# Action of Correspondences on Filtrations on Cohomology and 0-cycles of Abelian Varieties

## TL;DR

This paper establishes a link between correspondences acting trivially on certain cohomology quotients and their action on filtrations of the Chow group of 0-cycles in abelian varieties, with applications to automorphisms and Kummer varieties.

## Contribution

It proves that symmetrically distinguished correspondences vanishing on specific cohomology quotients also vanish on deep filtrations of the Chow group, extending understanding of automorphism actions.

## Key findings

- Correspondences vanishing on cohomology quotients also vanish on Chow group filtrations.
- Automorphisms acting trivially on cohomology quotients act as identity on Chow group filtrations.
- Automorphisms of Generalized Kummer varieties act as identity on certain Chow subgroup.

## Abstract

We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which include any abelian variety of dimension atmost 5, powers of complex elliptic curves, etc.) which vanishes as a morphism on a certain quotient of its middle singular cohomology, then it vanishes as a morphism on the deepest part of a particular filtration on the Chow group of 0-cycles of the abelian variety. As a consequence, we prove that given an automorphism of such an abelian variety, which acts as the identity on a certain quotient of its middle singular cohomology, then it acts as the identity on the deepest part of this filtration on the Chow group of 0-cycles of the abelian variety. As an application, we prove that for the Generalized Kummer variety associated to a complex abelian surface and the automorphism induced from a symplectic automorphism of the complex abelian surface, the automorphism of the Generalized Kummer variety acts as the identity on a certain subgroup of its Chow group of 0-cycles.