Kodaira's projectivity criterion for surfaces

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MSc Mathematics

Master’s Thesis

Kodaira’s projectivity criterion for surfaces

Author: Supervisor:

Emma Brakkee

prof. dr. L.D.J. Taelman

Examination date:

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Abstract

Kodaira’s projectivity criterion for surfaces gives a necessary and sufficient condition for a 2-dimensional compact connected complex manifold to be projective. Our main goal is to give new presentations of two existing proofs of the criterion. We start by explaining some background in analytic geometry and intersection theory, needed to understand the statement of the theorem. Then we give the proofs, and we finish with studying projectivity of complex tori and Kummer surfaces.

Title: Kodaira’s projectivity criterion for surfaces

Author: Emma Brakkee, emma.brakkee@xs4all.nl, 10026045 Supervisor: prof. dr. L.D.J. Taelman

Second Examiner: dr. M. Shen Examination date: February 17, 2016

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

This thesis is about complex geometry, i.e. geometry over the complex numbers. One can study complex geometry both from the algebraic, and from the analytic perspective. In (classical) algebraic geometry, one studies of algebraic varieties. These are spaces which are locally defined as zero sets of polynomials and have the Zariski topology. The objects of analytic geometry are called complex spaces. Their underlying spaces have the Euclidean topology; locally, they are defined as zero sets of holomorphic functions.

Since all polynomials are holomorphic, it is not too surprising that algebraic varieties over C can also be looked at from the point of view of analytic geometry. In fact, there is a functor (−)an from the category of complex algebraic varieties to the category of complex spaces. Moreover, for every complex variety X there is a functor

(−)an: Coh(X) → Coh(Xan)

from the category of coherent OX-modules to the category of coherent OXan-modules.

In 1956, Serre published his famous paper Géométrie Algébrique et Géométrie Analytique (‘GAGA’, [26]), in which he showed that the functor (−)an is especially well-behaved when the variety X is projective:

Theorem (Serre, 1956). Let X be a projective variety over C. Then the functor (−)an: Coh(X) → Coh(Xan)

is an equivalence of categories. If F is a coherent sheaf on X, then for all i, the induced map Hi(X,F ) → Hi(Xan,Fan)

is an isomorphism.

Grothendieck extended the functor (−)anto the category of schemes which are locally of finite type over C (see [16, Expos´e XII]). He proved that the theorem above holds for proper schemes. He also showed, among others, the following properties of the functor (−)an.

1. The scheme X is regular if and only if Xan is smooth;

2. X is connected (reduced, irreducible) if and only if Xanis connected (reduced, irreducible); 3. X is separated if and only if Xan _{is Hausdorff}1_;

4. X has dimension n if and only if Xan has dimension n; 5. X is complete if and only if Xan is compact.

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We see that the algebraic and the analytic point of view are strongly related. The equivalences above imply that one can prove properties about a scheme X not just by using algebraic techniques, but also by passing to Xan and using analytic techniques. Especially when X is a smooth variety, so that Xan is a complex manifold, we will see that this is very useful.

Now it is a natural question if all complex spaces are the analytization of a scheme which is locally of finite type over C. The answer is negative. Moreover, even if a complex space comes from a scheme, this scheme might not be unique. For an example, see [18, p. 440]. If a complex space Y is the analytization of a unique scheme, locally of finite type over C, then we call X algebraic.

For a reduced compact complex space Y , Serre showed that there can be at most one variety X such that Xan ∼= Y . When Y is projective, the existence of X was proven already in 1949 by Chow [5].

Theorem (Chow, 1949). Every reduced compact complex subspace of Pn is algebraic.

The question that arises next is which complex spaces are projective. It is well-known that all 1-dimensional compact connected complex manifolds, i.e. Riemann surfaces, are algebraic. We call a 2-dimensional compact connected complex manifold a surface. Kodaira’s projectivity criterion for surfaces gives a necessary and sufficient condition for a surface to be projective. Theorem (Kodaira, 1964). A surface is projective if and only if it has a line bundle L such that L2_{> 0.}

The term L2 in the theorem is an integer called the intersection product of L with itself. The intersection product is a symmetric bilinear product on the Picard group of the surface. It can be defined in terms of Euler characteristics as follows:

(L.M) = χ(X, OX) − χ(X, L−1) − χ(X, M−1) + χ(X, L−1⊗ M−1)

for all line bundles L and M. Alternatively, as we will see, it can be defined in terms of first Chern classes. The intersection product is a well-understood concept, which makes Kodaira’s criterion very useful.

We will see that not all 2-dimensional compact connected complex manifolds are projective. That these cannot be algebraic either is due to the following theorem by Zariski, see [28, p. 54], footnote 9.

Theorem (Zariski, 1958). Non-singular complete algebraic surfaces are projective.

Outline

The goal of this thesis is to present two existing proofs of Kodaira’s criterion in a more readable way. The first is Kodaira’s original proof, given in [24]. The second is a proof from the book Compact Complex Surfaces by Barth et al. [1, p. 160].

In chapter 1 we give some background in analytic geometry. We treat the basics of complex spaces and then move to complex manifolds, about which we prove some (differential) geometric results and for which we give the analytic analogue of some algebro-geometric concepts.

In chapter 2 we define the intersection product on surfaces. We give the definition in terms of first Chern classes. We show that it is equal to the definition in terms of Euler characteris-tics, give a specific formula for line bundles coming from curves, and prove some identities for intersection products on blow-ups.

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Chapter 3 has two main parts, corresponding to the two proofs of Kodaira’s projectivity criterion. The proofs are almost entirely different. We explain both of them in full detail.

We finish in chapter 4 with studying some examples, namely complex tori and Kummer surfaces. It is known that some of them are projective, but most of them are not. We show how one can give criteria for projectivity using Kodaira’s criterion and prove the existence of non-algebraic tori.

Acknowledgements

I want to thank my supervisor Lenny Taelman for his patience and his enthusiasm, even though my thesis topic is not his specialization. Thanks for all the advice and comments about both my thesis and my applications for PhD positions, and for reading my thesis drafts so thoroughly that even ugly spacing was pointed out.

Many thanks also to Ben Moonen, for advising me to take a seminar on complex geometry. I want to thank my fellow students for the nice time during our studies, especially Didier, for being there during the hard last weekends at Science Park.

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1 Basics of complex geometry

The objects studied in analytic geometry are called complex spaces. In this chapter, we define them and give some basic properties. After this we go more into detail about complex manifolds, which form an important class of complex spaces.

1.1 Complex spaces

This section is meant to summarize the basic theory of complex spaces. We mainly follow [13] and [14].

Consider Cn with the sheaf of holomorphic functions O_Cn. This is a sheaf of C-algebras.

For each point x ∈ Cn_{, the stalk O}

Cn,x consists of all convergent power series around x. It

is a unique factorization domain (see [14, p. 44]) and a local C-algebra: the maximal ideal m_x ⊂ O_Cn_,x consists of all functions vanishing in x. For x = 0, these are exactly the power

series with no constant term.

Let D ⊂ Cn be a connected open subset and let OD be the restricted sheaf OD = OCn|D. If

f1, . . . , fk are holomorphic functions on D, denote byI the sheaf

I (f1, . . . , fk) := f1OD+ . . . + fkOD ⊂ OD

and let Z =Z (f1, . . . , fk) be the zero set of f1, . . . , fk:

Z (f1, . . . , fk) = {x ∈ D | f1(x) = . . . = fk(x) = 0} = Supp(OD/I ).

This is a closed subset of D which we give the induced topology. The pair (Z, (OD/I )|Z) is a

ringed space, called a complex model space.

Definition 1.1. A complex model space is a ringed space of the form (Z, (OD/I )|Z) where

D ⊂ Cn is a connected open subset, f1, . . . , fk are holomorphic functions on D, and Z and I

are defined by Z =Z (f1, . . . , fk) and I = f1OD + . . . + fkOD.

Complex model spaces are examples of C-ringed spaces, which are defined as follows.

Definition 1.2. A C-ringed space is a ringed space (X, OX) such that OX is a sheaf of local

C-algebras. A morphism of C-ringed spaces (X, OX) → (Y, OY) is a morphism of ringed spaces

(f : X → Y, f#: OY → f∗OX) such that f# is a morphism of sheaves of algebras.

Definition 1.3. A complex space is a C-ringed space (X, OX) such that X is Hausdorff and

around every point x ∈ X, there exists an open neighbourhood U ⊂ X such that (U, OX|U) is

isomorphic, as a C-ringed space, to a complex model space.

The morphisms in the category of complex spaces are the morphisms of C-ringed spaces. They are called holomorphic maps.

We will usually denote a complex space (X, OX) just by X. The maximal ideal of a stalk

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Any open subset U ⊂ X, with the induced topology, forms a complex space (U, OX|U). We

call this an open complex subspace of X.

Let (X, OX) be a complex space. IfI ⊂ OX is a sheaf of ideals, we define its zero setZ (I )

as follows:

Z (I ) = {x ∈ X | Ix 6= OX,x} = {x ∈ X |Ix⊂ mx} = Supp(OX/I ).

