# On multiplicities of fibers of Fano fibrations

Guodu Chen and Chuyu Zhou  Institute for Theoretical Sciences, Westlake
University, Hangzhou, Zhejiang, 310024, China<EMAIL_ADDRESS>École
Polytechnique Fédérale de Lausanne (EPFL), MA C3 615, Station 8, 1015
Lausanne, Switzerland<EMAIL_ADDRESS>
###### Abstract.

In this note, we reduce various conjectures in birational geometry, including
Shokurov’s conjecture on singularities of the base of log Calabi-Yau
fibrations of Fano type and boundedness conjecture for rationally connected
Calabi-Yau varieties, to a conjecture on multiplicities of fibers of Fano
fibrations over curves.

###### 2020 Mathematics Subject Classification:

14E30, 14J17, 14J27, 14J32, 14J45

_Keywords_ : log Calabi-Yau fibration, Fano fibration, rationally connected
Calabi-Yau variety, boundedness.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Divisors
    2. 2.2 Singularities of g-pairs
    3. 2.3 Canonical bundle formulas
    4. 2.4 Boundedness
  3. 3 From multiplicities to singularities
    1. 3.1 Proof of Theorem 1.6
    2. 3.2 Generalized version of Conjecture 1.1
  4. 4 On boundedness of RC Calabi-Yau varieties
    1. 4.1 Conjectures on RC Calabi-Yau varieties
    2. 4.2 Proof of Theorem 1.7
    3. 4.3 Proof of Theorem 1.8
    4. 4.4 Conjecture 1.4 in dimension $\leq 3$
  5. 5 Further remark on Conjecture 4.2

##  1\. Introduction

We work over the complex number field $\mathbb{C}$.

As the outcome of minimal model program, log Calabi-Yau fibrations and Mori-
Fano fibrations play important roles in the classification of algebraic
varieties, and it is natural to ask how the singularities behave on the total
space and the base of these fibrations (e.g. [Bir18]). For Fano varieties,
people have recently established many powerful theory including boundedness,
complement theory (e.g. [Bir19, Bir21]), and K-moduli theory (e.g. [Xu21]).
For Calabi-Yau varieties, however, many aspects are still mysterious, e.g.
boundedness. The following conjecture due to Mckernan-Shokurov on the
singularities of the base of log Calabi-Yau fibrations of Fano type is still
widely open, we refer the readers to [Bir16, Bir18, HJL22] for the recent
process.

######  Conjecture 1.1 (Mckernan-Shokurov).

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then
there exists a positive real number $\delta$ depending only on $d$ and
$\epsilon$ satisfying the following.

Assume that $(X,B)$ is a log pair of dimension $d$ and $\pi:X\to Z$ is a
contraction between projective normal varieties such that

  1. (1)

$(X,B)$ is $\epsilon$-lc,

  2. (2)

$X$ is of Fano type over $Z$, and

  3. (3)

$K_{X}+B\sim_{\mathbb{R},Z}0$.

Then $B_{Z}\leq 1-\delta$, where $B_{Z}$ is the discriminant part of the
canonical bundle formula for $(X,B)$ over $Z$.

The above conjecture is known in some special cases. For example, if one
requires that the coefficients of the boundary $B$ are bounded from below,
then the general fibers of the log Calabi-Yau fibrations lie in a log bounded
family, and this leads to a positive answer to the conjecture (see [Bir16,
Theorem 1.4]). In Birkar’s solution to this case, the key idea is to show that
the multiplicities of fibers are bounded. Then it is also natural to ask the
following question.

######  Conjecture 1.2.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then
there exists a positive integer $m$ depending only on $d$ and $\epsilon$
satisfying the following.

Assume that $(X,B)$ is a log pair of dimension $d$ and $\pi:X\to C$ is a
contraction such that

  1. (1)

$(X,B)$ is $\epsilon$-lc,

  2. (2)

$X$ is of Fano type over $C$ where $C$ is a curve, and

  3. (3)

$K_{X}+B\sim_{\mathbb{R},C}0$.

Then $m_{i}\leq m$ for every $i$, where $\pi^{*}z=\sum_{i}m_{i}F_{i}$, $F_{i}$
are the irreducible components of $\pi^{*}z$, and $z\in C$ is a closed point.

It is clear that Conjecture 1.2 is a direct corollary of Conjecture 1.1. As we
will see later, Conjecture 1.1 provides a suitable inductive frame to prove
boundedness results of some special kind of Calabi-Yau varieties (e.g.
Corollary 3.3).

We recall that a variety $X$ is said to be _rationally connected_ (RC for
short) if any two general points can be connected by the image of a rational
curve.

######  Conjecture 1.3.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then the
set of rationally connected projective varieties $X$ such that $\dim X=d$,
$(X,B)$ is $\epsilon$-lc and $K_{X}+B\sim_{\mathbb{R}}0$ for some boundary
$B$, is bounded up to flops.

In Conjecture 1.3, if we replace the condition $K_{X}+B\sim_{\mathbb{R}}0$
with $-K_{X}-B$ being nef, it is also expected that $X$ is bounded (e.g.
[MP04, Conjecture 3.9]). This case could be put in Calabi-Yau setting in the
generalized sense. Then we naturally ask the following boundedness conjecture
for generalized Calabi-Yau varieties.

######  Conjecture 1.4.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then the
set of rationally connected projective varieties $X$ such that $\dim X=d$,
$(X,B+\mathbf{M})$ is an $\epsilon$-lc g-pair and
$K_{X}+B+\mathbf{M}_{X}\sim_{\mathbb{R}}0$ for some $\mathbb{R}$-divisor $B$
and b-$\mathbb{R}$-divisor $\mathbf{M}$, is bounded in codimension one.

######  Remark 1.5.

For all above conjectures, we do not put any assumption on the coefficients of
$B$ like DCC condition or the coefficients are bounded from below. All these
conjectures are stated for pairs with $\mathbb{R}$ coefficients, however, up
to a perturbation, in the rest of this paper, we always assume that $\epsilon$
and the coefficients of $B$ are rational.

The purpose of this note is to reduce both Conjecture 1.1, Conjecture 1.3 and
Conjecture 1.4 to Conjecture 1.2. In Proposition 3.5, we will further reduce
Conjecture 1.2 to a even ‘simpler’ setting which is stated for Fano fibrations
(see Conjecture 3.4). Thus all conjectures stated in this note will be implied
by Conjecture 1.2 or Conjecture 3.4.

######  Theorem 1.6.

(=Proposition 3.1) Conjecture 1.2 in dimension $d$ implies Conjecture 1.1 in
dimension $d$.

######  Theorem 1.7.

(=Proposition 4.6) Conjecture 1.2 in dimension $d$ implies Conjecture 1.3 in
dimension $d$.

######  Theorem 1.8.

(=Proposition 4.10) Conjecture 1.2 in dimension $d$ implies Conjecture 1.4 in
dimension $d$.

In Section 4, we also confirm Conjecture 1.4 for surfaces and threefolds.
Moreover, we prove that those surfaces in Conjecture 1.4 are bounded. Remark
that Conjecture 1.4 in dimension $3$ is proved by Birkar, Di Cerbo, and Svaldi
[BDCS20, Theorem 1.6].

Acknowledgements. This work began when G. Chen visited C. Zhou at EPFL in
August of 2022. G. Chen would like to thank their hospitality. We would like
to thank Jingjun Han and Jihao Liu for valuable discussions and suggestions.
G. Chen is supported by the China post-doctoral grants BX2021269 and
2021M702925. C. Zhou is supported by grant European Research Council
(ERC-804334).

##  2\. Preliminaries

###  2.1. Divisors

Let $\mathbb{F}\in\\{\mathbb{Q},\mathbb{R}\\}$. Let $X$ be a normal variety
and $\operatorname{WDiv}(X)$ the free abelian group of Weil divisors on $X$.
Then an $\mathbb{F}$-divisor is defined to be an element of
$\operatorname{WDiv}(X)_{\mathbb{F}}:=\operatorname{WDiv}(X)\otimes\mathbb{F}$.

A _b-divisor_ on $X$ is an element of the projective limit

$\textbf{WDiv}(X)=\lim_{Y\to X}\operatorname{WDiv}(Y),$

where the limit is taken over all the pushforward homomorphisms
$f_{*}:\operatorname{WDiv}(Y)\to\operatorname{WDiv}(X)$ induced by proper
birational morphisms $f:Y\to X$. In other words, a b-divisor D on $X$ is a
collection of Weil divisors $\textbf{D}_{Y}$ on higher models $Y$ of $X$ that
are compatible under pushforward. The divisors $\textbf{D}_{Y}$ are called the
_traces_ of D on the birational models $Y.$ A b-$\mathbb{F}$-divisor is
defined to be an element of $\textbf{WDiv}(X)\otimes\mathbb{F}$.

The _Cartier closure_ of an $\mathbb{F}$-Cartier $\mathbb{F}$-divisor $D$ on
$X$ is the b-$\mathbb{F}$-divisor $\overline{D}$ with trace
$\overline{D}_{Y}=f^{*}D$ for any proper birational morphism $f:Y\to X$. A
b-$\mathbb{F}$-divisor D on $X$ is _b- $\mathbb{F}$-Cartier_ if
$\textbf{D}=\overline{D_{Y}}$ where $\overline{D_{Y}}$ is an
$\mathbb{F}$-Cartier $\mathbb{F}$-divisor on a birational model $Y$ of $X$, in
this situation, we say D _descends_ to $Y$. We say D is _b-nef_ if it descends
to a model $Y$ over $X$, such that $\textbf{D}_{Y}$ is nef. We say D is _b-
abundant_ if it descends to a model $Y$ over $X$ such that $\textbf{D}_{Y}$ is
abundant, i.e., $\kappa(Y,\textbf{D}_{Y})=\kappa_{\sigma}(Y,\textbf{D}_{Y})$,
see [Nak04, Chapters II,V] for the definitions.

###  2.2. Singularities of g-pairs

We adopt the standard notation and definitions in [KM98], and will freely use
them.

Here, we discuss very briefly the analogous concepts for generalized pairs,
while for further details we refer the readers to [BZ16].

######  Definition 2.1.

A _generalized sub-pair_ (g-sub-pair for short) $(X,B+\mathbf{M})$ consists of
an $\mathbb{R}$-divisor $B$ on $X$, and a nef b-$\mathbb{R}$-divisor
$\mathbf{M}$ on $X$, such that $K_{X}+B+\mathbf{M}_{X}$ is
$\mathbb{R}$-Cartier. If $B$ is effective, then we call $(X,B+\mathbf{M})$ a
_generalized pair_ (g-pair for short).

Let $(X,B+\mathbf{M})$ be a g-(sub-)pair and $f:W\to X$ a log resolution of
$(X,\operatorname{Supp}B)$ to which $\mathbf{M}$ descends. We may write

$K_{W}+B_{W}+\mathbf{M}_{W}=f^{*}(K_{X}+B+\mathbf{M}_{X})$

for some $\mathbb{R}$-divisor $B_{W}$ on $W$. Let $E$ be a prime divisor on
$W$. The _log discrepancy_ of $E$ with respect to $(X,B+\mathbf{M})$ is
defined as

$a(E,X,B+\mathbf{M}):=1-\operatorname{mult}_{E}B_{W}.$

We say $(X,B+\mathbf{M})$ is _$\epsilon$ -lc_ (resp., _lc, klt_) for some non-
negative real number $\epsilon$ if $a(E,X,B+\mathbf{M})\geq\epsilon$ (resp.,
$\geq 0,\ >0$) for any prime divisor $E$ over $X$.

