$\mathscr{M}_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)$ admits a projective good
coarse moduli space $M_{slc}(d,\Phi_{c},\Gamma,\sigma)$.

###### Proof.

See the proof of [Birkar2022, Theorem 1.14]. Following the notations in
[Birkar2022, §10.7], we have

$\mathscr{M}_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)=\left[M_{\rm
slc}^{e}(d,\Phi_{c},\Gamma,\sigma,a,r,\mathbb{P}^{n})/{\rm
PGL}_{n+1}(\mathbb{C})\right]$

where the right hand side is the stacky quotient. ∎

A stable minimal model $(X,B),A$ is called a lc stable minimal model if
$(X,B)$ is a lc pair. Let $\mathscr{M}_{\rm
lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)\subset\mathscr{M}_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$ denote the open substack consisting of
$(d,\Phi_{c},\Gamma,\sigma)$-lc stable minimal models $(X,B),A$ such that the
coefficients of $B$ lie in $(0,1)$. Denote $M_{\rm
lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)$ to be the quasi-projective coarse moduli
spaces of $\mathscr{M}_{\rm lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)$.

###  6.2. Polarization on $M_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)$

In this part we consider some natural ample line bundles on $M_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$. Their constructions are implicit in the proof
of [Birkar2022, Theorem 1.14], based on the ampleness criterion of Kollár
[Kollar1990]. Fix a data $d,\Phi_{c},\Gamma,\sigma$. Since $\mathscr{M}_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$ is of finite type, there are constants

$(a,r,j)\in\mathbb{Q}^{\geq 0}\times(\mathbb{Z}^{>0})^{2}$

depending only on $d,\Phi_{c},\Gamma,\sigma$ such that the following hold for
every $(d,\Phi_{c},\Gamma,\sigma)$-stable minimal model $(X,B),A$ (c.f.
[Birkar2022, Lemma 10.2]).

  * •

$X+B+aA$ is slc,

  * •

$r(K_{X}+B+aA)$ is a very ample integral Cartier divisor with

$H^{i}(X,kr(K_{X}+B+aA))=0,\quad\forall i>0,\forall k>0,$

  * •

the embedding $X\hookrightarrow\mathbb{P}(H^{0}(X,r(K_{X}+B+aA)))$ is defined
by degree $\leq j$ equations,

  * •

the multiplication map

$S^{j}(H^{0}(X,r(K_{X}+B+aA)))\to H^{0}(X,jr(K_{X}+B+aA))$

is surjective.

######  Definition 6.2.

$(a,r,j)\in\mathbb{Q}^{\geq 0}\times(\mathbb{Z}^{>0})^{2}$ that satisfies the
conditions above is called a $(d,\Phi_{c},\Gamma,\sigma)$-polarization data.

Let $(a,r,j)\in\mathbb{Q}^{\geq 0}\times(\mathbb{Z}^{>0})^{2}$ be a
$(d,\Phi_{c},\Gamma,\sigma)$-polarization data. Let $(X,B),A\to S$ be a family
of $(d,\Phi_{c},\Gamma,\sigma)$-stable minimal models. Then
$f_{\ast}(r(K_{X/S}+B+aA))$ is locally free and commutes with an arbitrary
base change. Therefore the assignment

$f:(X,B),A\to S\in\mathscr{M}_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)(S)\mapsto
f_{\ast}(r(K_{X/S}+B+aA))$

gives a locally free coherent sheaf on the stack $\mathscr{M}_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$, which is denoted by $\Lambda_{a,r}$. Let
$\lambda_{a,r}:=\det(\Lambda_{a,r})$. Since $\mathscr{M}_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$ is Deligne-Mumford, some power
$\lambda_{a,r}^{\otimes k}$ descends to a line bundle on $M_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$. For this reason we regard $\lambda_{a,r}$ as
a $\mathbb{Q}$-line bundle on $M_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)$.

