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 Φ m : G → G is regular for some m 1 and that there exists no non-constant homomorphism τ : G −→ G 0 of semiabelian varieties such that τ • Φ mk = 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.