Characterizing smooth affine spherical varieties via the automorphism group

Regeta, Andriy; van Santen, Immanuel

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 379-414.

Soit $G$ un groupe réductif connexe. Nous montrons que le monoïde des poids d’une variété $G$-sphérique quasi-affine est déterminé par les poids de ses ${𝔾}_a$-actions non triviales homogènes sous l’action d’un sous-groupe de Borel de $G$. Comme application, nous obtenons qu’une variété sphérique affine lisse non isomorphe à un tore est déterminée par son groupe des automorphismes (considéré comme un ind-groupe) dans la catégorie des variétés irréductibles affines lisses.

Let $G$ be a connected reductive algebraic group. We prove that for a quasi-affine $G$-spherical variety the weight monoid is determined by the weights of its non-trivial ${𝔾}_a$-actions that are homogeneous with respect to a Borel subgroup of $G$. As an application we get that a smooth affine spherical variety that is non-isomorphic to a torus is determined by its automorphism group (considered as an ind-group) inside the category of smooth affine irreducible varieties.

Classification : 14R20, 14M27, 14J50, 22F50
Keywords: Automorphism groups of quasi-affine varieties, quasi-affine spherical varieties, root subgroups, quasi-affine toric varieties
Mot clés : Groupes des automorphismes des variétés quasi-affines, variétés sphériques quasi-affines, sous-groupes de racines, variétés toriques quasi-affines

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