Fibered algebraic surfaces over the projective line

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Universit`

a degli Studi di Pisa

FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea magistrale in Matematica

Tesi di laurea magistrale

Fibered algebraic surfaces over the projective line

Candidato:

Federico Cesare Giorgio Conti

Matricola 540126

Relatore:

Prof.ssa Rita Pardini

Controrelatore:

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Abstract

This thesis is dedicated to the article of Beauville [6] which tries to answer a classical problem first studied by Szpiro: given a fibered algebraic surfaces over the projective line which is the minimum number of singular fibres that such a fibration must have? In the first chapter we give some preliminary results about algebraic varieties, line bundles, divisors as in [17].

In the second chapter the theory of complex algebraic tori and abelian varieties is developed as in [9] and [17]. In particular it is proven that the Siegel upper half space, which parametrizes the polarized abelian varieties with a given symplectic basis, is biholomorphic to a bounded domain (proven by Siegel in [33]). This statement is fundamental in the proof of the final theorem of chapter 4.

In the third chapter, some other tools concerning complex tori are ex-plained. In particular we define, as in [5], the Albanese and the Picard torus on a variety X, and we give conditions for them to be abelian varieties. At the en of the chapter there is a sketch of the proof of Torelli’s Theorem.

In the fourth chapter we give some properties concerning algebraic sur-faces and curves on it.At the end of the chapter the proof of the main theo-rem concerning fibered surfaces is given: excluding the isotrivial case, every fibered surface has least 3 singular fibres.

In the fifth chapter we recall the theory of abelian covers as in [29]. This is very useful in the construction of examples of fibered surfaces. In particular at the end of the chapter an example of fibered surface with exactly 3 singular fibres for every fixed genus g of the general fibre is given.

In the last chapter we study the case of semi-stable fibrations, i.e. fi-brations whose fibres have only nodal singularities. In this case the result of chapter 4 can be improved, showing that such a fibration has at least 4 singular fibres (proved by Beauville in [6]). At the end of the chapter some examples of semi-stable fibrations are given, in particular a semi-stable fibration with general fibre of genus 1 with exactly 4 singular fibres.

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Chapter 1

Preliminaries

In this chapter we collect, sometimes without proof, proposition and defi-nitions which we will need later. In this chapter we give some information about complex manifold with analytic structure, but many of these results remain valid in the algebraic case: thanks to the general Theorems of Serre (cf. [32]) there exists a bijection between algebraic and analytic coherent sheaves which preserves exactness and cohomology (in particular we have a bijection between regular (rational) and holomorphic (meromorphic) func-tions). Moreover the Chow’s Theorem ( [5] Theorem I.19.2) assures that every analytic subvariety of the projective space is algebraic.

1.1 Complex Manifold

For a deepening of the next four sections cf. [17] chapter 0 and 1.

Definition 1.1.1. A complex manifold X is a differentiable manifold admit-ting an atlas of charts φα to open sets in Cn such that the transition maps

φα◦ φ−1_β are holomorphic.

A function f is holomorphic if f ◦ φα is in every chart. A collection

z= (z1, . . . , zn) is a holomorphic coordinate system if φα◦ z−1 and z ◦ φ−1α

are holomorphic. A map f : X → Y between two complex manifolds is holomorphic if it is given in terms of local coordinates on Y by holomorphic functions.

Example 1.1.2. 1. A one dimensional complex manifold is usually called a Riemann surface.

2. The simplest example of complex manifold which needs more than one single chart is the projective space. The charts are φi: Ui= {[z0 : . . . :

zn]| zi 6= 0} → Cndefined as [z1 : . . . : zn] 7→ (z0/zi, . . . , [zi/zi, . . . , zn/zi).

3. Other important examples of complex manifold are the complex tori, which will be studied in the second chapter.

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In a complex manifold there are three different notions of tangent space for a given point p ∈ X:

1. TR,p(X) is the usual real tangent space to X at p, where we consider

X as a real manifold of dimension 2n. It can be realized as the space of R-linear derivations on the ring of real-valued C∞ _{functions in a}

neighbourhood of p. If z = (z1, . . . , zn) is a holomorphic coordinate

system with zi = xi+ iyi, it can be proved that

T_R,p(X) = R ∂ ∂xi , ∂ ∂yi .

2. TC,p(X) = TR,p(X) ⊗RC is called the complexified tangent space to X

at p. It can be seen as the space of C-linear derivations in the ring of complex-valued C∞ _{functions on X around p. We can write}

T_C,p(X) = C ∂ ∂xi , ∂ ∂yi = C ∂ ∂zi , ∂ ∂zi where ∂ ∂zi = 1 2 ∂ ∂xi − i ∂ ∂yi , ∂ ∂zi = 1 2 ∂ ∂xi + i ∂ ∂yi . 3. T0

p(X) = C{∂/∂zi} ⊂ TC,p(X)is called the holomorphic tangent space

to X at p. It can be realized as the subspace of TC,p(X)consisting of

derivations that vanish on antiholomorphic functions.

Usually we will deal with this last notion of tangent space. In particular a C∞ _{map f : X → Y is holomorphic if and only if f}

∗(Tp0(X)) ⊆ Tf (p)0 (Y ).

1.2 Sheaves and cohomology

Definition 1.2.1. Given a topological space X, a sheaf F assoicates to each open set U ⊆ X a group F(U), called the sections of F over U, and to each pair U ⊂ V of open sets a map rV,U: F(V ) → F(U ), called the restriction

map, satisfying:

1. rU,U = Id and for any triple U ⊂ V ⊂ W of open sets

rW,U = rV,U · rW,V.

Because of this, it makes sense to write σ

U instead of rV,U(σ);

2. for any pair of open sets U, V and sections σ ∈ F(U), τ ∈ F(V ) such that σ

U ∩V = τ

U ∩V there exists a unique section ρ ∈ F(U ∪ V ) with

ρ

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Given a point x ∈ X, we define the stalk of F in x to be Fx= lim_−→

U 3x

F(U ). Example 1.2.2. Examples of sheaves are:

• the sheaf of smooth functions C∞_;

• the sheaf of C∞ _{p-forms P}p_;

• the sheaf of holomorphic (regular) functions OX;

• the sheaf of nonzero hoomorphic (regular) functions O∗ X;

• the sheaf of holomorphic (regular) p-forms Ωp

X (if the dimension of X

is n, we sometimes write KX instead of Ωn_X);

• the sheaf of C∞ _{forms of type (p, q) P}p,q_;

• the sheaf of holomorphic (regular) sections of a holomorphic vector bundle E (which will be defined later) OX(E);

• the sheaf of meromorphic (rational) function MX;

• the sheaf of meromorphic (rational) functions not identically zero M∗_.

Definition 1.2.3. A map of sheaves F −→ Gα is given by a collection of homomorphism {αU: F(U ) → G(U )}U ⊆X such that the maps αU commutes

with the restriction maps. It is then possible to define ker(α) and Im(α) (cf. [17] pages 36-37).

Let F be a sheaf on X, and U = {Uα} a locally finite open cover. We

define C0(U , F) =Q αF(Uα), C1(U , F) =Q α6=βF(Uα,β) ... Cp(U , F) =Q α06=α16=···6=αpF(Uα0,...,αp)

where F(Uα0,...,αp) means F(Uα0 ∩ . . . ∩ Uαp). It is defined a coboundary operator δ: Cp(U , F) → Cp+1(U , F) by the formula (δσ)i0,...,ip+1 = p+1 X j0 (−1)j_σ i0,..., bij,...,ip+1.

It is easy to see that δ2 _{= 0}_{. We define}

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and

Bp(U , F) = Im δ,

then the p-th cohomology group related to the cover U is Hp(U , F) = Z

p_{(U , F)}

Bp_{(U , F)}.

Finally we define the p-th cohomology group of the sheaf F on X to be the direct limit of the Hp_{(U , F)’s as U becomes finer:}

Hp(X, F) = lim_−→

U

Hp(U , F)

(cf. [17] pages 38-39). From now on, we will denote with h0_{(X, F)} _the

di-mension as complex vector space of H0_{(X, F).}

The following properties are satisfied (cf. [17] page 40). • If the covering U is acyclic for the sheaf F (i.e. Hq_{(U , F)(U}

i1,...,ip, F) = 0 for every q > 0 and i1, . . . , ip) then H∗(U , F) = H∗(X, F).

• For every short exact sequence of sheave 0 → E → F → G → 0 there exists a long exact sequence

0 → H0(X, E → H0(X,F) → H0(X, G) → H1(X, E → H1(X,F) → H1(X, G) → . . .

...

· · · → Hp(X, E → Hp(X,F) → Hp(X, G) → . . . .

