Rational self-maps with a regular iterate on a semiabelian variety

Abstract

Let G be a semiabelian variety defined over an algebraically closed field K of characteristic 0. Let Φ:G⇢G be a dominant rational self-map. Assume that an iterate Φm:G→G is regular for some m⩾1 and that there exists no non-constant homomorphism τ:G⟶G0 of semiabelian varieties such that τ∘Φmk=τ for some k⩾1. We show that under these assumptions Φ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp.