# Okounkov bodies associated to abundant divisors
and Iitaka fibrations

Sung Rak Choi  Department of Mathematics, Yonsei University, Seoul, Korea
<EMAIL_ADDRESS>,  Jinhyung Park  School of Mathematics, Korea Institute
for Advanced Study, Seoul, Korea<EMAIL_ADDRESS>and  Joonyeong Won
Center for Geometry and Physics, Institute for Basic Science, Pohang, Korea
<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

The aim of this paper is to study the Okounkov bodies associated to abundant
divisors. First, we extend a result of Lazarsfeld-Mustaţă [LM] and Jow [J] to
the valuative Okounkov bodies of abundant divisors: two abundant divisors are
numerically equivalent if and only if they have the same valuative Okounkov
bodies with respect to all admissible flags. Thus the valuative Okounkov
bodies encode all numerical information of an abundant divisor. We then
recover the restricted base locus of an abundant divisor from the valuative
Okounkov bodies. Note that these statements were proven for limiting Okounkov
bodies in [CHPW1], [CHPW2], but do not hold true in general for valuative
Okounkov bodies of arbitrary effective divisors. Finally, we show a criterion
of when the valuative and limiting Okounkov bodies of an abundant divisor
coincide by comparing their Euclidean volumes. To achieve these theorems, we
prove some versions of Fujita’s approximations for Okounkov bodies, and use
results on Nakayama subvarieties and Iitaka fibrations.

###### Key words and phrases:

Okounkov body, abundant divisor, Fujita’s approximation, Iitaka fibration

###### 2010 Mathematics Subject Classification:

14C20, 52A20

S. Choi and J. Park were partially supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government (Ministry of
Science and ICT) (No. NRF-2016R1C1B2011446). J. Won was partially supported by
IBS-R003-D1, Institute for Basic Science in Korea.

##  1\. Introduction

Inspired by the works of Okounkov [O1], [O2], Lazarsfeld-Mustaţă [LM] and
Kaveh-Khovanskii [KK] independently introduced and studied the Okounkov bodies
associated to big divisors. Following their philosophy, there has been a
number of attempts to understand the asymptotic properties of divisors by
analyzing the convex sets called the Okounkov bodies. An Okounkov body
$\Delta_{Y_{\bullet}}(D)$ of a divisor $D$ on a smooth projective variety $X$
of dimension $n$ with respect to an admissible flag $Y_{\bullet}$ is defined
as a convex subset of the Euclidean space $\mathbb{R}^{n}$ (see Definition
2.1). It is expected that one can extract the positivity information of the
divisor $D$ from the structure of the Okounkov bodies. In [CHPW1], [CPW1],
[CPW2], we extended the study of Okounkov bodies to pseudoeffective divisors
by introducing the _valuative Okounkov body_
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ and the _limiting Okounkov
body_ $\Delta^{\lim}_{Y_{\bullet}}(D)$ of a pseudoeffective divisor $D$ with
respect to an admissible flag $Y_{\bullet}$ (see Definition 2.2). By
definition,
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\Delta^{\lim}_{Y_{\bullet}}(D)$
holds in general and
$\Delta_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(D)$
when $D$ is a big divisor. See Subsection 2.6 for more details.

By [LM, Proposition 4.1 (i)], it is known that Okounkov bodies are numerical
in nature, i.e., if two big divisors $D,D^{\prime}$ on a smooth projective
variety $X$ are numerically equivalent, then
$\Delta_{Y_{\bullet}}(D)=\Delta_{Y_{\bullet}}(D^{\prime})$ for every
admissible flag $Y_{\bullet}$ on $X$. The converse also holds by [J, Theorem
A]. These results were extended to the pseudoeffective divisors using the
limiting Okounkov bodies in [CHPW1, Theorem C]. Thus theoretically one should
be able to read off all the numerical information of a given pseudoeffective
divisor from its limiting Okounkov bodies. However, we note that the valuative
Okounkov bodies are not numerical in nature as we observed in [CHPW1, Remark
3.13].

The first aim of this paper is to show that as is often the case, imposing the
abundance conditions on divisors turns the valuative Okounkov bodies into the
numerical objects (see Subsection 2.7 for definition and properties of
abundant divisors). More precisely, we prove the following theorem, which is
an extension of [LM, Proposition 4.1 (i)] and [J, Theorem A] to valuative
Okounkov bodies of abundant divisors.

######  Theorem A (=Corollary 4.11).

Let $D,D^{\prime}$ be pseudoeffective abundant $\mathbb{R}$-divisors on a
smooth projective variety $X$. Then we have:

$D\equiv D^{\prime}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ if and
only if }\leavevmode\nobreak\ \leavevmode\nobreak\
\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})\leavevmode\nobreak\
\text{ for every admissible flag $Y_{\bullet}$ on $X$}.$

We remark that the ‘if’ direction of Theorem A does not hold when
$D,D^{\prime}$ are not abundant. It is because we may possibly have
$\kappa(D)\neq\kappa(D^{\prime})$ even when $D\equiv D^{\prime}$ so that
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\neq\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})$
for every admissible flag $Y_{\bullet}$ (see [CHPW1, Remark 3.13]). However,
the ‘only if’ direction of Theorem A holds without the abundance assumption on
$D,D^{\prime}$ (see Proposition 4.9). As a consequence, we will also show in
Corollary 4.12 that if $\operatorname{Pic}(X)$ is finitely generated, then for
any divisors $D,D^{\prime}$ with $\kappa(D),\kappa(D^{\prime})\geq 0$, we
have: __

$D\sim_{\mathbb{R}}D^{\prime}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{
if and only if }\leavevmode\nobreak\ \leavevmode\nobreak\
\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})\leavevmode\nobreak\
\text{ for every admissible flag $Y_{\bullet}$ on $X$}.$

It is natural to ask how to extract the numerical properties of abundant
divisors from the valuative Okounkov bodies. The next aim is to study the
restricted base locus $\mathbf{B}_{-}(D)$ (see Subsection 2.2 for the
definition) of an abundant divisor $D$ using the valuative Okounkov bodies.
The following statement for limiting Okounkov bodies was shown in [CHPW2,
Theorem A] (see also [KL1], [KL2], [KL3]).

######  Theorem B (=Theorem 5.1).

Let $D$ be a pseudoeffective abundant $\mathbb{R}$-divisor on a smooth
projective variety $X$ of dimension $n$, and $x\in X$ be a point. Then the
following are equivalent:

  1. $(1)$

$x\in\mathbf{B}_{-}(D)$

  2. $(2)$

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ does not contain the origin of
$\mathbb{R}^{n}$ for some admissible flag $Y_{\bullet}$ on $X$ centered at
$x$.

  3. $(3)$

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ does not contain the origin of
$\mathbb{R}^{n}$ for every admissible flag $Y_{\bullet}$ on $X$ centered at
$x$.

As we observed in [CPW1, Remark 4.10],
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ may not contain the origin
$\mathbb{R}^{n}$ for some admissible flag $Y_{\bullet}$ even if $D$ is nef.
Note that the analogous statements concerning $\mathbf{B}_{+}(D)$ as in
[CHPW2, Theorem C] for an abundant divisor $D$ easily follow from [CHPW2,
Theorem 6.5] since big divisors are abundant and $\mathbf{B}_{+}(D)=X$ holds
if $D$ is not big.

In [CHPW1], we prove that the Okounkov bodies
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ and
$\Delta^{\lim}_{Y_{\bullet}}(D)$ encode nice asymptotic properties of the
divisor $D$ if the given admissible flag $Y_{\bullet}$ contains a Nakayama
subvariety or a positive volume subvariety of $D$, respectively (see
Subsection 2.6 for the definitions of these special subvarieties). We show
that for an abundant divisor $D$ on a smooth projective variety $X$, a
subvariety is a Nakayama subvariety of $D$ if and only if it is a positive
volume subvariety of $D$ (Proposition 2.13). However, this is still not enough
to guarantee that the two Okounkov bodies
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$,
$\Delta^{\lim}_{Y_{\bullet}}(D)$ coincide as our last main result below shows.

######  Theorem C (=Theorem 6.1).

Let $D$ be a pseudoeffective abundant $\mathbb{R}$-divisor on an
$n$-dimensional smooth projective variety $X$ with $\kappa(D)>0$. Fix an
admissible flag $Y_{\bullet}$ on $X$ such that $V=Y_{n-\kappa(D)}$ is a
Nakayama subvariety of $D$ and $Y_{n}$ is a general point in $X$. Consider the
Iitaka fibration $\phi\colon X^{\prime}\to Z$ of $D$ and the strict transform
$V^{\prime}$ of $V$ on $X^{\prime}$. Then we have

$\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\lim}_{Y_{\bullet}}(D))=\deg(\phi|_{V^{\prime}}\colon
V^{\prime}\to
Z)\cdot\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)).$

In particular,
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(D)$
if and only if the map $\phi|_{V^{\prime}}\colon V^{\prime}\to Z$ is
generically injective.

Recall that Nakayama subvarieties and positive volume subvarieties always
exist by [CHPW1, Propositions 2.13 and 2.22]. We remark that even if $D$ is an
abundant $\mathbb{R}$-divisor with $\kappa(D)>0$, there may not exist Nakayama
subvarieties $V$ giving rise to a generically injective map
$\phi|_{V^{\prime}}\colon V^{\prime}\to Z$ as in Theorem C (see Example 6.3).
See also Section 4 of [CHPW1] for more related results.

To prove the above main theorems, we use results on Nakayama subvarieties and
Iitaka fibrations (see Subsection 2.7). Another main ingredient is some
versions of Fujita’s approximations for the valuative Okounkov bodies
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ of an effective divisor $D$
(Lemma 3.1) and for the limiting Okounkov bodies
$\Delta^{\lim}_{Y_{\bullet}}(D)$ of an abundant divisor $D$ (Lemma 3.6). These
results may be regarded as alternative constructions of Okounkov bodies
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ and
$\Delta^{\lim}_{Y_{\bullet}}(D)$.

The organization of the paper is as follows. We will first collect relevant
basic facts on various asymptotic invariants, Iitaka fibrations, Zariski
decompositions, Okounkov bodies, etc. in Section 2. In Section 3, we show the
main ingredient of the proofs of Theorems A and C. Sections 4, 5, and 6 are
devoted to proving Theorems A, B, and C, respectively.

##  2\. Preliminaries

In this section, we collect relevant facts which will be used later.

###  2.1. Conventions

By a _$($ sub$)$variety_, we mean an irreducible (sub)variety, and $X$ denotes
a smooth projective variety of dimension $n$. Unless otherwise stated, a
_divisor_ means an $\mathbb{R}$-Cartier $\mathbb{R}$-divisor. A divisor $D$ on
$X$ is _pseudoeffective_ if its numerical equivalence class $[D]\in
N^{1}(X)_{\mathbb{R}}$ lies in the pseudoeffective cone
$\overline{\operatorname{Eff}}(X)$, the closure of the cone spanned by
effective divisor classes. A divisor $D$ on $X$ is _big_ if $[D]$ lies in the
interior $\text{Big}(X)$ of $\overline{\operatorname{Eff}}(X)$. Throughout the
paper, we work over the field $\mathbb{C}$ of complex numbers.

###  2.2. Asymptotic invariants

Let $D$ be a divisor on $X$. The _stable base locus_ of $D$ is defined as
$\operatorname{SB}(D):=\bigcap_{D\sim_{\mathbb{R}}D^{\prime}\geq
0}\operatorname{Supp}(D^{\prime})$. The _augmented base locus_ of $D$ is
defined as $\mathbf{B}_{+}(D):=\bigcap_{A}\text{SB}(D-A)$ where the
intersection is taken over all ample divisors $A$. The _restricted base locus_
of $D$ is defined as $\mathbf{B}_{-}(D):=\bigcup_{A}\operatorname{SB}(D+A)$
where the union is taken over all ample divisors $A$. It is easy to check that
$\mathbf{B}_{+}(D)$ and $\mathbf{B}_{-}(D)$ depend only on the numerical class
of $D$. Note that $\mathbf{B}_{-}(D)=X$ (resp. $\mathbf{B}_{+}(D)=X$) if and
only if $D$ is not pseudoeffective (resp. not big), and
$\mathbf{B}_{-}(D)=\emptyset$ (resp. $\mathbf{B}_{+}(D)=\emptyset$) if and
only if $D$ is nef (resp. ample). For more details, see [ELMNP1].

Consider a subvariety $V\subseteq X$ of dimension $v$. The _restricted volume_
of $D$ along $V$ is defined as
$\operatorname{vol}_{X|V}(D):=\limsup_{m\to\infty}\frac{h^{0}(X|V,\lfloor
mD\rfloor)}{m^{v}/v!}$ where $h^{0}(X|V,\lfloor mD\rfloor)$ is the dimension
of the image of the natural restriction map $\varphi\colon
H^{0}(X,\mathcal{O}_{X}(\lfloor mD\rfloor))\to H^{0}(V,\mathcal{O}_{V}(\lfloor
mD\rfloor|_{V}))$. If $V\not\subseteq\mathbf{B}_{+}(D)$, then the restricted
volume $\operatorname{vol}_{X|V}(D)$ depends only on the numerical class of
$D$, and it uniquely extends to a continuous function
$\operatorname{vol}_{X|V}\colon\text{Big}^{V}(X)\to\mathbb{R}$ where
$\text{Big}^{V}(X)$ is the set of all $\mathbb{R}$-divisor classes $\xi$ such
that $V$ is not properly contained in any irreducible component of
$\mathbf{B}_{+}(\xi)$. When $V=X$, we simply let
$\operatorname{vol}_{X}(D):=\operatorname{vol}_{X|X}(D)$, and we call it the
_volume_ of $D$. For more details, we refer to [La, Section 2.2.C], [ELMNP2].
See also [CHPW1, Subsection 2.3].

Now, assume that $V\not\subseteq\mathbf{B}_{-}(D)$. The _augmented restricted
volume_ of $D$ along $V$ is defined as
$\operatorname{vol}_{X|V}^{+}(D):=\lim_{\varepsilon\to
0+}\operatorname{vol}_{X|V}(D+\varepsilon A)$ where $A$ is an ample divisor on
$X$. The definition is independent of the choice of $A$. Note that
$\operatorname{vol}_{X|V}^{+}(D)=\operatorname{vol}_{X|V}(D)$ for
$D\in\text{Big}^{V}(X)$. This also extends uniquely to a continuous function
$\operatorname{vol}_{X|V}^{+}\colon\overline{\text{Eff}}^{V}(X)\to\mathbb{R}$
where
$\overline{\text{Eff}}^{V}(X):=\text{Big}^{V}(X)\cup\\{\xi\in\overline{\text{Eff}}(X)\setminus\text{Big}(X)\mid
V\not\subseteq\mathbf{B}_{-}(\xi)\\}$. For $D\in\overline{\text{Eff}}^{V}(X)$,
we have
$\operatorname{vol}_{X|V}(D)\leq\operatorname{vol}_{X|V}^{+}(D)\leq\operatorname{vol}_{V}(D|_{V})$,
and both inequalities can be strict in general. For more details, see [CHPW1,
Subsection 2.3].

