Complex manifolds

Michael Groechenig

March 23, 2016

Contents

1 Motivation 21.1 Real and complex differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 A proof of the Cauchy formula using “real techniques” . . . . . . . . . . . . . . . . . 31.3 Holomorphic functions on complex manifolds . . . . . . . . . . . . . . . . . . . . . . 41.4 Complex projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Complex projective manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Holomorphic functions of several variables 112.1 The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 The Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 ∂-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Hartog’s Extension Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Complex vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Complex differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Complex manifolds 203.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Holomorphic functions, complex vector fields, and forms . . . . . . . . . . . . . . . . 213.3 Smooth complex projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Holomorphic maps between manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Vector bundles 284.1 The basics of holomorphic vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Line bundles on projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Line bundles and maps to projective space . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Sheaves 385.1 Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Locally free sheaves and vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . 425.4 Exact sequences of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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6 Sheaf cohomology 476.1 Overview: why do we need sheaf cohomology? . . . . . . . . . . . . . . . . . . . . . . 486.2 The exponential sequence revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Sheaf cohomology as cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.4 Construction of sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 GAGA 637.1 Projective algebraic geometry in a nutshell . . . . . . . . . . . . . . . . . . . . . . . 637.2 Coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.3 A short exact sequence for twisting sheaves . . . . . . . . . . . . . . . . . . . . . . . 747.4 The proof of the first GAGA theorem (modulo basic cohomology computations) . . 767.5 Holomorphic sections of coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . 79

8 Computing Cohomology: Dolbeault and Cech 808.1 The Dolbeault resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.2 Partitions of unity and sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . 848.3 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.4 Cohomology of complex projective space . . . . . . . . . . . . . . . . . . . . . . . . . 908.5 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 isidentical to the one of smooth manifolds. It is only necessary to replace open subsets of Rn by opensubsets of Cn, and smooth functions by holomorphic functions.

This might lead you to think that the notion of complex manifolds has been created as a mereintellectual 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 realworlds are fundamentally different. The complex analogues of “real” definitions often yields morerestrictive, and rigid mathematical objects.

To illustrate this principle we take a look at the notion of differentiability. Let U ⊂ R be someopen subset, and f : U // R a function. We say that f is differentiable, if for every x ∈ U the limit

f ′(x) = limy→x

f(y)− f(x)

y − x(1)

exists. It is well-known that the thereby defined derivative f ′(x) can be quite pathological, inparticularly it doesn’t even have to be continuous (for instance the continuous extension of thefunction x 7→ x2 sin( 1

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 thatf is complex differentiable, or holomorphic, if and only if the aforementioned limit exists for everyx ∈ 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 thefollowing is true.

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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.That is, for every z0 ∈ U there exists ε > 0, and (an)n∈N ∈ C, such that f(z) =

∑n≥0 an(z − z0)n

for |z − z0| < ε.

These two theorems demonstrate the philosophical principle mentioned earlier: “The complexanalogues of “real” definitions often yields more restrictive, and rigid mathematical objects.” Whileanalytic functions (that is, those locally expressible by converging power series) are examples ofdifferentiable functions in both the real and complex world, they are revealed to be the only complexexample thereof, while being considered special amongst real differentiable functions.

Pretty much everything that is known about holomorphic functions is a consequence of anidentity of Cauchy. For a holomorphic function f as above, z0 ∈ U , and ε > 0, such that Uε =z| |z − z0| < ε ⊂ U , we have ∫

|z−z0|=εf(z)dz = 0. (2)

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 theoryto 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∫

M

dω =

∫∂M

ω.

We will consider the disc Dε(z0) = z ∈ C | |z − z0| ≤ ε as a bounded manifold, such thatthe boundary S1 is oriented in a counterclockwise manner. We will then attach to a holomorphicfunction f(z) a complex 1-form ω = f(z)dz, and apply Stokes Theorem in order to deduce Cauchy’sformula (2). Before getting there, we need some preparatory work.

We use the standard bijection C ∼= R2, obtained by decomposing a complex number z = x+ iyinto real and imaginary part, z 7→ (x, y). In other words, we view C as a smooth manifold ofdimension 2, and similarly this allows us to view Dε(z0) as a smooth manifold of dimension 2 withboundary.

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 vare differentiable and satisfy the Cauchy–Riemann differential equations ∂u

∂x = ∂v∂y and ∂v

∂x = −∂u∂y .

This theorem follows easily from the following observation in linear algebra. We can embed thefield of complex numbers C into the ring of (2× 2)-matrices, by assigning to a+ ib the matrix(

a −bb a

).

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The Cauchy–Riemann Equations can now be rephrased as stipulating that the Jacobi matrix(∂u∂x

∂u∂y

∂v∂x

∂v∂y

)

is of the shape above, that is, corresponds to multiplication of (x+ iy) by a complex number.On the manifold with boundary Dε(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. Wedefine 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 differ-entiable, 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:

d[(udx− vdy) + i(udy + vdx)] =∂u

∂ydy ∧ dx− ∂v

∂xdx ∧ dy + i

∂u

∂xdx ∧ dy + i

∂v

∂ydy ∧ dx.

Using that dx ∧ dy = −dy ∧ dx, we obtain the following result:

dω = [(∂u

∂y+∂v

∂x) + i(

∂u

∂x− ∂v

∂y)]dx ∧ dy.

The equation dω = 0 is equivalent to vanishing of both the real and imaginary part, which corre-sponds to the Cauchy–Riemann Equations.

Now we apply the Stokes Theorem to the manifold with boundary Dε(z0), and the complex1-form ω = f(z)dz. For f a holomorphic function, we obtain∫

∂ Dε(z0)

f(z)dz =

∫Dε(z0)

dω = 0.

1.3 Holomorphic functions on complex manifolds

In this paragraph we will give a sloppy definition of complex manifolds, which will be revised ata later point. We will ignore some technical points, like that the topological space underlying acomplex manifold should be Hausdorff and second-countable. Moreover, we pretend that we alreadyknow what a holomorphic function in several variables is. Moreover, we will ignore differencesbetween atlases and maximal atlases, etc.

Definition 1.6. Let X be a topological space, together with an open covering Uii∈I , and home-

omorphisms φi : Ui' // U ′i , where U ′i ⊂ Cn is an open subset. If for every pair (i, j) the induced

bijection

φi(Ui ∩ Uj)φjφ−1

i // φj(Ui ∩ Uj)

is holomorphic, we say that X is endowed with the structure of a complex manifold of complexdimension n.

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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 aRiemann surface.

Definition 1.7. Let X be a continuous manifold, and f : X → C be a continuous map. We saythat f is holomorphic, if for every chart (U, φ) as above, the composition f φ−1 : U ′ // C is aholomorphic function on U ′.

To emphasise the difference between complex and real manifolds, we will show that a compactcomplex manifold has only very few holomorphic functions. We will deduce this from a lemmaabout holomorphic functions, which states that for a non-constant holomorphic function g definedon an open subset U ′ ⊂ Cn, the image g(U ′) ⊂ C is an open subset.

Lemma 1.8. If X is a compact, connected, complex manifold, then every holomorphic functionf : X // C is constant.

Proof. Let us assume that f is not constant. For every chart (Ui, φi) we obtain a holomorphicfunction gi = f φi. We see that f(X) =

⋃i∈I g(U ′i) ⊂ 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. Theonly open and closed subsets of C are ∅ and C. The second case is not possible, because C is notcompact.

Corollary 1.9. A compact complex manifold X cannot be embedded into any Cn, unless X is0-dimensional.

Proof. Let us assume that we have an injective holomorphic map h : X → Cn. Let pi : Cn // Cthe projection to the i-th coordinate, that is, the map sending (z0, . . . , zn) ∈ Cn to zi. The mapspi h are holomorphic, and therefore constant on every connected component of X. This impliesthat 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. Everysmooth manifold can be embedded into Rn, for n large enough!

1.4 Complex projective space

In the preceding paragraph we saw that a compact complex manifold cannot be holomorphicallyembedded into Cn. This means the theory of smooth manifolds favourite strategy to produceexamples, an application of the Regular Value Theorem to produce smooth submanifolds of Rn,will not directly apply to the theory of complex manifolds. However, at this stage we haven’t evenseen a single example of a compact complex manifold.

Example 1.10. Consider the 2-sphere S2 as a topological space, and let x ∈ S2 be an arbitrarypoint. We denote by N = (0, 0, 1) the “north pole”, and by S = −N the “south pole”. In themanifolds course, stereographic projection was used to produce a homeomorphism between S2 \xwith R2. If we further use the standard identification C ∼= R2, we obtain a homeomorphism (or

chart) φ1 : S2 \N ' // C. We also denote S2 by C, and refer to it as the Riemann sphere, and Nas ∞.

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We will now construct a second chart, endowing C with the structure of a complex manifold.Since the underlying topological space is homeomorphic to S2, it is our first example of a compactcomplex manifold.

Lemma 1.11. Consider the map z 7→ 1z , which is well-defined on C \0. It extends to a homeomor-

phism 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 everypoint belonging to C \0 ∼= S2 \N,S, so it remains to verify continuity at 0 and ∞. We use that0 has a neighbourhood basis given by ε-neighbourhoods Uε = z ∈ C | |z| < ε, while ∞ has aneighbourhood basis by VR = z ∈ C | |z| > R∪∞. So to show continuity at 0, we need to showthat for every R > 0, there exists an ε > 0, such that i(Uε) ⊂ VR. This is automatically satisfied forz = 0, so we may focus on z ∈ Uε \ 0. Since |i(z)| = | 1z | =

1|z| , we have that for ε = 1

R the property

i(Uε) = VR is satisfied. Continuity at ∞ is verified similarly.We have i2 = idC, since this property is satisfied in the dense subset C \0, and the functions i2

and idC are continuous. Therefore, we see that i is its own continuous inverse, in particular it is ahomeomorphism.

This lemma now directly implies that the Riemann sphere C is a complex manifold of dimension1, that is, it is a Riemann surface. As charts we use U0 = C ⊂ C, with the identity map φ : U =

C id // C, and V = C \ 0 with the map ψ = i|V : V → C. Since V = i(U), and ψ = φ i, we obtainfrom Lemma 1.11 that ψ is a homeomorphism.

The change of coordinates map is given by

U ∩ Vψ

##

C \0z 7→ 1

z //

φ−1

;;

C \0

which is a bijective holomorphic map, with holomorphic inverse (that is, a biholomorphic map).This implies that C is a complex manifold.

The Riemann sphere C is also often denoted by P1. It belongs to a family of complex manifoldsPn, known as complex projective spaces.

Definition 1.12. For n ≥ 0, we denote by Pn the set of equivalence classes in Cn+1 \0, with respectto the equivalence relation (z0, . . . , zn) ∼ (w0, . . . , wn), if and only if there exists λ ∈ C× = C \0,such that (z0, . . . , zn) = (λw0, . . . , λwn). We have a canonical map Cn+1 \0 // Pn. We endow Pnwith the quotient topology, that is, consider a subset U ⊂ Pn as open, if and only if the preimagep−1(U) ⊂ Cn+1 \0 is open.

The equivalence class [(z0, . . . , zn)] will be denoted by (z0 : . . . , zn). As a next step, we constructcharts on Pn, and verify that they define the structure of a complex manifold.

For 0 ≤ i ≤ n we let Ui ⊂ Pn be the subset consisting of all (z0 : . . . : zn), such that zi 6= 0. Thepreimage p−1(Ui) is (z0, . . . , zn) ∈ Cn+1 |zi 6= 0 is an open subset, hence Ui is by definition openin the quotient topology. Moreover, we know that

⋃Ui = Pn, since for every (z0, . . . , zn) ∈ Cn+1 \0,

at least one coordinate zi must be non-zero. Hence, we have defined an open covering (Ui)i=0,...,n

of Pn.

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We define a continuous map φi : p−1(Ui) // Cn, by sending (z0, . . . , zn) to 1

zi(z0, . . . , zi, . . . , zn)

(omitting the i-th component). Since the value of φi is the same for every representative of a givenequivalence class (z0 : . . . : zn) ∈ Ui, we obtain a well-defined continuous map φi : Ui // Cn.

We have a continuous map ψi : Cn // p−1(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 ψi definesa continuous map ψi : Cn → Ui. Since φi ψi = idCn , and ψi φi = idUi , we have that φi is ahomeomorphism.

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 relabellingcoordinates. Let’s assume that (z0 : . . . : zn) belongs to Ui ∩Uj , that is, we have zi 6= 0 and zj 6= 0.The image φi(Ui∩Uj) is equal to (w1, . . . , wn) ∈ Cn |wj 6= 0. The resulting change of coordinatesmap is now given by the map

φi(Ui ∩ Uj) = (w1, . . . , wn) ∈ Cn |wj 6= 0 // Cn,

sending (w1, . . . , wn) to φj(ψi(w1, . . . , wn)) = (w1

wj, . . . , wiwj ,

1wj, wi+1

wj, . . . ,

wj−1

wj,wj+1

wj, . . . , wnwj ). Since

the components of this map are rational functions, they are in particular holomorphic. Therefore,we have established the following result.

Proposition 1.13. The topological space Pn, together with the charts defined above, is a complexmanifold.

One can also show that the topological space underlying the complex manifold Pn is compact.The easiest way to see this is by noting that every equivalence class [z0 : . . . : zn] has a repre-sentative (z0, . . . , zn) satisfying |z0|2 + · · · + |zn|2 = 1 (simply rescale appropriately). The subset(z0, . . . , zn) ∈ Cn+1 | |z0|2 + · · · + |zn|2 = 1 is by definition equivalent to S2n+1, that is, definesa compact topological space. Since Pn = p(S2n+1), we see that Pn is the image of a compacttopological 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∑i0,...,in≥0 ai1...inz

i00 · · · zinn , and refer to max(i0 + · · · + in|ai0...in 6= 0) as the degree degG of

G.

Definition 1.14. The polynomial G is called homogeneous of degree d if for every λ ∈ C, andevery z = (z0, . . . , zn) ∈ Cn+1 we have G(λz) = λdG(z).

This definition can be reformulated in more explicit terms, as the following lemma shows. Theproof is left as an exercise.

Lemma 1.15. A polynomial G =∑i0,...,in≥0 ai1...inz

i00 · · · zinn is homogeneous of degree d if and

only if ai1...in = 0 for i0 + . . . in 6= d. In particular we see that for a non-zero homogeneouspolynomial of degree d, we have d = degG.

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 polynomialG(z0, . . . , zn)of degree d, such that g(z0, . . . , zn−1) = G(z0, . . . , zn−1, 1). Take for example g(z0, z1) = z2

0z1 +

7

5z0 − 3z21 . It is easy to see that G(z0, z1, z2) = z2

0z1 + 5z0z22 − 3z2

1z2 satisfies the requirement. Wesay that G is the homogenisation of g.

The reason to introduce homogeneous polynomials is that they can be used to define subsetsof Pn. 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), thenthere exists λ ∈ C× = C \0, such that (w0, . . . , wn) = λ(z0, . . . , zn). Thus, we have G(w0, . . . , wn) =λdG(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(z0 : . . . : zn) ∈ Pn |Gi(z0, . . . , zn) = 0 ∀i = 1, . . . , k is called a complex projective subvariety ofPn, 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 thatthis is true generically. The criterion for it to be a manifold is reminiscent of the Regular ValueTheorem in the theory of smooth manifolds and will be proven using an analogue of this result forholomorphic functions.

Proposition 1.17. Let G1, . . . , Gk be non-zero homogeneous polynomials of degrees di, and as-sume that the Jacobi matrix (∂Gi∂zj

)i,j is regular for every value of (z0, . . . , zn) ∈ Cn+1 \0, such that

Gi(z0, . . . , zn) = 0 for all i = 1, . . . , k. Then there exists a natural structure of a complex manifoldon V (G0, . . . , Gk).

We can now state one of the first big theorems. We will learn more about its proof at a laterpoint.

Theorem 1.18 (Chow). Let X be a compact complex manifold, such that there exists a holomorphicmap f : X // Pn. Then, there exist non-zero homogeneous polynomials G1, . . . , Gk, such thatf(X) = V (G0, . . . , Gn).

Chow’s theorem implies in particular that every compact complex submanifold of Pn can bedescribed 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 biggerthan just algebraic ones. At a later point we will deduce this result from a theorem of Serre, whichis referred to as GAGA (abbreviating the title of Serre’s article Geometrie algebrique et geometrieanalytique). 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(submanifolds, vector bundles, etc.), which we can define on Pn is in fact algebraic, that is, definedin 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 produceexamples of compact complex manifolds. There is another, even easier strategy to produce a lot ofexamples.

Definition 1.19. A lattice in V = Rn is a subgroup Γ ⊂ (Rn,+), such that V/Γ is a compacttopological 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, thecomplex exponential map induces a homeomorphism R /Z ⊂ S1, and the circle is known to be acompact topological space. There is a more general class of example.

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Example 1.20. Let V be a finite-dimensional vector space with a basis ω1, . . . , ωn. Then thesubgroup Γ(ω1, . . . , ωn) = Zω1 + · · ·+ Zωn is a lattice.

It is clear that the subgroup defined above is discrete, in the lemma below we show that thequotient is indeed compact.

Lemma 1.21. Let V be a finite-dimensional real vector space, and Γ = Γ(ω1, . . . , ωn) ⊂ V a latticeas in the example above. Then, the topological space V/Γ is a torus, that is, homeomorphic to afinite product of copies of S1 (and in particular it is compact).

Proof. The complex exponential map induces a homeomorphism R/Z ∼= S1. We choose a basisω1, . . . , ωn, such that Γ = Γ(ω1, . . . , ωn). Choose the unique isomorphism of vector spaces V ∼= Rn,transforming (ωi) to the standard basis (ei). This induces a homeomorphism V/Γ ∼= Rn /Zn ∼=(S1)n.

We will show first that every lattice Γ can be written as Γ(ω1, . . . , ωn) = λ1ω1 + · · · +λnωn|(λ1, . . . , λn) ∈ Zn,where (ω1, . . . , ωn) is a basis of V = Rn viewed as a real vector spaceof dimension n. Afterwards, we will verify that V/Γ is a complex manifold, if V is a complex vectorspace.

Lemma 1.22. Let Γ ⊂ V ∼= Rn be a lattice, then there exists a real basis ω1, . . . , ωn, such thatΓ = 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,there exists an isomorphism A ∼= Zn.

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 continuousmap V/Γ // V/W , sending [v]Γ to [v]W . The right hand side V/W is also a finite-dimensionalvector space. Therefore, not compact, unless V/W = 0. Since images of compact spaces undercontinuous maps are compact, we must have V/W = 0, which implies W = V .

Linear algebra tells us, that the set Γn contains tuples (v1, . . . , vn), forming a real basis of thevector space V . We denote this subset of Γn by BΓ.

We know that Γ ⊃ Γ(ω1, . . . , ωn). Consider now the quotient Γ/Γ(ω1, . . . , ωn) ⊂ V/Γ(ω1, . . . , ωn),which is closed with respect to the quotient topology, since its preimage p−1(Γ/Γ(ω1, . . . , ωn) = Γ inV is closed as a discrete subspace of V . Closed subsets of compact topological spaces are themselvescompact, 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 im-plies 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 representa-tives 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 generatedabelian 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 asa 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 complexcharts, and verify that the change of coordinates maps are holomorphic.

In order to define the charts, we choose ε > 0, such that v ∈ Γ and |v| < 2ε implies v = 0. Thisis possible, because Γ ⊂ V is discrete, that is, 0 ∈ Γ must possess a neighbourhood, which doesn’tcontain another element of Γ.

Let p : V → V/Γ be the canonical projection. If we restrict p to Uε(v) = w ∈ V | |w − v| < ε,we obtain an injective map. Indeed if p(w1) = p(w2), then w1 − w2 ∈ Γ. But this is impossible forw1, w2 ∈ Uε(v), since we also have |w1 − w2| < 2ε by the triangle inequality.

Moreover, for every open subset V ⊂ Uε(v), the image p(V ) is open in the quotient topology,since p−1(p(V )) =

⋃w∈Γ(w + V ) is a union of open subsets. We have therefore defined a bijec-

tive continuous map ψε,v : Uε(v) // p(Uε(v)), mapping open sets to open sets. Hence ψε,v is ahomeomorphism with continuous inverse denoted by φε,v.

We define charts by (p(Uε(v)), φε,v), choosing ε once and for all. It only remains to verify that thechange of coordinates maps are holomorphic. Let v, w ∈ V be two element of V , the composition

φv(Uε(v) ∩ Uε(w))φ−1v // Uε(v) ∩ Uε(w)

φw // φw(Uε(v) ∩ Uε(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 andΓ′ ⊂W be lattices. Then every holomorphic map f : V/Γ //W/Γ′, is induced by a complex linearmap φ : V // W, such that φ(Γ) ⊂ Γ′, 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 yetgiven a complete definition of the notions of complex manifolds, and holomorphic maps betweenthem.

Proof. We choose complex isomorphisms V ∼= Cn and W ∼= Cm. We denote by Cm×n the complexvector space of linear maps Cn → Cm. At first we will show that we have a well-defined holomorphicmap Df : V/Γ→ Cm×n.

This map is defined as follows: for [v] ∈ V/Γ, we choose a representative v ∈ V , and also arepresentative w for f([v]). Choosing ε and δ appropriately, we may assume that for every v ∈ Vwe have f(Uε(v)) ⊂ Uδ(w). We define Df(v) to be the Jacobi matrix of φw f φ−1

v . This iswell-defined, since for another choice of representatives v′ and w′, the map φw′ f φ−1

v′ differs fromthe 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 ofthe function Df we have for the chart (p(Uε(v)), φv) that Df φ−1

v is the Jacobi matrix of theholomorphic map Uε(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 aconstant function. We denote the corresponding matrix by A. This implies that φw f φ−1

v

is equivalent to z 7→ Az + bv,w, where bv,w is a constant depending on the charts labelled by vrespectively w.

If Uε(v) ∩ Uε(v′) 6= ∅, then bvw − bv′w ∈ Γ′. This follows from [Az + bvw] = [Az + bv′w] forz ∈ Uε(v) ∩ Uε(v′). Similarly, if Uδ(w) ∩ Uδ(w′) 6= ∅, we must have bvw − bvw′ ∈ Γ′. Since forevery v, v′ ∈ V , there exists a sequence of points v0, . . . , vk ∈ V , such that v0 = v, vk = v′,and Uε(vi) ∩ Uε(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 havef(z) = [Az + b] = f(0) = [b]. Hence, f(z) ∈ Γ′.

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) = Γ′.

It is an interesting question when a complex torus V/Γ can be realised as a complex submanifoldof a projective space Pn. For dimC V = 1 this turns out to be always the case. Those complex toriare 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 ofthe 1 variable case. The most important example being Cauchy’s formula below. However, thereare 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 Dε(z0) := z ∈ C | |z − z0| < ε ⊂ U is a closed disc fully contained in U , then for everyz ∈ Uε(z0), we have

f(z) =1

2πi

∫∂ Dε(z0)

f(w)

w − zdw.

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 Aε,δ theannulus Dε(z0) \ Uδ(z), for δ small enough. It is a manifold with boundary being a union of twocircles.

As in 1.2, we define a complex 1-form f(z)z−z0 dz by decomposing into real and imaginary parts:

f = u+ iv, and dz = dx+ idv. This 1-form is well-defined on Aε,δ, because the singularity at z = z0

lies outside of Aε,δ. As in 1.2 we have that d( f(z)z−z0 dz) = 0, since f(z)

z−z0 is holomorphic. Thereforethe Stokes Theorem implies∫

∂ Dε

f(w)

w − zdw −

∫∂ Dδ

f(w)

w − zdw =

∫∂Aε,δ

f(w)

w − zdw = 0.

This also implies that the value of∫∂ Dδ(z)

f(w)w−z dw does not depend on δ. We will therefore study

what happens when δ → 0.To compute the integral we parametrise ∂ Dε(z0) by the path γ : [0, 2π] // C, which sends

t 7→ z + δeit. By substitution we obtain∫∂ Dε

f(w)

w − zdw =

∫ 2π

0

f(z + δeit)

δeitδieitdt =

∫ 2π

0

f(z + δeit)idt

which tends to 2πif(z), as required.

