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arXiv · 2209.14720

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

Kyusik Hong·DongSeon Hwang·Kyeong-Dong Park
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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 $m = 4, 5$.

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Kyusik Hong, DongSeon Hwang, Kyeong-Dong Park. 2022-09-29. K\"ahler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII. https://doi.org/10.1142/s0129167x2450023x

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