Kähler-Einstein metrics on group compactifications
Résumé
We obtain a necessary and sufficient condition of existence of a Kähler-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-Ampère equation, using the invariance under the action of a maximal compact subgroup $K\times K$.
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