Realisation of abelian varieties as automorphism groups

TL;DR

This paper proves that an abelian variety over a field can be realized as the automorphism group scheme of a smooth projective variety if and only if it has finitely many automorphisms over an algebraic closure, extending known results.

Contribution

It establishes a necessary and sufficient condition for realizing abelian varieties as automorphism groups of smooth projective varieties over any field.

Findings

Realization is possible iff the abelian variety has finitely many automorphisms over an algebraic closure.

The result generalizes previous work over complex and algebraically closed fields.

Provides a criterion linking automorphism finiteness to realizability as automorphism groups.

Abstract

Let $A/F$ be an abelian variety over a field. Does there exist a smooth projective $F$-variety $X$, such that $A$ is isomorphic to the automorphism group scheme of $X/F$? We show that the answer is positive, if and only if $A$ has only finitely many automorphisms, over an algebraic closure of $F$. When $F=\mathbb C$, this result is due to Lombardo and Maffei. When $F$ is algebraically closed, it was obtained independently by Blanc and Brion.