We have introduced three equivalence relations in the previous two sections, which can be defined as follows for cycles in every codimension.

For simplicity, we restrict ourselves to the case
*k* = *C*. Now

(*X*) (*X*) (*X*) (*X*).

One of Grothendieck's
(*X*) = {*x* (*X*) | *nx* (*X*) for
some positive integer *n*}.

Equivalently,
(*X*) *Q* = (*X*) *Q*.

One way of proving this could be to attempt to show that
(
The terminology is because of the famous example of Griffiths [16]
showing that
(*X*) 0 for a general hypersurface *X* of degree 5
in
*P*^{4}_{C}, and in fact
(*X*) has an element of infinite order.
Later, Clemens [10] showed that
(*X*) has infinite rank
in this case. Other examples of the non-triviality of the Griffiths group
were given by Ceresa [9], who showed that if *C* is a generic curve
of genus 3 embedded in its Jacobian variety *X*, and *i*(*C*) is the
image under multiplication by -1 on *X*, then
[*C*] - [*i*(*C*)] gives an element
of infinite order in
(*X*); for *g* = 3, Nori [28] noted
that using the action of Hecke correspondences, this in fact implies that
(*X*) has infinite rank in that case. Further examples of
non-triviality or infinite dimensionality of
(*X*) *Q* were
obtained by Bardelli [3], Voisin [42] and Paranjape
[30].

B. Harris [18] showed that if *C* is the Fermat quartic curve
*U*^{4} + *V*^{4} + *W*^{4} = 0 in
*P*^{2}_{C}, then Ceresa's cycle
[*C*] - [*i*(*C*)] is
non-trivial in
(*X*), where *X* is the Jacobian of *C*, by reducing
this via iterated integrals to the observation that

In all of these examples, the ambient variety has trivial canonical bundle
(tangent bundle, in Ceresa's situation), and one uses image of the cycle
under the Abel-Jacobi homomorphism to the intermediate Jacobian of Griffiths'
(explained in Section 6). For example, in B. Harris' example, the number
whose non-integrality is asserted is essentially an integral of holomorphic
3-form (an element of
(*X*)), whose value is not a period of that
3-form.

There is a new class of examples of non-triviality of
(*X*) *Q*
constructed by M. Nori [29], in which the canonical bundle of the
variety is ample, and the intermediate Jacobian in question is 0. Nori has
introduced a filtration of the Griffiths' group and one can show that every
associated graded term in this filtration can be non-zero (Albano and
Collino [1] have shown that it can even be of infinite rank).
We discuss this further below in the context of conjectural Lefschetz
theorems for Chow groups.

Bloch had asked if the Griffiths group is always divisible (for
varieties over algebraically closed fields); very recently, Bloch and
Esnault have found a counter-example [5]. Schoen
[34] has an example (in positive characteristic) of a smooth
variety *X* such that
(*X*) contains a (non-zero) divisible
subgroup, for some *p*.

Other equivalences have been recently introduced and studied on Chow groups with the idea of settling the standard conjectures and also the Bloch conjecture.