bundle. ` n We define E to be the quotient ( i∈I Ui × C )/ ', where (x, z) is equivalent to (y, w) if and 2 only if ∃(i, j) ∈ I , such that x ∈ Ui, y ∈ Uj, x = y, and αij(w) = z. The map π sends the equivalence class of (x, z) to x. The universal property of the quotient topology guarantees that it is continuous. −1 n By construction, π (Ui) is in bijection with Ui × C , by means of the continuous map Ui × n −1 C / π (Ui) sending (x, z) to the equivalence class [(x, z)]. We call the inverse of this homeomorphism αi (it’s continuous by the universal property of the quotient topology). The resulting topological space is second countable, because there exists a countable covering S of X by open subsets: X = n∈ Vn, such that every Vn is contained in one of the Ui (since X N −1 n is second countable). The preimage π (Vn) is thus homeomorphic to Vi × C , and thus second countable itself. Lemma 3.4 implies that E is second countable. Exercise 4.4. Show that E is Hausdorff. It remains to construct an atlas for E. We may assume without loss of generality that every Ui belongs to a chart (Ui, φi) of the complex manifold X. This is achieved by replacing the open subsets Ui by smaller ones, if necessary. We then have homeomorphisms

−1 0 n n π (Ui) / Ui × C = φi(Ui) × C , which serve as our charts. For i, j ∈ I2 the change of coordinates maps are given by (x, z) 7→ (φji(x), αji · z), where φji denotes the change of coordinates map of (Ui, φi) and (Uj, φj). Since this map is constructed from holomorphic functions, it is holomorphic itself.

29 We can now use abstract cocycle data to deﬁne new vector bundles. Deﬁnition 4.5. Let {Ui}i∈I , (αij)(i,j)∈I2 be a cocycle datum for a vector bundle E on a complex T −1 ∨ manifold X. Then, {Ui}i∈I , ((αij) )(i,j)∈I2 is a cocycle datum for the dual vector bundle E . We leave it as an exercise to the reader to check that the isomorphism class of E∨ only depends on the isomorphism class of E.

Deﬁnition 4.6. Let X be a complex manifold.

0 (a) An isomorphism (or equivalence) between two vector bundle E π /X and E0 π /X is a pair of mutually inverse holomorphic maps f : E / E0, g : E0 / E, which are ﬁbrewise linear, such that E / E0

~ X commutes.

n p1 (b) The trivial rank n vector bundle on X is X×C /X, where p1 :(x, z) 7→ x is the projection to the ﬁrst component.

(c) We say that a rank n vector bundle E on X is trivial, if it is isomorphic to the trivial rank n n vector bundle X × C . We will see many examples of non-trivial vector bundles. But how can one actually prove that a vector bundle is non-trivial, or in general that two vector bundles are not isomorphic? As a ﬁrst tool to do so we introduce the vector space of global sections of a vector bundle. It turns out to be an isomorphism invariant, and also explicitly computable. By computing the dimension of this vector space we can compare and distinguish between the examples introduced below.

Deﬁnition 4.7. A global section of a vector bundle E π / X is a holomorphic map s: X / E, such that π ◦ s = idX , that is, the diagram

X s / E

π idX X commutes. We denote the set of global sections of E by Γ(X,E). As we already hinted at above, the set of global sections is actually canonically endowed with the structure of a C-vector space. The reason is that s: X / E is a holomorphic map, which assigns −1 to every x ∈ X an element in the vector space π (x). Given two sections s1, s2 ∈ Γ(X,E) we can −1 therefore deﬁne s1 + s2 : x 7→ s1(x) + s2(x) ∈ π (x). In order for this to make sense however, we have to verify that the resulting map is holomorphic. Similarly, given a complex number λ ∈ C we will check that λ · s: x 7→ λ · s(x) is a holomorphic function.

30 Lemma 4.8. Let E be a vector bundle on X, and s1, s2 ∈ Γ(X,E) two global sections. Then the functions s1 + s2 : X / E defined by fibrewise addition is holomorphic, that is, defines a global section of E. Similarly, for every λ ∈ C, the function λ·s: X /E is holomorphic, and thus defines a global section of E. The triple (Γ(X,E), +, ·) satisfies the axioms of a complex vector space.

Proof. Since E π / X is a vector bundle, every point x ∈ X has an open neighbourhood U ⊂ X, n such that π−1(U) =∼ U × C is equivalent to the trivial vector bundle. The restriction of a section n s ∈ Γ(X,E) is given by a holomorphic function U 7→s|U U ×C , which is of the shape x 7→ (x, f(x)), n where f : U / C is a holomorphic function. Therefore, we see that s1 + s2 is presented by the holomorphic function x 7→ (x, f1(x) + f2(x)). The sum f1 + f2 of two holomorphic functions is of course again holomorphic. After all it can be written as a composition of the holomorphic maps (f1,f2) n n n n n U / C × C , and + · C × C / C . A similar analysis applies to the map λ · s. It only remains to show that with respect to the operations + and · the set Γ(X,E) really is a vector space. −1 The zero element is represented by the zero section 0X : X / E, which maps x / 0 ∈ π (X). It is clear that the axioms of a vector space are satisﬁed, because they hold ﬁbrewise by assumption.

In Deﬁnition 4.2 we introduced so-called cocycle data for vector bundles. One can also express global sections in terms of the cocycle data. Lemma 4.9. If {Ui}i∈I , (αij)(i,j)∈I2 is a cocycle datum for a vector bundle E on X, then we have n a canonical bijection between Γ(X,E) and the collection of holomorphic functions {fj : Uj / C }j∈I , 2 satisfying the condition fi|Ui∩Uj = αijfj|Ui∩Uj for all (i, j) ∈ I .

−1 ' n Proof. By assumption we have an isomorphism of vector bundles αi : π (Ui) /Ui ×C for every / i ∈ I. If s: X E is a global section, we can use the isomorphisms αi to see that s|Ui is of the n shape x 7→ (x, fi(x)), where fi : Ui / C is a holomorphic function. Since two trivialisations differ by αij, we obtain the relation fi(x) = αij(x)fj(x) for all x ∈ Ui ∩ Uj. n Vice versa, given {fj : Uj / C }j∈I , we can define a section s: X 7→ E by sending x ∈ Ui to −1 αi (fi(x)). This does not depend on the choice of i ∈ I, such that x ∈ Ui, due to the relation / fi|Ui∩Uj = αijfj|Ui∩Uj . Since the resulting map X E is holomorphic on each Ui, we see that it defines indeed a global section.

4.2 Line bundles A vector bundle of rank 1 is called a line bundle. Here we use the convention that a line is a 1-dimensional complex vector space. A line bundle is therefore quite literally a bundle of lines over a complex manifold X. Lemma 4.10. We say that a section s of a line bundle L on X is nowhere vanishing if for all x ∈ X we have s(x) 6= 0. There is a canonical bijection between nowhere vanishing sections and ' isomorphisms α: L / X × C. Proof. The trivial bundle X × C has a canonical section given by the constant map x 7→ 1. It is ' clear that this section is nowhere vanishing. Hence, if α: L / X × C is an isomorphism with the trivial bundle, then we obtain an induced section of L.

31 Vice versa, if s ∈ Γ(X,L) is nowhere vanishing, then every y ∈ π−1(x) is a multiple of s(x); y = λs(x). Sending y to (x, λ) defines a bijective map α: L / X × C. To see that this map is biholomorphic, we use that every line bundle is locally trivial, and sections are given by holomorphic functions (the details are left to the reader). In Definition 4.2 we have defined cocycle data to describe vector bundles. The special case of × rank 1 is given by pairs {Ui}i∈I , (αij)i,j∈I2 , where αij : Ui ∩ Uj / C is a holomorphic function which is nowhere zero.

0 Definition 4.11. Let L and L be line bundles on X, given by the cocycle data {Ui}i∈I , (αij)i,j∈I2 0 0 and {Ui}i∈I , (αij)i,j∈I2 . We define the tensor product L ⊗ L to be the line bundle given by the 0 cocycle datum {Ui}i∈I , (αijαij)i,j∈I2 Our current definition of tensor products is depending on the representation of a line bundle through a cocycle datum. Moreover, it doesn’t directly expand to higher ranks. Exercise 4.12. Verify that the isomorphism class of the tensor product of two line bundles is 0 independent of the chosen cocycle datum. Let {Ui}i∈I , (αij)i,j∈I2 , {Ui}i∈I , (αij)i,j∈I2 be cocycle 0 data for rank n vector bundles. Explain why in general {Ui}i∈I , (αijαij)i,j∈I2 won’t be a cocycle datum, unless n = 1. Can you think of an alternative way to define a tensor product for higher rank vector bundles?

After having defined sheaves we will find a more general definition of tensor products, which is not depending on any choices, and works for vector bundles of general rank. Definition 4.13. For a complex manifold X one denotes by Pic(X) the set of isomorphism classes of line bundles on X. With respect to the operation given by ⊗, Pic(X) can be viewed as a group. The inverse of a line bundle L is the dual L∨, and the unit is represented by the trivial line bundle X × C. This group is an important invariant of complex manifolds. The next section is devoted to con- n structing line bundle Ld on complex projective spaces P , such that we get a group homomorphism Z / Pic(X), sending d to the isomorphism class of Ld. For n ≥ 1 we will show that this is really an injective map. Later, after we developed sheaf cohomology, we will be able to prove that this n group homomorphism is in fact an isomorphism, and therefore that Pic(P ) =∼ Z for n ≥ 1.

4.3 Line bundles on projective space n For every integer d ∈ Z we will deﬁne a line bundle Ld on complex projective space P , n ≥ 1. n n+1 × Recall that P can be described as a quotient (C \0)/ C . We will therefore deﬁne Ld as a n+1 × quotient (C \0) × C / C .

n+1 × Definition 4.14. Consider the topological space (C \0) × C with the action of C given by 0 n 0 n d n+1 × λ · (z , . . . , z ; w) = (λz , . . . , λz ; λ w). We define Ld to be the quotient (C \0)/ C , and n 0 n 0 n π : Ld / P the continuous map induced by (z , . . . , z ; w) 7→ (z , . . . , z ). n Recall that we have standard charts (Ui, φi) for P . We define a homeomorphism

−1 αi : π (Ui) / Ui × C

32 by the formula w [(z0, . . . , zn; w)] 7→ (z0 : ··· : zn), . (zi)d Using the deﬁnition of the quotient topology it is easy to see that this is a well-deﬁned continuous −1 0 n 0 n i d map. The inverse map αi sends (z : ... : z ), u to [(z , . . . , z , (z ) u)]. As before, the universal property of the quotient topology implies that this map is continuous. This shows that αi is indeed a homeomorphism. It remains to check that the transition functions αij = αi ◦ αj :(Ui ∩ Uj) × C / (Ui ∩ Uj) × C are holomorphic, and linear in the second component C. Using the formulae given above, we see j d 0 n 0 n z −1 n ' / that αij (z : ... : z ), u = (z : ... : z ), zi u . Composing with (φi) = ψi : C Ui from the right and φi from the left, we obtain the function

1 n 1 n d (w , . . . , w ; u) 7→ (w , . . . , w ; wj u), if i < j. This is certainly a holomorphic function, which is linear in the variable u. We will now prove the following theorem. It’s another example of Serre’s GAGA principle.

0 n Theorem 4.15. The vector space of holomorphic sections of Ld is canonically equivalent to C[z , . . . , z ]d, that is, the vector space of degree d homogeneous polynomials in n+1 variables. In particular, there is no non-zero holomorphic section of Ld for d < 0. We will deduce the theorem from a lemma, which is intuitively clear.

n Lemma 4.16. A holomorphic section s: P / Ld corresponds to a holomorphic section

n+1 s˜ n+1 (C \0) / (C \0) × C ,

× which is C -equivariant. × n+1 Proof of Theorem 4.15 using Lemma 4.16. Let’s determine the C -equivariant sections of (C \0) × C . n+1 n+1 By deﬁnition, (C \0) × C is the trivial line bundle on (C \0). Its holomorphic sections are n+1 therefore the same as holomorphic functions f on (C \0). By Hartog’s Extension Theorem 2.14, n+1 n+1 a holomorphic function on (C \0) extends uniquely to a holomorphic function on C , since we assumed n ≥ 1. We know that holomorphic functions are analytic, that is, there exists an -neighbourhood of n+1 0 ∈ C , where f can be represented by a converging power series

n X 0 d0 n d f(z) = ad0...dn (z ) ··· (z ) ,

d0...,dn≥0

× for z ∈ U(0). The condition that the section is C -equivariant translates into the property f(λz) = d × λ f(z) for λ ∈ C . This implies that the coeﬃcients above vanish for d0 + ··· + dn 6= d, that is, f is a homogeneous polynomial of degree d. It remains to prove the lemma.

33 π n Proof of Lemma 4.16. Let s be a holomorphic section of Ld / P . Using the bundle charts −1 ' αi : π (Ui) / Ui × C we obtain a holomorphic section αi(s) of the trivial bundle Ui × C, corresponding to a holomorphic function fi. On the overlaps Ui ∩Uj the resulting holomorphic functions are related according to d zj fi = fj, zi i d j d −1 n+1 n or equivalently (z ) fi ◦ p = (z ) fj ◦ p on p (Ui ∩ Uj), where p: C \0 / P is the map (z0, . . . , zn) 7→ (z0 : ... : zn). n+1 This implies that there exists a well-defined holomorphic function f : C \0 / C, satisfying i d 0 n −1 f|p (Ui) = (z ) fi ◦ p. Multiplying (z , . . . , z ) with λ therefore has to change the value of f with d × n+1 n+1 λ , that is, f defines a C -equivariant section of (C \0) × C / (C \0). × n+1 Vice versa, any C -equivariant section corresponds to a holomorphic function f : C \0 / C, d × 0 n satisfying f(λz) = λ f(z) for λ ∈ C . We have a function fi : Ui / C, which sends (z : ··· : z ) to 1 0 n (zi)d f(z , . . . , z ). This is a well-defined continuous function by definition of the quotient topology. −1 1 n 1 1 n Precomposing it with φi = ψi we obtain fi ◦ ψi(w , . . . , w ) = 1d f(w ,..., 1, . . . , w ), which is by definition a holomorphic function. For the intersection Ui ∩ Uj of two charts we have the relation d zj n fi = fj, and hence gives rise to a section of Ld / . zi P

0 n Corollary 4.17. For d 6= d the line bundles Ld and Ld0 on P are not equivalent. n ∼ 0 n Proof. For d ≥ 0 we have determined the vector space of global sections Γ(P ,Ld) = C[z , . . . , z ]d 0 0 n in Theorem 4.15. For 0 ≤ d < d we have a non-canonical embedding of vector spaces C[z , . . . , z ]d ,→ 0 n 0 d0−d [z , . . . , z ]d0 , which sends a degree d homogeneous polynomial G to (z ) G. This map is in- C 0 jective, but cannot be surjective, since the degree d0 homogeneous polynomial (z1)d doesn’t lie in its image. This implies that for non-negative numbers d 6= d0 the vector space of global sections are ∼ of diﬀerent dimensions, therefore Ld =6 Ld0 . d zj We have seen that Ld is given by the cocycle (αij) = zi . This implies that L−d is equivalent ∼ 0 to the dual of Ld (see Deﬁnition 4.5). This implies that Ld =6 Ld0 for negative integers d 6= d . 0 ∼ 0 For d < 0 < d we know that Ld = Ld is impossible, because Ld does not have any global sections.

n After having developed the appropriate tools we will show that every line bundle on P is 1 equivalent to one of the line bundles Ld. For P even more is true. 1 Theorem 4.18 (Birkhoff, Grothendieck). Every rank m vector bundle E on P can be uniquely written as a direct sum of line bundle Ld1 ⊕ · · · ⊕ Ldm . Birkhoff didn’t formulate his theorem in terms of line bundles on projective space. He proved a factorisation theorem for matrices in Laurent polynomials, known as Birkhoff factorisation. Grothendieck recognised the geometric content of Birkhoff’s result and reproved the result (in higher generality) using sheaf cohomology.

34 4.4 Line bundles and maps to projective space n Now that we have studied line bundles on complex projective space P , we will try a different approach. Starting with a line bundle L on a complex manifold X, and sections s0, . . . , sn of L, we n will use this datum to construct a holomorphic map X / P . Definition 4.19. For a line bundle L on a complex manifold X we say that a tuple of sections (s0, . . . , sn) ∈ Γ(X,L) generates L, if for every x ∈ X there exists at least one i ∈ {0, . . . , n}, such n that si(x) 6= 0. In this case, we define a map f : X / P by stipulating x 7→ (s0(x): ··· : sn(x)).

This construction is as simple, as it is brilliant. It doesn’t make any sense to think of si(x) as an −1 element of C, but rather si(x) belongs to the 1-dimensional complex vector space π (x). But all × 1-dimensional complex vector spaces are isomorphic to C, and the set of automorphisms of C is C . In order to define (s0(x): ··· : sn(x)) we are formally forced to choose an arbitrary identification of −1 π (x) with C. However, as we are only interested in (s0(x), . . . , sn(x)) up to rescaling, we obtain n a well-defined element of P . The assumption that the tuple (s0, . . . , sn) generates L was needed, n because (0 : ··· : 0) does not define a point in P . Lemma 4.20. The map f defined above in 4.19 is holomorphic.

' Proof. Let (V, α) be a bundle chart of L. That is, α: π−1(V ) / V × C as a line bundle bundle. The sections (α(sj))j=0,...,n can then be understood to be holomorphic functions g0, ··· , gn, that is, n+1 in particular we have a holomorphic map g = (g0, . . . , gn): V / C . By assumption, the image n+1 of this map is contained in C \0, and it is therefore meaningful to form the composition p ◦ g, n+1 n where p: C \0 / P is the canonical projection. Since the composite of two holomorphic maps is again holomorphic, we see that p ◦ g is holomorphic. However, by deﬁnition, p ◦ g = f|V . The n complex manifold X can be covered by bundles charts (V, α), and this implies that f : X / P is holomorphic everywhere.

n n ∼ 0 n Lemma 4.21. Consider the line bundle L1 on P and recall that Γ(P ,L1) = C[z , . . . , z ]1 is equivalent to the space of homogeneous polynomials in n + 1 variables. The tuple of sections (z0, . . . , zn) n n / n is generating and the resulting holomorphic map f : P P is the identity map idP .

−1 n ' Proof. We can use the local computations above to identify the map f◦ψ0, where ψ0 = φ0 : C U/ 0 ⊂ n P . The sections (z0, . . . , zn) correspond to the holomorphic functions (w1, . . . , wn) 7→ (1, w1, . . . , wn). And since the constant function 1 is never 0, we see that at least for every point x ∈ U0 there exists one section which is never 0. Permuting coordinates we see that this is true for any chart (Ui, φi) n of P . 1 n 1 n The holomorphic map f ◦ ψ0 is given by (w , . . . , w ) 7→ (1 : w : ··· : w ), which is equal to n ψ0. This implies that f|U0 is equal to the identity map. Since U0 is dense in P (or by permuting n coordinates), we obtain that f = idP .

f n ∗ Vice versa, any holomorphic map X / P induces a line bundle L = f L1, generated by n+1 ∗ sections si = f xi. And of course, f is precisely the holomorphic map corresponding to this datum. The construction ﬁts into a more general context of pulling back vector bundles and sections. We begin with the general notion of ﬁbre products.

35 f g Definition 4.22. Let X / Y and Z / Y be continuous maps of topological spaces. The fibre product X ×Y Z is defined to be the following subset of the product {(x, y) ∈ X × Y |f(x) = g(z)} with the subspace topology.

The projections p1 : X × Z / X and p2 : X × Z / Z restrict to maps from X ×Y Z to X respectively Y . By construction of the ﬁbre product they belong to a commutative square

X ×Y Z / Z

X / Y. In fact, this commutative square is the universal such diagram. That is, if you have a space W and h k maps W / X, W / Y , such that f ◦ h = g ◦ k, then there exists a unique map W / X ×Z Y , w 7→ (h(w), k(w)), ﬁtting into a commutative diagram W k ∃! $ & X ×Y Z / Z h

X / Y.

The intuition behind fibre products or pullbacks is as follows: X ×Y Z / X is a space with “the −1 −1 same fibres” as Z / Y . That is, for x ∈ X, the fibre of p1 (x) is the same as g (f(x)). This follows directly from the set-theoretic definition of the fibre product (but also from the universal property). Since fibre products leave the fibres of a map un-affected, they are the perfect device to pullback vector bundles! If the fibres of Z / Y have a vector space structure, the same is true for the fibres of X ×Y Z / X, because every fibre of the second map is a fibre of the first one. Lemma 4.23. For π : E / Y a vector bundle over a complex manifold Y , and f : X / Y a ∗ holomorphic map, we define the pullback f E to be the fibre product X ×Y E together with the continuous map πf ∗E : X ×Y E / X. This construction can be refined to define a holomorphic vector bundle on X. Proof. The proof can be divided into several steps, using only the formal definition of fibre products. n ∼ n (1) The pullback of the trivial vector bundle: X ×Y (Y × C ) = X × C . This follows directly n n from the definition of the fibre product: X ×Y (Y × C ) = {(x, y, z) ∈ X × Y × C |f(x) = y} =∼ X × Y , because y = f(x) means that y is redundant.

(2) X ×Y (?) deﬁnes a functor from topological spaces with a map to Y , Z / Y . That is, in particular if Z ' / Z0 is a homeomorphism ﬁtting into a commutative diagram Z ' / Z0

~ Y,

36 ' 0 then we get an induced homeomorphism X ×Y Z / X ×Y Z .

