K-stability of Gorenstein Fano group compactifications with rank two

TL;DR

This paper classifies rank two Gorenstein Fano bi-equivariant compactifications of semisimple complex Lie groups, identifying which are K-stable and admit Kähler-Einstein metrics, and computes their Ricci bounds, providing new examples of K-stable Fano varieties.

Contribution

It provides a classification of rank two Gorenstein Fano group compactifications and determines their K-stability and existence of Kähler-Einstein metrics, with explicit examples.

Findings

Identified K-stable Fano varieties with Kähler-Einstein metrics.

Computed greatest Ricci lower bounds for K-unstable varieties.

Provided new examples of solutions to the Kähler-Ricci flow of type II.

Abstract

We give a classification of Gorenstein Fano bi-equivariant compactifications of semisimple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) K\"{a}hler-Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) K\"{a}hler-Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the K\"{a}hler-Ricci flow is of type II.