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Complex Manifolds

Lecture Notes, Winter Term 2018/19

Janko Bohm

February 17, 2019

Contents

1 Introduction 11.1 The key idea . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Affine algebraic sets . . . . . . . . . . . . . . . . . . . . 21.3 Projective algebraic sets . . . . . . . . . . . . . . . . . . 41.4 My first complex manifold . . . . . . . . . . . . . . . . 121.5 Chow’s Theorem . . . . . . . . . . . . . . . . . . . . . . 131.6 What about singularities? . . . . . . . . . . . . . . . . 14

2 Multivariate holomorphic functions 162.1 A quick reminder of the one variable case . . . . . . . 17

2.1.1 Holomorphic functions . . . . . . . . . . . . . . 172.1.2 Convergent power series and analytic functions 192.1.3 Analytic equals holomorphic . . . . . . . . . . . 202.1.4 Features of holomorphic functions . . . . . . . 21

2.2 How to generalize to several variables? . . . . . . . . . 222.3 Multivariate holomphic functions:

What is the same and what not? . . . . . . . . . . . . 272.4 Analytic subsets . . . . . . . . . . . . . . . . . . . . . . 30

3 Complex Manifolds 363.1 Some notes on the underlying topological space . . . 363.2 Charts and atlas . . . . . . . . . . . . . . . . . . . . . . 383.3 Holomorphic functions on manifolds . . . . . . . . . . 403.4 Analytic subsets of manifolds . . . . . . . . . . . . . . 43

4 Sheaves 554.1 The definition and where it comes from . . . . . . . . 554.2 Stalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4 Short exact sequences of sheaves . . . . . . . . . . . . . 714.5 Sheaves on complex manifolds . . . . . . . . . . . . . . 73

1

CONTENTS 2

5 Vector bundles 775.1 Vector bundles and transition functions . . . . . . . . 775.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Vector fields and differential forms . . . . . . . . . . . 895.5 Vector bundles and locally free sheaves . . . . . . . . . 93

6 Cech Cohomology 966.1 Cochainoperator . . . . . . . . . . . . . . . . . . . . . . 966.2 Refining open covers . . . . . . . . . . . . . . . . . . . . 1006.3 Long exact cohomology sequence . . . . . . . . . . . . 1026.4 Lemma of Leray . . . . . . . . . . . . . . . . . . . . . . 1066.5 Cohomology from resolutions . . . . . . . . . . . . . . . 108

7 Cohomology of Differential forms 1107.1 De Rham Cohomology . . . . . . . . . . . . . . . . . . . 1107.2 Dolbeault Cohomology . . . . . . . . . . . . . . . . . . 1117.3 Poincare Duality . . . . . . . . . . . . . . . . . . . . . . 1127.4 Serre Duality . . . . . . . . . . . . . . . . . . . . . . . . 1137.5 Hodge Decomposition . . . . . . . . . . . . . . . . . . . 114

List of Figures

1.1 Charts of a complex manifold . . . . . . . . . . . . . . 21.2 Graph of a rational function. . . . . . . . . . . . . . . . 31.3 Kummer quartic . . . . . . . . . . . . . . . . . . . . . . 41.4 Elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Group structure on an elliptic curve. . . . . . . . . . . 61.6 Projective space P2(R). . . . . . . . . . . . . . . . . . . 71.7 Mapping A2(R) into P2(R) by stereographic projec-

tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Parabola x1 − x2

2 = 0 in A2(R). . . . . . . . . . . . . . . 91.9 Projective parabola . . . . . . . . . . . . . . . . . . . . 101.10 Projective parabola as a subset of the unit disc. . . . 111.11 Hyperboloids and double cone . . . . . . . . . . . . . . 14

2.1 Proof of the identity theorem. . . . . . . . . . . . . . . 282.2 Proof of the Open Mapping Theorem. . . . . . . . . . 292.3 Analytic subset as the a zeroset of holomorphic func-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Determining the codimension of an analytic subset. . 312.5 The set M in Hartogs’ theorem. . . . . . . . . . . . . . 322.6 Filling in the complement of two concentric poly-

cylinders. . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Application of Kugelsatz for a codimension 2 ana-

lytic subset. . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Covering of torus. . . . . . . . . . . . . . . . . . . . . . 403.2 Holomorphic function on a manifold. . . . . . . . . . . 413.3 Holomorphic functions on manifolds and coordinate

change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Holomorphic map between manifolds . . . . . . . . . . 433.5 Analytic subset of a complex manifold. . . . . . . . . . 443.6 Submanifold: Definition and interpretation as a man-

ifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.7 Implicit function theorem . . . . . . . . . . . . . . . . . 51

3

LIST OF FIGURES i

3.8 Submanifold as an algebraic set. . . . . . . . . . . . . . 54

4.1 Function with vanishing Taylor series. . . . . . . . . . 59

5.1 Local trvialization of a vector bundle . . . . . . . . . . 785.2 Section of a vector bundle . . . . . . . . . . . . . . . . 87

List of Symbols

An(K) Affine space of dimension n over the field K 2V (f1, ..., fn) Affine algebraic set define by f1, ..., fn . . . . 2Γ(g) Graph of g . . . . . . . . . . . . . . . . . . . . 3Pn(K) Projective space of dimension n over the

field K . . . . . . . . . . . . . . . . . . . . . . 4

ii

LIST OF SYMBOLS iii

1

Introduction

What is a complex manifold? The goal of this chapter is to give animpression of that (we will postpone most of the technical detailsand proofs).

1.1 The key idea

The idea of a complex manifold is that of a space which can bepatched together from open subsets of Cn. We start out with atopological space X (which has a sufficiently nice topology). Thepatching is realized by charts which are homeomorphisms

ϕ ∶ U → V

with U ⊂ X open and V ⊂ Cn open. If X can be covered by a(sufficiently nice) collection of charts

ϕα ∶ Uα → Vα

such thatX = ⋃

αUα

and for any two charts ϕ ∶ U → V and ϕ′ ∶ U ′ → V ′ the change ofcoordinate map

ϕ′ ϕ−1 ∶ ϕ(U ∩U ′)→ ϕ′(U ∩U ′)

is biholomorphic (see Figure 1.1), then we call X a complex man-ifold.

This concept has a close connection to algebraic geometry, andthis is one of the reasons why it is so interesting. Let us first recapthe key definitions there:

1

1. INTRODUCTION 2

X

U

j

j

j‘

j‘V

V‘

U‘

-1

Figure 1.1: Charts of a complex manifold

1.2 Affine algebraic sets

Algebraic geometry studies the set of solutions of polynomial sys-tems of equations.

Definition 1.2.1 The affine space of dimension n over the fieldK is defined as

An(K) =Kn

By this notation we express that we (usually) do not care aboutthe structure of Kn as a K-vector space.

Definition 1.2.2 An affine algebraic set is the common zeroset

V (f1, . . . , fr) = p ∈ An(K) ∣ f1(p) = 0, . . . , fr(p) = 0

of polynomials f1, . . . , fr ∈K[x1, . . . , xn].An affine algebraic set X is called irreducible, if it cannot be

written as X =X1 ∪X2 with affine algebraic sets Xi ⫋X. Then wealso call X an affine algebraic variety.

1. INTRODUCTION 3

Example 1.2.3 Affine algebraic varieties commonly known alsooutside algebraic geometry are V (1) = ∅, V (0) = Kn, the set ofsolutions of a linear system of equations

A ⋅ x − b = 0

or the graph

Γ (g) = V (x2 ⋅ b (x1) − a (x1)) ⊂ A2(K)

of a rational function

g = ab∈K(x1).

For example, the graph of g(x1) = x31−1

x1is

V (x2x1 − x31 + 1) ⊂ A2(K),

as depicted in Figure 1.2 for K = R.

Figure 1.2: Graph of a rational function.

Example 1.2.4 If f ∈K[x1, . . . , xn] is non-zero and non-constant,then V (f) ⊂ An(K) is called a hypersurface. For example, thesurface shown in Figure 1.3 is a hypersurface in A3(R).

Note, that V (f) is irreducible, if and only if f is irreducible.

Example 1.2.5 Not every curve in A2(K) is a graph, for example,the circle

V (x21 + x2

2 − 1) ⊂ A2(R)or the elliptic curve

V (x22 − x3

1 − x21 + 2x1 − 1) ⊂ A2(R)

are not, see Figure 1.4.

1. INTRODUCTION 4

Figure 1.3: Kummer quartic

Example 1.2.6 In general, an elliptic curve is of the form E =V (f) where f ∈ K[x1, x2] is an irreducible polynomial of degree 3and V (f, ∂f∂x1 ,

∂f∂x2

) = ∅, which guarantees that the curve does nothave a singularity. The set of points of an elliptic curve has thefollowing remarkable property: It comes with the structure of anabelian group with the operation as specified in Figure 1.5. Theneutral element is the point at infinity in the direction of the x2-axis. This point is missing in the affine picture. The solution is topass to projective space.

1.3 Projective algebraic sets

Similar to the situation of the elliptic curve, many results in al-gebraic geometry improve, if we replace the affine space An

K byprojective space PnK . For example, any line in A2

K gets added itspoint at infinity, which leads to the desirable fact that any two dis-tinct lines intersect in a point: Two parallel lines intersect at theircommon point at infinity.

Definition 1.3.1 The n-dimensional projective space over K is

Pn(K) = 1-dimensional vector subspaces of Kn+1

If ⟨(p0, . . . , pn)⟩ ∈ Pn(K), so in particular (p0, . . . , pn) ≠ 0, then we

1. INTRODUCTION 5

Figure 1.4: Elliptic curve

write(p0 ∶ ... ∶ pn) ∶= ⟨(p0, . . . , pn)⟩ .

The quotient symbol ∶ indicates that the generator of the 1-dimension vector space is only unique up to multiplication with anon-zero constant:

(p0 ∶ ... ∶ pn) = (q0 ∶ ... ∶ qn)⇐⇒ ∃λ ≠ 0 ∶ λ ⋅ (p0, . . . , pn) = (q0, . . . , qn)

With the group action

K× ×Kn+1 →Kn+1, (λ, (p0, . . . , pn))↦ λ ⋅ (p0, . . . , pn)

we havePn(K) = (Kn+1 − 0) /K×.

This quotient construction has a generalization in the setting oftoric geometry and geometric invariant theory.

For example, we can think of P2(R) as a half sphere with oppo-site boundary points identified (since they generate the 1-dimensionvector space), see Figure 1.6.

To specify algebraic subsets of Pn(K) we have to consider ho-mogeneous polynomials:

Definition 1.3.2 A graded ring is a commutative ring R with 1together with a fixed decomposition

R =⊕α∈GRα

1. INTRODUCTION 6

Figure 1.5: Group structure on an elliptic curve.

as a direct sum of abelian groups, where G is a monoid (that issemigroup with unit), such that

RαRβ ⊂ Rα+β.

If f ∈ Rα then f is called homogeneous of degree α.If f ∈ R then f = ∑α∈G, finite fα with the uniquely determined

homogeneous components fα ∈ Rα.A homogeneous ideal is an ideal I ⊂ R which can be generated

by homogeneous elements.

Note that 1 ∈ R0 (since for any r ∈ Rβ we have r = 1 ⋅ r ∈ Rα+β),and 0 ∈ Rα for all α.

Definition 1.3.3 The standard grading on K[x0, . . . , xn] by N0

is given by setting degxi = 1∀i. Then

K[x0, . . . , xn] =⊕∞d=0K[x0, . . . , xn]d

whereK[x0, . . . , xn]d = ∑∣α∣=dcαxα ∈K[x0, . . . , xn]

Example 1.3.4 In the standard grading on K[x, y] the polynomi-als 3x2y and 3x2y+5xy2 are homogeneous, whereas 3x2y+5x is nothomogeneous.

However if you consider the grading of K[x, y] by N20 with degx =

e1 and deg y = e2, then the degree of a monomial is its exponent vec-tor. Hence 3x2y is homogeneous of degree (2,1), however 3x2y +

1. INTRODUCTION 7

Figure 1.6: Projective space P2(R).

5xy2 and 3x2y+5x are not. The homogeneous polynomials are pre-cisely the constant multiples of monomials (that is, terms).

If not mentioned otherwise, in what follows, we will always con-sider the standard grading on

R =K[x0, . . . , xn].

The main idea to define projective algebraic sets is the following:

If f ∈ R is homogeneous of degree d, then

f (λ ⋅ (x0, . . . , xn)) = λd ⋅ f(x0, . . . , xn)

hence the condition f(x0 ∶ ... ∶ xn) = 0 for (x0 ∶ ... ∶ xn) ∈ Pn(K) iswell-defined (whereas the value of f at a point is not).

Definition 1.3.5 If I ⊂ R is homogeneous, then the projectivealgebraic set defined by I is

V (I) = x ∈ Pn(K) ∣ f(x) = 0 ∀ homogeneous f ∈ I

1. INTRODUCTION 8

and if X ⊂ Pn(K)

I(X) = ⟨f ∈ R homogeneous ∣ f(x) = 0 ∀x ∈X⟩

is the homogeneous zero ideal of X.

Remark 1.3.6 Consider the hyperplane at infinity

H = x ∈ Pn(K) ∣ x0 = 0

and the corresponding affine chart

U = x ∈ Pn(K) ∣ x0 ≠ 0

so Pn(K) = U ∪H. The map

ψ ∶ An(K) → U(x1, . . . , xn) ↦ (1 ∶ x1 ∶ ... ∶ xn)(x1x0, . . . , xnx0 ) ←[ (x0 ∶ x1 ∶ ... ∶ xn)

is a well-defined bijection. Figure 1.7 depicts the identification ofA2(R) with U ⊂ P2(R). The horizontal lines correspond to thepoints of H.

Figure 1.7: Mapping A2(R) into P2(R) by stereographic projection.

If F ∈K[x0, . . . , xn], then we define

F a = F (1, x1, . . . , xn).

1. INTRODUCTION 9

Given a homogeneous ideal I ⊂K[x0, . . . , xn], the ideal

Ia = F a ∣ F ∈ I ⊂K[x1, . . . , xn]

is called the dehomogenization of I. In this way we can associateto the projective algebraic set X = V (I) ⊂ Pn(K) the affine algebraicset

ϕ−1(X ∩U) = V (Ia).

Note that, more generally, any choice of a homogeneous linearform l yields an affine chart U = l(x) ≠ 0.

Example 1.3.7 The projective parabola X = V (x1x0 −x22) ⊂ P2(R)

consists of the points (1 ∶ x1 ∶ x2) corresponding to the point (x1, x2) ∈V (x1−x2

2) ⊂ A2(R) of the affine parabola (Figure 1.8) and one pointat infinity (0 ∶ 1 ∶ 0), so

U = (x0 ∶ x1 ∶ x2) ∈ P2(R) ∣ x0 ≠ 0 → A2(R)∪ ∪

X = V (x1x0 − x22) → V (x1 − x2

2)(1 ∶ x1 ∶ x2) ↦ (x1, x2)

In Figure 1.8 the projective parabola is depicted by identifying a1-dimensional subspaces of A3(R) with its generator on the upperhalf sphere. Projecting the upper half sphere into the (x1, x2)-planemaps the projective parabola into the unit disc, see Figure 1.10.

Figure 1.8: Parabola x1 − x22 = 0 in A2(R).

1. INTRODUCTION 10

Figure 1.9: Projective parabola

Theorem 1.3.8 If K =K then there is a 1 ∶ 1-map

projective algebraic setsX ⊂ Pn(K)

IV

homogeneous ideals J ⊂ Rwith J =

√J and J ≠ ⟨x0, . . . , xn⟩

Note that there are precisely two homogeneous radical idealsI with V (I) = ∅ namely J = ⟨x0, . . . , xn⟩ and J = ⟨1⟩. Finally weobserve, that to any affine variety we can associate a projective one:

Definition 1.3.9 Given f ∈ K[x1, . . . , xn], the homogenizationof f is

fh = xdeg f0 ⋅ f (x1

x0

, . . . ,xnx0

) ∈K[x0, . . . , xn]

which clearly is a homogeneous polynomial.If I ⊂K[x1, . . . , xn], then the homogenization of I is defined as

Ih = ⟨fh ∣ f ∈ I⟩ ⊂K[x0, . . . , xn],

which, by definition, is a homogeneous ideal.

Example 1.3.10 For F = x30 + x2

0x1 we get F a = 1 + x1, hence

(F a)h = x0 + x1. So if we dehomogenize and then homogenize, we

1. INTRODUCTION 11

Figure 1.10: Projective parabola as a subset of the unit disc.

remove any common x0-power of the terms. On the other hand, bydefinition it is clear that for f ∈K[x1, . . . , xn] it holds (fh)a = f .

Definition 1.3.11 Let X ⊂ An(K) be an affine algebraic set. Wedefine the projective closure of X as

pc(X) = V (I(X)h) ⊂ Pn(K),

the vanishing locus of the homogenization of the zero ideal of X.

Theorem 1.3.12 The projective closure pc(X) of an affine alge-braic set X is the smallest projective algebraic set containing ϕ(X)with the affine chart map ϕ defined as above.

Remark 1.3.13 We define the Zariski topology on Pn(K) byconsidering as closed sets the projective algebraic sets. The theoremshows that the projective closure of X is the Zariski closure of ϕ(X),that is,

pc(X) = ϕ(X).

Example 1.3.14 If we homogenize the defining ideal I of an affinevariety V (I), then pc(V (I)) = V (Ih), however only if K = K. Asa counterexample in case K ≠K, consider I = ⟨x2

1 + x42⟩ ∈ R[x1, x2].

Then V (I) = (0,0), hence I(V (I)) = ⟨x1, x2⟩, which coincideswith its homogenization, whereas Ih = ⟨x2

0x21 + x4

2⟩, so

(1 ∶ 0 ∶ 0) = pc(V (I)) ⫋ V (Ih) = (1 ∶ 0 ∶ 0), (0 ∶ 1 ∶ 0).

1. INTRODUCTION 12

Example 1.3.15 Consider the affine twisted cubic

C = V (I) ⊂ A3(K)

defined byI = ⟨x2

1 − x2, x31 − x3⟩ ⊂K[x1, x2, x3].

Hence, We can compute the homogenization in Singular by deter-mining a Grobner basis with respect to a monomial ordering refiningthe total degree:ring R = 0,(x(0..3)),dp;

ideal I = x(1)^2-x(2), x(1)^3-x(3);

std(I);

[1]=x(2)^2-x(1)*x(3)

[2]=x(1)*x(2)-x(3)

[3]=x(1)^2-x(2)

SoIh = ⟨x2

2 − x1x3, x1x2 − x0x3, x21 − x2x0⟩ .

Note that homogenizing the given set of generators of I will notyield the correct result:

V (x21 − x2x0, x

31 − x2

0x3) = pc(C) ∪ V (x0, x1).

1.4 My first complex manifold

Obiously the affine space An(C) is a complex manifold, the sameis true for any finite-dimensional complex vector space. The firstreally non-trivial example, in the sense that we are talking abouta compact complex manifold, is projective space Pn(C). We willcheck that in detail (with regard to various topological technicali-ties), but for now note the we already know a collection of chartscovering Pn(C) given by

Ui = (x0 ∶ . . . ∶ xn) ∈ Pn(C) ∣ xi ≠ 0

and

ϕi ∶ Ui → Vi = An(C)(x0 ∶ x1 ∶ ... ∶ xn) ↦ (x0

xi, . . . , xi−1xi

, xi+1xi, . . . , xnxi )

We determine the change of coordinate maps for the charts ϕ0 andϕ1 (but exactly the same formula works with ϕi and ϕj for anyi < j). First we observe that

ϕ0(U0 ∩U1) = (w1, . . . ,wn) ∈ Cn ∣ w1 ≠ 0ϕ1(U0 ∩U1) = (w1, . . . ,wn) ∈ Cn ∣ w1 ≠ 0

1. INTRODUCTION 13

The biholomorphic change of coordinate map is then the composi-tion of

ϕ−10 ∶ ϕ0(U0 ∩U1) → U0 ∩U1

(w1, . . . ,wn) ↦ (1 ∶ w1 ∶ . . . ∶ wn)and

ϕ1 ∶ U0 ∩U1 → ϕ1(U0 ∩U1)(x0 ∶ x1 ∶ ... ∶ xn) ↦ (x0

x1, x2x1 , . . . ,

xnx1

)

which is

ϕ1 ϕ−10 ∶ ϕ0(U0 ∩U1) → ϕ1(U0 ∩U1)

(w1, . . . ,wn) ↦ ( 1w1, w2

w1, . . . , wnw1

)

Remark 1.4.1 If we think now of a holomorphic function on acomplex manifold X as a holomorphic function defined in termsof charts X → U → C, then it is e.g. a very interesting question,how many (C-linearly independent) such function exist. Similarquestions can be asked also for differential forms. We will adressthese questions when discussing Hodge theory. The number of inde-pendent global holomorphic 1-forms, usually denoted as h0(X,Ω1

X)allows us, e.g., to distinguish beween P1 and and elliptic curve E:We have

h0(P1,Ω1P1) = 0

whileh0(E,Ω1

E) = 1.