11

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

1

w − z=

1

(w − z0)− (z − z0)=

1

(w − z0)(1− z−z0w−z0 )

=1

(w − z0)

∑k≥n

(z − z0

w − z0))k.

Therefore, we deduce from Cauchy’s formula that

f(z) =1

2πi

∫∂ Dε(z0)

f(w)

w − zdw =

1

2πi

∫∂ Dε(z0)

∑k≥0

f(w)(z − z0)k

(w − z0)k + 1) =

∑k≥0

[1

2πi

∫∂ Dε(z0)

f(w)

(w − z)k+1dw](z−z0)k,

which converges uniformly for |z−z0||w−z0| ≤ ε′ < 1.

The most direct way to define what it means for a function in several variables to be holomorphicis the following.

Definition 2.2. Let U ⊂ Cn be an open subset, and f : U // C a continuous function. Wesay that f is holomorphic, if for each (z1, . . . , zn) ∈ U , and each i = 1, . . . , n, the functionf(z1, . . . , zi−1, w, zi+1, . . . , zn) is holomorphic in the variable w, defined in a small neighbourhoodof w = zi.

As a first step, we will derive a multivariable version of Cauchy’s integral formula.

Proposition 2.3. Let f : U // C be a holomorphic function, defined on an open subset U ⊂ Cn.For a point z = (z1, . . . , zn) ∈ U we choose ε1, . . . , εn > 0, such that the polydisc Dε1,...,εn(z) =Dεn(z1)× · · · × Dεn(zn) is a subset of U . Then, for each w = (w1, . . . , wn) ∈ Dε1,...,εn(z) we have

f(w′1, . . . , w′n) =

1

(2πi)n

∫∂ Dε1 (z1)

· · ·∫∂ Dεn (zn)

f(w)

(w1 − w′1) · · · (wn − w′n)dw1 · · · dwn.

Proof. We prove the formula by induction on n, the anchor case being n = 1, for which it is theclassical integral formula of Proposition 2.1.

By induction we assume that the statement of the proposition has already been proven forholomorphic functions in n− 1 variables.

The definition of holomorphic functions in several variables tells us that for a fixed tuplew′1, . . . , w

′n−1, the function w1 7→ f(w1, w

′2, . . . , w

′n) is holomorphic in w1. Therefore, applying

the classical Cauchy formula, we obtain

f(w′1, . . . , w′n) =

1

2πi

∫∂ Dε1 (z1)

f(w1, w′2, . . . , w

′n)

(w1 − w′1)dw′1.

Inserting the induction hypothesis for the holomorphic function in n−1 variables (w′2, . . . , w′n) 7→

f(w1, w′2, . . . , w

′n), we obtain the asserted identity.

Corollary 2.4. A holomorphic function f : U // C, for U ⊂ Cn an open subset, is analytic. Thatis, for a small enough open neighbourhood V of each point (z1, . . . , zn) ∈ U , the there exists apower sum

∑m1,...,mn≥0 am(w1 − z1)m1 · · · (wn − zn)mn , convergent for w ∈ V , such that f(w) =∑

m1,...,mn≥0 am(w1 − z1)m1 · · · (wn − zn)mn for each point w of V .

12

The strategy of the proof is the same as in the 1-dimensional case, that is, uses the geometricseries in each variable. We avoid proliferation of indices by leaving the details to the reader.

Given an open subset U ⊂ Cn, we may view it as an open subset of R2n. With respect to thistransition, the complex coordinates (z1, . . . , zn) get replaced by 2n real coordinates (x1, yn, . . . , xn, yn),satisfying the relation zi = xi + iyi.

Let A : Cn // Cn 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 A. It is represented

by a (2n× 2n)-matrix A = (ak,j)k,j=1,...,2n.

Remark 2.5. The map A 7→ A defines an injective morphism of rings EndC(Cn) // EndR(R2n).

This implies that if B = A is invertible, then also its inverse B−1 is induced by a complex matrix(namely A−1). A real (2n × 2n)-matrix C = (ck,l)k,j=1,...,2n ∈ EndR(R2n) lies in the image, ifand only if for every i, j = 1, . . . , n, the “linear Cauchy–Riemann equations” are satisfies, that isc2i,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 derivativeswith respect to the real variables xi, yi satisfy for each i = 1, . . . , n the Cauchy–Riemann equations∂u∂xi

= ∂v∂yi

, and ∂u∂yi

= − ∂v∂xi

(see 1.4 for more details on the Cauchy–Riemann equations).

Given a partially differentiable function f = (f1, . . . , fk) : V // Rk, for V ⊂ Rm an opensubset, we denote the Jacobi matrix by DRf . Likewise, if U ⊂ Cn is an open subset, and f =

(f1, . . . , fk) : U // Ck we have a Jacobi matrix of complex partial derivatives Df =(∂fi∂zj

)i=1,...,k

j=1,...,n.

With respect to the standard identification Cn ∼= R2n we have the following remark.

Remark 2.6. A function f : U // Ck is holomorphic, if and only if it is differentiable as a functiondefined on an open subset of R2n, and for every z ∈ U the real Jacobi matrix DRf(z) is of the shape

A for A ∈ EndC(Cn). 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.

Theorem 2.7 (Inverse Function Theorem). Let U ⊂ Cn be an open subset, and f = (f1, . . . , fn) : U →Cn a holomorphic function. Assume that w ∈ U is a point, such that the complex Jacobi matrix( ∂fi∂zj

) is invertible at z = w. Then, there exist open neighbourhoods V of w, and W of f(w), as well

as a holomorphic function g : W // V , such that g f = idV , and f g = idW .

Proof. We identify Cn with R2n, 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 // Vwith 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 equiv-alent to the inverse of the Jacobi-matrix of f with respect to (x1, y1, . . . , xn, yn). Since the Jacobi-matrix of f corresponds to a complex matrix, by means of the isomorphism Cn ∼= Rn, this mustalso be true for the Jacobi-matrix of g (by Remark 2.5). Therefore we deduce from Remark 2.6that 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 ofholomorphic equations, can be parametrised by holomorphic functions, near a given solution. Wewill use it in the future to construct several examples of complex manifolds, for instance smoothcomplex projective manifolds, which arise as simultaneous zero sets to a system of homogeneouspolynomials.

Theorem 2.8 (Implicit Function Theorem). Let U ⊂ Cn be an open subset, and f = (f1, . . . , fk) : U //Cka holomorphic function. If f(0) = 0, and the determinant | ∂fi∂zj

(0)|0≤i,j≤k 6= 0 is non-zero. Then

there exists a holomorphic function g = (g1, . . . , gk) : V // Ck, for some open subset V ⊂ Cn−k,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 .

Proof. Consider the function F : U // Cn, given by (z1, . . . , zn) 7→ (f1, . . . , fk, zk+1, . . . , zn). Sinceall components are holomorphic functions, F is holomorphic as well. Moreover, we have |∂Fi∂zj

(0)|0≤i,j≤n =

| ∂fi∂zj(0)|0≤i,j≤k 6= 0, since the Jacobi matrix of F has a block upper triangular shape, with invertible

diagonal blocks ( (∂fi∂zj

(0))i,j=1,...,k

∗

0 1

).

We can therefore apply the Inverse Function Theorem 2.7 to deduce the existence of a locallydefined holomorphic inverse G : V ′ //U , where V ′ ⊂ Cn is an open neighbourhood of F (0) = 0 (byassumption). We denote the components of G by (g′1, . . . , g

′n). We let g1, . . . , gn be the holomorphic

functions on V = V ′ ∩ 0k × Cn−k 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 fori = k + 1, . . . , n

The relation FG = 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 ′), and z ∈W , such that f(z) = 0, we obtain from GF = 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 importantto remember the existence of a biholomorphic map G as constructed in the proof.

Porism 2.9. Let U ⊂ Cn be an open subset, and f = (f1, . . . , fk) : U // Ck a holomorphic functionas in the statement of Theorem 2.8. Then there exist open neighbhourhoods W,W ′ ⊂ Cn of 0, suchthat W ⊂ U , and a biholomorphic map G : W ′ // W , such that (f1, . . . , fk) G(z1, . . . , zn) =(z1, . . . , zk).

2.4 ∂-derivatives

There is a formal device, which simplifies many arguments concerning multivariable holomorphicfunctions. We fix an open subset U ⊂ Cn, and a continuous function f : U // C. Using thestandard identification Cn ∼= R2n, which on the level of coordinates works by decomposing eachcomplex component zi into real and imaginary part, zi = xi + iyi, we may ask that f is partiallydifferentiable with respect to the 2n real coordinates xi and yi.

14

Definition 2.10. For f as above, we define a differential operator

∂f

∂zi=

1

2(∂f

∂xi+ i

∂f

∂yi).

We also define∂f

∂zi=

1

2(∂f

∂xi− i ∂f

∂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 thecoordinates (xi, yi) if and only if ∂f

∂zi= 0.

Proof. Let us compute ∂f∂zi

, with respect to the decomposition f = u+ iv. We have

1

2(∂f

∂xi+∂f

∂yi) =

1

2(∂u

∂xi− ∂v

∂yi+ i

∂u

∂yi+ i

∂v

∂xi).

And by considering real and imaginary part separately, we see that the Cauchy–Riemann equationswith respect to (xi, yi) are satisfied, if and only if ∂f

∂zi= 0.

This differential operator allows us to check that certain functions, defined by integration, areholomorphic.

Lemma 2.12. For a smooth function f : U // C, with U ⊂ C2 and open subset, and γ : [0, 1] // Ca continuous path, we have

∂

∂w

∫γ

f(z, w)dz =

∫γ

∂

∂wf(z, w)dz,

whenever well-defined. In particular, if f is holomorphic in w, then so is w 7→∫γf(z, w)dz,

whenever well-defined.

2.5 Hartog’s Extension Theorems

The result of this paragraph highlights an interesting difference between holomorphic functionsin 1 and several variables. If f is a holomorphic function on an open subset of 2-dimensionalcomplex 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 orempty. Moreover, the proof given below, also gives us an explicit method to construct a holomorphicextension.

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, andU ⊂ Cn a connected open subset. If there exists an open subset ∅ 6= V ⊂ U , such that f |V = 0,then f = 0.

We denote by Uε1,ε2 = (z, w) ∈ C2 | |z| < ε1, |w| < ε2 the polydisk and by Dε1,ε2 its closure.

15

Theorem 2.14 (Hartog). Let εi > δi > 0 for i = 1, 2, and f : Uε1,ε2 \Dδ1,δ2 // C be a holomorphicfunction, then f extends to a holomorphic function on Uε1,ε2 .

Proof. Choose r between ε2 and δ2, and define F : Uε to be

F (z1, z2) =1

2πi

∫|w2|=r

f(z1, w2)dw2

w2 − z2,

which is well-defined for all (z1, z2) ∈ Uε(0). Moreover, F agrees with f , whenever |zi| > δi, byCauchy’s integral formula. A 3-variable version of Lemma 2.12 tells us that ∂

∂ziapplied to the

above integral is zero, because we may exchange the order of this differential operator with theintegral.

2.6 Complex vector fields

Let’s fix an open subset U ⊂ Cn of a complex space. Our goal in this paragraph is to define complexvector fields on U . A careful study of this local situation will allow us later to give a definition ofcomplex vector fields on complex manifolds. From now on we modify our notation slightly, turningsome 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∑i=1

Vi ∂

∂zi, (3)

where (V1, . . . ,Vn) : U // Cn is a holomorphic function on U .

Equivalently, we could have defined a complex vector field V as a holomorphic function U // Cn.Since the differential operators ∂

∂zi are linearly independent (which can be seen by using the relation∂zi

∂zj = δij), 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 thinkthis through in some detail: consider a biholomorphic map φ : V // U , with inverse ψ : U // V .A holomorphic function g : U // Ck 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 wouldbe inconsistent with the viewpoint of a complex vector field V as a differential operator.

Indeed, if U ′ ⊂ U is an open subset, and f : U // C a holomorphic function, then we obtain aholomorphic function h = f φ : ψ(U ′) // C. Equivalently, we can write f = h ψ, and use thechain rule to compute

V f(z0) = V(h ψ)(z0) =

n∑j=0

Vj ∂(h ψ)

∂zj(z0) =

n∑i=0

n∑j=0

Vj ∂h

∂wi(ψ(z0))

∂ψj

∂zj(z0).

This defines a holomorphic function on U ′, but by composing with φ, we see that we have defineda first order differential operator

h 7→n∑i=0

n∑j=0

∂ψi

∂zj(Vj φ) φ

∂h

∂wi.

16

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

n∑i=1

Vi ∂

∂zi=

n∑i=1

n∑j=1

(Vj φ)∂wi

∂zj∂

∂wi.

This inspires the following definition.

Definition 2.16. For open subset U, V ⊂ Cn, a biholomorphic map ψ : U // V , and a complexvector field V =

∑ni=1 V

i ∂∂zi on U , we define

ψ∗ V = Dψ

V1 φ...

Vn φ

=

n∑i=0

n∑j=0

∂ψi

∂zj(Vj φ)

∂

∂wi.

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

2.7 Complex differential forms

As before we fix an open subset U ⊂ Cn. A complex m-form ω on U is an object, which assignsto m complex vector fields V1, . . . ,Vm, defined on an open subset U ′ ⊂ U , a holomorphic functionω(V1, . . . ,Vm) on U ′. Several properties need to be satisfied. For instance we want this constructionto be compatible with restriction, that is, for an open subset U ′′ ⊂ U ′ we want

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

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

ω(V1, . . . ,Vi−1,Vi +fW,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 functionon U given by

ω(∂

∂zi1, . . . ,

∂

∂zim).

Since every complex vector field can be expressed as a linear combination (3) of the coordinatevector fields ∂

∂zi , the multilinearity (5) and anti-symmetry conditions (6) of m-forms guaranteethat 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

(dzi1 ∧ · · · ∧ dzim)(∂

∂zj1, . . . ,

∂

∂zjm) =

1, if (i1, . . . , im) = (j1, . . . , jn)

0, else.

It is then clear that we have

ω =∑

1≤i1<···<im≤n

ωi1...imdzi1 ∧ · · · ∧ dzim (7)

for every complex m-form.

Definition 2.17. The complex vector space of complex m-forms on U ⊂ Cn will be denoted byΩm(U). By convention we agree that a complex 0-form is a holomorphic function. We will denotethe 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 theexterior, 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 uniquecollection of maps satisfying for ω, ω′ ∈ Ωm(U), η ∈ Ωk(U), and f ∈ O(U)

(a) ω ∧ η = (−1)mkη ∧ ω

(b) (ω + ω′) ∧ η = ω ∧ η + ω′ ∧ η

(c) (fω) ∧ η = f(ω ∧ η).

(d) for 1 ≤ i1 < . . . im ≤ n, (dzi1) ∧ · · · ∧ (dzim) agrees with the m-form dzi1 ∧ · · · ∧ dzim definedabove.

(e) For U ′ ⊂ U an open subset we have (ω ∧ η)|U ′ = (ω|U ′) ∧ (η|U ′).

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, ω′ ∈ Ωm(U), η ∈ Ωk(U), and f ∈ O(U)

(a) d(ω + ω′) = dω + dω′

(b) d(fω) =∑ni=1

∂f∂zi dz

i ∧ ω + fdω

(c) d(ω ∧ η) = dω ∧ η + (−1)mω ∧ dη.

(e) For an open subset U ′ ⊂ U we have (dω)|U ′ = d(ω|U ′).

Differentiating a complex m-form twice leads to a vanishing (m+ 2)-form.

Lemma 2.21. For a complex m-form ω ∈ Ωm(U) defined on an open subset U ⊂ Cn we haved2ω = 0.

18

Proof. Since ω can be written as∑

1≤i1<···<im≤n ωi1...imdzi1 ∧ · · · ∧ dzim it suffices to check this

identity for 0-forms, that is a holomorphic function f ∈ O(U). We have

d2 = d

n∑i=1

∂f

∂zidzi =

n∑j=1

n∑i=1

∂2f

∂zj∂zidzj ∧ dzi.

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 acollection of

(nm

)holomorphic functions. However, their definition as alternating multi-linear forms

on holomorphic vector fields, and equation (7) are more than mere bookkeeping devices. They helpus to pin down the behaviour of m-forms with respect to biholomorphic coordinate changes. Thekey difference to the theory of complex vector fields is that complex m-forms can be pulled backalong an arbitrary holomorphic map, without imposing the existence of a holomorphic inverse.

Definition 2.22. For open subsets U ⊂ Cn1 , V ⊂ Cn2 , a holomorphic map ψ : U // V , and acomplex m-form ω on V , we define

ψ∗ω =∑

1≤i1<···<im≤n2

(ωi1<···<imψ)(ψ∗dz1)∧· · ·∧(ψ∗dzm) =∑

1≤i1<···<im≤n2

(ωi1<···<imψ)dψ1∧· · ·∧dψm.

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

Lemma 2.23. Let U ⊂ Cn1 , V ⊂ Cn2 be open subsets, and ω ∈ Ωm(V ). For a holomorphicfunction φ : U // V we have dφ∗ω = φ∗dω.

Proof. Since φ∗ commutes with sums and wedge products, and d is compatible with wedge productsby means of a sort of Leibniz rule (see (c) of Definition 2.20), it suffices to prove the identityfor 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 writeω =

∑n2

i=1 ωidzi. We have φ∗ω =

∑n2

i=1(ωi φ)dφi. Therefore, dφ∗ω =∑n2

i=1(d(ωi φ) ∧ dφi +(ωi φ) ∧ d2φi) = φ∗(

∑n2

i=1

∑n1

j=1 dωi ∧ dzj). In our computation we used the identity d2 = 0 fromLemma 2.21.

Lemma 2.24. For open subsets Ui ⊂ Cni for i = 1, 2, 3, and holomorphic maps φ : U1// U2,

ψ : U2// U3, we have an equality of linear maps

(ψ φ)∗ = φ∗ψ∗ : Ωm(U3) // Ωm(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 .

Remark 2.25. For m > n and U ⊂ Cn, ever complex m-form ω on U vanishes. As we have seenin (7) we have ω =

∑1≤i1<···<im≤n ωi1...imdz

i1 ∧ · · · ∧ dzim . But for m > n there is no orderedtuple 1 ≤ i1 < · · · < im ≤ n.

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

φ : U'−→ U ′, where U ′ ⊂ Rn 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 ∩ Vψ

yy

φ

%%

ψ(U ∩ V )φ|U∩V ψ−1|ψ(U∩V )

// φ(U ∩ V )

In the definition of a (real) manifold one demands that the transition map φ|U∩V ψ−1|ψ(U∩V ) issmooth, that is infinitely many times differentiable.

The definition of a complex manifold is similar. We demand that each chart (U, φ) defines ahomeomorphism between U and an open subset U ′ ⊂ Cn. The transition map above is required tobe holomorphic.

3.1 The definition

Let’s repeat the definition of a complex manifolds with the necessary technical details. To preparethe 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

φ : U' // U ′, where U ′ ⊂ Cn is an open subset of a complex space.

(b) Two charts (U, φ), (V, ψ) are called compatible, if the change-of-coordinates map

φ|U∩V ψ−1|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 iscompatible.

(d) An atlas is called maximal, if it is not strictly contained in another atlas (that is, it containsevery 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, togetherwith 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” amaximal atlas, trivial cases excluded. In concrete examples, we define the structure of a complexmanifold 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 ofchoice?

Let’s take another look at the examples covered in Section 1. We have defined atlases for Pnand V/Γ, where Γ ⊂ V is a lattice in a complex vector space. To complete the proof that theseare complex manifolds it remains to verify that the topological spaces are Hausdorff and secondcountable.

Lemma 3.4. If X =⋃i∈N Ui is a countable covering by open subsets, such that each Ui is second

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 subsetsVijj∈N, such that every open subset W ⊂ Ui can be expressed as the union

⋃Vij⊂W W = W .

Since N×N is countable, we have produced a countable collection of open subsets Vij(i,j)∈N2 ofX with an analogous property: let W ⊂ X be an arbitrary open subset, then we have that⋃

Vij⊂WVij =

⋃i∈N

⋃Vij⊂W∩Ui

Vij =⋃i∈N

(W ∩ Ui) = W.

This shows that X is also second countable.

As we know from our construction of the charts, Pn admits a finite covering my topologicalspaces homeomorphic to Cn, that is, Pn is second countable by the Lemma above. Similarly, wehave remarked that the spaces V/Γ are compact and covered by topological spaces homeomorphicto open subsets of Cn. Compactness tells us that a finite number of these is already a covering, andthe Lemma above implies that complex tori are second countable too. Since V/Γ is homeomorphicto (S1)2n, we see that complex tori are also Hausdorff spaces. It remains to conclude that Pn is acomplex manifold.

Exercise 3.5. The topological space Pn 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 defineholomorphic functions on them. From now on we fix a complex manifold X, and denote themaximal atlas, inducing the complex structure on X, by U .

Definition 3.6. A continuous function f : X // Ck is called holomorphic, if and only if for everychart (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 saythat f is holomorphic, if and only if for every x ∈ X, there exists a chart (U, φ), containing x, suchthat 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 ofa complex manifold that f φ−1

1 is holomorphic at φ1(x) if and only if f φ−12 is holomorphic at

φ2(x). Indeed, we have f φ−11 = (f φ−1

2 ) (φ2 φ−11 ), and the change of coordinates map φ2 φ−1

1

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 acomplex vector field V(U,φ) on U ′ ⊂ Cn, such that for (U1, φ1), (U2, φ1) ∈ U we have

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

where φ21 : φ1(U1 ∩ U2) // φ2(U1 ∩ U2) is the biholomorphic change of coordinates map φ2 φ−11 .

Let’s unravel this definition a little bit. The expression (φ21)∗ refers to the push-forward oper-ation of complex vector fields that we introduced in Definition 2.16. More concretely, we demandthat

V(U2,φ2) |φ2(U1∩U2) = Dφ21

V1(U1,φ1) φ−1

21...

Vn(U1,φ1)φ−1

21

|φ2(U1∩U2) =

n∑i=0

n∑j=0

(∂φi21

∂zjVj φ−1

21 )

∂

∂wi.

Unlike in the definition of a holomorphic function, this definition produces a global object fromscratch, by assembling local data (the complex vector fields V(U,φ)) in a consistent way. Whenwe will have introduced the notion of sheaves, we will see that this definition fits very well into asheaf-theoretic framework.

Definition 3.8. A complex m-form on X is a rule which assigns to a chart (U, φ) an m-formω(U,φ), such that for every pair of charts (U1, φ1), (U2, φ2), and the change of coordinates mapφ21 : 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 characterisedby the fact that ψ∗dω = dψ∗ω, ψ∗(ω ∧ η) = ψ∗ω ∧ ψ∗η, and for 0-forms (that is, holomorphicfunctions) 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 workwith complex m-forms or vector fields. The next lemma addresses this question, and as we’ll see inits proof, holomorphic compatibility implies that we don’t have to work with all possible charts todefine those objects.

Lemma 3.9. Let⋃i∈I Ui = X be an open covering of X, such that (Ui, φi) ∈ U are charts. Assume

that for every i ∈ I we have a complex m-form ωi ∈ Ωm(U ′i) (respectively a complex vector field Vion U ′i), 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 fieldV), 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 vectorfields to the reader. Given an arbitrary chart (U, φ), we need to define a complex m-form ω(U,φ).To do so, we observe that Vi = φ(Ui ∩ U) ⊂ U ′ defines an open covering of U ′ ⊂ Cn, and that the

22

restrictions ωi|Vi still define a compatible collection of locally defined m-forms for U ′. Therefore, inorder to define ω(U,φ) ∈ Ωm(U ′), we may temporarily assume that X = U ′ ⊂ Cn is an open subsetof X.

Using the temporary assumption that X ⊂ Cn, we define for every 1 ≤ j1 < · · · < jm ≤ n, andevery i ∈ I, a holomorphic function ωj1...jm;i ∈ O(Vi). The functions are defined by means of thedecomposition of the complex m-form on Vi

φ∗iωi =∑

1≤j1<···<jm≤n

ωj1...jm;idzj1 ∧ · · · ∧ dzjm .

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

ωj1...jm;i|Vi∩Vi′ = ωj1...jm;i′ .