π (3) If ({Ui}i∈I , (αi)i∈I ) is a cocycle datum for E / Y , that is, we have that αij = αi ◦ αj is a holomorphic function Ui ∩ Uj / GLn(C), then we want to obtain from this a cocycle datum ∗ S S −1 for f E. Since i∈I Ui = Y we see that i∈I f (Ui) = X. The definition of fibre products −1 −1 −1 −1 readily implies that πf ∗E(f (Ui)) = f (Ui) ×Ui π (E). By (2), the homeomorphism −1 ' n ∗ −1 ' −1 n αi : f (Ui) /Ui×C induces therefore a homeomorphism f αi : f (Ui) /f (Ui)×C . ∗ ∗ ∗ −1 −1 It remains to check that the maps f αij = f αi ◦ f αj are holomorphic maps f (Ui) ∩ f (Ui) = −1 ∗ f (Ui ∩ Uj) / GLn(C). Since f αij = αij ◦ f, and αij and f are supposed to be holomorphic, we are done. We know that holomorphic sections s ∈ Γ(Y,E) correspond to families of holomorphic functions / n −1 / n hi : Ui C , such that αij ·(hj|Ui∩Uj ) = hi|Ui∩Uj . The compositions hi ◦f : f (Ui) C define −1 −1 holomorphic functions, satisfying the relations (αij ◦ f) · (hj ◦ f|f (Ui∩Uj )) = hi ◦ f|f (Ui∩Uj ). Hence we obtain a holomorphic section f ∗s ∈ Γ(X, f ∗E). f ∗s Using the same arguments as in the proof above, one sees that the continuous map X /f ∗E = ∗ X ×Y E is given by x 7→ (f(x), s(f(x))). In particular, the section f s does not depend on the choice of a cocycle datum for E.

i i Deﬁnition 4.24. Let L1,L2 be line bundles on a complex manifold X, and (s0, . . . , sn) ∈ Γ(X,Li) 1 1 2 2 be a generating set of sections. We say that the tuples (L1, s0, . . . , sn) and (L2, s0, . . . , sn) are ∼ equivalent if there exists an isomorphism of line bundles L1 = L2, such that the induced isomorphism ∼ 1 1 2 2 Γ(X,L1) = Γ(X,L2) maps (s0, . . . , sn) to (s0, . . . , sn).

f n Theorem 4.25. Let X be a complex manifolds. The set of holomorphic maps X / P is in bijection with the set of equivalence classes of tuples (L, s0, . . . , sn), where L is a line bundle on X, and (s0, . . . , sn) is a generating tuple of sections of L. n n Proof. Let’s denote the set of holomorphic maps X / P by Hom(X, P ), and the set of isomorphism classes of tuples (L, s0, . . . , sn) by Ln+1(X). n We have already constructed a map A: Ln+1(X) / Hom(X, P ), by sending (L, s0, . . . , sn) to the map f : x 7→ (s0(x): ··· : sn(x)). n f n ∗ ∗ 0 ∗ n There’s also a map B : Hom(X, P ) L/ n+1(X), which sends X / P to the tuple (f L1, f (z ), . . . , f (z )). To decipher this notation, recall that Γ(X,L1) is equivalent to the vector space of degree 1 homo- i 0 n geneous polynomials. That is, every (z ) ∈ C[z , . . . , z ]1 gives rise to a global section of L1. It remains to verify that A and B are mutually inverse maps. We’ll first take a look at A ◦ B = n n / idHom(X,P ). Let f : X P be a holomorphic map. We need to check that f is equal to the ∗ ∗ 0 holomorphic map g assigned to the tuple (f L1, s0, . . . , sn). By definition, g : x 7→ (f (z )(x): ··· : f ∗(zn)(x)). ∗ i ∗ i We have seen above that f (z ): X /f L1 is equal to the map x 7→ (f(x), (z ◦f)). In Lemma n n 0 n 4.21 we showed that the holomorphic map P / P associated to the tuple (L1, z , . . . , z ) is the identity map. This implies that this map is equal to f. f / n It remains to show that B ◦ A = idLn+1(X). Let (L, s0, . . . , sn) ∈ Ln+1(X), and X P ∗ ∼ the associated holomorphic map. We have to prove that f L1 = L, and with respect to this ∗ i identification we have si = f (z ).

37 Sn n Consider the standard open covering i=0 Ui = P . We can deﬁne Ui as the locus where the sec- i −1 tion z 6= 0 of L1. Therefore, the preimage f (Ui) is equal to the open subset where si 6= 0, respec- ∗ i ' −1 −1 / tively f (z ) 6= 0. Using Lemma 4.10, we see that we have trivialisations L|f (Ui) f (Ui) × ∗ ' −1 −1 / C × C and f L1|f (Ui) f (Ui)×C, induced by the nowhere vanishing sections si respectively ∗ i ∗ −1 ∼ −1 f (z ). By composition we obtain an isomorphism βi : L|f (Ui) = f L1|f (Ui). It suﬃces to check that on Ui ∩ Uj the two isomorphisms βi and βj agree. Replacing the sj ∗ i ∗ j section si by the section sj introduces the factor . Similarly, changing f (z ) by f (z ) the factor si i z ◦f si j = . And we see that both changes cancel each other out. z ◦f sj

Deﬁnition 4.26. Let X be a complex manifold and L a line bundle on X. We say that L is very ample, if there exist sections s0, . . . , sn ∈ Γ(X,L) that generate L and yield an embedding into n projective space X,→ P as a submanifold.

5 Sheaves

Few, light, and worthless,-yet their triﬂing weight Through all my frame a weary aching leaves; For long I struggled with my hapless fate, And staid and toiled till it was dark and late, Yet these are all my sheaves.

Elizabeth Akers Allen

This section is devoted to determining the common denominator of the following constructions we have encountered in this class: holomorphic functions, complex vector fields, complex m-forms, and sections of vector bundles. Locally, all these objects can be described in terms of (a tuple of) holomorphic functions on an open subset of a complex vector space. However, globally, they exhibit novel features, and for example the section of a line bundle should not be confused with a holomorphic function. We have seen in Theorem 4.15 that for d ≥ 0, the vector space of global 1 sections of Ld on P is at least 3-dimensional. That’s quite different from the 1-dimensional vector 1 space of holomorphic functions on P (all of which are constant). For a complex manifold X, equipped with a line bundle L, we temporarily denote for an open subset U ⊂ X the vector space of holomorphic functions, sections of line bundles, complex m-forms, or complex vector fields on U by F(U). For U ⊂ V we have a linear restriction map F(V ) / F(U), which takes a element f ∈ F(V ) (think of f as a holomorphic function, section of a line bundle, complex vector field, or complex m-form on V ), and sends it to the restriction f|U . Restriction of functions, etc., satisfies a few unsurprising identities. For example, considering the tautological inclusion U ⊂ U, the map f 7→ f|U is the identity map. Similarly, for a triple of inclusion U ⊂ V ⊂ W , we have (f|V )|U = f|U . We can summarise our discussion so far as observing that functions, sections of line bundles, etc. have in common that they can be restricted to subsets of their domain of definitions. Using technical jargon, we have just seen our first examples of presheaves. However, there are a lot of examples of presheaves, which don’t possess the same formal properties as holomorphic functions and sections of line bundles. To sift chaff from the wheat1, we 1Agricultural terminology is very popular in this area.

38 stipulate one further condition in order to deﬁne sheaves. It’s known as the sheaf condition, but might as well be called the jigsaw puzzle principle: matching local fragments can be assembled into a global picture. Given an open covering {Ui}i∈I of X, and elements fi ∈ F(Ui) for every i ∈ I (that is, holomorphic functions, sections of line bundles, etc.), such that on the overlaps Uij = Ui ∩ Uj we have fi|Uij = fj|Uij , there exists a unique f ∈ F(X), such that f|Ui = fi.

5.1 Presheaves The notion of sheaves intends to provide an abstract generalisation of functions on a topological space. As a predecessor one has to study presheaves, which aim to extract the most basic property of functions. Deﬁnition 5.1. Let X be a topological space, a presheaf F on X consists of the following data: - an abelian group F(U) for every open subset U ⊂ X,

V - a homomorphism rU : F(V ) / F(U) for every inclusion of open subsets U ⊂ V ⊂ X, - such that for every triple of inclusions of open subsets U ⊂ V ⊂ W ⊂ X we have a commutative diagram rW F(W ) V / F(V )

V rU rW U $ F(U),

W W V that is, we have rU = rV ◦ rU . V We will often denote the effect of rU by using classical notation for restriction of functions. E.g., V for s ∈ F(V ) we write rU (s) = s|U . In order to put this definition into context, let us observe that functions on a topological space X form a presheaf. If V ⊂ X is an open subset, and f : V / C a continuous function, then for every open subset U ⊂ X we obtain a continuous function f|U . If f : W / C is continuous, and we have U ⊂ V ⊂ W , then by definition

f|U = (f|V )|U . There are a lot more examples, using complex-valued functions satisfying stricter conditions than just continuity. Example 5.2. We ﬁx a topological space X. In some of the examples, e.g. (a), (d), (e), (f), and (g) we will assume that X is endowed with the structure of a complex manifold.

(a) (The presheaf of holomorphic functions on C.) For U ⊂ X deﬁne OX (U) to be the abelian group of holomorphic functions f : U / C. For U ⊂ V ⊂ C deﬁne rV,U : OX (V ) / OX (U) to be the map sending f : V / C to f|U . pre (b) We denote by Z the presheaf of constant, integer-valued functions. That is, to an open subset U ⊂ X we assign the abelian group Z, whose elements we interpret as constant functions on U.

39 (c) We denote by Z the presheaf of locally constant integer-valued functions. To an open subset f U ⊂ X we assign the abelian group of continuous functions U / Z. Since Z is a discrete topological space, we see that every x ∈ X has an open neighbourhood U = f −1(f(x)), such that f|U is constant. However, f will not be globally constant. For example, if X = [0, 1] ∪ [2, 3], we could consider the function, which is equal to 1 on [0, 1], and equal to 2 on [2, 3].

× (d) We also have the presheaf OX , which assigns to U ⊂ X the abelian group of holomorphic functions f : U / C, which are nowhere zero. m (e) We denote by ΩX the presheaf of complex m-forms, that is, to an open subset U ⊂ X we assign the vector space of complex m-forms.

(f) The presheaf of complex vector ﬁelds on X is denote by VX . (g) Let E be a vector bundle on X. We have a presheaf of sections E, which maps an open subset U ⊂ X to the vector space of sections s deﬁned over U:

> E s π U / X.

Presheaves on a fixed topological space X form a category. In order to see this we have to define first what we mean by a morphism of presheaves. Definition 5.3. Let X be a topological space, and F and G be presheaves on X.A morphism of f f presheaves F / G is given by a homomorphism of abelian groups F(U) U / G(U) for every open subset U ⊂ X, such that for every inclusion U ⊂ V ⊂ X of open subsets, we have a commutative diagram f F(V ) V / G(V )

V V rU rU f F(U) U / G(U).

For instance we have a morphism Z / OX , which takes a locally-constant Z-valued function, and views it as a holomorphic function on X. Or we could take f ∈ OX (U) and send it to the × exp × nowhere vanishing holomorphic function exp(f) ∈ OX (U). This deﬁnes a morphism OX / OX .

f Deﬁnition 5.4. A morphism F / G of presheaves on a topological space X is called an isomorphism, if there exists a morphism g : G / F, such that f ◦ g = idG, and g ◦ f = idF . As we are already used to from the world of sets, groups, and vector spaces, we can describe isomorphisms also as bijective morphisms of presheaves.

f Deﬁnition 5.5. We denote by F / G a morphism of presheaves on X.

(a) We say that f is injective, if for every open U ⊂ X, the map fU : F(U) / G(U) is injective.

40 (b) ... f is surjective, if ... the map fU is surjective. (c) We say that f is bijective if it is injective and surjective.

f It is clear that if f is bijective, then for every open U ⊂ X the map F(U) U / G(U) is bijective. −1 In particular, there exists an inverse map fU , and it remains to see that the collection of such maps forms a morphism of presheaves. For every inclusion of open subsets U ⊂ V ⊂ X, we have V V −1 −1 that rU ◦ fV = fU ◦ rU . Applying fV from the right, and fU from the left, we obtain the relation −1 V V −1 −1 fU ◦ rU = rU ◦ fV . That is, f is indeed a morphism of presheaves.

5.2 Sheaves A presheaf is called a sheaf, it a compatible collection of local sections can be assembled into a global section. Definition 5.6. Let F be a presheaf on a topological space X, such that the following condition is S satisfied: for every open covering U = i∈I Ui, and a collection of sections si ∈ F(Ui), such that si|Ui∩Uj = sj|Ui∩Uj for every pair of indices (i, j), there exists a unique section s ∈ F(U), such that s|Ui = si. We then say that F is a sheaf on X. The definition of a sheaf is also often stated by imposing the following two conditions. As before S U = i∈I Ui is an open covering.

(S1) If s, t ∈ F(U) are two sections, such that s|Ui = t|Ui for all i ∈ I, then s = t.

(S2) If si ∈ F(Ui) is a collection of sections, such that si|Ui∩Ui = sj|Ui∩Uj for all pairs of indices,

then there exists a sections s ∈ F(U), such that s|Ui = si for all i ∈ I. The condition (S2) corresponds to ∃ in our definition of a sheaf, while (S1) is equivalent to unicity in the definition above. Most of the examples of presheaves we’ve encountered so far, are actually sheaves. Example 5.7. We fix a topological space X. In some of the examples, e.g. (a), (d), (e), (f), and (g) we will assume that X is endowed with the structure of a complex manifold.

(a) The presheaf OX of holomorphic functions on X is a sheaf. Indeed, if U ⊂ X is an open S subset, and U = i∈I Ui an open covering of U, and fi : Ui / C are holomorphic functions, f / such that fi|Ui∩Uj = fj|Ui∩Uj , then there exists a unique holomorphic function U C, such

that f|Ui = fi. Indeed, we can deﬁne f by the following rule: f(x) = fi(x), if x ∈ Ui. It only remains to check that this expression does not depend on the choice of an Ui containing x. If x ∈ Ui and x ∈ Uj, then x ∈ Uij = Ui ∩ Uj. That is, we have fi(x) = fj(x). This yields / a well-deﬁned map of sets f : U C. Since holomorphicity is a local property, and f|Ui is holomorphic for every i ∈ I, we see that f is locally holomorphic, hence holomorphic.

pre (b) The presheaf of constant integer-valued functions Z is not a sheaf in general. To see this for a generic topological space, we choose an open subset U ⊂ X, such that U = U0 ∪ U1, with U0 ∩ U1 = ∅. In other words, U has (at least) two connected components. We deﬁne f0 : U0 / Z to be the constant function with value 0, and f1 : U1 / Z to be the constant pre function with value 1. If Z was a sheaf, there would be a constant function f : U / Z,

such that f|U0 = f0 and f|U1 = f1. Since f has to be constant, this would imply 0 = 1.

41 (c) In (b) we’ve seen a ﬁrst example of a presheaf which is not a sheaf. However, it is clear that this can be remedied by considering the presheaf of locally constant integer-valued functions Z, which is always a sheaf. Indeed, a function U / Z is locally constant, if and only if it is continuous, where we endow Z with the discrete topology. One can then argue similarly to (a) to verify the sheaf condition.

× (d) The presheaf of nowhere zero holomorphic functions Ox is a sheaf. If fi : Ui / C are nowhere

zero holomorphic functions, such that fi|Uij = fj|Uij , then we’ve seen in (a) that there exists a unique holomorphic function f : U / C, agreeing with the locally deﬁned functions fi on each Ui. Since the functions fi is nowhere zero, we obtain f(x) 6= 0 for every x, which belongs S to a Ui. But U = i∈I Ui, hence f is also nowhere 0. m (e) The presheaf ΩX of complex m-forms is a sheaf. We have veriﬁed the sheaf condition in Lemma 3.9.

(f) The presheaf of complex vector fields VX on X is a sheaf. This has been shown in Lemma 3.9. (g) Let E be a vector bundle on X. The presheaf of sections E, which maps an open subset U ⊂ X to the vector space of sections s defined over U, is a sheaf. To see this, we recall that a section is given by a holomorphic map U / E, fitting into a commutative diagram.

> E s π U / X.

/ / If si : Ui E is a collection of sections, such that si|Uij = sj|Uij , then we can deﬁne s: U E by s(x) = si(x), if x ∈ Ui. As in (a) we see that s is a well-deﬁned holomorphic function s U / E. We have π(s(x)) = π(si(x)) = x, hence the diagram above commutes, and s is indeed a section.

5.3 Locally free sheaves and vector bundles In this paragraph we’ll see that vector bundles on complex manifolds can be described as so-called locally free sheaves (of finite rank). In order to justify the nomenclature we recall the notion of an R-module, and what it means for an R-module M to be free, and finitely generated. Definition 5.8. Let R be a ring (with a unit 1 ∈ R). An R-module M consists of an abelian group (M, +), together with a scalar multiplication ·: R × M / M, such that for every λ ∈ R, the map m 7→ λ · m defines a homomorphism of abelian groups, and the resulting map R / End(M) is a ring homomorphism. That is, we have λ · (m + n) = λ · m + λ · n, 1 · m = m for every m ∈ M, and (λ + µ) · m = λ˙m + µ · m, as well as (λµ) · m = λ · (µ · m). If k is a field, then a k-module is simply a k-vector space. For every non-negative integer n ≥ 0, we have the R-module Rn. We say that an R-module M is free of rank n, if it is isomorphic as an R-module to Rn. We say that an R-module M is finitely m generated, if there exists a surjection of R-modules R M.

42 Every free R-module of rank n is finitely generated, but the converse is not true in general (unless R is a field). Let R be a commutative ring with a unit, which is not a field. Then there exists a non-zero ideal I ⊂ R. The quotient R/I is a finitely generated R-module, since we have a surjection R R/I. We leave it to the reader to conclude the proof that R/I cannot be free. Now we are ready to define sheaf of rings, and sheaves of modules. In order to give compact definitions (in a non-topological sense), we use that there is a natural notion of products of two presheaves F × G on X, which respects the sheaf condition. Definition 5.9. Let X be a topological space, and F, G two presheaves on X. We define F × G to be the presheaf which assigns to an open subset U ⊂ X the abelian group F(U) × G(U). The product of two sheaves is again a sheaf. This follows from the definition, since the sheaf condition holds componentwise. And sections of (F × G)(U) correspond to an ordered tuple (s, t), where s ∈ F(U) and t ∈ G(U). Definition 5.10. Let X be a topological space. (a) A sheaf of rings R on X is given by a sheaf R, together with a multiplication map · R × R / R of sheaves, such that for every open subset U ⊂ X the abelian group R(U) is a ring, with respect to the multiplication ·: R(U) × R(U) / R(U). (b) Let R be a sheaf of rings on X. A sheaf of R-modules M is a sheaf M on X, together with a multiplication map ·: R × M / M of sheaves, such that for every U ⊂ X, the map R(U) × M(U) / M(U) endows M(U) with the structure of an R(U)-module. It’s illustrative to unpack the definition of sheaves of rings, and sheaves of modules. A sheaf of rings R is given by a sheaf, such that for every open U ⊂ X, the abelian group R(U) is additionally V endowed with the structure of a ring, and all the restriction maps rU : R(V ) / R(U), for every inclusion of open subsets U ⊂ V , are ring homomorphisms. If M is a sheaf of R-modules, then for every open U ⊂ X we have the structure of an R(U)- module on M(U). Moreover, if U ⊂ V is an inclusion of open subsets, we have for f ∈ R(V ), and m ∈ M(V ) the compatibility condition

(f · m)|U = (f|U ) · (m|U ).

We are now almost ready to define locally free sheaves of modules. The definition below requires us to restrict sheaves on X to open subsets U ⊂ X. This makes sense, because an open subset U 0 ⊂ U of an open subset U ⊂ X, also defines an open subset U 0 ⊂ X of X. A sheaf F on X gives therefore rise to a sheaf F |U on every open subset U ⊂ X. Definition 5.11. Let X be a topological space, and R a sheaf of rings on X. A sheaf of R-modules S M is said to be locally free of rank n, if there exists an open covering X = i∈I Ui, such that ∼ n M |Ui = R |Ui as a sheaf of R-modules. The goal of this paragraph is to prove the following theorem. Theorem 5.12. Suppose that X is a complex manifold. There is a one-to-one correspondence between complex vector bundles E of rank n on X, and locally free sheaves of OX -modules M of rank n.

43 The proof is broken down into several steps. At ﬁrst we have to explain how one can pass from a complex vector bundle E to a locally free sheaf of OX -modules.

Lemma 5.13. The sheaf of sections E of a rank n complex vector bundle is a locally free OX -module of rank n.

Proof. If U ⊂ X is an open subset, and s: U /E a section of E over U, then for every holomorphic function f ∈ OX (U) we have a section f ·s: U /E. For every x ∈ U,(f ·s)(x) = f(x)s(x), and it’s not difficult to check that this defines a holomorphic section (see for instance the proof of Lemma 4.8 for similar arguments). Hence E is naturally endowed with the structure of an OX -module. S ∼ n By definition, there exists an open covering X = i∈I Ui, such that E|Ui = Ui × C as a vector n bundle. This implies that E|Ui is equivalent to OX , because a section of a trivial vector bundle n U × C / U, corresponds to an n-tuple of holomorphic functions over U. n n Lemma 5.14. The set of isomorphisms of sheaves of OX -modules, OX / OX is equivalent to the set of holomorphic maps X / GLn(C). Proof. We begin by considering the abstract situation of automorphisms of free R-modules, where f R is a ring with unit. Every R-linear map Rn / Rn is given by an (n × n)-matrix A ∈ Rn×n. It is an isomorphism, if and only if A ∈ GLn(R). Hence, for every open subset U ⊂ X, we obtain a n n matrix AU ∈ GLn(OX (U)). Since AU represents fU , where f : OX / OX is a map of sheaves, we have for every inclusion U ⊂ V of open subsets the relation AV |U = AU . The matrix AX corresponds to a holomorphic function X / GLn(C), and as we have seen, it defines the map of n n sheaves f : OX / OX . Proof of Theorem 5.12. We have already seen that the sheaf of sections E of a complex vector bundle of rank n is a locally free OX -module of rank n. As a next step, we have to show that every rank n locally free sheaf of OX -modules M is equivalent to E, for some rank n complex vector bundle E. S ' / n Given M, we choose an open covering X = i∈I Ui, and isomorphisms of sheaves φi : M |Ui (OUi ) . −1 φi◦φ Over the overlaps U = U ∩U we obtain an isomorphism of free sheaves φ : On j / On . ij i j ij Uij Uij By virtue of Lemma 5.14, φij corresponds to a holomorphic function Uij / GLn(C). Moreover, the definition of φij implies immediately that the cocycle condition φij · φjk = φik is satisfied on triple intersections Uijk. We have therefore constructed a cocycle datum {Ui}i∈I , (φij)(i,j)∈I2 . Let us denote by E the corresponding complex vector bundle. For an open subset V ⊂ X, a section s: V /E corresponds to a tuple of holomorphic functions n 2 fi : V ∩Ui / C , satisfying fi = φijfj for every (i, j) ∈ I (see Lemma 4.9). However, such a tuple, also defines an element of M(V ): indeed, we have f ∈ On , and hence s = φ−1(f ) ∈ M(U ). i Ui i i i i −1 −1 −1 −1 On the overlaps Uij, we have si|Uij = φi (fi)|Uij = φi (φijfj)|Uij = φi (φi ◦ φj fj)|Uij = sj|Uij .

By the sheaf condition, there exists a unique section s ∈ M(Ui), such that s|Ui = si. This shows M =∼ E. To conclude the proof, it remains to show that for two vector bundle E, F on X. the existence ∼ of an isomorphism of OX -modules E = F implies the existence of an isomorphism of vector bundles E =∼ F . S Indeed, let X = i∈I Ui be an open covering of X, such that both E and F can be trivialised over the open subsets Ui. We denote the cocycle datum for E by (φij), and for F by (ψij). An

44 ' isomorphism of OX -modules E / F yields for every Ui:

β On i / On . Ui Ui

By Lemma 5.14, we may understand βi to be a holomorphic function Ui / GLn(C). The definition of βi implies the relation φijβj = βiφij on every overlap Uij. Using a construction similar to Lemma 4.9 we obtain a well-defined isomorphism of vector bundles E / F . The details are left to the reader as an exercise. Corollary 5.15. Let X be a complex manifold of (complex) dimension n. There exists a rank n vector bundle TX / X, called the tangent bundle, such that the sheaf of sections TX is equiv- n alent to the sheaf of complex vector fields VX . Similarly, there exists a rank m vector bundle Vm ∗ Vm ∗ m T X / X, such that T X / X is equivalent to the sheaf of complex m-forms ΩX . m Proof. We only have to show that the sheaves VX and ΩX are locally free. It is clear that they are OX -modules, because vector fields and m-forms can be multiplied with a holomorphic function. By definition, a complex manifold X is covered by open subsets, which are biiholomorphic to n open subsets U of C . We may therefore without loss of generality assume that X = U is an n open subset of C (due to the local nature of the statement we are interested in). We have seen in Definition 2.15 that a complex vector field on U is given by an n-tuple of holomorphic functions

n X ∂ V = Vi . ∂zi i=1

n In (7) we expressed an m-form by means of an m -tuple of holomorphic functions

X i0 im ω = ωi0···im dz ∧ · · · ∧ dz .

0≤i0<···

m These considerations show that the sheaves VX and ΩX are locally free.

5.4 Exact sequences of sheaves

f In this paragraph we denote by F / G a map of sheaves on a topological space X. We will study the local nature of injectivity and surjectivity. It will turn out that injectivity of f is a local property, while the same cannot be said for surjectivity.

Lemma 5.16. If every x ∈ X has an open neighbourhood Ux, such that fU : F(U) / G(U) is injective, then the map F(X) / G(X) is injective.

g Proof. Recall that a map of abelian groups A /B is injective, if and only if g(x) = 0 is equivalent to x = 0. Translated to our situation this means that we want to show for every s ∈ F(X), such that fX (s) = 0, we already have s = 0. / By assumption we have an open covering {Ux}x∈X of X, such that fUx : F(Ux) G(Ux) is injective for every x ∈ X. The assumption fX (s) = 0 implies 0 = fX (s)|Us = fUx (s|Ux ). Hence, we have s|Ux = 0 for every x ∈ X. The sheaf property implies now that s = 0, since the two sections 2 s and 0 both solve the glueing problem for 0 ∈ F(Uxy) for (x, y) ∈ X .

45 Definition 5.17. We say that f is locally surjective, if for every open subset U ⊂ X, and every S s ∈ G(U), there exists an open covering U = i∈I Ui, and sections ti ∈ F(Ui), such that fUi (ti) = s|Ui . It is important to emphasise the difference between local surjectivity and surjectivity. The definition above does not imply that there exists a section t ∈ F(U), such that fU (t) = s. Indeed, we know that the equation f(t) = s can only locally be solved in t. The problem is that there is no unique element in the preimage, and hence, the local solutions ti, don’t glue - that is, they won’t necessarily agree on the overlaps: ti|Uij 6= tj|Uij in general. As we have seen in Lemma 5.16, the same problem does not arise for injective maps. Probably the first example, where this phenomenon has been observed in the history of mathe- matics, is in the study of complex logarithms.

× Example 5.18. Let X = C \{0}, we have a morphism of sheaves exp: OX / OX , which sends f a holomorphic function U / C to the invertible holomorphic function exp(f). × If g : X / C is an invertible holomorphic function, we may ask if there exists a holomorphic × f : X / C , such that exp(f) = g. It is clear that this equation doesn’t have a unique solution. Indeed, for every such f, also f + 2πik, with k ∈ Z is a solution. Nonetheless, there exists an open covering {Ui}i∈I of X, such that we have a holomorphic / function fi : Ui C, satisfying exp(fi) = g|Ui for every i ∈ I. To construct such an open × covering can be constructed as follows. Observe that every z ∈ C has a small -neighbhourhood U(z), where a complex logarithm log:f U(z) / C can be defined (non-canonically). Hence, over −1 Ui = g (Ui (zi)) it makes sense to define fi = logf ◦ g|Ui , where we assume that zi is a family of S × points in X, such that i∈I Ui (Zi) = C . We have exp(fi) = exp(logf ◦ g|Ui ) = gUi , and hence × have shown that exp: OX / OX is a locally surjective morphism of sheaves. exp × × However, it is clear that OX (X) / OX (X) is not a surjective map. Indeed, let g : X / C be the holomorphic function z 7→ z. If there was a holomorphic function f : X / C, such that 0 g0 1 exp(f) = g, we would have f (z) = g (z) = z (this is a consequence of the chain rule). Therefore, the integral Z dz z ∂ D(0) would be zero. But we know that this is not true, since the value of this integral can be computed to be 2πi. A pessimist would probably abandon the theory of sheaves at this point. Next week, we will carefully study the obstruction for a locally surjective map of sheaves to be surjective. The answer turns out to be given by sheaf cohomology. As a preparation we need to define exact sequences of sheaves.

f g Definition 5.19. Let A, B, C be abelian groups, and A / B / C homomorphisms. We say that this sequence of maps is exact, if ker g = image f. A more concrete definition of exactness in terms of elements is the following: for b ∈ B, we have g(b) = 0 if and only if ∃a ∈ A, such that f(a) = b. This motivates the following definition of exactness for sheaves.

46 f g Definition 5.20. A sequence of sheaves F / G / H on a topological space X is called exact, if and only if for every open U ⊂ X, and b ∈ G(U), we have g(b) = 0, if and only if there exists an S open covering U = i∈I Ui, and sections ai ∈ F(Ui), such that f(ai) = b|Ui for all i ∈ I. Our definition of an exact sequence of sheaves is inspired by the definition of locally surjective maps of sheaves. Indeed, we see that a map of sheaves G / H is locally surjective, if and only if G / H / 0 is exact. Lemma 5.21 (Exponential sequence). Let X be a complex manifold, the sequence of sheaves

2πi· exp × 0 / Z / OX / OX / 0 is exact. Proof. The statement above can be broken down into three assertions. Exactness at Z: we have to show that the map Z / OX is injective. But this is clear, since 2πin = 0 is equivalent to n = 0. Exactness at OX : If f ∈ OX (U) is a holomorphic function on U ⊂ X, such that exp(f) = 1, then f(z) = 2πn(z), where n: U / Z is a holomorphic function, taking values in Z. Since holomorphic functions are continuous, we see that n is locally constant. Thus, it defines a section n ∈ Z(U). × × Exactness at OX : We have to show that the map exp is locally surjective. We cover C by an S × open covering i∈I Vi = C , such that each Vi supports a branch of complex logarithm log:f Vi / C. × −1 Consider a holomorphic function g : U / C , where U ⊂ X is an open subset. On Ui = g (Vi) we have well-defined holomorphic functions fi = logf ◦ g|Ui . By definition, exp(fi) = g|Ui . Definition 5.22. Let X be a topological space, and F a sheaf on X. We write Γ(X, F) for the abelian group F(X) of global sections. We have seen that an exact sequence of sheaves does not induce an exact sequence of global sections in general. However, the following is true.

f g Lemma 5.23. Let 0 / F / G / H / 0 be a short exact sequence of sheaves. The sequence of abelian groups 0 / Γ(X, F) / Γ(X, G) / Γ(X, H) is exact. Proof. Exactness at Γ(X, F) is equivalent to injectivity of the map F(X) / G(X). We have seen this in Lemma 5.16. Exactness at Γ(X, G) can be shown as follows: let s ∈ G(X), such that g(s) = 0. By assumption S there exists an open covering X = i∈I Ui, and sections ti ∈ F(Ui), such that f(ti) = s|Ui . We 2 claim that the glueing condition is satisﬁed, that is ti|Uij = tj|Uij for all (i, j) ∈ I . This is true, since f(ti|Uij) = s|Uij = f(tj|Uij ), but f is injective, which implies what we want. Therefore, there exists a section t ∈ F(X), such that f(t)|Ui = s|Ui for all i ∈ I. We conclude that f(t) = s.

6 Sheaf cohomology

In this section we will introduce sheaf cohomology. We will take an axiomatic approach, and assume at the beginning that a theory with such properties exists. At the end of the section we will verify by a simple inductive procedure that this is indeed the case.

47 6.1 Overview: why do we need sheaf cohomology? We have seen in Lemma 5.23 that the functor Γ(X, −): Sh(X) / AbGrp doesn’t send short exact sequences of sheaves to short exact sequences of abelian groups. Instead,

0 / F / G / H / 0

gives rise to an exact sequence falling short of being short exact.

0 / Γ(X, F) / Γ(X, G) / Γ(X, H).

× The example of the exponential sequence 0 / Z / OX / OX / 0 shows that the map Γ(X, G) / Γ(X, H) is not surjective in general. Our best hope is therefore that we can continue the above sequence to a long exact sequence

g 0 / Γ(X, F) / Γ(X, G) / Γ(X, H) δ / H1(X, F) / H1(X, G) / H1(X, H)

with the help of mysterious functors H1(X, −): Sh(X) / AbGrp. If we had such a formalism at our disposal, we could analyse precisely which global sections s ∈ Γ(X, H) = H(X) can be lifted to global section t of G. Indeed, exactness of the sequence above asserts that there exists a t ∈ G(X), such that g(t) = s, if and only if δ(s) = 0. That is, δ(s) ∈ H1(X, F) measures the obstruction for a global section of H to stem from a global section of G. Deﬁnition 6.1 (Sheaf cohomology). A sheaf cohomology theory is a collection of functors

Hi(X, −): Sh(X) / AbGrp

for i ≥ 0, such that: (A1) H0(X, F) = Γ(X, F) as functors, (A2) for every short exact sequence of sheaves 0 / F / G / H / 0 we have a long exact sequence ··· / Hi(X, F) / Hi(X, G) / Hi(X, H) / Hi+1(F) / ···

(A3) (...) The third axiom will be stated at a later point. We will see that there is essentially only one theory of sheaf cohomology up to a unique isomorphism. At ﬁrst we take another look at our favourite example, and try to guess what the space of obstructions H1(X, Z) amounts to in geometric terms.

6.2 The exponential sequence revisited

For the purpose of this discussion we let X be C \0. The motivated reader may work with an arbitrary complex manifold X. We consider the exponential sequence

2πi· exp × 0 / Z / OX / OX / 0,

48 × and ask when a nowhere vanishing holomorphic function f : X / C has a global logarithm logf f. That is, when does there exists a holomorphic function g : X / C such that f = exp(g)? We know that for for f(z) = z such a function g = logf z cannot exist. However, if f(z) = const = c 6= 0, we can choose d ∈ C, such that exp(d) = c, and define g(z) = const = d. How × can we decide if a general f ∈ Γ(X, OX ) admits a global logarithm? We will utilise a notion from topology: covering spaces. As the attentive reader will confirm, their definition is reminiscent of the definition of vector bundles. Definition 6.2. Let G be a group, endowed with the discrete topology. A continuous map π : Y /X is called a G-Galois covering, if

(a) the group G acts on Y by homeomorphisms, and the map π is G-invariant, that is π(gy) = π(y) for all g ∈ G, and y ∈ Y , (b) for every x ∈ X, there exists a y ∈ π−1(x), such that G / π−1(x), g 7→ g · y is a bijection, (c) every x ∈ X has an open neighbourhood U, such that there exists a G-equivariant homeomorphism φ: π−1(U) ' / U × G, ﬁtting into a commutative diagram

φ π−1(U) / U × G

π p1 # | U.

Recall that a map of spaces U /V carrying a G-action is called G-equivariant, if f(gu) = gf(u) for all g ∈ G, and u ∈ U. exp × An example of a Z-Galois coverning is the exponential map C / C . This is the reason, why we introduced this notion. There is a correspondence between G-Galois coverings of X, and group homomorphisms π1(X, x) G/ , as long as X is a nice enough space (for instance, a manifold). The idea is the following: given a G-Galois covering as above, and a closed path γ : [0, 1] / X, with γ(0) = γ(1) = x. We can construct a continuous pathγ ˜ : [0, 1] / Y , such that γ(0) = y ∈ π−1(Y ), and π ◦ γ˜ = γ. There is a unique such pathγ ˜, since every x0 ∈ X has an open neighbourhood U, over which π−1(U) =∼ U × G is a trivial fibration (hence lifts exist, and are unique if one point on the curve is specified as a sort of boundary condition). × Take for instance γ(t) = e2πit as a path in X = C . By definition a lift is given byγ ˜(t) = 2πit. And we see that 0 =γ ˜(0) 6= 2πi =γ ˜(1). However, since π(˜γ(0)) = γ(0) = γ(1) = π(˜γ(1)), we have thatγ ˜(1) ∈ π−1(x). The definition of a G-Galois covering implies now that there exists a unique g ∈ G, such thatγ ˜(1) = g · y =γ ˜(0). Moreover, this g depends only on the homotopy class of the path γ, leaving the end points fixed. This allows one to associate to a G-Galois covering π : Y / X a group homomorphism π1(X, x) / G. It follows from basic theory of covering spaces that every homomorphism π1(X, x) / G arises in this way. The details are left to the reader (see exercise sheet). Definition 6.3. A G-Galois covering π : Y / X is trivial, if there exists a G-equivariant homo-

49 morphism Y =∼ X × G, such that

φ π−1(X) / X × G

π p1 # { X.

A G-Galois covering is trivial, if and only if the corresponding homomorphism π1(X, x) / G is the constant map to the unit of G. We will be interested in the case G = Z, since the existence of logarithms is closely related to Z-Galois coverings. × Proposition 6.4. Let X be a complex manifold, and f : X / C be a nowhere vanishing holomorphic function. The ﬁbre product

Y = X × × = {(x, z) ∈ X × |f(x) = exp(z)} / X f C C C is a Z-Galois covering, with respect to the action k·(x, z) = (x, 2πik+z). There exists a holomorphic function g : X / C, such that f = exp(g), if and only if the Z-Galois covering constructed above is trivial. π S Proof. Let’s show ﬁrst that Yf / is a Z-Galois covering. There exists an open covering i∈I Ui = × / C , such that we have holomorphic functions logf i : Ui C, such that exp(logf i) = idUi . Therefore, S −1 i∈I f (Ui) = X is an open covering of X, and the map

z − logf (z) (x, z) 7→ (x, i ) 2πi −1 −1 −1 defines a Z-equivariant homeomorphism between π (f (Ui)) and f (Ui) × Z. This proves that π Yf / X is a Z-Galois covering. Next we will show that Yf / X is the trivial Z-Galois covering, if and only if f = exp(g), for some continuous function g / C (it is then clear that g is also holomorphic). The trivial Z-Galois covering X × Z / X has at least one continuous section s: X / X × Z, for instance x 7→ (x, 0). If Yf / X is the trivial covering, then there exists a continuous map s: X / Yf , such that π ◦ s = id . By definition of Y = X × × , we see that g = p ◦ s: X / is a continuous X C C 2 C map, such that exp(g) = f. Vice versa, assume that there’s a holomorphic map f : X / C, with the property exp(f) = g. Then, we can define a Z-equivariant homomorphism Yf / X × Z by the formula z − f(x) (x, z) 7→ (x, ). 2πi This concludes the proof. Summarising the discussion above, we see that we associated to every nowhere vanishing holo- × morphic function f : X / C a Z-Galois covering, which is trivial, if and only if f = exp(g), for some holomorphic function g : X / C. The set of isomorphism classes of Z-Galois covering is equal to the set of group homomorphisms ρ: π1(X, x) / Z. Since two Z-valued functions can be multiplied, we see that this set actually carries a natural structure of an abelian group, which we will denote by Hˇ 1(X, Z). That is, we’ve verified the following theorem.

50 Theorem 6.5. Let X be a connected complex manifold, there exists an exact sequence

× δ ˇ 1 0 / Γ(X, Z) / Γ(X, OX ) / Γ(X, OX ) / H (X, Z). We see that the map δ fulﬁls the goal of determining the obstruction to the existence of a holomorphic function g with exp(g) = f.

6.3 Sheaf cohomology as cohomology Probably you’ve already encountered the notions of cohomology (or homology) of chain complexes in other lectures (diﬀerential or algebraic topology). In those courses, cohomology refers in ﬁrst instance to the cohomology of a chain complex of abelian groups. Recall that a chain complex is a sequence of homomorphisms of abelian groups

0 1 2 3 A• = [A0 d / A1 d / A2 d / A3 d / ··· ], such that the relation di+1 ◦ di = 0 is satisﬁed for all i ≥ 0. The abelian group Hn(A•) is deﬁned to be ker dn , image dn−1 and referred to as degree n cohomology of the chain complex A•. Note that A• is exact at An, if and only if Hn(A•) = 0. Thus, we think of Hn(A•) as measuring the failure of A• being exact at An. We have also seen that sheaf cohomology measures the failure of exactness of the global sections functor Γ(X, −). It is therefore not unreasonable to ask, if the cohomology groups Hi(X, F) of a sheaf F on X can be computed as cohomology groups of a chain complex. It turns out that the answer is yes. There will be more than one such complex computing sheaf cohomology, and every instance of this phenomenon translates into an interesting comparison theorem. For example:

i Theorem 6.6 (de Rham). Let X be a smooth manifold, we denote by Hsing(X, R) the singular i cohomology of X with real coeﬃcients, and by HdR(X, R) the so-called de Rham cohomology of X. There is a canonical isomorphism

i ∼ i Hsing(X, R) = HdR(X, R). We will recall the definitions of singular and de Rham cohomology in more detail. For singular cohomology this requires some preparation, but the rough idea is to cove X by simplices (basic geometric shapes, generalising an interval of bounded length, a triangle, a tetrahedron, etc), and construct a chain complex from these decompositions. De Rham cohomology is computed using the real vector spaces of smooth m-forms Ωm(X), and the fact that exterior differentiation R d:Ωm(X) / Ωm+1(X) satisfies the relation d ◦ d = 0. Hence, differential forms give rise to a chain R R i complex in vector spaces, and the resulting cohomology groups, are denoted by HdR(X, R). 3 Example 6.7. Assume that X is an open subset of R . A smooth (real) m-form on X is then expressible as a linear combination P f dxi1 ∧ · · · ∧ dxim . Only the cases m = 0≤i1<···

51 exterior differentiation, and battling with signs, one sees that d0 :Ω0 (X) / Ω1(X) is basically R computing the gradient vector field of a smooth function, while d1 corresponds to the curl operator, and d2 to divergence of vector fields. It is well-known that gradient vector fields are curl-free, but what about the converse? De Rham’s theorem tells us that the quotient vector space

{V smooth vector ﬁeld on X|∇ × V = 0} = H1 (X, ) {∇ · f|f smooth function on X} sing R is only depending on the topology of X. Hence, if there are curl-free vector ﬁelds, only the topology of X obstructs us from expressing them globally as a gradient.

Exercise 6.8. (a) In the movie “A beautiful mind”, Russell Crowe asks the students of his multivariable calculus class to solve a supposedly very difficult problem about vector fields. Solve it or argue why Russell’s exercise might have been too difficult for an introductory course.

3 (b) Remind yourself that on a star-shaped open subset of R , every curl-free smooth vector field can be written as a gradient of a smooth function. Deduce that the following is a short exact ∞ ∇· ∞ ∇× ∞ sequence of sheaves 0 / R / CX / VX / 0 on X. Here, CX is the sheaf, assigning ∞ to an open subset U, the vector space of smooth R-valued functions, and VX denotes the sheaf of smooth real vector fields on X. Mimic our analysis of the existence of complex logarithms (Theorem 6.5), using R-Galois covers, to understand when a curl-free vector field on X can be written as a gradient of a smooth function.

The comparison between de Rham and singular cohomology is achieved by comparing both sides to the cohomology of the sheaf R on X, which assigns to an open subset U ⊂ X the real vector space of locally constant maps U / R (we could also view R as a discrete topological space, and disc look at continuous maps U / R ). The so-called de Rham complex (Ω• (X), d) can be viewed as the chain complex of global sections R m Γ(X, −) of an exact sequence in sheaves. Let us denote by ΩX the sheaf, which assigns to an open subset U ⊂ X the sheaf of smooth (real) m-forms on U. In diﬀerential geometry or topology, the following lemma should have been proven.

0 d 1 d 2 Lemma 6.9. The sequence of sheaves 0 / R / ΩX / ΩX / ΩX / ··· is exact. i We will deduce from this lemma, and the Universal Comparison Theorem below that HdR(X, R) agrees with the sheaf cohomology group Hi(X, R). Theorem 6.10 (Universal Comparison Theorem). Let F be a sheaf on a topological space X, and

0 1 2 0 / F / G0 d / G1 d / G2 d / / ··· an exact sequence of sheaves, such that Hi(X, Gj) = 0 for all i ≥ 1 (one also says that the sheaf Gj is acyclic). Then, Hi(X, F) is isomorphic to the degree i cohomology group of the chain complex

0 1 2 Γ(X, G0) d / Γ(X, G1) d / Γ(X, G2) d / ··· .

52 Proof. We will prove this statement by induction on i. In the course of the proof, we denote by Ki = ker di, the subsheaf of Gi, assigning to an open subset U ⊂ X the subgroup {s ∈ Gi(U)|di(s) = 0}. 0 Our proof is anchored to the case i = 0. Exactness of 0 / F / G0 d / G1 yields a short exact sequence 0 / F / G0 / K1 / 0. Applying Γ(X, −), we obtain from Lemma 5.23 that 0 / Γ(X, F) / Γ(X, G0) / Γ(X,K1) is exact. We have a commutative diagram

Γ(X, G0) / Γ(X,K1) _

0 d % Γ(X, G1), where injectivity of the vertical arrow follows from Lemma 5.23 applied to the short exact sequence 0 / K1 / G1 / K2 / 0 of sheaves (also a consequence of exactness). Therefore, we have

H0(Γ(X, G•)) = ker(d0 : Γ(X, G0) / Γ(X, G1)) = Γ(X, F).

Let’s assume that the claim had already been proven for a given i ≥ 0, and see why the assertion must also hold for degree i + 1 cohomology groups. By exactness of the sequence G• we have a short exact sequence i 0 / Ki / Gi d / Li+1 / 0. The associated long exact sequence in cohomology contains the following interesting part

H0(X, Gi) / H0(X,Ki+1) / H1(X,Ki) / H1(X, Gi) = 0.

i+1 1 i Γ(X,K ) i+1 • Exactness implies that H (X, K ) = image(Γ(X,Gi) / Γ(X,Ki+1)) = H (Γ(X, G )). The short exact sequences 0 / Kj / Gj / Kj+1 / 0 imply now inductively that we have

H1(X,Ki) = H2(X,Ki−1) = ··· = Hi+1(X, F).

This concludes the proof of the argument. The Universal Comparison Theorem implies de Rahm’s theorem, once we’ve shown that the sheaves of smooth m-forms don’t have any higher cohomology; and we found a way of expressing singular cohomology via a resolution of sheaves. We defer the proof of the vanishing of higher cohomology to a later section, and devote the rest of this paragraph to an explanation how singular cohomology arises from a resolution of sheaves.

n Deﬁnition 6.11. (a) For every positive integer n ≥ 0, we denote by ∆ = {(λ0, . . . , λn) ∈ n+1 n+1 [0, 1] |λ0 + ··· + λn = 1} ⊂ R , and view it as a topological space with respect to the subset topology. (b) Let X be a topological space. A singular n-simplex is a continuous map σ : ∆n / X. The set of singular n-simplices on X will be denoted by Sn(X).

53 n−1 n (c) For 0 ≤ i ≤ n we denote by ∂i : ∆ /∆ , the unique aﬃne map (linear + translation), which preserves the vertices with respect to their standard ordering:

e0 7→ e0, e1 7→ e1, ··· , ei−1 7→ ei−1, ei 7→ ei+1, ··· en−1 7→ en.

n c (d) For an abelian group A, we let C (X,A) be the set of maps of sets Sn(X) /A. We will denote R c(σ) by σ c, and deﬁne an A-linear map by

dn : Cn(X,A) / Cn+1(X,A)

by the formula R d(c) = R c = Pn+1(−1)i R c. σ ∂σ i=0 σ◦∂i It is an important exercise to check that dn+1 ◦ dn = 0 for every n ≥ 0. Therefore, we have constructed a chain complex

0 / C0(X,A) / C1(X,A) / C2(X,A) / ··· .

The degree i cohomology of this chain complex is called the i-th singular cohomology group i Hsing(X,A) with coeﬃcients in A.

i Theorem 6.12. If X is a locally contractible topological space, then Hsing(X,A) is equivalent to Hi(X,A), where A is the sheaf of locally constant A-valued functions on X.

n Sketch. The main idea is to consider the presheaf CX , assigning to an open subset U ⊂ X the abelian group U 7→ Cn(U, A). n We have a map of presheaves A / CX , which assigns to a locally constant function f : U / A, 0 R the element c ∈ C (U, A), given by σ c = f(σ(1)) (a singular 0-simplex is just a single point in X). If U is contractible as a topological space, it is shown in algebraic topology that the complex 0 / A / C0(U, A) / C1(U, A) / ··· is exact. If there was any justice in the world, we could n pretend that CX was a sheaf, and hence assert that

0 1 0 / A / CX / CX / ···

i j j was a resolution by sheaves, and that H (X,CX ) = 0 for i ≥ 1. Unfortunately, CX is not a sheaf, n and one needs to invoke the sheafification process discussed in the exercises. Let us denote by CX the j j j j+1 j j j+1 sheafification. What saves the day is that K = ker(d : CX / CX ) = ker(d : CX / CX ), and the proof of the universal comparison theorem still applies to our situation pre-sheafification. Exercise 6.13. Turn the sketch above into a complete proof.

6.4 Construction of sheaf cohomology We start with some diagram chasing lemmas. Recall that for a morphism f : A / D we denote by coker f the quotient D/image f.

54 Exercise 6.14. (Snake Lemma) Given a commutative diagram

A / B / C / 0

f g h 0 / D / E / F with exact rows, show that there is an exact sequence

ker f / ker g / ker h −→δ coker f / coker g / coker h.

An important corollary of the Snake Lemma is the tic-tac-toe lemma.

Lemma 6.15 (Tic-Tac-Toe). Assume that the following diagram is commutative, and its columns are exact. 0 0 0

0 / A / B / C / 0

0 / D / E / F / 0

0 / G / H / I / 0

0 0 0 If the ﬁrst two rows are exact, then so is the third row.

Proof. Take a thin strip of paper and cover the last row of the commutative diagram above. The resulting diagram contains the one of the Snake Lemma as a subdiagram. The morphism C / F has kernel 0 and cokernel I. Therefore we obtain directly that

0 / 0 / 0 / G / H / I is exact. it remains to show that H / I / 0 is exact, that is H / I is surjective. To see this, we zoom into the commutative diagram above, to isolate the commuting square

E / / F

H / I.

The arrows marked by are surjective by assumption. Commutativity of the diagram implies that H / I is surjective too.

55 The tic-tac-toe lemma also holds for sheaves on a topological space X. This is true, because exactness of a sequence of sheaves F / G / H can be tested by verifying that for every x ∈ X the sequence of stalks F x / Gx / Hx is an exact sequence of abelian groups (see E7). As a next step we have to deﬁne injective sheaves. As warm-up we discuss injectivity for abelian groups. Deﬁnition 6.16. An abelian group I is called injective, if the following is true: let A,→ B be an injective morphism of abelian groups, and f : A / I a homomorphism, then there exists a homomorphism B / I, such that the diagram A / B

f ∃ I commutes. It’s important to emphasise that this is not a universal property. An extension B / I as the one above, is in general not unique. The next lemma shows that injectivity is quite easy to verify for abelian groups, and implies that Q and Q/ Z are injective. Lemma 6.17. An abelian group M is called divisible, if for every m ∈ M, and every integer n ≥ 1, there exists an m0 ∈ M, such that n · m0 = m. An abelian group is injective if and only if it is divisible. Proof. Let’s assume ﬁrst that I is injective, and show it’s divisible. For m ∈ I and an integer n ≥ 1, we consider the map n Z / M n Z / Z

n7→m ~ ∃? M which sends the generator n to m. Since n Z ⊂ Z is an injective homomorphism of abelian groups, / and I is injective, there exists a homomorphism g : Z I, such that g|n Z = f. Therefore, m0 = f(1) satisfies the property n · m0 = n · g(1) = g(n) = f(n) = m. This shows that I is divisible. Vice versa, let M be a divisible abelian group. We have to show that M is injective. We will prove this using the axiom of choice. Let A,→ B be an injective homomorphism of abelian groups, and f : A / M a homomorphism. We have to prove that f can be extended to B. We consider the set of pairs (A0, g), where A ⊂ A0 ⊂ B is a subgroup, and g : A0 / M is a 0 00 homomorphism, such that g|A0 = f. There is an ordering on this set. We say (A , g) ≤ (A , h) if 0 00 A ⊂ A , and h|A0 = g. This set is partially ordered, because if I = {(Ai, fi)} is totally ordered S subset, then there exists an upper bound, given by ( i∈I Ai, g), with g(a) = fi(a) if a ∈ Ai. Zorn’s lemma implies that there exists a maximal element (A,¯ f¯). It remains to show that B = A¯. Assume by contradiction that there exists x ∈ B \ A¯. We claim that f¯ can be extended to the subgroup A¯ + Z x / M. We only need to define f(x) in a suitable way, in order to make sense of it. If Z x∩(A¯\0) = ∅, then we can define f(x) = 0 or any other element of M. If the intersection is not empty, then there exists a minimal integer n ≥ 1, such that nx ∈ A¯. Since M is divisible, there exists an m0 ∈ M, such that nm0 = f¯(nx). We define f(x) = m0. This is a contradiction, and we conclude A¯ = B.

56 Deﬁnition 6.18. A sheaf I is called injective, if the following is true: let F ,→ G be an injective morphism of sheaves, and f : F / I a morphism of sheaves. Then there exists a momorphism G / I, such that the diagram F / G

f ∃ I commutes. Again we emphasise that injectivity is a property, and not a universal property! We are now ready to state the third axiom of sheaf cohomology, and prove the existence of such a theory. Deﬁnition 6.19 (revised version of 6.1). A sheaf cohomology theory is a collection of functors

Hi(X, −): Sh(X) / AbGrp

for i ≥ 0, such that: (A1) H0(X, F) = Γ(X, F) as functors, (A2) for every short exact sequence of sheaves 0 / F / G / H / 0 we have a long exact sequence

··· / Hi(X, F) / Hi(X, G) / Hi(X, H) / Hi+1(F) / ··· .

And for every commutative diagram with exact rows

0 / F 1 / G1 / H1 / 0

0 / F 2 / G2 / H2 / 0

we have a commutative diagram whose rows are the aforementioned long exact sequences.

i i i i+1 ··· / H (X, F 1) / H (X, G1) / H (X, H1) / H (X, F 1) / ···

i i i i+1 ··· / H (X, F 2) / H (X, G2) / H (X, H2) / H (X, F 2) / ···

(A3) If I is an injective sheaf, then Hi(X, I) = 0 for i ≥ 1. What did (A3) buy us? We’ve shown in the exercises (see E7) that every sheaf F can be embedded into an injective sheaf I. Therefore, there exists a short exact sequence

0 / F / I / I / F / 0.

The stalks of the sheaf (I / F)x are equal to the cokernel Ix / F x. The sections of I / F(U) are Q S given by elements (sx)x∈U ∈ x∈U Ix / F x, such that there exists an open covering U = i∈I Ui, sections ti ∈ I(Ui), such that sx = [(ti)x] for every x ∈ Ui.

57 Assuming that there is a theory of sheaf cohomology satisfying the axioms (A1-3), we obtain for i ≥ 0 an exact sequence ··· / Hi(X, I) / Hi(X, I / F) / Hi+1(F) / 0 = Hi+1(X, I), where we use that Hi+1(X, I) = 0, since I is injective. This implies H1(X, F) =∼ coker(H0(X, I) / H0(X, I / F)), and for i ≥ 1: Hi+1(X, F) =∼ Hi(X, I / F). We will turn this observation into an inductive definition. For this to make sense, we have to verify the resulting higher cohomology groups, are independent of the choice of the embedding F ,→ I. Lemma 6.20. The following defines a functor H1(X, −): Sh(X) / AbGrp. For every F ∈ Sh(X) we choose an embedding F ,→ I, where I is an injective sheaf, and define H1(X, F) = coker(H0(X, I) / H0(X, I / F)).

f For a morphism F / G we choose a commutative diagram F / I

f ∃ G / J and deﬁne H1(f) to be the map

H0(X, I) / H0(X, I / F) / H1(X, F) / 0

H1(f) H0(X, J ) / H0(X, J / G) / H1(X, G) / 0. The resulting functor is independent (up to a unique natural isomorphism) of the chosen embeddings F / I. Proof. We begin the proof by verifying that the resulting map H1(f) is independent of the choices. That is, if we have two morphisms g and h giving rise to a commutative diagram F / I

f g h G / J we want to prove that the induced maps α, β : H1(X, F) / H1(X, G) agree. We will show that α − β = 0. H0(X, I) / H0(X, I / F) / H1(X, F) / 0

g h g h α β H0(X, J ) / H0(X, J / F) / H1(X, F) / 0.

58 The map g − h: I / I satisﬁes (g − h)|F = 0 by deﬁnition. Therefore, we obtain a factorisation as indicated by the dotted arrow I / I / F

g−h } J / J / G, and we see that g − h: H0(X, I / F) / H0(X, I / F) factors through H0(X, J ). Since H1(X, G) = coker(H0(X, J ) / H0(X, J / G)), we obtain α − β = 0. Applying this to the commutative diagram F / I

f g h F / I,

1 1 we see that H (idF ) is the identity map of H (X, F). Applying the observation to F / I

f F / J

F / I,

(and also switching the roles of I and J ) we see that the abelian group H1(X, F) is independent of the chosen embedding F ,→ I. f g It remains to see that for composable morphisms of sheaves F / G /H we have H1(g◦f) = H1(g) ◦ H1(f). Consider the commutative diagram F / I

f ∃ G / J

∃ H / K . We know that the induced maps H1(f),H1(g),H1(g◦) are independent of the chosen extensions presented by the dashed arrows. Therefore H1(g ◦ f) = H1(g) ◦ H1(f), because we can simply form the composition of two successive extension. The next lemma can be understood as a consistency check of the third axiom of sheaf cohomology (A3). The condition H1(X, I) = 0 for an injective sheaf I implies in particular (using the long exact sequence) that every short exact sequence of sheaves 0 / I / G / H / 0 gives rise to a short exact sequence of global sections. This can be veriﬁed directly, and will be used in the proof of existence of sheaf cohomology.

59 Lemma 6.21. Let 0 / I / G / H / 0 be a short exact sequence of sheaves, such that I is injective. Then the sequence of global sections

0 / Γ(X, I) / Γ(X, G) / Γ(X, H) / 0 is exact. Proof. Using injectivity of I, we obtain a morphism of sheaves r : F / I, such that I / G

id ∃s I commutes. This implies that the short exact sequence 0 / I / G / H / 0 splits (see E5, 1(d) for a similar exercise), that is G =∼ I ⊕H = I ×H. But H0(X, I ⊕H) =∼ H0(X, I) ⊕ H0(X, H), and therefore we see that the map H0(X, G) / H0(X, H) is indeed surjective, and the sequence above thus exact. Lemma 6.22. For every short exact sequence of sheaves 0 / F / G / H / 0 we have a long exact sequence 0 /H0(X, F) /H0(X, G) /H0(X, H) /H1(X, F) /H1(X, G) /H1(X, H), where H1(X, −) is the functor deﬁned in Lemma 6.20. Proof. In the exercises (see E7, ex. 5) it is shown that there exist embeddings into injective sheaves F ,→ I, G ,→ J , and H ,→ K, such that we have the following commutative diagram with exact rows and exact columns.

0 0 0

0 / F / G / H / 0

0 / I / J / K / 0

0 / I/F / J /G / K/H / 0

0 0 0

Applying the functor H0(X, −) to the lower two rows we obtain the commutative diagram with exact rows

0 / H0(X, I) / H0(X, J ) / H0(X, H) / 0

0 / H0(X, I/F) / H0(X, J /G) / H0(X, K/H).

60 The ﬁrst row is exact by virtue of Lemma 6.21. The Snake Lemma gives rise to the required long exact sequence 0 /H0(X, F) /H0(X, G) /H0(X, H) /H1(X, F) /H1(X, G) /H1(X, H).

Theorem 6.23. There exists a unique (up to a unique isomorphism) formalism of sheaf cohomology as in Definition 6.19. Proof. We will prove by induction on the degree i, that there exists a family of functors Hi(X, −): Sh(X) / AbGrp, verifying the axioms (A1 − 3) up to the given degree. For i ≤ 1 we know that such a family exists by virtue of Lemma 6.20. In this lemma we verified explicitly that (A1-2) are satisfied, and (A3) holds by definition of the functor H1(X, −). Indeed, if I is an injective sheaf, we may consider the 1 1 id 0 trivial embedding idI : I ,→ I, and hence obtain H (X, I) = coker(H (X, I) / H (X, I)) = 0. For i ≥ 1 we define Hi+1(X, F) = Hi(X, I / F), where F ,→ I is an embedding into an injective f sheaf (according to E7, ex. 5 this is always possible). For a morphism of sheaves F / G we choose a commutative diagram F / I (11)

f G / J . as in our construction of the functor H1(X, −) in Lemma 6.20. We will use the same strategy as in loc. cit. to verify that the induced map

Hi+1(X, F) / Hi+1(X, G)

Hi(X, I / F) / Hi(X, J / G) is independent of the choice of the dashed morphism. Again we denote by g, h: I / J two morphisms of sheaves, fitting into the commutative diagram (11). Their difference g − h satisfies (g − h)|F = 0 by commutativity. Therefore, we obtain a factorisation

I / I / F (12)

g−h g−h | J / J / G . as indicated by the dotted arrow. Applying the functor Hi(X, −), we obtain a commutative diagram

H(X, I) / Hi(I / F) (13)

g−h g−h w Hi(X, J ) / Hi(X, J / G).

By the induction hypothesis we have Hi(X, J ) = 0. Therefore, we see that g − h induces the zero morphism. As in the proof of Lemma 6.20 we conclude that Hi+1(X, −) is a functor.

61 It remains to verify the axioms (A2 − 3). If I is an injective sheaf, then we obtain for i ≥ 1, Hi+1(X, I) = Hi(X, I / I) = 0. This shows that our family of functors satisﬁes (A3). Let 0 / F / G / H / 0 be a short exact sequence of sheaves. It has been shown in the exercises (E7, ex. 5) that there exists a commutative diagram with exact rows and columns, such that I, J , and K are injective sheaves. 0 0 0 (14)

0 / F / G / H / 0

0 / I / J / K / 0

0 / I/F / J /G / K/H / 0

0 0 0 For i ≥ 2 we may simply apply the existence of the long exact sequence for Hi−1(X, −) and Hi(X, −), to obtain

Hi−1(X, I / F) / Hi−1(X, J / G) / Hi−1(X, K /H) / Hi(X, I / F) / Hi(X, J / G) / Hi(X, K /H)

Hi(X, F) / Hi(X, G) / Hi(X, H) / Hi+1(X, F) / Hi+1(X, G) / Hi+1(X, H).

For i = 1 we have to be more careful, because H1(X, F) is not equal to H0(X, I / F) in general, but rather to coker(H0(X, I) / H0(X, I / F)). We obtain the corresponding part of the long exact sequence as follows. Consider the bottom two rows of (14). Lemma 6.20 implies that we have a commutative diagram

0 / H0(X, I) / H0(X, J ) / H0(X, K) / 0 / 0 / 0

α β γ 0 / H0(X, I/F) / H0(X, J /G) / H0(X, K/H) / H1(X, I/F) / H1(X, J /G) / H1(X, K/H) with exact rows. Taking cokernels of the vertical arrows, a simple diagram chase veriﬁes that the sequence

coker α / coker β / coker γ / H1(X, I/F) / H1(X, J /G) / H1(X, K/H)

H1(X, F) / H1(X, G) / H1(X, H) / H2(X, F) / H2(X, G) / H2(X, H) is exact as required.

62 7 GAGA

In this section we will finally state and (partially) prove Serre’s GAGA theorems. GAGA, the abbreviation of Géométriealgébriqueet géométrieanalytique, usually refers to the following three theorems (which we state in vague terms, and in a slightly less general form for the purpose of this introduction) and their corollaries. Theorem 7.1 (Serre’s GAGA). Let X be a complex projective manifold.

(a) Let E be an algebraic vector bundle on X, then every global holomorphic section of E is algebraic. (b) If E and F are algebraic vector bundles on X, then every ﬁbrewise linear holomorphic morphism of vector bundles E / F is algebraic.

(c) Every holomorphic vector bundle E on X arises from an algebraic vector bundle in an essentially unique way. We will prove (a) and (b) completely in this class. The third result (c) requires deeper analytic results, but we will nonetheless try to gain an understanding why it is true. Combining the three statements above, one sees that the category of algebraic and holomorphic vector bundles on X are equivalent. This has the curious interpretation that somehow that compactness of a complex projective manifold X forces holomorphic vector bundles and their sections to be algebraic. In order to understand this theorem we have to make precise what we mean by algebraic. That is the content of the first paragraph below. We will show that for so-called vector bundles E, there i exist complex vector spaces Halg(X,E) of algebraic degree i cohomology classes. For i = 0 this vector space agrees with the space of algebraic (or regular) sections of E. Serre proves the first part of the theorem above by a downward induction on the degree i. We will show that for i > dimC X i i the cohomology spaces H (X,E) and Halg(X,E) will be both 0. The inductive comparison is anchored to this case. The long exact sequence of sheaf cohomology, and an ingenious application i ∼ i of the Five Lemma allow us to deduce that H (X,E) = Halg(X,E) the two types of cohomology groups agree canonically. The argument contains a second inductive strand: over the complex dimension. In fact, it turns out to be easier to prove a more general comparison theorem, where vector bundles are replaced by n so-called coherent sheaves. This way one can reduce everything to the case X = P . Recall that vector bundles on X correspond to locally free sheaves of OX -modules on X. A coherent sheaf is n a locally finitely generated sheaf of OX -modules. If E is a vector bundle on X, and i: X,→ P n denotes the embedding into projective space, then the sheaf i∗E is not locally free, unless X = P or X = ∅. However, i∗E is always coherent. While this argument is formally very pleasing, we’ll try to stay within the realm of complex manifolds and vector bundles as long as possible.

7.1 Projective algebraic geometry in a nutshell

Let G1,...,Gk ∈ C[z0, . . . , zn] be homogeneous polynomials of degree di in n+1 variables z0, . . . , zn. n We denote by V (G1,...,Gk) ⊂ P the subset

n {(z0 : ... : zn) ∈ P |Gi(z0, . . . , zn) = 0 ∀i = 1, . . . , k},

63 n and refer to subsets of P of this type as complex projective varieties. In Proposition 3.11 we proved that if i=,1,...,k ∂Gi j (w) is a surjective linear map, (15) ∂z j=0,...,m then V (G1,...,Gk) is a complex submanifold. n In this section we aim to turn the subset V (G1,...,Gk) ⊂ P into a mathematical object (an algebraic variety) itself. We will see that there exists a topology (the Zariski topology) on n V (G1,...,Gk), which is coarser than the standard topology on subsets of P , but better adapted reg to algebraic geometry. The Zariski topology carries a sheaf of rings OX , the so-called regular sections of OX . Regular sections can be described in terms of polynomials and rational functions, and are therefore of algebraic nature.