###  2.3. Iitaka dimensions

Let $D$ be a divisor on $X$. The _Iitaka dimension_ of $D$ is defined as

$\kappa(D):=\max\left\\{k\in\mathbb{Z}_{\geq
0}\left|\;\limsup_{m\to\infty}\frac{h^{0}(X,\mathcal{O}_{X}(\lfloor
mD\rfloor))}{m^{k}}>0\right.\right\\}$

if $h^{0}(X,\mathcal{O}_{X}(\lfloor mD\rfloor))\neq 0$ for some $m>0$, and
$\kappa(D):=-\infty$ otherwise. Similarly, the _numerical Iitaka dimension_ of
$D$ is defined as

$\kappa_{\nu}(D):=\max\left\\{k\in\mathbb{Z}_{\geq
0}\left|\;\limsup_{m\to\infty}\frac{h^{0}(X,\mathcal{O}_{X}(\lfloor
mD\rfloor+A))}{m^{k}}>0\right.\right\\}$

for some fixed ample Cartier divisor $A$ if $h^{0}(X,\mathcal{O}_{X}(\lfloor
mD\rfloor+A))\neq 0$ for some $m>0$, and $\kappa_{\nu}(D):=-\infty$ otherwise.
It is independent of the choice of the ample divisor $A$.

We remark that $\kappa(D)$ is not an invariant of the $\mathbb{R}$-linear
equivalence class of $D$. Nonetheless, it satisfies the property that
$\kappa(D)=\kappa(D^{\prime})$ when $\kappa(D),\kappa(D^{\prime})\geq 0$ and
$D\sim_{\mathbb{R}}D^{\prime}$ (see [CHPW1, Remark 2.8]). On the other hand,
$\kappa_{\nu}(D)$ is a numerical invariant, i.e., it depends only on the
numerical class $[D]\in N^{1}(X)_{\mathbb{R}}$. By definition,
$\kappa(D)\leq\kappa_{\nu}(D)$ holds and the inequality can be strict in
general. However, $\kappa_{\nu}(D)=\dim X$ if and only if $\kappa(D)=\dim X$
if and only if $D$ is big. We refer to [E], [Le1], [N] for more detailed
properties.

###  2.4. Iitaka fibration

Let $D$ be a divisor on $X$ with $\kappa(D)>0$. Then there exists a morphism
$\phi\colon X^{\prime}\to Z$ of smooth projective varieties $X^{\prime},Z$
with connected fibers such that for all sufficiently large integers $m>0$, the
rational maps $\phi_{mD}\colon X\dashrightarrow Z_{m}$ given by $|\lfloor
mD\rfloor|$ are birationally equivalent to $\phi$, i.e., there exists a
commutative diagram

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of a rational map $\phi_{mD}$ and morphisms $f,\phi,g_{m}$ with connected
fibers, where the horizontal maps are birational, $\dim Z=\kappa(D)$, and
$\kappa(f^{*}D|_{F})=0$, where $F$ is a very general fiber of $\phi$. (see
e.g., [La, Theorem 2.1.33], [N, Theorem-Definition II.3.14]). Such a fibration
is called an _Iitaka fibration_ of $D$. It is unique up to birational
equivalence.

###  2.5. Zariski decompositions

To define the divisorial Zariski decomposition, we first consider a divisorial
valuation $\sigma$ on $X$ with the center $V:=\operatorname{Cent}_{X}\sigma$
on $X$. If $D$ is a big divisor on $X$, we define _the asymptotic valuation_
of $\sigma$ at $D$ as
$\operatorname{ord}_{V}(||D||):=\inf\\{\sigma(D^{\prime})\mid D\equiv
D^{\prime}\geq 0\\}$. If $D$ is only a pseudoeffective divisor on $X$, we
define $\operatorname{ord}_{V}(||D||):=\lim_{\varepsilon\to
0+}\operatorname{ord}_{V}(||D+\varepsilon A||)$ for some ample divisor $A$ on
$X$. This definition is independent of the choice of $A$. Note that
$\operatorname{ord}_{V}(||D||)$ is a numerical invariant of $D$. The
_divisorial Zariski decomposition_ of a pseudoeffective divisor $D$ is the
decomposition

$D=P_{\sigma}+N_{\sigma}=P_{\sigma}(D)+N_{\sigma}(D)$

into the _negative part_
$N_{\sigma}=N_{\sigma}(D):=\sum_{E}\operatorname{ord}_{E}(||D||)E$ where the
summation is over all the finitely many prime divisors $E$ of $X$ such that
$\operatorname{ord}_{E}(||D||)>0$ and the _positive part_
$P_{\sigma}=P_{\sigma}(D):=D-N_{\sigma}$. The positive part $P_{\sigma}(D)$ is
characterized as the maximal divisor such that $P_{\sigma}\leq D$ and
$P_{\sigma}(D)$ is movable (see [N, Proposition III.1.14]). Note that by
construction $N_{\sigma}(D)$ is a numerical invariant of $D$. For more
details, see [B1], [N], [P].

Let $D$ be a divisor on $X$ with $\kappa(D)\geq 0$. The _$s$ -decomposition_
of $D$ is the decomposition

$D=P_{s}+N_{s}=P_{s}(D)+N_{s}(D)$

into the _negative part_ $N_{s}=N_{s}(D):=\inf\\{L\mid
L\sim_{\mathbb{R}}D,L\geq 0\\}$ and the _positive part_
$P_{s}=P_{s}(D):=D-N_{s}$. The positive part $P_{s}(D)$ is characterized as
the smallest divisor such that $P_{s}\leq D$ and $R(X,P_{s})\simeq R(X,D)$
(see [P, Proposition 4.8]). Note that $N_{s}(D)$ is an $\mathbb{R}$-linear
equivalence invariant of $D$. Note that $P_{s}(D)\leq P_{\sigma}(D)$ and
$P_{s}(D),P_{\sigma}(D)$ do not coincide in general. If $D$ is an abundant
divisor, then $P_{s}(D)=P_{\sigma}(D)$ so that $P_{s}(D),N_{s}(D)$ become
numerical invariants of $D$ (see Theorem 2.11 (2)). For more details, see [P].

###  2.6. Okounkov bodies

In this subsection, we recall the construction and basic properties of
Okounkov bodies associated to pseudoeffective divisors in [LM], [KK], [CHPW1].

Throughout this subsection, we fix an _admissible flag_ $Y_{\bullet}$ on $X$,
which by definition is a sequence of subvarieties

$Y_{\bullet}:X=Y_{0}\supseteq Y_{1}\supseteq\cdots\supseteq Y_{n-1}\supseteq
Y_{n}=\\{x\\}$

where each $Y_{i}$ is an irreducible subvariety of codimension $i$ in $X$ and
is nonsingular at $x$. Let $D$ be a divisor on $X$ with
$|D|_{\mathbb{R}}:=\\{D^{\prime}\mid D\sim_{\mathbb{R}}D^{\prime}\geq
0\\}\neq\emptyset$. We define a valuation-like function

$\nu_{Y_{\bullet}}:|D|_{\mathbb{R}}\to\mathbb{R}_{\geq 0}^{n}$

as follows: For $D^{\prime}\in|D|_{\mathbb{R}}$, let
$\nu_{1}=\nu_{1}(D^{\prime}):=\operatorname{ord}_{Y_{1}}(D^{\prime})$. Since
$D^{\prime}-\nu_{1}(D^{\prime})Y_{1}$ is effective and does not contain
$Y_{2}$, we define
$\nu_{2}=\nu_{2}(D^{\prime}):=\operatorname{ord}_{Y_{2}}((D^{\prime}-\nu_{1}Y_{1})|_{Y_{1}})$.
We then inductively define
$\nu_{i+1}=\nu_{i+1}(D^{\prime}):=\operatorname{ord}_{Y_{i+1}}((\cdots((D^{\prime}-\nu_{1}Y_{1})|_{Y_{1}}-\nu_{2}Y_{2})|_{Y_{2}}-\cdots-\nu_{i}Y_{i})|_{Y_{i}})$.
Thus we finally obtain

$\nu_{Y_{\bullet}}(D^{\prime}):=(\nu_{1}(D^{\prime}),\nu_{2}(D^{\prime}),\cdots,\nu_{n}(D^{\prime})).$

######  Definition 2.1.

When $|D|_{\mathbb{R}}\neq\emptyset$, the _Okounkov body_
$\Delta_{Y_{\bullet}}(D)$ of a divisor $D$ on $X$ with respect to an
admdissible flag $Y_{\bullet}$ on $X$ is defined as the closure of the convex
hull of $\nu_{Y_{\bullet}}(|D|_{\mathbb{R}})$ in $\mathbb{R}^{n}_{\geq 0}$.
When $|D|_{\mathbb{R}}=\emptyset$, we set
$\Delta_{Y_{\bullet}}(D):=\emptyset$.

More generally, a similar construction can be applied to a graded linear
series $W_{\bullet}$ associated to a $\mathbb{Z}$-divisor on $X$ to construct
the Okounkov body $\Delta_{Y_{\bullet}}(W_{\bullet})$ of $W_{\bullet}$ with
respect to $Y_{\bullet}$. For more details, we refer to [LM].

In [LM], [KK], the Okounkov bodies $\Delta_{Y_{\bullet}}(D)$ were mainly
studied for big divisors. When $D$ is not big, the following extension was
introduced in [CHPW1].

######  Definition 2.2 ([CHPW1, Definition 1.1]).

  1. (1)

For a divisor $D$ which is effective up to $\sim_{\mathbb{R}}$, i.e.,
$|D|_{\mathbb{R}}\neq\emptyset$, the _valuative Okounkov body_
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ of $D$ with respect to an
admissible flag $Y_{\bullet}$ is defined as the closure of the convex hull of
$\nu_{Y_{\bullet}}(|D|_{\mathbb{R}})$ in $\mathbb{R}^{n}_{\geq 0}$. If
$|D|_{\mathbb{R}}=\emptyset$, then we set
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D):=\emptyset$.

  2. (2)

For a pseudoeffective divisor $D$, the _limiting Okounkov body_
$\Delta^{\lim}_{Y_{\bullet}}(D)$ of $D$ with respect to an admissible flag
$Y_{\bullet}$ is defined as

$\Delta^{\lim}_{Y_{\bullet}}(D):=\lim_{\varepsilon\to
0+}\Delta_{Y_{\bullet}}(D+\varepsilon
A)=\bigcap_{\varepsilon>0}\Delta_{Y_{\bullet}}(D+\varepsilon
A)\subseteq\mathbb{R}^{n}_{\geq 0}$

where $A$ is an ample divisor on $X$. (Note that
$\Delta^{\lim}_{Y_{\bullet}}(D)$ is independent of the choice of $A$.) If $D$
is not pseudoeffective, then we set
$\Delta^{\lim}_{Y_{\bullet}}(D):=\emptyset$.

Note that we actually have
$\Delta_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ for any
divisor $D$ and any admissible flag $Y_{\bullet}$. However, we will only use
the notation $\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ unless $D$ is
known to be big in order to distinguish our results from the well known cases
for big divisors. Boucksom’s _numerical Okounkov body_
$\Delta^{\text{num}}_{Y_{\bullet}}(D)$ in [B2] coincides with our limiting
Okounkov body $\Delta^{\lim}_{Y_{\bullet}}(D)$.

By definition,
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\Delta^{\lim}_{Y_{\bullet}}(D)$,
and the inclusion can be strict in general (see [CHPW1, Examples 4.2 and
4.3]). For a divisor $D$ with $\kappa(D)\geq 0$, by [B2, Proposition 3.3 and
Lemma 4.8], we have
$\dim\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\kappa(D)\leq\dim\Delta^{\lim}_{Y_{\bullet}}(D)\leq\kappa_{\nu}(D)$
for any admissible flag $Y_{\bullet}$. If $D$ is big, then
$\Delta_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(D)$.

The valuative Okounkov body $\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ is
an $\mathbb{R}$-linear invariant of $D$ but not a numerical invariant of $D$
(see [CHPW1, Remark 3.13 and Proposition 3.15]). The limiting Okounkov body
$\Delta^{\lim}_{Y_{\bullet}}(D)$ is a numerical invariant of $D$, and $D\equiv
D^{\prime}$ if and only if
$\Delta^{\lim}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(D^{\prime})$ for
every admissible flag $Y_{\bullet}$ on $X$ when $D,D^{\prime}$ are
pseudoeffective divisors (see [CHPW1, Theorem C]).

######  Lemma 2.3 (cf. [CHPW2, Lemma 3.4], [CPW2, Lemma 3.4]).

Let $D$ be a divisor on $X$. Consider a birational morphism
$f:\widetilde{X}\to X$ with $\widetilde{X}$ smooth and an admissible flag

$\widetilde{Y}_{\bullet}:\widetilde{X}=\widetilde{Y}_{0}\supseteq\widetilde{Y}_{1}\supseteq\cdots\supseteq\widetilde{Y}_{n-1}\supseteq\widetilde{Y}_{n}=\\{x^{\prime}\\}$

on $\widetilde{X}$. Suppose that $f$ is isomorphic over a neighborhood of
$f(x^{\prime})$ and

$Y_{\bullet}:=f(\widetilde{Y}_{\bullet}):X=f(\widetilde{Y}_{0})\supseteq
f(\widetilde{Y}_{1})\supseteq\cdots\supseteq f(\widetilde{Y}_{n-1})\supseteq
f(\widetilde{Y}_{n})=\\{f(x^{\prime})\\}$

is an admissible flag on $X$. Then we have
$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(f^{*}D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$
and
$\Delta^{\lim}_{\widetilde{Y}_{\bullet}}(f^{*}D)=\Delta^{\lim}_{Y_{\bullet}}(D)$.

###### Proof.

The limiting Okounkov body case is shown in [CHPW2, Lemma 3.4]. The proof for
the valuative Okounkov body case is almost identical, so we leave the details
to the readers. ∎

######  Remark 2.4.

By Lemma 2.3 and [CHPW2, Lemma 3.5], we can assume that each $Y_{i}$ in the
admissible flag $Y_{\bullet}$ on $X$ is smooth (see also [CHPW2, Remark 3.6]).

######  Lemma 2.5 (cf. [CHPW2, Lemma 3.9], [CPW2, Lemma 3.5]).

Let $D$ be a divisor on $X$ with the $s$-decomposition $D=P_{s}+N_{s}$ and the
divisorial Zariski decomposition $D=P_{\sigma}+N_{\sigma}$. Fix an admissible
flag $Y_{\bullet}$ on $X$. Then we have
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(P_{s})+\Delta^{\operatorname{val}}_{Y_{\bullet}}(N_{s})$
and
$\Delta^{\lim}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(P_{\sigma})+\Delta^{\lim}_{Y_{\bullet}}(N_{\sigma})$.
If $Y_{n}$ is a general point (i.e.,
$Y_{n}\not\subseteq\operatorname{Supp}(N)$), then
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(P_{s})$
and $\Delta^{\lim}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(P_{\sigma})$.

