vanishing at p. In terms of trivialization of L, s becomes a function on

a small disk containing p. Then, m is just the order of this function

at p. We denote m = vp (s) and de¬ne deg(L) = p vp (s). It does

not depend on a choice of non-zero meromorphic section. To see it,

16 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

we take another section s , which, of course, can be written as s = f s

for some meromorphic function f on X. Then we have p vp (s ) =

p vp (s) + p vp (f ). It is a well-known fact that the second term in the

rignt hand side vanishes. (One may prove it by considering the surface

with boundary Σ = X \∪i Ui , where Ui are small disks containing points

pi in which vpi (f ) = 0. Then one uses Stokes™ theorem to have

√

df df

0= d( )= = 2π ’1 vpi (f ).)

f f

Σ ‚Σ i

Example. We take X = CP1 , L = T — CP1 L—’2 . As before, we

have a local coordinate z on U0 and CP1 = U0 ∪ {∞}. A meromorphic

section of L may be taken as the meromorphic 1-form σ = dz which

does not vanish on U0 but has a pole of order 2 at ∞, which makes

deg(L) = ’2.

The degree is additive with respect to the tensor product: deg(L1 —

L2 ) = deg L1 + deg L2 .

The next example we consider involves the tangent bundle T X

to a Riemann surface X of the genus g. It is true that deg(T X) =

χ(X) - the Euler characteristic of the surface. By de¬nition, χ(X) =

dim H 0 (X) ’ dim H 1 (X) + dim H 2 (X) = 2 ’ 2g. The integer g such

that dim H 1 (X) = 2g is called the genus of the surface. Then one no-

tices that if g ≥ 2, there exists no non-zero holomorphic vector ¬eld

on X, because if there existed some it would imply deg(T X) ≥ 0 or

2 ’ 2g ≥ 0. In the case g = 0 we have X = CP1 and “hol (T X) has

dimension 3 and is isomorphic to sl(2, C) as the Lie algebra with the

basis d/dz, z.d/dz, and z 2 .d/dz. (Here z again stands for the holomor-

phic coordinate on U0 ‚ CP1 ). In the case g = 1 the line bundle T X is

trivial as X is a complex Lie group.

The degree of a line bundle is a good invariant, but we need a ¬ner

invariant which is given by the notion of divisor. Formally a divisor on

a Riemann surface X is a ¬nite sum i mi [pi ] with integer coe¬cients

mi . For a meromorphic function f on X we de¬ne its divisor div(f ) =

p vp (f )[p] and for any meromorphic section σ of a holomorphic line

bundle L we similarly put div(σ) = p vp (σ)[p]. We say that a divisor

1.5. LINE BUNDLES OVER COMPLEX MANIFOLDS 17

D is a principal divisor if there exists a meromorphic function f such

that D = div(f ).

The set of divisors on X form an abelian group with respect to the

obvious operation of addition. This group is denoted as Div(X). Inside

this group we have a subgroup of principal divisors P (X). The quotient

group is a group of classes of divisors Cl(X) :=Div(X)/P (X). For a

holomorphic line bundle L we have a well-de¬ned element in Cl(X). It

is the image of div(σ) under the projection Div(X) ’Div(X)/P (X),

where σ is a meromorphic section of L. Clearly, we have a homomor-

div

phism P ic(X) ’ Cl(X) sending the isomorphism class of a line bundle

to the class of its divisor. We will see in the next section that this is

actually an isomorphism.

1.5 Holomorphic line bundles over a com-

plex manifold

We ¬x a complex manifold X and consider all holomorphic line bundles

over X which have some non-zero meromorphic section. If X is a

projective manifold, i.e., X ‚ CPN is a closed complex submanifold

of CPN for some N , then it follows from a theorem of Serre that any

line bundle over X satis¬es this condition. In our study we always

identify two holomorphic bundles which are isomorphic. Using the

tensor product it is possible to get a group structure on the set of line

bundles. This group is called the Picard group of X and is denoted

P ic(X).