• Given a sheaf F and a resolution of it (i.e. a long exact sequence of sheaves of the shape 0 → F → F0 d0

−→ F1 d1

−→ . . .−−−→ Fdp−1 p dp

−→ . . .) we have that Hp_{(X, F) = ker d}

p/Im dp−1. In particular we can prove the

de Rham and Dolbeault Theorems. Indeed by the Poincaré Lemmas ( [17] page 25) we have that the following sequences are resolution of sheaves: 0 → R → P0 d0 −→ P1 d1 −→ P2→ . . . 0 → Ωp _{→ P}p,0 ∂p,0 −−→ Pp,1 ∂p,1 −−→ Pp,2_{→ . . . .}

In particular by the previous argument it follows that H_dRp (X) ' Hp(X, R) and H_∂p,q(X) ' Hq_{(X, Ω}p₎ where Hp dR(X) = ker dp/Im dp−1 and H p,q ∂ (X) = ker ∂p,q/Im ∂p,q.

Definition 1.2.4. Given a sheaf F on X the Euler-Poincaré characteristic of F is χ(F) = n X i=0 (−1)ihi(X, F).

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1.3 Line bundles and divisors

Definition 1.3.1. Let X be a complex manifold. A holomorphic vector bundle is a complex manifold E with a holomorphic map π : E → X such that for every x0 ∈ X there exists an open set U in M containing x0 and a

biholomorphic map

φU: π−1(U ) → U × Ck

where k is a fixed integer called the rank of the vector bundle E. In particular a vector bundle of rank 1 is called a line bundle.

Remark 1.3.2. For every complex vector bundle π : E → X and two open sets U, V ⊆ X we have maps gU V : U ∩ V → GL(k) given by gU V(x) =

(φu◦ φ−1_V )

{x}×Ck which are holomorphic functions (called the transition functions) satisfying the following identities

gU V(x)gV U(x) = Id gU V(x)gV W(x)gW U(x) = Id. (1.1)

Conversely given an open cover U = {Uα} of X and holomorphic

func-tions gα,β: Uα ∩ Uβ → GL(k) satisfying realations 1.1 there is a unique

holomorphic vector bundle π : E → X with transitions functions {gαβ}: this

is just the space

[

α

Uα× Ck/ ∼

where the equivalence is given by (x, λ) ∼ (x, gαβ(x) · λ) (cf. [17] page 66).

In particular, in the case of line bundles, we have an equivalence between the set of line bundles and the cohomology group H1_{(X, O}∗

X) given by the

transition functions. Indeed it can be proved (cf. [17] page 133) that two line bundles with transition functions gαβ and g_αβ0 are equivalent if and only if

there exist nonzero functions fα∈ Uα such that

g0_αβ = fα fβ

gαβ.

Remark 1.3.3. As a general rule, it is possible to induce operations on vector bundles by operations on vector spaces. Given two line bundles E and F on X of rank k and l with transition functions gαβ and hαβ, we can construct

(cf. [17] page 67)

• the dual bundle E∗ _{given by transition functions j}

αβ = gαβ ∈ GL(k);

• the direct sum bundle E⊕F with transition functions jαβ =

gαβ 0 0 hαβ ∈ GL(k + l);

• the tensor bundle E ⊗ F with transition functions jαβ = gαβ⊗ hαβ ∈

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• the alternating product bundle Vr _{with transition functions j} αβ =

∧r_g

αβ ∈ GL(∧rCk).

In particular in the case of the line bundles, the tensor products and the dual bundle allow us to give the set of line bundles a group structure. Given two line bundles we have that the product is given by the tensor bundle, while the inverse is given by the dual bundle. This group structure is the same of H1_{(X, O}∗

X) considering the equivalence of set exposed in Remark

1.3.2.

Definition 1.3.4. We define the Picard group P ic(X) the set of all line bundles on X with the group structure as in Remark 1.3.3 (or, equivalently, the cohomology group H1_{(X, O}∗

X)).

Definition 1.3.5. Given a holomorphic map f : X → Y and a holomorphic vector bundle E on Y it is possible to define the pullback bundle f∗_E_{. If}

φU: EU → U × Ck is a set of trivializations of E, then φU◦ f : f∗Ef−1_{(U )}→ f−1(U ) × Ck _{is a set of trivializations for f}∗_E_.

Another characterization of the Picard group, in the case of complex manifolds, is given by the group of divisors modulo principal divisors. Definition 1.3.6. A divisor D on a complex manifold X is a locally finite formal linear combination

D=XaiVi

of irreducible analytic hypersurfaces of X where the ai are integers. A divisor

is effective if every ai ≥ 0. The set of divisors has a natural structure of group

and is called Div(X).

Remark 1.3.7. An analytic hypersurface in X is a subspace locally given by the zeros of a holomorpic function. Given a meromorphic function defined on X it is possible to associate a divisor (f) = PV ordV(f ) (cf. [17] page

130-131): such a divisor is called principal. Moreover one can prove that this group is isomorphic to the cohomology group H0_{(X, M}∗_/O∗₎ _{(cf. [17] page}

132).

Remark 1.3.8. By the exact sequence of sheaves 0 → O∗ → M∗ → M∗/O∗ → 0,

considering the corresponding long exact sequence in cohomology, we obtain an identification

H0(X, M∗/O∗) H0_{(X, M}∗₎ ' H

1_{(X, O}∗

).

In particular this means that there is a correspondence between the Picard group and the group Div(X) modulo principal divisors.

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Remark 1.3.9. Notice that the group of sections of a vector bundle π : E → X (i.e. the morphisms σ : U ⊆ X → E such that π ◦ σ = Id) forms a sheaf on X. In particular it makes sense to calculate the cohomology of a line bundle or of a divisor. Given a line bundle L (equivalently a divisor D), we will write Hi_{(X, L)}_{(equivalently H}i_{(X, D)}_{) for the cohommology groups of the}

associated sheaf, and we will write hi_{(X, L)}_{(equivalently h}i_{(X, D)}_{) for their}

complex dimensions.

Remark 1.3.10. Let X be a compact complex manifold of dimension n. The exact sequence of sheaves

0 → ZX → OX exp

−−→ O_X∗ → 0 gives a long exact cohomology sequence

· · · → H1_{(X, Z}X) → H1(X, OX) → H1(X, O∗X) →

→ H2_{(X, Z}_X) → H2(X, OX) → H2(X, OX∗ ) → . . . (1.2)

In particular we have the following short exact sequence 0 → H 1_{(X, O} X) H1_{(X, Z} X) → H1_{(X, O}∗ X) δ − → N S(X) → 0 where, as we will show later (Reamrk 3.2.3), H1_{(X, O}

X)/H1(X, ZX) is a

complex torus and NS(X) is a subgroup of H2_{(X, Z}

X) called the

Néron-Severi group. For a line bundle we define the first Chern class c1(L) of L to

be δ(L) ∈ H2_{(X, Z}

X) ⊆ HdR2 (X). In a similar way we define the first Chern

class of D to be c1([D]). It is obvious by the definition that c1 is a morphism

of groups (i.e. c1(L ⊗ L0) = c1(L) ⊗ c1(L0)and c1(L∗) = −c1(L)) and that it

commutes with f∗_{for a given morphism f : X → Y (i.e. c}

1(f∗L) = f∗c1(L)).

1.4 Kähler manifolds and Hodge decomposition

Definition 1.4.1. Let X be a complex manifold of dimension n. A hermitian metric ds2 _{on X is given by a positive definite hermitian inner product}

( , )z: Tz0(M ) ⊗ Tz0(M ) → C

on the holomorphic tangent space at z for each z ∈ X, depending smoothly on z. That is for local coordinates the functions

hij(z) = ∂ ∂zi , ∂ ∂zj z

are C∞ _{and is given by}

ds2=X

ij

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Definition 1.4.2.Given a hermitian metric on X we define ω = −1/2 Im ds2 as the associated (1, 1)-form of the metric.

Remark 1.4.3. If ds2 _{is locally defined as ds}2 ₌ P

ijhij(z)dzi ⊗ dzj then

ω is locally defined as ω = −i/2 P_ijhijdzi∧ dzj. Moreover there is a 1-1

correspondence between the set of hermitian metric on X and the set of real (1, 1)-forms such that, for every z ∈ X and v ∈ T_z0(X), i<ω(z), v ∧ v> > 0, which are called positive (cf. [17] page 29).

Example 1.4.4. • The hermitian metric on Cn _{given by}

ds2 =

n

X

i=1

dzi⊗ dzi

is called the Euclidean or standard metric.

• If Λ is a lattice of maximal rank, then the metric on Cn_/Λ _{given by}

ds2 =X

i

dzi⊗ dzi

is again called the Euclidean metric on Cn_/Λ.

• Let z0, . . . , zn be coordinates of Cn+1 and denote by π : Cn+1\ {0} →

Pn

C the standard projection map. Let U ⊆ P n

C be an open subset and

Z: U → Cn+1_{\ {0}} _{a holomorphic map such that π ◦ Z = Id. Then}

the differential form

ω= i

2∂∂logkZk

2

is a globally defined positive (1, 1)-form and hence defines a hermitian form on Pn

Ccalled the Fubini-Study metric (cf. [17] page 31).

