dim}Z\leq 4$ and $K_{X}+\Delta\sim_{\mathbb{R},Z}0$. If $K_{X}+\Delta$ is
pseudo-effective, then $(X,\Delta)$ has a log minimal model.

###### Proof.

By taking the Stein factorization of $\pi$, we may assume that $Z$ is normal
and $\pi$ is a contraction. By Ambro’s canonical bundle formula [15], we can
find a generalized lc pair $(Z,B_{Z}+M_{Z})$ such that
$K_{X}+\Delta\sim_{\mathbb{R}}\pi^{*}(K_{Z}+B_{Z}+M_{Z})$ and the nef part of
$(Z,B_{Z}+M_{Z})$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier b-divisors. By [34, Corollary 3.4] and [24, Theorem
1.5] (see also [21, Corollary 1]), we see that $(Z,B_{Z}+M_{Z})$ has a minimal
model as a generalized lc pair. Let
$(Z^{\prime},B_{Z^{\prime}}+M_{Z^{\prime}})$ be a minimal model of
$(Z,B_{Z}+M_{Z})$ with the boundary part $B_{Z^{\prime}}$ and the nef part
$M_{Z^{\prime}}$. By taking a common resolution of $Z\dashrightarrow
Z^{\prime}$ and the negativity lemma, we see that $K_{Z}+B_{Z}+M_{Z}$
birationally has the Nakayama–Zariski decomposition with nef positive part.
Let $g\colon Z^{\prime}\to Z$ be a resolution such that
$g^{*}(K_{Z}+B_{Z}+M_{Z})$ has the Nakayama–Zariski decomposition with nef
positive part. We take a resolution $f\colon X^{\prime}\to X$ such that the
induced rational map $\pi^{\prime}\colon X^{\prime}\dashrightarrow Z^{\prime}$
is a morphism. Since

$f^{*}(K_{X}+\Delta)\sim_{\mathbb{R}}\pi^{\prime*}g^{*}(K_{Z}+B_{Z}+M_{Z}),$

by [35, 5.17 Corollary], the divisor $f^{*}(K_{X}+\Delta)$ has the
Nakayama–Zariski decomposition with nef positive part. By Theorem 2.23, the
pair $(X,\Delta)$ has a log minimal model. ∎

######  Corollary 3.12.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties, and let $(X,\Delta)$ be an lc pair. Suppose that

  * •

${\rm dim}X=6$,

  * •

$\kappa_{\iota}(X/Z,K_{X}+\Delta)\geq 3-{\rm dim}\,\pi(X)$, and

  * •

$\llcorner\Delta\lrcorner=0$.

Then $(X,\Delta)$ has a log minimal model over $Z$. In particular, all
projective lc pairs $(X,0)$ satisfying ${\rm dim}X=6$ and $\kappa(X,K_{X})\geq
3$ have a log minimal model.

###### Proof.

We first show that we may assume that $Z$ is a point. By [26, Corollary 1.3],
there is an lc closure $(\bar{X},\bar{\Delta})$ of $(X,\Delta)$, that is, a
projective lc pair $(\bar{X},\bar{\Delta})$ and a projective morphism
$\bar{\pi}\colon\bar{X}\to\bar{Z}$ such that $X$ (resp. $Z$) is an open subset
of $\bar{X}$ (resp. $\bar{Z}$), $\bar{\pi}|_{X}=\pi$,
$(\bar{X}|_{X},\bar{\Delta}|_{X})=(X,\Delta)$ and all lc centers of
$(\bar{X},\bar{\Delta})$ intersects $X$. Let $A_{\bar{Z}}$ be a general
sufficiently ample $\mathbb{R}$-divisor on $\bar{Z}$. Because
$\llcorner\Delta\lrcorner=0$ and $A_{\bar{Z}}$ is general, we may assume
$\llcorner(\bar{\Delta}+\bar{\pi}^{*}A_{\bar{Z}})\lrcorner=0$. Furthermore,
the condition $\kappa_{\iota}(X/Z,K_{X}+\Delta)\geq 3-{\rm dim}\,\pi(X)$
implies that
$\kappa_{\iota}(\bar{X},K_{\bar{X}}+\bar{\Delta}+\bar{\pi}^{*}A_{\bar{Z}})\geq
3$. Since $A_{\bar{Z}}$ is sufficiently ample, the existence of a log minimal
model of $(X,\Delta)$ over $Z$ follows from that of
$(\bar{X},\bar{\Delta}+\bar{\pi}^{*}A_{\bar{Z}})$. Thus, by replacing
$(X,\Delta)\to Z$ with $(\bar{X},\bar{\Delta}+\bar{\pi}^{*}A_{\bar{Z}})\to{\rm
Spec}\mathbb{C}$, we may assume that $Z$ is a point.

Next, we prove that $K_{X}+\Delta$ is abundant. Note that we do not need the
condition $\llcorner\Delta\lrcorner=0$ to prove the fact. Therefore, in this
paragraph we will freely replace $(X,\Delta)$ by a log smooth model. Since
$\kappa_{\iota}(X,K_{X}+\Delta)\geq 3$, by replacing $(X,\Delta)$ we may
assume that the Iitaka fibration $X\to V$ associated to $K_{X}+\Delta$ is a
morphism. Then ${\rm dim}V\geq 3$, so general fibers $F$ satisfy ${\rm
dim}F\leq 3$. Then $(F,\Delta|_{F})$ has a good minimal model. Therefore
$\kappa_{\sigma}(X/V,K_{X}+\Delta)=\kappa_{\iota}(F,K_{X}+\Delta|_{F})=0$. By
[35, 2.7 Proposition (9)], we have

$\kappa_{\sigma}(X,K_{X}+\Delta)\leq\kappa_{\sigma}(X/V,K_{X}+\Delta)+{\rm
dim}V={\rm dim}V.$

Since ${\rm dim}V=\kappa_{\iota}(X,K_{X}+\Delta)$, we have
$\kappa_{\sigma}(X,K_{X}+\Delta)\leq\kappa_{\iota}(X,K_{X}+\Delta)$. This
implies that $K_{X}+\Delta$ is abundant.

