# Invariance of plurigenera for generalized polarized pairs with abundant nef
parts

Zhan Li  Department of Mathematics, Southern University of Science and
Technology, 1088 Xueyuan Rd, Shenzhen 518055, China<EMAIL_ADDRESS>and
Zhiwei Wang  School of Mathematical Sciences, Beijing Normal University,
Beijing 100875, China<EMAIL_ADDRESS>
###### Abstract.

We show the invariance of plurigenera for generalized polarized pairs with
abundant nef parts and generalized canonical singularities. This is obtained
by investigating a type of newly introduced multiplier ideal sheaf which is of
bimeromorphic nature.

###### 2010 Mathematics Subject Classification:

32Q99, 14E99

###### Contents

  1. 1 Introduction
  2. 2 Generalized polarized pairs
  3. 3 Metrics on complex spaces
  4. 4 Metrics for the abundant nef parts
  5. 5 Extension theorems and invariance of plurigenera

##  1\. Introduction

Let $X\to\Delta$ be a smooth projective family of complex manifolds over a
disc. A celebrated result of Siu (see [Siu98, Siu02]) asserts that the
plurigenera $h^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}))$ is a constant
function of $t\in\Delta$ for each positive integer $m$. Results of such type
have been investigated for a long time (see [Kaw99b] for a survey) and have
been generalized in many directions after Siu’s work (see [Pău07, Tak07, BP12,
RT20], etc.). Invariance of plurigenera is an important ingredient in the
proof of the boundedness of moduli of general type varieties (see [HMX13,
Theorem 1.8], [HMX18, §2.3]). Moreover, Siu’s method has a profound impact on
problems such as deformations of canonical singularities (see [Kaw99a]) and
the existence of good minimal models (see [DHP13]).

Let $Y$ be a smooth projective variety and $\mu:Y\dashrightarrow X$ be the
canonical model of $Y$ (see [BCHM10, Definition 3.6.5], [Li22, Definition
2.1]). It is natural to ask whether $X$ (birationally) belongs to a bounded
moduli space given $Y$ satisfying certain restrictions (see [FS20], [Li20,
Conjecture 1.2]). When $Y$ is of general type, then the answer is affirmative
if $Y$ has a fixed dimension and a bounded volume (see [HMX13, HMX18]). To
study the general situation, one can adopt the same strategy as [HMX13,
HMX18]. In this case, the base variety $X$ admits a generalized polarized pair
structure instead of a log pair structure (see Definition 2.1). Roughly
speaking, there is a triple $(X,B+M)$ with $B,M$ divisors on $X$. Here $B\geq
0$ accounts for the singularities of the fibers, and $M$ is the push-forward
of a nef and abundant divisor which accounts for the moduli of the fibers. It
is this additional $M$ that ruins the straightforward applications of the
results in [HMX13, HMX18]. For one thing, $M$ is only well-defined up to
linear equivalence.

More generally, if $Y\to X$ is a $K_{Y}$-trivial fibration, then $X$ also
admits a generalized polarized pair structure. In fact, the concept of
generalized polarized pairs originates from such observations (see [Bir21]).
Note that in the definition of generalized polarized pairs, the nef part is
just assumed to be the push-forward of a nef divisor instead of a nef and
abundant divisor. However, the latter case is more meaningful in geometry, and
our main motivation is to study the invariance of plurigenera in such setting:

######  Theorem 1.1.

Let $\pi:X\to\Delta$ be a projective contraction from a complex space $X$ to
the disc $\Delta$. Assume that $X$ has canonical singularities. Let $L$ be a
Cartier divisor on $X$ and $h$ be a metric for $\mathcal{O}_{X}(L)$ with non-
negative curvature current. Let $X_{0}\subset X$ be the fiber over
$0\in\Delta$. Suppose that $X_{0}$ has canonical singularities and
$h|_{X_{0}}$ is well-defined. Then for each $m\in\mathbb{N}$, any section of

$H^{0}(X_{0},\mathcal{O}_{X_{0}}(mK_{X_{0}}+L|_{X_{0}})\otimes\mathcal{G}_{m}(h|_{X_{0}}))$

extends over $X$ (i.e. any section has a preimage in
$H^{0}(X,\mathcal{O}_{X}(mK+L))$ under the map in Lemma 5.8).

The newly introduced ideal sheaf $\mathcal{G}_{m}(h)$ is of the birational (or
bimeromorphic) nature (see Section 5.3). Comparing with previous studies, this
birational point of view gives new ingredients even in the smooth case (see
Example 5.6, Remark 5.16). For relevant notions of metrics on complex spaces
which are adapted to this perspective, see Section 3.

Theorem 1.1 involves with metrics and $L^{2}$-conditions. In order to use it
in algebraic geometry, we need to obtain such analytic requirements by natural
algebro-geometric conditions. The following statement is metric-free.

######  Theorem 1.2.

Let $\pi:X\to\Delta$ be a projective contraction from a complex space $X$ to
the disc $\Delta$. Let $X^{\prime}\xrightarrow{f}X\xrightarrow{\pi}\Delta$ be
a generalized polarized pair with the boundary part $B$ and the abundant nef
part $M$. Let $X_{0}\subset X$ be the fiber over $0\in\Delta$ and
$(X_{0},B_{0}+M_{0})$ be the generalized polarized pair obtained by
restricting to $X_{0}$. Assume that $(X_{0},B_{0}+M_{0})$ has g-canonical
singularities (in particular, $B_{0}=0$). Then for each $m\in\mathbb{N}$ such
that $mM$ is Cartier, any section of

$H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}})))$

extends over $X$.