• If Y ⊆ X is a subvariety, then for every z ∈ Y we have a natural inclusion

T_z0(Y ) ⊆ T_z0(X)

and, consequently, a hermitian metric on X induces a hermitian met-ric on Y . In particular the Fubini-Study metmet-ric is defined for every projective variety.

Definition 1.4.5. Given a hermitian metric ds2 on a manifold X we say that it is Kähler if its associated (1, 1)-form ω is d closed (where d is the map of the de Rham resolution). A complex manifold which can be equipped with a Kähler metric is called a Kähler manifold.

Remark 1.4.6. Every projective variety is a Kähler one. Indeed it is possible to prove that the Fubini-Study metric is Kähler (cf. [17] page 109). Other examples of Kähler manifolds are complex tori and complex vector spaces with the Euclidean metric. This will be very important because most of the varieties we will use later on are projective and, as we will see soon, in the Kähler varieties the Hodge decomposition holds.

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Proposition 1.4.7(Hodge Decomposition). For a compact Kähler manifold X, the complex cohomology satisfies

Hr(X, C) ' M

p+q=r

Hp,q(X),

Hp,q(X) ' Hp,q_(X).

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Proof. See [17] page 116.

Proposition 1.4.8(Kodaira-Serre Duality). For a complex manifold X and a coherent sheaf F we have

1. Hn_{(X, K} X)

∼

−_{→ C;}

2. there is a perfect pairing

Hq(X, F) ⊗ Hn−q(X, F−1⊗ KX) → Hn(X, KX).

Proof. For a proof see [17] pages 102-103 or [23] section III.7.

Remark 1.4.9. By the previous proposition we obtain that Hq_{(X, F) '}

Hn−q(X, F−1 ⊗ K_X)∗ and in particular hq_{(X, F) = h}n−q_{(X, F}−1 _{⊗ K} X).

In the case F = Ωp _{we obtain}

Hp,q(X) ' Hn−p,n−q(X)∗

and in particular

hp,q(X) = hn−p,n−q(X).

Remark 1.4.10 (Hodge diamond). We can put the cohomology groups of a compact Kähler manifold diagrammatically in the Hodge diamond, so that the k-th cohomology group of X can be read off as the sum of the groups in the k-th orizontal row. The Serre duality gives a simmetry about the center of the diamond, while conjugation gives simmetry about the center vertical

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line. If, for example, X has dimension 3 the Hodge diamond looks as follows. h3,3 h3,2 h2,3 h3,1 h2,2 h1,3 h3,0 h2,1 h1,2 h0,3 h2,0 h1,1 h0,2 h1,0 h0,1 h0,0

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Chapter 2

Complex Tori

Let V be a complex vector space of dimension g and Λ be a lattice in V (i.e. a free Z-module of rank 2g whose generators span V as a real vector space). The lattice Λ acts on V by addition.

Definition 2.0.1. A complex torus of dimension g is the quotient V/Λ where V and Λ are as before.

Remark 2.0.2. A complex torus X is a connected, compact, complex mani-fold and has a natural structure of an abelian group. Moreover V can be seen as the universal covering space of X through the quotient map π : V → X, thus Λ (which is the kernel of π, seen as a group morphism) is canonically isomorphic to π1(X) and, because it is abelian, it is also canonically

iso-morphic to H1(X, Z). The canonical isomorphism h : Λ → H1(X, Z) is given

explicitly as follows: to every λ ∈ Λ we associate the path t 7→ tλ(0 ≤ t ≤ 1). Moreover the torus is locally biholomorphic to V thus, for every point x ∈ X, the tangent space at x is identified by translation with the tangent space at the origin, which is canonically isomorphic to V . Hence the tangent (and the cotangent) sheaf of X is canonically isomorphic to the free sheaf V ⊗COX

(respectively V∗ _⊗

COX where V

∗ _{is the dual of V ). In particular there}

is an isomorphism δ : V∗ _{→ H}0_{(X, Ω}1

X), given explicitly as follows: a form

x∗ ∈ V∗ defines a function on V satisfying x∗(v + λ) = x∗(v) + constant, for all v ∈ V and λ ∈ Λ. This ensures that the differential dx∗ _{defines a form}

δx∗ ∈ H0_{(X, Ω}1

X). In particular we have that

Z

hλ

δx∗ = <x∗, λ>.

If we choose bases e1, . . . , eg of V and λ1, . . . , λ2g of the lattice Λ we then

can write λi = Pgj=1λjiej obtaining a matrix Π ∈ Mg,2g(C) whose entries

are given by λji. This matrix is called the period matrix of the complex torus

and completely determines X but it depends on the choice of the bases. A question which immediately arises is: given a matrix Π ∈ Mg,2g(C) is this

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Proposition 2.0.3. Π ∈ Mg,2g(C) is the matrix of a complex torus if and

only if P = Π

Π ∈ M2g(C) is non singular

Proof. We first observe that Π is a period matrix if and only if its columns span V as a real vector space if and only if they are linearly independent.

Now suppose that the columns of Π are linearly dependant over R, then there exist 0 6= x ∈ R2g _{such that Πx = 0, hence P x = 0, showing that P is}

singular.

Conversely, if P is singular, then there exist x, y ∈ R2g_{, not both zero,}

such that P (x+iy) = 0. Hence Π(x+iy) = 0 and Π(x−iy) = Π(x + iy) = 0 and consequently Πx = Πy = 0 showing that the column of Π are linearly dependant over R.

We now study the homomorphisms between complex tori. There are two obvious types of holomorphic maps between complex tori, which are homomorphisms and translations and we’ll soon show that all holomorphic maps are composition of such functions.

Let X = V/Λ and X0_{= V}0_/Λ0 _{be complex tori of dimension g and g}0_.

Definition 2.0.4. A homomorphism of X to X0 is a holomorphic map f: X → X0 which is also a group homomorphism.

A translation by an element x0 ∈ X is defined by x 7→ x + x0 and the

map is denoted by tx0.

Proposition 2.0.5. Let h: X → X0 be a holomorphic map. Then: • there is a unique homomorphism f : X → X0 _{such that h = t}

h(0)◦ f;

• there is a unique C-linear map F : V → V0 _{with F (Λ) ⊂ Λ}0 _inducing

f;

• f is determined by f∗_{: H}0_(X0_,_Ω1

X0) → H0(X, Ω1_X).

Proof. Let’s define f = t−h(0)h, then, because f ◦ π(Λ) = 0, we can lift the

map f ◦ π to a map F : V → V0 _{such that F (0) = 0 and f ◦ π = π}0_{◦ F}_.

V V0

X X0

F π π0

f

The last equivalence implies that F (v + λ) − F (v) ∈ Λ0 _{and thus, fixing}

λ ∈Λ, the continuous map v 7→ F (v + λ) − F (v) is constant and we obtain that F (v + λ) = F (v) + K(λ) where K(λ) is a constant depending only on λ. So the partial derivatives of F are 2g-fold periodic and then constant for the Liouville’s Theorem. It then follows that F is C-linear and and f is a homomorphism and the uniqueness follows by the construction. Clearly f∗

is identified, via the identifications δ : V∗ _{→ H}0_{(X, Ω}1

X), with the transpose

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Thanks to the previous proposition we have an inclusion of the set of homomorphisms between two complex tori (denoted by Hom(X, X0₎_{) in the}

set HomC(V, V

0₎ _{given by f 7→ F .}

Definition 2.0.6. The analytic representation of Hom(X, X0) is the map defined above.

ρa: Hom(X, X0) → HomC(V, V 0

) f 7→ F.

Let FΛ denote the restriction of F to Λ: this is a Z-linear map and

completely determines F and f. Thus we get an injective homomorphism ρr: Hom(X, X0) → HomZ(Λ, Λ

0₎ _{which sends f to F} Λ

Definition 2.0.7. The rational representation of Hom(X, X0)is the exten-sion of ρr to to HomQ(X, X 0_{) = Hom(X, X}0_{) ⊗} ZQ. ρr: HomQ(X, X 0 ) → Hom_Z(Λ, Λ0) ⊗_ZQ f → FΛ.

Suppose Π ∈ Mg,2g(C) and Π0 ∈ Mg0_,2g0(C) are period matrices for X and X0 _{with respect to some bases of V , Λ, V}0 _{and Λ}0_{. Let f : X → X}0

be a homomorphism. With respect to the chosen bases the two representa-tions ρaand ρrare given respectively by the two matrices A ∈ Mg0_,g(C) and R ∈ M2g0_,2g(Z). The condition ρ_a(f )(Λ) ⊂ Λ0 means AΠ = Π0R and con-versely given any two matrices A and R satisfying this condition is defined a homomorphism f : X → X0_.

2.1 Cohomology of complex tori

Let X = V/Λ be a complex torus. As seen before we have that π1(X) =

H1(X, Z) = Λ and so, by the universal coefficient Theorem, we have a natural

isomorphism H1_{(X, Z) = Hom(Λ, Z).}

Lemma 2.1.1. The canonical map VnH1(X, Z) → Hn_{(X, Z) induced by}

the cup product is an isomorphism for every n ≥ 1.