Or, in other words, the family of holomorphic functions ωj1,...,jm;i agree on overlaps of the opencovering Vii∈I of X ⊂ Cn. Therefore, there exist holomorphic functions ωj1...jm on X, such thatωj1...jm |Vi = ωj1...jm;i. We let ω = (ωU,φ) be

∑1≤j1<···<jm≤n ωj1...jmdz

j1 ∧ · · · ∧ dzjm .From now on we discard the temporary assumption X ⊂ Cn, and let X be a general complex

manifold, like in the statement of the Lemma. For every (U, φ) ∈ U we have defined an ω(U,φ) ∈Ωm(U ′), and it remains to check compatibility of the complex m-forms.

For (U1, φ1), (U2, φ) ∈ U we denote the biholomorphic change of coordinates map by

φ21 = φ2 φ−11 : φ1(U1) // φ2(U2).

We have to show thatφ∗21ω(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 acommutative triangle of change of coordinates maps

φ1(U1 ∩ U2 ∩ U3)φ31 //

φ21 ))

φ3(U1 ∩ U2 ∩ U3)

φ2(U1 ∩ U2 ∩ U3).

φ32

55

By definition of ωU1,φ1we have ω(U1,φ1)|φ1(U1∩U2∩U3) = φ∗31ω3, where ω3 = ωi|φ1(U1∩U2∩U3). And

similarly, ω(U2,φ2)|φ2(U1∩U2∩U3) = φ∗21ω3. This implies that φ∗21ω2 = ω1, since φ∗31 = (φ32 φ21)∗ =φ∗21 φ∗23 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 intoCn. Nonetheless, the notion of complex submanifolds plays as much of a role in the theory ofcomplex manifolds to construct examples, as it does in the theory of smooth manifolds. The formaldefinition of complex submanifolds is as follows.

23

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

φ|Y ∩U : Y ∩ U ' // U ′ ∩ 0n−m × Cm = (z1, . . . , zn) ∈ 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 . Weonly need to consider φ|Y ∩U as a homeomorphism from Y ∩U to an open subset of Cm. This givesus a covering of compatible charts for Y , and we simply pass to the maximal atlas containing thesecharts.

Since X is a complex manifold, the underlying topological space of X is Hausdorff and secondcountable. This implies that those topological properties are also held by the underlying space ofY . Summarising this discussion we see that a complex submanifold carries indeed the structure ofa complex manifold.

We fix k homogeneous polynomials Gi = Gi(z0, . . . , zn) of degree di in n+1 variables. Moreover,we assume that w = (w0, . . . , wn) ∈ Cn+1 \0 is a vector, such that Gi(z0, . . . , zn) = 0 for everyi = 1, . . . , k, the condition (

∂Gi∂zj

(w)

)i=,1,...,kj=0,...,m

is a surjective linear map (9)

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

Proposition 3.11. If the homogeneous polynomials G1, . . . , Gk satisfy condition (9) for everyw ∈ 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 ValueTheorem. At first we need to explain what it means for a map between complex manifolds to beholomorphic.

If X and Y are complex manifolds, and f : X // Y is a continuous map we are going to definenext 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 weobtain a commutative diagram

Uf//

φ

V

ψ

U ′ψfφ−1

// V ′.

Definition 3.12. A continuous map Xf//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 , suchthat 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 situationsthat 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⋃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

only required to show that for every i ∈ I that the resulting map ψj(i) f φ−1i is holomorphic.

Theorem 3.13 (Regular Value Theorem). Let Xf// 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)

containing x, we have that (∂(fφ−1)i

∂zj )i,j is a surjective linear map. Then, f−1(y) ⊂ X is a complexsubmanifold.

Proof. Without loss of generality one can assume that Y = Ck is a complex space. This is possiblebecause y ∈ Y belongs to a chart (V, ψ), which is by means of ψ biholmorphically equivalent to an

open subset V ′ ⊂ Ck. The preimage f−1(y) ⊂ X only depends on f−1(V )f |f−1(V )

//V . Henceforth,we replace X by the open subset f−1(V ) and Y by Ck.

Let (U, φ) be a complex chart containing x ∈ f−1(x). We have a holomorphic function f φ−1

on U ′ ⊂ Cn. We use the Implicit Function Theorem 2.8 to construct a smaller open neighbourhoodW ⊂ U of x, belonging to a chart (W,ψ), such that we have a homeomorphism ψ : f−1(y)∩W with0k × Cn−k ∩W ′ for W ′ an open subset of Cn.

Using the chart φ we identify U with U ′ ⊂ Cn. To simplify the notation we will refrain fromdistinguishing U and U ′. By translating the open subset we may assume without loss of generalitythat x = 0 ∈ U and f(0) = 0. We now have an open neighbourhood of 0 ∈ U ⊂ Cn and a

holomorphic function f : U //Ck, such that the Jacobi matrix(∂fi

∂zk(0))i=1,...,k

j=1,...,nis a surjective linear

map. This implies that there exist 0 ≤ i1 < · · · < ik ≤ n, such that(∂fi

∂zij(0))i=1,...,k

j=1,...,kis surjective.

By permuting coordinates we may assume ij = j for every j = 1, . . . , k. By Porism 2.9 we see thatthere exist open neighbhourhoods W , W ′ of 0 in Cn, and a biholomorphic map h : W //W ′, suchthat (f φ)(z1, . . . , zn) = (z1, . . . , zk). Thus, by restriction we have a homeomorphism

h : f−1(0) ∩W ' // (z1, . . . , zn) ∈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.Despite having holomorphic functions Gi : Cn+1 \0 // C, we cannot directly define a holomorphicfunction Gi : (z1 : · · · : zn) 7→ Gi(z

0, . . . , zn), because choosing a different representative (e.g.λ(z0, . . . , zn)) modifies the value of Gi by λdi .

Proof of Proposition 3.11. Let G be a homogeneous polynomial of degree d. By expressing G as alinear combination of degree d monomials (z0)d0 · · · (zn)dn , we see that

n∑i=0

zi∂G

∂zi= d ·G(z0, . . . , zn). (10)

In particular if (z0, · · · , zn) ∈ Cn+1 \0 is a non-trivial zero of G, that is G(z0, . . . , zn) = 0, then∑ni=0 z

i ∂G∂zi = d ·G(z0, . . . , zn) = 0.

25

Recall the charts (Ui, φ) of Pn, and the inverses ψi : Cn //Pn, presented by a map ψi : Cn //Cn+1 \0.

We will apply the Regular Value Theorem 3.13 to (G1, . . . , Gk)ψi, to see that V (G1, . . . , Gk)∩Ui isa submanifold for every i = 0, . . . , n. This will imply that V (G1, . . . , Gk) ⊂ Pn+1 is a submanifold.Without loss of generality we may assume i = 0, because the general case may be reduced to thisone by relabelling the coordinates z0, . . . , zn.

The map ψ0 : Cn // Pn is given by (w1, . . . , wn) 7→ (1, w1 : · · · , wn). Hence we have

(G1, . . . , Gk) ψi : (w1, . . . , wn) 7→ (G1(1, w1, . . . , wn), . . . , Gk(1, w1, . . . , wn)).

The chain rule implies(∂(Gi ψ0)

∂wj

)i=1,...,k

j=0,...,n

=

(∂Gi∂zj

)i=1,...,k

j=0,...,n

·

(∂(ψ0)i

∂wj

)j=0,...,n

`=1,...,n

.

This implies that the matrix on the left hand side is obtained from the Jacobi matrix A(z0, . . . , zn) =(∂Gi∂zj

)i=1,...,k

j=0,...,nby setting the variable z0 = 0 and omitting the 0-th column. Omitting the 0-th column

corresponds to considering a complex linear subspace Cn with the basis (e1, . . . , en) rather than(e0, . . . , en) for Cn+1. It remains to show that the resulting linear map A|Cn is still surjective.

To see this we consider the basis of Cn+1

(

1w1

...wn

, e1, . . . , en).

As we have seen above, we have∑ni=0 z

i ∂G∂zi |z0=1,zi=wi = d ·G(1, w1, . . . , wn) = 0, and therefore the

matrix A vanishes on the first vector of this basis. Since A is surjective, we see that A|Cn has to besurjective 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.

Example 3.14. The canonical map p : Cn+1 \0 // Pn, sending (z0, . . . , zn) to (z0 : . . . : zn) isholomorphic.

Complex projective space Pn is covered by the n + 1 standard charts (Ui, φi). The preimagep−1(Ui) is by definition equal to the open subset (z0, . . . , zn)|zi 6= 0. We choose the family ofcharts (p−1(Ui), id)i=0,...,n for Cn+1 \0, and (Ui, φi)i=0,...,n for Pn. The map φi p id−1 is

given by (z0, . . . , zn) 7→ ( z0

zi , . . . ,zi−1

zi ,zi+1

zi , . . . ,zn

zi ), which is indeed holomorphic for zi 6= 0.

Definition 3.15. Let Xf//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 imagef(U) ⊂ V for a chart (V, ψ) of Y , the form (f∗ω)(U,φ) = (ψ f φ−1)∗ω(V,ψ).

In principle it is possible that X has a chart (U, φ), such that f(U) 6⊂ V for every complexchart (V, ψ) of Y . Take for instance the holomorphic map p : Cn+1 \0 // Pn. The left hand side

26

is already an open subset of Cn+1, that is, we have a tautological chart (U, φ) = (Cn+1 \0, id). Butthe image p(U) = Pn 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 ofthe atlas U . However, since f is continuous, the collection of these charts covers X. As we havechecked in Lemma 3.9, this is sufficient to define a differential form.

In this section we will study an interesting example of a holomorphic map: a blow-up Bl0(Cn)f//Cn.

The complex manifold Bl0(Cn) has the property that f−1(C \0) // C \0 is a biholomorphic map,and f−1(0) is equivalent to a projective space Pn−1. This sudden chance in dimension of the fibreis the reason why this construction is referred to as blow-up.

Definition 3.16. Remember that as a set Pn−1 is the set of 1-dimensional complex subspaces (orlines) ` ⊂ Cn. As a topological space, we define Bl0(Cn) to be the subset of Cn×Pn−1, as definedby the incidence relation z ∈ `:

Bl0(Cn) = (z, `) ∈ Cn×Pn−1 |z ∈ `.

We let f : Bl0(Cn) // Cn be the map induced by the projection (z, `) 7→ z.

Since Bl0(Cn) is defined as a topological subspace of Cn×Pn−1 it is clear that it is Hausdorffand second countable. For the same reason we see that f is a continuous map.

There’s another continuous map, g : Bl0(Cn) //Pn−1, mapping (z, `) to ` ∈ Pn−1. By definition,we have the interesting relation g` = ` ⊂ Cn for every point of ` ∈ Pn−1.

We’ve already introduced standard charts (Ui, φi) for Pn−1, and will use a similar constructionto define charts for Bl0(Cn). Since g is continuous, g−1(Ui) ⊂ Bl0(Cn) is open. Changing back toour usual notation ` = (z0 : · · · : zn−1) we have a continuous map φ′i : g

−1(Ui) // Cn

(u, (z0 : · · · : zn−1)) 7→ (ui,z0

zi, . . . ,

zi−1

zi,zi+1

zi, . . . ,

zn

zi)

with a continuous inverse Cn // g−1(Ui)(w0(w1, · · · , wi, 1, wi+1, . . . , wn−1), (w1 : · · · , wi : 1 : wi+1 : . . . , wn−1)

).

Exercise 3.17. Verify that coordinate changes are holomorphic, and hence conclude that Bl0(Cn)is a complex manifold.

Using these charts we can check that f : Bl0(Cn) // Cn and g : Bl0(Cn) // Pn−1 are holomor-phic maps. We have f (φ′i)

−1(w0, . . . , wn) = w0(w1, · · · , wi, 1, wi+1, . . . , wn−1), which is certainlyholomorphic. Similarly, φi g(φ′i)

−1(w0, . . . , wn) = w1, · · · , wi, 1, wi+1, . . . , wn−1 is a holomorphicmap.

As we have observed above, the fibres of g are naturally 1-dimensional complex vector spaces.Moreover, we have produced biholomorphic maps

p−1(Ui)' //

g##

Ui × C

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 simplifythe 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

Eπ //X, such that

(a) every fibre π−1(x) has the structure of a finite-dimensional complex vector space,

(b) there exists an open covering⋃i∈I Ui = X and biholomorphic maps π−1(Ui)

αi // Ui × Cn,

(c) such that p1 αi = π, where p1 : Ui × Cn // Ui is the canonical projection to the secondcomponent. That is, the following diagram commutes

π−1(Ui) //

##

Ui × Cn

Ui,

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

In heuristic terms we can think of a vector bundle Eπ // 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 notsupposed to behave pathologically, hence we stipulate that X can be covered by open subsets overwhich the vector bundle is trivial, that is, of the shape Ui × Cn. Vector bundles are locally trivial,but often allow to encode interesting global phenomena.

Let Eπ //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 terminologyis 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 referto 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(Uii∈I , (αij)ij∈I2

),

where X =⋃i∈I Ui is an open covering for X, and αij : Ui∩Uj //GLn(C) ⊂ Cn×n 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.

Note that taking (i, j, k) = (i, i, i) we obtain αii = α2ii, that is, since αii ∈ GLn(C), we must

have αii = 1. Similarly, taking (i, j, k) = (i, j, i), we obtain αii = 1 = αijαji, that is, αji = (αij)−1

(taking the inverse matrix).

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If E is a vector bundle of rank n on X, then there exists an open covering⋃i∈I Ui = X, and

fibrewise linear biholomorphic maps αi : π−1(Ui)

' //Ui×Cn. Over the intersections Uij = Ui∩Ujwe therefore obtain a fibrewise linear map αiα−1

j : Uij×Cn //Uij×Cn, defined as the compositionof the top row of the following commutative diagram

Uij × Cnα−1j//

p1&&

π−1(Uij)αi //

π

Uij × Cn

p1xx

Uij .

By commutativity of this diagram, and fibrewise linearity, we see that αi α−1j is of the shape

x 7→ (x, αij(x)), where αij : U // GLn(C) is a holomorphic function taking values in the opensubset GLn(C) ⊂ Cn×n. This process defines a cocycle datum. To see this, it suffices to check thatfor every triple of indices (i, j, k), the relation αik = αij · αjk is satisfied on Uijk = Ui ∩ Uj ∩ Uk.This follows from the fact that x 7→ (x, αij) is by definition equal to αi α−1

j . Hence, we have

αi α−1j αj α

−1k = αi α−1

k 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

Eπ //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 bundlecharts to show it is indeed a vector bundle.

We define E to be the quotient (∐i∈I Ui × Cn)/ ', where (x, z) is equivalent to (y, w) if and

only if ∃(i, j) ∈ I2, such that x ∈ Ui, y ∈ Uj , x = y, and αij(w) = z. The map π sends theequivalence class of (x, z) to x. The universal property of the quotient topology guarantees that itis continuous.

By construction, π−1(Ui) is in bijection with Ui × Cn, by means of the continuous map Ui ×Cn // π−1(Ui) sending (x, z) to the equivalence class [(x, z)]. We call the inverse of this homeo-morphism αi (it’s continuous by the universal property of the quotient topology).

The resulting topological space is second countable, because there exists a countable coveringof X by open subsets: X =

⋃n∈N Vn, such that every Vn is contained in one of the Ui (since X

is second countable). The preimage π−1(Vn) is thus homeomorphic to Vi × Cn, and thus secondcountable 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 everyUi belongs to a chart (Ui, φi) of the complex manifold X. This is achieved by replacing the opensubsets Ui by smaller ones, if necessary. We then have homeomorphisms

π−1(Ui) // U ′i × Cn = φi(Ui)× Cn,

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 thismap 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(Uii∈I , (αij)(i,j)∈I2

)be a cocycle datum for a vector bundle E on a complex

manifold X. Then,(Uii∈I , ((αT

ij)−1)(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 dependson the isomorphism class of E.

Definition 4.6. Let X be a complex manifold.

(a) An isomorphism (or equivalence) between two vector bundle Eπ //X and E′

π′ //X is a pairof mutually inverse holomorphic maps f : E // E′, g : E′ // E, which are fibrewise linear,such that

E //

E′

~~

X

commutes.

(b) The trivial rank n vector bundle on X is X×Cnp1 //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 nvector bundle X × Cn.

We will see many examples of non-trivial vector bundles. But how can one actually prove thata vector bundle is non-trivial, or in general that two vector bundles are not isomorphic? As a firsttool to do so we introduce the vector space of global sections of a vector bundle. It turns out tobe an isomorphism invariant, and also explicitly computable. By computing the dimension of thisvector space we can compare and distinguish between the examples introduced below.

Definition 4.7. A global section of a vector bundle Eπ //X is a holomorphic map s : X // E,

such that π s = idX , that is, the diagram

Xs //

idX

E

π

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 thestructure of a C-vector space. The reason is that s : X // E is a holomorphic map, which assignsto every x ∈ X an element in the vector space π−1(x). Given two sections s1, s2 ∈ Γ(X,E) we cantherefore define s1 + s2 : x 7→ s1(x) + s2(x) ∈ π−1(x). In order for this to make sense however, wehave to verify that the resulting map is holomorphic. Similarly, given a complex number λ ∈ C wewill 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 thefunctions s1 + s2 : X // E defined by fibrewise addition is holomorphic, that is, defines a globalsection of E. Similarly, for every λ ∈ C, the function λ ·s : X //E is holomorphic, and thus definesa global section of E. The triple (Γ(X,E),+, ·) satisfies the axioms of a complex vector space.

Proof. Since Eπ //X is a vector bundle, every point x ∈ X has an open neighbourhood U ⊂ X,

such that π−1(U) ∼= U × Cn is equivalent to the trivial vector bundle. The restriction of a sections ∈ Γ(X,E) is given by a holomorphic function U 7→s|U U×Cn, which is of the shape x 7→ (x, f(x)),where f : U // Cn is a holomorphic function. Therefore, we see that s1 + s2 is presented by theholomorphic function x 7→ (x, f1(x) + f2(x)). The sum f1 + f2 of two holomorphic functions is ofcourse again holomorphic. After all it can be written as a composition of the holomorphic maps

U(f1,f2)

// Cn×Cn, and + · Cn×Cn // Cn.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.The zero element is represented by the zero section 0X : X //E, which maps x //0 ∈ π−1(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 expressglobal sections in terms of the cocycle data.

Lemma 4.9. If(Uii∈I , (αij)(i,j)∈I2

)is a cocycle datum for a vector bundle E on X, then we have

a canonical bijection between Γ(X,E) and the collection of holomorphic functions fj : Uj //Cnj∈I ,satisfying the condition fi|Ui∩Uj = αijfj |Ui∩Uj for all (i, j) ∈ I2.

Proof. By assumption we have an isomorphism of vector bundles αi : π−1(Ui)

' //Ui×Cn for everyi ∈ I. If s : X // E is a global section, we can use the isomorphisms αi to see that s|Ui is of theshape x 7→ (x, fi(x)), where fi : Ui // Cn 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 .

Vice versa, given fj : Uj // Cnj∈I , we can define a section s : X 7→ E by sending x ∈ Ui toα−1i (fi(x)). This does not depend on the choice of i ∈ I, such that x ∈ Ui, due to the relationfi|Ui∩Uj = αijfj |Ui∩Uj . Since the resulting map X // E is holomorphic on each Ui, we see that itdefines 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 a1-dimensional complex vector space. A line bundle is therefore quite literally a bundle of lines overa complex manifold X.

Lemma 4.10. We say that a section s of a line bundle L on X is nowhere vanishing if for allx ∈ 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 isbiholomorphic, we use that every line bundle is locally trivial, and sections are given by holomorphicfunctions (the details are left to the reader).

In Definition 4.2 we have defined cocycle data to describe vector bundles. The special case ofrank 1 is given by pairs

(Uii∈I , (αij)i,j∈I2

), where αij : Ui ∩Uj // C× is a holomorphic function

which is nowhere zero.

Definition 4.11. Let L and L′ be line bundles on X, given by the cocycle data(Uii∈I , (αij)i,j∈I2

)and

(Uii∈I , (α′ij)i,j∈I2

). We define the tensor product L ⊗ L′ to be the line bundle given by the

cocycle datum(Uii∈I , (αijα′ij)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 isindependent of the chosen cocycle datum. Let

(Uii∈I , (αij)i,j∈I2

),(Uii∈I , (α′ij)i,j∈I2

)be cocycle

data for rank n vector bundles. Explain why in general(Uii∈I , (αijα′ij)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 higherrank vector bundles?

After having defined sheaves we will find a more general definition of tensor products, which isnot 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 classesof 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 bundleX × C.

This group is an important invariant of complex manifolds. The next section is devoted to con-structing line bundle Ld on complex projective spaces Pn, such that we get a group homomorphismZ // Pic(X), sending d to the isomorphism class of Ld. For n ≥ 1 we will show that this is reallyan injective map. Later, after we developed sheaf cohomology, we will be able to prove that thisgroup homomorphism is in fact an isomorphism, and therefore that Pic(Pn) ∼= Z for n ≥ 1.

4.3 Line bundles on projective space

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

((Cn+1 \0)× C

)/C×.

Definition 4.14. Consider the topological space (Cn+1 \0) × C with the action of C× given byλ · (z0, . . . , zn;w) = (λz0, . . . , λzn;λdw). We define Ld to be the quotient (Cn+1 \0)/C×, andπ : Ld // Pn the continuous map induced by (z0, . . . , zn;w) 7→ (z0, . . . , zn).

Recall that we have standard charts (Ui, φi) for Pn. We define a homeomorphism

αi : π−1(Ui) // Ui × C

32

by the formula

[(z0, . . . , zn;w)] 7→(

(z0 : · · · : zn),w

(zi)d

).

Using the definition of the quotient topology it is easy to see that this is a well-defined continuousmap. The inverse map α−1

i sends((z0 : . . . : zn), u

)to [(z0, . . . , zn, (zi)du)]. As before, the universal

property of the quotient topology implies that this map is continuous. This shows that αi is indeeda homeomorphism.

It remains to check that the transition functions αij = αi αj : (Ui ∩Uj)×C // (Ui ∩Uj)×Care holomorphic, and linear in the second component C. Using the formulae given above, we see

that αij((z0 : . . . : zn), u

)=

((z0 : . . . : zn),

(zj

zi

)du

). Composing with (φi)

−1 = ψi : Cn ' // Ui

from the right and φi from the left, we obtain the function

(w1, . . . , wn;u) 7→ (w1, . . . , wn;wdju),

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.

Theorem 4.15. The vector space of holomorphic sections of Ld is canonically equivalent to C[z0, . . . , zn]d,that is, the vector space of degree d homogeneous polynomials in n+1 variables. In particular, thereis no non-zero holomorphic section of Ld for d < 0.

We will deduce the theorem from a lemma, which is intuitively clear.

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

(Cn+1 \0)s //((Cn+1 \0)× C

),

which is C×-equivariant.

Proof of Theorem 4.15 using Lemma 4.16. Let’s determine the C×-equivariant sections of((Cn+1 \0)× C

).

By definition,((Cn+1 \0)× C

)is the trivial line bundle on (Cn+1 \0). Its holomorphic sections are

therefore the same as holomorphic functions f on (Cn+1 \0). By Hartog’s Extension Theorem 2.14,a holomorphic function on (Cn+1 \0) extends uniquely to a holomorphic function on Cn+1, sincewe assumed n ≥ 1.

We know that holomorphic functions are analytic, that is, there exists an ε-neighbourhood of0 ∈ Cn+1, where f can be represented by a converging power series

f(z) =∑

d0...,dn≥0

ad0...dn(z0)d0 · · · (zn)dn

,

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

It remains to prove the lemma.

33

Proof of Lemma 4.16. Let s be a holomorphic section of Ldπ // Pn. Using the bundle charts

αi : π−1(Ui)

' //Ui ×C we obtain a holomorphic section αi(s) of the trivial bundle Ui ×C, corre-sponding to a holomorphic function fi. On the overlaps Ui∩Uj the resulting holomorphic functionsare related according to

fi =

(zjzi

)dfj ,

or equivalently (zi)dfi p = (zj)dfj p on p−1(Ui ∩ Uj), where p : Cn+1 \0 // Pn is the map(z0, . . . , zn) 7→ (z0 : . . . : zn).

