Classification of Fiber Bundlesover the Riemann Sphere

Joao Pedro Correia Matias

Dissertacao para obtencao do Grau de Mestre em

Matematica e Aplicacoes

Juri

Presidente: Professor Doutor Miguel Abreu

Orientador: Professor Doutor Carlos Florentino

Vogal: Professora Doutora Margarida Mendes Lopes

Julho de 2012

Acknowledgments

The first person I would like to thank is my advisor Professor Carlos Florentino for

proposing this subject to me and for guiding me throughout this work. His motivation and

the time he took with me were very important for me to finish this thesis.

I would like to thank Professor Margarida Mendes Lopes for introducing me to Algebraic

Geometry and for motivating me to work harder to achieve success.

I also wish to thank my parents and my family for providing me with a good atmosphere

to work and for caring about me and always being around. Obrigado Pai, Mae, Joana e

Rui!

Last but not least, I would like to thank my friends and “brothers in arms” for being be-

side me and for their advice and motivation. It is thanks to them that I haven’t (completely)

despaired!

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Resumo

Esta tese trata de fibrados holomorfos sobre superfıcies de Riemann e, em particular,

a esfera de Riemann P1. Baseando-nos no Teorema de Grothendieck sobre classificacao de

fibrados principais sobre P1 e depois de introduzir varios resultados e tecnicas importantes

da teoria de fibrados vectoriais, apresentamos e provamos uma classificacao de fibrados

principais com grupos de estrutura ortogonal ou simplectico sobre a esfera de Riemann.

Palavras-Chave: Superfıcie de Riemann, Esfera de Riemann, Fibrado Vectorial, Fi-

brado Principal, Cohomologia, Grupo de Lie.

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iv

Abstract

This thesis deals with holomorphic fiber bundles over Riemann surfaces and, in par-

ticular, over the Riemann sphere P1. Basing ourselves on Grothendieck’s Theorem on the

classification of principal bundles over P1 and after introducing many important results and

techniques in the theory of vector bundles, we present and prove a classification of principal

bundles with orthogonal or symplectic structure groups over the Riemann sphere.

Keywords: Riemann Surface, Riemann Sphere, Vector Bundle, Principal Bundle, Co-

homology, Lie Group.

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Contents

Acknowledgments i

Resumo iii

Abstract v

Introduction 1

1 Vector Bundles 3

1.1 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Presheaves and Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Fundamental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Grothendieck’s Theorem for Vector Bundles . . . . . . . . . . . . . . . . . . . 15

2 Principal Bundles 19

2.1 Cohomology of non-Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Complex Reductive Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Grothendieck’s Theorem for Principal Bundles . . . . . . . . . . . . . . . . . 22

2.4 Application to Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 A lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 When G = O(n,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Orthogonal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.8 When G = Sp(2n,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.9 Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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Introduction

In short, a Riemann surface is an object that locally behaves as C and where we can

define coherently holomorphic functions. One of the most famous examples of such is the

Riemann sphere, that may be denoted by P1 and is homeomorphic to the surface of a sphere,

hence the name.

A fiber bundle is an object defined over a topological space, in particular a Riemann

surface, by connecting a fixed set (fiber) to every point of the latter and satisfying a local

trivialization condition. The vast amount of different types of conditions for this local

trivialization, provides many different structures that one can study. In particular, we will

focus on vector bundles, where the fibers are a vector space, and principal bundles, where

we consider the action of a group on the fibers.

The main purpose of this work is to study a result on classification of vector bundles

and principal bundles over the Riemann sphere, due to Grothendieck. The vector bundle

version is a very famous result in Algebraic Geometry and is known simply as Grothendieck’s

Theorem. However, on the original paper it was a given a more general result for principal

bundles with structure group a complex Lie group G, from which the previous result arises

as a corollary, when we take GL(n,C) as the structure group.

Our goal is to present the more general case in a clear way, going through the appropriate

basic concepts that are necessary to understand it. We will also prove it for the particular

cases of vector bundles with an holomorphic quadratic form, or a symplectic form defined

on its fibers.

Through the way, we will study basic results on Riemann surfaces and introduce other

concepts such as sheaves, its cohomology groups, and Lie groups.

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2

Chapter 1

Vector Bundles

Throughout this chapter we will focus on Riemann surfaces and vector bundles. First

we will define properly a Riemann surface and give some examples. We will then take a

look at the analogous case for greater dimensions. By then, we will be able to give a proper

definition of vector bundle, along with the first few examples. Then, we define an important

tool to study vector bundles, sheaves, taking the sheaf of sections of a vector bundle as our

main example.

Furthermore, we define the cohomology groups of a sheaf and we see the long exact

sequence they generate from a short exact sequence of sheaves. Then, we present two

fundamental results: Riemann-Roch’s Theorem and Serre’s Duality.

The chapter culminates with Grothendieck’s Theorem for vector bundles (Theorem 3).

1.1 Riemann Surfaces

A Riemann surface is, in a sense, a generalization of the complex plane. It is a 2-

dimensional real manifold that is locally homeomorphic to an open set of C and where we

can define holomorphic functions coherently.

To study a surface locally, we describe it through a well known object, for example the

complex plane C, using coordinate charts.

Definition 1. A complex coordinate chart, or simply chart, on a topological space S is a

homeomorphism φ : U →W , where U ⊂ S and W ⊂ C are open sets.

Remark. It is also usual to use z to refer to the image of a point p ∈ S through the map φ.

We write z(p) = φ(p) to be more precise, or z = φ(p).

Definition 2. A Riemann surface S is a topological space, together with a covering of

coordinate charts φα : Uα → Wα ⊂ C, S = ∪αUα, such that for every pair α, β, φα φ−1β

3

is an invertible holomorphic function.

Remark. From now on, when omitted, S will designate a Riemann surface.

Example 1. The projective line P1, also known as the Riemann sphere, is defined as the

quotient (C2 \ (0, 0))/ ∼, with (x, y) ∼ λ(x, y) = (λx, λy) for any λ ∈ C∗. A point in P1

is denoted by (x : y), when (x, y) is a representative of the class it defines through ∼.

To see that this is a Riemann surface, we use two coordinate charts to cover P1. They

are:

U0 = (1 : y) : y ∈ C U1 = (x : 1) : x ∈ C

φ0 : U0 → C φ1 : U1 → C

(1 : y) 7→ y (x : 1) 7→ x

The change of chart is an inversion:

φ0 φ−11 (x) = φ0((x : 1)) = φ0

((1 :

1

x

))=

1

x, x ∈ C∗,

which is a holomorphic automorphism on C∗.

Example 2. The torus T2 ∼= S1 × S1 is also a Riemann surface. To see this, we consider

the lattice generated by 1 and i, (Z + iZ) ⊂ C. Then, we set the torus as the quotient

T2 ∼= R2/Z2 ∼= C/(Z + iZ).