######  Proposition 6.3.

Let $(a,r,j)\in\mathbb{Q}^{\geq 0}\times(\mathbb{Z}^{>0})^{2}$ be a
$(d,\Phi_{c},\Gamma,\sigma)$-polarization data. Then $\lambda_{a,r}$ is ample
on $M_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)$.

###### Proof.

By the same arguments as in [Kollar1990, §2.9], it suffices to show that
$f_{\ast}(r(K_{X/S}+B+aA))$ is nef when $S$ is a smooth projective curve. This
is accomplished by Fujino [Fujino2018] and Kovács-Patakfalvi [Kovacs2017]. ∎

###  6.3. Numerical bound of $\lambda_{a,r}$

Recall that a family $f:(X,\Delta)\to S$ is log smooth if $X$ is smooth
projective over $S$ and $\Delta$ is a simple normal crossing
$\mathbb{Q}$-divisor whose strata are all smooth over $S$.

######  Definition 6.4.

Let $f:(X,B),A\to S$ be a family of stable minimal model. A log smooth
birational model of $f$ is a log smooth family
$(X^{\prime},\Delta^{\prime})\to S$ together with a birational map
$g:X^{\prime}\dashrightarrow X$ over $S$ such that $g$ is defined on a dense
Zariski open subset of $\Delta^{\prime}$ and $g_{\ast}(\Delta^{\prime})=A+B$.
$f$ is called admissible if it admits a log smooth birational model and the
coefficients of $B$ lie in $(0,1)$.

The following lemma will be used in the proof of Theorem 6.6.

######  Lemma 6.5 (Relative Kawamata’s covering).

Let $X\to S$ be a morphism between smooth projective varieties and $D$ a
simple normal crossing divisor whose coefficients lie in $(0,1)$. Let
$S^{o}\subset S$ be a Zariski open subset such that $(f^{-1}(S^{o}),D\cap
f^{-1}(S^{o}))\to S^{o}$ is a log smooth family. Then there is a finite
ramified covering $h:Y\to X$ such that the following hold.

  1. (1)

$Y$ is a smooth projective variety,

  2. (2)

$f\circ h$ is a smooth morphism over $S^{o}$,

  3. (3)

there is a $\mathbb{Q}$-divisor $F\geq 0$ such that

(6.1) $\displaystyle h^{\ast}(K_{X}+D)=K_{Y}-F$

###### Proof.

The proof is the same as [Kawamata1981, Theorem 17] except two modifications.
The first is that the general hyperplanes $H_{1},\dots,H_{d}$ in loc. cit.
should satisfy that

$\left(H_{1}\cup\dots\cup H_{d}\cup D_{1}\cup\dots\cup D_{l}\right)\cap
f^{-1}(S^{o})$

is a relative simple normal crossing divisor over $S^{o}$. The second is that
one should let $m_{i}$ be sufficiently large in loc. cit. in order to ensure
the validity of (6.1). ∎

######  Theorem 6.6 (Uniform numerical bound of the polarization).

Let $f^{o}:(X^{o},B^{o}),A^{o}\to S^{o}$ be an admissible family of
$(d,\Phi_{c},\Gamma,\sigma)$-lc stable minimal models over a smooth quasi-
projective variety $S^{o}$. Let $S$ be a smooth projective variety containing
$S^{o}$ as a Zariski open subset. Assume that $D:=S\backslash S^{o}$ is a
divisor and the morphism $\xi^{o}:S^{o}\to M_{\rm
lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)$ induced from the family $f^{o}$ extends
to a morphism $\xi:S\to M_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)$. Let
$(a,r,j)\in\mathbb{Q}^{\geq 0}\times(\mathbb{Z}^{>0})^{2}$ be a
$(d,\Phi_{c},\Gamma,\sigma)$-polarization data satisfying that
$B^{o}+aA^{o}<(A^{o}+B^{o})_{\rm red}$. Assume that $K_{S}+D$ is pseudo-
effective. Then the following inequalities hold.

