On the Chow group of zero-cycles of a generalized Kummer variety

TL;DR

This paper constructs specific subvarieties in generalized Kummer varieties to show that the rational orbit filtration aligns with the Beauville decomposition, providing insights into the structure of zero-cycle Chow groups.

Contribution

It introduces co-isotropic subvarieties foliated by constant cycle subvarieties in generalized Kummer varieties and proves the equivalence of two filtrations on their Chow groups.

Findings

Rational orbit filtration coincides with Beauville decomposition.

Rational orbit filtration is opposite to Bloch-Beilinson filtration.

Constructs co-isotropic subvarieties in generalized Kummer varieties.

Abstract

For a generalized Kummer variety X of dimension 2n, we will construct for each 0 < i < n some co-isotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit filtration introduced by Voisin on the Chow group of zero-cycles of a generalized Kummer variety coincides with the induced Beauville decomposition from the Chow ring of abelian varieties. As a consequence, the rational orbit filtration is opposite to the conjectural Bloch-Beilinson filtration for generalized Kummer varieties.