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

l

mi [Yi ],

i=1

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 ].

168 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

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

algebraic.

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

div

A1 (X)

P ic(X) ’’

c1 β

ˇ

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.

4.5. ALGEBRAIC CYCLES 169

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,

namely

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

170 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

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

a

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

Jacobian

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

where

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

4.6. OPERATIONS ON ALGEBRAIC CYCLES 171

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.

e

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

172 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

p

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

rule

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

f—

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

4.6. OPERATIONS ON ALGEBRAIC CYCLES 173

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)

J(X)

f— “ “ f—

A1 (Y )

J(Y )

A1 (Y )

J(Y )

f— “ “ f—

A1 (X)

J(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