scaling of an effective $\mathbb{R}$-divisor $A$. Let $\\{Y_{i}\\}_{i\geq 0}$
and $\\{\phi_{i}\colon X_{i}\dashrightarrow X^{\prime}_{i}\\}_{i\geq 0}$ be
the data of the log MMP. Let $S$ be an lc center of $(X,\Delta)$ and
$(S,\Delta_{S})$ the dlt pair defined by adjunction
$K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$. Suppose that for every $i\geq 0$,
after we shrink $Y$ to $Y_{i}$ so that the $(K_{X_{0}}+\Delta_{0})$-MMP
$(X_{0},\Delta_{0})\dashrightarrow(X_{i},\Delta_{i})$ is represented by a
bimeromorphic contraction $X_{0}\dashrightarrow X_{i}$, the isomorphic locus
of $X_{0}\dashrightarrow X_{i}$ intersects general points of $S$. Let $S_{i}$
be the corresponding lc center of $(X_{i},\Delta_{i})$, and let
$(S_{i},\Delta_{S_{i}})$ be the dlt pair defined by adjunction. Suppose the
following conditions:

  * •

For any $i$ and lc center $C_{i}$ of $(X_{i},\Delta_{i})$ whose dimension is
less than ${\rm dim}\,S$, the restriction of $\phi_{i}$ to $C_{i}$ is
biholomorphic to its image, and

  * •

if we put $\lambda_{i}:={\rm inf}\\{t\in\mathbb{R}_{\geq
0}\,|\,\text{$K_{X_{i}}+\Delta_{i}+tA_{i}$ {\rm is nef over} $W$}\\}$, then
${\rm lim}_{i\to\infty}\lambda_{i}=0$.

Then there exists an $i_{0}$ satisfying the following; If
$(S_{i_{0}},\Delta_{i_{0}})$ has a log minimal model over $Y$ around $W$ after
shrinking $Y$, then the $(i+1)$-th step of the $(K_{X}+\Delta)$-MMP does not
modify a Zariski open neighborhood of $S_{i}$ for all $i\gg i_{0}$.

###  3.7. Lift of MMP

In this subsection, we discuss the lift of MMP ([B12, Remark 2.9]).

Let $\pi\colon X\to Y$ be a contraction from a normal analytic variety $X$ to
a Stein space $Y$, and let $W\subset Y$ be a connected compact subset such
that $\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc pair, and let

$(X_{0},\Delta_{0})\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{n},\Delta_{n})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{X}+\Delta)$-MMP over $Y$ around $W$ with
scaling of an effective $\mathbb{R}$-divisor $A$.

By shrinking $Y$ around $W$, we may assume that the first step of the MMP is
represented by a bimeromorphic contraction. Over $Y$, we have a diagram

---
$\textstyle{(X_{0},\Delta_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\scriptstyle{f_{0}}$$\textstyle{(X_{1},\Delta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}_{0}}$$\textstyle{Z_{0}}$

where $f_{0}$ is the $(K_{X_{0}}+\Delta_{0})$-negative extremal contraction.
Note that $K_{X_{1}}+\Delta_{1}$ is ample over $Z_{0}$. By [F22a, Theorem
1.21] and shrinking $Y$ around $W$, we get a dlt blow-up
$g_{0}\colon(\widetilde{X}_{0},\widetilde{\Delta}_{0})\to(X_{0},\Delta_{0})$
of $(X_{0},\Delta_{0})$ such that $\widetilde{X}_{0}$ is
$\mathbb{Q}$-factorial over $W$. Let $H_{Z_{0}}$ be a Cartier divisor on
$Z_{0}$ which is ample over $Y$, and let $H_{0}\geq 0$ be a
$\mathbb{Q}$-Cartier divisor on $X_{0}$ such that
$H_{0}\sim_{\mathbb{R}}(3\cdot{\rm dim}\,X)f_{0}^{*}H_{Z_{0}}$. We put
$H_{{1}}=\phi_{0*}H_{0}$. By choosing $H_{0}$ appropriately, we may assume
that $(X_{0},\Delta_{0}+H_{0})$ and $(X_{1},\Delta_{1}+H_{1})$ are lc pairs.
Shrinking $Y$ around $W$, we may assume that $K_{X_{1}}+\Delta_{1}+H_{1}$ is
ample over $Y$, and $(X_{1},\Delta_{1}+H_{1})$ is a weak lc model of both
$(X_{0},\Delta_{0}+H_{0})$ and
$(\widetilde{X}_{0},\widetilde{\Delta}_{0}+g_{0}^{*}H_{0})$.

Put $\widetilde{H}_{0}:=g_{0}^{*}H_{0}$. We run a
$(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0}+\widetilde{H}_{0})$-MMP over
$Y$ around $W$ with scaling of an ample divisor. By Theorem 3.13 and Theorem
3.15 and shrinking $Y$ around $W$, the
$(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0}+\widetilde{H}_{0})$-MMP
terminates with a log minimal model
$(\widetilde{X}_{k_{1}},\widetilde{\Delta}_{k_{1}}+\widetilde{H}_{k_{1}})$
over $Y$ around $W$. By Lemma 3.3,
$K_{\widetilde{X}_{k_{1}}}+\widetilde{\Delta}_{k_{1}}+\widetilde{H}_{k_{1}}$
is semi-ample over $Y$ around $W$. Therefore, the divisor defines a
contraction over $Y$, and the target of the contraction is biholomorphic to
$X_{1}$ because $K_{X_{1}}+\Delta_{1}+H_{1}$ is ample over $Y$. Let
$g_{1}\colon\widetilde{X}_{k_{1}}\to X_{1}$ be the contraction. Then
$K_{\widetilde{X}_{k_{1}}}+\widetilde{\Delta}_{k_{1}}+\widetilde{H}_{k_{1}}=g_{1}^{*}(K_{X_{1}}+\Delta_{1}+H_{1})$
and $\widetilde{H}_{k_{1}}=g_{1}^{*}H_{1}$. Thus we have
$K_{\widetilde{X}_{k_{1}}}+\widetilde{\Delta}_{k_{1}}=g_{1}^{*}(K_{X_{1}}+\Delta_{1})$.
Moreover, $(\widetilde{X}_{k_{1}},\widetilde{\Delta}_{k_{1}})$ is dlt and
$\widetilde{X}_{k_{1}}$ is $\mathbb{Q}$-factorial over $W$. In this way, we
get a diagram

---
$\textstyle{(\widetilde{X}_{0},\widetilde{\Delta}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\textstyle{(\widetilde{X}_{k_{1}},\widetilde{\Delta}_{k_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{(X_{0},\Delta_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\scriptstyle{f_{0}}$$\textstyle{(X_{1},\Delta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}_{0}}$$\textstyle{Z_{0},}$

such that $\widetilde{X}_{0}\dashrightarrow\widetilde{X}_{k_{1}}$ is a
sequence of steps of a
$(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0}+\widetilde{H}_{0})$-MMP over
$Y$ around $W$ with scaling of an ample divisor.

We set $\widetilde{A}_{0}:=g_{0}^{*}A_{0}$. We check that
$(\widetilde{X}_{0},\widetilde{\Delta}_{0})\dashrightarrow(\widetilde{X}_{k_{1}},\widetilde{\Delta}_{k_{1}})$
is a sequence of steps of a
$(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0})$-MMP over $Y$ around $W$ with
scaling of $\widetilde{A}_{0}$. By applying Lemma 3.14 to
$\widetilde{X}_{0}\to Z_{0}\to Y$, we see that the
$(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0}+\widetilde{H}_{0})$-MMP is also
a $(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0})$-MMP over $Y$ around $W$.
Put

$\lambda_{0}:={\rm inf}\\{t\in\mathbb{R}_{\geq
0}\,|\,\text{$K_{X_{0}}+\Delta_{0}+tA_{0}$ {\rm is nef over} $W$}\\}.$

Then $K_{X_{0}}+\Delta_{0}+\lambda_{0}A_{0}$ is nef over $W$, and we get
$K_{X_{0}}+\Delta_{0}+\lambda_{0}A_{0}\sim_{\mathbb{R},Z_{0}}0$ after
shrinking $Y$ around $W$. This fact implies that
$(\widetilde{X}_{0},\widetilde{\Delta}_{0})\dashrightarrow(\widetilde{X}_{k_{1}},\widetilde{\Delta}_{k_{1}})$
is a sequence of steps of a
$(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0})$-MMP over $Y$ around $W$ with
scaling of $\widetilde{A}_{0}$.

Repeating the above discussion, we get

$\textstyle{(\widetilde{X}_{0},\widetilde{\Delta}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\textstyle{(\widetilde{X}_{k_{1}},\widetilde{\Delta}_{k_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{X}_{k_{i}},\widetilde{\Delta}_{k_{i}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{i}}$$\textstyle{\cdots}$$\textstyle{(X_{0},\Delta_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X_{1},\Delta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X_{i},\Delta_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

such that

  * •

the sequence of upper horizontal arrows is a sequence of steps of a
$(K_{\widetilde{X}_{0}}+\widetilde{\Delta}_{0})$-MMP over $Y$ around $W$ with
scaling of $\widetilde{A}_{0}$, and

  * •

each $g_{i}\colon\widetilde{X}_{k_{i}}\to X_{i}$ is a dlt blow-up of
$(X_{i},\Delta_{i})$ and $\widetilde{X}_{k_{i}}$ are $\mathbb{Q}$-factorial
over $W$.

Note that the arrows are not necessarily a bimeromorphic maps (see Definition
3.5) and
$(\widetilde{X}_{k_{i}},\Delta_{k_{i}})\dashrightarrow(\widetilde{X}_{k_{i+1}},\Delta_{k_{i+1}})$
is not necessarily one step of a
$(K_{\widetilde{X}_{k_{i}}}+\Delta_{k_{i}})$-MMP. We call it the lift of the
$(K_{X}+\Delta)$-MMP.

##  4\. Asymptotic vanishing order and Nakayama–Zariski decomposition

In this section, we study relations between the Nakayama–Zariski decomposition
and existence of log minimal models for lc pairs. For the statements in the
algebraic case, see [BH14] and [H20, Subsection 2.4].

###  4.1. Definitions and basic properties

In this subsection, we collect the definition and basic properties of
Nakayama–Zariski decomposition. We freely use the results in [Na04, III, §4].
For the algebraic case, see [LX23].

######  Definition 4.1.

Let $\pi\colon X\to Y$ be a projective morphism from a non-singular analytic
variety $X$ to an analytic space $Y$. Let $D$ be a $\pi$-big
$\mathbb{R}$-Cartier divisor on $X$. Let $P$ be a prime divisor on $X$. We
define $m_{P}$ to be $\infty$ if $\pi_{*}\mathcal{O}_{X}(\lfloor D\rfloor)=0$,
and otherwise

$m_{P}(D):={\rm max}\set{n\in\mathbb{Z}_{\geq
0}}{\text{$\pi_{*}\mathcal{O}_{X}(\lfloor D\rfloor-
nP)\hookrightarrow\pi_{*}\mathcal{O}_{X}(\lfloor D\rfloor)$ is an
isomorphism}}.$

We set $\sigma_{P}(D;X/Y)_{\mathbb{Z}}:=m_{P}(D)+{\rm coeff}_{P}(\\{D\\})$,
and we define

$\sigma_{P}(D;X/Y):=\underset{m\to\infty}{\rm
liminf}\frac{1}{m}\sigma_{P}(mD;X/Y)_{\mathbb{Z}}.$

######  Remark 4.2.

By the definition of $\pi$-bigness ([F22a, Definition 2.46]) and the standard
argument of divisors (cf. [Na04, II, 3.17. Corollary]), there exists
$m_{0}\in\mathbb{Z}_{>0}$ such that $\pi_{*}\mathcal{O}_{X}(\lfloor
mD\rfloor)\neq 0$ for all $m\geq m_{0}$. Thus the equality

$\sigma_{P}(D;X/Y):=\underset{m\to\infty}{\rm
lim}\frac{1}{m}\sigma_{P}(mD;X/Y)_{\mathbb{Z}}$

holds.

######  Lemma 4.3.

Let $\pi\colon X\to Y$ be a projective morphism from a non-singular analytic
variety $X$ to an analytic space $Y$. Let $D$ be a $\pi$-big
$\mathbb{R}$-Cartier divisor on $X$ and let $P$ be a prime divisor on $X$. If
$Y$ is Stein, then the equalities

$\sigma_{P}(D;X/Y)={\rm inf}\set{{\rm
coeff}_{P}(E)}{E\in|D/Y|_{\mathbb{R}}}={\rm inf}\set{{\rm
coeff}_{P}(E)}{E\in|D|_{\mathbb{R}}}$

hold.

###### Proof.

By Cartan’s Theorem A ([F22a, Theorem 2.6 (1)]), for any $m\in\mathbb{Z}_{>0}$
such that $\pi_{*}\mathcal{O}_{X}(\lfloor mD\rfloor)\neq 0$, the equalities

$\begin{split}\sigma_{P}(mD;X/Y)_{\mathbb{Z}}=&{\rm inf}\\{{\rm
coeff}_{P}(E)\,|\,E\geq 0,\,E\sim\lfloor mD\rfloor\\}+{\rm
coeff}_{P}(\\{mD\\})\\\ =&{\rm inf}\\{{\rm
coeff}_{P}(E^{\prime})\,|\,E^{\prime}\geq 0,\,E^{\prime}\sim mD\\}\end{split}$

hold (see also the discussion in [Na04, III, §4.a]). From this, we have

$\begin{split}\sigma_{P}(D;X/Y)=&{\rm inf}\\{{\rm coeff}_{P}(E)\,|\,E\geq
0,\,E\sim_{\mathbb{Q}}D\\}.\end{split}$

This implies that the following properties hold:

  * •

For any $t\in\mathbb{Q}_{>0}$, we have
$\sigma_{P}(tD;X/Y)=t\sigma_{P}(D;X/Y)$.

  * •

If $D=D^{\prime}+B$, where $D^{\prime}$ is a $\pi$-big $\mathbb{R}$-Cartier
divisor and $B$ is effective, then

$\sigma_{P}(D;X/Y)\leq\sigma_{P}(D^{\prime};X/Y)+{\rm coeff}_{P}(B).$

  * •

If $D=D_{1}+D_{2}$, where $D_{1}$ and $D_{2}$ are $\pi$-big
$\mathbb{R}$-Cartier divisors, then

$\sigma_{P}(D;X/Y)\leq\sigma_{P}(D_{1};X/Y)+\sigma_{P}(D_{2};X/Y).$

We will check that $\sigma_{P}(D;X/Y)=0$ when $D$ is $\pi$-ample. When $D$ is
$\pi$-ample, we can write $D=\sum_{i=1}^{l}r_{i}A_{i}$ for some
$r_{i}\in\mathbb{R}_{>0}$ and $\pi$-ample Cartier divisors $A_{i}$ on $X$. By
the third property, it is sufficient to check that
$\sigma_{P}(r_{i}A_{i};X/Y)=0$ holds for every $1\leq i\leq l$. By [Na04, III,
4.1. Lemma] and shrinking $Y$ around a point of $\pi(P)$, we may assume that
every $A_{i}$ has only finitely many components. Then the equality
$\sigma_{P}(r_{i}A_{i};X/Y)=0$ follows from the standard argument. In this
way, we have $\sigma_{P}(D;X/Y)=0$ when $D$ is $\pi$-ample.