We will also use another point of view, when we have an algebraic

manifold X (for instance, X ‚ CN or X ‚ CPN de¬ned by algebraic

equations). In this situation we consider algebraic line bundles L over

X. This means that the total space L is an algebraic variety, the

projection map L ’ X is an algebraic mapping, and the C-action

on L is given by an algebraic mapping C — L ’ L. For instance, the

tautological line bundle E over CPN is an algebraic line bundle, because

E ‚ CN +1 —CPN is an algebraic manifold de¬ned by algebraic equations.

Indeed if (w0 , · · · , wN ) are the coordinates on CN +1 and [z0 : · · · : zN ]

are the homogeneous coordinates on CPN , then E is de¬ned by the

18 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

equations

wi zj = wj zi for all i, j ∈ {0, · · · , N }

which express that the vectors (w0 , · · · , wN ) and (z0 , · · · , zN ) are paral-

lel.

Example.(To be proved later.) P ic(CPN ) = Z = {L—k }.

Example. We consider a Riemann surface X of genus 1. It is known

that X = C/Λ is isomorphic to the quotient of C by the lattice Λ

spanned by 1 ∈ C and „ ∈ C. The complex number „ cannot be real

and we may as well assume that Im(„ ) > 0 (otherwise, replace „ by

’„ ). The surface X is a complex Lie group. We will see later that

P ic(X) = Z • X. In general, the Picard group ususally is computed

using various kinds of cohomological apparatus.

Now we return to a general complex manifold X. We need the

notion of an irreducible complex-analytic subvariety Y ‚ X of codi-

mension 1 on X (with possible singularities). Locally Y is de¬ned by

1 holomorphic equation f = 0. The subvariety Y may be not smooth.

For example, the origin is a singular point of suvariety Y ‚ C4 as we

2 2 2 2

de¬ne Y = {(z1 , z2 , z3 , z4 ); z1 + z2 + z3 + z4 = 0}. We say that a subva-

riety Y ‚ X is irreducible if for any two subvarieties Y1 , Y2 ‚ X such

that Y1 ∪ Y2 = Y it follows that Y1 = Y or Y2 = Y . The following result

is basic, simple and well-known.

THEOREM 1.5.1 Given a compact algebraic variety X and Y ‚ X

a complex-analytic subvariety of codimension 1 in X, there exists a

unique decomposition of Y as a ¬nite union of irreducible subvarieties.

Now we de¬ne a divisor on X as a formal linear combination with

integer coe¬cients D = n mi [Yi ] with Yi an irreducible complex-

i=1

analytic subvarieties of codimension 1. Suppose next that we are given

a holomorphic function f on X which is not identically zero. For any

irreducible complex-analytic subvariety Y ‚ X we can de¬ne the order

vY (f ) of f along Y . We denote the zero locus of f by (f = 0). First,

if Y is not completely contained in (f = 0), then we put vY (f ) = 0.

Otherwise we have (f = 0) = Y ∪ Y2 ∪ · · · ∪ Yl . We take a point x ∈ Y

such that Y is smooth at x and x ∈ (f = 0) \ (Y2 ∪ · · · ∪ Yl ). We

pick a neighbourhood U x and a 1-dimensional complex submanifold

C ‚ U such that C is transverse to Y at x. (Meaning that Tx C does

1.5. LINE BUNDLES OVER COMPLEX MANIFOLDS 19

not belong to Tx Y .) Now we restrict f to C and de¬ne vY (f ) = vx (f|C ).

It is a fact that this de¬nition does not depend upon √ choice of x

the

and C. Basically it is due to the fact that vx (f|C ) = (2π ’1)’1 γ df /f

(where γ is a circle in C around x) doesn™t change if we move the point

x along the variety Y \ (Y © (Y2 ∪ · · · ∪ Yl )), which is connected by a

theorem of Whitney. Obviously, we have vY (f g) = vY (f ) + vY (g) and

vY (f + g) ≥ min(vY (f ), vY (g)).

Now we introduce the concept of a meromorphic function f on X.

This means that f is a holomorphic mapping U ’ CP1 on some dense

open subset U of X, and that for every open set V in X there exist two

holomorphic functions g, h on V with h not identically zero and f g = h

over U © V . In other words, f is locally a quotient of two holomorphic

functions.

Using the identity vY (f g) = vY (f )+vY (g) for holomorphic functions

one can extend the de¬nition of vY (f ) to the case of a meromorphic

function f , in such a way that vY (f g) = vY (f ) + vY (g) for all mero-

morphic functions f, g.

