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 $\hat{I}¦:G⇢G$ be a dominant rational self-map. Assume that an iterate $\hat{I}{¦}^m:G\to G$ is regular for some $m⩾1$ and that there exists no non-constant homomorphism $\ddot{I}„:G\to G_0$ of semiabelian varieties such that $\ddot{I}„\circ \hat{I}{¦}^{mk}=\ddot{I}„$ for some $k⩾1$. We show that under these assumptions $\hat{I}¦$ 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.

15 pages