Stability of Valuations: Higher Rational Rank

Abstract

Given a klt singularity \(x\in (X, D)\), we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized volume function \({\widehat{\text{vol}}}_{(X,D),x}\), if and only if v induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity \(x\in X\) on the Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of \(x\in X\), hence confirming a conjecture by Donaldson–Sun.

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Notes

  1. 1.

    The following calculations and arguments do not depend on the choice of reference metrics, i.e. remain valid for any choice of reference metric.

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Acknowledgements

We want to thank Harold Blum, Yuchen Liu, Mircea Mustaţǎ and Gang Tian for helpful discussions and comments. CL is partially supported by NSF DMS-1405936 and Alfred P. Sloan research fellowship. Part of this work was done during CX’s visiting of the Department of Mathematics in MIT, to which he wants to thank the inspiring environment. CX is partially sponsored by ‘The National Science Fund for Distinguished Young Scholars (11425101)’.

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Correspondence to Chi Li.

Additional information

The original version of this article was revised: Example 4.2 as well as Lemma 3.31 and Lemma 2.51 have been incorrectly referenced by error. Furthermore, the multi-letter variables such as gr, div, spec have been set in italic instead in roman.

The original version of this article was revised: for details see article https://doi.org/10.1007/s42543-019-00009-y.

Appendix: Example: \(D_{k+1}\)-Singularities

Appendix: Example: \(D_{k+1}\)-Singularities

In this section, we verify that the candidate minimizers computed in [38] for \(D_{k+1}\) singularities induced by monomial valuations on the ambient spaces are indeed the unique quasi-monomial minimizers of \({\widehat{{\text{vol}}}}\), except possibly for the case of four-dimensional \(D_4\) singularity for which we cannot confirm yet.

Example 4.1

Consider the three-dimensional \(D_{k+1}\) singularity for \(k\ge 4\):

$$\begin{aligned} o:=\{0,0,0,0\}\in X=\left\{ f(z_1,\dots , z_4):=z_1z_2+z_3^2z_4+z_4^k=0\right\} . \end{aligned}$$

\(X={\text{Spec}}(R)\) with \(R=\mathbb {C}[z_1, \dots , z_4]/(f(z))\). In [38], we calculated the candidate minimizing valuation \(v_0\) of \({\widehat{{\text{vol}}}}_{X,x}\). \(v_0\) is induced by the weight

$$\begin{aligned} w_0=(1, 1, \sqrt{3}-1, 4-2\sqrt{3}). \end{aligned}$$

We verify here that this is indeed a global minimizer of \({\widehat{{\text{vol}}}}_{X,x}\). First notice that the weight \(w_0\) degenerates X to the following klt singularity:

$$\begin{aligned} X_0=\left\{ z_1 z_2+z_3^2 z_4=0 \right\} . \end{aligned}$$
(84)

\(X_0\) is called the suspended pinch point in [50]. It is a toric singularity. Indeed it admits an effective action by \(T=(\mathbb {C}^*)^3\) given by

$$\begin{aligned} (t_1, t_2, t_3)\circ (z_1, \dots , z_4)=(t_1 z_1, t_2 z_2, t_3 z_3, t_1 t_2 t_3^{-2} z_4). \end{aligned}$$

It is easy to see that the polyhedral cone \(\sigma\) and its dual (moment cone) \(\sigma ^{\vee }\) are given by

$$\begin{aligned} \sigma=\, & {} {\text{Span}}\left\{ \left( \begin{array}{c} 1\\ 0\\ 0 \end{array}\right) , \left( \begin{array}{c} 0\\ 1\\ 0 \end{array}\right) , \left( \begin{array}{c} 2\\ 0\\ 1 \end{array}\right) , \left( \begin{array}{c} 0\\ 2\\ 1 \end{array}\right) \right\} ;\\ \sigma ^{\vee }=\, & {} {\text{Span}}\left\{ \left( \begin{array}{c} 1\\ 0\\ 0 \end{array} \right) ,\left( \begin{array}{c} 0\\ 1\\ 0 \end{array}\right) , \left( \begin{array}{c} 0\\ 0\\ 1 \end{array}\right) , \left( \begin{array}{c} 1\\ 1\\ -2 \end{array}\right) \right\} . \end{aligned}$$

Moreover, it was known that there exists a Sasaki–Einstein metric on \(X_0\) and its Reeb vector field can be calculated explicitly (see [50]). Here we can calculate the Reeb vector field using the above combinatorial data. \(J(r\partial _r)=2 {\text{Im}}(\xi _0)\) where the holomorphic vector field \(\xi _0\) corresponds to an element \(\xi _0\in {\mathfrak {t}}^{+}_\mathbb {R}\) which satisfies two conditions: (1) \(A_X(\xi _0)=3\), (2) \(\xi _0\) minimizes \({\widehat{{\text{vol}}}}(\xi )\) among all \(\xi \in {\mathfrak {t}}^{+}_\mathbb {R}\). Notice that \(X_0\) is a Gorenstein singularity and \(A(\xi )=\langle u_0, \xi \rangle\) with \(u_0=(1,1,-1)\). By using the \(\mathbb {Z}_2\) symmetry of the cones, it is elementary to get the unique minimizer

$$\begin{aligned} \xi _0=\left( \frac{3+\sqrt{3}}{2}, \frac{3+\sqrt{3}}{2}, \sqrt{3}\right) . \end{aligned}$$
(85)

Now the weight corresponding to \(\xi _0\) on the \((z_1, \dots , z_4)\) is equal to

$$\begin{aligned} \left( \frac{3+\sqrt{3}}{2}, \frac{3+\sqrt{3}}{2}, \sqrt{3}, 3-\sqrt{3} \right) =\frac{3+\sqrt{3}}{2}\left( 1, 1, \sqrt{3}-1, 4-2\sqrt{3} \right) =\frac{3+\sqrt{3}}{2}w_0. \end{aligned}$$
(86)

So \(w_0\) is indeed a global minimizer of \({\widehat{{\text{vol}}}}_{X,x}\).

Example 4.2

Consider the four-dimensional \(D_{k+1}\) singularity for \(k\ge 4\):

$$\begin{aligned} X=\left\{ z_1z_2+z_3^2+z_4^2z_5+z_5^k=0\right\} . \end{aligned}$$

The candidate minimizing valuation calculated in [38] is induced by the following weight:

$$\begin{aligned} w_0=\left( 1,1, \frac{-3+\sqrt{33}}{4}, \frac{7-\sqrt{33}}{2}\right) . \end{aligned}$$

The weight \(w_0\) degenerates X to the non-isolated singularity:

$$\begin{aligned} X_0=\left\{ z_1 z_2+z_3^2+z_4^2z_5=0\right\} . \end{aligned}$$

We observe that \(X_0\) is a T-variety of complexity one. \(T=(\mathbb {C}^*)^3\) acts by

$$\begin{aligned} t\cdot z=(t_1 z_1, t_1^{-1} t_2^2 z_2, t_2 z_3, t_3 z_4, t_2^2 t_3^{-2} z_5). \end{aligned}$$

We want to show that \((X_0, \xi _0)\) is K-semistable by using the theory of T-varieties as has been used in [16] which is based on the study of T-equivariant special test configurations in [30]. Notice that because we have been studying the question purely algebraically, we can indeed deal with K-semistability of general (non-isolated) klt singularities like \(X_0\).

Using the process in [1, Section 11], we can write down the polyhedral divisor determining \(X_0\). First we write down the polyhedral divisor for \(\mathbb {C}^5\) as the T-variety. Following [1], for the above T-action, we have the exact sequence:

$$\begin{aligned} 0\longrightarrow N_1:=\mathbb {Z}^3{\mathop {\longrightarrow }\limits ^{F}} N_2:=\mathbb {Z}^5 {\mathop {\longrightarrow }\limits ^{P}} N_3:=\mathbb {Z}^2\longrightarrow 0, \end{aligned}$$
(87)

where F and P are given by the following matrices:

$$\begin{aligned} F=\left( \begin{array}{ccc} 1&{}0&{}0\\ -1&{}2&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1\\ 0&{}2&{}-2 \end{array} \right) , \quad P= \left( \begin{array}{ccccc} -1&{}-1&{}0&{}2&{}1\\ -1&{}-1&{}2&{}0&{}0 \end{array} \right) \end{aligned}$$
(88)

We then find \(s: N_2\rightarrow N_1\) satisfying \(s\circ F={\text{id}}_{N_1}\). s can be chosen simply to be

$$\begin{aligned} s=\left( \begin{array}{ccccc} 1&{}0&{}0&{}0&{}0\\ 0&{}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}1&{}0 \end{array} \right) . \end{aligned}$$
(89)

The generic fiber of \(\widetilde{\mathbb {C}^5}\rightarrow Y_{\text{toric}}\) is the toric variety associated with the following cone:

$$\begin{aligned} \sigma=\, & {} s\left( {\mathbb {Q}}^5_{\ge 0}\cap F({\mathbb {Q}}^3)\right) =\left\{ x\ge 0, y\ge 0, z\ge 0, -x+2y\ge 0, y-z\ge 0 \right\} \\=\, & {} {\text{Span}}_{\mathbb {R}_{\ge 0}}\left\{ \left( \begin{array}{c} 0\\ 1\\ 0 \end{array}\right) , \left( \begin{array}{c} 2\\ 1\\ 0 \end{array}\right) , \left( \begin{array}{c} 2\\ 1\\ 1 \end{array}\right) , \left( \begin{array}{c} 0\\ 1\\ 1 \end{array}\right) \right\} . \end{aligned}$$