$\displaystyle c_{1}(\xi^{\ast}\lambda_{a,r})A_{1}A_{2}\cdots A_{\dim
S-1}\leq\frac{rd{\rm rank}(\Lambda_{a,r})}{2}(K_{S}+D)A_{1}A_{2}\cdots A_{\dim
S-1}$

for any semiample effective divisors $A_{1},\dots,A_{\dim S-1}$ on $S$, and

$\displaystyle c_{1}(\xi^{\ast}\lambda_{a,r})\cdot\alpha\leq rd{\rm
rank}(\Lambda_{a,r})\left(K_{S}+D\right)\cdot\alpha+D\cdot\alpha$

for every movable class $\alpha\in N_{1}(S)$. If in particular $\dim S=1$,
then

(6.2) $\displaystyle\deg(\xi^{\ast}\lambda_{a,r})\leq\frac{rd{\rm
rank}(\Lambda_{a,r})}{2}\deg(K_{S}+D).$

###### Proof.

Step 1 (Compactifying the family): By the properness of $\mathscr{M}_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$ there are data as follows.

  * •

A proper generically finite morphism $\sigma:\widetilde{S}\to S$ from a smooth
projective variety $\widetilde{S}$. $\sigma$ is a combination of smooth
blowups and a finite flat morphism such that $\sigma^{-1}(D)$ is a simple
normal crossing divisor. Note that $\sigma$ might be ramified over $S^{o}$.

  * •

Denote $\widetilde{S}^{o}:=\sigma^{-1}(S^{o})$,
$\widetilde{X}^{o}:=\widetilde{S}^{o}\times_{S^{o}}X^{o}$. Denote
$\widetilde{A}^{o}$ and $\widetilde{B}^{o}$ to be the divisorial pullbacks of
$A^{o}$ and $B^{o}$ respectively. There is a compactification
$\widetilde{f}:(\widetilde{X},\widetilde{B}),\widetilde{A}\to\widetilde{S}$ of
the base change family
$(\widetilde{X}^{o},\widetilde{B}^{o}),\widetilde{A}^{o}\to\widetilde{S}^{o}$
such that $\widetilde{f}\in\mathscr{M}_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)(\widetilde{S})$.

Step 2 (Taking the log smooth birational models): Consider the following
commutative diagram

where the arrows are explained as follows.

  * •

$f^{\prime o}:X^{\prime o}\to S^{o}$ is a projective smooth morphism and
$\rho^{o}:X^{\prime o}\to X^{o}$ is a rational map over $S$ whose image
contains an dense Zariski open subset of $A^{o}+B^{o}$. Denote the
$\mathbb{Q}$-divisor $A^{\prime o}$ and $B^{\prime o}$ on $X^{\prime o}$ to be
the birational transforms of $A^{o}$ and $B^{o}$ respectively. $(X^{\prime
o},A^{\prime o}+B^{\prime o})\to S^{o}$ is a log smooth birational model of
$f^{o}:(X^{o},B^{o}),A^{o}\to S^{o}$.

  * •

$\widetilde{f}^{\prime o}$ and $\widetilde{\rho}^{o}$ are the base changes of
$f^{\prime o}$ and $\rho^{o}$ respectively. Denote $\widetilde{A}^{\prime o}$
and $\widetilde{B}^{\prime o}$ to be the birational transforms of
$\widetilde{A}^{o}$ and $\widetilde{B}^{o}$. Then $\widetilde{f}^{\prime
o}:(\widetilde{X}^{\prime o},\widetilde{A}^{\prime o}+\widetilde{B}^{\prime
o})\to\widetilde{S}^{o}$ is a log smooth birational model of
$(\widetilde{X}^{o},\widetilde{B}^{o}),\widetilde{A}^{o}\to\widetilde{S}^{o}$.