This is a closed subset of X. It has a structure of a C-ringed space Z = Z (I ), (OX/I )|_{Z (I )}.

The sheaf I is called of finite type if for every x ∈ X, there is an open neighbourhood U of X and elements f1, . . . , fk∈ OX(U ) such that I |U is generated by f1, . . . , fk. Then locally, Z

is of the form (Z (f1, . . . , fk),I (f1, . . . , fk)) for finitely many holomorphic functions f1, . . . , fk

on an open subset U ⊂ X. So Z is a complex space. We call it a closed complex subspace of X. The sheafI is called the ideal sheaf of Z.

If I is not of finite type, Z may not be a complex space. For an example, see [14, p. 14]. A subset A ⊂ X is called analytic if A is the underlying set of a closed complex subspace. This holds if and only if for every x ∈ A, there exists an open neighbourhood U ⊂ X and holomorphic functions f1, . . . , fkon U such that A ∩ U =Z (f1, . . . , fk). The set A can be given

a unique structure of a reduced closed complex subspace of X, whose ideal sheafIA is defined

by IA(U ) = {f ∈ OX(U ) | A ∩ U ⊂Z (f)} for all open U ⊂ X.

The underlying set of a complex space X is always analytic, as it is the zero set of the zero ideal. In a complex model space (Z =Z (f1, . . . , fk), (OD/I (f1, . . . , fk))|Z) with D ⊂ Cn, each

point x forms an analytic set. Namely, let Fi: D → C; z 7→ zi− xi for i = 1, . . . , n. Then

{x} =Z (F₁|_Z, . . . , Fn|Z). It follows that the set {x} is analytic for every complex space X and

every x ∈ X, and that fibers of holomorphic maps are analytic.

An analytic subset A ⊂ X is called reducible if there are non-empty proper analytic subsets A1, A2 ⊂ A such that A = A1 ∪ A2. If these do not exist, A is called irreducible. For some

equivalent statements, see [14, p. 168]. The empty set is reducible by convention.

It can be shown, see [14, p. 44], that for every complex space (X, OX), all stalks OX,x are

noetherian. We distinguish the following other properties: Definition 1.4. Let X be a complex space and let x ∈ X.

(i) x ∈ X is called a smooth or regular point if there is an open neighbourhood U ⊂ X of x which is isomorphic to (D, O_Cn|_D) for some open subset D ⊂ Cn. If x is not smooth, we

call it a singular point. If all points of X are smooth, we call X a (complex) manifold. (ii) x is a reduced point if OX,xdoes not contain any non-zero nilpotent elements. If this holds

for all x ∈ X, then X is called reduced.

(iii) x is a normal point if OX,xis a normal ring, i.e. it is an integral domain which is integrally

closed in its quotient ring. If this holds for all x ∈ X, then X is called normal.

If x is normal, then x is reduced. If x is smooth, then the local ring OX,x is isomorphic to

O_Cn_,0 for some n. So O_X,x is a unique factorization domain. This implies that x is normal.

We denote by S(X) the set of singular points in X. It can be shown, see [14, p. 117], that it is an analytic subset, and if X is reduced then S(X) is nowhere dense. If X is reduced then also the set of non-normal points in X is analytic, and it is contained in S(X) (see [14, p. 128]). If S(X) is empty (so X is a manifold) and X is connected, then X is irreducible.

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It is easy to show, see e.g. [13, p. 28], that a holomorphic map between two smooth complex spaces is uniquely defined by its underlying map of topological spaces, which is a holomorphic map of manifolds in the usual sense.

We next introduce the concept of dimension. We give the analytic definition. For other equivalent definitions, see [14].

Definition 1.5. Let X be a complex space and let x ∈ X. The dimension dimxX of X at x

is the minimal number k for which there is an open neighbourhood U of X and holomorphic functions f1, . . . , fkon U such that {x} =Z (f1, . . . , fk). The dimension of X is the supremum

sup{dimxX | x ∈ X}.

Note that the dimension of X can be infinite. If dimxX is the same for all x ∈ X, then we

call X pure-dimensional. All irreducible spaces are pure-dimensional. The space (Cn, O_Cn) is

pure-dimensional of dimension n. A pure-dimensional complex space of dimension 1 is called a curve.

If A ⊂ X is an analytic subset, its dimension is defined as the dimension of the unique reduced complex space with underlying set A. The codimension of A is dim X − dim A.

An irreducible complex space X has the following two useful properties (see [14, p. 168-170]). If f ∈ OX(X) is nonzero, then the analytic setZ (f) is empty or of pure dimension dim X − 1.

For any proper analytic subset A ⊂ X, the space X\A is irreducible, so in particular connected. If X is also smooth, we know the following.

Proposition 1.6. Let X be a connected complex manifold and let A ⊂ X be an analytic subset. Then either A = X, or A is nowhere dense and X\A is irreducible.

Proof. See [10], Proposition IV.1.6.

For example, the proper analytic subsets of C are unions of isolated points. Any normal complex space is almost a manifold, in the following sense.

Proposition 1.7. Let X be a normal complex space. Then the singular locus S(X) of X has codimension at least 2.

Proof. See [14, p. 128].

In particular, normal complex spaces of dimension less than two are smooth.

If the complex space X is reduced, but not necessarily normal, then we can normalize X. Normalizations are finite maps. A continuous map between topological spaces f : X → Y is called finite if f is closed and f−1(y) is finite for all y ∈ Y . We need one more notion:

Definition 1.8. Let X be a complex space. A subset T ⊂ X is called thin if every point x ∈ T has an open neighbourhood U such that T ∩ U is contained in a proper analytic subset of X.

In particular, proper analytic subsets are thin.

Definition 1.9. Let X be a reduced complex space. A finite holomorphic map ν : ˆX → X is called a normalization of X if ˆX is normal and there exists a thin set T ⊂ X such that ν|_X\νˆ −1_{(T )}: ˆX\ν−1(T ) → X\T is biholomorphic.

Theorem 1.10 (Normalization). For every reduced complex space X there exists a normaliza-tion ν : ˆX → X. Moreover, it is unique up to unique isomorphism: if ν1: ˆX1 → X is another

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Proof. See [22, p. 304] or [14, p. 161 and p. 164].

So we can talk about the normalization of a reduced complex space. It has the following property:

Proposition 1.11. Let X be a reduced complex space and let N (X) ⊂ X be the subset of non-normal points. Let ν : ˆX → X be the normalization of X. Then ν induces a biholomorphic map ˆX\ν−1(N (X)) → X\N (X).

Proof. See [14, p. 164].

It follows from Proposition 1.7 that the normalization ˜C of a 1-dimensional reduced complex space C is smooth.

We finish this section with a list of theorems that we will need later.

Theorem 1.12 (Identity theorem for holomorphic maps). Let f, g : X → Y be two holomorphic maps between complex spaces and assume that X is reduced and irreducible. If there is a non-empty open subset U ⊂ X such that f and g coincide on U , then f = g.

Proof. See [22, p. 195].

Theorem 1.13 (Riemann extension theorem). Let n ≥ 2 and let U ⊂ Cn be a connected open set. Let A ⊂ U be an analytic subset of codimension at least 2. Then every holomorphic function on U \A can be extended uniquely to a holomorphic function on U .

Proof. See [14, p. 132].

Theorem 1.14 (L¨uroth’s theorem). If f : P1 → C is any non-constant holomorphic map from P1 to a smooth connected curve C, then C ∼= P1.

Proof. This follows from Theorem V.4 in [2].

If X is a topological space, a function f : X → Z is called upper semi-continuous if for every x ∈ X, there is an open neighbourhood U ⊂ X such that f (y) ≤ f (x) for all y ∈ U .

Theorem 1.15 (Semi-continuity of fibre dimension). Let f : X → Y be a holomorphic map between complex spaces. Then the function X → Z sending x to dimxf−1({f (x)}) is upper

semi-continuous. Proof. See [22, p. 196].

Let f : X → Y be a continuous map between topological spaces. We call f proper if f is closed and the inverse image under f of any compact subset of Y is compact. This holds if and only if f is closed and the pre-image of every point is compact.

If X is compact and Y Hausdorff, then every map X → Y is proper. If Y is Hausdorff and locally compact, then the condition that f−1(K) is compact for compact subsets K implies that f is closed. Every complex space is Hausdorff (by definition) and locally compact.

Theorem 1.16 (Proper mapping theorem). Let f : X → Y be a proper holomorphic map between complex spaces. If A ⊂ X is an analytic subset, then f (A) is an analytic subset of Y . Proof. See [22, p. 170].

Theorem 1.17 (Stein factorization). Let X and Y be complex spaces and f : X → Y a proper holomorphic map. Then f has a unique factorization f = h ◦ g where g : X → Z is a proper surjection unto a complex space Z and h : Z → Y is finite. Moreover, g satisfies g∗OX = OY,

in particular, g has connected fibers. Finally, if X is normal, then Z is normal as well. Proof. See [14, p. 213].

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1.2 Real Hodge structures

In the next section we will need some Hodge theory on complex manifolds. We first give a short introduction to the theory of real Hodge structures.

Let V be a finite-dimensional real vector space and let V_C= V ⊗_RC be its complexification. We define complex conjugation on V_C by v ⊗ λ = v ⊗ λ.