###### Proof.

The assertion for $\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ follows from
the fact that $R(X,D)\simeq R(X,P_{s})$ and the construction of the valuative
Okounkov body. The assertion for $\Delta^{\lim}_{Y_{\bullet}}(D)$ is nothing
but [CHPW2, Lemma 3.9]. ∎

In [CHPW1], we introduced a Nakayama subvariety and positive volume subvariety
of a divisor $D$ to extract asymptotic invariants of $D$ from the Okounkov
bodies.

######  Definition 2.6 ([CHPW1, Definitions 2.12 and 2.19], [CPW2, Definition
4.1]).

  1. (1)

For a divisor $D$ such that $\kappa(D)\geq 0$, a _Nakayama subvariety of $D$_
is defined as an irreducible subvariety $U\subseteq X$ such that $\dim
U=\kappa(D)$ and for every integer $m\geq 0$ the natural map

$H^{0}(X,\mathcal{O}_{X}(\lfloor mD\rfloor))\to
H^{0}(U,\mathcal{O}_{X}(\lfloor mD|_{U}\rfloor))$

is injective (or equivalently,
$H^{0}(X,\mathcal{I}_{U}\otimes\mathcal{O}_{X}(\lfloor mD\rfloor))=0$ where
$\mathcal{I}_{U}$ is an ideal sheaf of $U$ in $X$).

  2. (2)

For a divisor $D$ with $\kappa_{\nu}(D)\geq 0$, a _positive volume subvariety
of $D$_ is defined as an irreducible subvariety $V\subseteq X$ such that $\dim
V=\kappa_{\nu}(D)$ and $\operatorname{vol}_{X|V}^{+}(D)>0$.

We have the following characterization of a Nakayama subvariety and a positive
volume subvariety in terms of Okounkov bodies.

######  Theorem 2.7 ([CPW2, Theorem 1.2]).

Let $D$ be a divisor on $X$. Fix an admissible flag $Y_{\bullet}$ such that
$Y_{n}$ is a general point in $X$. Then we have the following:

  1. $(1)$

If $D$ is effective, then $Y_{\bullet}$ contains a Nakayama subvariety of $D$
if and only if
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa(D)}\times\mathbb{R}^{\kappa(D)}$.

  2. $(2)$

If $D$ is pseudoeffective, then $Y_{\bullet}$ contains a positive volume
subvariety of $D$ if and only if
$\Delta^{\lim}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa_{\nu}(D)}\times\mathbb{R}^{\kappa_{\nu}(D)}$
and $\dim\Delta^{\lim}_{Y_{\bullet}}(D)=\kappa_{\nu}(D)$.

The following is the main result of [CHPW1].

######  Theorem 2.8 ([CHPW1, Theorems A and B]).

  1. $(1)$

Let $D$ be a divisor on $X$ with $\kappa(D)\geq 0$. Fix an admissible flag
$Y_{\bullet}$ containing a Nakayama subvariety $U$ of $D$ such that $Y_{n}$ is
a general point in $X$. Then
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa(D)}\times\mathbb{R}^{\kappa(D)}$
so that one can regard
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\mathbb{R}^{\kappa(D)}$.
Furthermore, we have

$\dim\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\kappa(D)\text{ and
}\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D))=\frac{1}{\kappa(D)!}\operatorname{vol}_{X|U}(D).$

  2. $(2)$

Let $D$ be a pseudoeffective divisor on $X$, and fix an admissible flag
$Y_{\bullet}$ containing a positive volume subvariety $V$ of $D$. Then
$\Delta^{\lim}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa_{\nu}(D)}\times\mathbb{R}^{\kappa_{\nu}(D)}$
so that one can regard
$\Delta^{\lim}_{Y_{\bullet}}(D)\subseteq\mathbb{R}^{\kappa_{\nu}(D)}$.
Furthermore, we have

$\dim\Delta^{\lim}_{Y_{\bullet}}(D)=\kappa_{\nu}(D)\text{ and
}\operatorname{vol}_{\mathbb{R}^{\kappa_{\nu}(D)}}(\Delta^{\lim}_{Y_{\bullet}}(D))=\frac{1}{\kappa_{\nu}(D)!}\operatorname{vol}_{X|V}^{+}(D).$

######  Remark 2.9.

As in [CHPW1], [CPW2], when considering
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ (resp.
$\Delta^{\lim}_{Y_{\bullet}}(D)$), we say that $Y_{n}$ is _general_ if it is
not contained in $\text{SB}(D)$ (resp. $\mathbf{B}_{-}(D)$) (see [CPW2, Remark
4.7]).

###  2.7. Abundant divisor

######  Definition 2.10.

A pseudoeffective divisor $D$ on $X$ is said to be _abundant_ if
$\kappa(D)=\kappa_{\nu}(D)$.

The following theorem essentially due to Lehmann will play crucial roles in
proving our main results, Theorems A, B, and C.

######  Theorem 2.11.

Let $D$ be a pseudoeffective abundant divisor on $X$. Then the following hold:

  1. $(1)$

If $D^{\prime}$ is a divisor on $X$ such that $D\equiv D^{\prime}$, then
$D^{\prime}$ is also an abundant divisor.

  2. $(2)$

For any divisorial valuation $\sigma$ on $X$ with the center
$V=\operatorname{Cent}_{X}\sigma$ on $X$, we have

$\operatorname{ord}_{V}(||D||)=\inf\\{\sigma(D^{\prime})\mid
D\sim_{\mathbb{R}}D^{\prime}\geq 0\\}.$

In particular, $P_{\sigma}(D)=P_{s}(D)$.

  3. $(3)$

There is a smooth projective variety $W$ admitting a birational map $\mu\colon
W\to X$ and a morphism $g\colon W\to T$ with connected fibers such that
$P_{\sigma}(\mu^{*}D)\sim_{\mathbb{Q}}P_{\sigma}(g^{*}B)$ for some big divisor
$B$ on $T$.

###### Proof.

See [Le2, Corollary 6.3] for (1), [Le2, Proposition 6.4] and [CPW2, Lemma 2.3]
for (2), and [Le2, Theorem 6.1] for (3). ∎

Note that the morphism $g$ in Theorem 2.11 (3) is birationally equivalent to
an Iitaka fibration of $D$ (see [Le2, Proof of Theorem 6.1]).

######  Lemma 2.12.

Let $D$ be a pseudoeffective abundant divisor on $X$. If $V$ is a Nakayama
subvariety of $D$ or a positive volume subvariety of $D$, then
$V\not\subseteq\operatorname{SB}(D)$.

###### Proof.

If $V$ is a Nakayama subvariety of $D$, then the assertion follows from
definition. Assume that $V$ is a positive volume subvariety of $D$. We can
take an admissible flag $Y_{\bullet}$ containing $V$. By Theorem 2.8 (2),
$\Delta^{\lim}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa_{\nu}(D)}\times\mathbb{R}^{\kappa_{\nu}(D)}$.
Since
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\Delta^{\lim}_{Y_{\bullet}}(D)$,
it follows that $\operatorname{ord}_{V}(D^{\prime})=0$ for every effective
divisor $D^{\prime}\sim_{\mathbb{R}}D$. Thus
$V\not\subseteq\operatorname{Supp}(D^{\prime})$, so we are done. ∎

######  Proposition 2.13.

Let $D$ be a pseudoeffective abundant divisor on $X$. A subvariety $V$ of $X$
is a Nakayama subvariety of $D$ if and only if it is a positive volume
subvariety of $D$.

###### Proof.

We can always construct an admissible flag $Y_{\bullet}$ on $X$ containing a
given Nakayama subvariety of $D$ or a given positive volume subvariety of $D$.
By Lemma 2.12, we may assume that $Y_{n}$ is a general point in $X$.

Let $V\subseteq X$ be a Nakayama subvariety of $D$. Then for an admissible
flag $Y_{\bullet}$ containing $V$ with general $Y_{n}$, we have
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa(D)}\times\mathbb{R}^{\kappa(D)}$
by Theorem 2.7 (1). Recall that
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\Delta^{\lim}_{Y_{\bullet}}(D)$
and
$\dim\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\kappa(D)=\kappa_{\nu}(D)=\dim\Delta^{\lim}_{Y_{\bullet}}(D)$.
Arguing as in the proof of [CHPW1, Proposition 3.21], we can easily prove that
$\Delta^{\lim}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa(D)}\times\mathbb{R}^{\kappa(D)}$.
Theorem 2.7 (2) implies that $V$ is a positive volume subvariety of $D$.

Let $V\subseteq X$ be a positive volume subvariety of $D$, and $Y_{\bullet}$
be an admissible flag containing $V$ with general $Y_{n}$. Then by Theorem 2.7
(2), we have
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\Delta^{\lim}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa(D)}\times\mathbb{R}^{\kappa(D)}$.
Theorem 2.7 (1) implies that $V$ is a Nakayama subvariety of $D$. ∎

##  3\. Fujita’s approximations for Okounkov bodies

The aim of this section is to prove some versions of Fujita’s approximations
for Okounkov bodies, which may be regarded as alternative constructions of
valuative and limiting Okounkov bodies (see Lemmas 3.1 and 3.6). This will be
used in the course of the proofs of Theorems A and C. Throughout the section,
$X$ is a smooth projective variety of dimension $n$.

###  3.1. Valuative Okounkov body case

We fix notations used throughout this subsection. Let $D$ be a divisor on $X$
with $\kappa(D)>0$. We do not impose the abundant condition on $D$ in this
subsection. Fix an admissible flag $Y_{\bullet}$ on $X$ containing a Nakayama
subvariety $U$ of $D$ such that $Y_{n}=\\{x\\}$ is general in $X$ so that
$x\not\in\operatorname{SB}(D)$. We can regard the valuative Okounkov body
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\\{0\\}^{n-\kappa(D)}\times\mathbb{R}^{\kappa(D)}$
as a subset of $\mathbb{R}^{\kappa(D)}$ (see Theorem 2.8).

Now, for a sufficiently large integer $m>0$, we take a log resolution
$f_{m}\colon X_{m}\to X$ of the base ideal $\mathfrak{b}(\lfloor mD\rfloor)$
so that we obtain a decomposition $f_{m}^{*}(\lfloor
mD\rfloor)=M_{m}^{\prime}+F_{m}^{\prime}$ into a base point free divisor
$M_{m}^{\prime}$ and the fixed part $F_{m}^{\prime}$ of $|f_{m}^{*}(\lfloor
mD\rfloor)|$. Let $M_{m}:=\frac{1}{m}M_{m}^{\prime}$ and
$F_{m}:=\frac{1}{m}F_{m}^{\prime}$. We may assume that $f_{m}\colon X_{m}\to
X$ is isomorphic over a neighborhood of $x$. Let $f_{m}^{*}D=P_{m}+N_{m}$ be
the $s$-decomposition.

Since $Y_{n}$ is general, by taking the strict transforms $Y_{i}^{m}$ of
$Y_{i}$ on $X_{m}$, we obtain an admissible flag
$Y_{\bullet}^{m}:Y_{0}^{m}\supseteq\cdots\supseteq Y_{n}^{m}$ on $X_{m}$. We
note that $U_{m}:=Y_{n-\kappa(D)}^{m}$ is also a Nakayama subvariety of
$f^{*}_{m}D$ since $f_{m}$ is $U$-birational (see [CHPW1, Proposition 2.15]).
By definition, we see that $U_{m}$ is also a Nakayama subvariety of $M_{m}$.

Let $W_{\bullet}$ be a graded linear series on $U$ associated to $D|_{U}$
where $W_{k}$ is the image of the natural injective map $H^{0}(X,\lfloor
kD\rfloor)\to H^{0}(U,\lfloor kD\rfloor|_{U})$. We also consider a graded
linear series $W_{\bullet}^{m}$ on $U_{m}$ associated to $M_{m}|_{U_{m}}$
where $W_{k}^{m}$ is the image of the natural injective map
$H^{0}(X_{m},\lfloor kM_{m}\rfloor)\to H^{0}(U_{m},\lfloor
kM_{m}\rfloor|_{U_{m}})$. Note that $\dim W_{m}=\dim W_{m}^{m}$. Let
$\phi_{m}\colon X_{m}\to Z_{m}$ be the morphism defined by $|M_{m}^{\prime}|$.
Then there is an ample divisor $H_{m}$ on $Z_{m}$ such that
$\phi_{m}^{*}H_{m}=M_{m}$. Note that $\phi_{m}|_{U_{m}}\colon U_{m}\to Z_{m}$
is a surjective morphism of projective varieties of the same dimension
$\kappa(D)$. Since $Y_{n}$ is general, we can assume that
$\overline{Y}_{\bullet}^{m}:Z_{m}=\phi_{m}(Y_{n-\kappa(D)}^{m})\supseteq\cdots\supseteq\phi_{m}(Y_{n}^{m})$
is an admissible flag on $Z_{m}$.

The following lemma is the main result of this subsection.

######  Lemma 3.1.

Under the same notations as above, we have

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\lim_{m\to\infty}\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(M_{m})=\lim_{m\to\infty}\Delta_{\overline{Y}_{\bullet}^{m}}(H_{m}).$

###### Proof.

As we noted above, we treat
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D),\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(M_{m}),$
and $\Delta_{\overline{Y}_{\bullet}^{m}}(H_{m})$ as the subsets of the same
fixed space $\mathbb{R}^{\kappa(D)}$. By Lemmas 2.3 and 2.5, we have

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(f_{m}^{*}D)=\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(P_{m}),$

and by [CHPW1, Remark 3.11] and [CPW2, Lemma 5.1], we have

$\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(M_{m})=\Delta_{Y^{m}_{n-\kappa(D)\bullet}}(W_{\bullet}^{m})=\Delta_{\overline{Y}_{\bullet}^{m}}(H_{m}).$

Note that
$\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(M_{m})\subseteq\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(P_{m})$.
By [CHPW1, Remark 3.11], we also have

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta_{Y_{n-\kappa(D)\bullet}}(W_{\bullet}).$

By applying [LM, Theorems A and D], we see that

$\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta_{Y_{n-\kappa(D)\bullet}}(W_{\bullet}))=\lim_{m\to\infty}\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta_{Y^{m}_{n-\kappa(D)\bullet}}(W_{\bullet}^{m})),$

and hence, we obtain

$\Delta_{Y_{n-\kappa(D)\bullet}}(W_{\bullet})=\lim_{m\to\infty}\Delta_{Y^{m}_{n-\kappa(D)\bullet}}(W_{\bullet}^{m}).$

Thus the assertion now follows. ∎

######  Remark 3.2.

When $D$ is a big divisor, Lemma 3.1 is the same as [LM, Theorem D]. See [LM,
Remark 3.4] for the explanation on how this statement implies the classical
statement of Fujita’s approximation (see also [La, Theorem 11.4.4]).