  * •

$\widetilde{f}^{\prime}:\widetilde{X}^{\prime}\to\widetilde{S}$ is a
completion of $\widetilde{f}^{\prime o}$. Since $\widetilde{A}$ and
$\widetilde{B}$ do not contain any component of a fiber of $\widetilde{f}$,
$\widetilde{\rho}^{o}$ extends naturally to a rational map
$\widetilde{\rho}:\widetilde{X}^{\prime}\to\widetilde{X}$ whose image contains
a dense Zariski open subset of $\widetilde{A}+\widetilde{B}$. Hence we can
define the birational transforms of $\widetilde{A}$ and $\widetilde{B}$,
denoted by $\widetilde{A}^{\prime}$ and $\widetilde{B}^{\prime}$ respectively.
By blowing up along the centers over $\widetilde{f}^{\prime-1}\sigma^{-1}(D)$
we may assume the following.

    * –

$\widetilde{X}^{\prime}$ is a smooth projective variety,

    * –

$\widetilde{f}^{\prime-1}\sigma^{-1}(D)+\widetilde{A}^{\prime}+\widetilde{B}^{\prime}$
is a simple normal crossing divisor, and

    * –

$\widetilde{f}^{\prime}:\widetilde{X}^{\prime}\to\widetilde{S}$ is smooth over
$\widetilde{S}^{o}$.

The constructions yields that

(6.3)
$\displaystyle\widetilde{f}_{\ast}(r(K_{\widetilde{X}/\widetilde{S}}+\widetilde{B}+a\widetilde{A}))\simeq\widetilde{f}^{\prime}_{\ast}(r(K_{\widetilde{X}^{\prime}/\widetilde{S}}+\widetilde{B}^{\prime}+a\widetilde{A}^{\prime}))$

whenever $r(K_{\widetilde{X}/\widetilde{S}}+\widetilde{B}+a\widetilde{A})$ is
integral.

Step 3 (Kawamata’s covering): Let $(a,r,j)\in\mathbb{Q}^{\geq
0}\times(\mathbb{Z}^{>0})^{2}$ be a $(d,\Phi_{c},\Gamma,\sigma)$-polarization
data with $0<a\ll 1$ so that $\widetilde{B}^{\prime}+a\widetilde{A}^{\prime}$
is a simple normal crossing divisor with coefficients lie in $(0,1)$. By Lemma
6.5 there is a finite ramified covering
$\varrho:\widetilde{Y}^{\prime}\to\widetilde{X}^{\prime}$ such that the
following hold.

  1. (1)

$\widetilde{Y}^{\prime}$ is a smooth projective variety,

  2. (2)

$\widetilde{f}^{\prime}\varrho$ is smooth over $S^{o}$,

  3. (3)

there is a non-negative $\mathbb{Q}$-divisor $F$ such that

(6.4)
$\displaystyle\varrho^{\ast}(K_{\widetilde{X}^{\prime}}+\widetilde{B}^{\prime}+a\widetilde{A}^{\prime})=K_{\widetilde{Y}^{\prime}}-F.$

The same construction is valid on $X^{\prime o}$. One can choose a suitable
Kawamata covering of $Y^{o}\to X^{\prime o}$ such that
$\widetilde{f}^{\prime}\varrho:\widetilde{Y}^{\prime}\to\widetilde{S}$ is a
compactification of the base change morphism
$Y^{o}\times_{S^{o}}\widetilde{S}^{o}\to\widetilde{S}^{o}$. Consider the
following diagram

