K\"ahler-Einstein metrics on group compactifications

TL;DR

This paper establishes a precise criterion for when a Kähler-Einstein metric exists on certain compactifications of complex reductive groups, linking geometric conditions to associated polytopes.

Contribution

It provides a necessary and sufficient condition for the existence of Kähler-Einstein metrics on G×G-equivariant Fano compactifications, independent of the Futaki invariant.

Findings

Derived a polytope-based criterion for existence

Connected the problem to a real Monge-Ampère equation

Used the continuity method and symmetry invariance

Abstract

We obtain a necessary and sufficient condition of existence of a K{\"a}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{\`e}re equation, using the invariance under the action of a maximal compact subgroup $K\times K$.