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
The terminology is because of the famous example of Griffiths  showing that (X) 0 for a general hypersurface X of degree 5 in P4C, and in fact (X) has an element of infinite order. Later, Clemens  showed that (X) has infinite rank in this case. Other examples of the non-triviality of the Griffiths group were given by Ceresa , 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  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 , Voisin  and Paranjape .
B. Harris  showed that if C is the Fermat quartic curve U4 + V4 + W4 = 0 in P2C, 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 , 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  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 . Schoen  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.