Automorphism Groups of Affine Varieties and a Characterization of Affine n-Space

TL;DR

This paper proves that the automorphism group of affine n-space uniquely characterizes the space itself, explores which groups can appear as automorphism groups of affine varieties, and analyzes the structure of these automorphism groups.

Contribution

It establishes that the automorphism group of affine n-space determines the space up to isomorphism and characterizes the possible automorphism groups of affine varieties.

Findings

Automorphism group of affine n-space uniquely determines the space.

Every finite group and torus can appear as an automorphism group of some affine variety.

Automorphism groups cannot be isomorphic to semisimple groups.

Abstract

We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus. Concerning the structure of $Aut(A^n)$ we prove that any homomorphism $Aut(A^n) \to G$ of ind-groups either factors through the Jacobian determinant $jac\colon Aut(A^n) \to k^*$, or it is a closed immersion. For $SAut(A^n):=\ker(jac)$ we show that every nontrivial homomorphism $SAut(A^n) \to G$ is a closed immersion. Finally, we prove that every non-trivial homomorphism $SAut(A^n) \to SAut(A^n)$ is an automorphism, and is given by conjugation with an element from $Aut(A^n)$.