For any smooth projective variety we can form a countable collection
of projective schemes *Hilb*_{n} (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:

- 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*? - Does this induce an isomorphism of (
*X*) with*N*(*X*)×*A*(*k*)?

(Here*A*(*k*) denotes the group of*k*-rational points of*A*.) - Is there a variety (or scheme)
*H*parametrising effective cycles on*X*so that the morphism*H**A*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 *p*th *intermediate Jacobian* of *X*, and is defined by

(*X*)*J*^{p}(*X*).

However, this does not have as good properties as the Picard and Albanese
maps, in general. The image of
(
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.