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

TL;DR

This paper extends a theorem on the structure of automorphism groups to projective varieties over arbitrary characteristic, showing that zero entropy automorphisms form a virtually unipotent group with bounded derived length.

Contribution

It generalizes a known result from Kähler manifolds to all projective varieties in arbitrary characteristic, establishing a bound on the derived length of zero entropy automorphism groups.

Findings

The automorphism group modulo isotopic to identity elements is virtually unipotent.

The derived length of this unipotent group is at most the dimension minus one.

The result holds over fields of arbitrary characteristic.

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.