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# 6 Chow groups and Abelian varieties

For any smooth projective variety we can form a countable collection of projective schemes Hilbn (the Hilbert or Chow schemes) that parametrise effective cycles of codimension p on X. Let C(X) be a subgroup of (X). A homomorphism of groups from C(X) to the group of rational points of a group variety A is called regular if for any non-singular variety H parametrising cycles on X lying in C(X), the resulting set maps from the set of rational points on H to the set of rational points of A is induced by a morphism of varieties.

In analogy with the picture for divisors one may ask the following questions:

1. Is there a finitely generated group N(X), a surjective map : (X)N(X), and a regular map from C(X) = ker to the group of points on an Abelian variety A, so that any regular map from C(X) to an Abelian variety factors through A?
2. Does this induce an isomorphism of (X) with N(XA(k)?
(Here A(k) denotes the group of k-rational points of A.)
3. Is there a variety (or scheme) H parametrising effective cycles on X so that the morphism HA is surjective? Or at least can we arrange H = (X)?

There is no general geometric construction of the required Abelian variety. There is a complex torus associated to codimension p cycles, defined by Griffiths, which generalizes the Picard and Albanese varieties. This is called the pth intermediate Jacobian of X, and is defined by

Jp(X) = .

Here

Fp(X,C) = (X)

is the pth piece of the Hodge filtration of (X,C). We can take the map : (X)(X,C) and let C(X) = (X) = ker as before. There is a map, which Griffiths calls the Abel-Jacobi map,

(X)Jp(X).

However, this does not have as good properties as the Picard and Albanese maps, in general. The image of (X) is an Abelian subvariety of J2(X) whose Lie algebra is contained in (X) (X, C)/Fp((X,C).

The Griffiths group (X) is always countable, since all effective algebraic cycles of a fixed degree are parametrized by the points of a (possibly reducible) Chow variety of X; taking the union over all degrees, all effective algebraic cycles lie in a countble collection of connected algebraic families, so that (X)/(X) is countable. Hence if (X) 0 for some i > p, then the Abel-Jacobi map cannot be surjective. The restriction of the Abel-Jacobi map to (X) is a regular homomorphism onto the Abelian variety which is its image; conjecturally, this is the universal regular homomorphism, as in the case of the Albanese map. One also expects the Abel-Jacobi map to be injective on torsion, in general. The injectivity of the Abel-Jacobi map on torsion is known for codimension 2 cycles, from work of Merkurjev and Suslin on the K-theory of division algebras, combined with results of Bloch and Ogus; we discuss this below. The universality of the Abel-Jacobi map on (X) has been proved by Murre [27] using the injectivity on torsion.

Next: 7 Relation with cohomology Up: Algebraic Cycles Previous: 5 Equivalence relations
Kapil Hari Paranjape 2002-11-21