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

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