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

Ziquan Zhuang  Department of Mathematics, Johns Hopkins University, Baltimore,
MD 21218, USA<EMAIL_ADDRESS>
###### Abstract.

We show that a set of K-semistable log Fano cone singularities is bounded if
and only if their local volumes are bounded away from zero, and their minimal
log discrepancies of Kollár components are bounded from above. As corollaries,
we confirm the boundedness conjecture for K-semistable log Fano cone
singularities in dimension three, and show that local volumes of
$3$-dimensional klt singularities only accumulate at zero.

##  1\. Introduction

Following the recent study on K-stability of Fano varieties (see [Xu-K-
stability-survey] for a comprehensive survey), there has been growing interest
in establishing a parallel stability theory for klt singularities, which are
the local analog of Fano varieties. In the global theory, boundedness of Fano
varieties plays an important role. Building on the seminal work of Birkar
[Birkar-bab-1, Birkar-bab-2], Jiang [Jia-Kss-Fano-bdd] proved that (in any
fixed dimension) K-semistable Fano varieties with anti-canonical volumes
bounded away from zero form a bounded family.111Several different proofs were
later found in [LLX-nv-survey, XZ-minimizer-unique]. This is the first step in
the general construction of the K-moduli space of Fano varieties; it is also a
key ingredient in the proof [LXZ-HRFG] of the global version of the Higher
Rank Finite Generation Conjecture.

To advance the local stability theory, it is therefore natural to investigate
the boundedness of klt singularities. Several years ago, Li [Li-normalized-
volume] introduced an interesting invariant of klt singularities called the
local volume. It has become clear that the stability theory of klt
singularities should be built around this invariant. In particular, it is
speculated that (in any fixed dimension) klt singularities with local volumes
bounded away from zero are specially bounded, i.e. they isotrivially
degenerate to a bounded family. This is known in some special cases [HLQ-vol-
ACC, MS-bdd-toric, MS-bdd-complexity-one, Z-mld^K-1], but the general
situation is still quite mysterious.

By the recent solution of the Stable Degeneration Conjecture [Blu-minimizer-
exist, LX-stability-higher-rank, Xu-quasi-monomial, XZ-minimizer-unique, LWX-
metric-tangent-cone, XZ-SDC] (see also [LX-stability-kc, BLQ-convexity]),
every klt singularity has a canonical “stable degeneration” to a K-semistable
log Fano cone singularity (see Section 2.4 for the precise definition; roughly
speaking, K-semistable log Fano cone singularities are generalizations of
cones over K-semistable Fano varieties). This suggests a more precise
boundedness conjecture ([XZ-SDC, Conjecture 1.7], see also [SZ-no-
semistability, Problem 6.9]):

######  Conjecture 1.1.

Fix $n\in\mathbb{N}$, $\varepsilon>0$ and a finite set $I\subseteq[0,1]$. Then
the set

$\Big{\\{}(X,\Delta)\ \Big{|}\begin{array}[]{l}x\in(X,\Delta)\mbox{ is a
K-semistable log Fano cone singularity},\\\ \dim X=n,\,{\rm
Coef}(\Delta)\subseteq I,\,\mbox{and }\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon\end{array}\Big{\\}}$

is bounded.

Here $\widehat{\rm vol}(x,X,\Delta)$ denotes the local volume of the
singularity $x\in(X,\Delta)$. This conjecture has been verified for toric
singularities [MS-bdd-toric, Z-mld^K-1], for hypersurface singularities, and
for singularities with torus actions of complexity one [MS-bdd-complexity-
one].

In this paper, we study Conjecture 1.1 using the minimal log discrepancies of
Kollár components (or simply $\mathrm{mld}^{\mathrm{K}}$), see Section 2.2.
Our main result gives a boundedness criterion in terms of
$\mathrm{mld}^{\mathrm{K}}$ and the local volume.

######  Theorem 1.2 (=Corollary 3.10).

Fix $n\in\mathbb{N}$ and consider a set $\mathcal{S}$ of $n$-dimensional
K-semistable log Fano cone singularities with coefficients in a fixed finite
set $I\subseteq[0,1]$. Then $\mathcal{S}$ is bounded if and only if there
exist some $\varepsilon,A>0$ such that

$\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon\quad\mathit{and}\quad\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq
A$

for all $x\in(X,\Delta)$ in $\mathcal{S}$.

This upgrades the special boundedness result from our previous work
[Z-mld^K-1] to actual boundedness. We also prove boundedness results for more
general log Fano cone singularities, replacing the K-semistability requirement
by lower bounds on stability thresholds (as introduced in [Hua-thesis]), see
Corollary 3.14.

Comparing Conjecture 1.1 and Theorem 1.2, we are naturally led to the
following conjecture, already raised in [Z-mld^K-1]:

######  Conjecture 1.3 ([Z-mld^K-1, Conjecture 1.7]).

Let $n\in\mathbb{N}$, $\varepsilon>0$ and let $I\subseteq[0,1]$ be a finite
set. Then there exists some constant $A$ depending only on $n,\varepsilon,I$
such that

$\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq A$

for any $n$-dimensional klt singularity $x\in(X,\Delta)$ with
$\mathrm{Coef}(\Delta)\subseteq I$ and $\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon$.

This is known in dimension up to three [Z-mld^K-1]. By Theorem 1.2, we then
get a complete solution of Conjecture 1.1 in dimension three (the surface case
was already treated in [HLQ-vol-ACC]). The same result is also independently
proved in [LMS-bdd-dim-3] using a different method.

######  Corollary 1.4 (=Corollaries 3.16+3.17).

Conjecture 1.1 holds in dimension $3$. Moreover, for any fixed finite
coefficient set $I\subseteq[0,1]$, the set of possible local volumes of
$3$-dimensional klt singularities is discrete away from zero.

###  1.1. Strategy of the proof

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 $X$.

###  1.2. Structure of the article

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.

### Acknowledgement

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.

##  2\. Preliminaries

###  2.1. Notation and conventions

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 $n,\varepsilon,\cdots$.

###  2.2. Kollár components

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

######  Definition 2.1 ([KM98, Definition 2.34]).

We say a pair $(X,\Delta)$ is klt if $K_{X}+\Delta$ is $\mathbb{R}$-Cartier,
and for any prime divisor $F$ over $X$ we have $A_{X,\Delta}(F)>0$. We say
$x\in(X,\Delta)$ is a klt singularity if $(X,\Delta)$ is klt.

######  Definition 2.2 ([Xu-pi_1-finite]).

Let $x\in(X,\Delta)$ be a klt singularity and let $E$ be a prime divisor over
$X$. If there exists a proper birational morphism $\pi\colon Y\to X$ such that
$E=\pi^{-1}(x)$ is the unique exceptional divisor, $(Y,E+\Delta_{Y})$ is plt
and $-(K_{Y}+\Delta_{Y}+E)$ is $\pi$-ample, we call $E$ a _Kollár component_
over $x\in(X,\Delta)$ and $\pi\colon Y\to X$ the plt blowup of $E$.

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.

######  Definition 2.3 ([Z-mld^K-1]).

Let $x\in(X,\Delta)$ be a klt singularity. The minimal log discrepancy of
Kollár components, denoted $\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)$, is the
infimum of the log discrepancies $A_{X,\Delta}(E)$ as $E$ varies among all
Kollár components over $x\in(X,\Delta)$.

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.

######  Lemma 2.4.

Let $E$ be a Kollár component over a klt singularity $x\in(X,\Delta)$. Then
for any effective $\mathbb{R}$-Cartier divisor $D$ on $X$, we have

$\mathrm{lct}_{x}(X,\Delta;D)\geq\min\\{1,\alpha(E,\Delta_{E})\\}\cdot\frac{A_{X,\Delta}(E)}{\mathrm{ord}_{E}(D)}.$

###### Proof.

Let $\alpha=\min\\{1,\alpha(E,\Delta_{E})\\}$, let
$t=\frac{A_{X,\Delta}(E)}{\mathrm{ord}_{E}(D)}$ and let $\pi\colon Y\to X$ be
the plt blowup of $E$. Then we have

$\pi^{*}(K_{X}+\Delta+tD)=K_{Y}+\Delta_{Y}+tD_{Y}+E,$

which gives
$tD_{Y}|_{E}\sim_{\mathbb{R}}-(K_{Y}+\Delta_{Y}+E)|_{E}\sim_{\mathbb{R}}-(K_{E}+\Delta_{E})$.
By the definition of alpha invariants, the pair $(E,\Delta_{E}+\alpha
tD_{Y}|_{E})$ is log canonical, hence by inversion of adjunction [KM98,
Theorem 5.50] we know that $(Y,\Delta_{Y}+\alpha tD_{Y}+E)$ is lc around $E$.
As $\alpha\leq 1$, we also have

$\pi^{*}(K_{X}+\Delta+\alpha tD)=K_{Y}+\Delta_{E}+\alpha tD_{Y}+sE$

for some $s\leq 1$, thus by the above discussion we deduce that
$(Y,\Delta_{Y}+\alpha tD_{Y}+sE)$ is sub-lc around $E$, and hence
$(X,\Delta+\alpha tD)$ is lc at $x$. In other words,
$\mathrm{lct}_{x}(X,\Delta;D)\geq\alpha t$ as desired. ∎

###  2.3. Local volumes

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

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

######  Definition 2.5.

Let $x\in(X,\Delta)$ be an $n$-dimensional klt singularity. For any
$v\in\mathrm{Val}^{*}_{X,x}$, we define the _normalized volume_ of $v$ as

$\widehat{\rm
vol}_{X,\Delta}(v):=A_{X,\Delta}(v)^{n}\cdot\mathrm{vol}_{X,x}(v).$

The _local volume_ of $x\in(X,\Delta)$ is defined as

$\widehat{\rm vol}(x,X,\Delta):=\inf_{v\in\mathrm{Val}^{*}_{X,x}}\widehat{\rm
vol}_{X,\Delta}(v).$

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.

######  Lemma 2.6 ([LX-cubic-3fold, Theorem 1.6]).

Let $x\in(X,\Delta)$ be a klt singularity of dimension $n$. Then $\widehat{\rm
vol}(x,X,\Delta)\leq n^{n}$.