######  Definition 2.2.

Let $(X,B+\mathbf{M})$ be a g-pair which is $a$-lc for some $a\geq 0$. Assume
that $D$ is an effective $\mathbb{R}$-divisor. The _$a$ -lc threshold_ of
$(X,B+\mathbf{M})$ with respect to $D$ is defined as

$\displaystyle a\text{-}\operatorname{lct}(X,B+\mathbf{M};D):=\sup\\{t\geq
0\mid$ $\displaystyle(X,(B+tD)+\mathbf{M})\text{ is $a$-lc}\\}.$

######  Definition 2.3.

We say $\pi:X\to Z$ is a _contraction_ if $X$ and $Z$ are normal quasi-
projective varieties, $\pi$ is a projective morphism, and
$\pi_{*}\mathcal{O}_{X}=\mathcal{O}_{Z}$ $(\pi$ is not necessarily
birational$)$.

######  Definition 2.4 ([PS09, Definition 2.3]).

Let $\pi:X\rightarrow Z$ be a contraction between normal varieties, $X$ is
said to be of _Fano type_ over $Z$ if one of the following equivalent
conditions holds:

  1. (1)

there exists a klt pair $(X,B)$ such that $-(K_{X}+B)$ is ample over $Z$;

  2. (2)

there exists a klt pair $(X,B^{\prime})$ such that
$K_{X}+B^{\prime}\sim_{\mathbb{R},Z}0$ and $-K_{X}$ is big over $Z$.

######  Lemma 2.5.

Assume that $(X,B+\mathbf{M})$ is an $\epsilon$-lc g-pair for some positive
real number $\epsilon$, such that $\mathbf{M}$ is b-abundant. Then there
exists a boundary $\Delta$ such that $(X,\Delta)$ is $\frac{\epsilon}{2}$-lc
and $\Delta\sim_{\mathbb{R}}B+\mathbf{M}_{X}$.

###### Proof.

By assumption, there exist a higher model $f:X^{\prime}\to X$ such that
$\mathbf{M}_{X^{\prime}}$ is abundant. We may write

$K_{X^{\prime}}+B^{\prime}+\mathbf{M}_{X^{\prime}}=f^{*}(K_{X}+B+\mathbf{M}_{X})$

for some $\mathbb{R}$-divisor $B^{\prime}$. Since $\mathbf{M}_{X^{\prime}}$ is
abundant, we can choose a decomposition
$\mathbf{M}_{X^{\prime}}\sim_{\mathbb{R}}E^{\prime}+A^{\prime}$ such that

  1. (1)

$E^{\prime}\geq 0$ and $A^{\prime}\geq 0$ is semi-ample, and

  2. (2)

$(X,B^{\prime}+E^{\prime}+A^{\prime})$ is $\frac{\epsilon}{2}$-sub-lc.

Set $\Delta:=f_{*}(B^{\prime}+E^{\prime}+A^{\prime})$, then
$0\leq\Delta\sim_{\mathbb{R}}B+\mathbf{M}_{X}$. Moreover, by the negativity
lemma, we see that

$f^{*}(K_{X}+\Delta)=K_{X^{\prime}}+B^{\prime}+E^{\prime}+A^{\prime}.$

It follows that $(X,\Delta)$ is $\frac{\epsilon}{2}$-lc. ∎

###  2.3. Canonical bundle formulas

Recall that for a sub-lc g-sub-pair $(X,B+\mathbf{M})$ and a contraction
$\pi:X\to Z$ such that $(X,B+\mathbf{M})$ is a g-pair over the generic point
of $Z$, $\dim Z>0$ and $K_{X}+B+\mathbf{M}_{X}\sim_{\mathbb{R}}\pi^{*}L_{Z}$
for some $\mathbb{R}$-Cartier $\mathbb{R}$-divisor $L_{Z}$ on $Z$, there
exists a g-sub-pair $(Z,B_{Z}+\mathbf{M}_{\pi})$ such that

$K_{X}+B+\mathbf{M}_{X}\sim_{\mathbb{R}}\pi^{*}(K_{Z}+B_{Z}+\mathbf{M}_{\pi,Z}),$

where $B_{Z}$ is defined as

$B_{Z}:=\sum_{P}(1-\operatorname{lct}(X/{Z}\ni\eta_{P},B+\mathbf{M};\pi^{*}P))P.$

We call $B_{Z}$ and $\mathbf{M}_{\pi}$ the _discriminant divisor_ and _moduli
b-divisor_ of _the canonical bundle formula_ for $(X,B+\mathbf{M})$ over $Z$
respectively, see [Fil20] for more details. If $(X,B+\mathbf{M})$ is an lc
(resp., klt) g-pair, then $(Z,B_{Z}+\mathbf{M}_{\pi})$ is lc (resp., klt). We
may refer the readers to [Bir19, Fil20, HL21, JLX22] for the recent process
towards the canonical bundle formulas.

The following result is well-known [Amb05].

######  Proposition 2.6.

Notation as above. Assume that $\mathbf{M}={\bf 0}$ and $(X,B)$ is klt. Then
$\mathbf{M}_{\pi}$ is b-nef and b-abundant.

###  2.4. Boundedness

A collection $\mathcal{P}$ of projective varieties is said to be _bounded_
(resp., _bounded in codimension one_) if there exists a projective morphism
$h:\mathcal{X}\to S$ between schemes of finite type such that each
$X\in\mathcal{P}$ is isomorphic (resp., isomorphic in codimension one) to
$\mathcal{X}_{s}$ for some closed point $s\in S$.

We say that a collection of projective log pairs $\mathcal{P}$ is _log
bounded_ (resp., _log bounded in codimension one_) if there is a quasi-
projective scheme $\mathcal{X}$, a reduced divisor $\mathcal{B}$ on
$\mathcal{X}$, and a projective morphism $h:\mathcal{X}\rightarrow S$, where
$S$ is of finite type and $\mathcal{B}$ does not contain any fiber, such that
for every $(X,B)\in\mathcal{P}$, there is a closed point $s\in S$ and an
isomorphism (resp., isomorphism in codimension one)
$f:\mathcal{X}_{s}\rightarrow X$ such that the support of
$\mathcal{B}_{s}:=\mathcal{B}|_{\mathcal{X}_{s}}$ coincides with the support
of $f_{*}^{-1}B$. Moreover, if $\mathcal{P}$ is a set of (log) Calabi-Yau
pairs, we also say $\mathcal{P}$ is _(log) bounded modulo flops_ when
$\mathcal{P}$ is (log) bounded in codimension one.

######  Lemma 2.7.

Assume that $\mathcal{P}$ is a bounded family of projective RC varieties. Then
there exists a positive integer $n$ depending only on $\mathcal{P}$ such that
for any $X\in\mathcal{P}$ and any Cartier divisor $D$ on $X$ with $D\equiv 0$,
we have that $nD\sim 0$.

###### Proof.

By assumption, there exists a projective morphism $h:\mathcal{X}\to S$ between
schemes of finite type such that each $X\in\mathcal{P}$ is isomorphic to
$X_{s}$ for some closed point $s\in S$. Then we can stratify $S$ into a union
of finitely many constructible subsets $S=\cup_{\alpha}S_{\alpha}$, such that
over each $S_{\alpha}$, $\mathcal{X}\times_{S}S_{\alpha}$ admits a resolution
$\mu_{\alpha}:\mathcal{Y}_{\alpha}\rightarrow\mathcal{X}\times_{S}S_{\alpha}$,
i.e., $(\mathcal{Y}_{\alpha},E_{\alpha}:=\operatorname{Ex}(\mu_{\alpha}))$ is
a simple normal crossing, and each stratum is log smooth over $S_{\alpha}$. In
particular, $(\mathcal{Y}_{s},E_{s})\rightarrow X_{s}$ is a resolution for
each $s\in S_{\alpha}$.

Fix an arbitrary $s\in S_{\alpha}$, and let $U_{\alpha}\subseteq S_{\alpha}$
be a contractible analytic neighborhood of $s$. Let
$f_{U_{\alpha}}:\mathcal{Y}_{U_{\alpha}}:=f^{-1}(U_{\alpha})\rightarrow
U_{\alpha}$ be the restriction of $f_{\alpha}:\mathcal{Y}_{\alpha}\to
S_{\alpha}$. Since $f_{U_{\alpha}}$ is topologically locally trivial and
$U_{\alpha}$ is contractible, the restriction map
$H^{2}(\mathcal{Y}_{U_{\alpha}},\mathbb{Z})\rightarrow
H^{2}(\mathcal{Y}_{s},\mathbb{Z})$ is an isomorphism. Hence we see that
$H^{2}(\mathcal{Y}_{s},\mathbb{Z})$ is independent of the choice of $s\in S$.
Since $X$ is rationally connected, ${\rm Pic}(\mathcal{Y}_{s})\cong
H^{2}(\mathcal{Y}_{s},\mathbb{Z})$. Moreover, as
$H^{2}(\mathcal{Y}_{s},\mathbb{Z})$ is a finitely generated
$\mathbb{Z}$-module, we can find a positive integer $n$ which only depends on
$H^{2}(\mathcal{Y}_{s},\mathbb{Z})$ such that if $D_{s}$ is a numerically
trivial Cartier divisor on $\mathcal{Y}_{s}$, then $nD_{s}\sim 0$. We can
conclude that $n$ has the required property. This finishes the proof. ∎

The following lemma is well-known.

######  Lemma 2.8.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Assume
that $(X,B+\mathbf{M})$ is a projective $\epsilon$-lc log Calabi-Yau g-pair of
dimension $d$ and $\phi:X\dashrightarrow X^{\prime}$ is a birational
contraction. Suppose that $X^{\prime}$ is bounded in codimension one. Then $X$
is bounded in codimension one.

###### Proof.

Replacing $X^{\prime}$, we may assume that $X^{\prime}$ is bounded. Denote
$B^{\prime}$ to be the strict transform of $B$ on $X^{\prime}$, then it is
obvious that $(X^{\prime},B^{\prime}+\mathbf{M})$ is an $\epsilon$-lc log
Calabi-Yau g-pair. For any prime divisor $E$ on $X$ which is exceptional over
$X^{\prime},$ $a(E,X^{\prime},B^{\prime}+\mathbf{M})=a(E,X,B+\mathbf{M})\leq
1$. So we can take a birational morphism $g:X^{\prime\prime}\rightarrow
X^{\prime}$ extracting all prime divisors on $X$ which are exceptional over
$X^{\prime}$. In particular, $X^{\prime\prime}$ is $\mathbb{Q}$-factorial and
isomorphic to $X$ in codimension 1 . We may write

$K_{X^{\prime\prime}}+B^{\prime\prime}+\mathbf{M}_{X^{\prime\prime}}=g^{*}(K_{X^{\prime}}+B^{\prime}+\mathbf{M}_{X^{\prime}}).$

Then $(X^{\prime\prime},B^{\prime\prime}+\mathbf{M})$ is also an $\epsilon$-lc
log Calabi-Yau g-pair and $X^{\prime\prime}\rightarrow X^{\prime}$ is of Fano
type. Apply [Bir18, Theorem 2.2] (see also [HJ22, Theorem 3.1]) to
$X^{\prime\prime}\rightarrow X^{\prime}$, such $X^{\prime\prime}$ belongs to a
bounded family, and hence $X$ is bounded in codimension one. ∎

The following lemma is a small modification of [Bir18, Theorem 2.2].

######  Lemma 2.9.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Assume
that $(X,B+\mathbf{M})$ is a projective g-pair and $\pi:X\to Z$ is a
contraction between $\mathbb{Q}$-factorial varieties such that

  * •

$(X,B+\mathbf{M})$ is $\epsilon$-lc of dimension $d$,

  * •

$\pi:X\to Z$ is a Mori fiber space, and

  * •

$K_{X}+B+\mathbf{M}_{X}\sim_{\mathbb{R}}0$.

Suppose that $Z$ is bounded in codimension one. Then $X$ is bounded in
codimension one.

###### Proof.

By assumption, $B+\mathbf{M}_{X}\not\equiv 0$. As $Z$ is bounded in
codimension one, there exists a projective normal variety $Z^{\prime}$ that is
bounded, such that $Z^{\prime}$ is isomorphic to $Z$ in codimension one. By
the same arguments as in [DCS21, Lemma 3.4], there exists a projective normal
variety $X^{\prime}$ such that

  * •

$\phi:X\dashrightarrow X^{\prime}$ is isomorphic in codimension one, and

  * •

$X^{\prime}\rightarrow Z^{\prime}$ is a Mori fiber space.

Let $B^{\prime}$ be the strict transform of $B$ on $X^{\prime}$, it is clear
that $(X^{\prime},B^{\prime}+\mathbf{M})$ is $\epsilon$-lc and
$K_{X^{\prime}}+B^{\prime}+\mathbf{M}_{X^{\prime}}\sim_{\mathbb{R}}0$. Since
$Z^{\prime}$ is bounded, $X^{\prime}$ is bounded according to [Bir18, Theorem
2.2]. Thus $X$ is bounded in codimension one. This completes the proof. ∎

The following lemma is a small modification of [Bir18, Theorem 1.4].

######  Lemma 2.10.

Let $d$ be a positive integer, and $\epsilon,\delta$ two positive real
numbers. Let $\mathcal{P}$ be a set of $\mathbb{Q}$-factorial projective
varieties. Consider the set $\mathcal{P}^{\prime}$ consisting of
$\mathbb{Q}$-factorial projective pairs $(X,B)$ such that

  1. (1)

$(X,B)$ is $\epsilon$-lc log Calabi-Yau of dimension $d$,

  2. (2)

the non-zero coefficients of $B$ are $\geq\delta$,

  3. (3)

there is a contraction $\pi:X\to Z$ with $Z\in\mathcal{P}$, and

  4. (4)

$X\rightarrow Z$ factors as a tower of Fano fibrations such that $X_{i}\to
X_{i+1}$ is a Mori fiber space for each $0\leq i\leq l-1$.

If $\mathcal{P}$ is bounded modulo flops, then $\mathcal{P}^{\prime}$ is log
bounded modulo flops.

###### Proof.

By assumption, there exists a tower of Fano fibrations

$X=:X_{0}\stackrel{{\scriptstyle\phi_{0}}}{{\longrightarrow}}X_{1}\stackrel{{\scriptstyle\phi_{1}}}{{\longrightarrow}}X_{2}\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle}}{{\longrightarrow}}X_{l}:=Z,$

i.e., the morphism $\phi_{i}:X_{i}\rightarrow X_{i+1}$ is a Mori fiber space
for any $0\leq i<l$. As $(X,B)$ is $\epsilon$-lc log Calabi-Yau, there exists
a boundary $B_{i}$ on $X_{i}$ such that $(X_{i},B_{i})$ is klt log Calabi-Yau,
and $K_{X_{i-1}}+B_{i-1}\sim_{\mathbb{R}}\phi_{i-1}^{*}(K_{X_{i}}+B_{i})$ for
each $1\leq i\leq l$, see [Amb05]. Note here that $B_{i}\neq 0$ for any $1\leq
i\leq l-1$, as $\phi_{i}$ are Fano fibrations. As $Z\in\mathcal{P}$ is bounded
modulo flops, there exist a projective normal variety $Z^{\prime}$ which is
bounded and a small birational contraction $Z\dashrightarrow Z^{\prime}$.
Moreover, by [DCS21, Lemma 3.4], there exist projective normal varieties
$X_{0}^{\prime},\dots,X_{l}^{\prime}:=Z^{\prime}$ such that for any $0\leq
i\leq l-1$, it holds that

  * •

$X_{i}\dashrightarrow X_{i}^{\prime}$ is a small birational contraction, and

  * •

$X_{i}^{\prime}\rightarrow X_{i+1}^{\prime}$ is a Mori fiber space.

${X_{0}}$${X_{1}}$${X_{2}}$${\cdots}$${X_{l}:=Z}$${X_{0}^{\prime}}$${X_{1}^{\prime}}$${X_{2}^{\prime}}$${\cdots}$${X_{l}^{\prime}:=Z^{\prime}}$

Denote by $B_{0}^{\prime}$ the push-down of $B$ on $X_{0}^{\prime}$. It is
clear that $(X_{0}^{\prime},B_{0}^{\prime})$ is also $\epsilon$-lc log Calabi-
Yau and $B_{0}^{\prime}\geq\delta$. According to [Bir18, Theorem 1.4],
$(X_{0}^{\prime},B_{0}^{\prime})$ is log bounded. Therefore $X$ is bounded
modulo flops and thus $(X,B)$ is log bounded modulo flops. The proof is
finished. ∎

##  3\. From multiplicities to singularities

###  3.1. Proof of Theorem 1.6

######  Proposition 3.1.

Conjecture 1.2 in dimension d implies Conjecture 1.1 in dimension d.

###### Proof.

To prove Conjecture 1.1, we first assume that $Z$ is a curve. Then it is
enough to show that

$\operatorname{lct}(X,B;\pi^{*}z)\geq\delta$

for some positive real number $\delta$ which only depends on $d$ and
$\epsilon$. Put

$t:=\frac{\epsilon}{2}\text{-}\operatorname{lct}(X,B;\pi^{*}z).$

Take $\phi:W\to X$ a log resulotion of $(X,B+\pi^{*}z)$ such that we can write

$K_{W}+B_{W}+t(\pi^{*}z)_{W}+(1-\frac{\epsilon}{2})\sum_{i}E_{i}=\phi^{*}(K_{X}+B+t\pi^{*}z)+\sum_{j}b_{j}F_{j},$

where $B_{W}$ and $(\pi^{*}z)_{W}$ are the strict transforms of $B$ and
$\pi^{*}z$ on $W$ respectively, $E_{i}\neq F_{j}$ are the $\phi$-exceptional
prime divisors, and $b_{j}\geq\frac{\epsilon}{2}-1$ are real numbers. We may
also write

$K_{W}+B_{W}=\phi^{*}(K_{X}+B)+\sum_{i}a_{i}E_{i}+\sum_{j}c_{j}F_{j},$

for some real numbers $a_{i},c_{j}\geq\epsilon-1$.

Pick a small enough positive real number $\mu$ such that

$(1-\mu)(1-\frac{\epsilon}{2})-\mu\max_{i}{a_{i}}>0.$

It follows that

$\Delta_{W}:=B_{W}+(1-\mu)t(\pi^{*}z)_{W}+\sum_{i}((1-\mu)(1-\frac{\epsilon}{2})-\mu
a_{i})E_{i}\geq 0.$

Note that $(X,B+(1-\mu)t\pi^{*}z)$ is $\frac{\epsilon}{2}$-lc, and

$K_{W}+\Delta_{W}=\phi^{*}(K_{X}+B+(1-\mu)t\pi^{*}z)+\sum_{j}((1-\mu)b_{j}+\mu
c_{j})F_{j},$

where $(1-\mu)b_{j}+\mu c_{j}\geq\frac{\epsilon}{2}-1$. As
$(W,\Delta_{W}+\sum_{j}F_{j})$ is dlt and

$K_{W}+\Delta_{W}+\sum_{j}F_{j}\sim_{X,\mathbb{Q}}\sum_{j}(1+(1-\mu)b_{j}+\mu
c_{j})F_{j}\geq\frac{\epsilon}{2}\sum_{j}F_{j},$

we could run an MMP on $K_{W}+\Delta_{W}+\sum_{j}F_{j}$ over $X$ and it
terminates with a model $X^{\prime}$ over $X$. Note that the MMP contracts
only $\sum_{j}F_{j}$. Therefore we have

$K_{X^{\prime}}+\Delta^{\prime}=\phi^{\prime*}(K_{X}+B+(1-\mu)t\pi^{*}z)\sim_{\mathbb{Q},Z}0,$

where $\Delta^{\prime}$ is the push-forward of $\Delta_{W}$ on $X^{\prime}$
and we denote by $\phi^{\prime}:X^{\prime}\to X$.

Now $(X^{\prime},\Delta^{\prime})$ and $\pi^{\prime}:X^{\prime}\to Z$ satisfy
the assumptions in Conjecture 1.2 by replacing $\epsilon$ with
$\frac{\epsilon}{2}$, thus $n_{i}\leq m$ for some positive integer $m$
depending only on $d$ and $\epsilon$, where

$\pi^{\prime*}z=\sum_{i}n_{i}E_{i}^{\prime}\ +\ \text{(other components)}$

and $E_{i}^{\prime}$ are the push-forwards of $E_{i}$ on $X^{\prime}$. It is
not hard to see that, for each $i$, we have

$tn_{i}-a_{i}=1-\frac{\epsilon}{2}.$

Therefore we can see that

$t=\frac{1-\frac{\epsilon}{2}+a_{i}}{n_{i}}\geq\frac{\epsilon}{2m}=:\delta.$

We conclude that $\delta$ has the required property as

$\operatorname{lct}(X,B;\pi^{*}z)>\frac{\epsilon}{2}\text{-}\operatorname{lct}(X,B;\pi^{*}z)\geq\delta.$

When $\dim Z>1$, by the proof of [Bir16, Lemma 3.2], one can always reduce it
to the case when the base is of dimension one. The proof is finished. ∎

The idea of the proof of the following result follows from [Bir16, Proposition
3.5].

######  Proposition 3.2.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Assume
Conjecuture 1.2 holds in dimension $d$. Suppose that $(X,B)$ and $\pi:X\to Z$
satisfy the assumptions in Conjecture 1.1, then the generalized pair
$(Z,B_{Z}+\mathbf{M}_{\pi})$ is $\delta$-lc for some positive real number
$\delta$ depending only on $d$ and $\epsilon$. Here $\mathbf{M}_{\pi}$ is the
moduli b-divisor of the canonical bundle formula for $(X,B)$ over $Z$.

###### Proof.

By Theorem 1.6, Conjecture 1.1 also holds in dimension $d$. Let $\delta$ be a
positive real number satisfying the property of Conjecture 1.1 with data $d$
and $\frac{\epsilon}{2}$. We will show that $\delta$ has the required
property.

Let $Z^{\prime}\to Z$ be a log resolution of $(Z,\operatorname{Supp}B_{Z})$
such that $\mathbf{M}_{\pi}$ descends to $Z^{\prime}$. It suffices to show
that $B_{Z^{\prime}}\leq 1-\delta.$ Let $\phi:W\rightarrow X$ be a log
resolution such that the induced rational map $W\rightarrow Z^{\prime}$ is a
morphism. Let

$\Delta_{W}:=B_{W}+(1-\frac{\epsilon}{2})G$

where $B_{W}$ is the strict transform of $B$ on $W$ and $G$ is the reduced sum
of exceptional divisor of $\phi$. The pair $(W,\Delta_{W})$ is
$\frac{\epsilon}{2}$-lc and we can write

$K_{W}+\Delta_{W}=\phi^{*}(K_{X}+B)+E$

where $E\geq 0$ and $\operatorname{Supp}E=\operatorname{Supp}G$.

Let $T$ be the graph of the rational map $X\dashrightarrow Z^{\prime}$, that
is, $T$ is the closure in $X\times Z^{\prime}$ of the graph of
$X_{0}\rightarrow Z^{\prime}$ where $X_{0}\subseteq X$ is the domain of
$X\dashrightarrow Z^{\prime}$. Since $W$ maps to both $X$ and $Z^{\prime}$, we
get an induced morphism $W\rightarrow T$. Let $U\subseteq Z$ be a non-empty
open set over which $Z^{\prime}\rightarrow Z$ is an isomorphism. We can run an
MMP on $K_{W}+\Delta_{W}$ with scaling of some ample divisor over $T$ and it
terminates with a model $Y$ on which $K_{Y}+\Delta_{Y}$ is nef over $T$. Since
$\pi^{-1}U\subseteq X_{0}$, the morphism $T\rightarrow X$ is an isomorphism
over $U$. We may also assume that $U$ intersects nothing with the image of $G$
under $W\to Z$, then the morphism $Y\rightarrow T$ is also an isomorphism over
$U$. Note that $K_{X}+B\sim_{\mathbb{Q},Z}0,$ we see that
$K_{Y}+\Delta_{Y}\sim_{\mathbb{Q},U}0$.

${W}$${Y}$${X^{\prime}}$${X}$${T}$${Z}$${Z^{\prime}}$${Z^{\prime\prime}}$$\scriptstyle{\phi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$

Now, by [BCHM10], we could run an MMP on $K_{Y}+\Delta_{Y}$ with scaling of
some ample divisor over $Z^{\prime}$ such that it ends up with a model
$X^{\prime}$, on which $K_{X^{\prime}}+\Delta_{X^{\prime}}$ is semi-ample over
$Z^{\prime}$ as $\Delta_{Y}$ is big over $Z^{\prime}$, where
$\Delta_{X^{\prime}}$ is the push-forward of $\Delta_{Y}$ on $X^{\prime}$.
Recall that $K_{Y}+\Delta_{Y}\sim_{\mathbb{Q},U}0$, we see that
$Y\dashrightarrow X^{\prime}$ is an isomorphism over $U$. Let
$\pi^{\prime}:X^{\prime}\rightarrow Z^{\prime\prime}$ be the canonical model
of $K_{X^{\prime}}+\Delta_{X^{\prime}}$ over $Z^{\prime}$. Then the morphism
$Z^{\prime\prime}\rightarrow Z^{\prime}$ is birational. Moreover, we have that
$(X^{\prime},\Delta_{X^{\prime}})$ is $\frac{\epsilon}{2}$-lc, and
$K_{X^{\prime}}+\Delta_{X^{\prime}}\sim_{\mathbb{Q},Z^{\prime\prime}}0$. Let
$\Delta_{Z^{\prime\prime}}$ be the discriminant part on $Z^{\prime\prime}$ of
the canonical bundle formula for $(X^{\prime},\Delta_{X^{\prime}})$ over
$Z^{\prime\prime}$. By our choice of $\delta$, we see that
$\Delta_{Z^{\prime\prime}}\leq 1-\delta$. Denote by $\tau:Z^{\prime\prime}\to
Z$. It is enough to show that
$B_{Z^{\prime\prime}}\leq\Delta_{Z^{\prime\prime}}$, where
$B_{Z^{\prime\prime}}$ is given by

$K_{Z^{\prime\prime}}+B_{Z^{\prime\prime}}+\mathbf{M}_{\pi,Z^{\prime\prime}}=\tau^{*}(K_{Z}+B_{Z}+\mathbf{M}_{\pi,Z}).$

Let $p:V\rightarrow X$ and $q:V\rightarrow X^{\prime}$ be a common resolution
of $X$ and $X^{\prime}$, and let

$P:=K_{X^{\prime}}+\Delta_{X^{\prime}}-q_{*}p^{*}(K_{X}+B).$

As mentioned above we have

$K_{W}+\Delta_{W}-\phi^{*}(K_{X}+B)=E\geq 0.$

Now $P$ is just the pushdown of $K_{W}+\Delta_{W}-\phi^{*}(K_{X}+B)$ via the
rational map $W\dashrightarrow X^{\prime}$. Therefore, $P\geq 0$. From
$K_{X}+B\sim_{\mathbb{Q},Z}0$ we get

$p^{*}(K_{X}+B)=q^{*}q_{*}p^{*}(K_{X}+B),$

and this combined with $P\geq 0$ results in

$\displaystyle
q^{*}(K_{X^{\prime}}+\Delta_{X^{\prime}})-p^{*}(K_{X}+B)=q^{*}(K_{X^{\prime}}+\Delta_{X^{\prime}})-q^{*}q_{*}p^{*}(K_{X}+B)=q^{*}P\geq
0.$

It follows that $B_{Z^{\prime\prime}}\leq\Delta_{Z^{\prime\prime}}$ by the
definition of the canonical bundle formula. The proof is finished. ∎

######  Corollary 3.3 (cf. [Amb05, FG12]).

In the same setting as in Proposition 3.2, there exists a boundary
$\Delta_{Z}$ on $Z$, such that $(Z,\Delta_{Z})$ is $\frac{\delta}{2}$-lc.

###### Proof.

The result follows from Lemma 2.5, Proposition 2.6 and Proposition 3.2
immediately. ∎

Corollary 3.3 essentially tells us that, under the assumption of Conjecture
1.2, for any given fibration $(X,B)\to Z$ as in Conjecture 1.1, there exists a
boundary $\Delta_{Z}$ such that $(Z,\Delta_{Z})$ is $\frac{\delta}{2}$-lc,
where $\delta$ depends only on $d,\epsilon$. As we do not put any assumption
on the coefficients on the boundary (e.g. DCC condition), this provides a
suitable inductive frame.

######  Conjecture 3.4.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then
there exists a positive integer $m$ depending only on $d$ and $\epsilon$
satisfying the following.

Assume that $X$ is a normal variety of dimension $d$ and $\pi:X\to C$ is a
contraction such that

  1. (1)

$X$ is $\epsilon$-lc,

  2. (2)

$-K_{X}$ is ample over $C$ where $C$ is a curve, and

  3. (3)

every fiber of $X\to C$ is irreducible.

Then $m_{z}\leq m$, where $\pi^{*}z=m_{z}F_{z}$, $F_{z}$ is the irreducible
component of $\pi^{*}z$, and $z\in C$ is a closed point.

######  Proposition 3.5.

Conjecture 3.4 in dimension $d$ implies Conjecture 1.2 in dimension $d$.

###### Proof.

Let $\pi:(X,B)\to C$ be a contraction given in Conjecture 1.2. Possibly
replacing $X$ with a small $\mathbb{Q}$-factorialization, we may assume that
$X$ is $\mathbb{Q}$-factorial. Write $\pi^{*}z=\sum m_{i}F_{i}$, we aim to
show that there exists a positive integer $m$ depending only on $d$ and
$\epsilon$ such that $m_{i}\leq m$ for every $i$. We focus on $i_{0}$ among
these $i$. Run an MMP on $K_{X}+B+\eta\sum_{i\neq i_{0}}F_{i}$ over $C$ for
some $0<\eta\ll 1$, we get a good minimal model $\phi:X\dashrightarrow
X^{\prime}/C$ such that $F_{i_{0}}^{\prime}:=\phi_{*}F_{i_{0}}$ is the only
component in the fiber over $z$ for $\pi^{\prime}:X^{\prime}\to C$. Denote by
$B^{\prime}:=\phi_{*}B$. Further run an MMP on
$K_{X^{\prime}}+(1+\eta^{\prime})B^{\prime}$ over $C$ for some
$0<\eta^{\prime}\ll 1$, we get a good minimal model $X^{\prime}\dashrightarrow
X^{\prime\prime}/C$ such that $-K_{X^{\prime\prime}}$ is big and nef over $C$.
Replace $X^{\prime\prime}$ with its ample model with respect to
$-K_{X^{\prime\prime}}$ over $C$, we may assume $-K_{X^{\prime\prime}}$ is
ample over $C$. Denote by $\pi^{\prime\prime}:X^{\prime\prime}\to C$ and
$B^{\prime\prime}$ the push-forward of $B^{\prime}$ on $X^{\prime\prime}$, we
see that $\pi^{\prime\prime}:(X^{\prime\prime},B^{\prime\prime})\to C$
satisfies all the conditions in Conjecture 3.4 and
$\pi^{\prime\prime*}z=m_{i_{0}}F_{i_{0}}^{\prime\prime}$, where
$F_{i_{0}}^{\prime\prime}$ is the push-forward of $F_{i_{0}}^{\prime}$ on
$X^{\prime\prime}$. By Conjecture 3.4, we see that there exists a required $m$
such that $m_{i_{0}}\leq m$. The proof is finished. ∎

######  Corollary 3.6.

Conjecture 3.4 in dimension d implies Conjecture 1.1 in dimension d.

###### Proof.

Combine Proposition 3.1 with Proposition 3.5. ∎

###  3.2. Generalized version of Conjecture 1.1

######  Conjecture 3.7.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then
there exists a positive real number $\delta$ depending only on $d$ and
$\epsilon$ satisfying the following.

Assume that $(X,B+\mathbf{M})$ is a g-pair of dimension $d$ and $\pi:X\to Z$
is a contraction between projective normal varieties such that

  1. (1)

$(X,B+\mathbf{M})$ is $\epsilon$-lc,

  2. (2)

$X$ is of Fano type over $Z$, and

  3. (3)

$K_{X}+B+\mathbf{M}_{X}\sim_{\mathbb{R},Z}0$.

Then $B_{Z}\leq 1-\delta$, where $B_{Z}$ is the discriminant part of the
canonical bundle formula for $(X,B+\mathbf{M})$ over $Z$.

######  Conjecture 3.8.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then
there exists a positive integer $m$ depending only on $d$ and $\epsilon$
satisfying the following.

Assume that $(X,B+\mathbf{M})$ is g-pair of dimension $d$ and $\pi:X\to C$ is
a contraction such that

  1. (1)

$(X,B+\mathbf{M})$ is $\epsilon$-lc,

  2. (2)

$X$ is of Fano type over $C$ where $C$ is a curve, and

  3. (3)

$K_{X}+B+\mathbf{M}_{X}\sim_{\mathbb{R},C}0$.

Then $m_{i}\leq m$ for every $i$, where $\pi^{*}z=\sum_{i}m_{i}F_{i}$, $F_{i}$
are the irreducible components of $\pi^{*}z$, and $z\in C$ is a closed point.

######  Proposition 3.9.

Conjecture 1.2 in dimension $d$ implies Conjectures 3.7 and 3.8 in dimension
$d$.

###### Proof.

We first show that Conjecture 1.2 implies Conjecture 3.8.

Let $(X,B+\mathbf{M})$ be a g-pair and $\pi:X\to C$ a contraction as in
Conjecture 3.8. Possibly replacing $X$ with a $\mathbb{Q}$-factorialization,
one can assume that $X$ is $\mathbb{Q}$-factorial. Write $\pi^{*}z=\sum
m_{i}F_{i}$. We focus on $i_{0}$ among these $i$. One can run an MMP on
$K_{X}+B+\mu\sum_{i\neq i_{0}}F_{i}+\mathbf{M}_{X}$ over $C$ for some
$0<\mu\ll 1$, which ends with a good minimal model $X\dashrightarrow
X^{\prime}$ over $C$, cf. [BZ16, Lemma 4.4]. This MMP contracts all components
in $\pi^{*}z$ except $F_{i_{0}}$. Replace $X$ with $X^{\prime}$ we may assume
that $\pi^{*}z$ is irreducible.

Then we could run MMP on $K_{X}+B+(1+\mu)\mathbf{M}_{X}$ over $C$ for some
$0<\mu\ll 1$ and it ends with a good minimal model $X\dashrightarrow
X^{\prime\prime}$ over $C$ (e.g. [BZ16, Lemma 4.4]). In other words,
$\mathbf{M}_{X^{\prime\prime}}$ is semi-ample over $C$. Note that
$(X,B+\mathbf{M})$ and $(X^{\prime\prime},B^{\prime\prime}+\mathbf{M})$ are
crepant to each other, thus we may assume $\mathbf{M}_{X}$ is semi-ample over
$C$ via replacing $X$ with $X^{\prime\prime}$. Pick a general element $A$ in a
very ample linear system on $C$ such that

  1. (1)

$\mathbf{M}_{X}+\pi^{*}A$ is semi-ample, and

  2. (2)

the g-pair $(X,B+(\mathbf{M}+\overline{\pi^{*}A}))$ is still $\epsilon$-lc
with nef part $\mathbf{M}+\overline{\pi^{*}A}$.

