6 - On Finite-dimensional Motives and Murre's Conjecture

Summary

Introduction

The conjectures of Bloch, Beilinson, and Murre predict the existence of a certain functorial filtration on the Chow groups (with ℚ-coefficients) of all smooth projective varieties, whose graded quotients only depend on cycles modulo homological equivalence. This filtration would offer a rather good understanding of these Chow groups, and would allow to prove several other conjectures, like Bloch's conjecture on surfaces of geometric genus 0. In Murre's formulation (cf. 6.5.1 below) one can check the validity of the conjecture for particular smooth projective varieties, and in fact, a slightly weaker form of the conjecture has been proved for several cases, e.g., for surfaces [Mu1] and several threefolds [GM] (proving parts (A), (B) and (D) of the conjecture, and giving evidence for (C)). But to my knowledge, there are few results for higher-dimensional varieties, and the strongest form of Murre's conjecture (including part (C)) is only known for curves, rational surfaces, and, trivially, for Brauer-Severi varieties.

The first aim of this paper is to exhibit some cases, where the full Murre conjecture can be shown. The positive aspect is that we get this for some non-trivial cases of varieties of higher (in fact arbitrarily high) dimension, the negative aspect is that we get this just for some special varieties and special ground fields. In particular, not over some universal domain.