Proof. First of all we notice that this is a topological result and hence we can look at X as a real manifold and we obtain X ' S2g

1 . Now, using induction

on m, we’ll show that γ_mn:

n

^

H1(S₁m, Z) → Hn(S₁m, Z)

is an isomorphism. This is obviously true for m = 1. Suppose now that γn p is

bijective for every p < m and for all n. Thanks to the Künneth’s formula and to its compatibility with the cup product the following diagram commutes

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L p+q=n Vp H1(S₁m−1, Z) ⊗Vq H1(S1, Z) L p+q=n Hp(S₁m−1, Z) ⊗ Hq(S1, Z) Vn H1(S₁m−1, Z) ⊕ H1(S1, Z) Vn H1(Sm 1 , Z) Hn(S1m, Z). γ βm αm γn m Because γ = ⊕p+q=nγm−1p ⊗ γ q

1 is an isomorphism for hypothesis and αm

and βm are too (they are the usual Künneth morphism) we obtain that also

γ_mn is an isomorphism.

Corollary 2.1.2. There is a canonical isomorphism between Hn_{(X, Z) and} Altn(Λ, Z) for every n ≥ 1 where Altn_{(Λ, Z) =}Vn

Hom(Λ, Z).

Proof. As we have said before one has H1_{(X, Z) = Hom(Λ, Z) and so the}

thesis follows.

Corollary 2.1.3. For every n ≥ 1 there are canonical isomorphisms Hn_{(X, C) ' Alt}n_R= n ^ Hom_R_{(V, C) '} n ^ H1_{(X, C).}

Proof. This is just a consequence of the universal coefficients theorem, which says that Hn_{(X, C) = H}n_{(X, Z)⊗C, and of Alt}n_{(Λ, Z)⊗C = Alt}n

R(V, C).

2.2 Line bundles on complex tori I

We recall that the set of line bundles P ic(X) can be identified with H1_{(X, O}∗ X)

in an natural way and that through the short exact sequence of the exponen-tial map (0 → Z → OX → O∗X → 1) we get its long cohomology sequence

→ H1_{(X, Z) → H}1(OX) → H1(O∗X) c1

−→ H2_{(X, Z) → .}

According to Corollary 2.1.2 we can identify the image of a line bundle through c1 as an element of Alt2(Λ, Z) and thus, via R-linear extension,

we can associate an alternating real form V × V → R tensoring with R. Conversely we ask when such a alternating real form represents the Chern class of a line bundle.

Proposition 2.2.1. For an alternating form E : V × V → R the following conditions are equivalent:

• there is a holomorphic line bundle L on X such that E represents c1(L);

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Remark 2.2.2. Before the proof of this proposition we recall some results without proof that can be found in [9] chapter 1.4. For complex tori we have a Hodge decomposition of the cohomology groups Hn_{(X, C) = ⊕}

p+q=nHp,q(X)

and it can be proven that Hp,q_{(X) = H}q_(Ωp X) =

Vp

Ω ⊗ Vq

Ω where Ω = Hom_C(V, C) and Ω = Hom_C(V, C) the group of C-antilinear forms on V .

Proof. We consider the following diagram H1_(O∗ X) H2(X, Z) H2(OX) H2(X, C) H0,2(X) V2 Ω ⊕ (Ω ⊗ Ω) ⊕V2 Ω V2 Ω c1 i p β γ p (2.1)

where the first line of the diagram is part of the long cohomological sequence of the exponential map, i is the natural embedding, p the projection map and β and γ are the isomorphisms told in Remark 2.2.2. We claim that the diagram commutes as we will see in theorem 3.2.1.

Now suppose that E comes from the Chern class of a line bundle, we already know that E(Λ, Λ) ⊆ Z, we only need to show that E(iv, iw) = E(v, w). For the Hodge decomposition we have E = E1 + E2+ E3 with

E1 ∈V2Ω, E2∈ Ω ⊕ Ω and E3∈V2Ω, because E takes values in R we have

E = E and so E1 = E3. The above diagram assures that E3= 0and hence

E= E2∈ Ω⊗Ω. In conclusion we have E(iv, iw) = i(−i)E(v, w) = E(v, w).

Conversely if we have E alternating form with values in R with E(iv, iw) = E(v, w), E is forced to stay in Ω ⊗ Ω and the fact that E(Λ, Λ) ⊆ Z assures that E actually lies in H2_{(X, Z) and is sent in 0 ∈ H}2_(O

X). The

exact-ness of the first line of 2.1 says that E represents the Chern class of a line bundle.

Definition 2.2.3. An hermitian form H on V complex vector space is a map H : V × V → C which is C-linear in the first argument and satisfies H(v, w) = H(v, w)for all v, w ∈ V .

Lemma 2.2.4. There is a 1−1 correspondence between the set of hermitian forms H on V and the set of real valued alternating forms E on V satisfying E(iv, iw) = E(v, w) given by

E(v, w) = Im H(v, w) and H(v, w) = E(iv, w) + iE(v, w) for all v, w ∈ V .

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Proof. Let E be a real valued alternating forms satisfying E(iv, iw) = E(v, w). Then H(v, w) = E(iv, w) + iE(v, w) = −E(iw, −v) − iE(w, v) = H(w, v) and is obviously C-linear in the first argument.

Conversely given any H hermitian form, then E = Im H is alternating and E(iv, iw) = Im H(iv, iw) = Im H(v, w) = E(v, w).

To summarize we have found an identification between the Néron-Severi group NS(X) and the group of the R-valued alternating forms E on V satisfying E(Λ, Λ) ⊆ Z and E(iv, iw) = E(v, w), or equivalently with the group of hermitian forms H on V such that Im H(Λ, Λ) ⊆ Z.

We now recall a proposition about alternating forms over a free module over a principal ideal domain which will be useful in characterizing alternat-ing forms.

Theorem 2.2.5. Let A be a principal ideal domain and V a free A-module of finite rank n and let E be an alternating bilinear form over V . Then there exist a basis {ei}1≤i≤n and an integer r ≤ n/2 such that

• E(e1, e2) = δ1, E(e3, e4) = δ2, . . . , E(e2r−1, e2r) = δr for every 1 ≤ i ≤

r where δi 6= 0 and δi|δi+1;

• E(ei, ej) = 0 for all other couples of i and j.

In addiction the ideals Aδi are univocally determined by the previous

con-ditions and the submodule V0 _{orthogonal to V is generated by {e}

i}2r+1≤i≤n.

Proof. We will prove the theorem by induction on n: it is obviously true if n= 0, 1and also if E = 0. So let’s assume that E 6= 0. Let f ∈ Hom(V, V∗) be such that E(v, w) = (f(w))(v). Then there exist δ0

1, . . . , δn0 such that

δ_i0|δ0_i+1 and f(V ) is generated by δ₁0e0₁, δ0₂a₂0 . . . , δ0_na0_n where e0₁, a0₂, . . . , a0_n is a basis of V∗_{. Let e}

1, a2, . . . , an be the correspondent basis of V (identified

with its bidual V∗∗_{). Then there exist e}

2 such that f(e2) = δ01e01 so that

E(e1, e2) = −E(e2, e1) = (f (e2))(e1) = δ01. (2.2)

Let P = Ae1⊕ Ae2, we want to show that V = Ae1⊕ Ae2⊕ P0 where

P0 is the orthogonal to P in V . It is sufficient to prove that for every x ∈ V there exist ξ1, ξ2 ∈ Aunivocally determined such that x − ξ1e1− ξ2e2 ∈ P0,

i. e.

E(e1, x − ξ1e1− ξ2e2) = 0 E(e2, x − ξ1e1− ξ2e2) = 0.

Thanks to equations 2.2 these become

(f (x))(e1) = ξ2δ01 (f (x))(e2) = −ξ1δ20.

Moreover we know that (f(V ))(V ) ⊆ Aδ0

1and then existence and

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there exist a basis {e3, . . . , en} satisfying the conditions of the theorem. It

is now enough to prove that δ0

1 = δ1|δ2 but this follows from the fact that

δ1 divides E(v, w) for all v, w ∈ V . Moreover if {e0i} is the dual basis of

{ei} then we have f(e2j−1) = −δje2j0 and f(e2j) = δje02j−1 for j = 1, . . . , r

and f(ej) = 0 for 2r < j ≤ n. This proves that Aδ1, Aδ1, . . . , Aδr, Aδr

are the invariant factors of f(V ) respect to V∗ _{and so they are univocally}

determined.

Remark 2.2.6. Returning to the case of complex tori we have proved that, given a complex torus X of dimension g and a line bundle L with first Chern class H (seen as a hermitian form) whose alternating form E = Im H is integer-valued on the lattice Λ, there is a basis λ1, . . . , λg, µ1, . . . , µg of the

lattice Λ with respect to which E is given by the matrix

0 D

−D 0,

where D = diag(d1, . . . , dg) with integers di≥ 0 satisfying di|di+1.