1.5 Chow’s Theorem

The most important complex manifolds are submanifolds of projec-tive space. Here we are in for a big surprise.

Definition 1.5.1 Let X be a complex manifold of dimension n andY ⊂ X a closed subset. Then Y is called a submanifold of codi-mension k if for all y ∈ Y there is a chart ϕ ∶ U → V around y ∈ Xsuch that

ϕ(U ∩ Y ) = V ∩ (Cn−k × 0)

with 0 ∈ Ck.

Remark 1.5.2 That means that the inclusion of Y ⊂ X looks lo-cally like the inclusion of Cn−k ⊂ Cn. The submanifold Y is an(n − k)-dimensional complex manifold. So any submanifold can lo-cally be given a the zero-locus of k holomorphic functions.

1. INTRODUCTION 14

Figure 1.11: Hyperboloids and double cone

That looks very similar to the concept of algebraic varieties, ex-cept that here we use holomorphic (i.e. analytic) equations insteadof polynomial equations to define the submanifold. However, thereis the following big theorem, which shows a very deep connectionbeween the concept of complex manifolds and algebraic geometry.

Theorem 1.5.3 Every submanifold of Pn(C) is a projective alge-braic set.

What about the converse? Affine or projective algebraic setscan have singularities.

1.6 What about singularities?

An interesting feature algebraic varieties can have are singulari-ties, which are points that do not have a well-defined tangent plane.Considering as an example the hyperboloids Ht ⊂ R3 given

x2 + y2 − z2 + t = 0

varying with the parameter t ∈ R, the tangent plane at the point(x, y, z) is given by the affine space

(x, y, z) + ker(x, y,−z),

since (x, y,−z) is the normal vector of the surface at that point.So every point of Ht has a well-defined tangent plane, providedt ≠ 0. However, if t = 0, then (0,0,0) lies on the hyperboloid (whichnow becomes a double cone) and at this point the dimension of thekernel is not 2 but 3. So Ht is non-singular for t ≠ 0, but has asingularity at (0,0,0) for t = 0, see Figure 1.11. These plots weregenerated by the visualization tool Surfer [6], which provides an

1. INTRODUCTION 15

easy-to-use graphical interface to the raytracing program Surf [4]for plotting algebraic surfaces in 3-space. It can also be called fromSingular.

Complex manifolds in our setting cannot have singularities sincethey look at every point locally like a Cn. There is a generalizationof the notion of a complex manifold, which also handles spaces withsingularities, but we will not deal much with that.

Remark 1.6.1 One can show that any algebraic subvariety Y ⊂An(C) which is smooth at every point is locally a complete inter-section, that is, it can locally be given by codimension many polyno-mial equations. Using the implicit function theorem it follows thenthat Y is a complex manifold. Since this is a local definition, thesame works of course also for projective algebraic subvarieties. Sowe know a lot of complex manifolds!

Remark 1.6.2 Combining Remark 1.6.1 with Theorem 1.5.3 weobtain: Submanifolds of Pn(C) and smooth projective algebraic setsare the same!

2

Multivariate holomorphicfunctions

Passing from the real numbers to the complex numbers has impor-tant advantages. The key observation is the fundamental theo-rem of algebra, which states that any non-constant polynomialf ∈ C[x] has a zero and, hence, can be factored into degree manylinear polynomials. This is not true over R, e.g.

x2 + 1 = (x − i) ⋅ (x + i)

is irreducible over R. The statement generalizes to several variablesin Hilbert’s Nullstellensatz, which states that if K = K and I ⊂K[x1, . . . , xn] is an ideal, then

V (I) = ∅⇐⇒ I =K[x1, . . . , xn].

The Nullstellensatz is used in an essential way in the proof of Theo-rem 1.3.8. So it makes a lot of sense to do calculus over the complexnumbers.

What about continuity of functions f ∶ C → C? The open ballsin any metric space form a base for the topology. The complexnumbers are a metric space with d(z1, z2) = ∣z1 − z2∣ where for

z = x + iy

with x, y ∈ R, we set

∣z∣ =√z ⋅ z =

√x2 + y2

(denoting by z = x− iy the complex conjugate). This is exactly theEuclidean metric on R2. Hence C comes with the same topology asthe real plane R2. As a result the notions of continuity using limits

16

2. MULTIVARIATE HOLOMORPHIC FUNCTIONS 17

and the Euclidean distance are the same for functions R2 → R2 andC → C. Since R2 has the product topology of R, such a functionis continuous if and only if both coordinate functions (i.e. thereal and the imaginary part) are continuous. Things become moreinteresting if we talk about differentiability.

2.1 A quick reminder of the one vari-

able case

2.1.1 Holomorphic functions

Definition 2.1.1 Let U ⊂ C be open. We say that a function f ∶U → C is complex differentiable at z if the limit

f ′(z) = dfdz

(z) = limh→0

f(z + h) − f(z)h

exists. We say that f is complex differentiable or holomorphic inU if it is complex differentiable in every z ∈ U .

This notation satisfies all the usual rules of differentiation likethe sum, product, quotient and chain rule. The obvious question isthen, how this notion relates to real differentiability.

The first important observation on holomorphic functions is thatthe real and imaginary part satisfy the Cauchy-Riemann differentialequations:

Proposition 2.1.2 Let U ⊂ C be open and f ∶ U → C holomorphic.If we write

z = x + iy

andf = g + i ⋅ h

with g, h ∶ U → R, then

∂g

∂x= −∂h

∂yand

∂g

∂y= ∂h∂x

.

This means that the gradient vectors of g(x, y) und h(x, y) areorthogonal, e.g. for f(z) = z we have the gradients

( 10

) ⊥ ( 01

)

of g(x, y) = x respectively h(x, y) = y.


Remark 2.1.3 With the Wirtinger derivatives

∂f

∂z= 1

2(∂f∂x

− i∂f∂y

)

∂f

∂z= 1

2(∂f∂x

+ i∂f∂y

).

one can reformulate the Cauchy-Riemann equations as

∂g∂x =

∂h∂y

∂g∂y = −

∂h∂x

⇐⇒ ∂f

∂z= 0.

Using the Cauchy-Riemann equations and the Gauss divergencetheorem one can prove:

Proposition 2.1.4 Let U ⊂ C be open and Γ ⊂ U a (sufficientlysmooth) closed curve which does not intersect with itself and suchthat the interior of the curve is completely contained in U . Letf be complex differentiable on U . Then the integral of f along Γvanishes:

∮Γ

f(z)dz = 0.

Using this we can derive Cauchy’s integral formula:

Theorem 2.1.5 If Dρ(w) ⊂ C is an open disc with radius ρ around

w ∈ C and f is holomorphic in an open neighborhood of Dρ(w) thenfor every z ∈Dρ(w) we have

f(z) = 1

2πi ∫∣u−w∣=ρ

f(u)u − z

du.

As a corollary we conclude that f is infinitely often complexdifferentiable and obtain an explicit formula for the derivatives:

Corollar 2.1.6 With the conditions as in Theorem 2.1.5, the func-tion f is infinitely complex differentiable and

f (n)(z) = 1

2πi ∫∣z−w∣=ρ

f(u)(u − z)n+1

du.

Moreover we have the estimate

∣f (n)(z)∣ ≤ n!

ρn∥f∥ρ

where ∥f∥ρ denotes the sup norm of f on the circle ∣z −w∣ = ρ.

This formula can be used to describe a holomorphic functionlocally in terms of a convergent power series. Let us first recallsome facts on convergent power series:


2.1.2 Convergent power series and analytic func-tions

Theorem 2.1.7 For a power series

P (z) =∞∑n=0

an(z −w)n

with an ∈ C there is a number r > 0 (which may be ∞), the conver-gence radius of P , such that P (z) converges absolutely and com-pactly (the latter means uniformly on compact sets) for ∣z −w∣ < rand diverges for ∣z −w∣ > r.

Theorem 2.1.8 If

P (z) =∞∑n=0

an(z −w)n

has convergence radius r, then P (z) is holomorphic on Dr(w), theseries ∑∞

n=1 nan(z − w)n−1 has the same radius of convergence andthe derivative is given by

dP (z)dz

=∞∑n=1

nan(z −w)n−1

that is, by summand-wise differentiation.

Definition 2.1.9 Let U ⊂ C be open. We say that a function isanalytic on U if for every w ∈ U there is an r > 0 such that thereis a power series

P (z) =∞∑n=0

an(z −w)n

which converges absolutely and satisfies

P (z) = f(z)

for all z ∈Dr(w).

For two analytic functions their sum, product, quotient (wheredefined) and composition (where it makes sense) are again analytic.

By shifting the development point w of a power series to anarbitrary w0 inside the convergence radius one proves that con-vergent power series give analytic functions. For the proof oneassumes without loss of generality that w = 0 and then expressz = w0 + (z −w0).

Theorem 2.1.10 If P (z) = ∑∞n=0 an(z −w)n is a power series with

radius of convergence r, then f is analytic on Dr(w).


2.1.3 Analytic equals holomorphic

What is the relation between the notations complex differentiableand locally expandable into a power series? First of all from The-orem 2.1.8 we obtain:

Corollar 2.1.11 Analytic functions are holomorphic.

On the other hand, using Corollary 2.1.6 we can expand anyholomorphic function locally in its Taylor series:

Theorem 2.1.12 Let U ⊂ C be open and f holomorphic in U . IfPr(w) ⊂ U then

f(z) =∞∑n=0

an(z −w)n

with

an =1

n!f (n)(w).

By the estimate in Corollary 2.1.6, the radius of convergence of thepower series expansion is ≥ r.

This proves the converse of Theorem 2.1.11:

Corollar 2.1.13 Holomorphic functions are analytic.

Remark 2.1.14 We summarize our observations: Let U ⊂ C beopen. The following are equivalent:

1) f is holomorphic (i.e., complex differentiable) on U .

2) f is analytic.

3) f is infinitely often complex differentiable on U .

4) The real and imaginary parts g, h of f are continuously par-tially differentiable on U (in the real sense) and the Cauchy-Riemann differential equations are satisfied.

5) f is totally differentiable (in the real sense) on U and theCauchy-Riemann differential equations are satisfied.

Condition (1) is our definition, (5) is the right equivalent state-ment from the real point of view. The easier to check condition (4)is equivalent to it, since continuously partially differentiable impliestotally differentiable, and on the other hand by (3) a holomorphicfunction is infinitely often differentiable (which follows from (2),


which again we got from (1) by the Taylor expansion). Recall thata partially differentiable function does not have to be totally differ-entiable, consider for example

f ∶ R2 → R

f(x, y) = x3

x2+y2 (x, y) ≠ (0,0)0 (x, y) = (0,0)

In fact, a partially differentiable function also does not have to becontinuous.

Holomorphic functions have a couple of striking properties, whichare not true at all for real functions:

2.1.4 Features of holomorphic functions

We first give tworesults which can be obtained as an application ofthe Cauchy integral formula:

Theorem 2.1.15 (Liouville) Every bounded holomorphic functionf ∶ C→ C is constant.

Here, a complex function is called bounded if its absolute valueis bounded. A holomorphic function defined on the whole of C, asin the theorem, is called an entire function. The theorem impliesthe fundamental theorem of algebra:

Corollar 2.1.16 Every non-constant polynomial f ∈ C[x] has azero in C. In particular, C is algebraically closed.

Proof. Assume that f(z) = anzn + . . . + a0 ≠ 0 for all z ∈ C. Theng = 1

f is an entire function. Moreover, for any sequence (zj) in Cwith limj→∞ ∣zj ∣ = ∞ we have limj→∞ ∣f(zj)∣ = limj→∞ ∣aj ∣ ∣znj ∣ = ∞,hence g is bounded. So by Theorem 2.1.15, g and, hence, f isconstant, which is a contradiction.

Theorem 2.1.17 (Riemann extension theorem) Let U ⊂ C beopen and w ∈ U . If f is holomorphic on U/w and is bounded insome neighborhood of w then f can be extended in a unique way toa holomorphic function on U .

Using the representation of holomorphic functions in terms ofpower series, one can show that the zeroes of holomorphic functionsin one variable are discrete (that is, the induced topology on theset of zeroes is the discrete topology). This implies:


Theorem 2.1.18 (Identity theorem) Let f and g be holomor-phic functions on a domain G ⊂ C (that is, an open and connectedsubset). Then the following are equivalent:

1) f = g,

2) The set of all z ∈ G with f(z) = g(z) is infinite and containsan accumulation point,

3) f(z) = g(z) for all z in a non-empty open subset of G,

4) There is a w ∈ G with f (n)(w) = g(n)(w) for all n ∈ N0.

Theorem 2.1.19 (Open mapping theorem) Let f be a holo-mophic function on a domain G ⊂ C. If f is not constant, thenit is open (that is, it maps open subsets of G to open subsets of C).

This is implies:

Theorem 2.1.20 (Maximum modulus principle) Let G ⊂ Cn

be a domain and f holomorphic on G. If f has a local maximumin G, that is, there is a z ∈ G with

∣f(w)∣ ≤ ∣f(z)∣

for all w in a neighborhood of z, then f is constant on G.

Remark 2.1.21 The term modulus refers to the absolute value∣f ∣ of f .

2.2 How to generalize to several vari-

ables?

Definition 2.2.1 Let U ⊂ Cn be open. A function f ∶ U → Cis called holomorphic if f is continuous and for all w ∈ U thefunction

z ↦ f(w1, . . . ,wj−1, z,wj+1, . . . ,wn)

is holomorphic. We write

O(U) = f ∶ U → C ∣ f holomorphic

In fact one does not have to require the condition continuous,it follows from the other conditions by a theorem of Hartogs.


Remark 2.2.2 Let U ⊂ Cn be open. By the characterization ofholomorphic functions in one variable, any continuously partiallydifferentiable f ∶ U → C (where we consider U ⊂ Cn = R2n and C =R2) is holomorphic if and only if the Cauchy-Riemann differentialequations

∂f

∂zj(z) = 0

for all z ∈ U and j = 1, . . . , n are satisfied.

Definition 2.2.3 Let U ⊂ Cn be open. A function f = (f1, . . . , fm) ∶U → Cm is called holomorphic if all fj are holomorphic.

Notation 2.2.4 For ρ ∈ Rn with all entries ρj > 0 and w ∈ Cn wewrite

Pρ(w) = z ∣ ∣zj −wj ∣ < ρi for all j = 1, . . . , nfor the polydisc around z0 with polyradius ρ, and

Tρ(w) = z ∣ ∣zj −wj ∣ = ρi for all j = 1, . . . , n .

Note that Tρ(w) ⫋ ∂Pρ(w) if n ≥ 2. For example

P(1,2)(0) = z ∈ C2 ∣ ∣z1∣ < 1 and ∣z2∣ < 1

hence∂P(1,2)(0) = z ∈ C2 ∣ ∣z1∣ = 1 or ∣z2∣ = 1

butT(1,2)(0) = z ∈ C2 ∣ ∣z1∣ = 1 and ∣z2∣ = 1 .

Remark 2.2.5 Note that the polydisc Pρ(w) is the cartesian prod-uct

Pρ(w) =Dρ1(w1) × . . . ×Dρn(wn)of the discs Dρi(wi) = z ∈ C ∣ ∣z −wi∣ < ρi.

Theorem 2.2.6 (Cauchy integral formula) Let f ∶ Tρ(w) → Cbe continuous. For z ∈ Pρ(w) define

h(z) ∶= 1

(2πi)n ∫Tρ(w)

f(u)u − z

du

∶= 1

(2πi)n ∫∣zn−wn∣=ρn

⋯ ∫∣z1−w1∣=ρ1

f(u)u − z

du1⋯dun

wheref(u)u − z

∶= f(u)(u1 − z1) ⋅ . . . ⋅ (un − zn)


Then h ∶ Pρ(w) → C is continuous. If f is holomorphic in an open

neighbourhood of Pρ(w) then h ∈ O(Pρ(w)) and

f ∣Pρ(w)= h,

i.e. on the polydisc the holomorphic function f is given by theCauchy integral formula, which uses only values of f on Tρ(w).

Proof. The function

Φ ∶ Tρ(w) × Pρ(w) → C(u, z) ↦ f(u)

u−z

is continuous, hence by elementary multivariate calculus the func-tion h is continuous. We now use induction on n. For n = 1 we havethe Cauchy integral formula in one variable. For the induction stepn − 1↦ n we use this formula again: If z ∈ Pρ(w), we have

f(z1, . . . , zn−1, zn) =1

2πi ∫∣un−wn∣=ρn

f(z1, . . . , zn−1, un)un − zn

dun.

Moreover, by the induction hypothesis, we have

f(z1, . . . , zn−1, un) =1

(2πi)n ∫∣un−1−wn−1∣=ρn−1

⋯ ∫∣u1−w1∣=ρ1

f(z1, . . . , zn−1, un)(u1 − z1) ⋅ . . . ⋅ (un−1 − zn−1)

du1⋯dun−1

so the claim follows by Fubini’s theorem.

Corollar 2.2.7 Let U ⊂ Cn be open, f ∈ O(U) and Pρ(w) ⊂ U .Then

DJf(z) = J !

(2πi)n ∫Tρ(w)

f(u)(u − z)J+1

du

for all z ∈ Pρ(w), and we have the Cauchy inequality

∣DJf(w)∣ ≤ J !

ρJ∥f∥Tρ(w) .

Here we use the usual multi-index notation

DJf(z) ∶= ∂ ∣J ∣f

∂zJ11 . . . ∂zJnn

zJ ∶= zJ11 ⋅ . . . ⋅ zJnnJ + 1 ∶= (J1 + 1, . . . , Jn + 1)J ! = J1! ⋅ . . . ⋅ Jn!.


and ∥f∥M denotes the sup norm on M , i.e., the supremum over all∣f(u)∣ with u ∈M .Proof. The proof of the first statement is essentially the same as inone variable (interchange integration and differentiation), we justhave to introduce an ordering on (N0)n and then use induction onJ . For the second statement, we use again Fubini.

Definition 2.2.8 Let w ∈ Cn we write a formal power series inz = (z1, . . . , zn) in the usual multi-index notation

∞∑J=0

aJ(z −w)J ∶=∞∑

j1,...,jn=0

aJ(z1 −w1)j1 ⋅ . . . ⋅ (zn −wn)jn

with aJ ∈ C. The power series is called convergent if ∀ε > 0 thereis an N0 ⊂ (N0)n such that

RRRRRRRRRRR∑

J∈A/BaJ(z −w)J

RRRRRRRRRRR< ε

for all A,B ⊂ (N0)n with N0 ⊂ A and N0 ⊂ B.

Things become easier, if we limit ourselfs to the notion of ab-solute convergence, that is, convergence of the real non-negativeseries ∑J ∣aJ(z −w)J ∣, which implies convergence and is usually okfor power series, see Lemma 2.2.10 below.

Similar as in the univariate case one proves:

Theorem 2.2.9 Let U ⊂ Cn be open, and let (fk) be a sequencein O(U). If (fk) converges compactly (that is, uniformly on ev-ery compact subset) to f ∶ U → C, then f ∈ O(U). In particular,convergent power series define holomorphic functions (since theyconverge absolutely and compactly).

Lemma 2.2.10 (Abel) Let ∑J

aJzJ be a formal power series and

u ∈ Cn/0.

1) If the setaJuJ ∣ J ∈ (N0)n ⊂ C

is bounded, then ∑J

∣aJzJ ∣ converges compactly in

P∣u∣(0),

where ∣u∣ ∶= (∣u1∣ , . . . , ∣un∣).


2) If the set aJuJ ∣ J ∈ (N0)n ⊂ C is not bounded, then ∑J

aJzJ

is divergent for all z ∈ Cn with

∣zi∣ > ∣ui∣ for all i.

Proof. Let z ∈ P∣u∣(0). Then

∣zj ∣∣uj ∣

< 1

for all j. Let ε > 0. Since

∑J∈Nn

zJ

is compactly convergent in P(1,...,1)(0) and

∑J

zJ =n

∏j=1

1

1 − zjthere is a k ∈ N such that

∑∣J ∣≥k

∣zJ ∣∣uJ ∣

< ε

So if C is a bound for the ∣aJuJ ∣ then

RRRRRRRRRRR∑∣J ∣≥k

aJzJ

RRRRRRRRRRR≤ ∑∣J ∣≥k

∣aJ ∣ ∣z∣J = ∑∣J ∣≥k

∣aJwJ ∣∣z∣J

∣w∣J≤ C ∑

∣J ∣≥k

∣z∣J

∣w∣J< C ⋅ ε.

The second claim is trivial.

Corollar 2.2.11 Let f ∈ O(Pρ(w)). Then the Taylor series

∑J∈Nn

1

J !DJf(w)(z −w)J

converges on Pρ(w) to f .

Proof. By translation we can assume that w = 0.

1) Convergence: Let z ∈ Pρ(0). By Cauchy’s inequality (Corol-lary 2.2.7), the set

1

J !DJf(0)zJ ∣ J ∈ Nn

is bounded by∣z∣J

ρJ∥f∥Tρ(w)

with ∣zj ∣ < ρj < ρj for all j. Hence Lemma 2.2.10 gives conver-gence in P∣z∣(0). Since z ∈ Pρ(0) was arbitrary, this impliesconvergence in Pρ(0).


