Singularités canoniques et actions horosphériques

Langlois, Kevin

Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 365-369.

Soit G un groupe algébrique linéaire réductif connexe. Nous considérons les G-variétés normales avec orbites horosphériques. Dans cette courte note, nous donnons un critère pour déterminer si ces variétés ont au plus des singularités canoniques, log canoniques ou terminales dans le cas où elles admettent une courbe algébrique comme quotient rationnel. Ce résutat semble nouveau pour le cas spécial des actions de tores algébriques avec orbites générales de codimension 1. Pour la G-variété considérée X, notre critère est exprimé en terme d'une fonction de poids ωX qui est construite à partir de l'ensemble des valuations G-invariantes du corps des fonctions k(X). Dans le cas log terminal, la fonction génératrice de ωX correspond au volume motivique des cordes de X. Comme application, nous traitons le cas des k⋆-surfaces normales.

Let G be a connected reductive linear algebraic group. We consider the normal G-varieties with horospherical orbits. In this short note, we provide a criterion to determine whether these varieties have at most canonical, log canonical or terminal singularities in the case where they admit an algebraic curve as rational quotient. This result seems to be new in the special setting of torus actions with general orbits of codimension 1. For the given G-variety X, our criterion is expressed in terms of a weight function ωX that is constructed from the set of G-invariant valuations of the function field k(X). In the log terminal case, the generating function of ωX coincides with the stringy motivic volume of X. As an application, we discuss the case of normal k⋆-surfaces.

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