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

The wonderful compactification Formula: see text of a symmetric homogeneous space of type AIIIFormula: 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.