######  Lemma 2.7 ([XZ-minimizer-unique, Corollary 1.4]).

Let $x\in(X,\Delta)$ be a klt singularity of dimension $n$ and let $D$ be a
$\mathbb{Q}$-Cartier Weil divisor on $X$. Then the Cartier index of $D$ is at
most $\frac{n^{n}}{\widehat{\rm vol}(x,X,\Delta)}$.

###  2.4. Log Fano cone singularities

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].

######  Definition 2.8.

Let $X=\mathbf{Spec}(R)$ be a normal affine variety and
$\mathbb{T}=\mathbb{G}_{m}^{r}$ ($r>0$) an algebraic torus. We say that a
$\mathbb{T}$-action 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 $\mathbb{T}$-orbit.
We call $x$ the vertex of the $\mathbb{T}$-variety $X$, and call the
corresponding singularity $x\in X$ a $\mathbb{T}$-singularity.

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.

######  Definition 2.9.

A _Reeb vector_ on $X$ is a vector $\xi\in N_{\mathbb{R}}$ such that
$\langle\xi,\alpha\rangle>0$ for all $0\neq\alpha\in M$ with $R_{\alpha}\neq
0$. The set $\mathfrak{t}^{+}_{\mathbb{R}}$ of Reeb vectors is called the Reeb
cone.

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
$\mathrm{wt}_{\xi}\in\mathrm{Val}_{X,x}$.

######  Definition 2.10.

A log Fano cone singularities is a klt singularities that admits a nontrivial
good torus action. A polarized log Fano cone singularity $x\in(X,\Delta;\xi)$
consists of a log Fano cone singularity $x\in(X,\Delta)$ together with a Reeb
vector $\xi$ (called a polarization).

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.

######  Lemma 2.11.

The function $\xi\mapsto\widehat{\rm vol}_{X,\Delta}(\mathrm{wt}_{\xi})$
defined on the Reeb cone is continuous and has a minimum.

###### Proof.

This follows from [LX-stability-higher-rank, Theorem 2.15(3) and Proposition
2.39]. ∎

######  Definition 2.12.

We say a polarized log Fano cone singularity $x\in(X,\Delta;\xi)$ is _K-
semistable_ if

$\widehat{\rm vol}(x,X,\Delta)=\widehat{\rm
vol}_{X,\Delta}(\mathrm{wt}_{\xi}).$

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.

######  Definition 2.13.

The _volume ratio_ of a polarized log Fano cone singularity
$x\in(X,\Delta;\xi)$ is defined to be

$\Theta(X,\Delta;\xi):=\frac{\widehat{\rm vol}(x,X,\Delta)}{\widehat{\rm
vol}_{X,\Delta}(\mathrm{wt}_{\xi})}.$

The _volume ratio_ of a log Fano cone singularity $x\in(X,\Delta)$ is defined
to be

$\Theta(x,X,\Delta):=\sup_{\xi}\Theta(X,\Delta;\xi),$

where the supremum runs over all polarizations on $X$.

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.

###  2.5. Bounded family of singularities

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.

######  Definition 2.14.

We call $B\subseteq(\mathcal{X},\mathcal{D})\to B$ an $\mathbb{R}$-Gorenstein
family of klt singularities (over a normal but possibly disconnected base $B$)
if

  1. (1)

$\mathcal{X}$ is flat over $B$, and $B\subseteq\mathcal{X}$ is a section of
the projection,

  2. (2)

For any closed point $b\in B$, $\mathcal{X}_{b}$ is connected, normal and is
not contained in $\mathrm{Supp}(\mathcal{D})$,

  3. (3)

$K_{\mathcal{X}/B}+\mathcal{D}$ is $\mathbb{R}$-Cartier and
$b\in(\mathcal{X}_{b},\mathcal{D}_{b})$ is a klt singularity for any $b\in B$.

######  Lemma 2.15.

Let $B\subseteq(\mathcal{X},\mathcal{D})\to B$ be an $\mathbb{R}$-Gorenstein
family of klt singularities. Then there exist constants $\varepsilon,A>0$ such
that

$\widehat{\rm
vol}(b,\mathcal{X}_{b},\mathcal{D}_{b})\geq\varepsilon\quad\mathrm{and}\quad\mathrm{mld}^{\mathrm{K}}(b,\mathcal{X}_{b},\mathcal{D}_{b})\leq
A$

for all closed point $b\in B$.

###### Proof.

The uniform lower bound of the local volumes follows from their lower semi-
continuity [BL-vol-lsc] in families together with Noetherian induction. Note
that although the main result of [BL-uks-openness] is stated for families with
$\mathbb{Q}$-coefficients, it remains valid for real coefficients. This is
because there exist $\mathbb{Q}$-divisors
$\mathcal{D}^{\prime}\leq\mathcal{D}$ such that
$K_{\mathcal{X}/B}+\mathcal{D}^{\prime}$ is $\mathbb{Q}$-Cartier and the
coefficients of $\mathcal{D}-\mathcal{D}^{\prime}$ are arbitrarily small; in
particular, $\widehat{\rm
vol}(b,\mathcal{X}_{b},\mathcal{D}_{b})\geq\widehat{\rm
vol}(b,\mathcal{X}_{b},\mathcal{D}^{\prime}_{b})$ and their difference can be
made arbitrarily small for any fixed $b\in B$. Thus the lower semicontinuity
of $\widehat{\rm vol}(b,\mathcal{X}_{b},\mathcal{D}_{b})$ in $b\in B$ follows
from the lower semicontinuity of $\widehat{\rm
vol}(b,\mathcal{X}_{b},\mathcal{D}^{\prime}_{b})$. Alternatively, the local
volume lower bound follows from the constructibility of the volume function
[Xu-quasi-monomial, Theorem 1.3], which also holds for families with real
coefficients by the discussions in Theorem 2.19.

The uniform upper bound of $\mathrm{mld}^{\mathrm{K}}$ is a direct consequence
of [HLQ-vol-ACC, Theorem 2.34]: after a (quasi-finite and surjective) base
change, the family admits a flat family of Kollár components (in the sense of
[HLQ-vol-ACC, Definition 2.32]), whose log discrepancy is locally constant and
hence uniformly bounded. ∎

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).

######  Definition 2.16.

We say that a set $\mathcal{S}$ of polarized log Fano cone singularities is
bounded if there exists finitely many $\mathbb{R}$-Gorenstein families
$B_{i}\subseteq(\mathcal{X}_{i},\mathcal{D}_{i})\to B_{i}$ of klt
singularities, each with a fiberwise good $\mathbb{T}_{i}$-action for some
nontrivial algebraic torus $\mathbb{T}_{i}$, such that every
$x\in(X,\Delta;\xi)$ in $\mathcal{S}$ is isomorphic to
$b\in(\mathcal{X}_{i,b},\mathcal{D}_{i,b};\xi_{b})$ for some $i$, some $b\in
B_{i}$ and some $\xi_{b}\in N(\mathbb{T}_{i})_{\mathbb{R}}$. We say
$\mathcal{S}$ is log bounded if we only ask $x\in(X;\xi)$ to be isomorphic to
$b\in(\mathcal{X}_{i,b};\xi_{b})$ and that
$\mathrm{Supp}(\Delta)\subseteq\mathrm{Supp}(\mathcal{D}_{i,b})$ under this
isomorphism.

We say that a set $\mathcal{C}$ of log Fano cone singularities is bounded if
it’s the underlying set of singularities from a bounded set of polarized log
Fano cone singularities.

######  Lemma 2.17.

Let $\mathcal{C}$ be a bounded set of log Fano cone singularities. Then there
exists some constant $\theta>0$ such that $\Theta(x,X,\Delta)\geq\theta$ for
all $x\in(X,\Delta)$ in $\mathcal{C}$.

###### Proof.

By definition, it suffices to show that for any $\mathbb{R}$-Gorenstein family
of log Fano cone singularities $B\subseteq(\mathcal{X},\mathcal{D})\to B$ with
a fiberwise good torus $\mathbb{T}$-action, we have a uniform positive lower
bound of $\Theta(b,\mathcal{X}_{b},\mathcal{D}_{b})$. We may assume that $B$
is connected. Replacing $\mathcal{X}$ by $\mathbb{T}$-invariant affine subset,
We may also assume that $\mathcal{X}=\mathbf{Spec}(\mathcal{R})$ is affine. We
have a weight decomposition $\mathcal{R}=\oplus_{\alpha\in
M}\mathcal{R}_{\alpha}$ which reduces to the weight decomposition on the
fibers. By flatness of $\mathcal{X}\to B$, each $\mathcal{R}_{\alpha}$ is flat
over $B$. It follows that the Reeb cone
$\mathfrak{t}^{+}_{\mathbb{R},b}\subseteq N_{\mathbb{R}}$ is locally constant
in $b\in B$, hence is constant as $B$ is connected. Choose any primitive
$\xi\in N$ that lies in the (constant) Reeb cone. It suffices to show that
$\Theta(b,\mathcal{X}_{b},\mathcal{D}_{b};\xi)$ is uniformly bounded from
below. Since the $\mathbb{T}$-action is good, the $\mathcal{R}_{\alpha}$’s are
also finite over $B$, hence locally free by flatness. From the defition of
volume, this easily implies that $\mathrm{vol}(\mathrm{wt}_{\xi})$ is constant
as $b\in B$ varies. Since $\xi\in N$ comes from a $\mathbb{G}_{m}$-action,
there is a divisor $\mathcal{E}$ over $\mathcal{X}$ such that
$\mathrm{ord}_{\mathcal{E}}=\mathrm{wt}_{\xi}$ as valuations over
$\mathcal{X}$ and hence
$A_{\mathcal{X}_{b},\mathcal{D}_{b}}(\mathrm{wt}_{\xi})=A_{\mathcal{X}_{b},\mathcal{D}_{b}}(\mathrm{ord}_{\mathcal{E}_{b}})=A_{\mathcal{X},\mathcal{D}}(\mathcal{E})$
for general $b\in B$. By Noetherian induction, this implies that
$A_{\mathcal{X}_{b},\mathcal{D}_{b}}(\mathrm{wt}_{\xi})$ is constructible in
$b\in B$ and thus uniformly bounded from above. Finally, by Lemma 2.15 we know
that $\widehat{\rm vol}(b,\mathcal{X}_{b},\mathcal{D}_{b})\geq\varepsilon$ for
some constant $\varepsilon>0$. Putting these together we deduce from the
definition that $\Theta(b,\mathcal{X}_{b},\mathcal{D}_{b};\xi)$ is uniformly
bounded from below. ∎

###  2.6. Real coefficients

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.

