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

Abstract

The wonderful compactification [Formula: see text] of a symmetric homogeneous space of type AIII[Formula: see text] for each [Formula: see text] is Fano, and its blowup [Formula: see text] along the unique closed orbit is Fano if [Formula: see text] and Calabi–Yau if [Formula: see text]. Using a combinatorial criterion for K-polystability of smooth Fano spherical varieties obtained by Delcroix, we prove that [Formula: see text] admits a Kähler–Einstein metric for each [Formula: see text] and [Formula: see text] admits a Kähler–Einstein metric if and only if [Formula: see text].