A direct consequence of Theorem 1.2 is

######  Corollary 1.3.

Let $\pi:X\to\Delta$ be a projective contraction from a complex space $X$ to
the disc $\Delta$. Let $X^{\prime}\xrightarrow{f}X\xrightarrow{\pi}\Delta$ be
a generalized polarized pair with the boundary part $B$ and the abundant nef
part $M$. Let $X_{t}\subset X$ be the fiber over $t\in\Delta$ and
$(X_{t},B_{t}+M_{t})$ be the generalized polarized pair obtained by
restricting to $X_{t}$. Assume that for any $t\in\Delta$,
$(X_{t},B_{t}+M_{t})$ has g-canonical singularities (in particular,
$B_{t}=0$). Then for each $m\in\mathbb{N}$ such that $mM$ is Cartier,

$h^{0}(X_{t},\mathcal{O}_{X_{t}}(m(K_{X_{t}}+M|_{X_{t}})))$

is independent of $t\in\Delta$.

Both the abundant assumption and the singularity assumption are indispensable.
In fact, Corollary 1.3 is false if $M$ is just assumed to be nef (see Example
5.17). An example of Kawamata shows that the extension of local sections fails
if the singularities are klt (see [Kaw99b, Example 4.3]).

It turns out that even in the smooth case, Păun’s twisted version of
invariance of plurigenera (see [Pău07, Theorem 1]) is not enough to obtain
Theorem 1.2 (see Remark 5.16). Instead, we need Theorem 1.1 which allows
potentially more sections to be extended.

Besides, [FS20, Theorem 4.1] establishes a version of invariance of
plurigenera assuming that $K_{X}+B+M$ or $B+M$ is big over $\Delta$ and $M$ is
nef over $\Delta$. The argument of [FS20, Theorem 4.1] follows from [HMX13,
Theorem 4.2] which relies on the minimal model program for varieties of
general type.

We discuss the structure of the paper. Section 2 gives the background material
on generalized polarized pairs for complex spaces. Section 3 discusses metrics
of $\mathbb{Q}$-Cartier divisors on complex spaces. Section 4 gives the
construction of metrics for the abundant nef parts of generalized polarized
pairs. Section 5 introduces a new type multiplier ideal sheaf and proves the
aforementioned theorems and corollary.

Acknowledgements. We thank Hanlong Fang, Sheng Rao and Lei Zhang for many
discussions and for answering questions. Z. L. is partially supported by a
grant from SUSTech. Z. W. is partially supported by National Key R&D Program
of China (No.2021YFA1002600) and NSFC grant (No.12071035).

##  2\. Generalized polarized pairs

###  2.1. Notation and conventions

Let $\mathbb{Z}$ be the set of integers and $\mathbb{N}=\mathbb{Z}_{>0}$ be
the set of natural numbers.

A holomorphic map $f:X\to Y$ between complex spaces is called a morphism, and
a morphism is called a contraction if it is surjective with connected fibers.
The morphism $f$ is called a proper modification if (1) $f$ is proper and
surjective, and (2) there exists a nowhere dense analytic subset $Z\subset Y$
such that $f|_{X-f^{-1}(Z)}:X-f^{-1}(Z)\to Y-Z$ is an isomorphism. A
bimeromorphic map $f:X\dashrightarrow Y$ between complex spaces is a
meromorphic map such that the graph $\Gamma_{f}$ is an irreducible analytic
subset in $X\times Y$, and the natural maps $\Gamma_{f}\to X,\Gamma_{f}\to Y$
are proper modifications (see [Uen75, Definitions 2.1, 2.2, 2.7]). A proper
modification $\mu:W\to X$ is called a resolution if $W$ is smooth. A proper
morphism $f:X\to Y$ between complex spaces is called projective if for any
relatively compact open set $U\subset Y$, there is an embedding
$j:f^{-1}(U)\hookrightarrow\mathbb{P}_{U}^{n}\coloneqq\mathbb{P}_{\mathbb{C}}^{n}\times
U$ such that $f|_{f^{-1}(U)}=p_{2}\circ j$, where $p_{2}:\mathbb{P}_{U}^{n}\to
U$ is the natural projection map (see [Pet94b, Chapter V]).

A divisor means a Weil divisor. A finite $\mathbb{Q}$-linear combination of
divisors gives $\mathbb{Q}$-divisors. A $\mathbb{Q}$-divisor is called
$\mathbb{Q}$-Cartier if it is a finite $\mathbb{Q}$-linear combination of
Cartier divisors.

Let $D$ be a $\mathbb{Q}$-divisor on $X$. Then $(X,D)$ is called a log pair. A
resolution $\mu:W\to X$ is called a log resolution of $(X,D)$ if $W$ is smooth
and
$\operatorname{Supp}\mu_{*}^{-1}D\cup\operatorname{Supp}\operatorname{Exc}(\mu)$
is a simple normal crossing divisor, where $\mu_{*}^{-1}D$ is the strict
transform of $D$ and $\operatorname{Exc}(\mu)$ is the exceptional locus of
$\mu$. When $X$ is a reduced complex space, $(X,D)$ always admits a projective
log resolution (see [Wło09, Theorem 2.0.2]).

Let $X$ be a normal complex space. Let $X\to S$ be a morphism to a complex
space $S$. We write the relative property over $S$ by $/S$. Let
$k=\mathbb{Z},\mathbb{Q}$. For two $k$-divisors $B$ and $D$, we use
$B\sim_{k}D/S$ to denote that $B$ and $D$ are $k$-linearly equivalent over
$S$.