(6.9) $\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.54778pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\widetilde{Y}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
1.88812pt\raise-26.4947pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.48613pt\hbox{$\scriptstyle{\widetilde{f}^{\prime}\varrho}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
30.54778pt\raise-31.96268pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
30.54778pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.10335pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
6.54779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.10335pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.11723pt\raise-28.16669pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.54778pt\raise-38.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
49.72145pt\raise-33.49306pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\sigma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 66.76308pt\raise-38.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.76308pt\raise-38.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{S}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $Y\to S$ is a completion of $Y^{o}\to S^{o}$ with $Y$ smooth and
projective, $\widetilde{Y}\to\widetilde{Y}^{\prime}$ is some modification
which is biholomorphic over $Y^{o}\times_{S^{o}}\widetilde{S}^{o}$ so that
there is a morphism $\widetilde{Y}\to Y$ making the diagram commutative. We
may assume that $\sigma$ is sufficiently ramified so that
$\widetilde{Y}\to\widetilde{S}$ is semistable in codimension one. Denote
$g:Y\to S$ and $\widetilde{g}:\widetilde{Y}\to\widetilde{S}$ to be the maps we
have just constructed. According to (6.3) and (6.4) there is an inclusion

(6.10)
$\displaystyle\widetilde{f}_{\ast}(rK_{\widetilde{X}/\widetilde{S}}+r\widetilde{B}+ra\widetilde{A})\subset\widetilde{g}_{\ast}(rK_{\widetilde{Y}/\widetilde{S}}).$

By the same argument we could obtain an inclusion

(6.11) $\displaystyle f^{o}_{\ast}(rK_{X^{o}/S^{o}}+rB^{o}+raA^{o})\subset
g^{o}_{\ast}(rK_{Y^{o}/S^{o}})$

where $g^{o}:Y^{o}\to X^{\prime o}\to S^{o}$ is the composition map. The two
inclusions (6.10) and (6.11) are compatible via the base change.

Last step: We finish the proof by applying Theorem 3.5. The proof is parallel
to the proof of Theorem 4.7. Denote

$W:=\widetilde{f}_{\ast}(rK_{\widetilde{X}/\widetilde{S}}+r\widetilde{B}+ra\widetilde{A})\quad\textrm{and}\quad
l:={\rm rank}(W).$

By the construction of $\lambda_{a,r}$ one has

$\displaystyle(\xi\circ\sigma)^{\ast}\lambda_{a,r}\simeq\det(W).$

Since $\xi^{\ast}(\lambda_{a,r})$ is a $\mathbb{Q}$-line bundle, there is
$k\in\mathbb{Z}^{>0}$ such that $\xi^{\ast}(\lambda_{a,r})^{\otimes k}$ is a
line bundle. Let $\widetilde{Y}^{(klr)}$ denote a functorial desingularization
of the main component of the $klr$-fiber product
$\widetilde{Y}\times_{\widetilde{S}}\times\cdots\times_{\widetilde{S}}\widetilde{Y}$
and denote $\widetilde{g}^{(klr)}:\widetilde{Y}^{(klr)}\to\widetilde{S}$ to be
the projection. Define $g^{(klr)}:Y^{(klr)}\to S$ similarly. We may assume
that there is a morphism $\widetilde{Y}^{(klr)}\to Y^{(klr)}$ making the
diagram

(6.16)

commutative. According to (6.10) there is an inclusion
$W\subset\widetilde{g}_{\ast}(rK_{\widetilde{Y}/\widetilde{S}})$. Since
$\widetilde{Y}\to\widetilde{S}$ is smooth over $\widetilde{S}^{o}$ and
$\widetilde{Y}\to\widetilde{S}$ is semistable in codimension one, by
Proposition 4.2 we get a natural inclusion

$\displaystyle\det(W)^{\otimes kr}\otimes
I_{\widetilde{Z}}\to\left(\widetilde{g}_{\ast}(rK_{\widetilde{Y}/\widetilde{S}})^{\otimes
klr}\right)^{\vee\vee}\otimes
I_{\widetilde{Z}}\subset\widetilde{g}^{(klr)}_{\ast}(rK_{\widetilde{Y}^{(klr)}/\widetilde{S}}),$

where $I_{\widetilde{Z}}$ is an ideal sheaf whose co-support $\widetilde{Z}$
lies in $\sigma^{-1}(D)$ and ${\rm codim}_{\widetilde{S}}(\widetilde{Z})\geq
2$. By [Viehweg1983, Lemma 3.2] (see also [Fujino2020, Lemma 3.1.20]), there
is an inclusion

$\displaystyle\widetilde{g}^{(klr)}_{\ast}(rK_{\widetilde{Y}^{(klr)}/\widetilde{S}})\subset\sigma^{\ast}g_{\ast}^{(klr)}(rK_{Y^{(klr)}/S}).$

Let $I_{Z}$ be an ideal sheaf such that the map
$\sigma^{\ast}(I_{Z})\to\mathscr{O}_{\widetilde{X}}$ factors through
$I_{\widetilde{Z}}$. Taking the composition of the maps above we get a
morphism

$\sigma^{\ast}\left(\xi^{\ast}(\lambda_{a,r})^{\otimes k}\otimes
I_{Z}\right)^{\otimes r}\to\sigma^{\ast}g^{(klr)}_{\ast}(rK_{Y^{(klr)}/S}).$

Taking the adjoint we have a morphism

$\alpha:\left(\xi^{\ast}(\lambda_{a,r})^{\otimes k}\otimes
I_{Z}\right)^{\otimes
r}\to\sigma_{\ast}\sigma^{\ast}g^{(klr)}_{\ast}(rK_{Y^{(klr)}/S})\stackrel{{\scriptstyle\rm
trace}}{{\to}}g^{(klr)}_{\ast}(rK_{Y^{(klr)}/S}).$

By the constructions, $\alpha|_{S^{o}}$ is the composition of the maps

$\left(\det f^{o}_{\ast}(rK_{X^{o}/S^{o}}+rB^{o}+raA^{o})\right)^{\otimes
kr}\stackrel{{\scriptstyle(\ref{align_put_polarization_to_pushforward2})}}{{\subset}}g^{o}_{\ast}(rK_{Y^{o}/S^{o}})^{\otimes
klr}\simeq g^{o(klr)}_{\ast}(rK_{Y^{o(klr)}/S^{o}}).$

Hence $\alpha|_{S^{o}}$ is injective. Since
$g^{(klr)}_{\ast}(rK_{Y^{(klr)}/S})$ is torsion free, $\alpha$ is an injective
map. Notice that $Y^{(klr)}\to S$ is smooth over $S^{o}$. Applying Theorem 3.5
to the morphism $Y^{(klr)}\to S$ (which is smooth over $S^{o}$) and the
torsion free sheaf $\xi^{\ast}(\lambda_{a,r})^{\otimes k}\otimes I_{Z}$ we
obtain the theorem. ∎

###  6.4. Deformation boundedness of family of lc stable minimal models

Throughout this part we fix a data

$d\in\mathbb{N},c\in\mathbb{Q}^{\geq 0},\Gamma\subset\mathbb{Q}^{\geq
0}\textrm{ a finite set and }\sigma\in\mathbb{Q}[t].$

######  Theorem 6.7.

Let $S$ be an algebraic variety such that $S_{\rm sing}$ is compact. Then
there is a scheme of finite type ${\bf M}$ and a morphism $S\times{\bf M}\to
M_{\rm lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)$ that contains all maps $S\to
M_{\rm lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)$ which is induced from an
admissible family of $(d,\Phi_{c},\Gamma,\sigma)$-stable minimal models over
$S$.

