COMPLEX MANIFOLDS, VECTOR BUNDLES AND HODGE THEORY

JEAN-LUC BRYLINSKI

PHILIP FOTH

c Birkhauser Boston 1998. All print and electronic rights and use

rights reserved. Personal, non-commerical use only, for individuals with

permission from author or publisher.

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Contents

1 Holomorphic vector bundles 5

1.1 Vector bundles over smooth manifolds . . . . . . . . . . 5

1.2 Complex manifolds . . . . . . . . . . . . . . . . . . . . . 8

1.3 Holomorphic line bundles . . . . . . . . . . . . . . . . . . 10

1.4 Divisors on Riemann surfaces . . . . . . . . . . . . . . . 15

1.5 Line bundles over complex manifolds . . . . . . . . . . . 17

1.6 Intersection of curves inside a surface . . . . . . . . . . . 24

1.7 Theta function and Picard group . . . . . . . . . . . . . 28

2 Cohomology of vector bundles 31

ˇ

2.1 Cech cohomology for vector bundles . . . . . . . . . . . . 31

2.2 Extensions of vector bundles . . . . . . . . . . . . . . . . 38

2.3 Cohomology of projective space . . . . . . . . . . . . . . 41

2.4 Chern classes of complex vector bundles . . . . . . . . . 47

2.5 Construction of the Chern character . . . . . . . . . . . . 52

2.6 Riemann-Roch-Hirzebruch theorem . . . . . . . . . . . . 53

2.7 Connections, curvature and Chern-Weil . . . . . . . . . . 54

2.8 The case of holomorphic vector bundles . . . . . . . . . . 64

2.9 Riemann-Roch-Hirzebruch theorem for CPn . . . . . . . . 66

2.10 RRH for a hypersurface in projective space . . . . . . . . 69

2.11 Applications of Riemann-Roch-Hirzebruch . . . . . . . . 71

2.12 Dolbeault cohomology . . . . . . . . . . . . . . . . . . . 75

2.13 Grothendieck group . . . . . . . . . . . . . . . . . . . . . 83

2.14 Algebraic bundles over CPn . . . . . . . . . . . . . . . . . 84

3 Hodge theory 89

3.1 Complex and Riemannian structures . . . . . . . . . . . 89

3

4 CONTENTS

3.2 K¨hler manifolds . . . . . . . . . . . . . . .

a . . . . . . . 93

3.3 The moduli space of polygons is K¨hler . .

a . . . . . . . 104

3.4 Hodge decomposition in dimension 1 . . . . . . . . . . . 107

3.5 Harmonic forms on compact manifolds . . . . . . . . . . 109

3.6 Hodge theory on K¨hler manifolds . . . . . .

a . . . . . . . 112

3.7 Hodge Conjecture . . . . . . . . . . . . . . . . . . . . . . 128

3.8 Hodge decomposition and sheaf cohomology . . . . . . . 131

3.9 Formality of cohomology . . . . . . . . . . . . . . . . . . 132

4 Complex algebraic varieties 137

4.1 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.2 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.3 Examples and Siegel space . . . . . . . . . . . . . . . . . 142

4.4 Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.5 Algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . 161

4.6 Operations on algebraic cycles . . . . . . . . . . . . . . . 171

4.7 Abel-Jacobi theorem . . . . . . . . . . . . . . . . . . . . 175

4.8 K3 surface . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.9 Compact complex surfaces . . . . . . . . . . . . . . . . . 181

4.10 Cohomology of a quadric . . . . . . . . . . . . . . . . . . 186

4.11 Lefschetz theorem . . . . . . . . . . . . . . . . . . . . . . 190

5 Families and moduli spaces 199

5.1 Families of algebraic projective manifolds . . . . . . . . . 199

5.2 The Legendre family of elliptic curves . . . . . . . . . . . 208

5.3 Deformation of complex structures . . . . . . . . . . . . 214

5.4 Vector bundles over an elliptic curve . . . . . . . . . . . 219

5.5 Moduli spaces of vector bundles . . . . . . . . . . . . . . 221

5.6 Unitary bundles and representations of π1 . . . . . . . . 224

5.7 Symplectic structure on moduli spaces . . . . . . . . . . 232

5.8 Verlinde formula . . . . . . . . . . . . . . . . . . . . . . 237

5.9 Non-abelian Hodge theory . . . . . . . . . . . . . . . . . 243

5.10 Hyper-K¨hler manifolds . . . . . . . . . .

a . . . . . . . . 252

5.11 Monodromy groups . . . . . . . . . . . . . . . . . . . . . 255

Chapter 1

Holomorphic vector bundles

Weaseling out of things - this is

what separates us from the animals

(except for weasel). Homer Simpson

1.1 Real and complex vector bundles over

smooth manifolds

Vector bundles arise in geometry in several contexts. One may remem-

ber from the study of smooth manifolds that the notion of tangent

bundle inevitably appeared as a powerful tool of di¬erential geometry.

If the dimension of a manifold M is k then the dimension of the total

space of the tangent bundle to M is twice as big. The ¬rst and simple

example arises when we take M = Rk . Here the tangent bundle is just

the direct product of two copies of Rk . So, T Rk = {(x, y); x, y ∈ Rk }.

A vector bundle always comes with the projection map p to the

manifold. In turn, the manifold is imbedded into the bundle as its

zero-section σ0 :

TM

σ0 ‘“ p

M

In fact, for every point x ∈ M , the ¬ber p’1 (x) is the tangent space

Tx M .

5

6 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

To de¬ne a vector bundle properly, we also need the local triviality

condition. The map p is a submersion and represents a locally triv-

ial ¬bration meaning the following. Any point x ∈ M has an open

neighbourhood U , such that we have a trivialization:

p’1 (U ) U — Rk

U

Next we introduce the important notion of a section of the tangent

bundle. A section of T M is a smooth mapping v : M ’ T M such that

p · v = IdM . A section of T M is exactly a smooth vector ¬eld on the

manifold M . We denote by “(T M ) the space of all smooth sections of

T M . Apparently, it has the structure of a vector space. Besides, if we

take a smooth function f ∈ C ∞ (M ) and a section v ∈ “(T M ), then

f.v is a section of T M too, so “(T M ) also is a module over C ∞ (M ).

In addition, “(T M ) has a Lie algebra structure under the bracket of

vector ¬elds.

Further, one meets the ¬rst example of a dual vector bundle as one

consideres the cotangent bundle T — M , which is dual to T M . The ¬ber

of T — M over a point x ∈ M is the cotangent space Tx M = (Tx M )— .

—

The sections of T — M are the smooth 1-forms on M . An interesting

fact is that the manifold T — M has a canonical structure of a symplectic

manifold. It means that there exists a two-form ω on M such that

dω = 0 and ω § ... § ω is a volume form. To get ω explicitly, we take

k

the Liouville 1-form ± on T — M de¬ned as follows. For x ∈ M , ξ ∈ Tx M —

and v, a tangent vector to T — M at (x, ξ) we let ±(x,ξ) , v = ξ, dp(v) ,

where dp(v) ∈ Tx M . In local coordinates (x1 , ..., xk , ξ1 , ..., ξk ) on T — U

with U - small open subest of M di¬eomorphic to Rk we have ± =

k

i=1 ξi dxi . The symplectic form is now taken as ω = d±. (In local

coordinates, ω = k dξi dxi . It is easy to see that dω = 0 and ω k =

i=1

k

(’1) k!dx1 § · · · § dxk § dξ1 § · · · dξk .)

Let give rigorous de¬nition of a real vector bundle E over a manifold

M . First, we require the existence of a smooth map p:

E

“p

M

1.1. VECTOR BUNDLES OVER SMOOTH MANIFOLDS 7

Next, we de¬ne a manifold E —M E ‚ E — E consisting of pairs (v1 , v2 )

in the same ¬ber: E —M E = {(v1 , v2 ) ∈ E — E, such that pv1 = pv2 }.

We must have the smooth addition map:

+

E —M E ’ E

M

and the smooth dilation map R — E ’ E. We impose the requirement

that each ¬ber has a vector space structure. Besides, we need the local

triviality condition as in the case with the tangent bundle:

p’1 (U ) U — Rk

U

It follows that E —M E is a closed submanifold of E — E. We can put

the ¬eld of complex numbers C instead of R to obtain the de¬nition of

a smooth complex vector bundle over M . We notice that the number

k in the de¬nition is usually referred to as the rank of the bundle E.

A section of a vector bundle p : E ’ M is a smooth map s : M ’ E

such that p —¦ s = IdM .

