On a conjectural filtration on the Chow groups of an algebraic variety

Abstract In this paper we describe a conjectural filtration on the Chow groups of a projective, smooth variety. This filtration is suggested by, and based upon, Grothendieck's theory of motives provided one uses the so-called category of Chow motives. This category is constructed by using as intersection ring the full Chow ring tensored with Q. We discuss some evidence for the conjectures. Finally we mention a recent result of U. Jannsen saying that this filtration coincides with conjectral filtration of Bloch and Beilinson.