to $(X,\Delta)$, by taking an appropriate resolution of the graph of $\phi$,
we get a common log resolution $f\colon\tilde{X}\to X$ and
$f^{\prime}\colon\tilde{X}\to X^{\prime}$ of
$(X,\Delta)\dashrightarrow(X^{\prime},\Delta^{\prime})$ such that $f^{\prime}$
and $U^{\prime}$ satisfy the final condition of Theorem 4.5. Since
$K_{X}+\Delta+M$ is not big, we have ${\rm dim}Z<{\rm dim}X^{\prime}$. Let $F$
be the general fiber of $\pi$, and we put $\tilde{F}=f^{\prime-1}(F)$. We
recall the hypothesis that $M=\sum_{i}\mu_{i}M_{i}$ and $M_{i}$ are log big
with respect to $(X,\Delta)$. Since ${\rm dim}F>0$, we see that
$(f^{*}M_{i})|_{\tilde{F}}$ is not numerically trivial for all $i$. Applying
the global ACC [7, Theorem 1.6] to the restriction of
$(X^{\prime},\Delta^{\prime},\overline{M})$ to $F$, we see that all $\mu_{i}$
belong to a finite set $\Phi^{\prime}\subset\Phi$ depending only on $d$ and
$\Phi$. Therefore, there exists a positive integer $p$, depending only on $d$
and $\Phi$, such that $p\mu_{i}$ are integers. Then $p\overline{M}$ is
b-Cartier. Clearly, the coefficients of $\Delta^{\prime}$ belong to $\Phi$.
From these facts, we can apply Theorem 4.5 to
$(X^{\prime},\Delta^{\prime},\overline{M})\to Z$. By Theorem 4.5, there exist
positive integers $n$ and $q$ and a DCC set $\Omega\subset\mathbb{Q}$,
depending only on $d$, $p$, and $\Phi$, such that
$n(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})\sim\pi^{*}D$ for some
$\mathbb{Q}$-Cartier divisor $D$ on $Z$ and there is a generalized lc pair
$(Z,\Delta_{Z},\boldsymbol{\rm N})$ such that
$K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z}=\frac{1}{n}D$, the coefficients of
$\Delta_{Z}$ belong to $\Omega$, and $q\boldsymbol{\rm N}$ is b-Cartier. Then
$n$, $q$, and $\Omega$ depends only on $d$ and $\Phi$ because $p$ depends only
on $d$ and $\Phi$. Since we have

$\begin{split}H^{0}(X,\mathcal{O}_{X}(\lfloor
ln(K_{X}+\Delta+M)\rfloor))&\simeq
H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(\lfloor
ln(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})\rfloor))\\\ &\simeq
H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(\lfloor l\pi^{*}D\rfloor))\\\
&\simeq H^{0}(Z,\mathcal{O}_{Z}(\lfloor ln(K_{Z}+\Delta_{Z}+\boldsymbol{\rm
N}_{Z})\rfloor))\end{split}$

for every positive integer $l$, we get the desired $n$, $p$, and $\Omega$. ∎

###### Proof of Theorem 1.2.

The theorem follows from Theorem 1.3 and [7, Theorem 1.3]. ∎

######  Example 4.6.

We fix $d$, $m$, and $p$. We consider the category $\mathcal{C}$ whose objects
are generalized dlt pairs $(X,\Delta,\overline{M})$ such that ${\rm dim}X=d$,
$m\Delta$ is a Weil divisor, $pM$ is Cartier, $M$ is nef and log big with
respect to $(X,\Delta)$, and $K_{X}+\Delta+M\sim_{\mathbb{Q}}0$. We show that
the set

$\mathcal{D}\subset\\{X\,|\,\text{$(X,\Delta,\overline{M})\in\mathcal{C}$ for
some $\Delta$ and $M$}\\}$

is unbounded for some $\mathcal{D}$.

We put $V=\mathbb{P}^{1}$. We fix a very ample Cartier divisor $H_{V}$ such
that $\mathcal{O}_{V}(H_{V})=\mathcal{O}_{V}(1)$. We construct a
$\mathbb{P}^{1}$-bundle
$X_{n}:=\mathbb{P}_{V}(\mathcal{O}_{V}\oplus\mathcal{O}_{V}(-nH_{V}))\overset{f}{\longrightarrow}V.$
Let $\Delta_{n}$ be the unique section corresponding to
$\mathcal{O}_{X_{n}}(1)$. Define $M_{n}:=\Delta_{n}+(n+2)f^{*}H_{V}$. By
construction, we have $K_{X_{n}}+\Delta_{n}+M_{n}\sim 0.$ We can easily check
that $(X_{n},\Delta_{n})$ is a dlt pair and $M_{n}$ is an ample Cartier
divisor. We also see that $(-K_{X_{n}}\cdot A_{n})\geq n$ for every $n$ and
every ample Cartier divisor $A_{n}$ on $X_{n}$. Indeed, we may write
$A_{n}\sim a_{n}\Delta_{n}+b_{n}f^{*}H_{V}$ for some $a_{n}\geq 1$ and
$b_{n}\geq 1$ (cf. [27, III, Exercise 12.5]). Then

$(-K_{X_{n}}\cdot A_{n})=(\Delta_{n}+M_{n})\cdot A_{n}>(nf^{*}H_{V}\cdot
A_{n})\geq(nf^{*}H_{V}\cdot a_{n}\Delta_{n})\geq n.$

This fact shows that the set $\\{X_{n}\\}_{n\geq 1}$ is unbounded.

##  5\. Boundedness results

In this section, we prove some boundedness results. Before we start the main
parts of this section, we recall the notion of the Iitaka volume for
$\mathbb{Q}$-Cartier divisors and we prove Theorem 1.4.

######  Definition 5.1 (Iitaka volume, [39, Definition 1.1]).

Let $X$ be a normal projective variety, and let $D$ be a $\mathbb{Q}$-Cartier
divisor on $X$ such that the Iitaka dimension $\kappa(X,D)$ is nonnegative.
Then the Iitaka volume of $D$, denoted by ${\rm Ivol(D)}$, is defined by

${\rm Ivol(D)}:=\underset{m\to\infty}{\rm
lim\,sup}\frac{H^{0}(X,\mathcal{O}_{X}(\lfloor
mD\rfloor))}{m^{\kappa(X,D)}/\kappa(X,D)!}.$

###### Proof of Theorem 1.4.

The theorem follows from Theorem 1.3 and [5, Theorem 1.3]. With notations as
in Theorem 1.3, the divisor $K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z}$ is ample
and $\mathbb{Q}$-Cartier. Therefore, ${\rm
Ivol}(K_{X}+\Delta+M)=(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})^{{\rm
dim}Z}\in\mathbb{Q}_{>0}$. ∎

From now on, we prove some boundedness results, and we prove Theorems 1.6 and
Theorem 1.7.

###  5.1. Boundedness of complements

In this subsection, we prove Theorem 1.6.

######  Lemma 5.2.

Let $\mathfrak{R}\subset[0,1]$ be a finite set of rational numbers, and let
$\Phi(\mathfrak{R})$ be the hyperstandard set associated to $\mathfrak{R}$.
Let $q$ be a positive integer such that $qa\in\mathbb{Z}$ for every
$a\in\mathfrak{R}$. Then $b-\\{-qmb\\}\geq 0$ for every
$b\in\Phi(\mathfrak{R})$ and every positive integer $m$.

###### Proof.

We may write $b=1-\frac{a}{r}$ for some $a\in\mathfrak{R}$ and
$r\in\mathbb{Z}_{>0}$. Then

$b-\\{-qmb\\}=1-\frac{a}{r}-\left\\{-qm\left(1-\frac{a}{r}\right)\right\\}=1-\frac{a}{r}-\left\\{\frac{qma}{r}\right\\}.$

Since $qa\in\mathbb{Z}$ and $m\in\mathbb{Z}$, we have
$\\{-\frac{qma}{r}\\}\leq\frac{r-1}{r}$. Since $a\leq 1$, we have

$1-\frac{a}{r}-\left\\{\frac{qma}{r}\right\\}\geq
1-\frac{1}{r}-\frac{r-1}{r}\geq 0.$

Therefore $b-\\{-qmb\\}\geq 0$. ∎

######  Lemma 5.3.

Let $(X,\Delta)$ be a sub-pair such that $X$ is projective, and let $D$ be a
Cartier divisor on $X$. Let $n$ be a positive integer, and let
$V_{1},\cdots,V_{l}$ be subvarieties of $X$. Suppose that for each $1\leq
i\leq l$, there is $D_{i}\in|D|$ such that $(X,\Delta+\frac{1}{n}D_{i})$ is
sub-lc on a neighborhood of $V_{i}$. Then, for every general member
$D^{\prime}\in|D|$ the sub-pair $(X,\Delta+\frac{1}{n}D^{\prime})$ is sub-lc
on a neighborhood of $\bigcup_{i=1}^{l}V_{i}$.

###### Proof.

We take a resolution $f\colon Y\to X$ of the linear system $|D|$, i.e.,
$f^{*}|D|=|M|+F$ where $M$ is a base point Cartier divisor and $F$ is the
fixed divisor. Replacing $X$ by $Y$ and replacing $\Delta$, $D$, and
$V_{1},\cdots,V_{l}$ accordingly, we may assume that the movable part of $|D|$
is base point free. By the hypothesis, $F$ satisfies the property that
$(X,\Delta+\frac{1}{n}F)$ is sub-lc on a neighborhood of
$\bigcup_{i=1}^{l}V_{i}$. Thus, Lemma 5.3 holds true. ∎

The following theorem is the main result of this subsection.

######  Theorem 5.4.

Let $d$ be a positive integer. Let $\mathfrak{R}\subset[0,1]$ be a finite set
of rational numbers and $\Phi(\mathfrak{R})$ the hyperstandard set associated
to $\mathfrak{R}$. Then there exists a positive integer $n$, depending only on
$d$ and $\mathfrak{R}$, satisfying the following. Let $(X,\Delta)$ be a
projective dlt pair such that

  * •

${\rm dim}X=d$,

  * •

the coefficients of $\Delta$ belong to $\Phi(\mathfrak{R})$, and

  * •

$-(K_{X}+\Delta)$ is nef and log big.

Then there is an effective $\mathbb{Q}$-divisor $\Xi$ such that
$(X,\Delta+\Xi)$ is lc and $n(K_{X}+\Delta+\Xi)\sim 0$.

###### Proof.

We prove the theorem by induction on the dimension of $X$. If $(X,\Delta)$ is
klt, then the existence of $n$ is proved by Birkar [4]. So we may assume that
$(X,\Delta)$ is not klt. Let $p$ be the minimum positive integer such that
$pa\in\mathbb{Z}$ for all $a\in\mathfrak{R}$.

Since $(X,\Delta)$ is dlt, there is a log resolution $f\colon\tilde{X}\to X$
of $(X,\Delta)$ such that $f$ is an isomorphism over an open subset $U\subset
X$ containing all the generic points of lc centers of $(X,\Delta)$. We put
$\tilde{D}=f^{*}(K_{X}+\Delta)$ and $\tilde{S}=f_{*}^{-1}S$. We prove Theorem
5.4 in two steps.

######  Step 1.

In this step. we prove that the morphsim

$H^{0}(\tilde{X},\mathcal{O}_{\tilde{X}}(\lfloor-
mp\tilde{D}\rfloor))\longrightarrow
H^{0}(\tilde{S},\mathcal{O}_{\tilde{S}}(\lfloor-
mp\tilde{D}\rfloor|_{\tilde{S}}))$

defined with Definition 2.9 is surjective for every component $S$ of
$\lfloor\Delta\rfloor$ and every positive integer $m$. We closely follow the
proof of Lemma 2.16.

