“ — Q = [Λ — Q] • W,

“ — R = [Λ — R] • [W — R].

Moreover, W — R is a complex subspace of “ — R. Now let us de¬ne a

complex torus B = W — R/(W © “) and it is clearly a sub-torus of A.

Besides, we clearly have a map

ψ :A —B ’A

given by addition, which is a Lie group homomorphism and which is

holomorphic. The kernel of the map ψ is the ¬nite group “/[Λ + (“ ©

W )], giving us the desired isogeny.

Let X, Y be two compact Riemann surfaces and let f : X ’ Y be

a non-constant holomorphic mapping. (Thus, the genus of Y is less or

equal to the genus of X, the ¬bers of f are ¬nite and f is onto.) Then

up to isogeny

J(X) ∼ J(Y ) — Af ,

where the abelian variety Af is not completely understood even now-

days.

4.5 Algebraic cycles

Algebraic cycles appear in algebraic geometry for purposes similar to

singular homology in algebraic topology. Let X be an arbitrary alge-

braic variety, then a ¬nite sum with integer coe¬cients i ni [Zi ] is an

162 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

algebraic cycle provided that Zi is an irreducible (complex) algebraic

subvariety. For example, zero-cycles on X are just ¬nite combinations

of points with integer coe¬cients. The degree of the zero-cycle i ni [Zi ]

is equal to i ni

Let us consider in detail the situation in dimension 1, when X is a

compact Riemann surface. Here, the group of zero-cycles C0 (X) has as

a subgroup

Z0 (X) = { ni [pi ], ni = 0},

i i

the group of zero-cycles of degree 0. There is the Abel-Jacobi map

± : Z0 (X) ’ J(X),

which is a group homomorphism. For a zero cycle [q]’[p] we pick a path

γ : [0, 1] ’ X, γ(0) = p, γ(1) = q, a smooth 1-chain. By de¬nition,

J(X) = „¦1 (X) /H1 (X, Z), and thus to de¬ne ± we shall ¬nd a linear

functional fγ on „¦1 (X) corresponding to γ. An obvious candidate is

fγ (ω) = ω.

γ

To see that made in such a way ± is well-de¬ned, let γ = γ + δ be

another 1-chain with the same boundary. Clearly, ‚δ = 0, meaning

that δ is a 1-cycle, thus it corresponds to an element of H1 (X, Z), which

we factored out. Since any element of Z0 (X) is a linear combination of

di¬erences [q] ’ [p], this de¬nes ±.

The map ± has the property that it is onto, and, its kernel has

a nice description which we give in a moment. Recall, that given a

meromorphic function f on X, there is a well-de¬ned notion of its

divisor

div(f ) = vp (f )[p],

p∈X

where vp (f ) is the order of f at the point p. The divisor of a meromor-

phic function is a zero-cycle of degree 0. Moreover, we have

THEOREM 4.5.1 A zero-cycle c is in the kernel of ± if and only if

c = div(f )

for some meromorphic function f .

4.5. ALGEBRAIC CYCLES 163

Let Σ be a compact Riemann surface of genus g. The Poincar´ e

duality allows us to identify the Jacobian variety J(Σ) with the quotient

H 0,1 (Σ)/H 1 (Σ, Z),

where the group H 0,1 (Σ) is the Dolbeault cohomology group, which,

by de¬nition, is A0,1 (Σ)/d C ∞ (M ). Given ± ∈ A0,1 we will construct a

ˇ

Cech 1-cocycle fij with values in holomorphic functions. Let us choose

a ¬nite covering Σ = ∪i Ui such that Ui ∆ - the standard unit dics in

C. The fij are holomorphic function over Ui © Uj , satisfying the cocycle

condition fik = fij + fjk over Ui © Uj © Uk . The main fact is that over

Ui one can solve the equation ± = d hi , hi ∈ C ∞ (Ui ) and then we put

fij = hj ’ hi over Ui © Uj . We have d fij = d hj ’ d hi = 0, hence fij

is holomorphic. Also

fik = hk ’ hi = hk ’ hj + hj ’ hi = fjk + fij ,

hence fij is a cocycle.

Now of course we could change hi to hi +gi , assuming that d gi = 0,

which means that gi is holomorphic. Then fij is changed into fij +gj ’gi .

