Stability of Valuations: Higher Rational Rank
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Abstract
Given a klt singularity \(x\in (X, D)\), we show that a quasimonomial 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 Ksemistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasimonomial 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 Ksemistable 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.
Keywords
Quasimonomial valuation Normalized volume Kstability Metric tangent cone1 Introduction
Throughout this paper, we work over the field \(\mathbb {C}\) of complex numbers. In [38], the normalized volume function was defined for any klt singularity \(o\in (X,D)\) and the question of studying the geometry of its minimizer was proposed. See [8, 40, 41, 42, 47] for the results obtained recently. In particular, the existence of a minimizer of the normalized volume conjectured in [40] was confirmed in [8]. On the other hand, in [47], we intensively studied the case when the minimizer is a divisorial valuation. In the current paper, we want to investigate the case when the minimizing valuations are quasimonomial of rational rank possibly greater than one. We note that this kind of cases do occur (see, e.g. [8] or the Appendix to the current paper) and it was conjectured in [38] that any minimizer is quasimonomial.
1.1 The Strategy of Studying a Minimizer
After the existence being settled in [8], the remaining work of the theoretic study of a minimizer of the normalized volume function is to understand its geometry (see the conjectures in [38, 47]). Our method does not say much about the part of the conjecture saying that the minimizer must be quasimonomial. Therefore, in the following we will always just assume that the minimizer is quasimonomial. Partly inspired by the differential geometrical theory on the metric tangent cone, the strategy of understanding it consists of two steps, for which we apply quite different techniques:
In the first step, we deal with a special case of a log Fano cone singularities. Such a singularity has a good torus action and a valuation induced by an element \(\xi \in {\mathfrak {t}}_{\mathbb {R}}^+\). This is indeed the case that has been studied in Sasakian–Einstein geometry on the link of a cone singularity (see, e.g. [16, 51] etc.). In particular, when an isolated Fano cone singularity \((X_0, \xi _0)\) admits a Ricciflat Kähler cone metric, it was shown in [51] that the normalized volume achieves its minimum at the associated Reeb vector field \(\xi _0\) among all \(\xi \in {\mathfrak {t}}^{+}_{\mathbb {R}}\). Here we work on the algebraic side and only assume Ksemistability instead of the existence of a Ricciflat Kähler cone metric. We also improve this result by removing the isolated condition on the singularity and more importantly showing that \(\xi\) is indeed the only minimizer among all quasimonomial valuations centered at o, which form a much more complicated space than just the Reeb cone. In the rank 1 case, namely the case of a cone over a Fano variety, this question was investigated in [40, 42] which used arguments from [24]. Here to treat the higher rank case, we work along a somewhat different approach using more ingredients from the convex geometry inspired by a circle of ideas from the Newton–Okounkov body construction (see, e.g. [37, 54]) (see Sect. 2.2).
In the second step, given a quasimonomial valuation in \({\text{Val}}_{X,x}\) which minimizes the normalized volume function \({\widehat{\text{vol}}}_{(X,D)}\), we aim to obtain a degeneration from the klt singularity \(o\in (X,D)\) to a log Fano cone singularity \((X_0,D_0, \xi )\), and then we can study this degeneration family to deduce results for general klt singularities from the results for log Fano cone singularities. Our study of the degeneration heavily relies on recent developments in the minimal model program (MMP) based on [7] (cf. e.g. [46, 69] etc.). In the rank 1 case, i.e., when the minimizing valuation is divisorial, we showed in [47] that the valuation yields a Kollár component (see also [8]). With such a Kollár component, we can indeed complete the picture. For the case that the quasimonomial minimizing valuation has a higher rational rank, the birational models we construct should be considered as asymptotic approximations. In particular, unlike the rank 1 case, we cannot conclude the conjectural finite generation of the associated graded ring. Thus we have to post it as an assumption. Nevertheless, once we have the finite generation, then we can degenerate both the approximating models and the sequences of ideals to establish a process of passing the results obtained in the Step 1 for the log Fano cone singularity in the central fiber to prove results for the general fiber (see Sect. 2.3).
1.2 Geometry of Minimizers
Now we give a precise statement of our results. Assume \(x\in X\) is a singularity defined by the local ring \((R, \mathfrak {m})\). Denote by \({\text{Val}}_{X,x}\) the set of real valuations on R that are centered at \(\mathfrak {m}\). For any \(v\in {\text{Val}}_{X,x}\), we denote by \({\text{gr}}_v(R)\) the associated graded ring. First we prove the following result, which partially generalizes [47, Theorem 1.2] to the higher rank case. We refer to Sect. 2.1 for other notations used in the statements.
Theorem 1.1
 1.There is a natural divisor \(D_0\) defined as the degeneration of D such thatis a klt singularity;$$\begin{aligned} \big (X_0:={\text{Spec}}\big ({\text{gr}}_v(R)\big ), D_0\big ) \end{aligned}$$
 2.
v is a Ksemistable valuation;
 3.
Let\(v'\)be another quasimonomial valuation in\({\text{Val}}_{X,x}\)that minimizes\({\widehat{{\text{vol}}}}_{(X,D)}\). Then\(v'\)is a rescaling ofv.
For the definition of Ksemistable valuations, see Definition 2.63. The definition uses the notion of Ksemistability of log Fano cone singularities (see [15]), which in turn generalizes the original Ksemistability introduced by Tian [65] and Donaldson [19]. In fact, this leads to a natural refinement of [38, Conjecture 6.1].
Conjecture 1.2
We shall also prove the following converse to Theorem 1.1.2, which was known in the rank 1 case by [47, Theorem 1.2].
Theorem 1.3
If (X, D) admits aKsemistable valuationvovero, thenvminimizes\({\widehat{{\text{vol}}}}_{(X,D)}\). In particular, if\((X,D,\xi )\)is a Ksemistable Fano cone singularity, then\({\text{wt}}_{\xi }\)is a minimizer of\({\widehat{{\text{vol}}}}_{(X,D)}\).
1.3 Applications to Singularities on GH Limits
As proposed in [38], one main application of our work is to study a singularity \(x\in X\) appearing on any Gromov–Hausdorff limit (GH) of Kähler–Einstein Fano manifolds. By the work of Donaldson–Sun and Tian [21, 64, 66], we know that \(M_\infty\) is homeomorphic to a normal algebraic variety. Donaldson and Sun [21] also proved that \(M_\infty\) has at worst klt singularities. For any point on the Gromov–Hausdorff limit, we can consider the metric tangent cone C using the limiting metric (see [12, 13]). In [22], Donaldson and Sun described C as a degeneration of a cone W and they conjectured that both C and W only depend on the algebraic structure of the singularity. Intuitively, W and C should be considered, respectively, as a ‘canonical’ Ksemistable and a Kpolystable degeneration of \(o\in M_{\infty }\). We refer to Sect. 3.1 for more details of this conjecture.
Here we want to verify the semistable part of the conjecture, namely we will show that the valuation used to construct W in [22] is Ksemistable, and since such a valuation is unique by Theorem 1.1, it does not depend on what kind of metric it carries but only the algebraic structure. We note that all the assumptions in Theorem 1.1 are automatically satisfied in this situation, which essentially follows from the work of [22].
Theorem 1.4
(See [22, Conjecture 3.22]) Denote by\({\text{Spec}}(R)\)the germ of a singularityoon a Gromov–Hausdorff limit\(M_{\infty }\)of a sequence of Kähler–Einstein Fano manifolds. Using the notation in [22] (see Sect. 3.1.1), the coneWassociated with the metric tangent cone is isomorphic to\({\text{Spec}}({\text{gr}}_v(R))\)wherevis the unique Ksemistable valuation in\({\text{Val}}_{M_{\infty },o}\). In particular, it is uniquely determined by the algebraic structure of the singularity.
A standard point of the proof of Theorem 1.4 is to deduce the stability from the metric. For now, we only need the following statement, which generalize [15] to the case of nonisolated singularities (see Remark 3.6).
Theorem 1.5
(= Theorem 3.5) If\((X, \xi _0)\)admits a Ricciflat Kähler cone metric, then\(A_X(\xi _0)=n\)and\((X, \xi _0)\)is Ksemistable.
Remark 1.6
In a forthcoming work [45], we plan to complete the proof of [22, Conjecture 3.22] by showing that the metric tangent cone C is also uniquely determined by the algebraic structure of the germ \((o\in M_{\infty })\). In fact, after Theorem 1.4, what remains to show is that C only depends on the algebraic structure of the Ksemistable Fano cone singularity \((W,\xi _v)\). It follows from a combination of two wellexpected speculations. The first one is an improvement of Theorem 1.5 which says that a Fano cone C with a Ricci flat Kähler cone metric is indeed Kpolystable. This was solved in [4] for the quasiregular case. In [45], we plan to extend [4] to the irregular case. The second one is that any Ksemistable log Fano cone has a unique Kpolystable log Fano cone degeneration. In the case of smoothable Fano varieties (a regular case), this was proved independently in [44, 60] using analytic tools. In [45], we will use the algebraic tools developed in [40, 47] and the current paper, especially the ones related to Tequivariant degeneration, to investigate the problem again and give a purely algebrogeometric treatment.
As a consequence of our work, we also obtain the following formula for singularities appearing on the Gromov–Hausdorff limits (GH) of Kähler–Einstein Fano manifolds, which sharpens [59, Proposition 3.10] (see Corollary 3.7) as well as partially [48, Theorem 1.3.4]:
Theorem 1.7
In fact, for a quasiétale finite covering \((Y,y)\rightarrow (X,x)\) between any klt singularities, as we expect the minimizer of \(y\in Y\) is unique and thus Ginvariant, such a formula should also hold. However, for now we cannot prove this in the full generality.
1.4 Outline of the Paper
In this section, we give an outline as well as the organization of the paper. The paper is divided into two parts. In Part I, we study the geometry of the minimizer of a klt singularity in general. We note that this part is completely algebraic.
In Sect. 2.1, we recall a few concepts and establish some background results, especially on valuations and Tvarieties (i.e. varieties with a torus action).
In Sect. 2.2, we focus on studying log Fano cone singularity (see Definition 2.23) and put it into our framework as mentioned in Sect. 1.1. In particular, we want to show that a Ksemistable log Fano cone singularity does not only minimize the normalized volumes among the valuations in the Reeb cone, but indeed also among all valuations in \({\text{Val}}_{X,x}\). Furthermore, it is unique among all quasimonomial valuations. Our main approach is to use the ideas from the construction of Newton–Okounkov body to reduce the volume of valuations to the volume of convex bodies and then apply the known convexity of the volume function in such setting. This is obtained by three steps with increasing generality: we first consider toric singularities with toric valuations (see Sect. 2.2.2.1) where we set up the convex geometry problem; then general Tsingularities with toric valuations (see Sect. 2.2.2.2); and eventually Tsingularities with Tinvariant valuations (see Sect. 2.2.2.3).