Definition 1.18. A Hodge decomposition of V is a direct sum decomposition V_C= M

p,q∈Z

Vp,q

such that Vp,q _{= V}q,p_{. A (real) Hodge structure is a finite-dimensional vector space with a}

Hodge decomposition.

Suppose that we have an R-linear action of C∗ on V , i.e. a homomorphism ρ : C∗ → Aut_R(V ) of algebraic groups over R. Then we get an action ρC: C

∗_{→ Aut}

C(VC) of C ∗ _{on V}

Cby C-linear

extension, so for v ∈ V and λ ∈ C, we have ρC(z)(v ⊗ λ) = ρ(z)(v) ⊗ λ. Note that

ρ_C(z)(v ⊗ λ) = ρ_C(z)(v ⊗ λ) = ρ(z)(v) ⊗ λ = ρ(z)(v) ⊗ λ = ρ_C(z)(v ⊗ λ). Now let p, q ∈ Z and define

Vp,q= {v ∈ V_C| ρ_C(v) = zpzqv for all z ∈ C∗} ⊂ V_C. Suppose that v =P ivi⊗ λi∈ Vp,q, so ρC(z)(v) = zpzqv = P ivi⊗ zpzqλi. Then zqzpv =X i vi⊗ zqzpλi =X i vi⊗ zpzqλi =X i vi⊗ zpzqλi = ρ_C(v) = ρ_C(v).

This shows that Vp,q _{= V}q,p_{. The following theorem shows that the V}p,q _{give a Hodge}

decom-position of V .

Proposition 1.19. Let V be a finite-dimensional real vector space with an algebraic group action of C∗ over R. For all p, q ∈ Z, let Vp,q_{⊂ V}

C be defined as above. There is a direct sum

decomposition

V_C∼= M

p,q∈Z

Vp,q.

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In fact, it is a well-known theorem that all real Hodge structures come from a algebraic group action over of C∗ over R on a real vector space, see [8, p. 25].

Example 1.20. Let V be an n-dimensional complex vector space. Its underlying 2n-dimensional real vector space has a natural action of C∗ by ρ(z)(v) = zv. The map V → V_C sending v to v ⊗ i + iv ⊗ 1 induces an isomorphism V ∼= V1,0. We find that

V_C∼= V1,0⊕ V0,1.

Next, we work out some linear algebra operations on Hodge structures. Let V and W be two finite-dimensional real vector spaces with actions ρV and ρW of C∗.

(i) There is a natural action ρ of C∗ on V ⊕ W given by ρ(z)(v, w) = (ρV(z)(v), ρW(z)(w)).

We have (V ⊕ W )_C= V_C⊕ W_C, and we find that (V ⊕ W )p,q equals

{(v, w) ∈ V_C⊕ W_C| (ρ_V(z)(v), ρW(z)(w)) = zpzq(v, w) = (zpzqv, zpzqw) for all z ∈ C∗}

which is Vp,q⊕ Wp,q_.

(ii) On the tensor product V ⊗_RW , we have an action of C∗ given by ρ(z)(v ⊗ w) = ρV(z)(v) ⊗ ρW(z)(w)

for all v ∈ V and w ∈ W . Similarly as above, one computes that (V ⊗_RW )p,q =M p1+p2=p q1+q2=q Vp1,q1 _⊗ CW p2,q2 for all p, q ∈ Z.

(iii) We denote by V∨ the dual vector space. There is an action of C∗ on V∨ given by ρ(z)(α)(v) = α(ρV(z−1)(v)) for all α ∈ V∨ and v ∈ V . The Hodge decomposition we

get is given by

(V∨)p,q= (V−p,−q)∨. For example, if V and W are complex vector spaces, we find

(V ⊗_RW )_C∼= (V ⊗RW ) 2,0_{⊕ (V ⊗} RW ) 1,1_{⊕ (V ⊗} RW ) 0,2 where (V ⊗_RW )2,0 = V1,0⊗_CW1,0 (V ⊗_RW )0,2 = V0,1⊗_CW0,1 (V ⊗_RW )1,1 = (V1,0⊗_CW0,1) ⊕ (V0,1⊗_CW1,0).

We give one more example of a linear algebra operation on the Hodge structure of a complex vector space: the exterior algebra. Let k ≤ 2 dim(V ). The natural action ρ of C∗ on Vk

RV is

given by ρ(z)(v1∧ . . . ∧ vk) = ρV(z)(v1) ∧ . . . ∧ ρV(z)(vk). We have (Vk_RV )C=

Vk

CVC. We get

the Hodge decomposition

^k C V_C∼= M p+q=k ^p,q C V whereVp,q C V = Vp CV 1,0_⊗ C Vq CV 0,1_.

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1.3 Differential forms on complex manifolds

In the following section X will always be an n-dimensional complex manifold. We first study the underlying real manifold of X.

We denote by T X the real tangent bundle of X, by T∨X the cotangent bundle and by Akthe sheaf of differential k-forms on X, so this is the sheaf of sections of the vector bundleVk

RT ∨_X.

We denote by H∗_dR(X, R) the real de Rham cohomology of X, the cohomology of the cochain complex

A• : A0(X)→ Ad 1(X)→ Ad 2(X) → · · ·

Tensoring with C, we get the complexified tangent bundle TCX = T X ⊗RC, the complexified

cotangent bundle

T_C∨X = (T_CX)∨ ∼= T∨X ⊗_RC, the vector bundles

^k

C

T_C∨X ∼= (^kT∨X) ⊗_RC and the sheaves Ak

C = A

k _⊗

RC of complex-valued differential forms, so Ak_C is the sheaf of

sections of Vk

CT ∨

CX. The complex de Rham cohomology H ∗

dR(X, C) is the cohomology of the

cochain complex

A•_C: A0_C(X)→ Ad 1_C(X)→ Ad 2_C(X) → · · ·

where with d we denote the complex linear extension of the real d. There are isomorphisms Hi_dR(X, C) ∼= Hi_dR(X, R) ⊗ C.

We also have natural isomorphisms between de Rham and singular cohomology Hi_dR(X, R) ∼= Hi(X, R) and Hi_dR(X, C) ∼= Hi(X, C)

given as follows: a class [ω] ∈ Hi_dR(X, R) is sent to the class in Hi(, R) represented by the i-cocycle which sends an i-cycle σ to R_σω. This isomorphism respects the cup product structure on both sides (see [27, p. 205-214]).

Finally, we have isomorphisms

Hi_dR(X, R) ∼= Hi(X, R) and Hi_dR(X, C) ∼= Hi(X, C)

between de Rham cohomology and the cohomology of the constant sheaves R and C with values in R and C, respectively. This follows for example from the isomorphism

Hi(X, Z) ∼= Hi(X, Z) for singular cohomology (see [27, p. 191-200]).

If X is compact, then these cohomology spaces are finite-dimensional, see e.g. [4, p. 43] for a proof for de Rham cohomology. We define the i-th Betti number of X by bi(X) = dimRH

i

(X, R). It is zero for all i > 2 dim_CX.

Because X is a complex manifold, it has a canonical almost complex structure J : T X → T X defined by multiplication with i. So as in example 1.20, we get a Hodge decomposition

T_C,xX ∼= Tx1,0X ⊕ Tx0,1X

for each x ∈ X. As J is smooth, this gives a direct sum decomposition of T_CX into subbundles T1,0X and T0,1X.

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Proposition 1.21. Let X be a complex manifold. There is a direct sum decomposition T_CX ∼= T1,0X ⊕ T0,1X

where T1,0X = {v ∈ T_CX | J_C(v) = i · v} and T0,1X = {v ∈ T_CX | J_C(v) = −i · v}.

The bundle T1,0X is called the holomorphic tangent bundle of X. As is shown in [15, p. 71], it is indeed a holomorphic vector bundle. We get induced Hodge decompositions of the cotangent bundle

T_C∨X ∼= (T1,0X)∨⊕ (T0,1_X)∨

and of the bundles Vk

CT ∨ CX for all k ≤ 2n: ^k CT ∨ CX ∼= M p+q=k ^p,q T_CX where ^p,q T_C∨X :=^p C (T1,0X)∨⊗^q C (T0,1X)∨. Let Ap,q be the sheaf of sections of Vp,q

T∨

CX. Elements of A

p,q_{(X) are called (p, q)-forms.}

We usually write ΩX for A1,0 and ΩpX = Ap,0 for every p. The sheaf ωX := Ωdim XX is called the

canonical sheaf of X. For every k we have a decomposition Ak C ∼ = M p+q=k Ap,q. (1.1)

Denote by Πp,q the projection Ak

C(X) → A

p,q_{(X) and by ∂ and ∂ the differential operators}

∂ := Πp+1,q◦ d : Ap,q → Ap+1,q and ∂ := Πp,q+1◦ d : Ap,q→ Ap,q+1. These maps give us cochain complexes

A•,q: A0,q ∂→ A1,q ∂→ A2,q→ · · · Ap,•: Ap,0 ∂→ Ap,1 ∂→ Ap,2→ · · ·

for all p and q. The following proposition states that if f : X → Y is a holomorphic map, then the pullback f∗ gives a cochain map for each of the complexes above.