ˇ

The cocycle gj ’ gi is called a coboundary. The ¬rst Cech cohomology

group with coe¬cients in holomorphic functions is then

ˇ

H 1 (Σ, O) = {1 ’ cocycles fij }/coboundaries.

ˇ

PROPOSITION 4.5.2 H 0,1 (Σ) H 1 (Σ, O).

ˇ

Given such a Cech cocycle (fij ) we can construct a holomorphic line

√

bundle. We have gij = exp(2π ’1fij ) a (non-vanishing) holomorphic

cocycle, since gik = gij gjk is satis¬ed. Now we use (gij ) as transition

functions for a holomorphic line bundle L over Σ:

L= (Ui — C)/identi¬cations.

Here we identify (x, ») ∈ Ui — C with (x, »gij (x)) ∈ (Uj — C). We notice

that if fij ∈ Z then gij = 1 and therefore the bundle L is trivial. In this

fashion we get a map

J(Σ) = H 0,1 (Σ)/H 1 (Σ, Z) ’ P ic(Σ),

where P ic(Σ) is the group of isomorphism classes of holomorphic line

bundles.

164 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

P ic0 (Σ) - the connected component of

THEOREM 4.5.3 J(Σ)

P ic(Σ).

To each line bundle over Σ, as we know, one can assign the integer

- its degree, or the ¬rst Chern class. The group P ic0 (Σ) is exactly

the group of isomorphism classes of bundles of degree zero. If one

picks a meromorphic section s of a holomorphic bundle L, which is not

identically zero, one gets deg(L) = deg(s) = p∈Σ vp (s), where vp (s) is

the order of s at the point p. This integer is independent upon a choice

of a section s.

Before we make an attempt to classify holomorphic line bundles over

a general manifold, let us understand smooth complex line bundles ¬rst.

Let P ic∞ (X) be the group of isomorphism classes of line bundles over

a manifold X.

PROPOSITION 4.5.4 P ic∞ (X) = H 2 (X, Z).

Proof. Let us take an nice open covering (Ui ) (meaning that all the

intersections Ui1 © · · · © Uij are empty or contractible) of X and let si

be a non-vanishing section of a smooth line bundle L over Ui . Then we

form a transition cocycle gij = si /sj : Ui © Uj ’ C— .

Now let fij be a holomorphic function ovewr Uij such that gij =

exp(fij ). Then it easily follows from the cocycle condition that fij +

√

fjk ’ fik ∈ 2π ’1Z.

ˇ

On the other hand side, H 2 (X, Z) is described in terms of Cech

ˇ

cohomology as follows. A Cech 2-cocycle νijk ∈ Z is de¬ned when

Ui © Uj © Uk is not empty and satis¬es νjkl ’ νikl + νijl ’ νijk = 0. To get

the cohomology group H 2 (X, Z) we must factor out the coboundaries

νijk = ajk ’ aik + aij , aij ∈ Z. Let us put

fij + fjk ’ fik

√

νijk = .

2π ’1

√

Of course, we can change fij to fij + 2π ’1aij , aij ∈ Z. Doing this, we

change νijk by aij + ajk ’ aik , i.e. it is changed by a coboundary. One

can use a homotopy argument to see that if the cohomology class of

the cocycle νijk is trivial, then the line bundle arising from the cocycle

(gij ) is trivial as well. Indeed, let t ∈ [0, 1] be our parameter, then the

4.5. ALGEBRAIC CYCLES 165

homotopy etfij connects the trivial line bundle with the line bundle L,

provided that fik = fij +fjk . At least over compact manifolds, we know

that two homotopic vector bundles are isomorphic. Another way to look

at it is to ¬nd over each Ui a non-vanishing section si of L and a smooth

function hi such that over Ui ©Uj one has σi := ehi si = ehj sj =: σj . This

amounts to solving for hj ’ hi = fij , which can easily be done using a

partition of unity. Then σi gives a global non-vanishing section, which

trivializes L.

Therefore, we have proved that P ic∞ (X) injects into H 2 (X, Z). To

prove the surjectivity, we use homotopy theory which says that

H 2 (X, Z) = [X, CPN ],

the group of homotopy classes of maps from X to CPN for N big enough.