Let $M\in|\mathbf{M}_{X}+\pi^{*}A|_{\mathbb{Q}}$ be a general element, then it
is not hard to see that the fibration $(X,B+M)\to C$ satisfies all the
conditions in Conjecture 1.2. Note that $\pi^{*}z=m_{i_{0}}F_{i_{0}}$, by
Conjecture 1.2, there exists a positive integer $m$ depending only on $d$ and
$\epsilon$ such that $m_{i_{0}}\leq m$.

Next we turn to show that Conjecture 1.2 implies Conjecture 3.7. It is enough
to show that Conjecture 3.8 implies Conjecture 3.7. Indeed, the idea of the
proof is essentially the same as the proof of Proposition 3.1 but one need to
use the termination of MMP for g-pairs developed in [BZ16], e.g. [BZ16, Lemma
4.4]. ∎

Using very similar arguments as in Proposition 3.2 and the tools developed in
[BZ16], e.g. [BZ16, Lemma 4.4], we can show the following result. We left the
proof to the reader.

######  Proposition 3.10.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Assume
Conjecuture 1.2 holds in dimension $d$. Suppose that $(X,B+\mathbf{M})$ and
$\pi:X\to Z$ satisfy the assumptions in Conjecture 3.7, then the g-pair
$(Z,B_{Z}+\mathbf{M}_{\pi})$ is $\delta$-lc for some positive real number
$\delta$ depending only on $d$ and $\epsilon$. Here $\mathbf{M}_{\pi}$ is the
moduli b-divisor of the canonical bundle formula for $(X,B+\mathbf{M})$ over
$Z$.

######  Corollary 3.11.

Conjecture 3.4 in dimension $d$ implies Conjecture 3.7 in dimension $d$.

###### Proof.

The proof is a combination of Propositions 3.9 and 3.5. ∎

######  Corollary 3.12.

Conjecture 3.7 holds in dimension 2. More generally, Conjecture 3.7 holds for
relative dimension one fibrations, and the g-pair $(Z,B_{Z}+\mathbf{M}_{\pi})$
is $\delta$-lc.

###### Proof.

By [Bir16], Conjecture 1.2 holds in dimension 2. Thus by Proposition 3.10,
Conjecture 3.7 holds in dimension 2. By the same argument as the proof of
[Bir16, Corollary 1.7], the “More generally” part holds true. ∎

##  4\. On boundedness of RC Calabi-Yau varieties

###  4.1. Conjectures on RC Calabi-Yau varieties

######  Conjecture 4.1.

Let $d$ be a positive integer. Then $1$ is not an accumulation point from
below for the set of minimal log discrepancies of all normal projective
Calabi-Yau rationally connected varieties of dimension $d$.

######  Conjecture 4.2.

Let $d$ be a positive integer. Then there exists a positive integer $n$
depending only on $d$ such that for any $d$-dimensional klt Calabi-Yau
rationally connected projective variety $X$, $nK_{X}\sim 0$.

######  Conjecture 4.3.

Let $d$ be a positive integer. Then all $d$-dimensional klt Calabi-Yau
rationally connected projective varieties form a bounded family modulo flops.

######  Proposition 4.4.

The above three conjectures are equivalent.

###### Proof.

The equivalence between Conjecture 4.2 and Conjecture 4.3 is obtained by
[BDCS20, Corollary 4.3]. The implication $4.2\Rightarrow 4.1$ is clear. The
rest is devoted to the implication $4.1\Rightarrow 4.3$.

First observe that $X$ is non-canonical (e.g. [Jia21, Lemma 6.3]), that is
there exists a prime divisor $E$ over $X$ such that $a(E,X,0)<1$. Then we can
take a crepant model $(Y,B_{Y})$ of $X$ that extracts only $E$ and $Y$ is
$\mathbb{Q}$-factorial. By Conjecture 4.1, $B_{Y}\geq\delta$ for some positve
real number $\delta$ which only depends on $d$. By global ACC (e.g. [HMX14]),
$(Y,B_{Y})$ is $\epsilon$-lc for some $0<\epsilon<\frac{1}{2}$ depending only
on $d$.

We claim that $(Y,B_{Y})$ is log bounded in codimension one. In fact, by
[DCS21, Theorem 3.2], there exists a birational contraction
$\psi:Y\dashrightarrow Y^{\prime}$ to a $\mathbb{Q}$-factorial log Calabi-Yau
pair $(Y^{\prime},B_{Y^{\prime}}:=\psi_{*}B_{Y}),$ $B_{Y^{\prime}}\neq 0$ and
a tower of morphisms

$Y^{\prime}=:X_{0}\stackrel{{\scriptstyle}}{{\longrightarrow}}X_{1}\stackrel{{\scriptstyle}}{{\longrightarrow}}X_{2}\stackrel{{\scriptstyle}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle}}{{\longrightarrow}}X_{k}$

such that

  * •

the morphism $X_{i}\rightarrow X_{i+1}$ is a Mori fibre space for any $0\leq
i<k$, and

  * •

either $\operatorname{dim}X_{k}=0$, or $\operatorname{dim}X_{k}>0$ and $X_{k}$
is $\mathbb{Q}$-factorial klt Calabi-Yau.

If $\dim X_{k}=0$, then $Y^{\prime}$ is bounded according to [Bir18, Theorem
1.4]. It implies that $Y$ is bounded in codimension one by Lemma 2.8 and thus
$(Y,B_{Y})$ is log bounded in codimension one as $B_{Y}\geq\delta$. If $\dim
X_{k}>0$ and $X_{k}$ is $\mathbb{Q}$-factorial klt Calabi-Yau, then by
induction on dimension, $X_{k}$ is bounded modulo flops. Applying Lemma 2.10,
$(Y,B_{Y})$ is log bounded modulo flops. Therefore the claim holds.

We conclude that $X$ is also bounded in codimension one by the next Lemma 4.5.
∎

The following lemma is useful in order to show that if a set of log Calabi-Yau
pairs is bounded in codimension one, then the set of certain birational models
of them remains bounded in codimension one. The proof is well-known to experts
(cf. [CDCH+21, Theorem 5.1]).

######  Lemma 4.5.

Let $\epsilon$ be a positive real number. Let $\mathcal{P}$ be a set of
projective $\epsilon$-lc log Calabi-Yau pairs $(Y,B_{Y})$ such that
$\operatorname{Supp}B_{Y}$ is a prime divisor. Denote $\mathcal{P}^{\prime}$
to be the set consisting of Calabi-Yau projective varieties $X$ such that
there exists a birational morphism $h:Y\rightarrow X$, and
$h^{*}K_{X}=K_{Y}+B_{Y}$, and $(Y,B_{Y})\in\mathcal{P}$. Suppose that
$\mathcal{P}$ is log bounded modulo flops. Then $\mathcal{P}^{\prime}$ is
bounded modulo flops.

###### Proof.

By assumption $\mathcal{P}$ is bounded modulo flops, there is a quasi-
projective scheme $\mathcal{Y}$, a prime divisor $\mathcal{B}$ that does not
contain any fiber of the projective morphism $\mathcal{Y}\to S$, where $S$ is
a disjoint union of finitely many normal varieties such that for every
$(Y,B_{Y})\in\mathcal{P}$, there is a closed point $s\in S$ and a small
birational contraction $f:\mathcal{Y}_{s}\dashrightarrow Y$ such that
$\mathcal{B}_{s}:=\mathcal{B}|_{\mathcal{Y}_{s}}=f_{*}^{-1}B_{Y}$. We may
assume that the set of those points $s$ is dense in $S$. After decomposing
$S$, we may also assume that $S$ is smooth.

Fix a point $s\in S$ corresponding to some $(Y,B_{Y})$,

$K_{\mathcal{Y}_{s}}+f_{*}^{-1}B_{Y}\sim_{\mathbb{Q}}f_{*}^{-1}(K_{Y}+B_{Y})\sim_{\mathbb{Q}}0$

and therefore $(\mathcal{Y}_{s},f_{*}^{-1}B_{Y})$ is an $\epsilon$-lc log
Calabi-Yau pair. Now consider a log resolution
$g:\mathcal{W}\rightarrow\mathcal{Y}$ of $(\mathcal{Y},\mathcal{B})$ and
denote by $\mathcal{B}_{\mathcal{W}}$ the reduced sum of strict transform of
$\mathcal{B}$ and all $g$-exceptional divisors on $\mathcal{W}$. Consider the
$\frac{\epsilon}{2}$-lc pair
$(\mathcal{W},(1-\frac{\epsilon}{2})\mathcal{B}_{\mathcal{W}})$. Up to
shrinking $S$, we can suppose that
$g_{s^{\prime}}:\mathcal{W}_{s^{\prime}}\rightarrow\mathcal{Y}_{s^{\prime}}$
is a log resolution for any $s^{\prime}\in S$. Write

$K_{\mathcal{W}_{s}}+B_{s}=g_{s}^{*}(K_{\mathcal{Y}_{s}}+f_{*}^{-1}B_{Y})\sim_{\mathbb{Q}}0$

where $B_{s}\leq 1-\epsilon$ and
$\operatorname{Supp}B_{s}\subseteq\mathcal{B}_{\mathcal{W}_{s}}:=\mathcal{B}_{\mathcal{W}}|_{\mathcal{W}_{s}}$.
We have

$(K_{\mathcal{W}}+(1-\frac{\epsilon}{2})\mathcal{B}_{\mathcal{W}})|_{\mathcal{W}_{s}}=K_{\mathcal{W}_{s}}+(1-\frac{\epsilon}{2})\mathcal{B}_{\mathcal{W}_{s}}\sim_{\mathbb{Q}}(1-\frac{\epsilon}{2})\mathcal{B}_{\mathcal{W}_{s}}-B_{s}.$

Note that the support of
$(1-\frac{\epsilon}{2})\mathcal{B}_{\mathcal{W}_{s}}-B_{s}$ coincides with the
support of $\mathcal{B}_{\mathcal{W}_{s}}$ which is precisely the divisors on
$\mathcal{W}_{s}$ exceptional over $X$. Hence
$(K_{\mathcal{W}}+(1-\frac{\epsilon}{2})\mathcal{B}_{\mathcal{W}})|_{\mathcal{W}_{s}}$
has a good minimal model $X$ and thus we can run a
$(K_{\mathcal{W}}+(1-\frac{\epsilon}{2})\mathcal{B}_{\mathcal{W}})$-MMP with
scaling of an ample divisor over $S$ to get a relative good minimal model
$\mathcal{W}^{\prime}$ over $S$ by [HMX18, Theorem 1.2]. Note that
$\mathcal{B}_{\mathcal{W}_{s}}$ is contracted and hence
$\mathcal{W}^{\prime}_{s}$ is isomorphic to $X$ in codimension one. Moreover,
up to shrinking $S$, this MMP contracts $\mathcal{B}_{\mathcal{W}}$ only, and
$\mathcal{W}^{\prime}_{s^{\prime}}$ is a good minimal model of
$(\mathcal{W}_{s^{\prime}},\mathcal{B}_{\mathcal{W}}|_{\mathcal{W}_{s^{\prime}}})$
by [HMX18, Lemma 6.1]. It implies that $\mathcal{W}^{\prime}_{s^{\prime}}$ is
isomorphic in codimension one to $X^{\prime}$ which is descend from
$(Y^{\prime},B_{Y^{\prime}})$, where $(Y^{\prime},B_{Y^{\prime}})$ is the pair
corresponding to $s^{\prime}$ for any $s^{\prime}\in S$. Applying Noetherian
induction, the lemma holds. ∎

###  4.2. Proof of Theorem 1.7

######  Proposition 4.6.

Conjecture 1.2 in dimension d implies Conjecture 1.3 in dimension d.

###### Proof.

To prove Conjecture 1.3 in dimension $d$, we assume Conjecture 1.3 holds up to
dimension $d-1$.