Having a bundle, we can de¬ne the dual vector bundle, its symmetric

and exterior powers. Also, we can have the direct sum of two bundles

as well as the tensor product of them. Though it is more or less clear in

the former case, we claim that there exists unique manifold structure

on E —F , such that if v, w are smooth sections of E and F respectively,

then v — w is a smooth section of E — F . In terms of trivializations we

have E|U U — Rp and F|U U — Rq imply (E — F )|U U — Rpq .

One can form from two bundles E and F on M a ”hom bundle”.

By de¬nition, Hom(E, F ) = E — — F . There exists a vector bundle map

Hom(E, F ) — E ’ F . It is formed via

E— — F — E (E — E — ) — F ’ R — F

Hom(E, F ) — E F.

So, one naturally has the mapping “(Hom(E, F )) ’ Hom(“(E), “(F )).

8 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

1.2 Complex manifolds

The basic di¬erence with the real case is that the transition functions

are biholomorphic. For any open set U ‚ M we have an algebra H(U )

of holomorphic functions over U . The complex structure on M is com-

pletely de¬ned by a linear map J : T M ’ T M of its tangent bundle,

such that J 2 = ’IdT M . So, T M becomes a complex vector bundle.

Furthemore, J must satisfy some integrability condition.

If M is a complex manifold we introduce the notion of holomorphic

vector bundle E.

E

“p

M

First, E is a complex vector bundle and the total space is a complex

manifold. We require that the map p, the addition and dilation maps

are holomorphic.

Example. We consider in detail the complex projective space CPn ,

because it has a lot of structures and interesting holomorphic vector

bundles.

The complex manifold CPn is de¬ned as the set of lines in Cn+1

through the origin and is covered by n + 1 open sets U0 , U1 , ..., Un .

Each Ui is biholomorphic to Cn and is de¬ned as the set of lines in Cn+1

spanned by a vector (z0 , z1 , ..., zn ) with zi = 0. The map ψi : Ui ’ Cn

may be viewed as the one sending the line passing through the point

(z0 , z1 , ..., zn ) ∈ Cn+1 to the point ( z0 , ..., zi’1 , zi+1 , ..., zn ) ∈ Cn . The

zi zi zi zi

’1 n

inverse map ψi : C ’ Ui sends the point (u1 , ..., un ) to the line in

CPn+1 passing through the point (u1 , ..., ui , 1, ui+1 , ..., un ).

It is useful to introduce the homogeneous coordinate notation. A

point on CPn is denoted by [z0 : z1 : · · · : zn ], where at least one

coordinate is not zero, and represents the line passing through the point

(z0 , ..., zn ) in Cn+1 . In our new notation the point [z0 : z1 : · · · : zn ] is

the same as the point [»z0 : »z1 : · · · : »zn ] for any » = 0.

The topology on CPn as well as the complex manifold structure is

determined by those on Ui . Each Ui is an open set and we carry over

the topology from Cn to Ui . The complex manifold structure is given

by the atlas ψ0 , ..., ψn . Let us show it explicitly in low dimensions.

1.2. COMPLEX MANIFOLDS 9

ψ0 ψ1

In the case n = 1 we have CP1 = U0 © U1 , U0 C, U1 C,

U0,1 := U0 © U1 ‚ CP1 :

ψ

C—

0

U0,1 ’

ψ1 φ

C—

The map φ has to be compatible with the complex manifold structure.

Now we have ψ0 [z0 : z1 ] = z1 /z0 ∈ C— and ψ1 [z0 : z1 ] = z0 /z1 ∈

C— . This means that φ is a holomorphic map from C— to C— : φ(u) =

u’1 . The manifold CP1 is called also the ”Riemann sphere”, partly

because it is homeomorphic to the usual sphere S 2 and Riemann was

among the ¬rst who treated it as a complex manifold. We see that

it is possible to represent CP1 as the union of U0 and the point ”at

in¬nity”. Also there exists a well-known stereographic projection which

holomorphically identi¬es U0 with C.

This discussion gives us the idea how to prove that CPn = (Cn+1 \

{0})/C— with the quotient topology is compact in general. The space

Cn+1 \ {0} contains the sphere S 2n+1 = {(z0 , ..., zn ) ∈ Cn+1 ; |zo |2 +

... + |zn |2 = 1}. Besides, we have the direct product decompositions

Cn+1 \ {0} = R— S 2n+1 and C— = R— T, where T = {z ∈ C, |z| = 1} -

+ +