Forms of almost homogeneous varieties over perfect fields

We study the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -forms of almost homogeneous varieties over perfect base fields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> . First, we discuss criteria for the existence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -forms in the homogeneous case. Then, we extend the Luna-Vust theory from algebraically closed fields to perfect fields to determine when a given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -form of the open orbit of an almost homogeneous variety extends to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -form of the entire variety. Finally, in the last section, we apply these results to determine the real forms of complex almost homogeneous <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo form="prefix">SL</mml:mo> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -threefolds.