Since $\sigma_{P}(D;X/Y)={\rm inf}\\{{\rm coeff}_{P}(E)\,|\,E\geq
0,\,E\sim_{\mathbb{Q}}D\\}$, it is easy to see that

$\sigma_{P}(D;X/Y)\geq{\rm inf}\\{{\rm
coeff}_{P}(E)\,|\,E\in|D|_{\mathbb{R}}\\}.$

We also have

${\rm inf}\\{{\rm coeff}_{P}(E)\,|\,E\in|D|_{\mathbb{R}}\\}\geq{\rm
inf}\\{{\rm coeff}_{P}(E)\,|\,E\in|D/Y|_{\mathbb{R}}\\}.$

Pick $G\geq 0$ such that $D-G$ is $\pi$-ample. For any
$E\in|D/Y|_{\mathbb{R}}$ and $\epsilon\in\mathbb{Q}_{>0}$, the divisor
$D-E+\epsilon(D-G)$ is $\pi$-ample, and therefore we have

$\begin{split}(1+\epsilon)\sigma_{P}(D;X/Y)=&\sigma_{P}((1+\epsilon)D;X/Y)\\\
=&\sigma_{P}(D-E+\epsilon(D-G)+(E+\epsilon G);X/Y)\\\
\leq&\sigma_{P}((D-E+\epsilon(D-G));X/Y)+{\rm coeff}_{P}(E)+\epsilon\cdot{\rm
coeff}_{P}(G)\\\ =&{\rm coeff}_{P}(E)+\epsilon\cdot{\rm
coeff}_{P}(G).\end{split}$

Taking the limit $\epsilon\to 0$, we have $\sigma_{P}(D;X/Y)\leq{\rm
coeff}_{P}(E)$ for any $E\in|D/Y|_{\mathbb{R}}$. Hence

${\rm inf}\\{{\rm
coeff}_{P}(E)\,|\,E\in|D/Y|_{\mathbb{R}}\\}\geq\sigma_{P}(D;X/Y).$

Therefore Lemma 4.3 holds. ∎

######  Remark 4.4.

In Lemma 4.3, the Steinness of base space is crucial. Even in the algebraic
case [LX23, Section 3], the quasi-projectivity of the base variety is crucial.
For example, consider a complete but not projective variety $Y$ with the
trivial Picard group (see, for example, [CLS11, Example 4.2.13]). Let $X\to Y$
be a blow-up at a smooth point, and let $E$ be the exceptional divisor. Then
$-E$ is clearly big over $Y$, but there is no effective $\mathbb{R}$-divisor
$E^{\prime}$ such that $E^{\prime}\sim_{\mathbb{R},\,Y}-E$ because $Y$ has the
trivial Picard group. In this case, the invariant

${\rm inf}\set{{\rm coeff}_{P}(E^{\prime})}{E^{\prime}\in|-E/Y|_{\mathbb{R}}}$

cannot be defined since $|-E/Y|_{\mathbb{R}}$ is empty.

######  Lemma 4.5.

Let $\pi\colon X\to Y$ be a projective morphism from a non-singular analytic
variety $X$ to an analytic space $Y$. Let $D$ be a $\pi$-big
$\mathbb{R}$-Cartier divisor on $X$, $P$ a prime divisor on $X$, and $A$ a
$\pi$-big $\mathbb{R}$-divisor on $X$. If $Y$ is Stein, then the equality

$\sigma_{P}(D;X/Y)=\underset{\epsilon\to 0+}{\rm lim}\sigma_{P}(D+\epsilon
A;X/Y)$

holds.

###### Proof.

By Lemma 4.3, for any $\epsilon\in\mathbb{R}_{>0}$ we have

$\sigma_{P}(D+\epsilon A;X/Y)\leq\sigma_{P}(D;X/Y)+\epsilon\sigma_{P}(A;X/Y).$

By taking the limit, we have

$\underset{\epsilon\to 0+}{\rm lim}\sigma_{P}(D+\epsilon
A;X/Y)\leq\sigma_{P}(D;X/Y).$

On the other hand, take $r\in\mathbb{R}_{>0}$ such that
$rD-A\sim_{\mathbb{R}}E$ for some effective $\mathbb{R}$-divisor $E$ on $X$
([F22a, Lemma 2.53]). Then, for any $\epsilon\in\mathbb{R}_{>0}$ we have

$\begin{split}\sigma_{P}(D;X/Y)\leq(1+r\epsilon)\sigma_{P}(D;X/Y)=&\sigma_{P}(D+\epsilon
A+\epsilon(rD-A);X/Y)\\\ \leq&\sigma_{P}(D+\epsilon A;X/Y)+\epsilon\cdot{\rm
coeff}_{P}(E).\end{split}$

From this, we have

$\sigma_{P}(D;X/Y)\leq\underset{\epsilon\to 0+}{\rm lim}\sigma_{P}(D+\epsilon
A;X/Y).$

Therefore Lemma 4.5 holds true. ∎

######  Definition 4.6 (Asymptotic vanishing order, [Na04, III, §4]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$. Let $D$ be a $\pi$-pseudo-effective
$\mathbb{R}$-Cartier divisor on $X$, and let $P$ be a prime divisor over $X$.
Then the asymptotic vanishing order of $P$ over $Y$, denoted by
$\sigma_{P}(D;X/Y)$, is defined as follows: We take a resolution $f\colon
X^{\prime}\to X$ of $X$ such that $P$ appears as a prime divisor on
$X^{\prime}$, and we take a $(\pi\circ f)$-ample $\mathbb{R}$-divisor
$A^{\prime}$ on $X^{\prime}$. Then we define

$\sigma_{P}(D;X/Y):=\underset{\epsilon\to 0+}{\rm
lim}\sigma_{P}(f^{*}D+\epsilon A^{\prime};X^{\prime}/Y).$

As the algebraic case, we can check that $\sigma_{P}(D;X/Y)$ does not depend
on $f\colon X^{\prime}\to X$ and $A^{\prime}$. However, we may have
$\sigma_{P}(D;X/Y)=\infty$.

######  Theorem 4.7.

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $P$ be a prime divisor over $X$. Then the
following properties hold.

  1. (1)

For a $\pi$-pseudo-effective $\mathbb{R}$-Cartier divisor $D$ on $X$ and a
positive real number $r$, we have

$\sigma_{P}(rD;X/Y)=r\sigma_{P}(D;X/Y).$

  2. (2)

Let $W\subset Y$ be a subset. Suppose that $P$ is a prime divisor on $X$ and
$\pi(P)\cap W\neq\emptyset$. For any $\mathbb{R}$-Cartier divisor $D$ on $X$,
if $D$ is a limit of movable divisors over $W$ then

$\sigma_{P}(D;X/Y)=0.$

  3. (3)

For two $\pi$-pseudo-effective $\mathbb{R}$-Cartier divisors $D_{1}$ and
$D_{2}$ on $X$, we have

$\sigma_{P}(D_{1}+D_{2};X/Y)\leq\sigma_{P}(D_{1};X/Y)+\sigma_{P}(D_{2};X/Y).$

  4. (4)

For two $\pi$-pseudo-effective $\mathbb{R}$-Cartier divisors $D_{1}$ and
$D_{2}$ on $X$, we have

$\sigma_{P}(D_{1};X/Y)=\underset{\epsilon\to 0+}{\rm
lim}\sigma_{P}(D_{1}+\epsilon D_{2};X/Y).$

  5. (5)

Let $f\colon X^{\prime}\to X$ be a projective bimeromorphism from a normal
analytic variety $X^{\prime}$. For a $\pi$-pseudo-effective
$\mathbb{R}$-Cartier divisor $D$ on $X$ and an effective $f$-exceptional
$\mathbb{R}$-Cartier divisor $E$ on $X^{\prime}$, we have

$\sigma_{P}(f^{*}D+E;X^{\prime}/Y)=\sigma_{P}(D;X/Y)+{\rm ord}_{P}(E).$

  6. (6)

Suppose that $P$ is a prime divisor on $X$. Let $U\subset Y$ be a Stein open
subset such that $U\cap\pi(P)\neq\emptyset$. We put $X_{U}:=\pi^{-1}(U)$ and
let $P|_{X_{U}}=\bigcup_{j}P_{j}$ be the irreducible decomposition. For any
$\pi$-pseudo-effective $\mathbb{R}$-Cartier divisor $D$ on $X$, all $P_{j}$
satisfy

$\sigma_{P}(D;X/Y)=\sigma_{P_{j}}(D|_{X_{U}};X_{U}/U).$

###### Proof.

The properties (1)–(5) easily follow as in the algebraic case. Thus we only
need to check (6). By Definition 4.6, we may assume that $D$ is $\pi$-big. By
replacing $X$ with a resolution of $X$, we may assume that $X$ is non-
singular. By Definition 4.1, it is sufficient to prove that the equality
$m_{P}(lD)=m_{P_{j}}(lD|_{X_{U}})$ holds for all $j$ and $l\gg 0$. By Cartan’s
Theorem A ([F22a, Theorem 2.6 (1)]), for any $l\in\mathbb{Z}_{>0}$ such that
$\pi_{*}\mathcal{O}_{X}(\lfloor lD\rfloor)\neq 0$, the equalities

$\begin{split}m_{P}(lD)=&{\rm inf}\\{{\rm coeff}_{P}(E)\,|\,E\geq
0,\,E\sim\lfloor lD\rfloor\\},\qquad{\rm and}\\\ m_{P_{j}}(lD|_{X_{U}})=&{\rm
inf}\\{{\rm coeff}_{P_{j}}(E^{\prime})\,|\,E^{\prime}\geq
0,\,E^{\prime}\sim\lfloor lD|_{X_{U}}\rfloor\\}.\end{split}$

We fix an index $j$. It is easy to check that

$m_{P}(lD)\geq m_{P_{j}}(lD|_{X_{U}}).$

We take an effective divisor $E^{\prime}\sim\lfloor lD|_{X_{U}}\rfloor$ on
$X_{U}$ such that ${\rm coeff}_{P_{j}}(E^{\prime})=m_{P_{j}}(lD|_{X_{U}})$. By
Cartan’s Theorem A ([F22a, Theorem 2.6 (1)]), there is an open subset
$V\subset U$, an effective divisor $E\sim\lfloor lD\rfloor$, and
$\varphi\in\mathcal{O}_{Y}(V)$ such that
$V\cap(\pi|_{X_{U}})(P_{j})\neq\emptyset$ and if we put $X_{V}:=\pi^{-1}(V)$
then $E|_{X_{V}}+{\rm div}(\pi^{*}\varphi)=E^{\prime}|_{X_{V}}$. Then ${\rm
coeff}_{P}(E)\leq{\rm coeff}_{P_{j}}(E^{\prime})$, and therefore we have

$m_{P}(lD)\leq m_{P_{j}}(lD|_{X_{j}}).$

Thus we have $m_{P}(lD)=m_{P_{j}}(lD|_{X_{j}})$. Therefore, (6) holds. ∎

######  Lemma 4.8 ([Na04, III, 4.2. Lemma]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$. Let $D$ be a $\pi$-pseudo-effective
$\mathbb{R}$-Cartier divisor on $X$ and let $P_{1},\,\cdots,\,P_{l}$ be
mutually distinct prime divisors on $X$. Then the following properties hold.

  * •

For any $1\leq i\leq l$ and any real number $s_{i}$ such that $0\leq
s_{i}\leq\sigma_{P_{i}}(D;X/Y)$, we have

$\sigma_{P_{i}}\Bigl{(}D-\sum_{j=1}^{l}s_{j}P_{j};X/Y\Bigr{)}=\sigma_{P_{i}}(D;X/Y)-s_{i}.$

  * •

If $\sigma_{P_{i}}(D;X/Y)>0$ for all $1\leq i\leq l$, then

$\sigma_{P_{i}}\Bigl{(}\sum_{j=1}^{l}r_{j}P_{j};X/Y\Bigr{)}=r_{i}$

for any nonnegative real numbers $r_{1},\,\cdots,\,r_{l}$.

######  Theorem 4.9.

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$. Let $W\subset Y$ be a compact subset. Fix a
resolution $\widetilde{X}\to X$ of $X$. Let $D$ be a $\pi$-pseudo-effective
$\mathbb{R}$-Cartier divisor on $X$ and let $l$ be (possibly infinite) the
number of prime divisors $P$ on $X$ satisfying $\pi(P)\cap W\neq\emptyset$ and
$\sigma_{P}(D;X/Y)>0$. Then the inequality $l\leq\rho(\widetilde{X}/Y;W)$
holds. In particular, if $W\cap Z$ has only finitely many connected components
for any analytic subset $Z\subset Y$ which is defined over an open
neighborhood of $W$ (see also the condition $({\rm P}4)$ in [F22a]), then
$l<\infty$.

###### Proof.

By replacing $X$ with $\widetilde{X}$, we may assume that $X$ is non-singular.
We may assume $\rho(\widetilde{X}/Y;W)<\infty$ because otherwise there is
nothing to prove. If $l>\rho(\widetilde{X}/Y;W)$, then there are prime
divisors $P_{1},\,\cdots,\,P_{l}$ on $X$ and non-zero elements
$a_{1},\,\cdots,\,a_{l}\in\mathbb{R}$ such that $\pi(P_{i})\cap
W\neq\emptyset$, $\sigma_{P_{i}}(D;X/Y)>0$ for all $1\leq i\leq l$, and
$\sum_{i=1}^{l}a_{i}P_{i}$ is numerically trivial over $W$. We put

$Q:=\sum_{a_{i}>0}a_{i}P_{i}\qquad{\rm and}\qquad
Q^{\prime}:=-\sum_{a_{i}<0}a_{i}P_{i}.$

Then neither $Q$ nor $Q^{\prime}$ is a zero divisor and
$Q\equiv_{W}Q^{\prime}$. By Theorem 4.7 (2), we have
$\sigma_{P_{1}}(Q-Q^{\prime};X/Y)=0$. By Theorem 4.7 (3) and Lemma 4.8, we
have

$\begin{split}0<&\sigma_{P_{1}}(Q;X/Y)=\sigma_{P_{1}}(Q^{\prime}+(Q-Q^{\prime});X/Y)\\\
\leq&\sigma_{P_{1}}(Q^{\prime};X/Y)+\sigma_{P_{1}}(Q-Q^{\prime};X/Y)=0+0=0.\end{split}$

Thus we get a contradiction. From this, Theorem 4.9 follows. ∎

######  Definition 4.10 (cf. [Na04, III, §4.a]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a subset. Let $D$ be a
$\pi$-pseudo-effective $\mathbb{R}$-Cartier divisor on $X$. Then
$N_{\sigma}(D;X/Y,W)$ is defined by the formal sum

$N_{\sigma}(D;X/Y,W):=\sum_{\begin{subarray}{c}P\,:\,{\rm
prime\,divisor\,on\,}X\\\
W\cap\pi(P)\neq\emptyset\end{subarray}}\sigma_{P}(D;X/Y)P.$

######  Lemma 4.11.

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a subset. Let $D$ be a
$\pi$-pseudo-effective $\mathbb{R}$-Cartier divisor on $X$. Then the following
properties hold.

  * •

If $D\sim_{\mathbb{R},Y}E$ for some effective $\mathbb{R}$-divisor on $X$,
then we have

$\sigma_{P}(D;X/Y)\leq{\rm coeff}_{P}(E)$

for all prime divisors $P$ on $X$. In particular, $N_{\sigma}(D;X/Y,W)$ is an
$\mathbb{R}$-divisor on $X$ and $N_{\sigma}(D;X/Y,W)\leq E$.

