Niveau and coniveau filtrations on cohomology groups and Chow groups

The Bloch–Beilinson–Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology – i.e. the cohomology – of smooth projective varieties. In this paper, given a smooth projective complex variety X, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of X has an incidence on the Chow groups of X. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low-dimensional cases or when a variety is known to have small Chow groups. For instance, we give a new example of a fourfold of general type with trivial Chow group of zero-cycles and we prove Murre’s conjectures for threefolds dominated by a product of curves, for threefolds rationally dominated by the product of three curves, for rationally connected fourfolds and for complete intersections of low degree. The BBM conjectures are closely related to Kimura–O’Sullivan’s notion of finite-dimensionality. Assuming the standard conjectures on algebraic cycles the former is known to imply the latter. We show that the missing ingredient for finite-dimensionality to imply the BBM conjectures is the coincidence of a certain niveau filtration with the coniveau filtration on Chow groups