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where the section σ is a holomorphic section in this neighbourhood not
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

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