Existence of equivariant models of spherical varieties and other G-varieties

TL;DR

This paper establishes necessary and sufficient conditions for the existence of equivariant models of spherical varieties over different fields, advancing understanding of their algebraic and geometric structures.

Contribution

It provides a complete characterization of when spherical varieties admit models compatible with given group forms over different fields.

Findings

Characterization of equivariant models for spherical varieties

Conditions depend on the field and group forms

Advances classification of spherical varieties over various fields

Abstract

Let $k_0$ be a field of characteristic $0$ with algebraic closure $k$. Let $G$ be a connected reductive $k$-group, and let $Y$ be a spherical variety over $k$ (a spherical homogeneous space or a spherical embedding). Let $G_0$ be a $k_0$-model ($k_0$-form) of $G$. We give necessary and sufficient conditions for the existence of a $G_0$-equivariant $k_0$-model of $Y$.