K\"{a}hler-Einstein metrics on smooth Fano symmetric varieties with Picard number one

TL;DR

This paper proves that all smooth Fano symmetric varieties with Picard number one admit K"{a}hler-Einstein metrics by applying a combinatorial criterion for K-stability, involving algebraic moment polytopes and barycenter computations.

Contribution

It establishes the existence of K"{a}hler-Einstein metrics on a class of Fano symmetric varieties using combinatorial and algebraic methods, extending previous results in the field.

Findings

All smooth Fano symmetric varieties with Picard number one admit K"{a}hler-Einstein metrics.

Explicit algebraic moment polytopes are constructed for these varieties.

Barycenters of the moment polytopes are computed to verify K-stability.

Abstract

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit K\"{a}hler-Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure.