  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.
