Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic -- A remark to a paper of Dinh-Oguiso-Zhang

Abstract

Let $X$ be a projective variety of dimension $n\ge1$ over an algebraically closed field of arbitrary characteristic. We prove a Fujiki-Lieberman type theorem on the structure of the automorphism group of $X$. Let $G$ be a group of zero entropy automorphisms of $X$ and $G_0$ the set of elements in $G$ which are isotopic to the identity. We show that after replacing $G$ by a suitable finite-index subgroup, $G/G_0$ is a unipotent group of the derived length at most $n-1$. This result was first proved by Dinh, Oguiso and Zhang for compact K\"ahler manifolds.