Kähler-Einstein metrics on group compactifications

Abstract

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation into a real Monge-Amp{è}re equation, using the invariance under the action of a maximal compact subgroup $K\times K$.

References in corpus (7)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• K-stability and Kähler-Einstein metrics
• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• K-Stability of Fano spherical varieties
• Kähler-Einstein metrics along the smooth continuity method
• Kaehler-Ricci solitons on homogeneous toric bundles (I)

Cited by in corpus (21)

• K-Stability of Fano spherical varieties
• Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds
• K{ä}hler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functional
• Coupled Kähler-Ricci solitons on toric Fano manifolds
• Kähler-Ricci solitons on horospherical variety
• Kähler-Einstein metrics on smooth Fano symmetric varieties with Picard number one
• Log canonical thresholds on group compactifications
• Singular limits of Kähler-Ricci flow on Fano $G$-manifolds
• Greatest Lower Bounds on Ricci Curvature for Fano $T$-manifolds of Complexity $1$
• Ricci flat Kähler metrics on rank two complex symmetric spaces
• Examples of K-unstable Fano manifolds
• K-energy on polarized compactifications of Lie groups
• K-stability of Gorenstein Fano group compactifications with rank two
• Canonical blow-ups of Grassmann manifolds
• K-stability and polystable degenerations of polarized spherical varieties
• Kähler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII
• Limits of conical Kähler-Einstein metrics on rank one horosymmetric spaces
• The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations
• Finiteness of $\mathbb{Q}$-Fano compactifications of semisimple group with Kähler-Einstein metrics
• Kähler-Einstein metrics and Ding functional on $\mathbb{Q}$-Fano group compactifications
• Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one