Finiteness of $\mathbb Q$-Fano compactifications of semisimple group with K\"ahler-Einstein metrics

TL;DR

This paper proves that only finitely many $Q$-Fano compactifications of a semisimple group admit (possibly singular) K"ahler-Einstein metrics, providing a classification approach and improving previous results.

Contribution

It establishes the finiteness of $Q$-Fano compactifications of semisimple groups with K"ahler-Einstein metrics, advancing classification methods in complex geometry.

Findings

Finiteness of such compactifications proven

Provides a classification method for $Q$-Fano compactifications

Improves previous results on K"ahler-Einstein metrics

Abstract

In this note, we give a way to classify $\mathbb Q$-Fano compactifications of a semisimple group $G$. We will prove that there are only finitely many such $\mathbb Q$-Fano $G$-compactifications, which admits (singular) K\"ahler-Einstein metrics. As an application, this improves a former result in arXiv:2001.11320.