Let $(X,B)$ be a pair in Conjecture 1.3. We first assume $B\neq 0$. Up to
small modification, we may assume that $X$ is $\mathbb{Q}$-factorial. In this
case, we could run an MMP on $K_{X}$ to get a Mori fiber space
$\pi^{\prime}:(X^{\prime},B^{\prime})\to Z$, where $B^{\prime}$ is the push-
forward of $B$ via $X\dashrightarrow X^{\prime}$. By Corollary 3.3, there
exists a boundary $\Delta_{Z}$ on $Z$ such that $(Z,\Delta_{Z})$ is an RC
Calabi-Yau variety with $\delta$-lc singularities, where $\delta$ depends only
on $d$ and $\epsilon$. Since Conjecture 1.3 holds up to dimension up to $d-1$,
we see that $Z$ is bounded in codimension one. Thus $X^{\prime}$ is bounded in
codimension one by Lemma 2.9. By Lemma 2.8, $X$ is also bounded in codimension
one.

We now turn to the case when $B=0$. In this case, $X$ is non-canonical, and
there exists a prime divisor $E$ over $X$ such that $a(E,X,0)<1$. Then we can
take a crepant model $(Y,B_{Y})$ of $X$ that extracts only $E$ and $Y$ is
$\mathbb{Q}$-factorial. By what we have proved, we see that $Y$ is bounded in
codimension one. We aim to show that $X$ is also bounded in codimension one.
Now the situation is a little different with the setting in Lemma 4.5, as
$(Y,B_{Y})$ is log bounded up to flops there but we only have boundedness
modulo flops for $Y$ here. That is to say, we cannot control $B_{Y}$ in our
situation. This difficulty is overcome by Lemma 4.7. ∎

Notice that in the following lemma, we only assume that $\mathcal{P}$ is
bounded modulo flops rather than log bounded modulo flops.

######  Lemma 4.7.

Let $\epsilon$ be a positive real number. Let $\mathcal{P}$ be a set of
projective $\epsilon$-lc RC log Calabi-Yau pairs $(Y,B_{Y})$ such that
$\operatorname{Supp}B_{Y}$ is a prime divisor. Denote $\mathcal{P}^{\prime}$
to be the set consisting of Calabi-Yau projective varieties $X$ such that
there exists a birational morphism $h:Y\rightarrow X$, and
$h^{*}K_{X}=K_{Y}+B_{Y}$, and $(Y,B_{Y})\in\mathcal{P}$. Suppose that
$\mathcal{P}$ is bounded modulo flops. Then $\mathcal{P}^{\prime}$ is also
bounded modulo flops.

###### Proof.

By assumption that $\mathcal{P}$ is bounded modulo flops, then there is a
quasi-projective scheme $\mathcal{Y}$ and a projective morphism
$\mathcal{Y}\to S$, where $S$ is a disjoint union of finitely many normal
varieties such that for every $(Y,B_{Y})\in\mathcal{P}$, there is a closed
point $s\in S$ and a small birational map $f:\mathcal{Y}_{s}\dashrightarrow
Y$. We may assume that the set of those points $s$ is dense in $S$. After
decomposing $S$, we may also assume that $S$ is smooth.

We may fix a $(Y,B_{Y})\in\mathcal{P}$ and let $s\in S$ be the corresponding
point. We first claim that up to shrinking $S$ and an étale cover, there is a
$\mathbb{Q}$-divisor $\mathcal{B}$ on $\mathcal{Y}$ such that
$\mathcal{B}|_{\mathcal{Y}_{s}}=f_{*}^{-1}B_{Y}$. Indeed, up to shrinking $S$,
there exists a fiberwise resolution $g:\mathcal{W}\to\mathcal{Y}$ such that
$\mathcal{W}_{s}\to\mathcal{Y}_{s}$ is a log resolution of $(Y,B_{Y})$. Let
$B_{\mathcal{W}_{s}}$ be the strict transform of $B_{Y}$ on $\mathcal{W}_{s}$.
Then by Lemma 4.9, up to an étale cover of $S$, there exists a
$\mathbb{Q}$-divisor $\mathcal{B}_{\mathcal{W}}$ on $\mathcal{W}$ such that
$\mathcal{B}_{\mathcal{W}}|_{\mathcal{W}_{s}}=B_{\mathcal{W}_{s}}$. Denote by
$\mathcal{B}$ the push-forward of $\mathcal{B}_{\mathcal{W}}$ on
$\mathcal{Y}$, then we have that
$\mathcal{B}|_{\mathcal{Y}_{s}}=f_{*}^{-1}B_{Y}$.

######  Claim 4.8.

Up to shrinking $S$,
$(\mathcal{Y}_{s^{\prime}},\mathcal{B}_{s^{\prime}}):=(\mathcal{Y},\mathcal{B})|_{\mathcal{Y}_{s^{\prime}}}$
is an $\frac{\epsilon}{2}$-lc log Calabi-Yau pair for any $s^{\prime}\in S$.

Assume the claim holds now. Then we could conclude our lemma by Lemma 4.5 and
Noetherian induction. It suffices to show Claim 4.8.

###### Proof of Claim 4.8.

Let $\mathcal{E}$ be the reduced sum of all $g$-exceptional divisors on
$\mathcal{W}$. Up to shrinking $S$, one can assume that
$(\mathcal{W},\mathcal{B}_{\mathcal{W}}+(1-\frac{\epsilon}{2})\mathcal{E})$ is
log smooth over $S$. Since
$(\mathcal{W},\mathcal{B}_{\mathcal{W}}+(1-\frac{\epsilon}{2})\mathcal{E})|_{\mathcal{W}_{s}}$
has a good minimal model, we could run an MMP on
$K_{\mathcal{W}}+\mathcal{B}_{\mathcal{W}}+(1-\frac{\epsilon}{2})\mathcal{E}$
over $S$ which terminates with a good minimal model $\mathcal{W}^{\prime}$
over $S$ (see [HMX18, Theorem 1.2]). Denote by $\mathcal{B}^{\prime}$ the
push-forward of $\mathcal{B}_{\mathcal{W}}$ via
$\mathcal{W}\dashrightarrow\mathcal{W}^{\prime}$. Up to shrinking $S$, this
MMP contracts exactly all components of $\mathcal{E}$, as $(Y,B_{Y})$ is a
good minimal model of
$(\mathcal{W}_{s},\mathcal{B}_{\mathcal{W}_{s}}+(1-\frac{\epsilon}{2})\mathcal{E}_{s})$.
Moreover,
$(\mathcal{W}^{\prime}_{s^{\prime}},\mathcal{B}^{\prime}_{s^{\prime}})$ is a
good minimal model of
$(\mathcal{W},\mathcal{B}_{\mathcal{W}}+(1-\frac{\epsilon}{2})\mathcal{E})|_{\mathcal{W}_{s^{\prime}}}$
for every $s^{\prime}\in S$ by [HMX18, Lemma 6.1]. Therefore by the invariance
of plurigenera [HMX18, Theorem 4.2],
$(\mathcal{W}^{\prime}_{s^{\prime}},\mathcal{B}^{\prime}_{s^{\prime}})$ is
$\frac{\epsilon}{2}$-lc log Calabi-Yau for any $s^{\prime}\in S$. It follows
that $(\mathcal{Y}_{s^{\prime}},\mathcal{B}_{s^{\prime}})$ is also
$\frac{\epsilon}{2}$-lc log Calabi-Yau as $\mathcal{Y}_{s^{\prime}}$ is
isomorphic in codimension one to $\mathcal{W}^{\prime}_{s^{\prime}}$ for any
$s^{\prime}\in S$. ∎

∎

In the above proof, we need the following lemma on extension of divisors (e.g.
[FHS21, Theorem 5.1]), which is a variant of [Tot12, Theorem 4.1].

######  Lemma 4.9.

Assume that $\mathcal{X}\to(S,s)$ is a smooth family of projective varieties
such that the special fiber $\mathcal{X}_{s}$ is rationally connected, then up
to shrinking $S$ around $s$ and an étale cover of $S$,
$\operatorname{Pic}(\mathcal{X}/S)\to\operatorname{Pic}(\mathcal{X}_{s^{\prime}})$
is surjective for any $s^{\prime}\in S$.

###  4.3. Proof of Theorem 1.8

######  Proposition 4.10.

Conjecture 1.2 in dimension $d$ implies Conjecture 1.4 in dimension $d$.

###### Proof.

Possibly replacing $X$ by a small $\mathbb{Q}$-factorialization, we may assume
that $X$ is $\mathbb{Q}$-factorial.

We first assume $B+\mathbf{M}_{X}\not\equiv 0$. In this case, we could run an
MMP on $K_{X}$ to get a Mori fiber space
$\pi^{\prime}:(X^{\prime},B^{\prime}+\mathbf{M})\to Z$, where $B^{\prime}$ is
the push-forward of $B$ via $X\dashrightarrow X^{\prime}$. By Proposition
3.10, there is a g-pair $(Z,\Delta_{Z}+\mathbf{M}_{\pi^{\prime}})$ which is
$\delta$-lc Calabi-Yau and $Z$ is RC, and $\delta$ depends only on $d$ and
$\epsilon$. By induction, we see that $Z$ is bounded in codimension one, thus
$X^{\prime}$ is bounded in codimension one by Lemma 2.9. By Lemma 2.8, $X$ is
also bounded in codimension one.

If $B+\mathbf{M}_{X}\equiv 0$, it follows from Theorem 1.7. ∎

###  4.4. Conjecture 1.4 in dimension $\leq 3$

######  Theorem 4.11.

Conjecture 1.4 holds in dimension $2$. Moreover, those $X$ are bounded.

###### Proof.

By [HL20, Lemma 2.4], $(X,B)$ is klt and thus $X$ is $\mathbb{Q}$-factorial.
Then we can pick a positive real number $\epsilon^{\prime}$ such that
$(X,B+(1+\epsilon^{\prime})\mathbf{M})$ is klt. We may run an MMP on
$K_{X}+B+(1+\epsilon^{\prime})\mathbf{M}_{X}\sim_{\mathbb{R}}-\epsilon^{\prime}(K_{X}+B)$
and reach a model $X^{\prime}$ on which $-(K_{X^{\prime}}+B^{\prime})$ is nef,
where $B^{\prime}$ is the strict transform of $B$ on $X^{\prime}$. According
to [Ale94], $X^{\prime}$ is bounded. From [Bir18, Theorem 2.2], we see that
$X$ is also bounded. ∎

######  Theorem 4.12.

Conjecture 1.4 holds in dimension $3$.

###### Proof.

If $B+\mathbf{M}_{X}\equiv 0$, then we are done by [Jia21, Theorem 1.6]. If
$B+\mathbf{M}_{X}\not\equiv 0$, then we could run an MMP on $K_{X}$ to get a
Mori fiber space $\pi^{\prime}:(X^{\prime},B^{\prime}+\mathbf{M})\to Z$, where
$B^{\prime}$ is the push-forward of $B$ via $X\dashrightarrow X^{\prime}$. We
claim that $Z$ is bounded. Assume the claim holds, then $X$ is bounded in
codimension one by Lemmas 2.8 and 2.9. We only need to prove the claim. If
$\dim Z=0$, there is nothing to prove. If $\dim Z=1$, then $Z$ is either a
rational curve or an elliptic curve. If $\dim Z=2$, then there is a g-pair
$(Z,B_{Z}+\mathbf{M}_{\pi^{\prime}})$ which is $\delta$-lc for some $\delta>0$
depending only on $d$ and $\epsilon$ by Corollary 3.12. It follows that $Z$ is
bounded by Theorem 4.11. This finishes the proof. ∎

##  5\. Further remark on Conjecture 4.2

This section is devoted to Conjecture 4.2. Since g-pair structure appears
naturally if we want to prove Conjecture 4.2 by induction on the dimension via
canonical bundle formula, the following conjecture is also expected.