This implies that there exists a well-defined holomorphic function f : Cn+1 \0 // C, satisfyingf |p−1(Ui) = (zi)dfi p. Multiplying (z0, . . . , zn) with λ therefore has to change the value of f with

λd, that is, f defines a C×-equivariant section of((Cn+1 \0)× C

)// (Cn+1 \0).

Vice versa, any C×-equivariant section corresponds to a holomorphic function f : Cn+1 \0 // C,satisfying f(λz) = λdf(z) for λ ∈ C×. We have a function fi : Ui // C, which sends (z0 : · · · : zn) to

1(zi)d

f(z0, . . . , zn). This is a well-defined continuous function by definition of the quotient topology.

Precomposing it with φ−1i = ψi we obtain fi ψi(w1, . . . , wn) = 1

1df(w1, . . . , 1, . . . , wn), which is by

definition a holomorphic function. For the intersection Ui ∩ Uj of two charts we have the relation

fi =(zjzi

)dfj , and hence gives rise to a section of Ld // Pn.

Corollary 4.17. For d 6= d′ the line bundles Ld and Ld′ on Pn are not equivalent.

Proof. For d ≥ 0 we have determined the vector space of global sections Γ(Pn, Ld) ∼= C[z0, . . . , zn]din Theorem 4.15. For 0 ≤ d < d′ we have a non-canonical embedding of vector spaces C[z0, . . . , zn]d →C[z0, . . . , zn]d′ , which sends a degree d homogeneous polynomial G to (z0)d

′−dG. This map is in-jective, but cannot be surjective, since the degree d′ homogeneous polynomial (z1)d

′doesn’t lie in

its image. This implies that for non-negative numbers d 6= d′ the vector space of global sections areof different dimensions, therefore Ld 6∼= Ld′ .

We have seen that Ld is given by the cocycle (αij) =(zj

zi

)d. This implies that L−d is equivalent

to the dual of Ld (see Definition 4.5). This implies that Ld 6∼= Ld′ for negative integers d 6= d′.For d < 0 < d′ we know that Ld ∼= L′d is impossible, because Ld does not have any global

sections.

After having developed the appropriate tools we will show that every line bundle on Pn isequivalent to one of the line bundles Ld. For P1 even more is true.

Theorem 4.18 (Birkhoff, Grothendieck). Every rank m vector bundle E on P1 can be uniquelywritten as a direct sum of line bundle Ld1 ⊕ · · · ⊕ Ldm .

Birkhoff didn’t formulate his theorem in terms of line bundles on projective space. He proveda factorisation theorem for matrices in Laurent polynomials, known as Birkhoff factorisation.Grothendieck recognised the geometric content of Birkhoff’s result and reproved the result (inhigher generality) using sheaf cohomology.

34

4.4 Line bundles and maps to projective space

Now that we have studied line bundles on complex projective space Pn, we will try a differentapproach. Starting with a line bundle L on a complex manifold X, and sections s0, . . . , sn of L, wewill use this datum to construct a holomorphic map X // Pn.

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, suchthat si(x) 6= 0. In this case, we define a map f : X // Pn 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 anelement of C, but rather si(x) belongs to the 1-dimensional complex vector space π−1(x). But all1-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 ofπ−1(x) with C. However, as we are only interested in (s0(x), . . . , sn(x)) up to rescaling, we obtaina well-defined element of Pn. The assumption that the tuple (s0, . . . , sn) generates L was needed,because (0 : · · · : 0) does not define a point in Pn.

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,in particular we have a holomorphic map g = (g0, . . . , gn) : V // Cn+1. By assumption, the imageof this map is contained in Cn+1 \0, and it is therefore meaningful to form the composition p g,where p : Cn+1 \0 // Pn is the canonical projection. Since the composite of two holomorphic mapsis again holomorphic, we see that p g is holomorphic. However, by definition, p g = f |V . Thecomplex manifold X can be covered by bundles charts (V, α), and this implies that f : X // Pn isholomorphic everywhere.

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

Proof. We can use the local computations above to identify the map fψ0, where ψ0 = φ−10 : Cn ' //U0 ⊂

Pn. 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 existsone section which is never 0. Permuting coordinates we see that this is true for any chart (Ui, φi)of Pn.

The holomorphic map f ψ0 is given by (w1, . . . , wn) 7→ (1 : w1 : · · · : wn), which is equal toψ0. This implies that f |U0 is equal to the identity map. Since U0 is dense in Pn (or by permutingcoordinates), we obtain that f = idPn .

Vice versa, any holomorphic map Xf// Pn 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. Webegin with the general notion of fibre products.

35

Definition 4.22. Let Xf// Y and Z

g// 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 Xrespectively 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

maps Wh //X, W

k // 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

∃!

$$

h

k

&&X ×Y Z //

Z

X // Y.

The intuition behind fibre products or pullbacks is as follows: X ×Y Z //X is a space with “thesame fibres” as Z // Y . That is, for x ∈ X, the fibre of p−1

1 (x) is the same as g−1(f(x)). Thisfollows directly from the set-theoretic definition of the fibre product (but also from the universalproperty). Since fibre products leave the fibres of a map un-affected, they are the perfect device topullback vector bundles! If the fibres of Z // Y have a vector space structure, the same is true forthe 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 aholomorphic map, we define the pullback f∗E to be the fibre product X ×Y E together with thecontinuous map πf∗E : X ×Y E // X. This construction can be refined to define a holomorphicvector bundle on X.

Proof. The proof can be divided into several steps, using only the formal definition of fibre products.

(1) The pullback of the trivial vector bundle: X ×Y (Y × Cn) ∼= X × Cn. This follows directlyfrom the definition of the fibre product: X ×Y (Y × Cn) = (x, y, z) ∈ X × Y × Cn |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' // Z ′ is a homeomorphism fitting into a commutative diagram

Z' //

Z ′

~~

Y,

36

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

(3) If (Uii∈I , (αi)i∈I) is a cocycle datum for Eπ // 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 datumfor f∗E. Since

⋃i∈I Ui = Y we see that

⋃i∈I f

−1(Ui) = X. The definition of fibre products

readily implies that π−1f∗E(f−1(Ui)) = f−1(Ui) ×Ui π−1(E). By (2), the homeomorphism

αi : f−1(Ui)

' //Ui×Cn induces therefore a homeomorphism f∗αi : f−1(Ui)

' //f−1(Ui)×Cn.

It remains to check that the maps f∗αij = f∗αi f∗αj are holomorphic maps f−1(Ui)∩ f−1(Ui) =f−1(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 functionshi : Ui // Cn, such that αij · (hj |Ui∩Uj ) = hi|Ui∩Uj . The compositions hi f : f−1(Ui) // Cn defineholomorphic functions, satisfying the relations (αij f) · (hj f |f−1(Ui∩Uj)) = hi f |f−1(Ui∩Uj).Hence we obtain a holomorphic section f∗s ∈ Γ(X, f∗E).

Using the same arguments as in the proof above, one sees that the continuous mapXf∗s

//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 thechoice of a cocycle datum for E.

Definition 4.24. Let L1, L2 be line bundles on a complex manifold X, and (si0, . . . , sin) ∈ Γ(X,Li)

be a generating set of sections. We say that the tuples (L1, s10, . . . , s

1n) and (L2, s

20, . . . , s

2n) are

equivalent if there exists an isomorphism of line bundles L1∼= L2, such that the induced isomorphism

Γ(X,L1) ∼= Γ(X,L2) maps (s10, . . . , s

1n) to (s2

0, . . . , s2n).

Theorem 4.25. Let X be a complex manifolds. The set of holomorphic maps Xf// Pn 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.

Proof. Let’s denote the set of holomorphic maps X // Pn by Hom(X,Pn), and the set of isomor-phism classes of tuples (L, s0, . . . , sn) by Ln+1(X).

We have already constructed a map A : Ln+1(X) // Hom(X,Pn), by sending (L, s0, . . . , sn) tothe map f : x 7→ (s0(x) : · · · : sn(x)).

There’s also a mapB : Hom(X,Pn) //Ln+1(X), which sendsXf//Pn to the tuple (f∗L1, f

∗(z0), . . . , f∗(zn)).To decipher this notation, recall that Γ(X,L1) is equivalent to the vector space of degree 1 homo-geneous polynomials. That is, every (zi) ∈ C[z0, . . . , zn]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 =idHom(X,Pn). Let f : X // Pn be a holomorphic map. We need to check that f is equal to theholomorphic map g assigned to the tuple (f∗L1, s0, . . . , sn). By definition, g : x 7→ (f∗(z0)(x) : · · · :f∗(zn)(x)).

We have seen above that f∗(zi) : X //f∗L1 is equal to the map x 7→ (f(x), (zi f)). In Lemma4.21 we showed that the holomorphic map Pn // Pn associated to the tuple (L1, z

0, . . . , zn) is theidentity map. This implies that this map is equal to f .

It remains to show that B A = idLn+1(X). Let (L, s0, . . . , sn) ∈ Ln+1(X), and Xf// Pn

the associated holomorphic map. We have to prove that f∗L1∼= L, and with respect to this

identification we have si = f∗(zi).

37

Consider the standard open covering⋃ni=0 Ui = Pn. We can define Ui as the locus where the sec-

tion zi 6= 0 of L1. Therefore, the preimage f−1(Ui) is equal to the open subset where si 6= 0, respec-

tively f∗(zi) 6= 0. Using Lemma 4.10, we see that we have trivialisations L|f−1(Ui)' // f−1(Ui)×

C×C and f∗L1|f−1(Ui)' //f−1(Ui)×C, induced by the nowhere vanishing sections si respectively

f∗(zi). By composition we obtain an isomorphism βi : L|f−1(Ui)∼= f∗L1|f−1(Ui).

It suffices to check that on Ui ∩ Uj the two isomorphisms βi and βj agree. Replacing thesection si by the section sj introduces the factor

sjsi

. Similarly, changing f∗(zi) by f∗(zj) the factorzifzjf = si

sj. And we see that both changes cancel each other out.

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

5 Sheaves

Few, light, and worthless,-yet their trifling weightThrough 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 constructionswe 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 tupleof) holomorphic functions on an open subset of a complex vector space. However, globally, theyexhibit novel features, and for example the section of a line bundle should not be confused witha holomorphic function. We have seen in Theorem 4.15 that for d ≥ 0, the vector space of globalsections of Ld on P1 is at least 3-dimensional. That’s quite different from the 1-dimensional vectorspace of holomorphic functions on P1 (all of which are constant).

For a complex manifold X, equipped with a line bundle L, we temporarily denote for an opensubset 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, consideringthe tautological inclusion U ⊂ U , the map f 7→ f |U is the identity map. Similarly, for a triple ofinclusion 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. Usingtechnical 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 prop-erties 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, butmight as well be called the jigsaw puzzle principle: matching local fragments can be assembled intoa global picture.

Given an open covering Uii∈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 wehave 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 topologicalspace. As a predecessor one has to study presheaves, which aim to extract the most basic propertyof 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,

- a homomorphism rVU : 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 commu-tative diagram

F(W )rWV //

rWU $$

F(V )

rVU

F(U),

that is, we have rWU = rWV rVU .

We will often denote the effect of rVU by using classical notation for restriction of functions. E.g.,for s ∈ F(V ) we write rVU (s) = s|U .

In order to put this definition into context, let us observe that functions on a topological spaceX form a presheaf. If V ⊂ X is an open subset, and f : V // C a continuous function, then forevery 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 conditionsthan 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 abeliangroup 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 .

(b) We denote by Zpre the presheaf of constant, integer-valued functions. That is, to an opensubset U ⊂ X we assign the abelian group Z, whose elements we interpret as constant functionson U .

39

(c) We denote by Z the presheaf of locally constant integer-valued functions. To an open subset

U ⊂ X we assign the abelian group of continuous functions Uf// Z. Since Z is a discrete

topological space, we see that every x ∈ X has an open neighbourhood U = f−1(f(x)), suchthat 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 O×X , which assigns to U ⊂ X the abelian group of holomorphicfunctions f : U // C, which are nowhere zero.

(e) We denote by ΩmX the presheaf of complex m-forms, that is, to an open subset U ⊂ X weassign 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 subsetU ⊂ X to the vector space of sections s defined over U :

E

π

U

//

s

>>

X.

Presheaves on a fixed topological space X form a category. In order to see this we have to definefirst 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

presheaves Ff// G is given by a homomorphism of abelian groups F(U)

fU // G(U) for every opensubset U ⊂ X, such that for every inclusion U ⊂ V ⊂ X of open subsets, we have a commutativediagram

F(V )fV //

rVU

G(V )

rVU

F(U)fU // 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

nowhere vanishing holomorphic function exp(f) ∈ O×X(U). This defines a morphism OXexp//O×X .

Definition 5.4. A morphism Ff// G of presheaves on a topological space X is called an isomor-

phism, 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 describeisomorphisms also as bijective morphisms of presheaves.

Definition 5.5. We denote by Ff// 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.

It is clear that if f is bijective, then for every open U ⊂ X the map F(U)fU // G(U) is bijective.

In particular, there exists an inverse map f−1U , 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 havethat rVU fV = fU rVU . Applying f−1

V from the right, and f−1U from the left, we obtain the relation

f−1U rVU = rVU f

−1V . That is, f−1 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 aglobal section.

Definition 5.6. Let F be a presheaf on a topological space X, such that the following condition issatisfied: 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), suchthat 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 beforeU =

⋃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 tounicity 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 opensubset, and U =

⋃i∈I Ui an open covering of U , and fi : Ui // C are holomorphic functions,

such that fi|Ui∩Uj = fj |Ui∩Uj , then there exists a unique holomorphic function Uf// C, such

that f |Ui = fi. Indeed, we can define f by the following rule: f(x) = fi(x), if x ∈ Ui. It onlyremains 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 yieldsa well-defined map of sets f : U // C. Since holomorphicity is a local property, and f |Ui isholomorphic for every i ∈ I, we see that f is locally holomorphic, hence holomorphic.

(b) The presheaf of constant integer-valued functions Zpre is not a sheaf in general. To see thisfor 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 definef0 : U0

// Z to be the constant function with value 0, and f1 : U1// Z to be the constant

function with value 1. If Zpre 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 thatthis can be remedied by considering the presheaf of locally constant integer-valued functionsZ, which is always a sheaf. Indeed, a function U // Z is locally constant, if and only if it iscontinuous, 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 O×x is a sheaf. If fi : Ui // C are nowherezero holomorphic functions, such that fi|Uij = fj |Uij , then we’ve seen in (a) that there existsa unique holomorphic function f : U // C, agreeing with the locally defined functions fi oneach Ui. Since the functions fi is nowhere zero, we obtain f(x) 6= 0 for every x, which belongsto a Ui. But U =

⋃i∈I Ui, hence f is also nowhere 0.

(e) The presheaf ΩmX of complex m-forms is a sheaf. We have verified the sheaf condition inLemma 3.9.

(f) The presheaf of complex vector fields VX on X is a sheaf. This has been shown in Lemma3.9.

(g) Let E be a vector bundle on X. The presheaf of sections E, which maps an open subsetU ⊂ X to the vector space of sections s defined over U , is a sheaf. To see this, we recall thata section is given by a holomorphic map U // E, fitting into a commutative diagram.

E

π

U

//

s

>>

X.

If si : Ui //E is a collection of sections, such that si|Uij = sj |Uij , then we can define s : U //Eby s(x) = si(x), if x ∈ Ui. As in (a) we see that s is a well-defined holomorphic function

Us // 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-calledlocally free sheaves (of finite rank). In order to justify the nomenclature we recall the notion of anR-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 mapm 7→ λ ·m defines a homomorphism of abelian groups, and the resulting map R // End(M) is aring homomorphism. That is, we have λ · (m+ n) = λ ·m+ λ · n, 1 ·m = m for every m ∈M , and(λ+ µ) ·m = λm+ 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 finitelygenerated, if there exists a surjection of R-modules Rm 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 thereexists a non-zero ideal I ⊂ R. The quotient R/I is a finitely generated R-module, since we have asurjection 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 compactdefinitions (in a non-topological sense), we use that there is a natural notion of products of twopresheaves 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 tobe 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 sheafcondition 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 //Rof sheaves, such that for every open subset U ⊂ X the abelian group R(U) is a ring, withrespect 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, togetherwith a multiplication map · : R×M // M of sheaves, such that for every U ⊂ X, the mapR(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 ofrings R is given by a sheaf, such that for every open U ⊂ X, the abelian group R(U) is additionallyendowed with the structure of a ring, and all the restriction maps rVU : R(V ) // R(U), for everyinclusion 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 ), andm ∈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 requiresus to restrict sheaves on X to open subsets U ⊂ X. This makes sense, because an open subsetU ′ ⊂ U of an open subset U ⊂ X, also defines an open subset U ′ ⊂ X of X. A sheaf F on X givestherefore 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-modulesM is said to be locally free of rank n, if there exists an open covering X =

⋃i∈I Ui, such that

M|Ui ∼= Rn |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 correspondencebetween complex vector bundles E of rank n on X, and locally free sheaves of OX-modules M ofrank n.

43

The proof is broken down into several steps. At first we have to explain how one can pass froma 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-moduleof rank n.

Proof. If U ⊂ X is an open subset, and s : U //E a section of E over U , then for every holomorphicfunction f ∈ OX(U) we have a section f ·s : U //E. For every x ∈ U , (f ·s)(x) = f(x)s(x), and it’snot difficult to check that this defines a holomorphic section (see for instance the proof of Lemma4.8 for similar arguments). Hence E is naturally endowed with the structure of an OX -module.

By definition, there exists an open covering X =⋃i∈I Ui, such that E|Ui ∼= Ui ×Cn as a vector

bundle. This implies that E|Ui is equivalent to OnX , because a section of a trivial vector bundleU × Cn // U , corresponds to an n-tuple of holomorphic functions over U .

Lemma 5.14. The set of isomorphisms of sheaves of OX-modules, OnX // OnX is equivalent tothe set of holomorphic maps X // GLn(C).

Proof. We begin by considering the abstract situation of automorphisms of free R-modules, where

R is a ring with unit. Every R-linear map Rnf//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 amatrix AU ∈ GLn(OX(U)). Since AU represents fU , where f : OnX // OnX is a map of sheaves,we have for every inclusion U ⊂ V of open subsets the relation AV |U = AU . The matrix AXcorresponds to a holomorphic function X // GLn(C), and as we have seen, it defines the map ofsheaves f : OnX // OnX .

Proof of Theorem 5.12. We have already seen that the sheaf of sections E of a complex vectorbundle of rank n is a locally free OX -module of rank n. As a next step, we have to show that everyrank n locally free sheaf of OX -modules M is equivalent to E, for some rank n complex vectorbundle E.

GivenM, we choose an open coveringX =⋃i∈I Ui, and isomorphisms of sheaves φi : M|Ui

' //(OUi)n.

Over the overlaps Uij = Ui∩Uj we obtain an isomorphism of free sheaves φij : OnUijφiφ−1

j// OnUij .

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 ontriple intersections Uijk. We have therefore constructed a cocycle datum

(Uii∈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

fi : V ∩Ui // Cn, satisfying fi = φijfj for every (i, j) ∈ I2 (see Lemma 4.9). However, such a tuple,also defines an element of M(V ): indeed, we have fi ∈ OnUi , and hence si = φ−1

i (fi) ∈ M(Ui).

On the overlaps Uij , we have si|Uij = φ−1i (fi)|Uij = φ−1

i (φijfj)|Uij = φ−1i (φi φ−1

j fj)|Uij = sj |Uij .By the sheaf condition, there exists a unique section s ∈ M(Ui), such that s|Ui = si. This showsM∼= E.

To conclude the proof, it remains to show that for two vector bundle E, F on X. the existenceof an isomorphism of OX -modules E ∼= F implies the existence of an isomorphism of vector bundlesE ∼= F .

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:

OnUiβi // OnUi .

By Lemma 5.14, we may understand βi to be a holomorphic function Ui // GLn(C). The definitionof βi implies the relation φijβj = βiφij on every overlap Uij . Using a construction similar to Lemma4.9 we obtain a well-defined isomorphism of vector bundles E // F . The details are left to thereader as an exercise.

Corollary 5.15. Let X be a complex manifold of (complex) dimension n. There exists a rank nvector bundle TX // X, called the tangent bundle, such that the sheaf of sections TX is equiv-alent to the sheaf of complex vector fields VX . Similarly, there exists a rank

(nm

)vector bundle∧m

T ∗X //X, such that∧m

T ∗X //X is equivalent to the sheaf of complex m-forms ΩmX .

Proof. We only have to show that the sheaves VX and ΩmX are locally free. It is clear that they areOX -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 toopen subsets U of Cn. We may therefore without loss of generality assume that X = U is anopen subset of Cn (due to the local nature of the statement we are interested in). We have seen inDefinition 2.15 that a complex vector field on U is given by an n-tuple of holomorphic functions

V =

n∑i=1

Vi ∂

∂zi.

In (7) we expressed an m-form by means of an(nm

)-tuple of holomorphic functions

ω =∑

0≤i0<···<im≤n

ωi0···imdzi0 ∧ · · · ∧ dzim .

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

5.4 Exact sequences of sheaves

In this paragraph we denote by Ff// 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 localproperty, 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) isinjective, then the map F(X) // G(X) is injective.

Proof. Recall that a map of abelian groups Ag//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), suchthat fX(s) = 0, we already have s = 0.

By assumption we have an open covering Uxx∈X of X, such that fUx : F(Ux) // G(Ux) isinjective for every x ∈ X. The assumption fX(s) = 0 implies 0 = fX(s)|Us = fUx(s|Ux). Hence, wehave s|Ux = 0 for every x ∈ X. The sheaf property implies now that s = 0, since the two sectionss and 0 both solve the glueing problem for 0 ∈ F(Uxy) for (x, y) ∈ X2.

45

Definition 5.17. We say that f is locally surjective, if for every open subset U ⊂ X, and everys ∈ 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. Thedefinition 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 nounique element in the preimage, and hence, the local solutions ti, don’t glue - that is, they won’tnecessarily agree on the overlaps: ti|Uij 6= tj |Uij in general. As we have seen in Lemma 5.16, thesame 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 // O×X , which sends

a holomorphic function Uf// C to the invertible holomorphic function exp(f).

If g : X // C× is an invertible holomorphic function, we may ask if there exists a holomorphicf : 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 Uii∈I of X, such that we have a holomorphicfunction fi : Ui // C, satisfying exp(fi) = g|Ui for every i ∈ I. To construct such an opencovering can be constructed as follows. Observe that every z ∈ C× has a small ε-neighbhourhood

Uε(z), where a complex logarithm log : Uε(z) // C can be defined (non-canonically). Hence, over

Ui = g−1(Uεi(zi)) it makes sense to define fi = log g|Ui , where we assume that zi is a family of

points in X, such that⋃i∈I Uεi(Zi) = C×. We have exp(fi) = exp(log g|Ui) = gUi , and hence

have shown that exp: OX // O×X is a locally surjective morphism of sheaves.

However, it is clear that OX(X)exp// O×X(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

exp(f) = g, we would have f ′(z) = g′

g (z) = 1z (this is a consequence of the chain rule). Therefore,

the integral ∫∂ Dε(0)

dz

z

would be zero. But we know that this is not true, since the value of this integral can be computedto be 2πi.

A pessimist would probably abandon the theory of sheaves at this point. Next week, we willcarefully study the obstruction for a locally surjective map of sheaves to be surjective. The answerturns out to be given by sheaf cohomology. As a preparation we need to define exact sequences ofsheaves.

Definition 5.19. Let A, B, C be abelian groups, and Af// B

g// 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, wehave g(b) = 0 if and only if ∃a ∈ A, such that f(a) = b. This motivates the following definition ofexactness for sheaves.

46

Definition 5.20. A sequence of sheaves Ff// G

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 anopen 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 surjectivemaps of sheaves. Indeed, we see that a map of sheaves G //H is locally surjective, if and only ifG //H // 0 is exact.

Lemma 5.21 (Exponential sequence). Let X be a complex manifold, the sequence of sheaves

0 // Z 2πi· // OXexp// O×X // 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 holomorphicfunctions are continuous, we see that n is locally constant. Thus, it defines a section n ∈ Z(U).

