morphic function hi over Ui the result will be the same. The point

is that the cocycles gij and gij .(hi /hj ) de¬ne isomorphic line bundles.

Such cocycles are called homologous. The cocycle de¬ned by the col-

lection of invertible holomorphic functions hi /hj over Ui © Uj is called

a coboundary. .

So, what we are really interested in is the factorgroup

{cocycles}/{coboundaries}. This group is isomorphic to the group

P ic(X). Let us take any global holomorphic section σ of a line bundle

L over X. Also, assume that we have a nice covering of X by open

sets Ui . Over each Ui we can write σ = »i si , where »i : Ui ’ C— is a

holomorphic function and si is a non-vanishing holomorphic section of

L|Ui . Let gij = si /sj as usual. The sections »i si and »j sj have to agree

over Ui © Uj . It follows that »i gij = »j and thus we proved

PROPOSITION 1.5.5 A holomorphic section of L is the same thing

1.5. LINE BUNDLES OVER COMPLEX MANIFOLDS 23

as a family of holomorphic functions »i over Ui satisfying the glueing

conditions over Ui © Uj : »j = »i gij .

As an example we take the line bundle L over CP1 = U0 ∪ U1 , where as

usual, U0 = CP1 \ ∞ C (with the coordinate z) and U1 = CP1 \ 0 C

(with the coordinate w = 1/z). Let us have g01 = z n , so that »1 (w) =

z n »0 (z) over U0 © U1 and »i is as before a holomorphic function on

Ui . Representing »0 and »1 as everywhere convergent Taylor series

and comparing the coe¬cients we see that »1 (1/z) = ∞ bj z ’j = j=0

∞ i’m

i=0 ai z , where we let m = ’n. If m < 0, we have no solution.

Otherwise, »1 (w) = m bj wj and we see that g01 = z ’m corresponds

j=0

—m

to the line bundle L . Moreover, the space of holomorphic sections is

the space of polynomials P (w) of degree at most m. In any case, the

knowledge of a transition cocycle is su¬cient for the restoration of a

line bundle.

Roughly speaking there are three types of compact complex man-

ifolds. The ¬rst type which is not too common consists of manifolds

such that their canonical line bundle (the maximal exterior power of the

holomorphic cotangent bundle) doesn™t have non-zero global sections.

An example of a manifold of this type is provided by CPn as we saw.

The second type of compact complex manifold which is the most fre-

quent is called “manifold of general type”. The positive tensor powers

of canonical line bundle of a manifold of the second type have many

sections. (As an example one takes a Riemann surface of genus more

than 1.) Finally, the third type of manifolds is called Calabi-Yau mani-

folds and their canonical bundle is trivial (as in the case of elliptic curve

or any projective variety X ‚ CPn de¬ned by a minimal homogeneous

equation F = 0 of degree n + 1.) The classi¬cation of compact complex

manifolds in complex dimension 1 is a classical problem and basically

there is only one discrete invariant in the one-dimensional case, namely

the genus. A very nice classi¬cation in dimension 2 is due to Kodaira.

In higher dimensions this is still an unsolved problem, though much

progress has been made in the last twenty years, due to work of S.

Mori and many others.

As a ¬rst application of the above theorem we prove that the normal

bundle N to CPn embedded in CPn+1 is actually isomorphic to L -

the dual of the tautological line bundle. Let the embedding CPn ’

24 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

CPn+1 be given as a hyperplane [z0 , ..., zn ] ’ [z0 , ..., zn , 0]. The vector

¬eld ‚/‚zn+1 is a nowhere vanishing section of N — L—’1 as it is of

homogeneous degree ’1. It means that div(N — L—’1 ) = 0 and hence

using the injectivity of the morphism in the above theorem we see that

N — L—’1 is trivial and N L.

1.6 Intersection of curves inside a surface

Let us consider a compact complex surface X and two complex curves

C1 and C2 inside it. We give several de¬nitions of the intersection

number (C1 , C2 ). We start from the topological de¬nition. Both curves

de¬ne homology classes in H2 (X, Z) which we denote by [C1 ] and [C2 ].

We denote by , the intersection pairing in homology. So, our ¬rst

de¬nition is (C1 , C2 ) = C1 , C2 . Another way to look at things is

to consider the divisor [C1 ] which de¬nes the holomorphic line bundle

L1 over X. We may restrict it to C2 and then we make our second

de¬nition for the intersection number as (C1 , C2 ) = deg((L1 )|C2 ).

Later we will prove the following result.

PROPOSITION 1.6.1 These two de¬nitions agree.

