A Course in Complex Geometry and Holomorphic...

Hossein Movasati

A Course in Complex Geometryand Holomorphic Foliations

April 27, 2018

Publisher

v

Omar Khayyam, Robaiyat 93. Calligraphy: nastaliqonline.ir.

Dedicated to my masters.

Acknowledgements

The comments and questions of the students and colleagues who sat in my courseshave been extremely useful for me to improve the present text and put more details.For this, I would like to thank Enzo Aljovin, Alvaro Almeida, Jorge Duque, DanielLopez, Walter Paez and Roberto Villaflor.

ix

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The organization of the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Complex analysis in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Germs of analytic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Analytic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Embedding dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Extension theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 Dolbeault lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.9 Algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.10 Analytic schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.11 Analytic sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.12 Restriction of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.13 Homomorphism between sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.14 Rank of coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.15 Push-forward of a sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Covering and direct limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Acyclic sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 How to compute Cech cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . 313.6 Resolution of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.7 Cech cohomology and Eilenberg-Steenrod axioms . . . . . . . . . . . . . . . 343.8 Dolbeault cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.9 Cohomology of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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xii Contents

3.10 Short exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Stein varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Stein varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Some properties of Stein varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Stein covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Kahler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Positive forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Hermitian metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Fubini-Study metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.5 Torus as Kahler manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Strongly convex/plurisubharmonic functions . . . . . . . . . . . . . . . . . . . . . . 496.1 Strongly convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.4 Strongly pseudoconvex domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.5 Plurisubharmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Cohomological properties of pseudoconvex domains . . . . . . . . . . . . . . . 617.1 A theorem of Grauert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 Exceptional varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.3 Complementary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 Positive and negative line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.1 Line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.2 Chern classes in Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.3 Chern classes in de Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 728.4 The case of a Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.5 Positive line bundle in the sense of Kodaira . . . . . . . . . . . . . . . . . . . . . 778.6 Positive and negative bundles in the sense of Grauert . . . . . . . . . . . . . 788.7 The equivalence of Grauert and Kodaira positivity . . . . . . . . . . . . . . . 788.8 Complementary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

9 Vanishing theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.2 Homogeneous functions along fibers . . . . . . . . . . . . . . . . . . . . . . . . . . 829.3 Kodaira vanishing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.4 The case of a Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.5 Grauert vanishing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859.6 Main vanishing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869.7 Restriction of line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Contents xiii

9.8 A negative divisor of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10 Embedding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.1 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.2 Blow-up of a singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.3 An embedding theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9210.4 Blow up along a submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

11 Deformation of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.1 Kodaira-Spencer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

12 Foliated neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9912.1 Holomorphic foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9912.2 Construction of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10012.3 Equivalence of transverse foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10212.4 Construction of line fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10312.5 Construction of holomorphic foliations . . . . . . . . . . . . . . . . . . . . . . . . 10412.6 Grauert’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.7 Arbitrary codimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.8 Proof of Theorem 12.5, codimension greater than one . . . . . . . . . . . . 10612.9 Proof of Theorem 12.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.10Rational curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.11Complementary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

13 Few theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10913.1 Serre duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10913.2 Birkhoff-Grothendieck theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

14 Remmert reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11114.1 Proper mapping and direct image theorems . . . . . . . . . . . . . . . . . . . . . 11114.2 Equivalence relations in varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11214.3 Cartan’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11314.4 Stein factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11414.5 Remmert reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11514.6 Complementary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

15 Formal and finite neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11715.1 Formal and finite neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11815.2 Some propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11915.3 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12115.4 Obstructions to formal isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 12115.5 Calculating the obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12415.6 Breaking Aut(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12515.7 The case of a Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12615.8 Complementary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xiv Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 1Introduction

There are many theorems and statements in algebraic aeometry which are of com-plex nature and are not covered in a classical algebraic geometry course. The presentbook pretends to partially fill this gap and discuss all the necessary materials forproving one of these theorems. This is namely the following theorem of Grauert andits generalizations to higher dimensions:

Theorem 1 (H. Grauert [Gr62]) Let A be a Riemann surface of genus g embed-ded in a smooth complex surface X and with negative and less than 4− 4g self-intersection. Then a topological neighborhood of A is biholomorphic to the neigh-borhood of the zero section of the normal bundle of A in X.

See Figure 1.1. Having Grauert’s theorem in mind, we have tried to collect manyother theorem, most of them of algebraic geometric nature. However, we have notlimited ourselves to purely algebraic proofs, for instance, we have fairly used C∞ andplurisubharmonic functions. A typical example of the above theorem is a rationalcurve A = P1 obtained by a blow-up of the origin in C2. If π is the blow-downmap and ψ : C2 → R+, ψ(x,y) = |x|2 + |y|2, then A has strongly pseudo-convexneighborhoods x ∈ X | ψ π(x) < ε. This picture has been the guiding principlein Grauert’s paper [Gr62].

We have many reasons to choose such a theorem and its generalizations to higherdimensional varieties A and X , as a basic theme of our book. First, it uses the usualtopology of algebraic varieties over complex numbers. The same statement is triv-ially false with Zariski topology. Therefore, it cannot be stated in a purely algebraiccontext. The usage of formal neighborhood both in analytic or algebraic contextwould not imply the Grauert theorem. Second, in higher dimension the complexmanifold A must be necessarily algebraic and so we need the machinary of algebraicgeometry applied to A. This means that we will apply the machinery of algebraic andcomplex geometry in the same time. Third, in the way we learn its proof we learnmany other classical tools and theorems. This includes, Cech cohomology, positive(ample) and negative bundles, Kodaira vanishing theorem and embedding and so on.Fourth, we learn how the machinery of holomorphic foliations can be used to provestatements in complex Algebraic Geometry. Because of this, the book is also is an

1

2 1 Introduction

introduction to the theory of holomorphic foliations on complex manifolds. We willdeal with only foliation whose leaves are analytic varieties and so they will not haveany dynamics.

Fig. 1.1 A biholomorphism of two neighborhoods

We don’t prove all the theorems which appear in this text. Instead, we want thatthe reader feel himself comfortable with all such theorems, by explaining them inspecial cases and using examples. We provide the necessary references for miss-ing proofs so that the curious reader iterates back to the original proofs. The ba-sic idea is to prepare the reader for consulting the already published books like[GuI90, GuII90, GuIII90, GrRe79, GPR94] whenever he finds it necessary. Themathematics is growing fast and sometimes it is necessary to learn the art of reading,understanding and using many theorems, without going into details of their proof.

1.1 The organization of the text

The book is organized in the following way: In Chapter 2 we present all the nec-essary background in complex analysis and several complex variables in order toread and follow the book. In this chapter we explain concepts like germ of varieties,structural sheaf, Stein varieties, coherent sheaves and so on, and announce manytheorems related to all these concepts. Whenever it is possible, we explain theoremsby examples and particular cases and give the idea of the proof.

Chapter 3 is devoted to the basics of sheaf cohomology and in particular theconstruction of Cech cohomology. It is aimed to prepare the reader as fast as possiblein order to feel himself comfortable with the notation H i(X ,S ). We also explainline bundles and Chern classes in the framework of Cech cohomology.

In Chapter 4 we review basic facts and theorems on Stein varieties.Chapter 6 is devoted to pseudoconvex domains. For some technical reasons, we

have preferred to work with C2 convex functions instead of C2 plurisubharmonic

1.2 Notations 3

functions. A convex function carries just the convexity information of its level vari-eties and is easy to handle.

In Chapter 7 we prove that the cohomologies of strongly pseudoconvex domainswith coefficients in a coherent sheaf are finite dimensionals. The same is also true forcompact analytic varieties instead of strongly pseudoconvex domains. Using Rem-mert reduction theorem in Appendix 14 this leads us to the notion of exceptionalvarieties.

In Chapter 8 we apply the machinery of pseudoconvex domains to the neigh-borhoods of zero sections of line bundles and we get the notion of negative andpositive vector bundles. We prove that Grauert positivity of line bundles is the sameas Kodaira’s positivity and we give a proof of Kodaira vanishing theorem.

Chapter 9.5 is dedicated to one of the most important cohomological propertiesof strongly pseudoconvex domains. This is namely Theorem 9.4 which states that ifa coherent sheaf vanishes enough along the exceptional divisor of a strongly pseu-doconvex domain then its cohomologies are zero except the 0-th cohomology.

In Chapter 10 we review the notion of blow-up and we see that an exceptionalvariety can be obtained by a usual blow-up of a singularity. We also explain thenotion of a blow-up along a divisor. This plays a fundamental role in the study ofneighborhoods of varieties with codimension stricktly bigger than one.

In Chpater 12 we start the study of holomorphic foliations in neighborhoods ofvarieties. We first consider the most simple foliations which are those transversalto the variety. This will be used in order to give a geometric proof of Grauert’sTheorem. We also study foliations with tangencies.

In Appendix 14 we discuss Remmert reduction theorem. Using this one can showthat strongly pseudoconvex domains are the point modification of Stein varieties.This leads to the notion of exceptional or negatively embedded varieties.

In Appendix 15 we introduce the notion of a finite neighborhood and the exten-sion problems in this context. The content of this Appendix are part of the machinarywhich Grauert used to prove his theorem and we do not need them for our proof.However, it can be used as a nice collection of exercises to the concepts introducedin Chapter 2 and Chapter 3.

1.2 Notations

The letter A is mainly used to denote an analytic variety, complex manifold or analgebraic variety. It is usually embedded in another variety X . In this case we denoteby (X ,A) a neighborhood of A in X . This notation means that, if necessary, we cantake such a neighborhood smaller. In other words, we think of (X ,A) as a neigh-borhood of A in X , however, when (X ,A) appears in a statement or property, such aneighborhood can be smaller. For instance a function f : (X ,A)→ C is defined in aneighborhood U of Ain X . It satisfies a property P if there is another neighborhoodU ′ ⊂U of A in X such that f |U ′ satisfies P.

4 1 Introduction

We use the same letter i for the complex number√−1 and for indexing; being

clear in the text the distinction between them.For a C-vector space V we denote by V ∗ the dual vector space, that is, the vector

space of all linear maps V → C.For topological spaces A and B we write A ⊂⊂ B to denote that A is relatively

compact in B, i.e. the closure of A in B is compact in B.

Chapter 2Preliminaries

In this chapter we review the notion of analytic and algebraic varieties, coherentsheaves and basic facts about Stein varieties. The main references for contents ofthis chapter are [GuI90, GuII90, GuII90, GrRe79]. We assume that the readeris familiar with a basic knowledge in complex analysis in one variable and basictopology.

2.1 Complex analysis in one variable

There are many theorems which differentiate complex analysis from real analysis.Two of them are

1. Any holomorphic function f : (C,0)−0→ C has a Laurent series at 0.2. Cauchy’s integral formula.

In particular, Cauchy’s integral formula is one of the origins of homotopy theory intopology, and this might be an indication to the fact that complex analysis usuallytends to geometry rather than to pure analysis.

2.2 Holomorphic functions

In this section we remind some basic definitions and facts about holomorphic func-tions. For more details see [GuI90], Chapter A.

Definition 2.1 Let a ∈ Cn and r ∈ Rn+. An open polydisc in Cn is the set:

∆(a,r) :=

z ∈ Cn∣∣∣|zi−ai|< ri, for i = 1,2, . . . ,n

Definition 2.2 A function f : D→ C is called holomorphic if for each point a ∈ Dthere is an open polydisc ∆(a,r) ⊂ D such that the function f has a power series

5

6 2 Preliminaries

expansionf (z) = ∑

ifi(z−a)i

which converges to f (z) for all z ∈ ∆(a,r). Here, where i = (i1, i2, . . . , in) runsthrough (N∪0)n and (z− z)i = (z1−a1)

i1(z2−a2)i2 · · ·(zn−an)

in .

Proposition 2.1 (Osgood’s lemma) If a continous function f : D→ C defined inan open subset D⊂ Cn is holomorphic in each variable then it is holomorphic.

Proof. The main idea begind the proof is the Cauchy integral formula in one vari-able. First, we take a closed polydisc ∆(a,r)⊂D. We apply Cauchy integral formulan times and we get

f (z)=1

(2πi)n

∫|ζ1−a1|=r1

dζ1

ζ1− z1

∫|ζ2−a2|=r2

dζ2

ζ2− z2· · ·∫|ζn−an|=rn

f (ζ1,ζ2, . . . ,ζn)dζn

ζn− zn

Since f is continuous we get

f (z) =1

(2πi)n

∫f (ζ1,ζ2, . . . ,ζn)dζ1dζ2 · · ·dζn

(ζ1− z1)(ζ2− z2) · · ·(ζn− zn)(2.1)

where the integration takes place over the torus |z1−a1|= r1, |z2−a2|= r1, · · · , |zn−an|= rn. Using the geometric series

1ζi− zi

=∞

∑n=0

(zi−ai)n

(ζi−ai)n+1

we get the result. ut

Proposition 2.2 If f ,g : D→C are two holomorphic functions in a connected opensubset D⊂ Cn and if f = g in a nonempty open subset of D then f = g in D.

Proof. The proof is similar to the case of one variable, that is, n = 1. ut

We will use the notation:

zi = xi +√−1yi, i = 1,2, . . . ,n

and

∂

∂ zi:=

12(

∂

∂xi−√−1

∂

∂yi), i = 1,2, . . . ,n

∂

∂ zi:=

12(

∂

∂xi+√−1

∂

∂yi), i = 1,2, . . . ,n.

Proposition 2.3 A C1 function f : D→ C is holomorphic if and only if it satisfiesthe Cauchy-Riemann equations:

2.3 Local rings 7

∂ f∂ zi

= 0, i = 1,2, . . . ,n. (2.2)

Proof. This follows from the same theorem in one variable (n = 1) and Osgood’slemma. ut

2.3 Local rings

For a topological space X and a point x ∈ X we denote by (X ,x) a neighborhood ofx in X . This means that in our statements and arguments we fix a neighborhood of xin X but we can take it smaller if it is necessary. A C-algebra is a commutative ringcontaining the field C as a subring, with 1 ∈C as the identity element of the ring. Ahomomorphism between two C-algebras is a ring homomorphism that induces theidentity mapping on the subfield C. An example of C-algebra we use in this text is:

• OCn(U), the space of holomorphic functions in an open subset U of Cn.• OCn,x, the ring of germs of holomorphic functions in a neighborhood of x in Cn;

OCn,x = f ∈ OCn(U) for some open set U in Cn, x ∈U/∼

wheref1 ∈ OCn(U1), f2 ∈ OCn(U2), x ∈U1∩U2,

f1 ∼ f2⇔ f1 = f2 in some open set U3 ⊂U1∩U2,x ∈U3.

The maximal ideal of OCn,x is

MCn,x := f ∈ OCn,x | f (x) = 0.

We have OCn,x/MCn,x ∼= C which is given by evaluating functions at x. Therefore,OCn,x is a local ring. Slowly, the reader might get used to the notation of short exactsequences:

0→MCn,x→ OCn,x→ C→ 0 (2.3)

One can consider OCn,x as the ring of convergent power series ∑∞i ai(y− x)i,

where i = (i1, i2, . . . , in) runs through (N∪ 0)n and (y− x)i = (y1 − x1)i1(y2 −

x2)i2 · · ·(yn− xn)

in . Later, we will need the following theorems. For proofs see[GuII90], Theorem A8, Theorem A4.

Theorem 2.1 (Weierstrass preparation theorem) Let f ∈ OCn,0 be such that

f (0,0, · · · ,zn) 6= 0

and this has a zero of order m at zn = 0. There is a unique monic polynomial g =zm

n + am−1(z1,z2, . . . ,zn−1)zm−1n + · · ·+ a0(z1,z2, . . . ,zn−1) of degree m in zn, with

am−1(0) = · · ·= a0(0) = 0, and u ∈ OCn,0 with u(0) 6= 0 such that f = g ·u.

8 2 Preliminaries

Fig. 2.1 Weierstrass preparation theorem

Proof. The idea of the proof is depicted in Figure 2.1. We fix a polydisc ∆(0,r′)×∆(0,rn) around 0 such that for any z′ ∈ ∆(0,r′) the function f (z′,zn) has m zeroszn,i, i = 1,2, . . . ,m, counting with multilicity in ∆(0,rn). Let

g(z′,zn) := (zn− zn,1)(zn− zn,2) · · ·(zn− zn,m) = zmn +am−1(z′)zm−1

n + · · ·+a0(z′)(2.4)

The functions ai’s are holomorphic functions in z′ because by Cauchy’s integralformula we have:

m

∑i=1

zrn,i =

12πi

∫|ζ |=rn

ζr

∂ f (z′,ζ )∂ζ

f (z′,ζ )dζ . (2.5)

The function u := fg for fixed z′ is holomorphic in zn and it attains its maximum in

∂∆(0,rn). Therefore, it has maxmimum in ∆(0,r′)× ∂∆(0,rn) which is compact.By Riemann’s extension theorem we conclude that u is extends to a holomorphicfunction in ∆(0,r). ut

Theorem 2.2 The ring ring OCn,0 is Noetherian, that is, every ideal in OCn,0 isfinitely generated.

Proof. The proof is by induction on n. For n = 0 we have OCn,0 = C and so itNoetherian. Let us assume that OCn−1,0 is Noetherian and take an ideal I ⊂ OCn,0.After a linear change of coordinates in Cn, we can assume that for some f ∈ I it isregular in zn, that is f (0,zn) is not identically zero and, after a multilication by a unit,we can assume that f is in the Weierstrass format and so f ∈ I∩OCn−1,0[zn]. Usingthe hypothesis and Hilbert’s basis theorem we know that OCn−1,0[zn] is Noetherian,and so its ideal I∩OCn−1,0[zn] is generated by g1,g2, . . . ,gk. We claim that gi’s gen-

2.4 Germs of analytic varieties 9

erate I too. For an arbitrary element h ∈ I, if h is regular we proceed as we did by f ,otherwise we write h = h− f + f and we repeat the argument for h− f and f . ut

2.4 Germs of analytic varieties

Definition 2.3 A germ of an analytic subvariety (X ,x) of (Cn,x) is given by

(X ,x) := y ∈ (Cn,x)| f1(y) = 0, f2(y) = 0, · · · , fr(y) = 0,

where f1, f2, . . . , fr ∈MCn,x. The defining ideal of the germ (X ,x) is defined to be

IX ,x := f ∈ OCn,x | f |X= 0.

andOX ,x := OCn,x/IX ,x,

is called the ring of (germs) of holomorphic functions in a neighborhood of x on X .

One can view OX ,x as a set of functions f ∈ (X ,x)→C which extend to holomorphicfunctions in (Cn,x). The maximal ideal of OX ,x is:

MX ,x := f ∈ OX ,x | f (x) = 0,

We also define

M kX ,x := the sub C-algebra of MX ,x generated by Π

ki=1gi, gi ∈MX ,x.

Proposition 2.4 The following statements are true:

1. M mCn,0 is exactly the set of holomorphic functions with the leading term (in the

Taylor series) of degree greater than or to equal m;2. ∩∞

k=1MkCn,0 = 0;

Proof. We first prove the nontrivial part of the statement 1., that is, if f ∈ OCn,0with the leading term of degree ≥ m then f ∈M m

Cn,0. The proof is by induction onn. The case n = 1 is trivial. By a linear change of coordinates we can assume that fis regular in the variable x1, i.e. f (x1,0, . . . ,0) is not identically zero. Let us write

f = f (x1,0, . . . ,0) = xl1h(x1), h(0) 6= 0.

By Weierstrass preparation, see Theorem 2.1, we can write

f = u.(xl1 +a1xl−1

1 + · · ·+al−1x1 +al),

where a1,a2, . . . ,al are holomorphic functions in x2,x3, . . . ,xn with ai(0) = 0, i =1,2, . . . , l and u is a holomorphic function in x1,x2, . . . ,xn with u(0) 6= 0. Since f

10 2 Preliminaries

and fu have the same leading term up to multiplication by a constant, it is enough

to prove that xl1 +a1xl−1

1 + · · ·+al−1x1 +al ∈M mCn,0. By our hypothesis l ≥ m and

the degree of the leading term of ai is bigger than or equal m− (l− i). Now ourassertion follows from the hypothesis of induction for n−1. The second statementis a direct consequence of the first part.

For a holomorphic function f defined in some open subset U of Cn we will denoteby fx its classes in OCn,x, x∈U . Many times we use again f instead of fx and simplysay ” f in OCn,x”. We will also use this simplification later for sheaves on analyticvarieties.

Definition 2.4 A map τ : (X ,0) → (Y,0) between germs of analytic varieties iscalled holomorphic if the pull-back of functions induces a map

τ∗ : OY,0→ OX ,0

This is trivially a morphism of C-algebras and so the non-trivial part of the definitionis that the image of τ∗ is in OX ,0.

The following proposition is Theorem B14 of [GuII90]. We give its proof be-cause it is instructive.

Proposition 2.5 Let (X ,0)⊂ (Cn,0) and (Y,0)⊂ (Cm,0) be the germs of two ana-lytic varieties. Every holomorphic map τ : (X ,0)→ (Y,0) is induced by a holomor-phic map from (Cn,0) to (Cm,0).

Proof. We have a morphism τ∗ : OY,0→OX ,0 of C-algebras. Since it sends the unitsto units, it sends the maximal ideal MY,0 into the maximal ideal MX ,0 and so

τ∗(M k

Y,0)⊂M kX ,0, k = 1,2, . . . .

Let us denote the coordinate functions of (Cm,0) by y1,y2, . . . ,ym (∈MY,0) anddefine fi := τ∗(yi). The map f : (Cn,0)→ (Cm,0) defined by f = ( f1, f2, . . . , fm) isthe desired map. We consider the diagram

OCm,0f ∗→ OCn,0

j ↓ ı ↓OY,0

τ∗→ OX ,0

(2.6)

where ı, j are the canonical maps. We observe that the maps

ı f ∗,τ∗ j : OCm,0→ OX ,0

coincide on polynomials in yi’s. For an arbitrary k ∈ N, every g ∈ OCm,0 can bewritten as g1 + g2, where g1 is a polynomial in yi’s and g2 ∈M k

Cm,0 (here we haveused Proposition 2.4,1). Therefore (ı f ∗− τ∗ j)(g) ∈M k

X ,0 for all k = 1,2, . . ..Now Proposition 2.4, 3 implies that ı f ∗ = τ∗ j.

2.5 Analytic varieties 11

IY,0 is a subset of the kernel of τ∗ j and so of the kernel of ı f ∗. This impliesthat whenever a g∈OCm,0 is zero on (Y,0) then it is zero on f (X ,0) and so f (X ,0)⊂(Y,0). Since τ, f : X → Y induce the same map τ∗ = f ∗, the proof is finished. ut

2.5 Analytic varieties

In order to define an arbitrary variety we have to define the affine varieties and thentry to glue them.

Definition 2.5 Let D be an open domain in some Cn. A closed subset X of D iscalled an analytic subvariety of D if for any point a ∈ X there is an open neighbor-hood U of a in D and holomorphic functions f1, f2, . . . , fr in D such that

X ∩U := x ∈U | f1(x) = f2(x) = · · ·= fr(x) = 0.

We also call X an (analytic) affine variety. We look at X as a topological spaceequipped with a sheaf OX of C-algebras

OX (U) := f : U → C | f ∈ OX ,x, U open subset of X (2.7)

which is called the structural sheaf of X .

The definition (2.7) means that OX (U) contains all complex valued functions f :X → C such that for all x ∈ X we can find a holomorphic finction f : (D,x)→ Csuch that f restricted to X ∩D is equal to f . Let X and Y be two affine varieties. Acontinuous map τ : X→Y is called holomorphic if the pull-back of functions, givenby τ∗( f ) = f τ , defines a map τ∗ from OY,τ(x) into OX ,x, which is a morphismof C-algebras for all x ∈ X . The map τ is called a biholomorphism if there is aholomorphic map τ ′ : Y → X such that τ ′ τ and τ τ ′ are identity maps respectivelyon X and Y .

Definition 2.6 Let X be a second-countable Hausdorff topological space and CXbe the sheaf of complex valued continuous functions on X . We say that X witha sheaf of C-algebras OX ⊂ CX is an analytic variety if every point of X has anopen neighborhood U such that (U,OU ) is isomorphic to a (V,OV ), for some affinevariety V , i.e. there is a homeomorphism ψ : U →V such that

ψ∗ : OV → OU , ψ

∗( f ) = f ψ

is an isomorphism of sheaves of C-algebras.

In Figure 2.2 we have roughly depicted a variety X with three points pi, i = 1,2,3on it. Let X be a variety. For every point x ∈ X there exist an open set U around x,V a closed analytic subset of an open domain D in some Cn and a homeomorphismψ : U →V which induces an isomorphism between OV and OU . A rough picture ofthis definition is depicted in Figure 2.2 This is called a chart around x and we denoteit simply by

12 2 Preliminaries

Fig. 2.2 Analytic variety

• ψ : U →V ⊂ D⊂ Cn, a chart around x.

Given two such charts ψα : Uα →Vα ⊂ Dα ⊂ Cnα and ψβ : Uβ →Vβ ⊂ Dβ ⊂ Cnβ

around x, the first is called a subchart of the second if there is an embeddingem : (Dα ,ψα(x)) → (Dβ ,ψβ (x)) such that ψβ = emψα . They are called equivalentif one is a subchart of the other and nα = nβ . In this case the map em is a biholo-morphism. This is an equivalence relation. Note that n, the dimension of D, differschart by chart. For this reason it is better to define a variety using the language ofC-algebras rather than the formal definition by charts and transition functions, forinstance see [GuII90], Definition B16.

Definition 2.7 Let X be an analytic variety and Y a subset of X . We say that Y is an(analytic) subvariety of X if for every y ∈ Y there exists an open set U, y ∈U andf1, f2, . . . , fr ∈ OX (U) such that

U ∩Y = x ∈U | f1(x) = · · ·= fr(x) = 0.

It is easy to see that Y is an analytic variety in such a way that the inclusion Y → Xis analytic.

2.6 Embedding dimension 13

Fig. 2.3 Embedding dimension

2.6 Embedding dimension

Definition 2.8 For a germ of an analytic variety (X ,x) we define

• T ∗x X := MX ,x/M 2X ,x, the cotangent space of X at x;

• TxX := the dual of T ∗x X . TxX is called the tangent space of X at x.

A holomorphic map f : (X ,x)→ (Y,y) induces the map

T ∗x f : T ∗f (x)Y → T ∗x X

It would be instructive to check that the definition of the tangent space in the casewhere X is smooth coincides with the usual definition of tangent space with dif-ferential of transition maps of X . In the singular case the bundle of tangent spacesTxX ,x ∈ X has a natural structure of an analytic variety (see [GuII90] I, J) and sowe can define in a natural way the notion of a vector field in a variety. ParticularlyTheorem 16I claims that both notions of tangent space there and here are the same.In general, one has the notion of linear spaces over varieties, see [GPR94] chapter2 section 3.

Definition 2.9 Let X be a variety and x ∈ X . Using some chart around x we canidentify the germ of the singularity (X ,x) as an analytic subspace of Cn, for some n.The smallest integer n with this property is called the embedding dimension of X atx and is denoted by embxX .

In Figure 2.2 we have roughly depicted a germ of variety (X ,x) whose embeddingdimension is 2 and not 3. The following proposition can be also found in [GrRe84]p. 115.

14 2 Preliminaries

Proposition 2.6 We have embxX = dimCT ∗x X. More precisely, for a point x ∈ Xif x1,x2, . . . ,xm ∈ MX ,x form a basis for T ∗x X then ψ : (X ,x) → Cm given byψ = (x1,x2, . . . ,xm) is a chart map around x whose associated affine space is of di-mension dimCT ∗x X. Every two charts with the dimension of the affine spaces equalto dimCT ∗x X are equivalent and every chart has a subchart whose associated affinespace is of dimension m = dimCT ∗x X.

Proof. Since our statement is local, we can assume that X ⊂ (Cn,0) and x = 0. Let

λ : IX ,0→MCn,0/M2Cn,0

be the canonical map. Its coimage

(MCn,0/M2Cn,0)/Im(λ )

is isomorphic to MX ,0/M2X ,0. Therefore if

r := dimIm(λ ), m := dimCMX ,0/M2X ,0

then r+m = n. Let f1, f2, . . . , fr ∈IX ,0 such that their image by λ form a C-basisfor Im(λ ). This means that the linear part of the map f = ( f1, f2, . . . , fr) has themaximum rank r. Therefore f is a regular map and N = x ∈ (Cn,0) | f (x) = 0 is asmooth complex submanifold of (Cn,0) and dimCN = m. But we have also X ⊂ N.We have proved that each chart has a subchart whose associated affine space is ofdimension m = dimT ∗0 X .

Let us be given two charts for (X ,0) whose associated affine spaces are of di-mension T ∗0 X . This means that (X ,0) is embedded in two different ways in (Cm,0),say X1,X2. By Proposition 2.5 the map induced by the identity ı : (X1,0)→ (X2,0)can be extended to a holomorphic map f : (Cm,0)→ (Cm,0). Using the argumentof the previous paragraph and the dimension condition we have

T ∗0 Cm = T ∗0 Xi, i = 1,2 (2.8)

But we know that ı∗ : OX2,0 → OX1,0 is an isomorphism of C-algebras and so itinduces an isomorphism MX2,0/M

2X2,0→MX1,0/M

2X1,0. The equality (2.8) and the

inverse mapping theorem imply that f is a biholomorphism.Let x1,x2, . . . ,xm ∈MX ,0 form a basis for T ∗x X . The map ψ : (X ,x)→ Cm given

by ψ = (x1,x2, . . . ,xm) is a holomorphic map. Take an arbitrary embedding of (X ,0)in (Cm,0). According to Proposition 2.5 ψ is obtained by restriction of a holomor-phic map f : (Cm,0)→ (Cm,0). Since T ∗0 X = T ∗0 Cm, the map T ∗0 f : T ∗0 Cm→ T ∗0 Cm

is an isomorphism and so f is a biholomorphism. This proves that ψ is an embed-ding. ut

Definition 2.10 A point x of an analytic variety X is called a smooth point if a chartψ : (X ,x)→ (Cn,0) with n := dim(TxX) is surjective (and hence a biholomorphismof analytic varieties). Otherwise, x is called a singular point. An analytic varietywithout singular points is called a complex manifold. If a complex manifold is con-

2.7 Extension theorem 15

nected then the number n := dim(X) is independent of a chart and it is called thedimension of X .

We can reformulate the definition of a complex manifold in the following way. Acomplex manifold is a topological space X equipped with a sheaf of complex valuedfunctions OX such that it is covered by open sets Ui, i ∈ I such that each (Ui,OUi) isisomorphic to some (Vi,OVi) with Vi is an open subset of Cn equipped with structuralsheaf of holomorphic functions on Vi.

Proposition 2.7 For a holomorphic map f : (X ,x)→ (Y,y) if T ∗x f is surjective thenf is an embedding.

Proof. Let σ be the canonical map MY,y → T ∗y Y and n = dimCT ∗x X . Chooseg1,g2, . . . ,gn ∈MY,y such that their image by T ∗x f σ form a basis of T ∗x X . Themap

g = (g1,g2, . . . ,gn) : (Y,0)→ (Cn,0)

has the following property: g f is an embedding of X in (Cn,0), for this see thefirst part of Proposition 2.6. We identify X with its image by g f in (Cn,0). Theset X1 := f (X) is an analytic variety because it coincide with g−1(X). The inverseof f : X → X1 is given by g.

Proposition 2.8 For an analytic variety X the set Sing(X) of singular points of Xis a proper analytic subvariety of X.

Proof.

Definition 2.11 An analytic variety X is called irreducible if the complex manifoldX\Sing(X) is connected. In this case the dimension of X\Sing(X) is called thedimension of X.

Theorem 2.3 (Hironaka’s desingularization theorem) Let X be an analytic vari-ety. There is a complex manifold X and a holomorphic map f : X → X such that fis a biholomorphism in X\ f−1(Sing(X))→ X\Sing(X).

2.7 Extension theorem

In this section we state few theorem on extension of holomorphic functions beyondtheir domain of definitions. For proofs see [GuII90], Chapter D.

Theorem 2.4 (extened Riemann removable singularities theorem ) Let Y ⊂X bea proper analytic subvariety of the complex manifold X. If f is a bounded holomor-phic function in X\Y then there is a unique holomorphic function f in X such thatf and f coincide in X\Y .

16 2 Preliminaries

Proof. Since the theorem is of local nature, we can assume that X = (Cn,0) and Yis a germ of variety. Since Y is a proper subvariety, there is non-zero holomorphicfunction g ∈ OCn,0 which vanishes on Y . Now we can use Theorem 2 Chapter D[GuII90]. By a linear change in the coordinate system we can assume that g isregular at zn. Let us now consider the same notation as in the proof of Theorem 2.1.The function

f (z) =1

2πi

∫|ζ |=rn

f (z′,ζ )ζ − zn

dζ (2.9)

is holomorphic in the polydisc ∆(0,r) and it coincides with f in outside the zeroset of g. For this we use the fact that for fixd z′, f (z′,zn) extends to a holomorphicfunction in ∆(0,rn). ut

Theorem 2.4 is true for X singular. For this we have to use Hironaka’s desingular-ization theorem. I do not have any elementray proof in this case.

Corollary 2.1 Let Y ⊂ X be a proper analytic subvariety of a connected complexmanifold X. Then X\Y is connected.

Proof. If X\Y is not connected then we define a holomorphic function on it such thatin one component is identically one and in another component iz identically zero.This cannot be extended to a holomorphic function in X , which is in contradictionwith Theorem 2.4.

Theorem 2.5 Let X be a complex manifold and Y be an analytic subvariety of Xsuch that the codimension of its components is bigger than one. If f is a holomorphicfunction in X\Y then there is a unique holomorphic function f in X such that f andf coincide in X\Y .

Proof. The theorem for X and Y smooth complex manifolds follows from Theorem4 Chapter D [GuII90]. In this case, we can assume that X = (Cn,0) and Y is givenby zn = zn−1 = 0. The function

f (z) =1

(2πi)2

∫|ζn|=rn

∫|ζn−1|=rn−1

f (z1, · · · ,zn−2,ζn−1,ζn)

(ζn−1− zn−1)(ζn− zn)dζn−1dζn (2.10)

gives us the desired extension of f .In general, we proceed as follows. we first perform the extension for the com-

plex submanifold Y\Sing(Y ) ⊂ X , and so, we might replace Sing(Y ) with Y . Thedimension of Y drpos down and finally we get the full extension.

Theorem 2.6 (Hartogs extension theorem) Let C=Cn′×Cn′′ with n′,n′′≥ 1) andA= (A′,A′′)∈Cn′×Cn′′ , R= (R′,R′′)∈Rn′×Rn′′ and S = (S′,S′′)∈Rn′×Rn′′ suchthat ∆(A;S)⊆ ∆(A;R). Consider the closed set

D =

(∆(A′;R′)×∆(A′′;S′′)

)∪([

∆(A′;R′)\∆(A′;S′)]×∆(A′′;R′′)

)

2.7 Extension theorem 17

Fig. 2.4 Hartogs Domain

and let U ⊆ Cn be an open set containing D and f : U → C be a holomorphicfunction. Then there exists a unique holomorphic function f : ∆(A;R)→ C suchthat

f∣∣∣∣D= f∣∣∣∣D

Proof. Consider the function f : ∆(A;R)→ C defined by

f (Z) =(

12πi

)n′ ∫|ζ1−a1|=r1

. . .∫|ζn′−an′ |=rn′

f (ζ1, . . . ,ζn′ ,Z′′)dζ1 . . .dζn′

(ζ1− z1) . . .(ζn′ − zn′)

where Z = (Z′,Z′′) ∈ Cn′ ×Cn′′ . It is straightforward to check that given a pointZ ∈ ∆(A;R) is convergent and hence f is a well-defined function.We claim that fis holomorphic throughout ∆(A;R). Indeed, whenever |ζi− ai| = ri for 1 ≤ i ≤ n′

and Z′′ ∈ ∆(A′′;R′′), then there exists an open neighborhood Ui of (ζ1. . . . ,ζn′ ,Z′′)where f is holomorphic. It follows that f is holomorphic as desired.

On the other hand, whenever Z′′ ∈ ∆(A′′;S′′), then f (ζ1. . . . .ζn′ ,Z′′) as a func-tion of the variables ζ1. . . . .ζn′ , is holomorphic throughout ∆(A′;R′) by hypothesis.Thus, by the Cauchy integral formula, f and f coincides in that region. Since thatregion has nonempty interior and D is connected, follows by the identity theoremthat f

∣∣D = f

∣∣D. The uniqueness of such f is a direct consequence of the identity

theorem. This concludes the proof. ut

Corollary 2.2 If M is a connected complex manifold of dimension n−1 in an opensubset D ⊆ Cn, if f is a holomorphic function on D−M, and if f can be extended

18 2 Preliminaries

Fig. 2.5 Finding a Hartogs Domain

as a holomorphic function through at least one point of M, then f extends to a holo-morphic function in all of D.

Proof. The idea of the proof is depicted in Figure 2.5.

Corollary 2.3 Every function that is a holomorphic in an open neighborhood ofthe boundary of a polydisc in Cn for n > 1 has a holomorphic extension to the fullpolydisc.

2.8 Dolbeault lemma

In this section we present Dolbeault lemma. We follow Chapter E, Theorem 1,Lemma 2 of [GuI90].

Proposition 2.9 (Generalized Cauchy integral formula.) Let D be a open simplyconnected subset of the complex plane bounded by a smooth closed curve γ , and letf be a C∞ function defined in an open neighborhood of D. Then, for any z ∈ D,

2πi f (z) =∫

γ

f (w)w− z

dw+∫∫

D

∂ f (w)∂w

dw∧dww− z

.

Proposition 2.10 For every function f that is C∞ in an open set U ⊂ C, whichcontains a compact set K, there is a C∞ function g defined in an open subset V ⊂Uthat contains K such that ∂g

∂ z = f in V . If f is C∞ or holomorphic in some additionalparameters, then g has the same properties.

2.8 Dolbeault lemma 19

Theorem 2.7 (Dolbeaut’s Lemma) Let P ⊂ Q ⊂ Cn be polydisks around the ori-gin. For each ϕ ∈Ω (p,q)(Q) with q > 0 and ∂ϕ = 0 there exists a ψ ∈Ω (p,q−1)(P)such that ∂ψ = ϕ|P.

Proof. Let k ≤ n be the smallest integer such that ϕ dos not involve the differentialdzk+1, ...,dzn. The proof is by induction in k. If k = 0, since q > 0 then ϕ = 0, thecondition is held trivially. For k ≥ 1. We can write

ϕ = dzk ∧α +β , α ∈Ω(p,q−1)(Q), β ∈Ω

(p,q)(Q)

α and β are differential forms not involving dzk, ...,dzn. Suppose that

α =∗

∑|I|=p,|J|=q−1

aIJdzIdzJ

The notation ∑∗ means that dzk, ...,dzn are not in the sum. If we compute ∂ϕ we get

∂ϕ = ∂dzk ∧α−dzk ∧ ∂α + ∂ β =−dzk ∧ ∂α + ∂ β

= (n

∑j=1, j 6=k

−dzk ∧dz j ∧∗

∑|I|=p,|J|=q−1

∂aIJ

∂ zi∧dzId∧ zJ)+ ∂ β = 0

Note that for j > k, the terms ∂aIJ∂ zi

has to be zero. Then by the Proposition 3.7 thereexist smooth functions AIJ on P, which are holomorphic in zk+1, ..., zn such that

∂

∂ zkAIJ = aIJ . We define

σ =∗

∑|I|=p,|J|=q−1

AIJdzI ∧dzJ .

and we have

∂σ =n

∑j=1

∗

∑|I|=p,|J|=q−1

∂

∂ z jAIJdz j ∧dzI ∧dzJ

=dzk ∧∗

∑|I|=p,|J|=q−1

aIJdzIdzJ + ∑j<k

∗

∑|I|=p,|J|=q−1

∂

∂ z jAIJdz j ∧dzId∧ zJ

+ ∑j>k

∗

∑|I|=p,|J|=q−1

∂

∂ z jAIJdz j ∧dzId∧ zJ

However for k > j, ∂

∂ z jAIJ = 0 because AIJ is holomorphic in these directions, there-

fore∂σ = dzk ∧α + γ

where γ ∈ Ω (p,q)(P) and does not involve the terms dzk+1, ...,dzn. Then ϕ− ∂σ =(β − γ) = ∂ (ξ ) with ξ ∈ Ω (p,q−1)(P), the last equality is by induction hipothesys.Thus ϕ = ∂ (ξ −σ). ut

20 2 Preliminaries

An open polydisc in the extended sense has polyradius R = (r1, . . . ,rn) whereri ∈ R+∪+∞.

Theorem 2.8 If D⊆Cn is an open polydisc in the extended sense, then H p,q(D) = 0whenever q > 0.

Proof. See [GuI90], Chapter E Theorem 5.

Theorem 2.9 Let U ∈ Cn be an open subset for n > 1 and let K ⊆U be a compactset. If U \K is connected, then any function that is holomorphic in U \K extends toa function that is holomorphic in U.

Proof. See [GuI90] Chapter E, Theorem 6.

2.9 Algebraic varieties

All the terminology of the previous sections can be introduced in algebraic context.For simplicity we consider the field of complex numbers but the whole discussion isvalid for an algebraically closed field k. The reader is referred to [Ha77] for furtherdetails on algebraic varieties.

We define an affine variety V in Cn to be the zero set of polynomials inC[x1,x2, . . . ,xn] and we denote by OV (V ) the restriction of polynomials in C[x1,x2, . . . ,xn]to V . In V we consider the Zariski topology, i.e. the the complement of an open setis given by zeros of the elements in OV (V ). A function f : U →C in an open subsetU of V is regular if there are f1, f2 ∈ OV (V ) such that f2 does not have zeros inU and f = f1

f2. We denote by OV (U) the C-algebra of regular functions in U . We

have defined the algebraic structural sheaf OV of V . An algebraic variety (X ,OX )is a topological space equipped with a sheaf of C-algebras such that every point ofX has a neighborhood U such that (U,OX |U ) is isomorphic to some affine varietywith its canonical structural sheaf. The first example of algebraic varieties are pro-jective varieties Pn. A closed subvariety of a projective variety is called a projectivevariety. An algebraic variety X is an analytic variety in a canonical way. One usuallydenotes by Xan the underlying analytic variety. We may keep in mid the followingclassical theorem

Theorem 2.10 (Chow’s theorem) Any closed analytic suvariety X of Pn is an undr-laying variety of an algebraic variety Y ⊂ Pn, that is, Y an ∼= X.

For a proof of the above theorem see [GrHa78]. This is a part of a bigger the-orem called Serre’s GAGA principle, see [Se56] and also Grothendieck’s articles[Gro58], [Gro69].

2.11 Analytic sheaves 21

2.10 Analytic schemes

In this book we are not going to encounter the concept of an analytic scheme, how-ever, the reader who wish to immigrate to algebraic geometry might find this concepta good bridge between the concept of an analytic variety and an algebraic scheme.It appears in a natural way in the study of Hodge and Noether-Lefschetz loci, see[Voi03, Mov16].

Definition 2.12 Let D be an open domain in some Cn, I ⊂ OD be a sheaf of idealsand

X := x ∈ D | f (x) = 0, ∀ f ∈ Ix.

The pair (X ,OX ) with OX := OCn/I |X is called an (affine) analytic scheme

The C-algebra OCn,x,x ∈D is Noetherian and so we can find holomorphic functionsf1, f2, . . . , fr in a neighborhood of x such that Ix = 〈 f1, f2, . . . , fr〉 and so X becomesan analytic variety. However, note that we have replaced the usual sheaf of holo-morphic functions in X with OX which may have nilpotent elements, that is, f 6= 0with f k = 0. We look at X as a topological space equipped with the sheaf OX ofC-algebras. We have a canonical map

(X ,OX )→ (X ,OX )

which is the identity map in X and f ∈ OX is mapped to the restriction of f on X .By definition, there might be such an f which is mapped to zero but it is not zero inOX . We call OX the structural sheaf of X .

Definition 2.13 Let X be a second-countable Hausdorff topological space and OXbe a sheaf of C-algebras. We say that (X ,OX ) is an analytic scheme if every pointof X has an open neighborhood U such that (U,OU ) is isomorphic to a an affineanalytic scheme (V,OV ), that is, there is a homeomorphism ψ : U→V under whichOU becomes isomorphic to OV .

We usually remove the ˇ sign and write (X ,OX ) = (X ,OX ), being clear in thecontext whether X is an analytic scheme or variety.

2.11 Analytic sheaves

For an analytic variety X we have used the structural sheaf OX . It is the collection ofall holomorphic functions defined on some open subset of X . We have also used thestalk of OX over a point x ∈ X . Some other examples of sheaves are the following:

• For a holomorphic vector bundle π : E → X over a complex manifold X we cantalk about the sheaf of holomorphic sections of E, i.e to each open set U ⊂ X wecan associate the OX (U)-module

E(U) := s : U → E | s is holomorphic and π s = id.

22 2 Preliminaries

• For a complex manifold X we have the sheaf of holomorphic vector fields whichis the sheaf of holomorphic sections of the tangent bundle of M.

• For a complex manifold X we can talk about the sheaf Ω i of holomorphic differ-ential i-forms in X . This can be interpreted as the sheaf of holomorphic sectionsof the vector bundle T ∗X ∧ ·· ·∧T ∗X , i times. In a local chart U ⊂ X with coor-dinate functions z1,z2, . . . ,zn : U → C one can write:

Ωi(U) = ∑

j1, j2,..., ji

f j1, j2,..., jidz j1 ∧dz j2 · · ·∧dz ji | f j1, j2,..., ji ∈ OX (U).

• Let Y be a subvariety of an analytic variety X . The sheaf MY of ideals of Y isdefined in the following way:

MY (U) = f ∈ OX (U) | f |Y= 0, U ⊂ X open.

• For a natural number n we can define the sheaf OnX = OX ×OX × ·· · ×OX , n

times. Its elements in an open set U are the n-tuples ( f1, f2, . . . , fn), fi ∈OX (U), i = 1,2, . . . ,n.

It is natural to extend the notion of sheaf S of holomorphic forms or vector fieldsto the case of analytic varieties. First of all note that in all these examples we canmultiply the elements of OX (U) with the elements of S (U), i.e. S is an analyticsheaf. There are many other sheaves that are not analytic. For instance

• For a ring R and topological space X the sheaf of constants in X with coefficientsin R associates to each open set in X the ring R. One usually denotes again by Rthe sheaf of constants.

• The sheaf of real valued continuous functions on a topological space X .• The sheaf O∗X of invertible holomorphic functions on an analytic variety X :

O∗X (U) := f ∈ OX (U) | ∀x ∈U, f (x) 6= 0, U open in X .

In all these examples we note that: First, we have associated to open subsets ofX some algebraic structure. Second we have the restriction maps to smaller opensubsets. This leads us to the following abstract notion of a sheaf:

Definition 2.14 A sheaf S of abelian groups on a topological space X is a collec-tion of abelian groups

S (U), U open subset of X

For U1⊂U2 open sets in X , it is equipped with morphisms rU2,U1 : S (U2)→S (U1)of abelian groups such that they satisfy:

1. For U an open set of X , rU,U is the identity map;2. For open sets U1 ⊂U2 ⊂U3 ⊂ X we have rU2,U1 rU3,U2 = rU3,U1 ;3. If for U = ∪i∈IU j, U and U j’s open subsets of X , and f j ∈S (U) we have

rUi,U j∩Ui( fi) = rU j ,U j∩Ui( f j)

then there is a unique element f ∈S (U) such that

2.12 Restriction of sheaves 23

rU,Ui( f ) = fi.

rU2,U1 is called the restriction map from U2 to U1. The stalk Sx of S at a point x∈ Xis defined in the following way:

Sx = f ∈S (U), for some open set U in X , x ∈U/∼

wheref1 ∈S (U1), f2 ∈S (U2), x ∈U1∩U2,

f1 ∼ f2⇔ f1 = f2 in some open set U3 ⊂U1∩U2, x ∈U3.

Note that the uniqueness condition 3 in the above definition implies that if the re-striction of g ∈S (U) to each U j is zero (of the corresponding abelian group) theng must be zero. If in the definition of a sheaf we remove item 3 then we get thedefinition of a presheaf. It is easy to see that any presheaf can be enlarged into asheaf.

Remark 2.1 We will usually use f |U1 instead of rU2,U2( f ); being clear that f is anelement of S (U2) for some open set U2 which contains U1. By definition S ( /0) = /0and rU, /0 = /0.

Throughout the text, for a given sheaf S over a topological space X , when wewrite x ∈S we mean that x is a section of S in some open neighborhood in X or itis an element in a stalk of S over X , being clear from the text which we mean.

We now define an analytic sheaf.

Definition 2.15 An analytic sheaf S on an analytic variety X is a collection ofOX (U)-modules S (U) for all open sets of X such that

1. The groups (S (U),+) form a sheaf of abelian groups;2. The restriction maps are morphism of OX -modules.

2.12 Restriction of sheaves

Let us be given an analytic sheaf S on X and let Y be a subvariety of X . In thissection we introduce the notion of restriction of elements S to Y .

We can define the sheaf of abelian groups S |Y :

S |Y (U) = f ∈S (V ), for some open set V in X , V ∩Y =U/∼

wheref1 ∈S (V1), f2 ∈S (V2), V1∩Y =V2∩Y =U,

f1 ∼ f2⇔ f1 = f2 in some open set V3 ⊂V1∩V2,V3∩Y =U.

The obtained sheaf on Y is not analytic. Now consider the sheaf MY ·S

24 2 Preliminaries

(MY ·S )(U) := f ∈S (U) | ∀x ∈U ∩Y, ∃ f1 ∈Sx, f2 ∈MY,x, f = f1 f2 in Sx(2.11)

For all examples of analytic sheaves that we have seen the restriction of elements ofMY ·S to Y is zero. Therefore, we may define the quotient sheaf S /MY S as acandidate for the restriction of S to Y . This sheaf is zero outside Y , i.e S (U) = 0for open sets U in X which do not intersect Y . We finally get the following definition

Definition 2.16 The structural restriction of of an analytic sheaf S on X to its sub-variety Y is defined to be

res(S ) = S ||Y = (S /MY S )|Y .

It is left to the reader to check that S ||Y is an analytic sheaf in a canonical way.

Remark 2.2 Every definition and construction with sheaves has a local nature.Sometimes such a definition coincides with the ”trivial” definition or construction.For instance for two sheaves S1 ⊂ S2 on X we may define

(S2/S1)(U) = S2(U)/∼, f1 ∼ f2⇔∀x ∈U, ( f1− f2) = 0 in S2,x/S1,x

It turns out that this coincides with the trivial definition (S2/S1)(U)= S2(U)/S1(U).This is not always the case. For instance, consider the the definition of MY S in(2.11).

2.13 Homomorphism between sheaves

Definition 2.17 A homomorphism f between two sheaves S and S ′ of abeliangroups on a topological space is X is a collection of group homomorphisms

fU : S (U)→S ′(U), U ⊂ X open

which is compatible with the restriction maps, i.e. for U1 ⊂U2 open sets in X , wehave

fU1 rU2,U1 = rU2,U1 fU2 .

A homomorphism f is an isomorphism if all the fU ’s are isomorphisms. In the casewhere X ,S and S ′ are analytic, we say that f is analytic if each fU is a morphismof OX (U)-modules.

Definition 2.18 Let f : S →S ′ be a homomorphism of sheaves of abelian groups.The kernel Kernel( f ) and image Image( f ) of f are defined in the following way:

Kernel( f )(U) := a ∈S (U) | ∀x ∈U, ax ∈ kernel( fx : Sx→S ′x)

Image( f )(U) := a ∈S ′(U) | ∀x ∈U, ax ∈ Image( fx : Sx→S ′x)

2.14 Rank of coherent sheaves 25

f is called injective (resp.surjective) if Kernel( f ) = 0 (resp. Image( f ) =S ′). In thiscase we write

Sf→S ′→ 0 ( resp. 0→S

f→S ′)

It is easy to see that Kernel(f) and Image(f) are sheaves of abelian groups, re-spectively analytic sheaves if X ,S and S ′ are analytic. According to the prop-erties of restriction maps, the definition of Kernel( f ) does not change if we defineKernel( f )(U) = kernel fU . But this is not the case for Image( f ). This is anotherexample for the situation discussed in Remark 2.2.

Example 2.1 Let X be an analytic variety and Y be a subvariety of X . We have thefollowing exact sequence

0→MYi→ OX

r→ OY → 0.

where i is the inclusion and r is the restriction map. Here OY is the trivial extensionof the sheaf OY to X , i.e. OY (U) = 0 if U does not intersects Y and = OY (U ∩Y )if U intersects Y . We usually omit the tilde and simply write OY to denote OY .

Example 2.2 For an analytic variety X we have the following exact sequence:

0→ Z i→ OXe→ O∗X → 0

where i is the inclusion map and e( f ) = e2πi f .

Example 2.3 Let S be an analytic sheaf on X and [ fi j]i=1,2....,n, j=1,2,...,m be a matrixwith entries in OX (X). Then we have an analytic sheaf homomorphism

f : S m→S n, s 7→ [ fi, j]str, s = (s1,s2, . . . ,sm) ∈S m

2.14 Rank of coherent sheaves

The content of this section will be necessary for the proof of Grauert and Kodairaembedding theorem, see Theorem 10.1. Let X be an analytic space and S be acoherent sheaf on X .

Definition 2.19 The dimension of the structural resstriction of S at x ∈ X

rankSx := dimC(S

MxS)

is called the rank of S at x.

Proposition 2.11 If s1,s2, . . . ,sp ∈ Sx generate SMxS

then they generate Sx as aOX ,x-module. In other words we have

O pX ,x→Sx→ 0. (2.12)

26 2 Preliminaries

Proof. See [GPR94] page 48.

If we have (2.12) at a point then it is also valid for all points in a neighborhood of xand so the function

x 7→ rankSx

is upper semi-continuous.

2.15 Push-forward of a sheaf

Let f : X → Y be a continuous map between two topological spaces and S be asheaf of abelian groups in X .

Definition 2.20 The push-forward f∗S is a sheaf on Y such that for any open setU ⊂ Y its section in U are given by

f∗S (U) := S ( f−1(U))

It is easy to see that this defines a sheaf on Y , see Exercise 2.1. If X and Y are analyticvarieties, f is holomorphic and S is an analytic sheaf then f∗S is an analytic sheafin a canonical way:

a ·b := a ·b f , a ∈ f∗S (U), b ∈ OY (U)

where the second · comes from the OX -module structure of S and b f is thecomposition of functions b and f .

Exercise 2.1 Show that a push-forward sheaf defined in Section 2.15 satisfies allthe axioms of a sheaf.

Chapter 3Cech cohomology

Il faut faisceautiser. (The motto of french revolution in algebraic and complex ge-ometry, see [Rem95], page 6).

3.1 Introduction

In Chapter 4 [Mov16] we discussed the axiomatic approach to singular homologyand cohomology. These are the first examples of cohomology theories constructedin the first half of 20th century. Almost in the same time, other cohomolgy the-ories, such as De Rham and Cech cohomologies, and their properties were underconstruction and intensive investigation.

In this chapter we aim to introduce sheaf cohomologies and its explicit construc-tion using Cech cohomologies. Similar to the case of singular homology and coho-mology, the categorical approach to sheaf cohomology and the way that it is usedin mathematics, shows that in most of occasions we only need to know a bunch ofproperties of the sheaf cohomology and not its concrete construction. However, insome other occasions, mainly when we want to formulate some obstructions, weobtain elements in some sheaf cohomologies and so just axioms of Cech cohomol-ogy would not work. Therefore, we introduce some properties of sheaf chomologywhich can be taken as axioms and we also explain its explicit construction usingCech cohomology. We assume that the reader is familiar with sheaves of abeliangroups on topological spaces. The reader who is interested in a more elaboratedversion of this section may consult other books like [BT82], Section 10, [Voi02]Section 4.3.

27

28 3 Cech cohomology

3.2 Cech cohomology

A sheaf S of abelian groups on a topological space X is a collection of abeliangroups

S (U), U ⊂ X open

with restriction maps which satisfy certain properties. In particular, S (X) is calledComplete this.

the set of global sections of S . Some other equivalent notations for this are

S (X) = Γ (X ,S ) = H0(X ,S )

It is not difficult to see that for an exact sequence of sheaves of abelian groups

0→S1→S2→S3→ 0. (3.1)

we have0→S1(X)→S2(X)→S3(X)

and the last map is not necessarily surjective. In this section we want to constructabelian groups H i(X ,S ), i = 0,1,2 . . . , H0(X ,S ) = S (X) such that we have thelong exact sequence

0→H0(X ,S1)→H0(X ,S2)→H0(X ,S3)→H1(X ,S1)→H1(X ,S2)→H1(X ,S3)→

H2(X ,S1)→ ···

that is in each step the image and kernel of two consecutive maps are equal.Let us explain the basic idea behind H1(X ,S1). The elements of H1(X ,S1) are

considered as obstructions to the surjectivity of H0(X ,S2)→H0(X ,S3). This mapis not surjective, however, we can look at an element f ∈S3(X) locally and use thesurjectivity of S2→S3. We fix a covering U = Ui, i∈ I of X such that the exactsequences corresponding to global section of (3.1) over Ui hold, that is,

0→S1(Ui)→S2(Ui)→S3(Ui)→ 0.

This covering is taken so that we have exactness at S2(Ui) and for the exactness atS3(Ui) we need to assume that the set of small open sets giving exactness at S2and S3 in (3.1) is not emprty. This is the case for all examples of the short exactsequence (3.1) in this text (one may also justify this by assuming that H1(Ui,S1) =0).

We take fi ∈S2(Ui), i ∈ I such that fi is mapped to f under S2(Ui)→S3(Ui).This means that the elements f j− fi, which are defined in the intersections Ui∩U j’s,are mapped to zero and so there are elements fi j ∈ S1(Ui ∩U j) such that fi j ismapped to f j− fi. It is easy to check that different choices of fi’s lead us to elements

fi j + f j− fi, (3.2)

3.3 Covering and direct limit 29

where fi ∈S1(Ui). This lead us to define H1(U ,S1) to be the set of ( fi j, i, j ∈ I)modulo those of the form (3.2).

3.3 Covering and direct limit

Let X be a topological space, S a sheaf of abelian groups on X and U = Ui, i ∈I a covering of X by open sets. In this paragraph we want to define the Cechcohomology of the covering U with coefficients in the sheaf S . Let U p denotesthe set of (p+1)-tuples σ = (Ui0 , . . . ,Uip), i0, . . . , ip ∈ I and for σ ∈U p define

|σ |= ∩pj=0Ui j .

A p-cochain f = ( fσ )σ∈U p is an element in

Cp(U ,S ) := ∏σ∈U p

H0(|σ |,S )

Definition 3.1 Let π be the permutation group of the set 0,1,2, . . . , p. It actson U p in a canonical way and we say that f ∈ Cp(U ,S ) is skew-symmetric iffπσ = sign(π) fσ for all σ ∈ U p. The set of skew-symmetric cochains form anabelian subgroup Cs(U ,S ).

For σ ∈ U p and j = 0,1, . . . , p denote by σ j the element in U p−1 obtained byremoving the j-th entry of σ . We have |σ | ⊂ |σ j| and so the restriction maps fromH0(|σ j|,S ) to H0(|σ |,S ) is well-defined. We define the boundary mapping

δ : Cps (U ,S )→Cp+1

s (U ,S ), (δ f )σ =p+1

∑j=0

(−1) j fσ j ||σ |

We have to check that

Proposition 3.1 The above map is well-defined, that is, if f ∈Cp(U ,S ) is skew-symmetric then δ f is also skew-symmetric.

Proof. For simplicity, and without loss of generality we can assume that π is per-mutation of 0 and 1.

From now on we identify σ with i0i1 · · · ip and write a p-cochain as f =( fi0i1···ip , i j ∈I). For simplicity we also write

(δ f )i0i1···ip+1 :=p+1

∑j=0

(−1) j fi0i1···i j−1 i j i j+1···ip+1

where i j means that i j is removed.


Proposition 3.2 We haveδ δ = 0.

Proof. Let f ∈Cps (U ,S ). We have

(δ 2 f )i0i1·ip+2 =p+2

∑j=0

(−1) j(δ f )i0i1···i j ·ip+2

=p+2

∑j=0

p+2

∑k=1, k 6= j

(−1) j(−1)k fi0i1···ik···ip+2= 0

where k = k if k < j and k = k−1 if k > j. See [BT82] Proposition 8.3.

Now

0→C0s (U ,S )

δ→C1s (U ,S) δ→C2

s (U ,S )δ→C3

s (U ,S )δ→ ···

can be viewed as cochain complexes, i.e. the image of a map in the complex is insidethe kernel of the next map.

Definition 3.2 The Cech cohomology of the covering U with coefficients in thesheaf S is the cohomology groups

H p(U ,S ) :=Kernel(Cp

s (U ,S) δ→Cp+1s (U ,S ))

Image(Cp−1s (U ,S) δ→Cp

s (U ,S )).

The above definition depends on the covering and we wish to obtain cohomologiesH p(X ,S ) which depends only on X and S . We recall that the set of all coveringsU of X is directed:

Definition 3.3 For two coverings Ui = Ui, j, j ∈ Ii, i = 1,2 we write U1 ≤ U2and say that U1 is a refinement of U2, if there is a map from φ : I1 → I2 such thatU1,i ⊂U2,φ(i) for all i ∈ I1.

For two covering U1 and U2 there is another covering U3 such that U3 ≤ U1 andU3 ≤U2. It is not difficult to show that for U1 ≤U2 we have a well-defined map

H p(U2,S )→ H p(U1,S )

which is obtained by restriction from U2,φ(i) to U1,i. For details see [BT82], Lemma10.4.1 and Lemma 10.4.2.

Definition 3.4 The Cech cohomology of X with coefficients in S is defined in thefollowing way:

H p(X ,S ) := dir limU H p(U ,S ).

We may view H p(X ,S ) as the union of all H p(U ,S ) for all coverings U , quo-tiented by the following equivalence relation. Two elements α ∈ H p(U1,S ) and

3.5 How to compute Cech cohomologies 31

β ∈ H p(U2,S ) are equivalent if there is a covering U3 ≤ U1 and U3 ≤ U2 suchthat α and β are mapped to the same element in H p(U3,S ).

3.4 Acyclic sheaves

Definition 3.5 A sheaf S of abelian groups on a topological space X is calledacyclic if

Hk(X ,S ) = 0, k = 1,2, . . .

The main examples of acyclic sheaves that we have in mind are the following:

Proposition 3.3 Let M be a C∞ manifold. The sheaves Ω iM∞ of C∞ differential i-

forms on M are acyclic.

Proof. The proof is based on the partition of unity and is left to the reader.

Definition 3.6 A sheaf S is said to be flasque or fine if for every pair of open setsV ⊂U , the restriction map S (U)→S (V ) is surjective.

Proposition 3.4 Flasque sheaves are acyclic.

See [Voi02], p.103, Proposition 4.34.

3.5 How to compute Cech cohomologies

Definition 3.7 The covering U is called acyclic with respect to S if U is locallyfinite, i.e. each point of X has an open neighborhood which intersects a finite numberof open sets in U , and H p(Ui1 ∩·· ·∩Uik ,S ) = 0 for all Ui1 , . . . ,Uik ∈U and p≥ 1.

Theorem 3.1 (Leray lemma) Let U be an acyclic covering of a variety X. There isa natural isomorphism

Hµ(U ,S )∼= Hµ(X ,S ).

See [Voi02] Theorem 4.41 and Theorem 4.44. A full proof can be found in the bookof Godement 1958. See also [GrHa78] page 40. For a sheaf of abelian groups Sover a topological space X , we will mainly use H1(X ,S ). Recall that for an acycliccovering U of X an element of H1(X ,S ) is represented by

fi j ∈S (Ui∩U j), i, j ∈ I

fi j + f jk + fki = 0, fi j =− f ji, i, j,k ∈ I

It is zero in H1(X ,S ) if and only if there are fi ∈S (Ui), i∈ I such that fi j = f j− fi.

Remark 3.1 For sheaves of abelian groups Si, i = 1,2, . . . ,k over a variety X wehave:

H p(X ,⊕iSi) =⊕iH p(X ,Si), p = 0,1, . . . .


3.6 Resolution of sheaves

A complex of abelian sheaves is the following data:

S • : S 0 d0→S 1 d1→ ·· ·dk−1→ S k dk→ ·· ·

where S k’s are sheaves of abelian groups and S k → S k+1 are morphisms ofsheaves of abelian groups such that the composition of two consecutive morphismis zero, i.e

dk−1 dk = 0, k = 1,2, . . . .

A complex S k,k ∈ N0 is called a resolution of S if

Im(dk) = ker(dk+1), k = 0,1,2, . . .

and there exists an injective morphism i : S →S 0 such that Im(i) = ker(d0). Wewrite this simply in the form

S →S •

For simplicity we will write d = dk; being clear in the text the domain of the map d.

Definition 3.8 A resolution S →S • is called acyclic if all S k are acyclic.

Theorem 3.2 Let S be a sheaf of abelian groups on a topological space X andS →S • be an acyclic resolution of S then

Hk(X ,S )∼= Hk(Γ (X ,S •),d), k = 0,1,2, . . . . (3.3)

whereΓ (X ,S •) : Γ (S 0)

d0→ Γ (S 1)d1→ ···

dk−1→ Γ (S k)dk→ ···

and

Hk(Γ (X ,S •),d) :=ker(dk)

Im(dk−1).

Proof. Let U = Uii∈I be an acyclic covering of X and let

S ij :=C j

s (U ,S i), S j :=C js (U ,S ), Γ (S i) := Γ (X ,S i)

Consider the double complex

3.6 Resolution of sheaves 33

↑ ↑ ↑ ↑0→ Sn → S 0

n → S 1n → S 2

n → ·· · → S nn →

↑ ↑ ↑ ↑ ↑0→ Sn−1 → S 0

n−1 → S 1n−1 → S 2

n−1 → ·· · → S nn−1 →

↑ ↑ ↑ ↑ ↑...

......

......

↑ ↑ ↑ ↑ ↑0→ S2 → S 0

2 → S 12 → S 2

2 → ·· · → S n2 →

↑ ↑ ↑ ↑ ↑0→ S1 → S 0

1 → S 11 → S 2

1 → ·· · → S n1 →

↑ ↑ ↑ ↑ ↑0→ S0 → S 0

0 → S 10 → S 2

0 → ·· · → S n0 →

↑ ↑ ↑ ↑Γ (S 0)→ Γ (S 1)→ Γ (S 2)→ ·· · → Γ (S n)→↑ ↑ ↑ ↑0 0 0 0

(3.4)

The up arrows are δ and the left arrow at S pq is (−1)qd, that is, we have multiplied

the map d with (−1)q, where q denotes the index related to Cech cohomology. Letus define the map A : Hk(Γ (X ,S •),d)→ Hk(X ,S ). It sends a d-closed globalsection ω of S n to a δ -closed cocycle α ∈Cn

s (U ,S ) and the receipie is sketchedhere:

0↑

α → ω0 → 0↑ ↑

η0 → ω1 → 0↑

η1 . . . . . .. . . ωn−1 → 0

↑ ↑ηn−1 → ωn → 0

↑ω

(3.5)

An arrow a→ b means that a is mapped to b under the corresponding map in (3.4).ωn is the restriction of ω to opens sets Ui’s etc. The same diagram 3.5 can be usedto explain the map B : Hk(X ,S )→ Hk(Γ (X ,S •),d). In this case we start from α

and we reach β . We have to check that

Exercise 3.1 Show that

1. A and B are well-defined.2. AB and BA are identity maps.


If we do not care about using d or (−1)qd we will still get isomorphisms A and B,however, they are defined up to multiplication by −1. The mines sign in (−1)qd isinserted so that D := δ +(−1)qd becomes a differential operator, that is, DD = 0.For further details, see [BT82] Chapter 2. Another way to justify (−1)q is to see itin the double complex of differential (p,q)-forms in a complex manifold. ut

Let us come back to the sheaf of differential forms. Let M be a C∞ manifold. Thede Rham cohomology of M is defined to be

H idR(M) = Hn(Γ (M,Ω i

M∞),d) :=global closed i-forms on Mglobal exact i-forms on M

.

Theorem 3.3 (Poincare Lemma) If M is a unit ball then

H idR(M) =

R if i = 00 if i = 0

The Poincare lemma and Proposition 3.3 imply that

R→Ω•M

is the resolution of the constant sheaf R on the C∞ manifold M. By Proposition 3.2we conclude that

H i(M,R)∼= H idR(M), i = 0,1,2, . . .

where H i(M,R) is the Cech cohomology of the constant sheaf R on M.

3.7 Cech cohomology and Eilenberg-Steenrod axioms

Let G be an abelian group and M be a polyhedra. We can consider G as the sheaf ofconstants on M and hence we have the Cech cohomologies Hk(X ,G), k = 0,1,2, . . ..This notation is already used in Chapter 4 of [Mov16] to denote a cohomologytheory with coefficients group G which satisfies the Eilenberg-Steenrod axioms.The following theorem justifies the usage of the same notation.

Theorem 3.4 In the category of polyhedra the Cech cohomology of the sheaf ofconstants in G satisfies the Eilenberg-Steenrod axioms.

Therefore, by uniqueness theorem the Cech cohomology of the sheaf of constantsin G is isomorphic to the singular cohomology with coefficients in G. We presentthis isomorphism in the case G = R or C.

Recall the definition of integration

Hsingi (M,Z)×H i

dR(M)→ R, (δ ,ω) 7→∫

δ

ω

This gives us

3.8 Dolbeault cohomology 35

H idR(M)→ ˇHsing

i (M,R)∼= H ising(M,R)

whereˇmeans dual of vector space.

Theorem 3.5 The integration map gives us an isomorphism

H idR(M)∼= H i

sing(M,R)

Under this isomorphism the cup product corresponds to

H idR(M)×H j

dR(M)→ H i+ jdR (M), (ω1,ω2) 7→ ω1∧ω2, i, j = 0,1,2, . . .

where ∧ is the wedge product of differential forms.

If M is an oriented manifold of dimension n then we have the following bilinearmap

H idR(M)×Hn−i

dR (M)→ R, (ω1,ω2) 7→∫

Mω1∧ω2, i = 0,1,2, . . .

3.8 Dolbeault cohomology

Let M be a complex manifold and Ωp,qM∞ be the sheaf of C∞ (p,q)-forms on M. We

have the complex

Ωp,0M∞

∂→Ωp,1M∞

∂→ ··· ∂→Ωp,qM∞

∂→ ···

and the Dolbeault cohomology of M is defined to be

H p,q∂

(M) := Hq(Γ (M,Ω p,•M∞), ∂ ) =

global ∂ -closed (p,q)-forms on Mglobal ∂ -exact (p,q)-forms on M

Theorem 3.6 (Dolbeault Lemma) If M is a unit disk or a product of one dimen-sional disks then H p,q

∂(M) = 0

Let Ω p be the sheaf of holomorphic p-forms on M. In a similar way as in Proposition3.3 one can prove that Ω

p,qM∞ ’s are fine sheaves and so we have the resolution of Ω p:

Ωp→Ω

p,•M∞ .

By Proposition 3.2 we conclude that:

Theorem 3.7 (Dolbeault theorem) For M a complex manifold

Hq(M,Ω p)∼= H p,q∂

(M)

We give an example of a domain D in Cn such that H0,1∂

(D) 6= 0. See [GuII90],the end of Chapter E.


3.9 Cohomology of manifolds

The first natural sheaves are constant sheaves. For an abelian group G, the sheaf ofconstants on X with coeficients in G is a sheaf such that for any open set it associatesG and the restriction maps are the identity. We also denote by G the correspondingsheaf. Our main examples are (k,+), k = Z,Q,R,C. For a smooth manifold X , thecohomologies H i(X ,G) are isomorphic in a natural way to singular cohomologiesand de Rham cohomologies, see respectively [Voi02] Theorem 4.47 and [BT82]Proposition 10.6. We will need the following topological statements.

Proposition 3.5 Let X be a topological space which is contractible to a point. ThenH p(X ,G) = 0 for all p > 0.

This statement follows from another statement which says that two homotopic mapsinduce the same map in cohomologies.

Proposition 3.6 Let X be a manifold of dimension n. Then X has a covering U =Ui, i ∈ I such that

1. all Ui’s and their intersections are contractible to points.2. The intersection of any n+2 open sets Ui is empty.

Using both propositions we get an acyclic covering of a manifold and we prove.

Proposition 3.7 Let M be an orientable manifold of dimension m.

1. We have H i(M,Z) = 0 for i > m.2. If M is not compact then the top cohomology Hm(M,Z) is zero.3. If M is compact then we have a canonical isomorphism Hm(M,Z)∼= Z given by

the orientation of M.

If M is a complex manifold of dimension n, then it has a canonical orientation givenby the orientation of C and so we can apply the above proposition in this case. Notethat M is of real dimension m = 2n.

3.10 Short exact sequences

In this section we return back to one of the main motivations of the sheaf cohomol-ogy, namely, for an exact sequence of sheaves of abelian groups

0→S1→S2→S3→ 0.

we have a long exact sequence

· · · → H i(X ,S1)→ H i(X ,S2)→ H i(X ,S3)→ H i+1(X ,S1)→ ··· (3.6)

3.10 Short exact sequences 37

that is in each step the image and kernel of two consecutive maps are equal. All themaps in the above sequence are canonical except those from i-dimensional coho-mology to (i+1)-dimensional cohomology. In this section we explain how this mapis constructed in Cech cohomology. Let S i

j :=C js (U ,Si). The idea is to use:

0 0 0↓ ↓ ↓

· · · → S 1n−1 → S 1

n → S 1n+1 → ·· ·

↓ ↓ ↓· · · → S 2

n−1 → S 2n → S 2

n+1 → ·· ·↓ ↓ ↓

· · · → S 3n−1 → S 3

n → S 3n+1 → ·· ·

↓ ↓ ↓0 0 0

(3.7)

Let us take a covering Ui, i ∈ I and f = f j, j ∈ Ii+1 ∈ H i(X ,S3). By takingthe covering smaller, if necessary, we can assume that f is in the image of the mapS2→S3, that is, there is g = g j, j ∈ Ii+1 such that each g j is mapped to f j underS2 → S3. Now we have δ f = 0 and so δg is mapped to zero under S2 → S3.We conclude that there is h = h j, j ∈ Ii+2 such that h is mapped to δg underS1→S2. We have the map

H i(X ,S3)→ H i+1(X ,S1), f 7→ h

which is well-defined and gives us the long exact sequence (3.6).

Chapter 4Stein varieties

In this chapter we review all the basic fact about Stein varieties. For a completeaccount on this topic and all missing proofs the reader is referred to [GuIII90] and[GrRe79]. Let X be an analytic variety and U Ui, i ∈ I be a covering of X suchthat each Ui comes from a chart

ψi : Ui∼→Vi ⊂ B(r)

where B(r) is a polydisc of radius r = (r1,r2, . . . ,rni) and Vi is a closed subvarietyof B(r) (it is given by the zero set of holomorphic functions in B(r)). The objectiveof this section is to prove that U is an acyclic covering for coherent sheaves S andso in this way Hµ(X ,S ) = Hµ(U ,S ).

4.1 Stein varieties

Stein varieties share many properties with germs of varieties. In this section we listthe definition and some theorems about Stein varieties. For more detailed study thereader is referred to [GrRe79, GuIII90].

Definition 4.1 Let K be a subset of a variety X. The set

KX := x ∈ X | | f (x)| ≤ supy∈K | f (y)|, ∀ f ∈ OX (X)

is called the (holomorphic) convex hull of K in X. The variety X is called holomor-phically convex if for any compact set K ⊂ X the convex hull KX is also compact.

Definition 4.2 A subset Y of a topological space X is called nowhere discrete ifthere is no point p ∈ Y and a neighborhood U of p in X such that U ∩Y := p.

Theorem 4.1 (Definition) A holomorphically convex variety X is called Stein if oneof the following equivalent condition is satisfied:

39

40 4 Stein varieties

1. For any point x ∈ X there exist holomorphic functions f1, f2, . . . , fm on X suchthat x is an isolated point of the set x ∈ X | f1(x) = f2(x) = · · ·= fm(x) = 0;

2. Holomorphic functions on X separate the points of X, i.e. for any pair of pointsx and y in X there exists a holomorphic function on X such that f (x) 6= f (y);

3. X does not contain nowhere discrete compact analytic subsets;4. For any discrete sequence of points Av in X and any sequence av of complex

numbers there is a holomorphic function f on X such that f (Av) = av for all v.In particular, if av tends to infinity then

limv→∞| f (Av)|= ∞

The reader is referred to [GuIII90], Theorems 4M,5M,11M for the proof of theequivalences.

Definition 4.3 An open subset D ⊂ Cn is called domain of holomorphy if thereexists a holomorphic function f in D that cannot be extended as a holomorphicfunction to any boundary point of D.

Proposition 4.1 A polydisc ∆(a,r) is a domain of holomorphy.

Proof. See [GuI90] Chapter G.

Theorem 4.2 An open subset D of Cn is holomorphically convex if and only if it isa domain of holomorphy.

Proof. See [GuI90], Chapter G, Theorem 5.

Proposition 4.2 If U1 and U2 are two Stein open subsets of a variety X then U1∩U2is Stein.

Proof. Since holomorphic functions separate points in U1, this is the case also inevery open subset of U1. Therefore it is enough to prove that U1 ∩U2 is holomor-phically convex. For a compact set K ⊂U1∩U2 we have KU1∩U2 ⊂ KU1 ∩ KU2 . SinceKU1∩U2 is closed and is a subset of a compact set, it is compact.

Example 4.1 Let U be an open connected subset of Pn, n ≥ 2, satisfying the fol-lowing conditions:

1. The boundary ∂U of U is not empty.2. For any p ∈ ∂U there exists a holomorphic mapping fp : (Cn−1,0)→ Pn such

that fp(0) = p and fp(Cn−1,0)∩U = /0.

Then U is Stein. For the proof see [Li99] and the references mentioned there. Inparticular, the above statement implies that the complement of a minimal set inPn, n≥ 2 is Stein. The non-existence of such an object for n≥ 3 is proved in [Li99]and it is still an open problem for n = 2.

Remark 4.1 There are algebraic varieties over complex numbers such that their un-derlying analytic variety is Stein but they are not affine algebraic varieties. The mainreferences for this topic are Fritzsche and Grauert’s book [FrGr02], Hartshorne’slectures [Ha70] and [Ne89].

4.3 Some properties of Stein varieties 41

4.2 Coherent sheaves

Definition 4.4 An analytic sheaf over an analytic variety X is coherent if for eachpoint x of X there is an exact sequence of holomorphic sheaves over U of the form:

OmX ,x→ On

X ,x→Sx→ 0

for some n,m ∈ N, i.e the stalk Sx as a OX ,x-module is finitely generated, forinstance by f1, f2, . . . , fn ∈ Sx, and the OX ,x-module of relations between fi’s,α ∈ On

X ,x | ∑ni=1 αi fi = 0 is also finitely generated.

1. A freely generated sheaf is coherent. In particular the sheaf of holomorphic sec-tions of a vector bundle over a complex manifold is coherent. Examples of thosesheaves are the sheaf of holomorphic vector fields and holomorphic differentialforms over a complex manifold.

2. (Oka’s Theorem)The kernel of any homomorphism OmX → On

X is coherent3. (Cartan’s theorem) The sheaf of ideals MY of an analytic subvariety Y of an

analytic variety X is a coherent sheaf.4. (Grauert direct image theorem). Let φ : X → Y be a proper analytic map. For a

coherent sheaf S in X , the sheaves Rµ φ∗S , µ = 1,2, . . . are also coherent.

Why in the definition of a coherent sheaf S , the fact that each stalk of S isfinitely generated is not enough to prove Cartan’s A and B theorem?.

4.3 Some properties of Stein varieties

Here are some properties of Stein varieties.

Theorem 4.3 (Cartan’s Theorem A) For a Stein variety X and a coherent analyticsheaf S on X the stalk of S over a point x ∈ X is generated by global sections.

There is an equivalent definition of Stein varieties using the cohomology of coherentsheaves and this is in fact the starting point in [GuIII90].

Theorem 4.4 (Cartan’s Theorem B) A variety X is Stein if and only if for any co-herent analytic sheaf S on X we have

Hµ(X ,S ) = 0, µ ≥ 1

For the main purpose of this section we prove the following properties of Steinvarieties.

Theorem 4.5 We have

1. Every holomorphic function on a closed subvariety Y of a Stein variety X extendsto a holomorphic function on X

2. Any closed subvariety of a Stein variety is Stein.

42 4 Stein varieties

Proof. We first the first part. The second part follows immediately from the defini-tion of the convex hull and the first part.

By definition of an analytic variety we can extend f ∈ OY (Y ) locally, that is, wecan take a covering U of Y with open sets in X and gi ∈ OX (Ui) such that gi andf coincide in Y ∩Ui. We complete U into a covering of X and put gi = 0 for opensets Ui which do not intersect Y . Now

g = (gi j := fi− f j, i, j ∈ I) ∈ H1(U ,MY )

Since MY is coherent, by Cartan’s B theorem H1(X ,MY ) = 0 and so there is acovering U ′ such that U ′ < U and g is mapped to 0 under the restriction mapH1(U ,MY ) → H1(U ′,MY ). From now on we use U ′ instead of U . We havegi j = fi− f j for fi ∈ OX (MY ) and so the functions fi− fi glue together to give usthe desired function.

It is a remarakable fact that every Stein manifold of dimension n can be embed-ded holomorphically in C2n+1. Here is another example of a Stein variety.

4.4 Stein covering

A covering U of a variety X is called Stein if it is locally finite and each open setin U is Stein. Combining Proposition 4.2, Theorem B of Cartan we conclude that aStein covering is acyclic and so by Leray lemma Hµ(U ,S ) ∼= Hµ(X ,S ) for anycoherent analytic sheaf S on X and µ ≥ 1.

Chapter 5Kahler manifolds

5.1 Tangent space

Let M be a complex manifold of dimension n and zi = xi+√−1yi, i = 1,2, . . . ,n be

a system of coordinates around a point p of M. The real tangent space of M, TRMis an R-vector space of dimension 2n and by definition it has the following basis:

∂

∂x j,

∂

∂y j, j = 1,2, . . . ,n (5.1)

We tensor it with C over R and get a canonical decomposition:

TCp M := TR

p M⊗RC= T 10p M⊕T 01

p M (5.2)

where T 10p M and T 01

p M) are generated over C by the vectors

∂

∂ zi:=

12(

∂

∂x j− i

∂

∂y j), i = 1,2, . . . ,n (5.3)

∂

∂ zi:=

12(

∂

∂x j+ i

∂

∂y j), i = 1,2, . . . ,n. (5.4)

respectively. There is defined a complex conjugation:

TCp M→ TC

p M,

n

∑i=1

ai∂

∂ zi+bi

∂

∂ zi7→

n

∑i=1

ai∂

∂ zi+ bi

∂

∂ zi

which maps TRp M to itself. Moreover, it maps T 10

p M isomorphically, as a R-vectorspace, to T 01

p M. The decomposition and the conjugation are well-defined and doesnot depend on a particular choice of the coordinates zi’s. We have an isomorphismof R-vector spaces

43

44 5 Kahler manifolds

T 10p → TR

p M,12(a 7→ a+ a) (5.5)

whose inverse is given by the composition TRp M → TpMC→ T 10

p M, where the lastmap is the projection in the decomposition (5.2). The multiplication by i-linear mapin the left hand side gives us J : TR

p M→ TRp M which satisfies J2 = −I, where I is

the identity map. It is given by

J(∂

∂x j) =

∂

∂y j, J(

∂

∂y j) =− ∂

∂x j

The real cotangent space (TRp M)∨ has the basis dx1,dy1, · · · ,dxn,dyn which is

the dual basis of (5.1). Its complexification is

(TCp M)∨ := (TR

p M)∨⊗RC= (T 10p M)∨⊕ (T 01

p M)∨

The dual basis of (5.4) is given by

dz1,dz2, . . . , · · · ,dzn, dz1,dz2, · · · ,dzn.

Exercise 5.1 For v1,v2 ∈ TRp M, the vector v1+ iv2 is in T 10

p if and only if v2 =−Jv1.Hint: Under the identification (5.5), the vectors v1 + iv2 and i(v1 + iv2) are mappedto v1 and −v2, respectively.

5.2 Positive forms

Recall the notations introduced in Section 5.1. The following exercise will be helpfulfor understanding the content of this section.

Exercise 5.2 Show that every real differential form on a complex manifold whichis of type (1,1) can be written locally in the form

ω = i(n

∑i, j=1

gi jdzi∧dz j), gi j = g ji

Can one assume that gi j’s are real functions and gi j = g ji? Show that

−12

iω(v, v) = ω(Re(v),JRe(v)), v ∈ T 10p M

and so it is a real number. Hint: use Exercise 5.1.

Definition 5.1 Consider a real differential form ω on a complex manifold and as-sume that ω is of type (1,1). ω is called a non-negative form (resp. positive form)if one of the follwing equivalent conditions is satisfied:

5.3 Hermitian metric 45

1. In a local chart (z1,z2, . . . ,zn), iω can be written diagonally, as −iω = ∑i αidzi∧dzi with αi real and non-negative (resp. positive).

2. For any vector v ∈ T 10M,−iω(v, v)≥ 0(resp. > 0).3. For any real tangent vector v ∈ TRM,ω(v,Jv)≥ 0 (resp. > 0), where J : TRM 7→

TRM is the complex structure operator of TRM.

Exercise 5.3 Show that the three statements in the above definitions are equivalent.

Proposition 5.1 A C∞-function q on a complex manifold A is plurisubharmonic(resp. strongly plurisubharmonic) if and only if ω :=−i∂ ∂q is a non-negative (resp.positive) form.

Proof. The proof follows from

−iω(v, v) =n

∑i, j=1

∂ 2q∂ zi∂ z j

(x)viv j = Lx(q)(v), v,w ∈ T 10x A.

5.3 Hermitian metric

Definition 5.2 Let V → X be a vector bundle on a complex manifold. A Hermitianmetric on V is a C1-family of Hermitian inner products 〈·, ·〉 : Vx×Vx→ C, that is

1. 〈·, ·〉 is C-linear in the first coordinate, that is,

〈au+bv,w〉= a〈u,w〉+b〈v,w〉, ∀u,v,w ∈Vx, a,b ∈ C.

2. We have〈u,v〉= 〈v,u〉.

and hence 〈w,au+bv〉= a〈w,u〉+ b〈w,v〉.3. 〈u,u〉 ≥ 0, ∀u ∈Vx and the equality happens if and only if u = 0.

Exercise 5.4 Show that any holomorphic vector bundle over a complex manifoldhas a Hermitian metric. Hint: Use partition of unity.

Let us consider a Hermitian metric in the complex tangent space T 10 (and notin complexfied real tangent space which is T 10⊕T 01) of a complex manifold X . Inlocal charts we have 〈u,v〉= h(u, v), where

h :=n

∑i=1

n

∑j=1

hi jdzi⊗dz j

where A := [hi j] is a Hermitian matrix, that is, At = A, and it is positive definite:

vAv > 0, ∀v ∈ Cn, v 6= 0.

The corresponding Riemannian metric is given by

46 5 Kahler manifolds

g :=12(h+ h) =

12

n

∑i, j=1

hi j(dzi⊗dz j +dz j⊗dzi).

and a positive real (1,1)-form

ω :=i2(h− h) =

n

∑i, j=1

hi jdzi∧dz j.

which gives us h = g− iω .

5.4 Fubini-Study metric

In Pn with the coordinates [z0 : z1 : · · · : zn] we consider the following

ω := ∂ ∂ log(n

∑i=0|z2

i |). (5.6)

Note that ω is invariant under (z0,z1, · · · ,zn) 7→ (λ z0,λ z1, · · · ,λ zn) for all λ ∈ C−0, and so it induces a (1,1)-form in Pn. Let us consider the affine charts Ui =zi 6= 0 with the coordinate system (z0,z1, . . . ,zi−1,zi+1, · · · ,zn) and the chart map

(z0,z1, . . . ,zi−1,zi+1, · · · ,zn) 7→ [z0 : z1 : . . . : zi−1 : 1 : zi+1 : · · · ,zn].

Exercise 5.5 Show that

K j : Cn→ R+, K j(z) = log(1+n

∑i=0, i6= j

|z2i |)

is a strongly plurisubharmonic function.

The (1,1)-form ω in the affine chart Ui is given by ω = ∂ ∂K j and it is positive.From ω we can define the corresponding Hermitian metric and Riemanninin metricwhich are usually called the Fubini-Study metric. K is called the Kahler potential.

Definition 5.3 A connected compact complex manifold is called Kahler if it has aHermitian metric h whose associated (1,1)-form is closed.

Every smooth projective variety inherits the Fubini-Study metric and so it is Kahler.

Theorem 5.1 Any Kahler manifold X such that [ω] ∈ H2(X ,Z) is a projective va-riety.

Proof. By Lefschetz (1,1)-theorem any closed (1,1)-form ω such that its class inH2

dR(X) lies in H2(X ,Z) is the Chern class of a line bundle. Now the theorem followsfrom Kodaira embedding theorem, see Theorem 10.2.

5.5 Torus as Kahler manifold 47

Let O(1) be the dual of the tautological line bundle on Pn, see Section 8.1. Letalso si be trivializing sections of O(1) in Ui. We define a Hermitian metric in O(1).In the affine chart Ui = zi 6= 0 it is given by

pi := 〈si,si〉12 := (

n

∑k=0| zk

zi|2)−

12

Since p j := | z jzi|pi and we have (8.3), the local data gives us a global Hermitian

metric in O(1). We follow the recipe for computing the Chern class of O(1) inProposition 8.1 and we get

c(O(1)) =

[−1

2π√−1

∂ ∂ log(n

∑i=0|z2

i |)

]

5.5 Torus as Kahler manifold

In this section we give an explicite description of positive line bundles on a complextorus Cn/Λ .

Chapter 6Strongly convex/plurisubharmonic functions

The notions of plurisubharmonic functions and pseudoconvex domains appeared incomplex analysis after E.E. Levi discovered around 1910 that the boundary of adomain of holomorphy in Cn satisfies certain conditions of pseudoconvexity. Thequestion of whether conditions on the boundary might determine a domain of holo-morphy became known as the Levi problem. The first definitions were made by K.Oka [Ok42] and P. Lelong [Le45]. The reader is referred to T. Peternell survey in[GPR94] Chapter V and R. C. Gunning’s book [GuI90] Chapters K-R for morehistory and developments not treated here. In this text we will consider only the C2

category of plurisubharmonic functions. We start this chapter by introducing the no-tion of strongly convex functions. They just carry the convexity information of theirfibers and contain the class of strongly plurisubharmonic functions. Strongly convexfunctions are easy to handle and this is the main reason we have chosen them in-stead of strongly plurisubharmonic functions. Gunning’s book [GuI90], Chapter Rhas been our main source about these functions. We also define the notion of convexfunction parallel to plurisubharmonic functions. But this seems to be useless, sincethey do not satisfy the maximum principle!

6.1 Strongly convex functions

Let ψ : (Cn, p)→ R be a C2-function. Recall that

∂ψ

∂ z j=

12(

∂ψ

∂x j− i

∂ψ

∂y j),

∂ψ

∂ z j=

12(

∂ψ

∂x j+ i

∂ψ

∂y j)

The Levi form of ψ at p ∈ Cn is defined by

Lp(ψ)(v) :=n

∑i, j=1

∂ 2ψ

∂ zi∂ z j(p)viv j, v = (v1,v2, . . . ,vn) ∈ Cn

49

50 6 Strongly convex/plurisubharmonic functions

Let us introduce the differential (1,1)-form

ω :=−i∂ ∂ψ = in

∑i, j=1

∂ 2ψ

∂ zi∂ z j(p)dzi∧dzi

First of all note that the form ω is real because

ω = i∂ ∂ψ = ω

Note also that

−iω(v, v) =n

∑i, j=1

∂ 2ψ

∂ zi∂ z j(p)viv j = Lp(ψ)(v), v,w ∈ T 10

p Cn.

The following simple equalities will be useful in forthcoming arguments. For v∈Cn

we have

Lp(hψ)(v) = h(p)Lp(ψ)(v)+ψ(p)Lp(h)(v)+2Re(Dpψ(v).Dph(v))(6.1)

Lp(φ ψ)(v) = φ′′(ψ(p))|Dpψ(v)|2 +φ

′(ψ(p)) ·Lp(ψ)(v) (6.2)

Lp(− logψ)(v) =1

ψ(p)(|Dpψ(v)|2

ψ(p)−Lp(ψ)(v)) (6.3)

where ψ,h : (Cn, p)→ R are two C2 functions and φ is a R-valued C2 functiondefined in a neighborhood of the image of ψ .

Exercise 6.1 Prove the equalities in (6.1),(6.2),(6.3).

Definition 6.1 A C2 function ψ : (Cn, p)→ R is called convex (resp. strongly con-vex) at the point p in the sense of Levi if

Dpψ(v) = 0⇒ Lp(ψ)(v)≥ 0 ( resp. > 0), ∀v ∈ Cn,v 6= 0. (6.4)

Let G ⊂ Cn be an open domain and ψ : G→ R a C2 function. We say that ψ isconvex (resp. strongly convex) in G if it is convex (resp. strongly convex) at eachpoint p ∈ G.

Practically in the above definition we will assume the additional condition

v ∈ S2n−1 := v ∈ Cn | |v|= 1

to obtain a compact space for the parameter v. This does not change the definition.

Definition 6.2 A C2-function ψ : (Cn, p) → R is called plurisubharmonic (resp.strongly plurisubharmonic) at p if its Levi form at p is positive semidefinite (resp.positive definite), i.e.

Lp(ψ)(v) :=n

∑i, j=1

∂ 2ψ

∂ zi∂ z j(p)viv j ≥ 0 (resp. > 0) ∀v = (v1,v2, · · · ,vn) 6= 0 ∈ Cn,

6.2 Maximum principle 51

For a strongly plurisubharmonic function ψ , the associated (1,1)-form ∂ ∂ψ is pos-itive and vice versa. Let G ⊂ Cn be an open domain and ψ : G→ R a C2 function.We say that ψ is plurisubharmonic (resp. strongly plurisubharmonic) in G if it isplurisubharmonic (resp. strongly plurisubharmonic) at each point p ∈ G.

A strongly plurisubharmonic function at p satisfies (6.4) and so it is a strongly con-vex function. The most simple strongly plurisubharmonic function is

ψ(z) = |z|2 =n

∑i=1

ziz j, ψ : Cn−0→ R+.

Exercise 6.2 Show that the norm function

Cn−0→ R+,(z1,z2, . . . ,zn) 7→ |z1|2 + |z2|2 + · · ·+ |zn|2

is strongly pseudoconvex.

For a holomorphic function f on an open domain D⊂ Cn and without zeros thefunction log | f | is plurisubharmonic. In fact

2∂ 2log| f |∂ zi∂ z j

=∂

∂ f∂ z jf

∂ zi= 0

A sum ψ1+ψ2 of a strongly plurisubharmonic function ψ1 with a plurisubharmonicfunction ψ2 is still strongly plurisubharmonic. In particular, for f as above ψ1 +log | f | is strongly plurisubharmonic.

6.2 Maximum principle

We start this section with the definition of a maximum principle.

Definition 6.3 (Maxmimum principle) A C2 function ψ : G→ R defined in anopen set G ⊂ Cn satisfies the maximum principle if ψ(x) ≤ ψ(a), ∀x ∈ (G,a)implies that ψ is constant.

Let us consider the one dimensional case, that is a C2 function ψ : G→R, G⊂C.By definition it is called (strongly) convex if

∂ψ

∂ z(p) = 0⇒ ∂ 2ψ

∂ z∂ z(p)≥ 0 ( resp. > 0), ∀p ∈ G

This implies that

Proposition 6.1 If a C2-function ψ : G→R defined in an open set G⊂C is stronglyconvex then it satisfies the maximum principle.


Note that constant functions are not strongly convex and hence the above propositionimplies that strongly convex functions in one variable has no local maximum.

Proof. Let us assume that ψ has a local maximum at 0 ∈ G. Consider the path[−ε,ε]→ G, t 7→ (ta, tb) which crosses 0. We have that for all 0 6= (a,b) ∈ C2, thefunction ψ(ta, tb) has a local maxmimum at 0, and so, its derivative at 0 is zeroand its second derivative at 0 is not positive. This implies that ∂ψ

∂ z (p) = 0 and theHessian matrix of ψ at 0 is not positive definite and so its trace

∆ := 4∂ 2ψ

∂ z∂ z=

∂ 2ψ

∂x2 +∂ 2ψ

∂y2 (6.5)

which is the Lablacian operator is not positive which is a contradiction. ut

Unfortunately we cannot say a similar statement for ψ convex. For exampleψ(x+ iy) = −(x4 + y4) has a local maximum at 0 and is a convex function. Thesituation for subharmonic functions, that is plurisubharmonic functions in one vari-able, is different.

Proposition 6.2 A C2-function ψ : G→ R defined in an open set G⊂ C is subhar-monic if and only if for any harmonic function h in a connected open set of G, thedifference ψ−h satisfies the maximum principle in U

Note that for this statement ψ need not to be strongly subharmonic.

Proof. See [GuI90], Chapter J, Theorem 7 and Theorm 8.

6.3 Some properties

The following propositions reveal some important properties of strongly convexfunctions.

Proposition 6.3 Let ψ : (Cn, p)→ R, ψ(p) = 0 be strongly convex at p and h :(Cn, p)→ R+ be a C2 function. Then hψ is also strongly convex at p.

Proof. This follows from (6.1).

Proposition 6.4 If ψ : (Cn, p)→ R is strongly convex (resp. strongly plurisubhar-monic) at p then it is strongly convex (resp. strongly plurisubharmonic) in a neigh-borhood of p in Cn.

Proof. Figure 6.1 might help to understand the proof. Let us prove the case ofstrongly convex function. We define

Y := (v,x) ∈ S2n−1× (Cn, p) | Dxψ(v) = 0

The projection on the second coordinate π : Y → (Cn, p) is a continuous proper map.Now, L.(.) : Y →R is continuous and strictly positive on the fiber π−1(p). Therefore

6.3 Some properties 53

Fig. 6.1 Strongly convex functions

it must be strictly positive on the fibers π−1(x) for x in a neighborhood of p in Cn. Inthe case of strongly convex functions we use Y := S2n−1× (Cn, p) and the argumentis as before. ut

Proposition 6.5 Let ψ be a strongly convex (resp. strongly plurisubharmonic) func-tion in a neighborhood of a compact set K in Cn. There exists an ε > 0 such that if his a real-valued C2 function on a neighborhood of K in Cn and the absolute valuesof its first and second derivatives on this neighborhood are less than ε , then ψ +his strongly convex (resp. strongly plurisubharmonic) in a neighborhood of K in Cn.

Proof. Let ψ be strongly convex. L.(.) is strictly positive on the compact se

Y := (v,x) ∈ S2n−1×K | Dxψ(v) = 0.

Therefore it is positive in a compact neighborhood U of Y in S2n−1×Cn. Since theprojection on the second coordinate is a continuous proper map, we can choose aneighborhood U ′ of K in Cn such that for all (v,x) ∈ S2n−1×U ′ if Dψx(v) = 0 then(v,x)∈U . We take ε1 such that if for (v,x)∈ S2n−1×U ′ we have |Dxψ(v)|< ε1 then(v,x) ∈U for all x ∈U ′. We take also

ε2 := min(v,x)∈ULx(ψ)(v)(∑ | vi |)2 .

Now U ′ and ε := minε1,ε2 are the desired objects. If Dx(ψ +h)(v) = 0 then

|Dxψ(v)|= |Dxh(v)|< ε|v|= ε

and so (v,x) ∈U . Now

Lx(ψ +h)(v)≥ Lx(ψ)−|Lx(h)| ≥ Lx(ψ)− ε ∑ |vi||v j|=


Lx(ψ)− ε(∑ |vi|)2 > 0, (v,x) ∈U

Let us no consider the case in which ψ is strongly plurisubharmonic. We take

ε = minp∈U,v∈S2n−1Lp(ψ)(v)(∑ | vi |)2 ,

where U is a compact neighborhood of K in which ψ is strongly plurisubharmonic.In U we have

Lp(ψ +h)(v)≥ Lp(ψ)− | Lp(h) |≥ Lp(ψ)− ε ∑ | vi || v j |=

Lp(ψ)− ε(∑ | vi |)2 > 0, p ∈U, v ∈ S2n−1

ut

Exercise 6.3 Let f : (Cm, p)→ (Cn,q) be a holomorphic map such that D f (p) isan injective map (and hence m≤ n) and ψ : (Cn, p)→R be a strongly convex (resp.strongly plurisubharmonic) function. Then

Lp(ψ f )(v) = Lqψ(Dp f (v))

Hint: For coordinate system w ∈ Cm and z ∈ Cn we have

∂ 2ψ f∂wi∂ w j

(p) =n

∑r,s=1

∂ 2ψ

∂ zr∂ zs(q) · ∂ fr

∂wi· ∂ fs

∂w j.

Proposition 6.6 We have

1. The pull-back of a strongly convex (resp. strongly plurisubharmonic) function bya biholomorphic map is a strongly convex (strongly plurisunharmonic) function.

2. The restriction of a strongly convex function ψ : (Cn,0)→ R to

Cm = (z1,z2, . . . ,zm,0, . . . ,0) ∈ (Cn,0)

is a strongly convex function.

The first statement is not true when we replace biholomorphic with holomorphic,for instance take a constant function which is of course a holomorphic function.

Proof. For the first part we use Exercise 6.3. The second part is immediate from thedefinition.

We are in a position to extend the notion of strongly convex (resp. strongly plurisub-harmonic) functions to varieties.

Definition 6.4 Let X be an analytic variety and ψ : X → R a continuous function.Then ψ is called strongly convex (resp. strongly plurisubharmonic) if for every localchart φ : U → V ⊂ D ⊂ Cn, U an open subset of X and V a closed analytic subsetof the open subset D of some Cn, there exists a strongly convex (resp. stronglyplurisubharmonic) function ψ on D such that ψ = ψ φ .

6.4 Strongly pseudoconvex domains 55

Fig. 6.2 Strongly pseudoconvex domain

Proposition 2.6 and Proposition 6.6 imply that the above definition is independentof the choice of a local chart.

Proposition 6.7 [Maximum Principle] Let (X , p) be a germ of a variety and X 6= p.Let ψ : (X , p) be a strongly convex function. If either X is smooth at p or ψ isplurisubharmonic then ψ does not attain a local maximum at p, that is, ψ(y) ≤ψ(p), ∀y ∈ (X , p) cannot happen.

In other words, there does not exist a non-discrete analytic varieties Y ⊂ X such thatY ⊂ ψ(x) ≤ 0. We do not have a proof for Proposition 6.7 stated for arbitrarygerm of an analytic variety and a strongly convex function.

Proof. We take a holomorphic function γ : (C,0)→ (X , p). If X is smooth at 0 thenwe can take it with non-vanishing derivative ar 0. In general, we use Hironaka’sdesingularization theorem. The pullback ψ γ is a strongly convex function in thefirst case and it is subharmonic function in the second case. Both functions satisfythe maximum principle, see Proposition 6.1 and Proposition 6.2. ut

6.4 Strongly pseudoconvex domains

Let X be an analytic variety and G a relatively compact open subset of X .

Definition 6.5 We say that G is strongly pseudoconvex if for every point p in theboundary of G there exist a neighborhood Up of p and a real valued strongly convexC2-function ψ defined in Up such that

G∩Up = x ∈Up | ψ(x)< 0

(see Figure 6.2).


We will only consider the case in which X is a complex manifold of complex di-mension n and the boundary ∂G is a smooth real submanifold of dimension 2n−1.In this case, ψ’s used in the above definition are regular at the points of ∂G, that is,

Exercise 6.4 Let G ⊂ X be a strongly convex domain with smooth boundary. InDefinition 6.5 we have Dpψ 6= 0. Hint: Use (6.2).

Proposition 6.3 says that the property of being stongly convex for ψ is a actually anintrinsic property of the level surface ψ−1(0)⊂ (Cn, p).

Exercise 6.5 Let ψ : (Cn, p)→R be a C2 strongly convex function with Dp(ψ) 6= 0and hence ψ−1(0) is a smooth real manifold of codimension 2n−1 in (Cn, p). If wehave another C3-function ψ ′ : (Cn, p)→R with ψ−1(0) =ψ ′−1(0) then the quotientψ ′

ψis a C2-function in (Cn, p) with no zeros and hence by Proposition 6.3 ψ ′ is also

strongly convex at p.

The real tangent space of the boundary ∂G of G is given by

Tp∂G =

v ∈ Cn

∣∣∣∣∣Re

(n

∑i=1

vi∂ψ

∂ zi

)= 0

, p ∈ ∂G

which contains the following complex vector space

T 10p ∂G =

v ∈ Cn

∣∣∣∣∣ n

∑i=1

vi∂ψ

∂ zi= 0

, p ∈ ∂G.

Therefore, ψ is strongly convex at p if its Levi form restricted to T 10p ∂G is positive

definite. The next proposition shows that instead of local C2-functions ψ , we canchoose a global one.

Proposition 6.8 Let G⊂ X be a relatively compact strongly pseudoconvex domain.There exists a neighborhood U of ∂G and a strongly convex C2-function ψ in Usuch that

U ∩G = x ∈U | ψ(x)< 0

Proposition 6.8 is also true when we replace strongly convex with strongly plurisub-harmonic. This proposition is stated in [Gr62] p. 338 Satz 2 and [GPR94] p. 228.In the proof by Grauert one reads: Wie man leicht nachrechnet, ist the Levi formL(φ ∗) in z positiv definit. This easy calculation in Narasimahn’s paper [Na62] p.204 and Laufer’s book [La71] Lemma 4.12 takes form as a complicated argument.This was one of the main reasons for us to prefer strongly convex functions in-stead of plurisubharmonic functions. The reader who is interested to know the proofof Proposition 6.8 with a global plurisubharmonic ψ can also look at the articles[Ri68, Wa72]. We have an alternative proof for exceptional varieties using the no-tion of strongly convex functions and Remmert reduction, see Theorem 7.7.

Proof. Let p ∈ ∂G. We have a strongly convex function ψ : Up → R defined in aneighborhood Up of p such that Up∩G= x∈U |ψ(x)< 0. Let h : Up→R+∪0

6.5 Plurisubharmonic functions 57

be a C2-function on Up and V an open subset of Up such that p∈V ⊂⊂ supp(h)⊂⊂Up and h is positive in V (no zeros). Since ∂G is compact, we can cover it by a finitenumber of such V ‘s, say ∂G ⊂ ∪r

i=1Vi. Let hi be the associated function to Vi asabove. We claim that the function

ψ =r

∑i=1

hiψi

is the desired function. In fact ψ restricted to ∂G is zero and is strictly negative inU ∩G (because ψi are negative and at least one of them is strictly negative at eachpoint). At each point p ∈ Vi ∩ ∂G the quotient ψ/ψi is positive at p. This followsfrom the fact that all hi’s are non-negative, ψ j’s are regular at ∂G and hence ψ j

ψinever vanishes in the common domain of ψi and ψ j, and hi is posotive in Vi.

Using Proposition 6.3, we conclude that ψ is strongly convex at any point of ∂G.By Proposition 6.4 ψ is strongly convex in a neighborhood of p in X . Since ∂G iscompact, a finite union of these open sets gives us the desired neighborhood.

Proposition 6.9 Let G⊂ X be a relatively compact strongly pseudoconvex domainand ψ be the function defined in a neighborhood U of ∂G as in Proposition 6.8.There exists an ε such that if the values of a C2-function h defined in U and itsfirst and second derivatives are less than ε then x ∈U | ψ(x) < h(x) is stronglypseudoconvex.

Proof. This is a direct consequence of Proposition 6.5 and 6.8.

Theorem 6.1 Let G ⊂ X be a relatively compact strongly pseudoconvex domain.Then there exists a compact set K ⊂ G containing all nowhere discrete analyticcompact subsets of G.

Proof. Let ψ be as in Proposition 6.8 and

U1 = x ∈U | −ε < ψ(x)< 0

for a small ε . We claim that K = G−U1 is the desired compact set. Let A be ananalytic nowhere discrete compact subset of G and A 6⊂ K or equivalently A∩U1 isnot empty. Then ψ has a maximum greater than −ε in A∩U1. By Proposition 6.7this is a contradiction with the fact that ψ is strongly convex.

If K is analytic, compact and nowhere discrete we say that K is maximal.

6.5 Plurisubharmonic functions

Proposition 6.10 Let ψ : (Cn, p)→R with ψ(p) = 0 be a strongly convex function.There exists a C2 function h : (Cn, p)→ R+ such that hψ is strongly plurisubhar-monic.


Proof. Let h be a C∞ function in (Cn, p) such that

Dph = Dpψ, h(p) = ε > 0.

We claim that for an ε enough small hψ is strongly plurisubharmonic at p. Usingthe formula (6.1) we have

Lp(hψ)(v) = εLp(ψ)(v)+2 | Dpψ(v) |2 .

Let S2n−1 := v || v |= 1 and H = S2n−1∩v ∈ Cn | Dpψ(v) = 0. By hypothesisLp(hψ)(.) is strictly positive in H and hence in a compact neighborhood K1 of H inS2n−1. Let K2 be a compact subset of S2n−1 such that S2n−1 = K1 ∪K2 and K2 ∩His empty. On K2, A =

Lp(ψ)(.)

2|Dpψ(.)|2 is a well-defined function with a minimum c. If c is

positive then Lp(hψ) is already positive definite. If c is negative we can take 0< ε <−1c and conclude that hψ is strongly plurisubharmonic at p and so by the discussion

before Proposition 6.10 it is strongly plurisubharmonic in a neighborhood of p inCn. ut

Exercise 6.6 Use Proposition 6.10 and show that an smooth point of the boundaryof an open domain in C is given by the zero locus of a plurisubharmonic function.

Proposition 6.11 Let ψ : (Cn, p)→ R with ψ(p) = 0 be a strongly plurisubhar-monic function and h : (Cn, p)→ R+ be a C2 function such that Dp(h) = Dp(ψ).Then hψ is a strongly plurisubharmonic function in a neighborhood of p in Cn.

Proof. Using the hypothesis and 6.1 we get Lp(hψ)(v) = h(p)Lp(ψ)(v) + 2 |Dpψ(v) |2. Therefore hψ is strongly plurisubharmonic at p. Proposition 6.4 imp-ies that hψ is strongly plurisubharmonic in a neighborhood of p in Cn.

When X is a an open domain in the complex plane C then plurisubharmonic func-tions on X are precisely C2 subharmonic functions on X(see [GuI90], J Theorem 8).For the following proposition see Figure 6.3.

Proposition 6.12 Let ψ : (X ,0)→R,ψ(0) = 0 be a strongly convex function. Thenthere exists a stein neighborhood U of 0 such that for any point p∈U with ψ(p) = 0we have a holomorphic function f in U depending on p such that

f = 0∩x ∈ X | ψ(x)< 0= p

Proof. The theorem for (Cn,0) implies easily the general case (X ,0). So we assumethat X =Cn. By Proposition 6.10 we can assume that ψ is strongly plurisubharmonicin a polydisc neighborhood U of 0. The Taylor series of ψ at p reads

ψ(z)= 2Re

(∑

i

∂ψ

∂ zi(p)(zi− pi)+

12 ∑

i j

∂ 2ψ

∂ zi∂ z j(p)(zi− pi)(z j− p j)

)+Lp(ψ)(z− p)+· · ·

where · · ·= o(|zi− pi|2) for all i = 1,2, . . . ,n. Now

6.5 Plurisubharmonic functions 59

Fig. 6.3 Analytic varieties and psedoconvex domains

f (z) := ∑i

∂ψ

∂ zi(p)(zi− pi)+∑

i j

∂ 2ψ

∂ zi∂ z j(p)(zi− pi)(z j− p j)

is the desired function. ut

Proposition 6.13 Let ψ : (X , p)→R with ψ(p) = 0 be strongly convex at p. Thereis a Stein neighborhood X ′ of p in X such that U := x ∈ X ′ | ψ(x)< 0 is Stein.

Proposition 6.13 can be generalized as follows: Let X be a Stein variety and ψ areal valued C2 function such that U := x ∈ X | ψ(x)< 0 is convex at each point xwith ψ(x) = 0 then U is a Stein variety. The proof can be found in [Na60] section4, corollary 1.

Proof. Let X ′ be a Stein neighborhood of p such that for all p ∈ ψ−1(0)∩X ′ thereis a holomorphic function f defined on X ′ with the property mentioned in Proposi-tion 6.12. Then X ′ is the desired Stein open set. Since X ′ can be embedded in someaffine space Cn, it is enough to prove that U is holomorphically convex. Let K bea compact subset of U . We have KU ⊂ KX ′ and KX ′ is compact in X ′. Thereforeif KU is not compact in U , its closure in X ′ must have a point p ∈ ψ−1(0). Let fbe the holomorphic function in X ′ associated to the point p as in Proposition 6.12.The function 1

f is a holomorphic function on U such that limx→p| 1f | = +∞. But| 1f (y)| ≤ maxx∈K | 1f (x)| for all y ∈ KU . This leads to a contradiction.

Since the intersection of two Stein open sets is Stein again (see Proposition 4.2) theassertion of the above proposition is true for Stein open sets smaller than X ′.

Proposition 6.14 Let (z,zn+1) be the coordinate system of (Cn×C,(p, pn+1)), ψ :(Cn, p)→ R+ be a C2 function. The following statements are equivalent:

1. − logψ is strongly plurisubharmonic at p.


2. For all positive ε ∈ R+ the function |zn+1|2 − εψ(z) is strongly convex at(p, pn+1) with

|pn+1|2 = εψ(p). (6.6)

3. The function |zn+1|2ψ(z) is strongly convex in its domain of definition.

Proof. The equivalence of item 2 and 3 follows from Proposition 6.3:

|zn+1|2− εψ(z) = ψ(z)(|zn+1|2

ψ(z)− ε

).

We prove 1⇒ 2. Suppose that − logψ is strongly plurisubharmonic. Let (v,vn+1) ∈Cn+1 such that

D(p,pn+1)(|zn+1|2− εψ)(v,vn+1) = pn+1vn+1− εDpψ(v) = 0 (6.7)

We have

Lp(|zn+1|2− εψ)(v,vn+1) = |vn+1|2− εLp(ψ)(v) (6.8)

(6.7)=

ε2|Dpψ(v)|2

|pn+1|2− εLp(ψ)(v) (6.9)

(6.6)= ε(

|Dpψ(v)|2

ψ(p)−Lp(ψ)(v)) (6.10)

(6.3)= εψ(p) ·Lp(− logψ)(v)> 0 (6.11)

Now let us prove 2⇒ 1. For a fixed v ∈ Cn we define pn+1 and vn+1 using theequalities (6.7) and (6.6). In this way we have all the equalities in (6.8) and theaffirmation follows. ut

The Riemann extension theorems are valid for upper semi-continuous stronglyplurisubharmonic functions (see [GuII90], K for definition). The precise statementand proof can be recovered from [Gr56].

Chapter 7Cohomological properties of pseudoconvexdomains

Pseudoconvex domain are essentially Stein varieties blown-up in may points. Wealready know that the cohomology of Stein varieties with coefficients in coherentsheaves are all zero except in dimension zero. Roughly speaking, this tells us thatthe the cohomology of pseudoconex domains is concentrated in the blow-up divi-sors and they must be finite dimensional. This is in fact true and in this chapter wedescribe its precise statement.

7.1 A theorem of Grauert

Let us state an important theorem concerning the cohomology of strongly pseudo-convex domains with values in a coherent sheaf.

Theorem 7.1 ([Gr58]) Let G be a relatively compact strongly pseudoconvex do-main in a complex variety X and S a coherent analytic sheaf on G. Then the coho-mology groups Hµ(G,S ) are finite dimensional vector spaces for µ > 0.

This section is devoted to the proof of the above theorem. We will use the C2-function ψ defined in a neighborhood U of ∂G such that G∩U = x∈U |ψ(x)< 0and we assume that X =U ∪G. If U ′ is a small Stein open set in U then accordingto Proposition 6.13, the intersection U ′∩G is again Stein. Let us state two lemmaswhose proofs are just topological manipulations.

Lemma 7.1 Consider the situation of Theorem 7.1. If U ′ is a small Stein open set inX then the restriction map r : Hµ(G∪U ′,S )→ Hµ(G,S ) is surjective for µ > 0.

Proof. Consider an arbitrary Stein covering U of G containing the Stein open setG∩U ′. We have Zµ(G,U ) = Zµ(G∪U ′,U ∪U ′). This is due to the fact that theintersection of at least two open sets in U ∪U ′ is a subset of G. Since U ′∩G isStein, U ∪U is a Stein covering of G∪U ′. Leray Lemma finishes the proof.

Lemma 7.2 Consider the situation of Theorem 7.1. To each Stein covering U =Ui | i = 1,2, . . . ,r of G in X one can find a strongly pseudoconvex domain G′ such

61

62 7 Cohomological properties of pseudoconvex domains

Fig. 7.1 The idea of the Proof of Lemma 7.2

that 1. G ⊂⊂ G′ ⊂⊂ ∪ri=1Ui 2. The restriction map r : Hµ(G′,S )→ Hµ(G,S ) is

surjective for µ > 0.

Proof. Take Ki ⊂⊂ ∂G∩Ui such that ∂G = ∪ri=1Ki. According to Proposition 6.5

in each Ui there is εi > 0 such that if hi is a C2 function on Ui and the absolute valueof hi and its first and second derivatives are less than εi then x ∈Ui | ψ(x)< hi(x)is strongly pseudoconvex at each point of (ψ−hi)

−1(0) in a neighborhood of Ki inUi. We take ε = minεi

r and in each chart Ui we take a C2 function hi such that

1. hi has a compact support in Ui;2. hi restricted to Ki is strictly positive;3. the absolute value of hi and its first and second derivatives are less than ε .

Define

D j := x ∈U | ψ(x)<j

∑i=1

hi(x)∪G, j = 1,2, . . . ,r, D0 := G

We have D0 ⊂ D1 ⊂ ·· · ⊂ Dr. By the choice of ε and by Proposition 6.5 we canconclude that Di is a strongly pseudoconvex domain. Since hi is strictly positiveon Ki and Ki’s cover ∂G, we have G ⊂⊂ Dr. We claim that G′ := Dr satisfies ourlemma. We must check that the restriction map r : Hµ(Dr,S )→ Hµ(G,S ) is sur-jective. It is enough to check that r : Hµ(Di,S )→ Hµ(Di−1,S ) is surjective forall i = 1,2, . . . ,r. Since the support of hi is in Ui, we have Di = Di−1∪ (Di∩Ui). ByProposition 6.13 Di∩Ui is Stein. Lemma 7.1 finishes the proof.

7.1 A theorem of Grauert 63

Let F1 and F2 be two Frechet spaces (see [GuI90] F). Recall that a linear mappingbetween two topological vector spaces is called compact (or completely continuous)if some open neighborhood of the origin in the domain is mapped to a relativelycompact set in the range. A theorem of L. Schwarz says

Theorem 7.2 (L. Schwarz) Let u,v : F1 → F2 be two continuous linear maps. If uis compact and v is surjective then the C-vector space F2/Im(u+ v) is finite dimen-sional.

A proof of this statement can be found in [GuRo] App. B 12.Proof of Theorem 7.1. Let U = Ui, i = 1,2, . . . ,r and U ′ = U ′i , i = 1,2, . . . ,r

be two Stein coverings of G such that Ui ⊂⊂U ′i and U ∩G is a Stein covering ofG. By Lemma 7.2, we have a strongly pseudoconvex domain G′ such that G ⊂⊂G′ ⊂⊂∪r

i=1Ui and U ′∩G′,U ∩G′ are Stein coverings of G′. We consider the maps

u,v : Zµ(U ′,S )⊕Cµ−1(U ,S )→ Zµ(U ,S )

u(a,b) = r(a)+δ (b), v(a,b) =−r(a)

where r is the restriction and δ is the coboundary map. Since Hµ(G′,S ) =Hµ(U ′ ∩G′,S ) and Hµ(G,S ) = Hµ(U ∩G,S ), the map v is surjective. Thefollowing theorem finishes the proof.

Theorem 7.3 Let U ⊂⊂U ′ be two open domains in a variety X and S be a coher-ent analytic sheaf on X. Then one can endow S (U ′) and S (U) with Frechet spacestructures such that the restriction r : S (U ′)→S (U) is a compact mapping.

In the case S = OX this is Montel’s Theorem (see [GuI90]). For an arbitrary co-herent sheaf we refer to [KK83] Lemma 62.6. ut

The tools used in the proof of Theorem 7.1 provide us with a proof of the follow-ing theorem due to Serre and Cartan (see [Ma68]).

Theorem 7.4 Let A be a compact variety and S be a coherent sheaf on A. ThenHµ(A,S ),µ > 0, are finite dimensional C-vector spaces.

Now we are in a position to prove that a strongly convex domain is holomorphicallyconvex.

Theorem 7.5 (R. Narasimhan [Na60] ). Let G⊂X be a relatively compact stronglypseudoconvex domain. Then G is holomorphically convex.

Proof. We prove that for every boundary point p ∈ ∂G one can find a holomorphicfunction g on G such that limx→p|g| = +∞. This implies that G is holomorphicallyconvex. Let K be a compact subset of G and p ∈ ∂G be a boundary point . For ally ∈ K we have |g(y)| ≤maxx∈K |g(x)|<+∞. This means that K cannot have p in itsclosure. Since G itself is relatively compact in X , we conclude that K is compact.

According to Proposition 6.12 for every boundary point p ∈ ∂G one can find aholomorphic function f defined in a neighborhood of p in X such that f = 0∩G=p. Let U be a Stein open set around p. One can choose a strongly pseudoconvexdomain G′ enough near G such that D := f = 0∩G′ is closed in G′ and is relatively


Fig. 7.2 The idea of the Proof of Theorem 7.5

compact in U . Now U ∩G′ is a Stein open set in G′ and one can choose a Steincovering U = Ui, i = 1,2, . . . ,r of G′ such that U1 :=U ∩G′ and Ui, i = 2,3, . . . ,rdo not intersect D. Put fi = 0 if i = 2, . . . ,r and f1 = 1

f m . We have the cocycleδm := fi− f j ∈ H1(U ,S ) = H1(G′,S ). But by Theorem 7.1 this vector spaceis finite dimensional. Therefore there exist mi ∈ N,ci ∈ C, i = 1,2, . . . ,s such that∑ciδmi = 0. This means that there is a meromorphic function g on G′ with polesalong D and such that in a neighborhood of p g−∑

si=1

cif mi is holomorphic. Therefore

g is not holomorphic at p. Thus g |G is the desired holomorphic function in G.

7.2 Exceptional varieties

Let G⊂ X be a relatively compact strongly pseudoconvex domain. By Theorem 7.5G is holomorphically convex, and so, we can apply Remmert reduction theorem (seeAppendix 14) to G and obtain a Stein space Y and a holomorphic map φ : G→ Y ,see figure 7.3. The outcome is summarized in the following theorem.

Theorem 7.6 Let G ⊂ X be a relatively compact strongly pseudoconvex domain.Then there is a Stein variety Y and a proper surjective holomorphic map φ : G→Ywith connected fibers such that the degeneracy set

A = x ∈ G | x is not an isolated point of φ−1(φ(x))

is the maximal compact analytic nowhere discrete subset of G. Moreover, the inclu-sion

7.2 Exceptional varieties 65

Fig. 7.3 Contraction

OY ⊂ φ∗(OX ). (7.1)

is an isomorphism (of analytic sheaves).

The map φ : G→Y is usually called the Remmert reduction. The set A has a finitelymany components and so φ(A) is a finite set. If not, then either φ(A) accumulates ata point p ∈Y or it has points outside any compact subset of Y . The first case cannotoccur as any poin p ∈G with φ(q) = p has a Stein neighborhood and Stein varietiesdo not have nowhere discrete compact analytic subvarieties. By Theorem 6.1 thesecond case does not occur too.

For an open set U ⊂Y if f is a holomorphic function in φ−1(U) then f is constantin connected components of A∩ φ−1(U). Since such components are compact weconclude that f is constant restricted to each component, and so, we can define afunction g in U such that f is the pull-back of g. The last part of Theorem 7.6 saysthat g is actually a holomorphic function.

Proof. The subsets φ−1 φ(x),x ∈ G, are connected, and so by the definition, A isnowhere discrete. We prove that A is a closed analytic set. The set R = (x1,x2) ∈X ×X | φ(x1) = φ(x2) is an analytic set and the projection on the first coordinateπ : R→ X is analytic. By [Gr83] Proposition 1 p.138 we know that

A = x ∈ R | dim(π−1π(x))> 0

is a closed analytic set. Since A = π(A) and π is proper, A is also an analytic closedset. By Theorem 6.8, there exists a compact set K which contains all compact ana-lytic nowhere discrete subsets of G. For any x ∈ A, φ−1φ(x) is connected, and so bydefinition, is compact nowhere discrete subset of A. This implies that φ−1φ(x)⊂ Aand hence A⊂ K. Since A is a closed set in the compact set K, A is compact.

The Remmert reduction φ : G → Y is proper and A is compact so φ(A) is acompact analytic subset of Y . But Y is Stein, and so, φ(A) is discrete set and A


is a union of compact connected analytic subsets A1,A2, . . . ,Ar of G. In this caseRemmert reduction substitute each Ai with a point. This leads us to the definition ofexceptional sets.

