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algebraic map to CP1 . So let us take Y = CP1 and remember that
J(CP1 ) = 0. We have
Z 1 (CP1 )
f— “ “ f—
Z 1 (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“ ,
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 ,
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 = ∞.)
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

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
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 .

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

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
Abel-Jacobi map ± : Cl (Σ) ’ J(Σ) which allows us to form the
P ic0 (Σ) Cl0 (Σ)
β ±
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
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

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

(z ’ ·1 ) · · · (z ’ ·g )
(’1)j σj (·)z ’j .
g· (z) = =
z j=0

Now we can compute the di¬erential of the map φ and see that (dφ)0 ( ‚σj )
is equal to the O-valued Cech cocycle (’1)j z ’j for the covering (∆, Σ \
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,
‚± ·k
= [ ω].
‚σj ‚σj k=1 0

a1 a2
ω = a0 P1 (·) + P2 (·) + P3 (·) + · · · ,
2 3
k=0 0

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
as desired.

4.8 K3 surface
The type of surfaces is named after Kummer, K¨hler, and Klein. Let
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-
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
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
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:
(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 .


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