Note that we have de¬ned a meromorphic function as a function de¬ned

on some dense open set which locally looks like a ratio of two holomor-

phic functions. But in general, such a meromorphic function does not

de¬ne even a continuous map from X to CP1 = C ∪ {∞}. For example,

taking X = C2 and f = z1 /z2 we see that we have to remove the origin

from C2 to get a nice map to CP1 . In any case there exists an open

dense U ∈ X such that codim(X \ U ) ≥ 2 and a meromorphic function

f de¬nes a nice map f : U ’ CP1 . Another important observation is

that the set of meromorphic functions on X forms a ¬eld denoted by

M (X). For example, M (CP1 ) = {P (z)/Q(z); such that P (z) and Q(z)

are polynomials in z}.

We de¬ne the notion of divisor of a meromorphic function f as

div(f ) = i vYi (f )[Yi ]. We write div(f ) ≥ 0 if every mi ≥ 0. We have

PROPOSITION 1.5.2 div(f ) ≥ 0 if and only if f is a holomorphic

function.

COROLLARY 1.5.3 If X is a compact complex manifold and f is a

meromorphic function on X such that div(f ) ≥ 0 then f = const.

20 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

Let us consider a meromorphic function f on a compact complex

manifold X of dimension d and let div(f ) = i mi [Yi ]. Each Yi de-

¬nes a homology class [Yi ] ∈ H2d’2 (X, Z). It may be viewed as follows.

Any complex manifold comes naturally oriented. Next we have a push-

forward morphism in homology: Hl (Y, Z) ’ Hl (X, Z). The homology

class [Y ] is then just the image under this map of the orientation class

in H2d’2 (Y, Z). It makes sense even if Y has singularities, by results of

Bloom and Herrera [9]. One has i mi [Yi ] = 0 in H2d’2 (X, Z). This

generalizes the statement that the degree of div(f ) is zero on a Riemann

surface and may be spelled out as the statement that the cycle div(f ) is

homologous to zero on X. Let us exhibit this for X = CPd . One knows

that Hj (CPd , Z) = Z if 0 ¤ j ¤ d is even and Hj (CPd , Z) = 0 otherwise.

So, H2d’2 (CPd , Z) = Z and let us denote by H the hyperplane in CPd

de¬ned in homogeneous coordinates by a linear equation i ai zi = 0.

Clearly [H] = 1 ∈ H2d’2 (CPd , Z).

It is easy to see that any meromorphic function on CPd can be

written down as the ratio of two homogeneous polynomials of the

same degree. For an arbitrary meromorphic function f we have then

f = P (z0 , z1 , ..., zd )/Q(z0 , z1 , ..., zd ) for P, Q homogeneous. We repre-

sent P and Q as products of irreducible homogeneous polynomials and

we also may assume that they don™t have common factors (otherwise

we cancel them out): P = P1 1 P2 2 · · · Prar , Q = Qb1 Qb2 · · · Qbs . Let us

a a

1 2 s

say that Pi de¬nes a hypersurface Yi and Qj de¬nes a hypersurface Zj .

With the identi¬cation [H] = 1 we have [Yi ] = deg(Pi ) ∈ H2d’2 (CPd , Z)

and similarly [Zj ] = deg(Qj ) ∈ H2d’2 (CPd , Z). Besides, vYi (f ) = ai due

to the fact that vYi (Pj ) = δij and analogously vZj (f ) = ’bj . Finally, we

summarize: div(f ) = div(P/Q) = i vYi [Yi ] + j vZj [Zj ] = i ai [Yi ] ’

j bj [Zj ], which gives in homology i ai deg(Pi ) ’ j bj deg(Qj ) =

deg(P ) ’ deg(Q) = 0.

Now we return to the correspondence between the Picard group

P ic(X) of a complex manifold X and the group Cl(X) of divisor classes

of X. Just as in the 1- dimensional case, the latter group is de¬ned as

the quotient of the group Div(X) of all formal divisors on X by the sub-

group P (X) of principal divisors formed by all divisors of meromorphic

functions: Cl(X) = Div(X)/P (X).