  * •

If $W$ is compact and $W\cap Z$ has only finitely many connected components
for any analytic subset $Z$ which is defined over an open neighborhood of $W$
(see also the condition $({\rm P}4)$ in [F22a]), then $N_{\sigma}(D;X/Y,W)$
has only finitely many components.

  * •

Let $U\subset Y$ be a Stein open subset containing $W$, and we put
$X_{U}=\pi^{-1}(U)$. If $N_{\sigma}(D;X/Y,W)$ is well defined as an
$\mathbb{R}$-divisor, then there exists a Zariski open subset $U^{\prime}$ of
$U$ such that $U^{\prime}\supset W$ and putting
$X_{U^{\prime}}=\pi^{-1}(U^{\prime})$ then

$N_{\sigma}(D;X/Y,W)|_{X_{U^{\prime}}}=N_{\sigma}(D|_{X_{U}};X_{U}/U,W)|_{X_{U^{\prime}}}$

as $\mathbb{R}$-divisor on $X_{U}$.

###### Proof.

The first property follows from Definition 4.6. The second property follows
from Theorem 4.9. The third property follows from Theorem 4.7 (6) and the fact
that $\sigma_{Q}(D|_{X_{U}};X_{U}/U)=0$ for any prime divisor $Q$ on $X_{U}$
that is not a component of the restriction $P|_{U}$ of any prime divisor $P$
on $X$. ∎

######  Definition 4.12 (Nakayama–Zariski decomposition).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a subset. Let $D$ be a
$\pi$-pseudo-effective $\mathbb{R}$-Cartier divisor on $X$. When
$N_{\sigma}(D;X/Y,W)$ is well defined as an $\mathbb{R}$-divisor on $X$, we
call it the negative part of Nakayama–Zariski decomposition over $W$. We set

$P_{\sigma}(D;X/Y,W):=D-N_{\sigma}(D;X/Y,W),$

and we call it the positive part of Nakayama–Zariski decomposition over $W$.

######  Theorem 4.13.

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a subset. Let $D$ be a
$\pi$-pseudo-effective $\mathbb{R}$-Cartier divisor on $X$. Let $f\colon
X^{\prime}\to X$ be a projective bimeromorphism from a normal variety
$X^{\prime}$. If $N_{\sigma}(D;X/Y,W)$ is well defined as an
$\mathbb{R}$-divisor on $X$ and $P_{\sigma}(D;X/Y,W)$ is nef over $W$, then
$N_{\sigma}(D;X/Y,W)$ is $\mathbb{R}$-Carter and the relation

$N_{\sigma}(f^{*}D;X^{\prime}/Y,W)=f^{*}N_{\sigma}(D;X/Y,W)$

holds. In particular, $N_{\sigma}(f^{*}D;X^{\prime}/Y,W)$ is well defined as
an $\mathbb{R}$-Cartier divisor on $X^{\prime}$, the relation
$f^{*}P_{\sigma}(D;X/Y,W)=P_{\sigma}(f^{*}D;X^{\prime}/Y,W)$ holds, and
$P_{\sigma}(f^{*}D;X^{\prime}/Y,W)$ is nef over $W$.

###### Proof.

Since $D$ and $P_{\sigma}(D;X/Y,W)$ are $\mathbb{R}$-Cartier,
$N_{\sigma}(D;X/Y,W)$ is $\mathbb{R}$-Cartier. For any prime divisor
$Q^{\prime}$ on $X^{\prime}$, we have

$\begin{split}\sigma_{Q^{\prime}}(f^{*}D;X^{\prime}/Y)=&\sigma_{Q^{\prime}}(f^{*}P_{\sigma}(D;X/Y,W)+f^{*}N_{\sigma}(D;X/Y,W);X^{\prime}/Y)\\\
\leq&\sigma_{Q^{\prime}}(f^{*}P_{\sigma}(D;X/Y,W);X^{\prime}/Y)+\sigma_{Q^{\prime}}(f^{*}N_{\sigma}(D;X/Y,W);X^{\prime}/Y)\\\
\leq&0+{\rm coeff}_{Q^{\prime}}(f^{*}N_{\sigma}(D;X/Y,W)),\end{split}$

where the final inequality follows from Theorem 4.7 (2) and Lemma 4.11. This
shows that $N_{\sigma}(f^{*}D;X^{\prime}/Y,W)$ is well defined as an
$\mathbb{R}$-divisor on $X^{\prime}$ and we have

$N_{\sigma}(f^{*}D;X^{\prime}/Y,W)\leq f^{*}N_{\sigma}(D;X/Y,W).$

To prove the converse relation, we may replace $f\colon X^{\prime}\to X$ by
$f\circ g\colon X^{\prime\prime}\to X^{\prime}\to X$ for some projective
bimeromorphism $g\colon X^{\prime\prime}\to X^{\prime}$ because

$g_{*}N_{\sigma}(g^{*}f^{*}D;X^{\prime\prime}/Y,W)=N_{\sigma}(f^{*}D;X^{\prime}/Y,W)$

holds by definition. Therefore, by replacing $f$ with a resolution of $X$, we
may assume that $X^{\prime}$ is non-singular. We put

$E^{\prime}=N_{\sigma}(f^{*}D;X^{\prime}/Y,W)-f^{*}N_{\sigma}(D;X/Y,W).$

Then $E^{\prime}$ is $f$-exceptional. We also have

$\begin{split}E^{\prime}\sim_{\mathbb{R},\,X}&N_{\sigma}(f^{*}D;X^{\prime}/Y,W)-f^{*}D=-P_{\sigma}(f^{*}D;X^{\prime}/Y,W),\end{split}$

and $P_{\sigma}(f^{*}D;X^{\prime}/Y,W)$ is a limit of movable divisors over
$W$. By applying the negativity lemma (Corollary 2.15) to $f\colon
X^{\prime}\to X$, $\pi^{-1}(W)$, and $E$, there is $U\supset\pi^{-1}(W)$ an
open subset of $X$ such that $E^{\prime}|_{f^{-1}(U)}\geq 0$. By construction,
any component $E^{\prime}_{i}$ of $E^{\prime}$ satisfies $(\pi\circ
f)(E^{\prime}_{i})\cap W\neq\emptyset$. Therefore $E^{\prime}\geq 0$, from
which we have

$N_{\sigma}(f^{*}D;X^{\prime}/Y,W)\geq f^{*}N_{\sigma}(D;X/Y,W).$

In this way, we see that Theorem 4.13 holds. ∎

######  Lemma 4.14 (cf. [H20, Lemma 2.4]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a subset. Let $D$ and
$D^{\prime}$ be $\pi$-pseudo-effective $\mathbb{R}$-Cartier divisors on $X$
such that $N_{\sigma}(D;X/Y,W)$ and $N_{\sigma}(D^{\prime};X/Y,W)$ are well
defined as $\mathbb{R}$-divisors on $X$. Then there exists a positive real
number $t_{0}$ such that $N_{\sigma}(D+tD^{\prime};X/Y,W)$ is well defined as
$\mathbb{R}$-divisors on $X$ and the support of
$N_{\sigma}(D+tD^{\prime};X/Y,W)$ is independent of $t\in(0,t_{0}]$.

###### Proof.

The argument in [H20, Proof of Lemma 2.4] works with no changes. ∎

######  Lemma 4.15 (cf. [H20, Lemma 2.6]).

Let $Y$ be a Stein space and $W\subset Y$ a compact subset. Let $(X,\Delta)$
and $(X^{\prime},\Delta^{\prime})$ be dlt pairs with projective morphisms
$\pi\colon X\to Y$ and $\pi^{\prime}\colon X^{\prime}\to Y$, and let $S$ and
$S^{\prime}$ be lc centers of $(X,\Delta)$ and $(X^{\prime},\Delta^{\prime})$
respectively. Let $f\colon X\dashrightarrow X^{\prime}$ be a bimeromorphic map
over $Y$ such that $f$ is a biholomorphism on a Zariski open subset
intersecting $S$ and $f|_{S}$ induces a bimeromorphic map $f_{S}\colon
S\dashrightarrow S^{\prime}$ over $Y$. Suppose that $K_{X}+\Delta$ is
$\pi$-pseudo-effective. Suppose in addition that

  * •

the inequality $a(D^{\prime},X^{\prime},\Delta^{\prime})\leq
a(D^{\prime},X,\Delta)$ holds for all prime divisors $D^{\prime}$ on
$X^{\prime}$ such that $\pi^{\prime}(D^{\prime})\cap W\neq\emptyset$, and

  * •

$\sigma_{P}(K_{X}+\Delta;X/Y)=0$ for all prime divisor $P$ over $X$ such that
$a(P,X,\Delta)<0$, the center of $P$ on $X$ intersects $S$, and the image of
$P$ on $Y$ intersects $W$.

Let $(S,\Delta_{S})$ and $(S^{\prime},\Delta_{S^{\prime}})$ be the dlt pairs
defined by adjunctions $K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$ and
$K_{S^{\prime}}+\Delta_{S^{\prime}}=(K_{X^{\prime}}+\Delta^{\prime})|_{S^{\prime}}$,
respectively. Then the inequality

$a(Q^{\prime},S^{\prime},\Delta_{S^{\prime}})\leq a(Q^{\prime},S,\Delta_{S})$

holds for any prime divisor $Q^{\prime}$ on $S^{\prime}$ such that
$\pi^{\prime}(Q^{\prime})\cap W\neq\emptyset$.

###### Proof.

We may apply [H20, Proof of Lemma 2.6] to our situation. We only outline the
proof.

Since the problem is local (see also Theorem 4.7 (6)), we may freely shrink
$Y$ around $W$ without loss of generality. By shrinking $Y$ around $W$, we may
assume that there is a common log resolution $g\colon\overline{X}\to X$ and
$g^{\prime}\colon\overline{X}\to X^{\prime}$ of $f\colon X\dashrightarrow
X^{\prime}$ and a subvariety $T\subset\overline{X}$ such that $g$ and
$g^{\prime}$ induce bimeromorphisms $g_{T}\colon T\to S$ and
$g^{\prime}_{T}\colon T\to S^{\prime}$, respectively. We may write

$g^{*}(K_{X}+\Delta)=g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})+M-N$

with $M\geq 0$ and $N\geq 0$ having no common components. By the first
condition of Lemma 4.15 and the argument in [H20, Proof of Lemma 2.6], $M$ is
$g^{\prime}$-exceptional over the inverse image of an open neighborhood of
$W$, and furthermore we have ${\rm Supp}\,M\not\supset T$ and ${\rm
Supp}\,N\not\supset T$ over the inverse image of an open neighborhood of $W$.
From these facts, we see that $K_{X^{\prime}}+\Delta^{\prime}$ is
$\pi$-pseudo-effective. By shrinking $Y$ around $W$ we may write

$g^{*}_{T}(K_{S}+\Delta_{S})=g^{\prime*}_{T}(K_{S^{\prime}}+\Delta_{S^{\prime}})+M|_{T}-N|_{T},$

where $M_{T}\geq 0$ and $N|_{T}\geq 0$ have no common components. By the same
argument as in [H20, Proof of Lemma 2.6] using Theorem 4.7, we have

$\sigma_{P}(K_{X}+\Delta;X/Y)>0$

for any component $P$ of $M$ that intersects $T$. By the second condition of
Lemma 4.15, we have

$a(P,X^{\prime},\Delta^{\prime})>a(P,X,\Delta)\geq 0$

for any component $P$ of $M$ such that $P\cap T\neq\emptyset$ and $(\pi\circ
g)(P)\cap W\neq\emptyset$. By shrinking $Y$ around $W$ and the standard
argument as in the proof of [H20, Lemma 2.5], it follows that $M|_{T}$ is
exceptional over $S^{\prime}$. Then, for any prime divisor $Q^{\prime}$ on
$S^{\prime}$, we have ${\rm coeff}_{g^{\prime-1}_{T}Q^{\prime}}(M|_{T})=0$. In
particular,

$a(Q^{\prime},S^{\prime},\Delta_{S^{\prime}})\leq a(Q^{\prime},S,\Delta_{S})$

holds for any prime divisor $Q^{\prime}$ on $S^{\prime}$ such that
$\pi^{\prime}(Q^{\prime})\cap W\neq\emptyset$. ∎

###  4.2. Criterion for existence of log minimal model

In this subsection, we study the existence of log minimal models for lc pairs
using the asymptotic vanishing order and the negative part of Nakayama–Zariski
decomposition.

######  Theorem 4.16 (cf. [BH14, Theorem 1.1], [T21, Theorem 4.18]).

Let $\pi\colon X\to Y$ be a contraction from a normal analytic variety $X$ to
a Stein space $Y$, and let $W\subset Y$ be a connected compact subset such
that $\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc pair. Then the
following conditions are equivalent.

  * •

After shrinking $Y$ around $W$, the lc pair $(X,\Delta)$ has a log minimal
model over $Y$ around $W$.

  * •

After shrinking $Y$ around $W$, there exists a resolution
$f\colon\widetilde{X}\to X$ of $X$ such that
$N_{\sigma}(f^{*}(K_{X}+\Delta);\widetilde{X}/Y,W)$ is well defined as an
$\mathbb{R}$-divisor on $\widetilde{X}$ and the divisor
$P_{\sigma}(f^{*}(K_{X}+\Delta);\widetilde{X}/Y,W)$ is a finite
$\mathbb{R}_{>0}$-linear combination of $\mathbb{Q}$-Cartier divisors on
$\widetilde{X}$ that are nef over $W$.

###### Proof.

Assume the first condition of Theorem 4.16. After shrinking $Y$ around $W$,
there exists a log minimal model $(X^{\prime},\Delta^{\prime})$ of
$(X,\Delta)$ over $Y$ around $W$. By the argument using Shokurov’s polytope
[F22b, Theorem 14.3 (3)], we can write

$K_{X^{\prime}}+\Delta^{\prime}=\sum_{i=1}^{l}r_{i}D^{\prime}_{i},$

where $r_{i}$ are positive real number and $D^{\prime}_{i}$ are
$\mathbb{Q}$-Cartier divisors on $X^{\prime}$ that are nef over $W$. Let
$g\colon X^{\prime\prime}\to X$ be a resolution of $X$ which resolves the
indeterminacy of $X\dashrightarrow X^{\prime}$, and let $g^{\prime}\colon
X^{\prime\prime}\to X^{\prime}$ be the induced bimeromorphism. After shrinking
$Y$ around $W$, we can write

$g^{*}(K_{X}+\Delta)=g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})+E^{\prime\prime}$

such that $E^{\prime\prime}$ is an effective $g^{\prime}$-exceptional
$\mathbb{R}$-divisor on $X^{\prime\prime}$. By Theorem 4.7 (2) (6),
$N_{\sigma}(g^{*}(K_{X}+\Delta);X^{\prime\prime}/Y,W)$ is well defined and we
can find an open subset $U\subset Y$ containing $W$ such that

$N_{\sigma}(g^{*}(K_{X}+\Delta);X^{\prime\prime}/Y,W)|_{(\pi\circ
g)^{-1}(U)}=E^{\prime\prime}|_{(\pi\circ g)^{-1}(U)}.$

Then

$P_{\sigma}(g^{*}(K_{X}+\Delta);X^{\prime\prime}/Y,W)|_{(\pi\circ
g)^{-1}(U)}=g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})|_{(\pi\circ
g)^{-1}(U)}=\sum_{i=1}^{l}r_{i}g^{\prime*}D^{\prime}_{i}|_{(\pi\circ
g)^{-1}(U)}.$

Thus $N_{\sigma}(g^{*}(K_{X}+\Delta);X^{\prime\prime}/Y,W)$ is well defined as
an $\mathbb{R}$-divisor on $X^{\prime\prime}$ and the divisor
$P_{\sigma}(g^{*}(K_{X}+\Delta);X^{\prime\prime}/Y,W)$ is a finite
$\mathbb{R}_{>0}$-linear combination of $\mathbb{Q}$-Cartier divisors on
$X^{\prime\prime}$ that are nef over $W$. From this, we see that the first
condition of Theorem 4.16 implies the second condition of Theorem 4.16.