###  3.2. Limiting Okounkov body case

We fix notations used throughout this subsection. Let $D$ be a pseudoeffective
abundant divisor on $X$ with $\kappa(D)=\kappa_{\nu}(D)>0$. Fix an admissible
flag $Y_{\bullet}$ on $X$ containing a positive volume subvariety $V$ of $D$
such that $Y_{n}=\\{x\\}$ is general in $X$ so that
$x\not\in\operatorname{SB}(D)$. By Proposition 2.13, $V$ is also a Nakayama
subvariety of $D$. We can regard the limiting Okounkov body
$\Delta^{\lim}_{Y_{\bullet}}(D)$ in
$\\{0\\}^{n-\kappa(D)}\times\mathbb{R}^{\kappa(D)}$ as a subset of
$\mathbb{R}^{\kappa(D)}$ (see Theorem 2.8).

Now, for a sufficiently large integer $m>0$, we take a log resolution
$f_{m}\colon X_{m}\to X$ of the base ideal $\mathfrak{b}(\lfloor mD\rfloor)$
so that we obtain a decomposition $f_{m}^{*}(\lfloor
mD\rfloor)=M_{m}^{\prime}+F_{m}^{\prime}$ into a base point free divisor
$M_{m}^{\prime}$ and the fixed part $F_{m}^{\prime}$ of $|f_{m}^{*}(\lfloor
mD\rfloor)|$. Let $M_{m}:=\frac{1}{m}M_{m}^{\prime}$ and
$F_{m}:=\frac{1}{m}F_{m}^{\prime}$. We may assume that the $f_{m}\colon
X_{m}\to X$ is isomorphic over a neighborhood of $x$. Let
$f_{m}^{*}D=P_{m}+N_{m}$ be the divisorial Zariski decomposition. By Theorem
2.11 (2), it is also the $s$-decomposition.

Since $Y_{n}$ is general, by taking the strict transforms $Y_{i}^{m}$ of
$Y_{i}$ on $X_{m}$, we obtain an admissible flag
$Y_{\bullet}^{m}:Y_{0}^{m}\supseteq\cdots\supseteq Y_{n}^{m}$ on $X_{m}$. We
note that $V_{m}:=Y_{n-\kappa(D)}^{m}$ is also a positive volume subvariety of
$f^{*}_{m}D$ since $f_{m}$ is $V$-birational ([CHPW1, Proposition 2.24]). By
definition, we also see that $V_{m}$ is also a Nakayama subvariety of $M_{m}$.
Clearly, it is also a positive volume subvariety of $M_{m}$.

The following lemma is easy (cf. [N, Lemma II.2.11]).

######  Lemma 3.3.

Let $f\colon X\to Y$ be a surjective morphism with connected fibers between
smooth projective varieties, and $D$ be an effective divisor on $Y$. Then
$H^{0}(X,\lfloor f^{*}(mD)\rfloor)=H^{0}(Y,\lfloor mD\rfloor)$ for a
sufficiently large integer $m>0$.

###### Proof.

We write $\lfloor f^{*}(mD)\rfloor=f^{*}\lfloor mD\rfloor+\lfloor
f^{*}\\{mD\\}\rfloor$. Note that for every irreducible component $E$ of
$\operatorname{Supp}\lfloor f^{*}\\{mD\\}\rfloor$, we have codim$f(E)\geq 2$.
By the projection formula, we obtain $f_{*}\lfloor f^{*}(mD)\rfloor=\lfloor
mD\rfloor$, and the assertion follows. ∎

We now prove a version of Fujita’s approximation for an abundant divisor,
which is a generalization of [Le1, Proposition 3.7].

######  Lemma 3.4.

Under the same notations as above, for a sufficiently large integer $m>0$,
there exists an ample divisor $H$ on $X$ such that

$M_{m}\leq P_{m}\leq M_{m}+\frac{1}{m}f_{m}^{*}H.$

###### Proof.

By Theorem 2.11 (3), we can take a birational morphism $\mu\colon W\to X$ with
$W$ smooth and a contraction $g\colon W\to T$ such that for some big divisor
$B$ on $T$, we have $P^{\prime}\sim_{\mathbb{Q}}P^{\prime\prime}$ where
$\mu^{*}D=P^{\prime}+N^{\prime}$ and
$g^{*}B=P^{\prime\prime}+N^{\prime\prime}$ are the divisorial Zariski
decompositions. By taking further blow-ups of $T$, we may assume that $T$ is
smooth. For any sufficiently large integer $m>0$, as in [Le1, Proof of
Proposition 3.7], we consider a log resolution of $h_{m}\colon T_{m}\to T$ of
the base ideal $\mathfrak{b}(\lfloor mB\rfloor)$ and the asymptotic multiplier
ideal $\mathcal{J}(||mB||)$ so that we obtain a decomposition
$h_{m}^{*}(\lfloor
mB\rfloor)=M_{m}^{\prime\prime\prime}+F_{m}^{\prime\prime\prime}$ into a base
point free divisor $M_{m}^{\prime\prime\prime}$ and the fixed part
$F_{m}^{\prime\prime\prime}$ of $|h_{m}^{*}(\lfloor mB\rfloor)|$. Let
$M_{m}^{\prime\prime}:=\frac{1}{m}M_{m}^{\prime\prime\prime}$ and
$F_{m}^{\prime\prime}:=\frac{1}{m}F_{m}^{\prime\prime\prime}$. Now, for a
sufficiently large integer $m>0$, we take a log resolution $f_{m}^{W}\colon
X_{m}^{W}\to W$ of the base ideal $\mathfrak{b}(\lfloor m\mu^{*}D\rfloor)$ so
that we obtain a decomposition $(f_{m}^{W})^{*}(\lfloor
m\mu^{*}D\rfloor)={M_{m}^{W}}^{\prime}+{F_{m}^{W}}^{\prime}$ into a base point
free divisor ${M_{m}^{W}}^{\prime}$ and the fixed part ${F_{m}^{W}}^{\prime}$
of $|(f_{m}^{W})^{*}(\lfloor m\mu^{*}D\rfloor)|$. Let
$M_{m}^{W}:=\frac{1}{m}{M_{m}^{W}}^{\prime}$ and
$F_{m}^{W}:=\frac{1}{m}{F_{m}^{W}}^{\prime}$. Note that for a sufficiently
large $m^{\prime}>m$, we may take birational morphisms $h_{m^{\prime},m}\colon
T_{m^{\prime}}\to T_{m}$ and $f_{m^{\prime},m}^{W}\colon X_{m^{\prime}}^{W}\to
X_{m}^{W}$. We can also take contractions $g_{m}\colon X_{m}\to T_{m}$. Thus
we have the following commutative diagram:

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We now claim that

($!$) $M_{m}^{W}\sim_{\mathbb{Q}}g_{m}^{*}M_{m}^{\prime\prime}.$

We can assume that $D$ itself is an effective divisor. Since we may also
assume that $m>0$ is sufficiently large and divisible, we can apply Lemma 3.3
successively to obtain

$\begin{array}[]{l}H^{0}(X,\lfloor
mD\rfloor)=H^{0}(W,\lfloor\mu^{*}(mD)\rfloor)=H^{0}(W,\lfloor
mP^{\prime}\rfloor)\\\ =H^{0}(W,\lfloor
mP^{\prime\prime}\rfloor)=H^{0}(W,\lfloor g^{*}(mB)\rfloor)=H^{0}(T,\lfloor
mB\rfloor).\end{array}$

We also have

$H^{0}(X_{m}^{W},mM_{m}^{W})=H^{0}(X,\lfloor mD\rfloor)=H^{0}(T,\lfloor
mB\rfloor)=H^{0}(T_{m},mM_{m}^{\prime\prime})=H^{0}(X_{m}^{W},g_{m}^{*}(mM_{m}^{\prime\prime})).$

Furthermore, we have
$M_{m}^{W}\leq(f_{m}^{W})^{*}P^{\prime}\sim_{\mathbb{Q}}(f_{m}^{W})^{*}P^{\prime\prime}\geq
g_{m}^{*}M_{m}^{\prime\prime}$. Since both
$mM_{m}^{W},g_{m}^{*}(mM_{m}^{\prime\prime})$ are base point free, we obtain
$M_{m}^{W}\sim_{\mathbb{Q}}g_{m}^{*}M_{m}^{\prime\prime}$ as desired.

Let $h_{m}^{*}B=P_{m}^{\prime}+N_{m}^{\prime}$ be the divisorial Zariski
decomposition. By [Le1, Proposition 3.7], there exists an effective divisor
$E^{\prime}$ on $T$ such that $M_{m}^{\prime\prime}\leq P_{m}^{\prime}\leq
M_{m}^{\prime\prime}+\frac{1}{m}{h^{\prime}}_{m}^{*}E^{\prime}$. (Even though
this assertion is slightly different from the actual statement of [Le1,
Proposition 3.7], in its proof, Lehmann also proved this assertion.) Thus we
have

${h}^{*}_{m^{\prime},m}M_{m}^{\prime\prime}\leq
M_{m^{\prime}}^{\prime\prime}\leq
P_{m^{\prime}}^{\prime}\leq{h}^{*}_{m^{\prime},m}P_{m}^{\prime}\leq{h}^{*}_{m^{\prime},m}M_{m}^{\prime\prime}+\frac{1}{m}{h}_{m^{\prime},m}^{*}{h}_{m}^{*}E^{\prime}$

so that $0\leq
M_{m^{\prime}}^{\prime\prime}-{h}^{*}_{m^{\prime},m}M_{m}^{\prime\prime}\leq\frac{1}{m}{h}^{*}_{m^{\prime}}E^{\prime}$.
Let $E:=g^{*}E^{\prime}$. By taking pullback via $g_{m^{\prime}}$ and by
applying the claim ($!$ ‣ 3.2), we obtain $0\leq
M_{m^{\prime}}^{W}-(f_{m^{\prime},m}^{W})^{*}M_{m}^{W}\leq\frac{1}{m}(f_{m^{\prime}}^{W})^{*}E$.
By taking pushforward via $f_{m^{\prime},m}^{W}$, we then have $0\leq
f_{m^{\prime},m*}^{W}M_{m^{\prime}}^{W}-M_{m}^{W}\leq\frac{1}{m}(f_{m}^{W})^{*}E$.
Let $(f_{m}^{W})^{*}\mu^{*}D=P_{m}^{W}+N_{m}^{W}$ be the divisorial Zariski
decomposition, which is also the $s$-decomposition. By definition of the
$s$-decomposition,
$P_{m}^{W}=\lim_{m^{\prime}\to\infty}f_{m^{\prime},m*}^{W}M_{m^{\prime}}^{W}$.
Hence we obtain $0\leq P_{m}^{W}-M_{m}^{W}\leq\frac{1}{m}(f_{m}^{W})^{*}E$. We
can take an ample divisor $H$ on $X$ such that $\mu^{*}H\geq E$. Then we have

($*$) $M_{m}^{W}\leq P_{m}^{W}\leq
M_{m}^{W}+\frac{1}{m}(f_{m}^{W})^{*}\mu^{*}H.$

To finish the proof, we consider a common log resolution $f_{m}^{\prime}\colon
Z\to X$ of $\mu\circ f_{m}^{W}\colon X_{m}^{W}\to X$ and $f_{m}\colon X_{m}\to
X$ with the morphisms $p\colon Z\to X_{m}^{W}$ and $q\colon Z\to X_{m}$. Note
that $M_{m}^{Z}:=p^{*}M_{m}^{W}=q^{*}M_{m}$ is also a base point free divisor.
Let $(f_{m}^{\prime})^{*}D=P_{m}^{Z}+N_{m}^{Z}$ be the divisorial Zariski
decomposition. It is clear that $M_{m}^{Z}\leq P_{m}^{Z}$. On the other hand,
since $P_{m}^{Z}\leq p^{*}P_{m}^{W}$, it follows from ($*$ ‣ 3.2) that
$P_{m}^{Z}\leq M_{m}^{Z}+\frac{1}{m}(f_{m}^{\prime})^{*}H$. Recall that
$P_{m}=q_{*}P_{m}^{Z}$. Thus by taking pushforward via $q$, we finally obtain

$M_{m}\leq P_{m}\leq M_{m}+\frac{1}{m}f_{m}^{*}H.$

This completes the proof. ∎

######  Remark 3.5.

When $D$ is a big divisor, one can easily deduce the classical statement of
Fujita’s approximation (see e.g., [La, Theorem 11.4.4]) from Lemma 3.4.

The following is the main result of this subsection. This generalizes [LM,
Theorem D] to the limiting Okounkov body case.

######  Lemma 3.6.

Under the same notations as above, we have

$\Delta^{\lim}_{Y_{\bullet}}(D)=\lim_{m\to\infty}\Delta_{Y_{n-\kappa_{\nu}(D)\bullet}^{m}}(M_{m}|_{V_{m}}).$

###### Proof.

Similarly, we treat $\Delta^{\lim}_{Y_{\bullet}}(D)$ and
$\Delta_{Y_{n-\kappa_{\nu}(D)\bullet}^{m}}(M_{m}|_{V_{m}})$ in the statement
as the subsets of the same fixed space $\mathbb{R}^{\kappa_{\nu}(D)}$. For any
sufficiently large $m^{\prime}>0$, by Lemmas 2.3 and 2.5, we have

$\Delta^{\lim}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}^{m^{\prime}}}(f_{m^{\prime}}^{*}D)=\Delta^{\lim}_{Y_{\bullet}^{m^{\prime}}}(P_{m^{\prime}}).$

Note that $\Delta^{\lim}_{Y_{\bullet}^{m^{\prime}}}(P_{m^{\prime}})$ is
actually independent of ${m^{\prime}}$. By [CPW2, Lemma 5.5], for any $m>0$,
we have

$\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m})=\Delta^{\lim}_{Y_{n-\kappa_{\nu}(D)\bullet}^{m}}(M_{m}|_{V_{m}}).$

We will prove the statement by verifying
$\Delta^{\lim}_{Y_{\bullet}^{m^{\prime}}}(P_{m^{\prime}})=\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m})$.

By Lemma 3.4, for any sufficiently large integer $m>0$, we have

$M_{m}\leq P_{m}\leq M_{m}+\frac{1}{m}f_{m}^{*}H$

for some ample divisor $H$ on $X$. Since $x$ is a general point, we may assume
that $x\not\in\operatorname{Supp}(H)$. Note that
$\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m}-M_{m})+\Delta^{\lim}_{Y_{\bullet}^{m}}(\frac{1}{m}f_{m}^{*}H+M_{m}-P_{m})\subseteq\Delta^{\lim}_{Y_{\bullet}^{m}}(\frac{1}{m}f_{m}^{*}H)=\frac{1}{m}\Delta_{Y_{\bullet}}(H)$.
Since $\lim_{m\to\infty}\frac{1}{m}\Delta_{Y_{\bullet}}(H)=\\{0\\}$, it
follows that

$\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m}-M_{m})=\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}\left(\frac{1}{m}f_{m}^{*}H+M_{m}-P_{m}\right)=\\{0\\}.$

Note also that
$\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m})+\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m}-M_{m})\subseteq\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m})$
and
$\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m})+\Delta^{\lim}_{Y_{\bullet}^{m}}(\frac{1}{m}f_{m}^{*}H+M_{m}-P_{m})\subseteq\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m}+\frac{1}{m}f_{m}^{*}H)$.
Since
$\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m})\subseteq\mathbb{R}^{\kappa_{\nu}(D)}$
and
$Y^{m}_{n-\kappa_{\nu}(D)}\not\subseteq\mathbf{B}_{+}(M_{m}+\frac{1}{m}f_{m}^{*}H)$,
it follows from Lemma 3.4 and [CPW2, Theorem 1.1] that

$\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m})\subseteq\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m})\subseteq\lim_{m\to\infty}\Delta_{Y^{m}_{n-\kappa_{\nu}(D)\bullet}}\left(M_{m}+\frac{1}{m}f_{m}^{*}H\right).$

The existence of the limits is guaranteed by the following claim:

($\sharp$)
$\lim_{m\to\infty}\operatorname{vol}_{\mathbb{R}^{\kappa_{\nu}(D)}}(\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m}))=\lim_{m\to\infty}\operatorname{vol}_{\mathbb{R}^{\kappa_{\nu}(D)}}\left(\Delta_{Y^{m}_{n-\kappa_{\nu}(D)\bullet}}\left(M_{m}+\frac{1}{m}f_{m}^{*}H\right)\right).$

If this claim ($\sharp$ ‣ 3.2) holds, then

$\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m})=\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m})=\lim_{m\to\infty}\Delta_{Y^{m}_{n-\kappa_{\nu}(D)\bullet}}\left(M_{m}+\frac{1}{m}f_{m}^{*}H\right).$

Since $\Delta^{\lim}_{Y_{\bullet}^{m}}(P_{m})$ coincide with
$\Delta^{\lim}_{Y_{\bullet}}(D)$ for all sufficiently large $m>0$ (as we
recalled in the beginning of the proof), it follows that

$\Delta^{\lim}_{Y_{\bullet}}(D)=\lim_{m\to\infty}\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m}).$

It now remains to prove the equality in the claim ($\sharp$ ‣ 3.2). We may
assume that $V_{m}=Y_{n-\kappa_{\nu}(D)}^{m}$ is a smooth positive volume
subvariety of $M_{m}$, and $f_{m}|_{V_{m}}\colon V_{m}\to V$ is a birational
contraction. By [CPW2, Lemma 5.5], we have

$\operatorname{vol}_{\mathbb{R}^{\kappa_{\nu}(D)}}(\Delta^{\lim}_{Y_{\bullet}^{m}}(M_{m}))=\frac{1}{\kappa_{\nu}(D)!}\operatorname{vol}_{V_{m}}(M_{m}|_{V_{m}})=\frac{1}{\kappa_{\nu}(D)!}(M_{m}|_{V_{m}})^{\kappa_{\nu}(D)}.$

Similarly, by [LM, (2.7) in p.804], we also have

$\begin{array}[]{l}\operatorname{vol}_{\mathbb{R}^{\kappa_{\nu}(D)}}\left(\Delta_{Y^{m}_{n-\kappa_{\nu}(D)\bullet}}\left(M_{m}|_{V_{m}}+\frac{1}{m}(f_{m}^{*}H)|_{V_{m}}\right)\right)\\\
=\frac{1}{\kappa_{\nu}(D)!}\operatorname{vol}_{V_{m}}(M_{m}+\frac{1}{m}f_{m}^{*}H)\\\
=\frac{1}{\kappa_{\nu}(D)!}\left(M_{m}|_{V_{m}}+\frac{1}{m}(f_{m}^{*}H)|_{V_{m}}\right)^{\kappa_{\nu}(D)}\\\
=\frac{1}{\kappa_{\nu}(D)!}\left((M_{m}|_{V_{m}})^{\kappa_{\nu}(D)}+\sum_{k=0}^{\kappa_{\nu}(D)-1}\frac{1}{m^{\kappa_{\nu}(D)-k}}(M_{m}|_{V_{m}})^{k}\cdot((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)-k}\right).\end{array}$

To prove the claim ($\sharp$ ‣ 3.2), it is sufficient to show that for each
$0\leq k\leq\kappa(D)-1$, there exists a constant $C_{k}$ independent of $m$
such that

$\frac{1}{m^{\kappa_{\nu}(D)-k}}(M_{m}|_{V_{m}})^{k}\cdot((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)-k}\leq\frac{C_{k}}{m^{\kappa_{\nu}(D)-k}}.$

If $k=0$, then we have
$((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)}=((f_{m}|_{V_{m}})^{*}(H|_{V}))^{\kappa_{\nu}(D)}=(H|_{V})^{\kappa_{\nu}(D)}$,
which is independent of $m$. Now, suppose that $1\leq k\leq\kappa(D)-1$. Note
that $V_{m}\not\subseteq\operatorname{SB}(f_{m}^{*}D)$ and
$M_{m}|_{V_{m}}\leq(f_{m}^{*}D)|_{V_{m}}$. Thus
$M_{m}|_{V_{m}}\cdot((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)-1}\leq(f_{m}^{*}D)|_{V_{m}}\cdot((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)-1}=D|_{V}\cdot(H|_{V})^{\kappa_{\nu}(D)-1}$.
By [La, Corollary 1.6.3 (i)], we have

$(M_{m}|_{V_{m}})^{k}\cdot((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)-k}\leq\frac{(M_{m}|_{V_{m}}\cdot((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)-1})^{k}}{((f_{m}^{*}H)|_{V_{m}})^{\kappa_{\nu}(D)})^{k-1}}\leq\frac{(D|_{V}\cdot(H|_{V})^{\kappa_{\nu}(D)-1})^{k}}{((H|_{V})^{\kappa_{\nu}(D)})^{k-1}}.$

Note that the right hand side is independent of $m$. We have shown the claim
($\sharp$ ‣ 3.2), and hence, we complete the proof. ∎

##  4\. Numerical equivalence and Okounkov body

In this section, we prove Theorem A as Corollary 4.11. Throughout the section,
$X$ is a smooth projective variety of dimension $n$.

First, we need the following lemma.

######  Lemma 4.1.

Let $f\colon\widetilde{X}\to X$ be a birational morphism with $\widetilde{X}$
smooth, and $D$ be a divisor on $X$ with $\kappa(D)\geq 0$. Consider an
admissible flag

$\widetilde{Y}_{\bullet}:\widetilde{X}=\widetilde{Y}_{0}\supseteq\widetilde{Y}_{1}\supseteq\cdots\supseteq\widetilde{Y}_{n-1}\supseteq\widetilde{Y}_{n}=\\{x^{\prime}\\}$

on $\widetilde{X}$ and an admissible flag

$Y_{\bullet}:X=Y_{0}\supseteq Y_{1}\supseteq\cdots\supseteq Y_{n-1}\supseteq
Y_{n}=\\{x\\}$

on $X$ such that each restriction
$f|_{\widetilde{Y}_{i}}\colon\widetilde{Y}_{i}\to Y_{i}$ is a birational
morphism for $0\leq i\leq n$. Assume that $Y_{i}$ and $\widetilde{Y}_{i}$ are
smooth for $0\leq i\leq n$. For $1\leq i\leq n$, we write
$f|_{\widetilde{Y}_{i-1}}^{*}Y_{i}=\widetilde{Y}_{i}+E_{i}$ for some effective
$f|_{\widetilde{Y}_{i-1}}$-exceptional divisor $E_{i}$ on
$\widetilde{Y}_{i-1}$. Then we have

$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(f^{*}D)=\left\\{\mathbf{x}+\sum_{i=1}^{n-1}x_{i}\cdot\nu_{\widetilde{Y}_{i\bullet}}(E_{i}|_{\widetilde{Y}_{i}})\leavevmode\nobreak\
\left|\;\leavevmode\nobreak\
\mathbf{x}=(x_{1},\cdots,x_{n})\in\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\right.\right\\}$

where we regard $\nu_{\widetilde{Y}_{i\bullet}}(E_{i}|_{\widetilde{Y}_{i}})$
as a point in $\\{0\\}^{i}\times\mathbb{R}^{n-i}\subseteq\mathbb{R}^{n}$. In
particular, $\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ and
$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(f^{*}D)$ determine each
other.

###### Proof.

We can canonically identify $|D|_{\mathbb{R}}$ with $|f^{*}D|_{\mathbb{R}}$.
For any $D^{\prime}\in|D|_{\mathbb{R}}$, let

$\nu_{Y_{\bullet}}(D^{\prime})=(\nu_{1},\cdots,\nu_{n})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
\nu_{\widetilde{Y}_{\bullet}}(f^{*}D^{\prime})=(\widetilde{\nu}_{1},\cdots,\widetilde{\nu}_{n}).$

Since $\nu_{Y_{\bullet}}(|D|_{\mathbb{R}}|$ and
$\nu_{\widetilde{Y}_{\bullet}}(|f^{*}D|_{\mathbb{R}})$ are dense subsets of
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ and
$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(f^{*}D)$, respectively,
it is sufficient to show that

($*$)
$(\widetilde{\nu}_{1},\cdots,\widetilde{\nu}_{n})=(\nu_{1},\cdots,\nu_{n})+\sum_{i=1}^{n-1}\nu_{i}\cdot\nu_{\widetilde{Y}_{i\bullet}}(E_{i}|_{\widetilde{Y}_{i}}).$

Here we regard
$\nu_{\widetilde{Y}_{i\bullet}}(E_{i}|_{\widetilde{Y}_{i}})\in\\{0\\}^{i}\times\mathbb{R}^{n-i}\subseteq\mathbb{R}^{n}$.

Let $D_{1}^{\prime}:=D^{\prime}$ on $X=Y_{0}$ and define inductively
$D_{i}^{\prime}:=(D_{i-1}^{\prime}-\nu_{i-1}Y_{i-1})|_{Y_{i-1}}$ on $Y_{i-1}$
for $2\leq i\leq n$. Similarly, let
$\widetilde{D}_{1}^{\prime}:=f^{*}D^{\prime}$ on
$\widetilde{X}=\widetilde{Y}_{0}$ and define inductively
$\widetilde{D}_{i}^{\prime}:=(\widetilde{D}_{i-1}^{\prime}-\widetilde{\nu}_{i-1}\widetilde{Y}_{i-1})|_{\widetilde{Y}_{i-1}}$
on $\widetilde{Y}_{i-1}$ for $2\leq i\leq n$. Then
$\nu_{i}=\operatorname{ord}_{Y_{i}}D_{i}^{\prime}$ and
$\widetilde{\nu}_{i}=\operatorname{ord}_{\widetilde{Y}_{i}}\widetilde{D}_{i}^{\prime}$
for $1\leq i\leq n$. First of all, the first coordinates of both sides in ($*$
‣ 4) are $\widetilde{\nu}_{1}$ and $\nu_{1}$. It is easy to check that
$\widetilde{\nu}_{1}=\nu_{1}$. Since
$\widetilde{D}_{2}^{\prime}=(f^{*}D_{1}^{\prime}-\nu_{1}\widetilde{Y}_{1})|_{\widetilde{Y}_{1}}=f|_{\widetilde{Y}_{1}}^{*}D_{2}^{\prime}+\nu_{1}E_{1}|_{\widetilde{Y}_{1}}$,
we have

$\nu_{\widetilde{Y}_{1\bullet}}(\widetilde{D}_{2}^{\prime})=\nu_{\widetilde{Y}_{1\bullet}}(f|_{\widetilde{Y}_{1}}^{*}D_{2}^{\prime})+\nu_{1}\cdot\nu_{\widetilde{Y}_{1\bullet}}(E_{1}|_{\widetilde{Y}_{1}}).$

Note that
$\operatorname{ord}_{\widetilde{Y}_{2}}\widetilde{D}_{2}^{\prime}=\widetilde{\nu}_{2}$
and
$\operatorname{ord}_{\widetilde{Y}_{2}}f|_{\widetilde{Y}_{1}}^{*}D_{2}^{\prime}=\nu_{2}$.
Since
$(f|_{\widetilde{Y}_{1}}^{*}D_{2}^{\prime}-\nu_{2}\widetilde{Y}_{2})|_{\widetilde{Y}_{2}}=f|_{\widetilde{Y}_{2}}^{*}D_{3}^{\prime}+\nu_{2}E_{2}|_{\widetilde{Y}_{2}}$,
we have

$\nu_{\widetilde{Y}_{2\bullet}}((f|_{\widetilde{Y}_{1}}^{*}D_{2}^{\prime}-\nu_{2}\widetilde{Y}_{2})|_{\widetilde{Y}_{2}})=\nu_{\widetilde{Y}_{2\bullet}}(f|_{\widetilde{Y}_{2}}^{*}D_{3}^{\prime})+\nu_{2}\cdot\nu_{\widetilde{Y}_{2\bullet}}(E_{2}|_{\widetilde{Y}_{2}}).$

Note that
$\operatorname{ord}_{\widetilde{Y}_{3}}(f|_{\widetilde{Y}_{1}}^{*}D_{2}^{\prime}-\nu_{2}\widetilde{Y}_{2})|_{\widetilde{Y}_{2}}+$the
third coordinate of
$\nu_{1}\cdot\nu_{\widetilde{Y}_{1\bullet}}(E_{1}|_{\widetilde{Y}_{1}})=\widetilde{\nu}_{3}$
and
$\operatorname{ord}_{\widetilde{Y}_{3}}f|_{\widetilde{Y}_{2}}^{*}D_{3}^{\prime}=\nu_{3}$.
In general, for $2\leq i\leq n$, we have

$\nu_{\widetilde{Y}_{i\bullet}}((f|_{\widetilde{Y}_{i-1}}^{*}D_{i}^{\prime}-\nu_{i}\widetilde{Y}_{i})|_{\widetilde{Y}_{i}})=\nu_{\widetilde{Y}_{i\bullet}}(f|_{\widetilde{Y}_{i}}^{*}D_{i+1}^{\prime})+\nu_{i}\cdot\nu_{\widetilde{Y}_{i\bullet}}(E_{i}|_{\widetilde{Y}_{i}})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for $1\leq i\leq n-1$. }$

Note that
$\operatorname{ord}_{\widetilde{Y}_{i+1}}(f|_{\widetilde{Y}_{i-1}}^{*}D_{i}^{\prime}-\nu_{i}\widetilde{Y}_{i})|_{\widetilde{Y}_{i}}+$the
$(i+1)$-th coordinate of
$\sum_{j=1}^{i-1}\nu_{j}\cdot\nu_{\widetilde{Y}_{j\bullet}}(E_{j}|_{\widetilde{Y}_{j}})=\widetilde{\nu}_{i+1}$
and
$\operatorname{ord}_{\widetilde{Y}_{i+1}}f|_{\widetilde{Y}_{i}}^{*}D_{i+1}^{\prime}=\nu_{i+1}$.
Thus we obtain ($*$ ‣ 4). ∎

We first prove the ‘only if’ direction of Theorem A, which is a generalization
of [LM, Proposition 4.1 (i)] to non-big divisors that are abundant.

######  Proposition 4.2.

Let $D,D^{\prime}$ be divisors on $X$ with $\kappa(D),\kappa(D^{\prime})\geq
0$. Suppose that $D$ or $D^{\prime}$ is an abundant divisor. If $D\equiv
D^{\prime}$, then
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})$
for every admissible flag $Y_{\bullet}$ on $X$.

###### Proof.

By Theorem 2.11 (1), both $D,D^{\prime}$ are abundant divisors. Fix an
admissible flag $Y_{\bullet}$ on $X$. Possibly by taking a higher birational
model of $X$, we may assume that each subvariety $Y_{i}$ in $Y_{\bullet}$ is
smooth (see Remark 2.4). By Theorem 2.11 (3), there is a birational morphism
$\mu\colon W\to X$ and a morphism $g\colon W\to T$ with connected fibers such
that $P_{\sigma}(\mu^{*}D)\sim_{\mathbb{Q}}P_{\sigma}(g^{*}B)$ for some big
divisor $B$ on $T$. Thus $P_{\sigma}(\mu^{*}D^{\prime})|_{F}\equiv 0$ for a
general fiber $F$ of $g$, and hence,
$P_{\sigma}(\mu^{*}D^{\prime})|_{F}\sim_{\mathbb{Q}}0$ since
$\kappa(\mu^{*}D^{\prime})=\kappa(D^{\prime})\geq 0$. This implies that
$\kappa(P_{\sigma}(\mu^{*}D^{\prime})|_{F})=0$. By taking a higher birational
model of $W$ if necessary, by [N, Corollary V.2.26] (see also [Le2, Theorem
5.7]), we may assume that
$P_{\sigma}(\mu^{*}D^{\prime})\sim_{\mathbb{Q}}P_{\sigma}(g^{*}B^{\prime})$
for some divisor $B^{\prime}$ on $T$. Applying [N, Lemma III.5.15] (see also
[Le2, Proof of Corollary 6.3]), we see that $P_{\sigma}(B)\equiv
P_{\sigma}(B^{\prime})$ and $B^{\prime}$ is also a big divisor on $T$. We also
have $P_{\sigma}(g^{*}B)=P_{\sigma}(g^{*}P_{\sigma}(B))$ and
$P_{\sigma}(g^{*}B^{\prime})=P_{\sigma}(g^{*}P_{\sigma}(B^{\prime}))$. We
write $P_{\sigma}(B)=P_{\sigma}(B^{\prime})+N$ for some numerically trivial
divisor $N$ on $T$. Then we have

$P_{\sigma}(\mu^{*}D)\sim_{\mathbb{Q}}P_{\sigma}(g^{*}P_{\sigma}(B))=P_{\sigma}(g^{*}P_{\sigma}(B^{\prime}))+g^{*}N\sim_{\mathbb{Q}}P_{\sigma}(\mu^{*}D^{\prime})+g^{*}N.$

By successively taking strict transforms $\widetilde{Y}_{i}$ of $Y_{i}$ via
the birational morphisms
$\mu|_{\widetilde{Y}_{i-1}}\colon\widetilde{Y}_{i-1}\to Y_{i-1}$ for $1\leq
i\leq n$, we obtain an admissible flag
$\widetilde{Y}_{\bullet}:W=:\widetilde{Y}_{0}\supseteq\widetilde{Y}_{1}\supseteq\cdots\supseteq\widetilde{Y}_{n-1}\supseteq\widetilde{Y}_{n}$
on $W$. Possibly by taking a higher birational model of $W$, we may assume
that each subvariety of $\widetilde{Y}_{\bullet}$ is smooth. By Theorem 2.11
(2), $P_{\sigma}(\mu^{*}D)=P_{s}(\mu^{*}D)$ and
$P_{\sigma}(\mu^{*}D^{\prime})=P_{s}(\mu^{*}D^{\prime})$. By Lemmas 2.5 and
4.1, it is sufficient to show that

$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D))=\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D^{\prime})).$

Now, take an ample divisor $A$ on $T$ so that $A+kN$ is also an ample divisor
for every $k\in\mathbb{Z}$. Choose a large integer $a>0$ such that
$aP_{\sigma}(B^{\prime})-A\sim_{\mathbb{Q}}E^{\prime}$ for some effective
divisor $E^{\prime}$ on $T$. Then
$aP_{\sigma}(g^{*}P_{\sigma}(B^{\prime}))-g^{*}A\sim_{\mathbb{Q}}E$ for some
effective divisor $E$ on $W$. For any integer $m>0$, we have

$(m+a)P_{\sigma}(\mu^{*}D)\sim_{\mathbb{Q}}(m+a)(P_{\sigma}(\mu^{*}D^{\prime})+g^{*}N)\sim_{\mathbb{Q}}mP_{\sigma}(\mu^{*}D^{\prime})+E+g^{*}(A+(m+a)N).$

Note that $g^{*}(A+(m+a)N)$ is a semiample divisor on $W$ so that it is
abundant. By the subadditivity property, which follows immediately from the
construction of the valuative Okounkov body, we have

$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D))\supseteq\frac{m}{m+a}\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D^{\prime}))+\frac{1}{m+a}\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(E)+\frac{1}{m+a}\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}((g^{*}(A+(m+a)N))).$

By Theorem 2.11 (2), for any $\varepsilon>0$, we can find
$E^{\prime}\in|g^{*}(A+(m+a)N)|_{\mathbb{R}}$ such that
$\operatorname{ord}_{\widetilde{Y}_{i}}(E^{\prime})<\varepsilon$ for $1\leq
i\leq n$. Thus the origin of $\mathbb{R}^{n}$ is contained in
$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(g^{*}(A+(m+a)N))$ (cf.
Corollary 5.2). Hence we obtain

$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D))\supseteq\frac{m}{m+a}\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D^{\prime}))+\frac{1}{m+a}\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(E).$

By letting $m\to\infty$, we see that

$\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D))\supseteq\Delta^{\operatorname{val}}_{\widetilde{Y}_{\bullet}}(P_{\sigma}(\mu^{*}D^{\prime})).$

Similarly by replacing $D$ by $D^{\prime}$ and $N$ by $-N$, we can also obtain
the reverse inclusion. Therefore we complete the proof. ∎

######  Remark 4.3.

Obviously, Proposition 4.2 does not hold without the assumption that $D$ or
$D^{\prime}$ is an abundant divisor (see [CHPW1, Remark 3.13]).

For the converse of Proposition 4.2, we need several lemmas.

######  Lemma 4.4.

Consider two surjective morphisms $f_{1}\colon X\to Z_{1}$ and $f_{2}\colon
X\to Z_{2}$ with connected fibers. Suppose that $\dim Z_{1}=\dim Z_{2}>0$ and
$f_{1},f_{2}$ are not birationally equivalent. Then for a general member
$G\in|H|$ where $H$ is a very ample divisor on $Z_{1}$, the inverse image
$f_{1}^{-1}(G)$ dominates $Z_{2}$ via $f_{2}$, i.e., we have
$f_{2}(f_{1}^{-1}(G))=Z_{2}$.

###### Proof.

We may assume that $f_{1}^{-1}(G)=f_{1}^{*}G\in|f_{1}^{*}H|$ is a general
member in a base point free linear system so that $f_{1}^{-1}(G)$ is a prime
divisor on $X$. Suppose that $f_{2}(f_{1}^{-1}(G))$ does not dominate $Z_{2}$
via $f_{2}$. Then $D:=f_{2}(f_{1}^{-1}(G))$ is a prime divisor on $Z_{2}$. We
then have $f_{1}^{*}G\leq f_{2}^{*}D$, so
$H^{0}(X,\mathcal{O}_{X}(mf_{1}^{*}G))\subseteq
H^{0}(X,\mathcal{O}_{X}(mf_{2}^{*}D))$ for any integer $m>0$. In particular,
$D$ is a big divisor on $Z_{2}$. Consider a rational map $\varphi\colon
X\dashrightarrow Z^{\prime}$ given by $|mf_{2}^{*}D|$ for a sufficiently large
and divisible integer $m>0$. The rational map $\varphi$ factors through
$Z_{2}$ via a birational map $\varphi^{\prime}\colon Z_{2}\dashrightarrow
Z^{\prime}$ given by $|mD|$. Since
$H^{0}(X,\mathcal{O}_{X}(mf_{1}^{*}G))\subseteq
H^{0}(X,\mathcal{O}_{X}(mf_{2}^{*}D))$, there is a rational map $\pi\colon
Z^{\prime}\dashrightarrow Z_{1}$. Note that $\pi\circ\varphi\colon
X\dashrightarrow Z_{1}$ is birationally equivalent to $f_{1}\colon X\to
Z_{1}$. Thus $\pi$ is birational, and so is $\pi\circ\varphi^{\prime}\colon
Z_{2}\dashrightarrow Z_{1}$. This implies that $f_{1},f_{2}$ are birationally
equivalent, so we get a contradiction. ∎

######  Theorem 4.5.

Let $D,D^{\prime}$ be divisors on $X$ with $\kappa(D),\kappa(D^{\prime})>0$.
If
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})$
for every admissible flag $Y_{\bullet}$ on $X$, then $D,D^{\prime}$ have a
same Iitaka fibration.

###### Proof.

Let $\phi\colon X^{\prime}\to Z$ and $\phi^{\prime}\colon X^{\prime}\to
Z^{\prime}$ be Iitaka fibrations of $D$ and $D^{\prime}$, respectively, and
$f\colon X^{\prime}\to X$ be a birational morphism. By considering the
admissible flags $Y_{\bullet}$ containing the Nakayama subvarieties of
$D,D^{\prime}$, we have $\kappa(D)=\kappa(D^{\prime})$ so that $\dim Z=\dim
Z^{\prime}$. To derive a contradiction, suppose that $\phi,\phi^{\prime}$ are
not birationally equivalent. By Lemma 4.4, for a general member $G\in|H|$
where $H$ is a very ample divisor on $Z$, the inverse image $\phi^{-1}(G)$
dominates $Z^{\prime}$ via $\phi^{\prime}$. We can take a general subvariety
$V^{\prime}\subseteq\phi^{-1}(G)$ of dimension $\kappa(D^{\prime})$ such that
$f(V^{\prime})$ is a Nakayama subvariety of $D^{\prime}$. By Theorem 2.7,
$f(V^{\prime})$ is also a Nakayama subvariety of $D$. However,
$\phi(V^{\prime})\subseteq\phi(\phi^{-1}(G))=G$, so $V^{\prime}$ does not
dominate $Z$ via $\phi$. This is a contradiction, so we are done. ∎

The following lemma plays a crucial role in the proof of the converse of
Proposition 4.2. It is a generalization of [J, Corollary 3.3 and Theorem 3.4
(b)], but our proof is completely different from Jow’s proof in [J].

######  Lemma 4.6.

Let $D$ be a divisor on $X$ with $\kappa(D)>0$, and $D=P_{s}+N_{s}$ be the
$s$-decomposition. Consider an irreducible curve $C$ on $X$ obtained by a
transversal complete intersection of general effective very ample divisors on
$X$. We can choose an admissible flag $Y_{\bullet}:X=Y_{0}\supseteq
Y_{1}\supseteq\cdots\supseteq Y_{n-1}\supseteq Y_{n}=\\{x\\}$ on $X$ such that
$Y_{n-\kappa(D)}$ is a Nakayama subvariety of $D$, $Y_{n-1}=C$, and $x$ is a
general point on $C$. Fix an Iitaka fibration $\phi\colon X^{\prime}\to Z$ of
$D$, and let $C^{\prime}$ be the strict transform of $C$ on $X^{\prime}$. Then
we have

$P_{s}\cdot
C=\deg(C^{\prime}\to\phi(C^{\prime}))\cdot\operatorname{vol}_{\mathbb{R}^{1}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0}).$

###### Proof.

Let $A_{1},\cdots,A_{n-1}$ be general effective very ample divisors on $X$
such that $A_{1}\cap\cdots\cap A_{n-1}=C$. We may assume that
$Y_{i}:=A_{1}\cap\cdots\cap A_{i}$ is an irreducible subvariety of codimension
$i$ for each $1\leq i\leq n-1$. By letting $Y_{n}:=\\{x\\}$ where $x$ is a
general point on $C$, we obtain an admissible flag
$Y_{\bullet}:X=Y_{0}\supseteq Y_{1}\supseteq\cdots\supseteq Y_{n-1}\supseteq
Y_{n}=\\{x\\}$ on $X$. Since $A_{1},\cdots,A_{n-\kappa(D)}$ are general
effective very ample divisors, $Y_{n-\kappa(D)}$ is a Nakayama subvariety of
$D$ by [CHPW1, Proposition 2.13]. Thus this admissible flag $Y_{\bullet}$
satisfies the conditions in the statement.

For a sufficiently large integer $m>0$, take a log resolution $f_{m}\colon
X_{m}\to X$ of the base ideal $\mathfrak{b}(\lfloor mD\rfloor)$ so that we
obtain a decomposition $f_{m}^{*}(\lfloor
mD\rfloor)=M_{m}^{\prime}+F_{m}^{\prime}$ into a base point free divisor
$M_{m}^{\prime}$ on $X_{m}$ and the fixed part $F_{m}^{\prime}$ of
$|f_{m}^{*}(\lfloor mD\rfloor)|$. Let $M_{m}:=\frac{1}{m}M_{m}^{\prime}$ and
$F_{m}:=\frac{1}{m}F_{m}^{\prime}$. Let $\phi_{m}\colon X_{m}\to Z_{m}$ be a
morphism given by $|M_{m}^{\prime}|$. By taking a higher birational model of
$Z_{m}$, we may assume that $Z_{m}$ is a smooth variety. There exists a nef
and big divisor $H_{m}$ on $Z_{m}$ such that $M_{m}=\phi_{m}^{*}H_{m}$. Since
our choice of admissible flag $Y_{\bullet}$ is independent of this process, we
may assume that $f_{m}\colon X_{m}\to X$ is isomorphic over a neighborhood of
$x$. Let $C_{m}$ be the strict transform of $C$ on $X_{m}$. By taking strict
transforms $Y_{i}^{m}$ of $Y_{i}$ on $X_{m}$ for each $0\leq i\leq n-1$ (note
that $Y_{n-1}^{m}=C_{m}$), we obtain an admissible flag
$Y_{\bullet}^{m}:X_{m}=Y_{0}^{m}\supseteq Y_{1}^{m}\supseteq\cdots\supseteq
Y_{n-1}^{m}\supseteq Y_{n}^{m}=\\{f_{m}^{-1}(x)\\}$ on $X_{m}$. We may also
assume that
$\overline{Y}^{m}_{\bullet}:Z_{m}=\overline{Y}^{m}_{0}=\phi_{m}(Y_{n-\kappa(D)}^{m})\supseteq\overline{Y}^{m}_{1}=\phi_{m}(Y_{n-\kappa(D)-1}^{m})\supseteq\cdots\supseteq\overline{Y}^{m}_{\kappa(D)-1}=\phi_{m}(Y_{n-1}^{m})\supseteq\overline{Y}^{m}_{\kappa(D)}=\phi_{m}(Y_{n}^{m})$
is an admissible flag on $Z_{m}$. Note that
$d:=\deg(C^{\prime}\to\phi(C^{\prime}))=\deg(C_{m}\to\phi_{m}(C_{m}))$. Then
$M_{m}\cdot C_{m}=d\cdot(H_{m}.\phi_{m}(C_{m}))$. By [CPW2, Theorem 1.1], we
have