In Sect. 2.3, we investigate a quasimonomial minimizer of a general klt singularities by intensively using the minimal model program and degeneration techniques. First in Sect. 2.3.1, we show that given a quasimonomial minimizer \(v\in {\text{Val}}_{X}\) of \({\widehat{{\text{vol}}}}_{(X,D)}\), one can find birational models which can be considered to approximate the valuation. This construction will be used later if we assume that the degeneration exists, i.e., the associated graded ring is finitely generated. In particular, we conclude that such a degeneration is also klt. Then in Sect. 2.3.2, we show that any quasimonomial minimizer is Ksemistable, and in Sect. 2.3.3, we use the degeneration technique and the uniqueness of the quasimonomial minimizer for a log Fano cone singularity obtained in Sect. 2.2 to conclude the uniqueness of the quasimonomial minimizer for a general singularity if one minimizer has a finitely generated associated graded ring.
In Part II (= Sect. 3.1), we apply our theory to the singularities that appeared on the Gromov–Hausdorff limit \(M_{\infty }\) of Kähler–Einstein Fano manifolds. Obviously, we need to establish the results that connect the previous existing differential geometry work to our algebraic results in Part I. Our aim is to show that the semistable cone W associated with the metric tangent cone in Donaldson–Sun’s work depends only on the algebraic structure of the singularity. It follows from the work in [22] such a cone is induced by a valuation v and it always carries an (almost) Sasakian–Einstein metric. What remains to show is then such a cone is Ksemistable, i.e., the valuation v is a Ksemistable quasimonomial valuation. This is achieved in Sect. 3.1.2. Then the finite degree multiplication formula is deduced in Sect. 3.1.3.
In Appendix A, we illustrate our discussion on ndimensional \(D_{k+1}\) singularities. In particular, we verify that all the candidates calculated out in [38], including all those irregular ones, are indeed minimizers of \({\widehat{{\text{vol}}}}\) (except possibly for \(D_4\) in dimension 4).
2 Part I: Geometry of Minimizers
2.1 Preliminary and Background Results
Notation and conventions: We follow [35, 36] for the standard conventions in birational geometry. For a log pair (X, D), we also use \(a_l(E;\,X,D)\) to mean the log discrepancy of E with respect to (X, D), i.e., \(a_l(E;\,X,D)=a(E;\,X,D)+1\). Similarly, we also define \(a_l(E;\, X,D+c\cdot \mathfrak {a})\) for an ideal \(\mathfrak {a}\subset \mathcal {O}_X\) and \(c\ge 0\).
In this note, any singularity \(x\in X\) means the germ of an algebraic singularity, i.e., \(X={\text{Spec}}(R)\) where R is essentially of finite type over \(\mathbb {C}\). Let \(x\in (X,D)\) be a klt singularity. We call a divisorial valuation \({\text{ord}}_S\) in \({\text{Val}}_{X,x}\) gives a Kollár component if there is a model \(\mu :Y\rightarrow X\) isomorphic over \(X{\setminus } \{x\}\) with the exceptional divisor given by S such that \((Y,\mu _*^{1}D+S)\) is plt and \(K_Y\mu _*^{1}DS\) is ample over X. We denote by \({\text{Kol}}_{X,D,x}\) the set of Kollár components over \(x\in (X,D)\).
2.1.1 Normalized Volumes
In this section, we summarize some known results of the normalized volumes.
Definition 2.1
We often abbreviate it as \({\widehat{{\text{vol}}}}(X,D)\) if x is clear or \({\widehat{{\text{vol}}}}(X,x)\) if \(D=0\). We have the following description of \({\widehat{{\text{vol}}}}(X,D)\) using the (normalized) multiplicities:
Theorem 2.2
In [47], we showed that the right hand of (1) is obtained by a minimizer \(\mathfrak {a}\) if and only if the minimum is calculated by the valuative ideals of a Kollár component. In general, [8] showed that if we replace an ideal \(\mathfrak {a}\) by a graded ideal sequence \(\{\mathfrak {a}_{\bullet }\}\), the minimum can always be obtained. Then we can easily show that a valuation v that computes the log canonical threshold of such a graded ideal sequence \(\{\mathfrak {a}_{\bullet }\}\) satisfies the identity \({\widehat{{\text{vol}}}}_{(X,D)}(v)={\widehat{{\text{vol}}}}(X,D)\).
Theorem 2.3
Remark 2.4
In [32], it was conjectured that any valuation computing the log canonical threshold of a graded sequence of ideals at a smooth point is quasimonomial (see [32, Conjecture B]). We can naturally extend this conjecture from a smooth point to a klt pair (X, D) and then have the following fact observed in [8]: the strong version of Conjecture B for klt pair (X, D) in [32] implies that any minimizer v of \({\widehat{{\text{vol}}}}_{(X,D)}\) is quasimonomial.
Another characterization of the normalized volume is by using the volumes of models:
Theorem 2.5
2.1.2 Approximation
For the latter, we need the following result:
Lemma 2.6
Proof
We denote the norm \(\cdot \) on \(\mathbb {R}^r\) to be \(x=\max _{1\le i\le r}x_i\).
Lemma 2.7
 1.
\(q_{i}v_i\in \mathbb {Z}^r\),
 2.
v is in the convex cone generated by \(v_1\),…, \(v_r\), i.e., \(v=\sum a_iv_i\) for some\(a_i>0\); and
 3.
\(v_iv<\frac{\epsilon }{q_i}\).
Proof
After relabelling, we can assume that \(v=(\alpha _1,\ldots ,\alpha _r)\) with \((1, \alpha _1,\ldots ,\alpha _j)\) is linearly independent and span the space \({\text{span}} (1, \alpha _1,\ldots ,\alpha _r)\). Then possibly replacing \(\epsilon\) by a smaller one, once we could approximate \(\alpha _1\),…, \(\alpha _j\), we automatically get the approximation of all \(\alpha _i\)\((1\le i \le r)\). Therefore, we may and will assume that \((1, \alpha _1,\ldots ,\alpha _r)\) is linearly independent.
Applying Lemma 2.6 for all \(2^r\) choices of \(\delta _1,\ldots ,\delta _r\), we find \(v_1\),…, \(v_{2^r}\) vectors, it suffices to show that we can choose r vectors out of them so that the condition (2) is satisfied.
2.1.3 Valuations and Associated Graded Rings
The valuations that our approach can deal with are called quasimonomial valuations. It is known that it is the same as Abhyankar valuation (see, e.g. [23, Proposition 2.8]).
Definition 2.8
 1.
\(\bigcap ^r_{i=1} E_i\ne \emptyset\);
 2.
there exists a component \(C \subset \bigcap E_{i}\), such that around the generic point \(\eta\) of C, \(E_i\) is given by the equation \(z_i\), and
 3.for \(f\in \mathcal {O}_{X,\eta }\subset \hat{\mathcal {O}}_{X,\eta }\) can be written as \(f=\sum c_{\beta }z^{\beta }\), with either \(c_{\beta }=0\) or \(c_{\beta }(\eta )\) is a unit; then$$\begin{aligned} v_{\alpha }(f)=\min \left\{ \sum \alpha _i\beta _i\ c_{\beta }(\eta )\ne 0\right\} . \end{aligned}$$
Definition 2.9
 1.
\(\overline{\eta }\) is a component of the intersection of component of E;
 2.
\(v=v_{\alpha }\) for some \(\alpha \in \mathbb {R}^{r}_{>0}\).
Lemma 2.10
Let\(v_{\alpha }\) be a quasimonomial valuation as defined in Definition 2.8with the maximal rational rank r, whose associated graded ring\({\text{gr}}_{v_{\alpha }}(R)\) is finitely generated. Then we can choose\(\epsilon\) sufficiently small such that for any\(\alpha '\in \mathbb {Q}^r\) with\(\alpha \alpha '<\epsilon\), there is an isomorphism\({\text{gr}}_{v_{\alpha }}(R)\rightarrow {\text{gr}}_{v_{\alpha '}}(R)\) induced by a morphism sending a set of homogeneous generators of\({\text{gr}}_{v_{\alpha }}(R)\) to one of\({\text{gr}}_{v_{\alpha '}}(R)\).
Proof
Let \(h_1,\ldots ,h_m\) be elements in I whose degenerations \(h_1',\ldots , h_m'\) generate \(I_0\). Assume \(h_j=h'_j+h_j''\) with \(\deg (h'_j)=\sum ^r_{i=1} c^j_i\alpha _i\) and the monomials of \(h_j''\) has degree larger than \(\sum ^r_{i=1} c^j_i\alpha _i\). We can choose \(\epsilon\) sufficiently small such that if \(\alpha '=(\alpha '_1,\ldots , \alpha '_r)\) satisfies \(\alpha '\alpha <\epsilon\), then any monomial of \(h_j''(t_1,\ldots ,t_k)\) has a corresponding degree larger than \(c^j_i\alpha '_i\) where \(\deg (t_i)\) is set to be \(\sum ^r_{i=1}b^j_{i}\alpha '_i\). Then the condition that v has the maximal rational rank implies that \(h_j'\) is the leading term of \(h_j\) if \(\deg (t_i)=\sum ^r_{i=1}b^j_{i}\alpha '_i\).
Consider the filtration given by setting the degree of \(t_j\) to be \(\sum _i b^j_i\alpha '_i\) and it induces a filtration on R by its image. We denote the corresponding vector of the degeneration by \(\xi _{\alpha '}\). Our argument above says for the filtration induced by \(\xi _{\alpha '}\), the associated graded ring given by the filtration coincides with \({\text{gr}}_v(R)\). Since the graded ring is an integral domain, by Lemma 2.11(1) the filtration comes from a valuation \(v'\). Now we claim \(v'\) is the same as the one given by \(v_{\alpha '}\), which implies what we aim to prove.
 (i)

\({\mathcal {F}}^m\ne 0\) for every \(m\in \Phi ^g\), \({\mathcal {F}}^m=R\) for \(m\le 0\) and \(\bigcap _{m\ge 0}{\mathcal {F}}^m=(0)\);
 (ii)

\({\mathcal {F}}^{m_1}\cdot {\mathcal {F}}^{m_2}\subseteq {\mathcal {F}}^{m_1+m_2}\) for every \(m_1, m_2\in \Phi ^g\).
Lemma 2.11
 (1)
([62, Page 8]) If\({\text{gr}}_{{\mathcal {F}}}R\) is an integral domain, then\(v=v_{{\mathcal {F}}}\) is a valuation centered at\(o\in X\). In particular, \(v(fg)=v(f)+v(g)\) for any\(f,g\in R\).
 (2)
[57] A valuation v is quasimonomial if and only if the Krull dimension of \({\text{gr}}_v R\)is the same as the Krull dimension of R.
2.1.4 Singularities with Good Torus Actions
For general results of Tvarieties in algebraic geometry, see [1, 2, 43, 56]. Assume \(X={\text{Spec}}_{\mathbb {C}}(R)\) is an affine variety with \({\mathbb {Q}}\)Gorenstein klt singularities. Denote by T the complex torus \((\mathbb {C}^*)^r\). Assume X admits a good Taction in the following sense;
Definition 2.12
(See [43]) Let X be a normal affine variety. We say that a Taction on X is good if it is effective and there is a unique closed point \(x\in X\) that is in the orbit closure of any Torbit. We shall call x the vertex point of the Tvariety X.
For a singularity \(x\in X\) (sometimes also denote by \(o\in X\)) with a good Taction, we will also call it a Tsingularity for simplicity.
For the convenience and by comparison with Sasaki geometry, we will introduce:
Definition 2.13
With the above notations, a vector \(\xi \in {\mathfrak {t}}^+_\mathbb {R}\) will be called a Reeb vector on the Tvariety X.
We recall the following structure results for any Tvarieties:
Theorem 2.14
Here a variety Y being semiprojective variety means it is projective over an affine variety Z, which can be chosen to be \(Z={\text{Spec}}(H^0(Y,\mathcal {O}_Y))\).
Theorem 2.15
 1.For anyTinvariant quasimonomial valuation v, there exist a quasimonomial valuation\(v^{(0)}\)overYand\(\xi \in M_{\mathbb {R}}\)such that for any\(f\cdot \chi ^u\in R_u\), we have:We will use\((v^{(0)}, \xi )\)to denote such a valuation.$$\begin{aligned} v(f\cdot \chi ^u)=v^{(0)}(f)+\langle u, \xi \rangle . \end{aligned}$$
 2.
Tinvariant prime divisors onXare either vertical or horizontal. Any vertical divisor is determined by a divisorZonYand a vertexv of \({\mathfrak {D}}_Z\), and will be denoted by\(D_{(Z,v)}\). Any horizontal divisor is determined by a ray\(\rho\)of\(\sigma\)and will be denoted by\(E_\rho\).
 3.LetDbe aTinvariant vertical effective\(\mathbb {Q}\)divisor. If\(K_X+D\) is \({\mathbb {Q}}\)Cartier, then the log canonical divisor has a representation\(K_X+D=\pi ^*H+{\text{div}}(\chi ^{u_0})\)where\(H=\sum _Z a_Z\cdot Z\)is a principal\({\mathbb {Q}}\)divisor onYand\(u_0\in M_{{\mathbb {Q}}}\). Moreover, the log discrepancy of the horizontal divisor\(E_\rho\)is given by:where\(n_\rho\)is the primitive vector along the ray\(\rho\).$$\begin{aligned} A_{(X,D)}(E_\rho )=\langle u_0, n_\rho \rangle , \end{aligned}$$(3)
Proof
In the first statement, the case for valuations with rational rank 1 follows from [2, 11]. It can be extended to quasimonomial valuations trivially since any such valuation is a limit of valuation of rational rank 1. The second statement is in [56, Proposition 3.13] and the third statement is in [43, Section 4]. \(\square\)
Lemma 2.16
For any\(\xi \in \mathfrak {t}^+_{\mathbb {R}}\), \({\text{wt}}_\xi\) is a quasimonomial valuation of rational rank equal to the rank of\(\xi\). Moreover, the center of\({\text{wt}}_\xi\) is x.
Proof
Next we can also realize \({\text{wt}}_\xi\) on a log smooth model. Let \(\tilde{Z}\rightarrow Z\) be a fixed toric resolution of singularities. Then we can follow the construction in [43, Section 2] to obtain a toroidal resolution \(\mathscr {X}\rightarrow X\) that dominates \(\tilde{X}\) and its restriction over U is isomorphic to \(\tilde{Z}\times U\). Let \(q\in \tilde{Z}\) be a contracting point of the torus action generated by \(\xi\) and choose a point \(p\in U\). Then it is easy to see that \({\text{wt}}_{\xi }\) is realized as a monomial valuation with nonnegative weights at \((p, q)\in U\times \tilde{Z}\). \(\square\)
Remark 2.17
The quasimonomial property also follows from Lemma 2.11(2).
By the construction in the above proof, the log discrepancy of \({\text{wt}}_{\xi }\) can indeed be calculated in a similar way as in the toric case, and the toric case is wellknown (see, e.g. [3, 9, Proposition 7.2]). Assume X is a normal affine variety with \({\mathbb {Q}}\)Gorenstein klt singularities and a good Taction. Let D be a Tinvariant vertical divisor. As in [51, 2.7], we can solve for a nowherevanishing section Tequivariant section s of \(m(K_X+D)\) where m is sufficiently divisible (also see Remark 2.19).
Lemma 2.18
Proof
Next we discuss the second statement. Because the map \(\xi \mapsto {\mathcal {L}}_{\xi }(s)/s\) is linear, we just need to verify the statement for rational \(\xi\). Then in the case \(D=\emptyset\), this follows from what was already showed in [38] and [40, Proof of Proposition 6.16]. The same argument applies in the logarithmic case. \(\square\)
Remark 2.19
Using the structure theory of Tvarieties and under the assumption that \(K_X+D\) is \({\mathbb {Q}}\)Gorenstein, one can write down a nowherevanishing holomorphic section s explicitly by using [56, Theorem 3.21] and [43, Proposition 4.4]. So one can also directly verify the equality \(\langle u_0, \xi \rangle =\frac{1}{m}({\mathcal {L}}_\xi s/s)\).
As a consequence of the above lemma, we can extend \(A_{(X,D)}(\xi )\) to a linear function on \({\mathfrak {t}}_{\mathbb {R}}\).
Definition 2.20
By Lemma 2.18, \(A_{(X,D)}(\eta )=\frac{1}{m} {\mathcal {L}}_\eta s/s\) where s is a Tequivariant nowherevanishing holomorphic section of \(m(K_X+D)\).
We will need the following important convexity property originally discovered in [51] for cones with isolated singularities (see also [22] for the case of metric tangent cones).
Proposition 2.21
Since we allow any klt singularity with good torus action, this is a gentle generalization of Martelli–Sparks–Yau’s result. We will give an algebraic proof of the above result in Sect. 2.2.2. In particular, we will interpret this as a phenomenon in convex geometry.
For klt Tsingularities, we have the following improvement of Theorem 2.2 in the equivariant case:
Theorem 2.22
Proof
2.1.5 KSemistability of Log Fano Cone Singularity
For a Tequivariant singularity, the valuations induced by vector fields in the Reeb cone play a special role, so we give the following:
Definition 2.23
(See also [16]) Let (X, D) be an affine klt pair with a good T action (see Definition 2.12). For any \(\xi \in {\mathfrak {t}}^+_\mathbb {R}\), we say that the associated valuation \({\text{wt}}_{\xi }\) gives a toric valuation. For a fixed \(\xi\), we call the triple \((X,D,\xi )\) a klt singularity with a log Fano cone structure that is polarized by \(\xi\).
We proceed to study the Ksemistable log Fano cone singularity \((X,D, \xi )\) in the sense of Collins–Székelyhidi [15, 16], which generalizes the Ksemistability for Fano varieties (see [19, 65]). We first define the special test configurations of log Fano cone singularities.
Definition 2.24
(See [15, 16]) Let \((X, D, \xi _0)\) be a log Fano cone singularity and T be the torus generated by \(\xi _0\).
 1.
\(\pi\) is a flat family of log pairs such that the fibres away from 0 are isomorphic to (X, D) and \({\mathcal {X}}={\text{Spec}}({\mathcal {R}})\) is affine, where \({\mathcal {R}}\) is a finitely generate flat \(\mathbb {C}[t]\) algebra. The torus T acts on \({\mathcal {X}}\), and we write \(\mathcal {R}=\bigoplus _{\alpha }\mathcal {R}_{\alpha }\) as the decomposition into weight spaces;
 2.
\(\eta\) is an algebraic holomorphic vector field on \({\mathcal {X}}\) generating a \(\mathbb {C}^*\)action on \(({\mathcal {X}},{\mathcal {D}})\) such that \(\pi\) is \(\mathbb {C}^*\)equivariant where \(\mathbb {C}^*\) acts on the base \(\mathbb {C}\) by multiplication (so that \(\pi _*\eta =t\partial _t\) if t is the affine coordinate on \({\mathbb {A}}^1\)) and there is a \(\mathbb {C}^*\)equivariant isomorphism \(\phi : ({\mathcal {X}}, {\mathcal {D}}) \times\, _{\mathbb {C}}\mathbb {C}^*\cong (X,D) \times \mathbb {C}^*\);
 3.
the algebraic holomorphic vector field \(\xi _0\) on \({\mathcal {X}}\times _{\mathbb {C}}\mathbb {C}^*\) (via the isomorphism \(\phi\)) extends to a holomorphic vector field on \({\mathcal {X}}\) (still denoted by \(\xi _0\)) and generates a Taction on \(({\mathcal {X}}, {\mathcal {D}})\) that commutes with the \(\mathbb {C}^*\)action generated by \(\eta\) and preserves \((X_0, D_0)\);
 4.
\((X_0, D_0)\) has klt singularities and \((X_0, D_0, \xi _0_{X_0})\) is a log Fano cone singularity (see Definition 2.23).
By abuse of notation, we will denote \(\xi _0_{X_0}\) by \(\xi _0\). For simplicity, we will just say that \(({\mathcal {X}}, {\mathcal {D}})\) is a (Tequivariant) special test configuration if \(\xi _0\) and \(\eta\) are clear. We also say that \((X,D,\xi _0)\) specially degenerates to \((X_0,D_0,\xi _0; \eta )\) (or simply to \((X_0, D_0)\)).
If \(({\mathcal {X}}, {\mathcal {D}}, \xi _0;\eta )\) is a special test configuration, then under the base change \({\mathbb {A}}^1\rightarrow {\mathbb {A}}^1, t\mapsto t^d\), we can pull back it to get a new special test configuration \(({\mathcal {X}}\times _{{\mathbb {A}}^1,t^d}{\mathbb {A}}^1, {\mathcal {D}}\times _{{\mathbb {A}}^1, t^d}{\mathbb {A}}^1, \xi _0; d \cdot (t^d)^*(\eta ))\), which will be simply denoted by \(({\mathcal {X}}, {\mathcal {D}}, \xi _0;\eta )\times _{{\mathbb {A}}^1,t^d}{\mathbb {A}}^1\).
Let \(({\mathcal {X}}, {\mathcal {D}}, \xi _0; \eta )\) be a Tequivariant special test configuration of \((X, D, \xi _0)\). We can define the Futaki invariant \({\text{Fut}}(X_0, D_0, \xi _0; \eta )\) following [15, 16] where the index character was used. However, for our purpose, we reformulate the definition as the derivative of the normalized volume and we only consider special test configurations.
Remark 2.25
Definition 2.26
Remark 2.27
When \(\xi _0\) generates a onedimensional torus (i.e., \(T\cong \mathbb {C}^*\)), then taking quotient by T, we get a special test configuration \((\mathcal {Y},{\mathcal {E}})\) of the log Fano pair \((Y,E)=(X,D){\setminus }\{x\}/\langle \xi _0\rangle\), and we have \({\text{Fut}}({\mathcal {X}}, {\mathcal {D}}, \xi _0; \eta )\) is a rescaling of the Futaki invariant of \(({\mathcal {Y}}, {\mathcal {E}})\) defined in [19, 65] (see, e.g. [40, Lemma 6.20]). This also verifies that the definition coincides with the one in [16] (up to a constant) as any vector can be approximated by rational ones, and the Futaki invariants in both definitions are continuous and coincide when \(\xi _0\) is rational.
Definition 2.28
Applying the above discussion, we can then put the study of Ksemistablity of a local singularity in the framework of the minimization of normalized volumes.
Lemma 2.29
Proof
2.2 Normalized Volumes Over Log Fano Cone Singularities
In this section, we will study log Fano cone singularities \((X,D,\xi )\) (see Definition 2.23). In differential geometry, the stability theory in such settings has been investigated in the context of searching for a Sasakian–Einstein metric (see [16, 51], etc.). In particular, we will focus on the case that when the singularity is Ksemistable and show that in this case the natural toric valuation \({\text{wt}}_{\xi }\) is the only minimizer up to rescaling among all Tinvariant quasimonomial valuations. For invoking different tools, we divide the argument in Sect. 2.2.2.1 into three steps with increasing generality: we first consider toric singularities with toric valuations, then general Tsingularities with toric valuations and eventually Tsingularities with Tinvariant valuations.
2.2.1 Special Test Configurations from Kollár Components
In this section, we study the special test configuration of Tvarieties associated with Kollár components.
Definition 2.30
Assume that \(o\in (X,D)\) is a klt singularity with a good Taction and S is a Tequivariant Kollár component. Let \(\mathcal {X}\rightarrow \mathbb {A}^1\) be the associated degeneration which degenerates (X, D) to a \((X_0,D_0)\) and admits a \(T'=T\times \mathbb {C}^*\)action. For any \(f=\sum f_k\in {\mathcal {R}}'\), \({\text{ord}}_S(f)=\min \{k; \,f_k\ne 0\}\). Over \(X_0\), \({\text{ord}}_S\) corresponds to the \(\mathbb {C}^*\)action corresponding to the \(\mathbb {Z}\)grading. Denote the generating vector by \(\xi _S\in {\mathfrak {t}}'^+_\mathbb {R}\).
With the above notations, we say that \(({\mathcal {X}}, {\mathcal {D}}, \xi _0; \,\xi _S)\) is the special test configuration associated with the Kollár component S. If \(\xi _0\) and \(\xi _S\) are clear, we just use \(({\mathcal {X}}, {\mathcal {D}})\) to denote the special test configuration.
Lemma 2.31
 1.
\(A_{(X,D)}({\text{ord}}_S)=A_{(X_0, D_0)}({\text{wt}}_{\xi _S})\) and\({\text{vol}}_{(X,D)}({\text{ord}}_S)={\text{vol}}_{(X_0,D_0)}({\text{wt}}_{\xi _S})\);
 2.
\(A_{(X,D)}({\text{wt}}_{\xi _0})=A_{(X_0,D_0)}({\text{wt}}_{\xi _0})\) and\({\text{vol}}_{(X,D)}({\text{wt}}_{\xi _0})={\text{vol}}_{(X_0,D_0)}({\text{wt}}_{\xi _0})\) .
Proof
For a later purpose, we need a little more:
Proposition 2.32
LetS be a Tinvariant Kollár component and\(({\mathcal {X}}, {\mathcal {D}})\)be the special test configuration associated with S. Then there is a Tequivariant nowherevanishing holomorphic section\(\mathscr {S}\) of\(m(K_{{\mathcal {X}}/{\mathbb {A}}^1}+{\mathcal {D}})\)for m sufficiently divisible.
Proof
Remark 2.33
We can now generalize the minimization result in [40, 42, 47] to the higher rank case.
Theorem 2.34
\((X, D, \xi _0)\)is Ksemistable if and only if\({\text{wt}}_{\xi _0}\)is a minimizer of\({\widehat{{\text{vol}}}}_{(X,D)}\)in\({\text{Val}}_{X,x}\).
Proof
For the converse direction, we assume \({\widehat{{\text{vol}}}}_{(X,D)}\) obtains its minimum at \({\text{wt}}_{\xi _0}\) and let \(({\mathcal {X}}, {\mathcal {D}}, \xi _0; \eta )\) be any special test configuration.
By the construction in the above proof and Theorem 2.22, we also get the following:
Proposition 2.35
To test Ksemistability of a log Fano cone singularity\((X,D,\xi _0)\), we only need to test on the special test configurations associated with Kollár components, i.e., we only need to check for any Tequivariant Kollár component S, the generalized Futaki invariant\({\text{Fut}}(X_0,D_0,\xi _0;\,\eta )\ge 0\), where\((X_0,D_0)\)is the induced special degeneration by S and\(\eta =\xi _S\).
The following purely algebrogeometric statement can be seen as a generalization of a result in [39], which was proved there by an analytic method.
Proposition 2.36
Let\((X,D,\xi _0)\) be a log Fano cone singularity, which specially degenerates to\((X_0,D_0,\xi _0; \eta )\)via a Tequivariant special test configuration. Then\((X,D,\xi _0)\) is Ksemistable if\((X_0,D_0,\xi _0)\) is Ksemistable.
Proof
We use a degeneration argument which is similar to the one used in [8, 47].
2.2.2 Convexity and Uniqueness
The results in this section do not depend on other sections. Here we aim to show the volume function on the Reeb cone is strictly convex and hence conclude the uniqueness of the minimizer when the log Fano cone singularity is Ksemistable. Our main approach is to describe the volume of a valuation as the volume of a convex body and then reduce the question to a known result in convex geometry. This is standard in our first step where we treat toric singularities. Then we generalize it to an arbitrary log Fano cone singularity with only toric valuations, by considering valuations with real rank larger than one. In the last step, we study a Ksemistable log Fano cone singularity and compare its volume with those of Tinvariant quasimonomial valuations.
2.2.2.1 Toric Valuations on Toric Varieties
In this section we will consider the case of toric singularities, where the minimization problem can be reduced to a convex geometric problem as known from [51]. For the latter application, we will consider a more general setting of strictly convex cones. This problem was considered by Gigena [27] in a different context.
Lemma 2.37
 1.
The function\(\xi \mapsto {\text{vol}}(\Delta _\xi )\) is a strictly convex function for\(\xi \in {\text{relint}}(\sigma )\), where\({\text{relint}}(\sigma )=\sigma ^{\circ }\) denotes the relative interior of the cone\(\sigma\).
 2.
\({\text{vol}}_{\hat{H}_0}\) is a strictly convex proper function for\(\xi \in \hat{H}_0=H_{0}\cap \sigma\). As a consequence, there is a unique minimizer\(\xi _0\) of\({\text{vol}}_{\hat{H}_0}\). Moreover, \(\xi _0\in \hat{H}_0=H_0\cap \sigma\) is a minimizer of\({\text{vol}}(\Delta _\xi )_{\hat{H}_0}\) if and only if\(u_0\) is the barycenter of\(\Pi _{\xi _0}\).
For the reader’s convenience, we provide a proof of this result by deriving the volume formula and reducing the minimization to a calculus problem as in [27].
Proof
Since \(H_0\) is affine, \({\widehat{{\text{vol}}}}_{\hat{H}_0}\) is also strict convex function for \(\xi \in \hat{H}_0\cap H_0\cap \sigma\). As \(\xi \rightarrow \partial \hat{H}_0\), \(\Delta _{\xi }\) becomes unbounded, so \({\text{vol}}(\Delta _\xi )\) approaches \(+\infty\). So we see that \({\text{vol}}(\xi )_{\hat{H}_0}\) is a strictly proper function. As a consequence, there exists a unique minimizer of \({\text{vol}}_{\hat{H}_0}\).
Remark 2.38
We notice that there is a similarity of the volume formula in (16) with the formula for the volumes of \(\mathbb {C}^*\)invariant valuations derived in [40]. We will see in Proposition 2.45 that this is not a coincidence. The properness of \({\widehat{{\text{vol}}}}({\text{wt}}_\xi )\) with respect to \(\xi\) also follows from the properness estimate in [38].
2.2.2.2 Toric Valuations on TVarieties
In this section, we treat a general Fano cone singularity with varying toric valuations. By using the Newton–Okounkov body technique, we will show that the volume function on the space of toric valuations associated with elements from the Reeb cone can be interpreted as a volume function of convex bodies as considered in the previous section. By applying Lemma 2.37, we get the strict convexity (see Proposition 2.39). As mentioned before, this is a generalization, from the case of isolated singularities to general klt singularities with good torus actions, of the important convexity property originally discovered in [51]. Unlike their use of analytic tools, our proof is algebraic. Notice that Proposition 2.39 was used in the proof of Theorem 2.34.
Proposition 2.39
The function\({\text{vol}}:\xi \mapsto {\text{vol}}({\text{wt}}_{\xi })\) is a strictly convex function of\(\xi \in {\text{relint}}(\sigma )\). The function\({\text{vol}}_{\hat{H}_0}\) is a strictly convex and proper function of\(\xi \in \hat{H}_0=H_0\cap \sigma\). As a consequence, there exists a unique minimizer of\({\text{vol}}_{\hat{H}_0}\).
To prove Proposition 2.39, we apply the ideas from the theory of Newton–Okounkov body to realize the volumes of \({\text{wt}}_{\xi }\) as volumes of convex bodies, and then apply Lemma 2.37.
Proof
Lemma 2.40
 (P1)
\({\mathcal {S}}\cap \{y\  \ \langle P(y), \xi \rangle =0\}=\{0\}\);
 (P2)
Denote by\(e_i\)the ith standard vector of\(\mathbb {Z}^r\). There exist finitely many vectors \(\left( e_i, v^{(i)}_k\right)\) spanning a semigroup\(B\subset \mathbb {Z}^n\) such that\({\mathcal {S}}\subset B\);
 (P3)
\({\mathcal {S}}\) generates\(\mathbb {Z}^n\) as a group.
Proof
 1.
\({\mathfrak {D}}(u)+{\mathfrak {D}}(u')\le {\mathfrak {D}}(u+u')\) holds for any two elements \(u, u'\in \sigma ^{\vee }\),
 2.
\(u\rightarrow {\mathfrak {D}}(u)\) is piecewisely linear, i.e., there is a quasifan \(\Lambda\) in \(M_{{\mathbb {Q}}}\) having \(\sigma ^{\vee }\) as its support such that \({\mathfrak {D}}\) is linear on the cones of \(\Lambda\),
 3.
\({\mathfrak {D}}(u)\) is always semiample, and \({\mathfrak {D}}(u)\) is big for \(u\in {\text{relint}}(\sigma ^{\vee })\).
Lemma 2.41
Proof
We can complete the proof of Proposition 2.39 by using the similar argument as in the toric case. Indeed, Lemma 2.37.1 and Lemma 2.41 together imply that \({\text{vol}}({\text{wt}}_\xi )\) is a proper strictly convex function of \(\xi \in \hat{H}_0=H_0\cap \sigma\). So there exists a unique minimizer of \({\text{vol}}({\text{wt}}_\xi )\) among \(\xi \in \hat{H}_0=H_0\cap \sigma\). Similarly to the item 2 of Lemma 2.37, \(\xi _0\) is a minimizer of \({\text{vol}}({\text{wt}}_\xi )\) if and only if \(\xi _0\) is the barycenter of the measured convex domain \((\Pi _{\xi _0}, (P_{\Pi _{\xi _0}})_*(\mathrm{d}{\text{vol}}_{\Pi _{\xi _0}}))\) where \(\Pi _{\xi _0}:=\{y\in \hat{\sigma }; \langle P(y),\xi _0\rangle =1\}\) and \(\mathrm{d}{\text{vol}}_{\Pi _{\xi _0}}\) is the standard Euclidean volume form on \(\Pi _{\xi _0}\). \(\square\)
2.2.2.3 TInvariant QuasiMonomial Valuations on TVarieties
As in the previous section, we assume that there is an effective good \(T=(\mathbb {C}^*)^{r}\) action on an affine normal variety \(X={\text{Spec}}(R)\). In this section, we aim to show the following theorem:
Theorem 2.42
Let\((X,D,\xi )\)be a Fano cone singularity. If\({\text{wt}}_\xi\)is the minimizer of\({\widehat{{\text{vol}}}}_{(X,D)}\), then it is unique among all Tinvariant quasimonomial valuations.
Remark 2.43
 (i)
By Theorem 2.34, the assumption is indeed equivalent to \((X,D,\xi )\) being Ksemistable.
 (ii)
Let v be another Tinvariant minimizer. If we could show the associated graded ring \({\text{gr}}_v(R)\) is finitely generated, then similar to the argument in Theorem 2.34, we can degenerate X to \(X_0\) via v and both \(\xi\) and \(\xi _v\) would be the minimizers of \({\widehat{{\text{vol}}}}_{X_0,D_0}\), which is contradictory to Proposition 2.39. Using this method, we can give another proof of uniqueness of divisorial minimizers proved in [47] with a different argument. However, for general quasimonomial minimizers, since we do not know yet the finite generation of the associated graded ring, we have to adapt a different argument. We also note that later in Proposition 2.69 we will show any quasimonomial minimizer is automatically Tinvariant.
The idea of the proof is to first connect any Tinvariant quasimonomial valuation v with \(v_\xi\) by a family of Tinvariant quasimonomial valuations \(v_t\). This depends on the description of Tinvariant valuations in [2]. Next we extend v to a valuation \({\mathbb {V}}\) of rational rank n and prove that it satisfies properties as in Lemma 2.40. Then we can use the works of Newton–Okounkov bodies to realize the volumes of valuations as volumes of convex bodies as has been done in the previous subsection. Finally, we use the previous convex geometric result to get the strict convexity of the volumes \({\text{vol}}(v_t)\) with respect to t which implies the uniqueness of the minimizer.
Proof
Extend the set \(\{\beta _1, \dots , \beta _{s}\}\) to \(d=nr\)\({\mathbb {Q}}\)linearly independent positive real numbers \(\{\beta _1, \dots , \beta _{s}; \gamma _1, \dots , \gamma _{ds}\}\). Define \({\mathbb {V}}_3(f)=w_{\gamma }(\chi _{m^*}(z''))\) where \(w_{\gamma }\) is the quasimonomial valuation with respect to the coordinates \(z''\) and the \((ds)\) tuple \(\{\beta _1, \dots , \beta _{s}; \gamma _1, \dots , \gamma _{ds}\}\).
Remark 2.44
The construction of \({\mathbb {V}}\) is an example of composite of valuations (see [70, VI.16]).
Let \({\mathcal {S}}\) be the valuative semigroup of \({\mathbb {V}}\). Then \({\mathcal {S}}\) generates a cone \(\tilde{\sigma }\). Let \(P_1: \mathbb {R}^n\rightarrow \mathbb {R}^{r}\), \(P_2: \mathbb {R}^n\rightarrow \mathbb {R}^{s}\) and \(P=(P_1, P_2): \mathbb {R}^n\rightarrow \mathbb {R}^{r+s}\) be the natural projections. Then \(P_1(\tilde{\sigma })=\sigma \subset \mathbb {R}^r\).
Proposition 2.45
Proof
Next consider the weighted blow up of Y along the smooth subvariety \(Z=\bigcap _{i=1}^s \{z_i=0\}\) with weights \(\mathbf{a}=(a_1, \dots , a_{s}):=(q \beta '_1, \dots , q \beta '_{s})\) where q is the least common multiple of the denominators of \(\beta '\). We will denote this weighted blow up by \(\mu _{Y}=\mu _{Y, \beta '}: \tilde{Y}\rightarrow Y\) with the exceptional divisor denoted by \(E=E_{\beta '}\). Since Z is a smooth subvariety of Y, we have \(E={\mathbb {P}}(N_Z, \mathbf{a})=(N_Z{\setminus } \{Z\})/\mathbb {C}^*\), where \(N_Z\) is the normal bundle of \(Z\subset Y\) and \(\tau \in \mathbb {C}^*\) acts along the fiber of the normal bundle \(N_Z\rightarrow Z\) by \(\tau \circ (x_1, \dots , x_{s})= (\tau ^{a_1}x_1, \dots , \tau ^{a_{s}} x_{s})\). So we have a fibration \(\pi _E: E\rightarrow Z\) with each fiber being isomorphic to the weighted projective space \({\mathbb {P}}(\mathbf{a}):={\mathbb {P}}(a_1, \dots , a_s)\). In particular, the inverse image of \(p\in Z\subset Y\) under \(\mu _Y\), denoted by \(E_p\), is a fiber of \(\pi _E\) which is isomorphic to the weighted projective space \({\mathbb {P}}(\mathbf{a})\).
Remark 2.46
As mentioned above, this argument for the estimate of \(k_0\) is essentially the same as the the argument in the proof of Lemma 2.40.
Lemma 2.47
\(\epsilon _0\) and\(\epsilon _1\) can be chosen to be uniform with respect to\(\beta '\) that is close to\(\beta\). More precisely, there exists\(\delta =\delta (\beta )>0\), \(\epsilon _0=\epsilon _0(\beta )>0\), \(\epsilon _1=\epsilon _1(\beta )>0\)such that if\(\beta '\beta \le \delta\) then\(\tilde{H}_{\epsilon _0}:=\mu _{Y, \beta '}^*H\epsilon _0 q E_{\beta '}\) is ample on\(\tilde{Y}\) and\(\mu _E^*(\tilde{H}_{\epsilon _0}_E)\epsilon _1 F\) is ample on\(\tilde{E}\).
Proof
Lemma 2.48
The righthand side of (43) is uniformly bounded independent of\(\beta '\) if\(\beta '\beta \le \delta\) where\(\delta =\delta (\beta )\) is the same one as that in Lemma 2.47.
Proof
2.3 Models and Degenerations
In [47], we showed that a divisorial minimizer always comes from a Kollár component, and it can yield a degeneration which is the key for us to deduce results on general klt singularities from cone singularities. However, it is less clear, at least to us, what should be the corresponding construction for a higher rank quasimonomial valuations. Nevertheless, in this section, we try to develop an approach to use models to approximate a quasimonomial valuations, with possibly higher rank.
2.3.1 Weak lc Model of a QuasiMonomial Minimizer
Let us first fix some notation. Let \(v\in {\text{Val}}_{X,x}\) be a quasimonomial valuation. We know that there exists a log smooth model (Y, E) over X such that v is computed at its center \(\eta\) on (Y, E) (see Definition 2.9). Denote by \(E_j\) (\(j=1,\ldots ,r\)) the components of E containing \(\eta\). In the following, we look at valuations \(v_{\alpha }\) computed on \(\eta \in (Y,E)\) for \(\alpha \in \mathbb {R}^r_{\ge 0}\). In fact, if we rescale \(v_{\alpha }\) such that \(A_{X,D}(v_{\alpha })=1\), then all such points canonically form a simplex \(\Delta \subset {\text{Val}}_{X,x}\) with the vertices given by \(v_j={\text{ord}}_{E_j}/ A_{X,D}(E_j)\).
