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Let us have a projective manifold X ’ CPN . Then the holomorphic
line bundles over X are in one-to-one correspondence with the algebraic
cycles of codimension 1 in X. We reacll that an algebraic cycle on X
is a ¬nite formal sum
mi [Yi ],

where mi ∈ Z and Yi ’ X is an irreducible complex-analytic subvari-
ety. We can make an abelian group Z 1 (X) generated by all codimension
1 algebraic cycles and for every meromorphic function f on X we have
div(f ) ∈ Z 1 (X).

Example. Let us take X = CP2 with homogeneous coordinates [z0 :
z1 : z2 ]. Then u = z0 /z2 and v = z1 /z2 are meromorphic functions on
X as well as f = u2 + v 2 . Let Y1 , Y2 ,√and Y3 be the subvarieties of X

de¬ned by the linear equations u = ’1v, u = ’ ’1v and z2 = 0
respectively. Then one has

div(f ) = [Y1 ] + [Y2 ] ’ 2[Y3 ].

A very important and a basic fact about a complex projective man-
ifold X is the GAGA principle (the abbreviation comes from algebraic
geometry and analytical geometry) by J.-P. Serre which we state here.
1). (weak form) Every holomorphic vector bundle over X has a non-
zero meromorphic section.
2). (strong form) Every holomorphic vector bundle over X is in fact
Let us de¬ne the Chow group A1 in codimension 1. It is the quo-
tient Z 1 (X)/{div(f ), f ’ meromorphic}, where Z 1 (X) is the group
of algebraic cycles on X of codimension 1. It is a higher-dimensional
analogue of the group of divisor classes on a Riemann surface. For each
algebraic subvariety Y ∈ X of codimension 1 there is a well-de¬ned
class β(Y ) ∈ H 2 (X, Z) representing Y . Also, for each algebraic line
bundle over X, we can put into correspondence an element of A1 (X)
by picking a meromorphic section s and considering div(s) ∈ Z 1 (X).
In this way we get a commutative diagram
A1 (X)
P ic(X) ’’
c1 β
H 2 (X, Z)
We remark that the map β works also in a singular situation. If X
is a singular subvariety of X then we pick a Hironaka resolution of
singularities Z ’ Z which obviously maps to X and as the class β(Z)
˜ ˜ ˜
we take f— [Z], where [Z] ∈ H2n’2 (Z) is the fundamental class of Z, and
f : Z ’ X.
Let us return to the case of Riemann surface Σ and consider the
group Z0 of zero-cycles of degree 0, i.e. cycles i mi [pi ] such that
i mi = 0. It has as a subgroup the group of principal divisors P (Σ)
consisting of divisors of meromorphic functions. We recall the Abel-
Jacobi map
± : Z0 (Σ) ’ J(Σ).
We stated in Theorem 4.5.1 that the map ± is surjective with kernel
P (Σ). We need
LEMMA 4.5.8 Let X be a complex torus. Then any holomorphic
mapping φ : CP1 ’ X is constant.

Proof. Let X = Cm /Λ, where Λ is a lattice. Pick a coordinate system
(z1 , ..., zm ) and consider the pull-backs φ— dzi of the basis of the space of
the holomorphic one-forms on X. All those pull-backs are zero, since
H 1,0 (CP1 ) = 0. Then the di¬erential map is zero as well and φ is a
constant map.

Now we prove the theorem. Let f be a meromorphic function on Σ
and let φ : CP1 ’ J(Σ) be de¬ned by φ(» : µ) = ±(div(»f +µ)) ∈ J(Σ).
One can check that φ is a holomorphic map. Thus the previous Lemma
implies that φ is a constant map. In particular,

φ(1 : 0) = ±(div(f )) = φ(0 : 1) = ±(div(1)) = 0.

Let us have a non-constant holomorphic (hence surjective) map be-
tween two Riemann surfaces f : S ’ S . The degree of the map
f is de¬ned as the cardinality of the pre-image of a general point of
S . There are two maps between the Jacobians J(S) and J(S ): the
pull-back map F — and the pushforward map f— which we shall de¬ne.
Consider the following maps

f— : H1 (S, Z) ’ H1 (S , Z),

f — : „¦1 (S ) ’ „¦1 (S)
and its transpose
(f — ) : „¦1 (S) ’ „¦1 (S ) .
These maps de¬ne the push-forward map

f— : J(S) ’ J(S ),

because by de¬nition J(S) = „¦1 (S) /H1 (S, Z) and similarly for S .
To de¬ne the pull-back map

f — : J(S ) ’ J(S)

we need to use another description of J(S) which we already have seen,

J(S) = H 0,1 (S)/H 1 (S, Z) = H 1,0 \ H 1 (S, C)/H 1 (S, Z).