It is easy to obtain a covering of holomorphic coordinate charts for this case. As

an example, we suggest charts obtained from the projection into the torus of the sets in

B 12(0), B 1

2( 12 ), B 1

2( 12 i), B 1

2( 12 + 1

2 i), with B 12(z) = w ∈ C : |w − z| < 1

2. Notice that for

each of these sets the projection into the torus is a bijection, as the difference of any two

distinct points does not belong to the lattice (Z + iZ).

Definition 3. A function f : S → C is holomorphic if so are its representatives f φ−1α , for

a covering of coordinate charts φα : Uα → C.

Proposition 1. Let S be a connected compact Riemann surface, then the only holomorphic

functions on S are the constant functions.

Proof. Let f be an holomorphic function over S. So, |f | is continuous and as S is compact,

we conclude that |f | has a maximum point. By the Maximum Modulus Principle, this

maximum point may not be strict. Therefore, |f | is constant.

If |f | = 0, then we are done. If not, we can see that f = |f |2f is also holomorphic. Using

the Cauchy-Riemann equations for f and f , we conclude that their directional derivatives

must be 0. Hence, f is constant and we are done.

4

1.2 Complex Manifolds

Similarly to a Riemann surface, a n-dimensional complex manifold is an object that

locally behaves like Cn. Though its behavior is more complex than in the case of Riemann

surfaces, the basic definitions will be similar.

Definition 4. A n-dimensional complex coordinate chart, or simply chart, on a topological

space M is a homeomorphism φ : U →W , where U ⊂M and W ⊂ Cn are open sets.

Definition 5. A n-dimensional complex manifold M is a topological space, together with a

covering of coordinate charts φα : Uα →Wα ⊂ Cn, such that for every pair α, β, φα φ−1βis an invertible holomorphic function.

1.3 Vector Bundles

On this work we intend to study holomorphic vector bundles over Riemann surfaces (in

particular, over the Riemann sphere). So, let us define the objects of our study:

Definition 6. A holomorphic vector bundle of rank n, E, over a Riemann surface S is a

(n+1)-dimensional complex manifold together with a projection map π : E → S, such that:

1. for each point p ∈ S, π−1(p) is a C-vector space of dimension n;

2. for each point p ∈ S there is a neighborhood of p, U ⊂ S, and a homeomorphism ϕU ,

such that the following diagram commutes:

π−1(U)

U

U × Cn................................................................................................................................. ........

....

π

........................................................................................................................................................................ ............ϕU

....................................................................................................................................................

p1

(p1 is the usual projection on the first coordinate).

3. For every pair of homeomorphisms ϕU and ϕV as in 2. and whenever U ∩ V 6= ∅,

ϕU ϕ−1V |(U∩V )×Cn has the form

ϕU ϕ−1V |(U∩V )×Cn : (U ∩ V )× Cn → (U ∩ V )× Cn

(p, w) 7→ (p,AUV (p)w)

where AUV : U ∩ V → GL(n,C) defines a linear isomorphism (these are called transi-

tion functions).

5

Remark. Vector bundles of rank 1 are called line bundles. On the sequel, unless it is explicit

on the context, we will use n to denote the rank of a vector bundle.

We will use Ex to denote the fiber of a vector bundle E at a point x ∈ S.

Example 3. (1) The trivial vector bundle S × Cn over S.

(2) The simplest non-trivial line bundles over S are denoted by Lp, where p ∈ S. To define

these line bundles we take a chart φ0(•) = z(•) of a neighborhood U0 of p, with z(p) = 0,

and the open set U1 = S \ p. Furthermore, Lp is given by gluing U0 × C and U1 × Cthrough the transition function g01 = z.

Notice that g01 is invertible, i.e. never vanishes, because its domain is U0 ∩U1, thus not

containing p, which is the only zero of φ0 = z.

(3) When working on P1, using the standard cover U0, U1 of example 1, we may set a

transition function by g01 = zm, where z = φ0 are the coordinates on U0. This way, we

obtain a line bundle that we denote by O(m).

When E is a vector bundle over P1, we use E(m) to denote the vector bundle E⊗O(m).

(4) The canonical bundleK of a Riemann surface S is a line bundle given by the holomorphic

1-forms over it. Each holomorphic 1-form is locally given by ωU = f(z)dz, so when we

change coordinates to z = φV φ−1U (z), so does change its local expression:

ωV = f(z)dz

dzdz =

f(z)

(φU φ−1V )′(z)dz.

It immediately follows that the transition functions of K can be given by:

gV U (z) =1

(φU φ−1V )′(z)= (φV φ−1U )′(z)

We can obtain more examples of vector bundles through certain operations. We shall

consider: the direct sum, the dual, the tensor product and the determinant line bundle of

vector bundles.

(5) Consider E and F are vector bundles with rank m and n (respectively) and transition

functions Aij and Bij (respectively), for a covering Ui of S. We define the direct sum

of E and F as the vector bundle E ⊕ F with rank m+ n, whose transition function are

given by:

Cij :=

[Aij 0

0 Bij

].

6

(6) The dual of E, as set above, is the vector bundle E∗ with rank m, whose transition

functions are given by:

A′ij := (A−1ij )T .

(The superscript T is used for the transpose of a matrix.)

(7) The tensor product of E and F , as set above, is the vector bundle E⊗F with rank mn,

whose transition functions are given by:

Cij(uj ⊗ vj) := ui ⊗ vi = (Aijuj)⊗ (Bijvj)

This defines Cij(x) for a basis of Ex ⊗ Fx, so that we may define it everywhere by

linearity.

(8) The determinant line bundle of E, as set above, is the line bundle det(E), whose tran-

sition functions are given by:

cij := det(Aij)

Another motivation for this example is to use the highest exterior power forms on the

fibers of E, thus we may may also denote it as ∧mE ∼= det(E).

Definition 7. A holomorphic section of a vector bundle E, over a Riemann surface S, is a

holomorphic map s : S → E satisfying:

π s = idS

Example 4. (1) Every vector bundle has at least one section. Namely the one whose value

at every point of S is 0.

(2) The sections of a trivial line bundle over S are in one-to-one correspondence with the

holomorphic functions on S.

(3) Lp always has a non-zero holomorphic section denoted by sp, with only one zero at p.

It is defined by:

ϕ0 sp = (id, z), on U0

ϕ1 sp = (id,1), on U1

One can see that this agrees with the transition function:

ϕ0 sp = (ϕ0 ϕ−11 ) (ϕ1 sp)

= (ϕ0 ϕ−11 ) (id,1)

= (id, g01 · 1) = (id, z)

7

(4) The sections of the canonical line bundle K, are precisely the holomorphic 1-forms on

S.