Definition 7.1 Let X be an analytic variety and A be a compact connected subva-riety of X . A is exceptional in X if there exists an analytic variety X ′ and a propersurjective holomorphic map f : X → X ′ such that

• φ(A) = p is a single point;• φ : X−A→ X ′−p is an analytic isomorphism;• For small neighborhoods U ′ and U of p and A, respectively, OX ′(U ′)→ OX (U)

is an isomorphism.

We also say that A can be blown down to a point or is contractible or negativelyembedded.

Theorem 7.7 (Grauert,[Gr62] Satz 5 p. 340) Let A be a compact connected ana-lytic subset of X. Then A is an exceptional variety if and only if it has a stronglypseudoconvex neighborhood G in X such that A is the maximal compact analyticsubset of G.

Proof. Let us first suppose that A is exceptional. The analytic variety X ′ obtainedby definition can be embedded in a (Cn,0) (definition of analytic sets). The neigh-borhood of p in X ′ given by

U = x ∈ X ′ | z1(x)z1(x)+ · · ·+ zn(x)zn(x)< ε, ε a small positive number

is a pseudoconvex domain. Now it is easy to see that G = φ−1(U) is the desiredopen neighborhood of A.

Now let us suppose that A has a strongly pseudoconvex neighborhood G in Xsuch that A is the maximal compact analytic subset of G. Let φ : G→ X ′ be theRemmert reduction of G. We can see easily that A is the degeneracy set of φ andφ(A) is a single point p. Since the fibers φ−1φ(x) are connected, φ is one to one mapbetween G−A and X ′−p. Since φ∗OX = OY , we can conclude that φ induceslocal biholomorphims and so it is a biholomorphism between G−A and X ′−p.The third condition of an exceptional variety can be read directly from Remmertreduction theorem.

7.3 Complementary notes

In the proof of Theorem 7.1 we have used: For an small open relatively compact setU of a variety and a coherent sheaf S on X , H0(U,S ) has a canonical structure ofa Frechet space. The construction of such a canonical structure is done [GrRe79]Chapter VI, Section 3, [Ma68] Chapter 4 and [KK83] chapter 6.

The concept of being exceptional is contained in which neighborhood of A? LetA′ be the image of another embedding A → X ′ of A. The following theorem givesus an answer.

7.4 Exercises 67

Theorem 7.8 If A is exceptional and there exists an isomorphism φ(2) : A(2)→ A′(2)of 2-neighborhoods then A′ is also exceptional.

This is Theorem 4.9 (see also Theorem 6.12) of [La71], Satz 8 p.353 of [Gr62] andLemma 11 of [HiRo64]. The maim core of the proof is a geometric construction dueto Grauert (see [La71] p. 70-71). In the case where A is an exceptional curve in asmooth surface, M /M 2 is the nilpotent subsheaf of A(2) and so every isomorphismof 2-neighborhoods induce an isomorphism of M /M 2’s. Therefore A and A′ havethe same intersection matrix.

7.4 Exercises

Exercise 7.1 Let us take a sequence pn in (C2,0) which converges to 0. We per-form blow-ups at pn’s and obtain a sequence of complex surfaces with holomorphicmaps:

· · · →Mn+1→Mn→ ··· →M1→M0 := (C2,0)

Define M∞ = limn→∞ Mn. This is the union of all Mn’s modulo the equivalence rela-tion: x ∈Mm,y ∈Mn, m > n are equivalent if x is mapped to y under the canonicalmap Mm →Mn. Show that M∞ does not enjoy the structure of a complex manifoldsuch that the projections M∞→Mn are holomorphic.

7.1. Is there a holomorphic map φ : (X ,0)→ (Y,0) between germs of analytic vari-eties such that it is an isomorphism of sets but is not a biholomorphism?

Chapter 8Positive and negative line bundles

The notion of a positive line bundle is now classical in alegebraic geometry and itis mainly used under the name ample line bundle. Students learn that a line bundlebundle L is called ample if global holomorphic sections of a power of L give us anembedding of the underlying variety in a projective variety. This is actually the Ko-daira embedding theorem taken from complex geometry as a definition. Therefore,the whole machinary used to prove this theorem is missed in a course in algebraicgeometry. Another notion which is also missed is the concept of a negative linebundle which is simply the dual of a positive line bundle. The main reason mightbe that this concept, originally introduced by Grauert in [Gr62], uses tools fromanalysis like plurisubharmonic and convex functions. The main goal of this chapteris to cover all these topics.

8.1 Line bundles

Let A be an analytic variety and O∗A be the sheaf of invertible holomorphic func-tions in A. Elements of the cohomology H1(A,O∗A) are called line bundles in A. Itcorresponds to the geometric notion of a line bundle as follow.

Let us take the covering Ui, i ∈ I and assume that g ∈ H1(A,OA) is given byhi j ∈ O∗A(Ui∩U j). We can glue the data

Ui×C, i ∈ I

according to the biholomorphisms

fi j : (Ui∩U j)×C→ (Ui∩U j)×C

fi j((x,v)) = (x,hi jv) (8.1)

and obtain an analytic variety L with a canonical projection π : L→ A whose fibersLx := π−1(x) are isomorphic to C. The fact that δ (g) = 0, ensures us that the gluing

69

70 8 Positive and negative line bundles

process is consistent. An equivalent way of saying the equality (8.1) is the following:we take local holomorphic without zero sections of L, namely si : Ui→ L, i ∈ I, andwe have

s j = hi jsi, i, j ∈ I

A meromorphic (C∞, holomorphic etc.) section of a line bundle L = hi j is a col-lection fi, i ∈ I of meromorphic functions fi in Ui, such that

f j = h−1i j fi, i, j ∈ I. (8.2)

In particular, fi and f j have the same zero and polar set in their common domain ofdefinition Ui∩U j. See Figure 8.1. Note that

fisi = f js j, in Ui∩U j

and so we get a global section of L in a geometric sense.

Fig. 8.1 Line bundle

Exercise 8.1 For an analytic sheaf S and a line bundle L = hi j on A show that acocyle in H1(A,S ⊗OA L) is represented by si j, si j ∈S (Ui∩U j) such that

si j + s jkhi j + skihik = 0, i, j,k ∈ I.

Hint: Take sections si’s of L and then divide by si.

Let X be a complex manifold and D be a divisor in X . This is the formal sum

D :=s

∑i=1

niDi, ni ∈ Z

8.2 Chern classes in Cech cohomology 71

where Di is a codimension one subvariety of X . It might have singularities. Bydefinition, we can cover X by open sets Ui and find meromorphic functions fi in Uisuch that

div( fi) =s

∑i=1

ni(Di∩Ui).

The line bundle associated to the divisor D is the following cocycle:

L := hi j, hi j :=fi

f j.

By definition L has a global meromorphic section s such that div(s) = D.Here is another example of a line bundle. The tautological line bundle O(−1)

on the projective space Pn is defined in the following way: its fiber over a point[z0 : z: · · · : zn] is the line passing through 0 and (z0,z, · · · ,zn) in Cn+1. The dual ofO(−1) is denote by O(1).

Exercise 8.2 In the standard covering of Ui := zi 6= 0, i = 0,1, · · · ,n of Pn wehave:

O(1) = hi j, where hi j : Ui∩U j→ C∗, hi j(z) =z j

zi. (8.3)

Moreover, O(1) has a holomorphic section with a zero divisor of multiplicity onealong z0 = 0 ⊂ Pn.

Exercise 8.3 What is wrong with the following definition? Let L = hi j be a linebundle on a complex manifold A. For γ ∈ π1(A,b) we define aγ ∈C∗ in the followingway. We cover γ with open sets U0,U1, . . . ,Um,Um+1 =U0 such that in each Ui theline bundle L has a section si without zeros and Ui ∩Ui+1 6= /0. Therefore, in thisintersection we have si+1 = hi,i+1si. We define aγ := h01h12 · · ·hm0. This numberdoes not depend on the choice of si, as if we choose si := fisi, fi ∈ O∗A(Ui) thenhi j =

f jfi

hi j and fi’s cancel each other in the product of aγ .For which line bundles is it possible to make the above argument into a concrete

construction of a morphism of groups π(A,b)→ C∗.

8.2 Chern classes in Cech cohomology

Consider the short exact sequence

0→ Z→ OA→ O∗A→ 0 (8.4)

over a complex manifold A. The map OA→ O∗A is given by the exponential map

f 7→ e2πi f .

We write the corresponding long exact sequence


· · · → H1(A,O∗A)c→ H2(A,Z)→ ··· (8.5)

and call c the Chern class map. For a line bundle L ∈H1(A,O∗A) on A, c(L) is calledthe Chern class of L. Recall the general construction of the long exact sequence in(3.10).

Definition 8.1 A good covering of a complex manifold X is a covering U :=Uii∈I such that all Ui and Ui∩U j’s are simply connected, and hence, for any holo-morphic function with no zeros in these open sets we can find another holomorphicfunction g such that g = ln( f ).

From now on we will consider a good covering U of X such L over Ui is trivial. Itfollows that if L = (hi j, i, j ∈ I) then

c(L) = ci jki, j,k∈I , ci jk =lnhi j + lnh jk + lnhki

2πi∈ Z.

8.3 Chern classes in de Rham cohomology

There is a nice description of Chern classes in de Rham cohomology which we havedescribed it below.

Proposition 8.1 The Chern class of a line bundle L over a complex manifold Ain the de Rham cohomology H2

dR(A) is represented by a real (1,1)-form ω . Moreprecisely, in a covering U := Ui, i ∈ I of A by open sets we can write the Chernclass of L in the complex de Rham cohomology in the form

ω = ∂ ∂qi,

where qi is a pure imaginary C∞ function in Ui,

eπiq j = |hi j|eπiqi

and L is given by hi j ∈ H1(A,O∗A).

The converse of the above statement is known as Lefschetz (1,1)-theorem: everyelement of H2(A,Z) which is represented by a real (1,1)-form in the de Rhamcohomology H2

dR(A) is a Chern class of some line bundle over A.

Proof. In Cech cohomology the Chern class of L is obtained by δ fi j ∈H2(A,Z),where fi j := 1

2πi loghi j. To see this write the long exact sequence (8.5) associated tothe short exact sequence (8.4) and and recall the construction of the coboundary mapH1(A,O∗A)→ H2(A,Z) in Section 3.10. Now let us look at the diagram 3.4 whichproduces an isomorphism between Cech cohomology and de Rham cohomology.We are using this diagram for the resolution C→ Ω •A∞ , where Ω k

A∞ is the sheaf ofcomplex valued C∞ differential k-forms in A∞.

8.3 Chern classes in de Rham cohomology 73

0→ Ω 2A∞ → (Ω 2

A∞)0 → (Ω 2A∞)1 → (Ω 2

A∞)2↑ ↑ ↑ ↑

0→ Ω 1A∞ → (Ω 1

A∞)0 → (Ω 1A∞)1 → (Ω 1

A∞)2↑ ↑ ↑ ↑

0→ Ω 0A∞ → (Ω 0

A∞)0 → (Ω 0A∞)1 → (Ω 0

A∞)2↑ ↑ ↑

C0(U ,C)→ C1(U ,C)→ C2(U ,C)↑ ↑ ↑0 0 0

(8.6)

where the right arrow maps are δ ’s and the up arrow maps are (−1)p ·d : (Ω qA∞)p→

(Ω q+1A∞ )p’s. The sequence of elements we need to produce are depited in

0→ ω → dωi → 0 →↑ ↑ ↑ ↑

0→ → ωi → −d fi j → 0↑ ↑ ↑ ↑

0→ → → fi j → δ fi j↑ ↑ ↑→ → δ fi j

↑ ↑ ↑0 0 0

(8.7)

Here the arrows means the an element is mapped to another element under the cor-responding map in (8.6). We start constructing this diagram with

fi j ∈ (Ω 0A∞)1.

We have δ fi j ∈C2(U ,C). The equality dδ fi j = 0 implies δ (d fi j) = 0 andso there is a collection ωi of 1-forms such that

δωi= −d fi j (8.8)

The collection dωi defines a global closed form ω which represents the Chernclass c(L) in the de Rham cohomology H2

dR(A). If ωi = ω10i +ω01

i is the decompo-sition of ωi into (1,0) and (0,1) forms then δω01

i = 0 and so ω01i form a global

form and so it does not contribute to the cohomology class of ω . Replacing ωi withωi−ω01

i we can assume that ωi’s are (1,0)-forms. Now

ω = ∂ωi + ∂ωi

is the decomposition of ω into (2,0) and (1,1) forms. Now the main difficulty is tosay that ω plus some exact form on A is a real (1,1)-forms. Note that ∂ωi is globalform and we do not know whether it is d-closed, despite ω being d-closed. With theargument developed so far, it does not seem that we can change ωi so that ω has thenice properties we want.


From now no we put a Hermitian metric

〈·, ·〉 : L×L→ C.

on the line bundle L. The rest of the proof is inspired by a similar argument in[GrHa78] pages 71, 139. Let si : Ui → L be local holomorphic sections of L. Wedefine

pi := 〈si,si〉12 : Ui→ R+

qi := − ln(p2i )

2π√−1

=− ln(pi)

π√−1

ωi := ∂qi =−∂ ln(pi)

π√−1

It is easy to check that

p j = |hi j|pi because s j = hi jsi

ω j−ωi = −∂ ln(hi j)

2π√−1

=−d fi j becasue ∂ loghi j = 0.

Now in the diagram (8.6) we choose these specific ωi’s which give us ω with thedesired properties. ut

The following version of Proposition 8.1 is sometimes useful.

Proposition 8.2 Let L = hi j ∈H1(A,O∗A) be a line bundle and pi : Ui→R+ be acollection of C∞ functions such that

p j = |hi j|p j (8.9)

Then ∂ ∂− ln pi

πi coincide in the intersections of Ui’s and give us a global real closed(1,1) form ω in A such that

c(L) = [ω] ∈ H2dR(A). (8.10)

Conversely, if we have (8.10) for some closed real (1,1)-form ω then there are C∞

functions pi : Ui→ R+ (8.9) and ω = ∂ ∂− ln pi

πi in Ui.

Proof. The proof of ⇒ is as follows. We define qi =− log pi

πi and ∂ ∂qi form aglobal form ω because of (8.9) and

∂ ∂ log(|hi j|) = 0.

The rest of the proof follows from the construction of the Chern class in de Rhamcohomology in Proposition (8.1).

Now let us prove⇐. Assume that c(L) = [ω] and ω is a real differential (1,1)-form. Take a good covering Ui, i ∈ I of A. We claim that in each chart Ui there is a

8.4 The case of a Riemann surfaces 75

C∞ pure imaginary function qi such that ω = ∂ ∂qi. Since dω = 0 and ω is of type(1,1) we conclude that

∂ω = ∂ω = 0.

Therefore, in Ui we can write ω = ∂ωi, where ωi is a differential (1,0)-form in Ui.We have ∂ (∂ωi) = 0 and so ∂ωi is holomorphic 2-form. Since ∂ (∂ωi) = 0 we canwrite ∂ωi = ∂αi, where αi is a holomorphic 1-form. We substitute ωi with ωi−αi.It is still (1,0)-form and we have ∂ωi = 0. Therefore, there is a C∞-function qi onUi such that

ωi = ∂qi.

We substitute this in ω = ∂ωi and conclude that ω = ∂ ∂qi. Since ω is real we have∂ ∂Re(qi) = 0 and so we can replace qi with its imaginary part and we can assumethat qi is pure imaginary. The function

pi = e−πiqi

satisfy the property (8.10). The reasoning is as follows. Knowing ∂ ∂ (q j− qi) = 0and q j−qi is pure imaginary, we conclude that

q j−qi =−Im( fi j), (8.11)

where fi j is a holomorphic function in Ui∩U j, see Exercise 8.4. We have Im(δ fi j)=0 and δ fi j consists of holomorphic functions therefore it must consist of real con-tant functions, that is,

δ fi j ∈C2(U ,R).

The isomorphism between the de Rham cohomology and Cech cohomology givenin 8.6 tells us that this is the Chern class c(L) of L in Cech cohomology. We knowthat c(L) = ci jk ∈ H2(A,Z), therefore

δ fi j− ci jk = δpi j, pi j ∈ R.

We substitute fi j by fi j− pi j and we have L = e2πi fi j. Now, (8.10) folows from(8.11).

Exercise 8.4 If for some C∞-function f in (Cn,0) we have ∂ ∂ f = 0 then f canbe written in the form f = a+ b, where a and b are two holomorphic functions in(Cn,0). Moreover, if f is pure imaginary then a =−b.

8.4 The case of a Riemann surfaces

Now let us consider the case in which A is a compact Riemann surface. Using The-orem 3.7, part 3, we have an isomorphism H2(A,Z)∼= Z given by the orientation ofA. In de Rham cohomology this is the integration


H2dR(A)∼= C, ω 7→

∫A

ω. (8.12)

In this case we usually compose the Chern class map c with (8.12) and denote againby c the map

H1(A,O∗A)→ Z, L 7→∫

Ac(L).

being clear in the context whether c(L) is a number or a cohomology class.

Proposition 8.3 Let A be a Riemann surface and L be a line bundle over A. Take aglobal meromorphic section s of L which is not the zero section. We have

c(L) = ∑x∈pol(s)∪zero(s)

mulltiplicity(s,x)

Proof. As we have seen in Section 8.1 the data of a global meromorphic sectionof L is given by (8.2). Taking the logarithmic derivarive of this equality. Recall ournotation of the Chern class in de Rham cohomology in Section 8.3. We have

f ′jf j

dz− f ′ifi

dz =−h′i j

hi jdz =−d fi j = ω j−ωi.

Therefore,

ηi := ωi−f ′ifi

dz

coincide in the intersections Ui∩U j and give us a C∞ 1-form η defined in A minesthe pol(s)∪ zero(s) = z1,z2, . . . ,za. Moreover, it satisfies

dη = ω, where c(L) := [ω] ∈ H2dR(A).

Let γr, r = 1,2, . . . ,a be small circles oriented anticlockwise around zi. The setA\∪a

r=1 γi consists of discs around zi’s and an open set U . We have

c(L) :=∫

Aω =

∫U

ω +a

∑r=1

∫Di

ω

=∫

Udη +

a

∑r=1

∫Di

dωi

=a

∑r=1

∫γi

f ′ifi

dz

=a

∑i=1

mulltiplicity(s,zi)

In th third equality we have used Stokes theorem. Note that U and Di induce opositeorientation for γi. ut

8.5 Positive line bundle in the sense of Kodaira 77

The reader who is not familiar with the isomorphism H2(A,Z)∼=Z may take Propo-sition 8.3 as the definition of Chern class in the case of Riemann surfaces. Note thatfor two meromorphic section s1 and s2, the quotient s1

s2is a meromorphic function on

A and its corresponding sum of multilicities is zero. Therefore, this new definitiondoes not depend on the meromorphic section s.

Exercise 8.5 For a Riemann surface A and its canonical bundle Ω 1 and tangentbundle T show that

c(Ω 1) = 2g−2, c(T ) = 2−2g

Proposition 8.4 For a line bundle L on a Riemann surface A we have

1. We have c(L)≥ 0 if and only if L has a global holomorphic section. In particular,if c(L)< 0 then L does not have any global holomorphic section.

2. If L has a global holomorphic section s with at least one zero then c(L)> 0.3. If L has a section with no zero then L is the trivial line bundle.

Proof. This follows from Proposition 8.3. Note that in the isomorphisim is given by

A×C→ L, (x, t) 7→ ts(x).

ut

Using the short exact sequence (8.4) we know that if c(L) = 0 then

L = e2πi fi j, fi j ∈ H1(A,OA).

8.5 Positive line bundle in the sense of Kodaira

There is another definition in algebraic geometry for a positive line bundle as fol-lows:

Definition 8.2 The line bundle L→ A over a complex manifold is called positive(in the sense of Kodaira) if its Chern class in the de Rham cohomology H2

dR(A) isrepresented by a positive real (1,1) form. Note that the Chern class c(L) in the deRham cohomology H2

dR(A) is already represented by a real (1,1)-form ω .

For more information about this definition of positive line bundles the reader isreferred to [GrHa78].

Theorem 8.1 A line bundle L over A is positive in the sense of Kodaira if and onlyif there exist a covering Ui, i∈ I of A by open sets and a collection of C∞ functionspi : Ui→ R+, i ∈ I such that

1. − log pi is strongly plurisubharmonic for any i ∈ I;2. p j = |hi j|pi, where L is given by hi j ∈ H1(A,O∗A) in a covering.

Proof. This follows from Proposition 8.2 and Proposition 5.1.


Fig. 8.2 A negative vector bundle

8.6 Positive and negative bundles in the sense of Grauert

Recall the definition of an exceptional variety from S7.2. The definition of a negativevector bundle is given below.

Definition 8.3 The vector bundle V → A over a complex manifold A is called neg-ative (in the sense of Grauert) if its zero section is an exceptional variety in V .Naturally V → A is called positive if its dual is negative, see Figure 8.2.

8.7 The equivalence of Grauert and Kodaira positivity

Theorem 8.2 The line bundle π : L→ A is positive in the sense of Kodaira if andonly if it is positive in the sense of Grauert.

Proof. Let L = hi j be positive in the sense of Kodaira. We have the pi’s given byTheorem 8.1. Let si : Ui → L−1 be local non-zero sections and so s j = h−1

i j si. Theholomorphic function

zi : L−1 |Ui→ C,zi(p) :=psi

is called a coordinate system along the fibers of L−1 in Ui. Under the biholomor-phism Ui×C→ L−1 |Ui , (x,v) 7→ vsi(x) it is the projection on the second coordi-nate. We have

z j = hi jz j

8.8 Complementary notes 79

and so |zi|piπ : L−1 |Ui→R+∪0 glue to each other in L−1 |Ui ∩L−1 |U j ’s and give us

a holomorphic function in L−1. By Proposition 6.14, the square ψ of this function isstrongly convex and so we have a strongly pseudoconvex neighborhood of the zerosection which is give by x ∈ L−1 | ψ(x)< ε for any ε > 0.

Now suppose that the zero section of L−1 has a strongly pseudoconvex neighbor-hood. By Theorem 7.7 one can find a C∞ function ψ defined in a neighborhood Uof the zero section in L−1 such that

1. ψ is strongly plurisubharmonic in U−A;2. ψ ≥ 0 and ψ−1(0) = A.

Take V an open neighborhood of the zero section which is invariant under mul-tiplication by eiθ , 0≤ θ < 2π . For instance, V can be the interior of ∩0≤θ<2π eiθU .Define

ψ′(z) =

∫ 2π

0ψ(eiθ z)dθ , z ∈V

Since

Lz(ψ′)(v) =

∫ 2π

0Lzeiθ (ψ)(v)dθ ,

ψ ′ is also a strongly plurisubharmonic function. Here, v is obtained from v by mul-tiplying its last coordinate with eiθ . Let si and zi be as in the beginning of the proof.Fix an small ε we have

Vε := z ∈ L|Ui | ψ′(z)< ε= z ∈ L|Ui | |zi|< pi(π(z))

where π : L→ A is the bundle map and pi : Ui → R+ are C2-functions such thatpi(π(z)) is the radius of Vε ∩Lx. The function pi depends on ε , however, since ε isfixed we do not write it in the expression of pi. The derivative of |zi|− pi(π(z)) atany point of the boundary of Vε is not zero and this function together with ψ ′− ε

have the same zero set which is the boundary of Vε . This implies that ψ ′−ε

|zi|2−pi(π(z))2

has no zeros in a neighbourhood of any point of ∂Vε and so by Proposition 6.3|zi|2− pi(π(z))2 is strongly convex at any point of ∂Vε . Note that, the property of Vε

being strongly convex is a property of its boundary and not the function defining it.By proposition 6.14 this is equivalent to say that − log pi(x)2 is strongly plurisub-harmonic in Ui. The functions pi(x) are the desired functions. ut


The various definitions of positive line bundles coincide. However, for vector bun-dles whose fibers have dimension greater than one these definitions are not equiva-lent ( see [Gri69], [Gri65] and [Um73]).

A linear space L over a variety A is a natural generalization of a vector bundleover a manifold, for this see the survey [GPR94] chapter 1 section 3 and also [Gr62]


Definition 5 p. 351. L has a zero section biholomorphic to A and we say that a linearspace is negative if its zero section is exceptional. In Theorem 9.2 we have stronglyused the fact that the normal bundle of a an embedded variety has a structure of alinear space.

Chapter 9Vanishing theorems

In this chapter we state and prove the Kodaira and Grauert vanishing theorems. Forthe proofs we essentially follow Grauert’s article [Gr62]. Recall that a vector bundleis positive in the sense of Grauert if the zero section of its dual can be blow-down toa point, see Definition 8.3.

Theorem 9.1 (Kodaira vanishing theorem) Let A be a compact complex manifoldwith a positive vector bundle V . Let also S be a coherent sheaf on A. There existsa positive integer ν0 such that

Hµ(A,S ⊗OA V ν) = 0, µ ≥ 1, ν ≥ ν0

The case of V = L a line bundle with the definition of positivity in Definition 8.2is due to Kodaira. In this case note that Lν is a rank 1 analytic sheaf. The case ofarbitrary vector bundle with the definition of positivity in Definition 8.3 is due toGrauert.

9.1 Notations

We gather all necessary notations in order to work with neighborhoods of analyticvarieties. Let A be a subvariety of an analytic variety X . For simplicity, we mayassume that X is smooth.

• OX is the structural sheaf of X , that is, the sheaf of holomorphic functions on X .• M := MA is the subsheaf of OX consisting of elements that vanish at A.• The quotient

Qν := M ν/M ν+1 |Ais is a OA-module sheaf.

• For an analytic sheaf S on X we define the quotient

S (ν) := res(S )⊗OA Qν

81

82 9 Vanishing theorems

whereres(S ) := (S /MS ) |A

is the structural restriction of S to A.• For any vector bundle V → X over a complex manifold, V−1 denotes its dual and

V the sheaf of holomorphic sections of V . For simplicity we also use V to denoteV ; being clear from the context which we mean, either a vector bundle or thesheaf of its sections.

• T := T X , the sheaf of holomorphic vector fields in X (sections of the tangentbundle T X).

• TA, the subsheaf of T consisting of vector fields tangent to A.• For A smooth we define N := T X |A/TA the normal bundle of A in X . For singular

A we can define N without using TA:

N :=T

TA.

• For an analytic sheaf S on A we denote by the same letter S , the sheaf in Xobtained by extending S with zeros, that is, the new sheaf have the same stalkat the points of A and has zeros stalks at other points in X .

The reader is referred to [GuII90] I,J for the notion of tangent space of a singularvariety. Specially it is proved there that the bundle of tangent spaces of a variety hasa canonical structure of an analytic variety.

Exercise 9.1 Verify the details of the following statements:

1. There are natural isomorphisms of analytic sheaves in A

Q(1)∼= N−1,

2.Qν∼= Q(1)⊗Q(1)⊗·· ·⊗Q(1)(ν times), Qν

∼= (N−1)ν

3. There is a natural homomorphism S M ν →S (ν) for which we have the shortexact sequence

0→S M ν+1→S M ν →S (ν)→ 0

9.2 Homogeneous functions along fibers

Let N be a vector bundle over a smooth complex variety A. We consider N as acomplex manifold and A is embedded inside N by the zero section.

Definition 9.1 A holomorphic function defined in an open set in N is called homo-geneous of degree ν along the fibers of N if in a trivialization chart (x,z) ∈U ×Cn

it is a homogeneous polynomial of degree ν in the variable z.

9.3 Kodaira vanishing theorem 83

Since the transition functions are linear in z, this definition does not depend on thechart we choose.

Definition 9.2 Let Hν be the sheaf of homogeneous functions of degree ν alongthe fibers of N. The sheaf Hν has a natural structure of a π∗OA-module, whereπ : N→ A is the bundle map. The sheaf π∗OA is the sheaf of holomorphic functionsin N which are constant along the fibers of N.

Exercise 9.2 We have a natural isomorphism

Hν |A →Qν

obtained by the inclusion.

Definition 9.3 Let S be an analytic sheaf defined in A. The preimage (or pull-back)of S by the bundle map π : N→ A, namely π∗S , is a π∗OA-module:

π∗S (U) := S (π(U)).

Note that the bundle map π : N→ A sends open sets to open sets. We define

S := π∗S ⊗π∗OA OX (9.1)

It follows by definition that

Exercise 9.3 The sheaf S is a coherent if and only if S is coherent.

For simplicity we use S instead of S ; being clear from the context which we mean.One can define the homogeneous subsheaf of degree ν of S as

Sν := π∗S ⊗π∗OA Hν

Exercise 9.4 We haveSν |A∼= S (ν)

as OA-modules.

9.3 Kodaira vanishing theorem

Theorem 9.2 (Grauert, [Gr62], Hilfssatz 1, p. 344) Let S be a coherent analyticsheaf on a neighborhood of an exceptional variety A in X. Furthermore we assumethat N is negative, i.e. the zero section of N is an exceptional variety in N. Thereexists a positive integer ν0 such that

Hµ(A,S (ν)) = 0, µ ≥ 1, ν ≥ ν0


Proof. Let S := S /MS be the structural restriction of of S in A and then itsextention to N as an analytic sheaf in N, as in (9.1). We have S (ν)∼= S (ν) and soit is enough to prove the theorem for the zero section A of N and S bering a coherentsheaf of the form (9.1). Theorem 7.1 is the key of the proof of this theorem. We havethe maps

a : S →S1⊕S2⊕·· ·⊕Sν

b : S1⊕S2⊕·· ·⊕Sν →S

where a is the canonical map and b is the inclusion, with ab equal to the identity.Taking the µ-th cohomology from the above data we conclude that b∗ : Hµ(A,S1)⊕Hµ(A,S2)⊕ ·· ·⊕Hµ(A,Sν)→ Hµ(U,S ) is an injection, because a∗ b∗ is theidentity. Since by Theorem 7.1 Hµ(U,S ) is finite dimensional, we get the desirednumber in the theorem.

Proof (Proof of Theorem 9.1). When N :=V−1 is a negative line bundle over a mani-fold A, Theorem 9.2 is exactly Kodaira’s vanishing theorem. We have M ν/M ν+1'V ν and S (ν)'S ⊗OA V ν .

9.4 The case of a Riemann surface

In the case where A is a Riemann surface, S is the sheaf of section of a vector bundleV and L is a line bundle, the Kodaira vanishing theorem follows from from Serreduality in Section 15.4. In this case we can explicitly state the minimum number ν0with the property of Theorem 9.2. The precise computations are as follow.

Using the same notation as in Theorem 9.1, Serre duality implies that

H1(A,V ⊗Lν)∼= H0(A,Ω 1⊗V−1⊗L−ν)∗ (9.2)

We assume that the vector bundle V is a sum of line bundles V = ⊕ni=1Li. This is

always the case for vector bundles over the line P1, see Section 13.2. From Propo-sition 8.3 it follows thar the left hand side of (9.2) is zero if the Chern class of allΩ 1⊗L−1

i ⊗L−ν is negative. That is

−c(Li)+2g−2−ν · c(L), i = 1,2, . . . ,n.

Since L is positive we have c(L)> 0 and so

max−c(Li)+2g−2

c(L), i = 1,2, . . . ,n

< ν .

Exercise 9.5 A line bundle L on a Riemann surface A is positive if and only if ithas a holomorphic section with zeros. Note that if a line bundle has a holomrphicsection without zeros then it is trivial.

9.5 Grauert vanishing theorem 85

9.5 Grauert vanishing theorem

We have so far discussed Kodaira vanishing theorem for a variety A. The aim of thissection is to introduce vanishing theorems in a strongly convex neighborhood ofA. Theorem 9.4 is the main theorem in this chapter. We will follow Grauert’s article[Gr62], but our proof for Theorem 9.4 works for a general exceptional variety whileGrauert’s argument works for codimension one exceptional varieties in manifolds.

