vector bundle F in the unique way so that

1). Qf (F • G) = Qf (F )Qf (G);

2). It is compatible with pullbacks;

3). For a line bundle L one has Qf (L) = f (c1 (L)).

2.6 Riemann-Roch-Hirzebruch theorem

Let X be a compact complex manifold of complex dimension n and let

F be a holomorphic vector bundle on X. There exists a well-de¬ned

class [X] ∈ H2n (X, Z) called the orientation class. To state the theorem

we notice that as at the end of the previous subsection we can construct

for any complex vector bundle V over X and any formal power series

f (x) the corresponding characteristic class Qf (V ). Let us take f (x) =

x

and V = T X - the tangent bundle of X, then the resulting

1 ’ e’x

characteristic class is called the Todd class and is denoted by T d(X) :=

54 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

Q 1’exp(’x) (T X). We already have introduced the Euler characteristic

x

χ(X, F ) of X with coe¬cients in a holomorphic vector bundle F :

n

ˇ

dim H j (X, F ).

χ(X, F ) =

j=0

Now we are ready to state

THEOREM 2.6.1 (Riemann-Roch-Hirzebruch)

χ(X, F ) = ch(F ) ∪ T d(X), [X] ,

where the pairing , is the standard duality pairing between homology

and cohomology.

Remark. In the case X is a singular algebraic variety it is still possi-

ble to de¬ne the Todd class but now it will be an element of homol-

ogy of X rather then cohomology and the above theorem will have

ch(F ), T d(X) in the right hand side. The version of the Riemann-

Roch-Hirzebruch theorem for singular algebraic varieties is due to Baum,

Fulton and MacPherson.

2.7 Connections, curvature, and Chern-

Weil approach to the characteristic classes

Let F be a smooth complex vector bundle over a complex manifold X.

A connection on F is a way of attaching to a section s of F a 1-form

with values in the vector bundle F . More precisely, if s ∈ “(U, F ) for

some open U ‚ X, then s ∈ “(U, T — X — F ). Given a local basis

(f1 , ..., fr ) of F and local coordinates (x1 , ..., xn ) on X one can write

s= hij dxi — fj .

1¤i¤n

1¤j¤r

So one can think of as of an analogue of the exterior derivative d,

because in a trivial bundle d actually serves as a connection if we pick a

basis of Cr say f1 , ..., fr and identify this trivial bundle over X with the

2.7. CONNECTIONS, CURVATURE AND CHERN-WEIL 55

product X — Cr . Then each section of Cr — X can be written as j ej fj

with smooth functions ei and one can take = d with the action

d( j ej fj ) = j fj — dej . The standard rules of di¬erential calculus for

are

(1). (s1 + s2 ) = (s1 ) + (s2 ),

(2). (f s) = f s + df — s (Leibniz rule).

Claim. A connection is uniquely determined by fi , 1 ¤ i ¤ r,

which are arbitrary.

Proof. Each section s can be represented as s = gi fi and then

i

s = i (gi fi + dgi — fi ) .

Let us write down an r — r matrix A of complex-valued one-forms

such that fi is given by the i-th column Ai of the matrix A. The

matrix A is called the potential of the connection with respect to the

basis (f1 , ..., fr ).

Remark. If we change the basis of F by a matrix - valued function

g(x) ∈ GL(r, C), x ∈ X, then A gets transformed into g ’1 Ag +g ’1 (dg).

If a section s is given as a vector-valued function v in our local basis then

¯

of course v = d¯ + A¯. A change of basis given by the matrix-valued

¯ v v

function g(x) changes into

mg’1 mg = mg’1 (d + A)mg = mg’1 dmg + mg’1 Amg ,

where mg is the operator of multiplication by g. Due to the fact that

[d, mg ] = dg we obtain mg’1 mg = d + g ’1 dg + g ’1 Ag. Therefore

with respect to the new basis the potential of the connection is g ’1 dg +

g ’1 Ag.