######  Lemma 2.18.

Let $n\in\mathbb{N}$ and let $I\subseteq[0,1]$ be a finite set. Then there
exists a finite set $I^{\prime}\subseteq[0,1]\cap\mathbb{Q}$ depending only on
$n$ and $I$ such that the following holds.

For any $n$-dimensional klt singularity $x\in(X,\Delta)$ with coefficients in
$I$, there exists some effective $\mathbb{Q}$-divisor
$\Delta^{\prime}\geq\Delta$ on $X$ with coefficients in $I^{\prime}$, such
that $\mathrm{Supp}(\Delta)=\mathrm{Supp}(\Delta^{\prime})$,
$x\in(X,\Delta^{\prime})$ is klt, $\widehat{\rm vol}(x,X,\Delta^{\prime})\geq
2^{-n}\widehat{\rm vol}(x,X,\Delta)$ and
$\mathrm{mld}^{\mathrm{K}}(x,X,\Delta^{\prime})\leq
2\cdot\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)$. If in addition
$x\in(X,\Delta;\xi)$ is a log Fano cone singularity, then
$\Theta(x,X,\Delta^{\prime};\xi)\geq 4^{-n}\Theta(x,X,\Delta;\xi)$.

###### Proof.

This is essentially a consequence of [HLS-epsilon-plt-blowup, Theorem 5.6].
Without loss of generality we may assume $1\in I$. Write
$I=\\{a_{1},\dots,a_{m}\\}$ and let
$\mathbf{a}=(a_{1},\dots,a_{m})\in\mathbb{R}^{m}$. Let
$V\subseteq\mathbb{R}^{m}$ be the rational envelope of $\mathbf{a}$, i.e. the
smallest affine subspace defined over $\mathbb{Q}$ that contains $\mathbf{a}$.
By [HLS-epsilon-plt-blowup, Theorem 5.6], there exists an open neighbourhood
$U$ of $\mathbf{a}\in V$ depending only on $n$ and $I$ such that for any
$n$-dimensional singularity $x\in X$ and any Weil divisors
$\Delta_{1},\dots,\Delta_{m}\geq 0$, if
$x\in(X,\sum_{i=1}^{m}a_{i}\Delta_{i})$ is klt (resp. plt) then
$x\in(X,\sum_{i=1}^{m}a^{\prime}_{i}\Delta_{i})$ is klt (resp. plt) for any
$\mathbf{a}^{\prime}=(a^{\prime}_{1},\dots,a^{\prime}_{m})\in U$. Note that
the plt case implies that if $E$ is a Kollár component over
$x\in(X,\sum_{i=1}^{m}a_{i}\Delta_{i})$ then it is also a Kollár component
over $x\in(X,\sum_{i=1}^{m}a^{\prime}_{i}\Delta_{i})$.

Choose some $\mathbf{a}^{\prime}=(a^{\prime}_{1},\dots,a^{\prime}_{m})\in
U\cap\mathbb{Q}^{m}$ such that
$2\mathbf{a}^{\prime}-\mathbf{a},2\mathbf{a}-\mathbf{a}^{\prime}\in U$. We
claim that $I^{\prime}=\\{a^{\prime}_{1},\dots,a^{\prime}_{m}\\}$ satisfies
the required conditions. Indeed, for any $n$-dimensional klt singularity
$x\in(X,\Delta=\sum_{i=1}^{m}a_{i}\Delta_{i})$ (where the $\Delta_{i}$’s are
Weil divisors), if we set
$\Delta^{\prime}=\sum_{i=1}^{m}a^{\prime}_{i}\Delta_{i}$ and let
$\Delta_{1}=2\Delta^{\prime}-\Delta$, then by our choice of $U$ and
$\mathbf{a}^{\prime}$ we know that
$\mathrm{Supp}(\Delta^{\prime})=\mathrm{Supp}(\Delta)$,
$x\in(X,\Delta^{\prime})$ and $x\in(X,\Delta_{1})$ are both klt, and any
Kollár component over $x\in(X,\Delta)$ is also a Kollár component over
$x\in(X,\Delta^{\prime})$. Moreover, since
$\Delta^{\prime}=\frac{1}{2}(\Delta+\Delta_{1})$, we have
$A_{X,\Delta^{\prime}}(v)=\frac{1}{2}(A_{X,\Delta}(v)+A_{X,\Delta_{1}}(v))\geq\frac{1}{2}A_{X,\Delta}(v)$
for any valuation $v\in\mathrm{Val}_{X,x}^{*}$. Similarly using
$2\mathbf{a}-\mathbf{a}^{\prime}\in U$ we get $A_{X,\Delta^{\prime}}(v)\leq
2A_{X,\Delta}(v)$. In particular we see that $2^{-n}\widehat{\rm
vol}_{X,\Delta}(v)\leq\widehat{\rm vol}_{X,\Delta^{\prime}}(v)\leq
2^{n}\widehat{\rm vol}_{X,\Delta}(v)$. These imply the inequalities about
$\widehat{\rm vol}$, $\mathrm{mld}^{\mathrm{K}}$ and the volume ratio (all by
definition). ∎

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

######  Theorem 2.19 ([Blu-minimizer-exist, LX-stability-higher-rank, Xu-
quasi-monomial, XZ-minimizer-unique, XZ-SDC]).

Every klt singularity $x\in(X,\Delta)$ has a special degeneration to a
K-semistable log Fano cone singularity $x_{0}\in(X_{0},\Delta_{0})$ with
$\widehat{\rm vol}(x,X,\Delta)=\widehat{\rm vol}(x_{0},X_{0},\Delta_{0})$

###### Proof.

The arguments in [Blu-minimizer-exist, LX-stability-higher-rank, XZ-minimizer-
unique, XZ-SDC] extend directly to real coefficients, since they are not
sensitive to the coefficients. The proof in [Xu-quasi-monomial] uses the
existence of monotonic $N$-complement which requires
$\mathbb{Q}$-coefficients, but the main results in _loc. cit._ can be extended
to $\mathbb{R}$-coefficients by [HLQ-vol-ACC, Theorems 3.3 and 3.4]. Thus [XZ-
SDC, Theorem 1.2] holds for $\mathbb{R}$-pairs. In particular, the minimizer
$v$ of the normalized volume function $\widehat{\rm vol}_{X,\Delta}$ induces a
special degeneration of $x\in(X=\mathbf{Spec}(R),\Delta)$ to a K-semistable
log Fano cone
$x_{0}\in(X_{0}=\mathbf{Spec}(\mathrm{gr}_{v}R),\Delta_{0};\xi_{v})$. By [LX-
stability-higher-rank, Lemma 2.58], we have

$\widehat{\rm vol}_{X,\Delta}(v)=\widehat{\rm
vol}_{X_{0},\Delta_{0}}(\mathrm{wt}_{\xi_{v}}).$

Since $\widehat{\rm vol}(x,X,\Delta)=\widehat{\rm vol}_{X,\Delta}(v)$ (as $v$
is the minimizer of $\widehat{\rm vol}_{X,\Delta}$) and $\widehat{\rm
vol}(x_{0},X_{0},\Delta_{0})=\widehat{\rm
vol}_{X_{0},\Delta_{0}}(\mathrm{wt}_{\xi_{v}})$ (as
$x_{0}\in(X_{0},\Delta_{0};\xi_{v})$ is K-semistable), we see that the
degeneration preserves the local volume. ∎

###  2.7. Results from Part I

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.

######  Lemma 2.20.

Let $n\in\mathbb{N}$ and let $\varepsilon>0$. Then there exists some constant
$c=c(n,\varepsilon)>0$ such that for any $n$-dimensional klt singularity
$x\in(X,\Delta)$ with $\widehat{\rm vol}(x,X,\Delta)\geq\varepsilon$, we have
that $x\in(X,(1+c)\Delta)$ is of klt type.

###### Proof.

Since $x\in(X,\Delta)$ is klt, by [BCHM] (_cf._ [Z-mld^K-1, Lemma 4.7]) there
exists a small birational morphism $\pi\colon Y\to X$ such that $K_{Y}$ is
$\mathbb{Q}$-Cartier and relatively ample. Since $\pi$ is small, we have
$K_{Y}+\Delta_{Y}=\pi^{*}(K_{X}+\Delta)$, hence by [LX-cubic-3fold, Lemma
2.9(2)] (_cf._ the proof of [Z-mld^K-1, Lemma 2.10]) we have

$\widehat{\rm vol}(y,Y,\Delta_{Y})\geq\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon$

for all $y\in\pi^{-1}(x)$. By [Z-mld^K-1, Theorem 3.1], there exists some
constant $c=c(n,\varepsilon)>0$ such that the pair $(Y,(1+c)\Delta_{Y})$ is
klt for all $y\in\pi^{-1}(x)$. Note that
$-(K_{Y}+(1+c)\Delta_{Y})\sim_{\mathbb{Q},\pi}cK_{Y}$, hence $\pi$ is also the
ample model of $-(K_{X}+(1+c)\Delta)$. By [Z-direct-summand, Lemma 2.4], this
implies that $x\in(X,(1+c)\Delta)$ is of klt type. ∎

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

######  Lemma 2.21.

Let $n\in\mathbb{N}$, let $\varepsilon,A>0$, and let
$I\subseteq[0,1]\cap\mathbb{Q}$ be a finite set. Then there exists some
integer $N=N(n,\varepsilon,A,I)>0$ such that for any $n$-dimensional klt
singularity $x\in(X,\Delta)$ with coefficients in $I$ and $\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon$, and for any Kollár component $E$ over
$x\in(X,\Delta)$ with $A_{X,\Delta}(E)\leq A$, we have that $NE$ and
$N(K_{Y}+\Delta_{Y}+E)$ are Cartier on the plt blowup $Y\to X$ of $E$.