Let $D$ be a Weil divisor on a normal complex space $X$, then
$\mathcal{O}_{X}(D)$ denotes the sheaf associated with $D$. To be precise, let
$\mathscr{M}_{X}$ be the sheaf of germs of meromorphic functions on $X$ (see
[Dem12, Chapter II, §6.2]), then $\mathcal{O}_{X}(D)\subset\mathscr{M}_{X}$ is
the sheaf such that for any open set $U\subset X$,

$\mathcal{O}_{X}(D)(U)=\\{f\in\mathscr{M}_{X}(U)\mid\operatorname{div}(f)+D|_{U}\geq
0\\}.$

The sheaf $\mathcal{O}_{X}(D)$ is coherent and the subscript of
$\mathcal{O}_{X}(D)$ will be omitted if it is clear from the context.

Recall that for an $n$-dimensional normal complex space $X$, the canonical
divisor $K_{X}$ is any Weil divisor such that $j_{*}\Omega^{n}_{X_{\rm
reg}}\simeq\mathcal{O}_{X}(K_{X})$ for the open embedding $j:X_{\rm
reg}\hookrightarrow X$, where $X_{\rm reg}$ is the smooth locus of $X$ and
$\Omega^{n}_{X_{\rm reg}}$ is the sheaf of holomorphic $n$-forms on $X_{\rm
reg}$.

Let $f:X\to Y$ be a morphism and $B$ be a $\mathbb{Q}$-Cartier divisor on $Y$.
We write $f^{*}B$ for the $\mathbb{Q}$-Cartier divisor which is the pull-back
of $B$. If $D$ is a Cartier divisor on $X$, then we write $\mathcal{O}_{X}(D)$
for the corresponding line bundle, and $f_{*}\mathcal{O}_{X}(D)$ for the push-
forward of the sheaf $\mathcal{O}_{X}(D)$.

Let $f:X\to Y$ be a proper modification between normal complex spaces and $D$
be a prime divisor on $X$. We define the push-forward divisor

$f_{*}D\coloneqq\begin{cases}f(D),&\text{ if~{} }D\text{~{}is
not~{}}f\text{-exceptional},\\\ 0,&\text{ if~{}
}D\text{~{}is~{}}f\text{-exceptional}.\end{cases}$

This construction can be extended to $\mathbb{Q}$-divisors by linearity.
Moreover, for $k=\mathbb{Z},\mathbb{Q}$, if $B\sim_{k}D/S$, then
$f_{*}B\sim_{k}f_{*}D/S$ because the push-forward of a principal divisor is
still a principal divisor.

Finally, for a projective morphism $X\to S$, a $\mathbb{Q}$-Cartier divisor
$D$ on $X$ is nef$/S$ if there is an ample$/S$ divisor $H$ on $X$ such that
$D+\epsilon H$ is ample$/S$ for any $\epsilon\in\mathbb{Q}_{>0}$, and $D$ is
big$/S$ if $D|_{X_{t}}$ is big for a general $t\in S$.

###  2.2. Generalized polarized pairs with abundant nef part

Generalized polarized pairs originate from the canonical bundle formula (see
[Kaw98]). It was observed by [BZ16] that the generalized polarized pair
structure deserves to be studied separately and it behaves like the usual log
pair structure in many ways. We state the definition of generalized polarized
pairs for complex spaces (see [BZ16, Definition 1.4] for the original
definition in the algebraic category).

######  Definition 2.1.

Let $X^{\prime},X$ and $S$ be normal complex spaces. A generalized polarized
pair (g-pair) consists of projective morphisms $X^{\prime}\xrightarrow{f}X\to
S$ where $f$ is a modification, a $\mathbb{Q}$-divisor $B\geq 0$ on $X$, and a
$\mathbb{Q}$-Cartier divisor $M^{\prime}$ on $X^{\prime}$ which is nef$/S$
such that $K_{X}+{B}+{M}$ is $\mathbb{Q}$-Cartier, where $M\coloneqq
f_{*}M^{\prime}$. We call $B$ the boundary part and $M$ the nef part.

######  Remark 2.2.

In the definition of g-pairs, we can replace $X^{\prime}$ by any projective
modification of $X^{\prime}$ and $M^{\prime}$ by its pull-back. Therefore, we
can assume that $f$ is a projective log resolution of $(X,B)$.

######  Definition 2.3.

Under the notation of Definition 2.1, if there exists a proper contraction
$g:X^{\prime}\to Z/S$ such that $M^{\prime}\sim_{\mathbb{Q}}g^{*}H/S$ for some
nef and big$/S$ $\mathbb{Q}$-Cartier divisor $H$ on $Z$, then we call that
such g-pair has the abundant nef part.

Replacing $X^{\prime}$ by a projective modification and $M^{\prime}$ by its
pull-back (see Remark 2.2), the new g-pair still has the abundant nef part.

######  Remark 2.4.

A g-pair with the abundant nef part naturally appears in the canonical bundle
formula where the concept of g-pairs originates (see [Amb05]). The word
“abundant” comes from the fact that divisors satisfying the property in
Definition 2.3 is related to the abundance conjecture (see [Kaw85, Proposition
2.1]).

###  2.3. Singularities and adjunctions for g-pairs

######  Definition 2.5.

A normal complex space is called $\mathbb{Q}$-Gorenstein if $K_{X}$ is a
$\mathbb{Q}$-Cartier divisor.

######  Definition 2.6.

A normal complex space $X$ has canonical singularities if (1) $X$ is
$\mathbb{Q}$-Gorenstein, and (2) for any resolution $f:W\to X$, if
$K_{W}=f^{*}K_{X}+E$ with $E$ an $f$-exceptional divisor, then $E\geq 0$.