We can write $K_{\tilde{X}}+\tilde{\Delta}=f^{*}(K_{X}+\Delta)+\tilde{E},$
where $\tilde{\Delta}\geq 0$ and $\tilde{E}\geq 0$ have no common components.
Then $-\tilde{D}$ is nef and log big with respect to
$(\tilde{X},\tilde{\Delta})$, and we can write

$K_{\tilde{X}}+\tilde{\Delta}-(mp+1)\tilde{D}=-mp\tilde{D}+\tilde{E}.$

We can write $-mp\tilde{D}=\lfloor-
mp\tilde{D}\rfloor+\\{-mpf_{*}^{-1}\Delta\\}+\tilde{E}^{\prime}$ with an
effective $f$-exceptional $\mathbb{Q}$-divisor $\tilde{E}^{\prime}$. We put
$\tilde{\Delta}^{\prime}=\tilde{\Delta}-\\{-mpf_{*}^{-1}\Delta\\}-\tilde{S}$.
By Lemma 5.2, we have $\tilde{\Delta}^{\prime}\geq 0$ and

$K_{\tilde{X}}+\tilde{S}+\tilde{\Delta}^{\prime}-(mp+1)\tilde{D}=\lfloor-
mp\tilde{D}\rfloor+\tilde{E}+\tilde{E}^{\prime}.$

We define a Weil divisor $\tilde{F}$ on $\tilde{X}$ so that ${\rm
coeff}_{P}(\tilde{F})=0$ if ${\rm
coeff}_{P}(\tilde{\Delta}^{\prime}-\\{(\tilde{E}+\tilde{E}^{\prime})\\})\geq
0$ and ${\rm coeff}_{P}(\tilde{F})=1$ if ${\rm
coeff}_{P}(\tilde{\Delta}^{\prime}-\\{(\tilde{E}+\tilde{E}^{\prime})\\})<0$
for every prime divisor $P$. We put

$\tilde{B}=\tilde{\Delta}^{\prime}-\\{(\tilde{E}+\tilde{E}^{\prime})\\}+\tilde{F}\qquad{\rm
and}\qquad\tilde{G}=\lfloor(\tilde{E}+\tilde{E}^{\prime})\rfloor+\tilde{F}.$

Then we have

$K_{\tilde{X}}+\tilde{S}+\tilde{B}-(mp+1)\tilde{D}=\lfloor-
mp\tilde{D}\rfloor+\tilde{G}.$

By the definition, we can check that $\tilde{B}$ is a boundary divisor,
$\lfloor\tilde{B}\rfloor\leq\lfloor\tilde{\Delta}^{\prime}\rfloor$, and
$\tilde{G}$ is an effective $f$-exceptional Weil divisor. Then $-\tilde{D}$ is
nef and log big with respect to $(\tilde{X},\tilde{S}+\tilde{B})$. By the
Kodaira type vanishing theorem [12, Lemma 1.5], we see that

$H^{0}(\tilde{X},\mathcal{O}_{\tilde{X}}(\lfloor-
mp\tilde{D}\rfloor+\tilde{G}))\longrightarrow
H^{0}(\tilde{S},\mathcal{O}_{\tilde{S}}((\lfloor-
mp\tilde{D}\rfloor+\tilde{G})|_{\tilde{S}}))$

is surjective. Since $\tilde{G}$ is $f$-exceptional, there are natural
isomorphisms

$H^{0}(\tilde{X},\mathcal{O}_{\tilde{X}}(\lfloor-mp\tilde{D}\rfloor))\simeq
H^{0}(X,\mathcal{O}_{X}(\lfloor-mp(K_{X}+\Delta)\rfloor))\simeq
H^{0}(\tilde{X},\mathcal{O}_{\tilde{X}}(\lfloor-
mp\tilde{D}\rfloor+\tilde{G})).$

From the following diagram

$\textstyle{H^{0}(\tilde{X},\mathcal{O}_{\tilde{X}}(\lfloor-
mp\tilde{D}\rfloor+\tilde{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\tilde{S},\mathcal{O}_{\tilde{S}}((\lfloor-
mp\tilde{D}\rfloor+\tilde{G})|_{\tilde{S}}))}$$\textstyle{H^{0}(\tilde{X},\mathcal{O}_{\tilde{X}}(\lfloor-
mp\tilde{D}\rfloor))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{0}(\tilde{X},\mathcal{O}_{\tilde{X}}(\lfloor-
mp\tilde{D}\rfloor|_{\tilde{S}})),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

the lower horizontal morphism is surjective. We finish this step.

For each component $S$ of $\lfloor\Delta\rfloor$, we define a dlt pair
$(S,\Delta_{S})$ by divisorial adjunction
$K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$. By Remark 2.8, there exists a finite
set $\mathfrak{S}\subset[0,1]$ of rational numbers, depending only on $d$ and
$\mathfrak{R}$, such that all the coefficients of $\Delta_{S}$ belong to the
hyperstandard set associated to $\mathfrak{S}$. By the induction hypothesis of
Theorem 5.4, there is $n^{\prime}$, depending only on $d-1$ and
$\mathfrak{S}$, such that for every $(S,\Delta_{S})$ where $S$ is a component
of $\lfloor\Delta\rfloor$, there is $\Delta_{S}^{+}\geq\Delta_{S}$ such that
$(S,\Delta_{S}^{+})$ is lc and $n^{\prime}(K_{S}+\Delta_{S}^{+})\sim 0$.

######  Step 2.

We put $n=n^{\prime}p$. From now on, we prove that the $n$ satisfies the
property of Theorem 5.4.

Pick a component $S$ of $\lfloor\Delta\rfloor$, put $\tilde{S}=f_{*}^{-1}S$,
and take a $\mathbb{Q}$-divisor $\Delta_{S}^{+}\geq\Delta_{S}$ on $S$ such
that $(S,\Delta_{S}^{+})$ is lc and $n^{\prime}(K_{S}+\Delta_{S}^{+})\sim 0$.
We put $C_{S}=n(\Delta^{+}_{S}-\Delta_{S})\sim-n(K_{S}+\Delta_{S})$. We put
$C_{\tilde{S}}=f_{\tilde{S}}^{*}C_{S}$, where
$f_{\tilde{S}}=f|_{\tilde{S}}\colon\tilde{S}\to S$. Then
$-n\tilde{D}|_{\tilde{S}}\sim C_{\tilde{S}}$ because
$\tilde{D}=f^{*}(K_{X}+\Delta)$. Put $\tilde{R}=\\{-n\tilde{D}\\}$. Then
$-n\tilde{D}|_{\tilde{S}}=\lfloor-n\tilde{D}\rfloor|_{\tilde{S}}+\tilde{R}|_{\tilde{S}}$,
and $\tilde{R}|_{\tilde{S}}$ is well-defined as a $\mathbb{Q}$-divisor on
$\tilde{S}$ because $\tilde{S}$ is not a component of $\tilde{R}$. We have
$\lfloor\tilde{R}|_{\tilde{S}}\rfloor=0$ because ${\rm
Supp}(\tilde{R}+\tilde{S})$ is snc, hence we have
$\lfloor-n\tilde{D}\rfloor|_{\tilde{S}}=\lfloor-n\tilde{D}|_{\tilde{S}}\rfloor$
and $\tilde{R}|_{\tilde{S}}=\\{-n\tilde{D}|_{\tilde{S}}\\}$. Therefore, we
have

$\lfloor-n\tilde{D}\rfloor|_{\tilde{S}}\sim\lfloor
C_{\tilde{S}}\rfloor\qquad{\rm
and}\qquad\tilde{R}|_{\tilde{S}}=\\{C_{\tilde{S}}\\}.$

By Step 1, we can find an effective Weil divisor
$\tilde{C}\sim\lfloor-n\tilde{D}\rfloor$ such that
$\tilde{C}|_{\tilde{S}}=\lfloor C_{\tilde{S}}\rfloor$. We put
$C=f_{*}(\tilde{C}+\tilde{R})$. Since $\tilde{C}+\tilde{R}\sim-n\tilde{D}$ and
$\tilde{D}=f^{*}(K_{X}+\Delta)$, we can easily check $C\sim-n(K_{X}+\Delta)$
and $\tilde{C}+\tilde{R}=f^{*}C$. By definition of $C$, we see that

$C|_{S}=f_{\tilde{S}*}f^{*}_{\tilde{S}}(C|_{S})=f_{\tilde{S}*}((\tilde{C}+\tilde{R})|_{\tilde{S}})=f_{\tilde{S}*}C_{\tilde{S}}=C_{S}=n(\Delta^{+}_{S}-\Delta_{S}).$

From this and inversion of adjunction [35], the pair $(X,\Delta+\frac{1}{n}C)$
is lc on a neighborhood of $S$. We define $\tilde{\Gamma}$ to be
$K_{\tilde{X}}+\tilde{\Gamma}=f^{*}(K_{X}+\Delta)$. Since
$f^{*}C=\tilde{C}+\tilde{R}$, we see that the sub-pair
$(\tilde{X},\tilde{\Gamma}+\frac{1}{n}(\tilde{C}+\tilde{R}))$ is sub-lc on a
neighborhood of $f^{-1}(S)$.

In this way, for every component $S$ of $\lfloor\Delta\rfloor$, we may find
$\tilde{C}\in|\lfloor-n\tilde{D}\rfloor|$ such that the sub-pair
$(\tilde{X},\tilde{\Gamma}+\frac{1}{n}\tilde{R}+\frac{1}{n}\tilde{C})$ is sub-
lc on a neighborhood of $f^{-1}(S)$. By Lemma 5.3, there is
$\tilde{M}\in|\lfloor-n\tilde{D}\rfloor|$ such that
$(\tilde{X},\tilde{\Gamma}+\frac{1}{n}\tilde{R}+\frac{1}{n}\tilde{M})$ is sub-
lc on a neighborhood of $f^{-1}(\lfloor\Delta\rfloor)$. By construction, we
have $\tilde{M}+\tilde{R}\sim-nf^{*}(K_{X}+\Delta)$. By putting
$M=f_{*}(\tilde{M}+\tilde{R})$, we have $M\sim-n(K_{X}+\Delta)$ and the pair
$(X,\Delta+\frac{1}{n}M)$ is lc on a neighborhood of $\lfloor\Delta\rfloor$.

Finally, if $(X,\Delta+\frac{1}{n}M)$ is not lc, then there is
$\epsilon\in(0,\frac{1}{n})$ such that the non-klt locus of
$(X,\Delta+(\frac{1}{n}-\epsilon)M)$ has at least two connected comopnents.
Then we get a contradiction by the connectedness principle for non-klt locus
[37, 17.4 Theorem]. Thus, $(X,\Delta+\frac{1}{n}M)$ is lc, and therefore
$\Xi:=\frac{1}{n}M$ is the desired divisor. ∎

###### Proof of Theorem 1.6.

We will freely use the notations in Theorem 1.6. We may assume that
$K_{X}+\Delta+M$ is not big. By the argument similar to the proof of Theorem
1.3, the Global ACC for generalized pairs [7, Theorem 1.6] implies the
existence of a positive integer $m$, depending only on $d$ and $\Phi$, such
that $m\Delta^{\prime}_{F}$ is a Weil divisor and $mM$ is Cartier. Then
Theorem 4.1 shows $n_{1}(K_{F}+\Delta^{\prime}_{F}+M^{\prime}_{F})\sim 0$ for
some positive integer $n_{1}$ which depends only on $d$, $\Phi$, and $m$. If
there exists $n_{2}\in\mathbb{Z}_{>0}$ depending only on $d$ and $m$ such that
$(F,\Delta^{\prime}_{F}+\frac{1}{n_{2}}B^{\prime}_{F})$ is lc for some
$B^{\prime}_{F}\in|-n_{2}(K_{F}+\Delta^{\prime}_{F})|$, then $n:=n_{1}n_{2}$
is the desired positive integer. In this way, we only need to prove the
existence of the above $n_{2}\in\mathbb{Z}_{>0}$.

We run a $(K_{X}+\Delta+3dmM)$-MMP with scaling of an ample divisor. By the
length of extremal rays ([15, Section 18] and [17, Proposition 6.1]) and the
same argument as in the proof of Theorem 3.16, we get a sequence of steps of a
$(K_{X}+\Delta+3dmM)$-MMP to a good minimal model
$(\tilde{X},\tilde{\Delta},3dm\overline{M})$ such that

  * •

$X\dashrightarrow X^{\prime}$ is a sequence of steps of a
$(K_{X}+\Delta+M)$-MMP, and

  * •

defining $\tilde{M}$ to be the birational transform of $M$ on $\tilde{X}$,
then $m\tilde{M}$ is a nef Cartier divisor on $\tilde{X}$ that is log big with
respect to $(\tilde{X},\tilde{\Delta})$.