###### Proof.

By [Z-mld^K-1, (4.1)], the local volumes $\widehat{\rm vol}(y,Y,\Delta_{Y})$
(where $y\in E$) are bounded from below by some constants that only relies on
the given constants $n,\varepsilon,A$. Thus by Lemma 2.7 we get the desired
Cartier index bound. ∎

##  3\. Boundedness

In this section, we prove the following statement on boundedness of log Fano
cone singularities, which will imply the main results of this paper.

######  Theorem 3.1.

Let $n\in\mathbb{N}$ and let $I\subseteq[0,1]$ be a finite set. Let
$\varepsilon,\theta,A>0$. Let $\mathcal{S}$ be the set of $n$-dimensional
polarized log Fano cone singularities $x\in(X,\Delta;\xi)$ with coefficients
in $I$ such that

$\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon,\quad\Theta(X,\Delta;\xi)\geq\theta,\quad\mathrm{and}\quad\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq
A.$

Then $\mathcal{S}$ is bounded.

Recall that $\Theta(X,\Delta;\xi)$ is the volume ratio of the log Fano cone
singularity (Definition 2.13). We refer to Sections 2.2–2.5 for the other
relevant definitions.

###  3.1. Orbifold cones

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].

######  Definition 3.2.

Let $V$ be a normal projective variety. Let $L$ be an ample
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $V$.

  1. (1)

The _affine orbifold cone_ $C_{a}(V,L)$ is defined as

$C_{a}(V,L):=\mathbf{Spec}\bigoplus_{m=0}^{\infty}H^{0}(V,\mathcal{O}_{V}(\lfloor
mL\rfloor)).$

  2. (2)

The _projective orbifold cone_ $C_{p}(V,L)$ is defined as

$C_{p}(V,L):=\mathbf{Proj}\bigoplus_{m=0}^{\infty}\bigoplus_{i=0}^{\infty}H^{0}(V,\mathcal{O}_{V}(\lfloor
mL\rfloor)\cdot s^{i},$

where the grading of $H^{0}(V,\mathcal{O}_{V}(\lfloor mL\rfloor))$ and $s$ are
$m$ and $1$, respectively.

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).

######  Lemma 3.3.

The following conditions are equivalent:

  1. (1)

$x\in(X,\Delta)$ is klt;

  2. (2)

$(\overline{X},\overline{\Delta}+V_{\infty})$ is plt;

  3. (3)

$(V,\Delta_{V}+\Delta_{L})$ is a log Fano pair, and
$-(K_{V}+\Delta_{V}+\Delta_{L})\sim_{\mathbb{R}}rL$ for some $r>0$.

Moreover, when the above conditions are satisfied, we have
$K_{\overline{X}}+\overline{\Delta}\sim_{\mathbb{R}}-(1+r)V_{\infty}$ and
$A_{X,\Delta}(V_{0})=r$.

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.

######  Lemma 3.4.

Assume that $x\in(X,\Delta)$ is klt. Then under the notation of Lemma 3.3 we
have

$\widehat{\rm
vol}_{X,\Delta}(\mathrm{ord}_{V_{0}})=\mathrm{vol}(-(K_{\overline{X}}+\overline{\Delta}+V_{\infty}))=r\cdot\mathrm{vol}(-(K_{V}+\Delta_{V}+\Delta_{L})).$

###### Proof.

This follows from a direct calculation. First we have $A_{X,\Delta}(V_{0})=r$
by Lemma 3.3. We also have $-V_{0}|_{V_{0}}\cong L$, hence

$\widehat{\rm
vol}_{X,\Delta}(\mathrm{ord}_{V_{0}})=A_{X,\Delta}(V_{0})^{n}\cdot\mathrm{vol}_{X,\Delta}(\mathrm{ord}_{V_{0}})=r^{n}\mathrm{vol}(-V_{0}|_{V_{0}})=r^{n}L^{n-1}$

where $n=\dim X$. On the other hand, by Lemma 3.3 we get
$-(K_{\overline{X}}+\overline{\Delta}+V_{\infty})\sim_{\mathbb{R}}rV_{\infty}$;
recall also that
$-(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|_{V_{\infty}}\sim_{\mathbb{R}}-(K_{V}+\Delta_{V}+\Delta_{L})\sim_{\mathbb{Q}}rL$
(where we identify $V_{\infty}$ with $V$), thus

$\displaystyle\mathrm{vol}(-(K_{\overline{X}}+\overline{\Delta}+V_{\infty}))$
$\displaystyle=\left(-(K_{\overline{X}}+\overline{\Delta}+V_{\infty})\right)^{n-1}\cdot(rV_{\infty})$
$\displaystyle=r\cdot\mathrm{vol}(-(K_{V}+\Delta_{V}+\Delta_{L}))=r\cdot(rL)^{n-1}=r^{n}L^{n-1},$

which proves the desired equality. ∎

We also need a slight generalization of Lemma 3.3.

######  Lemma 3.5.

Notation as before. Assume that $x\in(X,\Delta)$ is of klt type. Then
$-(K_{V}+\Delta_{V}+\Delta_{L})$ is big.

###### Proof.

By assumption, there exists some effective $\mathbb{R}$-divisor $D$ such that
$x\in(X,\Delta+D)$ is klt (note that $D$ is not necessarily invariant under
the $\mathbb{G}_{m}$-action). In particular, $A_{X,\Delta+D}(V_{0})>0$ and we
have

$K_{Y}+\Delta_{Y}+D_{Y}+V_{0}\sim_{\mathbb{R}}A_{X,\Delta+D}(V_{0})\cdot
V_{0}$

over $X$. It follows that
$-(K_{Y}+\Delta_{Y}+D_{Y}+V_{0})|_{V_{0}}\sim_{\mathbb{R}}A_{X,\Delta+D}(V_{0})\cdot
L$ is ample. By adjunction, this implies that

$-(K_{V}+\Delta_{V}+\Delta_{L})=-(K_{Y}+\Delta_{Y}+D_{Y}+V_{0})|_{V_{0}}+D_{Y}|_{V_{0}}$

is the sum of an ample divisor and an effective divisor, hence is big. ∎

###  3.2. Effective birationality

Given Lemma 3.4, a naïve idea to prove Theorem 3.1 is to associate to each log
Fano cone singularity a projective orbifold cone and show that the
corresponding set of projective orbifold cones is bounded. In general, this is
too much to hope for, as the projective orbifold cone depends on the
(auxiliary) choice of a quasi-regular polarization. The following example
shows that even for a fixed singularity we can get an unbounded family of
projective orbifold cones by choosing different polarizations.

######  Example 3.6.

Let $a_{1},\dots,a_{n}\in\mathbb{N}$ be pairwise coprime positive integers.
Then $\xi=(a_{1},\dots,a_{n})\in\mathbb{N}^{n}$ gives a polarization of the
Fano cone singularity $0\in\mathbb{A}^{n}$; it generates the
$\mathbb{G}_{m}$-action with weights $a_{1},\dots,a_{n}$ on the coordinates.
This endows $\mathbb{A}^{n}$ with an affine orbifold cone structure
$C_{a}(V,L)$ where $V=\mathbb{P}(a_{1},\dots,a_{n})$ and
$L=\mathcal{O}_{V}(1)$. The associated projective orbifold cone is
$\overline{X}=\mathbb{P}(1,a_{1},\dots,a_{n})$, which is clearly unbounded as
the weights $a_{i}$’s vary. By choosing appropriate weights, we can even
ensure that the normalized volume $\widehat{\rm
vol}(\mathrm{wt}_{\xi})=\frac{(a_{1}+\dots+a_{n})^{n}}{a_{1}\dots a_{n}}$ is
fixed.

Nonetheless, we observe that in the above example the projective orbifold
cones we get satisfy the following interesting property: the linear system
$|-K_{\overline{X}}|$ always defines a birational map that is an embedding at
the vertex $[1:0:\dots:0]$. In fact, if $[s:x_{1}:\dots:x_{n}]$ are the
weighted homogeneous coordinates of $\overline{X}$, then for every
$i\in\\{1,\dots,n\\}$ there exists some $k_{i}\in\mathbb{N}$ such that
$s^{k_{i}}x_{i}\in H^{0}(-K_{\overline{X}})$ (this is possible because $s$ has
weight $1$); it is not hard to see that the sub-linear system spanned by
$s^{k_{i}}x_{i}$ ($i=0,\dots,n$) is base point free and restricts to an
embedding on the affine chart $\mathbb{A}^{n}=\overline{X}\setminus(s=0)$.

This motivates us to raise the following question.

######  Question 3.7.

Let $n\in\mathbb{N}$ and $\varepsilon>0$. Let $I\subseteq[0,1]\cap\mathbb{Q}$
be a finite set. Is there some integer $m=m(n,\varepsilon,I)>0$ such that for
any $n$-dimensional quasi-regular polarized log Fano cone $x\in(X,\Delta;\xi)$
with $\widehat{\rm vol}(x,X,\Delta)\geq\varepsilon$, we have that
$|-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|$ defines a birational map
that restricts to an embedding on $X$?

As before, $(\overline{X},\overline{\Delta})$ denotes the associated
projective orbifold cone, and $V_{\infty}=\overline{X}\setminus X$ is the
divisor at infinity.

The technical core of the proof of Theorem 3.1 is the answer to this question
after imposing upper bounds on the minimal log discrepancies of Kollár
components.

######  Proposition 3.8.

Let $n\in\mathbb{N}$ and $\varepsilon,A>0$. Let
$I\subseteq[0,1]\cap\mathbb{Q}$ be a finite set. Then there is some integer
$m=m(n,\varepsilon,A,I)>0$ such that for any $n$-dimensional quasi-regular
polarized log Fano cone $x\in(X,\Delta;\xi)$ with $\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon$ and
$\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq A$, we have that
$|-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|$ defines a birational map
that restricts to an embedding on $X$.

We remark that the integer $m$ is independent of the polarization $\xi$.