2) Limit: With the Cauchy integral formula (Theorem 2.2.6) wehave for all z ∈ Pρ(0)

f(z) = 1

(2πi)n ∫Tρ(w)

f(u)u − z

du = 1

(2πi)n ∫Tρ(w)

f(u)∑J

zJ

uJ+1du

=∑J

⎛⎜⎝

1

(2πi)n ∫Tρ(w)

f(u) zJ

uJ+1du

⎞⎟⎠=∑

J

1

J !DJf(0)zJ

where we can exchance summation and integration due tocompact convergence, and the last equality follows by Corol-lary 2.2.7.

So let us summarize: If U ⊂ Cn is open, then as in the univariatesetting the following are equivalent:

1) f ∈ O(U), that is, f is holomorphic, that is, f is complexdifferentiable in every coordinate,

2) f is continuously partially differentable and the Cauchy-Riemanndifferential equations are satisfied,

3) f is analytic, that is, f can locally be developed into a powerseries with positive convergence radius.

2.3 Multivariate holomphic functions:

What is the same and what not?

We now discuss the analogues of the key features of holomorphicfunctions in the multivariate case. There are a lot of similarities tothe situation of Section 2.1.4, but also some important differences.The mulitvariate identity theorem is a consequence of the univariateidentity theorem and is obtained by restricting the multivariateholomophic function f to a line segment L ⊂ Cn. Note that such arestriction is then, by parametrizing L by an open subset U ⊂ C, aunivariate holomphic function

U → Lf→ C

since the restriction of an analytic function is analytic.


Theorem 2.3.1 (Identity theorem) Let G ⊂ Cn be a domain(that is, an open and connected subset). Let f, g ∈ O(G), U ⊂ Gopen, U ≠ ∅. Then

f ∣U= g ∣UÔ⇒ f = g

Proof. The setV ∶= x ∈ G ∣ f(x) = g(x)

is not empty since U ⊂ V . Assume V is strictly smaller than G.Then there is an x0 ∈ ∂V ∩G. We can hence construct an open linesegment L (i.e. a line segment excluding the end points) throughx0 such that L ∩ V ≠ ∅ and the set L ∩ U ≠ ∅ is open inside L,see Figure 2.1.1 By the univariate Identity Theorem 2.1.18 applied

VG

L

Ux

x0

1

Figure 2.1: Proof of the identity theorem.

to U ∩ L ⊂ L ⊂ C, we get f ∣L= g ∣L. Since x0 ∈ ∂V , there is anx1 ∈ L/V ≠ ∅. So since x1 ∈ L we have f(x1) = g(x1), and sincex1 ∉ V we have f(x1) ≠ g(x1), a contradiction. We conclude thatV = G.

Remark 2.3.2 Unlike in the univariate case, equality on a se-quence with an accumulation point is not enough to get f = g: Con-sider the function f, g ∶ C2 → C with f(z) = z1 and g(z) = z1z2 andthe sequence zj = (1

j ,1).

The open mapping theorem we obtain from the multivariateidentity theorem and the univariate open mapping theorem. Firstrecall that a function f ∶X → Y of topological spaces is open if andonly if for every x ∈X and every neighborhood U of x, there existsa neighborhood V of f(x) such that V ⊂ f(U).

1A connected subset G ⊂ Cn is a set which cannot be written as the disjointunion of two non-empty open sets. If G is open then connected and path con-nected are the same. Path connected means, that any two points are connectedby a continuous path. This is used to construct the line segment L.


Theorem 2.3.3 (Open mapping theorem) Let G ⊂ Cn be a do-main and f ∈ O(G). If f is not constant, then f is open.

So f maps open sets to open sets, and domains to domains.Proof. Let a ∈ G and Uε(a) = z ∈ Cn ∣ ∥z − a∥ < ε ⊂ G a ballaround a. By the multivariate Identity Theorem 2.3.1, there is anelement b ∈ Uε(a) with f(a) ≠ f(b) (if not then f would be constanton G, contradicting our assumption). Consider the map

F ∶ U1(0)→ C, λ↦ f((1 − λ) ⋅ a + λ ⋅ b)

(where U1(0) ⊂ C is the unit ball around 0) obtained as the com-position of f with the parametrization of the open line segmentbetween a and b (see in Figure 2.2). By the univariate Open Map-

G

a

b

Figure 2.2: Proof of the Open Mapping Theorem.

ping Theorem 2.1.19 applied to the holomorphic function F , theimage L(U1(0)) is an open neighborhood of f(a) in C.

Remark 2.3.4 Note that it is sufficient to use one dimension ofthe source to generate an open neighborhood of the target.

The maximum modulus principle is then again a consequenceof the open mapping theorem:

Theorem 2.3.5 (Maximum modulus principle) Let G ⊂ Cn

be a domain and f ∈ O(G). If there is an element z ∈ G with

∣f(z)∣ = ∥f∥Gthen f is constant.

Proof. Suppose f is not constant. Then by Theorem 2.3.3 thesubset f(G) ⊂ C is a domain. Hence

∣f(w)∣ ∣ w ∈ G ⊂ R≥0

is open so it does not contain its supremum, hence there is no zwith ∣f(z)∣ = ∥f∥G, a contradiction.


2.4 Analytic subsets

Analytic subsets play an important role in multivariate complexanalysis, since they describe loci over which holomophic functionscan be extended. For this to work one needs, just like in the univari-ate setting, that the function is locally bounded around the analyticset. However, we will see that if the codimension of the analyticset is large enough, the boundedness condition is not necessary.Another important application of analytic sets is the description ofsubmanifolds in terms of local equations (see Remark 1.5.2).

Definition 2.4.1 Let U ⊂ Cn be open. A subset A ⊂ U is called ananalytic subset of U if for every z ∈ U there is an open neighbor-hood z ∈ V ⊂ U and f1, . . . , fm ∈ O(V ) such that

A ∩ V = z ∈ V ∣ f1(z) = . . . = fm(z) = 0

that is, A is locally the zeroset of a finite number of holomorphicfunctions, see Figure 2.3.

U m

Figure 2.3: Analytic subset as the a zeroset of holomorphic func-tions.

Remark 2.4.2 Note that if we require the condition in the defini-tion only for every z ∈ A, this leads to different notion (so calledanalytic sets): If A ⫋ U is a domain it is an analytic set howevernot an analytic subset.

Similar to the univariate setting, we have the following theoremon extending holomophic functions:

Theorem 2.4.3 (First Riemann extension theorem) Let G ⊂Cn be a domain, A ⊂ G analytic, f ∈ O(G/A). If f is locallybounded, that is, for every z ∈ A there is a an open neighborhoodz ∈ V ⊂ G such that ∥f∥V /A < ∞, then there is a unique f ∈ O(G)with

f ∣G/A= f.


We skip the proof since the situation is analogous to the uni-variate case. Note that uniqueness is clear by the Identity Theorem2.3.1.

Definition 2.4.4 Let U ⊂ Cn be open, A ⊂ U analytic, and a ∈ A.Then we say that the codimension codima(A) of A at a is equal tos if there is an s-dimensional, but not an (s+1)-dimensional affinecomplex subspace V ⊂ Cn (that is, a translate of a linear subspace)such that a is an isolated point of V ∩A, see Figure 2.3. We define

codimA ∶= mina∈G

codima(A)

A

a

L

Figure 2.4: Determining the codimension of an analytic subset.

Remark 2.4.5 Let G ⊂ Cn be a domain, A ⊂ G analytisch, andA ≠ G. Then

codimA ≥ 1.

Proof. We may assume that G is a ball. Let a ∈ A and b ∈ G/A,and let L be the line through a and b (which is then contained inG). So by the univariate Identity Theorem 2.1.18, the set A ∩ Lis discrete (otherwise we would have L ⊂ A, which contradicts theassumption). Hence codimaA ≥ 1.


Theorem 2.4.6 (Hartogs’ continuity theorem) Let n ≥ 2, G ⊂Cn−1 a domain, and U ⊂ G open. Let r > r > 0 be real numbers and

M = (U × Pr(0)) ∪ (G × (Pr(0)/Pr(0)))

(see Figure ) and f ∈ O(M). Then there is a unique f ∈ O(G ×Pr(0)) with

f ∣M= f .

Proof. For existence use the Cauchy integral formula. Uniquenessfollows from the identity theorem and since the components of G×(Pr(0)/Pr(0)) are connected through U × Pr(0).

G

n-1

U

Figure 2.5: The set M in Hartogs’ theorem.

Theorem 2.4.7 (Hartogs’ Kugelsatz) Let n ≥ 2 and r, r ∈ Rn

with rj > rj > 0 for all j. Then the restriction map

O(Pr(0))Ð→ O(Pr(0)/Pr(0))

is bijective, see Figure 2.6.

Proof. Injectivity follows with the Identity Theorem 2.3.1. Sur-jectivity is a special case of Hartogs’ theorem 2.4.6 for

U = P(r1,...,rn−1)(0)/P(r1,...,rn−1)(0)G = P(r1,...,rn−1)(0)r = rnr = rn


Figure 2.6: Filling in the complement of two concentric polycylin-ders.

Remark 2.4.8 This theorem is wrong for n = 1, for example thefunction 1

z cannot be extended holomorphically to z = 0.

The reason why this does not happen in higher dimension is thatthe zerosets of holomophic functions have always codimension 1, sofor n ≥ 2 they are not isolated. The same applies to singularities.

Corollar 2.4.9 Let n ≥ 2, U ⊂ Cn open and f ∈ O(U). Then fdoes not have any isolated zeroes or singularities.

Proof. If f ∈ O(U/a) then, by Theorem 2.4.7, the function fcan be extended to the point a. If a ∈ U is an isolated zero of f ,then a is also an isolated singularity of 1

f .

Corollar 2.4.10 Let G ⊂ Cn be a domain, A = f = 0 ⊂ G forf ∈ O(G) and f not constant. Then

codimA = 1.

Proof. By Remark 2.4.5, we have ≥. Suppose there is an a ∈ Awith codimaA ≥ 2. Then there is an affine complex subspace V ⊂ Cn

with k ∶= dimV ≥ 2, such that V ∩ f = 0 contains a as an isolatedpoint. Since V ≅ Ck and the function f ∣V has isolated zeroes, weget a contradiction to Corollary 2.4.9.

Theorem 2.4.11 (Second Riemann extension theorem) Let G ⊂Cn be a domain, A ⊂ G analytic of

codimA ≥ 2.


Then the restriction map

O(G)→ O(G/A)

is bijective.

Remark 2.4.12 Note that we do not need any boundedness condi-tion.

Proof. Injectivity follows with the Identity Theorem 2.3.1. Letf ∈ O(G/A). In order to apply the First Riemann Extension The-orem 2.4.3, we show that for every a ∈ A there is an open neigh-borhood a ∈ V ⊂ G such that f ∣V /A is bounded. Without loss ofgenerality we can assume that a = 0. Since codimA ≥ 2, the point0 can be isolated by a 2-dimensional affine space, so after a lineartransformation there is an r > 0 with

A ∩ ( 0 ×P(r,r)(0)) = 0∩ ∩ ∩

Cn−2 C2 Cn

Hence there is an ε > 0 such that

K ∶= Pε(0) × (P(r,r)(0)/P(r/2,r/2)(0)) ⊂ P(r,...,r)(0)/A,

see Figure 2.7 (note that this figure is actually in C3). Fix somew ∈ Pε(0) and define

fw ∶= f ∣w×P(r,r)(0)/P(r/2,r/2)(0) .

Note that

P(r,r)(0)/P(r/2,r/2)(0) ⊂ P(r,r)(0)/P(r/2,r/2)(0).

By the Kugelsatz 2.4.7 (applied in dimension 2), the holomorphicfunction fw has a holomorphic extension

gw ∈ O(w × P(r,r)(0)).

By the maximum modulus principle 2.3.5 (again in dimension 2),the non-constant function gw ∣w×P(r/2,r/2)(0) attains its supremumon the boundary of w × P(r/2,r/2)(0), that is, in K. So

∥gw∥P(r,r)(0) ≤ ∥f∥K <∞,

for all w, hence

∥f∥(Pε(0)×P(r,r)(0))/A ≤ ∥f∥K <∞.


e 0 -e

Figure 2.7: Application of Kugelsatz for a codimension 2 analyticsubset.

3

Complex Manifolds

Let us first recall some notations from topology.

3.1 Some notes on the underlying topo-

logical space

Definition 3.1.1 A topological space (X,A) is a set X togetherwith a subset A ⊂ 2X of the power set of X with the followingproperties:

• ∅ ∈ A and X ∈ A.

• If A1, . . . ,Ak ∈ A then

⋂ki=1Ai ∈ A

• If I is some set and Ai ∈ A for all i ∈ I then

⋃i∈IAi ∈ A

The set A is called a topology on X and the elements of A arecalled open.

Definition 3.1.2 Let (X,A) and (Y,B) be topological spaces. Amap f ∶ X → Y is called continuous if for all B ∈ B the preimagef−1(B) ∈ A.

Definition 3.1.3 A topological space (X,A) is called compact ifevery open cover of X has a finite subcover.

A subset U ⊂ X is called a neighborhood of x ∈ X if there isan A ∈ A with x ∈ A ⊂ U .

It is called locally compact if every x ∈ X has a compactneighborhood.

36

3. COMPLEX MANIFOLDS 37

Remark 3.1.4 The condition locally compact is a minimal require-ment for manifolds, since a manifold is locally Euclidean: It lo-cally looks like a Euclidean space (that is, for every point p thereis a open neighborhood U such that U is homoemorphic to an opensubset of Rn). Any locally Euclidean space is locally compact, sinceif φ ∶ U → V ⊂ Rn is a homeomorphism, then there is a Euclideanopen ball

Bε(φ(p)) = x ∈ Rn ∣ ∥x − φ(p)∥ < ε ⊂ V

and for the closed (and bounded, hence compact) ball, we have

Bε/2(φ(p)) ⊂ Bε(φ(p)) ⊂ V .

The image

φ−1 (Bε/2(φ(p))) ⊂ U

of this ball under the continuous map φ−1 is then a compact neigh-borhood of p.

Definition 3.1.5 A topological space (X,A) is called Hausdorffif for every x, y ∈ X with x ≠ y there are neighborhoods x ∈ U andy ∈ V with U ∩ V = ∅.

The condition Hausdorff is a reasonable requirement for man-ifolds, since it excludes examples like the line with two origins.Moreover, we have:

Remark 3.1.6 Locally compact Hausdorff spaces have the prop-erty that measures defined on them have similar properties to theLebesgue measure on the real line, so we can compute integrals.

Remark 3.1.7 Some definitions of a manifold include the require-ment that the topological space (X,A) is second-countable, thatis, the topology has a countable base (so every element of A can bewritten as the union of elements of some countable subfamily of A).It has been shown by Rado that every connected Riemann sur-face (that is, a 1-dimensional complex manifold) is already secondcountable. Also compact metric spaces are second countable. Onthe other hand, Calabi and Rosenlicht have proven, that there areconnected complex manifolds (in the sense of a locally Euclideanspace) which are not second countable. Hence, it is reasonable notto ask for X to be second countable: On the one hand this conditionis known to hold anyway for a lot of interesting cases, and on theother hand, there are some interesting cases, which one would blankout by asking for second countable.


3.2 Charts and atlas

So let X be a locally compact Hausdorff space.

Definition 3.2.1 An n-dimensional chart is a homeomorphismusϕ ∶ U → V with U ⊂ X open and V ⊂ Cn open. Two charts ϕ ∶ U →V and ϕ′ ∶ U ′ → V ′ are called biholomorphically compatible ifthe map

ϕ′ ϕ−1 ∶ ϕ(U ∩U ′)→ ϕ′(U ∩U ′)

is biholomorphic, that is, holomorphic, bijective and its inverse (ϕ′ϕ−1)−1 = ϕ ϕ′−1 is holomorphic.

An atlas on X is a set

A = ϕα ∶ Uα → Vα ∣ ϕα chart, α ∈ A

such that for all α,β ∈ A the charts ϕα and ϕβ are biholomorphicallycompatible, and

X = ⋃α∈AUαAn atlas is called maximal if for every atlas B such that all

charts of B are compatible with all charts of A we have B ⊂ A.If X admits a maximal atlas A as above, then (X,A) is called

an n-dimensional complex manifold. The atlas A is called acomplex structure on X.

Remark 3.2.2 Every atlas A determines a unique maximal atlasAmax. Hence, for practical purposes it is enough to just specify anatlas.

Proof. Recall Zorn’s lemma: If a poset (partially ordered set) Phas the property that every chain in P has an upper bound in P ,then P contains a maximal element.

Now the set P of all atlases containing a given atlas A is par-tially ordered by inclusion. Given a chain Aj in P the set ⋃αAα ∈ P .Hence Zorn’s lemma implies that there is a maximal atlas contain-ing A.

Example 3.2.3 If U ⊂ Cn is open, then U is an n-dimensionalcomplex manifold. The map

ϕ ∶= id ∶ U → U

is a chart, and A = ϕ is an atlas (not a maximal one).


Example 3.2.4 The complex projective space Pn(C) is an n-di-mensional complex manifold: In Section 1.4 we have constructedan atlas (again not a maximal one). We have described projectivespace as the quotient

Pn(C) = (Cn+1/0)/C∗

Via the quotient map

π ∶ Cn+1/0→ Pn(C)

Pn(C) inherits the quotient topology, that is, U ⊂ Pn(C) is con-sidered as open, if π−1(U) is open. This is the finest topology onPn(C) such that π is continuous. Taking the point of view of Section1.3 (but over the complex numbers and considering the completesphere), we can define the map p via

Cn+1/0 πÐ→ Pn(C)∪ p

Sn

with the sphereSn = z ∈ Cn+1 ∣ ∥z∥ = 1.

Since Sn is compact and p is continuous, also Pn(C) = p(Sn) iscompact, hence, locally compact. Moreover it is Hausdorff: Giventwo distinct points a, b ∈ Pn(C) the sets a∩Sn and b∩Sn are closedand disjoint in Sn. In the metric space Sn we can hence find smallenough balls around points in a ∩ Sn and b ∩ Sn, respectively, suchthat the images of these balls under p are disjoint.

Example 3.2.5 Let Γ ⊂ Cn be a lattice, that is

Γ = Za1 + . . . +Za2n

with a1, . . . , a2n ∈ Cn = R2n linearly independent over R. Then

T = Cn/Γ

is called a torus.For example, we could consider

T = C/Z ⟨2,1 + i⟩ .

Every torus is a complex manifold with the complex structureinduced by

π ∶ Cn → Cn/Γ


as follows: We consider T with the quotient topology. The mapπ is locally a homeomorphism, that is, for every x ∈ Cn there isa neighborhood U = U(x) ⊂ Cn such that π ∣U ∶ U → π(U) =∶ V isa homeomorphism. An atlas is given by the inverses ϕ ∶ V → U ,ϕ = (π ∣U)−1 of all these maps. If ϕ ∶ U → V and ϕ′ ∶ U ′ → V ′ aresuch charts, then

ϕ′ ϕ−1 ∶ V ∩ ϕ′(U ′)→ ϕ(U) ∩ V ′

is biholomorphic, since it is given by translation, that is, there is alattice vector γ ∈ Γ such that

(ϕ1 ϕ−12 )(z) = z + γ,

see Figure 3.1, which illustrates this map in the above example.

UV

U‘V‘

=

Re

Im

Figure 3.1: Covering of torus.

More generally, if π ∶M → N is a topological covering space, Mis a complex manifold and the deck tranformations (that is, home-omorphisms φ ∶ M → M with p φ = p) are holomorphic, then thecomplex structure on M induces via π a complex structure on N .

In the torus case, the deck transformations are the translationsby elements of Γ.

3.3 Holomorphic functions on manifolds

What are holomorphic functions on a complex manifold?

Definition 3.3.1 Let X be a complex manifold and A an atlas. Letf ∶X → C be a continuous map. Then f is called holomorphic iffor every (ϕ ∶ U → V ) ∈ A, the map

f ϕ−1 ∶ Vϕ−1→ U

f→ C∩ ∩Cn X


is holomorphic, see Figure 3.2.

X

U

j

jf

f

V-1

Figure 3.2: Holomorphic function on a manifold.

Remark 3.3.2 This definition is independent of the choice of at-las: Suppose f is holomophic with respect to A, and A′ is anotheratlas with Amax = A′max. Then we have to show that

f (ϕ′)−1 ∶ V ′ → C

is holomorphic for all (ϕ′ ∶ U ′ → V ′) ∈ A′. For all charts (ϕ ∶ U →V ) ∈ A, we have on ϕ′(U ∩U ′) that

f (ϕ′)−1 = f (ϕ−1 ϕ) (ϕ′)−1 = (f ϕ−1) (ϕ (ϕ′)−1),

see Figure 3.3. Since Amax = A′max, the charts ϕ and ϕ′ are biholo-morphically compatible, so

ϕ (ϕ′)−1 ∶ ϕ′(U ∩U ′)→ ϕ(U ∩U ′)

is holomorphic. Moreover, f ϕ−1 is holomorphic on ϕ(U ∩U ′) ⊂ Vby assumption. Hence, it follows by the above equality that also f (ϕ′)−1, which is the composition of these two maps, is holomophic.So our definition of biholomorphic compatiblity was exactly right!