$H_{m}\cdot\phi_{m}(C_{m})=\operatorname{vol}_{Z_{m}|\overline{Y}_{\kappa(D)-1}}(H_{m})=\operatorname{vol}_{\mathbb{R}^{1}}(\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}=0}).$

We now prove that $P_{s}\cdot C=\lim_{m\to\infty}M_{m}\cdot C_{m}$. Let
$E_{1},\cdots,E_{k}$ be divisorial components of $\operatorname{SB}(D)$. Since
the closure of $\operatorname{SB}(D)\setminus(E_{1}\cup\cdots\cup E_{k})$ has
codimension at least two in $X$, we may assume that
$C\cap\operatorname{SB}(D)\subseteq E_{1}\cup\cdots\cup E_{k}$. We can also
assume that $C$ is smooth and meets all $E_{i}$ transversally at smooth points
of $E_{i}$. Thus $C_{m}$ does not meet any effective $f_{m}$-exceptional
divisor. We write

$f_{m}^{*}\frac{\lfloor
mP_{s}\rfloor}{m}=M_{m}+e_{1}^{m}f_{m*}^{-1}E_{1}+\cdots+e_{k}^{m}f_{m*}^{-1}E_{k}+F_{m}^{\prime\prime}$

where $F_{m}^{\prime\prime}$ is an effective $f_{m}$-exceptional divisor. We
have

$\frac{\lfloor mP_{s}\rfloor}{m}\cdot C=f_{m}^{*}\frac{\lfloor
mP_{s}\rfloor}{m}\cdot C_{m}=M_{m}\cdot C_{m}+e_{1}^{m}E_{1}\cdot
C+\cdots+e_{k}^{m}E_{k}\cdot C.$

Since $\lim_{m\to\infty}e_{i}^{m}=0$ for each $1\leq i\leq k$ and
$\lim_{m\to\infty}\frac{\lfloor mP_{s}\rfloor}{m}\cdot C=P_{s}\cdot C$, we
obtain

$P_{s}\cdot C=\lim_{m\to\infty}M_{m}\cdot C_{m}.$

Combining everything together, we have

$P_{s}\cdot
C=d\cdot\lim_{m\to\infty}\operatorname{vol}_{\mathbb{R}^{1}}(\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}}).$

Thus it is sufficient to show that

($\\#$)
$\lim_{m\to\infty}\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}=0}=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0}.$

By definition,
$\lim_{m\to\infty}\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}=0}\subseteq\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0}$.
Suppose that

$\lim_{m\to\infty}\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}=0}\subsetneq\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0}.$

Fix a sufficiently large integer $m_{0}>0$, and choose an ample
$\mathbb{Q}$-divisor $A_{m_{0}}$ on $Z_{m_{0}}$ such that

$\operatorname{vol}_{\mathbb{R}^{1}}(\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}})+A_{m_{0}}\cdot\phi_{m_{0}}(C_{m_{0}})<\operatorname{vol}_{\mathbb{R}^{1}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0})-\varepsilon$

for any sufficiently small number $\varepsilon>0$ and any sufficiently large
integer $m>m_{0}$. There exists a sufficiently small number $\delta>0$ such
that the divisors

$\begin{array}[]{rcl}A_{m_{0}}^{1}=A_{m_{0}}^{1}(\delta_{1})&\sim_{\mathbb{Q}}&A_{m_{0}}+\delta_{1}\overline{Y}^{m_{0}}_{1},\\\
A_{m_{0}}^{2}=A_{m_{0}}^{2}(\delta_{1},\delta_{2})&\sim_{\mathbb{Q}}&A_{m_{0}}^{1}|_{\overline{Y}^{m_{0}}_{1}}+\delta_{2}\overline{Y}^{m_{0}}_{2},\\\
&\vdots&\\\
A_{m_{0}}^{\kappa(D)-1}=A_{m_{0}}^{\kappa(D)-1}(\delta_{1},\cdots,\delta_{\kappa(D)-1})&\sim_{\mathbb{Q}}&A_{m_{0}}^{\kappa(D)-2}|_{\overline{Y}_{\kappa(D)-2}^{m_{0}}}+\delta_{\kappa(D)-1}\overline{Y}^{m_{0}}_{\kappa(D)-1}\end{array}$

are successively ample $\mathbb{Q}$-divisors for nonnegative rational numbers
$\delta_{1},\delta_{2},\ldots,\delta_{\kappa(D)-1}\leq\delta$. By Lemma 3.1,
we know that
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\lim_{m\to\infty}\Delta^{\operatorname{val}}_{Y_{\bullet}^{m}}(M_{m})=\lim_{m\to\infty}\Delta_{\overline{Y}_{\bullet}^{m}}(H_{m})$.
Thus there exists a sufficiently large integer $m>0$ and an effective divisor
$H_{m}^{\prime}\sim_{\mathbb{Q}}H_{m}$ on $Z_{m}$ such that if we write
$\nu_{\overline{Y}^{m}_{\bullet}}(H_{m}^{\prime})=(\delta_{1},\cdots,\delta_{\kappa(D)-1},b)$,
then $\delta_{1},\delta_{2},\ldots,\delta_{\kappa(D)-1},b$ are nonnegative
rational numbers with
$\delta_{1},\delta_{2},\ldots,\delta_{\kappa(D)-1}\leq\delta$ and
$\operatorname{vol}_{\mathbb{R}^{1}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0})-\varepsilon\leq
b$. We can write

$\begin{array}[]{rcl}H_{m}^{\prime}&=&H_{m}^{1}+\delta_{1}\overline{Y}_{1}^{m},\\\
H_{m}^{1}|_{\overline{Y}_{1}^{m}}&=&H_{m}^{2}+\delta_{2}\overline{Y}_{2}^{m},\\\
&\vdots&\\\
H_{m}^{\kappa(D)-2}|_{\overline{Y}_{\kappa(D)-2}^{m}}&=&H_{m}^{\kappa(D)-1}+\delta_{\kappa(D)-1}\overline{Y}_{\kappa(D)-1}^{m}\end{array}$

where $H_{m}^{i}$ is an effective divisor on $\overline{Y}_{i-1}^{m}$ for
$1\leq i\leq\kappa(D)-1$. Note that
$H_{m}^{\kappa(D)-1}\cdot\phi_{m}(C_{m})=H_{m}^{\kappa(D)-1}\cdot\overline{Y}_{\kappa(D)-1}^{m}\geq
b$. By taking common resolution, we can assume that there is a birational
morphism $g_{m}\colon Z_{m}\to Z_{m_{0}}$ and
$\overline{Y}_{i}^{m}=g_{m}|_{\overline{Y}_{i-1}^{m}}^{*}\overline{Y}_{i}^{m_{0}}$
for $1\leq i\leq\kappa(D)$. Note that $H_{m}+g_{m}^{*}A_{m_{0}}+B$ is an ample
$\mathbb{Q}$-divisor on $Z_{m}$ for any ample $\mathbb{Q}$-divisor $B$ on
$Z_{m}$. We may assume that
$\overline{Y}_{\kappa(D)-2}^{m}\not\subseteq\operatorname{Supp}(B)$. Thus we
can find an effective divisor $E\sim_{\mathbb{Q}}H_{m}+g_{m}^{*}A_{m_{0}}+B$
such that
$E|_{\overline{Y}_{\kappa(D)-2}^{m}}=H_{m}^{\kappa(D)-1}+g_{m}|_{\overline{Y}_{\kappa(D)-2}^{m}}^{*}A_{m_{0}}^{\kappa(D)-1}+B|_{\overline{Y}_{\kappa(D)-2}^{m}}$
where
$A_{m_{0}}^{\kappa(D)-1}=A_{m_{0}}^{\kappa(D)-1}(\delta_{1},\cdots,\delta_{\kappa(D)-1})$.
Then we obtain

$(H_{m}+g_{m}^{*}A_{m_{0}}+B)\cdot\phi_{m}(C_{m})=E\cdot\phi_{m}(C_{m})=(H_{m}^{\kappa(D)-1}+g_{m}|_{\overline{Y}_{\kappa(D)-2}^{m}}^{*}A_{m_{0}}^{\kappa(D)-1}+B|_{\overline{Y}_{\kappa(D)-2}^{m}})\cdot\phi_{m}(C_{m})>b.$

Since $B$ can be arbitrarily small ample divisor, it follows that
$(H_{m}+g_{m}^{*}A_{m_{0}})\cdot\phi_{m}(C_{m})\geq b$. Note that
$\operatorname{vol}_{\mathbb{R}^{1}}(\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}})+A_{m_{0}}\cdot\phi_{m_{0}}(C_{m_{0}})=(H_{m}+g_{m}^{*}A_{m_{0}})\cdot\phi_{m}(C_{m})$.
Thus we have

$\operatorname{vol}_{\mathbb{R}^{1}}(\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}})+A_{m_{0}}\cdot\phi_{m_{0}}(C_{m_{0}})\geq
b\geq\operatorname{vol}_{\mathbb{R}^{1}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0})-\varepsilon.$

This is a contradiction. Therefore
$\lim_{m\to\infty}\Delta_{\overline{Y}^{m}_{\bullet}}(H_{m})_{x_{1}=\cdots=x_{\kappa(D)-1}=0}=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0}$,
which is ($\\#$ ‣ 4). ∎

######  Remark 4.7.

Here we explain why Lemma 4.6 generalizes [J, Corollary 3.3 and Theorem 3.4
(b)] to non-big divisor case. The actual statement of [J, Corollary 3.3 and
Theorem 3.4 (b)] states that if $D$ is a big divisor on $X$ and $Y_{\bullet}$
is an admissible flag on $X$ whose subvarieties are transversal complete
intersections of general effective very ample divisors, then

$\operatorname{vol}_{\mathbb{R}^{1}}(\Delta_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0})=D\cdot
Y_{n-1}-\sum_{i=1}^{k}\sum_{p\in Y_{n-1}\cap
E_{i}}\operatorname{ord}_{E_{i}}(||D||)$

where $E_{1},\cdots,E_{k}$ are irreducible components of
$\operatorname{SB}(D)$. Since $Y_{n-1}$ is a sufficiently general curve
obtained by a transversal complete intersection of general effective very
ample divisors, we may assume that $Y_{n-1}$ is smooth and meets all $E_{i}$
transversally at smooth points of $E_{i}$. Thus

$\sum_{i=1}^{k}\sum_{p\in Y_{n-1}\cap
E_{i}}\operatorname{ord}_{E_{i}}(||D||)=N_{\sigma}(D)\cdot Y_{n-1}$

so that
$\operatorname{vol}_{\mathbb{R}^{1}}(\Delta_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{n-1}=0})=P_{\sigma}(D)\cdot
Y_{n-1}$. Note that $P_{\sigma}(D)=P_{s}(D)$ and the identity
$\text{id}_{X}\colon X\to X$ is an Iitaka fibration of $D$ when $D$ is a big
divisor.

######  Lemma 4.8.

Let $D$ be a divisor on $X$ with $\kappa(D)>0$, and $D=P_{s}+N_{s}$ be the
$s$-decomposition. Let $E$ be an irreducible component of $N_{s}$. Then we
have

$\operatorname{mult}_{E}N_{s}=\inf\\{x_{1}\mid(x_{1},\cdots,x_{n})\in\Delta^{\operatorname{val}}_{Y_{\bullet}}(D),\text{
$Y_{\bullet}$ is an admissible flag such that $Y_{1}=E$}\\}.$

In particular, one can read off the negative part $N_{s}$ from
$\\{\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\mid Y_{\bullet}\text{ is an
admissible flag on $X$}\\}$.

###### Proof.

By the definition of the $s$-decomposition, we have

$\inf\\{x_{1}\mid(x_{1},\cdots,x_{n})\in\Delta^{\operatorname{val}}_{Y_{\bullet}}(P_{s}),\text{
$Y_{\bullet}$ contains $E$}\\}=0.$

Note also that $\Delta^{\operatorname{val}}_{Y_{\bullet}}(N_{s})$ consists of
a single point $(x_{1},\cdots,x_{n})$ with
$x_{1}=\operatorname{mult}_{E}N_{s}$. Thus the assertion now follows from
Lemma 2.5. ∎

We are now ready to complete the proof of Theorem A by proving the converse of
Proposition 4.2. The following result is a generalization of [J, Theorem A] to
possibly non-big divisor case.

######  Proposition 4.9.

Let $D,D^{\prime}$ be divisors on $X$ with $\kappa(D),\kappa(D^{\prime})\geq
0$. If
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})$
for every admissible flag $Y_{\bullet}$ on $X$, then $D\equiv D^{\prime}$.

###### Proof.

Recall that if $D$ is a divisor with $\kappa(D)\geq 0$, then any
$\kappa(D)$-dimensional general subvariety of $X$ is a Nakayama subvariety of
$D$. Thus we can take an admissible flag $Y_{\bullet}$ containing the Nakayama
subvarieties of $D,D^{\prime}$ with general $Y_{n}$. By the assumption, we can
deduce that $\kappa(D)=\kappa(D^{\prime})$. The assertion is trivial when
$\kappa(D)=\kappa(D^{\prime})=0$. Thus, from now on, we assume that
$\kappa(D),\kappa(D^{\prime})>0$. By Theorem 4.5, $D,D^{\prime}$ have a same
Iitaka fibration $f\colon X^{\prime}\to Z$. Let $D=P_{s}+N_{s}$ and
$D^{\prime}=P_{s}^{\prime}+N_{s}^{\prime}$ be the $s$-decompositions. By Lemma
4.8, we have $N_{s}=N_{s}^{\prime}$. Thus it is sufficient to show that
$P_{s}\equiv P_{s}^{\prime}$. By applying [J, Lemma 3.5], we can take
irreducible curves $C_{1},\cdots,C_{\rho}$ on $X$ obtained by transversal
complete intersections of general effective very ample divisors on $X$ in such
a way that they form a basis of $N_{1}(X)_{\mathbb{Q}}$. As in Lemma 4.6, for
each $1\leq i\leq\rho$, we can choose an admissible flag
$Y_{\bullet}^{i}:X=Y_{0}^{i}\supseteq Y_{1}^{i}\supseteq\cdots\supseteq
Y_{n-1}^{i}\supseteq Y_{n}^{i}=\\{x^{i}\\}$ on $X$ such that
$Y_{n-\kappa(D)}^{i}$ is a Nakayama subvariety of $D$, $Y_{n-1}^{i}=C_{i}$,
and $x^{i}$ is a very general point on $C_{i}$. For each $1\leq i\leq\rho$,
let $C_{i}^{\prime}$ be the strict transform of $C_{i}$ on $X^{\prime}$. Then
by Lemma 4.6 and the assumption, we have

$\begin{array}[]{rll}P_{s}\cdot
C_{i}&=\deg(C_{i}^{\prime}\to\phi(C_{i}^{\prime}))\cdot\operatorname{vol}_{\mathbb{R}^{1}}(\Delta^{\operatorname{val}}_{Y_{\bullet}^{i}}(D)_{x_{1}=\cdots=x_{n-1}=0})&\\\
&=\deg(C_{i}^{\prime}\to\phi(C_{i}^{\prime}))\cdot\operatorname{vol}_{\mathbb{R}^{1}}(\Delta^{\operatorname{val}}_{Y_{\bullet}^{i}}(D^{\prime})_{x_{1}=\cdots=x_{n-1}=0})&=P_{s}^{\prime}\cdot
C_{i}\end{array}$

for every $1\leq i\leq\rho$. Thus $P_{s}\equiv P_{s}^{\prime}$ as desired. ∎

Note that in Proposition 4.9, we do not assume that $D$ or $D^{\prime}$ is an
abundant divisor.

