Equivariant automorphism group and real forms of complexity-one varieties

TL;DR

This paper investigates the structure of equivariant automorphism groups of complexity-one G-varieties over perfect fields, establishing their representability and implications for the classification of real forms.

Contribution

It proves the representability of equivariant automorphism groups for broad classes of complexity-one G-varieties and links this to finiteness of real forms.

Findings

Equivariant automorphism groups are representable by group schemes in characteristic zero.

Almost homogeneous G-varieties have equivariant automorphism groups as linear algebraic groups.

Complexity-one G-varieties with such automorphism groups have finitely many real forms.

Abstract

Let G be a connected reductive algebraic group over a perfect field. We study the representability of the equivariant automorphism group of G-varieties. For a broad class of complexity-one G-varieties, we show that this group is representable by a group scheme locally of finite type when the base field has characteristic zero. We also establish representability by a linear algebraic group in the case of almost homogeneous G-varieties of arbitrary complexity. Finally, using an exact sequence description of the equivariant automorphism group, we deduce that complexity-one G-varieties with representable equivariant automorphism group admit only finitely many real forms.