Kähler–Einstein metrics on group compactifications

Abstract

We obtain a necessary and sufficient condition of existence of a Kähler–Einstein metric on a G × 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–Ampère equation, using the invariance under the action of a maximal compact subgroup K × K.