Action of correspondences on filtrations on cohomology and 0-cycles of Abelian varieties

Abstract

We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which includes any abelian variety of dimension at most 5, powers of complex elliptic curves, etc.) that 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 an automorphism of such an abelian variety that acts as the identity on a certain quotient of its middle singular cohomology 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.