Now the desired map f — is de¬ned by the pull-back maps of the coho-
mology groups.
The basic property of those maps is that f— f — = d.Id, where d is the
degree of the map f . It can be viewed as purely topological fact coming
from the observation that for two real tori X = Rp /“ and X = Rp /“
the group of continuous homomorphisms Hom(X, X ) is isomorphic to
the group Hom(“, “ ).
Now we will make a little step forward and consider the situation in
higher dimensions. Let X be a compact K¨hler manifold of dimension
n. Then the codimension d cycles are ¬nite sums i mi [Yi ], where mi ∈
Z and Yi are irreducible complex-analytic subvarieties (maybe singular)
of X. For every such cycle we put into correspondence an element of
H d (X, Z) in an obvious way. Let us consider the group Z d (X) consisting
of homologically trivial cycles, i.e. those which correspond to the zero
class in H d (X, Z).
When d ≥ 1 we can consider the degree d Gri¬ths intermediate

J d (X) = V d (X) \ H 2d’1 (X, C)/H 2d’1 (X, Z),

V d (X) = H 2d’1,0 (X) • · · · • H d,d’1 (X).
Since H 2d’1 (X, C) = V d (X) • V d (X), J d (X) is compact.
An advantage of Gri¬ths™ construction is the fact that V d (X) con-
sists of cohomology classes represented by di¬erential forms of holo-
morphic degree at least d. As a consequence, let us consider a proper
holomorphic submersion f : X ’ Y with projective ¬bers. Then there
exists a proper holomorphic submersion J d ’ Y whose ¬ber over y
the Gri¬ths intermediate Jacobian J d (f ’1 (y)). One can show that in
general J d (X) for d ≥ 2 is not an abelian variety. We should mention
that there is a generalized Abel-Jacobi map

Z d (X) ’ J d (X).

Let us consider a homologically trivial algebraic cycle Z = mi [Yi ]
in X of codimension p and let us pick a representative of the homology
class of Z which is a smooth singular (2n ’ 2p)-cycle γ supported on

the support |Z| = ∪Yi of Z. By assumption, the cycle γ is a boundary:
‚± = γ, where ± is a 2n ’ 2p + 1-chain over Z. Then ± de¬nes a linear
form ± on •s≥n’p+1 H s,2n’2p’s+1 (X) by:

w ∈ •s≥n’p+1 H s,2n’2p’s+1 (X) ’ w.

If we change ± to ± + ± , where ‚± = 0, then the new linear form
applied to w will change by

w = [X], [w] ∪ P D[± ] ,

where P D stands for the Poincar´ dual.
Also one can show the independence upon the representative γ. Let
γ be such that γ ’ γ = ‚β, where β is a singular (2n ’ 2p + 1)-chain on
∪i Yi . We have ‚(± ’ β) = γ , where ‚± = γ and β w = 0 for dimension
reasons, because w|Yi = 0. Thus we can de¬ne a map a by a(γ) = ± ˜
and we have
THEOREM 4.5.9 (i) Let X be a complex projective manifold. There
exists a canonical group homomorphism a : Z p (X) ’ J p (X).
(ii) Let the p-cycle Z be rationally (a.k.a. linearly) equivalent to zero,
(meaning that it a linear combination of divisors of meromorphic func-
tions on algebraic subvarieties Y ‚ X, dim(Y ) = p + 1). Then Z is in
the kernel of a.
It is an open problem to give a satisfactory description of the kernel
and the image of a.
To see (ii), we de¬ne the map ¦ : CP1 ’ J p (X) by ¦(» : µ) =
a(div(»f + µ)). The function ¦ is constant as a consequence of Lemma
4.5.8. As before, we notice that a(div(f )) = ¦(1 : 0) = ¦(0 : 1) = 0.