The dual cone \(\sigma ^{\vee }\) is given by:

$$\begin{aligned} \sigma ^{\vee }=\, & {} {\text{Span}}_{\mathbb {R}_{\ge 0}}\left\{ \left( \begin{array}{c} 1\\ 0\\ 0 \end{array} \right) , \left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right) , \left( \begin{array}{c} -1\\ 2\\ 0 \end{array} \right) , \left( \begin{array}{c} 0\\ 1 \\ -1 \end{array} \right) \right\} \\=\, & {} \left\{ y\ge 0, 2x+y\ge 0, 2x+y+z\ge 0, y+z\ge 0\right\} . \end{aligned}$$

The base of \(Y_{\text{toric}}\) of \(\mathbb {C}^5\) as the T-variety is given by the toric variety associated with the fan cutted out by the column vectors of P. So it is clear that \(Y_{\text{toric}}={\mathbb {P}}^2\). The associated polyhedral divisor, denoted by

$$\begin{aligned} {\mathfrak {D}}=\Delta _{(1,0)}\otimes \{w_0=0\}+\Delta _{(0,1)}\otimes \{w_1=0\}+\Delta _{(-1,-1)}\otimes \{w_2=0\}, \end{aligned}$$
(90)

can be calculated using the recipe from [1]:

$$\begin{aligned} \Delta _{(1,0)}&=s\left( {\mathbb {Q}}^5_{\ge 0} \cap P^{-1}(1,0)\right) =\{x\ge 0, y\ge 0, z\ge 0, -x+2y\ge 0, 2y-2z+1\ge 0\}\\&=\{(0,0,t); 0\le t\le 1/2\}+\sigma =: \Delta _0,\\ \Delta _{(0,1)}&=s\left( {\mathbb {Q}}^5_{\ge 0} \cap P^{-1}(0,1)\right) =\{x\ge 0, y\ge 0, z\ge 0, -x+2y-1\ge 0, 2y-2z-1\ge 0\}\\&=\{(0,t,0); 0\le t\le 1/2\}+\sigma =: \Delta _1,\\ \Delta _{(-1,-1)}&=s\left( {\mathbb {Q}}^5_{\ge 0}\cap P^{-1}(-1,-1)\right) =\{x\ge 0, y\ge 0, z\ge 0, -x+2y+1\ge 0, 2y-2z\ge 0\} \\&=\{(t,0,0); 0\le t\le 1\}+\sigma =: \Delta _2. \end{aligned}$$

Notice that \(\Delta _0\) and \(\Delta _1\) are non-integral while \(\Delta _2\) is integral. Now the base Y of X is the normalization of the closure of image of \(X\cap (\mathbb {C}^*)^5\) in \(Y_{\text{toric}}\). The map \((\mathbb {C}^*)^5\rightarrow (\mathbb {C}^*)^2\) is induced by the ring homomorphism \(\mathbb {C}[N_3^\vee ]\rightarrow \mathbb {C}[N_2^\vee ]\) and hence is given under the coordinate by

$$\begin{aligned} (\mathbb {C}^*)^5\rightarrow (\mathbb {C}^*)^2,\quad (z_1, z_2, z_3, z_4, z_5)=\left( \frac{z_4^2 z_5}{z_1z_2}, \frac{z_3^2}{z_1z_2}\right) . \end{aligned}$$

So Y is given by

$$\begin{aligned} Y=\{w_0+w_1+w_2=0\}\cong {\mathbb {P}}^1. \end{aligned}$$

We can restrict \({\mathfrak {D}}_{\text{toric}}\) to Y and thus obtain a proper polyhedral divisor for the T-variety X:

$$\begin{aligned} {\mathfrak {D}}=\Delta _0\otimes \{0\}+\Delta _1\otimes \{1\}+\Delta _\infty \otimes \{\infty \}. \end{aligned}$$

By the argument in [30], one knows that normal test configurations are determined by a triple (qvm) where \(q\in {\mathbb {P}}^1\), v is a vertex of \(\sigma \cap (N_1)_{\mathbb {Q}}\) and \(m\in \mathbb {Z}\) and they need to satisfy the following admissible condition ([30, Definition 3.8]): for all \(u\in \sigma ^\vee \cap N_1^\vee\), there is at most one \(p\in {\mathbb {P}}^1\) with \(p\ne q\) such that the function

$$\begin{aligned} \Delta _p(u)=\min _{v\in \Delta _p} \langle u, v\rangle \end{aligned}$$

is non-integral. In the current example, if we choose \(u=(0,1,-1)\in \sigma ^\vee \cap N_1^\vee\) then

$$\begin{aligned} \Delta _0(u)=-\frac{1}{2}, \quad \Delta _1(u)=\frac{1}{2}. \end{aligned}$$