Proposition 1.22. Let X and Y be complex manifolds and let f : X → Y be a holomorphic map. The pullback of differential forms f∗: Ak

C(Y ) → A k

C(Y ) respects the decomposition (1.1).

Moreover, the maps f∗: Ap,q_{(Y ) → A}p,q_{(X) are compatible with ∂ and ∂, i.e. f}∗_{◦ ∂}

Y = ∂X◦ f∗

and f∗◦ ∂Y = ∂X ◦ f∗.

Proof. See [20], Proposition 2.6.10.

Definition 1.23 (Dolbeault cohomology). The (p, q)-th Dolbeault cohomology space of a com-plex manifold X is the vector space Hp,q(X) := Hq(Ap,•(X), ∂).

Theorem 1.24 (Dolbeault Theorem). There is an isomorphism Hp,q(X) ∼= Hq(X, Ωp_X) for all p and q.

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If X is a compact K¨ahler manifold, or if X is a surface and k = 2, Dolbeault cohomology gives us a direct sum composition of Hk(X, C). This follows from the next theorem.

Theorem 1.25. Let X be a compact K¨ahler manifold. Then for all p and q, the space Hp,q(X) is isomorphic to the subspace of Hp+q_dR (X, C) which can be represented by a closed (p, q)-form. If X is a surface, not necessarily K¨ahler, then this isomorphism exists for p + q = 2.

Proof. See [1], Corollary I.13.4 and Theorem IV.2.10.

Corollary 1.26. Let X be a compact complex manifold and assume that X is K¨ahler. Then we have a direct sum decomposition

Hk(X, C) ∼= M

p+q=k

Hp,q(X)

for all k. If X is a surface, not necessarily K¨ahler, then this decomposition holds for k = 2. Let X be a surface. We define the space H1,1(X, R) ⊂ H2(X, R) by

H1,1(X, R) = Im H2(X, R) → H2(X, R) ∩ H1,1(X).

The next theorem is a statement about the bilinear form on this space given by the restriction of the cup product

^ : H2(X, R) × H2(X, R) → H4(X, R) = R,

where the identification H4(X, R) = R is the canonical isomorphism which we will describe in section 2.2.

Theorem 1.27 (Hodge index theorem). Let X be a surface. The cup product on H1,1(X, R) has signature (1, h1,1− 1) if b₁(X) is even and (0, h1,1) if b1(X) is odd.

Proof. See [1], Theorem IV.2.14.

We see that a subspace of H1,1(X) on which the cup product is positive definite has dimension at most 1.

We now turn to the cohomology of coherent OX-modules. We have the following important

theorem:

Theorem 1.28. Let X be a compact complex space and F a coherent OX-module. Then for

all p, the space Hp(X,F ) is finite-dimensional as a C-vector space. Proof. See [17, p. 245].

It follows from Theorem 1.24 that the Dolbeault cohomology spaces Hp,q(X) are finite-dimensional. We define hp,q = dim_CHp,q(X). For a general coherent OX-moduleF , we denote

dim_CHi(X,F ) by hi(X,F ). This number is zero for i > 2 dim_CX by the following general theorem.

Theorem 1.29. Let X be a topological manifold andF a sheaf of abelian groups on X. Then Hi(X,F ) = 0 for i > dim(X).

This implies that the Euler characteristic of coherent modules on compact complex manifolds is well-defined.

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Definition 1.30. Let X be a compact complex manifold and letF be a coherent OX-module.

The Euler characteristic of F is

χ(x,F ) :=X

i

hi(X,F ). We will now state the famous theorem of Serre duality.

Theorem 1.31 (Serre duality). Let X be an n-dimensional compact complex manifold and let F be a locally free OX-module. Then there are natural isomorphisms

Hi(X,F ) ∼= Hn−i(X,F∨⊗ ω_X)∨ for all i ∈ {0, . . . , n}.

Proof. See [20, p. 171].

In the rest of the section we study holomorphic maps between complex manifolds. Let X and Y be complex manifolds and let f : X → Y be a smooth map. Consider the derivative

df : T_CX → T_CY = T1,0Y ⊕ T0,1Y.

A simple calculation shows that f is holomorphic if and only if Π0,1◦ df = 0, see [20, p. 43]. Assume that f : X → Y is holomorphic. We call x ∈ X a regular point of f if the map dxf : Tx1,0X → T_{f (x)}1,0 Y is surjective. If x is not regular, we call it a critical point. If y ∈ Y

such that all x ∈ f−1({y}) are regular, then y is called a regular value of f . In particular, if f−1({y}) = ∅ then y is regular. If y is not regular, it is critical. Note that if dim X < dim Y , then there are no regular points.

Proposition 1.32. Let f : X → Y be a holomorphic map between complex manifolds. The set of critical points of f is an analytic subset of X. If f is proper, then the set of critical values of f is an analytic subset of Y .

Proof. Let n = dim X and m = dim Y . If n < m, then the assertion is trivial by the remark above. So suppose that n ≥ m.

Consider the map Vm

C df : Vm C T 1,0_{X →}Vm CT 1,0_{Y . Then d}

xf is surjective if and only if

(^m C df )x: ( ^m C T1,0X)x→ ( ^m C T1,0Y )f (x)∼= C

is non-zero. Let U ⊂ X an open subset such that T U is trivial and let s1, . . . , sm be linearly

independent nowhere vanishing sections of T_CU . Define a holomorphic function h : U → C by h(x) = (Vm

C df )x(s1(x) ∧ . . . ∧ sm(x)). Then x ∈ U is a critical point of f if and only if z ∈Z (h).

This shows that the subset of X of critical points of f is analytic.

It follows from the proper mapping theorem that if f is proper, then the critical values of f form an analytic subset of Y .

By Sard’s theorem the set of critical values of f always has measure 0 in Y . The following fundamental theorem is analogous to the real situation.

Theorem 1.33 (Inverse function theorem). Let f : X → Y be a holomorphic map between complex manifolds of the same dimension. If x ∈ X is a regular point of f , then there are open neighbourhoods U ⊂ X of x and V ⊂ Y of f (x) such that f |U: U → V is biholomorphic.

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Proof. This follows from the inverse function theorem for open subsets of Cn, see Proposition 1.1.10 in [20].

We state two more useful propositions.

Proposition 1.34. Let f : X → Y be a holomorphic map between complex manifolds which is bijective. Then f is biholomorphic.

Proof. This follows from the analogous statement for open subsets of Cn, see [15, p. 19]. Proposition 1.35. Let y ∈ Y be a regular value of the holomorphic map f : X → Y . Then the fibre f−1(y) is a closed submanifold of X.

Proof. See [20], Corollary 2.6.14.

1.4 Meromorphic functions and divisors

Meromorphic functions on complex spaces are the analytic analogue of rational functions on algebraic varieties. They can be defined for general complex spaces, see e.g. [14, p. 97 and p. 119], but we only need them for manifolds.

Let X be a complex manifold. For x ∈ X, we define a set

Acx := {f ∈ OX,x| f is not a zero divisor} ⊂ OX,x.

The elements of Acx are called active germs. For U ⊂ X open, define

Ac(U ) := {f ∈ OX(U ) | fx∈ OX,x for all x ∈ U }.

See [14, p. 97] for a proof that this gives a sheaf (but not an OX-module) Ac on X. We define

a presheaf of rings M_Xpre on X as follows: for every open U ⊂ X, let M_Xpre(U ) be the ring of fractions Mpre X (U ) = f g | f, g ∈ OX(U ), g ∈ Ac(U ) .

Definition 1.36. Let X be a complex manifold. The sheaf of meromorphic functions on X is the sheaf MX associated to M_Xpre.

The elements of MX(U ) are called meromorphic functions on U . They are holomorphic

functions outside a proper analytic subset of U . If X is connected, then M (X) = MX(X) is a

field (see [10, p. 197]), the field of meromorphic functions of X.

It can be shown (see [15, p. 168]) that every meromorphic function on a projective manifold X ⊂ Pnis rational, i.e. it is the restriction to X of a meromorphic function ϕ on Pnof the form ϕ = p_q with p and q homogeneous polynomials of the same degree.

We will assign to each meromorphic function two hypersurfaces.

Definition 1.37. Let X be a complex manifold. A hypersurface in X is a closed complex subspace of codimension 1.

This means that around every point x in a hypersurface Z ⊂ X, there exists an open neigh-bourhood U ⊂ X and a holomorphic function f : U → C such that Z ∩ U = Z (f ).

So every hypersurface can be given by a collection {(Ui, fi)} where the Ui form an open cover

of X, each fi is a holomorphic function on Ui and Z ∩ Ui = Z (f). The fi are unique up to

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Now if ϕ ∈M (X), then ϕ is given by a collection {(Ui,f_gi_i)} for an open covering X =SiUi

and elements fi, gi ∈ OX(Ui), gi ∈ Ac(Ui) such that _gfi_i and f_gj_j on Ui∩ Uj. It follows that

fi|Ui∩Uj = h · fj|Ui∩Uj for some holomorphic function h on Ui∩ Uj, and similar for gi and gj.

Assume that the functions fi and gi do not have any common factors. The collections

{(U_i, fi)} and {(Ui, gi)} define hypersurfaces on X which we denote by (ϕ)0 and (ϕ)∞,

re-spectively. If ϕ is invertible, then (ϕ)∞= (_ϕ1)0. The underlying set of (ϕ)0 is denoted byZ (ϕ)

and is called the zero set of ϕ. Note that if ϕ is holomorphic, then this definition agrees with the definition of Z (ϕ) that we gave in section 1.1.

We consider sums of hypersurfaces on X.

Definition 1.38. Let X be a complex manifold. A prime divisor on X is a reduced irreducible hypersurface Z ⊂ X. A divisor on X is a locally finite formal sum of prime divisors P

ZaZZ

with aZ ∈ Z.

The set of all divisors on X is denoted by Div X. It forms a group with addition as group operation and the zero divisor as neutral element. We say that a divisor D = P

ZaZZ is

effective and write D ≥ 0 if aZ ≥ 0 for all i. We write D ≥ D0 if D − D0 is effective.

By the Riemann extension theorem, divisors on X are determined by what they look like outside subsets of codimension at least 2.

Proposition 1.39. Let X be a complex manifold and V ⊂ X an analytic subset of codimension at least 2. Then restriction gives an isomorphism Div X−∼→ Div X\V .

Proof sketch. We define the inverse map: let Z ⊂ X\V be an irreducible hypersurface defined by {(Ui, fi)} for some open covering X\V = ∪iUi. Each Ui is of the form ˜Ui∩ (X\V ) for some

open ˜Ui⊂ X; by the Riemann extension theorem, fi can be extended uniquely to a holomorphic

function ˜fi on ˜Ui. We send Z to the (irreducible) hypersurface on X given by {( ˜Ui, ˜fi)}. Extend

to divisors by linearity.

LetM_X∗ be the sheaf of invertible meromorphic functions on X and O_X∗ the sheaf of invertible holomorphic functions, considered as a subsheaf of M_X∗. Consider the group homomorphism Ψ : Div X → H0(X,M_X∗/O_X∗) which is defined as follows: if Z ⊂ X is a prime divisor given by {(Ui, fi)}, then Ψ(Z) is the element of (MX∗/OX∗)(X) represented by the element ofM∗(X)

given on Ui by fi. For a divisor D, the element Ψ(D) is defined by linearity.

Proposition 1.40. The map Ψ : Div X → H0(X,M_X∗/O_X∗ ) defined above is an isomorphism. Proof. See e.g. [20], Proposition 2.3.9.

In particular, any hypersurface Z ⊂ X, given by {(Ui, fi)}, can be seen as an effective divisor

D whose prime divisors are the irreducible components of Z. Precisely, D = Ψ−1([f ]), where f ∈M_X∗/O∗_X(X) is given on Ui by fi.

From the short exact sequence 0 → M∗_X → O∗_X → M∗_X/O∗_X → 0 we get a long exact cohomology sequence, of which we consider the following part:

H0(X, M∗_X) = M(X)∗→ H0(X, M∗_X/O_X∗) → H1(X, O_X∗).

Just like in the algebro-geometric situation, we have an isomorphism between H1(X, O_X∗) and Pic X, the group of holomorphic invertible sheaves on X (see [20], Corollary 2.2.10). It follows from this and Proposition 1.40 that we have an exact sequence

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The image of a meromorphic function f under the first map will be denoted by (f ). It equals the divisor Ψ−1((f )0) − Ψ−1((f )∞). For an explicit description, see e.g. [20, p. 78].

The second map will be denoted by D 7→ OX(D). Explicitely, it is described as follows. If

α ∈ (M_X∗/O_X∗)(X), then α is given by a collection {(Ui, fi)} for some open covering X = ∪iUi

and non-zero meromorphic functions fi on Ui such that f_fi_j ∈ OX∗(Ui∩ Uj). The map OX(−)

sends α to the line bundle determined by the transition maps fi

fj.

Divisors of the form (f ) for some f ∈ M (X)∗ are called principal. We call two divisors D and D0 linearly equivalent and write D ≡ D0 if D − D0 is a principal divisor. By the identity (f g) = (f ) + (g) for all f, g ∈ M (X)∗, linear equivalence defines an equivalence relation on Div X.

It follows from the exactness of (1.2) that OX(D) ∼= OX(D0) if and only if D and D0 are

linearly equivalent. We get an injective homomorphism Div X/ ≡ ,→ Pic X. We will show at the end of the section, see Proposition 1.50, that it is surjective when X is a projective manifold. For an example of a manifold for which Div X → Pic X is not surjective, see section 4.1.

In general, we can say that a line bundle L is in the image of Div X → Pic X if and only if it has global sections. If s ∈ H0(X, L), consider s as an element ofM_X∗/O_X∗(X) via transition maps of L.

Proposition 1.41. The line bundle OX(Z (s)) is isomorphic to L. If D is an effective divisor

on X, then there is a non-zero s ∈ H0(X, OX(D)) such that Z (s) = D.

If D is any divisor on X, we introduce two sets associated with D. Let |D| be the set of effective divisors which are linearly equivalent to D and let L(D) = {f ∈M (X) | D + (f) ≥ 0}. There is a surjective map L(D) → |D| sending f to D + (f ). If X is compact and connected, then D + (f ) = D + (f0) if and only if f is a constant multiple of f0. So we get a bijection P(L(D))∼= |D|.

Definition 1.42. Let X be a compact connected complex manifold. A linear system of divisors on X is a family L of linearly equivalent effective divisors corresponding to a linear subspace V of P(L(D)) for some divisor D. The dimension of L is the dimension of V . A complete linear system is a linear system of the form |D|.

If L is a line bundle of the form OX(D) for some effective divisor D, then we define |L| = |D|.

By Proposition 1.41, |L| is in bijection with H0(X, L).

Definition 1.43. Let X be a compact connected complex manifold and let L be a linear system of divisors on X. The fixed part of L is the biggest effective divisor F such that D ≥ F for all D ∈ L.

Note that {D − F | D ∈ L} is again a linear system.

We will now define the pullback and the pushforward of a divisor. For simplicity of notation, we restrict ourselves to connected manifolds.

Definition 1.44. Let f : X → Y be a holomorphic map between connected complex manifolds. Let Z ⊂ Y be a reduced irreducible hypersurface, given by {(Ui, fi)} for some open cover

Y = ∪iUi. Assume that f (X) * Z. The pullback of Z is the (not necessarily irreducible)

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So Φ∗Z is a formal sum of the irreducible components of f−1(Z), with positive coefficients. The pullback of a divisor on Y such that f (X) is not contained in any of its prime divisors is defined by linearity.

If f is surjective, then the pullback f∗D is defined for any divisor. It gives a homomorphism f∗: Div Y → Div X. Clearly, we get a commutative diagram

0 //M∗ Y // f∗ O∗ Y // f∗ M∗ Y/OY∗ // f∗ 0 0 //M∗ X //OX∗ //M∗X/OX∗ //0

which implies that f∗OY(D) ∼= OX(f∗D) for all D ∈ Div(Y ), and that f induces a

homomor-phism Div(Y )/ ≡ → Div(X)/ ≡.

Now assume that X is connected and compact. If L ⊂ |D| is a linear system, the pullback of L is defined by f∗L = {f∗D0 | D0 ∈ L}. On the level of L(D), the pullback is given by the injective map ϕ 7→ ϕ ◦ f . This shows that f∗L is again a linear system. However, the pullback of a complete linear system does not need to be complete: in general, f∗|D| is a strict subset of |f∗D|. Equality does hold if the map M (Y ) → M (X), ϕ 7→ ϕ ◦ f is an isomorphism. This is for instance the case if f is bimeromorphic, see section 3.1.1.

The pushforward of divisors is defined for surjective holomorphic maps which are proper and generically finite.

Definition 1.45. A holomorphic map f : X → Y between complex spaces is called generically finite if there exists a proper analytic subset Z ⊂ Y such that for every y ∈ Y \Z, the fibre f−1({y}) is finite.

Suppose that f : X → Y is a proper and generically finite surjective holomorphic map between reduced, irreducible complex spaces. In particular, dim(X) = dim(Y ). Let Z ⊂ Y be a proper analytic subset such that f−1({y}) is finite for all y ∈ Y \Z. Let S(X) and S(Y ) be the singular loci of X and Y , respectively. These are proper analytic subsets of X and Y . By the proper mapping theorem, f (S(X)) is a proper analytic subset of Y .

So Z0 := Z ∪ f (S(X)) ∪ S(Y ) is also a proper analytic subset of Y . Let Y0 = Y \Z0 and X0 = f−1(Y0) = X0\f−1_(Z0_{). Then the map f}0 _{:= f |}

X0: X0 → Y0 is a finite holomorphic map

between connected manifolds. So above its regular values, which form a connected subset of Y by Proposition 1.32, it is a covering map (in the topological sense) of a certain degree d. We call d the degree of f .

Note that if f : X → Y is proper and generically finite and Z ⊂ X is a reduced irreducible hypersurface such that dim f (Z) = dim(Z), then f |Z: Z → f (Z) is again proper and generically

finite.

