algebraic map to CP1 . So let us take Y = CP1 and remember that

J(CP1 ) = 0. We have

Z 1 (CP1 )

0

f— “ “ f—

Z 1 (X)

J(X)

Now if f is a meromorphic function on X then div(f ) = f — ([0] ’ [∞]).

The third (and for now the last) operation on algebraic cycles will

be so-called specialization of a cycle. Let us have

Z ’X

f“ ,

C

where Z is an irreducible algebraic subvariety of codimension p in a

projective manifold X, and C is an algebraic curve. We assume that f

is ¬‚at, so all ¬bers of f have the same dimension. Let us pick a point

q ∈ C, which gives us a codimension 1 algebraic cycle f — (q) ∈ Z. We

can view f — (q) as a codimension (p + 1) algebraic cycle on X. Another

description of the same situation is given by

W ’ X —C

f p2 ,

C

where W is an algebraic subvariety of X — C, f is ¬‚at and p2 is the

projection onto the second factor. Here for any point q ∈ C we also get

a cycle Wq on X by intersecting W with X — {q}.

DEFINITION 4.6.2 The group Alg p (X) of cycles algebraically equiv-

alent to zero is the free abelian group generated by

[Wq ] ’ [Wq ], q, q ∈ C.

It is clear that we have a sequence of inclusions

Ratp (X) ‚ Alg p (X) ‚ Z p (X).

(To see the ¬rst inclusion one simply takes C = CP1 and q = 0, q = ∞.)

p

In general it is a non-trivial question how much bigger is Zo (X)

than Alg p (X). On the one hand side we have

4.7. ABEL-JACOBI THEOREM 175

THEOREM 4.6.3 (Clemens) There exists a projective variety X

for which the group Z 2 (X)/Alg 2 (X) is not ¬nitely generated.

The example is provided by a quintic hypersurface in CP4 with generic

moduli. Gri¬ths proved in the 60™s that there are algebraic cycles on

such a quntic threefold which are homologically equivalent to 0 but

not algebraically equivalent to 0. On the other hand, the Neron-Severi

group N S(X) = P ic(X)/P ic0 (X) is always ¬nitely generated (this fact

is still true over an arbitrary ¬eld). Let us look at more examples. Let

X be a compact Riemann surface of genus at least 3 with a ¬xed point

x0 ∈ X. There is a well-de¬ned map ± : X ’ J(X) which takes r ∈ X

to the image of the Abel-Jacobi map of the zero-cycle [r] ’ [x0 ]. Let

ι : J(X) ’ J(X) be the involution x ’ ’x, and let us consider the

˜

1-cycle X = ±(X) ’ ι±(X) ‚ Z 0 (J(X)). This cycle is homologically

trivial since ι acts as the identity map on the even-degree cohomology

groups H 2l (J(X), Z). However, we have

THEOREM 4.6.4 (Ceresa) If X has “generic moduli” then no mul-

˜

tiple of X is algebraically equivalent to zero.

”Generic moduli” mentioned in this theorem means that the class of

J(X) in the moduli space A3 of principally polarized abelian varieties

(we recall that A3 = H3 /Sp(6, Z)) does not belong to a proper algebraic

¯

subvariety of A3 de¬ned over the algebraic closure Q. There is another

result

THEOREM 4.6.5 (B. Harris) We have the same conclusion as in

4 4

the above theorem for the Fermat quartic curve (of genus 3) z0 + z1 +

z2 = 0 in CP2 .

4

Clearly the Fermat quartic curve does not have generic moduli. Fi-

nally, we mention that Fulton developed a theory of Chow groups for

quasi-projective varieties (i.e. complements of closed subvarieties of

projective varieties).

4.7 Abel-Jacobi theorem

Let Σ be a compact Riemann surface and let P ic0 (Σ) be the Picard

group of holomorphic line bundles of degree 0 on Σ. We recall the

176 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

div

isomorphism P ic0 (Σ) Cl0 (Σ) between the group P ic0 (Σ) and the

group Cl0 (Σ) of divisor classes on Σ of degree 0. The group Cl0 (Σ)

is de¬ned as the factor group (degree 0 algebraic cycles)/(principal

divisors). Besides, we know that

P ic0 (Σ) = Ker[deg = c1 : H 1 (Σ, OΣ ) ’ Z]

—

H 1 (Σ, OΣ )/H 1 (Σ, Z) = „¦1 (Σ) /H1 (Σ, Z) =: J(Σ).