###### Proof.

We start with some reductions. Let
$H:=-(K_{\overline{X}}+\overline{\Delta}+V_{\infty})$. We view $X$ as an
affine orbifold cone $C_{a}(V,L)$ and keep the notation (e.g.
$\Delta_{V},\Delta_{L}$, etc) from Section 3.1. First note that since the
coefficients of $\overline{\Delta}$ belong to a fixed finite set $I$ of
rational numbers, we can choose some $m_{0}$ depending only on $I$ such that
$m_{0}H$ has integer coefficients. Since $\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon$, by Lemma 2.7 we know that the Cartier index
of $m_{0}H$ at $x$ is bounded from above by $\frac{n^{n}}{\varepsilon}$. Thus
replacing $m_{0}$ by a sufficiently large fixed multiple (e.g.
$\lfloor\frac{n^{n}}{\varepsilon}\rfloor!$) we may further assume that
$m_{0}H$ is Cartier in a neighbourhood of $x$. By
$\mathbb{G}_{m}$-translation, this implies that $m_{0}H$ is Cartier on $X$.

The next step is to produce enough sections (of some multiple of $m_{0}H$)
that do not vanish at $x$. We do this by creating isolated non-klt centers at
$x$ and apply Nadel vanishing. Let $\varphi\colon\overline{Y}\to\overline{X}$
be the orbifold blowup of the vertex $x$, and let $\Delta_{\overline{Y}}$ be
the strict transform of $\overline{\Delta}$. Let $V_{0}$ be the exceptional
divisor as before. Recall that $\overline{Y}$ has an orbifold
$\mathbb{P}^{1}$-bundle structure $\pi\colon\overline{Y}\to V$. Then we have
(see [Kol-Seifert-bundle, Proposition 40 and Corollary 41])

$-(K_{\overline{Y}}+\Delta_{\overline{Y}}+V_{0}+V_{\infty})\sim_{\mathbb{Q}}-\pi^{*}(K_{V}+\Delta_{V}+\Delta_{L}).$

The left hand side is $\sim_{\mathbb{Q}}\varphi^{*}H-aV_{0}$ where
$a=A_{X,\Delta}(V_{0})>0$, while the right hand side is semiample since
$-(K_{V}+\Delta_{V}+\Delta_{L})$ is ample on $V$ by Lemma 3.3. From here we
deduce that $\varphi^{*}H-aV_{0}$ is semiample. For any positive integer $m$,
if we take $D$ to be a general member of the $\mathbb{Q}$-linear system
$\varphi_{*}|\varphi^{*}H-aV_{0}|_{\mathbb{Q}}$, then $D\sim_{\mathbb{Q}}H$
and $(\overline{X},\overline{\Delta}+mD)$ is klt away from $x$ (since this is
the only base point of the $\mathbb{Q}$-linear system). In particular, the
multiplier ideal $\mathcal{J}(\overline{X},\overline{\Delta}+mD)$ is co-
supported at $x$. Furthermore, we have

$\mathrm{ord}_{V_{0}}\mathcal{J}(\overline{X},\overline{\Delta}+mD)>m\cdot\mathrm{ord}_{V_{0}}(D)-A_{\overline{X},\overline{\Delta}}(V_{0})=(m-1)a$

by the definition of multiplier ideals. Thus we obtain

(3.1)
$\mathcal{J}(\overline{X},\overline{\Delta}+mD)\subseteq\mathfrak{a}_{(m-1)a},$

where $\mathfrak{a}_{(m-1)a}:=\mathfrak{a}_{(m-1)a}(\mathrm{ord}_{V_{0}})$
denotes the valuation ideals. Recall that $m_{0}H$ is Cartier at $x$, hence as

$mH-(K_{\overline{X}}+\overline{\Delta}+mD)\sim_{\mathbb{Q}}-(K_{\overline{X}}+\overline{\Delta})$

is ample, by the following Lemma 3.9 (Nadel vanishing for Weil divisor), we
have

$H^{1}(\overline{X},\mathcal{J}(\overline{X},\overline{\Delta}+mD)\otimes\mathcal{O}_{\overline{X}}(mH))=0$

for all $m\in\mathbb{N}$ that’s divisible by $m_{0}$. From the long exact
sequence we then deduce that the natural map

$H^{0}(\overline{X},\mathcal{O}_{\overline{X}}(mH))\to
H^{0}(\overline{X},\mathcal{O}_{\overline{X}}/\mathcal{J}(\overline{X},\overline{\Delta}+mD))$

is surjective. Combined with (3.1), we get a surjection

(3.2) $H^{0}(\overline{X},\mathcal{O}_{\overline{X}}(mH))\to
H^{0}(\overline{X},\mathcal{O}_{\overline{X}}/\mathfrak{a}_{(m-1)a})$

for all $m\in\mathbb{N}$ that’s divisible by $m_{0}$.

At this point, we have produced sections of $mH$ that separate the jets in
$\mathcal{O}_{\overline{X}}/\mathfrak{a}_{(m-1)a}$. Our goal is to make $|mH|$
into a birational map that restricts to an embedding on $X$. Clearly, a
necessary condition is that $|mH|$ separates tangent directions at $x$. Let us
first show that this latter condition can be achieved. In view of (3.2), it
suffices to show that

(3.3) $\mathfrak{a}_{(m-1)a}\subseteq\mathfrak{m}_{x}^{2},$

which becomes a local question. If $X$ is fixed, this is certainly true for
$m\gg 0$, so the main point is to make sure that the bound on $m$ depend only
on the given data $n,\varepsilon,A$ rather than on $X$. To this end, note that
one way to interpret the opposite condition
$\mathfrak{a}_{(m-1)a}\not\subseteq\mathfrak{m}_{x}^{2}$ is that there exists
some $f\in\mathfrak{m}_{x}\setminus\mathfrak{m}_{x}^{2}$ such that
$\mathrm{ord}_{V_{0}}(f)\geq(m-1)a$; in particular,
$\mathrm{lct}(f):=\mathrm{lct}(X,\Delta;(f=0))\leq\frac{1}{m-1}$. This
suggests to us that we should try to analyze the Izumi type constant

(3.4) $\inf\\{\mathrm{lct}(f)\mid
f\in\mathfrak{m}_{x}\setminus\mathfrak{m}_{x}^{2}\\}$

for the singularities in question.

Let $E$ be a Kollár component over $x\in(X,\Delta)$ such that
$A_{X,\Delta}(E)\leq A$, whose existence is guaranteed by our assumption that
$\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq A$. Let $f\colon Z\to X$ be the plt
blowup that extract $E$ and let $\Delta_{E}=\mathrm{Diff}_{E}(\Delta_{Z})$ be
the different. By Lemma 2.4, we have

(3.5)
$\mathrm{lct}(f)\geq\min\\{1,\alpha(E,\Delta_{E})\\}\cdot\frac{A_{X,\Delta}(E)}{\mathrm{ord}_{E}(f)}.$

In order to give uniform estimate of (3.4) using (3.5), we need further
information about the log discrepancy $A_{X,\Delta}(E)$, the
$\alpha$-invariant $\alpha(E,\Delta_{E})$, and the value of

$d_{E}:=\sup\\{\mathrm{ord}_{E}(f)\mid
f\in\mathfrak{m}_{x}\setminus\mathfrak{m}_{x}^{2}\\}$

(the last one can be thought of as measuring how well the valuation ideals of
$\mathrm{ord}_{E}$ approximate the first two powers of the maximal ideal
$\mathfrak{m}_{x}$).

For the log discrepancy, recall from the previous discussion that
$m_{0}(K_{X}+\Delta)$ is Cartier. Since $(X,\Delta)$ is klt, this implies that
$A_{X,\Delta}(E)\geq\frac{1}{m_{0}}$. For the $\alpha$-invariant, we know by
Lemma 2.21 that there exists some positive integer
$N_{1}=N_{1}(n,\varepsilon,A,I)$ divisible by $m_{0}$ such that $N_{1}E$ and
$N_{1}(K_{Z}+\Delta_{Z}+E)$ are Cartier. By adjunction, we see that
$N_{1}(K_{E}+\Delta_{E})$ is also Cartier. By [HMX-ACC, Corollary 1.8], the
log Fano pair $(E,\Delta_{E})$ belongs to a bounded family (which relies only
on $N_{1}$). This implies that $\alpha(E,\Delta_{E})$ is uniformly bounded
below. Finally, to give an upper bound for $d_{E}$, let $L=-E|_{E}$ be the
(ample) $\mathbb{Q}$-divisor defined as in [HLS-epsilon-plt-blowup, Definition
A.4] and let

$\mathfrak{b}_{k}:=\mathfrak{a}_{k}(\mathrm{ord}_{E})=f_{*}\mathcal{O}_{Z}(-kE)$

where $k=1,2,\dots$. Then $N_{1}L$ is Cartier as $N_{1}E$ is Cartier. Since
$-(K_{E}+\Delta_{E})\sim_{\mathbb{Q}}A_{X,\Delta}(E)\cdot L$ and
$A_{X,\Delta}(E)\geq\frac{1}{m_{0}}$, we see that the coefficient and degree
of $L$ are bounded, hence the triple $(E,\Delta,L)$ belongs to a bounded
family. Thus there exists an integer $N=N(n,\varepsilon,A,I)>0$ such that the
section ring $\oplus_{k\in\mathbb{N}}H^{0}(E,\lfloor kL\rfloor)$ is generated
in degree $\leq N$. We claim that

(3.6) $\mathfrak{b}_{N+1}\subseteq\mathfrak{m}_{x}^{2}.$

Taking this for granted, it follows immediately that $d_{E}\leq N$. Putting
these information together we deduce from (3.5) that (3.4) is bounded below by
some positive constant that only rely on $n,\varepsilon,A,I$. From the
discussion right above (3.4) we also know that (3.4) is bounded from above by
$\frac{1}{m-1}$ where $m$ is the largest integer such that
$\mathfrak{a}_{(m-1)a}\not\subseteq\mathfrak{m}_{x}^{2}$. Thus we see that
there is some fixed $m$ depending only on $n,\varepsilon,A,I$ such that $mH$
is Cartier and (3.3) holds. Combined with (3.2), we deduce that

(3.7) $H^{0}(\overline{X},\mathcal{O}_{\overline{X}}(mH))\to
H^{0}(\overline{X},\mathcal{O}_{\overline{X}}/\mathfrak{m}_{x}^{2})$

is surjective, i.e. $|mH|$ separates tangent direction at $x$.