By the arguments in [26, Proof of Lemma 2.14] (see also [28, Proof of Theorem
1.7]) with the length of extremal rays ([11, Section 18]) and [28, Proposition
6.1], we can construct a sequence of steps of a $(K_{X}+\Delta)$-MMP with
scaling of an ample divisor $H$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

such that if we define $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu H_{i}\text{\rm\, is nef}}$ and
$\lambda_{\infty}:={\rm lim}_{i\to\infty}\lambda_{i}$, where $H_{i}$ is the
birational transform of $H$ on $X_{i}$, then the $(K_{X}+\Delta)$-MMP
terminates or $\lambda_{\infty}=0$. By lifting this log MMP ([3, Remark 2.9]),
we get a $\mathbb{Q}$-factorial dlt model $f\colon(Y,\Gamma)\to(X,\Delta)$ and
a sequence of steps of a $(K_{Y}+\Gamma)$-MMP with scaling of $f^{*}H$

$(Y_{0}:=Y,\Gamma_{0}:=\Gamma)\dashrightarrow(Y_{k_{1}},\Gamma_{k_{1}})\dashrightarrow\cdots\dashrightarrow(Y_{k_{i}},\Gamma_{k_{i}})\dashrightarrow\cdots$

such that $(Y_{k_{i}},\Gamma_{k_{i}})$ is a $\mathbb{Q}$-factorial dlt model
of $(X_{i},\Delta_{i})$ for each $i$. For each $i$ and any lc center
$T_{k_{i}}$ of $(Y_{k_{i}},\Gamma_{k_{i}})$, we consider the dlt pair
$(T_{k_{i}},\Gamma_{T_{k_{i}}})$ constructed with adjunction
$K_{T_{k_{i}}}+\Gamma_{T_{k_{i}}}=(K_{Y_{k_{i}}}+\Gamma_{k_{i}})|_{T_{k_{i}}}$.
We prove that $(T_{k_{i}},\Gamma_{T_{k_{i}}})$ has a log minimal model or a
Mori fiber space. Since $\llcorner\Delta\lrcorner=0$ by hypothesis, we have
$\llcorner\Delta_{i}\lrcorner=0$. Since ${\rm dim}X_{i}=6$, all lc centers of
$(X_{i},\Delta_{i})$ have the dimension at most $4$. Therefore, the morphism
$Y_{k_{i}}\to X_{i}$ induces a morphism $T_{k_{i}}\to S_{i}$ to a variety
$S_{i}$ such that ${\rm dim}S_{i}\leq 4$ and
$K_{Y_{k_{i}}}+\Gamma_{k_{i}}\sim_{\mathbb{R},S_{i}}0$. By Lemma 3.11, the
pair $(T_{k_{i}},\Gamma_{T_{k_{i}}})$ has a log minimal model or a Mori fiber
space.

By the standard argument of the special termination ([10]), there is a
positive integer $m$ such that for any $i\geq k_{m}$, the non-isomorphic locus
of the birational map $Y_{k_{m}}\dashrightarrow Y_{i}$ does not intersect
$\llcorner\Gamma_{k_{m}}\lrcorner$. Since $\lambda_{\infty}=0$, by Lemma 2.22,
the restriction of $K_{Y_{k_{m}}}+\Gamma_{k_{m}}$ to any component of
$\llcorner\Gamma_{k_{m}}\lrcorner$ is nef, and
$\sigma_{P}(K_{Y_{k_{m}}}+\Gamma_{k_{m}})=0$ for all prime divisors $P$ over
$Y_{k_{m}}$ such that $c_{Y_{k_{m}}}(P)$ intersects an lc center of
$(Y_{k_{m}},\Gamma_{k_{m}})$. Because $K_{X}+\Delta$ is abundant, we see that
$K_{Y_{k_{m}}}+\Gamma_{k_{m}}$ is abundant. By Theorem 3.4, we see that
$(Y_{k_{m}},\Gamma_{k_{m}})$ has a log minimal model. Then $(Y,\Gamma)$ has a
log minimal model, thus, $(X,\Delta)$ has a log minimal model.

The second assertion directly follows from the first assertion. ∎

######  Corollary 3.13.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties and $(X,\Delta)$ an lc pair. Suppose that

  * •

$\kappa_{\iota}(X/Z,K_{X}+\Delta)\geq{\rm dim}X-3-{\rm dim}\,\pi(X)$, and

  * •

all lc centers $S$ of $(X,\Delta)$ satisfy ${\rm dim}S\leq 4$.

Then $(X,\Delta)$ has a log minimal model over $Z$.

Corollary 3.13 looks similar to the case of $n=3$ of [28, Theorem 1.2]. Main
differences between the two results are that Corollary 3.13 allows the
existence of $4$-dimensional lc centers but only shows the existence of a log
minimal model.

###### Proof of Corollary 3.13.

The proof is very similar to that of Corollary 3.12. So we only outline the
proof.

By [26, Corollary 1.3] and the same argument as in the first paragraph of the
proof of Corollary 3.12, we may assume that $Z$ is a point. Note that after
the reduction the two conditions of Corollary 3.13 are preserved.

We apply the argument in the second paragraph of the proof of Corollary 3.12.
By taking the Iitaka fibration associated to $K_{X}+\Delta$ and using [35, 2.7
Proposition (9)], we see that $K_{X}+\Delta$ is abundant.

By the arguments in [26, Proof of Lemma 2.14] (see also [28, Proof of Theorem
1.7]) with the length of extremal rays ([11, Section 18]) and [28, Proposition
6.1], we can construct a sequence of steps of a $(K_{X}+\Delta)$-MMP with
scaling of an ample divisor $H$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

such that if we define $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu H_{i}\text{\rm\, is nef}}$ and
$\lambda_{\infty}:={\rm lim}_{i\to\infty}\lambda_{i}$, where $H_{i}$ is the
birational transform of $H$ on $X_{i}$, then the $(K_{X}+\Delta)$-MMP
terminates or $\lambda_{\infty}=0$. By lifting this log MMP ([3, Remark 2.9]),
we get a $\mathbb{Q}$-factorial dlt model $f\colon(Y,\Gamma)\to(X,\Delta)$ and
a sequence of steps of a $(K_{Y}+\Gamma)$-MMP with scaling of $f^{*}H$