With the argument in [31, Proof of Lemma 2.14], we may construct a sequence of
steps of a $(K_{\tilde{X}}+\tilde{\Delta}+\tilde{M})$-MMP with scaling of
$(3dm-1)\tilde{M}$ terminating with a good minimal model
$(\tilde{X}^{\prime},\tilde{\Delta}^{\prime},\overline{M})$. Let
$\tilde{M}^{\prime}$ be the birational transform of $M$ on
$\tilde{X}^{\prime}$. By construction,
$K_{\tilde{X}^{\prime}}+\tilde{\Delta}^{\prime}+(1+t)\tilde{M}^{\prime}$ is
nef for some $t>0$. Moreover, taking a common resolution of
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ and using the negativity lemma,
we see that $\tilde{M}^{\prime}$ is log big with respect to
$(\tilde{X}^{\prime},\tilde{\Delta}^{\prime})$. Let
$\tilde{\pi}\colon\tilde{X}^{\prime}\to Z$ be the contraction induced by
$K_{\tilde{X}^{\prime}}+\tilde{\Delta}^{\prime}+\tilde{M}^{\prime}$, and let
$\tilde{F}$ be the general fiber of $\tilde{\pi}$. We put
$\tilde{\Delta}^{\prime}_{\tilde{F}}=\tilde{\Delta}^{\prime}|_{\tilde{F}}$ and
$\tilde{M}^{\prime}_{\tilde{F}}=\tilde{M}^{\prime}|_{\tilde{F}}$.

Since
$K_{\tilde{X}^{\prime}}+\tilde{\Delta}^{\prime}+\tilde{M}^{\prime}\sim_{\mathbb{Q},Z}0$
and $K_{\tilde{X}^{\prime}}+\tilde{\Delta}^{\prime}+(1+t)\tilde{M}^{\prime}$
is nef for some $t>0$, we see that
$\tilde{M}^{\prime}\sim_{\mathbb{Q},Z}-(K_{\tilde{X}^{\prime}}+\tilde{\Delta}^{\prime})$
is nef and log big over $Z$ with respect to
$(\tilde{X}^{\prime},\tilde{\Delta}^{\prime})$. By Theorem 5.4, there exists a
positive integer $n_{2}$, depending only on $d$ and $m$, such that
$(\tilde{F},\tilde{\Delta}^{\prime}_{\tilde{F}}+\frac{1}{n_{2}}\tilde{B}^{\prime}_{\tilde{F}})$
is lc for some
$\tilde{B}^{\prime}_{\tilde{F}}\in|-n_{2}(K_{\tilde{F}}+\tilde{\Delta}^{\prime}_{\tilde{F}})|$.
Because $X\dashrightarrow X^{\prime}$ and $X\dashrightarrow\tilde{X}^{\prime}$
are sequences of steps of $(K_{X}+\Delta+M)$-MMP to good minimal models, the
induced birational map $X^{\prime}\dashrightarrow\tilde{X}^{\prime}$ is
isomorphic in codimension one, and therefore $F\dashrightarrow\tilde{F}$ are
so. Then $(F,\Delta^{\prime}_{F}+\frac{1}{n_{2}}B^{\prime}_{F})$ is lc for
some $B^{\prime}_{F}\in|-n_{2}(K_{F}+\Delta^{\prime}_{F})|$. Since $m$ depends
only on $d$ and $\Phi$, we see that $n_{2}$ depends only on $d$ and $\Phi$.
Hence $n:=n_{1}n_{2}$ satisfies the conditions of Theorem 1.6. ∎

The following example shows that the condition of the log bigness of the nef
divisors in Theorem 1.2 and Theorem 1.6 cannot be relaxed to the bigness.

######  Example 5.5.

Let $(X,\Delta)$ be a projective log smooth dlt pair such that $\Delta$ is a
reduced divisor, $H^{1}(X,\mathcal{O}_{X})\neq 0$, and $-(K_{X}+\Delta)$ is
nef and big. For example, take an elliptic curve $E$ with a very ample divisor
$H$ and define $X:=\mathbb{P}_{E}(\mathcal{O}_{E}\oplus\mathcal{O}_{E}(H))$.
For each $n\geq 1$, let $D_{n}$ be a Cartier divisor on $X$ such that
$nD_{n}\sim 0$ and $lD_{n}\not\sim 0$ for all $0<l<n$. We define
$M_{n}:=-(K_{X}+\Delta)+D_{n}$, and we consider the set
$\\{(X,\Delta,\overline{M_{n}})\\}_{n\geq 1}$. If the log bigness of $M_{i}$
in Theorem 1.2 or Theorem 1.6 can be relaxed to the bigness, then there is a
positive integer $m$ such that $mD_{n}=m(K_{X}+\Delta+M_{n})\sim 0$ for all
$n$, which contradicts the definition of $D_{n}$. Therefore, the log bigness
of $M_{i}$ in Theorem 1.2 and Theorem 1.6 cannot be relaxed to the bigness.

###  5.2. Effective finite generation

In this subsection, we prove Theorem 1.7.

We will first prove the effective base point free theorem for special
generalized dlt pairs (Theorem 5.8). For the proof, we need the base point
free theorem for quasi-log schemes [17, Theorem 6.5.1] (see also [1, Theorem
5.1]).

######  Definition 5.6 (Quasi-log scheme, [17, Definition 6.2.2], see also [1,
Definition 4.1]).

A quasi-log scheme is a scheme $X$ endowed with an $\mathbb{R}$-Cartier
divisor (or $\mathbb{R}$-line bundle) $\omega$ on $X$, a closed subscheme
$X_{-\infty}\subsetneq X$, and a finite collection $\\{C\\}$ of reduced and
irreducible subschemes of $X$ such that there is a proper morphism
$f\colon(Y,B_{Y})\to X$ from a globally embedded simple normal crossing pair
satisfying the following properties:

  * •

$f^{*}\omega\sim_{\mathbb{R}}K_{Y}+B_{Y}$,

  * •

the natural map $\mathcal{O}_{X}\to
f_{*}\mathcal{O}_{Y}(\lceil-(B_{Y}^{<1})\rceil)$ induces an isomorphism

$\mathcal{I}_{X_{-\infty}}\overset{\simeq}{\longrightarrow}f_{*}\mathcal{O}_{Y}(\lceil-(B_{Y}^{<1})\rceil-\lfloor
B_{Y}^{>1}\rfloor),$

where $\mathcal{I}_{X_{-\infty}}$ is the defining ideal sheaf of
$X_{-\infty}$, and

  * •

the collection of reduced and irreducible subschemes $\\{C\\}$ coincides with
the images of the strata of $(Y,B_{Y})$ that are not included in
$X_{-\infty}$.

We simply write $[X,\omega]$ to denote the above data

$\left(X,\omega,f\colon(Y,B_{Y})\to X\right)$

if there is no risk of confusion. We sometimes use ${\rm Nqlc}(X,\omega)$ to
denote $X_{-\infty}$.

######  Theorem 5.7 ([15, Theorem 6.5.1]).

Let $[X,\omega]$ be a quasi-log scheme and $\pi\colon X\to S$ a projective
morphism to a scheme $S$. Let $D$ be a $\pi$-nef Cartier divisor on $X$ such
that $pD-\omega$ is $\pi$-ample for some positive integer $p$ and
$\mathcal{O}_{X_{-\infty}}(mD|_{X_{-\infty}})$ is
$\pi|_{X_{\infty}}$-generated for all $m\gg 0$. Then there is a positive
integer $m_{0}$ such that $\mathcal{O}_{X}(mD)$ is $\pi$-generated for every
integer $m$ such that $m\geq m_{0}$.

######  Theorem 5.8.

Let $d$ and $p$ be positive integers. Then there exists a positive integer
$m$, depending only on $d$ and $p$, satisfying the following. Let
$(X,\Delta,\boldsymbol{\rm M})$ be a generalized dlt pair such that

  * •

${\rm dim}X=d$,

  * •

$p\boldsymbol{\rm M}$ is b-Cartier and $p(K_{X}+\Delta+\boldsymbol{\rm
M}_{X})$ is nef and Cartier, and

  * •

there is a log resolution $f\colon\tilde{X}\to X$ of $(X,\Delta)$ such that

    * –

$f$ is an isomorphism over an open subset $U\subset X$ containing all the
generic points of generalized lc centers of $(X,\Delta,\boldsymbol{\rm M})$,

    * –

$\boldsymbol{\rm M}$ descends to $\tilde{X}$, and

    * –

writing $K_{\tilde{X}}+\tilde{\Delta}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+\Delta+\boldsymbol{\rm M}_{X})+\tilde{E}$, where
$\tilde{\Delta}\geq 0$ and $\tilde{E}\geq 0$ have no common components, then
$\boldsymbol{\rm M}_{\tilde{X}}$ is log big with respect to
$(\tilde{X},\tilde{\Delta})$.

Then $m(K_{\tilde{X}}+\tilde{\Delta}+\boldsymbol{\rm M}_{\tilde{X}})$ is base
point free.

###### Proof.

We prove it by induction on the dimension of $X$. Since $\boldsymbol{\rm
M}_{\tilde{X}}$ is nef and big, if $(X,\Delta,\boldsymbol{\rm M})$ is
generalized klt, then there is a $\mathbb{Q}$-divisor $E\geq 0$ on $X$ such
that $\boldsymbol{\rm M}_{X}-E$ is ample and $(X,\Delta+E)$ is klt. Then
Theorem 5.8 follows from the effective base point free theorem for klt pairs
[36]. Thus, we may assume that $(X,\Delta,\boldsymbol{\rm M})$ is not
generalized klt. Since $(X,\Delta,\boldsymbol{\rm M})$ is a weak generalized
lc model of $(\tilde{X},\tilde{\Delta},\boldsymbol{\rm M})$, by Theorem 1.1,
we see that $K_{X}+\Delta+\boldsymbol{\rm M}_{X}$ is semi-ample. Thus, the
divisor defines a contraction $\pi\colon X\to Z$.

######  Step 1.

In this step, we define the structure of a quasi-log scheme on $X$.

Let $A$ be an ample $\mathbb{Q}$-divisor on $X$. Since $\boldsymbol{\rm
M}_{\tilde{X}}$ is nef and big, rescaling $A$ if necessary, we can find
$\tilde{R}\in|\boldsymbol{\rm M}_{\tilde{X}}-f^{*}A|_{\mathbb{Q}}$ that is the
sum of an ample $\mathbb{Q}$-divisor $\tilde{R}_{1}$ and an effective
$\mathbb{Q}$-divisor $\tilde{R}_{2}$. We take a log resolution $g\colon
Y\to\tilde{X}$ of $(\tilde{X},\tilde{\Delta}+\tilde{E}+\tilde{R})$, where
$\tilde{\Delta}$ and $\tilde{E}$ are as in Theorem 5.8, such that $-E_{Y}$ is
$g$-ample for a $g$-exceptional $\mathbb{Q}$-divisor $E_{Y}\geq 0$. Rescaling
$E_{Y}$, we may assume that $-E_{Y}+g^{*}\tilde{R}_{1}$ is ample. We put
$h=f\circ g\colon Y\to X$. We define $\Delta_{Y}$ by
$K_{Y}+\Delta_{Y}=g^{*}(K_{\tilde{X}}+\tilde{\Delta}-\tilde{E})$. Then, for
every $0<t\ll 1$ we have

$g^{*}\boldsymbol{\rm
M}_{\tilde{X}}-th^{*}A\sim_{\mathbb{Q}}(1-t)g^{*}\boldsymbol{\rm
M}_{\tilde{X}}+t\tilde{R}=(1-t)g^{*}\boldsymbol{\rm
M}_{\tilde{X}}+t(g^{*}\tilde{R}_{1}-E_{Y})+tE_{Y}+tg^{*}\tilde{R}_{2}.$