Let A be an exceptional subvariety of a variety X . By Theorem 7.7 one can find aC∞ function ψ defined in a neighborhood of A in X such that ψ is strongly plurisub-harmonic outside A, ψ ≥ 0 and ψ−1(0) = A. Therefore we have a fundamentalsystem

Uε := ψ(x)< ε,0 < ε << 1

of relatively compact strongly pseudoconvex neighborhoods around A. Fix a Uε . LetS be an analytic sheaf on Uε and f be a holomorphic function on Uε . Since A iscompact connected, f restricted to A is constant. We denote this constant by f (A).Take a Stein covering U of Uε . We have Hµ(U ,S ) = Hµ(Uε ,S ), where µ > 0.The usual multiplication of f by cocycles in Zµ(Uε ,S ) yields a well-defined mapfrom Hµ(Uε ,S ) to itself.

Lemma 9.1 Let f be a holomorphic function in Uε and S be a coherent sheafdefined on a neighborhood of A in X. There exist a natural number n1 and a positivenumber ε1 such that

( f − f (A))nHµ(Uε ,S ) = 0, ∀n≥ n1,n ∈ N, 0 < ε ≤ ε1, µ ≥ 1

Proof. Without loosing the generality suppose that f (A) = 0, i.e. f vanishes onA. Let φ : (X ,A)→ (Y, p) be the Remmert reduction mapping (see Appendix 14).By Grauert direct image theorem Rµ φ∗S is a coherent sheaf. Since φ |X−A is abiholomorphism, Cartan’s theorem B implies that the support of Rµ φ∗S lies inp ∈ Y and so the stalk (Rµ φ∗S )p is a finite dimensional C-vector space. Now bythe property 1 listed in Remmert reduction theorem there is a holomorphic functiong in (Y, p) such that f = gφ . Multiplication by g with the stalk (Rµ φ∗S )p has noteigenvalue different from zero. Therefore it is unipotent and so there is n such thatgnRµ φ∗S is the zero sheaf.

Theorem 9.3 There exist a natural number ν0 and a positive number ε ′ such thatfor all ν ≥ ν0,ν ∈ N and 0 < ε < ε ′ the map induced by inclusion

Hµ(Uε ,S M ν)→ Hµ(Uε ,S )

is the zero map.

Our proof for this theorem is similar to Grauert’s proof. Grauert after proving thistheorem for pure codimension one A in a manifold X ([Gr62] Satz 1 p. 355) tells usthat for an arbitrary exceptional variety A in X this theorem follows from HauptsatzII of [Gr60]. This theorem is also proved for pure codimension one A in a manifoldX in [La71] Theorem 5.4.


Proof. One can blow down A to a point and obtain a singularity (Y, p). Let z1,z2, . . . ,znbe the coordinate functions of (Y, p) and f1, f2, . . . , fn be the pullback of zi’s by theblow down map. According to Lemma 9.1 there is a natural number ni and a positivenumber εi such that f m

i Hµ(Uε ,S )= 0, m≥ ni,0< ε < εi. Let m be the maximum ofni’s, ε ′ be the minimum of εi’s. From now on we write U =Uε for a fixed 0< ε < ε ′.

Let M be the subideal of OU generated by f mi ’s. The zero locus of f m

i ’s is A andso by Hilbert Nullstellensatz theorem (see [GuII90]) there exists a natural numberν1 such that

M ν1 ⊂ M

The proof of the theorem is by inverse induction on µ . If U is a finite Stein coveringof U with r open sets, then by Cech cohomology Hr(U,S M ν) = 0 for all naturalnumbers ν and for all sheaves S . Therefore our theorem is trivial for µ = r. Nowsuppose that it is true for µ +1. We want to prove that it is true for µ also.

Letπ : On

U → M

π(a1,a2, . . . ,an) =n

∑i=1

ai f mi

and R := Kerπ . We write the short exact sequence

0→R→ OnU → M → 0

and we make a tensor product of this short exact sequence with S (resp. S M ν2 ,where ν2 is an unknown natural number) over OU and then we write the associatedlong exact sequence. Since S n = S ⊗On

U and Hµ(U,S n)→ Hµ(U,S M ) is thezero map, we get the commutative diagram

0 → Hµ(U,S M ) → Hµ+1(U,S ⊗R) →↑ ↑

· · · → Hµ(U,S MM ν2)→ Hµ+1(U,S ⊗R⊗M ν2)→(9.3)

By induction for a big ν2 the second up arrow map is zero and so by the abovediagram the first is zero also. The map Hµ(U,S M ν)→ Hµ(U,S ),ν ≥ ν1 + ν2splits into

Hµ(U,S M ν)→ Hµ(U,S MM ν2)→ Hµ(U,S M )→ Hµ(U,S )

and so it is the zero map.

9.6 Main vanishing theorem

Let us be given a subvariety of a variety X . We say that A is strongly exceptional inX if A is exceptional and the normal bundle of A in X is negative.

9.7 Restriction of line bundles 87

Theorem 9.4 (Grauert [Gr62],Satz 2, p. 357) Let us be given a strongly excep-tional subvariety A of a variety X. There exists a positive integer ν0 such that

Hµ(U,S M ν) = 0, µ ≥ 1, ν ≥ ν0

where U is a small strongly pseudoconvex neighborhood of A in X.

Proof. Let ν0 be the number such that Hµ(A,S (ν)) = 0, ν ≥ ν0, µ ≥ 1. Considerthe short exact sequence

0→S M ν+1→S M ν →S (ν)→ 0

For ν ≥ ν0 the map Hµ(U,S M ν+1)→ Hµ(U,S M ν) is surjective and so forany k ≥ ν the map Hµ(U,S M k)→ Hµ(U,S M ν) is surjective. According toTheorem 9.3 for a large k this map is zero and so Hµ(U,S M ν) = 0, ν ≥ ν0.

Note that ν0 in the above theorem is the same number ν0 in Hµ(A,S (ν)) =0, ν ≥ ν0, µ ≥ 1.

9.7 Restriction of line bundles

From now on we use the letter U for a strongly pseudoconvex neighbourhood of Ain X such that the theorem 9.4 is valid. From now on we can substitute X by U . Inthis section we prove the following proposition:

Proposition 9.1 Let A be a complex manifold of dimension n negatively embeddedin a manifold X of dimension n+1. Moreover, suppose that

H1(A,N−ν) = 0, ν = 1,2,3, . . .

where N is the normal bundle of the embedding and N−1 is the dual bundle. Therestriction map

r : H1(U,O∗U )→ H1(A,O∗A)

is injective.

If A is a Riemann surface, i.e. n = 1, one uses the Serre duality

H1(A,N−ν) = H0(A,Ω 1⊗Nν)∗ = 0

and the negativity condition and H1(A,N∗) = 0 translates into

A ·A < 0 & A ·A < 2−2g

Note that the negativity condition does not imply H1(U,N−ν) = 0.

Proof. The sheaf of holomorphic sections of N−ν is isomorphic to M ν/M ν+1 andso we have


H1(A,M ν/M ν+1) = 0, ∀ν ∈ N

The variety A is negatively embeded and so by Theorem 9.4 applied to S = OX wehave

H1(U,M ) = 0

where U is a strongly pseudoconvex neighborhood of A in X . The diagram

0↓

M↓

0 → Z → OU → O∗U → 0↓ ↓ ↓

0 → Z → OA → O∗A → 0↓0

(9.4)

gives us

H1(U,M ) = 0↓

H1(U,Z)→ H1(U,OU ) → H1(U,O∗U )→ H2(U,Z)↓ ↓ ↓ ↓

H1(A,Z) → H1(A,OA) → H1(A,O∗A) → H2(A,Z)

(9.5)

By considering a smaller neighborhood U , if necessary, we can assume that A andU have the same topology and so the first and fourth column functions are isomor-phisms. In the argument which we are going to consider now we do not mention thename of mappings, being clear from the above diagram which mapping we mean.

Let us consider x1 ∈H1(U,O∗U ) which is mapped to the trivial bundle in H1(A,O∗A).Since the fourth column is an isomorphism, x1 maps to zero in H2(U,Z). Thismeans that there is a x2 ∈ H1(U,OU ) which maps to x1. Let x3 be the image ofx2 in H1(A,OA). Since the above diagram is commutative, x3 maps to the trivialbundle in H1(A,O∗A). Therefore there exists a x4 in H1(A,Z) which maps to x3.Since the first column is an isomorphism and the second is injective, we concludethat x4 ∈ H1(U,Z)∼= H1(A,Z) maps to x2 and so x2 maps to x1 = 0 in H1(U,O∗U ).

If U has a transverse foliation by curves then we have a holomorphic map σ :U → A which is constant along the leaves of the foliation. The pull-back of linebundles on A by the map σ shows that r is surjective.

Proposition 9.2 Let A be a complex manifold of dimension n negatively embeddedin a manifold X of dimension n+1. Assume that

H2(A,N−ν) = 0, ν = 1,2,3, . . . (9.6)

Then the restriction map

9.8 A negative divisor of curves 89

r : H1(U,O∗U )→ H1(A,O∗A)

surjective.

The condition (9.6) is automatically satisfied for the case in which A is a Riemannsurface. The proof follows again from the long exact sequnce of the diagram (9.4)and H2(U,M ) = 0. An alternative proof is:

Proof. We write the long exact sequence of

1→ J→ O∗U → O∗A→ 1

where J is the sheaf of holomorphic functions in U with value 1 at all points of A.We have

· → H1(U,O∗U )r→ H1(A,O∗A)→ H2(U,J)→ ···

The exponential map (M ,+)→ (J, ·) is an isomorphism and Theorem 9.4 and (9.6)imply that

H2(U,M ) = 0

Therefore, H2(U,J) = 0 and so r is surjective.

9.8 A negative divisor of curves

When A is a union of curves in a two dimensional manifold we have a numericalcriterion for contractablity of A.

Theorem 9.5 Let A be a compact connected one dimensional subvariety of a twodimension manifold X. Suppose that A contains only normal crossing singularities.Then A is exceptional in X if and only if the intersection matrix S = [Ai.A j] of Ain X is negative definite, where A = ∪Ai is the decomposition of A into irreduciblecomponents.

This theorem is due to Grauert in [Gra62], page 367, see also Theorem 4.9 of[La71].

Proof.

Chapter 10Embedding theorems

In this chapter we remind the classical notion of a blow-up of a point in the n+ 1-dimension space. We also describe the notion of blow-up along a variety. This willbe needed for the study of embeddings of codimension stricktly bigger than one.

10.1 Blow-up

The classical definition of blow up at 0 ∈Cn+1 goes as follows: the projective spacePn is the set of one dimensional sub vector spaces of Cn+1 and it has the followingline bundle

O(−1) = (x,y) ∈ Pn×Cn+1 | y ∈ x

which is called the tautological line bundle. It is a negative line bundle, because theprojection on the second coordinate π : O(−1)→ Cn+1 exhibits the zero section ofO(−1) as an exceptional variety. It is usual to write ˜Cn+1 := O(−1) and say thatπ : ˜Cn+1 → Cn+1 is the blow-up map of Cn+1 at 0. If no confusion is possible weidentify π−1(0) with Pn. The manifold ˜Cn+1 is covered by affine charts Ui := xi 6=0

Ui∼→ Cn+1

([x0 : x1 : · · · : xn],(y0,y1, . . . ,yn)) 7→

(t0, . . . , ti−1,yi, ti+1, . . . , tn) := (x0

xi, . . .

xi−1

xi,yi,

xi−1

xi, . . . ,

xn

xi)

10.2 Blow-up of a singularity

Let (X ,0)⊂ (Cn+1,0) be a germ of an analytic variety and I ⊂OCn+1,0 be the idealof holomorphic functions vanishing on X . We assume that Zero(I ) = (X ,0) but we

91

92 10 Embedding theorems

do not assume that I is radical. Therefore, (X ,0) with the structural sheaf OX ,0/Ican be considered as an analytic scheme which might have nilpotents.

Definition 10.1 For an element f ∈I , let f ∗ be the leading term of f and I ∗ bethe ideal in the polynomial ring C[x] generated by f ∗, f ∈I .

Definition 10.2 The variety

Zero(I ∗) :=

x ∈ Cn+1 | f (x) = 0, ∀ f ∈I

is called the tangent cone of X at 0.

It is a homogeneous variety, i.e. for all x ∈ Zero(I ∗) and λ ∈ C∗ we have λ · x ⊂Zero(I ∗). Therefore we can projectivize the tangent cone and obtain the projec-tivized tangent cone TC0X ⊂ Pn. Note that if X is given by f1 = 0, f2 = 0, . . . , fk = 0then not necessarily f ∗1 = 0, f ∗2 = 0, . . . , f ∗k = 0 defines the tangent cone of X at 0.We may need more leading terms of elements in I . Let π : ˜Cn+1 → Cn+1 be theblow-up map.

Proposition 10.1 The closure X of π−1(X −0) in ˜Cn+1 is an analytic varietyand X ∩Pn ∼= TC0X. In particular, TC0X is of pure codimension one in X , i.e. eachirreducible component of TC0X is of codimension one in X

Proof. In an affine chart (y0, t1, t2, . . . , tn) the set X is given by the zero set of poly-nomials

fm(1, t1, . . . , tn)+ y0 fm+1(1, t1, . . . , tm)+ · · · ,

for all f = fm + fm+1 + · · · ∈ I . This shows that X is an analytic variety. The in-tersection of X with Pn in this coordinate system is fm(1, t1, t2, . . . , tm) = 0, f ∈ Iwhich is TC0X in the coordinate system (t1, t2, . . . , tn) of Pn.

The dimension m of each irreducible component of X ∩Pn satisfies

dimX ≥ m≥ dim(X)+dimPn− (n+1) = dimX−1

see for instance [Ke] Theorem 3.6.1. Since X has no irreducible component in Pn,we conclude that m = dimX−1.

10.3 An embedding theorem

By definition the blow up variety X is embedded in Pn×Cn+1 and so we have theprojection on the second coordinate π : X → Cn+1 which is called the blow-up mapat 0 ∈ X , and the projection on the first coordinate π1 : X → Pn. Put A = TC0X andU a small neighborhood of A in X . We have

1. π induces a biholomorphism between U−A and π(U)−0;2. π1 |A is an embedding of A in Pn.

10.3 An embedding theorem 93

Theorem 10.1 Suppose the that all irreducible components of an exceptional va-riety A in a manifold X are of codimension one and the normal bundle of A in Xis negative. There is a positive integer ν1 and a neighborhood U of A in X suchthat for all k ≥ ν1 if s0,s1, . . . ,sn ∈ H0(U,M k) form a basis for the vector spaceH0(U,M k)/H0(U,M k+2) then

Fk : U → Pn×Cn+1

Fk(x) = ([s0(x) : s1(x) : · · · : sn(x)],(s0(x),s1(x), . . . ,sn(x)))

is a well-defined map and is an embedding of U with the properties 1,2 listed above.

Of course the number n depends on k.

Proof. We prove that there exists ν1 ∈ N such that for k ≥ ν1 the statements 1,2and 3 listed below are true. We take U a strongly pseudoconvex neighborhood ofA in X . In the following when we say that for big enough k ∈ N a property holds,this means that there is ν1 such that for k ≥ ν1 such a property holds. Withoutlosing the generality we can assume that s0,s1, . . . ,sm, 0 ≤ m ≤ n form a basis forH0(U,M k)/H0(U,M k+1).

1. Fk is well-defined. Let Zero(si),0 ≤ i ≤ m be the zero divisor of si. One canwrite

Zero(si) = Di + k ·A,

where Di is a divisor in U and it does not contain A. It is enough to prove that

∩mi=0 |Di|= /0 (10.1)

This is because if for a point x ∈U there is some Di such that x 6∈ Di and so s jsi, j =

1,2, . . . ,m, j 6= i is a holomorphic function near x. This means that

[s0(x) : s1(x) : . . . : sn(x)] =[

s0(x)si(x)

:s1(x)si(x)

: . . . :sn(x)si(x)

]is well-defined in a neighborhood of x. Recall that for a coherent sheaf S on X anda subvariety Y ⊂ X , the structural restriction of S to Y is ResY (S ) := S /S MY ,where MY is the zero ideal of Y . We will use this for Y consisting of a single ortwo points. If Y := x is a single point then resx(S ) is a finite dimensional vectorspace, and for S a non-zero sheaf, it is not a 0, see Proposition 2.11.

Now, let us prove (10.1). By Grauert’s theorem, see Theorem 9.4, for any pointx ∈U and big k ∈ N we have

H1(U,MxMk) = 0

and so0→ H0(U,MxM

k)→ H0(U,M k)→ Resx(Mk)→ 0 (10.2)

Now (10.2) is true for all points in a neighborhood of x, see Proposition 2.11. SinceU is compact, we can cover U by a finite number of such open sets. Therefore, for


big enough k, (10.2) is true for all x ∈U . If x ∈ ∩ni=0|Di| then

H0(U,M k)⊂ H0(U,MxMk).

By the above sequence we conclude that Resx(M k) is 0 which is a contradiction.2. Fk is one to one. Let x,y ∈U . We take k big enough such that

H1(U,Mx,yMk) = 0,

where by x,y we mean the set x,y. We have

H0(U,M k)→ Resx,y(Mk)→ 0 (10.3)

The above sequence is true in a neighborhood of (x,y) in U×U . Since U is compact,we can cover U × U by a finite number of such open sets. Therefore, for k bigenough, (10.3) is true for all x,y ∈U . Now for two points x,y ∈U we have

Resx,y(Mk) = Resx(M

k)⊕Resy(Mk).

This together with (10.3) implies that we have f ∈ H0(U,M k) such that f induces0 in Resx(M k) and a non-zero element of Resy(M k). This gives us Fk(x) 6= Fk(y).

3. Fk is a locally an embedding. In the above argument we can take Mx,x = M 2x

and so we have

H0(U,M k)α→ H0(U,M k/M kM 2

x )→ 0, ∀x ∈U

Since H0(U,M k+2)⊂ kerα we have

H0(U,M k)/H0(U,M k+2)β→ H0(U,M k/M kM 2

x )→ 0, ∀x ∈U. (10.4)

Fix a point x ∈ A. Since Fk is well-defined we can assume that the first coordinate in[s0(x) : s1(x) : · · · : sn(x)] is not zero. In other words, s0(x)

si(x)evaluated at x is not zero

for some i = 1,2, . . . ,n. The support of M k/M kM 2x is the point x and at this point

(M k/M kM 2x )x ∼= s0

OX ,x

M 2x.

The image of si by β is s0.sis0

and so by (10.4) the pullback of the coordinates func-tions xi

x0of Pn by Fk span OX ,x/M 2

x . This implies that the map T ∗Fk(x)Pn×Cn+1 →

T ∗x U is surjective and so by Proposition 2.7 Fk is an embedding in a neighborhoodof x.

Theorem 10.2 (Kodaira embedding theorem) Let A be a compact complex man-ifold with a positive line bundle L. There is ν0 ∈ N such that for ν ≥ ν0 ands0,s1, . . . ,sn global holomorphic sections of Lν we have the embedding

A → Pn, x 7→ [s0(x) : s1(x) : · · · : sn(x)]

10.4 Blow up along a submanifold 95

and hence A is a projective algebraic variety.

Proof. This is a direct consequence of Theorem 10.1 and Theorem 8.2. We have ashort exact sequence

0→ H0(U,M k+1)

H0(U,M k+2)→ H0(U,M k)

H0(U,M k+2)→ H0(U,M k)

H0(U,M k+1)→ 0

We can take a basis s0,s1, . . . ,sn of the second vector space such that s0,s1, . . . ,smform a basis of the first vector space and sm+1, · · · ,sn are mapped to a basis of thethird vector space. Now the map Fk in Theorem 10.1 restricted to A has its image in

x ∈ Pn | x0 = x1 = · · ·= xm = 0

and the embedding of A is given by sm+1,sm+1, · · · ,sn which gives a basis of

H0(U,M k)

H0(U,M k+1)∼= H0(A,

M k

M k+1 )∼= H0(A,Lk).

ut

10.4 Blow up along a submanifold

Let N be a vector bundle of rank m+ 1 over A and let A := P(N) be the projec-tivization of the fibers of N. We have a canonical projection map π : A→ A withfibers isomorphic to Pm. The space A carries a distinguished line bundle N which isdefined by:

Nx = the line representing x in the vector space Nπ(x), x ∈ A

In some books the notation OA(−1) is used to denote the sheaf of sections of Nbecause the line bundle N is the tautological bundle restricted to the fibers of π . Ithas the following properties:

π∗(O(N−ν))∼= O(N−ν), ν = 0,1,2, . . .

π∗(O(Nν)) = 0, ν = 1,2, . . .

Hq(A,π∗(S )⊗O(N−ν))∼= Hq(A,S ⊗O(N−ν)), ν = 1,2, . . .

for every locally free sheaf S on A (see [GPR94], p. 178). Here O of a bundlemeans the sheaf of its sections. When there is no ambiguity between a bundle andthe sheaf of its sections we do not write O . We will also use the following: if for asheaf of abelian groups S on A we have Riπ∗(S ) = 0 for all i = 1,2, . . ., then

H i(A,S )∼= H i(A,π∗S ), i = 0,1,2, . . . .


We will apply this for the sheaf of sections of TPm⊗ N−ν , ν = 1,2, . . ., where TPm

is the subbundle of T A corresponding to vectors tangent to the fibers of π .By definition N is a subbundle of π∗N and we have the short exact sequence:

0→ N→ π∗N→ TPm→ 0. (10.5)

We take O of the above sequence, make a tensor product with O(N−ν), ν = 1,2, . . .and apply π∗: we get

0→ N−ν+1→ N⊗N−ν → π∗(TPm⊗ N−ν)→ 0 (10.6)

(for simplicity we have not written O(· · ·)). Note that R1π∗O(N−ν+1) = 0, ν =1,2, . . .. Note also that if N is not a line bundle then N⊗N−1 may not be the trivialbundle.

The vector bundle TPm appears also in the short exact sequence:

0→ O(TPm)→ O(T A)→ π∗O(TA)→ 0, (10.7)

where O(T A)→ π∗O(TA) is the map obtained by derivation of A→ A and thenconsidering the pull-back of O(TA).

Let A be a compact submanifold of X with

n = dim(A), m+1 = dim(X)−n.

and let N = T X |A /TA be the normal bundle of A in X . We make the blow up of Xalong A:

π : X → X , A := π−1(A) = P(N).

The normal bundle of A in X is in fact:

N = NX/A∼= OA(−1).

Combining all these with Proposition 9.1, we get the same proposition without thecodimension restriction:

Proposition 10.2 Let A be a strongly exceptional complex submanifold of X. More-over, suppose that

H1(A,N−ν) = 0, ν = 1,2,3, . . .

where N is the normal bundle of the embedding and N−1 is the dual bundle. Therestriction map

r : H1(X ,O∗X )→ H1(A,O∗A)

is injective.

Chapter 11Deformation of hypersurfaces

For a given smooth hypersurface M of degree d in Pn+1 is there any deformationof M which is not embedded in Pn+1? We need the answer of this question becauseit would be essential to us to know that the fibers of a tame polynomial f form themost effective family of affine hypersurfaces. The answer to our question is givenby Kodaira-Spencer Theorem which we are going to explain it in this section. Forthe proof and more information on deformation of complex manifolds the reader isreferred to [Kod86], Chapter 5.

11.1 Kodaira-Spencer

Let M be a complex manifold and Mt , t ∈ B := (Cs,0), M0 = M be a deformationof M0 which is topologically trivial over B. We say that the parameter space B iseffective if the Kodaira-Spencer map

ρ0 : T0B→ H1(M,Θ)

is injective, where Θ is the sheaf of vector fields on M. It is called complete if anyother family which contain M is obtained from Mt , t ∈ B in a canonical way (see[Kod86], p. 228).

Theorem 11.1 If ρ0 is surjective at 0 then Mt , t ∈ B is complete.

Let m = dimC H1(M,Θ). If one finds an effective deformation of M with m = dimBthen ρ0 is surjective and so by the above theorem it is complete.

Let us now M be a smooth hypersurface of degree d in the projective space Pn+1.Let T be the projectivization of the coefficient space of smooth hypersurfaces inPn+1. In the definition of M one has already dimT =

(n+1+dd

)−1 parameters, from

which only

m :=(

n+1+dd

)− (n+2)2

97

98 11 Deformation of hypersurfaces

are not obtained by linear transformations of Pn+1.

Theorem 11.2 Assume that n ≥ 2, d ≥ 3 and (n,d) 6= (2,4). There exists a m-dimensional smooth subvariety of T through the parameter of M such that theKodaira-Spencer map is injective and so the corresponding deformation is com-plete.

For the proof see [Kod86] p. 234. Let us now discuss the exceptional cases. For(n,d) = (2,4) we have 19 effective parameter but dimH1(M,Θ) = 20. The differ-ence comes from a non algebraic deformation of M (see [Kod86] p. 247). In thiscase M is a K3 surface. For n = 1, we are talking about the deformation theory of aRiemann surface. According to Riemann’s well-known formula, the complex struc-ture of a Riemann surface of genus g ≥ 2 depends on 3g− 3 parameters which isagain dimH1(M,Θ) ([Kod86] p. 226).

Chapter 12Foliated neighborhoods

Let A be a complex compact manifold embedded in another manifold X . In whatfollows we use both X and (X ,A) to denote the germ of X around A.

12.1 Holomorphic foliations

A (holomorphic) foliation by curves in a complex manifold X is given by a col-lection of holomorphic vector fields Vα defined on Uα ,α ∈ I, where Uαα∈I is anopen covering of X , and such that

Vα = gαβVβ , α,β ∈ I,gαβ ∈ O∗X (Uα ∩Uβ ) (12.1)

where O∗X is the sheaf of holomorphic without zero functions in X . We further as-sume that the set of points in which Vα is zero has codimension greater than one.In other words Vα has not a zero divisor. Therefore, for any foliation F there isassociated a line bundle L given by the transition functions

L := TF = gαβα,β∈I ∈ H1(X ,O∗X )

The data (12.1) can be considered as a holomorphic section V ∈H0(X ,Θ⊗L) with-out zero divisor, where Θ is the sheaf of holomorphic vector fields. If instead of Θ

we use the sheaf of holomorphic differential 1-forms then we obtain the definitionof holomorphic codimension one foliations in X .

Let A be a complex compact manifold embedded in another manifold X . Weassume that the codimension of A in X is one.

Definition 12.1 A foliation by curves in (X ,A) is called a transverse foliation,if itis without singularity and each of its leaves is transverse to A.

Let us introduce some examples in the case A = P1 and X a two dimensionalcomplex manifold.

99

100 12 Foliated neighborhoods

x

y y

x

yx/y

y/xx/y

y/xnx / yn−1

Fig. 12.1 A projective line with self intersection −n

Example 12.1 By successive blow-ups at the origin of C2, we can get a A ∼= P1embedded in a two dimensional manifold and with A.A =−n. A neighborhood of Ais covered by coordinate systems

(u,y) = (XY,Y ), (x, t) = (

Xn

Y n−1 ,YX),

where X and Y are the pullback of a coordinates system at the origin of C2. Thechange of coordinates is given by

(x, t)→ (1t,xtn) = (u,y)

In this example we have a germ of transverse holomorphic foliation F given by the1-form

ω = XdY −Y dX = (xtn−1)2dt =−y2du

It is easy to check thatzer(ω) = 2.A+2(n−1)L

zer(Y ) = 1.A+n.L

zer(X) = 1.A+(n−1)L+L′

where zer() means the zero divisor and L (resp. L′) is the leaf of F given by t = 0(resp. u = 0 ) in the coordinates (x, t) (resp. (u,y)). It is the pullback of X-axis (resp.Y -axis).

12.2 Construction of functions

In this section we give an application of Proposition 9.1. Let us assume that (X ,A)has a transverse foliation namely F . The normal bundle N of A in X has a mero-morphic global section namely s. Let

div(s) = ∑niDi, ni ∈ Z

We define the divisor D in X as follows:

12.2 Construction of functions 101

D = A−∑niDi (12.2)

where Di is the saturation of Di by F . The line bundle LD associated to D restrictedto A is the trivial line bundle, and so by Proposition 9.1, LD is trivial or equivalently

Proposition 12.1 There exists a meromorphic function g on (X ,A) with

div(g) = D

where D is given by (12.2).

Now we give another application of Proposition 9.1.

Proposition 12.2 Consider the situation of Proposition 9.1. Let F be a foliation bycurves transversal to A in X. Then there exists a holomorphic vector field V defineda neighborhood of A in X with the following properties:

1. V is tangent to F ;2. The zero divisor of V is A.

Proof. Let Uα ,α ∈ I be an open covering of A in U such that in each Uα there isdefined a vector field Xα without zero locus and tangent to F . Then gαβ :=

Xβ

Xα∈

O∗(Uα ∩Uβ ) is a cocycle and hence

L := gαβ ∈ H1(U,O∗)

Since the Xα ’s are tangent to F , we can think of L as the tangent line bundle toF and consequently as the normal bundle of A in X when we restrict it to A. Nowlet us consider A as a divisor with coefficient +1 in X and let fα be a holomorphicfunction on Uα vanishing on A of order one (If it is necessary we can take a finercovering of A). The line bundle associated to A is given by

L′ := g′αβ, g′

αβ=

fα

fβ

L′ restricted to A is again the normal bundle N. Therefore LL′−1 restricted to A is thetrivial bundle and so by lemma 9.1 LL′−1 is the trivial bundle or equivalently thereare holomorphic functions sα ∈ O∗(Uα) such that

gαβ =sα

sβ

g′αβ⇒

Xβ

Xα

=sα

sβ

fα

fβ

Now the desired global vector field Uα is defined by V |Uα:= sα fα Xα .:


12.3 Equivalence of transverse foliations

The objective of this section is to prove the following theorem:

Theorem 12.1 Let A be a complex compact manifold embedded negatively as acodimension one subvariety in X. Further, assume that

H1(A,N−ν) = 0, ∀ν ∈ N (12.3)

Any transverse holomorphic foliation in (X ,A) is biholomorphic to the the canonicaltransverse foliation of (N,A) by the fibers of N. In particular, the germs of any twoholomorphic transverse foliations in (X ,A) are equivalent.

Let A be a Riemann surface of genus g embedded in a manifold X of dimension two.The negativity condition and (12.3) translate into

A ·A < 0, A ·A < 2−2g.

Proof (Proof of Theorem 12.1). Let F be the germ of a transverse foliation in (X ,A)and N the normal bundle of A in X . Let also F ′ be the canonical transverse foliationof (N,A). Let g (resp. g′) be the meromorphic function constructed in Proposition12.1 for the pair (X ,A) resp. (N,A). We claim that at each point a ∈ A there exists aunique biholomorphism

ψa : (X ,A,a)→ (N,A,a)

with the following properties:

1. ψ induces the identity map on A;2. ψ sends F to F ′;3. The pullback of g′ by ψ is g.

The uniqueness property implies that these local biholomorphisms are retriction ofa global biholomorphism ψ : (X ,A)→ (N,A) which send F to F ′.

Now we prove our claim. Fix a coordinates system x = (x1,x2, · · · ,xn) in a neigh-borhood of a in A. We extend x to a coordinates system (x,xn+1) of a neighborhoodof a in X such that A (resp. F ) in this coordinates system is given by xn+1 = 0 (resp.dxi = 0, i = 1,2, . . . ,n). In this coordinates system

g(x,xn+1) = Q(x)xn+1 f (x,xn+1),

where Q(x) is a meoromorphic function in a neighborhood of a in A and it does notdepend on the choice of an embedding of A and f is holomorphic function in (X ,a)without zeros. By changing the coordinates in xn+1 we can assume that f = 1. It iseasy to check that the coordinate system (x,xn+1) is unique and it gives us the localbiholomorphism ψa.

