Forms of almost homogeneous varieties over perfect fields

Moser-Jauslin, Lucy; Terpereau, Ronan

Annales Henri Lebesgue, Tome 7 (2024), pp. 357-407

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.

Nous étudions les $k$-formes des variétés presque homogènes sur des corps de base parfaits $k$. Tout d’abord, nous discutons de critères pour l’existence de $k$-formes dans le cas homogène. Ensuite, nous étendons la théorie de Luna-Vust des corps algébriquement clos aux corps parfaits pour déterminer quand une $k$-forme donnée de l’orbite ouverte d’une variété presque homogène s’étend à une $k$-forme de toute la variété. Enfin, dans la dernière section, nous appliquons ces résultats pour déterminer les formes réelles des ${SL}_2$-solides presque homogènes complexes.

Classification : 14M17, 20G15, 14M27, 14P99
Keywords: Homogeneous space, equivariant embedding, Luna-Vust theory, Galois descent, real structure, real form

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