| ABSTRACT |
For a Fano spherical G-variety, an algebraic moment polytope is a polytope encoding the structure of representation of G on the spaces of sections of tensor powers of the anticanonical line bundle. Many geometric properties of Fano spherical varieties can be described from the algebraic moment polytopes. In particular, Delcroix gave a combinatorial criterion for K-stability of a smooth Fano spherical variety in terms of the barycenter of its moment polytope with respect to the Duistermaat-Heckman measure and some data associated to the corresponding spherical homogeneous space. As a specific application, this criterion enables us to prove that all smooth Fano symmetric varieties with Picard number one admit Kaehler-Einstein metrics. For this purpose, I will present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure. |
