Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics

TL;DR

This paper characterizes when an abelian variety over an algebraically closed field can serve as the automorphism group scheme of a smooth projective variety, extending previous results from complex to arbitrary characteristics.

Contribution

It provides a necessary and sufficient condition for abelian varieties to be automorphism groups of smooth projective varieties in any characteristic, generalizing prior complex case results.

Findings

Characterization of abelian varieties as automorphism groups

Extension of previous complex results to arbitrary characteristics

Finiteness condition on automorphisms as a key criterion

Abstract

Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.