We note that this de¬nes an a¬ne action of the group of invertible

matrix-valued functions on the set of matrix-valued 1-forms. This fact

can be veri¬ed by a direct computation.

Let us see what is an obstruction to ¬nding a set of sections of F

which locally form a basis and which are horizontal, i.e. for each secion

s from this set one has s = 0. If s = ds + As = 0 then ds = ’As.

So one has 0 = dds = d(’As) = ’dA·s+A§ds = ’(dA+A§A)s = 0.

56 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

Here we use the exterior product for matrix-valued di¬erential forms

which is obtained as a natural extension of the exterior product of

di¬erential forms:

[B § C]ij = Bik § Ckj .

k

This implies that the curvature R = dA + A § A must be zero. To get

another formula for the curvature we notice that [A, A] = 2A § A and

hence R = dA + 1 [A, A]. This formula has an advantage: if A takes

2

values in a Lie subalgebra g of r by r matrices, for instance the Lie

algebra so(r, C) of skew-symmetric matrices, then R is a 2-form with

values in g.

Let us consider the case when F is just a line bundle L, which

corresponds to r = 1. We have considerable simpli¬cations here. First

of all, A § A = 0 and this gives R = dA. This means that R in this case

is a closed complex-valued 2-form (which is not exact in general since

A only exists locally). Let us pick a local non-vanishing section s of L,

then s = A — s, so morally A = “ ss ” and it has all the properties

(gs)

of a logarithmic derivative, because for instance, gs = ss + d log(g).

In fact, R is a ¬rst Chern class up to a numerical factor as we shall see

later.

Let us treat in detail the case of the tautological line bundle E over

the projective line CP1 . We saw that E is a subbundle of the trivial

bundle C2 — CP1 . Let p stand for the orthogonal projection from the

trivial bundle to E. Over the point [z0 : z1 ] ∈ CP1 this projection is

given by

w0 z0 + w1 z1

¯ ¯

p(w0 , w1 ) = (z0 , z1 ).

|z0 |2 + |z1 |2

This gives a connection on E if we take the exterior derivative d as

a connection on the trivial bundle followed by p. Let us take a section

s of E ((z0 , z1 ) at a point [z0 : z1 ]), then the one-form s with values

in E is given by

z0 dz0 + z1 dz1

¯ ¯

s = p(dz0 , dz1 ) = (z0 , z1 ).

|z0 |2 + |z1 |2

We can work over the open set U0 = {z0 = 0} and take z0 = 1, so we

2.7. CONNECTIONS, CURVATURE AND CHERN-WEIL 57

get in coordinates [z0 : z1 ] = [1 : z]:

z dz

¯

= d log(1 + |z|2 ),

A= 2

1 + |z|

where as usual for a complex valued function f we accept the notation

d f = ‚f dz, d f = ‚f d¯, and d = d + d . This notation is extended to

z

‚z ‚z

¯

all complex-valued di¬erential forms so that d and d are derivations

with respect to exterior product. We then have d = d + d in all

degrees. If ω is of type (p, q), then d ω is of type (p + 1, q) and d ω is

of type (p, q + 1). Now the curvature is given by

R = dA = dd log(1 + |z|2 ) = d d log(1 + |z|2 ),

so R is a 2-form of type (1, 1). At ¬rst sight it might seem that R will

blow up near the point z = ∞ of CP1 . However one can see that R

extends to a smooth 2-form in a neighborhood of the point at in¬nity

using the following trick. In local coordinates near ∞ ∈ CP1 one has

w = 1/z and 1 + |z|2 = |z|2 (1 + |w|2 ) and

d d log(1 + |z|2 ) = d d log(|z|2 ) + d d log(1 + |w|2 ).