So to get a normal test configuration, by the admissibility condition we are forced to choose either \(q=0\) or \(q=1\). On the other hand, the data (vm) only changes the action and does not change the total space of the test configuration. We can now easily guess the special test configurations whose special fibers are given by

$$\begin{aligned}&X'_0=\{z_1z_2+z_3^2=0\}=\mathbb {C}^2/\mathbb {Z}_2\times \mathbb {C}^2;\\&X''_0=\{z_1z_2+z_4^2z_5=0\}=\hat{X}^3 \times \mathbb {C}. \end{aligned}$$

Here \(\hat{X}^3\) is the three-dimensional suspended pinch point that appeared in (84). One can verify by the same calculation in [38] or [16] that these two special test configurations have positive Futaki invariants. So we conclude that \((X_0, \xi _0)\) is K-semistable and hence \(v_0\) induced by \(w_0\) is indeed a global minimizer of \({\widehat{{\text{vol}}}}\).

Theorem 4.3

For any \((n+1)\)-dimensional \(D_{k+1}\) singularity, except for four-dimensional \(D_4\) singularity, we know its unique quasi-monomial minimizer.

In fact, combining the above examples with the calculations in [38] and the arguments in [47], we have the following almost complete picture:

  1. 1.

    \(n+1=2\), then \(X\cong \{z_1^2+z_2^2z_3+z_3^k=0\}=\mathbb {C}^2/D_{k+1}\) where \(D_{k+1}\) is the \((k+1)\)-th binary dihedral group. By [42], the valuation \(v_0\) induced by the weight \(\left( 1,1-\frac{1}{k} , \frac{2}{k}\right)\) is a global minimizer of \({\widehat{{\text{vol}}}}\).

  2. 2.

    \(n+1=3\), \(k=3\). \(X=\{z_1z_2+z_3^2z_4+z_4^3=0\}\) is a T-variety of complexity one with an isolated singularity. By [16], X admits a quasi-regular Ricci flat Kähler cone metric whose Reeb vector field up to rescaling is associated with the natural weight (1, 1, 2/3, 2/3).

  3. 3.

    \(n+1=4\), \(k=3\). In this case, we expect that X admits a quasi-regular Ricci-flat Kähler cone metric whose Reeb vector field is associated with the natural weight (1, 1, 1, 2/3, 2/3).

  4. 4.

    \(n+1=5\), \(k=3\). \(X=\{z_1z_2+z_3^2+z_4^2+z_5^2z_6+z_6^3=0\}\). The minimizer \(v_0\) is induced by the weight \(w_0=(1,1,1,1,2/3,2/3)\) which preserves X. X is strictly semistable because it specially degenerates to \(X'=\{z_1z_2+z_3^2+z_4^2=0\}\cong A^3_1\times \mathbb {C}^2\) with zero Futaki invariant.

  5. 5.

    \(n+1=3\) or 4, and \(k\ge 4\). These are the examples considered above. The minimizers found are quasi-monomial valuations of rational rank 3.

  6. 6.

    \(n+1=5\) and \(k\ge 4\). The minimizer \(v_0\) is induced by the weight \(w_0=(1,1,1,2/3,2/3)\). \(w_0\) specially degenerates X to \(X_0=\{z_1z_2+z_3^2+z_4^2+z_5^2z_6=0\}\) which is strictly semistable since \(X_0\) further degenerates to \(X'=\{z_1 z_2+z_3^2+z_4^2=0\}\cong A^3_1\times \mathbb {C}^2\) with zero Futaki invariant.

  7. 7.

    \(n+1\ge 6\) and \(k\ge 3\). The minimizer \(v_0\) is induced by \(w_0=\left( 1, \ldots , 1, \frac{n-2}{n-1}, \frac{n-2}{n-3}\right)\). \(w_0\) degenerates X to \(\{z_1^2+\cdots +z_n^2=0\}=A^{n-1}_1\times \mathbb {C}^2\).

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Li, C., Xu, C. Stability of Valuations: Higher Rational Rank. Peking Math J 1, 1–79 (2018). https://doi.org/10.1007/s42543-018-0001-7

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Keywords

  • Quasi-monomial valuation
  • Normalized volume
  • K-stability
  • Metric tangent cone