If we have a coincidence C1 = C2 = C, then (L1 )|C identi¬es with

the normal bundle NC ’X to the curve C ’ X and hence (C, C) =

deg(NC ’X ). Geometric intuition tells us that a little deformation C

of the curve C in the direction of the normal vector ¬eld n has to have

the property that

(C, C) = (C , C) = local multiplicities =

p∈C©C

multiplicities = deg(normal bundle),

zeros of n

which is an accordance with our de¬nition of the self-intersection num-

ber (C, C). Next, we are interested in the question of computing the

genus g of a non-singular curve C ‚ CP2 given by a homogeneous equa-

tion F (z0 , z1 , z2 ) = 0 of degree d. We know the relation between the

1.6. INTERSECTION OF CURVES INSIDE A SURFACE 25

genus and the degree of the tangent bundle to C: deg(T C) = 2 ’ 2g.

Consider our basic exact sequence

0 ’ T C ’ (T CP2 )|C ’ N ’ 0,

where N is the normal bundle. Taking the maximum exterior powers

we see that (§2 T CP2 )|C T C — N . We know that §2 T CP2 L—3 and

hence

deg(§2 T CP2 )|C = deg(L—3 ) = 3 deg(L|C ) = 3(C, line) = 3d.

|C

Also, deg(N ) = (C, C) = (C , C) = d2 , therefore

2 ’ 2g = deg(T C) = deg(§2 T CP2 )|C ’ deg(N ) = 3d ’ d2 .

Thus, we obtain the desired relation

(d ’ 1)(d ’ 2)

g= .

2

Taking d = 3 we get an elliptic curve, d = 4 leaves us with a curve of

genus 3 and so on.

If C were a singular curve with δ ordinary double points, then the

genus of the normalized curve would be (d’1)(d’2) ’ δ.

2

Given a curve C inside a compact complex surface X we have de-

¬ned a corresponding line bundle LC such that div(LC ) = [C], where

div is our standard isomorphism P ic(X) Cl(X). This means that

there exists a meromorphic section s of LC such that div(s) = [C]. Any

holomorphic section of LC over some open set U is going to look like f s

for some meromorphic function f such that div(f ) ≥ ’[C], meaning

that f has at worst pole of order 1 along C. This provides us with

an intrinsic description of “hol (U, LC ). Consider the dual line bundle

L— . One has “hol (U, L— ) = { holomorphic functions f over U such that

C C

div(f ) ≥ [C]}. One also has the pairing “hol (U, LC ) — “hol (U, L— ) ’

C

holomorphic functions = “hol (U, 1) which is correctly de¬ned. Next we

consider the restriction of L— to the curve C. Any local equation f of

C

C will induce a section of (L— )|C©U . An equation f of C over U such

C

that df never vanishes will give a nowhere vanishing section of (L— )|C .

C

26 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

Now we would like to go back to the exact sequence of vector bundles

on C de¬ning the normal bundle:

0 ’ T C ’ (T X)|C ’ NC ’X ’ 0.

If f is an equation of C over U then dfx : Tx X ’ C vanishes on Tx C so

actually dfx gives a map NC ’X ’ C. It leaves us with a non-degenerate

pairing between (L— )|C and NC ’X and one sees that NC ’X (LC )|C .

C

Therefore we get (C, C) = deg(NC ’X ).

Example. We give another look at χ(C) for a compact Riemann

surface C. We may take C as the diagonal ∆C inside the complex

surface C — C. We know that (∆C , ∆C ) = deg(N∆C ’C—C ). But

N∆C ’C—C T ∆C T C, so we have deg(N∆C ’C—C ) = deg(T C) =

χ(C).

Let us have two distinct irreducible curves C1 and C2 in CP2 de-

¬ned by homogeneous equations F and G respectively. The inter-

section multiplicity (C1 , C2 ) = deg((LC2 )|C1 ) may be also de¬ned as

(C1 , C2 ) = p∈C1 ©C2 (C1 , C2 )p where (C1 , C2 )p is a local intersection

multiplicity at the point p. To compute (C1 , C2 ) we take the equa-

tion G of C2 , restrict it to C1 and compute vp (G|C1 ). Or we can come

up with more symmetric de¬nition of (C1 , C2 )p . Let A be the alge-

bra of functions holomorphic in some neighbourhood U of p in CP2 . If

B = A/F.A, then B is the algebra of functions holomorphic in some

neighbourhood of p in C1 . We have an inclusion of algebras B ’ C[[x]],

which exhibits C[[x]] as a completion of B. Now we have

vp (G|C1 ) = dim(B/G.B) = dim(C[[x]]/G(x).C[[x]]) and

(C1 , C2 )p = dim(Z/A.F + A.G),

which gives us a symmetric de¬nition of the intersection multiplicity.

This de¬nition also works when C1 and/or C2 are singular.