12.4 Construction of line fields 103

12.4 Construction of line fields

In this section we work with a vector bundle E of rank n+1 over a complex manifoldA of dimension n. We further assume that there is a fiber bundle injection TA → E.We call

N := E/TA

the normal bundle of A.The typical example of this situation is when A is embeddedas a codimension one subvariety in some other manifold X and E := T X |A.

We start this section by a definition.

Definition 12.2 Let L be a line bundle over A. The tangency divisor D of a bundlemorphism L→ E is the divisor of the composition L→ E→ N.

Note that composition map L→ N can be considered as a global section of L−1⊗Nand so L is uniquely determined by the divisor D:

L = L−1D ⊗N (12.4)

We call the image of L→ E a line field.

Theorem 12.2 Let D = ∑i niDi, ni > 0 be any divisor in A with positive coefficientsand let L be defined as in (12.4). Assume

H1(A,N−1⊗LD⊗TA) = 0. (12.5)

Then there exists a holomorphic bundle morphism Y : L→ E|A such that the tan-gency divisor of Y is D.

If A is a Riemann surface we use Serre duality and we conclude that if

c(N⊗L−1D ⊗Ω

1⊗Ω1)< 0 equivalently A ·A < 4−4g+∑ni

then (12.5) is satisfied.

Proof. Let L be an arbitrary line bundle on A. First, we construct Y locally, i.e. wefind Yi : L|Ui → E|Ui with the desired property for an open covering Ui, i ∈ I of A.Let Yi be the composition L|Ui → T X |Ui → N|Ui . Then Yi = ai jYj, where ai j ∈H1(A,O∗A) is a line bundle. Now, Yi’s are sections of L−1⊗N with the zero divisorD and so

ai j= L−1D ⊗L−1⊗N

By definition if L is a line bundle with local no where zero sections yi and yi = ai jy jthen L = ai j and for a divisor D in A the line bundle LD has a global s section withthe divisor D. If fi’s defines D locally then we define yi := s

fiwhich has no zeros or

poles in A and so LD := f jfi.

Since we have the identity (12.4), ai j is the trivial bundle we can assume thatYi = Yj. Now

Yi j := Yi−Yj ∈ H1(A,Hom(L,TA)).


Since Hom(L,TA)∼= L−1⊗TA∼=N−1⊗LD⊗TA, our assertion follows by vanishinghypothesis (12.5).

We are interested in the image of Y which can be considered as a line field.

Proposition 12.3 Let D = ∑i niDi, ni > 0 be any divisor in A with positive coeffi-cients and let L be defined as in (12.4). Assume that N−1⊗LD⊗TA is a negativevector bundle (i.e. its zero section can be blow down to a point) or equivalently fordim(A) = 1 assume that

A ·A > 2−2g+∑ni

For any two bundle morphism Y1,Y2 : L→ E with the tangency divisor D, there isa constant c ∈ C∗ such that Y1 = cY2 and so there is at most one line field with thetangency divisor D.

Proof. The induced maps Yi : L→N, i= 1,2 gives us holomorphic sections of LD =L−1⊗N with the same zero divisor and so Y1 = cY2 for some c ∈C∗. Now, Y1−cY2is a holomorphic bundle morphism from L→ TA. We get a global holomorphicnon-zero section of L−1⊗TA ∼= N−1⊗LD⊗TA which has contradiction with thenegativity hypothesis (such a section after the blow down of the zero section wouldgive us a compact variety in a neighborhood of a point).

12.5 Construction of holomorphic foliations

Let F be a non singular foliation by curves in (X ,A). We have the canonical map

α : L→ E, where L := TF |A

and the tangency divisor D of F with A. We call α the line field associated to F .In Section 12.4 we constructed L→ T X |A with a prescirbed tangency divisor and itis natural to ask whether it comes from a holomorphic foliation as above.

Theorem 12.3 Assume that A is a strongly exeptional codimension one submani-fold of X and

H1(A,N−ν−1⊗T X |A ⊗LD) = 0 (12.6)

Further if the divisor D is not zero assume that

H2(A,N−ν) = 0, ν = 1,2,3, . . .

Any line field L→ T X |A is a line field associated to a non singular foliation Fdefined in a neighborhood of A.

The hypothesis (12.6) follows from:

H1(A,N−v−1⊗TA⊗LD) = 0, H1(A,N−v⊗LD) = 0, ∀ν ∈ N (12.7)

If A is a Riemann surface then by Serre duality (12.7) is satisfied when

12.7 Arbitrary codimension 105

A ·A < 4−4g+∑ni,

Proof. We take local sections of L which trivialize L and have no zero point. Theimages of these sections under the line field L→ T X |A can be extended to vectorfields Xi, i ∈U defined in a covering Ui, i ∈ I of (X ,A). Therefore,

fi jXi|A = X j|A, L = fi j

Proposition 9.2 tells us that the restriction map H1(X ,O∗)→ H1(A,O∗) is surjec-tive. However, if D is the zero divisor then L = N and the normal bundle N of A in Xextends to a line bundle N in (X ,A) as follows: We take local holomorphic functionsfi in (X ,A) such that A = fi = 0. Now fi = fi j f j and N = fi j is a line bundle in(X ,A) which restricted to A is the normal bundle.

Now, we can take the line bundle L = fi j ∈ H1(A,O∗) such that L|A = L. Wedefine

Θi j= X j− fi jXi ∈ H1(X ,MA⊗Θ ⊗L−1)

where Θ is the sheaf of vector fields in (X ,A), see Exercise 8.1. By our hypothesisand theorem 9.4 the cohomology group in the right hand side is zero.

12.6 Grauert’s theorem

As an immediate corrolary of Theorem 12.1, Theorem 12.2 and Theorem 12.3 forthe trivial divisor D = 0 is the following:

Theorem 12.4 (Grauert’s theorem) Let A be negatively embedded codimensionone subvariety of X and

H1(A,N−ν) = H2(A,N−ν) = 0, H1(A,TA⊗N−ν−1) = 0, ν = 1,2,3, . . .

Then the germ of embedding of A in X is biholomorphic to the germ of embeddingof A in N.

12.7 Arbitrary codimension

In order to study the arbitrary codimension case we need the following cohomolog-ical conditions:

(I) Vanishing of cohomologies for arbitrary codimension of A on X :

H1(A,N−ν) = 0, H1(A,TA⊗N−ν) = 0, ν = 1,2, . . .

(II) If the codimension of A in X is greater than one, then:

H2(A,OA) = 0,


H1(A,N⊗N−ν) = 0, ν = 1,2, . . . .

The following theorem gives cohomological conditions for the existence of radialfoliations:

Theorem 12.5 Let (X ,A) be a germ of strongly exceptional manifold satisfying thecohomological conditions (I) and (II). Then there exists a germ of radial foliation in(X ,A)

The embedding theorem of Grauert [Gr62] states that under the cohomological con-dition (I) on a codimension one embedding there is a neighborhood of A⊂ X whichis biholomorphically equivalent to a neighborhood of the zero section in the normalbundle N to A in X . Combining Theorem ??, Theorem 12.1 and the Blow-up alonga submanufold we obtain the following generalization to any codimension of theembedding theorem of Grauert in [Gra62].

Theorem 12.6 Let (X ,A) be a germ of strongly exceptional manifold satisfying thecohomological conditions (I) and (II). Then, the germ of embedding of A in X isbiholomorphic to the germ of embedding of A in N.

The methods used in this paper give the following generalization of this theorem:

Theorem 12.7 Let F2 be a transverse regular foliation of dimension m+ 1 in agerm of strongly exceptional manifold (X ,A). Assume that (I) and (II) hold. Thenthere is a biholomorphic map (X ,A)→ (N,A), where N is the normal bundle of Ain X, which conjugates F2 with the foliation in (N,A) given by the fibers of N.

12.8 Proof of Theorem 12.5, codimension greater than one

We perform blow-up along A. Recall the notation introduced in Section 10.4. Wewould like to construct a transverse holomorphic foliation in (X , A). This is alreadydone in the previous section. We need the cohomological conditions:

H1(A, N−ν ⊗T A) = 0, H1(A, N−ν) = 0, ν = 1,2, . . . (12.8)

Now, we would like to translate all these in terms of the data of the embeddingA⊂ X . First, note that

H1(A, N−ν)∼= H1(A,N−ν).

We make the tensor product of the sequence (10.7) with Nν and write the long exactcohomology sequence. We conclude that if

H1(A,TPm⊗ N−ν) = 0, H1(A,TA⊗N−ν) = 0, ν = 1,2, . . . .

thenH1(A,T A⊗ N−ν) = 0, ν = 1,2, . . .

12.10 Rational curves 107

Since R1π∗(TPm⊗ N−ν) = 0, ν = 1,2, . . ., we have

H1(A,TPm⊗ N−ν) = H1(A,π∗(TPm⊗ N−ν)).

We write the long exact sequence of (10.6) and conclude that if

H1(A,N⊗N−ν) = 0, H2(A,N−ν+1) = 0, ν = 1,2, . . .

thenH1(A,TPm⊗ N−ν) = 0, ν = 1,2, . . .

Finally we conclude that if

H1(A,N⊗N−ν) = 0, H2(A,N−ν+1) = 0, H1(A,TA⊗N−ν) = 0, ν = 1,2, . . . .

thenH1(A,T A⊗ N−ν) = 0, ν = 1,2, . . . .

12.9 Proof of Theorem 12.7

Using Theorem ??, it is enough to construct a second foliation F1 such that(F1,F2) is a germ of radial bifoliation. In codimension one, we have F1 = F2and so we can assume that m > 0. After performing a blow-up along A our prob-lem is reduced to the following one: Let A be a codimension one submanifold ofX and let F2 be a (m+ 1)-dimensional regular foliation in X transverse to A. Thetransversality implies that F2∩ A is a regular foliation of dimension m in A. In factit is the foliation by the blow up divisors Pm. Its tangent bundle is denoted by TPm

in Section 10.4. We would like to construct a transverse to A foliation F1 of di-mension one such that its leaves are contained in the leaves of F2. The proof isa slight modification of Proposition 12.2 and Proposition 12.3. In both propositionT X |A is replaced with TF2|A and TA is replaced with TPm. In Proposition 12.2, thecohomological condition is

H1(A, N−1⊗TPm) = 0.

which follows from the condition (II).

12.10 Rational curves

Let us restrict to the case in which A is a Riemann surface and N is direct sum ofm+1 line bundles N = L1⊕L2⊕·· ·⊕Lm+1. In this case the Serre duality impliesthat the cohomological condition (I) is equivalent to say that Ω 1⊗Nν and Ω 1⊗Ω 1⊗Nν have no global sections, where Ω 1 is the cotangent bundle of A. We have


Nν =⊕i1+i2+···+im+1=ν , i j≥0 Li11 ⊗Li2

2 ⊗·· ·⊗Lim+1m+1

and so (I) together with the strongly exceptional property follows from

c(Li)< 0, c(Li)< 4−4g, i = 1,2, . . . ,m+1.

In a similar way the condition (II) is equivalent to say that A∼= P1 and:

|c(Li)− c(L j)| ≤ 1, i, j = 1,2, . . . ,m+1.

In this case the decomposition of the normal bundle is automatic and it is calledBirkhoff theorem. From this we obtain as a corollary the following result of Laufer[La81]:

Corollary 12.1 If P1 ⊂ X is strongly exceptional and c(Li) < 0, |c(Li)− c(L j)| ≤1, i, j = 1,2, . . .m+1, where Li’s are line bundles which appear in the decomposi-tion of the normal bundle of A in X, then the germ (X ,P1) is biholomorphic to thegerm (N,P1).

In the case in which the codimension of A in X is greater than one the condition (II)seems to be necessary for our theorem. It imposes conditions on the submanifoldA itself apart from negativity conditions on the normal bundle N. It would be ofinterest to show that, for instance, the Grauert theorem does not hold for Riemannsurfaces of genus greater than zero and codimension greater than one.


The study of Riemman surfaces embedded in two dimensional varieties with self-intersection equal to 0 is not discussed in this text. Arnold’s example [Ar76] seemsto be the first example of such an object in the literature. We have also Ueda’s article[Ue82] and lecture notes of Neeman [Ne89].

For formal pronciple and Artin’s theorem see [Art68] and Chapter 4 of [CM03].It provides us with an alternative proof of Grauert’s main theorem in this text.

Chapter 13Few theorems

13.1 Serre duality

Using Serre duality we can compute some Cech cohomologies easily. Let M be acomplex manifold, T M be its tangent bundle and T ∗M be its cotangent bundle. Thesections of Ω1 = Ω 1

M := T ∗M are called differential forms. The sheaf of differentialp-forms is the sheaf of sections of the vector bundle

Ωp = Ω

1∧Ω1∧·· ·∧Ω

1︸︷︷︸p times

Theorem 13.1 (Serre Duality) Let M be a complex manifold of complex dimen-sion n and V a holomorphic vector bundle over M. Then there exists a naturalC-isomorphism

Hq(M,Ω p⊗V )∼= (Hn−q(M,Ω n−p⊗V ∗))∗

For a proof of this theorem the reader is referred to [Ra65]. It is useful to considerthe following special cases:

1. V is the trivial line bundle

Hq(M,Ω p)∼= (Hn−q(M,Ω n−p))∗

The numbershq,p := dimC Hq(M,Ω p)

are called Hodge numbers and so

hq,p = hn−q,n−p.

2. q = n and p = 0.Hn(M,V )∼= (H0(M,Ω n⊗V ∗))∗

109

110 13 Few theorems

Note line bundle Ω n is called the canonical bundle of M. In particular, if rankV =dimM and the zero section of W := Ω n⊗V ∗ has a negative self intersection thenW does not have a global holomorphic section and so

Hn(M,(Ω n)∗⊗W ∗) = 0.

13.2 Birkhoff-Grothendieck theorem

In the previous section we have seen that if a vector bundle over a Riemann surfaceA is decomposed into a sum of line bundles then many cohomological conditionson V can be computed explicitly. If A is arational curve then this decomposition isautomatic and it follows from:

Theorem 13.2 (Birkhoff-Grothendieck) Every holomorphic bundle V of rank k overthe projective space P1 of dimension one is a direct sum of k line bundles:

V ∼= L1⊕L2⊕·· ·⊕Lk.

Note that by GAGA principle every holomorphic vector bundle over a projectivevariety is the complex vector bundle of an algebraic bundle.

Chapter 14Remmert reduction

Given a topological space T . We denote by CT the sheaf of continuous complexvalued functions on T . Let us be given a variety X and an equivalence relation Ron X . Let φ : X → X/R be the canonical map. We can define the sheaf OX/R ofC-algebras on X/R as follows: The data

U → f ∈ CX/R(U) | f φ ∈ OX (φ−1(U)), U an open subset of X/R

form the sheaf OX/R. In this section we want to answer the following question:When (X/R,OX/R) is an analytic variety? By definition of OX/R if (X/R,OX/R) isan analytic variety then φ : X → X/R is a holomorphic mapping. Cartan’s article[Ca60] is the main source for this section.

14.1 Proper mapping and direct image theorems

Theorem 14.1 (Remmert proper mapping theorem [Re57]) If f is a proper holo-morphic mapping of a variety X into a variety Y then the image f (X) is a subvarietyof Y .

Definition 14.1 The direct image f∗S is defined as follows: f∗S is the sheaf as-sociated to the presheaf U → OX ( f−1(U)), for open sets U in Y . One can definehigher order direct images Rµ f∗S ,µ ≥ 0 as the sheaf associated to the presheaf

U → Hµ( f−1(U),S )

Theorem 14.2 (Grauert direct image theorem [Gr60]) Let f be a proper holomor-phic mapping of a variety X into a variety Y . If S is a coherent sheaf on X thenRµ f∗S ,µ ≥ 0 is a coherent analytic sheaf on Y .

The reader is referred to [GrRe84] for the proof of the above classical theorems andtheir applications.

111

112 14 Remmert reduction

Exercise 14.1 Why in the above theorems we need f to be a proper? Construtcounter examples.

14.2 Equivalence relations in varieties

Let X ,Y be two varieties and f : X → Y be a holomorphic map. We can define theequivalence relation R f in X as follows:

∀x,y ∈ X , xR f y if and only if f (x) = f (y)

Theorem 14.3 If f : X→Y is a proper holomorphic map then (X/R f ,OX/R f ) is ananalytic variety.

Proof. By Remmert proper mapping theorem we can assume that f is a surjectivemap and then we can identify Y with X/R f pointwise. By this identification we de-note OX/R f by S . The structural sheaf OY of Y is a subsheaf of S . For a momentsuppose that S is a coherent (OY -module) sheaf. A part of the definition of a co-herent sheaf is the following: For every point y′ ∈ Y there is an open neighborhoodU of y′ in Y and sections s1,s2, . . . ,sk of S (U) such that s1y,s2y, . . . ,sky generateSy as a OY,y-module for all y ∈U . Now by definition fi := si f ’s are holomorphicfunctions on V := f−1(U). Define the map

g : V →U×Ck, g(x) = ( f (x), f1(x), f2(x), . . . , fk(x))

g is a proper holomorphic mapping and so we can apply Remmert proper mappingtheorem and obtain a subvariety Z := g(V ) of U ×Ck. Now the map f : V → Udecomposes into

Vg→ Z h→U

where h is the projection on the first coordinate and so it is a holomorphic map.Since the fi’s are constant along the fibers of f , h is a one to one map. Therefore wecan identify Z with U through h. By this identification, one can easily see that S onZ is nothing but the structural sheaf OZ of Z. We have proved that (U,OX/R f |U ) isisomorphic to the variety Z.

Now it remains to prove that S is a coherent sheaf on Y . Let T be the analyticvariety in X ×X given by the inverse image of the diagonal of Y ×Y by the mapf × f : X ×X → Y ×Y and πi : T → X , i = 1,2 be the projections on the first andsecond coordinates. We have a diagram

Tπi→ X

g ↓ fY

(14.1)

14.3 Cartan’s theorem 113

where g = f π1 = f π2. Now the maps πi, i = 1,2 induce the maps π∗i : OX →OTand so the maps

αi∗ : f∗OX → g∗OT , i = 1,2

and we haveS = ker(α1∗−α2∗)

To prove this equality, take an open set U in Y and r a holomorphic function onf−1(U). If r is constant on the fibers of f (in the case where f has disconnectedfibers this statement cannot be derived from the fact that r is holomorphic and f isproper) then α1∗(r) = α2∗(r). If α1∗(r) = α2∗(r) then the definition of T impliesthat r is constant on the fibers of f and so it is a section of S on U .

By Grauert direct image theorem g∗OT and f∗OX are coherent sheaves and so Sis a coherent sheaf.

Exercise 14.2 Give an example of two analytic variety X and Y and an analyticmap f : X → Y such that f is one to one and surjective but f−1OY & OX .

Now let us consider a family of proper holomorphic mappings fi : X → Yi, i ∈ I,where I is an index set. One can define the equivalence relation RI on X as follows:

∀x,y ∈ X , xRIy if and only if fi(x) = fi(y) ∀i ∈ I

In the case where I is finite the pair (X/RI ,OX/RI ) is an analytic variety becauseRI = R fI , where

fI := Πi∈I fi : X → YI , YI := Πi∈IYi

For an infinite family of holomorphic functions we have the following proposition:

Proposition 14.1 Let X and Yi, i ∈ I be varieties and fi : X →Yi, i ∈ I holomorphicfunctions. For any compact subset K of X there is a finite subset J ⊂ I such that RIand RJ induce the same relation on K.

Proof. For a finite set J ⊂ I let ∆J be the subset of X×X given by the inverse imageof the diagonal of YJ ×YJ by fJ × fJ : X ×X → YJ ×YJ . Each ∆J is a subvariety ofX ×X and if J ⊂ J′ be finite subsets of I then ∆J′ ⊂ ∆J . Such a family of varietiesbecomes constant on a given compact subset K of X×X . Take a point p ∈ K. Sincein the family ∆J the dimension of ∆J around p cannot drop infinitely many times,our claim is true locally. One can cover K by finitely many small open sets and getthe assertion for K.

Exercise 14.3 Give a counterexample to Proposition 14.1 whithout the compact-ness of K.

14.3 Cartan’s theorem

Let us first state a definition.


Definition 14.2 The equivalence relation R on a variety is called a proper equiv-alence relation if for any compact set K ⊂ X the K-saturated set, i.e. the union ofR-equivalence classes cutting K, is compact.

For a proper equivalence relation the set X/R is locally compact, X/R is Hausdorffand the continuous map X → X/R is proper.

Theorem 14.4 (H. Cartan [Ca60]) Let R be a proper equivalence relation on avariety X with the following property: Each point of x ∈ X/R has an open neigh-borhood U such that OX/R(U) separates the points of U, i.e. for any two pointsx1,x2 ∈U there is f ∈ OX/R(U) such that f (x1) 6= f (x2). Then (X/R,OX/R) is ananalytic variety.

Proof. Let U ⊂ X/R be the open set introduced in the theorem. Since OX/R(U)separates the points of U , the equivalence relation RI defined by the family I =φ ∗OX/R(U) in φ−1(U) is R. Therefore if U ′ is a relatively compact open subsetof U containing y, then by Proposition 14.1 and Theorem 14.3 (U ′,OX/R |U ′) is avariety.

Now as an application of Theorem 14.4 we state and prove Stein factorization andRemmert reduction theorems.

14.4 Stein factorization

Theorem 14.5 (Stein factorization) Let f : X →Y be a proper holomorphic map ofvarieties. Then there exist a variety Z and holomorphic maps

Xg→ Z h→ Y

such that

1. f = hg,2. h is a finite map,3. g∗OX = OZ .

The triple (h,g,Z) with properties 1,2 and 3 satisfies:

4. g has connected fibers5. It is unique up to biholomorphism, i.e. for any other triple (h′,g′,Z′) with the

properties 1,2,3 of the theorem there is a biholomorphic map a : Z → Z′ suchthat g′ = ag and h′ = ha−1.

Proof. We define the equivalence relation R in X as follows: For all x,y∈ X we havexRy if and only if f (x) = f (y) and x and y are in the same connected component off−1( f (x)). A simple topological argument shows that R is a proper equivalence

relation. The map f decomposes into Xg→ X/R h→ Y , where g and h are continu-

ous maps. For a Stein small open set U in Y , OY (U) separates the point of U and

14.5 Remmert reduction 115

h∗OY (U) ⊂ OX/R(h−1(U)). Therefore R satisfies the condition of Theorem 14.4and so Z := (X/R,OX/R) is a variety and g is a holomorphic map. The map h is alsoholomorphic because h∗OY ⊂ OZ . Since f is proper, a fiber of f has finitely manyconnected components and so h is a finite map. The condition 3 is true by definitionof OX/R.

Assume that g−1(x) is not connected and has two connected components A andB. In an open neighborhood of g−1(x) we can define a two valued function whichtakes 1 in a neighborhood of A and 0 in a neighborhood of B. This function is nota pullback of any holomorphic function in a neighborhood of x in Y , which is acontradiction with 3. The property 1,2 and 3 imply that the points of Z′ are in oneto one correspondence with connected components of the fibers of f . Therefore wehave a one to one map a : Z→ Z′. It can be easily seen that a is the desired map for5.

Remark 14.1 Let f : X → Y be a surjective proper holomorphic map of varietieswith f∗OX = OY . The argument which we used for 4. of Theorem 14.5 implies thatf has connected fibers. For any open set U ⊂ Y and a holomorphic function r inφ−1(U) there exists a holomorphic function s in U such that r = s f .

14.5 Remmert reduction

Theorem 14.6 (Remmert reduction [Re56]) Let X be a holomorphically convexspace. Then there exist a Stein space Y and a proper surjective holomorphic mapφ : X → Y such that

1. φ∗OX = OY .

Moreover the fact that Y is Stein and 1 imply

2. φ has connected fibers3. The map φ ∗ : OY (Y )→ OX (X) is an isomorphism4. The pair (φ ,Y ) is unique up to biholomorphism, i.e. for any other pair (φ ′,Y ′)

with Y ′ Stein and property 1, there is a biholomorphism a : Y → Y ′ such thatφ ′ = aφ .

Proof. Let R = RI be the relation in X defined by the family I = OX (X). For acompact set K in X the set ∪x∈KRx is closed and contained in the convex hull of Kin X . Since X is holomorphically convex, this means that R is a proper equivalencerelation. It satisfies also the condition of Theorem 14.4. Therefore (X/R,OX/R) isa variety. By definition X/R is holomorphically convex and holomorphic functionson X/R separate the points of X/R. This means that Y := X/R is a Stein variety.The canonical map φ : X → Y is the desired map. It is enough to prove that φ hasconnected fibers. If a fiber of φ has two connected components A and B then wecan use Stein factorization and obtain a holomorphic function f on X such thatf (A) 6= f (B). But this means that A and B are two distinct equivalence classes of Rwhich is a contradiction.


For any other pair (φ ′,Y ′) the existence of a bijective map a : Y → Y ′ followsfrom the fact that φ and φ ′ have the same fibers. The property 1. and remark afterTheorem 14.5 proves that a is a biholomorphism.


There are many contributions to complex analysis which are concerned with thefollowing problem: When the quotient space of an equivalence relation in a complexspace is again a complex space. Grauert’s direct image theorem plays an importantrole in these works. For a more detailed study in this direction we recommend thearticle [Gr83] and its references.

It would be nice if the proofs of Remmert proper mapping theorem and Grauertdirect image theorem would be discussed along the study of this text. These proofsand more applications of these classical theorems can be found in [GrRe84]

Chapter 15Formal and finite neighborhoods

Recall the notations in Section 9.1 and

• A(∗) := OX |A, A(∗) is called the neighborhood sheaf of A;• A(ν) := OX/M ν |A, A(ν) is called the ν-neighborhood of A. A(1) is the structural

sheaf of A;• M(ν) := M /M ν |A.

Exercise 15.1 Verify the details of the following statements:

1. We havenil(A(ν)) := x ∈ A(ν) | ∃n ∈ N,xn = 0= M /M ν (15.1)

2.Q(ν−1) = x ∈ A(ν) | x.nil(A(ν)) = 0 (15.2)

3. and a canonical short exact sequence

0→Q(ν−1)→ A(ν)→ A(ν−1)→ 0

Let us be given two embeddings A → X , A → X ′. If we denote the image of thefirst embedding by A and the second by A′ we have a natural biholomorphism

φ : A′→ A

which gives us an isomorphism

φ(1) : A(1)→ A′(1) (15.3)

This isomorphism is fixed from now on. We always assume that the pairs (X ,A)and (X ′,A′) have the same local structure, i.e. for any a′ ∈ A′ and its correspondinga = φ(a′) ∈ A there is a local biholomorphism

(X ′,A′,a′)→ (X ,A,a)

Notations related to A′ will be written by adding ′ to the notations of A.

117

118 15 Formal and finite neighborhoods

15.1 Formal and finite neighborhoods

The natural inclusions

· · · ⊂M ν+1 ⊂M ν ⊂M ν−1 ⊂ ·· · ⊂M

give us the natural chain of canonical functions:

· · · π→ A(ν+1)π→ A(ν)

π→ A(ν−1)π→ ··· π→ A(1)

Definition 15.1 We define

A(∞) := lim∞←ν A(ν)

In other words, every element of A(∞) is given by a sequence

. . . , fν+1, fν , fν−1, . . . , f1 fv ∈ A(ν)

π( fν+1) = fν

The C-algebra structure of A(∞) is defined naturally. A(∞) is called the formal neigh-borhood of A or the formal completion of X along A.

There exists a natural canonical homomorphism

A(∗)→ A(ν)

which extends to the inclusionA(∗) → A(∞)

Define in the setN= 1,2,3, · · · ,∞,∗

the order1 < 2 < 3 < · · ·< ∞ < ∗

we conclude that for any pair µ,ν ∈ N,µ ≤ ν there exists a natural homomorphism

π : A(ν)→ A(µ)

If no confusion is possible, we will not use any symbol for the homomorphismsconsidered above.

Remark 15.1 Let us analyze the global sections of the above sheaves. Every globalsection of A(∗) is a holomorphic function in a neighborhood of A. Let g be a globalsection of A(ν),ν < ∞. We can choose a collection of local charts Uαα∈I in Xcovering A and holomorphic functions gα in Uα such that g = gα in the sheaf A(ν).This means that

15.2 Some propositions 119

gα −gβ ∈M ν |Uα∩Uβ, α,β ∈ I

Conversely, every collection of gαα∈I satisfying the above conditions defines aglobal section of A(ν).

Definition 15.2 Let µ,ν ∈ N,µ ≤ ν . We say that the homomorphism φ(ν) : A(ν)→A′(ν) induces the homomorphism φ(µ) : A(µ) → A′(µ), if the following diagram iscommutative:

A(ν)

φ(ν)→ A′(ν)↓ ↓

A(µ)

φ(µ)→ A′(µ)

(15.4)

We also say that A(ν)→ A′(ν) extends A(µ)→ A′(µ).

Q(1) is the set of nilpotent elements of A(2) and so every homomorphism (isomor-phism) φ(2) : A(2) → A′(2) induces a homomorphism (isomorphism) φ() : Q(1) →Q(1). We also say that φ(2) extends φ().

Definition 15.3 The homomorphism φ(∞) : A(∞) → A′(∞) is called convergent if ittakes A(∗) into A′(∗).

15.2 Some propositions

Proposition 15.1 Every homomorphism (isomorphism) φ(ν) : A(ν)→ A′(ν),2≤ ν <

∞ induces natural homomorphisms (isomorphisms)

A(µ)→ A′(µ),µ ≤ ν

Proof. It is enough to prove our claim for µ = ν − 1. For an arbitrary µ onecan repeat the argument for the pair ν − 1,ν − 2 and so on. The kernel of π :A(ν)→ A(ν−1) is Q(ν−1) and Q(ν−1) is has the property (15.2). Therefore φ(ν) sendsQ(ν−1) to Q(ν ′−1). This implies that φ(ν) induces the desired map A(ν)/Q(ν−1)→A′(ν)/Q(ν ′−1), because A(ν−1) = A(ν)/Q(ν−1) and A′(ν−1) = A′(ν)/Q(ν ′−1).

The following proposition gives us the local information for analyzing a homo-morphism φ(ν) : A(ν)→ A′(ν), ν ∈ N.

Proposition 15.2 Let a ∈ A and U be a small neighborhood of a in A. Let alsoa′ = φ−1(a) and U ′ = φ−1(U). The following statements are true:

1. Every homomorphism (isomorphism) φ(∗) : A(∗) |U→ A′(∗) |U ′ which induces anisomorphism A(1) |U→ A′(1) |U ′ is induced by a unique holomorphic (biholomor-phic) map (X ′,A′,a′)→ (X ,A,a);

2. Every homomorphism φ(∗) : A(∗) |U→A′(∗) |U ′ which induces isomorphisms A(1) |U→A′(1) |U ′ and A(2) |U→ A′(2) |U ′ is an isomorphism also;


3. Every homomorphism (isomorphism) φ(ν) : A(ν) |U→ A′(ν) |U ′ , 2 ≤ ν < ∞ is in-duced by a homomorphism (isomorphism) A(∗) |U→ A′(∗) |U ′ .

In the case where a is a regular point of both A and X , the proof of this proposition iseasy. The proof in general uses simple properties of local rings and their homomor-phisms. The reader is referred to [Nag62] for more informations about local ringtheory.

Proof. By Proposition 2.5 the homomorphism φ(∗) : A(∗)a→ A′(∗)a′is induced by a

unique map (X ′,a′)→ (X ,a). We must prove that this map takes A′ to A. Since φ(∗)induces an isomorphism A(1) |U→ A′(1) |U ′ , it takes the ideal of A in X to the ideal ofA′ in X ′. This implies that (X ′,A′,a′)→ (X ,A,a).

