K\"ahler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII

TL;DR

This paper proves the existence of Kähler-Einstein metrics on certain smooth Fano symmetric varieties of type AIII, using combinatorial criteria for K-polystability, and identifies specific cases where these metrics exist.

Contribution

It establishes the existence of Kähler-Einstein metrics on the wonderful compactification and its blowups of type AIII symmetric varieties, extending previous stability criteria.

Findings

X_m admits a Kähler-Einstein metric for all m ≥ 4

Y_m admits a Kähler-Einstein metric if and only if m = 4, 5

Y_m is Fano for m ≥ 5 and Calabi-Yau for m = 4

Abstract

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 $m = 4, 5$.