######  Definition 2.7.

Under the notation of Definition 2.1, for a prime divisor $P$ over $X$, its
discrepancy with respect to (X, B+M) is defined to be

${\rm
discp}(P;X,B+M)\coloneqq\operatorname{mult}_{P}\left(K_{Y}+g^{*}M^{\prime}-(f\circ
g)^{*}(K_{X}+B+M)\right),$

where $g:Y\to X^{\prime}$ is a proper modification such that $P\subset Y$.

######  Definition 2.8.

A g-pair $(X,B+M)$ has g-canonical (resp. g-terminal, g-klt, g-lc)
singularities if ${\rm discp}(P;X,B+M)\geq 0$ (resp. $>0$, $>-1$, $\geq-1$)
for each prime divisor $P$ over $X$.

The adjunction formula for g-pairs is given by [BZ16, Definition 4.7] in the
algebraic setting. It can be naturally extended to the analytic setting.

######  Definition 2.9 (Adjunction formula for g-pairs).

Let $(X,B+M)$ be a g-pair with data $X^{\prime}\xrightarrow{f}X\to S$ and
$M^{\prime}$. Let $P$ be a normal irreducible component of $\lfloor B\rfloor$
and $P^{\prime}$ be its strict transform on $X^{\prime}$. We may assume that
$f$ is a projective log resolution of $(X,B+M)$. Write

$K_{X^{\prime}}+B^{\prime}+M^{\prime}=f^{*}(K_{X}+B+M),$

then

$K_{P^{\prime}}+B_{P^{\prime}}+M_{P^{\prime}}\coloneqq(K_{X^{\prime}}+B^{\prime}+M^{\prime})|_{P^{\prime}},$

where $B_{P^{\prime}}=(B^{\prime}-P^{\prime})|_{P^{\prime}}$ and
$M_{P^{\prime}}=M^{\prime}|_{P^{\prime}}$. Let $g$ be the induced morphism
$P^{\prime}\to P$. Set $B_{P}=g_{*}B_{P^{\prime}}$ and
$M_{P}=g_{*}M_{P^{\prime}}$. Then we get the equality

$K_{P}+B_{P}+M_{P}=(K_{X}+B+M)|_{P},$

which is referred as (generalized) adjunction formula.

In general, $M|_{P}\neq M_{P}$, and even if $(X,B+M)$ has the abundant nef
part, $(P,B_{P}+M_{P})$ may not have the abundant nef part. On the other hand,
in the category of algebraic varieties, if $(X,B+M)$ is g-lc, then
$(P,B_{P}+M_{P})$ is still g-lc with data $P^{\prime}\xrightarrow{f}P\to S$
and $M_{P^{\prime}}$ by [BZ16, Remark 4.8].

Let $X\to\Delta$ be a projective contraction to the disc $\Delta$. Let $X_{0}$
be the fiber over $0\in\Delta$. Suppose that $X_{0}$ is normal, then we define

(2.3.1) $(X_{0},B_{0}+M_{0})$

to be the g-pair by the adjunction of $(X,X_{0}+B+M)$ on $X_{0}$. In the
sequel, $(X_{0},B_{0}+M_{0})$ will be assumed to have g-canonical
singularities. Under this assumption, $B_{0}=0$ and $M_{0}=M|_{X_{0}}$.

##  3\. Metrics on complex spaces

###  3.1. Metrics and $\mathbb{Q}$-metrics

Let $X$ be a normal complex space, $\mathcal{L}$ be a holomorphic line bundle
on $X$. We define the notion of singular metrics in this setting.

Let $x\in X$ be a point and $\mathcal{L}_{x}$ be the stalk at $x$. Assume that
there is a function

$\|-\|_{h}:\mathcal{L}_{x}\to\mathbb{R}_{\geq 0}\cup\\{+\infty\\},$

such that if $\theta:\mathcal{L}|_{U}\simeq\mathcal{O}_{U}$ is a
trivialization on an open set $U\subset X$, then for any
$s\in\mathcal{L}_{x}$, we have

$\|s\|_{h}=|\theta(s)|e^{-\varphi}$

for a function $\varphi:U\to\mathbb{R}\cup\\{\pm\infty\\}$. This $\varphi$ is
called a local weight and it depends on $\theta$. Let $\mu:W\to X$ be a
resolution. Then
$\mu^{*}\theta:\mu^{*}\mathcal{L}|_{\mu^{-1}(U)}\simeq\mathcal{O}_{\mu^{-1}(U)}$
is a trivialization for $\mu^{*}\mathcal{L}$. For any
$\tilde{s}\in(\mu^{*}\mathcal{L})_{w}$ with $w\in\mu^{-1}(U)$, set

(3.1.1)
$\|\tilde{s}\|_{\mu^{*}h}\coloneqq|\mu^{*}\theta(\tilde{s})|e^{-\mu^{*}\varphi},$

where $\mu^{*}\varphi(w)=\varphi(\mu(w))$. Note that $\|-\|_{\mu^{*}h}$ is
independent of the choice of trivializations. In fact, for local
trivializations $\theta_{1},\theta_{2}$, there is a nowhere vanishing
holomorphic function $u$ such that $\theta_{1}=u\theta_{2}$. Thus
$|u|e^{-\varphi_{1}}=e^{-\varphi_{2}}$ and

(3.1.2) $|\mu^{*}u|e^{-\mu^{*}\varphi_{1}}=e^{-\mu^{*}\varphi_{2}}.$

Let $L^{1}_{\rm loc}(V)$ be the set of locally integrable functions on a
smooth open set $V$.

