Maximal commutative unipotent subgroups and a characterization of affine spherical varieties

TL;DR

This paper characterizes maximal commutative unipotent subgroups of automorphism groups of affine varieties and demonstrates that these groups can determine properties like sphericity and the variety itself, providing new insights into automorphism group structures.

Contribution

It introduces a description of maximal commutative unipotent subgroups of automorphism groups and shows these groups preserve unipotent elements under isomorphisms, aiding in classifying affine varieties.

Findings

Automorphism groups detect sphericity and weight-monoid.

Affine toric varieties are uniquely determined by their automorphism groups.

Smooth affine spherical varieties are characterized by their automorphism groups.

Abstract

We describe maximal commutative unipotent subgroups of the automorphism group $\mathrm{Aut}(X)$ of an irreducible affine variety $X$. Further we show that a group isomorphism $\mathrm{Aut}(X) \to \mathrm{Aut}(Y)$ maps unipotent elements to unipotent elements, where $Y$ is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight-monoid. As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties and we show that a smooth affine spherical variety different from an algebraic torus is determined by its automorphism group (up to an automorphism of the base field) among smooth irreducible affine varieties.