Before we proceed to show that $|mH|$ also defines a birational map, let us
finish the proof of the claim (3.6). In fact, we shall prove by descending
induction that $\mathfrak{b}_{k}\subseteq\mathfrak{m}_{x}^{2}$ for all $k\geq
N+1$. This is clear when $k\gg 0$. Suppose that
$\mathfrak{b}_{k+1}\subseteq\mathfrak{m}_{x}^{2}$ and $k\geq N+1$, then since
$\mathfrak{b}_{k+1}\subseteq\mathfrak{b}_{k}$ for all $k$ and since
$\oplus_{k\in\mathbb{N}}\mathfrak{b}_{k}/\mathfrak{b}_{k+1}\cong\oplus_{k\in\mathbb{N}}H^{0}(E,\lfloor
kL\rfloor)$ (see [LX-stability-kc, Section 2.4] or [LZ-Tian-sharp, Proposition
2.10]) is generated in degree $\leq N$, we see that

$\mathfrak{b}_{k}/\mathfrak{b}_{k+1}\subseteq\sum_{i=1}^{N}(\mathfrak{b}_{i}/\mathfrak{b}_{i+1})\cdot(\mathfrak{b}_{k-i}/\mathfrak{b}_{k-i+1})\subseteq\mathfrak{b}_{1}^{2}/\mathfrak{b}_{k+1};$

in other words,
$\mathfrak{b}_{k}\subseteq\mathfrak{b}_{k+1}+\mathfrak{b}_{1}^{2}$. As
$\mathfrak{b}_{k+1}\subseteq\mathfrak{m}_{x}^{2}$ by induction hypothesis and
clearly $\mathfrak{b}_{1}\subseteq\mathfrak{m}_{x}$, we obtain
$\mathfrak{b}_{k}\subseteq\mathfrak{m}_{x}^{2}$ as desired.

We are finally in a position to show that $|mH|$ defines a birational map that
restricts to an embedding on $X$. By (3.7), we already know that in a
neighbourhood of $x$, the linear system $|mH|$ is base point free and the
induced map is unramified. Since the base locus of $|mH|$ is closed and
invariant under the $\mathbb{G}_{m}$-action (coming from the orbifold cone
structure), we see that $|mH|$ has no base point in $X$. Similarly, as the
ramification locus of the induced map $\varphi$ on $X$ is a closed
$\mathbb{G}_{m}$-invariant subset, we see that $\varphi$ is unramified on $X$
and thus it is quasi-finite. This implies that
$\varphi(x^{\prime})\neq\varphi(x)$ for all $x^{\prime}\neq x\in X$, otherwise
by $\mathbb{G}_{m}$-translation we deduce that $\varphi$ contracts the closed
orbit $\overline{\mathbb{G}_{m}\cdot x^{\prime}}$. In particular,
$\varphi|_{X}^{-1}(\varphi(x))$ is supported at $x$; as $\varphi$ is also
unramified, the scheme-theoretic pre-image $\varphi|_{X}^{-1}(\varphi(x))$
equals $\\{x\\}$. By upper semi-continuity and the $\mathbb{G}_{m}$-action,
this implies that $\varphi|_{X}^{-1}(\varphi(x^{\prime}))$ has length $1$ and
thus consists of a single point for all $x^{\prime}\in X$. It follows that
$\varphi|_{X}$ is an embedding on $X$ and we finish the proof. ∎

We have used the following vanishing result in the above proof. This should be
well-known to experts, but we are unable to find a suitable reference.

######  Lemma 3.9.

Let $(X,\Delta)$ be a pair such that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier.
Let $L$ be a $\mathbb{Q}$-Cartier Weil divisor such that $L-(K_{X}+\Delta)$ is
nef and big. Assume that $(X,\Delta)$ is klt along the non-Cartier locus of
$L$. Then

$H^{i}(X,\mathcal{J}(X,\Delta)\otimes\mathcal{O}_{X}(L))=0$

for all $i>0$.

###### Proof.

If $L$ is a line bundle this is just the usual Nadel vanishing; in the general
case we follow the proof of Nadel vanishing. Let $f\colon Y\to X$ be a log
resolution. We may write

$\displaystyle K_{Y}+\Delta_{Y}$ $\displaystyle=f^{*}(K_{X}+\Delta)+\sum
a_{i}E_{i},$ $\displaystyle f^{*}L$ $\displaystyle=\lfloor f^{*}L\rfloor+\sum
b_{i}E_{i},$

where the $E_{i}$’s are the exceptional divisors, and $0\leq b_{i}<1$. Let

$\displaystyle L^{\prime}$ $\displaystyle=\lfloor
f^{*}L\rfloor-\lfloor\Delta_{Y}\rfloor+\sum\lceil a_{i}+b_{i}\rceil E_{i},$
$\displaystyle D$
$\displaystyle=\\{\Delta_{Y}\\}+\sum\\{-a_{i}-b_{i}\\}E_{i}.$

Then it’s easy to check that $(Y,D)$ is klt (i.e. $\\{D\\}=0$ since $Y$ is a
log resolution) and
$L^{\prime}-(K_{Y}+D)\sim_{\mathbb{Q}}f^{*}(L-K_{X}-\Delta)$ is nef and big.
By Kawamata-Viehweg vanishing we have $H^{i}(Y,\mathcal{O}_{Y}(L^{\prime}))=0$
and $R^{i}f_{*}\mathcal{O}_{Y}(L^{\prime})=0$ for all $i>0$. It follows that
$Rf_{*}\mathcal{O}_{Y}(L^{\prime})=f_{*}\mathcal{O}_{Y}(L^{\prime})$ and hence

$H^{i}(X,f_{*}\mathcal{O}_{Y}(L^{\prime}))=H^{i}(X,Rf_{*}\mathcal{O}_{Y}(L^{\prime}))=H^{i}(Y,\mathcal{O}_{Y}(L^{\prime}))=0.$

It remains to show that

(3.8)
$f_{*}\mathcal{O}_{Y}(L^{\prime})=\mathcal{J}(X,\Delta)\otimes\mathcal{O}_{X}(L).$

This is a local question, so it suffices to check the equality at any $x\in
X$. If $(X,\Delta)$ is klt at $x$, then locally
$\mathcal{J}(X,\Delta)=\mathcal{O}_{X}$, $\lfloor\Delta_{Y}\rfloor=0$, and
$a_{i}>-1$ which implies $\lceil a_{i}+b_{i}\rceil\geq 0$. It follows that

$f_{*}\mathcal{O}_{Y}(L^{\prime})=\mathcal{O}_{X}(L)=\mathcal{J}(X,\Delta)\otimes\mathcal{O}_{X}(L).$

If $(X,\Delta)$ is not klt at $x$, then $L$ is Cartier around $x$, which gives
$b_{i}=0$ for every $E_{i}$ whose image contains $x$, thus locally

$f_{*}\mathcal{O}_{Y}(L^{\prime})=f_{*}\mathcal{O}_{Y}(f^{*}L+\lceil-\Delta_{Y}+\sum
a_{i}E_{i}\rceil)=\mathcal{J}(X,\Delta)\otimes\mathcal{O}_{X}(L)$

by projection formula and the definition of the multiplier ideal. Thus we see
that (3.8) always holds. This completes the proof. ∎

###  3.3. Proof of Theorem 3.1

We are now in a position to prove Theorem 3.1.

###### Proof.

First assume that $I\subseteq[0,1]\cap\mathbb{Q}$. Let $x\in(X,\Delta;\xi)$ be
a polarized log Fano cone singularity in $\mathcal{S}$ and let $\mathbb{T}$ be
the torus generated by $\xi$. By Lemma 2.11, there exists some quasi-regular
polarization $\xi^{\prime}$ that is sufficiently close to $\xi$ in the Reeb
cone such that $\Theta(X,\Delta;\xi^{\prime})\geq\frac{\theta}{2}$. Using
$\xi^{\prime}$, we may realize $(X,\Delta)$ as an orbifold cone over some
polarized pair $(V,\Delta_{V};L)$. Moreover, if $V_{0}$ is the exceptional
divisor of the orbifold vertex blowup as in Section 3.1, then
$\mathrm{ord}_{V_{0}}$ is proportional to $\mathrm{wt}_{\xi^{\prime}}$. By
assumption and Lemma 2.6, we obtain

(3.9) $\widehat{\rm vol}_{X,\Delta}(\mathrm{ord}_{V_{0}})=\widehat{\rm
vol}_{X,\Delta}(\mathrm{wt}_{\xi^{\prime}})\leq 2\theta^{-1}\widehat{\rm
vol}(x,X,\Delta)\leq 2\theta^{-1}n^{n}.$

Let $(\overline{X},\overline{\Delta})$ be the projective orbifold cone over
$(V,\Delta_{V};L)$ and $V_{\infty}$ the divisor at infinity. By Proposition
3.8, there exists some integer $m$ depending only on $n,\varepsilon,A,I$ such
that $|-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|$ defines a
birational map $\varphi$ that restricts to an embedding on $X$. By (3.9) and
Lemma 3.4, we see that

$\mathrm{vol}(-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty}))=m^{n}\cdot\widehat{\rm
vol}_{X,\Delta}(\mathrm{ord}_{V_{0}})\leq 2\theta^{-1}(mn)^{n}$

is bounded above.