Remark. The set of all holomorphic sections of a vector bundle E over S is denoted by

H0(S,E).

On the next proposition we determine the set of holomorphic sections of the line bundle

O(m), m ∈ N.

Proposition 2. The set of holomorphic sections of the line bundle O(m), m ∈ N, over the

Riemann sphere satisfies:

dimC H0(P1,O(m)) =

0 , if m < 0

m+ 1 , if m ≥ 0

Proof. For simplicity, we denote by z the local coordinates in U0 and by z the local coordi-

nates in U1. So, for a point p ∈ P1 we write z(p) = φ0(p) and z(p) = φ1(p).

Let us consider a general section s ∈ H0(P1,O(m)). Write s0 : C → C for the local

coordinates of s in U0. As s0 is holomorphic it can be expressed by a Laurent series:

s0(z) =

+∞∑k=0

akzk

On the other hand, the local coordinates of s in U1, s1 : C→ C, should also describe an

holomorphic function. For z ∈ C∗ = φ1(U0 ∩ U1), we have:

s1(z) = g10(z)s0(z) =1

g01(z)s0(z)

=1

zm

+∞∑k=0

akzk = zm

+∞∑k=0

ak

(1

z

)k=

+∞∑k=0

akzm−k

The above series is the Laurent series of s1 around z = 0. As it has to be defined at z = 0,

it follows, only the terms with a non-negative power are admissible. Thus, ak = 0, k > m.

So, if m < 0, the section must vanish everywhere and if m ≤ 0, we have m + 1 degrees of

freedom for the numbers a0, . . . , am.

Below we define a morphism between vector bundles. It will be useful again when

classifying vector bundles up to isomorphism.

Definition 8. A morphism between the vector bundles π1 : E1 → S1 and π2 : E2 → S2 is a

pair of functions ϕ : E1 → E2 and f : S1 → S2, such that the following diagram commutes:

E1 E2

S1 S2

................................................................................................................................................................... ............ϕ

................................................................................................................................................................... ............f

.............................................................................................................................

π1

.............................................................................................................................

π2

8

and ϕ induces a linear map between the fibers π−11 (x) and π−12 (g(x)), ∀x ∈ S1.

When ϕ and f describe an invertible morphism we say that they are an isomorphism.

1.4 Presheaves and Sheaves

Another concept that is very useful when talking about vector bundles is sheaves. In

this context, we often use a particular sheaf, the sheaf of sections, which is obtained from

associating each open set U ⊆ S to the sections of the vector bundle that are holomorphic

in U .

As usual, we start by defining presheaf.

Definition 9. Let S be a Riemann surface. A presheaf of abelian groups F over S, is an

assignment of an abelian group F(U) to each open set U ⊆ S, together with morphisms

ρV U : F(V )→ F(U), whenever U ⊆ V , such that:

i) ρUU = idF(U);

ii) If U ⊆ V ⊆W , then ρV U ρWV = ρWU .

Remark. For an open set U ⊂ S, we generally designate the elements of F(U) as the sections

of F on U . We may also use Γ(U,F) to denote the group F(U).

When U ⊂ V ⊂ S are open sets and s ∈ F(V ) is a section over V , we may use s|U to

denote ρV U (s).

Having defined a presheaf, we impose some restrictions to it, which will give us a sheaf.

Definition 10. A presheaf F over S is a sheaf if, in addition to the conditions of a presheaf,

it satisfies:

iii) If s ∈ F(V ) and Ui is a covering of V , such that ρV Ui(s) = 0,∀i then s = 0;

iv) If Ui is a covering of V and si ∈ F(Ui) agree on the intersections, that is

si |Ui∩Uj= sj |Ui∩Uj , ∀i, j

then it exists s ∈ F(V ), such that s|Ui = si.

Example 5. Consider E a vector bundle over S. We obtain the sheaf of sections of E,

which is denoted by O(E), by associating to each open set U ⊆ S the set of holomorphic

sections on U . For the morphisms we simply use the restriction of the sections.

Example 6. In a similar way, we define the sheaf of holomorphic functions, OS , associating

to each open set of S the holomorphic functions on it. If G is a set with a complex structure

(or a complex manifold), we may also define a sheaf of the functions from open sets of S to

G that are locally holomorphic, which is denoted by OS(G). Again, for each of this cases,

the morphisms of the sheaves are the restriction of functions.

9

An important concept used to describe sheaves (or presheaves) are the stalks Fp, p ∈ S, of

a sheaf (or presheaf) F . These are defined through the sets of sections F(U) of successively

smaller neighborhoods U of p ∈ S:

Fp =∐p∈UF(U)

/∼

where for s0 ∈ F(U0) and s1 ∈ F(U1), s0 ∼ s1, if and only if it exists V ⊂ U0 ∩ U1, such

that s0 |V = s1 |V .

Other important fact is that for any presheaf F , there exists a sheaf associated to it,

F+, which is unique up to isomorphism. In particular, one may see that the group of stalks

for both a preasheaf F and the sheaf associated to F are the same.

For a more detailed description of this construction we recommend the reader to check

[3].

Example 7. One of the simplest examples of sheaf is called the skyscraper sheaf. Given a

group G and point p ∈ S, a skyscraper sheaf, G, is one whose sections are given by:

G(U) =

0 , if p 6∈ UG , if p ∈ U

Its easy to check that for every point q ∈ S different from p the stalk of G at q is Gq = 0,and the stalk at p is Gp = G.

1.5 Cohomology Groups

To construct the cohomology groups of a given sheaf F over S we will use a general

covering U = Uα. A p-cochain is an assignment of every intersection of p + 1 open sets

Uα0, . . . , Uαp

to an element gα0...αp∈ F(Uα0

∩ · · · ∩ Uαp). When the order of the sets is

changed, we also change gα0...αp = sgn(σ)·gσ(α0...αp) according to the sign of the permutation.

The set of all p-cochains for the covering U is denoted by Cp(U ,F).

We define the coboundary map ∂ : Cp(U ,F)→ Cp+1(U ,F) for g ∈ Cp(U ,F).

(∂g)α0...αp+1=

p+1∑k=0

(−1)kgα0...αk...αp+1

Inside the set of p-cochains we consider two subsets: the set of p-cocycles, Zp(U ,F), and

the set of p-coboundaries, Bp(U ,F). A p-cocycle is a p-cochain g, such that ∂g = 0. A p-

coboundary g which is in the image of ∂, that is it exists f ∈ Cp−1(U ,F) satisfying ∂f = g.

