A note on the Grothendieck group of an abelian variety

TL;DR

This paper revisits Beauville's conjecture on algebraic cycles in abelian varieties, reformulating it through the lens of filtrations on the Grothendieck group to explore underlying structural compatibilities and vanishings.

Contribution

It introduces a new reformulation of Beauville's conjecture using filtrations on the Grothendieck group, providing a different perspective on algebraic cycles in abelian varieties.

Findings

Reformulation of Beauville's conjecture in terms of Grothendieck group filtrations

Identification of compatibility conditions for filtrations on the Grothendieck group

Insights into vanishings related to algebraic cycles on abelian varieties

Abstract

We reformulate a conjecture of Beauville on algebraic cycles on an abelian variety in terms of certain compatibility and vanishings of some naturally defined filtrations on the Grothendieck group of the abelian variety.