Let us show that
$\mathrm{vol}(-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|_{\overline{\Delta}})$
is also bounded from above, so that the pair
$(\overline{X},\overline{\Delta})$ belongs to a birationally bounded family.
By Lemma 2.20, we know that $x\in(X,(1+c)\Delta)$ is of klt type for some
constant $c=c(n,\varepsilon)>0$. By Lemma 3.5, this implies that
$-(K_{V}+(1+c)\Delta_{V}+\Delta_{L})$ is big. By a similar calculation as in
Lemma 3.4, it follows that

$\displaystyle\mathrm{vol}(-(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|_{\overline{\Delta}})$
$\displaystyle=\left(-(K_{\overline{X}}+\overline{\Delta}+V_{\infty})\right)^{n-2}\cdot
rV_{\infty}\cdot\overline{\Delta}$
$\displaystyle=r\cdot\left(-(K_{V}+\Delta_{V}+\Delta_{L})\right)^{n-2}\cdot\Delta_{V}$
$\displaystyle\leq
c^{-1}r\cdot\left(-(K_{V}+\Delta_{V}+\Delta_{L})\right)^{n-1}$
$\displaystyle=c^{-1}\widehat{\rm vol}_{X,\Delta}(\mathrm{ord}_{V_{0}})\leq
2(c\theta)^{-1}n^{n},$

where the first equality is by Lemma 3.3, the second by adjunction along
$V_{\infty}\cong V$, the next inequality by the bigness of
$-(K_{V}+(1+c)\Delta_{V}+\Delta_{L})$, and the last equality by Lemma 3.4.
Thus
$\mathrm{vol}(-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|_{\overline{\Delta}})$
is also bounded from above as desired. Note that
$\varphi|_{\mathrm{Supp}(\overline{\Delta})}$ is also birational since
$\varphi$ is an embedding at $x$. Therefore, the image $(W,\Delta_{W})$ of
$(\overline{X},\overline{\Delta})$ under the birational map induced by
$|-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|$ belongs to a fixed
bounded family $(\mathcal{W},\overline{\mathcal{D}})\to B$.

By construction, $(\overline{X},\overline{\Delta})$ carries an effective
$\mathbb{T}$-action and $V_{\infty}$ is $\mathbb{T}$-invariant. It follows
that $|-m(K_{\overline{X}}+\overline{\Delta}+V_{\infty})|$ is a
$\mathbb{T}$-invariant linear system and the induced birational map $\varphi$
is $\mathbb{T}$-equivariant. In particular, the image $(W,\Delta_{W})$ also
carries an effective $\mathbb{T}$-action. We claim that there exists finitely
many morphisms $B_{i}\to B$ (depending only on the family
$(\mathcal{W},\overline{\mathcal{D}})\to B$) such that

  1. (1)

after base change, each family
$(\mathcal{W}_{i},\overline{\mathcal{D}}_{i})=(\mathcal{W},\overline{\mathcal{D}})\times_{B}B_{i}\to
B_{i}$ admits an effective fiberwise $\mathbb{T}_{i}$-action for some torus
$\mathbb{T}_{i}$,

  2. (2)

$(W,\Delta_{W})\cong(\mathcal{W}_{b_{i}},\overline{\mathcal{D}}_{b_{i}})$ for
some $b_{i}\in B_{i}$, and under this isomorphism, the $\mathbb{T}$-action on
$W$ is induced by some group homomorphism $\mathbb{T}\to\mathbb{T}_{i}$ (in
other words, the $\mathbb{T}$-action on $W$ is induced by the
$\mathbb{T}_{i}$-action on $\mathcal{W}_{b_{i}}$).

To see this, first observe that for any torus action on a projective variety,
the induced action on the Picard scheme is trivial. This is because both
$\mathrm{Pic}^{0}$ (an abelian variety) and $\mathrm{Pic}/\mathrm{Pic}^{0}$ (a
discrete group) have no $\mathbb{G}_{m}$-action. Thus if $\mathcal{L}$ is a
relatively ample line bundle on $\mathcal{W}$, then it is invariant under any
fiberwise torus action. Consider the relative automorphism group scheme
$\mathbb{G}$ over $B$, which parametrizes the automorphisms of the polarized
fibers $(\mathcal{W}_{b},\overline{\mathcal{D}}_{b};\mathcal{L}_{b})$. Note
that $\mathbb{G}$ is affine over $B$. Possibly after stratifying the base $B$,
we may also assume that $\mathbb{G}$ is smooth over $B$. By [SGA3-II, Exposé
XII, Théorèm 1.7(a)], after a further stratification of $B$, we may assume
that the dimension of the maximal torus of $\mathbb{G}_{b}$ is locally
constant in $b\in B$. We may discard the components of $B$ where this torus
dimension is zero. By [SGA3-II, Exposé XII, Théorèm 1.7(b)], it then follows
that there exists an (étale) cover $\cup B_{i}\to B$ of the remaining
components of $B$ and a subgroup scheme
$\mathbb{G}_{i}\subseteq\mathbb{G}\times_{B}B_{i}$ that reduces to the maximal
tori on the fibers. Passing to a further finite cover of the $B_{i}$’s, we may
assume that the $\mathbb{G}_{i}$’s are split, i.e.
$\mathbb{G}_{i}=\mathbb{T}_{i}\times B_{i}$ for some torus $\mathbb{T}_{i}$.
In particular, the $\mathbb{G}$-action on
$(\mathcal{W},\overline{\mathcal{D}})$ induces a fiberwise
$\mathbb{T}_{i}$-action on
$(\mathcal{W}_{i},\overline{\mathcal{D}}_{i}):=(\mathcal{W},\overline{\mathcal{D}})\times_{B}B_{i}$.
This gives the family in (1). Since $(W,\Delta_{W})$ has a non-trivial torus
action, we see that $(W,\Delta_{W})$ appears as a fiber of
$(\mathcal{W}_{i},\overline{\mathcal{D}}_{i})\to B_{i}$ for some $i$. Since
all maximal tori in $\mathrm{Aut}(W,\Delta_{W})$ are conjugate to each other,
we see that $\mathbb{T}$ is conjugate to a subtorus of the maximal torus
$\mathbb{T}_{i}$. In other words, there exists an isomorphism
$(W,\Delta_{W})\cong(\mathcal{W}_{b_{i}},\overline{\mathcal{D}}_{b_{i}})$ such
that (2) holds. This proves the claim.

Taking the fiberwise isolated $\mathbb{T}_{i}$-fixed points
$(\mathcal{W}_{i},\overline{\mathcal{D}}_{i})\to B_{i}$ for all $i$, we get
families $B_{i}\subseteq(\mathcal{X}_{i},\mathcal{D}_{i})\to B_{i}$ of
singularities (possibly after a refinement of the $B_{i}$’s), each with an
effective fiberwise torus $\mathbb{T}_{i}$-action. Since $\varphi$ restricts
to an embedding on $X$, by the second part of the above claim we see that
$x\in(X,\Delta;\xi)$ is isomorphic to
$b_{i}\in(\mathcal{W}_{b_{i}},\mathcal{D}_{b_{i}};\xi_{b_{i}})$ for some
$b_{i}\in B_{i}$ and some $\xi_{b_{i}}\in N(\mathbb{T}_{i})_{\mathbb{R}}$. In
particular, this gives log boundedness. Note that when
$I\not\subseteq\mathbb{Q}$, we can still apply the above argument to the
singularities and the coefficient set constructed from Lemma 2.18, thus the
same conclusion holds.

To get boundedness, we need to further stratify the family
$(\mathcal{W}_{i},\overline{\mathcal{D}}_{i})\to B_{i}$ so that it becomes
$\mathbb{R}$-Gorenstein klt. By [Kol-moduli-book, Lemma 4.44] and inversion of
adjunction, there exists a finite collection of locally closed subset $S_{j}$
of $\cup_{i}B_{i}$ such that the family becomes $\mathbb{R}$-Gorenstein after
base change to $\cup_{j}S_{j}$ and enumerates exactly all the klt fibers of
$\cup_{i}(\mathcal{W}_{i},\overline{\mathcal{D}}_{i})\to\cup_{i}B_{i}$ (note
that [Kol-moduli-book, Lemma 4.44] requires the family to be proper but the
proof applies to our situation, essentially because the section
$B_{i}\subseteq\mathcal{X}_{i}$ is proper over the base). Replacing the
$B_{i}$’s by the $S_{j}$’s, we obtain the desired family. The proof is now
complete. ∎

###  3.4. Applications

We now explain how to deduce the other main results of this paper from Theorem
3.1. First we prove the boundedness criterion for K-semistable log Fano cone
singularities.

######  Corollary 3.10.

Let $\mathcal{S}$ be a set of $n$-dimensional K-semistable log Fano cone
singularities with coefficients in a fixed finite set $I\subseteq[0,1]$. Then
$\mathcal{S}$ is bounded if and only if there exist some $\varepsilon,A>0$
such that

$\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon\quad\mathit{and}\quad\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq
A$

for all $x\in(X,\Delta)$ in $\mathcal{S}$.

###### Proof.

One direction follows from Lemma 2.15, while the other direction is implied by
Theorem 3.1 as the volume ratio of a K-semistable log Fano cone singularity
(with the K-semistable polarization) is equal to $1$. ∎

Similarly, we have an unpolarized version of Theorem 3.1.

######  Corollary 3.11.

Let $\mathcal{S}$ be a set of $n$-dimensional log Fano cone singularities with
coefficients in a fixed finite set $I\subseteq[0,1]$. Then $\mathcal{S}$ is
bounded if and only if there exist positive constants $\varepsilon,\theta,A>0$
such that

$\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon,\quad\Theta(x,X,\Delta)\geq\theta,\quad\mathrm{and}\quad\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq
A.$

for all $x\in(X,\Delta)$ in $\mathcal{S}$.

###### Proof.

One direction follows from Lemmas 2.15 and 2.17, while the other direction is
immediate by Theorem 3.1. ∎

We next prove a version of Theorem 3.1 that replaces the volume ratios with
the stability thresholds introduced in [Hua-thesis]. First we recall the
definition. Let $x\in(X=\mathbf{Spec}(R),\Delta;\xi)$ be a polarized log Fano
cone singularity and let $\mathbb{T}$ be the torus generated by $\xi$. For
this definition it would be more convenient to rescale the polarization so
that $A_{X,\Delta}(\mathrm{wt}_{\xi})=1$, which we will assume in what
follows. Using the weight decomposition $R=\oplus_{\alpha}R_{\alpha}$, we set

$R_{m}:=\bigoplus_{\alpha,\,m-1<\langle\alpha,\xi\rangle\leq m}R_{\alpha}.$

An $m$-basis type $\mathbb{Q}$-divisor of
$x\in(X=\mathbf{Spec}(R),\Delta;\xi)$ is defined to be a $\mathbb{Q}$-divisor
of the form

$D=\frac{1}{mN_{m}}\sum_{i=1}^{N_{m}}\\{s_{i}=0\\}$

where $N_{m}=\dim R_{m}$ and $s_{1},\dots,s_{N_{m}}$ form a basis of $R_{m}$.
Set $\delta_{m}=\inf_{D}\mathrm{lct}_{x}(X,\Delta;D)$ where the infimum runs
over all $m$-basis type $\mathbb{Q}$-divisors $D$. The stability threshold of
$x\in(X=\mathbf{Spec}(R),\Delta;\xi)$ is defined as

$\delta(X,\Delta;\xi):=\lim_{m\to\infty}\delta_{m}.$

If $(X,\Delta;\xi)$ is the cone over a log Fano pair $(V,\Delta_{V})$, then
this definition is closely related to the stability threshold of
$(V,\Delta_{V})$ introduced in [FO-delta] (see also [BJ-delta]). In fact,
using inversion of adjunction it is not hard to show that
$\delta(X,\Delta;\xi)=\min\\{1,\delta(V,\Delta_{V})\\}$, _cf._ [XZ-minimizer-
unique, Theorem 3.6].

