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 $\chi (X)\ne 0$ 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.

18 pages, comments welcome! In this version, we generalize the main theorem and simplify many proofs