Action of Correspondences on Filtrations on Cohomology and 0-cycles of Abelian Varieties
Rakesh Pawar
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.
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…
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Action of Correspondences on Filtrations on Cohomology and 0-cycles of Abelian Varieties
Rakesh R. Pawar
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.