$(Y_{0}:=Y,\Gamma_{0}:=\Gamma)\dashrightarrow(Y_{k_{1}},\Gamma_{k_{1}})\dashrightarrow\cdots\dashrightarrow(Y_{k_{i}},\Gamma_{k_{i}})\dashrightarrow\cdots$

such that $(Y_{k_{i}},\Gamma_{k_{i}})$ is a $\mathbb{Q}$-factorial dlt model
of $(X_{i},\Delta_{i})$ for each $i$. For each $i$ and any lc center
$T_{k_{i}}$ of $(Y_{k_{i}},\Gamma_{k_{i}})$, we consider the dlt pair
$(T_{k_{i}},\Gamma_{T_{k_{i}}})$ constructed with adjunction
$K_{T_{k_{i}}}+\Gamma_{T_{k_{i}}}=(K_{Y_{k_{i}}}+\Gamma_{k_{i}})|_{T_{k_{i}}}$.
Let $S_{i}$ be the image of $T_{k_{i}}$ by the morphism $Y_{k_{i}}\to X_{i}$.
Then $S_{i}$ is an lc center of $(X_{i},\Delta_{i})$, and we can find an lc
center $S$ of $(X,\Delta)$ such that the birational map $X\dashrightarrow
X_{i}$ induces a birational map $S\dashrightarrow S_{i}$. In particular, we
have ${\rm dim}S_{i}={\rm dim}S\leq 4$, where the second inequality follows
from the second condition of Corollary 3.13. Therefore, the induced morphism
$T_{k_{i}}\to S_{i}$ satisfies ${\rm dim}S_{i}\leq 4$ and
$K_{Y_{k_{i}}}+\Gamma_{k_{i}}\sim_{\mathbb{R},S_{i}}0$. By applying Lemma
3.11, the dlt pair $(T_{k_{i}},\Gamma_{T_{k_{i}}})$ has a log minimal model or
a Mori fiber space.

By the standard argument of the special termination ([10], Remark 2.21), we
can find a positive integer $m$ such that for any $i\geq k_{m}$, the non-
isomorphic locus of $Y_{k_{m}}\dashrightarrow Y_{i}$ is disjoint from
$\llcorner\Gamma_{k_{m}}\lrcorner$. By the same argument as in the proof of
Corollary 3.12, we can check that $(Y_{k_{m}},\Gamma_{k_{m}})$ satisfies all
the conditions of Theorem 3.4. By Theorem 3.4, the pair
$(Y_{k_{m}},\Gamma_{k_{m}})$ has a log minimal model. Then $(Y,\Gamma)$ has a
log minimal model, thus, $(X,\Delta)$ has a log minimal model. ∎

##  4\. On log MMP for semi log canonical pairs

In this section, by applying Corollary 3.10 and [2], we discuss log MMP for
semi log canonical (slc, for short) pairs. We assume that all schemes in this
section are reduced separated schemes of finite type over ${\rm
Spec}\mathbb{C}$. See [14, Definition 4.13.3] for definition of slc pairs. We
only deal with projective slc pairs, that is, slc pairs $(X,\Delta)$ such that
$X$ is projective over ${\rm Spec}\mathbb{C}$.

The goal of this section is to prove the following generalization of Corollary
3.10.

######  Theorem 4.1.

Let $(X,\Delta)$ be an slc pair such that $X$ is projective and $\Delta$ is a
$\mathbb{Q}$-divisor. Let $\nu\colon(\bar{X},\bar{\Delta})\to(X,\Delta)$ be
the normalization, where $K_{\bar{X}}+\bar{\Delta}=\nu^{*}(K_{X}+\Delta)$.
Suppose that every irreducible component $(\bar{X}^{(j)},\bar{\Delta}^{(j)})$
of $(\bar{X},\bar{\Delta})$ satisfies the following conditions.

  * •

$(\bar{X}^{(j)},\bar{\Delta}^{(j)})$ is log abundant and
$K_{\bar{X}^{(j)}}+\bar{\Delta}^{(j)}$ is pseudo-effective, and

  * •

the stable base locus of $K_{\bar{X}^{(j)}}+\bar{\Delta}^{(j)}$ does not
contain the image of any prime divisor $P$ over $\bar{X}^{(j)}$ satisfying
$a(P,\bar{X}^{(j)},\bar{\Delta}^{(j)})<0$.

Then there is a sequence of MMP steps for $(X,\Delta)$

$(X,\Delta)\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots\dashrightarrow(X_{m},\Delta_{m})$

such that $K_{X_{m}}+\Delta_{m}$ is semi-ample.

We recall definition of an MMP step for projective slc pairs.

######  Definition 4.2 (An MMP step for slc pairs, [2, Definition 11]).

Let $(X,\Delta)$ be an slc pair such that $X$ is projective and $\Delta$ is a
$\mathbb{Q}$-divisor. Then, an MMP step for $(X,\Delta)$ is the following
diagram

---
$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\textstyle{(X^{\prime},\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Z}$

such that

  1. (1)

$(X^{\prime},\Delta^{\prime})$ is an slc pair such that $X^{\prime}$ is
projective,

  2. (2)

$\phi$ is birational and it is isomorphic on an open dense subset containing
all generic points of codimension one singular locus,

  3. (3)

$\Delta^{\prime}=\phi_{*}\Delta$,

  4. (4)

$Z$ is a projective scheme, $f$ and $f^{\prime}$ are projective and
generically finite, and $f^{\prime}$ has no exceptional divisors, and

  5. (5)

$-(K_{X}+\Delta)$ and $K_{X^{\prime}}+\Delta^{\prime}$ are ample over $Z$.

Let $H$ be a $\mathbb{Q}$-Cartier divisor on $X$. Then the above diagram is an
MMP step with scaling of $H$ if the diagram satisfies (1)–(5) stated above and

  1. (6)

$H$ is $f$-ample and $-H^{\prime}:=-\phi_{*}H$ is an $f^{\prime}$-ample
$\mathbb{Q}$-Cartier divisor,

  2. (7)

$K_{X}+\Delta+\lambda H$ is numerically trivial over $Z$ for some
$\lambda\in\mathbb{Q}$, and

  3. (8)

for every proper curve $C$ on $X$, if $C$ is not contracted by $f$ then the
strict inequality $(K_{X}+\Delta+\lambda H)\cdot C>0$ holds.

The above definition is different from the usual log MMP even if $(X,\Delta)$
is lc. The following example illustrates the difference between the usual log
MMP and log MMP in the sense of Definition 4.2.

######  Example 4.3.

Let $(X,\Delta)$ be a projective lc pair such that $\Delta$ is a
$\mathbb{Q}$-divisor and $K_{X}+\Delta$ is pseudo-effective but not nef. Let
$A$ be an ample $\mathbb{Q}$-divisor on $X$ such that $K_{X}+\Delta+A$ is nef
but not ample. Then $K_{X}+\Delta+A$ is semi-ample ([11, Theorem 13.1]) and
big, so it induces a generically finite morphism $f\colon X\to Z$ to a
projective variety $Z$. Then $-(K_{X}+\Delta)\sim_{\mathbb{Q},Z}A$ is ample
over $Z$ and any curve $C$ on $X$ is contracted by $f$ if and only if
$(K_{X}+\Delta+A)\cdot C=0$. By [3, Theorem 1.1] and [28, Theorem 1.7], we may
run a $(K_{X}+\Delta)$-MMP over $Z$ and we may get a good minimal model
$(X_{l},\Delta_{l})$ over $Z$. Let $X_{l}\to X^{\prime}$ be the birational
morphism over $Z$ induced by $K_{X_{l}}+\Delta_{l}$. Let $\Delta^{\prime}$
(resp. $A^{\prime}$) be the birational transform of $\Delta$ (resp. $A$) on
$X^{\prime}$. Then $(X^{\prime},\Delta^{\prime})$ is lc,
$K_{X^{\prime}}+\Delta^{\prime}$ is ample over $Z$ and
$K_{X^{\prime}}+\Delta^{\prime}+A^{\prime}\sim_{\mathbb{Q},Z}0$.

We can directly check that the birational map
$(X,\Delta)\dashrightarrow(X^{\prime},\Delta^{\prime})$ over $Z$ is an MMP
step with scaling of $A$ for $(X,\Delta)$ as in Definition 4.2, but in general
this birational map is not necessarily a step of the usual
$(K_{X}+\Delta)$-MMP unless $f$ is birational and the relative Picard number
$\rho(X/Z)$ is $1$.

In general, we cannot construct MMP steps for slc pairs inductively. See, for
example, [2, Example 4 and Example 5] and [13, Example 5.4]. But, thanks to
[2, Theorem 9], we can construct MMP steps for slc pairs whose normalization
satisfies the two conditions of Corollary 3.10. We prove this fact in Lemma
4.4 below.

######  Lemma 4.4.

Let $(X,\Delta)$ be an slc pair such that $X$ is projective and $\Delta$ is a
$\mathbb{Q}$-divisor. Let $\nu\colon(\bar{X},\bar{\Delta})\to(X,\Delta)$ be
the normalization, where $K_{\bar{X}}+\bar{\Delta}=\nu^{*}(K_{X}+\Delta)$.
Suppose that every irreducible component $(\bar{X}^{(j)},\bar{\Delta}^{(j)})$
of $(\bar{X},\bar{\Delta})$ satisfies the following.

  * •

$|K_{\bar{X}^{(j)}}+\bar{\Delta}^{(j)}|_{\mathbb{R}}\neq\emptyset$ and the
stable base locus of $K_{\bar{X}^{(j)}}+\bar{\Delta}^{(j)}$ does not contain
the image of any prime divisor $P$ over $\bar{X}^{(j)}$ satisfying
$a(P,\bar{X}^{(j)},\bar{\Delta}^{(j)})<0$.

Let $H$ be a $\mathbb{Q}$-Cartier divisor such that $K_{X}+\Delta+cH$ is ample
for some $c>0$, and set $\lambda={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}}{\text{$K_{X}+\Delta+\mu H$ is nef}}$. If $\lambda>0$, then we may
construct an MMP step with scaling of $H$ for $(X,\Delta)$ as in Definition
4.2

---
$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\textstyle{(X^{\prime},\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Z}$

such that $K_{X^{\prime}}+\Delta^{\prime}+c^{\prime}H^{\prime}$ is ample for
some $0<c^{\prime}<\lambda$, where $H^{\prime}=\phi_{*}H$. Furthermore, for
the normalization
$\nu^{\prime}\colon(\bar{X}^{\prime},\bar{\Delta}^{\prime})\to(X^{\prime},\Delta^{\prime})$
where
$K_{\bar{X}^{\prime}}+\bar{\Delta}^{\prime}=\nu^{\prime*}(K_{X}+\Delta)$,
every irreducible component $(\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})$
satisfies the following.

  * •

$|K_{\bar{X}^{\prime(j)}}+\bar{\Delta}^{\prime(j)}|_{\mathbb{R}}\neq\emptyset$
and the stable base locus of
$K_{\bar{X}^{\prime(j)}}+\bar{\Delta}^{\prime(j)}$ does not contain the image
of any prime divisor $P^{\prime}$ over $\bar{X}^{\prime(j)}$ satisfying
$a(P^{\prime},\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})<0$.

###### Proof.

We have $\lambda<c$, and, by rationality theorem [13, Theorem 1.18] we see
that $\lambda$ is a rational number. We have $K_{X}+\Delta+\lambda
H=(1-\tfrac{\lambda}{c})(K_{X}+\Delta)+\tfrac{\lambda}{c}(K_{X}+\Delta+cH)$.
Since $K_{X}+\Delta+cH$ is ample, $K_{X}+\Delta+\lambda H$ is semi-ample by
[13, Theorem 1.15]. Thus, $K_{X}+\Delta+\lambda H$ induces a projective
morphism $f\colon X\to Z$ to a projective scheme $Z$. Since $K_{X}+\Delta+cH$
is ample and $K_{X}+\Delta+\lambda H$ is $\mathbb{Q}$-linearly equivalent to
the pullback of an ample $\mathbb{Q}$-divisor on $Z$, the divisors $H$ and
$-(K_{X}+\Delta)$ are ample over $Z$. Furthermore, any proper curve $C$ on $X$
is contracted by $f$ if and only if $(K_{X}+\Delta+\lambda H)\cdot C=0$. We
put $\bar{H}=\nu^{*}H$. For any irreducible component $\bar{X}^{(j)}$ of the
normalization $\bar{X}\to X$, since $K_{\bar{X}}+\bar{\Delta}+c\bar{H}$ is
ample and $|K_{\bar{X}^{(j)}}+\bar{\Delta}^{(j)}|_{\mathbb{R}}\neq\emptyset$,
the pullback of $K_{X}+\Delta+\lambda H$ to $\bar{X}^{(j)}$ is big. This
implies that $f$ is generically finite. In this way, $f\colon X\to Z$,
$K_{X}+\Delta$ and $H$ satisfy conditions of (4), (5), (6), (7) and (8) of
Definition 4.2.

