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 $\mathrm{Aut}(X)$ and $\mathrm{Aut}(\mathbb{A}^n)$ 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 $\mathrm{Pic}(X)$, 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 $\mathrm{Pic}(X)$. 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 $\mathrm{Cent}_{\mathrm{Aut}(X)}( (\mathbb{Z} / p \mathbb{Z})^n)$ is a torus.