Motive, Zhou Qun-related Issues

In this paper, we mainly study Murre's famous conjectures on Chow groups and the finite-dimensionality of Chow groups. Firstly, we give some results for the cases of some special product varieties for Murre's conjectures. More precisely, it is proved that:1) if Murre's conjectures (A), (B), (C) and (D) hold for a smooth projective variety X over a field k, then so do for X×Pr; 2) if k is algbraic closed and C is a smooth projective curve over k, and if Murre's conjectures (A) and (B) hold for X and CHalgj (X) (?) Ker(π2j) for any j, then Murre's conjectures (A) and (B) hold for X×C. In particular, it is proved that if X is an abelian variety of dimension at most 4 and CH0j(X)∩CHalgj(X)=0 for any j, then Murre's conjectures (A) and (B) hold for X×C. Then, it is proved that the Albanese kernel of the product of arbitrary finitely many smooth projective curves is the Chow group of some Kimura finite-dimensional Chow motive and hence is finite-dimensional in some sense, which generalizes a result of Kimura.