Kähler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functional

Géométrie Kählérienne des variétés horosymétriques, et application à la fonctionnelle de K-énergie de Mabuchi

Thibaut Delcroix (1, 2, 3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IRMA - Institut de Recherche Mathématique Avancée
3 IMAG - Institut Montpelliérain Alexander Grothendieck

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-Donaldson geometry of toric varieties. Namely we associate convex functions with hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge-Ampère volume form and scalar curvature. We then provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by Li, Zhou and Zhu on group compactifications.

Mots clés: moment polytope; complete symmetric variety; wonderful compactification; horosymmetric variety; horospherical variety; log Kahler-Einstein metric; Mabuchi K-energy functional; cscK metric; scalar curvature

Domaines: Géométrie différentielle [math.DG]; Géométrie algébrique [math.AG]

Thibaut Delcroix. Kähler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functional. Journal für die reine und angewandte Mathematik, 2020, 763, pp.129-199. ⟨10.1515/crelle-2018-0040⟩. ⟨hal-01652952v2⟩