Exactness at O×X : We have to show that the map exp is locally surjective. We cover C× by an

open covering⋃i∈I Vi = C×, such that each Vi supports a branch of complex logarithm log : Vi //C.

Consider a holomorphic function g : U // C×, where U ⊂ X is an open subset. On Ui = g−1(Vi)

we have well-defined holomorphic functions fi = log 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 theabelian group F(X) of global sections.

We have seen that an exact sequence of sheaves does not induce an exact sequence of globalsections in general. However, the following is true.

Lemma 5.23. Let 0 // Ff// G

g//H // 0 be a short exact sequence of sheaves. The sequence

of abelian groups0 // 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 seenthis in Lemma 5.16.

Exactness at Γ(X,G) can be shown as follows: let s ∈ G(X), such that g(s) = 0. By assumptionthere exists an open covering X =

⋃i∈I Ui, and sections ti ∈ F(Ui), such that f(ti) = s|Ui . We

claim that the glueing condition is satisfied, that is ti|Uij = tj |Uij for all (i, j) ∈ I2. This is true,since f(ti|Uij) = s|Uij = f(tj |Uij ), but f is injective, which implies what we want. Therefore, thereexists 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 assumeat the beginning that a theory with such properties exists. At the end of the section we will verifyby a simple inductive procedure that this is indeed the case.

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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 exactsequences 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 // O×X // 0 shows that the mapΓ(X,G) // Γ(X,H) is not surjective in general. Our best hope is therefore that we can continuethe above sequence to a long exact sequence

0 // Γ(X,F) // Γ(X,G)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 sequenceabove 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 sectionof 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 exactsequence

· · · //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 onlyone theory of sheaf cohomology up to a unique isomorphism. At first we take another look atour favourite example, and try to guess what the space of obstructions H1(X,Z) amounts to ingeometric 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 anarbitrary complex manifold X. We consider the exponential sequence

0 // Z 2πi· // OXexp// O×X // 0,

48

and ask when a nowhere vanishing holomorphic function f : X // C× has a global logarithm logf .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 = logz 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. Howcan we decide if a general f ∈ Γ(X,O×X) admits a global logarithm? We will utilise a notion fromtopology: covering spaces. As the attentive reader will confirm, their definition is reminiscent ofthe definition of vector bundles.

Definition 6.2. Let G be a group, endowed with the discrete topology. A continuous map π : Y //Xis 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 homeomor-

phism φ : π−1(U)' // U ×G, fitting into a commutative diagram

π−1(U)

π##

φ// U ×G

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 .

An example of a Z-Galois coverning is the exponential map Cexp// C×. This is the reason,

why we introduced this notion.There is a correspondence betweenG-Galois coverings ofX, 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: givena G-Galois covering as above, and a closed path γ : [0, 1] // X, with γ(0) = γ(1) = x. We canconstruct a continuous path γ : [0, 1] //Y , such that γ(0) = y ∈ π−1(Y ), and π γ = γ. There is aunique such path γ, since every x′ ∈ X has an open neighbourhood U , over which π−1(U) ∼= U ×Gis a trivial fibration (hence lifts exist, and are unique if one point on the curve is specified as a sortof boundary condition).

Take for instance γ(t) = e2πit as a path in X = C×. By definition a lift is given by γ(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 that γ(1) ∈ π−1(x). The definition ofa G-Galois covering implies now that there exists a unique g ∈ G, such that γ(1) = g · y = γ(0).Moreover, this g depends only on the homotopy class of the path γ, leaving the end points fixed. Thisallows 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 inthis 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

π−1(X)

π##

φ// X ×G

p1

X.

A G-Galois covering is trivial, if and only if the corresponding homomorphism π1(X,x) // Gis 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 toZ-Galois coverings.

Proposition 6.4. Let X be a complex manifold, and f : X // C× be a nowhere vanishing holo-morphic function. The fibre product

Yf = X ×C× C = (x, z) ∈ X × C |f(x) = exp(z) //X

is a Z-Galois covering, with respect to the action k·(x, z) = (x, 2πik+z). There exists a holomorphicfunction g : X // C, such that f = exp(g), if and only if the Z-Galois covering constructed aboveis trivial.

Proof. Let’s show first that Yfπ // is a Z-Galois covering. There exists an open covering

⋃i∈I Ui =

C×, such that we have holomorphic functions logi : Ui // C, such that exp(logi) = idUi . Therefore,⋃i∈I f

−1(Ui) = X is an open covering of X, and the map

(x, z) 7→ (x,z − logi(z)

2πi)

defines a Z-equivariant homeomorphism between π−1(f−1(Ui)) and f−1(Ui)×Z. This proves that

Yfπ //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 trivialZ-Galois covering X × Z // X has at least one continuous section s : X // X × Z, for instancex 7→ (x, 0). If Yf //X is the trivial covering, then there exists a continuous map s : X //Yf , suchthat π s = idX . By definition of Y = X ×C× C, we see that g = p2 s : X // C is a continuousmap, 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

(x, z) 7→ (x,z − f(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), forsome holomorphic function g : X // C. The set of isomorphism classes of Z-Galois covering isequal to the set of group homomorphisms ρ : π1(X,x) // Z. Since two Z-valued functions can bemultiplied, we see that this set actually carries a natural structure of an abelian group, which wewill denote by H1(X,Z). That is, we’ve verified the following theorem.

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

0 // Γ(X,Z) // Γ(X,OX) // Γ(X,O×X)δ // H1(X,Z).

We see that the map δ fulfils the goal of determining the obstruction to the existence of aholomorphic 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 complexesin other lectures (differential or algebraic topology). In those courses, cohomology refers in firstinstance to the cohomology of a chain complex of abelian groups. Recall that a chain complex is asequence of homomorphisms of abelian groups

A• = [A0 d0 //A1 d1 //A2 d2 //A3 d3 // · · · ],

such that the relation di+1 di = 0 is satisfied for all i ≥ 0. The abelian group Hn(A•) is definedto 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, ifand only if Hn(A•) = 0. Thus, we think of Hn(A•) as measuring the failure of A• being exact atAn.

We have also seen that sheaf cohomology measures the failure of exactness of the global sectionsfunctor Γ(X,−). It is therefore not unreasonable to ask, if the cohomology groups Hi(X,F) of asheaf F on X can be computed as cohomology groups of a chain complex. It turns out that theanswer is yes. There will be more than one such complex computing sheaf cohomology, and everyinstance of this phenomenon translates into an interesting comparison theorem. For example:

Theorem 6.6 (de Rham). Let X be a smooth manifold, we denote by Hising(X,R) the singular

cohomology of X with real coefficients, and by HidR(X,R) the so-called de Rham cohomology of X.

There is a canonical isomorphism

Hising(X,R) ∼= Hi

dR(X,R).

We will recall the definitions of singular and de Rham cohomology in more detail. For sin-gular 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 computedusing the real vector spaces of smooth m-forms ΩmR (X), and the fact that exterior differentiationd : ΩmR (X) //Ωm+1

R (X) satisfies the relation dd = 0. Hence, differential forms give rise to a chaincomplex in vector spaces, and the resulting cohomology groups, are denoted by Hi

dR(X,R).

Example 6.7. Assume that X is an open subset of R3. A smooth (real) m-form on X is thenexpressible as a linear combination

∑0≤i1<···<im≤m fi1...imdx

i1 ∧ · · · ∧ dxim . Only the cases m =0, 1, 2, 3 are non-trivial. A 0-form or 3-form corresponds to a single smooth function f , and 1-formsand 2-forms correspond to a triple of smooth functions (f1, f2, f3). Analysing the formulae for

51

exterior differentiation, and battling with signs, one sees that d0 : Ω0R(X) // Ω1(X) is basically

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? DeRham’s theorem tells us that the quotient vector space

V smooth vector field on X|∇ × V = 0∇ · f |f smooth function on X

= H1sing(X,R)

is only depending on the topology of X. Hence, if there are curl-free vector fields, only the topologyof 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 mul-tivariable calculus class to solve a supposedly very difficult problem about vector fields. Solveit or argue why Russell’s exercise might have been too difficult for an introductory course.

(b) Remind yourself that on a star-shaped open subset of R3, every curl-free smooth vector fieldcan be written as a gradient of a smooth function. Deduce that the following is a short exact

sequence of sheaves 0 //R //C∞X∇· // V∞X

∇×//0 on X. Here, C∞X is the sheaf, assigning

to an open subset U , the vector space of smooth R-valued functions, and V∞X denotes the sheafof 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 canbe written as a gradient of a smooth function.

The comparison between de Rham and singular cohomology is achieved by comparing both sidesto the cohomology of the sheaf R on X, which assigns to an open subset U ⊂ X the real vectorspace of locally constant maps U // R (we could also view R as a discrete topological space, andlook at continuous maps U // Rdisc).

The so-called de Rham complex (Ω•R(X), d) can be viewed as the chain complex of global sectionsΓ(X,−) of an exact sequence in sheaves. Let us denote by ΩmX the sheaf, which assigns to an opensubset U ⊂ X the sheaf of smooth (real) m-forms on U . In differential geometry or topology, thefollowing lemma should have been proven.

Lemma 6.9. The sequence of sheaves 0 // R // Ω0X

d // Ω1X

d // Ω2X

// · · · is exact.

We will deduce from this lemma, and the Universal Comparison Theorem below that HidR(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 // F // G0 d0 // G1 d1 // G2 d2 // // · · ·

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

Γ(X,G0)d0 // Γ(X,G1)

d1 // Γ(X,G2)d2 // · · · .

52

Proof. We will prove this statement by induction on i. In the course of the proof, we denote byKi = ker di, the subsheaf of Gi, assigning to an open subset U ⊂ X the subgroup s ∈ Gi(U)|di(s) =0.

Our proof is anchored to the case i = 0. Exactness of 0 // F // G0 d0 // G1 yields a shortexact 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) isexact. We have a commutative diagram

Γ(X,G0)

d0 %%

// Γ(X,K1) _

Γ(X,G1),

where injectivity of the vertical arrow follows from Lemma 5.23 applied to the short exact sequence0 //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 assertionmust also hold for degree i + 1 cohomology groups. By exactness of the sequence G• we have ashort exact sequence

0 //Ki // Gi di // 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.

Exactness implies that H1(X,Ki) = Γ(X,Ki+1)image(Γ(X,Gi) // Γ(X,Ki+1))

= Hi+1(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 thesheaves of smooth m-forms don’t have any higher cohomology; and we found a way of expressingsingular cohomology via a resolution of sheaves. We defer the proof of the vanishing of highercohomology to a later section, and devote the rest of this paragraph to an explanation how singularcohomology arises from a resolution of sheaves.

Definition 6.11. (a) For every positive integer n ≥ 0, we denote by ∆n = (λ0, . . . , λn) ∈[0, 1]n+1|λ0 + · · · + λn = 1 ⊂ Rn+1, and view it as a topological space with respect to thesubset topology.

(b) Let X be a topological space. A singular n-simplex is a continuous map σ : ∆n // X. Theset of singular n-simplices on X will be denoted by Sn(X).

53

(c) For 0 ≤ i ≤ n we denote by ∂i : ∆n−1 //∆n, the unique affine map (linear + translation), whichpreserves 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.

(d) For an abelian group A, we let Cn(X,A) be the set of maps of sets Sn(X)c //A. We will denote

c(σ) by∫σc, and define an A-linear map by

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

by the formula∫σd(c) =

∫∂σc =

∑n+1i=0 (−1)i

∫σ∂i c.

It is an important exercise to check that dn+1 dn = 0 for every n ≥ 0. Therefore, we haveconstructed 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 groupHi

sing(X,A) with coefficients in A.

Theorem 6.12. If X is a locally contractible topological space, then Hising(X,A) is equivalent to

Hi(X,A), where A is the sheaf of locally constant A-valued functions on X.

Sketch. The main idea is to consider the presheaf CnX , assigning to an open subset U ⊂ X theabelian group

U 7→ Cn(U,A).

We have a map of presheaves A // CnX , which assigns to a locally constant function f : U // A,the element c ∈ C0(U,A), given by

∫σc = 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 complex0 // A // C0(U,A) // C1(U,A) // · · · is exact. If there was any justice in the world, we couldpretend that CnX was a sheaf, and hence assert that

0 //A // C0X

// C1X

// · · ·

was a resolution by sheaves, and that Hi(X,CjX) = 0 for i ≥ 1. Unfortunately, CjX is not a sheaf,and one needs to invoke the sheafification process discussed in the exercises. Let us denote by CnX thesheafification. What saves the day is that Kj = ker(dj : CjX

//Cj+1X ) = ker(dj : CjX

//Cj+1X ), 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 bycoker f the quotient D/image f .

54

Exercise 6.14. (Snake Lemma) Given a commutative diagram

A //

f

B //

g

C //

h

0

0 // D // E // F

with exact rows, show that there is an exact sequence

ker f // ker g // kerhδ−→ 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 columnsare 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. Theresulting diagram contains the one of the Snake Lemma as a subdiagram. The morphism C // Fhas 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 thatH // I is surjective too.

55

The tic-tac-toe lemma also holds for sheaves on a topological space X. This is true, becauseexactness of a sequence of sheaves F // G //H can be tested by verifying that for every x ∈ Xthe sequence of stalks Fx // 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 abeliangroups.

Definition 6.16. An abelian group I is called injective, if the following is true: let A → B bean injective morphism of abelian groups, and f : A // I a homomorphism, then there exists ahomomorphism B // I, such that the diagram

A //

f

B

∃

I

commutes.

It’s important to emphasise that this is not a universal property. An extension B // I as theone above, is in general not unique. The next lemma shows that injectivity is quite easy to verifyfor 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 m′ ∈ M , such that n · m′ = m. An abelian group is injective if and only if it isdivisible.

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 nZ //M

nZ

//

n 7→m

Z

∃?~~

M

which sends the generator n to m. Since nZ ⊂ Z is an injective homomorphism of abelian groups,and I is injective, there exists a homomorphism g : Z // I, such that g|nZ = f . Therefore,m′ = f(1) satisfies the property n ·m′ = 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 willprove 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 (A′, g), where A ⊂ A′ ⊂ B is a subgroup, and g : A′ //M is ahomomorphism, such that g|A′ = f . There is an ordering on this set. We say (A′, g) ≤ (A′′, h) ifA′ ⊂ A′′, and h|A′ = g. This set is partially ordered, because if I = (Ai, fi) is totally orderedsubset, 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 thatB = A. Assume by contradiction that there exists x ∈ B \ A. We claim that f can be extended tothe subgroup A+Zx //M . We only need to define f(x) in a suitable way, in order to make senseof it. If Zx∩ (A\0) = ∅, then we can define f(x) = 0 or any other element of M . If the intersectionis not empty, then there exists a minimal integer n ≥ 1, such that nx ∈ A. Since M is divisible,there exists an m′ ∈M , such that nm′ = f(nx). We define f(x) = m′. This is a contradiction, andwe conclude A = B.

56

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

F

//

f

G

∃

Icommutes.

Again we emphasise that injectivity is a property, and not a universal property! We are nowready 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 exactsequence

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

And for every commutative diagram with exact rows

0 // F1//

G1//

H1//

0

0 // F2// G2

// H2// 0

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

· · · // Hi(X,F1) //

Hi(X,G1) //

Hi(X,H1) //

Hi+1(X,F1) //

· · ·

· · · // Hi(X,F2) // Hi(X,G2) // Hi(X,H2) // Hi+1(X,F2) // · · ·

(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 beembedded 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 /Fx. The sections of I /F(U) aregiven by elements (sx)x∈U ∈

∏x∈U Ix /Fx, 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 obtainfor 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 verifythe 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)).

For a morphism Ff// G we choose a commutative diagram

F

//

f

I

∃

G

// J

and define H1(f) to be the map

H0(X, I) //

H0(X, I /F)

// H1(X,F) //

H1(f)

0

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 embeddingsF // 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

//

f

Ig

h

G

// J

we want to prove that the induced maps α, β : H1(X,F) // H1(X,G) agree. We will show thatα− β = 0.

H0(X, I) //

g

h

H0(X, I /F)

g

h

// H1(X,F) //

α

β

0

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 factorisationas indicated by the dotted arrow

I //

g−h

I /F

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

//

f

I

g

h

F

// I,

we see that H1(idF ) is the identity map of H1(X,F). Applying the observation to

F

//

f

I

F

//

J

F

// I,

(and also switching the roles of I and J ) we see that the abelian group H1(X,F) is independentof the chosen embedding F → I.

It remains to see that for composable morphisms of sheaves Ff// G

g//H we have H1(gf) =

H1(g) H1(f). Consider the commutative diagram

F

//

f

I

∃

G

//

J

∃

H

// K .

We know that the induced maps H1(f), H1(g), H1(g) are independent of the chosen extensionspresented by the dashed arrows. Therefore H1(g f) = H1(g)H1(f), because we can simply formthe 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 longexact sequence) that every short exact sequence of sheaves 0 // I // G //H // 0 gives rise to ashort exact sequence of global sections. This can be verified directly, and will be used in the proofof existence of sheaf cohomology.

59

Lemma 6.21. Let 0 // I // G //H // 0 be a short exact sequence of sheaves, such that I isinjective. 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

id

// G

∃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 sequenceabove thus exact.

Lemma 6.22. For every short exact sequence of sheaves 0 // F // G //H // 0 we have a longexact 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 sheavesF → I, G → J , and H → K, such that we have the following commutative diagram with exactrows 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 withexact 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 longexact 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 cohomologyas in Definition 6.19.

Proof. We will prove by induction on the degree i, that there exists a family of functorsHi(X,−) : Sh(X) //AbGrp,verifying the axioms (A1− 3) up to the given degree. For i ≤ 1 we know that such a family existsby 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

trivial embedding idI : I → I, and hence obtain H1(X, I) = coker(H1(X, I)id //H0(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

sheaf (according to E7, ex. 5 this is always possible). For a morphism of sheaves Ff// G we

choose a commutative diagramF

//

f

I

G

// J .

(11)

as in our construction of the functor H1(X,−) in Lemma 6.20. We will use the same strategy asin 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 twomorphisms 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

g−h

// I /F

g−h||

J // J /G .

(12)

as indicated by the dotted arrow. Applying the functor Hi(X,−), we obtain a commutative diagram

H(X, I)

g−h

// Hi(I /F)

g−hww

Hi(X,J ) // Hi(X,J /G).

(13)

By the induction hypothesis we have Hi(X,J ) = 0. Therefore, we see that g − h induces the zeromorphism. 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 theexercises (E7, ex. 5) that there exists a commutative diagram with exact rows and columns, suchthat I, J , and K are injective sheaves.

0

0

0

0 // F //

G //

H //

0

0 // I //

J //

K //

0

0 // I/F //

J /G //

K/H //

0

0 0 0

(14)

For i ≥ 2 we may simply apply the existence of the long exact sequence for Hi−1(X,−) andHi(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, butrather to coker(H0(X, I) // H0(X, I /F)). We obtain the corresponding part of the long exactsequence as follows. Consider the bottom two rows of (14). Lemma 6.20 implies that we have acommutative diagram

0 // H0(X, I) //

α

H0(X,J ) //

β

H0(X,K) //

γ

0 //

0 //

0

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 thesequence

cokerα // cokerβ // 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.

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7 GAGA

In this section we will finally state and (partially) prove Serre’s GAGA theorems. GAGA, theabbreviation of Geometrie algebrique et geometrie analytique, usually refers to the following threetheorems (which we state in vague terms, and in a slightly less general form for the purpose of thisintroduction) 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 isalgebraic.

(b) If E and F are algebraic vector bundles on X, then every fibrewise linear holomorphic mor-phism of vector bundles E // F is algebraic.

(c) Every holomorphic vector bundle E on X arises from an algebraic vector bundle in an essen-tially unique way.

We will prove (a) and (b) completely in this class. The third result (c) requires deeper analyticresults, but we will nonetheless try to gain an understanding why it is true. Combining the threestatements above, one sees that the category of algebraic and holomorphic vector bundles on Xare equivalent. This has the curious interpretation that somehow that compactness of a complexprojective 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. Thatis the content of the first paragraph below. We will show that for so-called vector bundles E, thereexist complex vector spaces Hi

alg(X,E) of algebraic degree i cohomology classes. For i = 0 thisvector space agrees with the space of algebraic (or regular) sections of E. Serre proves the first partof the theorem above by a downward induction on the degree i. We will show that for i > dimCXthe cohomology spaces Hi(X,E) and Hi

alg(X,E) will be both 0. The inductive comparison isanchored to this case. The long exact sequence of sheaf cohomology, and an ingenious applicationof the Five Lemma allow us to deduce that Hi(X,E) ∼= Hi

alg(X,E) the two types of cohomologygroups agree canonically.

The argument contains a second inductive strand: over the complex dimension. In fact, it turnsout to be easier to prove a more general comparison theorem, where vector bundles are replaced byso-called coherent sheaves. This way one can reduce everything to the case X = Pn. Recall thatvector bundles on X correspond to locally free sheaves of OX -modules on X. A coherent sheaf isa locally finitely generated sheaf of OX -modules. If E is a vector bundle on X, and i : X → Pndenotes the embedding into projective space, then the sheaf i∗E is not locally free, unless X = Pnor X = ∅. However, i∗E is always coherent. While this argument is formally very pleasing, we’lltry 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.We denote by V (G1, . . . , Gk) ⊂ Pn the subset

(z0 : . . . : zn) ∈ Pn |Gi(z0, . . . , zn) = 0 ∀i = 1, . . . , k,

63

and refer to subsets of Pn of this type as complex projective varieties. In Proposition 3.11 we provedthat if (

∂Gi∂zj

(w)

)i=,1,...,kj=0,...,m

is a surjective linear map, (15)

then V (G1, . . . , Gk) is a complex submanifold.In this section we aim to turn the subset V (G1, . . . , Gk) ⊂ Pn into a mathematical object

(an algebraic variety) itself. We will see that there exists a topology (the Zariski topology) onV (G1, . . . , Gk), which is coarser than the standard topology on subsets of Pn, but better adaptedto algebraic geometry. The Zariski topology carries a sheaf of rings Oreg

X , the so-called regularsections 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 Zariskiopen, if there exists a projective variety W = W (H1, . . . ,H`), such that U = V \W .

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

Lemma 7.3. The set of Zariski open subsets defines a topology on the set V = V (G1, . . . , Gk) ⊂ Pn.

Proof. The empty subset ∅ ⊂ V is Zariski open, because ∅ = V \ V . Similarly, we have that V isZariski open, because V = V \∅ = V \V (1), where 1 denotes the constant homogeneous polynomialof degree 0.

If U1, U2 are Zariski open subsets, that is, Ui = V \ Wi, then U1 ∩ U2 = V \ (W1 ∪ W2).Writing Wi = V (H1i, . . . ,H`i) (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`ii) is a Zariski open subset. Then⋃i∈I Ui = V \

⋂i∈IWi. If I is a finite set, then it is clear that⋂

i∈IWi = V (H1i, . . . ,H`ii)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`ii)i∈I , and choose finitely many homogeneous generators H1, . . . ,H`. We then have⋂i∈IWi = V (H1, . . . ,H`).

If X = V (G1, . . . , Gk) is a projective variety, we denote the topological space induced by theZariski 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

be a Zariski open subset. A function Uf// C is called regular, if there exists an open covering

U =⋃i∈I Ui, such that for every i ∈ I, there exist homogeneous polynomials G, H ∈ C[z0, . . . , zn]d

of degree d (H 6= 0), such that ∀(z0 : . . . : zn) ∈ Ui we have f(z0 : . . . : zn) = GH (z0, . . . , zn). We

denote the sheaf of regular functions on XZar by OregX .

64

A regular function is precisely a holomorphic function, which can be locally expressed in termsof 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 subvarietyof zeroes of the denominator H.

We have seen that every Zariski open subset U ⊂ XZar is also open with respect to the standard(or manifold topology if X is smooth) of X ⊂ Pn. On U we can compare the rings OX(U) andOregXZar

(U). By definition, OregXZar

(U) ⊂ OX(U) is a subring. However, we are not allowed to callOregXZar

a subsheaf of OX , since the sheaves live on different topological spaces.

Definition 7.5. An algebraic vector bundle of rank n on a complex projective manifold X is alocally free OXreg

Zar-module E of rank n.