######  Conjecture 5.1.

Let $d,p$ be two positive integers and
$\mathcal{I}\subseteq[0,1]\cap\mathbb{Q}$ a DCC set. Then there exists a
positive integer $n$ depending only on $d,p$ and $\mathcal{I}$ satisfying the
following.

Assume that $(X,B+\mathbf{M})$ is a projective RC lc g-pair of dimension $d$
such that $B\in\mathcal{I}$ and $p\mathbf{M}$ is b-Cartier, and
$K_{X}+B+\mathbf{M}_{X}\equiv 0$. Then $n(K_{X}+B+\mathbf{M}_{X})\sim 0$.

It is worth mentioning that Conjecture 5.1 fails if we remove the “RC”
assumption. For example, one can take $X$ to be an elliptic curve, $B=0$ and
$\mathbf{M}_{X}$ a non-torsion numerically trivial Cartier divisor on $X$.

In the rest of this paper, we show that Conjecture 4.2 is equivalent to
Conjecture 5.1. Recall that we say a g-pair $(X,B+\mathbf{M})$ is _dlt_ if it
is lc and there is a closed subset $V\subseteq X$ ($V$ can be equal to $X$)
such that

  * •

$(X\backslash V,B|_{X\backslash V})$ is log smooth, and

  * •

if $a(E,X,B+\mathbf{M})=0$, then the center of $E$ satisfies
$\operatorname{Center}_{X}(E)\not\subset V$ and
$\operatorname{Center}_{X}(E)\backslash V$ is an lc center of $(X\backslash
V,B|_{X\backslash V})$.

######  Proposition 5.2.

Let $d$ be a positive integer. Assume Conjecture 5.1 in dimension $\leq d-1$.
Then Conjecture 5.1 holds in dimension $d$ if either $(X,B+\mathbf{M})$ is
non-klt lc or $B+\mathbf{M}_{X}\not\equiv 0$.

###### Proof.

Possibly replacing $(X,B+\mathbf{M})$ with a dlt modification (see [HL22,
Proposition 3.9]), we may assume that $(X,B+\mathbf{M})$ is
$\mathbb{Q}$-factorial and dlt. Moreover, $B+\mathbf{M}_{X}\not\equiv 0$ by
assumption. In particular, $K_{X}\equiv-(B+\mathbf{M}_{X})$ is not pseudo-
effective. Therefore we can run an MMP on $K_{X}$ which terminates with a Mori
fiber space $\pi:X^{\prime}\to Z$. Since $X$ is klt, so is $X^{\prime}$ and
hence $X^{\prime}$ is of Fano type over $Z$. Denote by $B^{\prime}$ the push-
forward of $B$ on $X^{\prime}$.

According to the Global ACC for g-pairs (see [BZ16, Theorem 1.6]), there
exists a finite set $\mathcal{I}_{0}\subseteq\mathcal{I}$ which only depends
on $d$ and $p$ such that $B\in\mathcal{I}_{0}$. Possibly replacing $p$ by a
multiple, we may assume that $p\mathcal{I}_{0}\subseteq\mathbb{Z}$.

If $Z$ is a point, then by [Bir19, Theorem 1.10], there exists a positive
integer $n_{1}$ which only depends on $d$ and $p$ such that
$n_{1}(K_{X^{\prime}}+B^{\prime}+\mathbf{M}_{X^{\prime}})\sim 0$. Therefore
$n_{1}(K_{X}+B+\mathbf{M}_{X})\sim 0$ as $(X,B+\mathbf{M})$ and
$(X^{\prime},B^{\prime}+\mathbf{M})$ are crepant. In the following, we may
assume that $\dim Z\geq 1$.

Now $K_{X^{\prime}}+B^{\prime}+\mathbf{M}_{X^{\prime}}\sim_{\mathbb{Q},Z}0$,
$pB^{\prime}$ is integral and $p\mathbf{M}$ is b-Cartier. By [HJ22,
Proposition 3.3], there exists a hyperstandard set
$\Phi\subseteq[0,1]\cap\mathbb{Q}$ and a positive integer $q$ depending only
on $d$ and $p$ such that there exists an lc g-pair
$(Z,B_{Z}+\mathbf{M}_{\pi})$ such that

$q(K_{X^{\prime}}+B^{\prime}+\mathbf{M}_{X^{\prime}})\sim
q\pi^{*}(K_{Z}+B_{Z}+\mathbf{M}_{\pi,Z})\equiv 0,$

$B_{Z}\in\Phi$ and $q\mathbf{M}_{\pi}$ is b-Cartier. Recall that $X$ is
rationally connected, $Z$ is also rationally connected. Now by our assumption,
one can find a positive integer $n_{2}$ depending only on $d-1,\Phi$ and $q$
such that $n_{2}(K_{Z}+B_{Z}+\mathbf{M}_{\pi,Z})\sim 0$. Then

$0\sim n_{2}q\pi^{*}(K_{Z}+B_{Z}+\mathbf{M}_{\pi,Z})\sim
n_{2}q(K_{X}+B+\mathbf{M}_{X}).$

Therefore $n:=n_{1}n_{2}q$ has the required property. The proof is finished. ∎

######  Proposition 5.3.

Conjecture 4.2 in dimension $d$ implies Conjecture 5.1 in dimension $d$.

###### Proof.

If $d=1$, then $X$ is indeed $\mathbb{P}^{1}$ as $X$ is rationally connected
by our assumption. In this case, Conjecture 5.1 is clear. In what follows, we
may assume that $d\geq 2$.

Possibly replacing $(X,B+\mathbf{M})$ by a dlt modification, we may assume
that $X$ is $\mathbb{Q}$-factorial. In particular, $X$ is klt. If
$B+\mathbf{M}_{X}\not\equiv 0$, then there exists a positive integer $n_{1}$
has the required property by Proposition 5.2.

Now assume that $B=0$ and $\mathbf{M}_{X}\equiv 0$, then
$K_{X}\sim_{\mathbb{Q}}0$. As we assume Conjecture 4.2 in dimension $d$, we
may find a positive integer $n_{2}$ which only depends on $d$ such that
$n_{2}K_{X}\sim 0$. According to [BDCS20, Theorem 1.4], there exists a bounded
family $\mathcal{P}$ depending only on $d$ and $n_{2}$ such that $X$ and
$X^{\prime}$ are isomorphic in codimenison one, for some
$X^{\prime}\in\mathcal{P}$. Possibly replacing $X$ with $X^{\prime}$, we may
assume that $X$ belongs to the bounded family $\mathcal{P}$. Since
$\mathbf{M}_{X}\equiv 0$, there exists a positive integer $l_{0}$ depending
only on $\mathcal{P}$ such that $l_{0}\mathbf{M}_{X}$ is Cartier, cf. [Bir19,
Lemma 2.25]. Moreover, by Lemma 2.7, there exists a positve integer $n_{3}$
which only depends on $\mathcal{P}$ such that $n_{3}l_{0}\mathbf{M}_{X}\sim
0$. It follows that $n_{2}n_{3}l_{0}(K_{X}+\mathbf{M}_{X})\sim 0$.

Therefore $n:=n_{1}n_{2}n_{3}l_{0}$ has the required property. ∎

######  Corollary 5.4.

Conjecture 5.1 holds in the following cases.

  1. (1)

$d\leq 3$.

  2. (2)

$d=4$ and either $(X,B+\mathbf{M})$ is non-klt lc or
$B+\mathbf{M}_{X}\not\equiv 0$.

###### Proof.

By [Jia21], Conjecture 4.2 holds in dimension $3$, thus we have (1). (2)
follows from (1) and Propostion 5.3 immediately. ∎

As a consequence, the following result holds. Note that in the following
corollary, we do not require that the DCC coefficients are rational numbers,
see [HLL22, Theorem 1.9] for a more general statement for threefold pairs.

######  Corollary 5.5.

Let $d,p$ be two positive integers, and $\mathcal{I}\subseteq[0,1]$ a DCC set.
Assume that either $d\leq 3$ or Conjecture 4.2 holds in dimension $d$. Then
the set of projective normal rationally connected varieties $X$ such that

  1. (1)

$(X,B+\mathbf{M})$ is klt g-pair of dimension $d$ for some boundary $B$ and
some nef part $\mathbf{M}$,

  2. (2)

$B\in\mathcal{I}$ and $p\mathbf{M}$ is b-Cartier, and

  3. (3)

$K_{X}+B+\mathbf{M}_{X}\equiv 0$,

is bounded in codimension one.

###### Proof.

By [CHL22, Remark 5.7] (cf. [HLS19]), there exists a finite set
$\mathcal{I}_{0}\subseteq[0,1]\cap\mathbb{Q}$ depending only on $d,p$ and
$\mathcal{I}$ such that we may find a boundary $B^{\prime}$ such that
$(X,B^{\prime}+\mathbf{M})$ is klt g-pair and
$K_{X}+B^{\prime}+\mathbf{M}_{X}\equiv 0$. Since we assume Conjecture 4.2 in
dimension $d$, Conjecture 5.1 also holds in dimensiond $d$ by Proposition 5.3.
Therefore one can find a positive integer $p|n$ which only on $d,p$ and
$\mathcal{I}_{0}$ such that $n(K_{X}+B^{\prime}+\mathbf{M}_{X})\sim 0$. We
conclude the statement by [HJ22, Theorem 1.4]. ∎

Finally, we remark here that the Conjecture 4.2 is expected for lc log Calabi-
Yau varieties, that is, we do not need the varieties to be RC, cf. [Xu19,
JL21].

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# The DialPort tools

Jessica Huynh ∗Equal contribution Shikib Mehri∗ Cathy Jiao∗ Maxine Eskenazi
Carnegie Mellon University
jhuynh, amehri, cljiao<EMAIL_ADDRESS>

###### Abstract

The DialPort project (http://dialport.org/), funded by the National Science
Foundation (NSF), covers a group of tools and services that aim at fulfilling
the needs of the dialog research community. Over the course of six years,
several offerings have been created, including the DialPort Portal and
DialCrowd. This paper describes these contributions, which will be demoed at
SIGDIAL, including implementation, prior studies, corresponding discoveries,
and the locations at which the tools will remain freely available to the
community going forward.

##  1 Introduction

The DialPort project111http://dialport.org/ has created tools and services
that respond to needs voiced by many in the dialog research community during
several workshops organized by the Principle Investigators (PIs). Its
offerings are available at no cost to the community with the goal of helping
researchers gather high quality data, and easily assess and compare their
dialog systems. This paper and its corresponding demos showcase the DialPort
Portal222https://dialport.org/portal and
DialCrowd333http://dialport.org/dialcrowd.html.

There is an increasing need for large amounts of natural dialog data that can
be obtained at reasonable cost and in an interactive manner. Static datasets
are ineffective for both evaluation and optimization. This has led to the
creation of the DialPort Portal, which facilitates the collection of flexible
and evolving data as well as interactive assessment with real users. Notably,
the Portal was used to connect systems and collect data for the Interactive
Evaluation of Dialog track (Mehri et al., 2021) at DSTC9 (Gunasekara et al.,
2020).

Another community need centers around how to gather high quality data when
using crowdsourcing platforms. DialCrowd has been constructed to facilitate
crowdsourcing by guiding researchers to give clear, understandable
explanations of the task to the workers who produce or annotate data. It also
aids in calculating the correct level of worker payment. Finally, it includes
several methods of data quality assessment.

The University of Southern California (USC) is a partner in DialPort. The team
at USC works on a tools repository444https://dialport.ict.usc.edu/ and the
REAL Challenge.

This paper gives background and describes in detail the parts of both the
Portal and DialCrowd. It also provides information on how to access and use
them. As the DialPort project draws to an end, the paper indicates the
permanent sites where these tools will reside.