Example. Take p = (0, 0) and two curves de¬ned by C1 : f1 =

x3 +y 2 and C2 : x2 ’y 4 . The curve C1 has a cusp at p and C2 is a union

of two smooth curves intersectiong at p. Apparently, C{x, y}/(f2 )

1.6. INTERSECTION OF CURVES INSIDE A SURFACE 27

C{y} • x.C{y} is an isomorphism of C{y}-modules. (Where C{y} =

germs of holomorphic functions in y at 0). We may treat f1 as an

operator acting on C{x, y}/f2 and hence it is true that (C1 , C2 )p is

the dimension of cokernel of this operator. The operator f1 acts by

the multiplication by x3 + y 2 and the matrix of this multiplication is

y2 y8

with det(M ) = y 4 ’ y 12 . So, the order of vanishing of

M= 4 2

yy

det(M ) at zero is 4 and hence the cokernel of ( multiplication by x3 +y 2 )

has dimension 4 too. It was intuitively clear right from the beginning

that it has to be even as we have the symmetry (x, y) ’ (’x, y), which

exchanges two branches of C2 . More generally if p ∈ Ci for i = 1, 2,

and 3, then (C1 , C2 ∪ C3 )p = (C1 , C2 )p + (C1 , C3 )p . This additivity

property of the intersection multiplicity follows from the fact that the

index of a product of Fredholm operators is the sum of the indices (see

for instance Atiyah™s book K-Theory).

Another thing we mention here which concerns the intersection mul-

tiplicity of a variety with itself. Let Y ‚ X be an inclusion of smooth

complex manifold Y into smooth complex manifold X such that Y is

of codimension 1 in X. We have seen that using the dual line bundle

one can show that (LY )|Y NY ’X . Let us say we have a smooth

hypersurface Y of degree d inside CPN . The exact sequence involving

normal bundle demonstrates that

§N ’1 (T — Y ) §N (T — CPN )|Y — NY ’CPN .

L—d and §N (T — CPN )|Y

But on the other hand we have NY ’CPN |Y

—’N ’1

, thus the left hand side of the above is L|Y ’1 . We saw that

—d’N

L|Y

L—k has many holomorphic sections for positive k and none for k < 0

that is why our discussion amounts to the following statements

1) when d > N + 1, the bundle (§N ’1 T — Y )—m has many sections;

2) when d < N + 1, the bundle (§N ’1 T — Y )—m has no non-zero

sections, and

3) when d = N + 1, the bundle (§N ’1 T — Y )—m is trivial (this is the

case of Calabi-Yau manifolds).

28 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

1.7 Theta function and Picard group of

an elliptic curve

An elliptic curve is identi¬ed with a 1-dimensional complex torus which

is the factor group X = C/Λ of the complex line by some lattice Λ.

We get an isomorphic Riemann surface by transforming Λ by complex

multiplication by some complex number, so we may as well assume that

Λ has basis (1, θ) with Im(θ) > 0. We intend to prove

THEOREM 1.7.1 P ic(X) Z • X.

Proof. We already saw the isomorphism P ic(X) Cl(X). We can

decompose Cl(X) = Z • Cl0 (X), where Cl0 (X) is the group of divisors

of degree zero. We recall that the degree of a divisor mi [qi ] is just

the integer mi . The group Z in this decomposition is assumed to

be generated by the divisor [O] - the identity of X if we consider it

as the group and take O to be the image of the origin in C in the

identi¬cation X = C/Λ. Any divisor D has now the decomposition

D = m.[O] + (D ’ m.[O]), where m = deg D. What actually we need

to show is that Cl0 (X) X. Let us consider a map A : Div 0 (X) ’ X

de¬ned by A( mi [qi ]) = mi qi , where in the right hand side we used

the group structure on X. Certainly A is onto, because for each q ∈ X

one has A([q] ’ [O]) = q ’ O = q. We will show in two steps that

P (X) = Ker(A):

Step 1: P (X) ‚ Ker(A). Let D = div(f ) = p∈X vp (f )[p]. We

must show that A(D) = p vp (f )p = 0 in X. Let us consider the

fundamental parallelogram P in C with vertices 0, 1, θ, 1 + θ. For each

pi ∈ X such that vpi (f ) = 0 we choose the representative zi of pi in P .

We may assume that none of zi is on the boundary ‚P , otherwise we

may move P a tiny bit. Let γi be the the i-th side of P numbered in the

counterclockwise fashion so that for any z ∈ γ1 we have Im(z) = 0. For

any meromorphic function f on X we have the corresponding function

f on C which is of course doubly periodic: f (z) = f (z + 1) = f (z + θ).

Now we have

√ df

2π ’1 zi vpi (f ) = z=

f

‚P

i