For any component $\bar{X}^{(j)}$ of the normalization $\bar{X}\to X$, let
$\bar{X}^{(j)}\to Z^{(j)}$ be the Stein factorization of $\bar{X}^{(j)}\to Z$.
This is a birational morphism. Since
$K_{\bar{X}}+\bar{\Delta}+\lambda\bar{H}\sim_{\mathbb{Q},Z}0$, applying [3,
Theorem 1.1], the lc pair $(\bar{X}^{(j)},\bar{\Delta}^{(j)})$ has a good
minimal model over $Z^{(j)}$ for all $j$. Therefore
$(\bar{X}^{(j)},\bar{\Delta}^{(j)})$ has the canonical model
$(\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})$ over $Z^{(j)}$, that is, a
weak lc model over $Z^{(j)}$ such that
$K_{\bar{X}^{\prime(j)}}+\bar{\Delta}^{\prime(j)}$ is ample over $Z^{(j)}$,
for all $j$. We put
$(\bar{X}^{\prime},\bar{\Delta}^{\prime})={\coprod_{j}}(\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})$,
and we consider the following diagram

---
$\textstyle{(\bar{X},\bar{\Delta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\bar{X}^{\prime},\bar{\Delta}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z}$

where the birational map $\bar{X}\dashrightarrow\bar{X}^{\prime}$ is defined
by each birational map $\bar{X}^{(j)}\dashrightarrow\bar{X}^{\prime(j)}$. By
construction, this diagram is an MMP step as in Definition 4.2 for the
disjoint union of lc pairs $(\bar{X},\bar{\Delta})$. Indeed, (1), (2), and (3)
are obvious, and (4) and (5) follow from that
$(\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})$ is the log canonical model
over $Z^{(j)}$ for all $j$. Since
$K_{\bar{X}}+\bar{\Delta}+\lambda\bar{H}\sim_{\mathbb{Q},Z}0$, we have
$K_{\bar{X}^{\prime}}+\bar{\Delta}^{\prime}+\lambda\bar{H}^{\prime}\sim_{\mathbb{Q},Z}0$.
Because $K_{\bar{X}^{\prime}}+\bar{\Delta}^{\prime}$ is ample over $Z$, we see
that $-\bar{H}^{\prime}$ is ample over $Z$, where $\bar{H}^{\prime}$ is the
birational transform of $\bar{H}$ on $\bar{X}^{\prime}$. Therefore (6) holds,
and we have already checked that (7) and (8) hold. From these discussion, the
above diagram is an MMP step with scaling of $\bar{H}$.

For all irreducible components
$(\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})$, the hypothesis of Lemma 4.4
on $(\bar{X},\bar{\Delta})$ and construction of the usual log MMP implies that
$|K_{\bar{X}^{\prime(j)}}+\bar{\Delta}^{\prime(j)}|_{\mathbb{R}}\neq\emptyset$
and its stable base locus does not contain the image of any prime divisor
$P^{\prime}$ over $\bar{X}^{\prime(j)}$ satisfying
$a(P^{\prime},\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})<0$. By taking a
common resolution of $\bar{X}^{(j)}\dashrightarrow\bar{X}^{\prime(j)}$ and
applying (5) and the negativity lemma, we see that the non-isomorphic locus of
$\bar{X}^{(j)}\dashrightarrow\bar{X}^{\prime(j)}$ is contained in the stable
base locus of $K_{\bar{X}^{(j)}}+\bar{\Delta}^{(j)}$. From these facts, the
above MMP step satisfies the condition of [2, Theorem 9], therefore we can
apply [2, Theorem 9]. We get a projective slc pair
$(X^{\prime},\Delta^{\prime})$ and a projective morphism $f^{\prime}\colon
X^{\prime}\to Z$ such that $(\bar{X}^{\prime},\bar{\Delta}^{\prime})$ is the
normalization of $(X^{\prime},\Delta^{\prime})$. By [2, Lemma 12], the induced
diagram

---
$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\textstyle{(X^{\prime},\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Z}$

is an MMP step with scaling of $H$ for $(X,\Delta)$.

By construction of $f$, the divisor $K_{X}+\Delta+\lambda H$ is
$\mathbb{Q}$-linearly equivalent to the pullback of an ample
$\mathbb{Q}$-divisor on $Z$, so is $K_{X^{\prime}}+\Delta^{\prime}+\lambda
H^{\prime}$, where $H^{\prime}=\phi_{*}H$. Since
$K_{X^{\prime}}+\Delta^{\prime}$ is ample over $Z$, we see that
$K_{X^{\prime}}+\Delta^{\prime}+\lambda
H^{\prime}+t(K_{X^{\prime}}+\Delta^{\prime})$ is ample for some $t>0$. Put
$c^{\prime}=\frac{\lambda}{1+t}$. Since
${\coprod_{j}}(\bar{X}^{\prime(j)},\bar{\Delta}^{\prime(j)})=(\bar{X}^{\prime},\bar{\Delta}^{\prime})$
satisfies the condition of Lemma 4.4, the diagram consists of $\phi$, $f$ and
$f^{\prime}$ satisfies all the condition of Lemma 4.4. ∎

From now on, we prove Theorem 4.1.

###### Proof of Theorem 4.1.

Put $X=X_{0}$ and $\Delta=\Delta_{0}$. Let $H_{0}$ be an ample
$\mathbb{Q}$-Cartier divisor on $X_{0}$. By Lemma 4.4, we may construct a
sequence of MMP steps with scaling of $H$

$(X_{0},\Delta_{0})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots.$

Put $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}}{\text{$K_{X_{i}}+\Delta_{i}+\mu H_{i}$ is nef}}$, where $H_{i}$ is the
birational transform of $H_{0}$ on $X_{i}$. By construction of each MMP step
(Lemma 4.4), we have $\lambda_{i-1}>\lambda_{i}$ for all $i\geq 1$. For each
$i$, let $(\bar{X}_{i},\bar{\Delta}_{i})$ be the normalization of
$(X_{i},\Delta_{i})$. Let $\bar{H}_{0}$ be the pullback of $H_{0}$ to
$\bar{X}_{0}$. Let $\bar{H}_{i}$ be the birational transform of $H_{0}$ on
$\bar{X}_{i}$. By construction of MMP steps (Lemma 4.4) and [2, Lemma 12], the
sequence of birational maps

$(\bar{X}_{0},\bar{\Delta}_{0})\dashrightarrow\cdots\dashrightarrow(\bar{X}_{i},\bar{\Delta}_{i})\dashrightarrow\cdots$

is MMP steps for $(\bar{X}_{0},\bar{\Delta}_{0})$ with scaling of
$\bar{H}_{0}$ and we see that

$\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}}{\text{$K_{\bar{X}_{i}}+\bar{\Delta}_{i}+\mu\bar{H}_{i}$ is nef}}$

because $\bar{H}_{i}$ is equal to the pullback of $H_{i}$ to $\bar{X}_{i}$. In
particular, $K_{\bar{X}_{i}}+\bar{\Delta}_{i}+u\bar{H}_{i}$ is nef for all
$u\in[\lambda_{i},\lambda_{i-1}]$.

