Stability of klt singularities

The local stability theory originates from questions in complex geometry. Recall that a Kähler-Einstein metric on a complex manifold is a Kähler metric $\omega$ with constant Ricci curvature. After appropriate rescaling, this means $\mathrm{Ric}(\omega)=\lambda\omega$ where $\lambda\in\{0,-1,1\}$. On a Fano manifold, we have $\lambda=1$. Consider a sequence of Kähler-Einstein Fano manifolds $(X_{k},\omega_{k})$ ($k=1,2,\dots$). By the convergence theory of Riemannian manifolds, specifically Gromov’s compactness theorem, one can pass to a subsequence and extract a Gromov-Hausdorff limit $X_{\infty}$. In this context, Donaldson and Sun [DS-GHlimit-1, DS-GHlimit-2] prove that the limit space $X_{\infty}$ is also a Kähler-Einstein Fano variety. In particular, it is algebraic (but may be singular). To analyze the singularities of $X_{\infty}$, they inspect the metric tangent cones of $X_{\infty}$, which are pointed Gromov-Hausdorff limits of $x\in(X_{\infty},r_{k}\omega_{\infty})$ for some fixed $x\in X_{\infty}$ and some increasing sequence of scaling factors $r_{k}\to\infty$. They find that the metric tangent cone again inherits some algebraic structure: it is a normal affine algebraic variety endowed with an effective torus action and a singular Ricci-flat Kähler cone metric. They also give a two-step degeneration description of the metric tangent cone, where the intermediate step (the K-semistable degeneration) is algebraic as well.

There should be an algebro-geometric explanation for the ubiquity of algebraic structures in these constructions, and this is achieved by the algebraic K-stability theory. The recent development in the (global) K-stability theory of Fano varieties culminates in the K-moduli theorem, which (among other things) provides an algebro-geometric construction of the Gromov-Hausdorff limit $X_{\infty}$. For those interested in this part of the theory, we recommend the survey [Xu-K-stability-survey] and the upcoming book [Xu-K-book] for a comprehensive and up-to-date account¹¹1The missing parts in [Xu-K-stability-survey] are the properness of the K-moduli space and the surrounding higher rank finite generation theory. These topics are covered in [Xu-K-book].. The local K-stability theory, which is the main topic of this survey article, will address Donaldson and Sun’s conjecture that the two-step degeneration of $x\in X_{\infty}$ to its metric tangent cone should only depend on the algebraic structure of the singularity (rather than the metric). More generally, as we will explain in subsequent sections, every Kawamata log terminal (klt) singularity has a two-step degeneration to a uniquely determined K-polystable Fano cone singularity, and it seems likely that there is a K-moduli of klt singularities.

Apart from Donaldson-Sun’s conjecture mentioned above, another source of inspiration for the development of the local stability theory is the question on the existence of Sasaki-Einstein metrics. In [MSY-SE-toric, MSY-SE-general], Martelli, Sparks and Yau set up a variational problem on Sasaki-Einstein manifolds whose critical point determines the Reeb vector field of the Sasaki-Einstein metric. The volume functional they considered and the minimization phenomenon they discovered may be seen as the first prototype of the local stability theory. Later, Collins and Székelyhidi [CS-Kss-Sasaki, CS-Sasaki-Einstein] proved a Yau-Tian-Donaldson type criterion for the existence of a Ricci flat Kähler cone metric on an isolated cone singularity, or equivalently, a Sasaki-Einstein metric on the link of the singularity. In particular, they defined K-semi/polystability for Fano cones (by mimicking the definitions in the global case [Tian-K-stability-defn, Don-K-stability-defn]), and related the existence of a Ricci flat Kähler cone metric to the algebro-geometric condition that the singularity is a K-polystable Fano cone.

The algebraic theory of local K-stability starts with Chi Li’s introduction of the normalized volumes of valuations [Li-normalized-volume]. Li’s insight (partly inspired by Martelli-Sparks-Yau’s work) is that valuations on the singularity represent algebraic “rescalings” of the singularity, and that the valuation with the smallest normalized volume represents an “optimal rescaling” that should be closely related to the metric tangent cone degeneration. Based on this philosophy, he proposed to attack Donaldson-Sun’s conjecture by solving a series of conjectures regarding the minimizer of the normalized volume function.

The theory is further investigated in [LX-stability-kc, LX-stability-higher-rank]. In particular, Li and Xu [LX-stability-higher-rank] show that the K-semistable degeneration step in Donaldson-Sun’s construction only depends on the algebraic structure of singularity, and is indeed induced by a minimizer of the normalized volume function. Later [LWX-metric-tangent-cone] completes the proof of Donaldson-Sun’s conjecture by proving the algebraicity of the other step (i.e. the K-polystable degeneration) of the metric tangent cone construction.

The proof [LX-stability-higher-rank] of the algebraicity of the K-semistable degenerations assumes the existence of such degenerations, which in turn relies on deep analytic results [CCT-sing-GH-limit, Tia-partial-C^0, DS-GHlimit-1, DS-GHlimit-2] and as such is restricted to singularities on Gromov-Hausdorff limits of Kähler-Einstein Fano manifolds. To give a purely algebraic construction of the two-step degeneration, and to extend the theory to arbitrary klt singularities, [LX-stability-higher-rank] refines Li’s original proposal, and put forth what is now called the Stable Degeneration Conjecture (see Section 2.4). It highlights a number of conjectural properties of the normalized volume minimizer, which, when put together, ensure that every klt singularity has a canonical stable degeneration induced by the said minimizer.

The Stable Degeneration Conjecture is subsequently proved by a series of works: the existence of the normalized volume minimizer is prove by Blum [Blu-minimizer-exist], the uniqueness is established in [XZ-minimizer-unique] (later [BLQ-convexity] gives another proof), Xu [Xu-quasi-monomial] proves that the minimizer is quasi-monomial (in loc. cit. he also gives another proof that the minimizer exists), while the finite generation property (known by itself as the local higher rank finite generation conjecture) is confirmed in [XZ-SDC]. It is proved in [LX-stability-higher-rank] that the induced degeneration is a K-semistable degeneration, and [LWX-metric-tangent-cone] further gives a recipe for constructing the K-polystable degeneration. These complete the algebro-geometric construction of the two-step degeneration.

The development of the local stability theory intertwines with the study on the K-stability of Fano varieties. The local and the global theory often draw inspirations from each other. The uniqueness of the normalized volume minimizer implies (through the cone construction) that equivariant K-semistability of a Fano variety is equivalent to K-semistability, and the proof of the uniqueness [XZ-minimizer-unique] is in turn inspired by the earlier work on equivariant K-stability of Fano varieties [Z-equivariant]. The idea behind Xu’s proof [Xu-quasi-monomial] of the quasi-monomial property of the minimizer led to the proof of the openness of K-semistability in families of Fano varieties [Xu-quasi-monomial, BLX-openness]. The finite generation part of the Stable Degeneration Conjecture is a local analog of the higher rank finite generation conjecture for Fano varieties, proved in [LXZ-HRFG]. In the setting of Fano varieties, there is also an algebro-geometric construction of canonical two-step degenerations to Fano varieties with Kähler-Ricci solitons [BLXZ-soliton].

Inspired by the K-moduli theory of Fano varieties, the focus of the local stability theory recently shifts towards the boundedness of singularities, an important missing ingredient for the local K-moduli theory. This topic has been intensively studied in [HLQ-vol-ACC, MS-bdd-toric, LMS-bdd-dim-3, Z-mld^K-1, Z-mld^K-2], yet the general case remains wide open.

Here is a roadmap for this survey. In section 2, we define some basic objects in the local stability theory and state the Stable Degeneration Conjecture. In Section 3, we introduce the notion of Kollár components, which plays an important role in the study of klt singularities. The entire Section 4 is devoted to explaining some key ingredients in the proof of the Stable Degeneration Conjecture. Section 5 surveys our current understanding on the boundedness of klt singularities. Finally we discuss some conjectures and open questions in Section 6.

Since our primary focus is on the stable degeneration and the boundedness of klt singularities, we have to leave out several other interesting topics such as the analytic aspect of the theory and further applications of the normalized volume. Some of these topics have been covered by the survey [LLX-nv-survey], which we recommend to the interested readers.

We always work over an algebraically closed field $\mathbbm{k}$ of characteristic $0$. A singularity $x\in X$ consists of a normal variety $X$ and a closed point $x\in X$. We will often assume that $X$ is affine and will freely shrink $X$ around $x$ as needed.

The author is partially supported by the NSF Grants DMS-2240926, DMS-2234736 and a Clay research fellowship. He would like to thank Harold Blum, Chi Li, Yuchen Liu, Xiaowei Wang and Chenyang Xu for many helpful comments and conversations.

We start with the notion of valuations.

Let us explain how valuations naturally arise in our context, at least in hindsight. For singularities appearing on Gromov-Hausdorff limits of Kähler-Einstein Fano manifolds, the stable degenerations are supposed to algebro-geometrically recover Donaldson-Sun’s two-step degeneration description of the metric tangent cones. Now there does exist a tangent cone construction in algebraic geometry: for any singularity $x\in X=\mathbf{Spec}(R)$, its tangent cone is defined as $X_{0}=\mathbf{Spec}(R_{0})$ where

As a typical example, if $X=(f=0)\subseteq\mathbb{A}^{n+1}$ is a hypersurface singularity of multiplicity $k$ at the origin, and we write

where $f_{k}$ is homogeneous of degree $k$, then the tangent cone to $0\in X$ is the hypersurface singularity $(f_{k}=0)\subseteq\mathbb{A}^{n+1}$. On the other hand, it is not hard to see that this is not the desired metric tangent cone in general. One reason is that the tangent cone can be reducible, while the metric tangent cone is always irreducible (it is an affine variety). In fact, the tangent cone of a (klt) hypersurface singularity $(f=0)\subseteq\mathbb{A}^{n+1}$ coincides with its metric tangent cone if and only if $(f_{k}=0)\subseteq\mathbb{P}^{n}$ is a K-polystable Fano variety (see Proposition 3.8). By the Yau-Tian-Donaldson correspondence, the latter condition is equivalent to the existence of a Kähler-Einstein metric on the Fano variety.

What are some variations of the “naïve” tangent cone construction? The first observation is that the same construction can be applied to any decreasing graded sequence of $\mathfrak{m}_{x}$-primary ideals.

Given a decreasing graded sequence $\mathfrak{a}_{\bullet}$ of $\mathfrak{m}_{x}$-primary ideals on $X=\mathbf{Spec}(R)$, we can form the associated graded algebra

When $\mathfrak{a}_{k}=\mathfrak{m}_{x}^{k}$, this recovers the graded algebra (2.1) that defines the tangent cone. In general, if the algebra $\bigoplus_{k\in\mathbb{N}}\mathfrak{a}_{k}$ is finitely generated, then so is $\mathrm{gr}_{\mathfrak{a}_{\bullet}}R$, and we get an isotrivial degeneration of $X$ to $\mathbf{Spec}(\mathrm{gr}_{\mathfrak{a}_{\bullet}}R)$ through the Rees construction. To see this, set $\mathfrak{a}_{k}=R$ for all $k<0$ and let

Then one can check that $\mathcal{X}=\mathbf{Spec}(\mathcal{R})\to\mathbb{A}^{1}_{t}$ is a flat family with general fiber $X$ and special fiber $\mathbf{Spec}(\mathrm{gr}_{\mathfrak{a}_{\bullet}}R)$. The $\mathbb{Z}$-grading of $\mathcal{R}$ also induces a $\mathbb{G}_{m}$-action on the total space $\mathcal{X}$ that commutes with the usual $\mathbb{G}_{m}$-action on $\mathbb{A}^{1}$. Such a family is also called a test configuration of the singularity $x\in X$. The $\mathfrak{m}_{x}$-primary condition further ensures that the closed point $x\in X$ on the general fiber specializes to a closed point in the central fiber (the fixed point of the $\mathbb{G}_{m}$-action). For us, such isotrivial degenerations serve as the algebraic analog of Gromov-Hausdorff limits.

As a slight generalization, we also allow graded sequences of ideals that are indexed by $\mathbb{R}_{\geq 0}$. These are called filtrations. A formal definition is as follows.

Denote by $\mathrm{Val}_{X}$ the set of valuations that has a center on $X$, and by $\mathrm{Val}_{X,x}$ the set of valuations centered at $x\in X=\mathbf{Spec}(R)$²²2We say that a valuation $v$ is centered at a scheme-theoretic point $\eta\in X$ if there is a local inclusion $\mathcal{O}_{X,\eta}\hookrightarrow\mathcal{O}_{v}$ into the valuation ring $\mathcal{O}_{v}:=\{f\in\mathbbm{k}(X)\mid v(f)\geq 0\}$ of $v$.. Every valuation $v\in\mathrm{Val}_{X,x}$ induces an $\mathfrak{m}_{x}$-primary filtration $\mathfrak{a}_{\bullet}(v)$ by setting

Similar to the case of graded sequences of ideals, for any filtration defined above we can form the associated graded algebra $\mathrm{gr}_{\mathfrak{a}_{\bullet}}R:=\bigoplus_{\lambda\in\mathbb{R}_{\geq 0}}\mathfrak{a}_{\lambda}/\mathfrak{a}_{>\lambda}$, where $\mathfrak{a}_{>\lambda}=\bigcup_{\mu>\lambda}\mathfrak{a}_{\mu}$. If $\mathfrak{a}_{\bullet}=\mathfrak{a}_{\bullet}(v)$ for some valuation $v\in\mathrm{Val}_{X,x}$, we further denote $\mathrm{gr}_{\mathfrak{a}_{\bullet}}R$ as $\mathrm{gr}_{v}R$. With this level of generality, Donaldson and Sun [DS-GHlimit-2] show that the two-step degenerations to the metric tangent cones are both induced by some $\mathfrak{m}_{x}$-primary filtration. Usually the first step is called the K-semistable degeneration, while the second step is the K-polystable degeneration. Philosophically, the two steps can be seen as analogous to the Harder-Narasimhan and Jordan-Hölder filtrations of vector bundles, where the graded pieces are (slope) semistable and polystable bundles, respectively. For now, we focus on the K-semistable degeneration, deferring the discussion of the K-polystable degeneration to Section 2.4.

Since the filtration in [DS-GHlimit-2] that induces the K-semistable degeneration is constructed using the (singular) Kähler-Einstein metric on the Gromov-Hausdorff limit, an immediate question, especially if we want to generalize the construction to arbitrary klt singularities, is how to identify the filtration using algebraic geometry. A simple but crucial observation is that the filtration is necessarily induced by some valuation, as the central fiber of the K-semistable degeneration is irreducible.

Because of this fact, we may restrict our search to valuations. Before we discuss further constraints, there are two important classes of valuations we shall keep in mind.

For a fixed pair $(Y,E)$ and a generic point $\eta$ of some stratum, we denote the corresponding set of quasi-monomial valuations by $\mathrm{QM}_{\eta}(Y,E)$. We also set $\mathrm{QM}(Y,E)=\cup_{\eta}\mathrm{QM}_{\eta}(Y,E)$, where $\eta$ varies over the smooth points of $Y$ at which $E$ is SNC. A general valuation can be thought of as limits of quasi-monomial valuations, c.f. [JM-val-ideal-seq]*Section 4. In fact, for any log resolution $\pi\colon Y\to X$ and any SNC divisor $E\subseteq Y$ (we will henceforth call such pair $(Y,E)$ a log smooth model of $X$), there is a natural retraction map $r_{Y,E}\colon\mathrm{Val}_{X}\to\mathrm{QM}(Y,E)$. As we vary the model $(Y,E)$, the images under the retraction maps give approximations of a given valuation.

The singularities that appear on Gromov-Hausdorff limits of Kähler-Einstein Fano manifolds are examples of Kawamata log terminal (klt) singularities. From an algebro-geometric perspective, it is more natural to set up a local stability theory for klt singularities. This class of singularities is particularly important in birational geometry, as they are also singularities of minimal models of algebraic varieties. The output of the stable degenerations belong to a special class of klt singularities called Fano cone singularities. We next review some basics on this class of singularities. For the readers’ convenience, we first recall some definitions for singularities of pairs. More details can be found in [KM98].

In a very rough sense, the klt (resp. lc) condition says that the singularities of holomorphic $n$-forms ($n=\dim X$) on $X$ are better (resp. not worse) than poles.

It will be convenient to reformulate the above definition using log discrepancies. For any pair $(X,D)$ and any prime divisor $E$ on some log resolution $\pi\colon Y\to X$, the log discrepancy $A_{X,D}(E)$ is defined to be

Then the pair $(X,D)$ is klt (resp. lc) if and only if $A_{X,D}(E)>0$ (resp. $\geq 0$) for all prime divisor $E$ over $X$.

Heuristically, klt singularities are the local analog of Fano varieties:

A more general construction that will play a key role in the local stability theory is given by Fano cone singularities. By definition, these are klt singularities with a nontrivial good torus action, together with the choice of a Reeb vector (also called a polarization). Let us explain the terminology.

We say a torus $\mathbb{T}=\mathbb{G}_{m}^{r}$-action on a singularity $x\in X=\mathbf{Spec}(R)$ is good if it is effective and $x$ is in the orbit closure of any $\mathbb{T}$-orbit. Let $N:=N(\mathbb{T})=\mathrm{Hom}(\mathbb{G}_{m},\mathbb{T})$ be the co-weight lattice and $M=N^{*}$ the weight lattice. We have a weight decomposition

and the action being good implies that $R_{0}=\mathbbm{k}$ and every $R_{\alpha}$ is finite dimensional. For $f\in R$, we denote by $f_{\alpha}$ the corresponding component in the above weight decomposition.

For later use, we also define the notion of toric valuations. For any singularity $x\in X=\mathbf{Spec}(R)$ with a good torus action as above and any $\xi\in\mathfrak{t}^{+}_{\mathbb{R}}$, we define a valuation $\mathrm{wt}_{\xi}$ (called a toric valuation) by setting

where $f\in R$. It is not hard to verify that $v:=\mathrm{wt}_{\xi}\in\mathrm{Val}_{X,x}$. We also see that $\mathrm{gr}_{v}R\cong R$, as both sides have the same weight decomposition. In other words, the toric valuation $v$ induces a degeneration of the singularity to itself.

A Fano cone singularity will be denoted as $x\in(X;\xi)$ where $\xi$ is the Reeb vector field. Through the inclusion $\mathbb{T}\subseteq\mathrm{Aut}(x,X)$, we often view the Reeb vector $\xi$ as an element of the Lie algebra of $\mathrm{Aut}(x,X)$. The subtorus in $\mathbb{T}$ generated by $\xi$ is independent of $\mathbb{T}$, and can be characterized as the smallest torus in $\mathrm{Aut}(x,X)$ whose Lie algebra contains $\xi$. If we assume that the torus $\mathbb{T}$ is generated by the Reeb vector $\xi$ (and we will often do), then we may recover $\mathbb{T}$ from the data $x\in(X;\xi)$. This justifies the absence of $\mathbb{T}$ in the notation. We will denote the torus generated by $\xi$ as $\langle\xi\rangle$.

Let us describe two extreme cases of Fano cone singularities in more details.

As we will see in Proposition 3.4, every klt singularity has a degeneration by test configuration to some Fano cone singularity (the proof relies on the notion of Kollár components). The local stability theory will allow us to find the “optimal” degeneration.

Chi Li observes that the K-semistable degeneration from Donaldson-Sun’s construction is induced by a valuation that minimizes what he calls the normalized volume. The definition involves two more classical invariants of valuations: the log discrepancy and the volume.

It can happen that $A_{X}(v)=+\infty$ for some valuation $v$. We denote by $\mathrm{Val}^{*}_{X,x}$ the set of valuations $v\in\mathrm{Val}_{X,x}$ with $A_{X}(v)<+\infty$.

A basic observation is that both log discrepancy and volume are homogeneous in the variable: if we rescale the valuation $v$ to $\lambda v$, we find

It follows that $A_{X}(v)^{n}\cdot\mathrm{vol}(v)$ is invariant under rescaling.

The local volume of a klt singularity can also be computed using normalized multiplicities of ideals, as observed in [Liu-vol-sing-KE]. This alternative approach offers a great deal of flexibility in the study of this invariant.

Here $\mathrm{lct}(\mathfrak{a})$ is the log canonical threshold (lct) of the ideal $\mathfrak{a}$, defined as

It is also the largest number $\lambda>0$ such that $(X,\mathfrak{a}^{\lambda})$ is log canonical. The log canonical threshold of a graded sequence $\mathfrak{a}_{\bullet}$ of ideals is defined in a similar manner, replacing $\mathfrak{a}$ by $\mathfrak{a}_{\bullet}$ in the above formula. By [JM-val-ideal-seq], the infimum is in fact a minimum.

The proof of the formula (2.2) is quite straightforward. On one hand, we have $A_{X}(v)\geq\mathrm{lct}(\mathfrak{a}_{\bullet}(v))$ and $\mathrm{vol}(v)=\mathrm{mult}(\mathfrak{a}_{\bullet}(v))$ for any valuation $v\in\mathrm{Val}_{X,x}^{*}$, hence

On the other hand, for any valuation $v\in\mathrm{Val}_{X,x}^{*}$ that computes $\mathrm{lct}(\mathfrak{a}_{\bullet})$ we may rescale it so that $v(\mathfrak{a}_{\bullet})=1$. By definition this gives $\mathrm{lct}(\mathfrak{a}_{\bullet})=A_{X}(v)$ and $\mathfrak{a}_{\bullet}\subseteq\mathfrak{a}_{\bullet}(v)$, hence $\mathrm{mult}(\mathfrak{a}_{\bullet})\geq\mathrm{vol}(v)$ and

It turns out that the local volume of a klt singularity is always positive [Li-normalized-volume] (we will sketch a proof in Section 3) and thus becomes an interesting invariant of the singularity. If $x\in X$ lives on some Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds, then the local volume $\widehat{\rm vol}(x,X)$ has the following differential geometric interpretation (see [LLX-nv-survey]*Theorem 5.6):

where the right hand side is the volume density (in the sense of geometric measure theory) of the Kähler-Einstein limit metric. An interesting question is the distribution of the possible values of local volumes, see Conjecture 6.5.

A guiding principle of the local stability theory, put forward by Li [Li-normalized-volume], is that the K-semistable degeneration of a klt singularity is induced by the valuation with the smallest normalized volume. For singularities on Gromov-Hausdorff limits of Kähler-Einstein Fano manifolds, this is confirmed in [LX-stability-higher-rank]*Section 3.1. Here we illustrate this connection through a few examples.

The local volumes of klt singularities also enjoy some nice properties. We only list some of them here, referring to [LLX-nv-survey] for a more thorough discussion.

Here we call $B\subseteq\mathcal{X}\to B$ a $\mathbb{Q}$-Gorenstein family of klt singularities if $\mathcal{X}$ is flat over $B$, $B\subseteq\mathcal{X}$ is a section of the projection, $K_{\mathcal{X}/B}$ is $\mathbb{Q}$-Cartier and $b\in\mathcal{X}_{b}$ is klt for any $b\in B$.

Note that the inequality part is also a consequence of the lower semi-continuity of local volumes, but the equality case requires more work.

In particular, local volumes are non-increasing under small birational morphisms. On the other hand, it is less clear how they behave under flips.

We now introduce the Stable Degeneration Conjecture, which gives a recipe for constructing the K-semistable degenerations of klt singularities using the minimizers of the normalized volume function.

Let us elaborate the various parts of the above conjecture. Since the K-semistable degeneration of a klt singularity eventually comes from the minimizing valuation of the normalized volume function, the existence of the minimizer is a necessary condition to begin with. The uniqueness part can be reformulated as saying that the normalized volume function has a unique minimizer on the normalized non-archimedean link. It implies the uniqueness of the K-semistable degeneration, since rescaling the valuation does not change the isomorphism class of the associated graded algebra.

Assuming that there exists a unique minimizer $v_{0}$, the natural candidate of the K-semistable degeneration (as we have discussed in Section 2.1) is $\mathbf{Spec}(\mathrm{gr}_{v_{0}}R)$. But there is a serious issue here, since a priori the algebra $\mathrm{gr}_{v_{0}}R$ need not be finitely generated. An obvious necessary condition is that the value semigroup $v_{0}(R\setminus\{0\})$ is finitely generated. With a bit more work, one can show that it is also necessary that the minimizer $v_{0}$ is a quasi-monomial valuation. This justifies the third item of the conjecture.

Unfortunately, there are still many quasi-monomial valuations whose associated graded algebras are not finitely generated, see Example 4.17. The finite generation part (also called the local higher rank finite generation conjecture) of the Stable Degeneration Conjecture turns out to be quite subtle.

Taking (1)–(4) for granted, let us elaborate the precise content of item (5). First we need to explain where the Reeb vector comes from. Denote by $r$ the rational rank of the quasi-monomial minimizer $v_{0}$. By choosing a (non-canonical) isomorphism $\Gamma\cong\mathbb{Z}^{r}$, we may replace the $\Gamma$-grading on $\mathrm{gr}_{v_{0}}R$ by a $\mathbb{Z}^{r}$-grading. In particular, we get a $\mathbb{T}=\mathbb{G}^{r}_{m}$-action on $\mathrm{gr}_{v_{0}}R$. Since $v_{0}$ takes the same positive value on each $\mathrm{gr}^{\lambda}_{v_{0}}R$, it induces a toric valuation on $\mathrm{gr}_{v_{0}}R$ and hence a Reeb vector $\xi_{0}$ on $X_{0}=\mathbf{Spec}(\mathrm{gr}_{v_{0}}R)$. The grading also determines a closed point $x_{0}$ that is the unique closed orbit of the torus action, thus we get a Fano cone singularity $x_{0}\in(X_{0};\xi_{0})$.

Next we shall define K-semistability for Fano cone singularities. The original definition from [CS-Kss-Sasaki, CS-Sasaki-Einstein] is via the non-negativity of generalized Futaki invariants. We choose the following definition, which is more convenient for our purpose.

Its equivalence with the original definition is shown in [LX-stability-higher-rank]*Theorem 2.34. Intuitively, the generalized Futaki invariants of the Fano cone singularity are “directional derivatives” of the normalized volume function at $\mathrm{wt}_{\xi}$, hence they are non-negative if $\mathrm{wt}_{\xi}$ is a minimizer. There is also a local-to-global correspondence: by [Li-equivariant, LL-vol-minimizer-KE, LX-stability-kc] (see the discussions in Example 2.20), a cone singularity $o\in C_{a}(V,L)$ is K-semistable if and only if the Fano base $V$ is K-semistable.

The stable degeneration of a klt singularity is a two-step process. Conjecture 2.24 takes care of the first step, the K-semistable degeneration. The other step, the K-polystable degeneration, can be done using the following theorem.

Let us clarify some of the terminologies in the above statement.

Next we define K-polystability. Again, the original definition involves generalized Futaki invariants, but we choose the following more convenient definition. They are equivalent by [LWX-metric-tangent-cone]. By [CS-Kss-Sasaki, CS-Sasaki-Einstein, Hua-thesis], we also know that a Fano cone singularity is K-polystable if and only if it admits a Ricci flat Kähler cone metric.

The intuition behind this definition is a notion of $S$-equivalence: two semistable objects are considered $S$-equivalent if one of them isotrivially degenerates to the other, and polystable objects are the ones without any further $S$-equivalent degenerations⁷⁷7The definition of polystable vector bundle and GIT-polystable point can both be formulated this way: two semistable vector bundles are $S$-equivalent if they have the same Jordan-Hölder factors, and a vector bundle is polystable if it is a direct sum of its Jordan-Hölder factors; similarly in GIT (geometric invariant theory), two GIT-semistable points are $S$-equivalent if their orbit closure intersect, and the GIT-polystable point represents the unique closed orbit in this $S$-equivalence class..

The proofs of Conjecture 2.24 and Theorem 2.26 will be sketched in Section 4.

Using Kollár component, we now discuss a special case of the Stable Degeneration Conjecture, namely when the minimizer is a divisorial valuation. It is worth noting that, in general, the minimizer can be a valuation of higher rank; one such example is the cone over $\mathrm{Bl}_{p}\mathbb{P}^{2}$, see [Blu-minimizer-exist]*Section 8.3. Nevertheless, the divisorial case will provide some intuition for our understanding of the higher rank case.

The key observation is that divisorial minimizers are necessarily Kollár components.