######  Definition 3.1 (Singular metrics on normal complex spaces).

Under the above notation, $h$ is a (singular) metric of $\mathcal{L}$ if for
any resolution $\mu:W\to X$, we have $\mu^{*}\varphi\in L_{\rm
loc}^{1}(\mu^{-1}(U))$ for any open set $U\subset X$.

The property of $\varphi$ in Definition 3.1 will be referred as the
$L^{1}_{\rm loc}$-property. By abuse of terminology, if $D$ is a Cartier
divisor, then we also call a metric of the corresponding line bundle
$\mathcal{O}(D)$ a metric of $D$. This is particularly convenient to treat
$\mathbb{Q}$-Cartier divisors.

######  Definition 3.2 ([Pet94b, Definition 1.4]).

Let $X$ be a complex space. A plurisubharmonic (psh for short) function on $X$
is a function $\varphi:X\rightarrow[-\infty,\infty)$ having the following
property. For every $x\in X$ there is an open neighborhood $U$ with a
biholomorphic map $h:U\rightarrow V$ onto a closed complex subspace $V$ of
some domain $G\subset\mathbb{C}^{m}$ and a plurisubharmonic function
$\tilde{\varphi}:G\rightarrow[-\infty,\infty)$ such that
$\varphi|_{U}=\tilde{\varphi}\circ h$.

If $f:X\rightarrow Y$ is a holomorphic map between two complex spaces, and
$\varphi:Y\rightarrow[-\infty,\infty)$ is a psh function on $Y$, then
$\varphi\circ f$ is a psh function on $X$ (see [Nar61, Page 356]).

######  Definition 3.3.

Under the notation of Definition 3.1, $(\mathcal{L},h)$ is said to have non-
negative curvature current if any local weight $\varphi$ on an open subset of
$X$ is a psh function.

######  Remark 3.4.

In Definition 3.1, the local $L_{\rm loc}^{1}$-property is needed to hold for
all resolutions. If the $L_{\rm loc}^{1}$-property holds for one resolution
$\mu:W\to X$, then it is not true that this property holds for another
resolution $\nu:V\to X$ even if $\nu$ factors through $\mu$. In fact, by the
change-of-variables formula,

$\int_{W}|\tilde{\varphi}|~{}dV_{W}=\int_{V}|\nu^{*}\tilde{\varphi}|\cdot|s|^{2}~{}dV_{V},$

where $s$ is the local equation for the effective exceptional divisor
$K_{V}-f^{*}K_{W}$ (hence $s$ is holomorphic). Hence $\tilde{\varphi}\in
L_{\rm loc}^{1}$ does not imply $\nu^{*}\tilde{\varphi}\in L_{\rm loc}^{1}$.

However, if the curvature current is non-negative (see Definition 3.3), then
it is enough to just consider one resolution (see Proposition 3.5).

The notion of psh functions is of bimeromorphic nature:

######  Proposition 3.5.

Let $\mu:W\to X$ be a resolution of a normal complex space $X$. Then a
function $\varphi:X\to[-\infty,\infty]$ is a psh function iff
$\varphi\circ\mu:Y\to[-\infty,\infty]$ is a psh function.

###### Proof.

One only needs to show the sufficient part. As $\mu$ is a proper modification,
$Z\coloneqq\mu(\operatorname{Exc}(\mu))$ is an analytic set of codimension
$\geq 2$. Then $\varphi$ is a psh function on $X-Z$. By [Pet94b, Theorem 1.5],
the definition of psh functions in [GR56] is the same as Definition 3.2. By
[GR56, Page 181, Satz 4], $\varphi$ extends to a psh function $\hat{\varphi}$
on $X$. As $\hat{\varphi}\circ\mu$ is a psh function on $W$ which coincides
with $\varphi\circ\mu$ on $W-\operatorname{Exc}(\mu)$, we have
$\hat{\varphi}\circ\mu=\varphi\circ\mu$ on $W$. Hence $\varphi=\hat{\varphi}$
is a psh function. ∎

For simplicity, we can formally extend the notion of metrics for
$\mathbb{Q}$-Cartier divisors (or $\mathbb{Q}$-line bundles).

######  Definition 3.6.

Let $D$ be a $\mathbb{Q}$-Cartier divisor on a complex space $X$. Then a
$\mathbb{Q}$-metric $h$ for $D$ is a triple $(m,D,h_{mD})$ such that
$m\in\mathbb{N}$ with $mD$ a Cartier divisor and $h_{mD}$ is a metric for
$mD$. We say that two $\mathbb{Q}$-metrics $(m,D,h_{mD})$ and $(n,D,h_{nD})$
are equal if $h_{mD}^{n}=h_{nD}^{m}$ as metrics for the Cartier divisor $mnD$.
For simplicity, $h$ is also called a metric.

######  Remark 3.7.

Given a metric $(m,D,h_{mD})$, for any $n\in\mathbb{N}$ such that $nD$ is
Cartier, there always exists a metric $(n,D,h_{nD})$ which is equal to
$(m,D,h_{mD})$. In fact, we can set $h_{nD}=h_{mD}^{\frac{n}{m}}$.

If $D=\sum D_{i}$ is a finite sum of $\mathbb{Q}$-Cartier divisors such that
each $D_{i}$ admits a metric $h_{i}$, then we define a metric
$h\coloneqq\prod_{i}h_{i}$ for $D$ as follows. Take $m\in\mathbb{N}$ such that
each $mD_{i}$ is Cartier, then

$h=(m,D,h_{mD})$

with $h_{mD}=\prod_{i}h_{mD_{i}}$ where $h_{i}=(m,D_{i},h_{mD_{i}})$ (see
Remark 3.7). It is straightforward to check that this definition is
independent of the choice of $m$. Let $D$ be a $\mathbb{Q}$-Cartier divisor
with a metric $h$, then for $r\in\mathbb{Q}$, we can similarly define a metric
$h^{r}$ for $rD$.