One can easily see that every Bp(U ,F) ⊆ Zp(U ,F), as for every g = ∂f ∈ Bp−1(U ,F), we

10

have:

(∂(∂f))α0...αp+1=

p+1∑k=0

(−1)k(∂f)α0...αk...αp+1

=

p+1∑k=0

(−1)k( k−1∑l=0

(−1)lfα0...αl...αk...αp+1+

p+1∑l=k+1

(−1)l−1fα0...αk...αl...αp+1

)

=

p+1∑k>l≥0

(−1)k+lfα0...αl...αk...αp+1+

p+1∑l>k≥0

(−1)k+l−1fα0...αk...αl...αp+1

= 0, ∀α0, . . . , αp+1

We define the p-th cohomology group of F with respect to the cover U as a quotient of

groups:

Hp(U ,F) :=Zp(U ,F)

Bp(U ,F)

This construction can be used to define a cohomology group on S, which does not depend

on the covering. For that we use a direct limit of the cohomology groups, using the refinement

of coverings:

Hp(S,F) := lim−→U

Hp(U ,F)

For the details of this construction, we recommend to the reader to check [6].

Remark. When considering the sheaf of sections of a vector bundle E over S, we write

Hp(S,E) instead of Hp(S,O(E)), for short.

Remark. When p = 0, note that the cohomology group H0(S,E) is just given by the vector

space of global sections of E over S. This is due to the fact that O(E)(S) ∼= H0(S,E).

As an example, we determine the first cohomology group of the skyscraper sheaf.

Proposition 3. If G is a skyscraper sheaf over S, then H1(S,G) = 0

Proof. We will prove that for any cover U = Uα we have H1(U ,G) = 0. To see this, we

can choose a neighborhood Uγ ∈ U of p ∈ S, and for any cochain g ∈ Ker(∂ : C1 → C2), we

define f ∈ C0 such that fα = (−1)gαγ . We can define f this way because for any open set

U , the group G(U) ∈ 0, G is determined by whether or not the set U contains p ∈ S, so

that G(U) = S(U ∩ Uγ). One can then see that ∂f = g:

(∂f)αβ = fβ − fα = −gαγ + gβγ + 0

= −gαγ + gβγ + (∂g)αβγ = gαβ , ∀α, β.

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1.6 Exact Sequences

In a similar way that we have exact sequences of groups, we may also have exact sequences

of sheaves. In short words, we say that a sequence of sheaves over S,

0→ R→ S → T → 0

is exact if for any p ∈ S there is a neighborhood V of p, such that whenever p ∈ U ⊆ V ,

0→ R(U)→ S(U)→ T (U)→ 0

is a short exact sequence. Moreover, any short exact sequence of sheaves defines a long exact

sequence with the cohomology groups,

0→ H0(S,R)→ H0(S,S)→ H0(S, T )→ H1(S,R)→ · · ·

The concept of exact sequence of sheaves comes in handy in many exercises and proofs,

and so it is important to have a good intuition for it.

Consider the following exact sequence of sheaves:

0→ OS(Z)→ OSexp(2πif)−−−−−−→ O∗S ∼= OS(C∗)→ 1 (1.1)

We obtain part of a long exact sequence:

· · · → H1(S,O∗S)δ−→ H2(S,OS(Z))→ · · ·

Using Poincare Duality we get H2(S,OS(Z)) ∼= H0(S,Z) ∼= Z:

Definition 11. The degree of a line bundle L, deg(L) is defined as the integer δ([L]) (it

may also be denoted by c1(L), the first Chern class).

For a vector bundle E, we define its degree as deg(det(E)).

Remark. We will see later that when S = P1, δ is an isomorphism. So, in the Riemann

sphere, a line bundle is uniquely defined by its degree.

Corollary 1. In the Riemann sphere Lp ∼= O(1) and, as a consequence, L⊗mp∼= O(m).

Proof. It is enough to check that both Lp and O(1) are line bundles with degree 1.

12

1.7 Fundamental Results

Before proceeding, we wish to introduce the following results. They are: Riemann-Roch’s

Theorem and Serre’s Duality. These are fundamental results in Algebraic Geometry and we

will use them a lot in the sequel.

Definition 12. The genus of a Riemann surface is the integer dimC H0(S,K) (where K is

the canonical bundle). In general, it is denoted by g.

Remark. When E is a vector bundle over S, for simplification, we use hp(S,E) to denote

the positive integer dimC Hp(S,E), with p ∈ N.

Theorem 1 (Riemann-Roch). Let S be a compact connected Riemann surface with genus

g and E a vector bundle over S. We have:

h0(S,E)− h1(S,E) = degE + (1− g) · rkE

Proof. We recommend the reader to check the proof on [4].

Theorem 2 (Serre’s Duality). Let E be a vector bundle over S. We have:

H1(S,E) ∼= H0(S,E∗ ⊗K)∗

Proof. For a proof, see, for example, [6].

Corollary 2 (Riemann-Roch, alternative form). Let S be a compact connected Riemann

surface with genus g and E a vector bundle over S. We have:

h0(S,E)− h0(S,E∗ ⊗K) = degE + (1− g) · rkE

Proof. It is a simple application of Serre’s Duality to Riemann-Roch’s Theorem. The term

h1(S,E) in the latter is exchanged by h0(S,E∗⊗K) because, using Serre’s Duality, we have:

dimC H1(S,E) = dimC H0(S,E∗ ⊗K)∗ = dimC H0(S,E∗ ⊗K)

We are now in condition of proving a result we mentioned before.

Proposition 4. The map δ that gives the degree of a line bundle over P1 is a bijection.

Proof. From the equation 1.1 we obtain the following sequence of cohomology groups:

H1(P1,OP1)→ H1(P1,O∗P1)δ−→ H2(P1,OP1(Z)) ∼= Z→ H2(P1,OP1)

and we will see that the extremities are zero. It is well known that Hp(P1,OP1) = 0, p ≥ 2.

This settles the term on the right.

13

As for the left term, we begin by using Serre’s Duality to obtain dimC H1(P1,OP1) =

dimC H0(P1,K). By definition, the last expression is the genus, g, which corresponds to

the number of holes of the Riemann sphere, and so is equal to zero. However, we will also

determine the genus according to the above definition. First, using the computations on

example 3.4, we see that K corresponds to the line bundle obtained from the standard two

chart covering U0, U1, and with transition function g01(z) = −z−2. Hence, K ∼= O(−2) and

by Proposition 2, g = 0.

Therefore, δ must be an isomorphism.

14

1.8 Grothendieck’s Theorem for Vector Bundles

We will now focus on our main result on vector bundles. It is Grothendieck’s Theorem

for vector bundles. We start with a lemma.

Lemma 1. Let E be a vector bundle of rank n over S. Then there is m ∈ N sufficiently

large, such that E ⊗ Lmp has a non-trivial holomorphic section.

Proof. Let us consider the following short exact sequence:

0→ O(E)smp−−→ O(E ⊗ Lmp )→ S → 0,

where S is the quotient sheaf O(E⊗Lmp )/O(E), which is a skyscraper sheaf with Sp = Cmn.

