Niveau and coniveau filtrations on cohomology groups and Chow groups

TL;DR

This paper investigates how the coniveau filtration on cohomology influences the structure of Chow groups, providing partial proofs of conjectures and new examples under minimal assumptions.

Contribution

It demonstrates the impact of coniveau filtration on Chow groups and proves some conjectures in low-dimensional cases or for varieties with small Chow groups.

Findings

Proved Murre's conjectures for certain threefolds and fourfolds.

Provided a new example of a fourfold of general type with trivial Chow group.

Connected the coniveau filtration with Kimura-O'Sullivan's finite-dimensionality conjecture.

Abstract

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 $X$. In this paper, 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.