Definition 2.49
 (a)
\({\text{Ex}} (\mu )=\mu ^{1}(x)=\sum ^r_{j=1} S_j\);
 (b)
\((Y^{\text{wlc}},\mu _*^{1}D+\sum ^r_{j=1} S_j)\) is log canonical;
 (c)
\(K_{Y^{\text{wlc}}}\mu _*^{1}D\sum ^r_{j=1} S_j\) is nef over X; and
 (d)
\((Y^{\text{wlc}},\mu _*^{1}D+\sum ^r_{j=1} S_j)\) is qdlt at the generic point \(\eta\) of a component of the intersection of \(S_j\)\((j=1,2,\ldots , r)\), where v can be computed (see Definition 2.9).
The main theorem of this section is the following:
Theorem 2.50
Let\(x\in (X,D)\)be a klt point. If\(v\in {\text{Val}}_{X,x}\)is a quasimonomial valuation which minimizes\({\widehat{{\text{vol}}}}_{(X,D)}\), then it admits a\(\mathbb {Q}\)factorial weak log canonical model.
Proof
We fix a model as in Definition 2.8 which computes \(v=v_{\alpha }\). We can further assume that the codimension of \(\eta\) in the model is the same as the rational rank of v. Let \(\mathfrak {a}_{\bullet }\) be the valuative ideals of v, i.e., \(\mathfrak {a}_{k}=\{f \ v(f)\ge k\}\). Let \(c={\text{lct}}(X,D, \mathfrak {a}_{\bullet })\).
Lemma 2.51
 1.
\((X, D+(1\epsilon _0)c\cdot \frac{1}{m}\mathfrak {a}_m)\) has a positive log discrepancy for any divisorE;
 2.
v is in the convex cone generated by\(s_1,\ldots ,s_r\), and
 3.
for any j, the log discrepancy\(a_l(S_j, X,D+(1\epsilon _0)c\cdot \frac{1}{m}\mathfrak {a}_m)<\epsilon\) for any\(m\gg 0\).
Proof
 1.the vector \(\alpha =(\alpha _i)\) can be written as$$\begin{aligned} \alpha =\sum ^r_{j=1} c_js_j \quad \hbox {with}\ c_j>0; \end{aligned}$$
 2.for any i and j,$$\begin{aligned} s_{ij} \alpha _j<\frac{\epsilon }{q_j}. \end{aligned}$$
 (a)
\(( X,D+(1\epsilon _0 )c\cdot \frac{1}{m}\mathfrak {a}_m)\) is klt,
 (b)
\(a_{l}(E_i, X, D+(1\epsilon _0)c \cdot \frac{1}{m} \mathfrak {a}_m)\le N\),
 1.
\(S_1\),…, \(S_r\) are the only exceptional divisors,
 2.there is an effective \(\mathbb {Q}\)divisor L on X, such that \((Y, \mu _*^{1}(D+L)+\sum ^r_{j=1} a_jS_j)\) is klt with \(1\epsilon<a_j<1\), and$$\begin{aligned} K_Y+\mu _*^{1}(D+L)+\sum ^r_{j=1} a_jS_j\sim _{\mathbb {Q},X}0. \end{aligned}$$
Therefore, \(Y_1\rightarrow X\) gives a weak log canonical model. \(\square\)
Remark 2.52
The above argument indeed implies that any quasimonomial valuation which computes the log canonical threshold of a graded sequence of ideals has a weak log canonical model.
In the following, we want to show that the volumes of the models produced in Theorem 2.50 converge to \({\widehat{{\text{vol}}}}(v)\) if the corresponding simplexes converge to v.
Lemma 2.53
Proof
Lemma 2.54
Proof
 1.
there exists \(\mu _i:Y_i\rightarrow X\) a \(\mathbb {Q}\)factorial weak lc model which precisely extracts the divisors \(S_{i,j}\) corresponding to the prime integral vector on \(\mathbb {R}_{>0}v_i^j\), and
 2.
for any \(\epsilon\), we can find i sufficiently large such that for any (i, j) there exists a constant \(q_j\) such that \({\text{ord}}_{S_{i,j}}q_j\cdot v< \epsilon\).
In the following, we take a detour to illustrate on how we use the models to understand the limiting process in [47, Theorem 1.3] when v is quasimonomial.
We will need a strengthening of [47, Theorem 1.3].
Proposition 2.55
Proof
 a)
\(a_l(T_i; Y^{\text{wlc}},\mu _*^{1}D+\sum ^r_{j=1} S_{j})=0\);
 b)
\(\{ c_i\cdot {\text{ord}}_{T_i }\}\) has a limit \(v'\ge v\);
 c)
\({\text{vol}}(v')={\text{vol}}(v)\).
Denote by \(\Delta _0\) the simplex in \(\Delta ^{\text{dlt}}\) generated by \(S_1\),…, \(S_r\) around \(\eta\). By Theorem 2.50, we can find a sequence of rational simplices \(\{\Delta _i\}_{i=1}^{\infty }\) with vertices \(S_{i,j}\)\((j=1,\ldots ,r)\) such that \(\Delta _{i+1}\subset \Delta _{i}\), \(\lim _i{\Delta _{i}}=v\) and for each \(\Delta _i\) we have a weak log canonical model \(\mu _i:Y^{\text{wlc}}_i\rightarrow X\). Furthermore, we can require the constant \(\epsilon _i\) and \(\epsilon _{0,i}\) in Lemma 2.51 converges to 0.
2.3.2 Minimizing Valuations and Ksemistability
In this section, we aim to prove the a quasimonomial valuation v is a minimizer only if it is Ksemistable. As we mentioned, we need to make the expected technical assumption that the associated graded ring \({\rm{gr}}_v(R)\) is finitely generated.
Definition 2.56
Let \(x\in X={\text{Spec}}(R)\) be a normal singularity. Let \(v\in {\text{Val}}_{X,x}\) be a valuation and we assume \({\rm{gr}}_v(R)\) is finitely generated. Denote \(X_0={\text{Spec}}({\rm{gr}}_vR)\). Let the rational rank of v be r. Then there is a \(T={(\mathbb {C}^*)^r}\)action on \(X_0\) induced by the \(\mathbb {Z}^r\)grading. We denote by \(\xi _v \in {\mathfrak {t}}_{\mathbb {R}}^+\) the natural vector given by the valuation v, namely \(\xi _v(f)= \min _{f_{\alpha }\ne 0}\{ \alpha \}\) for any \(f=\sum _{\alpha } f_{\alpha }\in {\rm{gr}}_v(R)\).
In this section, we will always consider the degeneration induced by a valuation in the following case: \(x\in (X,D)\) is a klt singularity, and v is a quasimonomial valuation over x whose associated graded ring is finitely generated. We denote \(X_0={\text{Spec}}({\rm{gr}}_v R)\). By Lemma 2.10, we can choose a sequence \(v_i\rightarrow v\), where \(v_i\) is a rescaling of a divisorial valuation, denoted by \(S_i\) over x and we have \({\rm{gr}}_vR\cong {\rm{gr}}_{v_i}R\).
Lemma 2.57
(See [31]) Under the above assumption, we can construct a model\(\mu :Y\rightarrow X\) such that the only exceptional divisor is\(S_i\) and\(S_i\)is ample over X.
Proof
Let \(\{\mathfrak {a}_{\bullet }\}\) be the valuative ideal sequence of \({\text{ord}}_{S_i}\). It suffices to prove that \(\bigoplus _i \mathfrak {a}_i\) is finitely generated given the associated graded ring is finitely generated. We lift generators \(\bar{f}_i\in \mathfrak {a}_{j_i}/\mathfrak {a}_{j_i+1}\)\((i=1,\ldots ,r)\) of \({\rm{gr}}_v(R)\) to elements \(f_i\in \mathfrak {a}_{j_i}\). Let \(d=\max j_i\), and let \(\{g_i\}\) (\(1\le i \le k\)) be a set of generators of \(\mathfrak {a}_j\) for \((0\le j \le d)\); then we show that \(\bigoplus _i \mathfrak {a}_i\) is generated by \(\{g_i\}\). We denote the graded ring generated by \(\{ g_i\}\) to be \(\bigoplus _i {\mathfrak {b}}_i \subset \bigoplus _i \mathfrak {a}_i\).
We assume that \(X_0\) is normal and define \(D_0\) to be the closure of D (as \(\mathbb {Q}\)divisor) in the following way: for each prime Weil divisor E on X with the ideal \(p_E\), we can consider the degeneration \(\mathbf{in}(p_E)\) and let the degeneration \(E_0\) to be its divisorial part. Then for a general \(\mathbb {Q}\)divisor \(D=\sum _i a_iE_i\), we define \(D_0=\sum _i a_iE_{i,0}\).
Lemma 2.58
With the above notation, we have\({\text{vol}}(v)={\text{vol}}(\mathrm{wt}_{\xi _v})\). Furthermore,\(K_{X_0}+D_0\) is\(\mathbb {Q}\)Cartier and\(A_{(X,D)}(v)=A_{(X_0,D_0)}(\mathrm{wt}_{\xi _v})\).
Proof
The first part is straightforward. For the second part, since the closure of a Cartier divisor \({\rm{div}}(f)\) is given by the Cartier divisor \({\rm{div}}(\mathbf{in}_v(f))\), we see that \(K_{X_0}+D_0\) is \(\mathbb {Q}\)Cartier as \(K_X+D\) is \(\mathbb {Q}\)Cartier.
If v is a minimizer such that the associated graded ring \({\rm{gr}}_{v}(R)\) is finitely generated, however, since the Kollár component produced in Proposition 2.55 may not be in interior of the simplex containing v, we cannot directly apply Lemma 2.10. We need to show that
Proposition 2.59
Let v be a minimizer of\({\widehat{{\text{vol}}}}\) such that the associated graded ring\({\rm{gr}}_{v}(R)\) is finitely generated. Let\(c_i\cdot {\text{ord}}_{S_i}\)be chosen sufficiently close to v as in Lemma 2.10; then\(S_i\) is a Kollár component.
Proof
By the proof of Theorem 2.50, we already know that \(S_i\) can be extracted alone to get a model \(\mu _i:Y_i\rightarrow X\) such that \((Y_i,\mu ^{1}_*(D)+S_i)\) is log canonical and \(S_i\) is \(\mu _i\)ample. Therefore, we can degenerate \(x\in (X,D)\) to \(o\in (X_0:={\text{Spec}}({\rm{gr}}_{S_i}(R)),D_0)\), where \(D_0\) is given by the adjunction. Recall that \({\text{Spec}}({\rm{gr}}_{S_i}(R)) \cong {\text{Spec}}({\rm{gr}}_{v}(R))\) and \((X_0,D_0)\) is aways semilogcanonical (slc), as so is \((S_i, D_{S_i})\) where \((K_{Y_i}+\mu _*^{1}D+S_i)_{S_i}:=K_{S_i}+D_{S_i}\). Here we use the fact that \(S_i\) is CM as Y is potentially klt. By Lemma 2.58, we know that \({\widehat{{\text{vol}}}}_{X,D}(v)={\widehat{{\text{vol}}}}_{X_0,D_0}(v_0:=\xi _v)\).
