Kähler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII

Abstract

The wonderful compactification Xm of a symmetric homogeneous space of type AIII(2,m) for each m >= 4 is Fano, and its blowup Ym along the unique closed orbit is Fano if m >= 5 and Calabi-Yau if m = 4. Using a combinatorial criterion for K-polystability of smooth Fano spherical varieties obtained by Delcroix, we prove that Xm admits a Kahler-Einstein metric for each m >= 4 and Ym admits a Kahler-Einstein metric if and only if m = 4, 5.