We end this section with a list of definition that will be useful.

Definition 2.2.7. The elements d1, . . . , dg are called elementary divisors

and are univocally determined by E and Λ (ans hence by L).

Definition 2.2.8. The vector (d₁, . . . , dg) or equivalently the matrix D are

called the type of the line bundle L.

Definition 2.2.9. The basis λ1, . . . , λg, µ1, . . . , µg is called a symplectic (or

canonical) basis of Λ for L (or H or E equivalently).

Definition 2.2.10. A line bundle L is nondegenerate if H is and is equiv-alent to ask that di > 0 for all i. Equivalently a line bundle L is positive

definite if H is.

2.3 Abelian varieties

Definition 2.3.1. A polarization on a complex torus X is the first Chern class H = c1(L)(or sometimes L itself) of a positive definite line bundle and

the type of L is called the type of the polarization. A polarization is called principal if it is of type (1, . . . , 1).

Definition 2.3.2. An abelian variety is a complex torus X admitting a polarization H = c1(L). The pair (X, H) (equivalently the pair (X, L)) is

called a polarized abelian variety.

We now give a more familiar characterization (without proof) of abelian varieties.

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Lemma 2.3.3. For a line bundle L on a complex torus X the following are equivalent:

1. L is ample;

2. L is positive definite.

Proof. For a detailed proof look at [9] Proposition 4.5.2.

Theorem 2.3.4. For a complex torus the following conditions are equivalent: i) X is an abelian variety;

ii) X is a projective variety.

Proof. i) ⇒ ii): This is a direct consequence of Lemma 2.3.3. ii) ⇒ i): see [9] Theorem 4.5.4.

2.4 Riemann relations

We now work out in terms of period matrices, what it means that a complex torus is an abelian variety. As usual, let X = V/Λ be a complex torus of dimension g, e1, . . . , eg a basis of V and λ1, . . . , λ2g a basis of Λ and

Π the corresponding period matrix. With respect to these bases we have X= Cg_/_ΠZ2g_.

We are now going to prove the following theorem.

Theorem 2.4.1. X is an abelian variety if and only if there exist a nonde-generate alternating matrix A ∈ M2g(Z) such that:

i) ΠA−1 tΠ = 0; ii) iΠA−1 tΠ > 0.

The conditions i) and ii) are called the Riemann relations. It will turn out that A is the matrix of the alternating form defining the polarization.

The proof of the theorem passes through the following two lemmas. First we consider an arbitrary nondegenerate alternating form E on Λ and denote again with E its extension to Λ ⊗ R = Cg_{. Let A be its matrix with respect}

to the basis λ1, . . . , λ2g and H as in Remark 2.2.4 then we have the following

lemma.

Lemma 2.4.2. H is an hermitian form on Cg if and only if ΠA−1 tΠ = 0. Proof. According to Lemma 2.2.4 H is hermitian if and only if E(iv, iw) = E(v, w) for all v, w ∈ Cg_{. In order to analyse this condition in terms of}

matrices we define I =Π Π −1 i Ig 0 0 −i Ig Π Π .

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The matrix I satisfies iΠ = ΠI. Since, by definition, E(Πx, Πy) = t_xAy

for all x, y ∈ R2g_{, the form H is hermitian if and only if} t_IAI _{= A} _or

equivalently iIg 0 0 −iIg Π Π A−1 tΠ t_Π −1 iIg 0 0 −iIg =Π Π A−1 tΠ t_Π −1 .

Taking the inverse and comparing the upper-left and lower-right g × g-blocks we see that it is equivalent to ΠA−1 t_{Π = 0.}

The next lemma gives the matrix of H with respect to the basis e1, . . . , eg

assuming that H is hermitian.

Lemma 2.4.3. Suppose that H is hermitian, then 2i(ΠA−1 tΠ)−1 is the matrix of H with respect to the given basis. In particular H is positive definite if and only if iΠA−1 t_{Π > 0}_.

Proof. Let v = Πx and w = Πy with x, y ∈ R2g_{. With the notation as in}

the proof of the previous lemma and using that ΠA−1 t_{Π = 0} _{we get:}

E(iv, w) = tx tIAy= tv v i Ig 0 0 −i I_g Π Π A−1 tΠ t_Π −1 w w = tv v 0 i(ΠA−1 tΠ)−1 −i(ΠA−1 tΠ)−1 0 w w = itv(ΠA−1 tΠ)−1w − itv (ΠA−1 tΠ)−1w. Similarly we can compute

E(v, w) = txAy= tv v Π Π A−1 tΠ t_Π −1 w w = tv v 0 (ΠA−1 tΠ)−1 (ΠA−1 tΠ)−1 0 w w = tv (ΠA−1 tΠ)−1w+ tv(ΠA−1 tΠ)−1w. In conclusion

H(v, w) = E(iv, w) + iE(v, w) = 2itv(ΠA−1 tΠ)−1w.

Remark 2.4.4. Finally, given a polarized abelian variety (X = V/Λ, L) we want to outline the Riemann relations in terms of the symplectic basis for L. Let Π = (Π1,Π2) (with Πi ∈ Mg(C)) be the period matrix of X with

respect to the basis e1, . . . , eg of V and a symplectic basis λ1, . . . , λ2g of Λ

for L. If the line bundle is of type D = diag(d1, . . . , dg), then by definition 0 D

−D 0

is the matrix of the alternating form E with respect to λ1, . . . , λ2g,

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i) Π2D−1 tΠ1− Π1D−1 tΠ2 = 0,

ii) iΠ2D−1 tΠ1− iΠ1D−1 tΠ2>0.

2.5 The Siegel upper half space

Definition 2.5.1. The set of complex matrices

H_g = {Z ∈ Mg(C)| tZ = Z, Im Z > 0}

is called the Siegel upper half space.

Remark 2.5.2. The set Hg is an open submanifold of the vector space of

symmetric matrices in Mg and hence has dimension 1₂g(g + 1). We’ll soon

show that this set parametrizes the set of polarized abelian varieties of a given type D with symplectic basis.

Lemma 2.5.3. With the notation as in Remark 2.2.6, both λ1, . . . , λg and

µ1, . . . , µg are a C − basis for V .

Proof. We prove it for µ1, . . . , µg, the other case is similar. Let V2= spanR<

µ1, . . . , µg >and consider V2∩ iV2 ⊆ V: this is a complex subvector space

of V where E (and hence H) vanishes identically. Because H is positive definite, it follows that V2∩ iV2 = 0 and so V = V2+ iV2 and the assertion

follows.

Remark 2.5.4. Thanks to the previous lemma, if we define ei= _d1_iµi the set

{ei}1≤i≤g is still a C-basis for V . With respect to these bases the period

matrix of X is of the form

Π = (Z, D), for some Z ∈ Mg(C).

We now see how look the Riemann relations in this case.

Proposition 2.5.5. With the notation as in Remark 2.5.4 the matrix Z satisfies

i) tZ= Z and Im Z > 0

ii) (Im Z)−1 is the matrix of the hermitian form H with respect to the basis e1, . . . , eg.

Proof. The first assertion simply follows from the Riemann relations substi-tuting Π1 with Z and Π2 with D. By Lemma 2.4.3 the matrix of H with

these bases is given by 2i (Z, D)( 0 D −D 0)−1(

t_Z

D)

−1

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Definition 2.5.6. An abelian variety of type D with symplectic basis is a triple

(X, H, {λ1, . . . , λg, µ1, . . . , µg}),

where X is an abelian variety, H a polarization of type D on X and {λ₁, . . . , λg, µ1, . . . , µg}a symplectic basis of Λ for H.

What we have shown is that a polarized abelian variety of type D with a symplectic basis determines a point Z in Hg.

Conversely, given a type D, any Z ∈ Hg determines a polarized abelian

variety with a symplectic basis. Indeed let ΛZ = (Z, D)Z2g be the

corre-sponding lattice in V = Cg_{, λ}

i= (Z, D)eiand µi= (Z, D)ei+g for 1 ≤ i ≤ g.

Then the quotient XZ = V /ΛZ is a complex torus and (Im Z)−1 defines a

positive definite hermitian form (all its entries are real).

We now show that HZ is a polarization of type D and that

{λ₁, . . . , λg, µ1, . . . , µg} is a standard basis for H. Obviously

{λ1, . . . , λg, µ1, . . . , µg}is a basis for ΛZand with respect to this basis Im HZ

is given by

Im t(Z, D)(Im Z)−1(Z, D) = ( 0 D −D 0).

Summing up we have proven the following proposition.

Proposition 2.5.7. There exist a 1 − 1 correspondence between the Siegel upper half space and the set of polarized abelian varieties of type D with a symplectic basis given by

Z ∈ Hg 7→ (XZ, HZ, {columns of (Z, D)})

(X, H, {λ1, . . . , λg, µ1, . . . , µg}) 7→ Z.