######  Remark 4.10.

In Proposition 4.9, we do not assume that $D$ or $D^{\prime}$ is an abundant
divisor. Clearly, Proposition 4.9 does not hold without the assumption that
$\kappa(D),\kappa(D^{\prime})\geq 0$. We have
$\kappa(D),\kappa(D^{\prime})=-\infty$ for any non-pseudoeffective divisors
$D$ and $D^{\prime}$. However,
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})=\emptyset$
for every admissible flag $Y_{\bullet}$ on $X$.

As a consequence of Propositions 4.2 and 4.9, we obtain Theorem A as Corollary
4.11.

######  Corollary 4.11.

Let $D,D^{\prime}$ be divisors on $X$ with $\kappa(D),\kappa(D^{\prime})\geq
0$. If $D$ or $D^{\prime}$ is an abundant divisor, then $D\equiv D^{\prime}$
if and only if
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})$
for every admissible flag $Y_{\bullet}$ on $X$.

###### Proof.

The assertion follows from Propositions 4.2 and 4.9. ∎

Finally, we prove the following.

######  Corollary 4.12.

Let $D,D^{\prime}$ be divisors on $X$ with $\kappa(D),\kappa(D^{\prime})\geq
0$. If $\operatorname{Pic}(X)$ is finitely generated, then
$D\sim_{\mathbb{R}}D^{\prime}$ if and only if
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\operatorname{val}}_{Y_{\bullet}}(D^{\prime})$
for every admissible flag $Y_{\bullet}$ on $X$.

###### Proof.

The ‘only if’ direction is trivial by definition (see also [CHPW1, Proposition
3.13]). For the converse, note that $D\equiv D^{\prime}$ if and only if
$D\sim_{\mathbb{R}}D^{\prime}$ under the assumption that
$\operatorname{Pic}(X)$ is finitely generated. Then the ‘if’ direction follows
from Proposition 4.9. ∎

##  5\. Restricted base locus via Okounkov body

We show Theorem B as Theorem 5.1 in this section. The proof is almost
identical to that of [CHPW2, Theorem A], but we include the whole proof for
reader’s convenience. Throughout the section, $X$ is a smooth projective
variety of dimension $n$.

######  Theorem 5.1.

Let $D$ be a pseudoeffective abundant divisor on $X$, and $x\in X$ be a point.
Then the following are equivalent:

  1. $(1)$

$x\in\mathbf{B}_{-}(D)$

  2. $(2)$

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ does not contain the origin of
$\mathbb{R}^{n}$ for some admissible flag $Y_{\bullet}$ on $X$ centered at
$x$.

  3. $(3)$

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ does not contain the origin of
$\mathbb{R}^{n}$ for every admissible flag $Y_{\bullet}$ on $X$ centered at
$x$.

###### Proof.

Note that since $D$ is an abundant divisor, for a divisorial valuation
$\sigma$ with the center $V$ on $X$, we have
$\operatorname{ord}_{V}(||D||)=\inf\\{\sigma(D^{\prime})\mid
D\sim_{\mathbb{R}}D^{\prime}\geq 0\\}$ by Theorem 2.11 (2).

$(1)\Rightarrow(2)$ Assume that $x\in\mathbf{B}_{-}(D)$ and fix an admissible
flag $Y_{\bullet}$ centered at $x$. For $D^{\prime}\in|D|_{\mathbb{R}}$, we
write
$\nu_{Y_{\bullet}}(D^{\prime})=(\nu_{1}(D^{\prime}),\cdots,\nu_{n}(D^{\prime}))$.
We first consider the case that $Y_{1}\subseteq\mathbf{B}_{-}(D)$. This is
equivalent to $\operatorname{ord}_{Y_{1}}(||D||)>0$. For any
$D^{\prime}\in|D|_{\mathbb{R}}$, we have

$\nu_{1}(D^{\prime})=\operatorname{ord}_{Y_{1}}(D^{\prime})\geq\operatorname{ord}_{Y_{1}}(||D||)>0,$

so
$\inf\\{x_{1}\mid(x_{1},\cdots,x_{n})\in\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\\}>0$.
Thus the origin of $\mathbb{R}^{n}$ is not contained in
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$.

Now we consider the case that $Y_{1}\not\subseteq\mathbf{B}_{-}(D)$. Take an
integer $k$ such that $Y_{i}\not\subseteq\mathbf{B}_{-}(D)$ for $1\leq i\leq
k$ and $Y_{k+1}\subseteq\mathbf{B}_{-}(D)$. For a sufficiently small number
$\varepsilon>0$, we can find $D^{\prime}\in|D|_{\mathbb{R}}$ such that
$\operatorname{ord}_{Y_{k}}(D^{\prime})<\varepsilon$. Then we have

$\nu_{i}(D^{\prime})\leq\operatorname{ord}_{Y_{i}}(D^{\prime})\leq\operatorname{ord}_{Y_{k}}(D^{\prime})<\varepsilon\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for $1\leq i\leq k$}.$

Thus
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{k}=0}\neq\emptyset$.
We then claim that

$\inf\\{x_{k+1}\mid(\underbrace{0,\cdots,0}_{k},x_{k+1},\cdots,x_{n})\in\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)_{x_{1}=\cdots=x_{k}=0}\\}>0.$

This claim implies that the origin of $\mathbb{R}^{n}$ is not contained in
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$. To show the claim, we let

$D_{1}^{\prime}:=D^{\prime},D_{2}^{\prime}:=(D_{1}^{\prime}-\nu_{1}(D^{\prime})Y_{1})|_{Y_{1}},\cdots,D_{i}^{\prime}:=(D_{i-1}^{\prime}-\nu_{i-1}(D^{\prime})Y_{i-1})|_{Y_{i-1}},\cdots.$

Then we obtain

$\nu_{k+1}(D^{\prime})=\operatorname{ord}_{Y_{k+1}}(D_{1}^{\prime})-\nu_{1}(D^{\prime})-\cdots-\nu_{k}(D^{\prime}).$

Since $\operatorname{ord}_{Y_{k+1}}(||D||)>0$, we may suppose that
$0<\varepsilon<\frac{1}{2k}\operatorname{ord}_{Y_{k+1}}(||D||)$. Thus we get

$\nu_{k+1}(D^{\prime})>\frac{1}{2}\operatorname{ord}_{Y_{k+1}}(||D||)>0.$

This proves the claim, and we complete the proof for $(1)\Rightarrow(2)$.

$(2)\Rightarrow(3)$ Trivial.

$(3)\Rightarrow(1)$ Assume that $x\not\in\mathbf{B}_{-}(D)$ and fix any
admissible flag $Y_{\bullet}$ centered at $x$. For any $\varepsilon>0$, we can
find $D^{\prime}\in|D|_{\mathbb{R}}$ such that
$\operatorname{ord}_{Y_{i}}(D^{\prime})<\varepsilon$ for $1\leq i\leq n$. Thus
the origin of $\mathbb{R}^{n}$ is contained in
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$. This proves
$(3)\Rightarrow(1)$. ∎

Recall that a divisor $D$ on $X$ is nef if and only if
$\mathbf{B}_{-}(D)=\emptyset$. Thus the following is immediate from Theorem
5.1.

######  Corollary 5.2.

Let $D$ be an abundant divisor on $X$. Then the following are equivalent:

  1. $(1)$

$D$ is nef.

  2. $(2)$

For every point $x\in X$, there exists an admissible flag $Y_{\bullet}$ on $X$
centered at $x$ such that $\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$
contain the origin of $\mathbb{R}^{n}$.

  3. $(3)$

$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ contain the origin of
$\mathbb{R}^{n}$ for every admissible flag $Y_{\bullet}$ on $X$.

######  Remark 5.3.

Note that Theorem 5.1 and Corollary 5.2 does not hold in general when $D$ is
not abundant (see [CPW1, Remark 4.10]. The main reason is that for a
divisorial valuation $\sigma$ with the center $V$ on $X$, we may have
$\operatorname{ord}_{V}(||D||)\neq\inf\\{\sigma(D^{\prime})\mid
D\sim_{\mathbb{R}}D^{\prime}\geq 0\\}$ in contrast to the abundant divisor
case (Theorem 2.11 (2)).

##  6\. Comparing two Okounkov bodies

In this section, we prove Theorem C as Theorem 6.1.

######  Theorem 6.1.

Let $D$ be a pseudoeffective abundant divisor on an $n$-dimensional smooth
projective variety $X$ with $\kappa(D)>0$. Fix an admissible flag
$Y_{\bullet}$ on $X$ such that $V:=Y_{n-\kappa(D)}$ is a Nakayama subvariety
of $D$ and $Y_{n}$ is a general point in $X$. Consider the Iitaka fibration
$\phi\colon X^{\prime}\to Z$ of $D$ and the strict transform $V^{\prime}$ of
$V$ on $X^{\prime}$. Then we have

$\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\lim}_{Y_{\bullet}}(D))=\deg(\phi|_{V^{\prime}}\colon
V^{\prime}\to
Z)\cdot\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)).$

In particular,
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(D)$
if and only if the map $\phi|_{V^{\prime}}\colon V^{\prime}\to Z$ is
generically injective.

###### Proof.

We use the notations in Section 3 (by abuse of notation, we assume that
$f_{m}\colon X_{m}\to X$ in Subsection 3.1 is the $\mu\circ f_{m}$ in
Subsection 3.2). By Proposition 2.13, $V$ is also a positive volume subvariety
of $D$. For a sufficiently large integer $m>0$, we have

$\deg(\phi_{m}|_{V_{m}}\colon V_{m}\to Z_{m})=\deg(\phi|_{V^{\prime}}\colon
V^{\prime}\to Z)=:d.$

Since $\phi_{m}|_{V_{m}}^{*}H_{m}=M_{m}$, it follows that
$\operatorname{vol}_{V_{m}}(M_{m}|_{V_{m}})=d\cdot\operatorname{vol}_{Z_{m}}(H_{m})$.
By Lemmas 3.1 and 3.6, Theorem 2.8 and [LM, Theorem A], we obtain

$\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\lim}_{Y_{\bullet}}(D))=\lim_{m\to\infty}\frac{1}{\kappa(D)!}\operatorname{vol}_{V_{m}}(M_{m}|_{V_{m}})\text{
and
}\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D))=\lim_{m\to\infty}\frac{1}{\kappa(D)!}\operatorname{vol}_{Z_{m}}(H_{m}).$

Thus the first assertion immediately follows.

Recall that
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)\subseteq\Delta^{\lim}_{Y_{\bullet}}(D)$.
Thus
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)=\Delta^{\lim}_{Y_{\bullet}}(D)$
if and only if
$\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\operatorname{val}}_{Y_{\bullet}}(D))=\operatorname{vol}_{\mathbb{R}^{\kappa(D)}}(\Delta^{\lim}_{Y_{\bullet}}(D))$.
Now the second assertion follows from the first assertion. ∎

######  Remark 6.2.

Upon obtaining Theorem 6.1, one may wonder whether under the same settings,
$\Delta^{\lim}_{Y_{\bullet}}(D)$ and
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$ coincide up to rescaling by a
constant, i.e., $\Delta^{\lim}_{Y_{\bullet}}(D)=(\deg(\phi|_{V^{\prime}}\colon
V^{\prime}\to
Z))^{\frac{1}{\kappa(D)}}\cdot\Delta^{\operatorname{val}}_{Y_{\bullet}}(D)$.
This is not true in general. For instance, consider a 3-fold
$X:=\mathbb{P}^{2}\times\mathbb{P}^{1}$ with the projections $f\colon
X\to\mathbb{P}^{2}$ and $g\colon X\to\mathbb{P}^{1}$. Let $H:=f^{*}L$ and
$F:=g^{*}P$ where $L$ is a line in $\mathbb{P}^{2}$ and $P$ is a point in
$\mathbb{P}^{1}$. Note that $f$ is the Iitaka fibration of $H$. Take general
members $H^{\prime}\in|H|$ and $S\in|H+2F|$ and a general point $x$ in
$H^{\prime}\cap S$. Note that $S$ is a Nakayama subvariety of an abundant
divisor $H$, and $\deg(f|_{S}\colon S\to\mathbb{P}^{2})=2$. We now fix an
admissible flag

$Y_{\bullet}:X\supseteq S\supseteq S\cap H^{\prime}\supseteq\\{x\\}$

on $X$. It is easy to check that
$\Delta^{\operatorname{val}}_{Y_{\bullet}}(H)$ is an isosceles right triangle
in $\\{0\\}\times\mathbb{R}_{\geq 0}^{2}$ and $\Delta^{\lim}_{Y_{\bullet}}(H)$
is a non-isosceles right triangle in $\\{0\\}\times\mathbb{R}_{\geq 0}^{2}$.
In particular, we see that
$\Delta^{\lim}_{Y_{\bullet}}(H)\neq\sqrt{2}\cdot\Delta^{\operatorname{val}}_{Y_{\bullet}}(H)$.

######  Example 6.3.

We construct an example of a variety with a pseudoeffective abundant divisor
which does not have any Nakayama subvariety $V$ giving rise to a generically
injective map $\phi|_{V^{\prime}}\colon V^{\prime}\to Z$ as in Theorem 6.1.
Let $S$ be a minimal surface with $\kappa(S)=1$. Then $K_{S}$ is semiample,
and $\kappa(K_{S})=\kappa_{\nu}(K_{S})=1$. Denote by $\pi\colon S\to C$ the
elliptic fibration induced by $|mK_{S}|$ for $m\gg 0$. Suppose that $\pi$ is
not a Jacobian fibration, i.e., $\pi$ has no section. Since $\pi$ is also the
Iitaka fibration of $K_{S}$, for any Nakayama subvariety $V$ of $K_{S}$, the
map $\pi|_{V}\colon V\to C$ is always not generically injective. In
particular, by Theorem 6.1, $\Delta^{\operatorname{val}}_{Y_{\bullet}}(K_{S})$
and $\Delta^{\lim}_{Y_{\bullet}}(K_{S})$ are different for every admissible
flag $Y_{\bullet}$ on $S$ containing a Nakayama subvariety of $K_{S}$.

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