(±§β) = (d±)§β+(’1)i ±§( β), ± ∈ “(§i T — X), β ∈ “(§p’i T — X—E).

Then the curvature R as a 2-form with values in End(E) is de¬ned by

R = 2 . More precisely,

(±) = R § ±, ∀± ∈ “(§p T — X — E).

Now we notice that when the curvature R vanishes then

· · · ’ “(§p T — X — E) ’ “(§p+1 T — X — E) ’ · · ·

is a complex.

Now we want to get ordinary di¬erential forms from an End(E)-

valued 2 - form R. We pick a polynomial function

P : Mr (C) ’ C

60 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

which is homogeneous of degree j and conjugation invariant: P (B ’1 CB) =

P (C). When we apply P to R ∈ “(§2 T — X —End(E)) we get a complex

valued 2j-form P (R). (Locally E Cn — U and R = (Rkl ), where the

matrix entries Rkl are 2-forms and we have P (R) = P (Rkl ).)

It is a well-known fact of invariant theory that the algebra of conjugacy-

invariant polynomial functions on Mr (C) is generated by

Pj (A) = T r§j Cr (§j A) = σj (»1 , ..., »r ),

where σj is the j-th elementary symmetric function and (»1 , ..., »r ) is

the set of eigenvalues of A. This is exactly the coe¬cient of tj in the

characteristic polynomial det(I + tA), where I is the identity matrix.

This may not be surprising because the Chern classes come from taking

symmetric polynomials in the Chern classes of the line bundles obtained

by applying the splitting principle. Now we are ready to de¬ne the

Chern-Weil di¬erential form

√

„¦j = (’2π ’1)’j Pj (R).

THEOREM 2.7.4 The form „¦j is closed and its cohomology class is

cj ∈ H 2j (X, C).

Note. With this description it is not clear why this cohomology class

is an integer cohomology class.

Proof. We will go in several steps:

(1) We shall show that „¦j is closed.

(2) We shall see that the class [„¦j ] ∈ H 2j (X, C) is independent of a

particular choice of connection.

(3) We shall prove that the classes [„¦j ] satisfy the Whitney sum for-

mula.

(4) We shall use the splitting principle to reduse the problem to line

bundles, for which we already know the statement.

Actually in the steps (1) and (2) we will only assume that a form

„¦ is de¬ned by some invariant polynomial P as the assertions remain

true in this generality.

(1) We trivialize E and deal with = d + A and R = dA + 1 [A, A].

2

We extend P to a symmetric multilinear functional

P (x1 , ..., xj ), xi ∈ Mr (C), P (x) = P (x, x, ..., x).

2.7. CONNECTIONS, CURVATURE AND CHERN-WEIL 61

LEMMA 2.7.5 For B ∈ Mr (C) the following identity holds:

j

P (x1 , ..., [B, xi ], ..., xj ) = 0.

i=1

Proof. This is the in¬nitesimal version of the invariance of P . One has

P (gx1 g ’1 , ..., gxj g ’1 ) = P (x1 , ..., xj ).

Now to see the result we take g = exp(tB) and take the derivative of

both sides at t = 0 using the identity

d

[exp(tB)xi exp(’tB)]|t=0 = [B, xi ].

dt

LEMMA 2.7.6 (Bianchi identity). dR = ’[A, R].

Proof. We have R = dA + 1 [A, A], so that

2

1 1

dR = [dA, A] ’ [A, dA] = ’[A, dA].

2 2

Now to ¬nish we apply the Jacobi identity which easily yields the equal-

ity [A, [A, A]] = 0. We obtain:

[A, R] = [A, dA] + [A, [A, A]] = [A, dA].

We now return to the proof of the theorem. For an invariant polynomial

function P and „¦ = P (R) = P (R, R, ..., R) we have

d„¦ = P (dR, R, ..., R) + P (R, dR, R, ..., R) + · · · = by Lemma 4.11

= ’P ([A, R], R, ..., R) ’ P (R, [A, R], ..., R) ’ · · · = 0 by Lemma 4.10.

We observe that the Bianchi identity has an intrinsic interpretation.

A connection on E gives rise to a connection on the dual bundle E —

as well as on the bundle End(E) E — — E. Now consider the map

: §2 T — X — End(E) ’ §3 T — X — End(E).

Now what Lemma 4.11 is saying is that R is covariantly closed with

respect to the connection : R = dR + [A, R] = 0.

62 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

(2) One can use di¬erent methods to see that the cohomology class

[P (R )] is actually independent of . The ¬rst method that is kind

of slow but produces so-called secondary characteristic classes consists

d

in representing the derivative dt P (R t ) as d(smth.), where t is a one-

parameter family of connections, 0 ¤ t ¤ 1 such that t lies on the

line segment connecting 0 and 1 . (We recall that the space of con-

nections is a¬ne.)

The second method is faster and uses a homotopy argument. First

we need to introduce the notion of pull-back of a connection. If we

have a smooth map f : Y ’ X and a vector bundle E on X then

there exists a unique connection f — on the pull-back bundle f — E

satisfying the following property. If s ∈ “(X, E) is a section of E

and f — s ∈ “(Y, f — E) is the pull-back of s then (f — )(f — s) = f — ( s),

where s ∈ “(X, T — X — E) and f — ( s) ∈ “(Y, T — Y — f — E) is the

pull-back of s. The point is that a section σ of f — E can be written as

σ = i gi f — si , where si is a local basis of sections of the bundle E and

gi are smooth functions. The Leibniz identity then implies the way of

writing down the unique expression for (f — )σ. The curvature of the

pull-back connection apparently is given by Rf — = f — R .