Definition 1.46. Let f : X → Y be a proper and generically finite surjective holomorphic map between connected complex manifolds of the same dimension. If Z ⊂ X is a reduced irreducible hypersurface, the pushforward f∗Z of Z under f is defined as deg(f |Z)f (Z) if dim f (Z) = dim Z,

and zero otherwise.

The pushforward of a divisor is defined by linearity. It defines a group homomorphism f∗: Div(X) → Div(Y ). We will prove that it preserves linear equivalence.

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Let f : X → Y be a proper and generically finite map between connected complex manifolds of the same dimension. We define the norm function N : M∗(X) →M∗(Y ) by

N (ϕ)(y) = Y

x∈f−1_({y})

ϕ(x)

for all regular values y of f . So this is a product of deg(f ) many functions. We claim that f∗(ϕ) = (N (ϕ)) for all ϕ ∈M∗(X).

First, note that it suffices to check this locally. Namely, if U ⊂ X is open, then the diagram M∗_(X) // (M_X∗/O∗_X)(X) ∼ // Div X M∗_{(U )} //₍_M∗ X/O ∗ X)(U ) ∼ //Div U

where the vertical arrows are all restriction maps, is commutative. For the right square, this follows directly from the definitions of the horizontal maps. The left square commutes because it comes from the short exact sheaf sequence

0 → O∗_X →M_X∗ →M_X∗/O∗_X → 0.

Now let ϕ ∈M∗(X) and let V ⊂ Y be an open subset such that on V , we have ϕ = α_β for some holomorphic functions α and β on V , and such that f |_f−1_{(V )}: f−1(V ) → V is a covering map

(in the topological sense). So we can write f−1(V ) = qiVi with each Vi isomorphic to V via f .

Note that it is enough to prove the claim for α. Let αi= α|Vi. Then N (αi) = αi◦ f |

−1 Vi. Since

f |Vi is biholomorphic, it is clear that (αi◦ f |

−1 Vi ) = ((f |Vi)∗(αi)). For α we have (N (α)) = Y i N (αi) ! =X i (N (αi))

which by the above equals X i (f |Vi)∗(αi) = (f |V)∗( X i (αi)) = (f |V)∗(α).

This proves the claim and hence, f∗ preserves linear equivalence.

So we can define the pushforward of a linear system L ⊂ |D| on X by f∗L = {f∗D | D ∈ L}.

This is a linear subsystem of |f∗D|, and if the map M (Y ) → M (X) sending ϕ to ϕ ◦ f is an

isomorphism, then we have f∗|D| = |f∗D|.

We finish this section by giving some statements with regard to projectivity.

Let X be a compact connected complex manifold and let L be a line bundle on X. The base points of L are defined as the points x ∈ X such that s(x) = 0 for all global sections s of L. If L = OX(D) for some divisor D, the base points of L are those x ∈ X lying on all D0∈ |D|.

Suppose that L has no base points and that k = h0(X, L) > 0. Let s1, . . . , sk∈ H0(X, L) be

linearly independent sections. We define a holomorphic map ϕL: X → Pk−1

x 7→ (s1(x) : . . . : sk(x)).

Then ϕ∗O

Pk−1(1)) ∼= L (see [20, p. 85]). If L = OX(D) for some divisor D, this means that

|D| = ϕ∗

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Definition 1.47. A line bundle L without base points on a connected compact complex man-ifold X is called very ample if the map ϕL is a closed embedding. We call L ample if there is a

p > 0 such that L⊗p is very ample.

If X ⊂ Pn is a projective manifold, then the pullback of O_Pn(1) under the embedding is a

very ample line bundle. So X is projective if and only if X has an ample line bundle. Ample line bundles have two important properties:

Theorem 1.48. Let L be an ample line bundle on an n-dimensional compact connected manifold X. There is an n0 such that Hq(X, L⊗n) = 0 for all n ≥ n0 and q > 0.

Proof. This follows from Theorem B in [15, p. 159], using that ample line bundles are positive (see section 3.3.1).

Theorem 1.49 (Serre’s theorem). Let X be a connected compact complex manifold. If L is an ample line bundle, then for any line bundle L1 on X there is a k0 such that L1⊗ L⊗k is very

ample for all k ≥ k0.

Proof. This follows from [15, p. 192], using that ample line bundles are positive (see section 3.3.1).

It follows that if L is ample, then L⊗n is very ample for all n bigger than some n0.

We can now prove the surjectivity of Div X → Pic X if X is projective.

Proposition 1.50. Let X be a connected compact complex manifold which is projective. Then the natural homomorphism OX(−) : Div X → Pic X is surjective.

Proof. First, note that every very ample line bundle L is in the image of OX(−). Namely, if

ϕL: X → Pn is the closed embedding associated to L, let D = ϕ∗LH) for some hypersurface

H ⊂ Pn. Then L = OX(D).

Now if L1 is any line bundle, let k such that L1(k) := L0⊗ L⊗k is very ample (here L again

denotes a very ample line bundle). So L1(k) = OX(ϕ∗L1(k)H1) for a hypersurface H1 in some

Pm. Let D1= ϕ∗L1(k)H1. We find that L1 = OX(D1− kD).

Lastly, we state an important theorem of Bertini. Whenever a family of objects A = {Ax}x∈X

is parametrized by a complex space X, we say that a property holds for a general Ax if there

is a proper analytic subset Z ⊂ X such that the property holds for all Ax with x /∈ Z.

Theorem 1.51 (Bertini’s theorem for complex manifolds). Let X be an n-dimensional compact connected complex manifold and let L be a linear system on X.

a) A general element of L is smooth away from the base points of L.

b) If f : X → PN is a holomorphic map such that dim(f (X)) ≥ 2, then the pullback of a general hyperplane in PN _{under f is an irreducible (n − 1)-dimensional submanifold of X.}

Proof. For (a), see [15, p. 137].

For (b), note that f∗|H| is a linear system on X without base points, so smoothness follows from (a). The statement about the dimension is clear. It remains to show connectedness.

Use Stein factorization to factor f as h ◦ g, where g : X → Z is a proper surjective map with connected fibres and h : Z → PN _{is finite. Because g is in particular a closed map we have, for}

all subsets X0 ⊂ X, that X0 is connected if and only if g−1(X0) is connected. So it suffices to

prove that the general element of h∗|H| is connected. This can be shown as in the proof of [11], using that since X is normal, the space Z is also normal and therefore locally irreducible (see e.g. [13, p. 27-28]).

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2 Intersection theory on surfaces

With a surface, we mean a 2-dimensional compact connected complex manifold S. Kodaira’s projectivity criterion for surfaces is stated in terms of the intersection product. This is a symmetric form ( . ) : Pic(S) × Pic(S) → Z which is bilinear with respect to the tensor product on Pic(S). We give the analytic definition, for which we first introduce the first Chern class of a line bundle.

2.1 The first Chern class

Recall that on every complex manifold X, there exists a short exact sequence of sheaves 0 → Z → OX → OX∗ → 0

called the exponential sequence. Here Z is the constant sheaf with values in Z and the map OX → O_X∗ sends f to exp(2πif ). Corresponding to the exponential sequence, we have a long

exact sequence in cohomology. We consider the connecting homomorphism δ : Pic X ∼= H1(X, O_X∗ ) → H2(X, Z).

Definition 2.1. Let L be a line bundle on a complex manifold X. The first Chern class c1(L)

of L is the cohomology class δ(L) ∈ H2(X, Z).

We will identify the sheaf cohomology H2(X, Z) with singular cohomology and just write c1(L) ∈ H2(X, Z). By tensoring with R and C, respectively, we can consider c1(L) as an

element in H2(X, R) ∼= H2_dR(X, R) or in H2(X, C) ∼= H2_dR(X, C).

Remark 2.2. Note that in particular, L 7→ c1(L) is a group homomorphism. Moreover, if

f : X → Y is a holomorphic map between complex manifolds and L is a line bundle on Y , then c1(f∗L)) = f∗c1(L).

The image of L 7→ c1(L) in H2(X, C) is called the N´eron-Severi group of X. It is denoted

by NS(X). For K¨ahler manifolds and surfaces, the Lefschetz theorem on (1,1)-classes tells us exactly what it looks like. Recall (see section 1.3) that if X is a compact K¨ahler manifold, or if X is a surface and k = 2, we have a Hodge decomposition

Hk(X, C) ∼= M

p+q=k

Hp,q(X).

Theorem 2.3 (Lefschetz theorem on (1,1)-classes). Let X be a compact K¨ahler manifold or a surface. Then

NS(X) = Im H2(X, Z) → H2(X, C) ∩ H1,1_(X).

Proof. See [20], Proposition 3.3.2 (note that the assumption that X is K¨ahler is not used in the proof).

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So a class in H2(X, C) is the first Chern class of a line bundle if and only if it is integral and can be represented by a closed (1,1)-form.

Now let X be a general compact complex manifold. Then if L ∼= OX(D) for some divisor D,

we can give a description of c1(L) in terms of D. For this we need to define the (co)homology

class of a divisor.

