Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One

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