Remark 3.3.3 To show that f ∶X → C is holomorphic it is enoughto show that for every x ∈ X there is a chart ϕ ∶ U → V withϕ ∈ Amax and x ∈ U such that f ϕ−1 is holomorphic.


X

U

j

j

j‘

(j‘)

(j‘)

VV‘

U‘

-1

-1

jf

f

f

-1

Figure 3.3: Holomorphic functions on manifolds and coordinatechange.

Definition 3.3.4 We write

O(X) = OX(X) = f ∶X → C ∣ f holomorphic

Remark 3.3.5 If X is a complex manifold with atlas A and U ⊂Xopen, then U is a complex manifold with atlas

AU = ϕ ∣U ∣ ϕ ∈ A .

We define thenOX(U) ∶= OU(U)

Remark 3.3.6 All local notions and theorems from Section 2 (Rie-mann extension theorems, maximum modulus principle, open map-ping theorem, identity theorem, etc.) transfer directly to complexmanifolds by using charts.

How to define now holomorphic maps between manifolds?

Definition 3.3.7 Suppose (X,A) and (Y,B) are complex mani-folds of dimension n and m, and f ∶X → Y continuous. Then f is


called holomorphic if for all (ϕ ∶ U → V ) ∈ A and (ψ ∶ U ′ → V ′) ∈B the map

ψ f ϕ−1 ∶ W → U → U ′ → W ′

∩ ∩Cn Cm

where W = ϕ(f−1(U ′)∩U) and W ′ = ψ(f(U)∩U ′) is holomorphic,see Figure 3.4. By doing a coordinate change just like in Remark

X X

U f(U)

j

j

y

yV

f

f

f

V‘

U‘(U‘)

-1

-1

Figure 3.4: Holomorphic map between manifolds

3.3.2, however both in source and target, we can again show thatthis notion is well-defined.

3.4 Analytic subsets of manifolds

Definition 3.4.1 Let X be a complex manifold of dimension n. Asubset A ⊂ X is called an analytic subset if there is an atlas Asuch that for all (ϕ ∶ U → V ) ∈ A the subset

ϕ(A ∩U) ⊂ V ⊂ Cn

is analytic.


This definition is independent of the choice of the atlas sincebiholomophic maps map analytic subsets of Cn to analytic subsetsof Cn.

Remark 3.4.2 According to our defintion of holomorphic functionon manifolds, the subset A ⊂ X is analytic if for ever x ∈ X (notethat again Remark 2.4.2 applies.) there is an open neighborhoodx ∈ W ⊂ X and f1, . . . , fm ∈ OX(W ) such that A ∩W = f1 = . . . =fm = 0, see Figure 3.5.

Vm

X

U

j

Figure 3.5: Analytic subset of a complex manifold.

Definition 3.4.3 In the notation of Definition 3.4.1 we define fora ∈ A

codima(A) ∶= codimϕ(a)ϕ(A ∩U)

andcodimA ∶= mina∈A codima(A)

With this notion we can formulate the second Riemann exten-sion theorem on manifolds:

Theorem 3.4.4 Let X be a complex manifold, A ⊂X analytic andcodim(A) ≥ 2. Let U ⊂ X be open and f ∈ OX(U/A). Then thereis a unique f ∈ OX(U) with

f ∣U/A= f .


Proof. Existence: It is enough to construct f locally, so we mayassume that ϕ ∶ U → V ⊂ Cn is a chart of X. By definition, B =ϕ(A) ⊂ V has codimB ≥ 2. Then by the second Riemann extensiontheorem on Cn (Theorem 2.4.11) applied to g ∶= f ϕ−1 ∈ O(V /B),there is a unique g ∈ O(V ) with g ∣V /B= g. If we define f ∶= g ϕ,then

f ∈ OX(U)and

f ∣U/A= g ϕ ∣U/A= g ∣V /B ϕ ∣U/A= g ϕ ∣U/A= f ϕ−1 ϕ ∣U/A= f .

It is important to note that analytic subsets are in general onlygiven locally by holomorphic functions:

Example 3.4.5 Consider the projective space P2(C) with homoge-neous coordinates (x0 ∶ x1 ∶ x2) and the projective subvariety

A = V (x1) ⊂X = P2(C).

The subset A ⊂X is an analytic subset: In the affine chart

ϕ0 ∶ U0 = x ∈ P2(C) ∣ x0 ≠ 0 → C2

(x0 ∶ x1 ∶ x2) ↦ (x1x0 ,x2x0

) = (x, y)

we haveϕ0(A ∩U0) = V (x) ⊂ C2

and in the affine chart

ϕ2 ∶ U2 = x ∈ P2(C) ∣ x2 ≠ 0 → C2

(x0 ∶ x1 ∶ x2) ↦ (x0x2 ,x1x2

) = (u, v)

we haveϕ2(A ∩U2) = V (v) ⊂ C2.

Note, however, that A cannot be given as the zero locus of a globalholomorphic function on P2(C). First note that, although x ∈ O(C2),this function does not extend to a global holomorphic function onP2(C), since it has a pole on the whole set x0 = 0: In U2 the functionx is

x = vu

hence it has poles along the subset V (u) as long as v ≠ 0. The setV (u) corresponds to the hyperplane at infinity of the chart ϕ0. Sowe can view x not as a holomorphic but as a meromorphic functionon P2(C).

Note that this observation does not contradict Theorem 3.4.4since the pole locus of x is of codimension 1.


Remark 3.4.6 In fact there is no non-constant global holomorphicfunction on Pn(C): As shown in Example 3.2.4, projective space iscompact. Hence any holomorphic (and thus continuous) function fassumes its maximum on Pn(C). By the maximum modulus princi-ple (Theorem 2.3.5), the function f must be constant. So we have

O(Pn(C)) ≅ C.

To describe analytic subsets again, if possible, as complex man-ifolds, we introduce the notation of a submanifold:

Definition 3.4.7 Let X be a complex manifold of dimension n andY ⊂ X a closed subset. Then Y is called a submanifold of codi-mension k if for all y ∈ Y there is a chart ϕ ∶ U → V around y ∈ Xsuch that

ϕ(U ∩ Y ) = V ∩ (Cn−k × 0)with 0 ∈ Ck, see the upper part of Figure 3.6.

Remark 3.4.8 The submanifold Y is an (n−k)-dimensional man-ifold: According to the definition of a submanifold, we can form anatlas A of X such that for all (ϕ ∶ U → V ) ∈ A we have

ϕ(U ∩ Y ) = V ∩ (Cn−k × 0).

Then the composition of the restriction ϕ ∣U∩Y of the chart mapwith the projection

π ∶ Cn−k × 0→ Cn−k

on the first n − k coordinates yields a chart map

ψ ∶ U ∩ Yϕ∣U∩YÐ→ Vα ∩ (Cn−k × 0) πÐ→ π(Vα ∩ (Cn−k × 0))

(see again Figure 3.6) and all such ψ form an atlas of Y . So ourdefinition of a submanifold was exactly as it should be!

Example 3.4.9 The analytic subset in Example 3.4.5 is indeed asubmanifold: Using again the affine charts of projective space, byRemark 3.4.8, we obtain an atlas of A with the charts

ϕ0 ∶ U0 ∩A = x ∈ P2(C) ∣ x1 = 0, x0 ≠ 0 → 0 ×C1 ⊂ C2

(x0 ∶ 0 ∶ x2) ↦ (0, x2x0 ) = (0, y)

and

ϕ2 ∶ U2 ∩A = x ∈ P2(C) ∣ x1 = 0, x2 ≠ 0 → C1 × 0 ⊂ C2

(x0 ∶ 0 ∶ x2) ↦ (x0x2 ,0) = (u,0)


Vn-k

n-k

n

X

U

j

Figure 3.6: Submanifold: Definition and interpretation as a mani-fold.

Note that the chart x2 ≠ 0 is not necessary to cover A, since it isdisjoint from A.

The manifold structure is then obtained by via projection

ϕ0 ∶ U0 ∩A → 0 ×C1 → C1

(x0 ∶ 0 ∶ x2) ↦ (0, x2x0 ) = (0, y) ↦ y

and

ϕ2 ∶ U2 ∩A → C1 × 0 → C1

(x0 ∶ 0 ∶ x2) ↦ (x0x2 ,0) = (u,0) ↦ u

So when, in general, is an analytic subset defined locally byf1, . . . , fm ∈ OX(U) actually a submanifold of X?


We will see that this is the case if and only if the rank of theJacobian matrix of partial derivatives of the fi is constant, wherewe will define the Jacobian matrix in terms of charts.

Definition 3.4.10 Let X ⊂ Cn be open, f ∶ X → Cm holomorphic,and a ∈X. Then the Jacobian matrix of f at a is

Jf(a) ∶=∂f

∂z(a) ∶= (∂fi

∂zj(a))

i=1,...,mj=1,...,n

Let X is an n-dimensional complex manifold, f ∶X → Cn holomor-phic, and a ∈X. Then we say that

detJf(a) ≠ 0

if there is a chart ϕ of X around a with

detJfϕ−1(ϕ(a)) ≠ 0

Remark 3.4.11 This definition is independent of the choice of ϕ ∶U → V since if ψ ∶ U ′ → V ′ is another chart around a, then by thechain rule and the product function for determinants

detJfϕ−1(ϕ(a)) = detJfψ−1ψϕ−1(ϕ(a))= det(Jfψ−1(ψ(a)) ⋅ Jψϕ−1(ϕ(a)))= detJfψ−1(ψ(a)) ⋅ detJψϕ−1(ϕ(a))

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶≠0

where detJψϕ−1(ϕ(a)) ≠ 0 since ψ ϕ−1 is biholomorphic:If β ∶W →M with W,M ⊂ Cn open is a biholomorphic map and

a ∈W , then by the chain rule

1 = detJid(a) = detJβ−1β(a) = detJβ−1(a) ⋅ detJβ(a).

Note that detJψϕ−1(ϕ(a)) is not necessarily equal to 1, so onlydetJf(a) ≠ 0 is well-defined, not the actual value of detJf(a).

Definition 3.4.12 More generally, we define for a holomorphicmap f ∶X → Y of complex manifolds of dimension n and m

ranka(f) ∶= rankϕ(a)(ψ f ϕ−1) ∶= rank(Jψfϕ−1(ϕ(a)))

for charts ϕ ∶ U → V around a and ψ ∶ U ′ → V ′ around f(a).By a similar argument as in Remark 3.4.11 one shows that this iswell-defined (using the characterization of rank via non-vanishingof minors of the respective size).

Similarly, for n =m we can define

detJf(a) ≠ 0 ∶⇔ detJψfϕ−1(ϕ(a)) ≠ 0

which is the case if and only if ranka(f) = n.


Theorem 3.4.13 (Inverse function theorem) Let X be a com-plex manifold of dimension n, let

f ∶X → Cn

be holomorphic and a ∈X. Then

detJf(a) ≠ 0

if and only if there is an open neighborhood U ⊂X of a such that

f ∣U ∶ U → f(U)

is biholomorphic.

Proof. By passing to a chart we can assume that X ⊂ Cn open.The conclusion ⇐ we have already seen in Remark 3.4.11. Forthe other direction ⇒ we use the well-known real version of thetheorem: This states that if

detJRf (a) ≠ 0

then there is an open neighborhood U ⊂ R2n = Cn of a such thatf ∣U ∶ U → f(U) is bijective and q = (f ∣U)−1 is real differentiable.This theorem can be applied since

∣detJf(a)∣2 = ∣detJRf (a)∣ ,

Note, that if n = 1 (for n > 1 the proof work analogously) and wewrite f = h + i ⋅ g, then

∣∂f∂z

∣2

= ∣12(∂f∂x

− i∂f∂y

)∣2

= 1

4∣∂h∂x

+ i∂g∂x

− i∂h∂y

+ ∂g∂y

∣2

= 1

4∣∂g∂y

+ i∂g∂x

+ i∂g∂x

+ ∂g∂y

∣2

= (∂g∂x

)2

+ (∂g∂y

)2

= det(∂g∂y

∂g∂x

− ∂g∂x∂g∂y

) = det(∂h∂x

∂g∂x

∂h∂y

∂g∂y

)

using twice the Cauchy-Riemann differential equations

∂g

∂x= −∂h

∂yand

∂g

∂y= ∂h∂x

.

In remains to show that the local inverse q is holomorphic. We have

0 = ∂ id

∂z= ∂(f q)

∂z= (∂f

∂z q) ⋅ ∂q

∂z+ (∂f

∂z q) ⋅ ∂q

∂z= (∂f

∂z q) ⋅ ∂q

∂z

Applying this equality to f(a) ∈ f(U) and using ∂f∂z (a) ≠ 0, it follows

that ∂q∂z(f(a)) = 0, that is, q is holomorphic.


Remark 3.4.14 Due to the local nature of Theorem 3.4.13, thesame statement works for a map of f ∶ X → Y between complexmanifolds of dimension n.

From the inverse function theorem, we obtain in the usual waythe implicit function theorem:

Corollar 3.4.15 (Implicit function theorem) Let U ⊂ Cn beopen, V ⊂ Cm open,

f ∶ U × V → Cm

holomorphic, and (a, b) ∈ U × V with

f(a, b) = 0,

write (z,w) for the coordinates on Cn+m, and assume that

rank( ∂f∂w

(a, b)) =m.

Then there are open neighborhoods a ∈ U1 ⊂ U , b ∈ V1 ⊂ V and aholomorphic function

g ∶ U1 → V1

such that for all (z,w) ∈ U1 × V1 we have

f(z,w) = 0⇔ w = g(z),

see Figure 3.7.

Proof. For the function

F ∶ U × V → Cn+m, (z,w)↦ (z, f(z,w))

we have

detJF (a, b) = det

⎛⎜⎜⎜⎝

1⋱

10

∂f∂z (a, b)

∂f∂w(a, b)

⎞⎟⎟⎟⎠≠ 0,

hence, Theorem 3.4.13 gives a ∈ U1 ⊂ U , b ∈ V1 ⊂ V , and W ⊂ Cn+m

open withF (a, b) = (a, f(a, b)) = (a,0) ∈W

such thatF ∣U1×V1 ∶ U1 × V1 →W


f(z,w)=0

n

m

Figure 3.7: Implicit function theorem

has a holomorphic inverse

F −1 ∶W → U1 × V1.

With the projection

π ∶ Cn+m → Cn, (z, s)↦ z

this function is of the form

F −1 = (π ∣W , g)

with a uniquely determined g and

(z, s) = (F F −1) (z, s) = (F (π ∣W , g)) (z, s) = (z, f(z, g(z, s))).

So we havef(z, g(z,0)) = 0

for all z ∈ π(W ) ⊂ Cn.We could have applied the argument also to arbitrary fixed s,

thus describing the level sets of g.

Corollar 3.4.16 (Rank theorem) Let X and Y be a complexmanifold of dimension n and m, let

f ∶X → Y


be holomorphic, a ∈X, and

rank (Jf(x)) = r

constant for all x ∈ U in an open neighborhood a ∈ U ⊂ X. Thenthere are open neighborhoods a ∈ U ⊂ U and f(a) ∈ V ⊂ Y , andpolydiscs P ⊂ Cn and Q ⊂ Cm around zero, and biholomorphic mapsϕ ∶ P → U and ψ ∶ Q → V with ϕ(0) = a and ψ(0) = b such that thediagram

Xf→ Y

∪ ∪a ∈ U → V ∋ f(a)

ϕ ↑ ↑ ψ0 ∈ P → Q ∋ 0

∩ ∩Cn π→ Cm

with the projection

π ∶ Cn → Cm, (z1, . . . , zn)→ (z1, . . . , zr,0, . . . ,0)

commutes.

Remark 3.4.17 This means that a holomorphic map of constantrank, by choice of appropriate charts, can be represented as thesimplest possible map of rank r.

We skip the proof of the theorem since it is more complicated,but similar to the proof of Theorem 3.4.15. Choosing the rightmaps to invert, it again relies on Theorem 3.4.13.

The rank theorem allows us to invesigate, when an analyticsubset is indeed a submanifold:

Theorem 3.4.18 Let X be an n-dimensional complex manifoldand Y ⊂ X closed. Then Y is a complex submanifold of X if andonly if for all y ∈ Y there is an open neighborhood y ∈ U ⊂X and aholomorphic function f ∶ U → Cm with

U ∩ Y = f = 0

andr ∶= rank (Jf(x))

is constant for all x ∈ U .In this case,

dimY = n − r.


Proof. We prove ⇒: Let Y be a submanifold of X, let y ∈ Y and

ϕ ∶ U → P

a chart around y withϕ(y) = 0,

where P ⊂ Cn is a polydisc around 0, such that

ϕ(Y ∩U) = Q × 0

with a polydisc 0 ∈ Q ⊂ Cn−r. Write P = Q × Q′ with a polydiscQ′ ⊂ Cr, denote the repspective coordinates as (x,w) and considerthe projection map

π ∶ P → Q′, (x,w)↦ w

to the transversal direction. With

f ∶= π ϕ ∶ U → Q′.

we then have

z ∈ U ∣ f(z) = 0 = z ∈ U ∣ π(ϕ(z)) = 0= ϕ−1((x,w) ∈ P ∣ w = π(x,w) = 0)= ϕ−1(Q × 0) = Y ∩U ,

see Figure 3.6. Note that we only used the definition of a subman-ifold.

For the converse⇐ we apply the Rank Theorem 3.4.16 at y ∈ U .After possibly making U smaller, we obtain a commutative diagram

Uf→ V

α ↑ ↑ βP →

πQ

(x,w) ↦ (0,w)

with polydiscsP = Q ×Q′ ⊂ Cn

Q ⊂ Cn−r

Q′ ⊂ Cr

Q ⊂ Cm

andπ(P ) = 0 ×Q′ ⊂ Q ⊂ Cm


n-r

r

n

X

U

P

j

p

Figure 3.8: Submanifold as an algebraic set.

Forϕ ∶= α−1

we then have

U ∩ Y = z ∈ U ∣ f(z) = 0= z ∈ U ∣ (β−1 f)(z) = 0= z ∈ U ∣ (π α−1)(z) = 0= z ∈ U ∣ (π ϕ)(z) = 0= ϕ−1((x,w) ∈ P ∣ w = π(x,w) = 0)= ϕ−1(Q × 0)

so Y is a complex submanfold of X (and hence manifold) of dimen-sion n − r.

4

Sheaves

4.1 The definition and where it comes

from

Functions have specific properties with regard to how they behaveunder restrictions. These properties are collected in the definitionof a presheaf:

Definition 4.1.1 Let X be a topological space. A presheaf F ofabelian groups consists of

1) abelian groups F(U) for every U ⊂X open,

2) group homomorphisms ρU,V ∶ F(U) → F(V ) for all V ⊂ Uopen in X with the properties

(a) F(∅) = 0(b) ρU,U = id for all U ⊂X,

(c) ρU,W = ρU,V ρV,W for all W ⊂ V ⊂ U open in X.

Remark 4.1.2 In the same way, one can define presheafs of rings,vector spaces, algebras, etc.

The ρU,V in the definition of a presheaf model the properties ofa restriction map of functions defined on the respective open sets.However functions have additional properties: If they restrict on aopen covering to zero then they have already been zero, and if wehave functions on an open covering which coincide on intersectionsthen we can glue these functions to a global function if the glueingstill satisfies whatever property we ask to the function. These ideasare captured in the definition of a sheaf:

55

4. SHEAVES 56

Definition 4.1.3 A presheaf (of abelian groups,...) is called asheaf (of abelian groups,...) if it satisfies:

(A1) (Locality) If U ⊂X is open,

U = ⋃i∈IVi

with Vi ⊂X open,s ∈ F(U)

withρU,Vi(s) = 0 for all i ∈ I

thens = 0.

(A2) (Gluing) If U ⊂X is open,

U = ⋃i∈IVi

with Vi ⊂X open, and

si ∈ F(Vi) for i ∈ I

withρVi,Vi∩Vj(si) = ρVj ,Vi∩Vj(sj) for all i, j ∈ I

then there is an s ∈ F(U) with

ρU,Vi(s) = si for all i ∈ I.

Example 4.1.4 Continuous functions on open subsets for a topo-logical space X form a presheaf

C(X) = f ∶ U → C ∣ f continous

withρU,V ∶ C(U)→ C(V ), f ↦ f ∣V

and C(∅) = 0. This presheaf is indeed a sheaf: If the restrictionof a function f to all Vi is zero, then f = 0. If we have functions fion Vi which agree on the Vi ∩ Vj, then we can glue these functionsto a continuous function on U .

Notation 4.1.5 In view of the example of sheafs of functions, weintroduce for a presheaf F and s ∈ F(U) the notation

s ∣V ∶= ρU,V (s)

We call s a section of F on U .Then the sheaf condition for U = ⋃i∈IVi can be written as:

4. SHEAVES 57

(A1) s ∣Vi= 0 for all iÔ⇒ s = 0

(A2) si ∈ F(Vi) with si ∣Vi∩Vj= sj ∣Vi∩Vj for all i, j Ô⇒ there iss ∈ F(U) with s ∣Vi= si for all i.

Example 4.1.6 Let X be a complex manifold. Since the conditionholomorphic is local,

OX(U) = f ∶ U → C ∣ f holomorphic

is a sheaf of C-algebras with

ρU,V ∶ OX(U)→ OX(V ), f ↦ f ∣Vthe sheaf of holomorphic functions on X.

Example 4.1.7 Let X be a complex manifold. Then

O∗X(U) = f ∶ U → C ∣ f has no zeros

is a sheaf of abelian groups with ρU,V again the restriction map.

Example 4.1.8 Bounded functions on open subsets of C form apresheaf F together with restriction. This presheaf is not a sheaf,since if we cover

C = ⋃i∈IViwith bounded open sets (keep in mind that the sheaf condition hasto be satisfied for all covers by open sets), and we have fi ∈ F(Vi)with fi ∣Vi∩Vj= fj ∣Vi∩Vj , then we can glue these fi to a function onC, however this function does not have to be bounded: Take forexample

fi ∶ Vi → C, z ↦ z.

So (A2) is not satisfied.

Example 4.1.9 Let X = 0,1 with the discrete topology (whereevery subset is defined to be open) and

F(∅) = 0

F(0) = CF(1) = CF(0,1) = 0

and ρU,V = 0 execpt for U = V where ρU,V = id, then this a presheaf,but not a sheaf, since again (A2) is not satisfied:

X = 0 ∪ 1

and if we take f0 = 1 ∈ F(0) and f1 = 1 ∈ F(1), then thesecannot be glued to an f ∈ F(0,1) since F(0,1) = 0.

4. SHEAVES 58

Example 4.1.10 Let X = 0,1 with the discrete topology and

F(∅) = 0

F(0) = 0

F(1) = 0

F(0,1) = C

and ρU,V = 0 execpt for U = V where ρU,V = id, then this a presheaf,but not a sheaf, since (A1) is not satisfied:

X = 0 ∪ 1

and if 0 ≠ f ∈ F(X) = C, then ρX,0(f) = 0 and ρX,1(f) = 0, so if(A1) would be satisfied we would have f = 0, a contradiction.

4.2 Stalks

Definition 4.2.1 If X is a topological space, F a presheaf (ofabelian groups,...) on X and x ∈ X, we define the stalk of Fat x as

Fx =⋅⋃U∋xF(U) / ∼

where the disjoint union is taken over all open neighborhoods of x,and for s ∈ F(U), t ∈ F(V ) we define

s ∼ t⇐⇒ ∃W ⊂ U ∩ V open neighborhood of x such that s ∣W= t ∣W

Note this indeed defines an equivalence relation (exercise).

Remark 4.2.2 Note that if F a presheaf of abelian groups, thenFx is a again an abelian group: If s ∈ F(U) and t ∈ F(V ) arerepresentatives of the respective classes [s] and [t], then we define

[s] + [t] ∶= [s ∣U∩V +t ∣U∩V ]

(show as an exercise that this gives a group structure, in particularthat it is well-defined).

Definition 4.2.3 There is a canonical map

F(U)→ Fxs↦ sx ∶= [s]

(as an exercise, show that this is a homomorphism). We call sx thegerm of s in x.

4. SHEAVES 59

Example 4.2.4 Let X be a complex manifold of dimension n andx ∈X. Then

OX,x ≅ Cz1, . . . , znis the ring of convergent power series: If ϕ ∶ U → V ⊂ Cn is achart around x with ϕ(x) = 0, then any fx ∈ Fx can be mappedto the Taylor series expansion of f ϕ−1. On the other hand, anyholomorphic function is locally determined by its Taylor series.

Note that this statement is not true for the sheaf AR of infinitelyoften differentiable functions on a real differentiable (in particular,on a complex) manifold: Taylor expansion again gives is a ringhomomorphism

ARX,x → Cx1, . . . , xn

however this homomorphism is not injective: There are a lot ofinfinitely often differentiable function with Taylor series equal tozero:

Example 4.2.5 For

f ∶ R → R

x ↦ exp(− 1x2 ) fur x ≠ 0

0 fur x = 0

(Figure 4.1) we have f (k)(0) = 0 for all k ≥ 0 (Exercise). Hence

–1

0

1

–2 –1 1 2

Figure 4.1: Function with vanishing Taylor series.

the Taylor series of f in x = 0 is identically zero. In particular,the radius of convergence of the Taylor series is ∞, but the Taylorseries does not converge to f for x ≠ 0.

4. SHEAVES 60

4.3 Morphisms

Definition 4.3.1 Let F and G be presheafs of abelian groups onX. A presheaf morphism φ ∶ F → G is a collection of grouphomomorphisms

φU ∶ F(U)→ G(U)for each U ⊂X open such that for all V ⊂ U open in X the diagram

F(U) φU→ G(U)ρU,V ↓ ↓ ρ′U,VF(V ) →

φVG(V )

commutes. A presheaf morphism of sheaves is called a sheaf mor-phism.

Note that in this way the category of presheaves is a full sub-category of the category of sheaves.

Definition 4.3.2 If φ ∶ F → G and ψ ∶ G → H are presheaf mor-phisms then we define the composition

ψ φ ∶ F → H

by(ψ φ)U ∶= ψU φU

Definition 4.3.3 A morphism of presheaves φ ∶ F → G is calledinjective or a monomorphism of presheaves (surjective oran epimorphism of presheaves) if φU is injective (surjective)for all U .

A presheaf morphism φ ∶ F → G is called bijective or an iso-morphism of presheaves if it is injective and surjective, that is,there is a presheaf morphism ψ ∶ G → F (the inverse) such that

φ ψ = idG and ψ φ = idF

(where idU = id for all U).

Definition 4.3.4 For a morphism of presheaves φ ∶ F → G thekernel presheaf kerφ is defined as the presheaf

U ↦ ker′ φU

the image presheaf im′ φ is the presheaf

U ↦ im′ φU

and the cokernel presheaf coker′ φ is

U ↦ coker′ φU

4. SHEAVES 61

Remark 4.3.5 For a presheaf morphism φ ∶ F → G it is equivalent:

1) φ is injective,

2) φU is injective for all U ,

3) ker′ φ = 0.

The analogous statements hold for surjective and bijective.

Remark 4.3.6 Life in presheafs is easy, but also boring: A se-quence of presheaf morphisms

0→ F1 → . . .→ Fn → 0

is exact if and only if

0→ F1(U)→ . . .→ Fn(U)→ 0

is exact for all U . From this point of view of exactness, it does noteven make sense to introduce the concept of a sheaf, which captureslocal properties of mathematical objects.

Remark 4.3.7 If φ is a morphism of sheaves, then ker′ φ is indeedagain a sheaf (exercise). The reason is that being zero is a localproperty. We then define the kernel sheaf as

kerφ ∶= ker′ φ.

A morphism of sheaves is called injective if

kerφ = 0

that is, the presheaf morphism φ is injective.

For the image (and then also the cokernel) the situation is morecomplicated in the sense that the image presheaf is in general nota sheaf, since being in the image of a presheaf morphism on someU is not a local property:

Example 4.3.8 For X = C we have an exact sequence of presheafs

0→ Z 2πi→ OCexp→ F → 1

with the image presheaf F = im′(exp). Note that

F(U)

is the set of of functions admitting a holomorphic logarithm. How-ever there are functions which do not have a global logarithm, hencethe gluing axiom (A2) is not satisfied. Consider for example U =C/0, U1 = U/R<0, U2 = U/R>0 so U1 ∪ U2 = U . The functionf(z) = z has a logarithm on U1 and U2 however not on U .

4. SHEAVES 62

Remark 4.3.9 So we have to reconsider the definition of the imagesheaf, and hence a surjective sheaf morphism is something differentthan a surjective presheaf morphism: Indeed we will see that if themorphism of sheaves

φ ∶ F → G

is surjective, the maps

φU ∶ F(U)→ G(U)

do not have to be surjective for all U . The defect of surjectivity ismeasured by the concept of cohomology.

Example 4.3.10 We will see that

0→ Z 2πi→ OCexp→ O∗

C → 1

is an exact sequence of sheaf morphisms (essentially since non-zeroholomorphic functions have locally a logarithm), but in

0→ Z 2πi→ OC(U) exp→ O∗C(U)

the exp-map does not have to be surjective since, for example, forU = C/0 and f(z) = z there is no global logarithm.

Example 4.3.11 We will also see that the sequence of sheaves

0→ C→ OC

ddz→ OC → 0

is exact, but the last map in the sequence

0→ C→ OC(U)ddz→ OC(U)

is in general not surjective: If U is simply connected then by Cauchy’sintegral theorem every element in OC(U) has an antiderivative.However, if U is not simply connected this is not true: Considerfor example U = C/0 and f(z) = 1

z .

Before we look into this more delicate question, let us first ex-plore what can be said on morphisms in terms of stalks. Most ofthis is true for sheaves but not for presheafs.

Remark 4.3.12 If F is a sheaf and s ∈ F(U) then s is determinedby all sx for x ∈ U . This follows from the locality axiom (A1), butdoes not use (A2).

4. SHEAVES 63

Definition 4.3.13 If φ ∶ F → G is a morphism of presheafs on Xand x ∈X, then we obtain a morphism

φx ∶ Fx → Gxsx ↦ φ(s)x

Proof. This is well-defined: Suppose s ∈ F(U) and s′ ∈ F(U ′) withsx = s′x. Then there is a neighborhood V ⊂ U ∩U ′ with x ∈ V and

s ∣V = s′ ∣V .

ThenφU(s) ∣V = φV (s ∣V ) = φV (s′ ∣V ) = φU ′(s′) ∣V

henceφU(s)x = φU ′(s′)x.

Theorem 4.3.14 A morphism φ ∶ F → G of sheafs on X is anisomorphism if and only if φx is an isomorphism for all x ∈X.

Proof. For ⇒ note that if ψ ∶ G → F is the inverse, then ψx is theinverse of φx for all x ∈X.

For ⇐ we have to show that φU ∶ F(U) → G(U) is an isomor-phism for all neighborhoods U ⊂X:

1) φU is injective: Suppose s ∈ F(U) and

φU(s) = 0.

For any x ∈X we then have

φx(sx) = φU(s)x = 0.

Since we assume that φx is injective, we have

sx = 0 for all x ∈ U

and thats = 0,

since we have a covering Ui of U and si ∈ F(Ui) with

0 = si = s ∣Ui

which implies by (A1) that s = 0.

4. SHEAVES 64

2) φU is surjective: Let t ∈ G(U). For any x ∈X there is a vx ∈ Fxwith

φx(vx) = txand for this a neighborhood V (x) ⊂ U of x and an s(x) ∈F(V (x)) with

(s(x))x = vxand hence

(φV (x)(s(x)))x = φx(vx) = txAfter shinking V (x) we can assume that

φV (x)(s(x)) = t ∣V (x) .

On intersections V (x) ∩ V (x′) we have

φV (x)∩V (x′)(s(x) ∣V (x)∩V (x′)) = φV (x)∩V (x′)(s(x′) ∣V (x)∩V (x′))

Since φ is injective, we have

s(x) ∣V (x)∩V (x′)= s(x′) ∣V (x)∩V (x′)

SinceU = ⋃

x∈UV (x)

by (A2) there is hence an s ∈ F(U) with

s ∣V (x)= s(x)

for all x ∈ U . It follows that

φU(s) ∣V (x)= φV (x)(s ∣V (x)) = φV (x)(s(x)) = t ∣V (x) .

which implies by (A1) that

φU(s) = t.

Remark 4.3.15 1) Note that the proof of surjectivity uses theinjectivity.

2) The proof uses both sheaf axioms A1 and A2 in an essentialway. This theorem is wrong for isomorphisms of presheaves.

4. SHEAVES 65

Remark 4.3.16 In a similar fashion as the injectivity, one provesthat a morphism of sheaves

φ ∶ F → G

on X is determined by the morphisms

φx ∶ Fx → Gx

of the stalks for all x ∈ X. This is again wrong for morphisms ofpresheaves.

Remark 4.3.17 Note that this only means that if we are givenφx which fit together to a sheaf morphism, then this morphism isuniquely determined by all the φx. On the other hand, if F and Gare sheaves on X with

Fx ≅ Gxfor all x ∈X, this does not necessarily imply that

F ≅ G,

since there need not be a global sheaf morphism between F and G.

To properly define what is a surjective sheaf morphism is, weneed the following, which is motivated by the fact that for sheavesany section is determined by its germs:

Definition and Theorem 4.3.18 Let F be a presheaf. Then thereis a sheaf F and a presheaf morphism α ∶ F → F with the followingproperties: For all sheaves G and presheaf morphisms φ ∶ F → Gthere is a unique morphism of sheaves

φ ∶ F → G

withφ = φ α,

that is, the diagram

F αÐ→ Fφ ↓ φ

G

commutes. In particular, F is unique up to a unique isomorphism.

4. SHEAVES 66

The proof consists out of an explicit construction:Proof. Let U ⊂X be open. We define F(U) as the set of all maps

s ∶ U → ⋃x∈UFx

such that

1) s(x) ∈ Fx for all x ∈ U , and

2) for all x ∈ U there is an open neighborhood V ⊂ U of x and at ∈ F(V ) such that

ty = s(y) for all y ∈ V .

We prove that this defines a sheaf:

1) We first observe that F becomes a presheaf with the restric-tion maps defined as the restriction of maps

ρU,V (s) = s ∣V

2) Moreover (A1) is satisfied: Suppose

U =⋃i∈IUi

and s ∣Ui= 0 for all i. Then sx = 0 for all x ∈ Ui and all i, hence

s = 0.

3) For (A2) write againU =⋃

i∈IUi

and let si ∈ F(Ui) with

si ∣Ui∩Uj= sj ∣Ui∩Uj

Then s ∈ F(U) defined by

s(x) ∶= (si)x if x ∈ Ui

is well-defined.

4) The morphism α ∶ F → F is defined by

αU ∶ F(U) → F(U)

s ↦⎛⎝U → ⋃

x∈UFx

x ↦ sx

⎞⎠

4. SHEAVES 67

5) Finally, if φ ∶ F → G is a morphism, then for U open we define

φ ∶ F → G

by mapping s ∈ F(U) to

(φ(s)) ∣V ∶= φV (t)

where for every x ∈ U we choose a neighborhood V ⊂ U of xand a t ∈ F(V ) with ty = s(y) for all y ∈ V (Exercise: this iswell-defined).

Uniqueness we leave as an exercise.

Example 4.3.19 If F is the presheaf of constant real valued func-tions on a topological space X, then F is the sheaf of locally constantreal valued functions.

Remark 4.3.20 If F is a presheaf on X and x ∈X, then

(F)x≅ Fx

sx ↦ s(x)

(where s is a representative of the equivalence class sx). We identifythen (F)

x= Fx and sx = s(x).

Proof. For


t ∶ V → ⋃x∈VFx

we have

s ∼ t⇐⇒ ∃W ⊂ U ∩V open neighborhood of x such that s ∣W= t ∣W

The morphism

(F)x→ Fx

[s]↦ s(x)

is an isomorphism: It is well-defined since s ∼ t implies that s(x) =t(x). It is surjective since sx represented by s ∈ F(U) is the image

αU(s)↦ sx

4. SHEAVES 68

and it is injective: Let s, t ∈ F(U) with

s(x) = t(x).

By definition of the sheafification, there is an open neighborhoodW ⊂ U of x and s′, t′ ∈ F(W ) such that

s′y = s(y) and t′y = t(y)

for all y ∈W . Moreover, since

s′x = s(x) = t(x) = t′x

there is an open neighborhood W ′ ⊂W of x with

s′ ∣W ′= t′ ∣W ′

Hence for all y ∈W ′,

s(y) = s′y = t′y = t(y),

thuss ∣W ′= t ∣W ′

that is,s ∼ t.

Remark 4.3.21 If F is a sheaf, then F ≅ F since any section isdetermined by its germs.

Remark 4.3.22 Consider X = 0,1 with the discrete topologyand the presheaf

F(∅) = 0

F(0) = 0

F(1) = 0

F(0,1) = C

which we have seen is not a sheaf. Then

F = 0

since all Fx = 0.

4. SHEAVES 69

Remark 4.3.23 Consider again X = 0,1 with the discrete topol-ogy and the presheaf

F(∅) = 0

F(0) = CF(1) = CF(0,1) = 0

Then

F(∅) = 0

F(0) = CF(1) = CF(0,1) = C2

since F0 = F1 = C.

Definition 4.3.24 Let φ ∶ F → G be a sheaf morphism. We define

imφ ∶= im′ φ

cokerφ ∶= coker′ φ

Then φ is called surjective if

imφ = G

Remark 4.3.25 Note that imφ is up to isomorphism a subsheafof G: For any s ∈ (imφ)(U) with U open there is, by the definitionof the image sheaf, locally in a neighborhood V ⊂ U of a point x, a

t ∈ imφV ⊂ G(V )

withsy = ty for all y ∈ V .

Since these t agree on intersections, we can glue them to a sectionin t ∈ G(U). The sheaf morphism

imφ→ G

defined by

(imφ)(U)→ G(U)s↦ t

for all U open, is injective: Since by Remark 4.3.12, sections ofsheaves are determined by their stalks, sy = ty = s′y for all y ∈ Uimplies that s = s′.

4. SHEAVES 70

The latter argument can be formulated more generally: UsingRemark 4.3.12 and the following observation, we can show that asheaf morphism is injective if it is injective on all stalks.

Remark 4.3.26 If φ ∶ F → G is a sheaf morphism, then

(kerφ)x = kerφx

for all x.

Theorem 4.3.27 A sheaf morphism φ ∶ F → G is injective if andonly if all φx ∶ Fx → Gx for x ∈X are injective.

Ee leave the proves an exercise, and give the the prove for theanalogous statement for surjective:

Theorem 4.3.28 A sheaf morphism φ ∶ F → G is surjective if andonly if all φx ∶ Fx → Gx for x ∈X are surjective.

We first observe:

Remark 4.3.29 If φ ∶ F → G is a sheaf morphism, then

(imφ)x = imφx

for all x.

Proof. To see this note

tx ∈ imφx ⇐⇒ ∃sx ∈ Fx with φx(sx) = tx⇐⇒ ∃U neighborhood of x and s ∈ F(U) with φU(s) = t⇐⇒ ∃U neighborhood of x with t ∈ im(φU)⇐⇒ tx ∈ (im′ φ)x = (imφ)x

where for the last equality we use Remark 4.3.20.We now prove 4.3.28:

Proof. Using Remark 4.3.29, the claim follows immediately: If allφx are surjective, we have

(imφ)x = imφx = Gxfor all x, hence the inclusion sheaf morphism imφ ⊂ G from Remark4.3.25 is by Theorem 4.3.14 an isomorphism, hence imφ = G. Viceversa, if φ is surjective, that is, imφ = G, then

imφx = (imφ)x = Gxfor all x, that is all φx are surjective.

However note:

Remark 4.3.30 If a sheaf morphism φ ∶ F → G is surjective thisdoes in general not imply that

φU ∶ F(U)→ G(U)is surjective for all U open, see Example 4.3.10.

4. SHEAVES 71

4.4 Short exact sequences of sheaves

Definition 4.4.1 A short exact sequence of sheaf morphism isa sequence of sheaf morphisms

0→ F ϕ→ G π→ H → 0

with ϕ injective, π surjective, and imφ = kerπ.

Remark 4.4.2 A sheaf morphism φ ∶ F → G is surjective if thesequence

F φ→ G → 0

is exact, it is injective if

0→ F φ→ G

is exact.

Although exactness of the following sequence is trivial by defi-nition, it is nevertheless a very important one, since it gives rise tothe homomorphism theorem:

Example 4.4.3 For a sheaf morphism φ ∶ F → G the sequence

0→ kerφ→ G φ→ imφ→ 0

is exact.

Corollar 4.4.4 Let

0→ F ϕ→ G π→ H → 0

be a sequence of sheaf morphisms. This sequence is exact if andonly if

0→ Fxϕx→ Gx

πx→ Hx → 0

is exact for all x.

Proof. Follows directly from Theorem 4.3.27 and 4.3.28, and Re-marks 4.3.29 and 4.3.26.

Corollar 4.4.5 If

0→ F ϕ→ G π→ H → 0

is an exact sequence of presheaves, then also the induced sequenceof sheaves

0→ F ϕ→ G π→ H → 0

is exact.

4. SHEAVES 72

Proof. Follows by Corollary 4.4.4 and Remark 4.3.20.

