Forms of almost homogeneous varieties over perfect fields

TL;DR

This paper investigates the existence and classification of k-forms of almost homogeneous varieties over perfect fields, extending Luna-Vust theory and applying results to real forms of complex SL(2)-threefolds.

Contribution

It extends Luna-Vust theory to perfect fields and provides criteria for the existence of k-forms of almost homogeneous varieties.

Findings

Criteria for existence of k-forms in the homogeneous case

Extension of Luna-Vust theory to perfect fields

Classification of real forms of complex SL(2)-threefolds

Abstract

We study the k-forms of almost homogeneous varieties over perfect base fields k. First, we discuss criteria for the existence of k-forms in the homogeneous case. Then, we extend the Luna-Vust theory from algebraically closed fields to perfect fields to determine when a given k-form of the open orbit of an almost homogeneous variety extends to a k-form of the entire variety. Finally, in the last section, we apply these results to determine the real forms of complex almost homogeneous SL(2)-threefolds.