Since $g^{*}\tilde{R}_{1}-E_{Y}$ is ample and $\boldsymbol{\rm M}_{\tilde{X}}$
is nef, choosing $t>0$ sufficiently small, we can find
$R_{Y}\sim_{\mathbb{Q}}g^{*}\boldsymbol{\rm M}_{\tilde{X}}-h^{*}(tA)$ such
that $R_{Y}$ is the sum of an effective ample $\mathbb{Q}$-divisor and an
effective $\mathbb{Q}$-divisor, $(Y,\Delta_{Y}+R_{Y})$ is log smooth, and
$\lfloor(\Delta_{Y}+R_{Y})\rfloor=\lfloor\Delta_{Y}\rfloor$. Since $R_{Y}$
contains an ample $\mathbb{Q}$-divisor, by perturbation of the coefficients of
$\Delta_{Y}+R_{Y}$, we may assume that no coefficient of $\Delta_{Y}+R_{Y}$ is
one. We put $B_{Y}=\Delta_{Y}+R_{Y}$ and $B=h_{*}B_{Y}$. Then
$K_{Y}+B_{Y}\sim_{\mathbb{Q}}(K_{Y}+\Delta_{Y})+g^{*}\boldsymbol{\rm
M}_{\tilde{X}}-h^{*}(tA)$, and therefore we have

$K_{Y}+B_{Y}\sim_{\mathbb{Q}}g^{*}(K_{\tilde{X}}+\tilde{\Delta}-\tilde{E})+g^{*}\boldsymbol{\rm
M}_{\tilde{X}}-h^{*}(tA)=h^{*}(K_{X}+\Delta+\boldsymbol{\rm M}_{X}-tA).$

From this, we have $K_{Y}+B_{Y}=h^{*}(K_{X}+B)$ and
$K_{X}+\Delta+\boldsymbol{\rm M}_{X}\sim_{\mathbb{Q}}K_{X}+B+tA$. Recalling
that no coefficient of $B_{Y}$ is one, we have

$\lceil(-B_{Y}^{<1})\rceil-\lfloor B_{Y}^{>1}\rfloor=-\lfloor
B_{Y}\rfloor=-\lfloor\Delta_{Y}\rfloor$

and the right hand side is the sum of $-g^{*}\lfloor\tilde{\Delta}\rfloor$ and
an effective $g$-exceptional divisor. Thus
$h_{*}\mathcal{O}_{Y}(\lceil(-B_{Y}^{<1})\rceil-\lfloor
B_{Y}^{>1}\rfloor)=f_{*}\mathcal{O}_{\tilde{X}}(-\lfloor\tilde{\Delta}\rfloor)=\mathcal{O}_{X}(-\lfloor\Delta\rfloor)$.

In this way, we defined an effective $\mathbb{Q}$-divisor $B$ on $X$, a log
resolution $g\colon Y\to X$ of $(X,\Delta+B)$, and an $\mathbb{Q}$-divisor
$B_{Y}$ defined with $K_{Y}+B_{Y}=h^{*}(K_{X}+B)$ such that

  * •

$(K_{X}+\Delta+\boldsymbol{\rm M}_{X})-(K_{X}+B)$ is ample, and

  * •

no coefficient of $B_{Y}$ is one and
$h_{*}\mathcal{O}_{Y}(\lceil(-B_{Y}^{<1})\rceil-\lfloor
B_{Y}^{>1}\rfloor)=\mathcal{O}_{X}(-\lfloor\Delta\rfloor)$.

From now on, we put $L=K_{X}+\Delta+\boldsymbol{\rm M}_{X}$.

######  Step 2.

In this step, we prove the existence of a positive integer $m^{\prime}$,
depending only on $d$ and $p$, such that $\mathcal{O}_{X}(m^{\prime}L)$ is
generated by global sections on a neighborhood of $\lfloor\Delta\rfloor$.

Pick any component $S$ of $\lfloor\Delta\rfloor$. By the induction hypothesis
of Theorem 5.8, there exists a positive integer $m^{\prime\prime}$, depending
only on $d$ and $p$, such that $m^{\prime\prime}L|_{S}$ is base point free. By
the projective case of Lemma 2.16, we see that the base locus of
$m^{\prime\prime}pL$ does not intersect $S$. Note that $p\Delta$ is a Weil
divisor because $p\boldsymbol{\rm M}$ is b-Cartier and $pL$ is Cartier. Since
$S$ is any component of $\lfloor\Delta\rfloor$, we see that
$m^{\prime}:=m^{\prime\prime}p$ is the desired positive integer.

######  Step 3.

We recall that $\pi\colon X\to Z$ is the contraction induced by $L$. In this
step, we prove that $m^{\prime}L\sim\pi^{*}D_{Z}$ for some ample Cartier
divisor $D_{Z}$ on $Z$.

By Step 1, we get a quasi-log scheme $(X,K_{X}+B,h\colon(Y,B_{Y})\to X)$.
Since $L$ is nef and $L-(K_{X}+B)$ is ample, $m^{\prime}L-(K_{X}+B)$ is ample.
By Step 2, it follows that $\mathcal{O}_{X}(m^{\prime}L)$ is generated by
global sections on a neighborhood of $\lfloor\Delta\rfloor={\rm
Nqlc}(X,K_{X}+B)$. In particular, $\mathcal{O}_{{\rm
Nqlc}(X,K_{X}+B)}(m^{\prime}L|_{{\rm Nqlc}(X,K_{X}+B)})$ is $\pi$-generated by
global sections. By applying Theorem 5.7 to $[X,K_{X}+B]$, $X\to S$, and
$m^{\prime}L$, we can find a positive integer $m_{0}$ such that both
$m_{0}m^{\prime}L$ and $(m_{0}+1)m^{\prime}L$ are the pullbacks of Cartier
divisors on $Z$. Therefore, $m^{\prime}L\sim\pi^{*}D_{Z}$ for some Cartier
divisor $D_{Z}$ on $Z$. Then $D_{Z}$ is ample because $\pi\colon X\to Z$ is
the contraction induced by $L$.

######  Step 4.

From this step, we prove the existence of $m$ as in Theorem 5.8. In this step,
we treat the case when there is a component $S^{\prime}$ of
$\lfloor\Delta\rfloor$ dominating $Z$.

By Lemma 2.16, the restriction $\pi_{S^{\prime}}:=\pi|_{S^{\prime}}\colon
S^{\prime}\to Z$ is a contraction. By Step 3, we have
$\pi_{S^{\prime}*}\mathcal{O}_{S^{\prime}}(m^{\prime}L|_{S^{\prime}})\simeq\mathcal{O}_{Z}(D_{Z})$.
By Step 2, it follows that $m^{\prime}L|_{S^{\prime}}$ is base point free.
Then $D_{Z}$ is base point free, and therefore $m^{\prime}L$ is base point
free.

######  Step 5.

Finally, we treat the case when $\lfloor\Delta\rfloor$ is vertical over $Z$.

We put $L^{\prime}=m^{\prime}L$. For each positive integer $l$, let
$Z_{l}\subset Z$ be the base locus of $D_{Z}$. By the relation
$L^{\prime}\sim\pi^{*}D_{Z}$, we see that $\pi^{-1}(Z_{l})$ is disjoint from
$\lfloor\Delta\rfloor$ for all $l$.

In the rest of the proof, we closely follow [14, Proof of Theorem 1.1]. We
recall the property that the non-klt locus of $(X,B)$ and the non-lc locus of
$(X,B)$ are the same and they coincide with $\lfloor
B\rfloor=\lfloor\Delta\rfloor$ (see Step 1). We put
$A^{\prime}_{l}=\frac{3}{2}lL^{\prime}-(K_{X}+B)$ for each positive integer
$l$. By Step 1, it follows that $A^{\prime}_{l}$ is ample for all $l$. Fix an
arbitrary positive integer $l$, and suppose $Z_{l}\neq\emptyset$. Let
$C_{0},C_{1},\cdots,C_{d}$ be general members of $|lD_{Z}|$, and we put
$C=\pi^{*}(\frac{1}{2}C_{0}+C_{1}+\cdots+C_{d})$. Then
$C\sim_{\mathbb{Q}}(d+\frac{1}{2})l\pi^{*}D_{Z}$, and the non-lc locus of
$(X,B+C)$ is the disjoint union of $\pi^{-1}(Z_{l})$ and $\lfloor B\rfloor$.
Pick an arbitrary irreducible component $V$ of $\pi^{-1}(Z_{l})$ such that
$\pi(V)$ has the maximal dimension. Then $V$ is disjoint from $\lfloor
B\rfloor$, and $\pi(V)\subsetneq Z$. Let $\widetilde{h}\colon\widetilde{Y}\to
X$ be a log resolution of $(X,B+C)$ such that ${\rm
Supp}\widetilde{h}^{-1}(\pi^{-1}(\pi(V))$ is an snc divisor. We may write

$K_{\widetilde{Y}}=\widetilde{h}^{*}(K_{X}+B)+\sum_{i}e_{i}\widetilde{E}_{i}$

for some $e_{i}\in\mathbb{Q}$. Let $\gamma$ be the log canonical threshold of
$C$ with respect to $(X,B)$ over the generic point of $\pi(V)$. It follows
that $c\in(0,1)\cap\mathbb{Q}$ because $(X,B)$ is klt on $X\setminus\lfloor
B\rfloor$ and $\pi(V)$ is disjoint from $\pi(\lfloor B\rfloor)$. We may write
$\gamma\widetilde{h}^{*}C=\sum_{i}\gamma_{i}\widetilde{E}_{i}$ for some
$\gamma_{i}\in\mathbb{Q}_{\geq 0}$. Then we have

$\begin{split}\widetilde{h}^{*}\pi^{*}((d+2)lD_{Z})\sim_{\mathbb{Q}}&\frac{3}{2}\widetilde{h}^{*}(lL^{\prime})+\widetilde{h}^{*}\pi^{*}((d+\frac{1}{2})lD_{Z})\\\
\sim_{\mathbb{Q}}&\widetilde{h}^{*}\bigl{(}K_{X}+B+A^{\prime}_{l}+\gamma\widetilde{h}^{*}C+(1-\gamma)\widetilde{h}^{*}C\bigr{)}\\\
\sim_{\mathbb{Q}}&K_{\widetilde{Y}}+\sum_{i}(\gamma_{i}-e_{i})\widetilde{E}_{i}+\widetilde{h}^{*}A^{\prime}_{l}+(1-\gamma)\widetilde{h}^{*}C.\end{split}$

We can write

$\sum_{i}\lfloor(\gamma_{i}-e_{i})\rfloor\widetilde{E}_{i}=\widetilde{F}+\widetilde{G}_{1}+\widetilde{G}_{2}-\widetilde{H}$

where $\widetilde{F}$, $\widetilde{G}_{1}$, $\widetilde{G}_{2}$, and
$\widetilde{H}$ are all effective and they have no common components each
other such that

  * •

the image of every component of $\widetilde{F}$ by $\pi\circ\widetilde{h}$ is
$\pi(V)$,

  * •

the image of every component of $\widetilde{G}_{1}$ by $\pi\circ\widetilde{h}$
does not contain $\pi(V)$,

  * •

the image of every component of $\widetilde{G}_{2}$ by $\pi\circ\widetilde{h}$
contains $\pi(V)$ but does not coincide with $\pi(V)$, and

  * •

$\widetilde{H}$ is $\widetilde{h}$-exceptional.

Note that $\widetilde{h}^{-1}_{*}\lfloor B\rfloor\subset{\rm
Supp}\widetilde{G}_{1}$, $\widetilde{F}\neq 0$ and $\widetilde{F}$ is reduced
by the definition of $\gamma$, and $\widetilde{G}_{2}$ is reduced because
every irreducible component of $\widetilde{h}(\widetilde{G}_{2})$ intersects
$X\setminus(\lfloor B\rfloor\cup\pi^{-1}(\pi(V)))$. By taking blow-ups along
lc centers of $(\widetilde{Y},\widetilde{G}_{2})$ if necessary, we may assume
that any lc center $T$ of $(\widetilde{Y},\widetilde{G}_{2})$ does not satisfy
$(\pi\circ\widetilde{h})(T)\subset\pi(V)$.

