Is the Affine Space Determined by Its Automorphism Group?

Abstract In this note we study the problem of characterizing the complex affine space ${\mathbb{A}}^n$ via its automorphism group. We prove the following. Let $X$ be an irreducible quasi-projective $n$-dimensional variety such that $\operatorname{Aut}(X)$ and $\operatorname{Aut}({\mathbb{A}}^n)$ are isomorphic as abstract groups. If $X$ is either quasi-affine and toric or $X$ is smooth with Euler characteristic $\chi (X) \neq 0$ and finite Picard group $\operatorname{Pic}(X)$, then $X$ is isomorphic to ${\mathbb{A}}^n$. The main ingredient is the following result. Let $X$ be a smooth irreducible quasi-projective variety of dimension $n$ with finite $\operatorname{Pic}(X)$. If $X$ admits a faithful $({\mathbb{Z}} / p {\mathbb{Z}})^n$-action for a prime $p$ and $\chi (X)$ is not divisible by $p$, then the identity component of the centralizer $\operatorname{Cent}_{\operatorname{Aut}(X)}( ({\mathbb{Z}} / p {\mathbb{Z}})^n)$ is a torus.