Now consider the diagram:

p— (E) ’ E

1

“ “

p1

X —R ’ X

On p— (E) we can construct the pull-back connections and look at the

1

connection

= tp— 1 + (1 ’ t)p— 0 ,

1 1

where t is the parameter on R. Then we have [P (R )] ∈ H 2j (X — R).

One has a family of the inclusions: it : X ’ X — R, it (x) = (x, t). Now

we see that

[P (R 0 )] = i— ([P (R )]) = i— ([P (R )]) = [P (R 1 )]

0 1

due to the homotopy invariance.

(3) To see that the Whitney formula is satis¬ed let us consider two

2.7. CONNECTIONS, CURVATURE AND CHERN-WEIL 63

vector bundles E and F on X. We should show that

j

[„¦j (E • F )] = [„¦k (E)] ∪ [„¦j’k (F )],

k=0

but this equality is actually satis¬ed on the level of di¬erential forms.

Let E and F be connections on E and F respectively. Then we

get a connection E + F on the direct sum E • F such that the

corresponding curvature matrix of 2-forms is given by

RE 0

R = .

E+

0 RF

F

A0

It is a simple fact which follows from the identity det(I + t )=

0B

det(I + tA) det(I + tB) that

j

A0

Pj = Pk (A)Pj’k (B).

0B k=0

This implies that „¦j (E + F ) = j „¦k (E) § „¦j’k (F ). √

k=0

(4) For a line bundle L we know that if „¦1 = (’2π ’1)’1 R then

its cohomology class [„¦1 ] = c1 (L) ∈ H 2 (X, C). Now to see that for

each j one has [„¦j ] = cj (E) one can easily see that the classes [„¦j ]

are compatible with pull-backs and then use splitting principle or the

uniqueness theorem for the Chern classes.

We conclude this section with some further remarks on connections.

Suppose that we have two vector bundles E1 and E2 over X equipped

with connections 1 and 2 respectively. Then one can take 1 — 1 +

1 — 2 as a connection on the tensor product E1 — E2 . This connection

acts like

( —1+1— 2 )(s1 — s2 ) = 1 s1 — s2 + s1 — 2 s2

1

on a section s1 — s2 of E1 — E2 .

Let us consider an example when E is a subbundle of a trivial bundle

on X:

i

E ’ Cn — X

.

“

X

64 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

On the trivial bundle Cn — X we have the exterior derivative d as a

connection and it gives the connection on the subbundle E according

to the rule s = p[d(is)], where p is a bundle projection p : Cn —X ’ E

and i is the inclusion. More generally in the presence of a hermitian

metric if we have a vector bundle with a connection we obtain induced

connections on each subbundle and each quotient bundle.

2.8 The case of holomorphic vector bun-

dles

DEFINITION 2.8.1 A hermitian form H on a complex vector space

V is a positive de¬nite sesquilinear form H(v, w) ∈ C, v, w ∈ V such

that

H(»v, µw) = »¯H(v, w),

µ

H(v, w) = H(w, v),

H(v, v) > 0, if v = 0,

where bar stands for the complex conjugation.

If one takes V = Cn then H is going to be represented by a self-

¯

adjoint matrix M = (H(ei , ej )): M = t M and the form H is given by

H(v, w) = v t M w for column-vectors v and w. One can view H as a map

¯

¯ ¯

of vector spaces V — V ’ C. Here V is the complex-conjugate vector

¯

space to V . This means that V = V as a set but the action of » ∈ C on

¯

¯

v ∈ V transforms it into »v. Now we de¬ne a hermitian inner product

H on a vector bundle E over X as follows. For each x ∈ X one has a

hermitian form Hx on the ¬ber Ex and this form varies smoothly with

x. Another way to say it is that H is a homomorphism of vector bundles

¯ ¯

H : E — E ’ 1X . Here E is the vector bundle complex-conjugate to

E.

Let us de¬ne further two important characteristics of a connection

. We notice that we can represent the exterior di¬erential d as the

sum d + d so that for a di¬erential form

ω = f dzi1 § dzi2 § · · · § dzip § d¯j1 § d¯j2 § · · · § d¯jq

z z z

2.8. THE CASE OF HOLOMORPHIC VECTOR BUNDLES 65

in local coordinates (z1 , ..., zk ) we have

‚f

dω = dzl § dzi1 § dzi2 § · · · § dzip § d¯j1 § d¯j2 § · · · § d¯jq

z z z

‚zl

l

and similarly for d . So if ω is a di¬erential form of pure type (p, q) then

d ω is of pure type (p + 1, q) and d ω is of type (p, q + 1). Analogously

to the exterior derivative d one can decompose a connection as =

+.

1). We say that is compatible with holomorphic structure if for a

holomorphic section s of E over an open set U ‚ X we have s = 0,

i.e. s is of pure type (1, 0).

2). We say that is compatible with hermitian structure H if for two

sections s1 and s2 of E one has

dH(s1 , s2 ) = H( s1 , s2 ) + H(s1 , s2 ).

In this equality H is extended to pairing between functions and 1-forms

which is complex-linear in the ¬rst variable and antilinear in the second

one.

PROPOSITION 2.8.2 (I) There exists unique connection com-

patible with both holomorphic and hermitian structures.

(II) The curvature of this connection is a 2-form of the type (1, 1) with

values in End(E).

Proof. Let us pick a local basis of E consisting of holomorphic sections

X — Cr . Denote by (e1 , ..., er ) the standard

of E so that locally E

basis of Cr . We denote H = (H(ei , ej )) the matrix of smooth functions