For each $j\in\mathbb{Z}_{\geq 0}$, we define
$\widetilde{N}_{j}:=\widetilde{h}^{*}\pi^{*}((j+d+2)lD_{Z})+\widetilde{H}-\widetilde{G}_{1}$,
and consider the exact sequence

$0\longrightarrow\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j}-\widetilde{F})\longrightarrow\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j})\longrightarrow\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}})\longrightarrow
0.$

By construction, we have

($\spadesuit$)
$\widetilde{N}_{j}-\widetilde{F}\sim_{\mathbb{Q}}K_{\widetilde{Y}}+\sum_{i}\\{\gamma_{i}-e_{i}\\}\widetilde{E}_{i}+\widetilde{G}_{2}+\widetilde{h}^{*}A^{\prime}_{l}+(1-\gamma)\widetilde{h}^{*}C+jl\widetilde{h}^{*}\pi^{*}D_{Z},$

and no lc center $T$ of the lc pair
$\bigl{(}\widetilde{Y},\sum_{i}\\{\gamma_{i}-e_{i}\\}\widetilde{E}_{i}+\widetilde{G}_{2}\bigr{)}$
satisfies $(\pi\circ\widetilde{h})(T)\subset\pi(V)$. Therefore, by [15,
Theorem 6.3 (i)], the connecting morphism

$(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}})\longrightarrow
R^{1}(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j}-\widetilde{F})$

is a zero map. Thus, we get the following exact sequence

$0\longrightarrow(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j}-\widetilde{F})\longrightarrow(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j})\longrightarrow(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}})\longrightarrow
0.$

Since $A^{\prime}_{l}$ is ample, there is
$\tilde{A}\sim_{\mathbb{Q}}\tilde{h}^{*}A^{\prime}_{l}$ such that
$\bigl{(}\tilde{Y},\sum_{i}\\{\gamma_{i}-e_{i}\\}\widetilde{E}_{i}+\widetilde{G}_{2}+\tilde{A}\bigr{)}$
is a log smooth lc pair. Because
$(1-\gamma)\widetilde{h}^{*}C+jl\widetilde{h}^{*}\pi^{*}D_{Z}$ is the pullback
of an ample $\mathbb{Q}$-divisor on $Z$, we have
$H^{1}(Z,(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j}-\widetilde{F}))=0$
by ($\spadesuit$ ‣ 5) and [15, Theorem 6.3 (ii)]. From this, we get a
surjective morphism

($\clubsuit$)
$H^{0}(\widetilde{Y},\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j}))\longrightarrow
H^{0}(\widetilde{F},\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}})).$

From the definitions of $\gamma$ and $\widetilde{F}$, it follows that no
component of $\widetilde{F}$ is a component of
$\sum_{i}\\{\gamma_{i}-e_{i}\\}\widetilde{E}_{i}$. By ($\spadesuit$ ‣ 5) and
construction of $\tilde{A}$, we have

$\widetilde{N}_{j}|_{\widetilde{F}}\sim_{\mathbb{Q}}K_{\tilde{F}}+\Bigl{(}\sum_{i}\\{\gamma_{i}-e_{i}\\}\widetilde{E}_{i}+\widetilde{G}_{2}+\tilde{A}\Bigr{)}\Big{|}_{\tilde{F}}+\Bigl{(}(1-\gamma)\widetilde{h}^{*}C+jl\widetilde{h}^{*}\pi^{*}D_{Z}\Bigr{)}\Big{|}_{\tilde{F}}$

and
$\bigl{(}\tilde{F},(\sum_{i}\\{\gamma_{i}-e_{i}\\}\widetilde{E}_{i}+\widetilde{G}_{2}+\tilde{A})|_{\tilde{F}}\bigr{)}$
is a simple normal crossing pair. Therefore, the vanishing theorem for simple
normal crossing pairs [14, Theorem 3.2 (b)], we see that
$H^{q}(\pi(V),(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}}))=0$
for all $q>0$. Thus, we have

$\begin{split}{\rm
dim}H^{0}(\widetilde{F},\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}}))=&{\rm
dim}H^{0}\bigl{(}\pi(V),(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}})\bigr{)}\\\
=&\chi\bigl{(}\pi(V),(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}})\bigr{)}.\end{split}$

By definition of $\widetilde{N}_{j}$, we have

$\begin{split}(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}})=(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(H|_{\widetilde{F}}-\widetilde{G}_{1}|_{\widetilde{F}})\otimes\mathcal{O}_{\pi(V)}\bigl{(}(j+d+2)D_{Z}|_{\pi(V)}\bigr{)},\end{split}$

and
$(\pi\circ\widetilde{h})_{*}\mathcal{O}_{\widetilde{F}}(H|_{\widetilde{F}}-\widetilde{G}_{1}|_{\widetilde{F}})$
is a non-zero sheaf because $\widetilde{G}_{1}|_{\widetilde{F}}$ is zero at
the generic point of $\pi(V)$ (see the definition of $\widetilde{G}_{1}$).
Therefore, ${\rm
dim}H^{0}(\widetilde{F},\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j}|_{\widetilde{F}}))$
is a non-zero polynomial of degree at most $d$ with one variable $j$. Then
there is $0\leq j^{\prime}\leq d$ such that
$H^{0}(\widetilde{F},\mathcal{O}_{\widetilde{F}}(\widetilde{N}_{j^{\prime}}|_{\widetilde{F}}))\neq
0$, so
$H^{0}(\widetilde{Y},\mathcal{O}_{\widetilde{Y}}(\widetilde{N}_{j^{\prime}}))$
has an element not vanishing on $\widetilde{F}$ by ($\clubsuit$ ‣ 5). Since
$\widetilde{N}_{j^{\prime}}=\widetilde{h}^{*}\pi^{*}((j^{\prime}+d+2)lD_{Z})+\widetilde{H}-\widetilde{G}_{1}$
and ${\rm Supp}\widetilde{G}_{1}$ does not contain $\widetilde{F}$, we see
that
$H^{0}(\widetilde{Y},\mathcal{O}_{\widetilde{Y}}(\widetilde{h}^{*}\pi^{*}((j^{\prime}+d+2)lD_{Z})+\widetilde{H}))$
has an element not vanishing on $\widetilde{F}$. Since $\widetilde{H}$ is
$\widetilde{h}$-exceptional,
$H^{0}(X,\mathcal{O}_{X}(\pi^{*}((j^{\prime}+d+2)lD_{Z})))$ has an element not
vanishing on $\widetilde{h}(\widetilde{F})$. This implies that the base locus
of $(j^{\prime}+d+2)lD_{Z}$ does not contain $\pi(V)$.

Recall that $V$ is an arbitrary irreducible component of $\pi^{-1}(Z_{l})$
such that $\pi(V)$ has the maximal dimension. From the above argument, there
is $0\leq j^{\prime}\leq d$ such that

${\rm dim}Z_{(j^{\prime}+d+2)l}<{\rm dim}Z_{l}.$

In particular, we have ${\rm dim}Z_{(2d+2)!\cdot l}<{\rm dim}Z_{l}$. The
relation holds true for all $l\in\mathbb{Z}_{>0}$. Therefore, $mL$ is base
point free where $m$ is defined to be $((2d+2)!)^{d}\cdot m^{\prime}$.

We finish the proof of Theorem 5.8. ∎

######  Remark 5.9.

By arguing inductively, for fixed integers $d$ and $p$ and every generalized
dlt pair $(X,\Delta,\boldsymbol{\rm M})$ as in Theorem 5.8, we can check that
$((2d+2)!)^{d^{2}}p(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is base point free.
However, compared to [14, Remark 1.2], it looks far from optimal.

We prepare the following lemma for the proof of Theorem 1.7.

######  Lemma 5.10.

Let $d$ and $p$ be positive integers, and let $v$ be a positive real number.
Then there exists a positive integer $m$, depending only on $d$, $p$, and $v$,
satisfying the following. Let $(X,\Delta,\boldsymbol{\rm M})$ be a generalized
lc pair such that

  * •

${\rm dim}X=d$,

  * •

$p\boldsymbol{\rm M}$ is b-Cartier,

  * •

$p(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is an ample and base point free
Cartier divisor, and

  * •

${\rm vol}(X,K_{X}+\Delta+\boldsymbol{\rm M}_{X})\leq v$.

Then $m(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is very ample, and putting
$R_{l}=H^{0}(X,\mathcal{O}_{X}(\lfloor l(K_{X}+\Delta+\boldsymbol{\rm
M}_{X})\rfloor))$ for every $l\in\mathbb{Z}_{\geq 0}$ then
$\bigoplus_{l\in\mathbb{Z}_{\geq 0}}R_{lm}$ is generated by $R_{m}$ as a
graded $\mathbb{C}$-algebra.

###### Proof.

If Lemma 5.10 does not hold, then there is a sequence of generalized lc pairs
$\\{X_{i},\Delta_{i},\boldsymbol{\rm M}^{i}\\}_{i\geq 1}$ as in Lemma 5.10
such that if $m_{i}$ is the smallest positive integer such that
$m_{i}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}^{i}_{X_{i}})$ is very ample and
$H^{0}(X_{i},\mathcal{O}_{X_{i}}(m_{i}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm
M}^{i}_{X_{i}})))$ generates $\bigoplus_{l\in\mathbb{Z}_{\geq
0}}H^{0}(X_{i},\mathcal{O}_{X_{i}}(lm_{i}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm
M}^{i}_{X_{i}})))$, then ${\rm lim}_{i\to\infty}m_{i}=\infty$.

Since $p\boldsymbol{\rm M}^{i}$ is b-Cartier and
$p(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}^{i}_{X_{i}})$ is Cartier,
$p\Delta_{i}$ is a Weil divisor for every $i$. By the effective birationality
for generalized pairs [7, Theorem 1.3], there exists a positive integer
$p^{\prime}$, depending only on $d$ and $p$, such that
$|pp^{\prime}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}^{i}_{X_{i}})|$ defines a
birational morphism. Then we may find
$G_{i}\in|pp^{\prime}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}^{i}_{X_{i}})|$.
Note that $G_{i}$ are Cartier and base point free, ${\rm vol}(X_{i},G_{i})\leq
p^{d}p^{\prime d}v$, and $G_{i}-(K_{X_{i}}+\Delta_{i})$ are the pushdowns of
pseudo-effective $\mathbb{Q}$-divisors. By [4, Proposition 4.4] and replacing
$\\{X_{i},\Delta_{i},\boldsymbol{\rm M}^{i}\\}_{i\geq 1}$ with a subsequence,
we may find a projective morphism $h\colon V\to T$ of varieties, a reduced
divisor $D$ on $V$, and a positive real number $c$, depending only on $d$,
$p$, and $v$, such that for every $i$, there is a closed point $t_{i}\in T$
with the fibers $V_{t_{i}}$ and $D_{t_{i}}$ satisfying the following.

  * •

$(V_{t_{i}},D_{t_{i}})$ is log smooth,

  * •

there is a birational map $\phi_{i}\colon X_{i}\dashrightarrow V_{t_{i}}$ such
that if $E_{i}$ is the sum of $\phi_{i*}G$ and the reduced
$\phi^{-1}$-exceptional divisor, then ${\rm Supp}E_{i}\subset D_{t_{i}}$, and

  * •

if $f_{i}\colon Y\to X_{i}$ and $f^{\prime}_{i}\colon Y\to V_{t_{i}}$ is a
common resolution of $\phi_{i}$, then
$f_{i}^{*}G_{i}\sim_{\mathbb{Q},V_{t_{i}}}0$ and all the coefficients of
$f^{\prime}_{i*}f_{i}^{*}G_{i}$ are at most $c$.

By shrinking $T$ and replacing $\\{X_{i},\Delta_{i},\boldsymbol{\rm
M}^{i}\\}_{i\geq 1}$ with a subsequence if necessary, we may assume that the
set $\\{t_{i}\\}_{i\geq 1}$ is dense in $T$. We put
$D^{\prime}_{i}=f^{\prime}_{i*}f_{i}^{*}G_{i}$. By the third property,
$f_{i}^{*}G_{i}=f^{\prime*}_{i}D^{\prime}_{i}$ for all $i$. Since $G_{i}$ is
ample, the birational map $\varphi_{i}:=\phi_{i}^{-1}\colon
V_{t_{i}}\dashrightarrow X_{i}$ is a morphism for all $i$. Taking a log
resolution of $(V,D)$ and shrinking $T$, we may assume that the morphism
$h\colon(V,D)\to T$ is log smooth and $T$ is affine. Moreover, taking an
appropriate étale base change of $h$, we may assume that $D|_{V_{t}}$ is
irreducible for every closed point $t\in T$ with the fiber $V_{t}$.