##  2 Background

###  2.1 Interactive Platforms for Dialog

As dialog models improve, it is imperative that they are evaluated in
interactive settings with real users. Mehri and Eskenazi (2020) show that
while pre-trained dialog systems excel at generating responses (Zhang et al.,
2019; Bao et al., 2020), they underperform in back-and-forth interactions.

The Alexa Prize challenge (Ram et al., 2018; Khatri et al., 2018) allows
university teams to build socialbots that are assessed in interactive settings
with Alexa users. In contrast, the DialPort Portal is accessible to the
broader research community. Furthermore, the Alexa Prize challenge primarily
relies on speech input from the user, which may result in speech recognition
errors. Though the DialPort Portal can accept speech input, its web interface
can also be used with text-only input.

###  2.2 Crowdsourcing

With the amount of dialog data available or able to be collected with systems
such as DialPort, it is important to have easy and accessible tools to create
detailed annotations of this data for different metrics. One method of
obtaining annotations is crowdsourcing with platforms such as Amazon
Mechanical Turk (AMT). However, it is sometimes difficult to obtain conclusive
results, and a survey of current natural language processing HITs has shown
the weaknesses of these HITs Huynh et al. (2021). Instructions Chandler et al.
(2013), examples Doroudi et al. (2016), and payment are some of the aspects
that need to be attended to in order for HITs to acquire higher quality data.

##  3 DialPort Portal

The DialPort Portal was initially conceived with the objective of listing many
dialog systems from a variety of sites. This type of platform, with
demonstrations, links, and references to various systems, is valuable to both
researchers and real users. The concept of the Portal evolved, and the
different systems were linked such that a user could interact with all of the
connected systems, transitioning seamlessly between systems, with the dialog
state (consisting of slots such as city or date) shared across systems (Zhao
et al., 2016; Lee et al., 2017). As dialog systems continued to improve,
especially with the advent of engaging response generation models (Zhang et
al., 2019; Bao et al., 2020), the Portal recruited real users through Facebook
advertising with the objective of providing researchers with a platform to
collect interactive dialogs with real users (Mehri et al., 2021).

###  3.1 Portal Version 1

The original version of the Portal grouped several dialog systems from
different sites (Cambridge, USC, CMU) and managed seamless switching amongst
(Zhao et al., 2016). For example, a user could ask for the weather in
Pittsburgh and get the CMU weather system, then ask the CMU system for the
weather in Cambridge, then ask for a restaurant and automatically switch to
the Cambridge restaurant system, then ask to play a game and get the USC
system.

This instance of the Portal serves as a platform to interact with different
systems over the course of one dialog (Zhao et al., 2016). To accomplish this,
the Portal needed to address several challenges (1) how to share information
across systems (e.g., remembering the city the user wanted the weather for
when interacting with the CMU system, and sharing that with the Cambridge
system when the user wants a restaurant recommendation), (2) how to gracefully
continue a dialog when a system is down, and (3) how to give two systems
addressing the same task (e.g., restaurants) equal time with the users.
Respectively, these problems were addressed by (1) maintaining a shared dialog
state across systems, (2) backing off to an equivalent system or changing the
topic, and (3) a pseudo-random system selection policy. In order to make the
system easy to use, an API was developed to facilitate connecting new systems
to the Portal. This version has pedagogical value as it can easily be
demonstrated for dialog classes.

###  3.2 Portal Version 2

Figure 1: DialPort Portal. This screenshot of the Portal displays (1) the
dialog history, shown in the center of the screen, (2) an input field for the
user to type their responses, and (3) a set of feedback buttons below the
dialog history (“Like”, “Dislike”, “Feedback?”, “Improve Response?” and “End
Conversation”). The interface clear and emphasizes the three important actions
that a user should perform while using the Portal: (1) reading the dialog
history, (2) responding to the dialog system, and (3) providing feedback.

With the advent of the API, the possibilities of use of the Portal greatly
expanded. The Portal was used for the DSTC9 Challenge (Mehri et al., 2021), as
a tool that enabled researchers to both compare their systems on one common
platform (with real users) and to gather considerable amounts of data. The
Portal was made available to DSTC9 participants. The idea was to connect
systems and have them tested by real (unpaid) users. The CMU DialPort team
advertised the Portal on Facebook and interested individuals tried it out
(with text only). Upon visiting the Portal, real users are randomly matched
with a dialog system, without knowledge of the specific system they are
interacting with. While some people left the site after only one or two turns
with a system, many actually continued to communicate with a system for a
substantial conversation, and were thus considered to be real users. Real
users consist of users who find some personal interest (getting information,
companionship, curiosity) in continuing a dialog. There were 11 participants
in the interactive part of the Challenge (Mehri et al., 2021). With an
advertising budget of $2500, we collect more than 4000 dialogs on the DialPort
portal (2960 dialogs with at least 4 turns or 8 utterances); thus the cost was
less than $1.00 per usable dialog. The DialPort portal, through funding from
the National Science Foundation, has been able to provide interactive
evaluation as a service free of charge to any dialog researchers. The Appendix
contains a sample dialog from the winning system of the DSTC9 track (Bao et
al., 2020).

DSTC9 demonstrated that the Portal could easily be used to both compare
systems and to gather data with real users. Besides challenges, another
potential use of the Portal would be for students to connect systems that they
build for a class project to see how well they do in real user interaction.

At the end of the DialPort project in the coming year, the Portal will move
from the Dialog Research Center at CMU to LDC at UPenn.

###  3.3 DialPort Dashboard

After collecting data from real users on the DialPort interface, a subsequent
task is to perform analysis on the gathered data. We provide the DialPort
dashboard which allows researchers to (i) analyze dialogs collected on their
system, (ii) interact with the dashboard to filter and organize dialogs based
on various criteria, and (iii) compare their system to other systems connected
to the DialPort Portal. Currently, the Dashboard contains over 7000 dialogs
from 28 systems. The Dashboard is connected to the DialPort Portal via API
calls, allowing dialogs to be quickly displayed on the Dashboard after being
collected from the Portal. The Dashboard code will soon be released, allowing
for use of the Dashboard in offline mode.

The Dashboard UI contains panels, tables, and charts. At both the system and
dialog level, attributes such as the number of utterances, likes, dislikes,
comments, corrections are displayed (see figure 3). In addition, the two
evaluation metrics of FED (Mehri and Eskenazi, 2020) and human ratings are
shown. Since the Dashboard is designed to be easily extended, additional
metrics can be added in the future. Users can interact with the dashboard by
filtering and ranking dialogs based on attributes and metrics. For example,
the provided toolbar can be used to find all conversations with a given user’s
system with more than n turns or rank conversations from most-to-least number
of likes. Users can also filter words and phrases in dialogs by their number
of occurrences from the perspective of both the system or human participant,
and thus view common phrases or words mentioned on either side of the
conversation. Finally, each system contains a progress monitor graph which
displays the number of dialogs being collected over time, allowing users to
actively observe data collection in the DialPort Portal.

Figure 2: DialCrowd Examples and Counterexamples with Explanations

##  4 DialCrowd

To address the many issues that present themselves when using crowdsourcing to
collect high quality data, DialCrowd was created. DialCrowd Lee et al. (2018)
is a dialog assessment toolkit which aids researchers with human intelligence
task (HIT) creation. Requesters follow templates on the DialCrowd site, which
generate a HIT that can be linked for a worker on any crowdsourcing site.

The second version of this tool Huynh et al. (2022) focuses on collecting
high-quality data with tools such as:

  * •

Links to create better instructions

  * •

Prompts to provide examples and counterexamples with explanations seen in
Figure 2

  * •

Functionality for adding golden data and duplicate data in each HIT

  * •

Payment suggestions

  * •

A feedback area

  * •

Overall statistics from the HIT (time, patterns in the responses, inter-
annotator agreement)

This allows for requesters to create a well-structured HIT which allows
workers to provide better quality annotations. Consequently, it makes it
easier to filter responses from potential bots. Additional tools include the
capability to include a mandatory consent form at the start of the HIT, and
detailed style changes for the HIT. Further description of the system along
with corresponding images can be found in Huynh et al. (2022).

One DialCrowd template, intent classification, has been merged into the new
home for DialCrowd, ParlAI
555https://github.com/facebookresearch/ParlAI/tree/main/parlai/crowdsourcing/tasks/dialcrowd,
and is now available for use.

##  5 The DialPort demo

The demos of the DialPort Portal and Dashboard and of DialCrowd at SIGDIAL
will include:

  * •

how to connect a system

  * •

what interaction with each tool looks like

  * •

advantages there are in using the tools, with examples (for example, what
resulting data looks like)

##  6 Conclusion and Future Directions

The tools presented in this demo help dialog researchers in data gathering and
assessment. As the community uses them, more types of applications will arise.
The tools have been created in a way that enable additions as the field and
the needs evolve.

##  7 Acknowledgements

This work is funded by National Science Foundation Grant Nos. CNS-1512973,
DGE1745016, and DGE2140739. The opinions expressed in this paper do not
necessarily reflect those of the National Science Foundation. The authors
would like to thank Tiancheng Zhao, Kyusong Lee, and Ting-Rui Chiang for their
contributions to these tools.

## References

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  * Chandler et al. (2013) Jesse Chandler, Gabriele Paolacci, and Pam Mueller. 2013.  Risks and rewards of crowdsourcing marketplaces.  In _Handbook of human computation_ , pages 377–392. Springer.
  * Doroudi et al. (2016) Shayan Doroudi, Ece Kamar, Emma Brunskill, and Eric Horvitz. 2016.  Toward a learning science for complex crowdsourcing tasks.  In _Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems_ , pages 2623–2634.
  * Gunasekara et al. (2020) Chulaka Gunasekara, Seokhwan Kim, Luis Fernando D’Haro, Abhinav Rastogi, Yun-Nung Chen, Mihail Eric, Behnam Hedayatnia, Karthik Gopalakrishnan, Yang Liu, Chao-Wei Huang, et al. 2020.  Overview of the ninth dialog system technology challenge: Dstc9.  _arXiv preprint arXiv:2011.06486_.
  * Huynh et al. (2021) Jessica Huynh, Jeffrey Bigham, and Maxine Eskenazi. 2021.  A survey of nlp-related crowdsourcing hits: what works and what does not.  _arXiv preprint arXiv:2111.05241_.
  * Huynh et al. (2022) Jessica Huynh, Ting-Rui Chiang, Jeffrey Bigham, and Maxine Eskenazi. 2022.  Dialcrowd 2.0: A quality-focused dialog system crowdsourcing toolkit.  In _Proceedings of the Language Resources and Evaluation Conference_ , pages 1256–1263, Marseille, France. European Language Resources Association.
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##  Appendix A Appendix

User: Hello
---
System: hello! hows your day going so far?
User: What are you up to?
System: just finished making coffee! what about you? what do you do?
User: i’m playing video games
System: oh, nice! what kind? and i’m just relaxing here and listening to some
classical music.
User: i’m playing league of legends
System: nice! what rank are you? i haven’t played in forever.
User: who is your favorite classical musician?
System: probably beethoven. i like classical music.
User: i’m plat 2
System: nice! i’d imagine you are better than that!
User: can you play an instrument?
System: i can play the piano. do you play any instruments?
User: thanks
System: i appreciate it! so what is it like being plat 2?
User: i can play the clarinet, but not very well haha
System: oh nice! i played clarinet in high school.
User: the games are difficult but rewarding
System: yes they are! i played the trumpet in college.
. . .
Table 1: An example of a conversation snippet from the DSTC9 top performing
system. The system is capable of maintaining two simultaneous threads of
dialog about League of Legends and music, and switch between them in a
coherent and engaging manner. Figure 3: The home page for a system on the
DialPort dashboard. General information about the conversations collected from
the system are displayed. Sections such as "Words and Phrases" and "Graphs"
can be expanded or collapsed to view additional information about the system.
Figure 4: Using the DialPort dashboard to find all conversations in a system
with more than 3 utterances