By Proposition 3, H1(P1,S) = 0. From the above sequence, we obtain a long exact

sequence:

0→ H0(P1, E)→ H0(P1, E ⊗ Lmp )→H0(S)→

→ H1(P1, E)→ H1(P1, E ⊗ Lmp )→ 0

As in an exact sequence, the alternating sum of the dimensions of the terms between two

zeros is zero, we obtain:

dimC H0(P1, E ⊗ Lmp )

= dimC H1(P1, E ⊗ Lmp ) + dimC H0(S) + dimC H0(P1, E)− dimC H1(P1, E)

≥ dimC H0(S) + dimC H0(P1, E)− dimC H1(P1, E)

= mn+ dimC H0(P1, E)− dimC H1(P1, E).

This way, choosing m sufficiently large we get dimC H0(P1, E ⊗ Lmp ) > 0. The proof is

complete.

We now prove the main theorem.

Theorem 3 (Grothendieck). For any vector bundle E with rank n over P1, there are integers

a1, . . . , an, unique up to permutation, such that E decomposes as

E ∼= O(a1)⊕ · · · ⊕ O(an).

Proof. Given m as in Lemma 1, we obtain a short exact sequence:

0→ O(E(m− 1))sp−→ O(E(m))→ T → 0,

where T is the quotient sheaf O(E(m))/O(E(m− 1)). From this we obtain:

0→ H0(P1, E(m− 1))sp−→ H0(P1, E(m))→ · · ·

15

Thus, multiplication by sp induces an injective map between the cohomology groups.

We can prove, though, that this multiplication is not an isomorphism, because such thing

would imply that every holomorphic section in E(m) would be zero at p. Yet, point p is

arbitrary, implying that H0(P1, E(m)) = 0. We get that:

dimC H0(P1, E(m− 1)) < dimC H0(P1, E(m)).

As a consequence, we may determine m, such that

0 = dimC H0(P1, E(m− 1)) < dimC H0(P1, E(m))

and every non-trivial holomorphic section s of E(m) will never vanish. If s had a zero at

p ∈ P1, then s·s−1p would be an holomorphic section of E(m)⊗L−1p ∼= E(m−1), contradicting

our choice for m. This allows us to define a trivial line bundle L ∼= P1 × C inside E(m),

whose inclusion is given in local coordinates by

(p, λ) 7→ (p, λs(p))

Furthermore, we define a quotient bundle Q with rank n− 1 and whose fibers are locally

given by Cn/〈s(z)〉 ∼= C(n−1). This gives us another short exact sequence:

0→ O(L) ∼= OP1 → O(E(m))→ O(Q)→ 0 (1.2)

Now, we will use induction on the rank of the vector bundle to finish the proof. So, we

may assume that there are integers b1, . . . , bn−1, such that Q decomposes as

Q ∼= O(b1)⊕ · · · ⊕ O(bn−1)

where b1. . . . , bn−1, are integers.

From the previous short exact sequence and doing the tensor product with O(−1) we

have

0→ O(L(−1))→ O(E(m− 1))→ O(Q(−1))→ 0

and we obtain the exact sequence

. . .→ H0(P1, E(m− 1))→ H0(P1, Q(−1))→ H1(P1,O(L(−1)))→ . . .

where H0(P1, E(m− 1)) = 0, by the way we defined m. By Riemann-Roch we also have

dimC H1(P1,O(−1)) = dimC H0(P1,O(−1))− degO(−1)− (1− g)

= 0 + 1− (1− 0) = 0.

This allows us to conclude that H0(P1, Q(−1)) = 0 and, hence, bi ≤ 0.

16

Now, we will see that E(m) ⊗ Q∗ has a non-trivial section. To do so, we do a tensor

product of a previous exact sequence with Q∗:

0→ O(Q∗)→ O(E(m)⊗Q∗) ∼=O(Hom(Q,E(m)))→

→ O(Q⊗Q∗) ∼= O(Hom(Q,Q))→ 0

and we obtain the following exact sequence:

. . . H0(P1,Hom(Q,E(m)))→ H0(P1,O(Hom(Q,Q))→ H1(P1, Q∗) . . . (1.3)

We can conclude that H1(P1, Q∗) = 0, using Riemann-Roch and Proposition 2 on its

factors:

dimC H1(P1,O(−bi)) = dimC H0(P1,O(−bi))− degO(−bi)− (1− g)

= (−bi + 1) + bi − (1− 0) = 0.

Therefore, H0(P1,Hom(Q,E(m))) → H0(P1,O(Hom(Q,Q))) ∼= C(n−1)×(n−1) is surjec-

tive, implying the existence of a section ι ∈ H0(P1,Hom(Q,E(m))) such that ι 7→ I, through

the map from equation (1.3) and where I is the identity section in Hom(Q,Q). Given that

the projection α : E(m)→ Q was in the base of the previous exact sequences, as in equation

(1.2), we get α ι = I. Hence, ι defines an inclusion of Q inside E(m) and we conclude that

E(m) ∼= L⊕Q. Finally, we get:

E ∼=(L⊕ (O(b1)⊕ · · · ⊕ O(bn−1))

)⊗O(−m)

∼= O(−m)⊕O(b1 −m)⊕ · · · ⊕ O(bn−1 −m).

This proves our induction hypothesis and we are done.

17

18

Chapter 2

Principal Bundles

On this chapter we present Grothendieck’s Theorem for principal bundles. We start by

defining fiber bundle and principal bundle. Then, we give a description of isomorphism

classes of principal bundles using cohomology. For the latter, we will first introduce a

description of the first cohomology group for sheaves of non-Abelian groups. After this, we

finally present Grothendieck’s Theorem and show the vector bundle case as a corollary of

the previous.

We end the chapter with a proof of Grothendieck’s Theorem for vector bundles that have

a holomorphic quadratic from, or symplectic form defined on its fibers.

Definition 13. A fiber bundle over a Riemann surface S is a topological space E, together

with a surjective map π : E → S, called projection, and a topological space F , called fiber,

such that for every p ∈ S there is a neighborhood U ⊂ S, such that the following diagram

commutes:

U

π−1(U) U × F.............................................................................................................................

π

...............................................................................................................................................................................................................................

p1

...................................................................................................................................................... ............φ

(φ is a homeomorphism and p1 is the usual projection on the first coordinate).

Definition 14. A principal bundle E with structure group G (or simply G-bundle) is a fiber

bundle with a variety E and fiber G (G is a topological group), together with a continuous

right action of G on E. The action of G preserves each fiber of E and its restriction to

π−1(p), p ∈ S, is free and transitive.

Example 8. (1) The trivial bundle S×F is the simplest example of a fiber bundle over S,

with fiber F .

19

(2) Vector bundles can also be given as principal bundles with structure group GL(n,C)

(see Definition 16).

