On boundedness of singularities and minimal log discrepancies of Kollár components, II

In addition to our previous work [Z-mld^K-1], the proof of Theorem 1.2 relies on several new ingredients. For simplicity, we assume $\Delta=0$, so that we are dealing with Fano cone singularities. Every such singularity $x\in X$ is an orbifold cone over some Fano variety $V$, so a natural idea is to prove Theorem 1.2 by showing the boundedness of the associated Fano variety $V$.

There are two main reasons why this naïve approach does not directly work. First, the orbifold base is highly non-unique; in fact, for a fixed Fano cone singularity the possible orbifold bases can be unbounded. For example, the simplest Fano cone singularity $0\in\mathbb{A}^{n}$ can be realized as an orbifold cone over any weighted projective space of dimension $n-1$, but without further constraint, weighted projective spaces do not form a bounded family.

Moreover, even when there is a canonical choice of the orbifold base (e.g. when the singularity has a unique $\mathbb{G}_{m}$-action), the anti-canonical volume of $V$ is only a fraction of the local volume of the singularity, where the factor is related to the Weil index of $V$ (defined as the largest number $q$ such that $-K_{V}\sim_{\mathbb{Q}}qL$ for some Weil divisor $L$). In particular, the volume of $V$ can a priori be arbitrarily small, which needs to be ruled out if we want any boundedness of this sort.

Our solution is to turn the local boundedness question into a global one by considering the projective orbifold cone $\overline{X}$ over $V$. A key observation is that by choosing the appropriate orbifold base $V$, the anti-canonical volume of $\overline{X}$ approximates the local volume of $x\in X$ and in particular is bounded from both above and below. This takes care of the second issue mentioned above.

Still, as we make different choices of $V$, the corresponding projective orbifold cones $\overline{X}$ can be unbounded. To circumvent this issue, we prove an effective birationality result for $\overline{X}$. More precisely, we show that regardless of the choice of $V$, there is some positive integer $m$ depending only on $\widehat{\rm vol}(x,X)$ and $\mathrm{mld}^{\mathrm{K}}(x,X)$ such that $|-mK_{\overline{X}}|$ induces a birational map that restricts to an embedding on $X$. This is the main technical part of the proof, and ultimately reduces to the construction of certain isolated non-klt centers and some careful analysis of certain Izumi type constants, see Section 3.2. Once the effective birationality is established, it is fairly straightforward to conclude the boundedness of $X$.

In Sections 2.2–2.5, we give the necessary background on $\mathrm{mld}^{\mathrm{K}}$, local volumes, log Fano cone singularities and their boundedness. In sections 2.6–2.7, we collect some useful results in previous works; these include results from [HLS-epsilon-plt-blowup] that tackles pairs with real coefficients, as well as modifications of some results from the prequel [Z-mld^K-1] of the present work. In Section 3, we present and prove a more general boundedness statement for polarized log Fano cone singularities, Theorem 3.1. Our main Theorem 1.2 will follow as an application of Theorem 3.1.

The author is partially supported by the NSF Grants DMS-2240926, DMS-2234736, a Clay research fellowship, as well as a Sloan fellowship. He would like to thank Harold Blum, János Kollár, Yuchen Liu and Chenyang Xu for helpful discussions and comments. He also likes to thank the referees for careful reading of the manuscript and several helpful suggestions.

We work over an algebraically closed field $\mathbbm{k}$ of characteristic $0$. We follow the standard terminology from [KM98, Kol13].

A pair $(X,\Delta)$ consists of a normal variety $X$ together with an effective $\mathbb{R}$-divisor $\Delta$ on $X$ (a priori, we do not require that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier). A singularity $x\in(X,\Delta)$ consists of a pair $(X,\Delta)$ and a closed point $x\in X$. We will always assume that $X$ is affine and $x\in\mathrm{Supp}(\Delta)$ (whenever $\Delta\neq 0$).

Suppose that $X$ is a normal variety. A prime divisor $F$ on some birational model $\pi\colon Y\to X$ (where $Y$ is normal and $\pi$ is proper) of $X$ is called a divisor over $X$. Given an $\mathbb{R}$-divisor $\Delta$ on $X$, we denote its strict transform on the birational model $Y$ by $\Delta_{Y}$. If $K_{X}+\Delta$ is $\mathbb{R}$-Cartier, the log discrepancy $A_{X,\Delta}(F)$ is defined to be

$$A_{X,\Delta}(F):=1+\mathrm{ord}_{F}(K_{Y}-\pi^{*}(K_{X}+\Delta)).$$

A valuation over a singularity $x\in X$ is an $\mathbb{R}$-valued valuation $v:K(X)^{*}\to\mathbb{R}$ (where $K(X)$ denotes the function field of $X$) such that $v$ is centered at $x$ (i.e. $v(f)>0$ if and only if $f\in\mathfrak{m}_{x}$) and $v|_{\mathbbm{k}^{*}}=0$. The set of such valuations is denoted as $\mathrm{Val}_{X,x}$. Given $\lambda\in\mathbb{R}$, the corresponding valuation ideal $\mathfrak{a}_{\lambda}(v)$ is

$$\mathfrak{a}_{\lambda}(v):=\{f\in\mathcal{O}_{X,x}\mid v(f)\geq\lambda\}.$$

When we refer to a constant $C$ as $C=C(n,\varepsilon,\cdots)$ it means $C$ only depends on $n,\varepsilon,\cdots$.

We first recall some definitions related to klt singularities and Kollár components.

By adjunction, we may write

$$(K_{Y}+\Delta_{Y}+E)|_{E}=K_{E}+\Delta_{E}$$

for some effective divisor $\Delta_{E}=\mathrm{Diff}_{E}(\Delta_{Y})$ (called the different) on $E$, and $(E,\Delta_{E})$ is a klt log Fano pair.

If $x\in(X,\Delta)$ is a klt singularity, then we can write $\Delta=\sum_{i=1}^{r}a_{i}\Delta_{i}$ as a convex combination of $\mathbb{Q}$-divisors $\Delta_{i}$ such that each $(X,\Delta_{i})$ is klt. Let $r>0$ be an integer such that $r(K_{X}+\Delta_{i})$ is Cartier for all $i$. Then as $A_{X,\Delta}(F)=\sum_{i=1}^{r}a_{i}A_{X,\Delta_{i}}(F)$, the possible values of log discrepancies $A_{X,\Delta}(F)$ belong to the discrete set $\left\{\left.\frac{1}{r}\sum_{i=1}^{r}a_{i}m_{i}\,\right|\,m_{i}\in\mathbb{N}\right\}$. This implies that the infimum in the above definition is also a minimum.

The following result will be useful later. Recall that the log canonical threshold $\mathrm{lct}(X,\Delta;D)$ of an effective $\mathbb{R}$-Cartier divisor $D$ with respect to a klt pair $(X,\Delta)$ is the largest number $t\geq 0$ such that $(X,\Delta+tD)$ is klt, and the $\alpha$-invariant of a log Fano pair $(X,\Delta)$ is defined as the infimum of the log canonical thresholds $\mathrm{lct}(X,\Delta;D)$ where $0\leq D\sim_{\mathbb{R}}-(K_{X}+\Delta)$. The log canonical threshold $\mathrm{lct}_{x}(X,\Delta;D)$ at a closed point $x\in X$ is defined analogously.

We next briefly recall the definition of the local volumes of klt singularities [Li-normalized-volume]. Let $x\in(X,\Delta)$ be a klt singularity and let $n=\dim X$. The log discrepancy function

$$A_{X,\Delta}\colon\mathrm{Val}_{X,x}\to\mathbb{R}\cup\{+\infty\},$$

is defined as in [JM-val-ideal-seq] and [BdFFU-log-discrepancy, Theorem 3.1]. It generalizes the usual log discrepancies of divisors; in particular, for divisorial valuations, i.e. valuations of the form $\lambda\cdot\mathrm{ord}_{F}$ where $\lambda>0$ and $F$ is a divisor over $X$, we have

$$A_{X,\Delta}(\lambda\cdot\mathrm{ord}_{F})=\lambda\cdot A_{X,\Delta}(F).$$

We denote by $\mathrm{Val}^{*}_{X,x}$ the set of valuations $v\in\mathrm{Val}_{X}$ with center $x$ and $A_{X,\Delta}(v)<+\infty$. The volume of a valuation $v\in\mathrm{Val}_{X,x}$ is defined as

$$\mathrm{vol}(v)=\mathrm{vol}_{X,x}(v)=\limsup_{m\to\infty}\frac{\ell(\mathcal{O}_{X,x}/\mathfrak{a}_{m}(v))}{m^{n}/n!}.$$

By [Li-normalized-volume, Theorem 1.2], the local volume of a klt singularity is always positive. We will frequently use the following properties of local volumes.

In this subsection we recall the definition of log Fano cone singularities and their K-semistability. These notions originally appear in the study of Sasaki-Einstein metrics [CS-Kss-Sasaki, CS-Sasaki-Einstein] and is further explored in works related to the Stable Degeneration Conjecture [LX-stability-higher-rank, LWX-metric-tangent-cone].

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

$$R=\oplus_{\alpha\in M}R_{\alpha},$$

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 any $\xi\in\mathfrak{t}^{+}_{\mathbb{R}}$, we can define a valuation $\mathrm{wt}_{\xi}$ (called a toric valuation) by setting

$$\mathrm{wt}_{\xi}(f):=\min\{\langle\xi,\alpha\rangle\mid\alpha\in M,f_{\alpha}\neq 0\}$$

where $f\in R$. It is not hard to verify that $\mathrm{wt}_{\xi}\in\mathrm{Val}_{X,x}$.

By abuse of convention, a good $\mathbb{T}$-action on a klt singularity $x\in(X,\Delta)$ means a good $\mathbb{T}$-action on $X$ such that $x$ is the vertex and $\Delta$ is $\mathbb{T}$-invariant. Using terminology from Sasakian geometry, we say a polarized log Fano cone $x\in(X,\Delta;\xi)$ is quasi-regular if $\xi$ generates a $\mathbb{G}_{m}$-action (i.e. $\xi$ is a real multiple of some element of $N$); otherwise, we say that $x\in(X,\Delta;\xi)$ is irregular.

We will often use the following result to perturb an irregular polarization to a quasi-regular one.

This definition differs from the original ones from [CS-Kss-Sasaki, CS-Sasaki-Einstein], but they are equivalent by [LX-stability-higher-rank, Theorem 2.34]. For our purpose, the above definition is more convenient. Since the minimizer of the normalized volume function is unique up to rescaling [XZ-minimizer-unique], the polarization $\xi$ is essentially determined by the K-semistability condition and hence we often omit the polarization and simply say $x\in(X,\Delta)$ is K-semistable.

By definition, $0<\Theta(X,\Delta;\xi)\leq 1$ and $\Theta(X,\Delta;\xi)=1$ if and only if $x\in(X,\Delta;\xi)$ is K-semistable. By Lemma 2.11, similar statement holds in the unpolarized case.

In this subsection we define boundedness of singularities and recall some properties of singularity invariants in bounded families. They will be useful in proving the easier direction of Theorem 1.2.

We next define the boundedness of log Fano cone singularities. Note that our definition is a priori stronger than their boundedness as klt singularities (it’s not clear to us whether they are equivalent).

The following result will help us reduce many questions about pairs with real coefficients to ones with rational coefficients. We will often use it without explicit mention when we refer to results that are originally stated for $\mathbb{Q}$-coefficients.

