complex then one gets the RRH theorem for a compact complex man-

ifold X.

Dolbeault cohomology is useful in many contexts. There are various

identities (Weitzenb¨ck type identities) involving curvature that imply

o

vanishing theorems under certain assumptions.

2.13. GROTHENDIECK GROUP 83

2.13 Grothendieck group of algebraic vec-

tor bundles

Let X be a complex algebraic manifold; we de¬ne the Grothendieck

group of algebraic vector bundles as a quotient

K 0 (X) = A/{[E2 ] ’ [E1 ] ’ [E3 ], 0 ’ E1 ’ E2 ’ E3 ’ 0 is exact}

where A is the free abelian group generated by algebraic vector bundles

E over X, and [E] ∈ A is the class of a such a bundle E. As a ¬rst

example consider the isomorphism

K 0 (Cn ) Z, E ’ rank(E),

where E is an algebraic vector bundle over Cn . In fact there was a

conjecture of Serre, proved in 1970-s by Quillen and Suslin (indepen-

dently), that any algebraic vector bundle over Cn is isomorphic to a

trivial algebraic bundle. The abelian group K 0 (X) is actually a ring

under the tensor product operation.

A basic fact about those groups is that if we have an inclusion

of a closed algebraic submanifold Y ’ X then there exists an exact

sequence

K 0 (Y ) ’ K 0 (X) ’ K 0 (U ) ’ 0,

where U := X \ Y is the complementary open set. When an algebraic

group G acts algebraically on an algebraic manifold X then we can

talk about equivariant vector bundles and we can form the equivariant

0

Grothendieck group KG (X). If we have a closed algebraic submanifold

Y of X which is invariant under the action of G then we have an exact

sequence

0 0 0

KG (Y ) ’ KG (X) ’ KG (U ) ’ 0,

For example, consider the following inclusion of the origin:

0 ’ Cn+1 ← Cn+1 \ {0}.

We have the action of C— on all those manifolds by dilations. Now we

consider an equivariant exact sequence:

m

KC— (point) ’ KC— (Cn+1 ) ’ KC— (Cn+1 \ {0}) ’ 0.

0 0 0

84 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

It is clear that KC— (point) is just the representation ring of C— which is

0

the ring of Laurent polynomials Z[t, t’1 ]. The ring KC— (Cn+1 ) turns out

0

to be isomorphic to Z[t, t’1 ] also and the map m is the multiplication by

(1 ’ t)n+1 . Note that KC— (Cn+1 ) is a module over KC— (point)’Z[t, t’1 ].

0 0

˜

n+1 i i — n+1

A generator for this module is the alternating sum i=0 (’1) [§ T C ],

where the vector bundles §i T — Cn+1 are C— -equivariant in a natural way.

This corresponds to the Koszul complex, whose cohomology is one-

dimensional and is concentrated at the origin. Finally we notice the

isomorphism

KC— (Cn+1 \ {0}) K 0 (CPn ).

0

This follows because the mapping Cn+1 \ {0} ’ CPn is a locally trivial

algebraic ¬bration with group C— , hence a C— -equivariant vector bundle

over Cn+1 \ {0} is the same thing as a vector bundle over CPn . This

gives rise to the following isomorphism

K 0 (CPn ) Z[t, t’1 ]/(1 ’ t)n+1 .

This group has basis (1, t, · · · , tn ).

n

2.14 Algebraic bundles over CP

Here we give a linear algebra description of algebraic vector bundles

over CPn following Bernstein-Gelfand-Gelfand [7]. Let W be an (n + 1)-

dimensional complex vector space Cn+1 and let

Λ = C • W • Λ2 W • · · · • Λn+1 W

be its exterior algebra. We endow this algebra with the structure of a

graded algebra by letting

deg(ξ) = ’1, ξ ∈ W.

Therefore, the degree of a homogeneous element w ∈ Λi W is equal to

’i.