This definition is inspired by Theorem 5.12, which showed that the theory of holomorphic vectorbundles is equivalent to locally free sheaves of OX -modules of rank n. There is another definition,which justifies calling them vector bundles.

Exercise 7.6. Show that every locally free OXregZar

-module Ereg of rank n, is given by the sheaf of

regular sections of a complex vector bundle Eπ // X, which is algebraic in the following sense.

There exists a cocycle datum (Uii∈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 itcorresponds to a tuple of regular functions fi : Ui // Cn, 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 importantexample of algebraic line bundles are the twisting line bundles on Pn.

Definition 7.7. Recall that the standard atlas of projective space consists only of Zariski open

subsets Ui = (z0 : . . . : zn) ∈ Pn |zi 6= 0. Moreover, the cocycle datum, (Uii=0,...,n, (zdjzdi

)i,j=1,...,n)

of the line bundles Ld, is the cocycle datum of an algebraic line bundle on Pn. We denote its sheafof regular sections by Oreg

Pn (d), and its sheaf of holomorphic sections by OPn(d).

We have already computed the complex vector space of global sections Γ(Pn,OPn(d)), andhave shown that it is canonically equivalent to the complex vector space of homogeneous degree dpolynomials. Given the algebraic nature of homogeneous polynomials, one suspects immediatelythat that is a first instance of the GAGA principle. To verify this suspicion we will compute thevector space of global sections Γ(Pn,Oreg

Pn (d)), which are regular.Note that a holomorphic section s ∈ Γ(Pn,OPd(d)) corresponds to a tuple of holomorphic

functions fi : Ui // C, for i = 0, . . . , n, such that fi =zdjzdifj for all i, j. A holomorphic function is

regular, if the functions fi are regular.

Lemma 7.8. The canonical map Γ(PnZar,OregPn (d)) // Γ(Pn,OPn(d)) is an equivalence.

Proof. It is clear that this map is injective. It remains therefore to show that it is surjective. Wehave computed the space of holomorphic global sections in Theorem 4.15. It is therefore sufficientto 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 holomorphicsection s : Pn //Ld as follows. Recall that Ld was defined to be the quotient space of (Cn+1 \0)×Cby the action of C×, where

λ · (z0, . . . , zn, w) = (λz0, . . . , λzn, λdw).

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The map sG : Pn // 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×

C, which sends [(z0, . . . , zn, w)] to [((z0 : . . . : zn), wzdi

)]. We obtain αi(sG)(z0 : . . . : zn) = G(z0,...,zn)

zdi.

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 globalsections. We begin a definition using cocycles, and subsequently we will reformulate it in terms oftensor products (which has the advantage of being formally better behaved).

Definition 7.9. Let E be an algebraic vector bundle on Pn, corresponding to a cocycle datum(Uii=0,...,n, (φi,j), where Uii=0,...,n is the standard open covering of Pn (it is a priori not clearthat such a presentation always exist, yet it is true). We define E(d) to be the algebraic vector

bundle corresponding to the twisted cocycle datum (Uii=0,...,n,zdjzdiφij)).

As we remarked above, it is a priori not clear that this definition can be applied to everyalgebraic vector bundle. Moreover, it is not clear that it is independent of the chosen presentationvia coycles. We will remedy this drawback by making a small detour through the world of sheavesand tensor products.

Lemma 7.10. Let R be a (commutative) ring (with unit), and M,N be R-modules. There existsan R-module M ⊗RN , and a bilinear map −⊗− : M ×N //M ⊗RN , such that for every bilinearmap of R-modules f : M ×N // P , there exists a unique linear map M ⊗R N // P , such that

M ×Nf

$$

−⊗−

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 formalsymbols m ⊗ n for every pair (m,n) ∈ M × N . The R-module S is defined to be the submoduleof F , generated by elements (m + m′) ⊗ n − m ⊗ n − m′ ⊗ n, m ⊗ (n + n′) − m ⊗ n − m ⊗ n′,(λm)⊗ n−m⊗ (λn), and λ(m⊗ n)− (λm)⊗ n for every possible choice of elements m,m′ ∈ M ,n, n′ ∈ N , and λ ∈ R. The map M ×N //F/S = M ⊗R S, sending (m,n) to m⊗n is by definitionbilinear. Moreover, if f : M ×N // P is a bilinear map, then we obtain a linear map F // P , bysending m⊗ n 7→ f(m,n). Since f |S = 0, there is an induced linear map g : F/S = M ⊗RN //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 commutativityof the diagram enforces the relation g(m ⊗ n) = f(m,n), and every element in M ⊗R N can bewritten 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, itsuffices to show that we have a universal bilinear map −⊗− : Rm × Rn // Rmn. We will denote

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the standard basis of Rm by e1, . . . , en, and of Rn by f1, . . . , fn. A basis of Rmn is given by formalsymbols ei ⊗ fj , and we define a bilinear map Rm ×Rn, 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 fis uniquely determined by the values f(ei, fj). We have a linear map g : Rmn // P , by sendingei ⊗ fj to f(ei, fj). By construction it verifies commutativity as above and is the only linear mapRmn // P with this property.

As a corollary we obtain a tensor product operation for sheaves of modules, preserving locallyfree ones of finite rank. For locally free sheaves of OX -modules of finite rank this can be understoodto 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 withunit). For two sheaves of R-modules M and N , there exists a unique (up to a unique isomor-phism) R-module M⊗RN , and a bilinear sheaf morphism − ⊗ − : M×N // M⊗RN , suchthat for every bilinear map of sheaves f : M×N // P, there exists a unique linear sheaf mor-phism g : M⊗RN // P, such that the diagram

M×Nf

$$

−⊗−

M⊗RN∃! // P

commutes. If M and N are locally free of finite rank, then so is M⊗RN .

Proof. We define a presheaf (M⊗RN )pre, 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 aunique linear maps gU : M(U)⊗R(U)N (U) //P(U), such that gU (m⊗n) = fU (m,n) for sectionsm ∈ M(U), n ∈ N (U). For an inclusion of open subsets U ⊂ V , we therefore see that thetwo maps rVU gV and gU must agree, because they both satisfy the commutativity condition(rVU gV )(m⊗ n) = (rVU fV )(m,n) = fU (m,n), and gU (m⊗ n) = fU (m,n). Therefore we obtaina commutative diagram of presheaves

M×Nf

%%

−⊗−

(M⊗RN )pre∃! // P.

Defining M⊗RN to be the sheafification of (M⊗RN )pre, and using the universal property ofsheafification, we see that we there is a unique morphism of sheavesM⊗RN //P, such that thefollowing diagram of presheaves

M×Nf

$$

−⊗−

−⊗−

ww

(M⊗RN )pre ∃! //M⊗RN∃! // P.

commutes.

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It remains to show that M⊗RN is locally free of finite rank, when M and N are. LetU ⊂ X be an open subset, such that M|U ∼= (R|U )m and N |U ∼= (R|U )n. We know that Xhas an open covering consisting of open subsets U with this property. By corollary 7.11 we havethat (M⊗RN )pre|U ∼= (R|U )mn. Since the right hand side is already a sheaf, it agrees with itssheafification M⊗RN . This shows that M⊗RN is locally free of finite rank.

? The reader might be curious if sheafification was really necessary. To see that this is indeedthe case, consider the example OP1(−1) ⊗ OP1(1). The cocycle pictures implies that this sheaf isequivalent to OP1 . Hence its space of global sections is 1-dimensional. However, OP1(−1) is knownnot to have any non-zero global sections. Therefore, the tensor product OP1(−1)(P1) ⊗OP1 (P1)

OP1(1) = 0.

Definition 7.13. Let E be an algebraic vector bundle on Pn, that is a locally free sheaf of OregPn -

modules of finite rank. For an integer d ∈ Z we define E(d) to be the tensor product E⊗OregPnOreg

Pn (d).

The reason to introduce this twisting operation is the following important result.

Theorem 7.14. Let E be an algebraic vector bundle on Pn. There exists an integer d0 ∈ Z, suchthat for d ≥ d0 the algebraic vector bundle E(d) is generated by finitely global (regular) sectionss1, . . . , sm ∈ Γreg(Pn, E(d)). That is, the map of sheaves Om // E(d), sending a tuple of regularfunctions (f1, . . . , fm) to f1s1 + · · ·+ fmsm is a surjective morphism of sheaves.

Proof. Since E is a locally free sheaf of OregPn -modules, there exist Zariski open subsets

⋃i∈I Vi = Pn,

such that E|Vi ∼= (OregVi

)m, for some integer m (the rank of E). The topological space Pn is compact,

hence we may assume that I is a finite set. For every i ∈ I we choose sections si1, . . . , sim which

form a basis for E|Ui .We will show that for each sij there exists a homogeneous polynomial G(z0, . . . , zn) of degree

d >> 0, such that the section sij⊗sG of E(d) extends from Ui to all of Pn. The intuitive idea is that

locally sij has a presentation as a rational function, and the zeroes of the denominator obstruct us

from extending sij beyond Ui. Since we are only dealing with finitely many sections, we can choose

d big enough, to find homogeneous polynomials Gij of degree d, such that sij ⊗ sGij always extends

to a global section of E(d).So let s ∈ E(U), where U is a Zariski open subset of Pn. We will show that there exists a

homogeneous polynomial G, such that s⊗ sG extends to a global section of E(d). Let x ∈ Pn \U ,and choose Zariski open neighbourhood V ⊂ Pn of x. By making V smaller, we can assume thatE|V ∼= (Oreg

Pn )m is a free OregPn -module. The section s|V can therefore be represented as a tuple of

regular functions (f1, . . . , fm) = ( F1

G1, . . . , FmGm ). This shows that s⊗ sG1···Gm extends uniquely from

a section over U ∩ V to a section over V .Using compactness of Pn, one covers Pn by finitely many open subsets where an extension across

U exists, after tensoring by a homogeneous polynomial. Taking the maximal degree, we can patchthese local extensions together.

At this point we can take a glimpse at the proof of the first assertion of GAGA. We will focuson the case of algebraic vector bundles on Pn. Let’s take the following facts for granted.

(a) For every algebraic vector bundle E on Pn we have a natural map Hi(PnZar, E) //Hi(Pn, E),mapping from the cohomology of the sheaf of regular sections to the cohomology of the sheafof holomorphic sections. (GAGA asserts that this map is an isomorphism.)

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(b) For the line bundles OregPn the map from (a) is an isomorphism (GAGA holds for these line

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 fordegrees i > m.

If E is an arbitrary algebraic vector bundle on Pn, then we have seen that there exists a d ∈ Z,such that we have a surjective map of sheaves (Oreg

Pn )m E(d). Tensoring with OregPn (−d) we obtain

a surjective morphism of sheaves (OregPn (−d))m E. Let’s denote the kernel by K. By fact (c) we

have a short exact sequence of algebraic vector bundles

0 //K // (OregPn (−d))m // E // 0.

Let’s take a look at the long exact sequences for cohomology of regular sections and holomorphicsections:

Hi(PnZar,K) //

f

Hi(PnZar, (OregPn (−d))m) //

g

Hi(PnZar, E) //

h

Hi+1(PnZar, E) //

k

Hi+1(PnZar,K)

`

Hi(Pn,K) // Hi(Pn, (OPn(−d))m) // Hi(Pn, E) // Hi+1(Pn, E) // Hi+1(Pn,K)

For very high degrees both types of cohomology groups vanish. We will therefore prove bydescending induction on i that Hi(PnZar, E) // Hi(Pn, E) is an isomorphism. Let’s assume thatthis has been proven for i+ 1 already. That is, the maps k and ` are isomorphisms. Since g is byassumption 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. Anotherapplication 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 theentire statement of GAGA to a statement about sheaves on Pn rather than a complex submanifoldX ⊂ Pn given by a smooth projective variety X. Let i : X → Pn 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 finiterank. We can push the sheaf E forward to Pn, that is, consider (i∗ E)(U) = E(i−1(U)) = E(U ∩X).

We have verified on E6 that this construction defines a sheaf. We know that i∗ E cannot belocally free in general, since for V ⊂ Pn \X we have (i∗ E)(V ) = E(∅) = 0. Therefore, i∗ E |Pn \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 coherentsheaves exist in both algebraic and analytic geometry. We will content ourself with establishing thebasic 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 theproof of GAGA to the case of Pn.

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Proposition 7.15. Let X ⊂ Y be a closed subspace of a topological space Y , we denote the inclusionmap X → Y by i. For every sheaf F on Y we have a canonical equivalence

Hi(X,F) ∼= Hi(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 shortexact 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 yis exact for every y ∈ Y . So let’s compute the stalks of (i∗ F)y. We claim that for y /∈ X thestalk is 0, and for x ∈ X, the stalk equals Fx. This then immediately verifies the claim that thepushforward (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, ands ∈ 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) equals0. 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 =U ′∩X, where U ′ is an open neighbourhood of x in Y . The resulting abelian group (i∗ F)(U ′) = F(U)is independent of the chosen U ′ with this property. This implies directly that (i∗ F)x = Fx.

It’s important to emphasise that for a general continuous map f : X // Y , the pushforwardfunctor f∗ is not exact. For Y = pt a singleton, Sh(Y ) = AbGrp, and f∗ agrees with the globalsections functor Γ(X,−) with respect to this identification. We know from the example of theexponential 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 leftadjoint f−1 : Sh(Y ) // Sh(X). That is, for every sheaf F ∈ Sh(Y ) and G ∈ Sh(X) we have

HomSh(X)(f−1 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’tmake any sense a priori. We will therefore interpret this formula by taking the stalk of F aroundf(U). That is we consider equivalence classes of pairs (V, s), where V ⊂ Y is an open subsetcontaining f(U), and s ∈ F(V ). The equivalence relation is defined as for stalks: if there exists anopen subsets W ⊂ V ∩ V ′, such that f(U) ⊂ W , such that t|W = s|W , then (V, s) and (V ′, t) areequivalent.

The resulting presheaf f−1pre 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.

Let’s check that every map Fg// f∗ G gives rise to a map f−1 F h // G. Given g, we have for

every V ⊂ Y a homomorphism F(V )gV // G(f−1(V )). For an open subset V ⊂ Y , containing f(U),

and an element [(V, s)] ∈ f−1 F(U) we can therefore look at the image gV (s) ∈ G(f−1(V )), andrestrict it further down to U . This defines a morphism of presheaves f−1

pre F // G. The universalproperty of sheafification yields a unique morphism f−1 F // G.

Vice versa, given h, we can construct a morphism of sheaves Fg// f∗ G. This is left as an

exercise to the reader.

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Proof of Proposition 7.15. We will show that for every short exact sequence 0 //F // I // I /F //0with 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 Lemma7.16. Let’s show that i∗ I is injective.

A diagramM

//

N

i∗ I

corresponds to a diagram

i−1M

//

i−1N

∃zzI

by Lemma 7.17. Since I is injective, there exists a dashed arrow as in the diagram above, renderingthe triangle commutative. Applying Lemma 7.17 again, we obtain

M

//

N

∃

i∗ I,

and therefore see that i∗ I is injective.

Coherent sheaves on PnZar are certain sheaves of OregPn -modules. The following examples are the

most important ones for us: locally free sheaves of finite rank E (algebraic vector bundles), andalso for every inclusion of a complex projective manifold i : X → Pn the sheaves i∗ E are coherent,where E is an algebraic vector bundle on X. There are however more examples of coherent sheaves,that is, not every coherent sheaf on Pn is of the form i∗ E . At first we define a related notion, theone of quasi-coherent sheaves.

Although Cn is not a complex projective manifold, it makes sense to speak of the Zariskitopology on Cn, and the sheaf of regular functions on Cn. The reason is that we can realise Cnas a Zariski-open subset of Pn. In fact, any of the standard charts Ui ⊂ Pn does the trick. TheZariski topology on Pn induces therefore a topology on the subsets Ui = Cn. Restriction of sheavesyields a sheaf of regular functions Oreg. The resulting topological space, with a sheaf of rings Oreg

is often denoted by An, to distinguish it from Cn with the standard topology and the sheaf ofholomorphic functions. It is called n-dimensional affine space. Its main premise is that An can bedefined without reference to Zariski open subsets of Pn, and regular functions on Pn.

Lemma 7.18. A subset U ⊂ Cn is Zariski-open, if and only if there exist polynomials h1, . . . , hk ∈C[x1, . . . , xn], such that U = (z1, . . . , zn) ∈ Cn |∃i ∈ 1, . . . , k : hi(z1, . . . , zn) 6= 0. A functionf : U // C is regular, if and only if there exists a Zariski-open covering U =

⋃mi=1 Ui, such that

f |Ui = gihi

for polynomials gi, hi ∈ C[x1, . . . , xm], hi 6= 0.

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Proof. We leave the details of the proof as an exercise to the reader. The main idea of the con-struction is the following. We can pass to rational functions g

h on Cn to rational functions on Pn

by homogenisation. Similarly, given a rational function GH on Pn with G and H two homogeneous

polynomials in n+ 1 variables of degree d. The restriction GH |Ui depends solely on the dehomogeni-

sations of G and H, obtained by specialising the variable zi to 1. Using this correspondence, andthe fact that both the Zariski topology, and the sheaf of regular functions, are defined in terms ofrational 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, andf ∈ R an element. The localisation Mf of M at f is the R-module given by formal symbols m

fn ,

where m ∈ M , and n ≥ 0. We stipulate that mfn = m′

fn′if and only if fd(fn

′m − fnm′) = 0 for

d >> 0.

Note that Rf is a ring itself, which could be defined to be R[x]/(xf − 1). It’s the ring obtainedfrom R by formally adjoining an inverse f−1 to f . The R-module Mf is in fact naturally anRf -module.

We can now define quasi-coherent sheaves, the immediate predecessor of coherent sheaves. Theredefinition is inspired by a property we observed for algebraic vector bundles E on Pn. If H ∈C[z0, . . . , zn]d is a non-zero homogeneous polynomial of degree d, then we have a Zariski opensubset V ⊂ Pn given by the locus where H is non-zero. Let U ⊂ Pn be another Zariski open subset.A regular section s ∈ E(U ∩V ) can be written as t

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 sacross the complement U \ V ⊂ U .

Definition 7.20. (a) A sheaf F of Oreg-modules on An is called quasi-coherent, if for everyZariski open subset Uf = z ∈ Cn |f(z) 6= 0 ⊂ An we have F(Uf ) = F(An)f = F(An)[f−1].

(b) A sheaf F of OregPn -modules on PnZar is called quasi-coherent, if for every i ∈ 0, . . . , n the

sheaf F |Ui is quasi-coherent.

Next we classify quasi-coherent sheaves on An. The proposition below tells us that the functor,sending a quasi-coherent sheaf F on An to the C[x1, . . . , xn]-module F(An) is an equivalence ofcategories.

Proposition 7.21. We have an equivalence of categories Γ(An,−) : QCoh(An) //Mod(C[x1, . . . , xn]),sending exact sequences of sheaves to exact sequences of modules. In other words, for every quasi-coherent sheaf F on An there exists a unique C[x1, . . . , xn]-module M , such that F(Uf ) = Mf , everymodule M arises in this way, and Oreg-linear morphisms of quasi-coherent sheaves, correspond tomorphisms of C[x1, . . . , xn]-modules.

Proof. The proof is formally very similar to the one of Proposition 7.26, which we will discussbelow. It will be broken down into several easy steps on an exercise sheet, and left to the reader asan exercise.

The fact that the global sections functor for quasi-coherent sheaves on An is exact, can bestrengthened2:

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

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Corollary 7.22. Let F be a quasi-coherent sheaf on An. Then for every i > 0 the cohomologygroups Hi(An,F) vanish.

Lemma 7.23. A quasi-coherent sheaf F on An 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 Zariskiopen covering An =

⋃kj=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.

Proof. For every j = 1, . . . , k we choose elements sj1, . . . , sjmj generating F(Ufj ). Since F is quasi-

coherent, we have F(Ufj ) = F(An)fj , and therefore there exists d >> 0, such that we can write

sji =tjifdj

with tji ∈ F(M). Consider the free R-module F of finite rank m generated by symbols

eij for j = 1, . . . , k and i = 1, . . . ,mj . Let F //M be the map sending eji to tji . By virtue ofProposition 7.21 we obtain a well-defined map (Oreg

An )m // F . Its cokernel has to be 0, since it iszero with respect to the open covering Ufj . This implies that the map is surjective, and thereforeM is finitely generated, and F is coherent.

Definition 7.24. A sheaf F of OregPn -modules on Pn is called coherent, if for every i = 0, . . . , n the

restriction F |Ui is a coherent sheaf on affine space.

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

Lemma 7.25. (a) The direct sum F1⊕F2 of two coherent sheaves on PnZar is again coherent.

(b) For a morphism of coherent sheaves F1// F2, kernel and cokernel are coherent.

(c) For every complex projective submanifold i : X → Pn, and an algebraic vector bundle E onX, i∗ E is a coherent sheaf on PnZar. In particular, ever algebraic vector bundle on PnZar iscoherent.

Proof. Assertion (a) is clear, its proof boils down to the fact that the direct sum of two finitelygenerated 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 thefinite generation of M2. Indeed, if Rn M2 is a surjective map, then the composition Rn M2 coker f is surjective and proves finite generation of coker f . The assertion that ker f isfinitely generated is not true for arbitrary commutative rings R (with unit). However, it is truefor so-called Noetherian ones, that is rings for which every ideal is finitely generated. Noetherianrings 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 polynomialrings happen to be Noetherian, we deduce that ker f is finitely generated too.

In order to prove (c) it suffices to check that i∗OregX is coherent on PnZar. Indeed, if E is locally

free of rank m, then there would exists a Zariski open covering Pn =⋃kj=1 Vj of Pn, such that

E |Vj∩X is free of rank m, that is isomorphic to (OregVj∩X)m (we can produce a finite cover, using

compactness of Pn). Therefore we see that (i∗ E)|Vi ∼= (i∗OVj∩X)m. Since coherence is a localproperty by Lemma 7.23, and we assume that i∗OVj∩X is coherent, we are done. It remains toshow that i∗Oreg

X is indeed coherent.

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If X = V (G1, . . . , Gk), then for every i = 0, . . . , n we have that X ∩ Uj is the zero set ofthe system of polynomial equations gi(w1, . . . , wn) = Gi(w1, . . . , wi, 1, wi+1, . . . , wn). The so-calledNullstellensatz in commutative algebra then implies that the ring of regular functions on X ∩Uj isequal to C[w1, . . . , wn]/(g1, . . . , gn), which is a module generated by one element.

Proposition 7.26. For every coherent sheaf F on PnZar, there exists a d ∈ Z, such that F(d) =F ⊗Oreg

PnOreg

Pn (d) is globally generated, that is, there exists a surjection of OregPn -modules (Oreg

Pn )m F(d).

Proof. For i = 0, . . . , n the restriction F |Ui corresponds to a finitely generated C[w1, . . . , wn] =C[(

zjzj

)j 6=i]-module Mi. Since F is quasi-coherent, we can express an element in M = F(Ui) over

Uj ∩Ui for j 6= i as a fraction s =tj

wdji(j)

=zdjj tj

zdji

, where tj ∈ F(Uj) and wi(j) = wi = zizj

if i > j, and

wj+1 if j < i.

On Uij` we have s|Uij` =zdkj tj

zdki

andzd`` t`

zd`i

that we want to compare. We can assume dk = d` = d′

by choosing the maximum. The definition of localisation implies that there exists an m ≥ 0, such

that zm+d′

i tj =zm+d′`

zm+d′j

zm+d′

i t`. Replacing tj by zm+di`i tj , and t` by zm+dik

i t`, we can therefore

assume that ti|Uij` and ti`|Uij` assume as sections of F(d), where d = m + d′. The sheaf property

of F allows one to patch the collection (tj)j 6=i, together with zdi s to a global section t ∈ F(d)(X),such that s = t

zdi.

We choose sections si1, . . . , simi (respectively elements of Mi), which generate F |Ui . For ev-

ery element of this list we choose an extension sik to a global section of F(d). The collection

(sik)j=1,...,mii=0,...,n generates F(d) as a sheaf, because for every Ui we have that F(d)|Ui is generated by

the elements (ski )k=1,...,mi , because ski |Ui = ski generates F |Ui by assumption. Here we use thenatural trivialisation Oreg

Pn (d)|Ui ∼= OregUi

.