Conversely, suppose that the second condition of Theorem 4.16 holds. After
shrinking $Y$ around $W$, we get a resolution $f\colon\widetilde{X}\to X$ of
$X$ such that $N_{\sigma}(f^{*}(K_{X}+\Delta);\widetilde{X}/Y,W)$ is well
defined as an $\mathbb{R}$-divisor on $\widetilde{X}$ and we can write

$P_{\sigma}(f^{*}(K_{X}+\Delta);\widetilde{X}/Y,W)=\sum_{i=1}^{k}\gamma_{i}M_{i},$

where $\gamma_{i}$ are positive real numbers and $M_{i}$ are
$\mathbb{Q}$-Cartier divisors on $\widetilde{X}$ that are nef over $W$. By
shrinking $Y$ around $W$, we get a bimeromorphism
$f^{\prime}\colon\widetilde{X}^{\prime}\to\widetilde{X}$ such that $f\circ
f^{\prime}\colon\widetilde{X}^{\prime}\to X$ is a log resolution of
$(X,\Delta)$. By Theorem 4.13 and replacing $f$ with $f\circ f^{\prime}$, we
may assume that $f\colon X^{\prime}\to X$ is a log resolution of $(X,\Delta)$.
By shrinking $Y$ around $W$ again, we may assume that there are only finitely
many $f$-exceptional prime divisors and $M_{i}$ are globally
$\mathbb{Q}$-Cartier. By replacing $\gamma_{i}$ with some multiple, we may
assume that $M_{i}$ are Cartier.

For the simplicity of notations, from now on we drop $\widetilde{X}/Y,W$ and
$\widetilde{X}^{\prime}/Y,W$ in the negative part and the positive part of the
Nakayama–Zariski decomposition.

Let $\widetilde{\Delta}$ be the sum of $f^{-1}_{*}\Delta$ and the reduced
$f$-exceptional divisor. Then we can write

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

where $E$ is effective and $f$-exceptional. By Theorem 4.7 (5), we have

$N_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta})=N_{\sigma}(f^{*}(K_{X}+\Delta))+E.$

Therefore, $N_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta})$ is well defined
as an $\mathbb{R}$-divisor on $\widetilde{X}$, and this shows

$P_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta})=P_{\sigma}(f^{*}(K_{X}+\Delta))=\sum_{i=1}^{k}\gamma_{i}M_{i}.$

Thus $P_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta})$ is a finite
$\mathbb{R}_{>0}$-linear combination of $\mathbb{Q}$-Cartier divisors on
$\widetilde{X}$ that are nef over $W$. Moreover, by Lemma 3.4, we may replace
$(X,\Delta)$ with $(\widetilde{X},\widetilde{\Delta})$. From these facts, by
replacing $(X,\Delta)$ with $(\widetilde{X},\widetilde{\Delta})$, we may
assume that

  * •

$(X,\Delta)$ is log smooth,

  * •

$N_{\sigma}(K_{X}+\Delta)$ is well defined as an $\mathbb{R}$-divisor on $X$,
and

  * •

$P_{\sigma}(K_{X}+\Delta)=\sum_{i=1}^{k}\gamma_{i}M_{i}$ for some Cartier
divisors $M_{i}$ on $X$ that are nef over $W$.

We follow the argument in [T21, Section 4.2]. We put
$M:=P_{\sigma}(K_{X}+\Delta)$. Let $\alpha$ be a positive real number such
that $\alpha\gamma_{i}>2\cdot{\rm dim}\,X$ for all $i$. We run a
$(K_{X}+\Delta+\alpha M)$-MMP over $Y$ around $W$ with scaling of an ample
divisor. Note that we may run the MMP since $M$ is nef over $W$. By the same
argument as in [F22a, Proof of Lemma 13.7], after shrinking $Y$ around $W$ we
get a bimeromorphic contraction

$(X,\Delta+\alpha
M)\overset{\phi}{\dashrightarrow}(\overline{X},\overline{\Delta}+\alpha\overline{M})$

such that $K_{\overline{X}}+\overline{\Delta}+\alpha\overline{M}$ is the limit
of movable divisors over a neighborhood of $W$. Then

$N_{\sigma}(K_{\overline{X}}+\overline{\Delta}+\alpha\overline{M})=0.$

By Lemma 4.8, we have

$N_{\sigma}(K_{X}+\Delta+\alpha M)=N_{\sigma}((\alpha+1)(K_{X}+\Delta)-\alpha
N_{\sigma}(K_{X}+\Delta))=N_{\sigma}(K_{X}+\Delta).$

This implies

$P_{\sigma}(K_{X}+\Delta+\alpha M)=K_{X}+\Delta+\alpha
M-N_{\sigma}(K_{X}+\Delta+\alpha M)=(\alpha+1)M.$

Therefore

$\begin{split}K_{X}+\Delta+\alpha M=(\alpha+1)M+N_{\sigma}(K_{X}+\Delta+\alpha
M).\end{split}$

By the definition of $\alpha$, the strict transform of $(\alpha+1)M$ is
trivial with respect to the extremal contraction in each step of the
$(K_{X}+\Delta+\alpha M)$-MMP. This shows that $(\alpha+1)\phi_{*}M$ is nef
over $W$ and $\phi\colon X\dashrightarrow\overline{X}$ is a sequence of steps
of a $(K_{X}+\Delta)$-MMP over $Y$ around $W$. By the same argument as in
[BH14, Proof of Lemma 4.1], we have

$P_{\sigma}(K_{\overline{X}}+\overline{\Delta}+\alpha\overline{M})=\phi_{*}P_{\sigma}(K_{X}+\Delta+\alpha
M)=(\alpha+1)\phi_{*}M.$

From these relations, we have

$K_{\overline{X}}+\overline{\Delta}+\alpha\overline{M}=P_{\sigma}(K_{\overline{X}}+\overline{\Delta}+\alpha\overline{M})+N_{\sigma}(K_{\overline{X}}+\overline{\Delta}+\alpha\overline{M})=(\alpha+1)\phi_{*}M,$

and therefore $K_{\overline{X}}+\overline{\Delta}=\phi_{*}M$ is nef over $W$.
Then

$(X,\Delta)\overset{\phi}{\dashrightarrow}(\overline{X},\overline{\Delta})$

is a sequence of steps of a $(K_{X}+\Delta)$-MMP over $Y$ around $W$ to a log
minimal model. In particular, $(X,\Delta)$ has a log minimal model over $Y$
around $W$.

By the above discussion, it follows that the second condition of Theorem 4.16
implies the first condition of Theorem 4.16. Therefore, Theorem 4.16 holds
true. ∎

By the same argument, we can prove the following.

######  Theorem 4.17 (cf. [H20, Theorem 2.23]).

Let $\pi\colon X\to Y$ be a contraction from a normal analytic variety $X$ to
a Stein space $Y$, and let $W\subset Y$ be a connected compact subset such
that $\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc pair. Then the
following conditions are equivalent.

  * •

After shrinking $Y$ around $W$, the lc pair $(X,\Delta)$ has a good minimal
model over $Y$ around $W$.

  * •

After shrinking $Y$ around $W$, there exists a resolution
$f\colon\widetilde{X}\to X$ of $X$ such that
$N_{\sigma}(f^{*}(K_{X}+\Delta);\widetilde{X}/Y,W)$ is well defined as an
$\mathbb{R}$-divisor on $\widetilde{X}$ and the divisor
$P_{\sigma}(f^{*}(K_{X}+\Delta);\widetilde{X}/Y,W)$ is semi-ample over a
neighborhood of $W$.

######  Theorem 4.18 (cf. [H20, Lemma 2.25]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc pair and let
$(X^{\prime},\Delta^{\prime})$ be an lc pair together with a projective
bimeromorphic morphism $f\colon X^{\prime}\to X$. Suppose that $K_{X}+\Delta$
is $\pi$-pseudo-effective and the relation

$0\leq
a(D,X^{\prime},\Delta^{\prime})-a(D,X,\Delta)\leq\sigma_{D}(K_{X}+\Delta;X/Y)$

holds for all prime divisors $D$ on $X^{\prime}$ such that $(\pi\circ
f)(D)\cap W\neq\emptyset$. Then the following conditions are equivalent.

  * •

After shrinking $Y$ around $W$, the lc pair $(X,\Delta)$ has a log minimal
model over $Y$ around $W$.

  * •

After shrinking $Y$ around $W$, the lc pair $(X^{\prime},\Delta^{\prime})$ has
a log minimal model over $Y$ around $W$.

###### Proof.

As in the discussion in Remark 3.7, we may assume that $\pi$ is a contraction
and $W$ is connected. By Theorem 4.7 (6), we may freely shrink $Y$ around $W$
if necessary without loosing the irreducibility of both $X$ and $X$’. By
shrinking $Y$ around $W$, we can take a log resolution
$g\colon\widetilde{X}\to X^{\prime}$ of $(X^{\prime},\Delta^{\prime})$ such
that $f\circ g\colon\widetilde{X}\to X$ is a log resolution of $(X,\Delta)$.
Let $\widetilde{\Delta}^{\prime}$ be the sum of $g^{-1}_{*}\Delta^{\prime}$
and the reduced $g$-exceptional divisor on $\widetilde{X}$, and let
$\widetilde{\Delta}$ be the sum of $(f\circ g)^{-1}_{*}\Delta$ and the reduced
$(f\circ g)$-exceptional divisor on $\widetilde{X}$. By the same argument as
in [H22b, Lemma 3.6], by simple computation of the asymptotic vanishing orders
using Theorem 4.7 (5), we have

$0\leq
a(\widetilde{D},\widetilde{X},\widetilde{\Delta}^{\prime})-a(\widetilde{D},\widetilde{X},\widetilde{\Delta})\leq\sigma_{D}(K_{\widetilde{X}}+\widetilde{\Delta};\widetilde{X}/Y)$

for every prime divisor $\widetilde{D}$ on $\widetilde{X}$ whose image on $Y$
intersects $W$. By Lemma 3.4, the lc pair $(X,\Delta)$ (resp.
$(X^{\prime},\Delta^{\prime})$) has a log minimal model over $Y$ around $W$ if
and only if the lc pair $(\widetilde{X},\widetilde{\Delta})$ (resp.
$(\widetilde{X},\widetilde{\Delta}^{\prime})$) has a log minimal model over
$Y$ around $W$. Replacing $(X,\Delta)$ and $(X^{\prime},\Delta^{\prime})$ by
$(\widetilde{X},\widetilde{\Delta})$ and
$(\widetilde{X},\widetilde{\Delta}^{\prime})$ respectively, we may assume
$X=X^{\prime}$ and that $(X,\Delta)$ and $(X^{\prime},\Delta^{\prime})$ are
log smooth. Then [H20, Proof of Lemma 2.25] works with minor changes because
we can use Theorem 4.16 instead of [H20, Theorem 2.23]. ∎

######  Theorem 4.19 (cf. [H19, Lemma 2.15]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc pair and let
$(X^{\prime},\Delta^{\prime})$ be an lc pair together with a projective
bimeromorphic morphism $f\colon X^{\prime}\to X$. Suppose that we can write

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

for some effective $f$-exceptional $\mathbb{R}$-divisor $E$ on $X^{\prime}$.
Then the following conditions are equivalent.

  * •

After shrinking $Y$ around $W$, the lc pair $(X,\Delta)$ has a log minimal
model over $Y$ around $W$.

  * •

After shrinking $Y$ around $W$, the lc pair $(X^{\prime},\Delta^{\prime})$ has
a log minimal model over $Y$ around $W$.

###### Proof.

The argument in the algebraic case (see [H22b, Proof of Lemma 3.7]) works with
no changes since we may apply Theorem 4.18. ∎

######  Lemma 4.20 (cf. [H20, Remark 2.9]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a subset. Let $(X,\Delta)$
be an lc pair. Let $(X^{\prime},\Delta^{\prime})$ be a weak lc model of
$(X,\Delta)$ over $Y$ around $W$. We denote the bimeromorphic map
$X\dashrightarrow X^{\prime}$ by $\phi$. Then the following properties hold.

  * •

For any prime divisor $P$ over $X$ whose image on $Y$ intersects $W$, we have

$\sigma_{P}(K_{X}+\Delta;X/Y)=a(P,X^{\prime},\Delta^{\prime})-a(P,X,\Delta).$

  * •

Suppose that $(X^{\prime},\Delta^{\prime})$ be a log minimal model of
$(X,\Delta)$ over $Y$ around $W$ and $\phi$ is a sequence of steps of a
$(K_{X}+\Delta)$-MMP over $Y$ around $W$. Let $D$ be a prime divisor on $X$
such that $\pi(D)\cap W\neq\emptyset$. Then $D$ is contracted by $\phi$ if and
only if $\sigma_{D}(K_{X}+\Delta;X/Y)>0$.

###### Proof.

Let $f\colon X^{\prime\prime}\to X$ be a bimeromorphism which resolves the
indeterminacy of $\phi\colon X\dashrightarrow X^{\prime}$, and let
$f^{\prime}\colon X^{\prime\prime}\to X^{\prime}$ be the induced
bimeromorphism. By shrinking $Y$ around $W$, we may assume that the divisor

$E:=f^{*}(K_{X}+\Delta)-f^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})$

is $f^{\prime}$-exceptional. Then

${\rm coeff}_{P}(E)=a(P,X^{\prime},\Delta^{\prime})-a(P,X,\Delta).$

On the other hand, by using Theorem 4.7, we obtain

$\begin{split}\sigma_{P}(K_{X}+\Delta;X/Y)=&\sigma_{P}(f^{*}(K_{X}+\Delta);X^{\prime\prime}/Y)\\\
=&\sigma_{P}(f^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})+E;X^{\prime\prime}/Y)\\\
=&\sigma_{P}(f^{\prime*}(K_{X^{\prime}}+\Delta^{\prime});X^{\prime\prime}/Y)+{\rm
coeff}_{P}(E)\\\ =&{\rm coeff}_{P}(E).\end{split}$

Therefore, the first statement holds true.

Next, suppose that $(X^{\prime},\Delta^{\prime})$ be a log minimal model of
$(X,\Delta)$ over $Y$ around $W$ and $\phi$ is a sequence of steps of a
$(K_{X}+\Delta)$-MMP over $Y$ around $W$. By the first statement,
$\sigma_{D}(K_{X}+\Delta;X/Y)>0$ if and only if $a(D,X,\Delta)\neq
a(D,X^{\prime},\Delta^{\prime})$, and the latter condition is equivalent to
that $D$ is contracted by $X\dashrightarrow X^{\prime}$ the sequence of steps
of the $(K_{X}+\Delta)$-MMP. Therefore, the second statement holds. ∎

######  Lemma 4.21 (cf. [H20, Lemma 2.26]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $\pi^{\prime}\colon X^{\prime}\to Y$ be a
projective morphism from a normal analytic variety $X^{\prime}$. Let
$(X,\Delta)$ and $(X^{\prime},\Delta^{\prime})$ be lc pairs and let
$X\dashrightarrow X^{\prime}$ be a bimeromorphic map over $Y$. Suppose that

  * •

$a(P,X,\Delta)\leq a(P,X^{\prime},\Delta^{\prime})$ for all prime divisors $P$
on $X$ whose image on $Y$ intersects $W$, and

  * •

$a(P^{\prime},X,\Delta)\geq a(P^{\prime},X^{\prime},\Delta^{\prime})$ for all
prime divisors $P^{\prime}$ on $X^{\prime}$ whose image on $Y$ intersects $W$.

Then the following conditions are equivalent.

  * •

After shrinking $Y$ around $W$, the lc pair $(X,\Delta)$ has a log minimal
model over $Y$ around $W$.

  * •

After shrinking $Y$ around $W$, the lc pair $(X^{\prime},\Delta^{\prime})$ has
a log minimal model over $Y$ around $W$.

###### Proof.

By shrinking $Y$ around $W$, we may assume that $a(P,X,\Delta)\leq
a(P,X^{\prime},\Delta^{\prime})$ for all prime divisors $P$ on $X$ and
$a(P^{\prime},X,\Delta)\geq a(P^{\prime},X^{\prime},\Delta^{\prime})$ for all
prime divisors $P^{\prime}$ on $X^{\prime}$. The argument of [H20, Proof of
Lemma 2.26] works with no changes because we can use Theorem 4.19. ∎

##  5\. Log MMP for log abundant lc pairs

In this section we study log MMP with scaling that contains log abundant lc
pairs.

###  5.1. Auxiliary results

In this subsection, we collect some results used to prove the main results of
this paper.

######  Lemma 5.1.

Let $\pi\colon X\to Y$ be a contraction from a normal complex variety $X$ to a
Stein space $Y$, and let $W\subset Y$ be a connected compact subset such that
$\pi$ and $W$ satisfy (P). Let $S$ be a subvariety of $X$ and let $(X,\Delta)$
be an lc pair. Let

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

be a sequence of steps of a $(K_{X}+\Delta)$-MMP over $Y$ around $W$ with
scaling of an effective $\mathbb{R}$-divisor $A$. Let $A_{i}$ be the
birational transform of $A$ on $X_{i}$. We define

$\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef over }W}.$

Suppose that each step of the $(K_{X}+\Delta)$-MMP is an biholomorphism on a
neighborhood of $S$ and ${\rm lim}_{i\to\infty}\lambda_{i}=0$. Then, for any
prime divisor $P$ over $X$ whose images on $X$ and $Y$ intersect $S$ and $W$
respectively, we have $\sigma_{P}(K_{X}+\Delta;X/Y)=0$.

###### Proof.

Since ${\rm lim}_{i\to\infty}\lambda_{i}=0$, it is sufficient to prove
$\sigma_{P}(K_{X}+\Delta+\lambda_{i}A;X/Y)=0$ for all $i\geq 0$. Let
$c_{X}(P)$ be the image of $P$ on $X$. By Theorem 4.7 (6), we may freely
shrink $Y$. By shrinking $Y$ around $W$, we may assume that $i$ steps of the
$(K_{X}+\Delta)$-MMP over $Y$ around $W$

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

is represented by a bimeromorphic contraction $X\dashrightarrow X_{i}$ over
$Y$. Let $f\colon X^{\prime}\to X$ and $f_{i}\colon X^{\prime}\to X_{i}$ be a
resolution of the indeterminacy of $X\dashrightarrow X_{i}$ such that $P$
appears as a prime divisor on $X^{\prime}$. By shrinking $Y$ around $W$, we
may write

$f^{*}(K_{X}+\Delta+\lambda_{i}A)=f_{i}^{*}(K_{X_{i}}+\Delta_{i}+\lambda_{i}A_{i})+E_{i},$

where $E_{i}$ is an effective $f_{i}$-exceptional $\mathbb{R}$-divisor on
$X^{\prime}$. By Theorem 4.7 and the facts that the $(K_{X}+\Delta)$-MMP is an
biholomorphism on a neighborhood of $S$ and that the images of $P$ on $X$ and
$Y$ intersect $S$ and $W$ respectively, we have

$\begin{split}\sigma_{P}(K_{X}+\Delta+\lambda_{i}A;X/Y)=&\sigma_{P}(f^{*}(K_{X}+\Delta+\lambda_{i}A);X^{\prime}/Y)\\\
=&\sigma_{P}(f_{i}^{*}(K_{X_{i}}+\Delta_{i}+\lambda_{i}A_{i})+E_{i};X^{\prime}/Y)\\\
=&\sigma_{P}(f_{i}^{*}(K_{X_{i}}+\Delta_{i}+\lambda_{i}A_{i});X^{\prime}/Y)+{\rm
coeff}_{P}(E_{i})=0.\end{split}$

Since ${\rm lim}_{i\to\infty}\lambda_{i}=0$, and thus
$\sigma_{P}(K_{X}+\Delta;X/Y)=0$. ∎

######  Lemma 5.2 (cf. [H18, Lemma 2.10]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
to an analytic space $Y$, and let $W\subset Y$ be a compact subset. Let
$(X,\Delta)$ be an lc pair and let $D$ be a reduced divisor on $X$. Then,
after shrinking $Y$ around $W$, there exists a log smooth model
$f\colon(\overline{X},\overline{\Delta})\to(X,\Delta)$ of $(X,\Delta)$
satisfying the following.

  1. (i)

$\overline{\Delta}=\overline{\Delta}^{\prime}+\overline{\Delta}^{\prime\prime}$,
where $\overline{\Delta}^{\prime}\geq 0$ and
$\overline{\Delta}^{\prime\prime}$ is zero or a reduced divisor,

  2. (ii)

$(\pi\circ f)({\rm Supp}\,\overline{\Delta}^{\prime\prime})\subsetneq Y$,

  3. (iii)

any lc center of
$(\overline{X},\overline{\Delta}-t\overline{\Delta}^{\prime\prime})$ dominates
$Y$ for all $t\in(0,1]$, and

  4. (iv)

the exceptional locus ${\rm Ex}(f)$ of $f$ is pure codimension one and ${\rm
Ex}(f)\cup f_{*}^{-1}(\Delta+D)$ has simple normal crossing support.

###### Proof.

By Remark 3.7, we may assume that $W$ is connected. By shrinking $Y$ around
$W$, we may assume that $\Delta+D$ has only finitely many components and
$(X,\Delta)$ has only finitely many lc centers. By replacing $(X,\Delta)$ with
a log smooth model and replacing $D$ accordingly, we may assume that
$(X,\Delta+D)$ is log smooth. For each lc center $S$ of $(X,\Delta)$ such that
$\pi(S)\subsetneq Y$, let $X_{S}\to X$ be the blow-up of $X$ along $S$. Let
$f\colon\overline{X}\to X$ be a log resolution of $(X,\Delta+D)$ that factors
through $X_{S}\to X$ for any lc center $S$ of $(X,\Delta)$ such that
$\pi(S)\subsetneq Y$. We construct a log smooth model
$f\colon(\overline{X},\overline{\Delta})\to(X,\Delta)$ of $(X,\Delta)$, we
define $\overline{\Delta}^{\prime\prime}$ to be the sum of all components of
$\lfloor\overline{\Delta}\rfloor$ that do not dominate $Y$, and we set
$\overline{\Delta}^{\prime}:=\overline{\Delta}-\overline{\Delta}^{\prime\prime}$.
By the same argument as in [H18, Proof of Lemma 2.10], it is easy to see that
$f\colon(\overline{X},\overline{\Delta})\to(X,\Delta)$ is the desired log
smooth model. ∎

######  Lemma 5.3 (cf. [HH20, Lemma 2.16], [K13, Corollary 1.37 and Corollary
1.38]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be a dlt pair. Let $\mathcal{T}$
be a (possibly empty) finite set of prime divisors $D$ over $X$ such that the
image of $D$ on $Y$ intersects $W$ and $-1<a(D,X,\Delta)<0$ for any
$D\in\mathcal{T}$. For any open subset $U\subset Y$ containing $W$, we put
$X_{U}=\pi^{-1}(U)$ and $\Delta_{U}=\Delta|_{X_{U}}$, and the restriction of
any divisor $D\in\mathcal{T}$ to the inverse image of $U$ is denoted by
$D_{U}$. Then there exist a Stein open subset $U\subset Y$ containing $W$, a
projective bimeromorphism $f\colon\widetilde{X}\to X_{U}$ from a normal
analytic variety $\widetilde{X}$, and $(\widetilde{X},\widetilde{\Delta})$ a
disjoint union of dlt pairs satisfying the following.

  * •

$\widetilde{X}$ is $\mathbb{Q}$-factorial over $W$,

  * •

$K_{\widetilde{X}}+\widetilde{\Delta}=f^{*}(K_{X_{U}}+\Delta_{U})$, and

  * •

the set of $f$-exceptional prime divisors on $\widetilde{X}$ is exactly the
same as the set of the components of $D_{U}$ for the divisors
$D\in\mathcal{T}$.

###### Proof.

The argument in the algebraic case (see [K13, Corollary 1.37 and Corollary
1.38]) works with no changes. ∎

######  Lemma 5.4 (cf. [H19, Proposition 3.3]).

Let $\pi\colon X\to Y$ be a projective surjective morphism from a normal
analytic variety $X$ to a Stein space $Y$, and let $W\subset Y$ be a compact
subset such that $\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc pair.
Suppose that there exists a contraction $\phi\colon X\to Z$ over $Y$, where
$Z$ is a normal analytic variety that is projective over $Y$, satisfying the
following.

  * •

$\kappa_{\sigma}(X/Z,K_{X}+\Delta)=0$,

  * •

$\kappa_{\sigma}(X/Y,K_{X}+\Delta)={\rm dim}\,Z-{\rm dim}\,Y$, and

  * •

any lc center of $(X,\Delta)$ dominates $Z$.

Then, after shrinking $Y$ around $W$, the lc pair $(X,\Delta)$ has a good
minimal model over $Y$ around $W$.

###### Proof.

The argument in [H19, Proof of Proposition 3.3] works with no changes because
we can use the canonical bundle formula in the analytic setting [F22a, Theorem
21.4] and the weak semistable reduction in the analytic setting [EH24]. So we
only outline the proof.

By Remark 3.7, we may assume that $\pi\colon X\to Y$ and $Z\to Y$ are
contractions and $W$ is connected. By taking a log resolution of $(X,\Delta)$
and applying weak semistable reduction ([EH24]), we may assume that $(X,0)$ is
$\mathbb{Q}$-factorial klt and all fibers of $\phi$ have the same dimensions.
Let $H_{Z}$ be a Cartier divisor on $Z$ which is ample over $Y$, and let
$H\geq 0$ be an $\mathbb{R}$-divisor on $X$ such that $(X,\Delta+H)$ is an lc
pair and $H\sim_{\mathbb{R}}r\phi^{*}H_{Z}$ for some $r>2\cdot{\rm dim}\,X$.
We run a $(K_{X}+\Delta+H)$-MMP over $Y$ around $W$ with scaling of an ample
divisor

$(X_{0},\Delta_{0}+H_{0})\dashrightarrow(X_{1},\Delta_{1}+H_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i}+H_{i})\dashrightarrow\cdots.$

By Lemma 3.14, the induced map $X_{i}\dashrightarrow Z$ is a morphism and the
$(K_{X}+\Delta+H)$-MMP is a $(K_{X}+\Delta)$-MMP over $Y$ around $W$. By Lemma
3.13 and replacing $(X,\Delta)$ with $(X_{i},\Delta_{i})$ for some $i\gg 0$,
we may assume that $K_{X}+\Delta+H$ is the limit of movable divisors over a
neighborhood of $W$ (however, we lose the property of being equi-dimensional
of $X\to Z$). Then $K_{X}+\Delta$ is the limit of movable divisors over a
neighborhood of the inverse image of $W$ to $Z$. By our hypothesis
$\kappa_{\sigma}(X/Z,K_{X}+\Delta)=0$ and the argument as in [H19, Proof of
Proposition 3.3], we have $K_{X}+\Delta\sim_{\mathbb{R},\,Z}E$ for some
effective $\mathbb{R}$-divisor $E$ on $X$ which is very exceptional over $Z$.
By the negativity lemma for very exceptional divisors (Lemma 2.14) and
shrinking $Y$ around $W$, we have $K_{X}+\Delta\sim_{\mathbb{R},Z}0$.

By the third condition of Lemma 5.4, the canonical bundle formula in the
analytic setting [F22a, Theorem 21.4] (and the analytic argument of [FG12]),
an argument from convex geometry, and the argument of the perturbation of
coefficients with the aid of $\kappa_{\sigma}(X/Y,K_{X}+\Delta)={\rm
dim}\,Z-{\rm dim}\,Y$, there is a klt pair $(Z,\Delta_{Z})$ such that

$K_{X}+\Delta\sim_{\mathbb{R}}\phi^{*}(K_{Z}+\Delta_{Z}).$

By $\kappa_{\sigma}(X/Y,K_{X}+\Delta)={\rm dim}\,Z-{\rm dim}\,Y$, we see that
$K_{Z}+\Delta_{Z}$ is big over $Z$. By [F22a] and shrinking $Y$ around $W$,
the klt pair $(Z,\Delta_{Z})$ has a good minimal model
$(Z^{\prime},\Delta_{Z^{\prime}})$ over $Y$ around $W$. Since $(Z,\Delta_{Z})$
is klt, the bimeromorphic map $Z\dashrightarrow Z^{\prime}$ over $Y$ is a
bimeromorphic contraction. Hence we can find a Zariski open subset
$U^{\prime}\subset Z^{\prime}$ such that ${\rm
codim}_{Z^{\prime}}(Z^{\prime}\setminus U^{\prime})\geq 2$ and
$Z^{\prime}\dashrightarrow Z$ is a biholomorphism on $U^{\prime}$.

Let $f\colon X^{\prime}\to X$ be a log resolution of $(X,\Delta)$ such that
the induced map $\phi^{\prime}\colon X^{\prime}\dashrightarrow Z^{\prime}$ is
a morphism. Let $(X^{\prime},\Delta^{\prime})$ be a log birational model of
$(X,\Delta)$ as in Definition 3.1. After shrinking $Y$ around $W$, we can
write

$f^{*}(K_{X}+\Delta)\sim_{\mathbb{R}}\phi^{\prime*}(K_{Z^{\prime}}+\Delta_{Z^{\prime}})+F$

for some effective $\mathbb{R}$-divisor $F$ on $X$ such that
$\phi^{\prime}({\rm Supp}F)\subset Z^{\prime}\setminus U^{\prime}$. Since
$K_{Z^{\prime}}+\Delta_{Z^{\prime}}$ is semi-ample over $Z$, we have
$N_{\sigma}(f^{*}(K_{X}+\Delta);X^{\prime}/Y,W)\leq F$, and the negativity
lemma for very exceptional divisors (Lemma 2.14) implies

$-P_{\sigma}(f^{*}(K_{X}+\Delta);X^{\prime}/Y,W)\sim_{\mathbb{R},\,Z^{\prime}}-(F-N_{\sigma}(f^{*}(K_{X}+\Delta);X^{\prime}/Y,W))\geq
0.$

Therefore, $N_{\sigma}(f^{*}(K_{X}+\Delta);X^{\prime}/Y,W)=F$, and the divisor

$P_{\sigma}(f^{*}(K_{X}+\Delta);X^{\prime}/Y,W)\sim_{\mathbb{R}}\phi^{\prime*}(K_{Z^{\prime}}+\Delta_{Z^{\prime}})$

is semi-ample over $Y$. By Theorem 4.17, after shrinking $Y$ around $W$ the
pair $(X,\Delta)$ has a good minimal model over $Y$ around $W$. ∎

######  Proposition 5.5 (cf. [H20, Proposition 3.2]).

Let $\pi\colon X\to Y$ be a projective surjective morphism from a normal
analytic variety $X$ to a Stein space $Y$, and let $W\subset Y$ be a connected
compact subset such that $\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc
pair such that $K_{X}+\Delta$ is $\pi$-pseudo-effective and $\pi$-abundant.
Then, after shrinking $Y$ around $W$, there exist a dlt blow-up
$\widetilde{f}\colon(\widetilde{X},\widetilde{\Delta})\to(X,\Delta)$, where
$\widetilde{X}$ is $\mathbb{Q}$-factorial over $W$, and effective
$\mathbb{R}$-Cartier divisors $\widetilde{G}$ and $\widetilde{H}$ on
$\widetilde{X}$ satisfying the following properties.