On the other hand d d log(|z|2 ) = 0 because log(|z|2 ) is a harmonic

function. Thus R is equal to d d log(1 + |w|2 ), which is smooth near

w = 0. We want to compare the cohomology class of R with the ¬rst

Chern class c1 (E) = deg(E) = ’1 ∈ H 2 (CP1 , Z) Z. For this purpose

we make the following computation. Let Dρ be the disk of the radius ρ

centered at the origin of C U0 and let Cρ be its boundary. Then we

get (using the fact that R = dA)

R = lim R = lim A=

ρ’∞ Dρ ρ’∞ Cρ

CP1

√

z dz

¯ z dz

¯ dz

lim = lim = lim = 2π ’1.

1 + |z|2 ρ’∞ |z|2 ρ’∞ z

ρ’∞ Cρ Cρ Cρ

Thus we proved that for the tautological line bundle √ the cohomology

E

class of R that we denote by [R] coinside with ’2π ’1c1 (E) for this

particular choice of connection.

58 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

In fact, for any line bundle L given a connection , any other

connection is of the form +β, where β is a complex-valued 1-form.

Note that for any 1-form β tensor product with β gives an operator

s ’ β — s from sections of L to sections of T — X — L. Thus + β is an

operator from sections of L to sections of L to sections of T — X—L, and it

satis¬es the leibniz identity, so it gives another connection. Conversely,

for two connections and , ’ gives a bundle homomorphism

from L to T — X — L, which must be given by tensoring with a 1-form.

If (locally) = d + A, and R = dA, then R +β = dA + dβ so the 2

forms R and R +β di¬er by an exact 2-form dβ and hence have the

same cohomology class.

THEOREM 2.7.1 For any line bundle L over X with a connection

√

we have [R ] = ’2π ’1c1 (L) ∈ H 2 (X, C).

Let us notice the following useful fact

LEMMA 2.7.2 (i) If φ : E ’ E is an endomorphism of a complex

vector bundle E over a manifold X equipped with a connection and ±

is a 1-from on X then +±—φ is a connection too, where (±—φ)(s) =

± — φ(s) ∈ “(T — X — E), s ∈ “(E). Equivalently, if ψ : E ’ T — X — E

is a bundle map then + ψ is a connection.

(ii) Conversly, any connection on E is of the form + ψ, where ψ :

E ’ T — X — E is a vector bundle map.

Remark. Usually, there is no preferred connection. But for instance

in Riemannian geometry there is a canonical Levi-Civita connection

on the tangent bundle, though the precise formulae for it are not very

intuitive.

Proof. (i) Direct computation shows that + ψ satis¬es Leibniz iden-

tity.

(ii) Let be a connection on E, then one has ( ’ )(f s) = f ( ’

)s, where s ∈ “(E) and f is a C ∞ complex valued function on X.

’ : “(E) ’ “(T — X — E) is linear over C ∞ (X, C) and thus

Thus

is a vector bundle map.

We note that a bundle homomrphism E ’ T — X — E is the same

thing a section of the bundle T — X — End(E).

2.7. CONNECTIONS, CURVATURE AND CHERN-WEIL 59

PROPOSITION 2.7.3 The space of all connections on E is an a¬ne

space whose underlying space of translations is equal to “(T — X—End(E))

(i.e. the space of End(E)-valued 1-forms.)

Proof. The only thing we have to show is that there exists at least

one connection on E. Let (ρ± ) be a partition of unity subordinate to a

locally ¬nite cover (U± ). We know that locally there exist connections

± and as a global connection one can take = ± ρ± ± .

We recall the concrete and less intrinsic description of the curvature

that we gave earlier. Locally a vector bundle E is trivial and the cur-

1

vature of a connection is the Mr (C)-valued 2-form R = dA + 2 [A, A],

where A = ’d is a Mr (C)-valued 1-form. One can check that under a

change of local trivialization given by a GL(r, C) - valued function g the

curvature R gets changed into g ’1 Rg. We constructed the curvature

as an obstruction, so when it is zero then it is possible to ¬nd a basis

of horizontal sections.

To give a more intrinsic description of the curvature, let us extend

our connection to an operator

: “(§p T — X — E) ’ “(§p+1 T — X — E)

by the rule