7.3 A short exact sequence for twisting sheaves

In a non-canonical manner, (n − 1)-dimensional complex projective space Pn−1 is a complex sub-manifold of Pn. Indeed, we can realise Pn−1 as the complex projective variety, corresponding tothe linear homogeneous equation z0 6= 0. Following Serre we refer to this particular copy of Pn−1

inside Pn with E = V (z0) ⊂ Pn. We write i : E → Pn for the inclusion morphism. The followingshort exact sequence will enable us to prove GAGA for the sheaves OPn(d) by induction on n.

Lemma 7.27. For every d ∈ Z we have a short exact sequence of sheaves

0 // OPn(d− 1) // OPn(d) // i∗OE(d) // 0,

respectively0 // Oreg

Pn (d− 1) // OregPn (d) // i∗Oreg

E (d) // 0.

Proof. We give the proof for sheaves of holomorphic sections (the first short exact sequence), sincethe proof for regular sections is the same. At first we prove the assertion for d = 0. We begin byconstructing a locally surjective morphism of sheaves p : OPn i∗OE . We let p be the restrictionmorphism, which sends a holomorphic function f : U // C, where U ⊂ Pn is an open subset, tof |E .

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In order to show that p is locally surjective, consider one the charts (Ui, φi). We will show thatthe maps OPn(Ui) // i∗OE(Ui) are surjective. For i = 0, the right hand side is equal to 0, becauseE = Pn \U0, and thus i∗OE(U0) = OE(E ∩ U0) = OE(∅) = 0.

For i 6= 0, E ∩ Ui corresponds to the closed subset of Cn given by w1 = 0. Therefore,i∗OE(Ui) = f(w2, . . . , wn) holomorphic function). But it is clear that we can extend f to aholomorphic function in (w1, . . . , wn) by defining g(w1, . . . , wn) = f(w2, . . . , wn).

What’s the kernel of the morphism of sheaves p? It is given by the map OPn(−1)z0· // OPn ,

which multiplies a local section s ∈ OPn(U) with the restriction of z0 ∈ OPn(Pn). To see that thismakes sense, recall our definition of the line bundles Ld, or use the identityOPn(−1)⊗OPn(1) ∼= OPn .

As before, we can check that this is the kernel of p, after restricting to the charts Ui. Over U0, z0

corresponds to the constant holomorphic function 1, and we see that OPn(−1)|U0

z0· // OPn |U0is

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 functionf on Ui, satisfying f |E = 0, can be written as f = w1g, where g is a holomorphic function. Itsuffices to show that f

w1is locally holomorphic. In order to see this we apply the Weierstrass division

theorem, to locally obtain f = hw1 + r, where r is a polynomial in w1 of degree 0, that is, it’sa 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 OPn(d), andusing the rule OPn(d1) ⊗OPn OPn(d2) ∼= OPn(d1 + d2). The only remaining step is to check that

(i∗OE)(d) ∼= i∗(OE(d)). This follows by recalling that OPn(d) is given by the cocycle (zdjzdi

)ij , which

restricts to the cocycle datum for OE(d) on E (we’re just omitting the variable z0).

These short exact sequences are relevant, because we’ve seen that Hi(E,F) ∼= Hi(Pn, i∗ F).The corresponding long exact sequence in cohomology will enable us to prove comparison resultsfor the cohomology of the sheaves OPn(d) by induction on n. Before getting to this, we needto introduce the comparison maps Hi(PnZar,O

regPn (d)) //Hi(PnZar,OPn(d)). Before getting to this,

we make a little detour via the Riemann-Roch Theorem for Riemann surfaces (that is, compactcomplex manifolds of complex dimension 1). We will use (without proof) that cohomology groupsof vector bundles on projective complex manifolds are finite dimensional vector spaces.

Theorem 7.28 (Riemann-Roch for P1). Let F be a sheaf on P1, we denote by χ(F ) ∈ Z the integergiven by dimH0(P1,F)−dimH0(P1,F) (assuming it is well-defined). We have χ(OPn(d)) = d+1.

Sketch. If 0 // A0d0 // · · · // An−1

dn−1// // An // 0 is an exact sequence of vector spaces,

then the alternating sum of dimensions∑ni=0(−1)i dimAi = 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

0 //A0d0 // · · · //An−1

dn−1// // ker dn−1

// 0, of length n = n+ 1− 1. Therefore, by virtueof induction,

∑n−2i=0 (−1)i dimAi + (−1)n−1 dim ker dn−1 = 0. Since dn : An−1

// An is surjective,we have dim ker dn−1 = dimAn−1 − dimAn, which concludes the proof of the claim.

We apply this to the long exact sequence corresponding to the short exact sequence 0 // O(d−1) // O(d) // OE(d) // 0. Note that E is a point, therefore H0(E,OE(d)) = C is the onlynon-zero cohomology group. Using the cohomology vanishing property (which we will discuss inthe future) that H2(P,O(d)) = 0, we obtain

0 //H0(P1,O(d− 1)) //H0(P1,O(d)) // C //H1(P1,O(d− 1)) //H1(P1,O(d)) // 0 // 0.

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The alternating sum of dimensions can be re-arranged into the identity χ(O(d)) = χ(O(d− 1)) + 1.Therefore it remains to prove the assertion for a single d ∈ Z. We already know that H0(P1,O) = 1,next week we will show that it’s also true that H1(P1,O) = 0.

Together with our previous computation that dimH0(P1,O(d)) = d+ 1, if d ≥ 0, the Riemann-Roch formula determines the dimensions of the cohomology groups of the sheaves O(d) on P1

completely. We can also observe the following interesting symmetry

dimH0(P1,O(d)) = dimH1(P1,O(−2− d)).

This fascinating identity can be explained through so-called Serre duality, the counterpart ofPoincare duality for coherent sheaves. One can show that on a smooth projective manifold Xof dimension n, and a vector bundle E, we have a canonical isomorphism of vector spaces

Hi(X,E) ∼= Hn−i(X,E∨ ⊗ ΩnX).

The Riemann-Roch theorem can be generalised to smooth projective manifolds X, but evenstating it properly would take us too far afield. For X a Riemann surface, this is possible. Forevery line bundle L on X, one can define its degree degL ∈ Z. For X = P1 we have degO(d) = d.The underlying topological space of X is an orientable surface of genus g. The Riemann-Rochformula reads

χ(L) = 1− g + degL.

7.4 The proof of the first GAGA theorem (modulo basic cohomologycomputations)

Let F be a coherent sheaf on PnZar, for example F could be a locally free sheaf of Oreg-modules,corresponding to an algebraic vector bundle. Serre defines a sheaf of holomorphic sections Fh onPn. For E an algebraic vector bundle, given by a regular cocycle datum (Uii∈I , (φij)), Eh isequal to the sheaf of holomorphic sections E. That is, to an open subset V ⊂ Pn, we assign thecollection of holomorphic functions fi : Ui ∩ V // C, such that fi = φijfj . This functor can begeneralised to coherent sheaves, by using the fact that Zariski locally, every coherent sheaf can bewritten 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 Fh = coker(Om // Ok). It is importantto check that this does not depend on the presentation of F as a quotient. We will say more aboutthis in a lecture devoted entirely to this process of taking holomorphic sections. The followinglemma is essential to the proof of GAGA. It asserts the existence of a comparison map betweenregular and holomorphic cohomology for coherent sheaves.

Lemma 7.29. (a) The functor F 7→ Fh from coherent sheaves on PnZar to sheaves on Pn sendsexact sequences of coherent sheaves to exact sequences.

(b) For every i ∈ N, and coherent sheaf F on Pn there exists a natural map Hi(PnZar,F) //Hi(Pn,Fh).Naturality asserts that for every morphism F // G of coherent sheaves we have a commutative

76

diagram

Hi(PnZar,F) //

Hi(PnZar,G)

Hi(Pn,Fh) // Hi(Pn,Gh),

and for every short exact sequence 0 // F // G //H // 0 of coherent sheaves we have acommutative diagram

Hi(PnZar,H) //

Hi+1(PnZar,F)

Hi(Pn,Hh) // Hi+1(Pn,Fh).

The proof of this lemma is deferred to another lecture. The proof of (a) makes use of resultsrelated to Weierstrass preparation, and is mostly algebraic. The proof of (b) relies on the fact thatfor every coherent sheaf F on An, the higher cohomology groups Hi(Cn,Fh) vanish (here we usethe fact that the sheaf of holomorphic sections can be defined locally on Pn, and that Pn can becovered by Zariski open subsets equivalent to An).

Next week we will learn the techniques to prove Hi(Pn,OPn) = 0 for i ≥ 1. Taking this forgranted, we obtain the following interesting special case of GAGA.

Proposition 7.30. If the natural map Hi(PnZar,Oreg) //Hi(Pn,O) is an isomorphism for every

i ∈ N, and every n ∈ N, then we have that also Hi(PnZar,Oreg(d)) //Hi(Pn,O(d)) is an isomorphism

for every d ∈ Z.

Proof. This follows from the short exact sequence 0 // OregPn (d− 1) // Oreg

Pn (d) // i∗OregE (d) // 0.

Applying the functor given by holomorphic sections, the resulting long exact sequence in cohomologygives us the following commutative diagram with exact rows

Hi(PnZar,OregPn (d))

// Hi(PnZar,OregPn (d− 1)) //

Hi(PnZar,OregPn (d)) //

Hi(EZar,OregE (d)) //

Hi+1(PnZar,OregPn (d− 1))

Hi(PnZar,OregPn (d)) // Hi(Pn,OPn(d− 1)) // Hi(Pn,OPn(d)) // Hi(E,OE(d)) // Hi+1(Pn,OPn(d− 1)),

and the Five Lemma implies together with induction on n, that Hi(PnZar,Oreg(d)) //Hi(Pn,O(d))

is an isomorphism, if it is an isomorphism for d − 1. Similarly, we can show that the case of d,implies the one of d− 1. This reduces the entire assertion of GAGA for the sheaves OPn(d) to thecase d = 0.

We are now ready to prove the entire proof of part (a) of the GAGA theorem, to the assertionthat Hi(PnZar,O

regPn ) //Hi(Pn,OPn) is an isomorphism. For i = 0 we already know this to be true.

Next week we will show that the higher cohomology groups Hi(Pn,O) vanish. Together with thedescending induction argument, this concludes the proof of GAGA (part (a)).

Proposition 7.31. Assume that the natural map Hi(PnZar,Oreg) //Hi(Pn,O) is an isomorphism

for every i ∈ N, and assume that higher cohomology groups of coherent sheaves on Pn vanish fori ≥ n + 1. Then, for every coherent sheaf F on PnZar, the map Hi(PnZar,F) // Hi(Pn,Fh) is anisomorphism.

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Corollary 7.32. Let X be a projective complex manifold, and E an algebraic vector bundle onX. Then the vector space of holomorphic section of E is isomorphic to the vector space of regularsections of E (assuming GAGA(a) for OPn).

Proof. Let E be the sheaf of regular sections of E on XZar. We denote by i : X → Pn an inclusionof X into complex projective space. We have H0(XZar, E) = H0(Pn, i∗ E). By GAGA part (a) thisvector space is equivalent to H0(X, i∗E), which agrees with the space of holomorphic sections ofE.

Proof of Proposition 7.31. According to Proposition 7.26 we can choose d >> 0, such that there is asurjection of sheaves (Oreg

Pn )m F(d). Untwisting, we obtain a surjection of sheaves (OregPn (−d))m

F . Let K be the sheaf given by the kernel of this morphism of sheaves. The resulting short exactsequence of sheaves yields the following commutative diagram with exact rows.

Hi(PnZar,K) //

f

Hi(PnZar, (OregPn (−d))m) //

g

Hi(PnZar,F) //

h

Hi+1(PnZar,K) //

k

Hi+1(PnZar, (OregPn (−d))m)

`

Hi(Pn,Kh) // Hi(Pn, (OPn(−d))m) // Hi(Pn,Fh) // Hi+1(Pn,Kh) // Hi+1(Pn, (OPn(−d))m)

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 alsoknow 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 impliesthat h is surjective. Since K is also a coherent sheaf this implies that f is surjective too, and hencea 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 timereasons.

Corollary 7.33. Let E1 and E2 be two algebraic vector bundles on Pn, or more generally a complexprojective manifold X. Then, every OX-linear map Eh1 // Eh2 is induced by a unique regularmorphism E1

// E2.

Proof. The sheaves E1, and E2 are OregX -locally free of finite ranks n1 and n2. Let Hom(E1, E2) be

the sheaf of OregX -modules, which assigns to a Zariski open subset U ⊂ X the Oreg

X -module of linearmaps Hom(E1(U), E2(U)). We leave the standard verification that Hom(E1, E2) is a sheaf to thereader.

We claim that Hom(E1, E2) is locally free of rank n1n2. By definition, its global sections areequivalent to the vector space of linear maps E1

// E2. GAGA part (a) implies therefore thatevery linear map between Eh1 and Eh2 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

Hom((OregX )n1 , (Oreg

X )n2) = (OregX )n1×n2 .

There is also a theory of analytic coherent sheaves on a complex manifold. An OX -linear sheafF 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

OmUi // OkUi // F // 0.

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The proof of GAGA (c) follows now from this result, and the following important theorem. Serrededuces it from the fact that Pn is compact, and that the cohomology groups Hi(Pn,F) are finite-dimensional.

Theorem 7.34. Let F be an analytic coherent sheaf on Pn, then there exists d >> 0, such that wehave a surjection OkPn F(d).

As before, this yields a short exact sequence 0 //K // (OPn(−d))k // F // 0, and using thefact that K is also an analytic coherent sheaf, we can express F as a cokernel of a linear morphism(OPn(−d′))m // (OPn(d))k. We have just seen that every such morphism is regular, hence F mustbe 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→ Fh, discuss why it is exact,and define a natural morphism Hi(PnZar,F) //Hi(Pn,Fh). Let U be a Zariski open subset of Pn.We will mostly be interested in the cases U = Pn or U = Ui. We denote by Φ: U // UZar thecontinuous map, which is the identity on underlying sets. It is continuous, since a Zariski opensubset is also open in the standard topology.

Definition 7.35. For a sheaf F of OregU -modules, we define Fh = Φ−1 F ⊗Φ−1Oreg

UOU .

Recall that Φ−1 : Sh(UZar) // Sh(U) is the functor we constructed in Lemma 7.17 for a gen-eral continuous map. We have a canonical map Φ−1Oreg

U// OU , which is adjoint to the map

OregU

// 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,and F //Φ∗ G an Oreg

U -linear map (where Φ∗ G is viewed as a sheaf of OregU through the canonical

map OregU

// Φ∗OU ). Then there exists a unique morphism of sheaves Fh // G, such that

F

##

Φ∗ Fh // Φ∗ G

commutes.

This universal property implies directly that holomorphisation commutes with cokernels, that is,coker(F1

//F2)h ∼= coker(Fh1 //Fh2 ). This shows that for a short exact sequence 0 //F //G //H //0of coherent sheaves, we have an exact sequence Fh // Gh //Hh // 0, but showing that the mapFh // Gh is injective is less obvious. Serre proves this using commutative algebra, the proof is notdifficult to understand, but unfortunately we don’t have the time to explain it. In the end it boilsdown to the Weierstrass Preparation Theorem and formal computations with power series. Thefollowing assertion is more generally known to be true for analytic coherent sheaves on Cn. Thisversion is known as Cartan’s Theorem B. The proof below uses that Hi(Cn,O) = 0 for i ≥ 1. Wewill prove this next week, using the Dolbeault resolution.

Proposition 7.37 (Weak Cartan B). Let F be an algebraic coherent sheaf on An, then for i ≥ 1we have Hi(Cn,Fh) = 0. The same statement holds when we replace An by a Zariski open subsetof the shape f 6= 0, where f ∈ C[x1, . . . , xn] is a polynomial.

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Proof. We prove this by descending induction, using the fact that Hi(Cn,Fh) = 0 for i > n, andHi(Cn,O) = 0 for i ≥ 1. We will see a proof of these statements very soon, using the formalism ofDolbeault cohomology.

Since F is coherent, there exists a short exact sequence 0 //K // (OregAn )m // F // 0, where

K is also coherent. The long exact sequence in cohomology implies Hi(Cn,Fh) = Hi+1(Cn,K) fori ≥ 1. Therefore, we obtain by descending induction on i the asserted vanishing statement.

We denote by Φ: Pn // PnZar the continuous map, which is the identity on underlying sets. Itis continuous, because every Zariski open subset is also open in the standard topology. Part (b)of Lemma 7.29, which asserted the existence of a natural map Hi(PnZar,F) //Hi(Pn,Fh) followsfrom the lemma below, and the fact that we have a natural map F // Φ∗ Fh.

Lemma 7.38. For every coherent sheaf F on PnZar, we have an equivalence Hi(Pn,Fh) ∼= Hi(PnZar,Φ∗ Fh),

that is, it doesn’t matter if we compute the cohomology of Fh with respect to the standard or theZariski topology.

Proof. Let (I•, d) be an injective resolution of F , that is a complex of injective sheaves, exact inpositive degrees, such that ker d0 = Fh. The pushforward Φ∗(I•) is also a chain complex. If weknew that the sheaves Φ∗ Ii were still acyclic (= no higher cohomology), then we would only haveto check that this still defines a resolution of Φ∗ F . The pushforward of injective sheaves is notinjective, but can be shown to be flasque as follows. We have seen in ex. 1, E8 that injectivesheaves are flasque, and that the pushforward of flasque sheaves is flasque. Hence, it remains toshow that Φ∗(I•) is exact in positive degrees. Since exactness for chain complexes of sheaves is alocal property, it suffices to show that for every standard chart Ui ⊂ Pn, the restriction Φ∗(I•)|Uiis exact in positive degrees. Let V = (Ui)f ⊂ Ui be a Zariski open subset of the form f 6= 0, for apolynomial f . The chain complex of global sections I0(V ) // I1(V ) // · · · computes H•(V,F |V ),which we have seen to vanish in positive degrees. Since every Zariski open subset of Ui is a unionof subsets (Ui)f , this implies the required exactness assertion in positive degrees.

8 Computing Cohomology: Dolbeault and Cech

In this section we will discuss two important tools to compute cohomology of sheaves. The Dolbeaultresolution and the Cech complex.

8.1 The Dolbeault resolution

The Dolbeault resolution is to the sheaf of holomorphic functions OX on a complex manifold, whatthe de Rham resolution is to the sheaf of locally constant functions R on a smooth manifold. Beforewe can define it, we need to introduce some notation, and recall the differential operator ∂

∂zifrom

2.4. For a C∞-function f : U // C, where U ⊂ Cn we defined

∂f

∂zi=

1

2(∂f

∂xi+ i

∂f

yi).

The function f is holomorphic, if and only if for every i = 0, . . . , n we have ∂f∂zi

= 0. This is simplya reformulation of the Cauchy-Riemann equations.

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The idea underlying the Dolbeault resolution is that there should be a sheaf A0,1, and a map

C∞X∂ // A0,1, such that ker ∂ = OX . Here, C∞X 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 ∂ // · · · .

Definition 8.1. Let U be an open subset of Cn. We define Ap,q(U) to be the complex vector spacewhose elements are formally denoted by∑

fi1...ip;j1...jqdzi1 ∧ · · · dzip ∧ dzj1 ∧ · · · ∧ dzjq ,

where the sum ranges over 0 ≤ i1 < · · · < ip ≤ n, 0 ≤ j1 < · · · < jq ≤ n, where fi1...ip;j1...jq denotesa complex-valued C∞-function on U . The resulting sheaf on Cn will be denoted by Ap,q

It is not difficult to see that Ap,q is closely related to the sheaf of smooth differential forms onCn.

Lemma 8.2. Denote by Am(U) the product∏p+q=mA

p,q(U). There is a canonical equivalencebetween Am(U), and the complex vector space of C-valued smooth m-forms∑

1≤i1<···<in≤2m

fi1···i2mdxi1 ∧ · · · ∧ dxin ∧ · · · ∧ dxim

on U (as a 2n-dimensional real manifold), where we use the notational shorthand xn+j = yj forj = 1, . . . , n.

Proof. An explicit isomorphism can be produced by making the substitutions: dzi = (dxi + idyi),dzi = dxi − idyi, dxi = 1

2 (dzi + dzi), and dyi = 12i (dz

i − dzi).

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 ∧ · · · dzip′ ∧ dzj1 ∧ · · · ∧ dzjq′ , andre-arranging terms. As usual we have the rule dzi ∧ dzi = 0, and dzi ∧ dzi = 0.

Next we define the operators ∂ and ∂.

Definition 8.3. The linear maps ∂p,q : Ap,q // Ap+1,q and ∂p,q

: Ap,q // Ap,q+1 are defined asthe unique linear maps, such that

d : Am // Am+1

for m = p+ q decomposes as d =∑p+q=m(∂p,q + ∂

p,q).

This implies directly the important relations ∂2 = 0, and ∂2

= 0.To obtain an explicit formula for ∂ and ∂, one uses that ∂

∂xi = ∂∂zi + ∂

∂zi, and ∂

∂yi = i( ∂∂zi −

∂∂zi

).Since we have

d(fdxi1∧· · ·∧dxim′∧dyim′+1∧· · ·∧dyi2m) =

n∑j=1

(∂f

∂xjdxj+

∂f

∂yjdyj)∧dxi1∧· · ·∧dxim′∧dyim′+1∧· · ·∧dyi2m ,

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we obtain from the identity

∂f

∂xjdxj +

∂f

∂yjdyj =

1

2(∂f

∂zj+∂f

∂zj)(dzj + dzj) +

1

2(∂f

∂zj− ∂f

∂zj)(dzj − dzj) =

∂f

∂zjdzj +

∂f

∂zjdzj ,

that

∂p,q(fdzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq ) =

n∑j=1

∂f

∂zjdzj ∧ dzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq ,

and

∂p,q

(fdzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq ) =

n∑j=1

∂f

∂zjdzj ∧ dzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq .

Lemma 8.4 (Dolbeault–Grothendieck). The complex of sheaves Ap,0 ∂ //Ap,1 ∂ //Ap,2 // · · · isexact in positive degrees. The kernel ker ∂0,0 is equivalent to the sheaf OCn 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 onlycare 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 forevery z0 ∈ U , and ε > 0, such that U ε(z0) ⊂ U , the formula

2πif(z) =

∫|w−z0|=ε

f(w)

w − zdw +

∫Uε(z0)

∂f

∂w(w)

dw ∧ dww − z

,

where z ∈ Uε(z0).

The proof of this lemma is similar to the way we deduced the Cauchy integral formula from theStokes 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 // · · · .

We already know that for a C∞-function f : U // C we have ∂f = ∂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 opencovering 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 ε-neighbourhood, such that U ε(z0) ⊂ U . We may assume thatfor |z − z0| ≥ ε

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 aroundeach z0 ∈ U . It has the added advantage that f can now be considered as a C∞-function on all ofC (it’s some kind of bump function itself).

For z ∈ Uε(z0) we define

g(z) =1

2πi

∫C

f(w)

w − zdw ∧ dw.

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Substituting w = w′ + z, we obtain

g(z) =1

2πi

∫C

f(w′ + z)

w′dw′ ∧ dw′.

We leave the straight-forward verification that g is of class C∞ to the reader. Using the last formula,we obtain

2πi∂

∂zg(z) =

∫C

∂f

∂w(w′ + z)

1

w′dw′ ∧ dw′ =

∫C

∂f

∂w(w)

1

w − zdw ∧ dw = 2πif(z).

And hence we see that ∂g = f , as required.

We apply the same strategy variable-by-variable to prove the n-dimensional version of theGrothendieck–Dolbeault Lemma.

Proof of the Grothendieck–Dolbeault Lemma. Let ω ∈ A0,q(U) be a (0, q)-form, such that ∂ω = 0.We claim that there for every z0 ∈ U there exists an ε > 0, such that we have an φ ∈ A0,q−1(Uε(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 theassertion is true.

Let’s assume that the lemma has already been proven for k−1. We can decompose ω as η∧dzk+ω′, where η and ω′ only contain the (0, 1)-forms dz1, . . . , dzk−1. We write η =

∑fi1...iq−1

dzi1 ∧· · · ∧ dziq−1 .