To a closed submanifold Y ⊂ X of dimension k we can assign a homology class [Y ] ∈ H2k(X, Z)

as follows. Triangulate Y and let σ1, . . . , σrbe the top dimensional simplices. These correspond

to a 2k-cycleP

iσi on Y and thus to a homology class [Piσi] ∈ H2k(Y, Z). Let j : Y ,→ X be

the inclusion and let [Y ] = j∗[P_iσi].

For Y = X this gives us a generator of H2n(X, Z) (see [19, p. 238]), which is often called the

fundamental class of X. It occurs in the Poincar´e duality theorem for singular homology. We recall this theorem and its version for de Rham cohomology:

Theorem 2.4 (Poincar´e duality). Let X be a compact orientable manifold of (real) dimension n. Then for each k ≤ n, there are isomorphisms

P_X: Hk(X, Z) → Hn−k(X, Z)

P_dRX : Hk_dR(X, R) → Hn−kdR (X, R) ∨

where PX sends α ∈ Hk(X, Z) to the cap product α _ [X], and PdRX ([η])([ω]) =

R

Xη ∧ ω for all

[η] ∈ Hk_dR(X, R) and [ω] ∈ Hn−kdR (X, R).

Proof. For the statement in singular (co)homology, see [19]. For the statement in de Rham cohomology, see [4, p. 44].

Remark 2.5. Let us denote by QX: H0(X, R) → R the canonical isomorphism sending the class

of a point to 1. If [η] ∈ Hn_dR(X, R), its singular cohomology class [α] ∈ Hn(X, R) is given by α(σ) =R

ση for all n-cycles σ. So P −1

X ([η]) = [α] ^ [X] =

R

Xη · [x], where [x] ∈ H0(X, R) is the

class of any point x ∈ X. It follows that QX(PX−1([η])) =

R

Xη.

Consider again a compact complex manifold X of dimension n and let Y ⊂ X be a closed submanifold of dimension k. The element of H2n−2k_dR (X, R)∨ corresponding to [Y ] by Poincar´e duality is given by [η] 7→ R

Y η. This last integral can also be defined if Y is reduced but not

necessarily smooth: let S(Y ) be its singular locus and define Z Y η := Z Y \S(Y ) η.

For details, see [15, p. 32]. Now using Poindar´e duality, we can define the homology and cohomology class of any irreducible subspace of X.

Definition 2.6. Let X be an n-dimensional compact complex manifold and let Y ⊂ X an irreducible subspace of dimension k. The homology class of Y is the class [Y ] ∈ H2k(X, R)

which is Poincar´e dual to the element α ∈ H2k_dR(X, R)∨ given by [η] 7→ R

Y η. The cohomology

class of Y is [Y ]coh= P_dRX (α) ∈ H2n−2k_dR (X, R).

For a divisor D the classes [D] ∈ H2n−2(X, R) and [D]coh ∈ H2dR(X, R) are defined by linearity.

We have the following description of [D]coh.

Proposition 2.7. Let X be a compact complex manifold and let D a divisor on X. Then [D]coh= c1(OX(D)).

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Proof. See [15, p. 141].

The degree of a line bundle on a compact reduced irreducible curve C is an important invariant defined in terms of the first Chern class. Let ν : ˆC → C be the normalization of C. So ˆC is smooth, and if C is smooth, then ν is an isomorphism.

Definition 2.8. Let C be a compact reduced irreducible curve, not necessarily smooth, and let ν : ˆC → C be its normalization. Let L be a line bundle on C. The degree deg L of L is R

ˆ Cc1(ν

∗_L).

Note that the degree is additive: if L and M are line bundles on C, then deg(L ⊗ M) equals deg L + deg M. Also, if C is smooth and L = OC(D) for a divisor D =Piaipi on C, it follows

from Remark 2.5 that deg L = QC([D]) =P_iai.

We can now state the important theorem of Riemann-Roch for line bundles on compact reduced irreducible curves.

Theorem 2.9 (Riemann-Roch for curves). Let L be a line bundle on a compact reduced irre-ducible curve X, not necessarily smooth. We have the following equality:

χ(C, L) = deg L + χ(C, OC).

Proof. If C is smooth, then this follows from Theorem 5.1.1 in [20]. If C is not smooth, let ν : ˆC → C be its normalization and consider the short exact sequences

0 → OC → ν∗ν∗OC → Q → 0

0 → L → ν∗ν∗L → Q0 → 0.

Here Q and Q0 are skyscraper sheaves concentrated in the critical values of ν. Because these are isolated points and L is locally free of rank 1, we have Q ∼= Q0. It follows that

χ(C, L) − χ(C, OC) = (χ(C, ν∗ν∗L − χ(C, Q0)) − (χ(C, ν∗ν∗OC− χ(C, Q))

= χ(C, ν∗ν∗L) − χ(C, ν∗ν∗OC).

Because ν is finite, the last expression equals χ( ˆC, ν∗L) − χ( ˆC, O_Cˆ) which is deg(ν∗L) = deg(L)

by Riemann-Roch on ˆC.

2.2 The intersection product on Pic S

If S is a surface, then we have an isomorphism H4(S, Z) ∼= Z via [X] 7→ 1. Tensoring with R and passing to de Rham cohomology, we get a canonical isomorphism

H4_dR(X, R)−_{→ R}∼ [η] 7→

Z

S

η.

If [η] is an integral class, thenR_Sη is an integer. In particular, this holds for a cup product of two first Chern classes. So we can define the intersection product on Pic S as follows.

Definition 2.10. Let S be a surface. The intersection product on Pic S is the symmetric bilinear map ( . ) : Pic S × Pic S → Z given by (L.M) =R

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We can now state the theorem of Riemann-Roch for surfaces. Write L2 := (L.L).

Theorem 2.11 (Riemann-Roch for surfaces). Let L be a line bundle on a surface S. Then

χ(S, L) = χ(S, OS) +

1 2 L

2_{− (L.ω} S) .

Proof. This follows from Theorem 5.1.1 in [20].

Corollary 2.12. For two line bundles L, M on a surface S, we have (L.M) = χ(OS) − χ(L−1) − χ(M−1) + χ(L−1⊗ M−1).

Proof. Substituting the formula of Theorem 2.11 into the right hand side of the equation gives

−1 2 (L −1₎2_{− (L}−1_.ω S) − 1 2 (M −1₎2_{− (M}−1_.ω S) + 1 2 (L −1_{⊗ M}−1₎2_{− (L}−1_{⊗ M}−1_.ω S) . This simplifies to −1 2 L 2_{+ (L.ω} S) + M2+ (M.ωS) + 1 2 L 2_{+ M}2_{+ 2(L.M) + (L.ω} S) + (M.ωS) . which equals (L.M).

If D and D0 are divisors on S, we set (D.D0) = (OS(D).OS(D0)) and D2 = (D.D).

Remark 2.13. If D = (f ) for a meromorphic function f on S, then OS(D) ∼= OS, and

substi-tuting this into the formula of Corollary 2.12 gives that (D.L) = 0 for all line bundles L. It follows from the bilinearity of the intersection product that if D and D0 are linearly equivalent divisors, then (D.L) = (D0.L) for all line bundles L.

It is useful to have yet another description of the intersection product, the Poincar´e dual of Definition 2.10. For a compact connected complex manifold X, denote by

∩ : Hk(X, Z) × Hl(X, Z) → Hk+l−4(X, Z)

the product induced by the cup product on cohomology via Poincar´e duality. It can be shown (see [4, p. 69]) that this coincides with the usual definition of the intersection product in singular homology. As in Remark 2.5, let QX: H0(X, Z)

∼

−_{→ Z be the canonical isomorphism, sending} the class of a point to 1. Analogous to the notation [D] for the homology class of a divisor, let [L] denote the homology class in H2(X, Z) which is Poincar´e dual to c1(L). If S is a surface,

the intersection product (L.M) of two line bundles L, M on S is given by QS([L] ∩ [M]).

Using the above description and the following lemma, we can prove two useful identities about the intersection product. If f : X → Y is a continous map between compact connected oriented manifolds, denote by f! the transfer homomorphism PY−1◦ f∗◦ PX: H

k

(X, Z) → Hk(Y, Z) for any k. So we have a commutative diagram

Hk(X, Z) f! // PX Hk(Y, Z) PY Hk(X, Z) _f ∗ //Hk(Y, Z)

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Lemma 2.14 (Projection formula). Let f : X → Y be a proper map between compact connected oriented manifolds. Then for all x ∈ Hk(X, Z) and y ∈ Hk(Y, Z),

f!(x ^ f∗y) = f!(x) ^ y.

Proof. See [9, p. 314].

We recall the definition of the degree of a smooth orientation-preserving map f : X → Y between compact connected oriented manifolds of the same dimension n. Both Hn(X, Z) and

Hn(Y, Z) are isomorphic to Z, so the pushforward f∗: Hn(X, Z) → Hn(Y, Z) is multiplication

with some integer. This integer is called the degree of f and is denoted by deg(f ).

It follows from Proposition VIII.4.7 in [9] that if X and Y are compact connected complex manifolds and f is holomorphic, then this definition agrees with the one we gave in section 1.4. Corollary 2.15. Let f : X → Y be a proper map between compact connected oriented manifolds of the same dimension. For any element y ∈ Hk(Y, Z), we have f!f∗y = deg(f )y.