To complete the proof, it is sufficient to prove that $\lambda_{m}=0$ for some
$m$ and $K_{X_{m}}+\Delta_{m}$ is semi-ample. By [23, Theorem 1.4] or [16,
Theorem 1.5], semi-ampleness of $K_{X_{m}}+\Delta_{m}$ follows from that of
$K_{\bar{X}_{m}}+\bar{\Delta}_{m}$. Since we only need to prove the semi-
ampleness in each irreducible component of $(\bar{X}_{m},\bar{\Delta}_{m})$,
replacing $(X_{0},\Delta_{0})$ by any irreducible component of
$(\bar{X}_{0},\bar{\Delta}_{0})$ and replacing $(X_{i},\Delta_{i})$ by the
corresponding irreducible component of $(\bar{X}_{i},\bar{\Delta}_{i})$ for
each $i$, we may assume that $(X_{i},\Delta_{i})$ are lc. Note that after we
replace slc pairs $(X_{i},\Delta_{i})$, the birational maps
$(X_{i},\Delta_{i})\dashrightarrow(X_{i+1},\Delta_{i+1})$ can be isomorphisms
for some $i$. By identifying all $(X_{i},\Delta_{i})$ such that $\lambda_{i}$
are the same values, we may assume that $\lambda_{i-1}>\lambda_{i}$ still
holds for all $i\geq 1$. Construction of MMP steps and the negativity lemma
show that $(X_{i},\Delta_{i}+uH_{i})$ is a weak lc model of
$(X_{0},\Delta_{0}+uH_{0})$ for any $u\in[\lambda_{i},\lambda_{i-1}]$.

By Corollary 3.10 and the hypothesis of Theorem 4.1, we see that the lc pair
$(X_{0},\Delta_{0})$ has a good minimal model. By [28, Theorem 1.7], we can
construct a sequence of steps of the usual $(K_{X_{0}}+\Delta_{0})$-MMP with
scaling of $H_{0}$

$(X^{\prime}_{0}:=X_{0},\Delta^{\prime}_{0}:=\Delta_{0})\dashrightarrow\cdots\dashrightarrow(X^{\prime}_{j},\Delta^{\prime}_{j})\dashrightarrow\cdots\dashrightarrow(X^{\prime}_{n},\Delta^{\prime}_{n})$

such that $(X^{\prime}_{n},\Delta^{\prime}_{n})$ is a good minimal model. We
set $H^{\prime}_{0}:=H_{0}$ and define

$\lambda^{\prime}_{j}:={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}}{\text{$K_{X^{\prime}_{j}}+\Delta^{\prime}_{j}+\mu H^{\prime}_{j}$ is
nef}},$

where $H^{\prime}_{j}$ is the birational transform of $H^{\prime}_{0}$ on
$X^{\prime}_{j}$. We show by induction that for any $0\leq l\leq n$, there
exists $k\geq 0$ such that $\lambda_{k}=\lambda^{\prime}_{l}$. It is clear
that $\lambda_{0}=\lambda^{\prime}_{0}$. We prove the case of $l+1$ assuming
$\lambda_{k}=\lambda^{\prime}_{l}$ for some $k$. If
$\lambda^{\prime}_{l+1}=\lambda^{\prime}_{l}$ then
$\lambda_{k}=\lambda^{\prime}_{l+1}$. If
$\lambda^{\prime}_{l+1}<\lambda^{\prime}_{l}$, then
$(X^{\prime}_{l+1},\Delta^{\prime}_{l+1}+u^{\prime}H^{\prime}_{l+1})$ is a
weak lc model of $(X_{0},\Delta_{0}+u^{\prime}H_{0})$ for all
$u^{\prime}\in[\lambda^{\prime}_{l+1},\lambda^{\prime}_{l}]$. Since
$(X_{k+1},\Delta_{k+1}+uH_{k+1})$ is a weak lc model of
$(X_{0},\Delta_{0}+uH_{0})$ for any $u\in[\lambda_{k+1},\lambda_{k}]$, by
taking a common resolution $g\colon W\to X_{k+1}$ and $g^{\prime}\colon W\to
X^{\prime}_{l+1}$ of the induced birational map $X_{k+1}\dashrightarrow
X^{\prime}_{l+1}$ and the negativity lemma (see also [3, Remark 2.7]), we have

$g^{*}(K_{X_{k+1}}+\Delta_{k+1}+u^{\prime}H_{k+1})=g^{\prime*}(K_{X^{\prime}_{l+1}}+\Delta^{\prime}_{l+1}+u^{\prime}H^{\prime}_{l+1})$

for any
$u^{\prime}\in[\lambda_{k+1},\lambda_{k}]\cap[\lambda^{\prime}_{l+1},\lambda^{\prime}_{l}]$.
Since $\lambda_{k}=\lambda^{\prime}_{l}$ and $\lambda_{k+1}<\lambda_{k}$, the
relation shows

$g^{*}(K_{X_{k+1}}+\Delta_{k+1}+tH_{k+1})=g^{\prime*}(K_{X^{\prime}_{l+1}}+\Delta^{\prime}_{l+1}+tH^{\prime}_{l+1})$

for any $t\in\mathbb{R}$. Thus, $K_{X_{k+1}}+\Delta_{k+1}+tH_{k+1}$ is nef if
and only if $K_{X^{\prime}_{l+1}}+\Delta^{\prime}_{l+1}+tH^{\prime}_{l+1}$ is
nef. By definitions of $\lambda_{k+1}$ and $\lambda^{\prime}_{l+1}$, we have
$\lambda_{k+1}=\lambda^{\prime}_{l+1}$.

In this way, for any $l$ there is $k$ such that
$\lambda_{k}=\lambda^{\prime}_{l}$. Therefore, there is $m$ such that
$\lambda_{m}=\lambda^{\prime}_{n}=0$. Thus $(X_{m},\Delta_{m})$ is a weak lc
model of $(X_{0},\Delta_{0})$. By taking a common resolution
$h\colon\tilde{W}\to X_{m}$ and $h^{\prime}\colon\tilde{W}\to X^{\prime}_{n}$
of the map $X_{m}\dashrightarrow X^{\prime}_{n}$ and the negativity lemma, we
have

$h^{*}(K_{X_{m}}+\Delta_{m})=h^{\prime*}(K_{X^{\prime}_{n}}+\Delta^{\prime}_{n}),$

hence $K_{X_{m}}+\Delta_{m}$ is semi-ample (see [26, Remark 2.10 (iii)] and
[3, Remark 2.7]). Thus, $K_{X_{m}}+\Delta_{m}$ is semi-ample for some $m$.
From this, Theorem 4.1 holds. ∎

##  5\. Varieties over an algebraically closed field of characteristic zero

In this section, we give a remark on generalizations of results of this paper
to varieties over an algebraically closed field of characteristic zero.

We go back to the definitions of the invariant Iitaka dimension (Definition
2.12) and the numerical dimension (Definition 2.13). Let ${{k}}$ be an
algebraically closed field of characteristic zero. For any normal projective
variety $X$ over ${{k}}$ and $\mathbb{R}$-Cartier divisor $D$ on $X$, we can
define $\kappa_{\iota}(X,D)$ and $\kappa_{\sigma}(X,D)$ by the same way as in
Definition 2.12 and Definition 2.13, respectively. A problem comes up when we
define the relative invariant Iitaka dimension and the relative numerical
dimension. For a given projective morphism from a normal variety to a variety
and an $\mathbb{R}$-Cartier divisor, the relative invariant Iitaka dimension
and the relative numerical dimension are defined with a sufficiently general
fiber of the Stein factorization of the morphism. But, when ${{k}}$ is
countable, we cannot always find a sufficiently general closed point, so
sufficiently general fibers do not always exist. Therefore, the base field is
not necessary the complex number field $\mathbb{C}$, we define the relative
invariant Iitaka dimension and the relative numerical dimension as follows.

######  Definition 5.1.

Let $X\to Z$ be a projective morphism from a normal variety to a variety over
an algebraically closed field ${{k}}$ of characteristic zero, and let
$X_{\bar{\eta}}$ be the geometric generic fiber of the Stein factorization of
$X\to Z$. Let $D$ be an $\mathbb{R}$-Cartier divisor on $X$. We define the
relative invariant Iitaka dimension of $D$ over $Z$, which we denote
$\hat{\kappa}_{\iota}(X/Z,D)$, by
$\kappa_{\iota}(X_{\bar{\eta}},D|_{X_{\bar{\eta}}})$. Similarly, we define the
relative numerical dimension of $D$ over $Z$, which we denote
$\hat{\kappa}_{\sigma}(X/Z,D)$, by
$\kappa_{\sigma}(X_{\bar{\eta}},D|_{X_{\bar{\eta}}})$.

######  Definition 5.2.

Let $\pi\colon X\to Z$ be a projective morphism from a normal variety to a
variety over an algebraically closed field ${{k}}$ of characteristic zero. For
any $\mathbb{R}$-Cartier divisor $D$ on $X$, we say that $D$ is $\pi$-abundant
(or abundant over $Z$) if the equality
$\hat{\kappa}_{\iota}(X/Z,D)=\hat{\kappa}_{\sigma}(X/Z,D)$ holds.

With Definition 5.2, for any lc pairs $(X,\Delta)\to Z$ and an
$\mathbb{R}$-Cartier divisor $D$ on $X$, we can define the property of being
log abundant.

When ${{k}}=\mathbb{C}$ and the base variety of a morphism is quasi-
projective, the following lemma shows that the new definitions of the relative
invariant Iitaka dimension and the relative numerical dimension coincide with
those of Definition 2.12 and Definition 2.13.

######  Lemma 5.3.

Let $\pi\colon X\to Z$ be a projective morphism from a normal variety to a
variety over $\mathbb{C}$, and let $D$ be an $\mathbb{R}$-Cartier divisor on
$X$. If $Z$ is quasi-projective, then
$\hat{\kappa}_{\iota}(X/Z,D)=\kappa_{\iota}(X/Z,D)$ and
$\hat{\kappa}_{\sigma}(X/Z,D)=\kappa_{\sigma}(X/Z,D)$.

###### Proof.

The equality $\hat{\kappa}_{\sigma}(X/Z,D)=\kappa_{\sigma}(X/Z,D)$ follows
from Definition 2.13, generic flatness and the semi-continuity of dimension of
cohomology of flat coherent sheaves. By definition, $\kappa_{\iota}(X/Z,D)\geq
0$ is equivalent to that $D\sim_{\mathbb{R},Z}E$ for some $E\geq 0$.

We show that the latter condition is equivalent to
$\hat{\kappa}_{\sigma}(X/Z,D)\geq 0$. It is obvious that the existence of an
effective $\mathbb{R}$-divisor $E$ such that $D\sim_{\mathbb{R},Z}E$ implies
$\hat{\kappa}_{\sigma}(X/Z,D)\geq 0$. Conversely, suppose that
$\hat{\kappa}_{\sigma}(X/Z,D)\geq 0$. Then, by the standard argument of convex
geometry, we can find $\mathbb{Q}$-Cartier divisors $D_{1},\cdots,D_{l}$,
positive real numbers $r_{1},\cdots,r_{l}$ and effective $\mathbb{R}$-divisors
$\bar{E}_{1},\cdots,\bar{E}_{l}$ on $X_{\bar{\eta}}$ such that
$\sum_{i=1}^{l}r_{i}=1$, $\sum_{i=1}^{l}r_{i}D_{i}=D$ and

$D_{i}|_{X_{\bar{\eta}}}\sim_{\mathbb{Q}}\bar{E}_{i}$

for each $1\leq i\leq l$ (see, for example, [28, Proof of Lemma 2.10]). Since
$D_{i}$ are $\mathbb{Q}$-Cartier divisors, for each $1\leq i\leq l$ we may
find $E_{i}\geq 0$ such that $D_{i}\sim_{\mathbb{Q},Z}E_{i}$. Then

$D=\sum_{i=1}^{l}r_{i}D_{i}\sim_{\mathbb{R},Z}\sum_{i=1}^{l}r_{i}E_{i}.$

Since all $r_{i}$ are positive, the right hand side is effective. In this way,
$\hat{\kappa}_{\sigma}(X/Z,D)\geq 0$ implies the existence of an effective
$\mathbb{R}$-divisor $E$ such that $D\sim_{\mathbb{R},Z}E$.