We also note the Stable Degeneration Conjecture holds for pairs with real coefficients.

In this subsection we collect slight modification of several results from [Z-mld^K-1] that we need in later proofs. For the first result, recall that we say a singularity $x\in(X,\Delta)$ is of klt type if there exists some effective $\mathbb{R}$-divisor $D$ on $X$ such that $x\in(X,\Delta+D)$ is klt.

The second result is extracted from the proof of [Z-mld^K-1, Theorem 4.1].

Since the volume ratio is continuous in the Reeb vector $\xi$ (Lemma 2.11), by perturbing the polarization, we see that it suffices to prove Theorem 3.1 when $\mathcal{S}$ consists of quasi-regular log Fano cones. Every quasi-regular log Fano cone singularity has a natural affine orbifold cone structure induced by the polarization. The proof of Theorem 3.1 relies on the associated projective orbifold cone construction. In this subsection, we first fix some notation and recall some basic properties of orbifold cones from [Kol-Seifert-bundle].

For ease of notation, denote $X=C_{a}(V,L)$, $\overline{X}=C_{p}(V,L)$ and let $x\in X$ be the vertex of the orbifold cone. On the projective orbifold cone, we also denote by $V_{\infty}$ the divisor at infinity, i.e., the divisor corresponding to $(s=0)$. We have $X\cong\overline{X}\setminus V_{\infty}$. Note that $X\setminus\{x\}$ is a Seifert $\mathbb{G}_{m}$-bundle over $V$ (in the sense of [Kol-Seifert-bundle]). Thus for any effective $\mathbb{R}$-divisor $\Delta_{V}$ on $V$, we can define the affine (resp. projective) orbifold cone over the polarized pair $(V,\Delta_{V};L)$ as the pair $(X,\Delta)$ (resp. $(\overline{X},\overline{\Delta})$), where $\Delta$ (resp. $\overline{\Delta}$) is the closure of the pullback of $\Delta_{V}$ to $X\setminus\{x\}$ (since the projection $X\setminus\{x\}\to V$ is equidimensional, the pullback of a Weil divisor is well-defined). Every quasi-regular polarized log Fano cone singularity $x\in(X,\Delta;\xi)$ arises in this way, so we can talk about its associated projective orbifold cone $(\overline{X},\overline{\Delta})$.

Alternatively, the Seifert $\mathbb{G}_{m}$-bundle $X\setminus\{x\}$ can be compactified by adding $V_{\infty}$ and the missing zero section $V_{0}$; the resulting space $\overline{Y}$ is also the orbifold blowup of $\overline{X}$ at $x$. Similarly, let $Y$ be the orbifold blowup of $X$ at $x$ (i.e. $Y=\overline{Y}\setminus V_{\infty}$). Write the fractional part of $L$ as $\{L\}=\sum_{i=1}^{k}\frac{a_{i}}{b_{i}}D_{i}$ for some prime divisors $D_{i}$ on $V$ and $0<a_{i}<b_{i}$ coprime integers. We denote $\Delta_{L}:=\sum_{i=1}^{k}\frac{b_{i}-1}{b_{i}}D_{i}$. Then the pairs $(V_{0},\mathrm{Diff}_{V_{0}}(0))$ and $(V_{\infty},\mathrm{Diff}_{V_{\infty}}(0))$ obtained by taking adjunction over some big open set of $V$ from the pair $(\overline{X},V_{0}+V_{\infty})$ are both isomorphic to $(V,\Delta_{L})$ (this is a local computation, see [Kol-Seifert-bundle, Section 4]).

Under these notation, we have the following well known result [Kol-Seifert-bundle] (cf. [Kol13, Section 3.1] for analogous statement for usual cones).

The next result is a key observation in the proof of Theorem 3.1. It expresses the normalized volume of a polarized log Fano cone singularities as the global volume of the associated projective orbifold cone.