Deﬁnition 7.2. Let V = V (G1,...,Gk) be a projective variety. A subset U ⊂ V is called Zariski open, if there exists a projective variety W = W (H1,...,H`), such that U = V \ W .

n Since a projective variety W (H1,...,H`) ⊂ P is closed in the standard topology, every Zariski open subset is open in the standard topology. However, Zariski open subsets are called open in their own right.

n Lemma 7.3. The set of Zariski open subsets defines a topology on the set V = V (G1,...,Gk) ⊂ P . Proof. The empty subset ∅ ⊂ V is Zariski open, because ∅ = V \ V . Similarly, we have that V is Zariski open, because V = V \∅ = V \V (1), where 1 denotes the constant homogeneous polynomial of degree 0. If U1, U2 are Zariski open subsets, that is, Ui = V \ Wi, then U1 ∩ U2 = V \ (W1 ∪ W2). Writing Wi = V (H1i,...,Hì) (possibly repeating polynomials several times), we have W1 ∪ W2 = V (Hi1 · Hj2)i,j=1,...,`. For i ∈ I we assume that Ui = V \ Wi = V (H1i,...,H` i) is a Zariski open subset. Then S T i i∈I Ui = V \ i∈I Wi. If I is a finite set, then it is clear that \ Wi = V (H1i,...,Hìi)i∈I i∈I is again a projective variety. For the general case, we have to use that the ring C[z0, . . . , zn] is Noetherian, that is, every ideal is finitely generated. We consider the ideal J generated by (H1i,...,H` i)i∈I , and choose finitely many homogeneous generators H1,...,H`. We then have T i i∈I Wi = V (H1,...,H`).

If X = V (G1,...,Gk) is a projective variety, we denote the topological space induced by the Zariski topology by XZar. There is a natural sheaf of rings on XZar.

Deﬁnition 7.4. Let XZar be the topological space associated to a projective variety, let U ⊂ X f be a Zariski open subset. A function U / C is called regular, if there exists an open covering S U = i∈I Ui, such that for every i ∈ I, there exist homogeneous polynomials G, H ∈ C[z0, . . . , zn]d G of degree d (H 6= 0), such that ∀(z0 : ... : zn) ∈ Ui we have f(z0 : ... : zn) = H (z0, . . . , zn). We reg denote the sheaf of regular functions on XZar by OX .

64 A regular function is precisely a holomorphic function, which can be locally expressed in terms of rational functions. It is important to emphasise that f is not strictly meromorphic, that is, doesn’t have any poles. The domain of definition of f excludes by definition the closed subvariety of zeroes of the denominator H. We have seen that every Zariski open subset U ⊂ XZar is also open with respect to the standard n (or manifold topology if X is smooth) of X ⊂ P . On U we can compare the rings OX (U) and Oreg (U). By definition, Oreg (U) ⊂ O (U) is a subring. However, we are not allowed to call XZar XZar X Oreg a subsheaf of O , since the sheaves live on different topological spaces. XZar X Definition 7.5. An algebraic vector bundle of rank n on a complex projective manifold X is a locally free O reg -module E of rank n. XZar This definition is inspired by Theorem 5.12, which showed that the theory of holomorphic vector bundles is equivalent to locally free sheaves of OX -modules of rank n. There is another definition, which justifies calling them vector bundles. reg Exercise 7.6. Show that every locally free O reg -module E of rank n, is given by the sheaf of XZar regular sections of a complex vector bundle E π / X, which is algebraic in the following sense. There exists a cocycle datum ({Ui}i∈I , (φij)(i,j)∈I2 ), such that every Ui is a Zariski open subset, and φij : Uij / GLn(C) is a regular morphism. A section s of E is then called regular, if it n corresponds to a tuple of regular functions fi : Ui / C , such that φijfj = fi for all i, j ∈ I. An algebraic vector bundle of rank 1 is also called an algebraic line bundle. The most important n example of algebraic line bundles are the twisting line bundles on P . Definition 7.7. Recall that the standard atlas of projective space consists only of Zariski open d n zj subsets Ui = {(z0 : ... : zn) ∈ P |zi 6= 0}. Moreover, the cocycle datum, ({Ui}i=0,...,n, ( zd )i,j=1,...,n) n i of the line bundles Ld, is the cocycle datum of an algebraic line bundle on P . We denote its sheaf reg of regular sections by O n (d), and its sheaf of holomorphic sections by O n (d). P P n n We have already computed the complex vector space of global sections Γ(P , OP (d)), and have shown that it is canonically equivalent to the complex vector space of homogeneous degree d polynomials. Given the algebraic nature of homogeneous polynomials, one suspects immediately that that is a first instance of the GAGA principle. To verify this suspicion we will compute the n reg vector space of global sections Γ(P , O n (d)), which are regular. P n Note that a holomorphic section s ∈ Γ(P , O d (d)) corresponds to a tuple of holomorphic P d zj functions fi : Ui / C, for i = 0, . . . , n, such that fi = d fj for all i, j. A holomorphic function is zi regular, if the functions fi are regular. n reg n Lemma 7.8. The canonical map Γ( , O n (d)) / Γ( , O n (d)) is an equivalence. PZar P P P Proof. It is clear that this map is injective. It remains therefore to show that it is surjective. We have computed the space of holomorphic global sections in Theorem 4.15. It is therefore sufficient to show that every holomorphic global section of Theorem 4.15 is regular. Given a degree d homogeneous polynomial G = G(z0, . . . , zn), one associates to G a holomorphic n n+1 section s: P /Ld as follows. Recall that Ld was defined to be the quotient space of (C \0)×C × by the action of C , where d λ · (z0, . . . , zn, w) = (λz0, . . . , λzn, λ w).

65 n The map sG : P / Ld sends (z0 : ... : zn) to [(z0, . . . , zn,G(z0, . . . , zn))]. Let us see what ' / holomorphic function this section corresponds to with respect to the trivialisation αi : Ld|Ui Ui× w G(z0,...,zn) C, which sends [(z0, . . . , zn, w)] to [((z0 : ... : zn), d )]. We obtain αi(sG)(z0 : ... : zn) = d . zi zi But these are regular functions. This concludes the proof. The next construction we will take a look at is twisting E(d) of algebraic vector bundles E. The reason is that an algebraic vector bundle E(d) might not have any (regular nor holomorphic) sections at all. By suﬃcient twisting of the vector bundle, we can ensure to obtain a lot of global sections. We begin a deﬁnition using cocycles, and subsequently we will reformulate it in terms of tensor products (which has the advantage of being formally better behaved).

n Definition 7.9. Let E be an algebraic vector bundle on P , corresponding to a cocycle datum n ({Ui}i=0,...,n, (φi,j), where {Ui}i=0,...,n is the standard open covering of P (it is a priori not clear that such a presentation always exist, yet it is true). We define E(d) to be the algebraic vector d zj bundle corresponding to the twisted cocycle datum ({Ui}i=0,...,n, d φij)). zi As we remarked above, it is a priori not clear that this definition can be applied to every algebraic vector bundle. Moreover, it is not clear that it is independent of the chosen presentation via coycles. We will remedy this drawback by making a small detour through the world of sheaves and tensor products. Lemma 7.10. Let R be a (commutative) ring (with unit), and M,N be R-modules. There exists an R-module M ⊗R N, and a bilinear map − ⊗ −: M × N / M ⊗R N, such that for every bilinear map of R-modules f : M × N / P , there exists a unique linear map M ⊗R N / P , such that

M × N f −⊗−

∃! $ M ⊗R N / P commutes.

Proof. We define M ⊗R N as a quotient F/S, where F is the free R-module generated by formal symbols m ⊗ n for every pair (m, n) ∈ M × N. The R-module S is defined to be the submodule of F , generated by elements (m + m0) ⊗ n − m ⊗ n − m0 ⊗ n, m ⊗ (n + n0) − m ⊗ n − m ⊗ n0, (λm) ⊗ n − m ⊗ (λn), and λ(m ⊗ n) − (λm) ⊗ n for every possible choice of elements m, m0 ∈ M, 0 n, n ∈ N, and λ ∈ R. The map M × N / F/S = M ⊗R S, sending (m, n) to m ⊗ n is by definition bilinear. Moreover, if f : M × N / P is a bilinear map, then we obtain a linear map F / P , by sending m ⊗ n 7→ f(m, n). Since f|S = 0, there is an induced linear map g : F/S = M ⊗R N / P . By construction it fits into the commutative diagram above, that is, we have g(m ⊗ n) = f(m, n). This takes care of the existence part of the assertion. Uniqueness is also clear, since commutativity of the diagram enforces the relation g(m ⊗ n) = f(m, n), and every element in M ⊗R N can be written as an R-linear combination of symbols m ⊗ n. Corollary 7.11. The tensor product of free R-modules of ranks m and n is free of rank mn. Proof. Since objects satisfying universal properties are unique up to a unique isomorphism, it suffices to show that we have a universal bilinear map − ⊗ −: Rm × Rn / Rmn. We will denote

66 m n mn the standard basis of R by e1, . . . , en, and of R by f1, . . . , fn. A basis of R is given by formal m n symbols ei ⊗ fj, and we deﬁne a bilinear map R × R , by sending (ei, fj) to ei ⊗ fj. Given a bilinear map f : Rm × Rn / P , where P is an arbitrary R-module, we use that f mn is uniquely determined by the values f(ei, fj). We have a linear map g : R / P , by sending ei ⊗ fj to f(ei, fj). By construction it veriﬁes commutativity as above and is the only linear map Rmn / P with this property.

As a corollary we obtain a tensor product operation for sheaves of modules, preserving locally free ones of ﬁnite rank. For locally free sheaves of OX -modules of ﬁnite rank this can be understood to be a tensor product operation for holomorphic vector bundles. Lemma 7.12. Let X be a topological space and R a sheaf of rings on X (commutative and with unit). For two sheaves of R-modules M and N , there exists a unique (up to a unique isomorphism) R-module M ⊗R N , and a bilinear sheaf morphism − ⊗ −: M × N / M ⊗R N , such that for every bilinear map of sheaves f : M × N / P, there exists a unique linear sheaf morphism g : M ⊗R N / P, such that the diagram

M × N f −⊗−

∃! $ M ⊗R N / P commutes. If M and N are locally free of ﬁnite rank, then so is M ⊗R N .

pre Proof. We deﬁne a presheaf (M ⊗R N ) , which sends an open subset U ⊂ X to the R(U)- module M(U) ⊗R(U) N (U). Applying Lemma 7.10 for every U ⊂ X, we see that we obtain a unique linear maps gU : M(U) ⊗R(U) N (U) / P(U), such that gU (m ⊗ n) = fU (m, n) for sections m ∈ M(U), n ∈ N (U). For an inclusion of open subsets U ⊂ V , we therefore see that the V two maps rU ◦ gV and gU must agree, because they both satisfy the commutativity condition V V (rU ◦ gV )(m ⊗ n) = (rU ◦ fV )(m, n) = fU (m, n), and gU (m ⊗ n) = fU (m, n). Therefore we obtain a commutative diagram of presheaves

M × N f −⊗− pre∃! % (M ⊗R N ) / P.

pre Defining M ⊗R N to be the sheafification of (M ⊗R N ) , and using the universal property of sheafification, we see that we there is a unique morphism of sheaves M ⊗R N / P, such that the following diagram of presheaves

M × N −⊗− f −⊗− w pre ∃! ∃! $ (M ⊗R N ) / M ⊗R N / P. commutes.

67 It remains to show that M ⊗R N is locally free of finite rank, when M and N are. Let ∼ m ∼ n U ⊂ X be an open subset, such that M |U = (R |U ) and N |U = (R |U ) . We know that X has an open covering consisting of open subsets U with this property. By corollary 7.11 we have pre ∼ mn that (M ⊗R N ) |U = (R |U ) . Since the right hand side is already a sheaf, it agrees with its sheafification M ⊗R N . This shows that M ⊗R N is locally free of finite rank. ? The reader might be curious if sheafification was really necessary. To see that this is indeed the case, consider the example O 1 (−1) ⊗ O 1 (1). The cocycle pictures implies that this sheaf is P P equivalent to O 1 . Hence its space of global sections is 1-dimensional. However, O 1 (−1) is known P P 1 1 1 not to have any non-zero global sections. Therefore, the tensor product O (−1)(P ) ⊗O 1 ( ) P P P O 1 (1) = 0. P n reg Definition 7.13. Let E be an algebraic vector bundle on P , that is a locally free sheaf of O n - reg P reg n modules of finite rank. For an integer d ∈ Z we define E(d) to be the tensor product E⊗O n O (d). P P The reason to introduce this twisting operation is the following important result.

n Theorem 7.14. Let E be an algebraic vector bundle on P . There exists an integer d0 ∈ Z, such that for d ≥ d0 the algebraic vector bundle E(d) is generated by finitely global (regular) sections reg n m s1, . . . , sm ∈ Γ (P ,E(d)). That is, the map of sheaves O / E(d), sending a tuple of regular functions (f1, . . . , fm) to f1s1 + ··· + fmsm is a surjective morphism of sheaves. reg S n Proof. Since E is a locally free sheaf of O n -modules, there exist Zariski open subsets Vi = , P i∈I P such that E| ∼ (Oreg)m, for some integer m (the rank of E). The topological space n is compact, Vi = Vi P i i hence we may assume that I is a finite set. For every i ∈ I we choose sections s1, . . . , sm which form a basis for E|Ui . i We will show that for each sj there exists a homogeneous polynomial G(z0, . . . , zn) of degree i n d >> 0, such that the section sj ⊗sG of E(d) extends from Ui to all of P . The intuitive idea is that i locally sj has a presentation as a rational function, and the zeroes of the denominator obstruct us i from extending sj beyond Ui. Since we are only dealing with finitely many sections, we can choose i i d big enough, to find homogeneous polynomials G of degree d, such that s ⊗ s i always extends j j Gj to a global section of E(d). n So let s ∈ E(U), where U is a Zariski open subset of P . We will show that there exists a n homogeneous polynomial G, such that s ⊗ sG extends to a global section of E(d). Let x ∈ P \U, n and choose Zariski open neighbourhood V ⊂ P of x. By making V smaller, we can assume that ∼ reg m reg E|V = (O n ) is a free O n -module. The section s|V can therefore be represented as a tuple of P P F1 Fm regular functions (f1, . . . , fm) = ( ,..., ). This shows that s ⊗ sG ···G extends uniquely from G1 Gm 1 m a section over U ∩ V to a section over V . n n Using compactness of P , one covers P by finitely many open subsets where an extension across U exists, after tensoring by a homogeneous polynomial. Taking the maximal degree, we can patch these local extensions together. At this point we can take a glimpse at the proof of the first assertion of GAGA. We will focus n on the case of algebraic vector bundles on P . Let’s take the following facts for granted.

n i n i n (a) For every algebraic vector bundle E on P we have a natural map H (PZar,E) / H (P ,E), mapping from the cohomology of the sheaf of regular sections to the cohomology of the sheaf of holomorphic sections. (GAGA asserts that this map is an isomorphism.)

68 reg (b) For the line bundles O n the map from (a) is an isomorphism (GAGA holds for these line P bundles). (c) For every short exact sequence 0 / K / E / F / 0, with E, F , algebraic vector bundles, K is an algebraic vector bundle too.

(d) Both algebraic, and holomorphic cohomology groups of algebraic vector bundles vanish for degrees i > m.

n If E is an arbitrary algebraic vector bundle on P , then we have seen that there exists a d ∈ Z, reg m reg such that we have a surjective map of sheaves (O n ) E(d). Tensoring with O n (−d) we obtain reg m P P a surjective morphism of sheaves (O n (−d)) E. Let’s denote the kernel by K. By fact (c) we P have a short exact sequence of algebraic vector bundles

reg m 0 / K / (O n (−d)) / E / 0. P Let’s take a look at the long exact sequences for cohomology of regular sections and holomorphic sections:

i n i n reg m i n i+1 n i+1 n H ( ,K) / H ( , (O n (−d)) ) / H ( ,E) / H ( ,E) / H ( ,K) PZar PZar P PZar PZar PZar

f g h k ` i n i n m i n i+1 n i+1 n / n / / / H (P ,K) H (P , (OP (−d)) ) H (P ,E) H (P ,E) H (P ,K)

For very high degrees both types of cohomology groups vanish. We will therefore prove by i n i n descending induction on i that H (PZar,E) / H (P ,E) is an isomorphism. Let’s assume that this has been proven for i + 1 already. That is, the maps k and ` are isomorphisms. Since g is by assumption also bijective, we see from the Five Lemma that h is surjective. But K is also a vector bundle, therefore we see that the map f has to be surjective too. Another application of the Five Lemma gives us that h is an isomorphism.

7.2 Coherent sheaves In this paragraph we discuss the theory algebraic coherent sheaves. This will allow us to reduce the n entire statement of GAGA to a statement about sheaves on P rather than a complex submanifold n n X ⊂ P given by a smooth projective variety X. Let i: X,→ P be the canonical inclusion map. A vector bundle on X can be understood as a locally free sheaf E of OX -modules on X of finite n −1 rank. We can push the sheaf E forward to P , that is, consider (i∗ E)(U) = E(i (U)) = E(U ∩ X). We have verified on E6 that this construction defines a sheaf. We know that i∗ E cannot be n n locally free in general, since for V ⊂ P \X we have (i∗ E)(V ) = E(∅) = 0. Therefore, i∗ E |P \X = 0, which could only happen for a locally free sheaf, if it was of rank 0. However, the pushforward i∗ E turns out to be coherent. Versions of the theory of coherent sheaves exist in both algebraic and analytic geometry. We will content ourself with establishing the basic properties of algebraic coherent sheaves. The analytic case requires more care. Before delving into this theory, we prove the following lemma, which will allow us to reduce the n proof of GAGA to the case of P .

69 Proposition 7.15. Let X ⊂ Y be a closed subspace of a topological space Y , we denote the inclusion map X,→ Y by i. For every sheaf F on Y we have a canonical equivalence i ∼ i H (X, F) = H (Y, i∗ F). The proof relies on a couple of lemmas.

Lemma 7.16. For i as above, the pushforward functor i∗ is exact, that is that it sends a short exact sequence 0 / F / G / H / 0 on X to a short exact sequence on Y . Proof. Recall that a sequence of sheaves on Y is exact, if and only if the sequence of stalks at y is exact for every y ∈ Y . So let’s compute the stalks of (i∗ F)y. We claim that for y∈ / X the stalk is 0, and for x ∈ X, the stalk equals F x. This then immediately veriﬁes the claim that the pushforward (along the embedding of a closed subspace) of an exact sequence is still exact. It remains to compute the stalks of i∗ F. Recall that the element of the stalk of a sheaf K at y, are given by equivalence classes of pairs (U, s), where U ⊂ Y is an open neighbourhood of x, and s ∈ K(U) is a section. Two pairs (U, s) and (V, t) are equivalent if ∃x ∈ W ⊂ U ∩ V , with W open, such that s|W = t|W . For y∈ / X, we can choose a neighbourhood V ⊂ Y \ X of y. Since every s ∈ (i∗ F)(U) equals 0. We conclude that (i∗ F)x = 0. For x ∈ X, every open neighbourhood U ⊂ X of x in X, can be realised as an intersection U = 0 0 0 U ∩X, where U is an open neighbourhood of x in Y . The resulting abelian group (i∗ F)(U ) = F(U) 0 is independent of the chosen U with this property. This implies directly that (i∗ F)x = F x. It’s important to emphasise that for a general continuous map f : X / Y , the pushforward functor f∗ is not exact. For Y = {pt} a singleton, Sh(Y ) = AbGrp, and f∗ agrees with the global sections functor Γ(X, −) with respect to this identiﬁcation. We know from the example of the exponential sequence that this functor is not exact!

Lemma 7.17. For a continuous map f : X / Y , the functor f∗ : Sh(X) / Sh(Y ) has a left adjoint f −1 : Sh(Y ) / Sh(X). That is, for every sheaf F ∈ Sh(Y ) and G ∈ Sh(X) we have

−1 HomSh(X)(f F, G) / HomSh(Y )(F, f∗ G). Proof. We would like to define (f −1 F)(U) = F(f(U)). But unless f is an open map, this doesn’t make any sense a priori. We will therefore interpret this formula by taking the stalk of F around f(U). That is we consider equivalence classes of pairs (V, s), where V ⊂ Y is an open subset containing f(U), and s ∈ F(V ). The equivalence relation is defined as for stalks: if there exists an 0 0 open subsets W ⊂ V ∩ V , such that f(U) ⊂ W , such that t|W = s|W , then (V, s) and (V , t) are equivalent. −1 The resulting presheaf fpre F is not a sheaf in general. But we know that this can be remedied by sheafification. This is how f −1 F is defined. g −1 h Let’s check that every map F / f∗ G gives rise to a map f F / G. Given g, we have for g every V ⊂ Y a homomorphism F(V ) V / G(f −1(V )). For an open subset V ⊂ Y , containing f(U), −1 −1 and an element [(V, s)] ∈ f F(U) we can therefore look at the image gV (s) ∈ G(f (V )), and −1 restrict it further down to U. This defines a morphism of presheaves fpre F / G. The universal property of sheafification yields a unique morphism f −1 F / G. g Vice versa, given h, we can construct a morphism of sheaves F / f∗ G. This is left as an exercise to the reader.

70 Proof of Proposition 7.15. We will show that for every short exact sequence 0 / F / I / I / F 0/ with I injective, the pushforward of this sequence

0 / i∗ F / i∗ I / i∗(I / F) / 0 is also a short exact sequence with i∗ I injective. Exactness has already been veriﬁed in Lemma 7.16. Let’s show that i∗ I is injective. A diagram M / N

i∗ I corresponds to a diagram i−1 M / i−1 N

∃ z I by Lemma 7.17. Since I is injective, there exists a dashed arrow as in the diagram above, rendering the triangle commutative. Applying Lemma 7.17 again, we obtain M / N

} ∃ i∗ I, and therefore see that i∗ I is injective.

n reg Coherent sheaves on are certain sheaves of O n -modules. The following examples are the PZar P most important ones for us: locally free sheaves of finite rank E (algebraic vector bundles), and n also for every inclusion of a complex projective manifold i: X,→ P the sheaves i∗ E are coherent, where E is an algebraic vector bundle on X. There are however more examples of coherent sheaves, n that is, not every coherent sheaf on P is of the form i∗ E. At first we define a related notion, the one of quasi-coherent sheaves. n Although C is not a complex projective manifold, it makes sense to speak of the Zariski n n n topology on C , and the sheaf of regular functions on C . The reason is that we can realise C n n as a Zariski-open subset of P . In fact, any of the standard charts Ui ⊂ P does the trick. The n n Zariski topology on P induces therefore a topology on the subsets Ui = C . Restriction of sheaves yields a sheaf of regular functions Oreg. The resulting topological space, with a sheaf of rings Oreg n n is often denoted by A , to distinguish it from C with the standard topology and the sheaf of n holomorphic functions. It is called n-dimensional affine space. Its main premise is that A can be n n defined without reference to Zariski open subsets of P , and regular functions on P . n Lemma 7.18. A subset U ⊂ C is Zariski-open, if and only if there exist polynomials h1, . . . , hk ∈ n C[x1, . . . , xn], such that U = {(z1, . . . , zn) ∈ C |∃i ∈ {1, . . . , k}: hi(z1, . . . , zn) 6= 0}. A function Sm f : U / C is regular, if and only if there exists a Zariski-open covering U = i=1 Ui, such that gi f|U = for polynomials gi, hi ∈ [x1, . . . , xm], hi 6= 0. i hi C

71 Proof. We leave the details of the proof as an exercise to the reader. The main idea of the con- g n n struction is the following. We can pass to rational functions h on C to rational functions on P G n by homogenisation. Similarly, given a rational function H on P with G and H two homogeneous G polynomials in n + 1 variables of degree d. The restriction H |Ui depends solely on the dehomogeni- sations of G and H, obtained by specialising the variable zi to 1. Using this correspondence, and the fact that both the Zariski topology, and the sheaf of regular functions, are deﬁned in terms of rational functions, one obtains the statement of the lemma. Lemma 7.19. We denote by R as always a commutative unital ring. Let M be an R-module, and m f ∈ R an element. The localisation Mf of M at f is the R-module given by formal symbols f n , m m0 d n0 n 0 where m ∈ M, and n ≥ 0. We stipulate that = 0 if and only if f (f m − f m ) = 0 for f n f n d >> 0.

