Equivariant $\mathbb R$-test configurations and semistable limits of $\mathbb Q$-Fano group compactifications

TL;DR

This paper classifies equivariant test configurations of G-compactifications, computes their invariants, and analyzes semistable limits of K-unstable Fano G-compactifications, linking algebraic and geometric stability.

Contribution

It provides a classification of equivariant R-test configurations and expresses their invariants, advancing understanding of stability and limits in Fano G-compactifications.

Findings

Classification of G×G-equivariant R-test configurations.

Expression of H-invariant in terms of combinatorial data.

Identification of semistable limits as Kähler-Ricci flow limits for specific cases.

Abstract

Let $G$ be a connected, complex reductive group. In this paper, we classify $G\times G$-equivariant normal $\mathbb R$-test configurations of a polarized $G$-compactification. Then for $\mathbb Q$-Fano $G$-compactifications, we express the H-invariant of its equivariant normal $\mathbb R$-test configurations in terms of the combinatory data. Based on \cite{Han-Li}, we compute the semistable limit of a K-unstable Fano $G$-compactification. As an application, we show that for the two K-unstable Fano $SO_4(\mathbb C)$-compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized K\"ahler-Ricci flow.