The second and third statements have a completely algebraic nature. To provethem we use the following notations

R := A(∗)a∼= A′(∗)a′

, I := MA,a,∼= MA′,a′ , τ := φ(∗), τν := φ(ν), ν ∈ N

( Note that (X ,A) and (X ′,A′) have the same local structure). Let us prove the secondstatement. Since τ2 : R/I2→ R/I2 is an isomorphism and the nilpotent set of R/I2

is the set I/I2, we have I = τ(I)+ I2. Let us prove that τ(I) = I. Put

R′ := I/τ(I)

We have IR′ = R′. Let a1,a2, . . . ,ar be a minimal set of generators for R′. We havear ∈ R′ = IR′ and so

ar =r

∑i=1

siai, si ∈ I

or (1−sr)ar lies in the ideal generated by a1,a2, . . . ,ar−1. Since 1−sr is a holomor-phic function in (X ,a) and its value in a is 1 it is invertible and so we get a contra-diction with this fact that no proper subset of a1,a2, . . . ,ar generates R′ (The usedargument is similar to the proof of Nakayama’s lemma (see [GuII90] A, Lemma9)).

We have proved that τ(I) = I. Since τ1 : R/I→R/I is an isomorphism and τ(I) =I, τ is surjective. Now let us prove that τ is injective. Define

Rn := x ∈ R | τn(x) = 0

τ induces a map from Rn to itself and the image of this map contains Rn−1 Since R isa Noetherian ring and we have an increasing sequence of ideals · · · ⊂ Rn ⊂ Rn+1 ⊂·· · , there is a natural number n0 such that Rn0 = Rn0+1 = · · ·= R∗. Now τ∗ = τ |R∗ isa surjective map from R∗ to R∗. But by definition of R∗, τ∗ must be zero. ThereforeR∗ = 0 and so R1 = 0. This means that τ is injective.

Now let us prove the third statement. Let x1,x2, . . . ,xn form a basis for the vectorspace MR

M 2R

, where MR denotes the maximal ideal of R. We have seen in Proposition

2.6 that (x1,x2, . . . ,xn) form an embedding of (X ,a) in (Cn,0). We can choose el-ements f1, f2, . . . , fn in R such that τν([xi]) = [ fi], i = 1,2, . . . ,n, where [.] denotes

15.4 Obstructions to formal isomorphism 121

the equivalence class. Now it is easy to verify that the homomorphism

τ : R→ R

f (x1,x2, . . . ,xn)→ f ( f1, f2, . . . , fn)

induces the desired map. If τν is an isomorphism then by the second part of theproposition τ is also an isomorphism.

15.3 Geometric interpretation

Now using Proposition 15.2 we can find geometrical interpretations of homomor-phisms A(µ)→ A′(µ), µ ∈ N as follows

1. There exists an isomorphism φ(∗) : A(∗) → A′(∗) if and only if there exists a bi-holomorphism of some neighborhood of A into some neighborhood of A′ in X ′

extending φ : A→ A′;2. Any isomorphism φ(ν) : A(ν) → A′(ν),1 < ν ∈ N is given by a collection of bi-

holomorphisms (Uα ,A)→ (U ′α ,A′), where Uαα∈I ( resp. U ′αα∈I) is an open

covering of A (resp. A′) in X (resp. X ′), and such that φα φ−1β

is the identity upto holomorphic functions vanishing on A of order ν ;

The first statement justifies the name neighborhood sheaf adopted for A(∗).Unfortunately an isomorphism

φ(∞) : A(∞)→ A′(∞) (15.5)

may not be given by a collection of isomorphisms φ(ν) : A(ν)→A′(ν),ν ∈N such thatfor ν ≥ µ , φ(ν) extends φ(µ). However, the φ(∞) which we will construct in the nextsection will have this property. For this reason when we talk about an isomorphism(15.5) we assume that it induces isomorphisms in finite neighborhoods.

15.4 Obstructions to formal isomorphism

In this section we will identify the obstructions for the existence of an isomorphismbetween formal neighborhoods of A and A′. We formulate our main problem in thissection as follows: Let A′ be the image of another embedding of A in a manifold X ′.

1. Given an isomorphism φ : Q(1) →Q(1)′. Under which conditions is it induced

by an isomorphism φ(2) : A(2)→ A′(2)?2. Given an isomorphism φ(ν) : A(ν)→ A′(ν),ν ≥ 2. Under which conditions does it

extend to φ(ν+1) : A(ν+1)→ A′(ν+1)?


In other words we want to describe the germ of an embedding A → X with minimaldata. The first elementary data of an embedding is its normal bundle (when A is notsmooth the sheaf Q(1) = M /M 2 plays the role of the normal bundle). The otherdata of an embedding are its finite neighborhoods.

Note that if all such conditions in the above questions are satisfied for A and A′,we get only an isomorphism of formal neighborhoods of A and A′. The applied meth-ods are quite formal and can be found in [Gr62, HiRo64, La71]. In what follows,every homomorphism A(ν) → A′(ν),ν ∈ N which we consider will be an extensionof the fixed isomorphism (15.3) (Note that A(1) is the structural sheaf of A).

Let a∈ A and a′ = φ−1(a) be its corresponding point in A′. The stalk of the sheafA(ν),ν ∈ N at a is denoted by A(ν)a. Any isomorphism

φ(ν)a : A(ν)a→ A′(ν)a′(15.6)

determines an isomorphism between A(ν) |Ua and A′(ν) |Ua′ , where Ua and Ua′ aretwo open neighborhood of a and a′ in A and A′, respectively (see Proposition 15.2).

The following proposition gives us the local solutions of our problem:

Proposition 15.3 Any isomorphism φ(ν)a : A(ν)a→ A′(ν)a′is induced by an isomor-

phismφ(∗)a : A(∗)a→ A(∗)a′ (15.7)

and hence extends toφ(ν+1)a : A(ν+1)a→ A(ν+1)a′ (15.8)

Proof. The above proposition is the third part of Proposition 15.2 in another form.Note that the isomorphism φ(∗) : A∗,a→ A′∗,a′ is not unique.

In the introduction of [GrRe84] we find the following statement of H. Cartan: lanotion de faisceau s’introduit parce qu’il s’agit de passer de donnees locales al’etude de proprietes globales. Like many other examples in complex analysis, theobstructions to glue the local solutions lie in a first cohomology group of a sheafover A. The precise identification of that sheaf and its first cohomology group is ourmain objective in this section.

Now, let us be given an isomorphism φ(ν) : A(ν)→ A′(ν). We want to extend φ(ν)

to φ(ν+1) : A(ν+1) → A′(ν+1), i.e. to find an isomorphism φ(ν+1) : A(ν+1) → A′(ν+1)such that the following diagram is commutative:

A(ν+1)φ(ν+1)→ A′(ν+1)

↓ ↓

A(ν)

φ(ν)→ A(ν)

(15.9)

Proposition 15.3 gives us the local solutions

15.4 Obstructions to formal isomorphism 123

A(ν+1)a

φ(ν+1)a→ A′(ν+1)a′↓ ↓

A(ν)a

φ(ν)a→ A′(ν)a′

(15.10)

where A(ν)a is the stalk of the sheaf A(ν) over the point a. Now, cover A with smallopen sets for which we have the diagrams of the type (15.10). Combining two dia-grams in the intersection of neighborhoods of the points a and b we get:

A(ν+1)a,b

φ(ν+1)a,b→ A(ν+1)a,b↓ ↓

A(ν)a,bid→ A(ν)a,b

(15.11)

whereφ(ν+1)a,b = φ(ν+1)

−1a φ(ν+1)b (15.12)

Note that we have used the notation φ(ν+1)a,b instead of φ(ν+1) |Ua∩Ub , φ(ν+1)a in-stead of φ(ν+1) |Ua and so on. The above transition elements are obstruction to ourextension problem. Now it is natural to define the following sheaf:

Aut(ν) is the sheaf of isomorphisms φ(ν+1) : A(ν+1)→ A(ν+1) inducing the iden-tity in A(ν), i.e. the following diagram is commutative

A(ν+1)φ(ν+1)→ A(ν+1)

↓ ↓A(ν)

id→ A(ν)

(15.13)

Later in Proposition 15.5 we will see that Aut(ν) is a sheaf of Abelian groups. Nowthe data in (15.12) form an element of

H1(A,Aut(ν))

The elements of H1(A,Aut(ν)) are obstructions to the extension problem.It is clear that the case ν = 1 needs an special treatment. A(1) is the structural

sheaf of A and the condition H1(A,Aut(1)) = 0 means that any two embeddings ofA have the same 2-neighborhood and in particular have isomorphic M /M 2’s. Thisimplies that the normal bundles of A and A′ are isomorphic! Therefore, the definitionof Aut(1) is not useful. We modify this definition as follows:

Aut(1) is the sheaf of isomorphisms φ(2) : A(2) → A(2) inducing the identity onM /M 2 and for which the following diagram is commutative

A(2)φ(2)→ A(2)

↓ ↓A(1)

id→ A(1)

(15.14)


Proposition 15.4 If H1(A,Aut(ν)) = 0 then any isomorphism

1. φ(ν) : A(ν)→ A′(ν) if ν > 12. φ() : Q(1)→Q(1)

′ if ν = 1

extends to an isomorphism φ(ν+1) : A(ν+1)→ A′(ν+1).

Proof. The obstruction to the above extension is obtained by diagram (15.11) andso is an element of H1(A,Aut(ν)).

15.5 Calculating the obstruction

Now we have to identify Aut(ν) and especially we have to verify when H1(A,Aut(ν))=0 is satisfied.

Proposition 15.5 Suppose that X is a smooth variety. For ν ≥ 2 we have

Aut(ν)∼= T (ν)(:= T ||A⊗OA Q(ν))

where T is the sheaf of holomorphic vector fields in X (sections of the tangentbundle of X); for the case ν = 1 we have

Aut(1)∼= TA(1)(:= TA||A⊗OA Q(1))

where TA is the sheaf of holomorphic vector fields in X tangent to A.

Proof. Let us introduce the function which will be our candidate for the desiredisomorphisms. First consider the case ν ≥ 2.

∗ : T (ν)→ Aut(ν)

For any ψ ∈T (ν) define

β ,β ′ : A(ν+1)→ A(ν+1)

β ( f ) = f +ψ.d f

β′( f ) = f −ψ.d f

we have

β β′( f ) = f −ψ.d f +ψd( f −ψ.d f ) = f −ψ.d(ψ.d f ) = f mod M 2ν−1

We have 2ν−1≥ ν +1 and so

β β′( f ) = f mod M ν+1 (15.15)

15.6 Breaking Aut(ν) 125

In other words β ′ is the inverse function of β . We define

∗(ψ) = β

Now it is enough to prove that ∗ is the desired isomorphism. Since X is nonsingular∗ is injective. Let β ∈ Aut(ν). We write

β ( f )− f = ψ′( f )

ψ ′( f ) = 0 mod M ν and so ψ ′ ∈ Hom(A(ν+1),Mν/M ν+1). Composing with

A(∗)→ A(ν+1) and without change in notations we can assume

ψ′ ∈ Hom(A(∗),M

ν/M ν+1)

Here Hom is the homomorphisims of abelian sheaves. We will use the fact that ψ ′

satisfies the Leibniz rule:

ψ′( f1 f2) = f1ψ

′( f2)+ f2ψ′( f1), f1, f2 ∈ A(∗). (15.16)

Let z1,z2, . . . ,zn be local coordinates. Define the vector field ψ by

ψ(dzi) = ψ′(zi)

Then ψ ∈ T (ν) and the mapping β → ψ is the inverse of ∗. This follows from thefact that both ψ ′ and ψd() coincide on zi’s and both satisfy the Leibniz rule.

The case ν = 1 is the same as previous one. We need to substitute TA for T toget the congruency (15.15). In a local chart U we write A= g1 = g2 = · · ·= gr = 0and ψ = ∑

ri=1 giψi, ψi ∈TA(U). We have

ψ.d(ψ.d f ) = ψ.(r

∑i=1

gi(ψi.d f ))

=r

∑i=1

dgi(ψ)(ψi.d f )+gid(ψi.d f )(ψ) ∈M 2

15.6 Breaking Aut(ν)

How can we calculate the cohomology groups H1(A,T (ν))? To do this, we breakT (ν) into two other simple sheaves as follows:There is a natural short exact sequence

0→TA→T →Q(1)∗→ 0

By tensorial multiplication over OA with Q(ν), we have


0→TA(ν)→T (ν)→Q(ν−1)→ 0

This gives us the long exact sequence

. . .→ H1(A,TA(ν))→ H1(A,T (ν))→ H1(A,Q(ν−1))→ . . .

We summarize the above arguments in the following proposition:

Theorem 15.1 If H1(A,TA(ν)) = 0 and H1(A,Q(ν−1)) = 0 then H1(A,T (ν)) = 0and so any isomorphism

φ(ν) : A(ν)→ A′(ν) if ν > 1

φ : Q(1)→Q(1)′ if ν = 1

extends to an isomorphism φ(ν+1) : A(ν+1)→ A′(ν+1).

15.7 The case of a Riemann surface

From now on we do not use the line under bundles (it denotes the sheaf of sections),for instance instead of H1(A,Ω 1) we write H1(A,Ω 1). Let A be a Riemann surface.Putting p = 0, q = 1 in Serre duality (see Chapter 13.1) we have

H1(A,V )∼= (H0(A,Ω 1⊗V ∗))∗

NowH1(A,Q(ν−1)) = H1(A,(N∗)ν−1)) = (H0(A,Ω 1⊗Nv−1))∗

Ω 1⊗Nv−1 has no global holomorphic section if

c(Ω 1⊗Nν−1) = 2g−2+(ν−1)A.A < 0 (15.17)

In the same way

H1(A,TA(ν)) = (H0(A,Ω 1⊗ (TA)∗⊗Nν))∗ = (H0(A,Ω 1⊗Ω1⊗Nν))∗ = 0

ifc(Ω 1⊗Ω

1⊗Nν) = 2(2g−2)+νA.A < 0 (15.18)

Finally we conclude that

Theorem 15.2 Let A be a Riemann surface of genus g embedded in a two dimen-sional manifold X. Suppose that

• A.A≤ 0 if g = 0;• A.A < 2(2−2g) if g≥ 1

Then the embedding A → X is formally equivalent with A′ → X ′, where the normalbundle of A′ in X ′ equals the normal bundle of A in X.

15.8 Complementary notes 127

Proof. Since the normal bundle of A′ in X ′ equals the normal bundle of A in X , thereexists an isomorphism φ() : Q(1)→Q(1)

′. To extend this isomorphism to a formalisomorphism of the neighborhoods of A and A′ in X and X ′, respectively, we musthave the inequalities (15.18) for all ν ≥ 1 and (15.17) for all ν > 1 satisfied. Thisimplies exactly A.A < 0 if g = 0 and A.A < 2(2−2g) if g≥ 1.


One can use [Gri66] in section 15.4 for more extension problems such as the exten-sion of fiber bundles, holomorphic maps and cohomology elements.


References

AGZV88. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singularities of differentiablemaps. Monodromy and asymptotics of integrals Vol. II, volume 83 of Monographs inMathematics. Birkhauser Boston Inc., Boston, MA, 1988.

BT82. Raoul Bott and Loring W. Tu. Differential forms in algebraic topology, volume 82 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1982.

GH94. Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Wiley ClassicsLibrary. John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.

GR65. Robert C. Gunning and Hugo Rossi. Analytic functions of several complex variables.Prentice-Hall Inc., Englewood Cliffs, N.J., 1965.

Gra62. Hans Grauert. Uber Modifikationen und exzeptionelle analytische Mengen. Math.Ann., 146:331–368, 1962.

Gro66. Alexander Grothendieck. On the de Rham cohomology of algebraic varieties. Inst.Hautes Etudes Sci. Publ. Math., (29):95–103, 1966.

Kod86. Kunihiko Kodaira. Complex manifolds and deformation of complex structures, volume283 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York,1986. Translated from the Japanese by Kazuo Akao, With an appendix by DaisukeFujiwara.

Mov16. Hossein Movasati. A course in Hodge theory:with emphasis on multiple integrals. Available at:http://w3.impa.br/∼hossein/myarticles/hodgetheory.pdf .2016.

Rem95. Reinhold Remmert. Complex analysis in “Sturm und Drang”. Math. Intelligencer,17(2):4–11, 1995.

Voi02. Claire Voisin. Hodge theory and complex algebraic geometry. I, volume 76 of Cam-bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,2002. Translated from the French original by Leila Schneps.

Voi03. Claire Voisin. Hodge theory and complex algebraic geometry. II, volume 77 of Cam-bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,2003. Translated from the French by Leila Schneps.

References

An80. Ancona, V. Sur l’quivalence des voisinages des espaces analytiques contractibles. Ann.Univ. Ferrara Sez. VII (N.S.) 26 (1980), 165–172 (1981).

Ar76. Arnold, V. I. Bifurcations of invariant manifolds of differential equations, and normal formsof neighborhoods of elliptic curves. Funkcional. Anal. i Prilo zen. 10 (1976), no. 4, 1–12.English translation: Functional Anal. Appl. 10 (1976), no. 4, 249–259 (1977).

Art62. Artin, M. Some numerical criteria for contractability of curves on algebraic surfaces.Amer. J. Math. 84 1962 485–496.

Art68. Artin, M. On the solutions of analytic equations. Invent. Math. 5 1968 277–291.BDLD79. Becker, J.; Denef, J.; Lipshitz, L.; van den Dries, L. Ultraproducts and approximations

in local rings. I. Invent. Math. 51 (1979), no. 2, 189–203.BT82. Bott, R.; Tu, L. W. Differential forms in algebraic topology. GTM, 82. Springer-Verlag,

New York-Berlin, 1982.Br02. Brunella M., Birational Geometry of Foliations.Ca60. Cartan, H. Quotients of complex analytic spaces. 1960 Contributions to function theory

(Internat. Colloq. Function Theory, Bombay, 1960) pp. 1–15 Tata Institute of FundamentalResearch, Bombay.

References 129

Ca88. Camacho, C. Quadratic forms and holomorphic foliations on singular surfaces. Math. Ann.282 (1988), no. 2, 177–184.

Ca00. Camacho, C. Dicritical singularities of holomorphic vector fields. Laminations and folia-tions in dynamics, geometry and topology, 39–45, Contemp. Math., 269, Amer. Math. Soc.,Providence, RI, 2001.

CM03. Camacho, C. Movasati, H. Sad, P. Neighborhoods of analytic varieties. Monografıas delInstituto de Matematica y Ciencias Afines, 35. Instituto de Matematica y Ciencıas Afines,IMCA, Lima, 2003.

CMS02. Camacho, C. Movasati, H. Sad, P. Fibered neighborhoods of curves in surfaces, Journalof Geometric Analysis, Vol. 13 No. 1.

CeMa82. Cerveau, D. ; Mattei, J.-F. Formes integrables holomorphes singulieres. Asterisque, 97.SocieteMathematique de France, Paris, 1982. 193 pp.

FrGr02. Fritzsche, Klaus; Grauert, Hans. From holomorphic functions to complex manifolds.Graduate Texts in Mathematics, 213. Springer-Verlag, New York, 2002.

Gr60. Grauert, H. Ein Theorem der analytischen Garbentheorie und die Modulraume komplexerStrukturen. Inst. Hautes Etudes Sci. Publ. Math. No. 5, 1960 64 pp. 57.00

Gr62. Grauert, H. Uber Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 1461962 331–368.

Gr83. Grauert, H. Set theoretic complex equivalence relations. Math. Ann. 265 (1983), no. 2,137–148

Gr81. Commichau, M. ; Grauert, H. Das formale Prinzip fr kompakte komplexe Untermannig-faltigkeiten mit 1-positivem Normalenbundel. , pp. 101–126, Ann. of Math. Stud., 100, Prince-ton Univ. Press, Princeton, N.J., 1981.

Gr58. Grauert, H. On Levi’s problem and the imbedding of real-analytic manifolds. Ann. of Math.(2) 68 1958 460–472.

Gr56. Grauert, H. ; Remmert, R. Plurisubharmonische Funktionen in komplexen Rumen. (Ger-man) Math. Z. 65 (1956), 175–194.

GrRe79. Grauert, H. ; Remmert, R. Theory of Stein spaces. Grundlehren der MathematischenWissenschaften, 236. Springer-Verlag, Berlin-New York, 1979

GrRe84. Grauert, H. ; Remmert, R. Coherent analytic sheaves. Grundlehren der MathematischenWissenschaften, 265. Springer-Verlag, Berlin-New York, 1984.

GPR94. Several complex variables. VII. Sheaf-theoretical methods in complex analysis. Editedby H. Grauert, Th. Peternell and R. Remmert. Encyclopaedia of Mathematical Sciences, 74.Springer-Verlag, Berlin, 1994.

GrHa78. Griffiths, P. ; Harris, J. Principles of algebraic geometry. Reprint of the 1978 original.Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994.

Gri69. Griffiths, P. A. Hermitian differential geometry, Chern classes, and positive vector bundles.1969 Global Analysis (Papers in Honor of K. Kodaira) pp. 185–251 Univ. Tokyo Press, Tokyo

Gri65. Griffiths, P. A. Hermitian differential geometry and the theory of positive and ample holo-morphic vector bundles. J. Math. Mech. 14 1965 117–140.

Gri66. Griffiths, P. A. The extension problem in complex analysis. II. Embeddings with positivenormal bundle. Amer. J. Math. 88 1966 366–446.

Gro69. Grothendieck, A. On the de Rham cohomology of algebraic varieties. Inst. Hautes tudesSci. Publ. Math. No. 29 1966 95–103.

Gro58. Grothendieck, A. Sur les faisceaux algbriques et les faisceaux analytiques cohrents.(French) Sminaire Henri Cartan; 9e anne: 1956/57. Quelques questions de topologie, Exposno. 2, 16 pp. Secretariat mathmatique, Paris 1958 73 pp. (mimeographed).

GuI90. Gunning, R. C. Introduction to holomorphic functions of several variables. Vol. I. Func-tion theory. The Wadsworth & Brooks/Cole Mathematics Series. Pacific Grove, CA, 1990

GuII90. Gunning, R. C. Introduction to holomorphic functions of several variables. Vol. II. Localtheory. The Wadsworth & Brooks/Cole Mathematics Series. Monterey, CA, 1990

GuIII90. Gunning, R. C. Introduction to holomorphic functions of several variables. Vol. III. Ho-mological theory. The Wadsworth & Brooks/Cole Mathematics Series. Monterey, CA, 1990

GuRo. Gunning, R. C.; Rossi, H. Analytic functions of several complex variables. Prentice-Hall,Inc., Englewood Cliffs, N.J. 1965


Ha77. Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

Ha70. R. Hartshorne: Ample subvarieties of algebraic varieties. Notes written in collaborationwith C. Musili. Lecture Notes in Mathematics, Vol. 156 Springer-Verlag, Berlin-New York1970 xiv+256 pp.

HiRo64. Hironaka, H.; Rossi, H. On the equivalence of imbeddings of exceptional complexspaces. Math. Ann. 156 1964 313–333.

Hi81. Hirschowitz, A. On the convergence of formal equivalence between embeddings. Ann. ofMath. (2) 113 (1981), no. 3, 501–514.

IlYa08. Yulij Ilyashenko, Sergei Yakovenko. Lectures on analytic differential equations. GraduateStudies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008.

Ke. Kempf, G. R. Algebraic varieties. London Mathematical Society Lecture Note Series, 172.Cambridge University Press, Cambridge, 1993.

KK83. Kaup, L.; Kaup, B. Holomorphic functions of several variables. An introduction to thefundamental theory. With the assistance of Gottfried Barthel. Translated from the German byMichael Bridgland. de Gruyter Studies in Mathematics, 3. Walter de Gruyter & Co., Berlin,1983.

Kos81. Kosarew, S. Das formale Prinzip und Modifikationen komplexer Raume. Math. Ann. 256(1981), no. 2, 249–254.

Kr73. Krasnov, V. A. Formal modifications. Existence theorems for modifications of complexmanifolds. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 848–882. English translation [Math.USSR-Izv. 7 (1973), 847–881].

Kr74. Krasnov, V. A. The equivalence of imbeddings of contractible complex spaces. Izv. Akad.Nauk SSSR Ser. Mat. 38 (1974), 1012–1036. English translation: Math. USSR-Izv. 8 (1974),no. 5, 1009–1033 (1975).

La71. Laufer, H. B. Normal two-dimensional singularities. Annals of Mathematics Studies, No.71. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971.

La81. Laufer, H. B. On CPS1 as an exceptional set. Recent developments in several complexvariables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979), pp. 261–275, Ann. of Math.Stud., 100, Princeton Univ. Press, Princeton, N.J., 1981.

Le66. Lehner, J. A short course in automorphic functions. Holt, Rinehart and Winston, New York-Toronto, Ont.-London 1966.

Le45. Lelong, P. Les fonctions plurisousharmoniques. Ann. Sci. cole Norm. Sup. (3) 62, (1945).301–338.

Li99. Lins Neto, Alcides. A note on projective Levi flats and minimal sets of algebraic foliations.Ann. Inst. Fourier (Grenoble) 49 (1999), no. 4, 1369–1385.

Ma68. Malgrange, Bernard Analytic spaces. Enseignement Math. (2) 14 1968 1–28.Ma88. Maskit, B. Kleinian groups. Grundlehren der Mathematischen Wissenschaften, 287.

Springer-Verlag, Berlin, 1988.MoSa. Hossein Movasati and Paulo Sad. Embedded curves and foliations. Publ. Mat. 55, 401–

411, 2011.Mi93. Mishustin, M. B. Neighborhoods of the Riemann sphere in complex surfaces. Funktsional.

Anal. i Prilozhen. 27 (1993), no. 3,29–41, 95; translation in Funct. Anal. Appl. 27 (1993), no.3, 176–185

Nag62. Nagata, M. Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13Interscience Publishers a division of John Wiley Sons New York-London 1962

Na68. Narasimhan, R. Compact analytical varieties. Enseignement Math. (2) 14 1968 75–98.Na60. Narasimhan, R. The Levi problem for complex spaces. Math. Ann. 142 1960/1961 355–

365.Na62. Narasimhan, R. The Levi problem for complex spaces II. Math. Ann. 146 1962, 195–216.Ne89. Neeman, Amnon, Ueda theory: theorems and problems. Mem. Amer. Math. Soc. 81 (1989),

no. 415.NiSp60. Nirenberg, L.; Spencer, D. C. On rigidity of holomorphic imbeddings. 1960 Contribu-

tions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) pp. 133–137 TataInstitute of Fundamental Research, Bombay

References 131

Ok42. Oka, K. Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudocon-vexes. Thoku Math. J. 49, (1942). 15–52.

Ra65. Ramabhadran, N. Serre’s duality theorem. Math. Student 33 1965 11–22.Re56. Remmert, R. Sur les espaces analytiques holomorphiquement separables et holomorphique-

ment convexes. C. R. Acad. Sci. Paris 243 (1956), 118–121.Re57. Remmert, R. Holomorphe und meromorphe Abbildungen komplexer Rume. (German)

Math. Ann. 133 (1957), 328–370.Re95. Remmert, R. Complex analysis in ”Sturm und Drang”. Math. Intelligencer 17 (1995), no.

2, 4–11.Ri68. Richberg, R. Stetige streng pseudokonvexe Funktionen. (German) Math. Ann. 175 1968

257–286.Ro69. Rossi, H. Strongly pseudoconvex manifolds. 1969 Lectures in Modern Analysis and Ap-

plications, I pp. 10–29 Springer, BerlinSa82. Savelev, V. I. Zero-type imbedding of a sphere into complex surfaces. (Russian. English

summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1982, no. 4, 28–32, 85Se56. J. P. Serre (1956), Geometrie algebrique et geometrie analytique. Annales de l’Institut

Fourier 6 (1956), 1-42.Su74. Suzuki, M. Sur les relations d’equivalence ouvertes dans les espaces analytiques. Ann. Sci.

Ecole Norm. Sup. (4) 7 (1974), 531–541 (1975).Su78. Suzuki, M. Sur les integrales premieres de certains feuilletages analytiques complexes.

Fonctions de plusieurs variables complexes, III (Sem. Francois Norguet, 1975–1977), pp. 53–79, 394, Lecture Notes in Math., 670, Springer, Berlin, 1978.

OSu75. Suzuki, O. Neighborhoods of a compact non-singular algebraic curve imbedded in a 2-dimensional complex manifold. Publ. Res. Inst. Math. Sci. 11 (1975/76), no. 1, 185–199.

Ue82. Ueda, Tetsuo. On the neighborhood of a compact complex curve with topologically trivialnormal bundle. J. Math. Kyoto Univ. 22 (1982/83), no. 4, 583–607.

Um73. Umemura, H. Some results in the theory of vector bundles. Nagoya Math. J. 52 (1973),97–128.

Wa72. Watari, S. A theorem on continuous strongly convex functions in complex spaces. Nederl.Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34 (1972), 62–67.

Wa75. Wavrik, J. J. A theorem on solutions of analytic equations with applications to deforma-tions of complex structures. Math. Ann. 216 (1975), no. 2, 127–142.

Index

(x,x), a neighborhood of x in X , 7TxX , tangent space, 13C-algebra , 7OX , the structural sheaf, 11, 21OCn (U), the space of holomorphic functions ,

7Qν , 81S (1),S (2),S (2), . . ., 82T , 82TA, 82T ∗x X , cotangent space, 13OX ,x, germs of holomorphic functions, 9OCn,x, the ring of germs of holomorphic

functions , 7MX ,x, the maximal ideal, 9MCn,x, a maximal ideal, 7A(∗), 117A(1),A(2),A(3), . . ., 117embxX , embedding dimension, 13CT , 111IX ,x, defining ideal, 9

KX , convex hull, 39MX ,x,M 2

X ,x,M3X ,x, . . ., 9

acyclic covering, 31analytic affine variety, 11, 21analytic scheme, 21analytic variety, 11

biholomorphism, 11blow down, 66

Cartan’s theorem B, 41Cauchy integral formula, 6Cauchy-Riemann equations, 6chart, 11

compact mapping, 63complete family, 97completely continuous, 63convergent homomorphism of formal

neighborhoods, 119convex function , 50convex hull, 39cotangent space, 13

direct image theorem, 111

effective family, 97embedding dimension, 13equivalence relation, 111equivalent charts, 12exceptional variety, 66

finite neighborhood, 117foliation, 99formal completion, 118formal neighborhood, 118Frechet space, 63

Hilbert Nullstellensatz theorem, 86holomorphic map, 11holomorphically convex, 39

Kodaira-Spencer map, 97Kodaira-Spencer Theorem, 97

Leray lemma, 31Levi form, 49line bundle, 69linear space, 79local ring, 7

maximal compact analytic set, 57

133

134 Index

Maximum principle, 55meromorphic section, 70

negative vector bundle, 78negatively embedded, 66neighborhood sheaf, 117normal bundle, 82nowhere discrete, 39

plurisubharmonic, 50positive vector bundle, 77, 78proper equivalence relation, 114Push-forward of a sheaf, 26

Remmert proper mapping theorem, 111Remmert reduction, 115restriction map, 23

Schwarz theorem, 63Stein factorization, 114Stein variety, 39strongly convex function, 50strongly convex functions on varieties, 54strongly exceptional, 86strongly plurisubharmonic, 50strongly pseudoconvex domain, 55structural restriction, 24structural sheaf, 11, 21

tangent cone, 92tangent space, 13transverse foliation, 99

variety, 11

Weierstrass preparation theorem, 7

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