Since $\varphi^{*}_{i}G_{i}$ is an effective Cartier divisor on $V_{t_{i}}$
such that ${\rm Supp}\varphi^{*}_{i}G_{i}\subset{\rm Supp}D|_{V_{t_{i}}}$ and
all the coefficients of $\varphi^{*}_{i}G_{i}$ is at most $c$, the
coefficients of $\varphi^{*}_{i}G_{i}$ belong to a finite set depending only
on $d$, $p$, and $v$. Therefore, replacing
$\\{X_{i},\Delta_{i},\boldsymbol{\rm M}^{i}\\}_{i\geq 1}$ with a subsequence,
we may find a Cartier divisor $\tilde{D}$ on $V$ such that for every $i$, we
have $\tilde{D}|_{V_{t_{i}}}=\varphi^{*}_{i}G_{i}$.

Consider the morphsim
$h^{*}h_{*}\mathcal{O}_{V}(\tilde{D})\to\mathcal{O}_{V}(\tilde{D}).$ Then
there is an open set $U\subset T$ such that for any $t^{\prime}\in U$, the
morphism $h_{*}\mathcal{O}_{V}(\tilde{D})\otimes k(t^{\prime})\to
H^{0}(V_{t^{\prime}},\mathcal{O}_{V_{t^{\prime}}}(\tilde{D}|_{V_{t^{\prime}}})$
is an isomorphism, where $k(t^{\prime})$ is the quotient field of
$\mathcal{O}_{T}$ by the maximal ideal corresponding to $t^{\prime}$ (see, for
example, [27, III, Theorem 12.8] and [27, III, Corollary 12.9]). Then the
morphism $h^{*}h_{*}\mathcal{O}_{V}(\tilde{D})\otimes
k(t^{\prime})\to\mathcal{O}_{V}(\tilde{D})\otimes k(t^{\prime})$ is surjective
for all $t^{\prime}\in U\cap\\{t_{i}\\}_{i\geq 1}$. Since $\\{t_{i}\\}_{i\geq
1}$ is dense in $T$, we see that $U\cap\\{t_{i}\\}_{i\geq 1}$ is not empty.
From this fact, shrinking $T$ and replacing
$\\{X_{i},\Delta_{i},\boldsymbol{\rm M}^{i}\\}_{i\geq 1}$ with a subsequence,
we may assume that $\tilde{D}$ is base point free over $T$.

Let $\tau\colon V\to W$ be the contraction over $T$ induced by $\tilde{D}$.
Then there is a Cartier divisor $H$, which is ample over $T$, such that
$\tilde{D}\sim\tau^{*}H$. By shrinking $T$ and replacing
$\\{X_{i},\Delta_{i},\boldsymbol{\rm M}^{i}\\}_{i\geq 1}$ with a subsequence,
we may assume that for every closed point $t\in T$ the fiber $W_{t}$ is
normal. We consider the morphisms $\tau_{i}:=\tau|_{V_{t_{i}}}\colon
V_{t_{i}}\to W_{t_{i}}$ and $\varphi_{i}\colon V_{t_{i}}\to X_{i}$ for each
$i$. Since
$\tau^{*}_{i}H|_{W_{t_{i}}}\sim\tilde{D}|_{V_{t_{i}}}=\varphi^{*}_{i}G_{i}$
and ${\rm dim}W_{t_{i}}\leq{\rm dim}V_{t_{i}}={\rm dim}X_{i}$, we see that
${\rm dim}W_{t_{i}}={\rm dim}X_{i}$. This shows that $\tau_{i}\colon
V_{t_{i}}\to W_{t_{i}}$ is birational, hence $\tau_{i}$ is a contraction.
Since $\tau^{*}_{i}H|_{W_{t_{i}}}\sim\varphi^{*}_{i}G_{i}$ and both
$H|_{W_{t_{i}}}$ and $G_{i}$ are ample, we see that $W_{t_{i}}\simeq X_{i}$.

By this discussion, we get a projective morphism $W\to T$ from a normal
varierty $W$ to an affine variety $T$ and an ample Cartier divisor $H$ such
that for every $i$, there is a closed point $t_{i}\in T$ and an isomorphism
$\psi_{i}\colon W_{t_{i}}\to X_{i}$ such that $\psi^{*}_{i}G_{i}\sim
H|_{W_{t_{i}}}$. By shrinking $T$ and replacing
$\\{X_{i},\Delta_{i},\boldsymbol{\rm M}^{i}\\}_{i\geq 1}$ with a subsequence,
we may assume that the morphism $W\to T$ is flat. Let $m$ be a positive real
number such that $mH$ is very ample over $T$, $H^{0}(W,\mathcal{O}_{W}(mH))$
generates $\bigoplus_{l\in\mathbb{Z}_{\geq 0}}H^{0}(W,\mathcal{O}_{W}(lmH))$,
and $H^{q}(W,\mathcal{O}_{W}(lH))=0$ for all $l\geq m$ and $q>0$. Then the
morphism

$H^{0}(W,\mathcal{O}_{W}(lmH))\otimes k(t)\to
H^{0}(W_{t},\mathcal{O}_{W_{t}}(lmH|_{W_{t}}))$

is surjective for every closed point $t\in T$. Therefore, for every closed
point $t\in T$ and every $l\in\mathbb{Z}_{\geq 0}$, we see that

$H^{0}(W_{t},\mathcal{O}_{W_{t}}(mH|_{W_{t}}))\otimes
H^{0}(W_{t},\mathcal{O}_{W_{t}}(lmH|_{W_{t}}))\longrightarrow
H^{0}(W_{t},\mathcal{O}_{W_{t}}((1+l)mH|_{W_{t}}))$

is surjective. In particular, $mG_{i}$ is very ample and
$H^{0}(X_{i},\mathcal{O}_{X_{i}}(mG_{i}))$ generates the graded ring
$\bigoplus_{l\in\mathbb{Z}_{\geq 0}}H^{0}(X_{i},\mathcal{O}_{X_{i}}(mlG_{i}))$
for every $i$. This is a contradiction since we have $mG_{i}\sim
mpp^{\prime}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}^{i}_{X_{i}})$ for all
$i$. Thus, Lemma 5.10 holds. ∎

We are ready to prove Theorem 1.7.

###### Proof of Theorem 1.7.

We will freely use the notations as in Theorem 1.7, and we put
$\Delta^{\prime}=\phi_{*}\Delta$ and $M^{\prime}=\phi_{*}M$. Since $M$ is nef
and log big with respect to $(X,\Delta)$, we may apply Theorem 5.8 to the
generalized dlt pair $(X^{\prime},\Delta^{\prime},\overline{M})$. By Theorem
5.8, there exists $m^{\prime}$, depending only on $d$ and $p$, such that
$m^{\prime}(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})$ is base point free.
Let $\pi\colon X\to Z$ be the Stein factorization of the morphism defined with
$|m^{\prime}(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})|$, where $Z$ is
nothing but ${\boldsymbol{\rm Proj}}\bigoplus_{l\in\mathbb{Z}_{\geq 0}}R_{l}$
in Theorem 1.7. Then there is an ample and base point free Cartier divisor
$H_{Z}$ on $Z$ such that
$m^{\prime}(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})\sim\pi^{*}H_{Z}$.
Because $pM^{\prime}$ and $p(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})$ are
Weil divisors, $p\Delta^{\prime}$ is a Weil divisor. Thus, we may apply
Theorem 4.5 to $\pi\colon(X^{\prime},\Delta^{\prime},\overline{M})\to Z$.
There exist positive integers $n$ and $q$, depending only on $d$ and $p$, such
that $Z$ has the structure of a generalized lc pair
$(Z,\Delta_{Z},\boldsymbol{\rm N})$ such that

  * •

$n(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})\sim
n(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})$, and

  * •

$q\boldsymbol{\rm N}$ is b-Cartier.

We have $nm^{\prime}\pi^{*}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})\sim
nm^{\prime}(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})\sim n\pi^{*}H_{Z}$ by
the first property. By [10, Proposition 5.3], we see that
$nm^{\prime}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})$ is Cartier and we have
$nm^{\prime}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})\sim nH_{Z}$. In
particular, $nm^{\prime}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})$ is ample
and base point free. Since we have ${\rm
vol}(Z,K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})={\rm Ivol}(K_{X}+\Delta+M)\leq
v$, we may apply Lemma 5.10 to $(Z,\Delta_{Z},\boldsymbol{\rm N})$. There is a
positive integer $m^{\prime\prime}$, depending only on $d$, $nm^{\prime}$, and
$v$, such that $m^{\prime\prime}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})$ is
very ample and
$H^{0}(Z,\mathcal{O}_{Z}(m^{\prime\prime}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm
N}_{Z})))$ generates $\bigoplus_{l\in\mathbb{Z}_{\geq
0}}H^{0}(Z,\mathcal{O}_{Z}(lm^{\prime\prime}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm
N}_{Z})))$.

We define $m=nm^{\prime}m^{\prime\prime}$. By construction, it is easy to
check that $m$ depends only on $d$, $p$, and $v$,
$m(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})$ is very ample, and
$H^{0}(Z,\mathcal{O}_{Z}(m(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})))$
generates $\bigoplus_{l\in\mathbb{Z}_{\geq
0}}H^{0}(Z,\mathcal{O}_{Z}(lm(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z})))$ as a
graded $\mathbb{C}$-algebra. Now we have

$\begin{split}H^{0}(X,\mathcal{O}_{X}(\lfloor
lm(K_{X}+\Delta+M)\rfloor))&\simeq
H^{0}(X,\mathcal{O}_{X^{\prime}}(lm(K_{X^{\prime}}+\Delta^{\prime}+M^{\prime})))\\\
&\simeq H^{0}(Z,\mathcal{O}_{Z}(lm(K_{Z}+\Delta_{Z}+\boldsymbol{\rm
N}_{Z})))\end{split}$

for every positive integer $l$, so $\bigoplus_{l\in\mathbb{Z}_{\geq 0}}R_{lm}$
is generated by $R_{m}$ as a graded $\mathbb{C}$-algebra, where
$R_{lm}=H^{0}(X,\mathcal{O}_{X}(\lfloor lm(K_{X}+\Delta+M)\rfloor))$.
Furthermore, the inequality

${\rm vol}(Z,m(K_{Z}+\Delta_{Z}+\boldsymbol{\rm N}_{Z}))\leq m^{d}v$

shows that $Z={\boldsymbol{\rm Proj}}\bigoplus_{l\in\mathbb{Z}_{\geq 0}}R_{l}$
belongs to a bounded family $\mathfrak{F}$ that depends only on $d$, $p$, and
$v$. We finish the proof. ∎

Finally, we introduce an application of Theorem 5.8.

######  Theorem 5.11.

Let $d$ and $p$ be positive integers. Then there exists a positive integer
$m$, depending only on $d$ and $p$, satisfying the following. Let
$(X,\Delta,\boldsymbol{\rm M})$ be a generalized dlt pair such that

  * •

${\rm dim}X=d$,

  * •

$p\boldsymbol{\rm M}$ is b-Cartier and $p(K_{X}+\Delta+\boldsymbol{\rm
M}_{X})$ is nef and Cartier, and

  * •

$K_{X}+\Delta+\boldsymbol{\rm M}_{X}$ is nef and log big with respect to
$(X,\Delta,\boldsymbol{\rm M})$.

Then $m(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is base point free.

