Chow--Kuenneth decomposition for special varieties

TL;DR

This paper explores Murre's conjecture by establishing Chow--K"unneth decompositions for universal families of smooth curves over certain moduli spaces and for representation varieties of finitely generated groups, using equivariant methods.

Contribution

It provides new cases where Murre's conjecture holds, specifically for certain moduli spaces of curves and representation varieties, employing equivariant Chow groups.

Findings

Existence of Chow--K"unneth decomposition for universal families over moduli spaces of genus up to 8.

Construction of equivariant Chow--K"unneth decompositions for representation varieties.

Advancement in understanding Murre's conjecture for specific algebraic varieties.

Abstract

In this paper we investigate Murre's conjecture on the Chow--K\"unneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space $\cM_g$, in genus at most 8 and show existence of a Chow--K\"unneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow--K\"unneth decompositions.