to the local trivialization the connection can be written as = d + A,

where A = (A1 |A2 | · · · |Ar ) is a matrix of one forms and its i-th column

is Ai = ei . If is compatible with holomorphic structure then Ai is

of pure type (1, 0). We have

dHij = H(Ai , ej ) + H(ei , Aj )

where H(Ai , ej ) is of pure type (1, 0) and H(ei , Aj ) is of pure type (0, 1).

This implies that

d Hij = H(Ai , ej ) = t Ai Hej = Aki Hkj = (t AH)ij .

k

66 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

So we have a matrix equality d H = t AH or A = (t H ’1 )d (t H). If we

denote M = t H the transposed matrix then A = M ’1 d M and we have

found the only possible connection. The curvature of this connection

R = dA+A§A = ’M ’1 (dM )§M ’1 d M +M ’1 dd M +M ’1 d M §M ’1 d M =

= ’M ’1 (d M ) § M ’1 (d M ) + M ’1 d d M = d (M ’1 d M ).

Thus we can see that R is of type (1, 1).

In the case of a line bundle H reduces to a function H = ||s||2 for

a local holomorphic section s. Then we have

R = 2d d log ||s||.

2.9 Riemann-Roch-Hirzebruch theorem for

n

CP

In this section we intend to prove the RRH theorem for the projective

space CPn . We shall proceed in two major steps:

1). We verify the theorem for the line bundle L—m (recall that L in our

notation is the dual bundle to the tautological line bundle).

2). We show that this implies RRH theorem for any holomorphic vector

bundle over CPn .

Let us consider an exact sequence of holomorphic vector bundles

over CPn :

0 ’ E1 ’ E2 ’ E3 ’ 0.

Claim. If RRH is true for any two of the vector bundles involved in

this exact sequence, it is also true for the third.

Proof. Obviously the equality

χ(CPn , E2 ) = χ(CPn , E1 ) + χ(CPn , E3 )

follows from the long exact sequence for cohomology. To see that the

r.h.s. of RRH is also additive it is enough to notice that ch(E2 ) =

ch(E1 )+ch(E3 ) because for this equality we can consider all the bundles

2.9. RIEMANN-ROCH-HIRZEBRUCH THEOREM FOR CPN 67

as just C ∞ complex vector bundles and split the exact sequence. So we

get

[CPn ], ch(E2 ) ∪ T d(T CPn ) =

[CPn ], ch(E1 ) ∪ T d(T CPn ) + [CPn ], ch(E3 ) ∪ T d(T CPn ) .

For each complex manifold X we de¬ne Grothendieck group K(X)

as the group generated by classes of isomorphisms [E] of all holomorphic

vector bundles subject to relations [E2 ]’[E1 ]’[E3 ] whenever E1 , E2 , E3

form an exact sequence like the one above. Actually K(X) is a ring if

we consider the tensor product operation. We will need the following

fact which we do not prove here.

PROPOSITION 2.9.1 The group K(CPn ) is generated by the classes

of line bundles [1CPn ], [L], ..., [L—n ]. Moreover, the relation ([L]’[1CPn ])n+1 =

0 is satis¬ed.

We already have seen the formula for the Euler characteristic of CPn

with coe¬cient in the holomorphic vector bundle L—m :

(m + 1)(m + 2) · · · (m + n)

χ(CPn , L—m ) = .

n!

With the notatiuon ξ = c1 (L) one has ch(L—m ) = exp(c1 (L—m )) = emξ .

We also recall the following exact sequence of vector bundles

0 ’ 1CPn ’ Cn+1 — L ’ T CPn ’ 0.

Taking into account that the total Chern class of a trivial bundle is just

1, by the Whitney sum formula we have

c(T CPn ) = c(Cn+1 — L) = (c(L))n+1 = (1 + ξ)n+1 .

Besides, we need the multiplicativity of the Todd class:

ξ

T d(T CPn ) = T d(L)n+1 = ( )n+1 .

1 ’ e’ξ

Now we can compute the right hand side of the RRH formula

ξ

r.h.s. = [CPn ], ch(L—m ) ∪ T d(T CPn ) = [CPn ], emξ ( )n+1

’ξ

1’e

68 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

and since [CPn ] is an element of degree 2n homology, this expression is

ξ

equal to the coe¬cient of ξ n in the Taylor series for emξ ( 1’e’ξ )n+1 . By

the residue theorem this coe¬cient is equal to

z n+1 emz

1 1

mz

z ’n’1 dz = √

√ e dz,

(1 ’ e’z )n+1 (1 ’ e’z )n+1

2π ’1 2π ’1

C C

where C a small circle around the origin in C. We introduce the new

2 dy

variable y = 1 ’ e’z = z ’ z2 + · · ·, so that dz = 1’y and this is an

invertible holomorphic change of coordinate and if a simple contour γ

around the origin is the image of C then we have

(1 ’ y)’m dy

1

√

r.h.s. =

y n+1 1 ’ y

2π ’1 γ

and now one can see that it is exactly the coe¬cient of y n in (1’y)’m’1

so it is equal to (m+1)(m+2)···(m+n) and we have proved RRH for the

n!

bundle L—m .

x

LEMMA 2.9.2 The function f (x) = is the only possible power

1’e’x

series in the RRH theorem.

This lemma is a consequence of

LEMMA 2.9.3 If Q(x) = 1 + ax + bx2 + · · · is a formal power series

such that the coe¬cient of xn in Q(x)n+1 is equal to 1 for all n then

x

Q(x) = 1’e’x .