Then given a class ± ∈ H 2 (X, Z) and f : X ’ CPN a map which

classi¬es it, we can pull back the tautological line bundle E to get a

line bundle over X mapping to ± ∈ H 2 (X, Z).

Let us take a connection on a line bundle L and let s be a non-

vanishing section over some open set U . Then (s)/s is a 1-form on U .

The curvature K is a closed complex-valued 2-form over X such that

K = d( (s)/s) over U . This is independent of the choice of s because

(s)

if we change s to f s for s : U ’ C— , then (s)/s is changed to s + df ,

f

df

and the 1-form f is closed.

THEOREM 4.5.5 (Chern, Weil, Kostant) The curvature form K

satis¬es the integrality condition

√

[K]/2π ’1 ∈ Im(H 2 (X, Z) ’ H 2 (X, C))

ˇ

This class coincides with the Cech cohomology class of (νijk ). Con-

versely, if K is a closed integral 2-form on a manifold X, then there

exists a line bundle L over X and a connection on L such that its

curvature is equal to K.

Example. Let ω be the K¨hler form on CPn which we have introduced

a

earlier. The tautological line bundle E has a natural connection with

√

the curvature equal to 2π ’1ω.

166 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

Now, let X be a complex manifold and let L be a holomorphic line

bundle over X. The Picard group P ic(X) is the group of isomorphism

ˇ

classes of holomorphic line bundles over X. Let us use Cech cohomology

to get a suitable description of P ic(X). As before, take an open cov-

ering (Ui ) of X where each Ui is biholomorphic to a disc. Then we can

associate to the line bundle a transition cocycle gij : Ui © Uj ’ C— . The

ˇ

Cech cohomology group H 1 (X, OX ) is therefore isomorphic to the the

—

Picard group P ic(X), because it classi¬es 1-cocycles and two cohomol-

ogous cocycles lead to isomorphic line bundles. Consider the so-called

exponential exact sequence

√ exp

1 ’ 2π ’1Z ’ C ’ C— ’ 0,

which gives rise to the following long exact sequence

c

H 1 (X, Z) ’ H 1 (X, OX ) ’ H 1 (X, OX ) ’ H 2 (X, Z)’H 2 (X, OX ),

— 1

where c1 maps a line bundle to its ¬rst Chern class.

If X is a compact K¨hler manifold, then H 1 (X, Z) ’ H 1 (X, OX )

a

is embedded as a cocompact lattice. In this case we get

0 ’ H 1 (X, OX )/H 1 (X, Z) ’ P ic(X) ’ Hdg 1 (X) ’ 0,

where Hdg p (X) consists by de¬nition of such γ ∈ H 2p (X, Z) that

γ — C ∈ H p,p (X). In our case Hdg 1 (X) is also Ker[H 2 (X, Z) ’

H 2 (X, OX )]. A class of ± in H 2 (X, Z) is in this kernel if and only

if ± is of type (1, 1). Introduce P ic0 (X) = H 1 (X, OX )/H 1 (X, Z): this

is a complex torus, called the Picard variety.

THEOREM 4.5.6 For any compact K¨hler manifold X the group

a

P ic0 (X) admits a polarization, hence it is an abelian variety.

Proof. Recall that the tangent space at the identity of this group is

H 1 (X, R) which admits a symplectic pairing

±§β§ω n’1 ,

B([±], [β]) =

X

4.5. ALGEBRAIC CYCLES 167

where ω is the K¨hler form on X, n = dimC (X), and [±], [β] ∈ H 1 (X, R).

a

The symplectic form B takes integral values on H 1 (X, Z). On the other

hand, it is the imaginary part of the hermitian form H where

±§β§ω n’1 ,

H(([±], [β]) = ¯

X

and we know that H is positive de¬nite.

Next we intend to establish the Hodge conjecture for degree 2 co-

homology classes. First we prove

THEOREM 4.5.7 Let X be a compact K¨hler manifold. Given any

a

γ ∈ Hdg 1 (X), there exists a holomorphic line bundle L over X such

that c1 (L) = γ.

Proof. As we just have seen there is an exact sequence

c1

ˇ

H 1 (X, OX ) ’ Hdg 1 (X) ’ 0,

—

ˇ

and the result follows from the fact that P ic(X) = H 1 (X, OX ).