Automorphism groups of affine varieties and a characterization of affine $n$-space

Abstract:

We show that the automorphism group of affine $n$-space ${\mathbb {A}^n}$ determines ${\mathbb {A}^n}$ up to isomorphism: If $X$ is a connected affine variety such that $\mathrm {Aut}(X) \simeq \mathrm {Aut}({\mathbb {A}^n})$ as ind-groups, then $X \simeq {\mathbb {A}^n}$ as varieties.

We also show that every torus appears as $\mathrm {Aut}(X)$ for a suitable irreducible affine variety $X$, but that $\mathrm {Aut}(X)$ cannot be isomorphic to a semisimple group. In fact, if $\mathrm {Aut}(X)$ is finite-dimensional and if $X \not \simeq {\mathbb {A}^1}$, then the connected component $\mathrm {Aut}(X)^{\circ }$ is a torus.

Concerning the structure of $\mathrm {Aut}({\mathbb {A}^n})$ we prove that any homomorphism $\mathrm {Aut}({\mathbb {A}^n}) \!\to \mathcal {G}$ of ind-groups either factors through $\mathrm {jac}\colon \mathrm {Aut}({\mathbb {A}^n}) \to \Bbbk ^*$ where $\mathrm {jac}$ is the Jacobian determinant, or it is a closed immersion. For $\mathrm {SAut}({\mathbb {A}^n}):=\mathrm {ker}(\mathrm {jac})\subseteq \mathrm {Aut}({\mathbb {A}^n})$ we show that every nontrivial homomorphism $\mathrm {SAut}({\mathbb {A}^n}) \to \mathcal {G}$ is a closed immersion.

Finally, we prove that every nontrivial homomorphism $\varphi \colon \mathrm {SAut}({\mathbb {A}^n}) \to \mathrm {SAut}({\mathbb {A}^n})$ is an automorphism, and that $\varphi$ is given by conjugation with an element from $\mathrm {Aut}({\mathbb {A}^n})$.