We denote this isomorphism β : P ic0 (Σ) J(Σ). We also recall the

0

Abel-Jacobi map ± : Cl (Σ) ’ J(Σ) which allows us to form the

diagram:

div

P ic0 (Σ) Cl0 (Σ)

β ±

J(Σ)

All three groups involved have natural topologies and all maps are

continuous. In fact, the original statement of the Abel-Jacobi theorem

is that the map ± is injective. This will follow from a stronger statement

which we present:

THEOREM 4.7.1 This diagram is commutative.

Proof. Let us denote by φ : Cl0 (Σ) ’ P ic0 (Σ) the map inverse to

div. We shall establish that the di¬erentials of the maps commute

and since we deal with tori the result will follow. Let us pick a point

x ∈ Σ and a point y ∈ Σ nearby and form an element [y] ’ [x] ∈

Cl0 (Σ). We would like to have an explicit description of the line bundle

ˇ

L[y]’[x] := φ([y] ’ [x]) in terms of a Cech cocycle gij = exp(fij ) de¬ning

an element of H 1 (Σ, OΣ ). Let us consider a small disk ∆ around x with

—

the coordinate z so that x corresponds to z = 0. Let also the point y,

which is close to x, be given by the value z = ·, · = 0 and belong to

the disk ∆. We cover Σ by two open sets U1 and U2 : Σ = U1 ∪ U2 ,

where U1 = ∆, and U2 = Σ \ {x, y}, so that U12 := U1 © U2 = ∆ \ {0, ·}.

We have the cocycle

z’· ·

g· : U12 ’ C— , g· (z) = =1’

z z

4.7. ABEL-JACOBI THEOREM 177

which has zero at y and pole at x both of ¬rst orders. The g· is a

1-cocycle for the line bundle L[y]’[x] . Now we take the derivative with

ˇ

respect to · and evaluate it at · = 0 and we ¬nd the O-valued Cech

cocycle ’1/z with respect to the covering (∆, Σ \ {x}).

Let S g (Σ) be a smooth manifold de¬ned as the gth symmetric power

of Σ. Let us consider the surjective map S g (Σ) ’ Cl0 (Σ) given by

·1 + · · · + ·g ’ [·1 ] + · · · [·g ] ’ g[x], ·i ∈ Σ.

This map is ´tale at x + · · · + x = g[x] ∈ S g (Σ) if we pick x to be a

e

g

general point of Σ (for example it is enough that x is not a Weierstraß

point of Σ). Let us identify a neighbourhood of the point g[x] of S g (Σ)

with a neighbourhood of the origin in Cg by

·1 + · · · + ·g ’ (σ1 (·), ..., σg (·)),

where · = (·1 , ..., ·g ) and σi is the i-th symmetric polynomial in (·1 , ..., ·g ).

ˇ

Now the Cech cocycle corresponding to g [·i ] ’ g[x] is given by

i=1

g

(z ’ ·1 ) · · · (z ’ ·g )

(’1)j σj (·)z ’j .

g· (z) = =

g

z j=0

‚

Now we can compute the di¬erential of the map φ and see that (dφ)0 ( ‚σj )

ˇ

is equal to the O-valued Cech cocycle (’1)j z ’j for the covering (∆, Σ \

{x}).

Next we pair the above cocycle with a holomorphic 1-form ω ∈

„¦1 (Σ) written locally as ω = dz(a0 + a1 z + · · ·):

[z ’j ], ω = Resz=0 (z ’j ω) = aj’1 .

On the other hand,

‚j

‚± ·k

= [ ω].

‚σj ‚σj k=1 0

Clearly,

g

a1 a2

·k

ω = a0 P1 (·) + P2 (·) + P3 (·) + · · · ,

2 3

k=0 0

178 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

i i

where Pi (·) = ·1 + · · · + ·g . As well as we did with the Chern classes

we may express Pi in terms of σ1 , ..., σi and then neglect quadratic and

higher order terms for the purposes of computing the di¬erential. It is

an easy exercise to show that

Pm (·) = (’1)m mσm (·) + higher order terms.