Lemma 2.60
Let\((S,D_S)\) be a slc pair with\(K_SD_S\) ample. Assume that\((S,D_S)\) is not klt. Let\(o\in (Y,D_Y):=C(S,D_S;r(K_S+D_S))\). Then\(\inf _{v} {\widehat{{\text{vol}}}}_o(v)=0\)where v runs through all \(\mathbb {C}^*\)invariant valuations.
Proof
Lemma 2.61
Let\(o\in (X,D)\) be a slc singularity such that\({mld}_{X,D}(o)>0\). Let I be a primary ideal cosupported on o. Then we can find a model\(\mu :Z\rightarrow X\)which extracts precisely a divisor G computing the log canonical threshold of I with respect to (X, D) such that\(G\) is\(\mu\)ample.
Proof
By [36, 10.56], there exists a semi log resolution \(\phi :X'\rightarrow X\) of \((X,D+cI)\) with the properties there, where \(c={\text{lct}}(I; X,D)\). By a tiebreak argument, using the fact that \(\phi _*(\mathcal {O}_{X'})=\mathcal {O}_X\), we can find a small \(\mathbb {Q}\)divisor H passing through o, such that there exists a unique divisor G over o with the log discrepancy 0 with respect to \((X,D+(c\epsilon )I+H)\).
Choose a sufficiently close \(\epsilon '<\epsilon\), then on \(X'\) the only divisor with negative log discrepancy with respect to \((X,D+(c\epsilon ')I+H)\) is G. By [55], we can construct the semilogcanonical modification \(\mu :Z\rightarrow X\) of \((X,D+(c\epsilon ')I+H)\), which thus only extracts G with \(G\) being \(\mu\)ample. \(\square\)
Theorem 2.62
Let \(X_0= {\rm{Spec}}({\rm{gr}}_vR)\) and denote by \(D_0\) the closure of D on \(X_0\). Then \((X_0,D_0)\) is klt.
Proof
By Lemma 2.10, under our assumption, we can choose a valuation divisorial \(v'\) such that \({\rm{gr}}_{v}R\) is isomorphic to \({\rm{gr}}_{v'}(R)\). Then Proposition 2.59 implies that \(v'\) indeed can be chosen to be a Kollár component. Thus by the argument in [47], we know \((X_0,D_0)\) is a klt pair, as it is an orbifold cone over a log Fano pair. \(\square\)
Definition 2.63
Under the above assumptions, we say that a quasimonomial valuation over \(x\in X\) is Ksemistable if \({\rm{gr}}_v(R)\) is finitely generated and the corresponding triple \((X_0, D_0,\xi _v)\) as in Definition 2.56 is Ksemistable in the sense of Definition 2.28.
Proof of Theorem 1.3
Theorem 2.64
If \(x\in (X,D)\) is a klt singularity and \(v\in {\text{Val}}_{X,x}\) which is a quasimonomial minimizer of \({\widehat{{\text{vol}}}}_{(X,D)}\) such that its associated graded ring \({\rm{gr}}_v(R)\) is finitely generated, then v is a Ksemistable valuation.
The proof of this theorem is similar to the case of Kollár component minimizer as in Section 6 of [47]. For reader’s convenience, we include a brief sketch here.
Proof
Now we lift \(f_1, \dots , f_p\) to generators \(F_1, \dots , F_p\) of R. Then we have: \(g_j(F_1, \dots , F_p)\) lies in \(R_{> n_j}\). By flatness of \({\mathcal {R}}\) over \(\mathbb {C}[\Phi ]\), there exist \(G_j\in g_j+P_{> n_j}\) such that \(G_j(F_1, \dots , F_p)=0\). Let I be the kernel of \(P\rightarrow R\); then \(\{G_j\}\) form a Gröbner basis with respect to the order function v and the associated initial ideal is J.
Given the above data, we know that there is an action of \(T:=(\mathbb {C}^*)^{r+1}=(\mathbb {C}^*)^r\times \mathbb {C}^*\)action on \(\mathbb {C}^p\). The valuation v corresponds to a linear holomorphic vector field \(\xi _0\) with an associated weight function denoted by \(\lambda _0\). \({\text{ord}}_S\) corresponds to another linear holomorphic vector field \(\xi _S\) on \(\mathbb {C}^p\) whose associated weight function will be denoted by \(\lambda _\infty\).
Since \(\lambda _\epsilon\) is linear with respect to \(\epsilon\), \(\{\xi _\epsilon \}\) is a ray in \({\mathfrak {t}}^+_\mathbb {R}\) emanating from \(\xi _0\). Denoting \(\eta =\frac{\mathrm{d}}{\mathrm{d}\epsilon }_{\epsilon =0}\xi _\epsilon \in {\mathfrak {t}}_{\mathbb {Q}}\), we then get a special test configuration \(({\mathcal {X}},{\mathcal {D}},\xi _0;\eta )\) of (X, D) to \((Y_0,D'_0)\), and also a special test configuration of \((X_0,D_0)\) to \((Y_0,D'_0)\).
Remark 2.65
The above argument works as long as we can find an equivariant degeneration \(Y_0\) of \(X_0={\text{Spec}}({\rm{gr}}_v (R))\), such that we can define the normalized volume on the toric valuations for \(\mathrm{wt}_{\xi }\) where \(\xi \in {\mathfrak {t}}_{\mathbb {R}}(Y)\), since we still have the convexity of the normalized volumes in this setting (see Proposition 2.39).
2.3.3 Uniqueness in General
In this section, we will verify the uniqueness of quasimonomial minimizer if we assume one of them has a finitely generated associated graded ring. This assumption is always fulfilled when the minimizer is divisorial [8, 47] and is conjectured to hold in general.
Lemma 2.66
There is a\(\mathbb {Q}\)factorial equivariant family\(\tilde{\mu }:\mathcal {Y}\rightarrow \mathcal {X}\) over\(\mathbb {A}^1\), whose general fiber gives\(Y\rightarrow X\). Furthermore, \((\mathcal {Y}, \mu _*^{1}\mathcal {D}+ \sum \mathcal {S}_i+Y_0)\) is log canonical and\(Y_0\) is irreducible.
Proof
It remains to show that \((\mathcal {Y}, \mu _*^{1}\mathcal {D}+ \sum \mathcal {S}_i+Y_0)\) is log canonical. Again by ACC of log canonical thresholds [28], it suffices to show that for the constant \(\beta\) chosen in Theorem 2.50, \((\mathcal {Y}, \mu _*^{1}\mathcal {D}+ \beta \cdot \sum \mathcal {S}_i+Y_0)\) is log canonical. But this is implied by the fact that the log pull back of \(K_\mathcal {Y}+\mu _*^{1}\mathcal {D}+\beta \sum \mathcal {S}_i\) is less or equal to the log pull back of \(K_\mathcal {X}+\mathcal {D}+(1\epsilon _0 )c\cdot \frac{1}{m} \overline{ \mathfrak {a}}_m\). \(\square\)
Then we can define a quasimonomial valuation \(w_0\) as in Definition–Proposition 2.67 over \(X_0\).
Definition–Proposition 2.67
Let \(\eta _0\) be the generic point of a component of \(\overline{\eta }\cap Y_0\). Let \(T_i\) be the (not necessarily irreducible) reduction divisor of \(\mathcal {S}_i\) at the generic point \(\eta _0\). Then \({\text{ord}}_{T_1}\), \({\text{ord}}_{T_2}\), …, and \({\text{ord}}_{T_r}\) generate a rank r sublattice in \({\text{Val}}_{Y_0,\eta _0}\). Furthermore, we can define a quasimonomial valuation \(v_0\) over \(\eta _0\) which is of rational rank r, such that \(v_0(T_i)=\alpha _i\).
Proof
By [18, Proposition 34] we know that \((\mathcal {Y}, Y_0+\sum \mathcal {S}_i+\tilde{\mu }_*^{1}\mathcal {D})\) is qdlt at the generic point of the log canonical center given by an irreducible component \(Z_0\) of \(Y_0\cap \mathcal {Z}\), so we can define such a quasimonomial valuation \(v_0\) over \(\eta\). \(\square\)
The following lemma implies that such a degeneration is indeed uniquely determined:
Lemma 2.68
Proof
We easily see \(w(f)\le w_0(\mathbf{in}(f))\) and now we assume \(w(f)< w_0(\mathbf{in}(f))\) from some f and we will argue this is contradictory to the fact that \({\text{vol}}(w)={\text{vol}}(w_0)\) as in the proof of [47, Proposition 2.3].
Now we prove the claim.
Step 2: Since taking the initial induces an isomorphism of \(\mathbb {C}\)linear spaces between \(R/{\mathfrak {b}}_k\) and \({\text{gr}}_vR/\mathbf{in}({\mathfrak {b}}_k)\) ( [47, Lemma 4.1]), to show \(\mathbf{in}(f^m\cdot g^{(j)}_{m})\) is linearly independent, it suffices to show that \(f^m\cdot g^{(j)}_{m}\) is linearly independent in \(R/{\mathfrak {b}}_k\). This is verbatim the same as Step 2 in the proof of [47, Proposition 2.3] once we replace v by w. \(\square\)
Proposition 2.69
Let\(x\in X\)be a Tsingularity. Assume a minimizer v of\({\widehat{{\text{vol}}}}_{X,x}\)is quasimonomial, then vis Tinvariant.
Proof
It suffices to prove this for \(T=\mathbb {C^*}\), since if a valuation v is \(\mathbb {C}^*\)equivariant for any \(\mathbb {C}^*\), then it is Tinvariant.
Thus \(\mathcal {Y}\) is indeed \(Y\times \mathbb {A}^1\) where \(Y=\mathcal {Y}\times _{\mathbb {A}^1}\{t\}\) for some \(t\ne 0\) as they are isomorphic in codimension 1, and both are the antiample model of the same divisorial valuation over \(X\times \mathbb {A}^1\). This implies that \(S_i\) is equivariant. \(\square\)
Now we can complete the proof of Theorem 1.1.
