Flexibility of affine spherical varieties

TL;DR

This paper investigates the automorphism groups of affine spherical varieties, proving transitivity on smooth points and establishing conditions under which the variety is flexible, meaning certain unipotent subgroups act transitively.

Contribution

It demonstrates that the automorphism group acts transitively on smooth points and characterizes when the variety is flexible based on invertible regular functions.

Findings

Automorphism group acts transitively on smooth points.

Variety is flexible if all invertible regular functions are constant.

Flexibility implies transitivity of the subgroup generated by all -subgroups.

Abstract

We prove that the automorphism group $\mathrm{Aut}(X)$ of an affine spherical variety $X$ acts transitively on the set of smooth points $X^{reg}.$ If every invertible regular function on $X$ is constant, we prove that $X$ is flexible, i.e., the subgroup of $\mathrm{Aut}(X)$ generated by all $\mathbb{G}_a$-subgroups acts transitively on $X^{reg}.$