Let $f:W\to X$ be a proper modification between normal complex spaces. Let $D$
be a $\mathbb{Q}$-Cartier divisor on $X$ with a metric $h$. We can define the
pull-back metric $f^{*}h$ for the $\mathbb{Q}$-Cartier divisor $f^{*}D$. In
fact, it is enough to assume that $D$ is a Cartier divisor by the definition
of metrics for $\mathbb{Q}$-Cartier divisors. If
$\theta:\mathcal{O}(D)|_{U}\simeq\mathcal{O}_{U}$ is a trivialization, then
$f^{*}\theta:\mathcal{O}(f^{*}D)|_{f^{-1}(U)}\simeq\mathcal{O}_{f^{-1}(U)}$.
If $\|s\|_{h}=|\theta(s)|e^{-\varphi}$ for $x\in U$ and
$s\in\mathcal{O}(D)_{x}$, then

$\|\tilde{s}\|_{f^{*}h}\coloneqq|f^{*}\theta(\tilde{s})|e^{-f^{*}\varphi}$

for $w\in f^{-1}(U)$ and $\tilde{s}\in\mathcal{O}(f^{*}D)_{w}$. This
definition is independent of the choice of $\theta$.

######  Remark 3.8.

It is more convenient to think of a metric in terms of a local weight: if
$\varphi$ is the local weight for $h_{mD}$ (under a given trivialization),
then $\frac{\varphi}{m}$ is defined to be the local weight of $(m,D,h_{mD})$.
The summation (resp. rational multiple, pull-back) of local weights
corresponds to the product (resp. rational exponent, pull-back) of metrics.

######  Remark 3.9.

Suppose that $B,D$ are $\mathbb{Q}$-Cartier divisors such that
$B\sim_{\mathbb{Q}}D$, then a metric of $B$ is also a metric of $D$. In fact,
local weights are identified under an isomorphism
$\mathcal{O}(mB)\simeq\mathcal{O}(mD)$ for some $m\in\mathbb{N}$.

On a complex manifold, a function $\varphi$ is called quasi-plurisubharmonic
(quasi-psh) if it is a summation of a psh function with a smooth function.

######  Definition 3.10 ([DPS00, Definition 1.4]).

For two quasi-psh functions $\varphi_{1},\varphi_{2}$, we say that
$\varphi_{1}$ is less singular than $\varphi_{2}$ (and write
$\varphi_{1}\preceq\varphi_{2}$) if $\varphi_{2}\leq\varphi_{1}+c$ for a
constant $c$. We write $\varphi_{1}\approx\varphi_{2}$ if
$\varphi_{1}\preceq\varphi_{2}$ and $\varphi_{2}\preceq\varphi_{1}$.

For two functions $\varphi_{i}:X\to\mathbb{R}\cup\\{-\infty\\},i=1,2$ on a
complex space $X$, we use the same notation as above if the corresponding
relations hold after pulling back $\varphi_{i}$ to all the resolutions.

###  3.2. Metrics defined by global sections

Let $X$ be a normal complex space and $L$ be a Cartier divisor on $X$. Suppose
that $\sigma_{i}\in H^{0}(X,\mathcal{O}(L)),1\leq i\leq k$ are global sections
(may not necessarily be distinct). They can be used to define a metric $h$ for
$L$ just as in the smooth case.

Let $U\subset X$ be an open set such that
$\theta:\mathcal{O}(L)|_{U}\simeq\mathcal{O}_{U}$ is a trivialization. Then
for any $s\in L_{x}$, set

(3.2.1) $\|s\|^{2}_{h}=\frac{|\theta(s)|^{2}}{\sum_{1\leq i\leq
k}|\theta(\sigma_{i})|^{2}}=|\theta(s)|^{2}e^{-2\varphi}.$

The local weight of $h$ (under the trivialization $\theta$) is

$\varphi=\frac{1}{2}\log(\sum_{1\leq i\leq k}|\theta(\sigma_{i})|^{2}).$

It satisfies the $L_{\rm loc}^{1}$-property for any resolution $\mu:W\to X$
because

$\mu^{*}\varphi=\frac{1}{2}\log(\sum_{1\leq i\leq
k}|\mu^{*}\theta(\mu^{*}\sigma_{i})|^{2})$

is a psh function and thus Proposition 3.5 applies to this situation.

More generally, for a $\mathbb{Q}$-Cartier divisor $D$ on $X$, suppose that
$mD$ is Cartier and $\sigma_{i}\in H^{0}(X,\mathcal{O}_{X}(mD)),1\leq i\leq k$
are global sections. If $\theta:\mathcal{O}(mD)|_{U}\simeq\mathcal{O}_{U}$ is
a trivialization, then set

$\varphi=\frac{1}{2m}\log(\sum_{1\leq i\leq k}|\theta(\sigma_{i})|^{2})$

as the local weight of $h$ (see Remark 3.8).

We call the metrics defined above the metrics defined by global sections.

For an effective Cartier divisor $D$, choose $\sigma=1\in
H^{0}(X,\mathcal{O}_{X}(D))$, then the metric as (3.2.1) is denoted by
$\hbar_{D}$.