###### Proof.

By Theorem 6.1, there is a log resolution $f\colon\tilde{X}\to X$ such that
$\boldsymbol{\rm M}$ descends to $\tilde{X}$ and $f$ is an isomorphism over an
open subset $U$ containing all the generic points of generalized lc centers of
$(X,\Delta,\boldsymbol{\rm M})$. Set $D=K_{X}+\Delta+\boldsymbol{\rm M}_{X}$
and consider the generalized dlt pair $(X,\Delta,\boldsymbol{\rm
M}+\overline{D})$. We can write

$K_{\tilde{X}}+\tilde{\Delta}+\boldsymbol{\rm
M}_{\tilde{X}}+f^{*}D=(K_{X}+\Delta+\boldsymbol{\rm M}_{X}+D)+E$

with $\tilde{\Delta}\geq 0$ and $\tilde{E}\geq 0$ having no common components.
Since $D$ is nef and log big with respect to $(X,\Delta,\boldsymbol{\rm M})$
and $f$ is an isomorphism over the genetic points of all generalized lc
centers of $(X,\Delta,\boldsymbol{\rm M})$, we see that $\boldsymbol{\rm
M}_{\tilde{X}}+f^{*}D$ is nef and log big with respect to
$(\tilde{X},\tilde{\Delta})$. Thus we may apply Theorem 5.8. There exists
$m^{\prime}$, depending only on $d$ and $p$, such that
$m^{\prime}(K_{X}+\Delta+\boldsymbol{\rm M}_{X}+D)$ is base point free. Then
$2m^{\prime}(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is base point free. Hence
$m:=2m^{\prime}$ is the desired positive integer. ∎

######  Corollary 5.12.

Let $d$ and $p$ be positive integers, and let $v$ be a positive real number.
Then there exists a positive integer $m$, depending only on $d$, $p$, and $v$,
satisfying the following. Let $(X,\Delta,\boldsymbol{\rm M})$ be a generalized
dlt pair such that

  * •

${\rm dim}X=d$,

  * •

$p\boldsymbol{\rm M}$ is b-Cartier and $p(K_{X}+\Delta+\boldsymbol{\rm
M}_{X})$ is Cartier,

  * •

$p(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is ample, and

  * •

${\rm vol}(X,K_{X}+\Delta+\boldsymbol{\rm M}_{X})\leq v$.

Then $m(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is very ample. In particular,
$(X,\Delta,\boldsymbol{\rm M})$ belongs to a bounded family $\mathfrak{F}$
depending only on $d$, $p$, and $v$ in the sense of Birkar [5].

###### Proof.

By Theorem 5.11, there exists a positive integer $n$, depending only on $d$
and $p$, such that $n(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$ is base point free
for every generalized dlt pair $(X,\Delta,\boldsymbol{\rm M})$ as in Corollary
5.12. Then Corollary 5.12 follows from Lemma 5.10. ∎

##  6\. Appendix: On definition of generalized dlt pairs

The goal of this appendix is to prove that the definition of generalized dlt
pairs by Birkar [4] and that by Han–Li [25] are equivalent when the nef part
is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors.

######  Theorem 6.1.

The following three conditions are equivalent for every generalized lc pair
$(X,\Delta,\boldsymbol{\rm M})/Z$ such that $\boldsymbol{\rm M}$ is a finite
$\mathbb{R}_{>0}$-linear combination of b-nef$/Z$ $\mathbb{Q}$-b-Cartier
$\mathbb{Q}$-b-divisors:

  1. (1)

For any generic point $\eta$ of any generalized lc center of
$(X,\Delta,\boldsymbol{\rm M})/Z$, $(X,\Delta)$ is log smooth near $\eta$ and
$\boldsymbol{\rm M}$ descends to $\boldsymbol{\rm M}_{X}$ on a neighborhood of
$\eta$ ([4, 2.13]).

  2. (2)

There is an open subset $U\subset X$ such that $(U,\Delta|_{U})$ is log
smooth, $U$ contains the generic point of any generalized lc center of
$(X,\Delta,\boldsymbol{\rm M})/Z$, and the generic point of any generalized lc
center of $(X,\Delta,\boldsymbol{\rm M})/Z$ is the generic point of an lc
center of $(U,\Delta|_{U})$ ([25, Definition 2.2]).

  3. (3)

There is a log resolution $f\colon\tilde{X}\to X$ of $(X,\Delta)$ and an open
subset $V\subset X$ such that $\boldsymbol{\rm M}$ descends to $\tilde{X}$,
$f$ is an isomorphism over $V$, and $V$ contains the generic point of any
generalized lc center of $(X,\Delta,\boldsymbol{\rm M})/Z$.

We note that we do not need to assume that $X$ or $Z$ is quasi-projective.

The following is the key ingredient for the proof.

######  Lemma 6.2.

Let $(X,\Delta,\boldsymbol{\rm M})/Z$ be a generalized klt pair such that
$\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier and $\boldsymbol{\rm M}$ is a
finite $\mathbb{R}_{>0}$-linear combination of b-nef$/Z$
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors. Then there is a projective
birational morphism $g\colon Y\to X$ such that $\boldsymbol{\rm M}$ descends
to $Y$ and the divisor $F:=g^{*}\boldsymbol{\rm M}_{X}-\boldsymbol{\rm M}_{Y}$
satisfies ${\rm Supp}F={\rm Ex}(g)$.

###### Proof.

Since $\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier, $(X,\Delta)$ is a klt
pair. Let $f\colon\bar{X}\to X$ be a log resolution of $(X,\Delta)$ such that
$\boldsymbol{\rm M}$ descends to $\bar{X}$. We can write

$K_{\bar{X}}+\bar{\Delta}=f^{*}(K_{X}+\Delta)+\bar{E}$

with $\bar{\Delta}\geq 0$ and $\bar{E}\geq 0$ with no common components. We
may write $\boldsymbol{\rm M}_{\bar{X}}=\sum_{i}\mu_{i}M_{i}$ where
$\mu_{i}\in\mathbb{R}_{>0}$ and $M_{i}$ are Cartier divisors that are nef over
$Z$. We fix $\alpha$ such that $\alpha\mu_{i}>2\cdot{\rm dim}X$ for all $i$.

We fix an open affine covering $\\{X_{j}\\}_{j}$ of $X$. In particular, all
$X_{j}$ are quasi-projective. We put $\bar{X}_{j}=f^{-1}(X_{j})$,
$\bar{\Delta}_{j}=\bar{\Delta}|_{\bar{X}_{j}}$, and $\boldsymbol{\rm
M}^{j}=\boldsymbol{\rm M}|_{X_{j}}$. In the rest of the proof, for any
projective birational morphism $W\to X$ and any open subset $V\subset W$, we
will denote $\boldsymbol{\rm M}_{W}|_{V}$ by $\boldsymbol{\rm M}_{V}$.

Since $(\bar{X}_{j},\bar{\Delta}_{j},\alpha\boldsymbol{\rm M}^{j})/X_{j}$ is
generalized klt, by applying [6] we get a sequence of steps of a
$(K_{\bar{X}_{j}}+\bar{\Delta}_{j}+\alpha\boldsymbol{\rm
M}_{\bar{X}_{j}})$-MMP over $X_{j}$ that terminates with a good minimal model
$(\bar{X}^{\prime}_{j},\bar{\Delta}^{\prime}_{j},\alpha\boldsymbol{\rm
M}^{j})/X_{j}$ of $(\bar{X}_{j},\bar{\Delta}_{j},\alpha\boldsymbol{\rm
M}^{j})/X_{j}$. By the length of extremal rays ([15, Section 18] and [17,
Proposition 6.1]), the birational transform of $\boldsymbol{\rm
M}_{\bar{X}_{j}}$ is trivial over the extremal contraction in each step of the
MMP. Therefore, we see that

  * •

$\boldsymbol{\rm M}^{j}$ descends to $\bar{X}^{\prime}_{j}$ (so
$K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+(\alpha+t)\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}$ is nef over $X_{j}$ for all $t\geq 0$), and

  * •

$\bar{X}_{j}\dashrightarrow\bar{X}^{\prime}_{j}$ is a sequence of steps of a
$(K_{\bar{X}_{j}}+\bar{\Delta}_{j}+(\alpha+t)\boldsymbol{\rm
M}_{\bar{X}_{j}})$-MMP over $X_{j}$ to a minimal model
$(\bar{X}^{\prime}_{j},\bar{\Delta}^{\prime}_{j},(\alpha+t)\boldsymbol{\rm
M}^{j})/X_{j}$ for all $t\geq 0$.

Then
$K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+(\alpha+t)\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}$ is semi-ample over $X_{j}$ for all $t\geq 0$ since
$\bar{X}^{\prime}_{j}\to X_{j}$ is birational which shows that all divisors on
$\bar{X}^{\prime}_{j}$ are big over $X_{j}$.

Let $\phi\colon\bar{X}^{\prime}_{j}\to Y_{j}$ be the contraction over $X_{j}$
induced by
$K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+2\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}$, and let $\phi\colon\bar{X}^{\prime}_{j}\to
Y^{\prime}_{j}$ be the contraction over $X_{j}$ induced by
$K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}$. Pick any curve $\xi\subset\bar{X}^{\prime}_{j}$
that is contracted by $\bar{X}^{\prime}_{j}\to X_{j}$. Since
$K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}$ and $\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}$ are nef over $X_{j}$, if $\xi$ is contracted by
$\phi\colon\bar{X}^{\prime}_{j}\to Y_{j}$ then

$0=\xi\cdot(K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+2\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}})\geq\xi\cdot(K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}})\geq 0.$

Thus, if $\xi$ is contracted by $\phi\colon\bar{X}^{\prime}_{j}\to Y_{j}$ then
$\xi$ is contracted by $\bar{X}^{\prime}_{j}\to Y^{\prime}_{j}$. In
particular, the induced birational map $Y_{j}\dashrightarrow Y^{\prime}_{j}$
is a morphism. Then we have

$\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}=(K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+2\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}})-(K_{\bar{X}^{\prime}_{j}}+\bar{\Delta}^{\prime}_{j}+\alpha\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}})\sim_{\mathbb{R},Y_{j}}0,$

so $\boldsymbol{\rm M}_{Y_{j}}$ is $\mathbb{R}$-Cartier and $\boldsymbol{\rm
M}_{\bar{X}^{\prime}_{j}}=\phi^{*}\boldsymbol{\rm M}_{Y_{j}}$. From this, we
see that $\boldsymbol{\rm M}^{j}$ descends to $Y_{j}$.

Put $\Delta_{Y_{j}}=\phi_{*}\bar{\Delta}^{\prime}_{j}$. By construction, the
generalized pair $(Y_{j},\Delta_{Y_{j}},2\alpha\boldsymbol{\rm M}^{j})/X_{j}$
is a weak generalized lc model of
$(\bar{X}_{j},\bar{\Delta}_{j},2\alpha\boldsymbol{\rm M}^{j})/X_{j}$ such that
$K_{Y_{j}}+\Delta_{Y_{j}}+2\alpha\boldsymbol{\rm M}_{Y_{j}}$ is ample over
$X_{j}$. This is a generalized pair analogue of the relative log canonical
models for lc pairs. Hence we may glue $Y_{j}$, and we get a projective
birational morphism $g\colon Y\to X$ and a birational contraction
$\psi\colon\bar{X}\dashrightarrow Y$ over $X$ such that

  * •

$\boldsymbol{\rm M}$ descends to $Y$, and

  * •

$K_{Y}+\psi_{*}\bar{\Delta}+2\alpha\boldsymbol{\rm M}_{Y}$ is ample over $X$.

Since $\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier, we see that
$K_{X}+\Delta+2\alpha\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier. Then
there is an effective $g$-exceptional $\mathbb{R}$-divisor $E_{Y}$ on $Y$ such
that

$K_{Y}+\psi_{*}\bar{\Delta}+2\alpha\boldsymbol{\rm
M}_{Y}=g^{*}(K_{X}+\Delta+2\alpha\boldsymbol{\rm M}_{X})-E_{Y}$

and ${\rm Supp}E_{Y}={\rm Ex}(g)$. We recall the relation
$K_{\bar{X}}+\bar{\Delta}=f^{*}(K_{X}+\Delta)+\bar{E}$, so we have
$K_{Y}+\psi_{*}\bar{\Delta}=g^{*}(K_{X}+\Delta)+\psi_{*}\bar{E}$. We put
$F=\tfrac{1}{2\alpha}(E_{Y}+\psi_{*}\bar{E})$. Then $F$ is effective,
$g$-exceptional, and ${\rm Supp}F\supset{\rm Ex}(g)$. We also have
$2\alpha\boldsymbol{\rm M}_{Y}=g^{*}(2\alpha\boldsymbol{\rm M}_{X})-2\alpha
F$. From these facts, we see that $g\colon Y\to X$ is the desired morphism. ∎

From now on, we prove Theorem 6.1.