The above discussion shows that $\kappa_{\iota}(X/Z,D)\geq 0$ is equivalent to
$\hat{\kappa}_{\sigma}(X/Z,D)\geq 0$. When $\kappa_{\iota}(X/Z,D)\geq 0$,
since there is an effective $\mathbb{R}$-divisor $E$ on $X$ such that
$D\sim_{\mathbb{R},Z}E$, the equality
$\hat{\kappa}_{\iota}(X/Z,D)=\kappa_{\iota}(X/Z,D)$ directly follows. From
these discussions, we have
$\hat{\kappa}_{\sigma}(X/Z,D)=\kappa_{\sigma}(X/Z,D)$ and
$\hat{\kappa}_{\iota}(X/Z,D)=\kappa_{\iota}(X/Z,D)$. We finish the proof. ∎

######  Remark 5.4.

Lemma 5.3 for $\mathbb{Q}$-divisors is well-known. On the other hand, when $D$
has irrational coefficients, we may have $\kappa_{\iota}(X/Z,D)\geq 0$ though
$H^{0}(F,\mathcal{O}_{F}(\llcorner mD|_{F}\lrcorner))=\\{0\\}$ for all
$m\in\mathbb{Z}_{>0}$ and all fibers $F$ of $X\to Z$ (for example, put
$X=\mathbb{P}^{1}_{\mathbb{C}}$, $Z={\rm Spec}\mathbb{C}$ and
$D=\sqrt{2}(p_{1}-p_{2})$ for distinct closed points $p_{1}$ and $p_{2}$),
hence the standard argument with
$H^{0}(X_{\bar{\eta}},\mathcal{O}_{X_{\bar{\eta}}}(\llcorner
mD|_{X_{\bar{\eta}}}\lrcorner))$ does not work well.

Thanks to Lemma 5.3, in the results of this paper we may assume that the
property of being log abundant is defined with Definition 5.2. Then, all
results of this paper hold for varieties over an algebraically closed field
${{k}}$ of characteristic zero.

## References

  * [1] D. Abramovich, K. Karu, Weak semistable reduction in characteristic $0$, Invent. math. 139 (2000), no. 2, 241–273.
  * [2] F. Ambro, J. Kollár, Minimal models of semi-log-canonical pairs, preprint (2017), arXiv:1709.03540.
  * [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, 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.
  * [5] C. Birkar, Z. Hu, Polarized pairs, log minimal models, and Zariski decompositions, Nagoya Math. J. 215 (2014), 203–224.
  * [6] C. Birkar, Z. Hu, Log canonical pairs with good augmented base loci, Compos. Math. 150 (2014), no. 4,579–592.
  * [7] S. R. Choi, The geography of log models and its applications, PhD Thesis, Johns Hopkins University, 2008.
  * [8] O. Fujino, Base Point Free Theorem of Reid-Fukuda Type, J. Math. Sci. Univ. Tokyo 7 (2000), 1–5.
  * [9] O. Fujino, What is log terminal? In Flips for $3$-folds and $4$-folds, Oxford University Press (2007).
  * [10] O. Fujino, Special termination and reduction to pl flips. In Flips for $3$-folds and $4$-folds, Oxford University Press (2007).
  * [11] O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789.
  * [12] O. Fujino, Basepoint-free theorems: saturation, $b$-divisors, and canonical bundle formula, Algebra Number Theory 6 (2012), no. 4, 797–823.
  * [13] O. Fujino, Fundamental theorems for semi log canonical pairs, Algebraic Geom. 1 (2014), no. 2, 194–228.
  * [14] O. Fujino, Foundations of the minimal model program, MSJ Mem. 35, Mathematical Society of Japan, Tokyo, 2017.
  * [15] O. Fujino, Y. Gongyo, On canonical bundle formulas and subadjunctions, Michigan Math. J. 61 (2012), no. 2, 255–264.
  * [16] O. Fujino, Y. Gongyo, Log pluricanonical representations and abundance conjecture, Compos. Math. 150 (2014) no. 4, 593–620.
  * [17] S. Fukuda, A base point free theorem of Reid type. II, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 3, 32–34.
  * [18] Y. Gongyo, On the minimal model theory for dlt pairs of numerical log kodaira dimension zero, Math. Res. Lett., 18 (2011) , no. 5, 991–1000.
  * [19] Y. Gongyo, B. Lehmann, Reduction maps and minimal model theory, Compos. Math. 149 (2013), no. 2, 295–308.
  * [20] 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.
  * [21] C. D. Hacon, J. Moraga, On weak Zariski decompositions and termination of flips, to appear in Math. Res. Lett.
  * [22] C. D. Hacon, C. Xu, Existence of log canonical closures, Invent. Math. 192 (2013), no. 1, 161–195.
  * [23] C. D. Hacon, C. Xu, On finiteness of $B$-representation and semi-log canonical abundance, Adv, Stud. Pure. Math. 70 (2016), Minimal Models and extremal rays–Kyoto, 2011, 361–378.
  * [24] J. Han, Z. Li, Weak Zariski decompositions and log terminal models for generalized polarized pairs, preprint (2018).
  * [25] K. Hashizume, Minimal model theory for relatively trivial log canonical pairs, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 2069–2107.
  * [26] K. Hashizume, Remarks on special kinds of the relative log minimal model program, Manuscripta Math. 160 (2019), no. 3, 285–314.
  * [27] K. Hashizume, A class of singularity of arbitrary pairs and log canonicalizations, Asian J. Math, 24 (2020), no. 2, 207–238.
  * [28] K. Hashizume, Z. Hu, On minimal model theory for log abundant lc pairs, J. Reine Angew. Math., 767 (2020), 109–159.
  * [29] Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567–588.
  * [30] J. Kollár, Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200. Cambridge University Press, Cambridge, 2013.
  * [31] J. Kollár, S. Kovács, Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), no. 3, 791–813.
  * [32] 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.
  * [33] C. J. Lai, Varieties fibered by good minimal models, Math. Ann. 350 (2011), no. 3, 533–547.
  * [34] V. Lazić, N. Tsakanikas, On the existence of minimal models for log canonical pairs, to appear in Publ. Res. Inst. Math. Sci.
  * [35] N. Nakayama, Zariski-decomposition and abundance, MSJ Mem. 14, Mathematical Society of Japan, Tokyo, 2004.
  * [36] M. Reid, Commentary by M. Reid, in 3-fold log flips, Appendix by Yujiro Kawamata: The minimal discrepancy coefficients of terminal singularities in dimension three, Russ. Acad. Sci., Izv., Math. 40 (1993), no. 1, 95–202.