In general, let $D$ be a $\mathbb{Q}$-Cartier divisor. If $D=\sum_{1\leq i\leq
n}r_{i}D_{i}-\sum_{1\leq j\leq m}s_{j}B_{j}$ is a decomposition such that
$r_{i},s_{j}\in\mathbb{Q}_{>0}$ and $D_{i},B_{j}$ are effective Cartier
divisors, then set

(3.2.2) $\hbar_{D}\coloneqq\left(\prod_{1\leq i\leq
n}\hbar_{D_{i}}^{r_{i}}\right)\cdot\left(\prod_{1\leq j\leq
m}(\hbar^{-1}_{B_{j}})^{s_{j}}\right),$

where $\hbar^{-1}_{B_{j}}$ is the dual metric for $\mathcal{O}(-B_{j})$. Note
that $\hbar_{D}$ is a metric as the corresponding weight still satisfies the
$L^{1}_{\rm loc}$-property.

The above $\hbar_{D}$ is independent of the choice of decompositions of $D$.
Indeed, suppose that $mD$ is Cartier for some $m\in\mathbb{N}$ and
$\theta:\mathcal{O}(mD)|_{U}\simeq\mathcal{O}_{U}$ is a trivialization. Then
for any $x\in U$ and $s\in\mathcal{O}(mD)_{x}$, we always have
$\|s\|^{2}_{\hbar_{D}^{m}}=|\theta(s)|^{2}/|\theta(1)|^{2}$.

Let $Z\subset Y$ be a normal complex subspace such that
$Z\not\subset\operatorname{Supp}D$, then $D|_{Z}$ is still a
$\mathbb{Q}$-Cartier divisor. Let $\hbar_{D}|_{Z}$ be the pull-back of
$\hbar_{D}$ to $Z$, then

(3.2.3) $\hbar_{D}|_{Z}=\hbar_{D|_{Z}}.$

######  Proposition 3.11.

Suppose that $h_{B},h_{D}$ are metrics defined by global sections for
$\mathbb{Q}$-Cartier divisors $B,D$ respectively. Then $h_{B}h_{D}$ is a
metric for $B+D$ which is also defined by global sections.

###### Proof.

Suppose that $h_{B}$ (resp. $h_{D}$) is defined by global sections of
$\mathcal{O}(rB)$ (resp. $\mathcal{O}(sD)$), and
$\theta_{rB}:\mathcal{O}(rB)|_{U}\simeq\mathcal{O}_{U}$ (resp.
$\theta_{sD}:\mathcal{O}(sD)|_{U}\simeq\mathcal{O}_{U}$) is a trivialization.
Then the local weights are

$\varphi_{B}=\frac{1}{2r}(\log\sum_{i=1}^{m_{B}}|\theta_{rB}(\sigma_{i})|^{2}),\quad\varphi_{D}=\frac{1}{2s}(\log\sum_{j=1}^{m_{D}}|\theta_{sD}(\tau_{j})|^{2}),$

where $\sigma_{i}\in H^{0}(U,\mathcal{O}(rB)),\tau_{j}\in
H^{0}(U,\mathcal{O}(sD))$. The local weight for $B+D$ is

$\begin{split}\varphi_{B}+\varphi_{D}&=\frac{1}{2rs}\log\left((\sum_{i=1}^{m_{B}}|\theta_{rB}(\sigma_{i})|^{2})^{s}\cdot(\sum_{j=1}^{m_{D}}|\theta_{sD}(\tau_{j})|^{2})^{r}\right)\\\
&=\frac{1}{2rs}\log\left(\sum_{\begin{subarray}{c}i_{1},\cdots,i_{s}\\\
j_{1}\cdots
j_{r}\end{subarray}}|\theta_{rB}(\sigma_{i_{1}})\cdots\theta_{rB}(\sigma_{i_{s}})\cdot\theta_{sD}(\tau_{j_{1}})\cdots\theta_{sD}(\tau_{j_{r}})|^{2}\right).\end{split}$

By $\sigma_{i_{1}}\cdots\sigma_{i_{s}}\cdot\tau_{j_{1}}\cdots\tau_{j_{r}}\in
H^{0}(U,\mathcal{O}(rs(B+D)))$, the claim follows. ∎

###  3.3. Push-forward metrics defined by global sections

Let $f:X^{\prime}\to X$ be a proper modification between normal complex spaces
and $D$ be a $\mathbb{Q}$-Cartier divisor on $X^{\prime}$. Suppose that
$B=f_{*}D$ is still a $\mathbb{Q}$-Cartier divisor on $X^{\prime}$. Suppose
that $D$ admits a metric $h^{\prime}$. In general, there is no push-forward
metric $f_{*}h^{\prime}$ of $h^{\prime}$. However, if $h^{\prime}$ is given by
global sections, then we can naturally define $f_{*}h^{\prime}$ as follows.

Recall that for a $\sigma\in\mathscr{M}_{X^{\prime}}(X^{\prime})$,
$f_{*}\sigma$ is defined as the meromorphic extension of
$\sigma|_{X-f(\operatorname{Exc}(f))}$ on $X$ (see [Dem12, Chapter II
(10.2)]). This is possible as
$\operatorname{codim}_{X}f(\operatorname{Exc}(f))\geq 2$. Hence
$f_{*}\sigma\in\mathscr{M}_{X}(X)$.

Suppose that $mD$ and $mB$ are Cartier divisors. If $\sigma\in
H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(mD))$, that is,
$\sigma\in\mathscr{M}_{X^{\prime}}(X^{\prime})$ such that
$\operatorname{div}(\sigma)+mD\geq 0$, then $f_{*}\sigma\in
H^{0}(X,\mathcal{O}_{X}(mB))$ because

$\operatorname{div}(f_{*}\sigma)+mB=f_{*}\operatorname{div}(\sigma)+f_{*}(mD)\geq
0.$

Therefore, if $h^{\prime}$ is the metric of $D$ defined by global sections as
(3.2.1), then the push-forwards of these sections also define a metric
$h\coloneqq f_{*}h^{\prime}$ for $B$. We call this metric the push-forward of
$h^{\prime}$.