Proof. The class [X]coh _{∈ H}0_{(X, Z) is of the form [α] with α(σ) = 1 for every 0-cycle σ. So}

[X]coh^ x = x for all x ∈ H4(X, Z). In particular, f!f∗y = f!([X]coh^ f∗y). By the projection

formula, this equals f!([X]coh) ^ y. Now

f!([X]coh) = PY−1f∗[X] = deg(f )P −1

Y [Y ] = deg(f )[Y ] coh

so f!f∗y = deg(f )([Y ]coh^ y) = deg(f )y.

Note that since f∗: H0(X, Z) → H0(Y, Z) sends the class of a point to the class of a point,

the map QY ◦ f∗◦ Q−1_X : Z → Z is the identity. Equivalently, QY ◦ f∗ = QX. We will use this

observation in the proof of the following proposition.

Proposition 2.16. Let X and Y be two surfaces and let f : X → Y be a surjective map. (i) If L and M are line bundles on Y , we have (f∗L.f∗M) = deg(f )(L.M);

(ii) If D is a divisor on X and L a line bundle on Y , then (D.f∗L) = (f∗D.L).

Proof. Note that f is proper, so we can use the projection formula. For (i), let x = c1(f∗L)

and y = c1(M). On the left hand side of the formula, we find

P_Y−1f∗PX(c1(f∗L) ^ c1(f∗M)) = PY−1f∗([f

∗_{L] ∩ [f}∗_M])

and on the right hand side, using Corollary 2.15, we have deg(f )c1(L) ^ c1(M). Applying PY

to both formulas we find that f∗([f∗L] ∩ [f∗M]) = deg(f )[L] ∩ [M]. It follows that

(f∗L.f∗M) = Q_X([f∗L] ∩ [f∗M]) = QY(f∗([f∗L] ∩ [f∗M]))

= QY(deg(f )[L] ∩ [M])

= deg(f )(L.M).

For (ii), let x = [D]coh and let y = c1(L). On the left hand side of the projection formula, we

find

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and on the right hand side

P_Y−1f∗PX([D]coh) ^ c1(L) = P_Y−1f∗[D] ^ c1(L).

Applying PY to both of them, we find that f∗([D] ∩ [f∗L]) = f∗[D] ∩ [L]. This implies that

(D.f∗L) = Q_X([D] ∩ [f∗L]) = QY(f∗([D] ∩ [f∗L]))

= QY(f∗[D] ∩ [L])

= (f∗D.L).

Finally, we study the intersection number (L.C) for L any line bundle and C a smooth irreducible curve on S.

Lemma 2.17. Let S be a surface, L a line bundle and C a smooth irreducible curve on S. Then (C.L) = deg(L|C).

Proof. Denote by [C]coh

C the cohomology class of C in H0(C, Z) and by [C]cohS ∈ H2(S, Z) and

[C]S ∈ H2(X, Z) the cohomology class and homology class, respectively, of C as a subspace of

S. The lemma again follows from the projection formula, with for f the inclusion j : C ,→ S, for x the class [C]coh_C and for y the class c1(L) ∈ H2(S, Z). On the right hand side of the projection

formula we have

j!([C]cohC ) ^ c1(L) = [C]cohS ^ c1(L)

and on the left hand side

P_S−1j∗PC([C]cohC ^ j∗c1(L)) = PS−1j∗PC(c1(j∗L))) = PS−1j∗[L|C].

Taking PS on both sides gives [C]S∩ [L] = j∗[L|C]. Now applying QS gives that

(C.L) = QS(j∗[L|C]) = QC([L|C]) =

Z

C

L|_C = deg(L|C).

2.3 Intersecting curves

If C and C0 are distinct reduced irreducible curves on a surface, then (C.C0) can be described in terms of the intersection C ∩ C0. This is where the name ‘intersection product’ comes from. Let p ∈ C ∩ C0 and let f, g ∈ OS,pbe local equations at p for C and C0, respectively. The ring

O_S,p is noetherian and, as S is smooth, a unique factorization domain. Because C and C0 are distinct, f and g have no common factor. Therefore the ring OS,p/(f, g) has Krull dimension

zero, which implies that it is finite-dimensional as a vector space over C. Definition 2.18. The intersection multiplicity of C and C0 at p is the number

mp(C, C0) = dimCOS,p/(f, g).

The curves C and C0 form a system of local coordinates at p if and only if there is an isomorphism of algebras OS,p → OC2,0 ⊂ C[[x, y]] sending f to x and g to y. This means that

f and g generate the maximal ideal mp ⊂ OS,p, so mp(C, C0) = 1. In this case, we will say that

C and C0 intersect transversally in p.

Because the curves C and C0 are compact, irreducible and distinct, the set C ∩ C0 consists of finitely many points. So summing up the intersection multiplicities at all the points in C ∩ C0 gives a non-negative integer, which turns out to be equal to (C.C0).

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Lemma 2.19. If C and C0 are two distinct irreducible curves on S, then (C.C0) = X

p∈C∩C0

mp(C, C0).

Proof. See [2], Theorem I.4.

Suppose that the surface S is projective and let H be the pullback of a hypersurface under some closed embedding of X into a projective space. Then we can ‘move H within its equiv-alence class’. This means that if C ⊂ X is any curve, we can assume that H and C intersect transversally.

By Serre’s theorem, if D is a divisor on S, then there is an n such that D + nH is linearly equivalent to the pullback of a hypersurface under a closed embedding in a projective space. So we can write D ≡ A − B, where A = D + nH and B = nH are pullbacks of hypersurfaces. Therefore we can always assume that divisors on projective surfaces intersect transversally.

2.4 Intersecting on blow-ups

In the next chapter the blow-up of a manifold in a point will play an important role. We recall some definitions. For details, see e.g. [15].

Let X be an n-dimensional complex manifold. Let p ∈ X and let ψ : U −∼→ U0 _{⊂ C}n _{be a}

system of local coordinates around p. Let V be the open subset of U × Pn−1 defined as follows: V = {(x, (y1 : . . . : yn)) | ψ(x)iyj = ψ(x)jyi for all i, j}.

The projection π0: V → U on the first coordinate is a surjective holomorphic map which is

biholomorphic when restricted to π₀−1(U \{p}). The blow-up of X in x is ˜X := (X\U ) ∪ V , the space obtained by replacing U with V . This is again an n-dimensional manifold. It comes with a surjective holomorphic map π : ˜X → X which equals π0on V and is the identity elsewhere. The

inverse image E := π−1({p}) is called the exceptional divisor. If Z ⊂ X is a closed subspace, we define the proper transform ˆZ of Z as the closure in ˜X of π−1(Z\{p}).

Remark 2.20. Via the map

E → P(TpX) (p, (y1 : . . . : yn)) 7→ y1 ∂ ∂z1 : . . . : yn ∂ ∂zn

which is biholomorphic, we can identify the points of E with the tangent directions at p. It follows that E ∼= Pn−1. As is proven in [15, p. 185], the line bundle O_X˜(E)|E can be identified

with the canonical bundle O(−1) on Pn−1.

Lemma 2.21. Let π : ˜X → X be the blow-up of a manifold X in a point p. Then π is proper. Proof. As the inverse image π−1({x}) is compact for all points x ∈ X, we only need to check that π is closed.

Let U be an open neighbourhood of p and let V ⊂ U × Pn−1 be as in the definition of the blow-up, so ˜X = (X\U ) ∪ V . Note that V is closed in U × Pn−1. Because Pn−1is compact, the projection map U × Pn−1 is closed, and thus π|V : V → U is closed. Let E = π−1({p}). Then

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In particular, if π : ˜S → S is the blow-up of a surface, then ˜S is compact. So it is a surface and we can consider the intersection product on Pic( ˜S).

Lemma 2.22. Let π : ˜S → S be a blow-up of the surface S in a point p, with exceptional divisor E = π−1({p}). For a divisor D and line bundles L, M on S, we have

i) (π∗D.E) = 0;

ii) (π∗L.π∗_{M) = (L.M).}

Proof. Both equalities follow from Proposition 2.16. For (i), note that π∗E is the zero divisor,

so we have (π∗D.E) = (D.π∗E) = 0. For (ii), use that deg(π) = 1.

It follows directly from Remark 2.20 and Lemma 2.17 that Corollary 2.23. (E.E) = deg(O_P1(−1)) = −1.

Now let C be an irreducible curve on S which passes the point p with multiplicity m (so it goes through p in m different tangent directions).

Lemma 2.24. π∗C = ˆC + mE.

Proof. Because π|E is biholomorphic, there is some k such that π∗C = ˆC + kE. By Lemma

2.22 and Corollary 2.23, we find that

0 = (π∗C.E) = ( ˆC + kE.E) = ( ˆC.E) − k.

Now the curve ˆC intersects E transversally in m points, namely, the points corresponding to the tangent directions in which C passes p. It follows that ( ˆC.E) = m and thus that k = m.

Kodaira's projectivity criterion for surfaces

MSc Mathematics

Master’s Thesis