Definition 15. A morphism between two G-bundles π1 : E1 → S1 and π2 : E2 → S2 is a

pair of functions ϕ : E1 → E2 and f : S1 → S2, such that the following diagram commutes:

E1 E2

S1 S2

................................................................................................................................................................... ............ϕ

................................................................................................................................................................... ............f

.............................................................................................................................

π1

.............................................................................................................................

π2

and the function ϕ induces a G-equivariant map between the fibers π−11 (p) and π−12 (f(p)),

∀p ∈ S1.

2.1 Cohomology of non-Abelian Groups

We wish to give another description of a vector bundle which involves cohomology. Re-

member that the transition functions of a vector bundles take values in GL(n,C), which is

not an Abelian group. That is why we want a more general description of cohomology group

(in fact, this will only apply directly for the first cohomology group).

Let us start by considering a sheaf of non-Abelian groups F over S and a covering

U = Ui. As mentioned before, a 1-cochain is an assignment of every intersection of 2

open sets Ui, Uj to an element gij ∈ F(Ui ∩ Uj). When the order of the sets is changed we

multiply gij = (−1)gji by −1. A 1-cocycle is a 1-cochain g, such that:

gij · gjk = gik ⇔ gjk · g−1ik · gij = 1, in Uijk, ∀i, j, k

The set of 1-cocycles is denoted by Z1(U ,F) and, in general, it is not a subgroup of the set

of 1-cochains, C1(U ,F).

We define an action of C0(U ,F) in C1(U ,F) by:

(D(g0) · g1)ij = gi · gij · g−1j , ∀i, j, where g0 = (gi) ∈ C0(U ,F)

g1 = (gij) ∈ C1(U ,F)

Doing a simple calculation, one concludes that Z1(U ,F) is stable under the action of

D(C0(U ,F)). So, we are able to define the first cohomology group of F with respect to

U , as the set of orbits of Z1(U ,F) under the action of D(C0(U ,F)):

H1(U ,F) := Z1(U ,F)/D(C0(U ,F))

20

Again, we can obtain the cohomology groups independently of the coverings using suc-

cessively finer coverings and taking the direct limit:

H1(S,F) := lim−→U

H1(U ,F)

Now, when we consider a vector bundle E, we can think of its transition functions

(Aij) ∈ C1(U ,OS(GL(n,C))) as 1-cochains, with respect to a covering U = Ui, which are

indeed 1-cocycles because, by definition, we have:

Aik = Aij ·Ajk, ∀i, j, k

Moreover, if we change the local trivializations multiplying them by (Ai) ∈ C0(U ,OS(GL(n,C))),

we are simply changing our local referential for the fibers, so we expect to obtain an home-

omorphic vector bundle E with transition functions given by:

Aij = Ai ·Aij ·A−1j

Thus, if we identify E ∼ E it is natural to relate vector bundles with the corresponding

class of H1(S,OS(GL(n,C))).

Definition 16 (alternative). A vector bundle of rank n over a Riemann surface S is a class

in H1(S,OS(GL(n,C))).

In particular a line bundle can also be given as a class in H1(S,OS(C∗)).A completely analogous approach is also valid for G-bundles in general, using its transi-

tion functions (gij) ∈ C1(U ,OS(G)), obtained from:

ϕi ϕ−1j |(Ui∩Uj)×G : (Ui ∩ Uj)×G→ (Ui ∩ Uj)×G

(p, g) 7→ (p, gij(p) · g)

Definition 17 (alternative). A principal bundle with structure group G over a Riemann

surface S is a class in H1(S,OS(G))).

21

2.2 Complex Reductive Lie Groups

We are now preparing to look at Grothendieck’s general classification theorem. Grothendieck’s

Theorem for Vector Bundles will be a consequence of this result, as any complex vector bun-

dle of rank n is a GL(n,C)-bundle.

This result provides a different description of the G-bundles over P1, given a complex

reductive Lie group G. A reductive Lie group is one whose Lie algebra is the direct sum of

its center with a semi-simple Lie algebra. For the theory of reductive Lie groups we refer to

[5]. However, in this work we concentrate on elementary examples of reductive groups such

as GL(n,C), O(n,C) and Sp(n,C).

Before continuing, we shall fix some notation. We will denote by G the Lie algebra of G

and h one of its Cartan algebras. H will denote the Cartan group associated to h. N will

denote the normalizer of H inside G and W will denote the Weyl group of G, which is the

discrete group given by W = N/H.

2.3 Grothendieck’s Theorem for Principal Bundles

We will compare principal bundles with structure group G with the ones with structure

group H over P1. To do this we use their description as classes from the cohomology groups

H1(P1,OP1(G)) and H1(P1,OP1(H)).

We can see that conjugation by an element of N stabilizes H ⊂ G, so it defines a map

on the classes of H1(P1,OP1(H)). However, from the construction of the cohomology groups

we see that conjugation by an element of H acts trivially on the classes of H1(P1,OP1(H)),

so we can say that the Weyl group W = N/H acts on H1(P1,OP1(H)) through conjugation.

The inclusion H → G also induces a map H1(P1,OP1(H)) → H1(P1,OP1(G)). Fur-

thermore, as N ⊂ G, conjugation with an element of N will act trivially on the classes of

H1(P1,OP1(G)) and so the following map is well defined:

H1(P1,OP1(H))/W → H1(P1,OP1(G)) (2.1)

The main result of the article is the following:

Theorem 4 (Grothendieck). The map defined in (2.1), induced by the inclusion of groups,

is bijective.

Here, we will not prove this theorem, and refer to [2].

2.4 Application to Vector Bundles

The classification of vector bundles over the Riemann sphere, as given in Theorem 3, can

be also obtained as a corollary of the previous, more general result.

22

Corollary 3. For any vector bundle E of rank n over P1, there are integers a1, . . . , an,

unique up to permutation, such that E decomposes as

E ∼= O(a1)⊕ · · · ⊕ O(an)

Proof. Remember that vector bundles of rank n are the principal bundles with structure

group GL(n,C). So, we take G = GL(n,C) on the Gothendieck’s general Theorem. The

Cartan Subgroup of this group is given by the set of invertible diagonal matrices, thus we

get H ∼= (C∗)n. Now, we apply the same Theorem:

H1(P1,OP1(GL(n,C))) ∼= H1(P1,OP1(H))/W

∼= (H1(P1,OP1(C∗))⊕n)/W

and the isomorphism is induced by the inclusion H → G.

Now, we will show that the Weyl group acts on H1(P1,OP1(C∗))⊕n through permutation

of the line bundles that we use on the direct sum. As the factors on the statement of

Grothendieck’s Theorem for Vector Bundles are unique up to permutation, this will conclude

our proof.