Definition 4.4.6 Let F ⊂ G be a subsheaf. Then the quotientsheaf G/F is the sheaf associated to the presheaf

U ↦ G(U)/F(U)

Proposition 4.4.7 Let F ⊂ G be a subsheaf. Then we have anexact sequence

0→ F → G → G/F → 0.

Proof. For a sheaf morphism

ϕ ∶ F → G

we have an exact sequence presheaves described by

0→ F(U)→ G(U)→ G(U)/F(U)→ 0.

Since by Remark 4.3.21 we have F = F and G = G, the claim followsfrom Corollary 4.4.5.

Remark 4.4.8 By Corollary 4.4.4 applied to the sequence in Propo-sition 4.4.7, the stalk of G/F is

(G/F)x = Gx/Fx

Theorem 4.4.9 From Example 4.4.3 we obtain the homomorphismtheorem: If φ ∶ F → G is a sheaf morphism, then

G/kerφ ≅ imφ

Proposition 4.4.10 If φ ∶ F → G is a sheaf morphism, then

cokerφ ≅ G/ imφ.

Proof. By im(φU) ⊂ im(φ)(U), we have exact sequences presheavesrepresented by

0→ im(φU)→ im(φ)(U)→ im(φ)(U)/ im(φU)→ 0

0→ im(φ)(U)/ im(φU)→ G(U)/ im(φU)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

coker(φU )

→ G(U)/ im(φ)(U)→ 0

Since im(φ) is the sheafification of the presheaf U ↦ im(φU), weget by sheafifying the first sequence that the sheafification of

U ↦ im(φ)(U)/ im(φU)

is zero, so by sheafifying the second sequence the claim follows.Using Proposition 4.4.7 for the subsheaf imφ ⊂ G and Proposi-

tion 4.4.10 we get:

4. SHEAVES 73

Corollar 4.4.11 For a sheaf morphism φ ∶ F → G the sequence

0→ imφ→ G → cokerφ→ 0

is exact.

We now turn to sheaves on complex manifolds.

4.5 Sheaves on complex manifolds

One exact sequence on a complex manifold we have already seen ina special case.

Example 4.5.1 If X is a complex manifold, the sequence

0 → Z → OX → O∗X → 0

f ↦ e2πif

is exact, since locally a logarithm exists, and

f ∈ ker(e2πi )⇐⇒ e2πif = 1⇐⇒ f locally constant integer.

Example 4.5.2 Let X be a complex manifold A ⊂X analytic. Theideal sheaf IA of A on X is defined by

IA(U) = f ∈ OX(U) ∣ f ∣A∩U= 0 .

Then by Proposition 4.4.7, the sequence

0→ IA → OX → OX/IA → 0

is exact.

Remark 4.5.3 Note that for x ∉ A we have IA,x = OX,x since thereis a neighborhood U of x with U ∩A = ∅. If x ∈ A ⫋ X and U is aneighborhood of x and fi ∈ OX(U) with A∩U = f1 = 0, . . . , fr = 0,then

IA,x = ⟨f1,x, . . . , fr,x⟩ ⫋ OX,xHence

(OX/IA)x ≠ 0⇐⇒ x ∈ A.

Remark 4.5.4 If X is a topological space, F is a sheaf on X, andA ⊂X, then there is a restriction sheaf F ∣A with

(F ∣A)x = Fx

4. SHEAVES 74

for all x ∈ A. Let U ⊂ X be open. We can construct F ∣A in asimilar way as the sheafification, defining for U ⊂ A open F ∣A (U)as the set of all maps


such that

1) s(x) ∈ Fx for all x ∈ U , and

2) for all x ∈ U there is an open neighborhood V ⊂X of x and at ∈ F(V ) such that

ty = s(y) for all y ∈ V ∩U .

Note that this construction has a generalization in form of achange of base construction called the inverse image (replacingthe inclusion A ⊂X by an arbitrary continous map).

Example 4.5.5 If A ⊂X is a submanifold, then for the restriction(OX/IA) ∣A of the sheaf OX/IA to A, we have an isomorphism

(OX/IA) ∣A≅ OA

Proof. Given a section s ∈ (OX/IA) ∣A (U) with U ⊂ A open thereis, locally in a neighborhood V of any point x ∈ U , a

t ∈ (OX/IA)(V )

withs(y) = ty for all y ∈ V ∩U .

Since by Remark 4.5.3, (OX/IA)x = 0 for all x ∉ A, these t agreeon intersections of the sets V , we can glue them to a section int ∈ (OX/IA)(V ′), where V ′ is a union of sets V which covers Ucompletely. The sheaf morphism

(OX/IA) ∣A→ OA

defined by

(OX/IA) ∣A (U)→ OA(U)s↦ t ∣A

for all U ⊂ A open, is an isomorphism. Injectivity is clear, sincesections are determined by their germs. Prove surjectivity as anexercise.

4. SHEAVES 75

Definition 4.5.6 Let X be a complex manifold. A sheaf of OX-modules on X is a a sheaf J on X such that J(U) is an OX(U)-module for U ⊂X open.

Example 4.5.7 For X be a complex manifold and A ⊂ X ana-lytic, OX and IA are sheaves of OX-modules. The same is true forthe sheaf A of complex valued C∞-functions on X and the sheafC of complex valued continuous functions, since the multiplicationof holomorphic functions with such functions retains the respectiveproperty.

Not sheaves of OX-modules are the constant sheaves Z, R, C,and the sheaf O∗

X of holomorphic functions without zeroes.

Remark 4.5.8 With the notation of OX-modules we can now dolinear algebra with sheaves, for example, for sheaves F , G of OX-modules we can define the direct sum sheaf

(F ⊕ G)(U) ∶= F(U)⊕ G(U)

and the tensor product sheaf F ⊗ G as the sheafification of

(F ⊗′ G)(U) ∶= F(U)⊗OX(U) G(U)

Example 4.5.9 We will introduce the sheaf OP2(1) of OP2-modules,which has as global sections on P2,

OP2(1)(P2) =C ⟨s0, s1, s2⟩ ≅ C3,

where the sections are define in terms of the standard charts

Ui = xi ≠ 0

by

sj ∣Ui=xjxi

.

If we consider now A = x2 = 0 then, by Example 4.5.5 and Re-mark 3.4.6,

(OP2/IA)(P2) ≅ OA(A) ≅ OP1(P1) ≅ C.

However, by tensoring with OP2/IA the section s2 becomes zero, andthus

(OP2(1)⊗ (OP2/IA)) (P2) ≅C ⟨s0, s1⟩ ≅ C2.

To see this, we check that s2⊗1 = 0 in all charts Ui: On U0 we have

s2 ⊗ 1 = x2

x0

⊗ 1 = x2

x0

⋅ (1⊗ 1) = 1y ⊗x2

x0

= 1y ⊗ 0 = 0

4. SHEAVES 76

using that

(OP2/IA)(U0) = C [x1

x0

,x2

x0

] / ⟨x2

x0

⟩ ,

and similarly on U1. By Remark 4.5.3, on U2 we have

s2 ⊗ 1 = 1⊗ 0 = 0.

Hence

(OP2(1)⊗ (OP2/IA)) (P2) ≅ C2 ≠ C3 = C3⊗C ≅ OP2(1)(P2)⊗(OP2/IA) (P2)

In particular,

OP2(1)⊗′ (OP2/IA) ≠ OP2(1)⊗ (OP2/IA)

which by Remark 4.3.21 implies that OP2(1) ⊗′ (OP2/IA) is not asheaf.

5

Vector bundles

The most important way sheaves arise in geometry is through sec-tions of vector bundles.

5.1 Vector bundles and transition func-

tions

Definition 5.1.1 Let X be a complex manifold. A holomophicvector bundle of rank r on X is a complex manifold E togetherwith a holomorphic map

π ∶ E →X

such that

1) For all x ∈X the fiber

Ex ∶= π−1(x)

is an r-dimensional C-vector space, and

2) π is locally trivial, that is, for all x ∈ X there is an openneighborhood U of x and a biholomorphic map

h ∶ π−1(U)→ U ×Cr

such thath(Ey) ⊂ y ×Cr

for all y ∈ U and

hy ∶= h ∣Ey ∶ Ey → y ×Cr

is a C-vector space isomorphism, see Figure 5.1.

77

5. VECTOR BUNDLES 78

XU

Figure 5.1: Local trvialization of a vector bundle

So withEU ∶= π−1(U)

and the projection maps πU ∶ π−1(U)→ U and U ×Cr → U we havea commutative diagram

EUhÐ→ U ×Cr

πU U

with linear maps hy on the fibers over y ∈ U . We call E the totalspace, (U,h) a local trivialization, and X the basis of the bun-dle. By abuse of notation, one sometimes denotes the bundle justby E.

Remark 5.1.2 In the same way, one can define real or complexdifferentiable vector bundles on differentiable manifolds, and realor complex topological vector bundles on a topological space.

Definition 5.1.3 Given two trivializations (Uα, hα) and (Uβ, hβ)of a rank r vector bundle E by writing

hα h−1β ∶ (Uα ∩Uβ) ×Cr → (Uα ∩Uβ) ×Cr

(x, v) ↦ (x, gαβ(x) ⋅ v)

we obtain the transition functions

gαβ ∶ Uα ∩Uβ → GL(r,C)

Remark 5.1.4 If E is a holomorphic vector bundle, then the gαβare holomorphic, that is,

gαβ ∈ GL(r,OX(Uα ∩Uβ))


Note that GL(r,OX) is a sheaf of non-abelian groups if r ≥ 2. Forr = 1, we have

GL(r,OX) = O∗X

Remark 5.1.5 On their domain of definition it holds

gαβ ⋅ gβγ = gαγgαα = id

for all α,β, γ, since for U = Uα ∩ Uβ ∩ Uγ we have a commutativediagram

EU

U ×Cr -

hα

U ×Cr

hβ

?- U ×Cr

hγ

-

(x, v) z→ (x, gαβ(x)v)

(x,w) z→ (x, gβγ(x)w)

Remark 5.1.6 From holomophic functions

gαβ ∶ Uα ∩Uβ → GL(r,C)

for an open covering (Uα)α of X with

gαβ ⋅ gβγ = gαγgαα = id

for all α,β, γ we can construct a holomorphic vector bundle E whichhas the gαβ as transition functions: On the (non-connected) complexmanifold

E =⋃α

α ×Uα ×Cr

we define the equivalence relation

(α,x, v) ∼ (β, y,w) ∶⇐⇒ x = y and v = gαβ(x) ⋅ v

ThenE ∶= E/ ∼

with the quotient topology is a holomorphic vector bundle with

π ∶ E →X, [(α,x, v)]↦ x


Example 5.1.7 The trivial bundle of rank r is the cartesian prod-uct

E =X ×Cr

with the projectionπ ∶ E →X

to the first factor.

Definition 5.1.8 If E and F are holomorphic vector bundles withtransition functions (gαβ) and (hαβ).

• The direct sum E⊕F is defined via the transition functions

( gαβ 00 hαβ

)

and the fibers(E ⊕ F )x = Ex ⊕ Fx

• The tensor product E⊗F is defined by the transition func-tions

gαβ ⊗ hαβ ∶ U → GL(n ⋅m,C)and has the fibers

(E ⊗ F )x = Ex ⊗ Fx

• The wedge power

⋀kE

defined by

⋀kgαβ ∶ U → GL((n

k),C)

and has the fibers

(⋀kE)

x=⋀k

Ex

• The symmetric power SkE has the transition functionsSkgαβ and has the fibers

(SkE)x = SkEx

• The dual E∗ with transition functions

gtαβ

and has the fibers(E∗)x = E∗

x


5.2 Tangent space

Definition 5.2.1 Let X be a real differentiable manifold, x ∈ Xand AR

X the sheaf of real C∞-functions. A derivation is an R-linear map

D ∶ ARX,x → R

such thatD(fg) = (Df)g(x) + f(x)Dg

for all f, g ∈ ARX,x.

The tangent space of X in x is then

TX,x = D ∶ ARX,x → R ∣D a real derivation

Definition 5.2.2 Let X be a complex manifold and x ∈ X. Wewrite

TRX,x

for the tangent space of X in x, regarding X as a differentiable realmanifold. Moreover, we write

TCX,x ∶= TR

X,x ⊗C

for the complexified tangent space.

Remark 5.2.3 We have

TCX,x = D ∶ AC

X,x → C ∣D a complex derivation

where AC is the sheaf of C∞ complex valued functions, and a com-plex derivation is a C-linear map D ∶ AC

X,x → C with

D(fg) = (Df)g(x) + f(x)Dg

for all f, g ∈ ACX,x.

Definition 5.2.4 Let X and Y be differentiable real manifolds, f ∶X → Y differentiable, and x ∈ X. Then the differential of f at xis the map

(Tf)x ∶ TX,x → TY,f(x)

D ↦ (ARY,f(x) → Rhf(x) ↦ D((h f)x)

)

What is the tangent space?


Lemma 5.2.5 1) Denoting by ( ∂∂xi

)0

the xi-derivative at 0

( ∂

∂x1

)0

, . . . ,( ∂

∂xn)

0

is a R-vector space basis of TRn,0.

2) Writing zj = xj + iyj,

( ∂

∂x1

)0

, . . . ,( ∂

∂xn)

0

,( ∂

∂y1

)0

, . . . ,( ∂

∂yn)

0

is a C-vector space basis of TCCn,0.

3) If X is an n-dimensional differentiable real manifold

dimTX,x = n

If ϕ is a chart of X around x0 with ϕ(x0) = 0, then

(Tϕ)x0 ∶ TX,x0 ≅ TRn,0

If X is an n-dimensional complex manifold

dimTCX,x = 2n

Proof. We first note, that (2) follows from (1), and (3) from (2).With regard to (1) we observe:

• The ∂∂xi

are derivations since for f0, g0 ∈ ARRn,0 we have by the

product rule

( ∂

∂xi)

0

(f0 ⋅ g0) =∂

∂xi(f ⋅ g)(0) = ∂f

∂xi(0)g(0) + f(0) ∂g

∂xi(0)

= (∂f0

∂xi)

0

g0(0) + f0(0) (∂g0

∂xi)

0

• They are linearly independent: If

∑i

λi (∂

∂xi)

0

= 0

then for all i

λi = λi (∂

∂xi)

0

xi = 0

• We leave it as an exercise to prove that they generate.


Definition 5.2.6 Let X be a complex manifold, and x ∈ X. Theholomorphic tangent space of X at x is defined as

TX,x = D ∶ OX,x → C ∣D derivation

Remark 5.2.7 The holomorphic tangent space TX,x is generatedby

( ∂

∂z1

)x

, . . . ,( ∂

∂zn)x

andTCX,x = TX,x ⊕ T ′

X,x

with T ′X,x generated by

( ∂

∂z1

)x

, . . . ,( ∂

∂zn)x

The conjugation operation on TCX,x = TR

X,x ⊗C is defined by

( ∂

∂zi)x

↦ ( ∂

∂zi)x

andT ′X,x = TX,x

soTCX,x = TX,x ⊕ TX,x

The mapTRX,x → TC

X,x → TX,x

is an R-linear isomorphism. In particular TRX,x does not give more

information about the complex manifold X at a than the complextangent space TX,x.

Definition 5.2.8 Let X be an n-dimensional differentiable realmanifold. The differentiable real tangent bundle TX is definedas follows: Consider an open covering U = Uα and correspondingcharts ϕβ and define the transitions functions

gαβ ∶ Uα ∩Uβ → GL(n,R)

by

(gαβ)i,j ∶=∂(ϕα ϕ−1

β )i∂xj


that isgαβ = Jϕαϕ−1β

is the Jacobian matrix. By the chain rule, the compatibility condi-tions are satisfied.

In the same way, we define for a complex manfiold X with opencovering U = Uα and corresponding charts ϕβ the holomorphictangent bundle TX by considering the holomorphic Jacobi matrix

gαβ = Jϕαϕ−1β = (∂(ϕα ϕ−1

β )i∂zj

)

with the fibers(TX)x = TX,x.

To distinguish the real tangent bundle of a complex manifold con-sidered as a real manifold from its holomorphic tangent bundle, wewrite TR

X the real tangent bundle, and also TCX for the complexified

version.

Remark 5.2.9 If X = Cn then the tangent bundle is trivial

TX =X ×Cn →X

and the fiber over x ∈X has basis

( ∂

∂z1

)x

, . . . ,( ∂

∂zn)x

Remark 5.2.10 Note that by definition of the tangent bundle us-ing an open covering of a manifold X, if ϕ ∶ U → V ⊂ Cn is a chart,then the tangent bundle restricted to U is trivial

TU = U ×Cn → U.

These trivial pieces get patched together via the transition functionsto form TX . So any chart domain trivializes the tangent bundle.

For an arbitrary bundle this is not true, one may have to choosethe chart domains small enough to trivialize the bundle.

Definition 5.2.11 Let E be a holomorphic vector bundle on Xwith projection map π ∶ E → X. A submanifold F ⊂ E is called asubbundle of rank r, if the following holds true:

1) For all x ∈XFx ∶= F ∩Ex ⊂ Ex

is a complex subvectorspace of dimension r.


2) For all x ∈ X there is an open neighborhoold U ⊂ X of x anda trivialization h such that the diagram

EUh→ U ×Cn = U × (Cr ⊕Ck−r)

∪ ∪ ↓ (idU ,prCr)FU → U ×Cr

commutes.

Remark 5.2.12 The subbundle F together with π ∣F is a holomor-phic vector bundle of rank r.

Example 5.2.13 Let Y ⊂X be a complex submanifold and dim(Y ) =m and dim(X) = n. Then

TY ⊂ TX ∣Y

is a subbundle.Here, for a vector bundle E on X the restriction E ∣Y is given

by restricting the transition functions, that is, if E is given by gαβon the covering Uα then TX ∣Y is given by gαβ ∣Y with fibers

(TX ∣Y )x = TX,x

for x ∈ Y .

Definition 5.2.14 Let E and F be holomorphic vector bundles onthe complex manifold X and f ∶ E → F a holomorphic map (betweencomplex manifolds). Then f is called a

1) vector bundle map if for all x ∈X we have

f(Ex) ⊂ Fx

andfx ∶= f ∣Ex ∶ Ex → Fx

is linear.

2) vector bundle morphism if in addition

rank(fx)

is constant for all x ∈X.

3) vector bundle isomorphism if f is a vector bundle mor-phism, f is bihomomorphic, and f−1 is a vector bundle map.Then we have a fiber-wise isomorphism Ex ≅ Fx for all x.


Theorem 5.2.15 Let f ∶ E → F be a vector bundle morphism.Then

ker(f) ∶= ⋃x∈X ker(fx) ⊂ Eim(f) ∶= ⋃x∈X im(fx) ⊂ F

are a subbundles (Exercise).

Definition 5.2.16 If f ∶ X → Y is a holomorphic map betweencomplex manifolds, and E is a holomorphic vector bundle on Ywith transition functions gαβ on the covering Uα. Then the pull-back bundle f∗E is given by fibers

(f∗E)x = Ef(x)for x ∈X and the pulled-back transition functions

f∗gαβ ∶= gαβ f

on the covering f−1(Uα) of X.

Example 5.2.17 If f ∶X → Y is a holomorphic map between com-plex manifolds, there is an induced vector bundle map

Tf ∶ TX → f∗TY

the tangent map, given by differential maps

(Tf)x ∶ TX,x → TY,f(x).

Remark 5.2.18 The tangent map Tf is in general not a mor-phism: Consider, for example

C→ Cz ↦ z2

then the differential has rank 1 except at z = 0 where it has rank 0.

5.3 Sections

Definition 5.3.1 Let X be a complex manifold, E a holomorphicvector bundle on X, and U ⊂ X open. A holomorphic sectionof E on U is a holomorphic map

s ∶ U → EU

such that for all x ∈ U we have

s(x) ∈ Exsee Figure 5.2.


s x( )0

Figure 5.2: Section of a vector bundle

Example 5.3.2 For any vector bundle E there is a canonical sec-tion, the zero section

0 ∶X → E

with0(x) = 0 ∈ Ex

Definition 5.3.3 We write

OX(E)(U) ∶= s ∶ U → EU ∣ s holomorphic section

This defines a sheaf OX(E), the sheaf of holomorphic sectionsof E.

In the same way we can define the sheaf of differentiablesections A(E) of a differentiable vector bundle E on a differen-tiable manifold X.

Remark 5.3.4 Let X be a complex manifold, E a holomorphicvector bundle on X, and U = Uα an open covering of X such thatwe have trivializations

EUα = π−1(Uα)hαÐ→ Uα ×Cr

Given a section s ∈ OX(E)(X) the maps

hα s ∣Uα ∶ Uα → Uα ×Cr

in the commutative diagram

EUαhαÐ→ Uα ×Cr

s ∣Uα (id, sα)Uα


are of the formhα s ∣Uα= (id, sα)

withsα ∶ Uα → Cr

holomorphic. So in the trivialization a section is given by an r-tupleof holomorphic functions.

On Uα ∩Uβ with the transition functions gαβ of E, we have

sα = gαβ ⋅ sβ

To see this, observe that with U = Uα ∩ Uβ we have a commutativediagram

U

EU

s

?