Note that Rf is a ring itself, which could be defined to be R[x]/(xf − 1). It’s the ring obtained −1 from R by formally adjoining an inverse f to f. The R-module Mf is in fact naturally an Rf -module. We can now define quasi-coherent sheaves, the immediate predecessor of coherent sheaves. There n definition is inspired by a property we observed for algebraic vector bundles E on P . If H ∈ C[z0, . . . , zn]d is a non-zero homogeneous polynomial of degree d, then we have a Zariski open n n subset V ⊂ P given by the locus where H is non-zero. Let U ⊂ P be another Zariski open subset. t A regular section s ∈ E(U ∩ V ) can be written as Hk , where k ∈ N and t ∈ E(U) is a regular section over U. It is precisely the appearance of H in the denominator which stops us from extending s across the complement U \ V ⊂ U.

reg n Deﬁnition 7.20. (a) A sheaf F of O -modules on A is called quasi-coherent, if for every n n n n −1 Zariski open subset Uf = {z ∈ C |f(z) 6= 0} ⊂ A we have F(Uf ) = F(A )f = F(A )[f ]. reg n (b) A sheaf F of O n -modules on is called quasi-coherent, if for every i ∈ {0, . . . , n} the P PZar sheaf F |Ui is quasi-coherent. n Next we classify quasi-coherent sheaves on A . The proposition below tells us that the functor, n n sending a quasi-coherent sheaf F on A to the C[x1, . . . , xn]-module F(A ) is an equivalence of categories.

n n Proposition 7.21. We have an equivalence of categories Γ(A , −): QCoh(A ) / Mod(C[x1, . . . , xn]), sending exact sequences of sheaves to exact sequences of modules. In other words, for every quasi- n coherent sheaf F on A there exists a unique C[x1, . . . , xn]-module M, such that F(Uf ) = Mf , every module M arises in this way, and Oreg-linear morphisms of quasi-coherent sheaves, correspond to morphisms of C[x1, . . . , xn]-modules. Proof. The proof is formally very similar to the one of Proposition 7.26, which we will discuss below. It will be broken down into several easy steps on an exercise sheet, and left to the reader as an exercise.

n The fact that the global sections functor for quasi-coherent sheaves on A is exact, can be strengthened2:

2Possibly we will include a proof of this Corollary in a non-examinable appendix in the future.

72 n Corollary 7.22. Let F be a quasi-coherent sheaf on A . Then for every i > 0 the cohomology n groups Hi(A , F) vanish. n Lemma 7.23. A quasi-coherent sheaf F on A is called coherent, if the corresponding C[x1, . . . , xn]- module M is finitely generated. This is a local property. That is, if there exists a finite Zariski n Sk open covering A = j=1 Ufj by Zariski open subsets Uf , such that F(Uj) is a finitely generated

C[x1, . . . , xn]fj -module for every j, then F is coherent.

j j Proof. For every j = 1, . . . , k we choose elements s1, . . . , sm generating F(Ufj ). Since F is quasi- n j coherent, we have F(Ufj ) = F(A )fj , and therefore there exists d >> 0, such that we can write j j ti j si = d with ti ∈ F(M). Consider the free R-module F of finite rank m generated by symbols fj i j j ej for j = 1, . . . , k and i = 1, . . . , mj. Let F / M be the map sending ei to ti . By virtue of reg m Proposition 7.21 we obtain a well-defined map (O n ) / F. Its cokernel has to be 0, since it is A zero with respect to the open covering Ufj . This implies that the map is surjective, and therefore M is finitely generated, and F is coherent.

reg n Deﬁnition 7.24. A sheaf F of O n -modules on is called coherent, if for every i = 0, . . . , n the P P restriction F |Ui is a coherent sheaf on aﬃne space.

Sometimes the standard covering {Ui}i∈I is too ﬁne to check coherence. In these cases, it is useful to remember Lemma 7.23, which allows one to check coherence on a ﬁner Zariski open covering. The basic properties of coherent sheaves listed below will be useful when proving GAGA.

n Lemma 7.25. (a) The direct sum F 1 ⊕ F 2 of two coherent sheaves on PZar is again coherent.

(b) For a morphism of coherent sheaves F 1 / F 2, kernel and cokernel are coherent. n (c) For every complex projective submanifold i: X,→ P , and an algebraic vector bundle E on n n X, i∗ E is a coherent sheaf on PZar. In particular, ever algebraic vector bundle on PZar is coherent. Proof. Assertion (a) is clear, its proof boils down to the fact that the direct sum of two finitely generated modules is again finitely generated. The same reasoning applies to the first part of assertion (b): if M1 / M2 is a map of finitely generated R-modules, then coker f = M2/ image f is finitely generated as a consequence of the n n finite generation of M2. Indeed, if R M2 is a surjective map, then the composition R M2 coker f is surjective and proves finite generation of coker f. The assertion that ker f is finitely generated is not true for arbitrary commutative rings R (with unit). However, it is true for so-called Noetherian ones, that is rings for which every ideal is finitely generated. Noetherian rings satisfy the property that every submodule of a finitely generated module is finitely generated (the proof is a simple inductive argument on the minimal number on generators). Since polynomial rings happen to be Noetherian, we deduce that ker f is finitely generated too. reg n In order to prove (c) it suffices to check that i∗ OX is coherent on PZar. Indeed, if E is locally n Sk n free of rank m, then there would exists a Zariski open covering P = j=1 Vj of P , such that E | is free of rank m, that is isomorphic to (Oreg )m (we can produce a finite cover, using Vj ∩X Vj ∩X n ∼ m compactness of P ). Therefore we see that (i∗ E)|Vi = (i∗ OVj ∩X ) . Since coherence is a local property by Lemma 7.23, and we assume that i∗ OVj ∩X is coherent, we are done. It remains to reg show that i∗ OX is indeed coherent.

73 If X = V (G1,...,Gk), then for every i = 0, . . . , n we have that X ∩ Uj is the zero set of the system of polynomial equations gi(w1, . . . , wn) = Gi(w1, . . . , wi, 1, wi+1, . . . , wn). The so-called Nullstellensatz in commutative algebra then implies that the ring of regular functions on X ∩ Uj is equal to C[w1, . . . , wn]/(g1, . . . , gn), which is a module generated by one element. n Proposition 7.26. For every coherent sheaf F on PZar, there exists a d ∈ Z, such that F(d) = reg reg reg m reg n n n F ⊗O n O (d) is globally generated, that is, there exists a surjection of O -modules (O ) P P P P F(d).

Proof. For i = 0, . . . , n the restriction F |Ui corresponds to a ﬁnitely generated C[w1, . . . , wn] = zj [( )j6=i]-module Mi. Since F is quasi-coherent, we can express an element in M = F(Ui) over C zj d z j t tj j j zi Uj ∩ Ui for j 6= i as a fraction s = d = d , where tj ∈ F(Uj) and w = wi = if i > j, and j j i(j) zj wi(j) zi wj+1 if j < i. d z k t d` j j z` t` 0 On Uij` we have s|Uij` = dk and d` that we want to compare. We can assume dk = d` = d zi zi by choosing the maximum. The deﬁnition of localisation implies that there exists an m ≥ 0, such 0 0 m+d 0 m+d z` m+d m+di` m+dik that zi tj = m+d0 zi t`. Replacing tj by zi tj, and t` by zi t`, we can therefore zj i i 0 assume that t |Uij` and t`|Uij` assume as sections of F(d), where d = m + d . The sheaf property d of F allows one to patch the collection (tj)j6=i, together with zi s to a global section t ∈ F(d)(X), t such that s = d . zi We choose sections si , . . . , si (respectively elements of M ), which generate F | . For ev- 1 mi i Ui i ery element of this list we choose an extensions ˜k to a global section of F(d). The collection i j=1,...,mi (˜sk)i=0,...,n generates F(d) as a sheaf, because for every Ui we have that F(d)|Ui is generated by k k=1,...,mi k k the elements (˜si ) , becauses ˜i |Ui = si generates F |Ui by assumption. Here we use the reg ∼ reg natural trivialisation O n (d)|U = O . P i Ui 7.3 A short exact sequence for twisting sheaves n−1 In a non-canonical manner, (n − 1)-dimensional complex projective space P is a complex sub- n n−1 manifold of P . Indeed, we can realise P as the complex projective variety, corresponding to n−1 the linear homogeneous equation z0 6= 0. Following Serre we refer to this particular copy of P n n n inside P with E = V (z0) ⊂ P . We write i: E,→ P for the inclusion morphism. The following n short exact sequence will enable us to prove GAGA for the sheaves OP (d) by induction on n. Lemma 7.27. For every d ∈ Z we have a short exact sequence of sheaves

/ n / n / / 0 OP (d − 1) OP (d) i∗ OE(d) 0, respectively reg reg reg 0 / O n (d − 1) / O n (d) / i∗ O (d) / 0. P P E Proof. We give the proof for sheaves of holomorphic sections (the ﬁrst short exact sequence), since the proof for regular sections is the same. At ﬁrst we prove the assertion for d = 0. We begin by

n constructing a locally surjective morphism of sheaves p: OP i∗ OE. We let p be the restriction n morphism, which sends a holomorphic function f : U / C, where U ⊂ P is an open subset, to f|E.

74 In order to show that p is locally surjective, consider one the charts (Ui, φi). We will show that n / the maps OP (Ui) i∗ OE(Ui) are surjective. For i = 0, the right hand side is equal to 0, because n E = P \U0, and thus i∗ OE(U0) = OE(E ∩ U0) = OE(∅) = 0. n For i 6= 0, E ∩ Ui corresponds to the closed subset of C given by {w1 = 0}. Therefore, i∗ OE(Ui) = {f(w2, . . . , wn) holomorphic function)}. But it is clear that we can extend f to a holomorphic function in (w1, . . . , wn) by defining g(w1, . . . , wn) = f(w2, . . . , wn). z0· n / n What’s the kernel of the morphism of sheaves p? It is given by the map OP (−1) OP , n n n which multiplies a local section s ∈ OP (U) with the restriction of z0 ∈ OP (P ). To see that this n n ∼ n makes sense, recall our definition of the line bundles Ld, or use the identity OP (−1)⊗OP (1) = OP . As before, we can check that this is the kernel of p, after restricting to the charts Ui. Over U0, z0 z0· n / n corresponds to the constant holomorphic function 1, and we see that OP (−1)|U0 OP |U0 is an isomorphism of sheaves, as we would have expected from OE |U0 = 0. For i 6= 0, z0 corresponds to the holomorphic function (w1, . . . , wn) 7→ w1. We have to show that every holomorphic function f on Ui, satisfying f|E = 0, can be written as f = w1g, where g is a holomorphic function. It suffices to show that f is locally holomorphic. In order to see this we apply the Weierstrass division w1 theorem, to locally obtain f = hw1 + r, where r is a polynomial in w1 of degree 0, that is, it’s a holomorphic function in the variables (w2, . . . , wn). Since f|E = w1|E = 0, we obtain r|E = 0. Since r is independent of w1, we see that r = 0. This proves the assertion. The general case is obtained by tensoring the short exact sequence above with O n (d), and ∼ P using the rule O n (d1) ⊗O n O n (d2) = O n (d1 + d2). The only remaining step is to check that P P P P zd ∼ j (i∗ OE)(d) = i∗(OE(d)). This follows by recalling that O n (d) is given by the cocycle ( d )ij, which P zi restricts to the cocycle datum for OE(d) on E (we’re just omitting the variable z0). i ∼ i n These short exact sequences are relevant, because we’ve seen that H (E, F) = H (P , i∗ F). The corresponding long exact sequence in cohomology will enable us to prove comparison results

n for the cohomology of the sheaves OP (d) by induction on n. Before getting to this, we need i n reg i n to introduce the comparison maps H ( , O n (d)) / H ( , O n (d)). Before getting to this, PZar P PZar P we make a little detour via the Riemann-Roch Theorem for Riemann surfaces (that is, compact complex manifolds of complex dimension 1). We will use (without proof) that cohomology groups of vector bundles on projective complex manifolds are ﬁnite dimensional vector spaces. 1 1 Theorem 7.28 (Riemann-Roch for P ). Let F be a sheaf on P , we denote by χ(F ) ∈ Z the integer 1 1 0 0 n given by dim H (P , F) − dim H (P , F) (assuming it is well-deﬁned). We have χ(OP (d)) = d + 1.

d0 dn−1 Sketch. If 0 / A0 / ··· / An−1 / / An / 0 is an exact sequence of vector spaces, Pn i then the alternating sum of dimensions i=0(−1) dim Ai = 0. This can be proven by induction on n. The anchoring cases of n = 0, or n = 1 are easy to prove. We also have an exact sequence

d0 dn−1 0 / A0 / ··· / An−1 / / ker dn−1 / 0, of length n = n + 1 − 1. Therefore, by virtue Pn−2 i n−1 of induction, i=0 (−1) dim Ai + (−1) dim ker dn−1 = 0. Since dn : An−1 / An is surjective, we have dim ker dn−1 = dim An−1 − dim An, which concludes the proof of the claim. We apply this to the long exact sequence corresponding to the short exact sequence 0 / O(d − 0 1) / O(d) / OE(d) / 0. Note that E is a point, therefore H (E, OE(d)) = C is the only non-zero cohomology group. Using the cohomology vanishing property (which we will discuss in the future) that H2(P, O(d)) = 0, we obtain 0 1 0 1 1 1 1 1 0 / H (P , O(d − 1)) / H (P , O(d)) / C / H (P , O(d − 1)) / H (P , O(d)) / 0 / 0.

75 The alternating sum of dimensions can be re-arranged into the identity χ(O(d)) = χ(O(d − 1)) + 1. 1 Therefore it remains to prove the assertion for a single d ∈ Z. We already know that H0(P , O) = 1, 1 next week we will show that it’s also true that H1(P , O) = 0. 1 Together with our previous computation that dim H0(P , O(d)) = d + 1, if d ≥ 0, the Riemann- 1 Roch formula determines the dimensions of the cohomology groups of the sheaves O(d) on P completely. We can also observe the following interesting symmetry

0 1 1 1 dim H (P , O(d)) = dim H (P , O(−2 − d)). This fascinating identity can be explained through so-called Serre duality, the counterpart of Poincar´eduality for coherent sheaves. One can show that on a smooth projective manifold X of dimension n, and a vector bundle E, we have a canonical isomorphism of vector spaces

i ∼ n−i ∨ n H (X,E) = H (X,E ⊗ ΩX ).

The Riemann-Roch theorem can be generalised to smooth projective manifolds X, but even stating it properly would take us too far aﬁeld. For X a Riemann surface, this is possible. For 1 every line bundle L on X, one can deﬁne its degree deg L ∈ Z. For X = P we have deg O(d) = d. The underlying topological space of X is an orientable surface of genus g. The Riemann-Roch formula reads χ(L) = 1 − g + deg L.

7.4 The proof of the ﬁrst GAGA theorem (modulo basic cohomology computations) n reg Let F be a coherent sheaf on PZar, for example F could be a locally free sheaf of O -modules, corresponding to an algebraic vector bundle. Serre deﬁnes a sheaf of holomorphic sections F h on n h P . For E an algebraic vector bundle, given by a regular cocycle datum ({Ui}i∈I , (φij)), E is n equal to the sheaf of holomorphic sections E. That is, to an open subset V ⊂ P , we assign the collection of holomorphic functions fi : Ui ∩ V / C, such that fi = φijfj. This functor can be generalised to coherent sheaves, by using the fact that Zariski locally, every coherent sheaf can be written as a cokernel of a map of locally free sheaves:

(Oreg)m / (Oreg)k / F / 0.

By deﬁnition, (Oreg)h = O, hence it makes sense to write F h = coker(Om / Ok). It is important to check that this does not depend on the presentation of F as a quotient. We will say more about this in a lecture devoted entirely to this process of taking holomorphic sections. The following lemma is essential to the proof of GAGA. It asserts the existence of a comparison map between regular and holomorphic cohomology for coherent sheaves.

h n n Lemma 7.29. (a) The functor F 7→ F from coherent sheaves on PZar to sheaves on P sends exact sequences of coherent sheaves to exact sequences.

n i n i n h (b) For every i ∈ N, and coherent sheaf F on P there exists a natural map H (PZar, F) H/ (P , F ). Naturality asserts that for every morphism F / G of coherent sheaves we have a commutative

76 diagram i n i n H (PZar, F) / H (PZar, G)

n h n h Hi(P , F ) / Hi(P , G ), and for every short exact sequence 0 / F / G / H / 0 of coherent sheaves we have a commutative diagram i n i+1 n H (PZar, H) / H (PZar, F)

n n h Hi(P , Hh) / Hi+1(P , F ). The proof of this lemma is deferred to another lecture. The proof of (a) makes use of results related to Weierstrass preparation, and is mostly algebraic. The proof of (b) relies on the fact that n n h for every coherent sheaf F on A , the higher cohomology groups Hi(C , F ) vanish (here we use n n the fact that the sheaf of holomorphic sections can be deﬁned locally on P , and that P can be n covered by Zariski open subsets equivalent to A ). n i n Next week we will learn the techniques to prove H (P , OP ) = 0 for i ≥ 1. Taking this for granted, we obtain the following interesting special case of GAGA. i n reg i n Proposition 7.30. If the natural map H (PZar, O ) / H (P , O) is an isomorphism for every i n reg i n i ∈ N, and every n ∈ N, then we have that also H (PZar, O (d)) /H (P , O(d)) is an isomorphism for every d ∈ Z. reg reg reg Proof. This follows from the short exact sequence 0 / O n (d − 1) / O n (d) / i∗ O (d) / 0. P P E Applying the functor given by holomorphic sections, the resulting long exact sequence in cohomology gives us the following commutative diagram with exact rows

i n reg i n reg i n reg i reg i+1 n reg H ( , O n (d)) / H ( , O n (d − 1)) / H ( , O n (d)) / H (EZar, O (d)) / H ( , O n (d − 1)) PZar P PZar P PZar P E PZar P

i n reg i n i n i i+1 n H ( , O n (d)) / H ( , O n (d − 1)) / H ( , O n (d)) / H (E, OE(d)) / H ( , O n (d − 1)), PZar P P P P P P P i n reg i n and the Five Lemma implies together with induction on n, that H (PZar, O (d)) / H (P , O(d)) is an isomorphism, if it is an isomorphism for d − 1. Similarly, we can show that the case of d,

n implies the one of d − 1. This reduces the entire assertion of GAGA for the sheaves OP (d) to the case d = 0. We are now ready to prove the entire proof of part (a) of the GAGA theorem, to the assertion i n reg i n / n that H (PZar, O n ) H (P , OP ) is an isomorphism. For i = 0 we already know this to be true. P n Next week we will show that the higher cohomology groups Hi(P , O) vanish. Together with the descending induction argument, this concludes the proof of GAGA (part (a)). i n reg i n Proposition 7.31. Assume that the natural map H (PZar, O ) / H (P , O) is an isomorphism n for every i ∈ N, and assume that higher cohomology groups of coherent sheaves on P vanish for n i n i n h i ≥ n + 1. Then, for every coherent sheaf F on PZar, the map H (PZar, F) / H (P , F ) is an isomorphism.

77 Corollary 7.32. Let X be a projective complex manifold, and E an algebraic vector bundle on X. Then the vector space of holomorphic section of E is isomorphic to the vector space of regular

n sections of E (assuming GAGA(a) for OP ). n Proof. Let E be the sheaf of regular sections of E on XZar. We denote by i: X,→ P an inclusion 0 0 n of X into complex projective space. We have H (XZar, E) = H (P , i∗ E). By GAGA part (a) this 0 vector space is equivalent to H (X, i∗E), which agrees with the space of holomorphic sections of E. Proof of Proposition 7.31. According to Proposition 7.26 we can choose d >> 0, such that there is a reg m reg m surjection of sheaves (O n ) F(d). Untwisting, we obtain a surjection of sheaves (O n (−d)) P P F. Let K be the sheaf given by the kernel of this morphism of sheaves. The resulting short exact sequence of sheaves yields the following commutative diagram with exact rows.

i n i n reg m i n i+1 n i+1 n reg m H ( ,K) / H ( , (O n (−d)) ) / H ( , F) / H ( ,K) / H ( , (O n (−d)) ) PZar PZar P PZar PZar PZar P

f g h k ` i n h i n m i n h i+1 n h i+1 n m / n / / / n H (P ,K ) H (P , (OP (−d)) ) H (P , F ) H (P ,K ) H (P , (OP (−d)) ) As planned, we argue by descending induction on i. For i > n, the cohomology groups vanish, and therefore the map comparing regular and holomorphic cohomology is an isomorphism. We also know that the maps g and ` are isomorphisms for all values of i. By descending induction we can assume that k is an isomorphism. The Five Lemma now implies that h is surjective. Since K is also a coherent sheaf this implies that f is surjective too, and hence a second application of the Five Lemma yields that h is bijective. Part (a) of GAGA quickly implies (b). We will also explain the idea underlying the proof of (c), but since the proof uses a lot more analysis, we will not be able to explain all the details for time reasons. n Corollary 7.33. Let E1 and E2 be two algebraic vector bundles on P , or more generally a complex h h projective manifold X. Then, every OX -linear map E1 / E2 is induced by a unique regular morphism E1 / E2. reg Proof. The sheaves E1, and E2 are OX -locally free of finite ranks n1 and n2. Let Hom(E1, E2) be reg reg the sheaf of OX -modules, which assigns to a Zariski open subset U ⊂ X the OX -module of linear maps Hom(E1(U), E2(U)). We leave the standard verification that Hom(E1, E2) is a sheaf to the reader. We claim that Hom(E1, E2) is locally free of rank n1n2. By definition, its global sections are equivalent to the vector space of linear maps E1 / E2. GAGA part (a) implies therefore that h h every linear map between E1 and E2 is induced by a unique linear map E1 / E2. Let U ⊂ X be an open subset, such that E1 |U and E2 |U are free of ranks n1 and n2. We have reg n1 reg n2 reg n1×n2 Hom((OX ) , (OX ) ) = (OX ) .

