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

Abstract

The wonderful compactification $X_m$ of a symmetric homogeneous space of type AIII$(2,m)$ for each $m\ge 4$ is Fano, and its blowup $Y_m$ along the unique closed orbit is Fano if $m\ge 5$ 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 Kähler-Einstein metric for each $m\ge 4$ and $Y_m$ admits a Kähler-Einstein metric if and only if $m=4,5$.

arXiv comment20 pages, 6 figures

References in corpus (13)

• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• Optimal destabilizing centers and equivariant K-stability
• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
• K-Stability of Fano spherical varieties
• Greatest lower bounds on the Ricci curvature of Fano manifolds
• The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality
• Kähler-Einstein metrics on group compactifications
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds
• K{ä}hler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functional
• Gorenstein spherical Fano varieties
• Delta-invariants for Fano varieties with large automorphism groups
• Kähler-Einstein metrics on smooth Fano symmetric varieties with Picard number one