Proof. We will show that if Q(x) and R(x) both satisfy this condition

for n ¤ k then Q(x) ’ R(x) = O(xk ). We will use induction on k.

For k = 1 this is certainly true. We assume next that the statement

is true for k = p: R(x) = Q(x) + xp S(x), where S(x) is a formal

power series. Now we raise both sides to the power p + 1: R(x)p+1 =

Q(x)p+1 + (p + 1)xp S(x)Q(x)p + O(x2p ). The comparison of coe¬cients

of xp gives us 1 = 1 + (p + 1)S(0) and it means that S(0) = 0 and we

see that R(x) ’ Q(x) = O(xp+1 ) as desired.

The assumption of this lemma is exactly the RRH formula for the

trivial line bundle over CPn and this is why Lemma 5.3 implies Lemma

5.2.

2.10. RRH FOR A HYPERSURFACE IN PROJECTIVE SPACE 69

2.10 RRH for a hypersurface in projec-

tive space

Let X ‚ CPn be a smooth hypersurface given by an irreducible ho-

mogeneous equation F = 0, where F is a homogeneous polynomial of

degree d. Assume that a holomorphic vector bundle E over X comes

˜

from a holomorphic vector bundle E over CPn :

˜

E ’ E

“ “

’ CPn

X

Let us consider the usual open covering (U0 , ..., Un ) of CPn . It gives us

an induced covering (U0 © X, ..., Un © X) of X such that Uj © X is Stein.

ˇ ˇ

The Cech cohomology of E is the cohomology of the Cech complex

··· ’ “(Ui0 ...ip © X, E) ’ · · · .

i0 <···<ip

We observe that a homogeneous polynomial F of degree d is the

same as a section of the bundle L—d so we have a natural map “(U, V —

F

L—’d ) ’ “(U, V ) for any holomorphic vector bundle V over an open

set U . The following result is straightforward.

LEMMA 2.10.1 The columns of the following diagram are exact.

0

“

˜

“(Ui0 ...ip , E — L—’d ) ’ · · ·

··· ’ i0 <···<ip

“F

˜

··· ’ i0 <···<ip “(Ui0 ...ip , E) ’ ···

“

··· ’ i0 <···<ip “(Ui0 ...ip , E) ’ ···

“

0

Thus we have a long exact sequence in cohomology:

ˇ ˜ Fˇ ˜ ˇ

· · · ’ H p (CPn , E — L—’d ) ’ H p (CPn , E) ’ H p (X, E) ’ · · ·

70 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

LEMMA 2.10.2 For any a ∈ H 2n’2 (CPn ) and its restriction a|X ∈

H 2n’2 (X) we have

[X], a|X = [CPn ], a ∪ (d · ξ) .

Proof. We note that d·ξ = cX is the cohomology class of X in H 2 (CPn ),

i

since X is the divisor of a holomorphic section of L—d . If X ’ CPn

denotes the inclusion then it follows that i— [X] = cX © [CPn ] and

[X], a|X = i— [X], a = cX © [CPn ], a = [CPn ], cX ∪ a

Now we are ready to do the main computational part.

χ(X, E) = by the long cohomology sequence =

˜ ˜

χ(CPn , E) ’ χ(CPn , E — L—’d ) =

˜ ˜

[CPn ], T d(T CPn ) ∪ {ch(E) ’ ch(E — L—’d )} =

˜

[CPn ], T d(T CPn ) ∪ ch(E)(1 ’ e’dξ ) =

by Lemma 5.5

1 ’ e’d·ξ

˜

n n

= [CP ], T d(T CP ) ∪ ch(E)( )(d · ξ) =

d·ξ

1 ’ e’d·ξ

[X], T d(T CPn )

= ∪ ch(E)( )=

|X

d·ξ

[X], ch(E) ∪ T d(T CPn )T d(L—d )’1 = [X], ch(E) ∪ T d(T X) ,

|X |X

because we alredy have seen the exact sequence for a submanifold X

0 ’ T X ’ (T CPn )|X ’ NX ’CPn ’ 0

and the Todd class is multiplicative in exact sequences. Also we used

the fact that we have established earlier that the normal bundle Nx ’CPn

L—d . We remark that it is possible to prove RRH for complete inter-

|X

sections using a slight generalization of the proposed method.

2.11. APPLICATIONS OF RIEMANN-ROCH-HIRZEBRUCH 71

2.11 Applications of Riemann-Roch-Hirzebruch

We recall that for a manifold X the Todd class T d(T X) was de¬ned

using the formal power series

x x2 x x2

x ’1

f (x) = = (1 ’ + + · · ·) = 1 + + + ···.

1 ’ e’x 2 6 2 12

If we formally decompose the total Chern class c(T X) as

c(T X) = (1 + ξ1 ) · · · (1 + ξd ),

where ξi are the ¬rst Chern classes of line bundles then

T d(T X) = f (ξ1 ) · · · f (ξd ), d = dim(X).

First, we consider the case of a compact Riemann surface X of genus

g. Here we have d = 1 and let E be a holomorphic vector bundle of

rank r over X. Let the total Chern class of E be written as

c(E) = (1 + ·1 ) · · · (1 + ·r ).

Then one has

1

ch(E) = e·1 + · · · + e·r = r + ( 2

·i ) + ( ·i ) + · · · .

2

Because of d = 1 we can write ch(E) = r + c1 (E) and

c1 (T X)

T d(T X) = f (c1 (T X)) = 1 + .

2