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to show that if we change fi by multiplying it by an invertible holo-
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

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