Therefore we conclude that

‚± al’1

(’1)l l = (’1)l al’1

|(0,...,0) =

‚σl l

and this is equal to

‚φ

|(0,...,0)

‚σl

as desired.

4.8 K3 surface

The type of surfaces is named after Kummer, K¨hler, and Klein. Let

a

X be a K3 surface, for example a Kummer surface obtained by blowing

up 16 points in A/ι, where A = C2 /Λ is a torus and ι is the involution

x ’ ’x. Another example of a K3 surface is given by a non-singular

quartic in CP3 , i.e. de¬ned by h(z0 , z1 , z2 , z3 ) = 0, where [z0 : z1 : z2 : z3 ]

are homogeneous coordinates in CP3 and h is a homogeneous polynomial

of degree 4.

We know that H 1 (X, R) = H 3 (X, R) = 0 and dim H 2 (X, R) = 22.

Since X is a compact K¨hler manifold, we use the Hodge decompo-

a

sition to exhibit the structure of cohomology of X as H 2 (X, C) =

H 2,0 (X) • H 1,1 (X) • H 0,2 (X), which are of dimensions 1, 20, and 1

respectively. There exists (unique up to a scalar multiple) holomor-

phic nowhere vanishing 2-form β on X, which can be written locally as

f (z1 , z2 )dz1 §dz2 and f (z1 , z2 ) = 0. If g(z1 , z2 )dz1 §dz2 is another such

form then f /g is globally de¬ned holomorphic function and therefore is

a constant. Thus the space H 2,0 (X) is a one-dimensional complex space

spanned by the form β, and H 0,2 (X) is its complex conjugate. We saw

that the signature of the intersection pairing on H 2 (X, R) (where it is

4.8. K3 SURFACE 179

symmetric bilinear) or on H 2 (X, C) (where it is hermitian) is equal to

’16.

We also recall that the three-dimensional subspace V of H 2 (X, C)

¯

spanned by β, β, and the K¨hler form ω is such that the intersection

a

pairing is positive de¬nite on V . The space V is clearly a real subspace.

Now we make the extra assumption that X is a projective manifold,

i.e. comes with a smooth embedding X ’ CPN . In this case we let

[ω] = i— ξ, where ξ is a cohomology class which generates H 2 (CPN , Z).

We introduce a necessary notation. Let Λ = [ω]⊥ ‚ H 2 (X, Z)/torsion

be the orthogonal complement of [ω] which is a lattice in E = [ω]⊥ =

Λ — R ‚ H 2 (X, R). The whole Hodge decomposition for E is then

determined by the complex line l ∈ H 2,0 ‚ E — C, because E — C =

H 2,0 (X) • E 1,1 (X) • H 0,2 (X), where the summands are of dimension

1, 19 and 1 respectively.

Let us denote by (, ) the intersection pairing on Λ, E and E —C. We

notice that (, ) is complex bilinear on E — C. There are some conditions

to which the line l mentioned above must obey. If v is any non-zero

vector v ∈ l then

(i) (v, v) = 0

(ii) (v, v ) > 0.

¯

Let M stand for the submanifold of the manifold P(E) comprised

of all the complex lines l in E — C, which satisfy (i) and (ii). Let us

√

write down the conditions on v = √ + ’1b, v ∈ l ∈ M:

a

(i) 0 = (v, v) = [(a, a) ’ (b, b)] + 2 ’1(a, b), i.e. ||a|| = ||b||, a ⊥ b.

(ii) 0 ¤ (v, v ) = ||a||2 + ||b||2 = 2||a||2 , i.e. ||a||2 > 0.

¯

Thus if we consider the real plane V ‚ E spanned by a and b then

the following is true.

LEMMA 4.8.1 The manifold M identi¬es with the set of oriented

real 2-planes V ‚ E such that the restriction (, )|V is positive de¬nite.

Let SO(2, 19) denote the connected component of the orthogonal group

preserving the quadratic form of signature ’17 in a 21-dimensional real

vector space R2,19 .

THEOREM 4.8.2 (Witt) The group SO(2, 19) acts transitively on

the set of such planes V .