###### Proof of Theorem 6.1.

Clearly, (3) implies (1), and (1) implies (2). Hence we only need to prove
that (2) implies (3). Let $U\subset X$ be the open subset as in (2). We will
prove the existence of a projective birational morphism $g\colon Y\to X$ and
an open subset $V\subset U$ such that $\boldsymbol{\rm M}$ descends to $Y$,
$g$ is an isomorphism over $V$, and $V$ contains the generic point of any
generalized lc center of $(X,\Delta,\boldsymbol{\rm M})/Z$. Assuming the
existence of such $g$, then we can construct the desired log resolution
$f\colon\tilde{X}\to X$ and the open subset $V\subset X$ by considering an
appropriate log resolution of $(X,\Delta)$ factoring through $g$. Thus, we
only need to prove the existence of $g\colon Y\to X$ and $V\subset U$ stated
above.

In this paragraph, we reduce the problem to the case where $K_{X}$ is
$\mathbb{Q}$-Cartier and $\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier. By
[30, Proposition 4.12], with notations as in [30], the pair $\langle
X,\Delta\rangle$ of $X$ and $\Delta$ is pseudo-lc in the sense of [30,
Definition 4.2]. By [30, Theorem 1.2], there exists a small lc modification of
$\langle X,\Delta\rangle$, that is, a projective small birational morphism
$h\colon W\to X$ from a normal variety $W$ such that putting
$\Delta_{W}=h^{-1}_{*}\Delta$ then $K_{W}+\Delta_{W}$ is an $h$-ample
$\mathbb{R}$-Cartier divisor on $W$. We note that [30, Theorem 1.2] can be
applied without the quasi-projectivity of the variety. Since $h$ is small,
$(K_{W}+\Delta_{W})|_{h^{-1}(U)}$ is the pullback of $(K_{X}+\Delta)|_{U}$,
hence $h$ is an isomorphism over $U$. Since $h$ is small, we can easily check
that $(W,\Delta_{W},\boldsymbol{\rm M})/Z$ is a generalized lc pair and
$h^{-1}(U)$ contains the generic point of any generalized lc center of
$(W,\Delta_{W},\boldsymbol{\rm M})/Z$. Then it is easy to see that $h^{-1}(U)$
satisfies the conditions of (2). Thus, by replacing $(X,\Delta,\boldsymbol{\rm
M})/Z$ with $(W,\Delta_{W},\boldsymbol{\rm M})/Z$, we may assume that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Then $\boldsymbol{\rm M}_{X}$ is
$\mathbb{R}$-Cartier. With notations as in [30], the log canonicity of
$(X,\Delta)$ and [30, Lemma 4.3] imply that the pair $\langle X,0\rangle$ is
pseudo-lc in the sense of [30, Definition 4.2]. By [30, Theorem 1.2], there is
a projective small birational morphism $h^{\prime}\colon W^{\prime}\to X$ from
a normal variety $W^{\prime}$ such that $K_{W^{\prime}}$ is an
$h^{\prime}$-ample $\mathbb{R}$-Cartier divisor on $W^{\prime}$. Then
$K_{W^{\prime}}|_{h^{\prime-1}(U)}$ is the pullback of $K_{X}|_{U}$, hence
$h^{\prime}$ is an isomorphism over $U$. It is easy to see that
$(W^{\prime},h^{\prime-1}_{*}\Delta,\boldsymbol{\rm M})/Z$ is a generalized lc
pair and $h^{\prime-1}(U)$ satisfies the conditions of (2). Therefore, by
replacing $(X,\Delta,\boldsymbol{\rm M})/Z$ with
$(W^{\prime},\Delta_{W^{\prime}},\boldsymbol{\rm M})/Z$, we may assume that
$K_{X}$ is $\mathbb{Q}$-Cartier. In this way, we may assume that $K_{X}$ is
$\mathbb{Q}$-Cartier and $\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier.

By the definition of $U$, it follows that $(X,\Delta)$ is dlt. Hence $(X,0)$
is klt. By applying Lemma 6.2 to the generalized klt pair
$(X,\tfrac{1}{2}\Delta,\tfrac{1}{2}\boldsymbol{\rm M})/Z$, we get a projective
birational morphism $g\colon Y\to X$ such that $\tfrac{1}{2}\boldsymbol{\rm
M}$ descends to $Y$ and the divisor $F=\tfrac{1}{2}g^{*}\boldsymbol{\rm
M}_{X}-\tfrac{1}{2}\boldsymbol{\rm M}_{Y}$ satisfies ${\rm Supp}F={\rm
Ex}(g)$. Note that $F$ is effective by the negativity lemma. Let $V_{0}\subset
X$ be the largest open subset over which $g$ is an isomorphsim. Let $\eta$ be
the generic point of a generalized lc center of $(X,\Delta,\boldsymbol{\rm
M})/Z$. Then $\eta$ is the generic point of an lc center of $(U,\Delta|_{U})$,
and $\eta$ is also the generic point of a generalized lc center of
$(U,\Delta|_{U},\boldsymbol{\rm M}|_{U})/U$ with the identity $U\to U$. Pick a
prime divisor $P$ over $U$ whose center is $\overline{\\{\eta\\}}\cap U$ and
the log discrepancy $a(P,U,\Delta|_{U})$ is zero. Then $0\leq
a(P,U,\Delta|_{U}+\boldsymbol{\rm M}_{U})\leq a(P,U,\Delta|_{U})=0$, in
particular, we have $a(P,U,\Delta|_{U}+\boldsymbol{\rm
M}_{U})=a(P,U,\Delta|_{U})$. Since ${\rm Supp}F={\rm Ex}(g)$, the center
$c_{g^{-1}(U)}(P)$ of $P$ on $g^{-1}(U)$ is not contained in ${\rm Ex}(g)$.
Then $c_{g^{-1}(U)}(P)$ intersects $g^{-1}(V_{0})$, hence we see that $\eta\in
V_{0}$.

We set $V=V_{0}\cap U$. Then $g$ is an isomorphism over $V$. By the above
discussion, $V$ contains the generic point of any generalized lc center of
$(X,\Delta,\boldsymbol{\rm M})/Z$. By the property of $U$ in (2), we see that
$(V,\Delta|_{V})$ is log smooth. In this way, we get a projective birational
morphism $g\colon Y\to X$ and an open subset $V\subset U$ such that
$\boldsymbol{\rm M}$ descends to $Y$, $g$ is an isomorphism over $V$, and $V$
contains the generic point of any generalized lc center of
$(X,\Delta,\boldsymbol{\rm M})/Z$. From this fact, (2) implies (3). We
complete the proof. ∎

## References

  * [1] F. Ambro, Quasi-log varieties, Trudy Mat. Inst. Steklova 240 (2003), 220–239.
  * [2] D. Abramovich, K. Karu, Weak semistable reduction in characteristic $0$, Invent. math. 139 (2000), no. 2, 241–273.
  * [3] C. Birkar, Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci. 115 (2012), no. 1, 325–368.
  * [4] C. Birkar, Anti-pluricanonical systems on Fano varieties, Ann. of Math., 19 (2019), no. 2, 345–463.
  * [5] C. Birkar, Boundedness and volume of generalised pairs, preprint (2021), arXiv:2103.14935v2.
  * [6] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468.
  * [7] C. Birkar, D. Q. Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publ. Math. Inst. Hautes Études Sci. 123 (2016), no. 1, 283–331.
  * [8] C. Cascini, D. Q. Zhang, Effective finite generation for adjoint rings, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 1, 127–144.
  * [9] S. R. Choi, The geography of log models and its applications, PhD Thesis, Johns Hopkins University, 2008.
  * [10] S. Filipazzi, J. Moraga, Strong $(\delta,n)$-Complements for Semi-Stable Morphisms, Doc. Math. 25 (2020), 1953–1996.
  * [11] E. Floris, V. Lazić, On the B-semiampleness conjecture, Épijournal de Géométrie Algébrique 3 (2019), epiga:5063.
  * [12] O. Fujino, Base Point Free Theorem of Reid-Fukuda Type, J. Math. Sci. Univ. Tokyo 7 (2000), 1–5.
  * [13] O. Fujino, Special termination and reduction to pl flips. In Flips for $3$-folds and $4$-folds, Oxford University Press (2007).
  * [14] O. Fujino, Effective base point free theorem for log canonical pairs-Kollár type theorem, Tohoku Math. J. 61 (2009), 475–481
  * [15] O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789.
  * [16] O. Fujino, Fundamental theorems for semi log canonical pairs, Algebraic Geom. 1 (2014), no. 2, 194–228.
  * [17] O. Fujino, Foundations of the minimal model program, MSJ Mem. 35, Mathematical Society of Japan, Tokyo, 2017.
  * [18] O. Fujino, Corrigendum: On subadditivity of the logarithmic Kodaira dimension, J. Math. Soc. Japan 72 (2020), no. 4, 1181–1187.
  * [19] O. Fujino, Y. Gongyo, On canonical bundle formulas and subadjunctions, Michigan Math. J. 61 (2012), no. 2, 255–264.
  * [20] O. Fujino, S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188.
  * [21] C. D. Hacon, J. McKernan, C. Xu, ACC for log canonical thresholds, Ann. of Math. (2) 180 (2014), no. 2, 523–571.
  * [22] C. D. Hacon, J. McKernan, C. Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. 20 (2018), no. 4, 865–901.
  * [23] C. D. Hacon, J. Liu, Existence of flips for generalized lc pairs, preprint (2021), arXiv:2105.13590v2.
  * [24] C. D. Hacon, C. Xu, Boundedness of log Calabi-Yau pairs of Fano type, Math. Res. Lett. 22 (2015), no. 6, 1699–1716.
  * [25] J. Han, Z. Li, Weak Zariski decompositions and log terminal models for generalized polarized pairs, preprint (2018), arXiv:1806.01234v2.
  * [26] J. Han, W. Liu, On a generalized canonical bundle formula for generically finite morphisms, to appear in Ann. Inst. Fourier (Grenoble).
  * [27] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
  * [28] K. Hashizume, Minimal model theory for relatively trivial log canonical pairs, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 2069–2107.
  * [29] K. Hashizume, Remarks on special kinds of the relative log minimal model program, Manuscripta Math. 160 (2019), no. 3, 285–314.
  * [30] K. Hashizume, A class of singularity of arbitrary pairs and log canonicalizations, Asian J. Math, 24 (2020), no. 2, 207–238.
  * [31] K. Hashizume, Finiteness of log abundant log canonical pairs in log minimal model program with scaling, preprint (2020), arXiv:2005.12253v3.
  * [32] K. Hashizume, Non-vanishing theorem for generalized log canonical pairs with a polarization, preprint (2020), arXiv:2012.15038v1.
  * [33] K. Hashizume, Z. Hu, On minimal model theory for log abundant lc pairs, J. Reine Angew. Math., 767 (2020), 109–159.
  * [34] Z. Hu, Log abundance of the moduli b-divisors of lc-trivial fibrations, arXiv:2003.14379v3.
  * [35] M. Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), no. 1, 129–133.
  * [36] J. Kollár, Effective base point freeness, Math. Ann. 296 (1993), no. 4, 595–605.
  * [37] J. Kollár, Adjunction and discrepancies, in Flips and abundance for algebraic threefolds, Astérisque 211 (1992), 183–192.
  * [38] J. Kollár, S. Mori, Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics 134. Cambridge University Press, Cambridge, 1998.
  * [39] Z. Li, Boundedness of the vase varieties of certain fibrations, preprint (2020), arXiv:2002.06565v2
  * [40] N. Nakayama, Zariski-decomposition and abundance, MSJ Mem. 14, Mathematical Society of Japan, Tokyo, 2004.
  * [41] E. Viehweg, D. Q. Zhang, Effective Iitaka fibrations, J. Algebraic Geom., 18 (2009), 711–730.