Given a meromorphic section s of a line bundle L over X we de¬ne

the divisor of s as: div(s) = i vYi (s)[Yi ]. The numbers vYi (s) are

1.5. LINE BUNDLES OVER COMPLEX MANIFOLDS 21

determined as usual using local trivializations since using such a local

trivialization a section becomes a meromorphic function. If we have a

meromorphic section s of L and a meromorphic function f on X, we see

that div(f s) = div(s)+div(f ), so that div(f s) and div(s) have the same

representative in Cl(X). In this way we obtain a group homomorphism

div : P ic(X) ’ Cl(X), where div sends the isomorphism class of L

to the class in Cl(X) representing div(s). As we just saw, the image

doesn™t depend on particular choice of a meromorphic section s as long

as it is not the zero section.

The picture becomes very clear in the case when X = CPd . We

Cd+1 = { linear

take L = L, which has the property that “hol (L)

functionals on Cd+1 }. Any linear form l = i ai zi de¬nes a holomorphic

section of L. When l = 0 it de¬nes a hyperplane H = div(l). So,

the holomorphic line bundle L maps to the class of this hyperplane

H. If another linear form l de¬nes a di¬erent hyperplane H , then

one has [H] ’ [H ] = div(l/l ), which is the divisor of a meromorphic

function. Moreover, Cl(CPd ) = Z and is generated by the class of a

hyperplane H. To see this we take an irreducible complex-analytic

subvariety Y ‚ CPd of degree k (de¬ned by a homogeneous polynomial

P of degree k). Then one ¬nds that [Y ] ’ k[H] is a principal divisor,

because [Y ] ’ k[H] = div(P/lk ), where l is a linear form de¬ning H.

So, we have P ic(CPd ) = Cl(CPd ) = Z. Now we prove the main result of

this section:

THEOREM 1.5.4 The map div is an isomorphism.

Proof. First, we establish injectivity. If L is in the kernel of div, then

for some meromorphic section s of L it is possible to ¬nd a meromorphic

function f on X such that div(s) = div(f ). Then we notice that the

section f ’1 s has the property that div(f ’1 s) = ’div(f ) + div(s) = 0.

Thus we have found a holomorphic nowhere vanishing section f ’1 s of

L. This implies that L is holomorphically trivial.

Now we want to show that div is actually surjective. For this we

provide an explicit construction of a line bundle corresponding to an

irreducible codimension one subvariety. To motivate this, consider the

line bundle E over CPn . As usual, we cover CPn by n + 1 open sets

U0 , ..., Un such that on Uj we have zj = 0. Now we have a non-

vanishing holomorphic section sj of E over Uj given by sj [z0 : · · · :

22 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

zn ] = (z0 /zj , ..., 1, ..., zn /zj ) where the entry is in position j. Of course,

si and sj do not agree over the intersection Ui © Uj and in fact we

have si = (zj /zi )sj . So, we denote by gij = zj /zi the corresponding in-

vertible holomorphic function over Uij = Ui © Uj , called the transition

cocycle. It satis¬es to the standard cocycle condition gij gjk gki = 1 over

Ui © Uj © Uk and we have si = gij sj . So, we introduced the transition

cocycle point of view where a holomorphic transition cocycle corre-

sponds to a holomorphic line bundle. Now we want to go backwards

and assume that X is covered by holomorphic charts {Ui } and over

each intersection Ui © Uj we are given a holomorphic functions gij satis-

fying to the cocycle condition. Now we represent the line bundle as the

the disjoint union of the manifolds (Ui — C) modulo some identi¬cation.

To be precise, we identify pairs of points (x, z) ∈ (Ui © Uj ) — C and

(x, gij (x).z) ∈ (Uj © Ui ) — C.

Consider an elementary divisor [Y ], where Y is an irreducible com-

plex analytic subvariety of X of codimension equal to 1. We can cover

X by a set of holomorphic charts {Ui } so that Y © Ui is de¬ned by

a holomorphic equation fi in Ui . We construct an invertible holomor-

phic function gij = fi /fj over Ui © Uj and gij gives us an allowable

cocycle and hence a line bundle L. If we patch fi together they give a

global holomorphic section s of L such that div(s) = [Y ]. It remains