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

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-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.

Appendix: An Algebraic Proof of Corollary 4.6

In this Appendix, we give an algebraic proof of Corollary 4.6. Recall that a ${\hat{G}}$-equivariant normal ${\mathbb {R}}$-test configuration ${\mathcal {F}}$ with reduced central fibre is special if and only if $\textrm{Gr}({\mathcal {F}})$ is an integral ring. It suffices to show

Proposition 7.1

Suppose that (4.28) holds. Then, the algebra $\textrm{Gr}({\mathcal {F}})$ defined by (2.3) is integral if and only if f is affine on $P_+$.

Proof

Assume that $\textrm{Gr}({\mathcal {F}})$ is integral. We show that f is affine. Otherwise, we can take two domains of linearity $Q_1,Q_2\subset P_+$, so that they intersect along a common facet. Take $\lambda _i\in Q_i\cap {\mathfrak {M}}_{\mathbb {Q}}$, so that the line segment $\overline{\lambda _1\lambda _2}\subset Q_1\cup Q_2$. Up to replacing $P_+$ by some $k_0P_+$, we can assume $\lambda _i\in Q_i\cap {\mathfrak {M}}$ for $i=1,2$.

Let $\sigma _i\in {\textrm{End}}(V_{\lambda _i})$ be a highest weight vector. Then, $\sigma _i$ has real weight $t^{-s_{\lambda _i}^{(1)}}$. Consequently, $\sigma _1\otimes \sigma _2\in {\textrm{End}}(V_{\lambda _1+\lambda _2})$ has real weight $t^{-s_{\lambda _1}^{(1)}-s_{\lambda _2}^{(1)}}$. On the other hand, for $\lambda _1+\lambda _2\in \overline{2P_+}\cap {\mathfrak {M}}$, by (4.28)

$$\begin{aligned} s_{\lambda _1+\lambda _2}^{(2)}=2f\left( \frac{1}{2}(\lambda _1+\lambda _2)\right) >s_{\lambda _1}^{(1)}+s_{\lambda _1}^{(2)}, \end{aligned}$$

where the last inequality follows from the concavity of f and the fact that $\lambda _i$’s lie in different domains of linearity. Hence, $\sigma _1\cdot \sigma _2=0$ in $\textrm{Gr}({\mathcal {F}})$. A contradiction to the assumption that $\textrm{Gr}({\mathcal {F}})$ is integral.

Conversely, assume that f is affine. We will show that $\textrm{Gr}({\mathcal {F}})$ is integral. Otherwise, there are $(0\not =)\sigma _i\in {\textrm{End}}(V_{\lambda _i}),i=1,2$, so that $\sigma _1\cdot \sigma _2=0$ in $\textrm{Gr}({\mathcal {F}})$. By Lemma 4.2, we can assume that each $\sigma _i$ is a highest weight vector. Assume that $\lambda _i\in \overline{k_iP_+}$. Then, by (4.28), $\sigma _i$ has real weight $t^{-k_if(\lambda _i/k_i)}$. Consequently, $\sigma _1\otimes \sigma _2\in {\textrm{End}}(V_{\lambda _1+\lambda _2})$ has real weight $t^{-k_1f(\lambda _1/k_1)-k_2f(\lambda _2/k_2)}$ with

$$\begin{aligned} k_1f\left( \frac{\lambda _1}{k_1}\right) +k_2f\left( \frac{\lambda _2}{k_2}\right) =(k_1+k_2)f\left( \frac{\lambda _1+\lambda _2}{k_2+k_2}\right) . \end{aligned}$$

Here, we used the fact that f is affine. Note that the right-hand side is just the real weight of the ${\textrm{End}}(V_{\lambda _1+\lambda _2})$-piece in $\textrm{Gr}({\mathcal {F}})$. Hence, $\sigma _1\cdot \sigma _2\not =0$, a contradiction. The proposition is proved. $\square$