By definition W = N/H, where N is the normalizer of the Cartan subgroup H. g ∈ Nif and only g−1Dg = D for every diagonal matrix D. Consider the case where the only

non-zero entry of D is the entry (i, i), assuming the value 1. We can see that all entries

of Dg, for the exception of the i-th line, are zero. As g is invertible and g−1(Dg) = D

(remember that D has only one non-zero entry), we conclude that exactly one coordinate of

the vector (Dg)i• = gi• is non-zero. Therefore, every matrix in N has the form PD, where

P is a permutation matrix and D is an invertible diagonal matrix. It is now obvious that

we may choose permutation matrices as representatives of the classes on W = N/H and so

W ∼= Sn, ending the proof.

23

2.5 A lemma

We now introduce a lemma that will reveal to be very useful to prove some results in

the sequel.

Lemma 2. Let f(z) ∈ C[z] be a complex polynomial, with f(0) 6= 0. There exists a polyno-

mial h(z) ∈ C[z], such that h(z)2 − z is divisible by f(z).

Proof. A simple approach using interpolation is enough. We shall provide the details for

this approach.

Consider λ1, . . . , λt the roots of f(z) and α1, . . . , αt the respective multiplicities. Our

idea is to determine h(z), such that (h(z)2−z)(k) |z=λr= 0, whenever 0 ≤ k < αr. Therefore,

we wish to choose h(z), such that

(h(z)2)(k) |z=λr=

λr , if k = 0

1 , if k = 1

0 , if 1 < k < αr

Now, notice that h(z)2 is the product of two polynomials, which happen to be the

same. So, we may obtain an expression for (h(z)2)(k) by choosing which of the k successive

derivatives are applied to the first term of the product. Thus, we have:

(h(z)2)(k) =

k∑l=0

(k

l

)h(l)(z)h(k−l)(z)

Consider vectors µr = (µ(0)r , . . . , µ

(αr−1)r ), 1 ≤ r ≤ t, satisfying

k∑l=0

(k

l

)µ(l)r µ

(k−l)r =

λr , if k = 0

1 , if k = 1

0 , if 1 < k < αr

, 0 ≤ k < αr, 1 ≤ r ≤ t

This choice is possible, because we choose µ(0)r 6= 0 to be a square root of λr 6= 0 and we

may set

µ(1)r =

1

2µ(0)r

, µ(k)r = −

∑k−1l=1 µ

(l)r µ

(l−k)r

2µ(0)r

, 1 < k < αr

Finally, we may use interpolation to obtain a polynomial h(z) satisfying:

h(k)(λr) = µ(k)r , 0 ≤ k < αr, 1 ≤ r ≤ t

Furthermore, we will have

(h(z)2 − z)(k) |z=λr= 0, 0 ≤ k < αr, 1 ≤ r ≤ t

and hence, f(z) must divide h(z)2 − z.

24

2.6 When G = O(n,C)

From the inclusion O(n,C) → GL(n,C) we may see that every principal bundle with

structure group O(n,C) also defines a vector bundle. Furthermore, a vector bundle has a

nondegenerate holomorphic symmetric bilinear form (quadratic form) on its fibers, if and

only if it can be reduced to a O(n,C)-bundle.

On this section we will provide a classification result for O(n,C)-bundles. The following

result shows that two distinct O(n,C)-bundles define distinct vector bundles, or equivalently,

a quadratic form on the fibers of a vector bundle is unique up to a linear transformation on

the fibers.

Proposition 5. The map H1(P1,OP1(O(n,C)))→ H1(P1,OP1(GL(n,C))) is injective.

Proof. Let us consider a fiber bundle E ∈ H1(P1,OP1(GL(n,C))) which is the image of two

fiber bundles A,B ∈ H1(P1,OP1(O(n,C))) through the inclusion map. Let us also denote

by Aij and Bij the transition functions of A and B, respectively. As the fiber bundles A

and B coincide as vector bundles, there are matrices Ci ∈ Γ(Ui,OP1(GL(n,C))) such that:

CiAijC−1j = Bij ⇔ Ci = BijCjA

−1ij (2.2)

Our objective is to prove that it also exists matrices C ′i ∈ Γ(Ui,OP1(O(n,C))) such that:

C ′iAij(C′j)−1 = Bij (2.3)

so that A and B will define the same class in H1(P1,OP1(O(n,C))).

Using the previous equation, we get:

CTi Ci = (BijCjA−1ij )TBijCjA

−1ij

= AijCTj B

TijBijCjA

−1ij

= AijCTj CjA

−1ij

We conclude that the CT• C• induces a map on the fibers of E, because it agrees with the

transition functions. The characteristic polynomial of CT• C• is also well defined as:

det(CTi Ci − z · I) = det(A−1ij (CTi Ci − z · I)Aij)

= det(CTj Cj − z · I)

Furthermore, as its coefficients define an holomorphic function on P1 and this surface is

compact, it follows that the characteristic polynomial is constant. We denote by p(z) this

polynomial and by h(z) a polynomial such that h(z)2 − z is divisible by p(z), according to

Lemma 2.

25

Now, set h(CTi Ci) = Qi. By the Cayley-Hamilton Theorem, we have p(CTi Ci) = 0, so

we may conclude that Q2i = h(CTi Ci)

2 = CTi Ci.

As CTi Ci is symmetric, its powers will also be symmetric, and so we may conclude that

Qi = h(CTi Ci) = (h(CTi Ci))T = Qi, i.e. Qi is symmetric.

Finally, we verify that C ′i = CiQ−1i ∈ Γ(P1,OP1(O(n,C))):

(C ′i)TC ′i = (Q−1i )TCTi CiQ

−1i

= Q−1i Q2iQ−1i = In

and using the fact that u• induces a map on the fibers of E, together with equation (2.2),

we have:

C ′iAij(C′j)−1 = (CiQ

−1i )Aij(QjC

−1j )

= CiQ−1i (AijQjA

−1ij )AijC

−1j

= CiQ−1i QiAijC

−1j = Bij

This ends the proof.

2.7 Orthogonal Structure

We now present a necessary and sufficient condition as to when it is possible to reduce

a vector bundle to a O(n,C)-bundle.

Theorem 5. Let E be a vector bundle. E has an orthogonal structure if and only if it is

isomorphic to its dual, E∗.

Proof. In the case that E has an orthogonal structure we obtain every homomorphism on

the fibers of E from the product by a section of E, using the quadratic form defined on the

fibers of E, so it easily follows that E is isomorphic to its dual.

In the case that E is isomorphic to E∗, we can use the factorization of E as line bundles

to get:

E ∼= O(m1)⊕ · · · ⊕ O(mn)

E∗ ∼= O(−m1)⊕ · · · ⊕ O(−mn)

As this factorization is unique up to isomorphism we conclude that we can pair any term

mi with −mi. Thus, by setting R =⊕

mi>0O(mi) we have:

E ∼= E0 ⊕R⊕R∗

Lemma 3. R⊕R∗ is endowed with a orthogonal structure.

