Is the affine space determined by its automorphism group?

TL;DR

This paper investigates whether the complex affine space $\\mathbb{A}^n$ can be uniquely characterized by its automorphism group, establishing conditions under which a variety with an isomorphic automorphism group to $\\\mathbb{A}^n$ must itself be isomorphic to it.

Contribution

The authors prove that certain varieties with automorphism groups isomorphic to that of $\\\mathbb{A}^n$ are indeed isomorphic to $\\\mathbb{A}^n$, extending the understanding of automorphism group characterizations.

Findings

Varieties with automorphism groups isomorphic to $\\\mathbb{A}^n$ are isomorphic to $\\\mathbb{A}^n$ under specified conditions.

The identity component of the centralizer of a $(\mathbb{Z}/p\mathbb{Z})^n$-action is a torus if certain topological and algebraic conditions are met.

The results connect automorphism group structures with geometric and topological properties of varieties.

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.