There is also a theory of analytic coherent sheaves on a complex manifold. An OX -linear sheaf S F on X is called an analytic coherent sheaf, if there exists an open covering X = i∈I Ui, such that for every i ∈ I we have an exact sequence Om / Ok / F / 0. Ui Ui

78 The proof of GAGA (c) follows now from this result, and the following important theorem. Serre n n deduces it from the fact that P is compact, and that the cohomology groups Hi(P , F) are ﬁnite- dimensional. n Theorem 7.34. Let F be an analytic coherent sheaf on P , then there exists d >> 0, such that we k have a surjection O n F(d). P k / / n / / As before, this yields a short exact sequence 0 K (OP (−d)) F 0, and using the fact that K is also an analytic coherent sheaf, we can express F as a cokernel of a linear morphism 0 m k n / n (OP (−d )) (OP (d)) . We have just seen that every such morphism is regular, hence F must be equal to the sheaf of holomorphic sections of an algebraic coherent sheaf.

7.5 Holomorphic sections of coherent sheaves In this paragraph we briefly discuss the definition of the functor F 7→ F h, discuss why it is exact, i n i n h n and define a natural morphism H (PZar, F) / H (P , F ). Let U be a Zariski open subset of P . n We will mostly be interested in the cases U = P or U = Ui. We denote by Φ: U / UZar the continuous map, which is the identity on underlying sets. It is continuous, since a Zariski open subset is also open in the standard topology.

reg h −1 Deﬁnition 7.35. For a sheaf F of O -modules, we deﬁne F = Φ F ⊗ −1 reg OU . U Φ OU −1 Recall that Φ : Sh(UZar) / Sh(U) is the functor we constructed in Lemma 7.17 for a gen- −1 reg eral continuous map. We have a canonical map Φ OU / OU , which is adjoint to the map reg OU / Φ∗ OU (see Lemma 7.17 for an explanation of the word adjoint).

Lemma 7.36. Holomorphisation satisﬁes a universal property. Let G be a sheaf of OU -modules, reg reg and F / Φ∗ G an OU -linear map (where Φ∗ G is viewed as a sheaf of OU through the canonical reg h map OU / Φ∗ OU ). Then there exists a unique morphism of sheaves F / G, such that F

h # Φ∗ F / Φ∗ G commutes. This universal property implies directly that holomorphisation commutes with cokernels, that is, h ∼ h h coker(F 1 / F 2) = coker(F 1 / F 2 ). This shows that for a short exact sequence 0 / F / G H/ 0/ of coherent sheaves, we have an exact sequence F h / Gh / Hh / 0, but showing that the map F h / Gh is injective is less obvious. Serre proves this using commutative algebra, the proof is not diﬃcult to understand, but unfortunately we don’t have the time to explain it. In the end it boils down to the Weierstrass Preparation Theorem and formal computations with power series. The n following assertion is more generally known to be true for analytic coherent sheaves on C . This n version is known as Cartan’s Theorem B. The proof below uses that Hi(C , O) = 0 for i ≥ 1. We will prove this next week, using the Dolbeault resolution. n Proposition 7.37 (Weak Cartan B). Let F be an algebraic coherent sheaf on A , then for i ≥ 1 n h n we have Hi(C , F ) = 0. The same statement holds when we replace A by a Zariski open subset of the shape {f 6= 0}, where f ∈ C[x1, . . . , xn] is a polynomial.

79 n h Proof. We prove this by descending induction, using the fact that Hi(C , F ) = 0 for i > n, and n Hi(C , O) = 0 for i ≥ 1. We will see a proof of these statements very soon, using the formalism of Dolbeault cohomology. reg m Since F is coherent, there exists a short exact sequence 0 / K / (O n ) / F / 0, where A n h n K is also coherent. The long exact sequence in cohomology implies Hi(C , F ) = Hi+1(C ,K) for i ≥ 1. Therefore, we obtain by descending induction on i the asserted vanishing statement.

n n We denote by Φ: P / PZar the continuous map, which is the identity on underlying sets. It is continuous, because every Zariski open subset is also open in the standard topology. Part (b) i n i n h of Lemma 7.29, which asserted the existence of a natural map H (PZar, F) / H (P , F ) follows h from the lemma below, and the fact that we have a natural map F / Φ∗ F . n i n h ∼ i n h Lemma 7.38. For every coherent sheaf F on PZar, we have an equivalence H (P , F ) = H (PZar, Φ∗ F ), that is, it doesn’t matter if we compute the cohomology of F h with respect to the standard or the Zariski topology.

Proof. Let (I•, d) be an injective resolution of F, that is a complex of injective sheaves, exact in h • positive degrees, such that ker d0 = F . The pushforward Φ∗(I ) is also a chain complex. If we i knew that the sheaves Φ∗ I were still acyclic (= no higher cohomology), then we would only have to check that this still defines a resolution of Φ∗ F. The pushforward of injective sheaves is not injective, but can be shown to be flasque as follows. We have seen in ex. 1, E8 that injective sheaves are flasque, and that the pushforward of flasque sheaves is flasque. Hence, it remains to • show that Φ∗(I ) is exact in positive degrees. Since exactness for chain complexes of sheaves is a n • local property, it suffices to show that for every standard chart Ui ⊂ P , the restriction Φ∗(I )|Ui is exact in positive degrees. Let V = (Ui)f ⊂ Ui be a Zariski open subset of the form {f 6= 0}, for a 0 1 • polynomial f. The chain complex of global sections I (V ) / I (V ) / ··· computes H (V, F |V ), which we have seen to vanish in positive degrees. Since every Zariski open subset of Ui is a union of subsets (Ui)f , this implies the required exactness assertion in positive degrees.

8 Computing Cohomology: Dolbeault and Cechˇ

In this section we will discuss two important tools to compute cohomology of sheaves. The Dolbeault resolution and the Cechˇ complex.

8.1 The Dolbeault resolution

The Dolbeault resolution is to the sheaf of holomorphic functions OX on a complex manifold, what the de Rham resolution is to the sheaf of locally constant functions R on a smooth manifold. Before ∂ we can define it, we need to introduce some notation, and recall the differential operator ∂z¯ from n i 2.4. For a C∞-function f : U / C, where U ⊂ C we defined ∂f 1 ∂f ∂f = ( + i ). ∂z¯i 2 ∂xi yi

The function f is holomorphic, if and only if for every i = 0, . . . , n we have ∂f = 0. This is simply ∂z¯i a reformulation of the Cauchy-Riemann equations.

80 The idea underlying the Dolbeault resolution is that there should be a sheaf A0,1, and a map ∞ ∂ 0,1 ∞ CX / A , such that ker ∂ = OX . Here, CX denotes the sheaf, sending an open subset U ⊂ X to the vector space of C∞-functions U / C. Of course we cannot stop there, we have to keep going and produce a complex

A0,1 ∂ / A0,2 ∂ / A0,3 ∂ / ··· .

n p,q Deﬁnition 8.1. Let U be an open subset of C . We deﬁne A (U) to be the complex vector space whose elements are formally denoted by

X i1 ip j1 jq fi1...ip;j1...jq dz ∧ · · · dz ∧ dz ∧ · · · ∧ dz ,

where the sum ranges over 0 ≤ i1 < ··· < ip ≤ n, 0 ≤ j1 < ··· < jq ≤ n, where fi1...ip;j1...jq denotes n p,q a complex-valued C∞-function on U. The resulting sheaf on C will be denoted by A It is not diﬃcult to see that Ap,q is closely related to the sheaf of smooth diﬀerential forms on n C . m Q p,q Lemma 8.2. Denote by A (U) the product p+q=m A (U). There is a canonical equivalence m between A (U), and the complex vector space of C-valued smooth m-forms

X i1 in im fi1···i2m dx ∧ · · · ∧ dx ∧ · · · ∧ dx

1≤i1<···

Proof. An explicit isomorphism can be produced by making the substitutions: dzi = (dxi + idyi), i i i i 1 i i i 1 i i dz = dx − idy , dx = 2 (dz + dz ), and dy = 2i (dz − dz ).

0 0 0 0 As for m-forms, we have a wedge product, Ap,q × Ap ,q / Ap+p ,q+q , which is given juxta- position of terms like dzi1 ∧ · · · dzip ∧ dzj1 ∧ · · · ∧ dzjq , and dzi1 ∧ · · · dzip0 ∧ dzj1 ∧ · · · ∧ dzjq0 , and re-arranging terms. As usual we have the rule dzi ∧ dzi = 0, and dzi ∧ dzi = 0. Next we define the operators ∂ and ∂. p,q Definition 8.3. The linear maps ∂p,q : Ap,q / Ap+1,q and ∂ : Ap,q / Ap,q+1 are defined as the unique linear maps, such that d: Am / Am+1 P p,q p,q for m = p + q decomposes as d = p+q=m(∂ + ∂ ).

2 This implies directly the important relations ∂2 = 0, and ∂ = 0. ∂ ∂ ∂ ∂ ∂ ∂ To obtain an explicit formula for ∂ and ∂, one uses that ∂xi = ∂zi + ∂zi , and ∂yi = i( ∂zi − ∂zi ). Since we have

n X ∂f ∂f d(fdxi1 ∧· · ·∧dxim0 ∧dyim0+1 ∧· · ·∧dyi2m ) = ( dxj+ dyj)∧dxi1 ∧· · ·∧dxim0 ∧dyim0+1 ∧· · ·∧dyi2m , ∂xj ∂yj j=1

81 we obtain from the identity ∂f ∂f 1 ∂f ∂f 1 ∂f ∂f ∂f ∂f dxj + dyj = ( + )(dzj + dzj) + ( − )(dzj − dzj) = dzj + dzj, ∂xj ∂yj 2 ∂zj ∂zj 2 ∂zj ∂zj ∂zj ∂zj that

n X ∂f ∂p,q(fdzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq ) = dzj ∧ dzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq , ∂zj j=1 and

n p,q X ∂f ∂ (fdzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq ) = dzj ∧ dzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq . ∂zj j=1

Lemma 8.4 (Dolbeault–Grothendieck). The complex of sheaves Ap,0 ∂ / Ap,1 ∂ / Ap,2 / ··· is

0,0 n exact in positive degrees. The kernel ker ∂ is equivalent to the sheaf OC of holomorphic functions. The proof of this result is not entirely straightforward. We begin by examining a special case. We may assume p = 0, because the terms dzi1 ∧ · · · ∧ dzip are merely decorative, as long as we only care about the ∂-operators. The proof uses the general Cauchy integral formula.

Lemma 8.5. Let U ⊂ C be an open subset, and f : U / C a C∞-function. Then, we have for every z0 ∈ U, and > 0, such that U (z0) ⊂ U, the formula Z f(w) Z ∂f dw ∧ dw 2πif(z) = dw + (w) , |w−z0|= w − z U (z0) ∂w w − z where z ∈ U(z0). The proof of this lemma is similar to the way we deduced the Cauchy integral formula from the Stokes theorem for smooth manifolds. The details are left to the reader as an exercise. Proof of the Dolbeault–Grothendieck Lemma for n = 1. The complex has precisely two non-zero terms:

A0,0 ∂ / A0,1 / 0 / ··· .

∞ / ∂f We already know that for a C -function f : U C we have ∂f = ∂z dz = 0 if and only if f is holomorphic. It remains to show that for every (0, 1)-form f(z)dz on U there exists an open S ∞ / covering U = i∈I Ui, such that f(z)dz|Ui = ∂gi, for a C -function gi : Ui C. For every z0 ∈ U, we choose an -neighbourhood, such that U (z0) ⊂ U. We may assume that for |z − z0| ≥ 2 we have f(z) = 0, by multiplying f with a suitable bump function. This does not restrict the generality, since we only care about producing g, such that f = ∂g = f, locally around ∞ each z0 ∈ U. It has the added advantage that f can now be considered as a C -function on all of C (it’s some kind of bump function itself). For z ∈ U(z0) we deﬁne 1 Z f(w) g(z) = dw ∧ dw. 2πi w − z C

82 Substituting w = w0 + z, we obtain

1 Z f(w0 + z) g(z) = dw0 ∧ dw0. 2πi w0 C We leave the straight-forward veriﬁcation that g is of class C∞ to the reader. Using the last formula, we obtain ∂ Z ∂f 1 Z ∂f 1 2πi g(z) = (w0 + z) dw0 ∧ dw0 = (w) dw ∧ dw = 2πif(z). ∂z ∂w w0 ∂w w − z C C And hence we see that ∂g = f, as required. We apply the same strategy variable-by-variable to prove the n-dimensional version of the Grothendieck–Dolbeault Lemma. Proof of the Grothendieck–Dolbeault Lemma. Let ω ∈ A0,q(U) be a (0, q)-form, such that ∂ω = 0. 0,q−1 We claim that there for every z0 ∈ U there exists an > 0, such that we have an φ ∈ A (U(z0) satisfying ω = ∂φ. The construction of φ is by induction. We assume that ω is built from the (0, 1)-forms dz1, . . . , dzk, where 0 ≤ k ≤ n, and we will prove the assertion by induction on k. If k = 0, then ω = 0, and the assertion is true. Let’s assume that the lemma has already been proven for k−1. We can decompose ω as η∧dzk + 0 0 1 k−1 P i1 ω , where η and ω only contain the (0, 1)-forms dz , . . . , dz . We write η = fi1...iq−1 dz ∧ · · · ∧ dziq−1 . The relation ∂ω = 0 implies for j > k ∂f i1...iq−1 = 0. (16) ∂zj We now use the same integration method as in the proof of the 1-dimensional case to deduce a ∂gi ...i 1 q−1 0 P i1 iq−1 smooth function gi1...iq−1 , such that ∂zk = fi1...iq−1 . Let φ = gi1...iq−1 dz ∧· · ·∧dz . By construction we have that ω−∂φ0 only contains the (0, 1)-forms dz1, . . . , dzk−1. By induction, there 00 0,q−1 0 00 0 00 exists a φ ∈ A in some -neighbourhood of z0, such that ω − ∂φ = ∂φ . Hence, φ = φ + φ does the trick. There is an interesting reﬁnement of the Grothendieck–Dolbeault Lemma, which applies to n (0, q)-forms on C . 0,q m Proposition 8.6. Let ω ∈ A (C ) be a (0, q)-form, such that ∂ω = 0. Then, there exists a 0,1 m (0, q − 1)-form φ ∈ A (C ), such that ω = ∂φ. m Proof. We begin with the case q = 1 which needs to be discussed separately. We write C = S m n≥1 Un(0), we claim that there exists a sequence of smooth functions φn : C / C, such that 1 for ∂φn| = ω| , and for all z ∈ U n(0) we have |φn+1(z) − φn(z)| ≤ n . The existence Un+ 1 (0) Un+ 1 (0) 2 2 2 of such a sequence follows inductively, the case n = 1 being a direct consequence of the proof of the Grothendieck–Dolbeault Lemma. Let’s assume by induction that φ1, . . . , φn have been constructed. We know that there exists 0 0 0 φ , such that ∂φ | = ω| . For z ∈ Un(0) we have ∂(φ − φ)(z) = 0 for n+1 n+1 Un+ 3 (0) Un+ 3 (0) n+1 2 2

83 z ∈ U 1 (0), that is, the difference is holomorphic on U 1 . Choose a polynomial g(z), such n+ 2 n+ 2 (0) 0 1 that |φn+1(z) − φn(z) − g(z)| < 2n for z ∈ U n(0). Since g is everywhere holomorphic, we may define φn+1 = φn − g, without affecting the property ∂φn+1| = ω| . Un+ 3 (0) Un+ 3 (0) 2 2 Defining φ = limn→∞ φn one sees directly that the resulting function is a continuous function, P since the convergence is uniform on the subsets U n(0). Writing φ = φ1 n≥1(φn+1 − φn), one obtains a presentation of φ, which restricts on each Un(0) to a finite sum of smooth functions, φ1, φ2 − φ1,. . . ,φn − φn−1, and a uniformly convergent infinite sum of holomorphic functions (thus it’s holomorphic itself). We conclude that φ is a smooth function, and that ∂φ = ω. Let q ≥ 2. We claim that there are (0, q −1)-forms φn, such that we have ∂φn| = ω|U , Un+ 1 (0) n+ 1 2 2 and that φ | = φ | . n+1 U n(0) n U n(0) 0 By induction we assume that φ1, . . . , φn have been constructed. We choose φn+1, such that 0 0 ∂φ | = ω|U . The difference φ − φn is sent to 0 by ∂ on the subset U 1 (0). n+1 Un+ 3 (0) n+ 3 n+1 n+ 2 2 2 Therefore, there exists a (0, q − 2)-form ψ, such that

0 (φ − φn)| = ∂ψ| 1 . n+1 Un+ 1 (0) n+ (0) 4 4 Therefore, we see that φ = φ0 − ∂ψ satisfies φ | = φ | , and ∂φ = ∂φ0 , n+1 n+1 n+1 U n(0) n U n(0) n+1 n+1 2 because ∂ = 0. Definition 8.7. Let X be a complex manifold. For an integer k we denote by Ak the sheaf of smooth, complex-valued k-forms on X (take the usual vector space of smooth k-forms and apply the p,q tensor product functor − ⊗R C). By choosing local coordinates, we define AX (for p + q = k) to be the subsheaf consisting to k-forms which are locally expressible as

X i1 ip j1 jq fi1...ip;j1...jq dz ∧ · · · dz ∧ dz ∧ · · · ∧ dz .

8.2 Partitions of unity and sheaf cohomology In this paragraph we will show that the Dolbeault resolution that we introduced earlier is a so-called acyclic resolution of the sheaves Ωp. This implies in particular that

ker(∂ : Ap,q / Ap,q+1) Hq(X, Ωp) =∼ . image(∂ : Ap,q−1 / Ap,q) This follows at once from the Universal Comparison Theorem 6.10, once we have shown that the sheaves Ap,q don’t have any higher cohomology themselves (that is, are acyclic). We will deduce this from the existence of partitions of unity. Deﬁnition 8.8. Let X be a topological space, and R a sheaf of rings on X. We denote by U ⊂ X S a closed subset of X, and X = i∈I Ui an open covering of X.

(a) The support, supp s, of a section s ∈ R(U) is defined to be the closure of the set {x ∈ X|sx ∈ Rx \0}. Alternatively, you can define supp s to be the complement of the maximal open subset V ⊂ X, such that s|V ∩U = 0. By definition, we have supp s ⊂ U.

J (b)A partition of unity, subordinate to {Ui}i∈I consists of a family of sections (sj)j∈J ∈ R(X) , such that

84 - for every j ∈ J there exists an i ∈ I, such that we have supp sj ⊂ Ui,

- every x ∈ X has a neighbourhood V , such that only finitely many sections sj|V are non-zero, P - and the identity j∈J sj = 0 holds (the infinite sum makes sense because all but finitely many sections vanish locally). (c) If R is a sheaf of rings which admits a partition of unity subordinate to any cover then we say that R is fine3. There are many examples of fine sheaves of rings, in fact most examples the reader encountered previously to studying complex manifolds were probably fine. Partitions of unity exist for continuous functions on paracompact spaces. On smooth manifolds partitions of unity can be constructed using bump functions. Holomorphic partitions of unity don’t exist however. The support condition supp sj ⊂ Ui would imply s = 0. Lemma 8.9. Let U ⊂ X be an open subset, and s ∈ R(X) a section, such that supp s ⊂ U. Given any sheaf of R-modules M, and any section m ∈ M(U), there is a unique section m0, denoted s · m 0 0 in future, such that m |U = s · m, and m |X\supp s = 0. Proof. This is a straight-forward application of the sheaf property for M and the open covering X = U ∪ (X \ supp s). Since (s · m)|U\supp s = 0, we see that there exists a unique such section. Lemma 8.10. If R is a fine sheaf of rings, then the higher cohomology groups of every sheaf of R-modules vanish. Proof. We prove this by our usual inductive trick. Let M be an R-module, and

0 / M / I / I / M / 0 a short exact sequence of R-modules, such that I is injective. Analysing our construction of injective sheaves containing a given sheaf, one sees that this also works for sheaves of R-modules. The long exact sequence (and vanishing of higher cohomology of injective sheaves) imply

Hi(X, M) =∼ Hi−1(X, I / M) for i ≥ 2, and H1(X, M) =∼ coker(H0(X, I) / H0(X, I / M)). By induction we have therefore reduced the proof to showing that the map H0(X, I) H/ 0(X, I / M) is surjective. Let s ∈ I / M(X) be a section. By definition of the quotient sheaf I / M there exists S an open covering X = i∈I Ui, and sections ti ∈ I(Ui), such that s|Ui equals the image of ti in I / M. Since R is a fine sheaf of rings, there exists a partition of unity (ej)j∈J , subordinate to {Ui}i∈I . Choose for every j ∈ J an i(j) ∈ I, such that supp ej ⊂ Ui(j). Define the section X t = ej · ti(j), j∈J

3In the literature one finds the definition of fine sheaves, which is similar but slightly different from the one given here. In the case of sheaves of rings it is an easy exercise to check that a sheaf is fine in our sense if and only if it is fine in the traditional one.

85 we have seen in Lemma 8.9 that this is well-deﬁned. We can check stalk-by-stalk that the section t is mapped to the section s in the quotient I / M. This shows surjectivity of the map H0(X, I) / H0(X, I / M) and hence provides an anchor for our inductive argument.

p,q Corollary 8.11. For any complex manifold X the sheaves AX are acyclic. In particular, one obtains that the Dolbeault complex

p,0 p,1 AX (X) / AX (X) / ···

q p computes the sheaf cohomology H (X, ΩX ). As a another consequence we obtain that the de Rham complex of a smooth manifold X

Ω0 / Ω1 / ··· X,R X,R computes the cohomology of the sheaf of locally constant R-valued functions RX . p Corollary 8.12. The sheaves O n , and Ω n have vanishing higher cohomology groups. C C Proof. This is a direct consequence of Proposition 8.6.

n × s This corollary can be generalised by replacing C by open subsets of the shape (C )r × C .