2.6 Boundedness of the Siegel upper half space

In this section we will prove that the Siegel upper half space is biholomorphic to the generalized unit disk Dg = {τ ∈ Mg(C)|tτ = τ, Ig−τ tτ >0}of degree

g and that Dg is a bounded domain.

Lemma 2.6.1. The set Dg is bounded in Cg

2 .

Proof. The condition Ig− τ tτ >0assures thattx(I¯ g− τ tτ)x > 0for every

x ∈ Cg_{; in particular for the elements e}

i of the canonical basis we get

(τ tτ)i,i<1.

Then it suffices to notice that (τ tτ)i,i= g X j=1 τi,jτi,j = g X j=1 |τ_i,j|2.

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Theorem 2.6.2. The Siegel upper half space Hg is biholomorphic to the

bounded domain Dg through the following maps

Z ∈ Hg7→ ψ(Z) = (Z − iIg)(Z + iIg)−1∈ Dg

W ∈ Dg 7→ φ(W ) = i(W + Ig)(Ig− W )−1∈ Hg.

Remark 2.6.3. Before the proof the theorem we shall recall two facts: • given two matrices A and B such that AB = BA, then we have (A +

B)(A − B)−1 = (A − B)−1(A + B). This follows immediately from the fact that, in this case, (A + B)(A − B) = (A − B)(A + B);

• given two symmetric matrices A and B such that AB = BA, then AB is symmetric. Indeedt_{(AB) =} t_B t_A_{= BA = AB}_.

Proof. First of all we prove that the two maps ψ and φ are well defined respectively in Hg and Dg. It is enough to prove that for all matrices in Hg,

−iis not an eigenvalue and that for all matrices in D_g, 1 is not an eigenvalue. If −i is an eigenvalue for Z ∈ Hg then Im Z is not positive definite, which

is a contradiction. Indeed, let v be an eigenvector for −i with euclidean norm equal to 1. Then we have that Zv = −iv, Zv = iv andt_vZ _{= −i} t_v

(notice that the last equivalence is true because Z i symmetric). Hence if we calculate t_v_{Im Zv =} t_v1 2i(Z − Z)v = 1 2i( t_{vZv −} t_{vZv) =} 1 2i(−i t_{vv − i}t_{vv) = −1.}

Similarly if 1 is an eigenvalue for W ∈ Dg, then there exist a vector v (with

euclidean norm equal to 1) such that W v = v and thent_vt_{W W v}_{= 1}_which

is a contradiction as well.

Now we prove that the functions are one the inverse of the other. We now calculate φ(ψ(Z)) = i(Z − iIg)(Z + iIg)−1+ Ig Ig− (Z − iIg)(Z + iIg)−1 −1 = =i(Z − iIg+ Z + iIg)(Z + iIg)−1 (Z + iIg− Z + iIg)(Z + iIg)−1 −1 = =2iZ(Z + Ig)−1(Z + Ig) 1 2iIg = Z. Analogously we have ψ(φ(W )) =i(W + Ig)(Ig− W )−1− iIg i(W + Ig)(Ig− W )−1+ iIg −1 = =(W + Ig− Ig+ W )(Ig− W )−1 (W + Ig+ Ig− W )(Ig− W )−1 −1 = =2W (Ig− W )−1(Ig− W ) 1 2Ig = W.

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At this point it remains only to show that ψ(Hg) ⊆ Dg and φ(Dg) ⊆ Hg

to finish the proof. If Z ∈ Hg we are going to prove that (Z −iIg)(Z + iIg)−1

is symmetric and that Θ = Ig− (Z − iIg)(Z + iIg)−1(Z − iIg)(Z + iIg)−1 is

positive definite. The first fact follows entirely from Remark 2.6.3. To prove that Θ is positive definite it’s enough to prove that all its eigenvalues are positive. So let y be an eigenvector of Θ with eigenvalue λ. Then we have

Ig− (Z − iIg)(Z + iIg)−1(Z + iIg)(Z − iIg)−1

y= λy which, using Remark 2.6.3, becomes

(Z + iIg)−1(Z + iIg)(Z − iIg)(Z − iIg)−1− −(Z + iIg)−1(Z − iIg)(Z + iIg)(Z − iIg)−1 y= = (Z + iIg)−1 (Z + iIg)(Z − iIg)−(Z − iIg)(Z + iIg) (Z − iIg)−1y = λy, which, if z = (Z − iIg)−1y, is equivalent to (Z + iIg)(Z − iIg) − (Z − iIg)(Z + iIg) z= λ(Z + iIg)(Z − iIg)z.

Now we have that

(Z + iIg)(Z − iIg) = ZZ − iZ + iZ + Ig = ZZ + 2 Im Z + Ig

and

(Z − iIg)(Z + iIg) = ZZ + iZ − iZ + Ig= ZZ − 2 Im Z + Ig.

So the previous equation becomes

4 Im Zz = λ(ZZ + 2 Im Z + Ig)z

and multiplying on the left byt_z _{we get}

4 tzIm Zz = λ tz(ZZ + 2 Im Z + Ig)z.

Because Im Z is positive definite (and obviously also the identity matrix is) and ZZ is positive semidefinite we have that necessarily λ is positive and we have thus proved that ψ(Z) ∈ Dg.

Conversely let now W ∈ Dg; with a similar argument as before we prove

that φ(W ) = t_{φ(W ). It remains only to prove that Im φ(W ) is positive}

definite. But

Im φ(W ) = Re(Ig+W )(Ig−W )−1=

1 2

(Ig+W )(Ig−W )−1+(Ig+W )(Ig−W )−1

,

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and then we want to show that all the eigenvalues of this matrix are positive. So we want to solve the following equation

(Ig+ W )(Ig− W )−1+ (Ig+ W )(Ig− W )−1

y= λy, which, using Remark 2.6.3, becomes

(Ig− W )−1(Ig+ W )(Ig− W )(Ig− W )−1+

+(Ig− W )−1(Ig− W )(Ig+ W )(Ig− W )−1

y= = (Ig− W )−1

(Ig+ W )(Ig− W )+(Ig− W )(Ig+ W )

(Ig− W )−1y = λy, which, if z = (Ig− W )−1y, is equivalent to (Ig+ W )(Ig− W) + (Ig− W )(Ig+ W ) z= λ((Ig− W )(Ig− W ))z.

Now we have that

(Ig+ W )(Ig− W ) = Ig+ W − W + W W

and

(Ig− W )(Ig+ W ) = Ig− W + W + W W .

So the previous equation become

2(Ig− W W)z = λ((Ig− W ) + (Ig− W ))z

and multiplying on the left byt_z _{we get}

2 tz(Ig− W W )z = λtz(Ig− W )(Ig− W )z.

Because (Ig− W W ) is positive definite and (Ig − W )(Ig − W ) is positive

semidefinite we have that necessarily λ is positive and the proof is concluded.

2.7 Line bundles on complex tori II

First of all we notice that any line bundle on V a C-vector space is trivial: this is just a direct consequence of the cohomology exact exponential sequence after noticing that H1_{(V, O}

V) = H2(V, Z) = 0. Thus if L is a line bundle

on X complex torus and π : V → X is the universal covering of X, then the pullback line bundle π∗_L_{is trivial and we can find a trivialization}

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This means that, for x ∈ X and λ ∈ Λ, the fibres of π∗_L_{in z and in z +λ}

are identified with the fibre of L at π(z) giving a linear automorphism of C: C←− (πφz ∗L)z= Lπ(z)= (π∗L)z+λ

φz+λ −−−→ C.

Such an automorphism is given as a multiplication by a non zero element of C, denoting this element by eλ(z), we obtain a collection of functions

{e_λ∈ O∗(V )}λ∈Λ

called a set of multipliers of L. This set of functions satisfies these relations eλ0(z + λ)e_λ(z) = e_λ0_+λ(z) = e_λ(z + λ0)e_λ0(z)

for all λ, λ0 _{∈ Λ}_{. Conversely, given such a set of non zero holomorphic}

functions {eλ}λ∈Λ satisfying these relations, we can construct a line bundle

Lon X having {eλ}λ∈Λas multipliers: it is enough to take L as the quotient

space of V × C under the identification

(z, x) ∼ (z + λ, eλ(z) · x).

Obviously, in order to define a line bundle L on X, it is enough to define the functions eλ for a base {λi}of Λ satisfying

eλi(z + λj)eλj(z) = eλj(z + λi)eλi(z).

One simplification of the set of multipliers is immediate: if {λ1, . . . , λg, µ1, . . . , µg}

is a basis of Λ such that {µ1, . . . , µg}is a C-basis of V we have

V

Z{µ1, . . . , µg} ' (C ∗₎g_,

and we can factor our projection map π : V → X by

V → V

Z{µ1, . . . , µg} π1 −→ X. It is known that (cf. [17] page 27) H1_((C∗₎g_{, O}

(C∗₎g) = H2((C∗)g, O_(C∗₎g) = 0 and hence, looking at the cohomology exact long sequence associated to the exponential sequence, any line bundle on (C∗₎g _{is uniquely determined}

by its Chern Class. For any line bundle L on X we can choose the basis {λ1, . . . , λg, µ1, . . . , µg}such that c1(L) is defined by the matrix

0 D

−D 0

where D is a diagonal matrix. Because H1_((C∗₎g_{, O}

(C∗₎g) = <µ₁, . . . , µ_g> and the restriction of E to these submodule is zero we have that

c1(π1∗L) = π ∗

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and consequently π∗

1Lis trivial. If we take a trivialization eφ: π1∗L →(C∗)g×

C and choose the trivialization φ of π∗L to extend eφ, we have eµi(z) ≡ 1, i= 1, . . . , g.