###### Proof.

Let $C$ be a smooth projective curve of genus $g$ and $D\subset C$ a reduced
divisor. Let $\xi^{o}:C^{o}:=C\backslash D\to M_{\rm
lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)$ be a morphism which is induced from an
admissible family of $(d,\Phi_{c},\Gamma,\sigma)$-stable minimal models over
$C^{o}$ and let $\xi:C\to M_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)$ be its
compactification. Let $(a,r,j)\in\mathbb{Q}^{\geq
0}\times(\mathbb{Z}^{>0})^{2}$ be a $(d,\Phi_{c},\Gamma,\sigma)$-polarization
data with $0\leq a\ll 1$. Then one has

(6.17) $\displaystyle\deg(\xi^{\ast}\lambda_{a,r})\leq\frac{rd{\rm
rank}(\Lambda_{a,r})}{2}(2g+\deg(D)).$

by Theorem 6.6. Notice that $K_{C}+D$ may not be pseudo-effective (e.g. when
$g=0$ and $\deg(D)=0,1$). We add two extra points $p_{1},p_{2}\in C\backslash
D$ so that $K_{C}+D^{\prime}$ is always pseudo-effective for
$D^{\prime}=D\cup\\{p_{1},p_{2}\\}$. (6.17) is deduced by applying Theorem 6.6
to the case when $S=C$ and $S^{o}=C\backslash(D\cup D^{\prime})$. According to
[Kovacs2009, Corollary 2.23], the theorem follows from (6.17). ∎

####  6.4.1. Strong boundedness

######  Definition 6.8.

A $(d,\Phi_{c},\Gamma,\sigma)$-klt stable minimal model $(X,B),A$ is called
log smooth if $X$ is smooth and $A+B$ is a simple normal crossing
$\mathbb{Q}$-divisor. A family $f:(X,B),A\to S$ of
$(d,\Phi_{c},\Gamma,\sigma)$-klt stable minimal models is called log smooth if
$X\to S$ is smooth and $A+B$ is an $f$-relative simple normal crossing
divisor.

The groupoids of log smooth families of $(d,\Phi_{c},\Gamma,\sigma)$-klt
stable minimal models form an open substack (denoted by $\mathscr{M}_{\rm
sm}(d,\Phi_{c},\Gamma,\sigma)$) of $\mathscr{M}_{\rm
slc}(d,\Phi_{c},\Gamma,\sigma)$. Its closure is a meaningful compactification
of $\mathscr{M}_{\rm sm}(d,\Phi_{c},\Gamma,\sigma)$.

######  Theorem 6.9.

Let $S$ be an algebraic variety such that $S_{\rm sing}$ is compact. Then
there is a scheme of finite type ${\bf M}$ and an admissible log smooth family
$F\in\mathscr{M}_{\rm sm}(d,\Phi_{c},\Gamma,\sigma)(S\times{\bf M})$ of stable
minimal models which contains all admissible log smooth families of
$(d,\Phi_{c},\Gamma,\sigma)$-stable minimal models over $S$.

###### Proof.

Let $C$ be a smooth projective curve of genus $g$ and $D\subset C$ a reduced
divisor. Let $\xi^{o}:C^{o}:=C\backslash D\to M_{\rm
lc,(0,1)}(d,\Phi_{c},\Gamma,\sigma)$ be a morphism which is induced from a log
smooth family of $(d,\Phi_{c},\Gamma,\sigma)$-stable minimal models over
$C^{o}$ and $\xi:C\to M_{\rm slc}(d,\Phi_{c},\Gamma,\sigma)$ its
compactification. Let $(a,r,j)\in\mathbb{Q}^{\geq
0}\times(\mathbb{Z}^{>0})^{2}$ be a $(d,\Phi_{c},\Gamma,\sigma)$-polarization
data such that $0\leq a\ll 1$. Then one has

(6.18) $\displaystyle\deg(\xi^{\ast}\lambda_{a,r})\leq\frac{rd{\rm
rank}(\Lambda_{a,r})}{2}(2g+\deg(D))$

by Theorem 6.6. Here we add two extra points to $D$ so that $K_{C}+D$ is
always pseudo-effective. According to [Kovacs2011, Theorem 1.7], the theorem
follows from (6.18). ∎

## References