26

Proof. We define a quadratic form on the fibers of R⊕R∗ given by:

(a+ a′, b+ b′) = 〈a, b′〉+ 〈b, a′〉, a, b ∈ Rx, a′, b′ ∈ R∗x, x ∈ P1

Notice that E0 can also be endowed with an orthogonal structure which can be defined

by (u, v) = uT v, u, v ∈ Ex, x ∈ P1, when considering the a chart where E0∼= P1 × Cr is

trivial.

Finally, adding the quadratic forms from both components of the direct sum gives us an

orthogonal structure on the global space E.

27

2.8 When G = Sp(2n,C)

As the previous case, from the inclusion Sp(2n,C) → GL(2n,C) we may see that every

principal bundle with structure group Sp(2n,C) also defines a vector bundle. Furthermore,

a vector bundle has a nondegenerate holomorphic antisymmetric bilinear form (symplectic

form) on its fibers, if and only if it can be reduced to a Sp(2n,C)-bundle.

On this section we will provide a classification result for Sp(2n,C)-bundles. The first re-

sult shows that two distinct Sp(2n,C)-bundles define distinct vector bundles, or equivalently,

a symplectic form on the fibers of a vector bundle is unique up to a linear transformation

on the fibers.

Definition 18. A matrix M ∈ GL(2n,C) is symplectic if

MTΩM = Ω, Ω =

[0 In

−In 0

]

The set of all (2n)× (2n) complex symplectic matrices is denoted by Sp(2n,C).

Remark. The matrix Ω satisfies the following relations:

1) ΩT = −Ω;

2) Ω−1 = −Ω = ΩT .

Proposition 6. The map H1(P1,OP1(Sp(2n,C)))→ H1(P1,OP1(GL(2n,C))) is injective.

Proof. We shall follow the proof of the previous proposition. Let us consider a vector

bundle E ∈ H1(P1,OP1(GL(2n,C))) which is the image of two principal bundles A,B ∈H1(P1,OP1(Sp(2n,C))), with transition functions Aij and Bij , respectively, for a given cover

U = Ui.As A and B coincide as vector bundles, there are matrices Ci ∈ Γ(P1,OP1(GL(2n,C))),

such that:

CiAijC−1j = Bij ⇔ Ci = BijCjA

−1ij

With some more calculations, we get:

−ΩCTi ΩCi = −Ω(BijCjA−1ij )TΩ(BijCjA

−1ij )

= (−Ω(A−1ij )TΩ)(−Ω)CTj Ω(−ΩBTijΩBij)CjA−1ij

= Aij(−ΩCTj ΩCj)A−1ij

This way, we may conclude that multiplication by −ΩCT• ΩC• is well defined on the fibers

of E (or A), as it agrees with the transition functions. In the same way, the characteristic

polynomial of −ΩCT• ΩC• is well defined and it is constant, as its coefficients define global

28

holomorphic functions on P1. We denote by p(z) this polynomial and by h(z) a polynomial

such that h(z)2 − z is divisible by p(z), according to Lemma 2.

Now, SetQi = h(−ΩCTi ΩCi). By the Cayley-Hamilton Theorem, we have p(−ΩCTi ΩCi) =

0, so we may conclude that Q2i = −ΩCTi ΩCi.

As −ΩCTi ΩCi satisfies −Ω(−ΩCTi ΩCi)TΩ = (−ΩCTi ΩCi), its powers will also satisfy

that equation. By linearity, it follows that −ΩQTi Ω = Qi.

Furthermore, we can see that C ′i = CiQ−1i ∈ Γ(P1,OP1(Sp(2n,C))) as:

(CiQ−1i )TΩCiQ

−1i = (Q−1i )TΩ(−ΩCTi ΩCi)Q

−1i

= (Q−1i )TΩQ2iQ−1i

= (QTi )−1(QTi Ω) = Ω

and

C ′iAij(C′j)−1 = (CiQ

−1i )Aij(QjC

−1j )

= CiQ−1i AijQjA

−1ij AijC

−1j

= CiQ−1i QiAijC

−1j

= CiAijC−1j = Bij

Finally, we conclude that A and B are the same principal bundle with group structure

Sp(2n,C).

2.9 Symplectic Structure

We now present a necessary and sufficient condition as to when it is possible to reduce

a vector bundle to a Sp(2n,C)-bundle.

Theorem 6. Let E be a vector bundle of rank 2n. E has a symplectic structure if and only

if it is isomorphic to its dual, E∗.

Proof. As in the case for the orthogonal structure, in the case that E has a symplectic

structure we obtain every homomorphism on the fibers of E from the product by a section

of E, using the symplectic form defined on the fibers of E, so it easily follows that E is

isomorphic to its dual.

In the case that E and E∗ are isomorphic, we can use the factorization of E as line

bundles to get:

E ∼= O(m1)⊕ · · · ⊕ O(m2n)

E∗ ∼= O(−m1)⊕ · · · ⊕ O(−m2n)

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As this factorization is unique up to isomorphism we conclude that we can pair any term

mi with −mi. Thus, by setting R =⊕

mi>0O(mi), we have:

E ∼= E0 ⊕R⊕R∗

Lemma 4. R⊕R∗ is endowed with a symplectic structure.

Proof. We define a symplectic form on the fibers of R⊕R∗ given by:

(a+ a′, b+ b′) = 〈a, b′〉 − 〈b, a′〉, a, b ∈ Rx, a′, b′ ∈ R∗x, x ∈ P1

Notice that E0 can also be endowed with a symplectic structure which can be defined

by (u, v) = uTΩv, u, v ∈ Ex, x ∈ P1, when considering the chart where E0∼= P1 × C2r.

Finally, adding the the symplectic forms from both components of the direct sum gives

us a symplectic on the global space E.

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Bibliography

[1] V. Balaji, Lectures on Principal Bundles, London Mathematical Soc. Lecture Note

Series 359, Cambridge University Press, 2009.

[2] A. Grothendieck, Sur la Classificationde Fibres Holomorphes sur la Sphere de Riemann,

American Journal of Mathematics 79, 1957.

[3] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-

Verlag, New York-Heidelberg, 1977.

[4] N. Hitchin, Riemann surfaces and integrable systems - Notes by Justin Sawon, Oxford

Graduate Texts on Mathematics 4, Integrable systems (Oxford, 1997), 1152, Oxford

University Press, New York, 1999

[5] A. Knapp, Lie groups beyond an introduction, Second edition, Progress in Mathematics

140, Birkhauser Boston, Inc., Boston, MA, 2002.

[6] R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics

5, American Mathematical Society, Providence, RI, 1995.

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