Now we choose an element E ∈ NS(X) of a positive line bundle with

matrix

0 D

−D 0

related to the basis {λ1, . . . , λg, µ1, . . . , µg}and we choose a C-basis {e1, . . . , eg}

as in remark 2.5.4 such that the period matrix as the form Π = (Z, D).

Lemma 2.7.1. The line bundle L given by multipliers eµi(z) ≡ 1 eλi(z) = e

−2πizi _i_{= 1, . . . , g,}

has Chern class c1(L) = E.

Proof. For a proof of this look at [17] page 310.

Proposition 2.7.2. Any positive line bundle on X is defined, up to trans-lation in X, by its Chern class.

Proof. Given an element ξ ∈ X the map τξ(z) = z + ξ is homotopic to

the identity, then, given a line bundle L, we have that c1(L) = c1(τµ∗L).

Moreover if L is given by multipliers eµi(z) ≡ 1 and eλi(z) = e

−2πizi, then τ_ξ∗Lis given by the multipliers e0_µ_i(z) = eµi(z + ξ) ≡ 1and e

0

λi(z) = eλi(z + ξ) = e−2πi(zi+ξi), i.e. the multipliers of τ∗

ξL will differ from those of L by

a multiplicative constant. Conversely if L0 _{is a line bundle with multipliers}

e0_µ_i(z) ≡ 1 and e0_λ i(z) = cieλi, with ci∈ C ∗_{, then setting} ξ =X i 2πlog ci· ei ∈ V then we have L0 _{= τ}∗ ξL.

Thus, to prove that any line bundle having the same Chern class as L must be a translate of L, it will suffices to show that any line bundle with Chern class 0 can be realized by constant multipliers. First of all we notice that the inclusion of exact sheaf sequences

0 Z OX OX∗ 0

0 Z C C∗ 0

exp

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induces H1(X, OX) H1(X, O∗_X) H2(X, Z) H1_{(X, C)} _H1_{(X, C}∗₎ _H2_{(X, Z).} c1 i∗ 1 i∗2 The map i∗

1 represents the projection of H1(X, C) = H1,0(X) ⊕ H0,1(X)

on the second factor (as we will prove in remark 3.2.3), so it is surjective. Let now γ ∈ H1_{(X, O}∗

X) be a cocycle in the kernel of c1. Then it follows

from the diagram above that is in the image of i∗

2 which means that any line

bundle on X with c1(X) = 0is given by constant transition functions.

If L is any line bundle on X with c1(X) = 0, we then can find an open

covering {Uα} of X such that for each α π−1(Uα) = ∪jUαj is a disjoint

union of open sets biholomorphic to Uα and a collection of trivialization

φα: LUα → Uα× C having constant transition functions {gαα0}. We then take hα0j0 ≡ 1for a certain α0j0 and set

hαj = hα0_j0g_αα0 for α, α0, j, j0 such that U_αj ∩ U_α0_j0 6= ∅.

By the cocycle condition one gets that it is well defined and the local trivi-alizations given by

φαj = hαj· π∗φα: π∗LUαj → Uαj× C are a trivialization of π∗_L _{having constant multipliers.}

2.8 Theta functions

We have proved that any positive line bundle on X = V/Λ is a quotient of the trivial bundle V × C and, in similar way, we want to construct global holomorphic sections of L as entire holomorphic functions on V .

Proposition 2.8.1. Let L be a positive line bundle on X with d₁, . . . , dg

elementary divisor of the polarization c1(L) of X. Then h0(X, O(L)) =

Q

idi.

Proof. As in remark 2.5.4, let {λ1, . . . , λg, µ1, . . . , µg}be a basis of the lattice

such that c1(L)is given by the matrix

0 D

−D 0

and let ei = d−1_i µi be a C-basis for V and z1, . . . , zg the corresponding

coordinates, such that the period matrix is given by Π = (Z, D),

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where Z is symmetric and has imaginary part positive definite.

thanks to Proposition 2.7.2 and to the fact that h0_(L) _{is obviously}

in-variant under translations, we can assume that L is given by the multipliers eµi(z) ≡ 1 eλi(z) = e

−2πizi−πiZii

and so a holomorphic section eθof L is given by an entire holomorphic function θon V such that

θ(z + µi) = θ(z) θ(z + λi) = e−2πizi−πiZii· θ(z).

Because of the first condition, such a function θ has a power series in the variables z∗

i = e2πid

−1

i zi and hence we can write θ(z) = X l∈Zg alz∗l11· · · z ∗lg g = X l∈Zg al· e2πi<l,D −1_z> .

We now use the second conditions to get a relation among the coefficients al. By definition we have θ(z + λi) = X l∈Zg al· e2πi<l,D −1_λ i>_{· e}2πi<l,D−1z>_,

while the second condition let us to write θ(z + λi) = e−2πizi−πiZii· θ(z) =

X

l∈Zg

al+Dei· e

−πiZii_{· e}2πi<l,D−1z>_.

Comparing these two equation we get al+diei = e

2πi<l,D−1λi>+πiZii_{· a}

l

implying that all the coefficients are determined as soon as one has the values of {al}l|0≤li<di implying that

h0(X, O(L)) ≤Ydi.

To prove the equality in the previous equation, it is enough to show that an arbitrary series with arbitrary coefficients converges and and a proof can be found in [17] page 319.

Remark 2.8.2. In the case that L gives a principal polarization of X we have proven that h0_{(X, O(L)) = 1}_{and so the space H}0_{(X, O(L))} _{is generated by}

the section eθ corresponding to the global function θ(z) = X

l∈Zg

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which satisfies

θ(z + µi) = θ(z)

θ(z + λi) = e−2πi(zi+ZZii/2)· θ(z)

θ(−z) = θ(z).

This is the so called Riemann θ-function of the principally polarized Abelian variety (X, c1(L)).

The corresponding divisor Θ = [eθ], determined up to translation by the cohomology class of c1(L), is called the Riemann Θ- divisor of the principally

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Chapter 3

Albanese and Picard torus

In this chapter we will define two complex tori related to a Kähler variety and will give some of their elementary properties. Then we will prove the Torelli’ Theorem.

3.1 The Albanese torus

To every connected compact Kähler manifold is associated a complex torus, of dimension g = h1,0_(X)_{, namely the Albanese torus Alb(X).}

Theorem 3.1.1. Let X be a connected compact Kähler manifold. Then there exist a complex torus A and a morphism α: X → A with the following universal property: for any complex torus T and any morphism f : X → T , there exist a unique morphism ef: A → T such that ef ◦ α= f.

X T

A

f

α fe

The complex torus A, determined up to isomorphism by the universal prop-erty, is called the Albanese variety of X and is written Alb(X). The mor-phism α induces an isomormor-phism α∗_{: H}0_{(A, Ω}1

A) → H0(X, Ω1X).

Proof. Let ω1, . . . , ωgbe a basis for H0(X, Ω1X); by the Hodge decomposition

it follows that ω1, . . . , ωg, ω1, . . . , ωg is a basis for H1(X, C). Let λ1, . . . , λ2g

be a basis for H1(X, Z) mod torsion and consider the vectors

vj = Z λj ω1, . . . , Z λj ωg j= 1, . . . , 2g which are coordinates of elements in the dual of H0_{(X, Ω}1

X) respect to the

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we claim that they are linearly independent over R. Indeed, if not, then the 2g vectors wj = Z λj ω1+ ω1, . . . , Z λj ωg+ ωg, i Z λj ω1− ω1, . . . , i Z λj ωg− ωg , would be dependent on R2g_{and thus, by duality, there exist α}

1, . . . , αg, β1, . . . , βg

not all 0 such that

α1(ω1+ ω1) + · · · + αg(ωg+ ωg) + iβ1(ω1− ω1) + · · · + βg(ωg− ωg)

is cohomologous to 0 implying that ω1, . . . , ωg, ω1, . . . , ωg are linearly

depen-dent over C which is a contradiction.

What we have proven is that, under the map i: H1(X, Z) → H0(X, Ω1X)∗

given by < i(λ), ω >= R_λω, the image of i is a lattice in H0_{(X, Ω}1 X) ∗_{, then} the quotient H0_{(X, Ω}1 X) ∗_/H 1(X, Z) is a complex torus.