The relation ∂ω = 0 implies for j > k

∂fi1...iq−1

∂zj= 0. (16)

We now use the same integration method as in the proof of the 1-dimensional case to deduce a

smooth function gi1...iq−1 , such that∂gi1...iq−1

∂zk= fi1...iq−1 . Let φ′ =

∑gi1...iq−1dz

i1∧· · ·∧dziq−1 . By

construction we have that ω−∂φ′ only contains the (0, 1)-forms dz1, . . . , dzk−1. By induction, thereexists a φ′′ ∈ A0,q−1 in some ε-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(0, q)-forms on Cn.

Proposition 8.6. Let ω ∈ A0,q(Cm) be a (0, q)-form, such that ∂ω = 0. Then, there exists a(0, q − 1)-form φ ∈ A0,1(Cm), such that ω = ∂φ.

Proof. We begin with the case q = 1 which needs to be discussed separately. We write Cm =⋃n≥1 Un(0), we claim that there exists a sequence of smooth functions φn : Cm // C, such that

for ∂φn|Un+1

2(0) = ω|U

n+12

(0), and for all z ∈ Un(0) we have |φn+1(z)− φn(z)| ≤ 12n . The existence

of such a sequence follows inductively, the case n = 1 being a direct consequence of the proof of theGrothendieck–Dolbeault Lemma.

Let’s assume by induction that φ1, . . . , φn have been constructed. We know that there existsφ′n+1, such that ∂φ′n+1|Un+3

2(0) = ω|U

n+32

(0). For z ∈ Un(0) we have ∂(φ′n+1 − φ)(z) = 0 for

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z ∈ Un+ 12(0), that is, the difference is holomorphic on Un+ 1

2 (0). Choose a polynomial g(z), such

that |φ′n+1(z) − φn(z) − g(z)| < 12n for z ∈ Un(0). Since g is everywhere holomorphic, we may

define φn+1 = φn − g, without affecting the property ∂φn+1|Un+3

2(0) = ω|U

n+32

(0).

Defining φ = limn→∞ φn one sees directly that the resulting function is a continuous function,since the convergence is uniform on the subsets Un(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’sholomorphic 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|Un+1

2(0) = ω|U

n+12

,

and that φn+1|Un(0) = φn|Un(0).

By induction we assume that φ1, . . . , φn have been constructed. We choose φ′n+1, such that

∂φ′n+1|Un+32

(0) = ω|Un+3

2

. The difference φ′n+1 − φn is sent to 0 by ∂ on the subset Un+ 12(0).

Therefore, there exists a (0, q − 2)-form ψ, such that

(φ′n+1 − φn)|Un+1

4(0) = ∂ψ|n+ 1

4 (0).

Therefore, we see that φn+1 = φ′n+1 − ∂ψ satisfies φn+1|Un(0) = φn|Un(0), and ∂φn+1 = ∂φ′n+1,

because ∂2

= 0.

Definition 8.7. Let X be a complex manifold. For an integer k we denote by Ak the sheaf ofsmooth, complex-valued k-forms on X (take the usual vector space of smooth k-forms and apply thetensor product functor −⊗R C). By choosing local coordinates, we define Ap,qX (for p+ q = k) to bethe subsheaf consisting to k-forms which are locally expressible as∑

fi1...ip;j1...jqdzi1 ∧ · · · dzip ∧ dzj1 ∧ · · · ∧ dzjq .

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-calledacyclic resolution of the sheaves Ωp. This implies in particular that

Hq(X,Ωp) ∼=ker(∂ : Ap,q // Ap,q+1)

image(∂ : Ap,q−1 // Ap,q).

This follows at once from the Universal Comparison Theorem 6.10, once we have shown that thesheaves Ap,q don’t have any higher cohomology themselves (that is, are acyclic). We will deducethis 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 ⊂ Xa 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 subsetV ⊂ X, such that s|V ∩U = 0. By definition, we have supp s ⊂ U .

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

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- 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,

- 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 wesay that R is fine3.

There are many examples of fine sheaves of rings, in fact most examples the reader encounteredpreviously to studying complex manifolds were probably fine. Partitions of unity exist for continuousfunctions on paracompact spaces. On smooth manifolds partitions of unity can be constructedusing bump functions. Holomorphic partitions of unity don’t exist however. The support conditionsupp 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 . Givenany sheaf of R-modules M, and any section m ∈M(U), there is a unique section m′, denoted s ·min 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 coveringX = 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 ofR-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 ofR-modules, such that I is injective. Analysing our construction of injectivesheaves containing a given sheaf, one sees that this also works for sheaves of R-modules. The longexact sequence (and vanishing of higher cohomology of injective sheaves) imply

Hi(X,M) ∼= Hi−1(X, I /M)

for i ≥ 2, andH1(X,M) ∼= coker(H0(X, I) //H0(X, I /M)).

By induction we have therefore reduced the proof to showing that the mapH0(X, I) //H0(X, I /M)is surjective. Let s ∈ I /M(X) be a section. By definition of the quotient sheaf I /M there existsan 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 Uii∈I .

Choose for every j ∈ J an i(j) ∈ I, such that supp ej ⊂ Ui(j). Define the section

t =∑j∈J

ej · ti(j),

3In the literature one finds the definition of fine sheaves, which is similar but slightly different from the one givenhere. 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 isfine in the traditional one.

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we have seen in Lemma 8.9 that this is well-defined. We can check stalk-by-stalk that the sec-tion t is mapped to the section s in the quotient I /M. This shows surjectivity of the mapH0(X, I) //H0(X, I /M) and hence provides an anchor for our inductive argument.

Corollary 8.11. For any complex manifold X the sheaves Ap,qX are acyclic. In particular, oneobtains that the Dolbeault complex

Ap,0X (X) // Ap,1X (X) // · · ·

computes the sheaf cohomology Hq(X,ΩpX).

As a another consequence we obtain that the de Rham complex of a smooth manifold X

Ω0X,R

// Ω1X,R

// · · ·

computes the cohomology of the sheaf of locally constant R-valued functions RX .

Corollary 8.12. The sheaves OCn , and ΩpCn have vanishing higher cohomology groups.

Proof. This is a direct consequence of Proposition 8.6.

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

8.3 Cech cohomology

In this paragraph we fix a topological space X, and a sheaf F ∈ Sh(X). We assume that we havea open covering X =

⋃i∈I Ui. The Cech cohomology of F with respect to the covering Uii∈I is

defined to be the cohomology of the following chain complex:∏i∈I

Γ(Ui,F) //∏

(i,j)∈I2Γ(Uij ,F) //

∏(i,j,k)∈I3

Γ(Uijk,F) // · · · ,

where the maps are given by

δ : Cp(X,F) =∏

(i0,...,ip)∈Ip+1

Γ(Ui0···ip ,F) //∏

(j0,...,jp+1)∈Ip+2

Γ(Uj0...jp+1,F) = Cp+1(X,F),

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

dj0...jp+1=

p+1∑i=0

(−1)i(cj0...ji...jp+1|Uj0...jp+1

).

One can verify by a direct computation that δ2 = 0, hence this defines a chain complex, knownas the Cech complex. The sheaf property of F implies directly that H0

Uii∈I (X,F) = F(X).

Definition 8.13. We will denote the cohomology group of this complex by HpUii∈I (X,F). It is

called the degree p Cech cohomology of F with respect to the open covering Uii∈I .

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Elements of HpUii∈I (X,F) are therefore presented by so-called Cech p-cocycles. That is, a

collection of local sections (ci0···ip)i0...ip , satisfying δ((ci0···ip)i0...ip) = 0. For p = 1 we have alreadyseen this in our discussion of line bundles and G-Galois covers.

In a way, Cech cohomology is amenable to direct computation, at least when working with afinite cover. In this case, one can also show that Hi

Uii∈I (X,F) vanishes for i > |I| − 1.

Lemma 8.14. On a complex manifold X, H1Uii∈I (X,O

×X) is canonically equivalent to the abelian

group of isomorphism classes of complex line bundles L, which are trivialisable when restricted tothe open subsets Ui for every i ∈ I.

Proof. We denote this set of line bundles (defined in the statement of the lemma) by PicUii∈I (X).

By definition, every L ∈ PicUii∈I (X) admits an abstract cocycle datum(Uii∈I , (φij)(i,j)∈I2)

).

By definition, the collection of nowhere vanishing holomorpic functions φij ∈ O×X(Uij) satisfies thecocycle condition

φik = φij · φjk,which (after switching from multiplicative to additive notation) amounts precisely to the conditionthat (φij) ∈ ker δ.

Two cocycle data for L with respect to the same covering Uii∈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 ∈ O×X(Ui). The resulting1-cocycle would be (φiφ

−1j )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, H1Uii∈I (X,A) is canonically equivalent

to the abelian group of isomorphism classes of A-Galois covers of X, which are trivialisable whenrestricted to the open subsets Ui for every i ∈ I.

Cech 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 thenotion of a filtered colimit. This construction is reminiscent of how we defined stalks of sheaves ata point x ∈ X.

Definition 8.16. Let I be a partially ordered set, such that every 2-element subset i, j ⊂ I hasa common upper bound (we also say that I is filtered). Let (Ai)i∈I a family of abelian groups (orvector spaces), such that for every i ≤ j we have a homomorphism f ij : Ai // Aj, satisfying the

relation f ii = idAi , and f jk f ij = f ik for every ordered triple i ≤ j ≤ k. The colimit (a. k. a. directlimit) of (Ai)i∈I is defined to be the abelian group (resp. vector space)

colimi∈I

Ai = lim−→i∈I

Ai,

consisting of equivalence classes of pairs (i, a). Here, i ∈ I, and a ∈ Ai. And we say that (i, a) isequivalent to (j, b) if and only if there exists a common upper bound k of i, j, such that we havean element c ∈ Ak, satisfying f ik(a) = c = f jk(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 Fx.

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Let U = Uii∈I , and V = Vjj∈J be two open coverings of X. We say that U ≤ V, if for everyj ∈ J , there exists an i(j) ∈ I, such that Vj ⊂ Ui. The resulting map J // I allows us to define amap between the resulting Cech cohomology groups

HpU (X,F) // Hp

V(X,F).

We define the Cech cohomology of F to be the direct limit of this system of cohomology groups,and denote it by Hp(X,F).

As an immediate corollary of the lemmas about line bundles and A-Galois covers, we obtain thefollowing assertions. We just use the fact that any line bundle, respectively A-Galois covering, istrivialised by some open covering, and hence they all get detected by the Cech cohomology groupH1(X,−).

Corollary 8.17. (a) On a complex manifold X, the abelian group H1(X,O×X) is equivalent tothe abelian group of isomorphism classes of line bundles Pic(X).

(b) On a connected manifold X, the abelian group H1(X,A) is equivalent to the abelian group ofisomorphism classes of A-Galois covers Hom(π1(X), A).

Our next goal is to replace Cech cohomology in the statement above by actual sheaf cohomology.Therefore we embark on a more detailed study of the relation between sheaf cohomology and Cechcohomology. Note that if X is a compact topological space, then every open covering is ≤ then afinite covering. Hence it’s sufficient to work with respect to finite coverings on compact spaces.

Proposition 8.18. Let U = Uii∈I be a covering of X. For every tuple (i0, . . . , ir) ∈ Ir+1 wedenote by ji0...ir : Ui0...ir = Ui0 ∩ · · · ∩Uir → X the inclusion. Let F be a sheaf on X, such that forevery (i0, . . . , ir) ∈ Ir+1 we have that (ji0...ir )∗(F |Ui0...ir ) is acyclic (that is, no higher cohomology),

then HpU (X,F) ∼= Hp(X,F).

Proof. The Cech resolution also makes can also be viewed as a resolution of sheaves, that is, wehave an exact sequence

0 // F // C0U (F) // C1

U (F) // · · · ,where CpU (F) =

∏(i0,...,ip)∈Ip+1(ji0...ir )∗(F |Ui0...ir ). By assumption, CpU (F) is a product of acyclic

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 Cechcomplex with respect to U .

Corollary 8.19. If F is a flasque sheaf on X, then its higher Cech cohomology vanishes withrespect to any open covering U . In particular Hi(X,F) = 0 for i > 0.

Proof. We have seen in the exercises that flasque sheaves are acyclic. Since restrictions to opensubsets, and pushforwards of flasque sheaves are flasque, we obtain that the sheaves Cp(F) areflasque. Therefore, they are acyclic, and hence assumptions of Proposition 8.18 are satisfied. Wededuce that 0 = Hi(X,F) = Hi

U (X,F) = Hi(X,F).

So far we have seen that Cech cohomology verifies the two axioms (A1) and (A3) of sheafcohomology. In general it is not true that every short exact sequence of sheaves yields a long exactsequence of Cech cohomology groups, and hence sheaf cohomology will not always agree with Cechcohomology. However, whenever this missing axiom is satisfied by the Cech cohomology groups, weobtain directly from our axiomatic approach that Cech cohomology agrees with sheaf cohomology.Nonetheless, we always have a long exact sequence up to degree 1.

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Lemma 8.20. Given a short exact sequence of sheaves 0 // F // G //H // 0 on X, we havean exact sequence of Cech cohomology groups

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

Proof. We only define the map δ, and check exactness of Γ(X,G) // Γ(X,H)δ // H1(X,F), and

leave the remaining parts as an exercise to the reader.We have a map δ : Γ(X,H) // H1(X,F), which is defined as follows. Let s ∈ Γ(X,H) be a

section, and let U = Uii∈I be an open covering, such that we have sections ti ∈ G(Ui), whichmap 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 Cech 1-cocycle withrespect 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

zero cocycle sii = 0. This implies that the composition Γ(X,G) // Γ(X,H)δ // H1(X,F) equals

the zero map.Vice versa, if δ(s) = 0, then there exists a Cech 1-cocycle (di)i∈I ∈

∏i∈I F(Ui), such that

cij = (di − dj)|Uij . We can then define t′i = ti − di, and obtain t′i|Uij = t′j |Uij . Hence, the sheafproperty implies that there exists a global section t′ ∈ G(X), such that t′|Ui = ti. This concludesthe proof of exactness.

Corollary 8.21. We have H1(X,F) ∼= H1(X,F). In particular, if X is a complex manifold,then Pic(X) ∼= H1(X,O×X). Moreover, we have for X a connected manifold that H1(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 injectivesheaves are flasque, we have H1(X, I) = H1(X, I) = 0. The long exact sequences imply

H1(X,F) = coker(Γ(X, I) // Γ(X, I /F)) ∼= H1(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), Cech cohomologygives the right answer, that is, agrees with sheaf cohomology. We have the following theorem:

Theorem 8.22 (Leray). Let F be a sheaf on X, and U an open covering, such that Hj(Ui0...ir ,F |U ) =0, for every tuple (i0, . . . , ir) ∈ Ir+1, where Ui0...ir = Ui0 ∩ · · · ∩ Uir . Then Hi

U (X,F) ∼= Hi(X,F).

In the proof of this theorem, we have to produce a suitable long exact sequence for Cechcohomology groups of sheaves satisfying the condition of the theorem. This is possible thanks tothe following lemma.

Lemma 8.23. Let C•, D•, and E• be chain complexes, and 0 // C• //D• // E• // 0 a shortexact sequence. Then, we have a long exact sequence of cohomology groups

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

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Proof. For a chain complex C• we writeBiC• for image(di−1 : Ci−1 //Ci), and ZiC• = ker(di : Ci //Ci+1).By definition, we have Hi(C•) = Zi(C•)/Bi(C•). We have a commutative diagram with exact rows

Ci/BiC•//

di

Di/BiD•//

di

Ei/BiE•

di

// 0

0 // Zi+1C•

// Zi+1D•

// Zi+1E•

and the Snake Lemma yields a long exact sequence

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

where we have used that the kernel of Ci/BC•di //Zi+1

C• is equivalent to Hi(C•), and its cokernelto Hi+1(C•).

Proof of Theorem 8.22. We can now prove Leray’s theorem. Let 0 // F // I // I /F // 0 bea short exact sequence with I exact. Since injective sheaves don’t have higher cohomology, we seethat F , I, and I /F satisfy the assumptions of the theorem (the quotient as a consequence of thelong exact sequence in cohomology). Moreover, vanishing of cohomology of these sheaves on thesubsets Ui0···ip guarantees that we have a long exact sequence of chain complexes in abelian groups

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

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

· · · // Hi(X,F) // Hi(X, I) // Hi(X, I /F) // Hi+1(X,F) // · · · .

Since injective sheaves are flasque, and flasque sheaves don’t have any higher Cech cohomologygroups by Corollary 8.19, we obtain

H1(X,F) ∼= coker(H0(X, I) //H0(X, I /F)) ∼= H1(X,F),

and for i ≥ 2 the equivalenceHi(X,F) ∼= Hi−1(X, I /F).

By induction we see that Hi(X,F) ∼= Hi(X,F).

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

8.4 Cohomology of complex projective space

In this paragraph we will explain two important results for the cohomology of sheaves on Pn.

Theorem 8.24 (Cohomology of Pn). Recall that for an abelian group A we denote by A the sheafof locally constant A-valued functions.

(a) We have H2i+1(Pn, A) = 0, and H2i(Pn, A) ∼= A, for 0 ≤ i ≤ n.

(b) The cohomology group Hi(Pn,OPn) = 0 vanishes for i ≥ 1, and H0(Pn,C) = C.

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Assertion (a) for 2i+1 = 1 follows from Corollary 8.21, because P1 is connected and 1-connected.The general case follows from a computation of the singular cohomology groups of Pn, using thatPn 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 thestandard covering Pn =

⋃ni=0 Ui of projective space, since OPn is known not to have any cohomology

over the intersections Ui0...ip by Cartan’s Theorem B (see Proposition 7.37 which is covering thecases 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),where we assume z 6= 0, and that U ⊂ Cn−1 is an open subset. There exist holomorphic functionsg : C×U // C, and h : C×U // C, such that we have

f(z, w1, . . . , wn−1) = g(z, w1, . . . , wn)− h(1

z, w1, . . . , wn−1),

for (z, w1, . . . , wn−1) ∈ C××Cn−1.

Proof. We define

g(z, w1, . . . , wn−1) =1

2πi

∫|u|=R

f(u,w1, . . . , wn−1)

(u− z)du,

where R is an arbitrary positive real number, satisfying R > |z|. Homotopy invariance of integralsof holomorphic functions implies that this is a well-defined holomorphic function on Cn. We cancompute f − g for z 6= 0, by using the Cauchy-integral formula for f :

1

2πi

∫|u|=R

f(u,w1, . . . , wn−1)

(u− z)du− 1

2πi

∫|u−z|=ε

f(u,w1, . . . , wn−1)

(u− z)du,

where ε is a sufficiently small positive number. Using the Residue Theorem we obtain

f − g = resu=0f(u,w1, . . . , wn−1)

u− z= − resu=0 f

z,

where resu=0 f is a holomorphic function in the variables (w1, . . . , wn−1). Since (f−g)( 1z , 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) satisfiesthe requirements.

As an immediate corollary we obtain vanishing of Hi(P1,OP1) for i ≥ 1. For i ≥ 2 this vanishingresult follows from the Dolbeault resolution, hence it is sufficient to prove vanishing for i = 1 inthis case.

Corollary 8.26. We have H1(P1,OP1) = 0.

Proof. We can apply Leray’s theorem to X = P1, and F = OP1 and the standard covering U0∪U1 =P1. The reason is that Ui ∼= C, and U0 ∩ U1

∼= C×, and by the Grothendieck–Dolbeault Lemmathere the sheaf of holomorphic functions O is acyclic on C, and C×. Therefore, the cohomologygroup Hi(P1,OP) is equivalent to the degree i cohomology of the Cech complex:

O(U0)×O(U1) // OU0×OU1

×OU0∩U1// · · · .

The definition of the boundary operator δ in Cech cohomology implies that δ(f, g) = (0, 0, f |U01−

g|U01), and that δ1 : C1U (P1,OP) // C2

U (P1,OP) is the zero map, and similarly in higher degrees.

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Therefore we are really computing the cohomology of the complex

O(U0)×O(U1) // OU0∩U1// 0 // · · · .

We already know H0(P1,O) ∼= C, and it remains to compute H1 as the cokernel of the mapO(U0)×O(U1) // OU0∩U1

, which sends (g, h) to g|U01− h|U01

. But we have seen in Lemma 8.25that every holomorphic function on U01 can be written as the difference of a holomorphic functionon U0 and on U1. This shows that the cokernel is 0, and therefore Hi(P1,O) = 0 for i ≥ 1.

In order to convey a feeling for the higher-dimensional case, we will show H2(P2,OP2) = 0.

Corollary 8.27. We have H2(P2,OP2) = 0.

Proof. We apply Leray’s Theorem to the standard covering P2 = U0 ∪ U1 ∪ U2. We have to showthat 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 holomorphicfunctions gij on the double intersections Uij .

We set g01 = 0, and choose g12, g02, such that g12 − g02 = f012, using Lemma 8.25. Now wechoose 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 tocheck that we have g23 − g13 + g12 = f123. But this follows directly from relation (17).

8.5 Line bundles on complex manifolds

The cohomology computation of OPn is an essential ingredient in the proof of GAGA, and allowsus to classify line bundles on Pn. Recall the exponential sequence

0 // Z 2πi // OPn // O×Pn // 0.

Using the equivalence H1(Pn,OPn ×) ∼= Pic(Pn) of Corollary 8.21, we can state the associated longexact sequence as

· · · //H1(Pn,OPn) // Pic(Pn) //H2(Pn,Z) //H2(Pn,OPn) // · · · .

We have seen that Hi(Pn,OPn) = 0 for i ≥ 2, and conclude that Pic(Pn) ∼= H2(Pn,Z) ∼= Z. Thelast equivalence uses Theorem 8.24(a).

One immediately guesses that the equivalence Pic(Pn) ∼= Z associated to the integer d ∈ Zthe line bundle O(d). This is indeed the case, but since we haven’t yet obtained a conceptualunderstanding of the group H2(Pn,Z) (e.g., we haven’t discussed methods to construct/describeelements), it is not easy to prove this assertion at this point. Instead, we will refer once more tothe GAGA Theorem, to prove that every line bundle on Pn is equivalent to a line bundle O(d). Inorder to keep our discussion as simple as possible, we limit ourselves to the case n = 1.

From now on we consider P1 with the Zariski topology. That is, a subset U ⊂ P1 is Zariskiopen, if and only if P1 \U consists of finitely many points (this is true, because we are workingin complex dimension 1). We denote by M the sheaf of rational functions, that is to U ⊂ P1 we

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associate the vector space of all rational functions V ⊂ C, where V ⊂ U . Let M× be the sheaf ofnon-zero rational functions, which form an abelian group with respect to multiplication.

We have an inclusion of sheaves O×reg →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 combinationn1 ·x1 + · · ·+nk ·xk, where xi ∈ U (this is what we call a divisor). We assign to a rational functionf 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

coker(M×(X) // Div(X)) ∼= Pic(P1).

The map Div(X) // Z sending∑ki=1 nixi to

∑ki=1 ni is surjective, because 1 · x is sent to 1 ∈ Z.

Moreover, since the sum of pole orders of a rational function f on P1 is 0, we get a well-definedmap

Pic(Pn) // Z .

Lemma 8.28. Let f : P1 // C be a non-zero meromorphic function on P1. The sum over all thepole orders, minus the zero orders is 0.

Proof. Without loss of generality we can assume that f does not have a pole or zero at∞ ∈ C ∼= P1.Otherwise, we apply a generic Mobius transformation to C to perturb a possible pole or zero awayfrom ∞. Choose a very small closed path γ around ∞, but which does not pass through ∞. Wecan view it as a non-trivial path in C, non-trivially circling the poles and zeroes of f . The ResidueTheorem implies that ∫

γ

f ′(z)

f(z)dz = m ·

∑x

ordz=xf,

where m is the non-zero winding number of f . Since γ is a path on P1 which can be contracted to∞, homotopy invariance of integration implies that the sum above vanishes.

Since the map Pic(P1) // Z is surjective, and we know that Pic(Pn) 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. Whichallows us to conclude that every line bundle on Pn is isomorphic to a line bundle O(d). This is aneasy computation, using the description of the boundary map in Cech cohomology.

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