  1. (I)

$K_{\widetilde{X}}+\widetilde{\Delta}\sim_{\mathbb{R}}\widetilde{G}+\widetilde{H}$,

  2. (II)

${\rm Supp}\,\widetilde{G}\subset{\rm
Supp}\,\llcorner\widetilde{\Delta}\lrcorner$, and

  3. (III)

there exists a positive real number $t_{0}$ such that for any $t\in(0,t_{0}]$,
the following properties hold:

    1. (III-a)

The pair $(\widetilde{X},\widetilde{\Delta}+t\widetilde{H})$ is dlt,
$N_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta}+t\widetilde{H};\widetilde{X}/Y,W)$
is well defined as an $\mathbb{R}$-divisor on $\widetilde{X}$, and the support
of
$N_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta}+t\widetilde{H};\widetilde{X}/Y,W)$
does not depend on $t$, and

    2. (III-b)

after shrinking $Y$ around $W$, the pair
$(\widetilde{X},\widetilde{\Delta}-t\widetilde{G})$ has a good minimal model
over $Y$ around $W$.

###### Proof.

By Lemma 5.2 and Lemma 5.4, we can apply [H20, Proof of Proposition 3.2] to
the situation of Proposition 5.5. Note that we only need to deal with finite
sequences of log MMP, and thus [H20, Proof of Proposition 3.2] works with no
changes. ∎

######  Proposition 5.6 (cf. [H20, Proposition 3.3]).

Let $\pi\colon X\to Y$ be a projective surjective morphism from a normal
analytic variety $X$ to a Stein space $Y$, and let $W\subset Y$ be a connected
compact subset such that $\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be a dlt
pair such that $X$ is $\mathbb{Q}$-factorial over $W$ and there are effective
$\mathbb{R}$-Cartier divisors $G$ and $H$ on $X$ satisfying (I), (II), (III),
(III-a), and (III-b) in Proposition 5.5. Then, after shrinking $Y$ around $W$,
there exist a real number $\lambda_{0}>0$, a bimeromorphic contraction
$X\dashrightarrow X_{1}$ over $Y$ with the strict transforms $\Delta_{1}$ and
$H_{1}$ of $\Delta$ and $H$ on $X_{1}$ respectively, and a sequence of steps
of a $(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$

$X_{1}\dashrightarrow X_{2}\dashrightarrow\cdots\dashrightarrow
X_{i}\dashrightarrow\cdots$

with the strict transforms $\Delta_{i}$ and $H_{i}$ of $\Delta$ and $H$ on
$X_{i}$ respectively, which satisfy the following properties.

  1. (1)

The pair $(X,\Delta+\lambda_{0}H)$ is dlt and all lc centers of
$(X,\Delta+\lambda_{0}H)$ are lc centers of $(X,\Delta)$,

  2. (2)

$X\dashrightarrow X_{1}$ is a sequence of steps of a
$(K_{X}+\Delta+\lambda_{0}H)$-MMP over $Y$ around $W$ that is represented by a
bimeromorphic contraction and $(X_{1},\Delta_{1}+\lambda_{0}H_{1})$ is a good
minimal model of $(X,\Delta+\lambda_{0}H)$ over $Y$,

  3. (3)

$X_{1}\dashrightarrow\cdots\dashrightarrow X_{i}\dashrightarrow\cdots$ is a
sequence of steps of a $(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$ with
scaling of $\lambda_{0}H_{1}$ 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 over $W$}}$

for each $i\geq 1$, then we have ${\rm lim}_{i\to\infty}\lambda_{i}=0$,

  4. (4)

for all $i\geq 1$ and all positive real numbers
$u\in[\lambda_{i},\lambda_{i-1}]$, after shrinking $Y$ around $W$, the pair
$(X_{i},\Delta_{i}+uH_{i})$ is a good minimal model of $(X,\Delta+uH)$ and
$(X_{1},\Delta_{1}+uH_{1})$ over $Y$ around $W$, and

  5. (5)

for any $i\geq 1$ and curve $C_{i}$ in the fiber of $X_{i}\to Y$ over $W$, if
$C_{i}$ is contracted by the extremal contraction in the $i$-th step of the
$(K_{X_{1}}+\Delta_{1})$-MMP, then $C_{i}\subset{\rm
Supp}\,\llcorner\Delta_{i}\lrcorner$.

###### Proof.

The argument in the algebraic case [H20, Proof of Proposition 3.3] works with
minor changes. We only write how to construct $X\dashrightarrow X_{1}$ and the
sequence of steps of the log MMP $X_{1}\dashrightarrow\cdots\dashrightarrow
X_{i}\dashrightarrow\cdots$.

If we fix $\lambda_{0}\in(0,t_{0})$, where $t_{0}$ is as in (III) in
Proposition 5.5, then $(X,\Delta+\lambda_{0}H)$ satisfies (1) of Proposition
5.6. By (I) in Proposition 5.5, for any $s\in\mathbb{R}_{>0}$, we have

$K_{X}+\Delta+sH\sim_{\mathbb{R},\,Y}(1+s)\left(K_{X}+\Delta-\frac{s}{1+s}G\right).$

By Theorem 3.15 and (III-b) in Proposition 5.5, for any $t\in(0,\lambda_{0}]$,
there exists a Stein open subset $U_{t}\subset Y$ containing $W$ such that
putting $X_{U_{t}}:=\pi^{-1}(U_{t})$ then there exists a sequence of steps of
a $(K_{X_{U_{t}}}+\Delta|_{X_{U_{t}}}+tH|_{X_{U_{t}}})$-MMP

$(X_{U_{t}},\Delta|_{X_{U_{t}}}+tH|_{X_{U_{t}}})\dashrightarrow(X^{\prime}_{t},\Delta^{\prime}_{t}+tH^{\prime}_{t})$

over $U_{t}$ around $W$ to a good minimal model, which we may assume that the
MMP is represented by a bimeromorphic contraction. By shrinking $Y$ to
$U_{\lambda_{0}}$ and putting $X_{1}:=X^{\prime}_{\lambda_{0}}$,
$\Delta_{1}:=\Delta^{\prime}_{\lambda_{0}}$, and
$H_{1}:=H^{\prime}_{\lambda_{0}}$, we get a bimeromorphic contraction over $Y$

$(X,\Delta+\lambda_{0}H)\dashrightarrow(X_{1},\Delta_{1}+\lambda_{0}H_{1}),$

where $(X_{1},\Delta_{1}+\lambda_{0}H_{1})$ is a good minimal model of
$(X,\Delta+\lambda_{0}H)$ over $Y$ around $W$. By construction,
$X\dashrightarrow X_{1}$ satisfies (2) of Proposition 5.6. Let $\pi_{1}\colon
X_{1}\to Y$ be the structure morphism. By (III-a) in Proposition 5.5 and Lemma
4.20, replacing the open sets $U_{t}\subset Y$ if necessary, we may assume
that the induced bimeromorphic map $\pi_{1}^{-1}(U_{t})\dashrightarrow
X^{\prime}_{t}$ is small for all $t\in(0,\lambda_{0}]$. Then
$(X^{\prime}_{t},\Delta^{\prime}_{t}+tH^{\prime}_{t})$ is a good minimal model
of $(\pi_{1}^{-1}(U_{t}),(\Delta_{1}+tH_{1})|_{\pi_{1}^{-1}(U_{t})})$ over
$U_{t}$ around $W$. By Theorem 3.16, we get a sequence of steps of a
$(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$ with scaling of
$\lambda_{0}H_{1}$

$(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

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 over $W$}}$

for each $i\geq 1$, then we have ${\rm lim}_{i\to\infty}\lambda_{i}=0$. By the
same argument as in the algebraic case [H20, Proof of Proposition 3.3], we can
check that the MMP satisfies (3)–(5) of Proposition 5.6. We note that the
property (II) in Proposition 5.5 is used to prove (5) of Proposition 5.6. For
details, see [H20, Proof of Proposition 3.3]. ∎

######  Theorem 5.7 (cf. [H20, Theorem 3.4]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be a dlt pair. Suppose that

  * •

$K_{X}+\Delta$ is $\pi$-pseudo-effective and $\pi$-abundant,

  * •

for any lc center $S$ of $(X,\Delta)$, the restriction $(K_{X}+\Delta)|_{S}$
is nef over $W$, and

  * •

$\sigma_{P}(K_{X}+\Delta;X/Y)=0$ for all prime divisor $P$ over $X$ such that
$a(P,X,\Delta)<0$, the center of $P$ on $X$ intersects an lc center of
$(X,\Delta)$, and the image of $P$ on $Y$ intersects $W$.

Then, after shrinking $Y$ around $W$, the dlt pair $(X,\Delta)$ has a log
minimal model over $Y$ around $W$.

###### Proof.

By Remark 3.7, we may assume that $\pi$ is a contraction and $W$ is connected.
We closely follow [H20, Proof of Theorem 3.4]. We divide the proof into
several steps.

######  Step 1.

In this step we will reduce Theorem 5.7 to a special termination of a log MMP
over $Y$ around $W$ with scaling. We follow [H20, Step 1 in the proof of
Theorem 3.4].

By Proposition 5.5 and shrinking $Y$ around $W$, we get a dlt blow-up
$(\widetilde{X},\widetilde{\Delta})\to(X,\Delta)$ and effective
$\mathbb{R}$-Cartier divisors $\widetilde{G}$ and $\widetilde{H}$ on
$\widetilde{X}$ such that $\widetilde{X}$ is $\mathbb{Q}$-factorial over $W$
and the properties (I), (II), (III), (III-a), and (III-b) in Proposition 5.5
hold. Then it is easy to check that we may replace $(X,\Delta)$ by
$(\widetilde{X},\widetilde{\Delta})$. Therefore, replacing $(X,\Delta)$ by
$(\widetilde{X},\widetilde{\Delta})$, we may assume that $X$ is
$\mathbb{Q}$-factorial over $W$ and there exist effective $\mathbb{R}$-Cartier
divisors $G$ and $H$ on $X$ satisfying (I), (II), (III), (III-a), and (III-b)
in Proposition 5.5. By Proposition 5.6 and shrinking $Y$ around $W$, we obtain
a real number $\lambda_{0}>0$, a bimeromorphic contraction

$X\dashrightarrow X_{1}$

over $Y$ with the strict transforms $\Delta_{1}$ and $H_{1}$ of $\Delta$ and
$H$ on $X_{1}$ respectively, and a sequence of steps of a
$(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$

$X_{1}\dashrightarrow X_{2}\dashrightarrow\cdots\dashrightarrow
X_{i}\dashrightarrow\cdots$

with the strict transforms $\Delta_{i}$ and $H_{i}$ of $\Delta$ and $H$ on
$X_{i}$ respectively, which satisfy the following properties.

  1. (1)

The pair $(X,\Delta+\lambda_{0}H)$ is dlt and all lc centers of
$(X,\Delta+\lambda_{0}H)$ are lc centers of $(X,\Delta)$,

  2. (2)

$X\dashrightarrow X_{1}$ is a sequence of steps of a
$(K_{X}+\Delta+\lambda_{0}H)$-MMP over $Y$ around $W$ that is represented by a
bimeromorphic contraction and $(X_{1},\Delta_{1}+\lambda_{0}H_{1})$ is a good
minimal model of $(X,\Delta+\lambda_{0}H)$ over $Y$,

  3. (3)

$X_{1}\dashrightarrow\cdots\dashrightarrow X_{i}\dashrightarrow\cdots$ is a
sequence of steps of a $(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$ with
scaling of $\lambda_{0}H_{1}$ 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 over $W$}}$

for each $i\geq 1$, then we have ${\rm lim}_{i\to\infty}\lambda_{i}=0$,

  4. (4)

for all $i\geq 1$ and all positive real numbers
$u\in[\lambda_{i},\lambda_{i-1}]$, after shrinking $Y$ around $W$, the pair
$(X_{i},\Delta_{i}+uH_{i})$ is a good minimal model of $(X,\Delta+uH)$ and
$(X_{1},\Delta_{1}+uH_{1})$ over $Y$ around $W$, and

  5. (5)

for any $i\geq 1$ and curve $C_{i}$ in the fiber of $X_{i}\to Y$ over $W$, if
$C_{i}$ is contracted by the extremal contraction in the $i$-th step of the
$(K_{X_{1}}+\Delta_{1})$-MMP, then $C_{i}\subset{\rm
Supp}\,\llcorner\Delta_{i}\lrcorner$.

By the argument in the algebraic case ([H20, Step1 in the proof of Theorem
3.4]), we see that Theorem 5.7 follows from the termination of the
$(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$. By (5), we only need to
prove that the $(K_{X_{1}}+\Delta_{1})$-MMP terminates on a neighborhood of
$\lfloor\Delta_{1}\rfloor$. In the rest of the proof, we will carry out the
argument of the special termination in Subsection 3.6.

Let $\\{Y_{i}\\}_{i\geq 1}$ and $\\{\phi_{i}\colon X_{i}\dashrightarrow
X^{\prime}_{i}\\}_{i\geq 1}$ be the data of the $(K_{X_{1}}+\Delta_{1})$-MMP.
There exists $m>0$ such that for any lc center $S_{m}$ of $(X_{m},\Delta_{m})$
and any $i\geq m$, the bimeromorphic map $X_{m}\times_{Y}Y_{i}\dashrightarrow
X_{i}$ induces a bimeromorphic map $S_{m}\dashrightarrow S_{i}$ to an lc
center $S_{i}$ of $(X_{i},\Delta_{i})$. For any $i\geq m$ and any lc center
$S_{i}$ of $(X_{i},\Delta_{i})$, we define $\mathbb{R}$-divisors
$\Delta_{S_{i}}$ and $H_{S_{i}}$ on $S_{i}$ by

$K_{S_{i}}+\Delta_{S_{i}}=(K_{X_{i}}+\Delta_{i})|_{S_{i}}\qquad{\rm and}\qquad
H_{S_{i}}=H_{i}|_{S_{i}},$

respectively. Similarly, on every lc center $S$ of $(X,\Delta)$, we define
$\mathbb{R}$-divisors $\Delta_{S}$ and $H_{S}$ by

$K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}\qquad{\rm and}\qquad H_{S}=H|_{S},$

respectively. Then $H_{S}$ and $H_{S_{i}}$ are effective $\mathbb{R}$-Cartier
divisors, and $(S_{i},\Delta_{S_{i}})$ is dlt for all $i\geq m$ and all
$S_{i}$.