3 Part II: Singularities on GH Limits
3.1 Canonicity of the Semistable Cone
3.1.1 Metric Tangent Cones and Valuations
Conjecture 3.1
(Donaldson–Sun) Both W and C depend only on the algebraic structure of \(M_\infty\)near o.
One goal of the project proposed in [38] is to prove this conjecture. We observed in [38, 47] that \(\{{\mathcal {F}}^{\lambda }\}\) comes from a valuation \(v_0\). This is due to the fact that W is an irreducible variety since it degenerates to the normal variety C. As mentioned in [29], this was implicit in [20, 22]. More explicitly, if we denote by \(X={\text{Spec}}(R)\) the germ of \(o\in M_{\infty }\), by the work in [22], one can embed both X and C into a common ambient space \(\mathbb {C}^N\), and \(v_0\) on X is induced by the monomial valuation \({\text{wt}}_{\xi _0}\) where \(\xi _0\) is the linear holomorphic vector field with \(2{\text{Im}}(\xi _0)\) being the Reeb vector field of the Ricci flat Kähler cone metric on C. By this construction, it is clear that the induced valuation by \(v_0\) on W is nothing but \({\text{wt}}_{\xi _0}\).
Here in this paper we also observe that \(v_0\) is a quasimonomial valuation. As pointed out in Lemma 2.11, this follows from a general fact due to Piltant ([57], see [61, Proposition 3.1]) that a valuation \(v_0\) is quasimonomial if and only if the associated graded ring has the same Krull dimension as \(\dim X\). See also Lemma 2.16 where the quasimonomial property of \({\text{wt}}_{\xi _0}\) on W and C is explained.
More importantly, we conjectured in [38] that \(v_0\) can be characterized as the unique minimizer of \({\widehat{{\text{vol}}}}_{M_\infty , o}\). For now we cannot prove this conjecture in the full generality. Nevertheless, as a corollary of the theory developed in this paper (and its predecessors [38, 40, 42, 47]), we can already prove Theorem 1.4 and confirm [22, Conjecture 3.22] for W:
Proof of Theorem 1.4
By the above discussion, for any valuation \(v_0\) as above, we already know that it is quasimonomial and centered at \(o\in M_\infty\) and the induced valuation on \(W={\text{Spec}}({{\text{gr}}_{v_0}}R)\) is equal to \({\text{wt}}_{\xi _0}\). By Theorem 3.5, we know that \((C, \xi _0)\) is Ksemistable.
We claim that \((W, \xi _0)\) is a klt Fano cone singularity and is Ksemistable. To see this, we first note that there exists a prime divisorial valuation S which has a finitely generated associated graded ring and degenerates X to W. As in the proof of Theorem 2.64, such an S can be obtained by perturbing \(\xi _0\) in the Reeb cone of C so that we can ensure that this perturbed vector generates a \(\mathbb {C}^*\) in the big torus that preserves the klt Fano cone singularity C. As a consequence, S is isomorphic to the quotient of C by the \(\mathbb {C}^*\)action. This implies that S is a Kollár component over C. By inversion of adjunction, we also conclude that W has klt singularities. Now since \((W, \xi _0)\) equivariantly degenerates to \((C, \xi _0)\), by Lemma 2.36, we know that \((W, \xi _0)\) is indeed Ksemistable (see also Remark 3.2).
By Theorem 1.3, \(v_0\) is a minimizer of \({\widehat{{\text{vol}}}}_{M_\infty ,o}\) and by Theorem 1.1.3, \(v_0\) is the unique minimizer of \({\widehat{{\text{vol}}}}_{M_\infty ,o}\) among all quasimonomial valuations in \({\text{Val}}_{M_{\infty }, o}\), and it only depends on the algebraic structure of R. Therefore, W only depends on the algebraic structure of the germ \(o\in M_{\infty }\). \(\square\)
Remark 3.2
There is an alternative but essentially equivalent way to show directly that \(v_0\) is a minimizer of \({\widehat{{\text{vol}}}}_{M_\infty , o}\). First, in the proof of Theorem 2.64, we have constructed a degeneration of \((M_\infty , o)\) to (C, o). On the other hand, we know that \({\widehat{{\text{vol}}}}_{M_\infty ,o}(v_0)={\widehat{{\text{vol}}}}_{C,o}({\text{wt}}_{\xi _0})\) and \({\text{wt}}_{\xi _0}\) minimizes \({\widehat{{\text{vol}}}}_{C,o}\). We can then use the same idealdegeneration argument as in the proof of Lemma 2.36 to conclude that \(v_0\) is the minimizer of \({\widehat{{\text{vol}}}}_{M_\infty ,o}\). Theorem 1.1 then also implies \((W, \xi _0)\) is a klt Fano cone singularity which is indeed Ksemistable by Lemma 2.36.
3.1.2 Minimizers from Ricci Flat Kähler Cone Metrics
Since T acts on X, T also acts on the set of functions on X by \(\tau \circ f(x)=f(\tau ^{1}x)\) for any \(\tau \in T\) and \(x\in X\). For convenience, we introduce the following:
Definition 3.3
 1.
\(\tau \circ {\varphi }={\varphi }\) for any \(\tau \in T\);
 2.
\(r^2_{\varphi }:=r^2 {\text {e}}^{{\varphi }}\) is a proper strictly plurisubharmonic function on X.
Definition 3.4
Theorem 3.5
If \((X, \xi _0)\) admits a Ricciflat Kähler cone metric, then \(A_X(\xi _0)=n\) and \((X, \xi _0)\) is Ksemistable.
Remark 3.6
 1.
In the case when X has isolated singularities at \(o\in X\), this was proved in [15] using an approximation by rational elements in \({\mathfrak {t}}_\mathbb {R}^+\) to reduce to the orbifold case studied in [58]. The proof given below for the general case is different and is a direct generalization of a corresponding proof in the usual Kähler case. We also depend heavily the calculations from [51] which have also appeared in different forms in [16, 22].
 2.
As already mentioned, after Berman’s work [4], it is natural to expect that \((X, \xi _0)\) should actually be Kpolystable. Since this requires more technical arguments involving geodesic rays and we do not need this stronger conclusion in this paper, we will leave its verification in [45].
Corollary 3.7
Proof
Remark 3.8
The rest of this section is devoted to the proof of Theorem 3.5. Since \(A_X(\xi _0)=n\) has been shown, we will focus on the second statement.
Following [16], we consider the following cone version of Ding energy:
Definition 3.9
By [16], \(E({\varphi })\) is a welldefined function of \({\varphi }\). Moreover by calculating in the polar coordinate with respect to \(r^2_{\varphi }\), one easily sees that \(D({\varphi }+c)=D({\varphi })\). The Euler–Lagrangian equation of \(D({\varphi })\) is the equation of Ricciflat Kähler cone metric in (57). The following lemma is a generalization of a wellknown fact in the regular case.
Lemma 3.10
\(E({\varphi }(t))\) is a concave function with respect to\(s=\log t^2 \in \mathbb {R}\).
Proof
We need the following important formula due to Martelli–Sparks–Yau (see also [22]):
Lemma 3.11
Since our notations may be different from that in the literature, for the reader’s convenience we provide a brief calculation.
Proof
Proposition 3.12
Proof
Lemma 3.13
The function\(\tilde{G}(t)\) in (79) is a bounded continuous function with respect to\(t\in \mathbb {C}\).
Proof
Proposition 3.14
Proof
Finally, we can complete the proof of Theorem 3.5.
Completion of the Proof of Theorem 3.5
If there exists a Ricciflat Kähler cone metric on \((X, \xi )\), then the Ding energy \(D({\varphi })\) is bounded from below. As pointed out in [22] this can be proved by following the same proof for the Kähler–Einstein case. Indeed, for any \({\varphi }\in PSH(X, \xi _0)\), we get transversal Kähler potential which is still denoted by \({\varphi }\). By using the same proof as in [5, pp. 156–157], there exists a bounded geodesic \({\varphi }_t\) connecting 0 and \({\varphi }\). On the other hand, adapting Berndtsson’s proof of subharmonicity to the Sasakian case, Donaldson–Sun showed that \(D({\varphi }_t)\) is convex with respect to t. Because \({\varphi }_0=0\) is a critical point of \(D({\varphi }_t)\), one knows that \(D({\varphi })\ge D(0)\).
Since \(D({\varphi }(s))\) is uniformly bounded from below, by (81), \({\text{Fut}}({\mathcal {X}}, \xi _0; \eta ) \ge 0\). As this holds for any special test configuration induced by any Kollár component, we get the conclusion. \(\square\)
3.1.3 Finite Degree Formula
Now we can verify the degree multiplication formula in Theorem 1.7.
Proof of Theorem 1.7
We first assume \(\pi\) is a Galois covering with the Galois group G.
Let \(v_0\) be the valuation defined in Sect. 3.1.1, which induces the degeneration of \((o\in M_{\infty })\) to W. We can fix a sequence of \(v_i\rightarrow v_0\) such that \(v_i\) is a rescaling of Kollár component. Since the pull back of a Kollár component is a Ginvariant Kollár component, we conclude that we can pull back \(v_0\) to get a Ginvariant valuation \(v'\in {\text{Val}}_{Y,y}\). It suffices to prove \(v'\) is a minimizer of \({\widehat{{\text{vol}}}}_{Y}\) (see, e.g., [48, Theorem 2.6]).
Since \(v'\) is Ginvariant, \(v'(f)=\frac{1}{G}v'({\text{Nm}}(f))\), we know that a Ginvariant element in \(R'\) has its valuation under \(v'\) is at least k if and only if it is an element in R whose valuation under v is at least k, i.e., \((\mathfrak {a}_{v'})^G_k=(\mathfrak {a}_v)_k\). In particular, \({\text{gr}}_{v'}R'\) is finitely generated as it is finite over \({\text{gr}}_{v}R\). We denote by \(W_Y={\text{Spec}}({\text{gr}}_{v'}R')\). Then \(\pi _W:W_Y\rightarrow W\) is quasiétale with Galois group G. In fact this is clear in the quasiregular case, and in the general case, we can use Lemma 2.10 to reduce to the quasiregular case. Furthermore, G commutes with the Taction on \(W_Y\) as it preserves the vdegree, and the vector associated with \(v'\) on \(W_Y\) is \(\pi ^*(\xi _0)\), which is denoted by \(\xi _0'\).
Footnotes
 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.
Notes
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 DMS1405936 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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