The stability threshold version of Theorem 3.1 is a direct consequence of the
following result.

######  Lemma 3.12.

Let $x\in(X=\mathbf{Spec}(R),\Delta;\xi)$ be a polarized log Fano cone
singularity of dimension $n$. Then

$\Theta(X,\Delta;\xi)\geq\delta(X,\Delta;\xi)^{n}.$

###### Proof.

Let $\delta:=\delta(X,\Delta;\xi)$. We first recall the valuative
interpretation of $\delta(X,\Delta;\xi)$ as explained in [Hua-thesis]. Fix
some $\mathbb{T}$-invariant valuation $v\in\mathrm{Val}_{X,x}^{*}$. Set
$S_{m}(v):=\sup_{D}v(D)$ where $D$ varies among $m$-basis type
$\mathbb{Q}$-divisors. By an Okounkov body argument [Hua-thesis, Section 4],
we know that $S(v):=\lim_{m\to\infty}S_{m}(v)$ exists and

(3.10) $A_{X,\Delta}(v)\geq\delta\cdot S(v)$

by [Hua-thesis, Theorem 4.3.5]. We next relate $S(v)$ to the “relative”
$S$-invariant $S(\mathrm{wt}_{\xi};v)$ defined in [XZ-minimizer-unique,
Section 3.1]. In our notation (and under our assumption that
$A_{X,\Delta}(\mathrm{wt}_{\xi})=1)$, we have
$S(\mathrm{wt}_{\xi};v)=\lim_{m\to\infty}S_{m}(\mathrm{wt}_{\xi};v)$ where

$S_{m}(\mathrm{wt}_{\xi};v):=\frac{\sum_{k=0}^{m}kN_{k}S_{k}(v)}{\sum_{k=0}^{m}kN_{k}}.$

Intuitively, the previous $S_{m}(v)$ is defined using basis type divisors for
$R_{m}$ while $S_{m}(\mathrm{wt}_{\xi};v)$ is defined via basis type divisors
for $\oplus_{k\leq m}R_{k}$ (the main reason for doing so in [XZ-minimizer-
unique] is that in the more general situation, the dimension of the analogous
space $\oplus_{k\leq m}R_{k}$ has an asymptotic expression, while the
individual $R_{m}$ does not). Note that our $S_{m}(\mathrm{wt}_{\xi};v)$
differs slightly from the one in [XZ-minimizer-unique, Section 3.1] by some
round-downs, but after taking the $m\to\infty$ limit we get the same value of
$S(\mathrm{wt}_{\xi};v)$. Since $S_{m}(v)\to S(v)$ as $m\to\infty$, from the
above expression we get $S(v)=S(\mathrm{wt}_{\xi};v)$. From the proof of [XZ-
minimizer-unique, Theorem 3.7] (especially the inequality after (3.8) in _loc.
cit._), we then obtain

$S(v)\geq\left(\frac{\mathrm{vol}(\mathrm{wt}_{\xi})}{\mathrm{vol}(v)}\right)^{\frac{1}{n}}.$

Combined with (3.10) and recall that $A_{X,\Delta}(\mathrm{wt}_{\xi})=1$, we
deduce

$\widehat{\rm
vol}_{X,\Delta}(v)=A_{X,\Delta}(v)^{n}\cdot\mathrm{vol}(v)\geq\delta^{n}S(v)^{n}\cdot\mathrm{vol}(v)\geq\delta^{n}\mathrm{vol}(\mathrm{wt}_{\xi})=\delta^{n}\widehat{\rm
vol}_{X,\Delta}(\mathrm{wt}_{\xi})$

for all $\mathbb{T}$-invariant valuation $v\in\mathrm{Val}_{X,x}^{*}$. Since
the local volume of $x\in(X,\Delta)$ is computed by some
$\mathbb{T}$-invariant valuation ([Blu-minimizer-exist] and [XZ-minimizer-
unique, Corollary 1.2]), this implies that $\widehat{\rm
vol}(x,X,\Delta)\geq\delta^{n}\widehat{\rm vol}_{X,\Delta}(\mathrm{wt}_{\xi})$
and hence $\Theta(X,\Delta;\xi)\geq\delta^{n}$. ∎

######  Remark 3.13.

We also have an inequality in the reverse direction, namely, there exists some
positive constant $c>0$ that only depends on the dimension such that

(3.11) $c\cdot\delta(X,\Delta;\xi)\geq\Theta(X,\Delta;\xi).$

To see this, let $c=c_{0}^{-1}$ where $c_{0}$ is the constant from [Z-mld^K-1,
Lemma 3.4] and let $D$ be any $m$-basis type $\mathbb{Q}$-divisor of
$(X,\Delta;\xi)$. Then by definition we have $\mathrm{wt}_{\xi}(D)\leq
1=A_{X,\Delta}(\mathrm{wt}_{\xi})$, hence by _loc. cit._ we see that

$\mathrm{lct}_{x}(X,\Delta;D)\geq c_{0}\cdot\Theta(X,\Delta;\xi).$

which gives (3.11). Note that the upper bound on the volume ratio is at least
linear in the stability threshold, as can be seen on the smooth toric
singularities $(X,\Delta)=(\mathbb{A}^{n},0)$: if
$\xi=(a_{1},\dots,a_{n})\in\mathfrak{t}_{\mathbb{R}}^{+}=\mathbb{R}^{n}_{>0}$
and $a_{1}\ll a_{2}=\dots=a_{n}$, then we have that
$\Theta(X,\Delta;\xi)=\frac{n^{n}a_{1}\cdots a_{n}}{(a_{1}+\dots+a_{n})^{n}}$
is roughly linear in $\delta(X,\Delta;\xi)=\frac{na_{1}}{a_{1}+\dots+a_{n}}$.

######  Corollary 3.14.

Let $n\in\mathbb{N}$ and let $I\subseteq[0,1]$ be a finite set. Let
$\varepsilon,\delta,A>0$. Let $\mathcal{S}$ be the set of $n$-dimensional
polarized log Fano cone singularities $x\in(X,\Delta;\xi)$ with coefficients
in $I$ such that

$\widehat{\rm
vol}(x,X,\Delta)\geq\varepsilon,\quad\delta(X,\Delta;\xi)\geq\delta,\quad\mathrm{and}\quad\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq
A.$

Then $\mathcal{S}$ is bounded.

###### Proof.

This is a direct consequence of Theorem 3.1 and Lemma 3.12. ∎

Finally we specialize our results to dimension three. For this we need the
following result from [Z-mld^K-1].

######  Proposition 3.15.

Let $\varepsilon>0$ and let $I\subseteq[0,1]$ be a finite set. Then there
exists some constant $A=A(\varepsilon,I)$ such that

$\mathrm{mld}^{\mathrm{K}}(x,X,\Delta)\leq A$

for all $3$-dimensional klt singularity $x\in(X,\Delta)$ with coefficients in
$I$ and $\widehat{\rm vol}(x,X,\Delta)\geq\varepsilon$.

###### Proof.

If $I\subseteq\mathbb{Q}$ this is [Z-mld^K-1, Corollary 6.11]; in general we
apply Lemma 2.18 to reduce to the rational coefficient case. ∎

######  Corollary 3.16.

For any finite set $I\subseteq[0,1]$ and any $\varepsilon>0$, the set of
$3$-dimensional K-semistable log Fano cone singularities with coefficients in
$I$ and with local volume at least $\varepsilon$ is bounded.

###### Proof.

Immediate from Corollary 3.10 and Proposition 3.15. ∎

The last application concerns the distribution of local volumes in dimension
$3$. For any $n\in\mathbb{N}$ and $I\subseteq[0,1]$, consider the set
$\mathrm{Vol}^{\mathrm{loc}}_{n,I}$ of all possible local volumes of
$n$-dimensional klt singularities $x\in(X,\Delta)$ with coefficients in $I$.

######  Corollary 3.17.

Let $I\subseteq[0,1]$ be a finite set. Then
$\mathrm{Vol}^{\mathrm{loc}}_{3,I}$ is discrete away from zero.

###### Proof.

By Theorem 2.19, it suffices to consider local volumes of K-semistable log
Fano cone singularities. Let $\varepsilon>0$. By Corollary 3.16, the set of
$3$-dimensional K-semistable log Fano cone singularities $x\in(X,\Delta)$ with
coefficients in $I$ and $\widehat{\rm vol}(x,X,\Delta)\geq\varepsilon$ is
bounded. On the other hand, the local volume function is constructible in
$\mathbb{R}$-Gorenstein families by [Xu-quasi-monomial, Theorem 1.3] (see
[HLQ-vol-ACC, Theorem 3.5] for the real coefficient case). In particular, the
local volumes only take finitely many possible values in a bounded family.
This implies $\mathrm{Vol}^{\mathrm{loc}}_{3,I}\cap[\varepsilon,+\infty)$ is a
finite set and we are done. ∎

######  Remark 3.18.

By combining the ideas in this work with some generalization of the proof of
[HLQ-vol-ACC, Theorem 1.2(2)], one should be able to show that if the set $I$
in Corollary 3.17 is not finite but satisfies DCC (descending chain
condition), then $\mathrm{Vol}^{\mathrm{loc}}_{3,I}$ satisfies ACC (ascending
chain condition). We leave the details to the interested readers.

## References