Holmstrom Murre's conjecture

Motives volumes, and stuff by Vial.

arXiv:1110.0668 Remarks on Murre’s conjecture on Chow groups from arXiv Front: math.AG by Kejian Xu, Ze Xu For certain product varieties, Murre’s conjecture on Chow groups is investigated. In particular, it is proved that Murre’s conjecture (B) is true for two kinds of four-folds. Precisely, if $C$ is a curve and $X$ is an elliptic modular threefold over $k$ (an algebraically closed field of characteristic 0) or an abelian variety of dimension 3, then Murre’s conjecture (B) is true for the fourfold $X\times C.$

nLab page on Murre's conjecture