U ×Cr -

(id, sβ)

hβ

U ×Cr

(id, sα)

-

hα

-

(x, v) z→ (x, gαβ(x) ⋅ v)

Remark 5.3.5 We can reverse the previous construction of thelocal description of sections by tuples of holomorphic functions: LetX be a complex manifold, E a holomorphic vector bundle on X,let U = Uα be an open covering of X with local trivializationshα ∶ EUα → Uα ×Cr of E and let gαβ be the corresponding transitionfunctions of E. If we are given holomorphic functions

sα ∶ Uα → Cr

withsα = gαβ ⋅ sβ

for all α,β, then for x ∈ X choose an i with x ∈ Uα and define thesection

s ∶X → E


bys(x) = h−1

α (x, sα(x))

(note that hα is an inverse). This is well-defined since if x ∈ Uα∩Uβwe have

h−1α (x, sα(x)) = h−1

α (x, gαβ(x) ⋅ sβ(x)) = (h−1α (hα h−1

β ))(x, sβ(x))= h−1

β (x, sβ(x)).

5.4 Vector fields and differential forms

Definition 5.4.1 Let X be a differentiable real manifold of dimen-sion n.

1) A vector field over U ⊂X is a section

s ∈ A(TX)(U)

Locally in a chart V ⊂ U , the section s can be described as

s =n

∑i=1

fi∂

∂xi

with fi ∈ A(V ), and

∂

∂xi(x) = ( ∂

∂xi)x

2) A differential form of degree r is a section

ω ∈ A(⋀r(TX)∗)(U)

Locally in a chart V ⊂ U , the section ω can be written as

ω = ∑i1<...<ir

fi1,...,irdxi1 ∧ . . . ∧ dxir =∶ ∑∣I ∣=r

fIdxI

with fI ∈ ARX(V ). Here we denote by (dx1)x, . . . , (dxn)x the

basis of (TX,x)∗ dual to the basis ( ∂∂x1

)x, . . . , ( ∂

∂xn)x

of TX,x,and by

dx1, . . . , dxn

the corresponding local sections of TX with

dxi(x) = (dxi)x.


Note thatdxi ∧ dxj = −dxj ∧ dxi

in particulardxi ∧ dxi = 0.

Definition 5.4.2 Let X be a complex manifold of dimension n. Aholomorphic vector field over U ⊂X is a section

ω ∈ OX(TX)(U)

Locally in a chart V ⊂ U , the section ω can be described as

ω =n

∑i=1

fi∂

∂zi

with fi ∈ OX(V ) using the analogous notation as in the preceedingdefinition.

Remark 5.4.3 Let X be a complex manifold of dimension n. Thedecomposition of the complexified tangent space induces a decompo-sition

(TCX,x)∗ = T ∗

X,x ⊕ (TX,x)∗

for all x ∈X. Denoting by (dzi)x the dual of ( ∂∂zi

)x

and by (dzi)x the

dual of ( ∂∂zi

)x

and again writing dzi(x) = (dzi)x and dzi(x) = (dzi)xfor the corresponding local sections of T ∗

X and (TX)∗, we have bases

T ∗X,x = ⟨dz1(x), . . . , dzn(x)⟩

(TX,x)∗ = ⟨dz1(x), . . . , dzn(x)⟩

and these vector spaces get identified by conjugation

T ∗X,x → (TX,x)

∗

dzi(x)↦ dzi(x)

Definition 5.4.4 Let X be a complex manifold of dimension n.

1) A complex diffenential form of degree r on U ⊂ X is asection

ω ∈ A(⋀r(TCX)∗)(U)

and can locally in a chart V ⊂ U be written

ω = ∑∣I ∣=r

fIdxI

with fI ∈ ACX(V ). The complex valued r-forms form a sheaf

which we denote by ArX .


2) We say that a complex valued r-form ω ∈ ArX(U) is of type(p, q) if it is a section of

Ap,qX ∶= A(⋀p(TX)∗ ⊗⋀q(TX)∗)

that is, it can locally in a holomorphic chart V ⊂ U be writtenas

ω = ∑∣I ∣=p∣J ∣=q

fIJdzI ∧ dzJ

with fIJ ∈ ACX(V ). We call Ap,qX the sheaf of (p, q)-forms.

Remark 5.4.5 If ω ∈ ArX(U), then there are unique ωp,q ∈ Ap,qX (U)with

ω = ∑p+q=r

ωp,q

hence there is a direct sum decomposition

ArX = ∑p+q=rAp,qX

We denote the corresponding projection maps by

prp,q ∶ ArX(U)→ Ap,qX (U), ω ↦ ωp,q.

Example 5.4.6 We give a form ω ∈ A2C3(C3):

ω = ω2,0 + ω1,1 + ω0,2

ω2,0 = f12dz1 ∧ dz2 + f13dz1 ∧ dz3 + f23dz2 ∧ dz3

ω1,1 = g11dz1 ∧ dz1 + g22dz2 ∧ dz2 + g33dz3 ∧ dz3

+ g12dz1 ∧ dz2 + g13dz1 ∧ dz3 + g23dz2 ∧ dz3

+ g21dz2 ∧ dz1 + g31dz3 ∧ dz1 + g32dz3 ∧ dz2

ω0,2 = f ′12dz1 ∧ dz2 + f ′13dz1 ∧ dz3 + f ′23dz2 ∧ dz3

Definition 5.4.7 A differential form ω ∈ Ap,0X (U) is called a holo-morphic differential form of degree p and can locally in a chartV ⊂ U be written as

ω = ∑∣I ∣=p

fIdzI

with holomorphic fi ∈ OX(V ). They form the sheaf ΩpX of holomor-

phic p-forms.


Definition 5.4.8 There is a family of operators

d ∶ ArX(U)→ Ar+1X (U)

(the individual operators depend on r and U , but we omit this infor-mation, since it can be inferred from the input) which are C-linear,satisfy

df =n

∑i=1

∂fi∂zi

dzi +n

∑i=1

∂fi∂zi

dzi

for all f ∈ ACX(U) and satisfy the Leibnitz rule

d(ϕ ∧ ψ) = dϕ ∧ ψ + (−1)rϕ ∧ dψ

for all ϕ ∈ ArX(U) and ψ ∈ AsX(U).


dϕ = dϕ

andd d = 0

as we have learned in our multivariate calculus class.

Definition 5.4.10 Moreover, we have Dolbeault operators

∂ ∶ Ap,qX (U)→ Ap+1,qX (U)

∂ ∶ Ap,qX (U)→ Ap,q+1X (U)

which are defined in terms of

d ∶ Ap,qX (U)→ Ap+q+1X (U)

and the projection maps as

∂ = prp+1,q d∂ = prp,q+1 d

Remark 5.4.11 It is clear from the definitions that

d = ∂ + ∂

From d d = 0 it follows that

∂ ∂ = 0

∂ ∂ = 0

∂ ∂ = −∂ ∂

since

d d = (∂ + ∂) (∂ + ∂) = ∂ ∂ + (∂ ∂ + ∂ ∂) + ∂ ∂


Remark 5.4.12 In local coordinates for

ω =∑I,J

fIJdzI ∧ dzJ

we havedω =∑

I,J

dfIJ ∧ dzI ∧ dzJ

using the Leibnitz rule and d d = 0. Writing

dfIJ =∑k

∂fIJ∂zk

dzk +∑k

∂fIJ∂zk

dzk

= ∂fIJ + ∂fIJ

thus

∂ω =∑I,J

∂fIJ ∧ dzI ∧ dzJ

∂ω =∑I,J


Remark 5.4.13 A complex p-form ω ∈ Ap,0X (U) is holomorphic ifand only if

∂ω = 0

Proof. Ifω =∑

I,J

fIJdzI ∧ dzJ

then0 = ∂ω =∑

I,J


if and only if∂fIJ = 0

for all I, J , that is, all fIJ are holomorphic.

5.5 Vector bundles and locally free sheaves

Definition 5.5.1 Let X be a complex manifold. A sheaf of OX-modules E is called locally free of rank r if for all x ∈ X there isan open neighborhood U ⊂X of x such that

E ∣U≅ O⊕rU

as modules.


Theorem 5.5.2 Let X be a complex manifold.

1) Let E be a holomomorphic vector bundle of rank r on X.Then OX(E) is locally free of rank r.

2) If E is locally free sheaf of OX-modules on X, then there is aholomorphic vector bundle E of rank r on X such that

OX(E) ≅ E

3) For holomorphic vector bundles E and E′ we have that

E ≅ E′⇐⇒ OX(E) ≅ OX(E′)

Proof.

1) In a trivialzation we have

EU ≅ U ×Cr

soOU(EU) ≅ O⊕r

U

since a section is an r-tuple of holomorphic functions on U .

2) Given an open covering Uα and isomorphisms

hα ∶ E ∣Uα→ O⊕rUα

we obtain holomorphic functions

gαβ = hα h−1β ∶ Uα ∩Uβ → GL(r,C)

(since isomorphisms between free modules are given by mul-tiplication with a matrix) which satisfy

gαβ ⋅ gβγ = gαγ

The gαβ define a holomorphic vector bundle of rank r, whichsatisfies

OX(E) ≅ E .

Note that any section s ∈ E(U), by Remark 5.3.5 correspondsto a section of E on U since for sα = hα s ∣U∩Uα we have

sα = hα h−1β sβ = gαβ ⋅ sβ


by construction of E. On the other hand, any section s ∈O(E)(U) is given by tuples of holomorphic functions sα ∶Uα ∩U → Cr which satisfy

sα = gαβ ⋅ sβ

Via the isomorphism OX(E)(Uα∩U) ≅ E(Uα∩U) they definesections

tα ∈ E(Uα ∩U)

withtα ∣Uα∩Uβ∩U= tβ ∣Uα∩Uβ∩U

which glue by the second sheaf axiom to a section t ∈ E(U).

6

Cech Cohomology

Let X be a topological space. In the following we will considersheaves of abelian groups (or anything else which is abelian, likemodules or rings) on X.

6.1 Cochainoperator

Definition 6.1.1 For a sheaf F on X define

H0(X,F) ∶= F(X)

and call it the 0-th cohomology group with values in F .

We would like to apply the global sections functor to exact se-quences. Unfortunately (or probably fortunately since otherwisethe theory of sheaves would be pretty useless) this functor is notexact due to the failure of surjectivity:

Proposition 6.1.2 Let

0→ F ϕ→ G ψ→ H → 0

be an exact sequence of sheaves. Then

0→H0(X,F) ϕX→ H0(X,G) ψX→ H0(X,H)

is exact (however, as we have seen e.g. in Example 4.3.10, ψX isin general not surjective).

Proof. Since 0 = (kerϕ)(X) = ker(ϕX), the homomorphism ϕX isinjective. We prove that

im(ϕX) = ker(ψX).

96

6. CECH COHOMOLOGY 97

We first show the inclusion ⊃: Suppose s ∈H0(X,H) with ψX(s) =0. By Corollary 4.4.4 for any x ∈X the sequence

0→ Fxϕx→ Gx

ψx→ Hx → 0

is exact. Hence there is an open neighborhood U(x) of x and asection

tx ∈ F(U(x))such that

ϕU(x)(tx) = s ∣U(x)Since X can be covered by open subsets U(x) and since by injec-tivity of ϕU(x)∩U(x′) we have

tx ∣U(x)∩U(x′)= tx′ ∣U(x)∩U(x′)

we can glue by the second sheaf axiom (A2) the tx to a section

t ∈H0(X,F).

ByϕX(t) ∣U(x)= ϕU(x)(t ∣U(x)) = ϕU(x)(tx) = s ∣U(x)

and the first sheaf axiom (A1) it holds

ϕX(t) = s.

We now show the inclusion ⊂, that is, we prove that ψX ϕX = 0.Let s = ϕX(t) ∈H0(X,G) with t ∈H0(X,F). We have to show that

ψX(t) = 0.

Since the sequence

0→ Fxϕx→ Gx

ψx→ Hx → 0

is exact, we have (by Remark 4.3.29)

(ψX(s))x = ψx(sx) = ψx(ϕx(tx)) = 0

Since, by Remark 4.3.12, sections of sheaves are determined by theirstalks, it follows that

ψX(s) = 0.

We now want to exactly describe the defect of surjectivity of

H0(X,G) ψX→ H0(X,H)

The amazing observation is that the image of ψX can be describedimplicitly as the kernel of a homomorphism with values in a groupwhich is again related to F .


Definition 6.1.3 Let U = (Ui)i∈I be an open covering of X with acountable index set I and let ≤ be a well ordering on I. Write

UI ∶= Ui0,...,iq ∶= Ui0 ∩ . . . ∩Uiqwhere I stands for the multi-index (i0, . . . , iq) with i0 < . . . < iq. LetF be a sheaf of abelian groups on X. For an integer q ≥ 0

Cq(U ,F) ∶= ∏i0<...<iq

(i0,...,iq)∈Iq+1

F(Ui0,...,iq) = ∏∣I ∣=q+1

F(UI)

is an abelian group (as a product of abelian groups, using the component-wise addition), and its elements are called q-cochains.

Remark 6.1.4 Recall that Zorn’s lemma states that on any par-tially ordered set such that every ordered chain of elements has anupper bound, there exists a maximal element.

From Zorn’s lemma, it follows that on any set I there exists awell-ordering (that is, an ordering such that every non-empty subsetof I has a smallest element).

Note that a well-ordering is also a total ordering, since any seta, b ⊂ I has a smallest element.

Definition and Theorem 6.1.5 The q-th cochain operator

dq ∶ Cq(U ,F)→ Cq+1(U ,F)

defined as

dq(α)i0,...,iq+1 ∶=q+1

∑k=0

(−1)kαi0,...,ik−1,ik+1,...,iq+1 ∣Ui0,...,iq+1

is a group homomorphism.

Lemma 6.1.6 dq+1 dq = 0.

Proof. Consider the case q = 0. Let α ∈ C0(U ,F), so

α = (αi) with αi ∈ F(Ui).

Then(d0α)i,j = (αj − αi) ∣Ui,j

and

(d1α)i,j,k = ((d0α)j,k − (d0α)i,k + (d0α)i,j) ∣Ui,j,k= ((αk − αj) − (αk − αi) + (αj − αi)) ∣Ui,j,k= 0

This easily generalizes to arbitrary q. We leave this as an exercise.


Definition 6.1.7 By taking the kernel of the cochain operator

Zq(U ,F) ∶= kerdq

we obtain an abelian group, whose elements we call cocyles. Takingthe image, we get for q ≥ 1 the abelian group

Bq(U ,F) ∶= imdq−1

and we set B0(U ,F) = 0. The elements of Bq(U ,F) are calledcoboundaries. According to Lemma 6.1.6, we have

Bq(U ,F) ⊂ Zq(U ,F)

and writeHq(U ,F) = Zq(U ,F)/Bq(U ,F)

for the quotient group, the Cech cohomology group on the opencovering U . We denote the class of α ∈ Zq(U ,F) in Hq(U ,F) by⟨α⟩.

Example 6.1.8 Let α = (αi,j)i<j ∈ C1(U ,F). Then

α ∈ Z1(U ,F) ⇐⇒ d1α = 0⇐⇒ (αi,j − αi,k + αj,k) ∣Ui,j,k= 0 for i < j < k

and

α ∈ B1(U ,F) ⇐⇒ ∃β = (βi) ∈ C0(U ,F) with α = d0β⇐⇒ αi,j = (βj − βi)Ui,j for i < j


H0(U ,F) =H0(X,F) = F(X)

Proof. Ifs = (si) ∈H0(U ,F) = Z0(U ,F)

thensj ∣Ui,j= si ∣Ui,j

so by the second sheaf axiom A2, there is a f ∈ F(X) with

f ∣Ui= si.

On the other hand, any f ∈ F(X) determines a cocycle f ∣Ui .


Example 6.1.10 Let X = P1 and U = U0, U1 with U0 = P1/∞ =C and U1 = P1/0 = C. Note that U0,1 = C∗. We have

B1(U ,O) = (f1 − f0) ∣ U0,1 ∈ C1(U ,O) ∣ f0 ∈ O(U0), f1 ∈ O(U1)

and

Z1(U ,O) = (f0,1) ∈ C1(U ,O) ∣ f0,1 ∈ O(U0,1)= O(C∗)

Any function f ∈ O(C∗) has a Laurent series expansion

f =∞∑i=−∞

aizi = P +R

with the principal part

P =−1

∑i=−∞

aizi ∈ O(U1)

and the regular part

R =∞∑i=0

aizi ∈ O(U0)

Sof = d0(fi)

withf0 = P and f1 = −R

that is,f ∈ B1(U ,O)

This implies that

H1(U ,O) = Z1(U ,O)/B1(U ,O) = 0

6.2 Refining open covers

We would now like to get rid of the choice of the open covering Uby using an appropriate equivalence relation.

Definition 6.2.1 Let U = (Ui)i∈I and V = (Vj)j∈J be open coveringsof X. We say that U is finer than V and write

U < V ∶⇐⇒ ∃τ ∶ I → J with Ui ⊂ Vτ(i)Then τ defines a refinement map

Cq(τ) ∶ Cq(V ,F)→ Cq(U ,F)Cq(τ)(α)i0,...,iq = ατ(i0),...,τ(iq) ∣Uτ(i0),...,τ(iq)


Lemma 6.2.2 We have

Cq+1(τ) dq = dq Cq(τ)


Cq(U ,F)dq→ Cq+1(U ,F)

Cq(τ) ↑ ↑ Cq+1(τ)Cq(V ,F)

dq→ Cq+1(V ,F)

commutes.

We leave the proof as an exercise.

Remark 6.2.3 The lemma implies that the refinement map mapscocycles to cocycles and coboundaries to coboundaries:

Cq(τ)(Bq(V ,F)) ⊂ Bq(U ,F)Cq(τ)(Zq(V ,F)) ⊂ Zq(U ,F)

Proof. Ifdqα = 0,

thendq(Cq(τ)(α)) = Cq+1(τ)(dqα) = 0,

and similarly for the coboundaries.

Remark 6.2.4 By the previous remark we obtain a well-definedgroup homomorphism

Hq(τ) ∶ Hq(V ,F) → Hq(U ,F)⟨α⟩ ↦ ⟨Cq(τ)(α)⟩

Definition 6.2.5 We define the q-th Cech-cohomology group ofX (which is again an abelian group) with values in F by

Hq(X,F) =⋅⋃

U an open cover of X

Hq(U ,F) / ∼

where for α ∈Hq(V ,F) and β ∈Hq(U ,F) we define an equivalencerelation by

α ∼ βif there is a covering W with W < U and W < V according to mapsτ1 and τ2 with

Hq(τ1)(α) =Hq(τ2)(β)We denote the equivalence class of α modulo ∼ by [α].


Remark 6.2.6 In order to be able to refine coverings as much aswe like, we will only consider topological spaces X which are para-compact. Recall that a topological space X is paracompact if everyopen cover has a locally finite open refinement. A cover is calledlocally finite if each point of the space has a neighbourhood thatintersects only finitely many elements of that cover.

Indeed we will only consider manifolds X with a countable topol-ogy (that is, there is a basis of the topology with countably manyelements). This implies that X is paracompact.

If you know about sheaf cohomology: On a paracompact space,sheaf cohomology and Cech-cohomology coincide, which is not truein general if the space is not paracompact.

6.3 Long exact cohomology sequence

Theorem 6.3.1 Let

0→ F ϕ→ G ψ→ H → 0

be exact. The there exists a long exact cohomology sequence

0 → H0(X,F) ϕX→ H0(X,G) ψX→ H0(X,H)δ→ H1(X,F)

H1(ϕ)→ H1(X,G)

H1(ψ)→ H1(X,H)

δ→ H2(X,F)H2(ϕ)→ H2(X,G)

H2(ψ)→ . . .

The homomorphisms δ, which we denote for simplicity all by thesame symbol, are called the connecting homomorphisms.

Proof.

1) We construct for every morphism ϕ ∶ F → G a group homo-morphism

Hq(ϕ) ∶Hq(X,F)→Hq(X,G)For the morphism

Cq(U , ϕ) ∶ Cq(U ,F) → Cq(U ,G)(αI)∣I ∣=q ↦ (ϕUI(αI))∣I ∣=q

we haveCq+1(U , ϕ) dq = dq Cq(U , ϕ)


Cq(U ,G)dq→ Cq+1(U ,G)

Cq(U , ϕ) ↑ ↑ Cq+1(U , ϕ)Cq(U ,F)

dq→ Cq+1(U ,F)


commutes (exercise!). Hence, Cq(U , ϕ) maps cocycles to co-cycles and coboundaries to coboundaries, so the morphism

Hq(U , ϕ) ∶ Hq(U ,F) → Hq(U ,G)⟨α⟩ ↦ ⟨Cq(U , ϕ)(α)⟩

is well-defined. We now define

Hq(ϕ) ∶ Hq(X,F) → Hq(X,G)[β] ↦ [Hq(U , ϕ)(β)]

where α ∈ Hq(U ,F). This is well-defined, since for coveringsU and V with U < V via τ the diagram

Cq(U ,F)Cq(U ,ϕ)→ Cq(U ,G)

Cq(τ) ↑ ↑ Cq(τ)Cq(V ,F)

Cq(U ,ϕ)→ Cq(V ,G)

commutes (exercise!).

