Real forms of minimal SL$_2$-threefolds

TL;DR

This paper completes the classification of real forms of minimal smooth complete SL$_2$-threefolds, analyzing their structure, rationality, and real points using Luna-Vust theory.

Contribution

It provides a comprehensive classification of real forms of minimal SL$_2$-threefolds and investigates their rationality and real points.

Findings

Classified all real forms of minimal smooth complete SL$_2$-threefolds.

Determined the rationality of these varieties.

Analyzed the set of real points on the classified varieties.

Abstract

We complete the classification of the real forms of almost homogeneous SL$_2$-threefolds. More precisely, we use the Luna-Vust theory to determine the real forms of minimal smooth complete SL$_2$-varieties containing an orbit isomorphic to SL$_2/H$, where $H$ is a finite cyclic subgroup of SL$_2$. Moreover, we study the rationality and the set of real points of those varieties.