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 $\Phi\colon G\dashrightarrow G$ be a dominant rational self-map. Assume that an iterate $\Phi^m \colon G \to G$ is regular for some $m \geqslant 1$ and that there exists no non-constant homomorphism $\tau: G\to G_0$ of semiabelian varieties such that $\tau\circ \Phi^{m k}=\tau$ for some $k \geqslant 1$. We show that under these assumptions $\Phi$ 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.