Let us consider a ¬nitely generated graded Λ-module V , that is

V= Vj

j

2.14. ALGEBRAIC BUNDLES OVER CPN 85

has such an action of Λ that

Λi W — Vj ’ Vj’i .

Further by a Λ-module we shall understand a ¬nitely generated graded

Λ-module. We denote by P the set of (graded) free Λ-modules, and we

say that two Λ-modules V and V are P-equivalent if and only if V • P

is isomorphic as a Λ-module to V • P for some P, P ∈ P.

Let us consider CPn - the projective space corresponding to W and

let us recall the line bundle L over CPn - dual to the tautological bundle.

We also recall that the space of sections of L—m identi¬es with the space

of homogeneous polynomials in ξ = (z0 , ..., zn ) of degree m for m ≥ 0.

Therefore the space of sections of the bundle L(j) := V’j — L—j is the

same as homogeneous polynomials f (ξ) of degree j with values in V’j .

In fact, we can de¬ne the di¬erential

d : L(j) ’ L(j+1) , df (ξ) = ξ · f (ξ).

When ξ = 0 we will think of ξ interchangeably as of a point in W as

well as of a point in CPn .

The complex of vector bundles L(•) (V ) over CPn has the following

(•)

¬ber Lξ (V ) (which is a complex of vector spaces) at a point ξ ∈ CPn :

ξ ξ ξ ξ

(•)

Lξ (V ) = (· · · ’ V1 ’ V0 ’ V’1 ’ · · ·).

We call a Λ-module V faithful if the cohomology groups of the complex

(•)

Lξ (V ) vanish except in degree zero for all ξ = 0:

Ker(ξ : Vi ’ Vi’1 )

(•)

H i (Lξ (V )) = = 0, i = 0.

Im(ξ : Vi+1 ’ Vi )

(•)

In a situation when this condition holds, the groups H 0 (Lξ (V )) form

a vector bundle ¦(V ) = H 0 (L(•) (V )) over CPn . For faithful Λ-modules

the map V ’ ¦(V ) respects tensor products, taking symmetric and

exterior powers and passing to dual modules.

We also notice the following useful property. If

0’V ’P ’V ’0

86 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

is an exact sequence of Λ-modules, where P ∈ P and V is faithful, then

V [1] is also a faithful Λ-module and ¦(V [1]) = ¦(V )—L. Here V [1] is

a standard notation for Λ-module obtained from V by shifting degrees

by 1:

(V [1])j = Vj+1 .

THEOREM 2.14.1 (Bernstein-Gelfand-Gelfand). Any algebraic

vector bundle over CPn has the form ¦(V ) for some faithful Λ-module

V . Moreover, ¦(V ) and ¦(V ) are isomorphic if and only if V and V

are P-equivalent.

In fact, the above theorem remains true if we replace a complex vector

space W by a vector space over any algebraically closed ¬eld. We shall

not prove this theorem and refer the reader to [7] for a proof that uses

derived categories. We shall prove much weaker result which can help

to understand the nature of these things.

LEMMA 2.14.2 If P ∈ P is a free ¬nitely generated Λ-module, then

¦(P ) = 0.

Proof. Let v1 , ..., vk be a basis of P as of free Λ-module. Assume now

that v ∈ V0 is in the kernel of ξ. Since v = »1 v1 + · · · »k vk for some

»i ∈ Λ and (v1 , ..., vk ) is a basis, we easily conclude that ξ§»i = 0 for

1 ¤ i ¤ k. The crucial point in our proof is the simple fact that for

ξ ∈ W , ξ = 0 the complex

ξ§ ξ§ ξ§ ξ§

0 ’ C ’ W ’ Λ2 W ’ · · · ’ Λn+1 W ’ 0

is exact. Therefore, »i = ξ§µi , where µi ∈ Λ and we see that v is in

the image of ξ as well. Thus all the cohomology groups of the complex

(•)

Lξ (V ) vanish and ¦(P ) = 0.

