Equivariant $\mathbb{R}$-Test Configurations and Semistable Limits of $\mathbb{Q}$-Fano Group Compactifications

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-invariants of their equivariant normal $\mathbb{R}$-test configurations in terms of the combinatory data. Based on Han and Li “Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties”, we compute the semistable limit of a K-unstable Fano G-compactification. As an application, we show that for the two smooth K-unstable Fano SO ${}_4(\mathbb{C})$-compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized Kähler–Ricci flow.