4.6 Operations on algebraic cycles
We need to introduce various operations on algebraic cycles. First of
all, we denote by Z p (X) the group of algebraic cycles of codimension

p on a complex projective manifold X, and by Z0 (X) the subgroup
of rationally trivial cycles. Dually, we use the notation Zn’p (X) =
Z p (X) if dimC (X) = n. Inside Z p (X) we consider the group Ratp (X)
of rationally trivial cycles of codimension p generated by divisors of
meromorphic functions on codimension (p ’ 1) irreducible subvarieties
in X. We de¬ne the Chow group Ap (X) = Z p (X)/ ∼, where ∼ is
the rational equivalence. We simply factor out all cycles which are
rationally equivalent to zero. We let An’p (X) = Ap (X). Now when
p = 1 then A1 (X) = P ic(X) is the group of isomorphism classes of line
bundles on X. Its connected component

P ic0 (X) = H 1,0 (X) \ H 1 (X, C)/H 1 (X, Z)

is an abelian variety.
The ¬rst type of operation we need is push-forward. Let f : X ’ Y
be a map between two projective manifolds X of dimension n and Y of
dimension k and let Z ’ X be an irreducible subvariety of codimension
p in X. We will always de¬ne all operations on irreducible cycles as
each cycle is a linear combination of such.
THEOREM 4.6.1 (Grothendieck-Grauert) The image f (Z) is an
algebraic subvariety of Y .
This allows us to obtain the map Zn’p (X) ’ Zn’p (Y ) given by the
f— [Z] = d.[f (Z)] if dim f (Z) = n ’ p,
f— [Z] = 0, otherwise,
where d is the degree of the ¬nite map f|Z . Passing to Chow groups we
get an induced map An’p (X) ’ An’p (Y ).
The second type of operation we consider is pull-back. The eas-
iest case when it works is when we have a submersion f : X ’ Y ,
which means that all ¬bers of f are smooth. Let Z ’ Y be an irre-
ducible cycle on Y , then we let f — [Z] = [f ’1 (Z)]. The variety f ’1 [Z]
is irreducible if f has connected ¬bers.
Before we continue to work with pull-backs we discuss the notion of
a ¬‚at morphism. Let f : X ’ Y be a morphism of projective varieties
as before and let x ∈ X, y ∈ Y be such that f (x) = y, and let OX,x and

OY,y be corresponding local rings. One can naturally consider Ox,X as
a module over OY,y . Now we pick any three modules M1 , M2 , M3 over
OY,y and let 0 ’ M1 ’ M2 ’ M3 ’ 0 be a short exact sequence of
OY,y -modules. We call f ¬‚at if for any such M1 , M2 , M3 and any x ∈ X
the sequence

0 ’ OX,x —OY,y M1 ’ OX,x —OY,y M2 ’ OX,x —OY,y M3 ’ 0

is exact as well. This is not a very visualizable notion, so we give a
short list of examples of ¬‚at morphisms.
- an open immersion is ¬‚at
- if f : X ’ Y is a submersive algebraic mapping, then f is ¬‚at
- if Y is a Riemann surface (identi¬ed, as usual with an algebraic curve)
and the ¬bers of f are irreducible then f is ¬‚at (e.g. when X is a
Riemann surface too)
Now let f : X ’ Y be a ¬‚at mapping with ¬nite ¬bers and Z ’ Y
is an irreducible cycle then we have f ’1 (Z) = ∪i Si , where Si is an
irreducible subvariety in X. We de¬ne then f — [Z] = mi [Si ], where
mi is the corresponding “rami¬cation index”. This index is de¬ned as
follows: let My ‚ OY,y be the maximal ideal. Let OX,Si be the local
ring of X at the generic point of Si . Let I be the ideal of OX,Si generated
by My . Then mi is the length of the artinian module OX,Si /I. In case
dim(X) = 1, and Z = q ∈ Y , for z a local parameter at q, mi is the
order of vanishing of z at the point Si of X.
Let us treat in detail the case dim(X) = dim(Y ) = 1. We will work
only with homologically trivial cycles we have the following commuta-
tive diagrams
A1 (X)
f— “ “ f—
A1 (Y )
J(Y )
A1 (Y )
J(Y )
f— “ “ f—
A1 (X)
The latter diagram explains, in particular, why the divisor of a mero-
morphic function on X maps to zero in J(X) by the Abel-Jacobi map.
The point is that any meromorphic function on X is the same as an


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