Let now ξ0 , ..., ξn be a basis of W , and let ω = ξ0 §ξ1 § · · · §ξn be

a basis vector in Λn+1 W ‚ Λ, which has the degree ’(n + 1). Each

Λ-module V can be written as V = V 0 • PV , where PV ∈ P is a free

Λ-module and V 0 is annihilated by ω: ωV 0 = 0. Moreover, one can

check that to P-equivalent Λ-modules correspond isomorphic modules

V 0 . Thus we conclude that algebraic vector bundles over CPn are in 1’1

2.14. ALGEBRAIC BUNDLES OVER CPN 87

correspondence with faithful modules over the algebra Λ/(Λn+1 W ) =

Λ/(ω).

¦

88 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

Chapter 3

Hodge theory

Graecum est: non legitur.1

In this course one of the main objects of discussion is K¨hler man-

a

ifolds. In short, a K¨hler manifold is a manifold with complex, sym-

a

plectic and Riemannian structures which are in a certain accordance

with each other. So, to start with, we review several important points

of the theory of complex manifolds as well as some related topics.

3.1 Complex and Riemannian structures

on a manifold.

It is possible to give several equivalent de¬nitions of a complex manifold.

A smooth manifold M is a complex manifold of dimension n if it is

possible to have a cover of M by a collection of open sets {Ui } together

with di¬eomorphisms

ψi : Ui ’ Vi ‚ Cn ,

such that the induced mapping

ψij : ψi (Ui © Uj ) ψj (Ui © Uj )

1

This is Greek: it is not read (placed against a Greek word in medieval

manuscripts; a permission to skip the hard words). Lat.

89

90 CHAPTER 3. HODGE THEORY

is a bi-holomorphic mapping between two open sets in Cn . The condi-

tion that ψi is holomorphic means that if

ψij (z1 , ..., zn ) = (w1 , ..., wn )

then for each j the corresponding function wj (z1 , ..., zn ) is a holomor-

phic function in (z1 , ..., zn ). For example, the Cauchy-Riemann equa-

tions

‚wi

=0

‚ zj

¯

are true for all i and j. An invertible mapping between open sets of

Cn is called biholomorphic it is holomorphic and the inverse mapping is

holomorphic. Here, since we assume that ψij is a di¬eomorphism, then

if ψij is holomorphic, it mus be biholomorphic by the inverse function

theorem.

Here is another de¬nition of a complex structure on a smooth C ∞

manifold M . Then we require that the tangent bundle T M has a

complex structure, i.e. a bundle homomorphism J : T M ’ T M which

satis¬es J —¦ Jv = ’v, v ∈ T M . Under this condition, the tangent

bundle T M becomes a complex bundle vector M . Each ¬ber Tx M has

√

a complex vector space structure given by (a + ’1b)v = av + bJv,

where v ∈ Tx M . We say that in this situation J de¬nes an almost

complex structure on M .

A basic example here is M = Cn √ an open set in Cn ) with com-

(or

pex coordinates (z1 , ..., zn ), zi = xi + ’1yi . In coordinates (x1 , ..., xn ,

y1 , ..., yn ) the tangent bundle T Cn is trivial with a basis (‚/‚x1 , ..., ‚/‚xn ,

‚/‚y1 , ..., ‚/‚yn ) and J acts as follows:

‚ ‚ ‚ ‚

J = ; J =’ .

‚xi ‚yi ‚yi ‚xi

The notion of a holomorphic mapping arises naturally in this con-

text. Let f : M ’ N be a smooth map between two open subsets

M ‚ Cn and N ‚ Cm . Then f is holomorphic if and only if for

any x ∈ M the di¬erential map dx f : Tx M ’ Tf (x) N is complex-

linear, i.e. two complex structures on M and N are compatible. The

√

simplest situation arises in complex dimension 1: z = x + ’1y,

3.1. COMPLEX AND RIEMANNIAN STRUCTURES 91