Next we define the map α: fix a point p ∈ X and let λx be a path joining

p to a point x of X, and let a(λx) ∈ H0(X, Ω1X)

∗ _{be the map defined by}

ω 7→ R

λxω. If we replace λx with another path λ

0

x joining p to x we are

changing a(λx) by an element of i(H1(X, Z)), so the class of a(λx) depends

only on x and we call it α(x).

We now show that α is analytic in a neighbourhood of a point x ∈ X. Choose a path λx from p to x and a neighbourhood U of x isomorphic to a

ball in Cn_{. For y ∈ U we consider the path λ}

y connecting p to y composed

by the path λx and the segment <x, y> ⊆ U. It is clear that the morphism

a: U → H0_{(X, Ω}1

X)∗ sending y to a(λy)is an analytic morphism: then, since

α

U = π ◦ a, where π is the natural projection of the torus, α is analytic in

U. Notice that α(p) = 0 and that changing p the morphism α is modified by a translation in A.

Let us prove the fact that α∗ _{is an isomorphism. Since, as in Remark}

2.0.2, we have an isomorphism δ : H0_{(X, Ω}1

X) → H0(A, Ω1A), it is enough

to show that α∗_{(δ(ω)) = ω} _{for all ω ∈ Ω. As above, locally we can write}

α= π ◦ aand then we have

α∗(δω) = a∗π∗(δω) = a∗d(<ω, ·>).

The value of this form at y ∈ X is d(<ω, a(y)>) = d(

Z y

p

(ω) = ω(y) which ends this part of the proof.

It only remains to prove the universal property of A. Let T = V/Λ be a complex torus and f : X → T a morphism. We now prove the uniqueness of the morphism ef through the following commutative diagram remembering

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that a morphism f between to complex tori is determined, up to translation, by f∗ _{(Proposition 2.0.5).} H0(X, Ω1 X) H0(T, Ω1T) H0_{(A, Ω}1 A) f∗ f f∗ α∗

But fixing ef(0) = f (p), we have proven the uniqueness.

Thanks to Proposition 2.0.5, to prove the existence of ef it is enough to prove that the composite homomorphism u: V∗ δ₋_{→ H}0_{(T, Ω}1

T) f∗

−→ H0_{(A, Ω}1 A)

satisfies t_u(i(H

1(X, Z))) ⊆ Λ. Let γ ∈ H1(X, Z), v∗ ∈ V∗; then (with the

notation of 2.0.5)

<tu(i(γ)), v∗ >=< i(γ), u(v∗) >= Z γ f∗(δv∗) = Z f∗γ δv∗ =< h−1(f∗γ), v∗ >

and thust_{u(i(γ)) = h}−1_(f

∗γ) ∈ Λand the theorem is proven.

Remark 3.1.2. In the case that X is smooth and projective it can be proven that the Albanese variety is an abelian one. In order to do this (cf. [25] chapter 11) it is enough to find an integral alternating form E on H1_{(X, C)}

such that E_H0_(X,Ω1

X)×H0(X,Ω 1 X)

≡ 0 and −iE(u, u) is positive definite on H0(X, Ω1

X). If ω is the (1, 1)-form associated to the kählerian manifold X,

this alternating form is given by E(σ1, σ2) =

R

Xσ1∧ σ2∧ ω

dimX−1_.

Remark 3.1.3. 1. We have that dimAlb(X) = dimH0_{(X, Ω}1

X). In

par-ticular if H0_{(X, Ω}1

X) = 0 we have that every morphism to a complex

torus is trivial. This holds in particular for X = Pn_.

2. From the universal property follows that the Albanese variety is func-torial: if f : X → Y is a morphism of kählerian varieties there exist a unique morphism F : Alb(X) → Alb(Y ) such that the following dia-gram is commutative. X Y Alb(X) Alb(Y ) f αX αY F

3. From the universal property we also deduce that Alb(X) is the torus generated by αX(X)(where the torus generated by αX(X) ∈ Alb(X)is

the smallest subtorus of Alb(X) containing αX(X)and this definition

works because the intersection of two subtorus is a subtorus as well), in particular if A 6= 0 then αX(X) is not reduced to a point. It also

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4. If X is a smooth curve, Alb(X) is nothing but the Jacobian J(X). Moreover it is possible to give to the Jacobian a natural structure of principally polarized abelian variety: it suffices to give an alternating form E on H1(X, Z) and a basis λ1, . . . , λg, µ1, . . . , µg for which E is

given by the matrix

0 Ig

−I_g 0

and such that the corresponding hermitian form is positive definite. This is just the intersection form defined in [17] section 0.4 and then with such a basis is possible to find a basis {ω1, . . . , ωg} of H0(X, Ω1X)

such that the period matrix of the dual basis in H0_{(X, Ω}1

X)∗, given by    R λ1ω1 . . . R µgω1 ... ... R λ1ωg . . . R µgωg,   

has the following form

Z Ig

where Z is a symmetric matrix with positive definite imaginary part proving that Jac(X) is a principally polarized abelian variety (for de-tails look also [17] section 2.2).

3.2 The Picard torus

Before defining the Picard torus it will be useful to prove the following the-orem.

Theorem 3.2.1 (Lefschetz Theorem on (1, 1)-classes). For every compact Kähler manifold X the morphism H2_{(X, Z) → H}2_{(X, O}

X), which arises

from the exponential sequence of sheaves, factorizes as

H2_{(X, Z)} _H2_{(X, O}

X)

H2(X, C),

k

i π0,2

where the map i is just the inclusion H2_{(X, Z) ⊆ H}2_{(X, C), while π}0,2 _is

the natural projection from H2_{(X, C) ' H}2,0_{(X) ⊕ H}1,1_{(X) ⊕ H}0,2_(X) _to

H2(X, OX) ' H0,2(X). In particular every class in i(H2(X, Z)) ∩ H1,1(X)

is the image of a line bundle in P ic(X).

Proof. Consider the exact exponential sequence of sheaves 0 → ZX → OX → O∗X → 0

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this gives the associated long exact cohomology sequence P ic(X) = H1(X, O∗_X) c1

−→ H2_{(X, Z}X) k

−→ H2(X, OX). (3.1)

What we want to show that the following diagram is commutative H2(X, ZX) H2(X, OX) H2(X, CX) H_DR2 (X, CX) H_∂0,2¯ (X), k i ' ' π0,2

where the equivalences are given by the de Rham and the Dolbeault Theo-rems (look at [17] pages from 43 to 46).

First of all we recall how the de Rham and Dolbeault isomorphisms are constructed using Čech cohomology. Let z = (zαβγ) ∈ Z2(X, ZX) be a

representative of a cohomology class in H2_{(X, Z}

X). The de Rham Theorem

says that there exist a set of functions fαβ defined in Uαβ such that zαβγ =

fαβ+ fβγ− fαγ in Uαβγ. Since zαβγ is constant we have that dfαβ is a well

defined global closed 1-cycle of closed 1-forms. Then, thanks to the de Rham Theorem exists a set of 1-forms ωα such that dfαβ = ωβ − ωα in Uαβ and

dωα defines a global 2-form which represents z inside H_DR2 (X, CX).

In a similar way, we want to see how is defined k(z) under the Dolbeault isomorphism. Using the Dolbeault isomorphism, we get in a similar way

zαβγ = fαβ + fβγ− fαγ

¯

∂fαβ = ω0,1_β − ωα0,1

and then ¯∂ω0,1

α represents i(z) inside H_∂0,2¯ (X) proving the first part of the

theorem.

Now if we take an element γ ∈ i(H2_{(X, Z)) ∩ H}1,1_(X) _{we have that}

k(γ) = 0 and then there exist an element L ∈ P ic(X) such that c1(L) =

γ.

Remark 3.2.2. The Néron-Severi group NS(X) is contained in H1,1_(X).

Indeed let c1(L) ∈ N S(X), because k ◦ c1 = 0, we have that i(c1(L)) ∈

ker π0,2 _{= H}2,0_{(X) ⊕ H}1,1_(X)_{that’s to say i(c}

1(L)) = (0, α, β) ∈ H0,2(X) ⊕

H1,1_{(X) ⊕ H}2,0_(X)_{. But an element in NS(X) must be real and then,}

recalling that H2,0_{(X) = H}0,2_(X)_{by Hodge duality, we obtain that}

(0, α, β) = (0, α, β) = (β, α, 0).

Fibered algebraic surfaces over the projective line

Universit`

a degli Studi di Pisa

Fibered algebraic surfaces over the projective line

Contents

Abstract

Chapter 1

Preliminaries

1.1

Complex Manifold

1.2

Sheaves and cohomology

1.3

Line bundles and divisors

1.4

Kähler manifolds and Hodge decomposition

Chapter 2

Complex Tori

2.1

Cohomology of complex tori

2.2

Line bundles on complex tori I

2.3

Abelian varieties

2.4

Riemann relations

2.5

The Siegel upper half space

2.6

Boundedness of the Siegel upper half space

2.7

Line bundles on complex tori II

2.8

Theta functions

Chapter 3

Albanese and Picard torus

3.1

The Albanese torus

3.2

The Picard torus