8.3 Cechˇ cohomology In this paragraph we ﬁx a topological space X, and a sheaf F ∈ Sh(X). We assume that we have S ˇ a open covering X = i∈I Ui. The Cech cohomology of F with respect to the covering {Ui}i∈I is deﬁned to be the cohomology of the following chain complex: Y Y Y Γ(Ui, F) / Γ(Uij, F) / Γ(Uijk, F) / ··· , i∈I (i,j)∈I2 (i,j,k)∈I3 where the maps are given by

ˇp Y / Y ˇp+1 δ : C (X, F) = Γ(Ui0···ip , F) Γ(Uj0...jp+1 , F) = C (X, F), p+1 p+2 (i0,...,ip)∈I (j0,...,jp+1)∈I

which sends (ci0···ip )i0...ip to the tuple of sections (dj0...jp+1 ), such that

p+1 X i dj ...j = (−1) (c |U ). 0 p+1 j0...jbi...jp+1 j0...jp+1 i=0

One can verify by a direct computation that δ2 = 0, hence this deﬁnes a chain complex, known as the Cechˇ complex. The sheaf property of F implies directly that Hˇ 0 (X, F) = F(X). {Ui}i∈I Deﬁnition 8.13. We will denote the cohomology group of this complex by Hˇ p (X, F). It is {Ui}i∈I called the degree p Cechˇ cohomology of F with respect to the open covering {Ui}i∈I .

86 Elements of Hp (X, F) are therefore presented by so-called Cechˇ p-cocycles. That is, a {Ui}i∈I collection of local sections (ci0···ip )i0...ip , satisfying δ((ci0···ip )i0...ip ) = 0. For p = 1 we have already seen this in our discussion of line bundles and G-Galois covers. In a way, Cechˇ cohomology is amenable to direct computation, at least when working with a ﬁnite cover. In this case, one can also show that Hˇ i (X, F) vanishes for i > |I| − 1. {Ui}i∈I Lemma 8.14. On a complex manifold X, Hˇ 1 (X, O× ) is canonically equivalent to the abelian {Ui}i∈I X group of isomorphism classes of complex line bundles L, which are trivialisable when restricted to the open subsets Ui for every i ∈ I.

Proof. We denote this set of line bundles (defined in the statement of the lemma) by Pic{U } (X). i i∈I 2 By definition, every L ∈ Pic{Ui}i∈I (X) admits an abstract cocycle datum {Ui}i∈I , (φij)(i,j)∈I ) . × By definition, the collection of nowhere vanishing holomorpic functions φij ∈ OX (Uij) satisfies the cocycle condition φik = φij · φjk, which (after switching from multiplicative to additive notation) amounts precisely to the condition that (φij) ∈ ker δ. Two cocycle data for L with respect to the same covering {Ui}i∈I can differ only to the effect ' / that different trivialisations φi : L|Ui Ui × C could have been chosen. Two such trivialisations × can be related by multiplying with a nowhere vanishing function φi ∈ OX (Ui). The resulting −1 ˇ 1-cocycle would be (φiφj )φij. That is, differs by precisely a Cech coboundary from (φij). Virtually the same argument shows the following. Lemma 8.15. For a manifold X, and an abelian group A, Hˇ 1 (X,A) is canonically equivalent {Ui}i∈I to the abelian group of isomorphism classes of A-Galois covers of X, which are trivialisable when restricted to the open subsets Ui for every i ∈ I. Cechˇ cohomology groups can also be defined without fixing a particular cover (or in other words, by considering all open coverings at once). In order to give this definition, we have to recall the notion of a filtered colimit. This construction is reminiscent of how we defined stalks of sheaves at a point x ∈ X. Definition 8.16. Let I be a partially ordered set, such that every 2-element subset {i, j} ⊂ I has a common upper bound (we also say that I is filtered). Let (Ai)i∈I a family of abelian groups (or i vector spaces), such that for every i ≤ j we have a homomorphism fj : Ai / Aj, satisfying the i j i i relation fi = idAi , and fk ◦ fj = fk for every ordered triple i ≤ j ≤ k. The colimit (a. k. a. direct limit) of (Ai)i∈I is defined to be the abelian group (resp. vector space)

colim Ai = lim Ai, i∈I −→ i∈I consisting of equivalence classes of pairs (i, a). Here, i ∈ I, and a ∈ Ai. And we say that (i, a) is equivalent to (j, b) if and only if there exists a common upper bound k of {i, j}, such that we have i j an element c ∈ Ak, satisfying fk(a) = c = fk (b). The set of open neighbourhoods U of x ∈ X, with respect to the partial ordering U ≤ V ⇔ V ⊂ U is a ﬁltered poset. Given a sheaf F on X, we obtain a directed system of abelian groups F(U), and the induced colimit is by deﬁnition equivalent to the stalk F x.

87 Let U = {Ui}i∈I , and V = {Vj}j∈J be two open coverings of X. We say that U ≤ V, if for every j ∈ J, there exists an i(j) ∈ I, such that Vj ⊂ Ui. The resulting map J / I allows us to define a map between the resulting Cechˇ cohomology groups ˇ p ˇ p HU (X, F) / HV (X, F). We define the Cechˇ cohomology of F to be the direct limit of this system of cohomology groups, and denote it by Hˇ p(X, F). As an immediate corollary of the lemmas about line bundles and A-Galois covers, we obtain the following assertions. We just use the fact that any line bundle, respectively A-Galois covering, is trivialised by some open covering, and hence they all get detected by the Cechˇ cohomology group Hˇ 1(X, −). ˇ 1 × Corollary 8.17. (a) On a complex manifold X, the abelian group H (X, OX ) is equivalent to the abelian group of isomorphism classes of line bundles Pic(X). (b) On a connected manifold X, the abelian group Hˇ 1(X,A) is equivalent to the abelian group of isomorphism classes of A-Galois covers Hom(π1(X),A). Our next goal is to replace Cechˇ cohomology in the statement above by actual sheaf cohomology. Therefore we embark on a more detailed study of the relation between sheaf cohomology and Cechˇ cohomology. Note that if X is a compact topological space, then every open covering is ≤ then a finite covering. Hence it’s sufficient to work with respect to finite coverings on compact spaces. r+1 Proposition 8.18. Let U = {Ui}i∈I be a covering of X. For every tuple (i0, . . . , ir) ∈ I we denote by ji0...ir : Ui0...ir = Ui0 ∩ · · · ∩ Uir ,→ X the inclusion. Let F be a sheaf on X, such that for every (i , . . . , i ) ∈ Ir+1 we have that (j ) (F | ) is acyclic (that is, no higher cohomology), 0 r i0...ir ∗ Ui0...ir ˇ p ∼ p then HU (X, F) = H (X, F). Proof. The Cechˇ resolution also makes can also be viewed as a resolution of sheaves, that is, we have an exact sequence ˇ0 ˇ1 0 / F / CU (F) / CU (F) / ··· , ˇp Q ˇp where C (F) = p+1 (j ) (F | ). By assumption, C (F) is a product of acyclic U (i0,...,ip)∈I i0...ir ∗ Ui0...ir U sheaves, and thus is acyclic itself. The Universal Comparison Theorem 6.10 implies that Hp(X, F) agrees with the degree p cohomology of the chain complex of global sections, which is the Cechˇ complex with respect to U. Corollary 8.19. If F is a flasque sheaf on X, then its higher Cechˇ cohomology vanishes with respect to any open covering U. In particular Hˇ i(X, F) = 0 for i > 0. Proof. We have seen in the exercises that flasque sheaves are acyclic. Since restrictions to open subsets, and pushforwards of flasque sheaves are flasque, we obtain that the sheaves Cˇp(F) are flasque. Therefore, they are acyclic, and hence assumptions of Proposition 8.18 are satisfied. We i ˇ i ˇ i deduce that 0 = H (X, F) = HU (X, F) = H (X, F). So far we have seen that Cechˇ cohomology verifies the two axioms (A1) and (A3) of sheaf cohomology. In general it is not true that every short exact sequence of sheaves yields a long exact sequence of Cechˇ cohomology groups, and hence sheaf cohomology will not always agree with Cechˇ cohomology. However, whenever this missing axiom is satisfied by the Cechˇ cohomology groups, we obtain directly from our axiomatic approach that Cechˇ cohomology agrees with sheaf cohomology. Nonetheless, we always have a long exact sequence up to degree 1.

88 Lemma 8.20. Given a short exact sequence of sheaves 0 / F / G / H / 0 on X, we have an exact sequence of Cechˇ cohomology groups

0 / Γ(X, F) / Γ(X, G) / Γ(X, H) δ / Hˇ 1(X, F) / Hˇ 1(X, G) / Hˇ 1(X, H).

Proof. We only define the map δ, and check exactness of Γ(X, G) / Γ(X, H) δ / Hˇ 1(X, F), and leave the remaining parts as an exercise to the reader. We have a map δ : Γ(X, H) / Hˇ 1(X, F), which is defined as follows. Let s ∈ Γ(X, H) be a section, and let U = {Ui}i∈I be an open covering, such that we have sections ti ∈ G(Ui), which map to the restrictions s|Ui for every i ∈ I. By exactness, (ti|Uij ) − (tj|Uij ) = cij, defines a section of F over Uij. The construction as a difference, reveals directly that cij is a Cechˇ 1-cocycle with respect to U. We define δ(s) = [(cij)]. If s has a global lift, then we can choose U to be the trivial cover X = X, and hence get the δ 1 zero cocycle sii = 0. This implies that the composition Γ(X, G) / Γ(X, H) / Hˇ (X, F) equals the zero map. ˇ Q Vice versa, if δ(s) = 0, then there exists a Cech 1-cocycle (di)i∈I ∈ i∈I F(Ui), such that 0 0 0 cij = (di − dj)|Uij . We can then define ti = ti − di, and obtain ti|Uij = tj|Uij . Hence, the sheaf 0 0 property implies that there exists a global section t ∈ G(X), such that t |Ui = ti. This concludes the proof of exactness.

Corollary 8.21. We have H1(X, F) =∼ Hˇ 1(X, F). In particular, if X is a complex manifold, ∼ 1 × 1 then Pic(X) = H (X, OX ). Moreover, we have for X a connected manifold that H (X,A) = Hom(π1(X),A) (that is, isomorphism classes of A-Galois covers). Proof. Choose a short exact sequence 0 / F / I / I / F /0 with I injective. Since injective sheaves are ﬂasque, we have H1(X, I) = Hˇ 1(X, I) = 0. The long exact sequences imply

H1(X, F) = coker(Γ(X, I) / Γ(X, I / F)) =∼ Hˇ 1(X, F).

The assertion about Pic(X) follows from Corollary 8.21 (a). However, in most cases (that is spaces or sheaves we actually care about), Cechˇ cohomology gives the right answer, that is, agrees with sheaf cohomology. We have the following theorem:

j Theorem 8.22 (Leray). Let F be a sheaf on X, and U an open covering, such that H (Ui0...ir , F |U ) = r+1 ˇ i ∼ i 0, for every tuple (i0, . . . , ir) ∈ I , where Ui0...ir = Ui0 ∩ · · · ∩ Uir . Then HU (X, F) = H (X, F). In the proof of this theorem, we have to produce a suitable long exact sequence for Cechˇ cohomology groups of sheaves satisfying the condition of the theorem. This is possible thanks to the following lemma. Lemma 8.23. Let C•, D•, and E• be chain complexes, and 0 / C• / D• / E• / 0 a short exact sequence. Then, we have a long exact sequence of cohomology groups

··· / Hi(C•) / Hi(D•) / Hi(E•) / Hi+1(C•) / ··· .

89 • i i−1 i−1 i i i i i+1 Proof. For a chain complex C we write BC• for image(d : C /C ), and ZC• = ker(d : C C/ ). By deﬁnition, we have Hi(C•) = Zi(C•)/Bi(C•). We have a commutative diagram with exact rows

i i i i i i C /BC• / D /BD• / E /BE• / 0

di di di i+1 i+1 i+1 0 / ZC• / ZD• / ZE• and the Snake Lemma yields a long exact sequence

Hi(C•) / Hi(D•) / Hi(E•) / Hi+1(C•) / Hi+1(D•) / Hi+1(E•),

i i d i+1 i • where we have used that the kernel of C /BC• / ZC• is equivalent to H (C ), and its cokernel to Hi+1(C•). Proof of Theorem 8.22. We can now prove Leray’s theorem. Let 0 / F / I / I / F / 0 be a short exact sequence with I exact. Since injective sheaves don’t have higher cohomology, we see that F, I, and I / F satisfy the assumptions of the theorem (the quotient as a consequence of the long exact sequence in cohomology). Moreover, vanishing of cohomology of these sheaves on the subsets Ui0···ip guarantees that we have a long exact sequence of chain complexes in abelian groups

0 / Cˇ•(X, F) / Cˇ•(X, I) / Cˇ•(X, I / F) / 0.

The previous lemma (8.23) implies that we have a long exact sequence of cohomology groups

··· / Hˇ i(X, F) / Hˇ i(X, I) / Hˇ i(X, I / F) / Hˇ i+1(X, F) / ··· .

Since injective sheaves are ﬂasque, and ﬂasque sheaves don’t have any higher Cechˇ cohomology groups by Corollary 8.19, we obtain

Hˇ 1(X, F) =∼ coker(H0(X, I) / H0(X, I / F)) =∼ H1(X, F), and for i ≥ 2 the equivalence Hˇ i(X, F) =∼ Hˇ i−1(X, I / F). By induction we see that Hˇ i(X, F) =∼ Hi(X, F).

n In the next section we will use Leray’s theorem to compute the cohomology of OP .

8.4 Cohomology of complex projective space n In this paragraph we will explain two important results for the cohomology of sheaves on P . n Theorem 8.24 (Cohomology of P ). Recall that for an abelian group A we denote by A the sheaf of locally constant A-valued functions.

n n (a) We have H2i+1(P ,A) = 0, and H2i(P ,A) =∼ A, for 0 ≤ i ≤ n. n n i n 0 (b) The cohomology group H (P , OP ) = 0 vanishes for i ≥ 1, and H (P , C) = C.

90 1 Assertion (a) for 2i+1 = 1 follows from Corollary 8.21, because P is connected and 1-connected. n The general case follows from a computation of the singular cohomology groups of P , using that n P has a CW decomposition with precisely one cell for every even number between 0 and 2n. We now turn to the details of the proof of (b). We can apply Leray’s Theorem 8.22 to the n n S n standard covering P = i=0 Ui of projective space, since OP is known not to have any cohomology over the intersections Ui0...ip by Cartan’s Theorem B (see Proposition 7.37 which is covering the cases of relevance to us, and also the strengthening of the Grothendieck-Dolbeault Lemma 8.6). × Lemma 8.25. Let f : C ×U / C be a holomorphic function in complex variables (z, w1, . . . , wn−1), n−1 where we assume z 6= 0, and that U ⊂ C is an open subset. There exist holomorphic functions g : C ×U / C, and h: C ×U / C, such that we have 1 f(z, w , . . . , w ) = g(z, w , . . . , w ) − h( , w , . . . , w ), 1 n−1 1 n z 1 n−1 × n−1 for (z, w1, . . . , wn−1) ∈ C × C . Proof. We define Z 1 f(u, w1, . . . , wn−1) g(z, w1, . . . , wn−1) = du, 2πi |u|=R (u − z) where R is an arbitrary positive real number, satisfying R > |z|. Homotopy invariance of integrals n of holomorphic functions implies that this is a well-defined holomorphic function on C . We can compute f − g for z 6= 0, by using the Cauchy-integral formula for f: 1 Z f(u, w , . . . , w ) 1 Z f(u, w , . . . , w ) 1 n−1 du − 1 n−1 du, 2πi |u|=R (u − z) 2πi |u−z|= (u − z) where is a sufficiently small positive number. Using the Residue Theorem we obtain f(u, w , . . . , w ) res f f − g = res 1 n−1 = − u=0 , u=0 u − z z

1 where resu=0 f is a holomorphic function in the variables (w1, . . . , wn−1). Since (f−g)( z , w1, . . . , wn−1) is deﬁned for all values of z, we see that h(z, w1, . . . , wn−1) = z resu=0 f(u, w1, . . . , wn−1) satisﬁes the requirements.

i 1 As an immediate corollary we obtain vanishing of H ( , O 1 ) for i ≥ 1. For i ≥ 2 this vanishing P P result follows from the Dolbeault resolution, hence it is suﬃcient to prove vanishing for i = 1 in this case.

1 1 Corollary 8.26. We have H ( , O 1 ) = 0. P P 1 Proof. We can apply Leray’s theorem to X = P , and F = O 1 and the standard covering U0 ∪U1 = 1 ∼ ∼ × P P . The reason is that Ui = C, and U0 ∩ U1 = C , and by the Grothendieck–Dolbeault Lemma × there the sheaf of holomorphic functions O is acyclic on C, and C . Therefore, the cohomology i 1 ˇ group H (P , OP) is equivalent to the degree i cohomology of the Cech complex: / / O(U0) × O(U1) OU0 × OU1 × OU0∩U1 ··· . ˇ The deﬁnition of the boundary operator δ in Cech cohomology implies that δ(f, g) = (0, 0, f|U01 − 1 ˇ1 1 / ˇ2 1 g|U01 ), and that δ : CU (P , OP) CU (P , OP) is the zero map, and similarly in higher degrees.

91 Therefore we are really computing the cohomology of the complex

/ / / O(U0) × O(U1) OU0∩U1 0 ··· .

1 We already know H0(P , O) =∼ C, and it remains to compute H1 as the cokernel of the map / O(U0) × O(U1) OU0∩U1 , which sends (g, h) to g|U01 − h|U01 . But we have seen in Lemma 8.25 that every holomorphic function on U01 can be written as the diﬀerence of a holomorphic function i 1 on U0 and on U1. This shows that the cokernel is 0, and therefore H (P , O) = 0 for i ≥ 1. 2 2 In order to convey a feeling for the higher-dimensional case, we will show H ( , O 2 ) = 0. P P 2 2 Corollary 8.27. We have H ( , O 2 ) = 0. P P 2 Proof. We apply Leray’s Theorem to the standard covering P = U0 ∪ U1 ∪ U2. We have to show that a collection of holomorphic functions (f012, f013, f023, f123), satisfying

f123 − f023 + f013 − f012 = 0, (17) can be expressed as (g12 − g02 + g01, g13 − g03 + g01, g23 − g03 + g02, g23 − g13 + g12) for holomorphic functions gij on the double intersections Uij. We set g01 = 0, and choose g12, g02, such that g12 − g02 = f012, using Lemma 8.25. Now we choose holomorphic functions g13 and g03, such that g13 − g03 = f013 − g01 = f013. Next, we choose holomorphic functions g23, g03, such that g23 − g03 = f023 − g02. It remains to check that we have g23 − g13 + g12 = f123. But this follows directly from relation (17).

8.5 Line bundles on complex manifolds

n The cohomology computation of OP is an essential ingredient in the proof of GAGA, and allows n us to classify line bundles on P . Recall the exponential sequence

2πi × 0 / / O n / O n / 0. Z P P n n 1 n ∼ Using the equivalence H (P , OP ×) = Pic(P ) of Corollary 8.21, we can state the associated long exact sequence as

1 n n 2 n 2 n / n / / / n / ··· H (P , OP ) Pic(P ) H (P , Z) H (P , OP ) ··· . n n n i n ∼ 2 ∼ We have seen that H (P , OP ) = 0 for i ≥ 2, and conclude that Pic(P ) = H (P , Z) = Z. The last equivalence uses Theorem 8.24(a). n One immediately guesses that the equivalence Pic(P ) =∼ Z associated to the integer d ∈ Z the line bundle O(d). This is indeed the case, but since we haven’t yet obtained a conceptual n understanding of the group H2(P , Z) (e.g., we haven’t discussed methods to construct/describe elements), it is not easy to prove this assertion at this point. Instead, we will refer once more to n the GAGA Theorem, to prove that every line bundle on P is equivalent to a line bundle O(d). In order to keep our discussion as simple as possible, we limit ourselves to the case n = 1. 1 1 From now on we consider P with the Zariski topology. That is, a subset U ⊂ P is Zariski 1 open, if and only if P \U consists of ﬁnitely many points (this is true, because we are working 1 in complex dimension 1). We denote by M the sheaf of rational functions, that is to U ⊂ P we

92 × associate the vector space of all rational functions V ⊂ C, where V ⊂ U. Let M be the sheaf of non-zero rational functions, which form an abelian group with respect to multiplication. × × We have an inclusion of sheaves Oreg ,→ M , and denote the sheaf of quotients by Div (we call it the sheaf of divisors). A section D ∈ Div(U) can be represented by a formal linear combination n1 · x1 + ··· + nk · xk, where xi ∈ U (this is what we call a divisor). We assign to a rational function f a divisor, by weighting every point x ∈ U with the pole order of f at x. The sheaves M× and Div are ﬂasque, and hence the long exact sequence implies

× 1 coker(M (X) / Div(X)) =∼ Pic(P ). Pk Pk The map Div(X) / Z sending i=1 nixi to i=1 ni is surjective, because 1 · x is sent to 1 ∈ Z. 1 Moreover, since the sum of pole orders of a rational function f on P is 0, we get a well-deﬁned map n Pic(P ) / Z . 1 1 Lemma 8.28. Let f : P / C be a non-zero meromorphic function on P . The sum over all the pole orders, minus the zero orders is 0.

1 Proof. Without loss of generality we can assume that f does not have a pole or zero at ∞ ∈ C =∼ P . Otherwise, we apply a generic M¨obiustransformation to C to perturb a possible pole or zero away from ∞. Choose a very small closed path γ around ∞, but which does not pass through ∞. We can view it as a non-trivial path in C, non-trivially circling the poles and zeroes of f. The Residue Theorem implies that Z f 0(z) X dz = m · ord f, f(z) z=x γ x 1 where m is the non-zero winding number of f. Since γ is a path on P which can be contracted to ∞, homotopy invariance of integration implies that the sum above vanishes.

1 n Since the map Pic(P ) / Z is surjective, and we know that Pic(P ) is an inﬁnite cyclic group, we conclude that this map is an isomorphism. It remains to show that O(d) is sent to d ∈ Z. Which n allows us to conclude that every line bundle on P is isomorphic to a line bundle O(d). This is an easy computation, using the description of the boundary map in Cechˇ cohomology.