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 Kähler–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.
Highlights
For the case of toric Fano manifolds, Wang and Zhu [14] proved that the existence of a Kähler–Einstein metric is equivalent to the vanishing of the Futaki invariant
By combining the above criterion and Ruzzi’s classification [18] of smooth Fano symmetric varieties with Picard number one, we prove the following
The normal equivariant embeddings of a given spherical homogeneous space are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. For the case of toric Fano manifolds, Wang and Zhu [14] proved that the existence of a Kähler–Einstein metric is equivalent to the vanishing of the Futaki invariant This was based on the theorem by Mabuchi [15], which says that the Futaki invariant vanishes if and only if the barycenter of the moment polytope is the origin. As a generalization of Wang and Zhu’s work, Delcroix [17] extended a combinatorial criterion for K-stability of Fano spherical manifolds, in terms of its moment polytope and spherical data This criterion is applicable to smooth Fano symmetric varieties (see Corollary 5.9 of [17]).
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