From now on, we prove that for any integer $d\geq 0$, there is $m_{d}\geq m$
such that after $m_{d}$ steps, the non-biholomorphic locus of the
$(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$ does not intersect any
$d$-dimensional lc center of $(X_{m_{d}},\Delta_{m_{d}})$ whose image on $Y$
intersects $W$. The case $d={\rm dim}\,X-1$ of this statement and (5) imply
the termination of the $(K_{X_{1}}+\Delta_{1})$-MMP over $Y$ around $W$.
Therefore, to prove Theorem 5.7 it is sufficient to prove the statement. We
will prove the statement by induction on $d$. In the case where $d=0$, we can
take $m$ as $m_{0}$. We assume the existence of $m_{d-1}$ in the statement. By
replacing $m$ with $m_{d-1}$, we may assume $m_{d-1}=m$. By Theorem 3.17 and
replacing $m$ again, we may assume that for any $d$-dimensional lc center
$S_{m}$ of $(X_{m},\Delta_{m})$ whose image on $Y$ intersects $W$, the
following condition holds: For every $i\geq m$, if we shrink $Y$ to $Y_{i}$ so
that the $(K_{X_{1}}+\Delta_{1})$-MMP
$(X_{1},\Delta_{1})\dashrightarrow(X_{i},\Delta_{i})$ is represented by a
bimeromorphic contraction $X_{1}\dashrightarrow X_{i}$, then the induced
bimeromorphic map $S_{m}\dashrightarrow S_{i}$ is small and the strict
transform of $\Delta_{S_{m}}$ (resp. $H_{S_{m}}$) on $S_{i}$ is equal to
$\Delta_{S_{i}}$ (resp. $H_{S_{i}}$). Since ${\rm
lim}_{i\to\infty}\lambda_{i}=0$, by Theorem 3.18, to prove the existence of
$m_{d}$ it is sufficient to prove that $(S_{m},\Delta_{S_{m}})$ has a log
minimal model over $Y$ around $W$.

######  Step 2.

In the rest of the proof, all indices $i$ are assumed to be greater than or
equal to $m$, unless otherwise stated. We will show that for any
$d$-dimensional lc center $S_{m}$ of $(X_{m},\Delta_{m})$ whose image on $Y$
intersects $W$, the pair $(S_{m},\Delta_{S_{m}})$ has a log minimal model, as
required.

In this step we apply [H20, Step 2 in the proof of Theorem 3.4] to our
setting. Fix a $d$-dimensional lc center $S_{m}$ of $(X_{m},\Delta_{m})$ whose
image on $Y$ intersects $W$. By replacing $Y$ with $Y_{m}$, we may assume that
the $(K_{X_{1}}+\Delta_{1})$-MMP
$(X_{1},\Delta_{1})\dashrightarrow(X_{m},\Delta_{m})$ is represented by a
bimeromorphic contraction $X_{1}\dashrightarrow X_{m}$. Then $X\dashrightarrow
X_{m}$ is also a bimeromorphic contraction. By construction of
$X\dashrightarrow X_{m}$ in (1)–(3) in Step 1, we can find an lc center $S$ of
$(X,\Delta)$ such that $X\dashrightarrow X_{m}$ induces a birmeromorphic map
$S\dashrightarrow S_{m}$. Using the lc center $S$, in the rest of this step,
we shrink $Y$ appropriately and we establish the following variety, divisor,
and inequalities.

  1. (a)

A projective bimeromorphism $\psi\colon T\to S_{m}$ such that $T$ is
$\mathbb{Q}$-factorial over $W$ and for any prime divisor $\bar{D}$ on $S$
whose image on $Y$ intersects $W$, if the strict inequality
$a(\bar{D},S_{m},\Delta_{S_{m}})<a(\bar{D},S,\Delta_{S})$ holds then $\bar{D}$
appears as a prime divisor on $T$,

  2. (b)

an effective $\mathbb{R}$-divisor $\Psi$ on $T$ defined by
$\Psi=-\sum_{\begin{subarray}{c}D\end{subarray}}a(D,S,\Delta_{S})D$, where $D$
runs over all prime divisors on $T$ whose image intersects $W$,

  3. (c)

for every $i\geq m$, if we replace $Y$ by $Y_{i}$ so that the log MMP
$(X_{1},\Delta_{1})\dashrightarrow(X_{i},\Delta_{i})$ is represented by a
bimeromorphic contraction $X_{1}\dashrightarrow X_{i}$, then

$a(Q,S,\Delta_{S}+\lambda_{i}H_{S})\leq
a(Q,S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})$

holds for any prime divisors $Q$ over $S$ whose image on $Y$ intersects $W$,
and

  4. (d)

$K_{T}+\Psi$ is $\mathbb{R}$-Cartier and the inequality
$a(Q^{\prime},S_{m},\Delta_{S_{m}})\leq a(Q^{\prime},T,\Psi)$ holds for all
prime divisors $Q^{\prime}$ over $S_{m}$, in particular, the pair $(T,\Psi)$
is lc.

Because [H20, Step 2 in the proof of Theorem 3.4] works with minor changes, we
only outline the proof.

We first show (c). We fix $i\geq m$ and we shrink $Y$ to $Y_{i}$ in this
paragraph. By taking a resolution $\overline{X}\to X$ of $X$ which resolves
the indeterminacy of $X\dashrightarrow X_{i}$, we can construct a common
resolution $\overline{X}\to X$ and $\overline{X}\to{X_{i}}$ and a subvariety
$\overline{S}\subset\overline{X}$ bimeromorphic to $S$ and $S_{i}$ such that
the induced bimeromorphisms $\overline{S}\to S$ and $\overline{S}\to S_{i}$
form a common resolution of $S\dashrightarrow S_{i}$. By (4) in Step 1,
comparing the coefficients of the pullbacks of
$(K_{X}+\Delta+\lambda_{i}H)|_{S}$ and
$(K_{X_{i}}+\Delta_{i}+\lambda_{i}H_{i})|_{S_{i}}$ to $\overline{S}$, we
obtain

$a(Q,S,\Delta_{S}+\lambda_{i}H_{S})\leq
a(Q,S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})$

for all prime divisors $Q$ over $S$ whose image on $Y$ intersects $W$. We have
proved (c).

Let $\widetilde{D}$ be a prime divisor on $S_{m}$ whose image on $Y$
intersects $W$. For every $i$, we put

$S_{m}^{(i)}:=S_{m}\times_{Y}Y_{i},\quad
S^{(i)}:=S\times_{Y}Y_{i},\quad\Delta_{S_{m}}^{(i)}:=\Delta_{S_{m}}|_{S_{m}^{(i)}},\quad{\rm
and}\quad\Delta_{S}^{(i)}:=\Delta_{S}|_{S^{(i)}}.$

Then there is an irreducible components $\widetilde{D}_{i}$ of
$\widetilde{D}|_{S_{m}^{(i)}}$ such that the image of $\widetilde{D}_{i}$ on
$Y$ intersects $W$ and

$\begin{split}a(\widetilde{D}_{i},S_{m}^{(i)},\Delta_{S_{m}}^{(i)})=&a(\widetilde{D},S_{m},\Delta_{S_{m}}),\quad{\rm
and}\\\
a(\widetilde{D}_{i},S^{(i)},\Delta_{S}^{(i)}+\lambda_{i}H_{S}|_{S^{(i)}})=&a(\widetilde{D},S,\Delta_{S}+\lambda_{i}H_{S}).\end{split}$

Since
$(S_{m}^{(i)},\Delta_{S_{m}}^{(i)})\dashrightarrow(S_{i},\Delta_{S_{i}})$ is
small, we have
$a(\widetilde{D}_{i},S_{m}^{(i)},\Delta_{S_{m}}^{(i)})=a(\widetilde{D}_{i},S_{i},\Delta_{S_{i}}).$
From these facts and (c), we have

$\begin{split}a(\widetilde{D},S,\Delta_{S}+\lambda_{i}H_{S})=&a(\widetilde{D}_{i},S^{(i)},\Delta_{S}^{(i)}+\lambda_{i}H_{S}|_{S^{(i)}})\leq
a(\widetilde{D}_{i},S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})\\\
\leq&a(\widetilde{D}_{i},S_{i},\Delta_{S_{i}})=a(\widetilde{D}_{i},S_{m}^{(i)},\Delta_{S_{m}}^{(i)})=a(\widetilde{D},S_{m},\Delta_{S_{m}}).\end{split}$

On the other hand, by applying Lemma 4.15 to
$(X,\Delta)\dashrightarrow(X_{m},\Delta_{m})$, $S$, and $S_{m}$, we have
$a(\widetilde{D},S_{m},\Delta_{S_{m}})\leq a(\widetilde{D},S,\Delta_{S}).$
Thus

$a(\widetilde{D},S,\Delta_{S}+\lambda_{i}H_{S})\leq
a(\widetilde{D},S_{m},\Delta_{S_{m}})\leq a(\widetilde{D},S,\Delta_{S})$

for every $i$. Since ${\rm lim}_{i\to\infty}\lambda_{i}=0$ by (3) in Step 1,
by considering the limit $i\to\infty$, the equality

($\star$)
$a(\widetilde{D},S_{m},\Delta_{S_{m}})=a(\widetilde{D},S,\Delta_{S})$

holds for all prime divisors $\widetilde{D}$ on $S_{m}$ whose image on $Y$
intersects $W$. This argument shows that ($\star$ ‣ 2) holds true even if we
shrink $Y$ to a Stein open subset containing $W$.

In this paragraph we will show (a). We set

$\mathcal{C}=\Set{\bar{D}}{\begin{array}[]{l}\\!\\!\text{$\bar{D}$ is a prime
divisor on $S$ such that the image of $\bar{D}$ on $Y$}\\\
\\!\\!\text{intersects $W$ and
$a(\bar{D},S_{m},\Delta_{S_{m}})<a(\bar{D},S,\Delta_{S})$}\end{array}\\!\\!}.$

By ($\star$ ‣ 2), all elements of $\mathcal{C}$ are exceptional over $S_{m}$.
By a basic property of discrepancies, we have
$a(\bar{D},S_{m},\Delta_{S_{m}}+\lambda_{m}H_{S_{m}})\leq
a(\bar{D},S_{m},\Delta_{S_{m}})$. Combining this with (c), we obtain

$a(\bar{D},S,\Delta_{S}+\lambda_{m}H_{S})\leq
a(\bar{D},S_{m},\Delta_{S_{m}})<a(\bar{D},S,\Delta_{S})\leq 0$

for all $\bar{D}\in\mathcal{C}$. Since every element of $\mathcal{C}$ is a
prime divisor on $S$, we see that all elements of $\mathcal{C}$ are components
of $H_{S}$. Thus $\mathcal{C}$ is a finite set, and furthermore, any
$\bar{D}\in\mathcal{C}$ satisfies
$-1<a(\bar{D},S,\Delta_{S}+\lambda_{m}H_{S})$. Thus, we obtain

$-1<a(\bar{D},S_{m},\Delta_{S_{m}})<0$

for all $\bar{D}\in\mathcal{C}$. By Lemma 5.3, after shrinking $Y$ around $W$
and replacing $\mathcal{C}$ accordingly, we get a projective bimeromorphism
$\psi\colon T\to S_{m}$ such that

  * •

$T$ is $\mathbb{Q}$-factorial over $W$,

  * •

any element of $\mathcal{C}$ appears as a $\psi$-exceptional prime divisor on
$T$, and

  * •

any $\psi$-exceptional prime divisor $\bar{E}$ on $T$ appears as a prime
divisor on $S$ and it satisfies
$a(\bar{E},S_{m},\Delta_{S_{m}})<a(\bar{E},S,\Delta_{S})$.

Then $\psi\colon T\to S_{m}$ is the desired bimeromorphism as in (a). We note
that these three properties hold even if we shrink $Y$ to a Stein open subset
containing $W$.

Next, we will show (b). Let $D$ be a prime divisor on $T$ whose image on $Y$
intersects $W$. If $D$ is $\psi$-exceptional, then
$a(D,S_{m},\Delta_{S_{m}})<a(D,S,\Delta_{S})\leq 0$ by the definition of
$\psi$. If $D$ is not $\psi$-exceptional, by ($\star$ ‣ 2) we obtain
$a(D,S,\Delta_{S})=a(D,S_{m},\Delta_{S_{m}})\leq 0$. In any case, the relation

($\star$$\star$) $a(D,S_{m},\Delta_{S_{m}})\leq a(D,S,\Delta_{S})\leq 0$

holds. By shrinking $Y$ around $W$, we may assume that only finitely many
prime divisors $D$ on $T$ satisfy $a(D,S,\Delta_{S})<0$ because
$a(D,S,\Delta_{S})=0$ when $D$ is a prime divisor on $S$ and not a component
of $\Delta_{S}$. Therefore, we may define an $\mathbb{R}$-divisor $\Psi\geq 0$
on $T$ by

$\Psi=-\sum_{\begin{subarray}{c}D\end{subarray}}a(D,S,\Delta_{S})D,$

where $D$ runs over all prime divisors on $T$ whose image on $Y$ intersects
$W$. This is the $\mathbb{R}$-divisor stated in (b).

Finally, we prove (d). Since $T$ is $\mathbb{Q}$-factorial over $W$, there
exist an Stein open subset $Y^{\prime}\subset Y$ containing $W$ such that
putting $S^{\prime}$ and $T^{\prime}$ as the inverse images of $Y^{\prime}$ by
$S\to Y$ and $T\to Y$ respectively, then $K_{T^{\prime}}$ is
$\mathbb{Q}$-Cartier and

$\Psi|_{T^{\prime}}=-\sum_{D}\left(\sum_{D|_{T^{\prime}}=\bigcup_{j}Q_{D,\,j}}a(Q_{D,\,j},S^{\prime},\Delta_{S}|_{S^{\prime}})Q_{j,\,D}\right)$

is $\mathbb{R}$-Cartier, where $D|_{T^{\prime}}=\bigcup_{j}Q_{D,\,j}$ is the
prime decomposition. Then the images of some prime divisors $Q_{D,\,j}$ on
$Y^{\prime}$ may not intersect $W$. However, by replacing $Y^{\prime}$ with a
smaller Stein open subset to remove such $Q_{D,\,j}$, we may assume that
$\Psi|_{T^{\prime}}$ is $\mathbb{R}$-Cartier. In particular, we may assume
that $K_{T}+\Psi$ is $\mathbb{R}$-Cartier. Moreover we may assume that the
image of any component of $K_{T}+\Psi-\psi^{*}(K_{S_{m}}+\Delta_{S_{m}})$
intersects $W$. By ($\star$$\star$ ‣ 2), we obtain
$K_{T}+\Psi\leq\psi^{*}(K_{S_{m}}+\Delta_{S_{m}})$. From this, we have

$-1\leq a(Q^{\prime},S_{m},\Delta_{S_{m}})\leq a(Q^{\prime},T,\Psi)$

for any prime divisor $Q^{\prime}$ over $S_{m}$. This shows (d).

######  Step 3.

In this step we follow [H20, Step 3 in the proof of Theorem 3.4]. We will
prove that after shrinking $Y$ around $W$ the lc pair $(T,\Psi)$ has a log
minimal model over $Y$ around $W$.

To apply Lemma 4.21 to the bimeromorphic map
$(S,\Delta_{S})\dashrightarrow(T,\Psi)$, we will prove the following claim.

