On the classification of normal $G$-varieties with spherical orbits

Langlois, Kevin

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 271-334.

Dans cet article, nous étudions la géométrie des opérations de groupes réductifs dans les variétés algébriques. Étant donné un groupe algébrique réductif connexe $G$, nous élaborons une approche géométrique et combinatoire basée sur la théorie de Luna–Vust pour décrire toute $G$-variété normale avec orbites sphériques. Cette description comprend le cas classique des variétés sphériques et la théorie des $𝕋$-variétés introduite récemment par Altmann, Hausen et Süss.

In this article, we investigate the geometry of reductive group actions on algebraic varieties. Given a connected reductive group $G$, we elaborate on a geometric and combinatorial approach based on Luna–Vust theory to describe every normal $G$-variety with spherical orbits. This description encompasses the classical case of spherical varieties and the theory of $𝕋$-varieties recently introduced by Altmann, Hausen, and Süss.

Classification : 14L30, 14M27, 14M25, 13A18
Mots clés : action of algebraic groups, Luna–Vust theory, homogeneous spaces, valuation theory

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