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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.
2
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} -
+ +

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