######  Claim.

Let $\tilde{Q}$ be a prime divisor over $S$ whose image on $Y$ intersects $W$.
Then the following two statements hold:

  * •

If $\tilde{Q}$ is a prime divisor on $S$, then $a(\tilde{Q},S,\Delta_{S})\leq
a(\tilde{Q},T,\Psi)$, and

  * •

if $\tilde{Q}$ is a prime divisor on $T$, then $a(\tilde{Q},T,\Psi)\leq
a(\tilde{Q},S,\Delta_{S})$.

###### Proof of Claim.

Let $\tilde{Q}$ be a prime divisor over $S$ whose image on $Y$ intersects $W$.
When $\tilde{Q}$ appears as a prime divisor on $T$ or a prime divisor on $S$
which is not exceptional over $T$, by (b) in Step 2 we have

$a(\tilde{Q},T,\Psi)=-{\rm
coeff}_{\tilde{Q}}(\Psi)=a(\tilde{Q},S,\Delta_{S}).$

Therefore, the second assertion of the claim holds. To prove the first
assertion of the claim, we may assume that $\tilde{Q}$ appears as a prime
divisor on $S$ that is exceptional over $T$. In this case, (a) in Step 2 shows
$a(\tilde{Q},S,\Delta_{S})\leq a(\tilde{Q},S_{m},\Delta_{S_{m}})$, then (d) in
Step 2 shows

$a(\tilde{Q},S,\Delta_{S})\leq a(\tilde{Q},S_{m},\Delta_{m})\leq
a(\tilde{Q},T,\Psi).$

Therefore, the first assertion of the claim also holds. In this way, the claim
holds. ∎

By the second condition of Theorem 5.7 and Lemma 4.21, after shrinking $Y$
around $W$ the lc pair $(T,\Psi)$ has a log minimal model over $Y$ around $W$.

######  Step 4.

In this step we follow [H20, Step 4 in the proof of Theorem 3.4]. We will
prove that after shrinking $Y$ around $W$ the lc pair $(S_{m},\Delta_{S_{m}})$
has a log minimal model over $Y$ around $W$.

For every $i$, we put

$\begin{split}S_{m}^{(i)}:=&S_{m}\times_{Y}Y_{i},\quad
S^{(i)}:=S\times_{Y}Y_{i},\quad T^{(i)}:=T\times_{Y}Y_{i},\\\
\Delta_{S_{m}}^{(i)}:=&\Delta_{S_{m}}|_{S_{m}^{(i)}},\quad\Delta_{S}^{(i)}:=\Delta_{S}|_{S^{(i)}},\quad
H_{S_{m}}^{(i)}:=H_{S_{m}}|_{S_{m}^{(i)}},\quad{\rm and}\quad
H_{S}^{(i)}:=H_{S}|_{S^{(i)}}\end{split}$

For each $i$, the pair $(S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})$ is a weak
lc model of $(S_{m}^{(i)},\Delta_{S_{m}}^{(i)}+\lambda_{i}H_{S_{m}}^{(i)})$
because the bimeromorphic map $S_{m}^{(i)}\dashrightarrow S_{i}$ is small,
$\Delta_{S_{i}}+\lambda_{i}H_{S_{i}}$ is equal to the strict transform of
$\Delta_{S_{m}}^{(i)}+\lambda_{i}H_{S_{m}}^{(i)}$ on $S_{i}$, and the divisor
$K_{S_{i}}+\Delta_{S_{i}}+\lambda_{i}H_{S_{i}}$ is nef over $W$. Pick any
prime divisor $D$ on $T$ whose image on $Y$ intersects $W$. For every $i$,
there is an irreducible components $D^{(i)}$ of $D|_{S_{m}^{(i)}}$ such that
the image of $D^{(i)}$ on $Y$ intersects $W$ and

$\begin{split}a(D^{(i)},S_{m}^{(i)},\Delta_{S_{m}}^{(i)}+\lambda_{i}H_{S_{m}}^{(i)})=&a(D,S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}}),\quad{\rm
and}\\\
a(D^{(i)},S^{(i)},\Delta_{S}^{(i)}+\lambda_{i}H_{S}^{(i)})=&a(D,S,\Delta_{S}+\lambda_{i}H_{S}).\end{split}$

By Lemma 4.20 and (c) in Step 2, we obtain

$\begin{split}\sigma_{D^{(i)}}(K_{S_{m}^{(i)}}+\Delta_{S_{m}}^{(i)}+\lambda_{i}H_{S_{m}}^{(i)})=a(D^{(i)},S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})-a(D^{(i)},S_{m}^{(i)},\Delta_{S_{m}}^{(i)}+\lambda_{i}H_{S_{m}}^{(i)})&\\\
\geq
a(D^{(i)},S^{(i)},\Delta_{S}^{(i)}+\lambda_{i}H_{S}^{(i)})-a(D^{(i)},S_{m}^{(i)},\Delta_{S_{m}}^{(i)}+\lambda_{i}H_{S_{m}}^{(i)})&.\end{split}$

By these relations and Theorem 4.7 (6), we obtain

$\sigma_{D}(K_{S_{m}}+\Delta_{S_{m}}+\lambda_{i}H_{S_{m}})\geq
a(D,S,\Delta_{S}+\lambda_{i}H_{S})-a(D,S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}}).$

By (b) in Step 2, we have $a(D,S,\Delta_{S})=a(D,T,\Psi)$. By Theorem 4.7 (4),
using the fact that ${\rm lim}_{i\to\infty}\lambda_{i}=0$ (see (3) in Step 1)
and taking the limit $i\to\infty$, we obtain

$\begin{split}\sigma_{D}(K_{S_{m}}+\Delta_{S_{m}})\geq&\underset{i\to\infty}{\rm
lim}\bigl{(}a(D,S,\Delta_{S}+\lambda_{i}H_{S})-a(D,S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}})\bigr{)}\\\
=&a(D,T,\Psi)-a(D,S_{m},\Delta_{S_{m}})\\\ \geq&0.\end{split}$

Here, the last inequality holds by (d) in Step 2. By Theorem 4.18 and the
existence of a log minimal model of $(T,\Psi)$ over a neighborhood of $W$ as
discussed in Step 3, after shrinking $Y$ around $W$ the lc pair
$(S_{m},\Delta_{S_{m}})$ has a log minimal model.

By Step 1 and Step 4, we complete the argument of the special termination as
in Subsection 3.6. Therefore, after shrinking $Y$ around $W$, the dlt pair
$(X,\Delta)$ has a log minimal model over $Y$ around $W$. We complete the
proof. ∎

###  5.2. Proof of main results

In this subsection we prove the main results of this paper.

######  Theorem 5.8 (cf. [H20, Theorem 3.5]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $(X,\Delta)$ be an lc pair. Let $A$ be an
effective $\mathbb{R}$-divisor on $X$ such that $(X,\Delta+A)$ is an lc pair
and $K_{X}+\Delta+A$ is nef over $W$. Then no infinite sequence of steps of a
$(K_{X}+\Delta)$-MMP over $Y$ around $W$ with scaling of $A$

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

satisfies the following properties.

  * •

If we define $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef over }W}$, where
$A_{i}$ is the strict transform of $A$ on $X_{i}$, then ${\rm
lim}_{i\to\infty}\lambda_{i}=0$, and

  * •

there are infinitely many $i$ such that $(X_{i},\Delta_{i})$ is log abundant
over $Y$.

###### Proof.

The argument in [H20, Proof of Theorem 3.5] works in our situation since we
may use the special termination (Subsection 3.6), the lift of MMP (Subsection
3.7), and Theorem 5.7. ∎

######  Theorem 5.9 (cf. [HH20, Theorem 1.5]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $(X,B+A)$ be an lc pair, where $B$ is an
effective $\mathbb{R}$-divisor on $X$ and $A$ is an effective $\pi$-ample
$\mathbb{R}$-divisor on $X$. Let $f\colon(\tilde{X},\tilde{B})\to(X,B)$ be a
dlt blow-up of $(X,B)$, and we put $\tilde{\Gamma}=\tilde{B}+f^{*}A$. Let
$\tilde{H}$ be a $(\pi\circ f)$-ample $\mathbb{R}$-divisor on $\tilde{X}$ such
that $(\tilde{X},\tilde{\Gamma}+\tilde{H})$ is lc and
$K_{\tilde{X}}+\tilde{\Gamma}+\tilde{H}$ is nef over $W$. Then there exists a
sequence of steps of a $(K_{\tilde{X}}+\tilde{\Gamma})$-MMP over $Y$ around
$W$ with scaling of $\tilde{H}$

$(\tilde{X}_{0}:=\tilde{X},\tilde{\Gamma}_{0}:=\tilde{\Gamma})\dashrightarrow(\tilde{X}_{1},\tilde{\Gamma}_{1})\dashrightarrow\cdots\dashrightarrow(\tilde{X}_{m},\tilde{\Gamma}_{m})$

such that after shrinking $Y$ around $W$ the lc pair
$(\tilde{X}_{m},\tilde{\Gamma}_{m})$ is a log minimal model or a Mori fiber
space of $(X,\Delta+A)$ over $Y$ around $W$.

###### Proof.

Let

$(\tilde{X}_{0}:=\tilde{X},\tilde{\Gamma}_{0}:=\tilde{\Gamma})\dashrightarrow(\tilde{X}_{1},\tilde{\Gamma}_{1})\dashrightarrow\cdots\dashrightarrow(\tilde{X}_{i},\tilde{\Gamma}_{i})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{\tilde{X}}+\tilde{\Gamma})$-MMP over $Y$
around $W$ with scaling of $\tilde{H}$. By the argument as in [H22a], we see
that all $(\tilde{X}_{i},\tilde{\Gamma}_{i})$ are log abundant over $Y$.
Moreover, if we define

$\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{\tilde{X}_{i}}+\tilde{\Gamma}_{i}+\mu\tilde{H}_{i}\text{\rm\, is
nef over }W},$

where $\tilde{H}_{i}$ is the strict transform of $\tilde{H}$ on
$\tilde{H}_{i}$, then ${\rm lim}_{i\to\infty}\lambda_{i}=0$. By Theorem 5.8,
the $(K_{\tilde{X}}+\tilde{\Gamma})$-MMP terminates. Therefore Theorem 5.9
holds. ∎

######  Theorem 5.10 (cf. [H19, Theorem 1.1], [B12, Theorem 1.1], [HX13,
Theorem 1.6]).

Let $\pi\colon X\to Y$ be a projective morphism from a normal analytic variety
$X$ to a Stein space $Y$, and let $W\subset Y$ be a compact subset such that
$\pi$ and $W$ satisfy (P). Let $(X,B)$ be a dlt pair and $A$ an effective
$\mathbb{R}$-Cartier divisor on $X$ such that $(X,B+A)$ is lc and
$K_{X}+B+A\sim_{\mathbb{R},\,Y}0$. Let $H$ be a $\pi$-ample
$\mathbb{R}$-divisor on $X$ such that $(X,B+H)$ is lc and $K_{X}+B+H$ is nef
over $W$. Then there exists a sequence of steps of a $(K_{X}+B)$-MMP over $Y$
around $W$ with scaling of $H$

$(X_{0}:=X,B_{0}:=B)\dashrightarrow(X_{1},B_{1})\dashrightarrow\cdots\dashrightarrow(X_{m},B_{m})$

such that after shrinking $Y$ around $W$ the lc pair $(X_{m},B_{m})$ is a log
minimal model or a Mori fiber space of $(X,B)$ over $Y$ around $W$.

###### Proof.

The argument in [H20, Remark 3.7] works with no changes. ∎

## References

  * [AK00] D. Abramovich, K. Karu, Weak semistable reduction in characteristic $0$, Invent. math. 139 (2000), no. 2, 241–273.
  * [BM97] E. Bierstone, P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302.
  * [B11] C. Birkar, On existence of log minimal models II, J. Reine Angew. Math. 658 (2011), 99–113.
  * [B12] C. Birkar, Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci. 115 (2012), no. 1, 325–368.
  * [BCHM10] 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.
  * [BH14] C. Birkar, Z. Hu, Polarized pairs, log minimal models, and Zariski decompositions, Nagoya Math. J. 215 (2014), 203–224.
  * [C08] S. R. Choi, The geography of log models and its applications, PhD Thesis, Johns Hopkins University, 2008.
  * [CHP16] F. Campana, A. Höring, T. Peternell, Abundance for Kähler threefolds, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 4, 971–1025.
  * [CLS11] D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, Graduate Studies in Mathematics 124, American Mathematical Society, Providence, RI, 2011
  * [EH24] M. Enokizono, K. Hashizume, Semistable reduction for complex analytic spaces, in preparation.
  * [F07] O. Fujino, Special termination and reduction to pl flips. In Flips for $3$-folds and $4$-folds, Oxford University Press (2007).
  * [F22a] O. Fujino, Minimal model program for projective morphisms between complex analytic spaces, preprint (2022), arXiv:2201.11315v1.
  * [F22b] O. Fujino, Cone and contraction theorem for projective morphisms between complex analytic spaces, preprint (2022), arXiv:2209.06382v3.
  * [FG12] O. Fujino, Y. Gongyo, On canonical bundle formulas and subadjunctions, Michigan Math. J. 61 (2012), no. 2, 255–264.
  * [G62] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., 146 (1962), 331–368.
  * [HX13] C. D. Hacon, C. Xu, Existence of log canonical closures, Invent. Math. 192 (2013), no. 1, 161–195.
  * [H18] K. Hashizume, Minimal model theory for relatively trivial log canonical pairs, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 2069–2107.
  * [H19] K. Hashizume, Remarks on special kinds of the relative log minimal model program, Manuscripta Math. 160 (2019), no. 3, 285–314.
  * [H20] K. Hashizume, Finiteness of log abundant log canonical pairs in log minimal model program with scaling, to appear in Michigan Math. J.
  * [H22a] K. Hashizume, Non-vanishing theorem for generalized log canonical pairs with a polarization, Selecta Math. New Ser. 28 (2022), article number 77.
  * [H22b] K. Hashizume, Iitaka fibrations for dlt pairs polarized by a nef and log big divisor, Forum Math. Sigma. 10 (2022), e85.
  * [HH20] K. Hashizume, Z. Hu, On minimal model theory for log abundant lc pairs, J. Reine Angew. Math., 767 (2020), 109–159.
  * [HP15] A. Höring, T. Peternell, Mori fibre spaces for Kähler threefolds, J. Math. Sci. Univ. Tokyo 22 (2015), no. 1, 219–246.
  * [HP16] A. Höring, T. Peternell, Minimal models for Kähler threefolds, Invent. Math. 203 (2016), no. 1, 217–264.
  * [K13] J. Kollár, Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200. Cambridge University Press, Cambridge, 2013.
  * [LM22] S. Lyu, T. Murayama, The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero, preprint (2022).
  * [LX23] J. Liu, L. Xie, Relative Nakayama–Zariski decomposition and minimal models of generalized pairs, Peking Math. J. (2023)
  * [M82] M. Mirella, Sard and Bertini Type Theorems for Complex Spaces, Annali di Matematica Pura ed Applicata, 131 (1982), 265–279.
  * [Na04] N. Nakayama, Zariski-decomposition and abundance, MSJ Mem. 14, Mathematical Society of Japan, Tokyo, 2004.
  * [No16] J. Noguchi, Analytic function theory of several variables. Elements of Oka’s coherence, Springer, Singapore, 2016.
  * [S93] V. V. Shokurov, 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.
  * [T21] N. Tsakanikas, On Minimal Models and the Termination of Flips for Generalized Pairs, Ph.D Thesis, Universität des Saarlandes, 2021.