2) We construct now the boundary operator

δ ∶Hq(X,H)→Hq+1(X,F)

Suppose [⟨α⟩] ∈Hq(X,H) is represented by α = (αI) ∈ Zq(U ,H).We construct a covering U ′ < U via τ and a β ∈ Cq(U ′,G) suchthat

Cq(U ′, ψ)(β) = Cq(τ)(α)

This is possible since

ψx ∶ Gx → Hx

is surjective, so forαI ∈ H(UI)

and all x ∈ UI there is an open neighborhood V (x) of x and a

β(x)I ∈ G(UI)

such thatψV (x)(β(x)I ) = αI ∣V (x)

Moreover we have

Cq+1(U ′, ψ)(dqβ) = dq(Cq(U ′, ψ)(β)) = dq(Cq(τ)(α))= Cq+1(τ)(dqα) = 0.


So we have a sequence of maps (in general not exact)

Cq+1(U ′,F)Cq+1(U ′,ϕ)Ð→ Cq+1(U ′,G)

Cq+1(U ′,ψ)Ð→ Cq+1(U ′,H)

dqβ z→ 0

So dqβ ≠ 0, but Cq+1(U ′, ψ)(dqβ) = 0.

With the analogous argument as above, there is a U ′′ < U ′ viaτ ′ and a

γ ∈ Cq+1(U ′′,F)

such thatCq+1(U ′′, ϕ)(γ) = Cq(τ ′)(dqβ)

Moreover we have

Cq+2(U ′′, ϕ)(dq+1γ) = dq+1(Cq+1(U ′′, ϕ)(γ)) = dq+1(Cq(τ ′)(dqβ))= dq+1dqC

q(τ ′)(β) = 0.

Since ϕ is injective, also ϕU is injective for all U ⊂ X open,hence Cq+2(U ′′, ϕ) is injective. Hence

dq+1γ = 0

that isγ ∈ Zq+1(U ′′,F)

and we defineδ([⟨α⟩]) ∶= [⟨γ⟩].

As an exercise show that δ(u) is independent of choices (α,U ′, U ′′, and γ).

3) We now show that the long sequence is exact. We write

ψ =Hq(ψ)ϕ =Hq(ϕ)

and showim(ψ) = ker(δ),

the remaining statements follow in a similar way (exercise!).Suppose

[⟨α⟩] ∈Hq(X,H) with δ([⟨α⟩]) = 0

and α ∈ Zq(U ,H). By the above construction we have a cov-ering U ′ < U via τ and a

β ∈ Cq(U ′,G)


such thatCq(U ′, ψ)(β) = Cq(τ)(α)

and a covering U ′′ < U ′ via τ ′ and a

γ ∈ Cq+1(U ′′,F)

such thatCq+1(U ′′, ϕ)(γ) = Cq(τ ′)(dqβ)

and0 = δ([⟨α⟩]) = [⟨γ⟩]

Hence there is a covering U ′′′ < U ′′ via τ ′′ and ε ∈ Cq(U ′′′,F)such that

Cq+1(τ ′′)(γ) = dqε.

Then we have

dq(Cq(τ ′′ τ ′)(β) −Cq(U ′′′, ϕ)(ε))= Cq+1(τ ′′)(Cq+1(U ′′, ϕ)(γ)) −Cq+1(U ′′′, ϕ)(Cq+1(τ ′′)(γ))= Cq+1(τ ′′)(Cq+1(U ′′, ϕ)(γ) −Cq+1(U ′′, ϕ)(γ)) = 0,

hence

Cq(τ ′′ τ ′)(β) −Cq(U ′′′, ϕ)(ε) ∈ Zq(U ′′′,F)

Applying Cq(U ′′′, ψ), we get

Cq(U ′′′, ψ)(Cq(τ ′′ τ ′)(β) −Cq(U ′′′, ϕ)(ε))= Cq(U ′′′, ψ)Cq(τ ′′ τ ′)(β) −Cq(U ′′′, ψ)Cq(U ′′′, ϕ)(ε)= Cq(U ′′′, ψ)Cq(τ ′′ τ ′)(β)= Cq(τ ′′ τ ′)(Cq(U ′, ψ)(β))= Cq(τ ′′ τ ′ τ)(α)

hence

ψ([⟨Cq(τ ′′ τ ′)(β) −Cq(U ′′′, ϕ)(ε)⟩]) = [⟨Cq(τ ′′ τ ′ τ)(α)⟩]= [⟨α⟩] ∈Hq(X,H)

thus [⟨α⟩] ∈ im(ψ). So we can lift α on cochain level, howeverthe lift is not β itself, but a corrected version of it.

For the other inclusion, suppose

[⟨α⟩] ∈ im(ψ)


with α ∈ Zq(U ,H), so if we assume the representative α ischose such that U is fine enough, there is a β ∈ Zq(U ,G) with

Cq(U , ψ)(β) = α.

We have to show that

δ([⟨α⟩]) = 0.

By construction of δ, we have

δ([⟨α⟩]) = [⟨γ⟩]

where (if we choose above U fine enough)

Cq+1(U , ϕ)(γ) = dqβ = 0

hence γ = 0, hence δ([⟨α⟩]) = 0.

6.4 Lemma of Leray

Definition 6.4.1 An open cover U is called Leray with respect tothe sheaf F if

Hq(UI ,F) = 0

for allq > 0

and all∣I ∣ ≥ 1

Lemma 6.4.2 (Leray) If U os Leray, then the canonical maps

κq ∶Hq(U ,F)→Hq(X,F)

are isomorphisms. If fact the lemma is still true, if we have Hq(UI ,F) =0 for all ∣I ∣ ≤ q.

Proof. We prove the lemma for q = 1: We have to show that ifV = (Vi) < U = (Ui) via τ , then

H1(τ) ∶H1(U ,F)→H1(V ,F)

is bijective.


1) H1(τ) is injective: Here we do not use the Leray property atall. Let

α = (αi,j) ∈ Z1(U ,F)

withH1(τ)(α) = 0.

Thenατ(k),τ(l) ∣Vk,l= (βl − βk) ∣Vk,l

with (βl) ∈ C1(V ,F). On Ui ∩ Vk ∩ Vl we have

βl − βk = ατ(k),τ(l) = ατ(k),i + αi,τ(l)

where we write αi,j = −αj,i if i ≥ j. So we have

βl + ατ(k),i = βk + ατ(l),i

hence by the sheaf axiom (A2) there are hi ∈ F(Ui) with

hi = ατ(k),i + βk

on Ui ∩ Vk. So on Ui ∩Uj ∩ Vk we have

αi,j = hj − hi

which by (A1) implies that this also holds on Ui ∩ Uj. Thismeans that

α = 0 ∈H1(U ,F).

2) H1(τ) is surjective: It surfices to use that

Hq(Ui,F) = 0

for all q ≥ 1 and i. For

(fαβ) ∈ Z1(V ,F)

we have to show that there is

(gi,j) ∈ Z1(U ,F)

with(gτ(α),τ(β) − fαβ) ∈ B1(U ,F)

For the covering W(i) = (Ui ∩ Vα) of Ui we have by (1) thatthe map

H1(W(i),F ∣Ui)→H1(Ui,F) = 0


is injective, hence,

H1(W(i),F ∣Ui) = 0

so there arehiα ∈ F(Ui ∩ Vα)

withfαβ = hiα − hiβ on Ui ∩ Vαβ

So on Uij ∩ Vαβ we have

hjα − hjβ = hiα − hiβ

so by (A2) there are gij ∈ F(Uij) with

gij = hjα − hjβ on Uij ∩ Vα

On Uijk we then have then

gij + gjk = gik

that is(gij) ∈ Z1(U ,F)

Moreover, we have

gτ(α),τ(β)−fαβ = gτ(β),α−gτ(α),α−(gτ(β),α−gτ(β),β) = gτ(β),β−gτ(α),α

So withhα ∶= gτ(α),α

we havegτ(α),τ(β) − fαβ = hβ − hα

6.5 Cohomology from resolutions

Definition 6.5.1 Let F be a sheaf on X. A resolution of F is anexact sequence of sheaf morphisms

0→ F ι→ F0 d0→ F1 d1→ F d2→ . . .

withdi+1 di = 0

for all i. The resolution is called acyclic if

Hq(X,F i) = 0 for all i ≥ 1 and q ≥ 1


Remark 6.5.2 Note that an acyclic resolution induces a complex(in general not exact)

0→ F(X) ι→ F0(X) d0→ F1(X) d1→ F(X) d2→ . . .

Lemma 6.5.3 (Formal de Rham lemma) If

0→ F ι→ F0 d0→ F1 d1→ F d2→ . . .

is acyclic, we have

Hq(X,F) ≅ ker(dq ∶ F q(X)→ F q+1(X))im(dq−1 ∶ F q−1(X)→ F q(X))

where we set F−1(X) = 0.

Proof. Decompose into short exact sequences

0→ F ι→ F0 d0→ imd0 → 0

0→ imd0 = kerd1 → F1 d1→ imd1 → 0

⋮

and use the long exact cohomology sequences.

7

Cohomology of Differentialforms

All manifolds in this chapter will be paracompact.

7.1 De Rham Cohomology

Theorem 7.1.1 If X is a differentiable manifold and U is an opencovering and A is the sheaf of differentiable functions on X, then

Hq(U ,A) = 0

for all q ≥ 1.

The proof uses the existence of a partition of 1.

Corollar 7.1.2 Hq(X,A) = 0 for all q ≥ 1.

The proof of Theorem 7.1.1 is also valid for any sheaf of A-modules.

Theorem 7.1.3 If X is a differentiable manifold and U is an opencovering and S is a sheaf of A-modules on X, then

Hq(X,S) = 0

for all q ≥ 1.

Corollar 7.1.4 If X is a complex manifold then

Hj(X,Ap,q) = 0

for j ≥ 1.

110

7. COHOMOLOGY OF DIFFERENTIAL FORMS 111

Definition 7.1.5 Let X be a differentiable manifold and

HpDR(X) = ϕ ∈ Ap(X) ∣ dϕ = 0

dAp−1(X)

where we set A−1(X) ∶= 0, is called the p-th de Rham cohomologygroup of X.

Theorem 7.1.6 (de Rham) If X is a (paracompact) differentiablemanifold, then

Hp(X,R) ≅HpDR(X)

Remark 7.1.7 If X is a complex manifold then

Hp(X,C) ≅HpDR(X)

where we consider complex valued differential forms.

Proof. A resolution of the constant sheaf R is given by

0→ R→ A(X) d0→ A1(X) d1→ . . .

since dr+1 dr = 0 and ker(dr) ⊂ im(dr−1) since by the Lemma ofPoincare for a star-shaped open subset (e.g. an open disc) for anyω ∈ Ar(U) with

dω = 0

there is a ψ ∈ Ar−1(U) with

dψ = ω.

By Theorem 7.1.3, the resolution is acyclic, hence the claim followsfrom Lemma 6.5.3.

This result has an analogon in the holomorphic world:

7.2 Dolbeault Cohomology

Definition 7.2.1 Let X be a complex manifold. We define

Hp,q

∂(X) =

ϕ ∈ Ap,q(X) ∣ ∂ω = 0∂Ap,q−1(X)

where we set Ap,−1(X) = 0, and we call

Hp,0(X) =HqDol(X)

the Dolbeault cohomology group of X.


Lemma 7.2.2 (Dolbeault) Let X ⊂ Cn be an open polycylinderand X ′ ⊂ X a concentric open polycyclinder (that is, all radii arestrictly smaller). Then for all q ≥ 1 and ω ∈ Ap,q(X) with ∂ω = 0,there is an σ ∈ Ap,q−1(X ′) with

ω ∣X′= ∂σ

Theorem 7.2.3 (Dolbeault) Let X be a complex manifold. Then

Hp,q

∂(X) ≅Hq(X,Ωp)

for all p, q.

Proof. An acyclic resolution of Ωp is given by

0→ Ωp ι→ Ap,0(X) ∂→ Ap,1(X) ∂→ . . .

For this note that:

1) We have ∂ ∂ = 0 by Remark 5.4.11.

2) Follows from the Lemma of Dolbeault 7.2.2 and Theorem4.3.28.

3) The resolution is acyclic by Corollary 7.1.4.

7.3 Poincare Duality

Theorem 7.3.1 Let X be a compact differentiable manifold of di-mension n. Then we have isomorphisms

Hr(X,R) ≅HrDR(X) φ→Hn−r

DR (X)∗ ≅Hn−r(X,R)∗ ≅Hn−r(X,R)

withφ([s]) ∶Hn−r

DR (X)→ R, [t]↦ ∫Xs ∧ t

Definition 7.3.2 We define the Hodge star operator

∗ ∶ AkX → An−kX

defined by the property

u ∧ ∗v = g(u, v)dx1 ∧ . . . ∧ dxn

for u, v ∈ AkX .


Example 7.3.3 The ∗ Operator is characterized by the property

∗(dx1 ∧ . . . ∧ dxj) = dxj+1 ∧ . . . ∧ dxn

Proof. The map φ is well-defined: If s′ = s+dα and t′ = t+dβ then

∫Xs′ ∧ t′ = ∫

Xs ∧ t + ∫

Xdα ∧ t + ∫

Xs ∧ dβ + ∫

Xdα ∧ dβ = ∫

Xs ∧ t

noting that

∫Xdα ∧ t = ∫

Xd(α ∧ t) − ∫

Xα ∧ dt = 0

by the Theorem of Stokes and by dt = 0, similarly

∫Xs ∧ dβ = 0,

and we have

∫Xdα ∧ dβ = ∫

Xd(α ∧ dβ) = 0

using d d = 0 and the Theorem of Stokes.We show that φ is injective: Using the ∗ operator, we have for

s with φ([s]) = 0, that

∫Xs ∧ ∗s = 0

hence s = 0.Exchanging r and n − r we obtain that φ is an isomorphism.

Remark 7.3.4 The same holds true for complex valued cohomol-ogy and differential forms.

Note that s ∧ t is an n-form on X.

7.4 Serre Duality

Theorem 7.4.1 Let X be a compact complex manifold. Then thepairing

ψ ∶Hp,q

∂(X) ×Hn−p,n−q

∂(X)→ C

([s], [t])↦ ∫Xs ∧ t

is non-degenerate, so

Hp,q

∂(X) ≅Hn−p,n−q

∂(X)

The proof is similar to that of Poincare duality, however moretechnical, so we skip it.


7.5 Hodge Decomposition

Definition 7.5.1 Let X be a compact Riemannian manifold, wedefine the operator

d∗ ∶ Ak(X)→ Ak+1(X)d∗ = −(−1)kn ∗ d∗

Remark 7.5.2 The operator ∗ is adjoint to d in the sense that

∫Xds ∧ t = ∫

Xs ∧ d∗t

Definition 7.5.3 The Laplace operator is defined as

∆ = dd∗ + d∗d

We define the space of harmonic forms as

Hk(X) = ω ∈ Ak(X) ∣ ∆ω = 0

Theorem 7.5.4 There is an orthogonal direct sum decomposition

Ak(X) = Hk(X)⊕ im ∆.

anddimRHk(X) <∞.

Definition 7.5.5 With the Laplace operator

∆∂ = ∂∂∗ + ∂∗∂

we defineHp,q(X) = ω ∈ Ak(X) ∣ ∆∂ω = 0

Theorem 7.5.6 Let X be a compact complex manifold. Then thereis a canonical isomorphism

Hp,q

∂(X)→ Hp,q(X)[ω]↦ ω

where ω is the unique harmonic representative of [ω].Under appropriate conditions on the manifold (the manifold

must be Kahler), we have a direct sum decomposition

Hk(X) = ⊕p+q=kHp,q(X)

and thus a decomposition

Hk(X,C) ≅HkDR(X) ≅

Hk(X) = ⊕p+q=kHp,q(X) ≅ ⊕

p+q=kHp,q

∂(X)


Remark 7.5.7 In terms of harmonic forms we can interpret Serreduality via harmonic forms and the ∗ operator as

s z→ ∫X s ∧Hp,q

∂(X) Ð→ Hn−p,n−q

∂(X)

↓ ↓Hp,q(X) ∗Ð→ Hn−p,n−q(X)

Remark 7.5.8 The Hodge diamond with entry (p, q) standingfor Hp,q

∂(X) is

(n,n). .

. .. .

(n,0) (0, n). .. .. .

(0,0)

hence has a star symmetry, plus a vertical symmetry via conjuga-tion.

Example 7.5.9 The Hodge diamond of projective space is given by

Hq(Pn,ΩpPn) =H

p,q

∂(Pn) = 0 for p ≠ q

Hp(Pn,ΩpPn) =H

p,p

∂(Pn) = C

so it is, e.g. for n = 3,

C0 0

0 C 00 0 0 0

0 C 0 00 0 0 0

0 C 00 0

C

This follows by the Hodge decomposition since for

H2k+1(Pn,Z) = 0

H2k(Pn,Z) = Z

Index

acyclic, 108affine algebraic set, 2affine algebraic variety, 2affine chart, 8affine space, 2analytic, 19, 27analytic set, 30analytic subset, 30, 43

bounded function, 21

Cech-cohomology, 101change of base construction, 74change of coordinate map, 1chart, 38charts, 1coboundaries, 99cochain operator, 98cochains, 98cocyle, 99codimension, 31compact, 36complex differentiable, 17complex manifold, 1, 38complex structure, 38composition, 60continuous, 36convergence radius, 19convergent, 25

de Rham cohomology, 111dehomogenization, 9derivation, 81differential, 81differential form, 89direct sum, 80Dolbeault cohomology, 111

Dolbeault operators, 92double cone, 14dual, 80

elliptic curve, 4entire, 21epimorphism of presheaves, 60

fiber, 77finer than, 100fundamental theorem of algebra,

16, 21

geometric invariant theory, 5graded ring, 5

harmonic forms, 114Hausdorff, 37Hodge diamond, 115holomophic vector bundle, 77holomorphic, 17, 22, 23, 40, 43holomorphic differential form, 91holomorphic section, 86holomorphic vector field, 90homogeneous ideal, 6homogeneous polynomial, 6homogeneous zero ideal, 8homogenization, 10hyperboloid, 14hyperplane at infinity, 8

ideal sheaf, 73injective presheaf morphism, 60inverse image sheaf, 74irreducible, 2isomorphism of presheaves, 60

Jacobian matrix, 48

116

INDEX 117

kernel, 60

Lemma of Poincare, 111Leray, 106local trivialization, 78locally compact, 36locally Euclidean, 37locally finite, 102locally free, 93long exact cohomology sequence,

102

maximal, 38modulus, 22monomorphism of presheaves, 60

neighborhood, 36

open sets, 36

paracompact, 102polydisc, 23presheaf, 55presheaf morphism, 60principal part, 100projective algebraic set, 7projective closure, 11projective space, 4pull-back bundle, 86

quotient sheaf, 72

rank, 77regular part, 100restriction sheaf, 73Riemann surface, 37

second-countable, 37section, 56sheaf, 56sheaf morphism, 60sheaf of $O X$-modules, 75sheaf of differentiable sections, 87sheaf of holomorphic functions,

57sheaf of holomorphic sections, 87

singularity, 14stalk, 58standard grading, 6submanifold, 13, 46Surfer, 14surjective presheaf morphism, 60symmetric power, 80

tangent bundle, 83tangent map, 86tangent space, 81Taylor series, 20tensor product, 75, 80Theorem of Stokes, 113topological space, 36toric geometry, 5torus, 39total space, 78transition functions, 78

vector bundle map, 85

wedge product, 80Wirtinger derivatives, 18

Zariski topology, 11zero section, 87

Bibliography

[1] Atiyah, M. F.; MacDonald, I. G.: Introduction to commutativealgebra (1969).

[2] Decker, W.; Greuel, G.-M.; Pfister, G.; Schonemann, H.: Sin-gular 4-1-3 — A computer algebra system for polynomial com-putations, available at http://www.singular.uni-kl.de (2017).

[3] Eisenbud, D.: Commutative Algebra with a View Toward Alge-braic Geometry. Springer (1995).

[4] Endrass, S.: Surf, http://surf.sourceforge.net

[5] Grayson, D. R.; Stillman, M. E.: Macaulay2, a soft-ware system for research in algebraic geometry, available athttp://www.math.uiuc.edu/Macaulay2/ (2017).

[6] Greuel, G.-M.; Meyer, H.; Stussak, Ch.: Surfer,http://www.imaginary-exhibition.com/surfer.php (2008).

[7] Greuel, G.-M.; Pfister, G: A Singular introduction to commu-tative algebra, Springer (2002).

[8] Maple (Waterloo Maple Inc.): Maple 2017,http://www.maplesoft.com/ (2017).

118

http://www.singular.uni-kl.de

http://surf.sourceforge.net

http://www.math.uiuc.edu/Macaulay2/

http://www.imaginary-exhibition.com/surfer.php

http://www.maplesoft.com/

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