Cohomological conditions on endomorphisms of projective varieties

TL;DR

This paper characterizes periodic subvarieties of complex abelian varieties under surjective endomorphisms using cohomological eigenvalues, extending prior work and analyzing dynamics on Albanese varieties.

Contribution

It introduces new cohomological criteria for periodic subvarieties and applies these to understand endomorphism dynamics on various projective varieties.

Findings

Eigenvalues of cohomological actions determine periodic subvarieties.

Characterization extends previous results by Pink and Roessler.

Insights into dynamics of endomorphisms on Albanese varieties.

Abstract

We characterize possible periodic subvarieties for surjective endomorphisms of complex abelian varieties in terms of the eigenvalues of the cohomological actions induced by the endomorphisms, extending previous work in this direction by Pink and Roessler. By applying our characterization to induced endomorphisms on Albanese varieties, we draw conclusions about the dynamics of surjective endomorphisms for a broad class of projective varieties. We also analyze several classes of surjective endomorphisms that are distinguished by properties of their cohomological actions.