######  Lemma 3.12.

Under the above notation and assumptions, let $D+F=f^{*}B$ with $F$ an
$f$-exceptional divisor ($F$ may not be effective). Then $f^{*}h$ is a metric
for $D+F$ such that

(3.3.1) $h^{\prime}\cdot\hbar_{F}=f^{*}h,$

where $\hbar_{F}$ is the metric defined as (3.2.2).

###### Proof.

Suppose that $h^{\prime}$ is defined by sections $\sigma_{i}\in
H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(mD)),i=1,\cdots,k$, then
$\sigma_{i}=f^{*}(f_{*}\sigma_{i})\in
H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(mD+mF)),i=1,\cdots,k$.

Let
$\theta:\mathcal{O}_{X^{\prime}}(mD)|_{U}\simeq\mathcal{O}_{U},\theta_{F}:\mathcal{O}_{X^{\prime}}(mF)|_{U}\simeq\mathcal{O}_{U}$
be trivializations on $U\subset X^{\prime}$, and
$\theta_{B}:\mathcal{O}_{X}(mB)|_{V}\simeq\mathcal{O}_{V}$ be a trivialization
on $V\subset X$ such that $U\subset f^{-1}(V)$ and
$f^{*}\theta_{B}=\theta\theta_{F}$. Then $h^{\prime}\cdot\hbar_{F}$ has the
local weight

$\frac{1}{2m}(\log\sum_{i}|\theta(\sigma_{i})|^{2}+\log|\theta_{F}(1)|^{2})=\frac{1}{2m}\log(\sum_{i}|\theta\theta_{F}(\sigma_{i})|^{2}),$

where $\sigma_{i}$ in $\theta(\sigma_{i})$ is a section with respect to the
divisor $mD$ while $\sigma_{i}$ in $\theta\theta_{F}(\sigma_{i})$ is a section
with respect to the divisor $mD+mF$.

The local weight for $h$ is

$\frac{1}{2m}\log(\sum_{i}|\theta_{B}(f_{*}\sigma_{i})|^{2}).$

Thus $f^{*}h$ has the local weight

$f^{*}\left(\frac{1}{2m}\log(\sum_{i}|\theta_{B}(f_{*}\sigma_{i})|^{2})\right)$

under the trivialization $f^{*}\theta_{B}$ (see (3.1.1)). The claim follows
from

$f^{*}(\theta_{B}(f_{*}\sigma_{i}))=(f^{*}\theta_{B})(f^{*}(f_{*}\sigma_{i}))=(\theta\theta_{F})(\sigma_{i}).$

∎

##  4\. Metrics for the abundant nef parts

###  4.1. Approximations for the abundant nef part

Let $\pi:X\to\Delta$ be a projective contraction to the disc $\Delta$. Suppose
that $f:X^{\prime}\to X/\Delta$ is a projective modification and $M^{\prime}$
is a $\mathbb{Q}$-divisor on $X^{\prime}$. Moreover, suppose that
$g:X^{\prime}\to Z/\Delta$ is a proper contraction such that
$M^{\prime}\sim_{\mathbb{Q}}g^{*}H^{\prime}/\Delta$, where $H^{\prime}$ is a
nef and big $\mathbb{Q}$-divisor over $\Delta$.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{f}$$\textstyle{\Delta}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$

Take projective resolutions $\tilde{f}:X^{\prime\prime}\to
X^{\prime},\tilde{\tau}:Z^{\prime\prime}\to Z$ such that
$\tilde{g}\coloneqq{\tilde{\tau}}^{-1}\circ
g\circ{\tilde{f}}:X^{\prime\prime}\to Z^{\prime\prime}$ is a morphism. By
Hironaka’s Chow lemma (see [Pet94a, Chapter VII, Theorem 2.8]), we can further
assume that $Z^{\prime\prime}\to\Delta$ and $\tilde{g}$ are projective. Let
$M^{\prime\prime}=\tilde{f}^{*}M^{\prime}$, then $M^{\prime\prime}$ is the
pull-back of the nef and big$/\Delta$ divisor $\tilde{\tau}^{*}H^{\prime}$.
Hence, replacing $X^{\prime},Z,M^{\prime}$ by
$X^{\prime\prime},Z^{\prime\prime},M^{\prime\prime}$, we can assume that $Z$
is smooth and $g,\tau$ are projective.

As the Picard group $\operatorname{Pic}(\Delta)$ is trivial, for
$k=\mathbb{Z},\mathbb{Q}$ and $k$-divisors $B,D$ on $X$, $B\sim_{k}D/\Delta$
is the same as $B\sim_{k}D$.

The following lemma is a direct consequence of the negativity lemma (see
[KM98, Lemma 3.39]). Note that such result also holds for normal complex
spaces (see [KM98, Proof of Lemma 3.40]).

######  Lemma 4.1.

Let $X^{\prime}\xrightarrow{f}X\to S$ be a g-pair with a nef$/S$ divisor
$M^{\prime}$ on $X^{\prime}$ and the nef part $M=f_{*}M^{\prime}$. Suppose
that $M$ is $\mathbb{Q}$-Cartier, then

(4.1.1) $f^{*}M=M^{\prime}+\Upxi,$

where $\Upxi\geq 0$ is an $f$-exceptional divisor.

######  Lemma 4.2.

Under the above notation and assumptions. For a g-pair
$X^{\prime}\xrightarrow{f}X\xrightarrow{\pi}\Delta$ with the abundant nef
part. Let $X_{0}\subset X$ be the fiber over $0\in\Delta$. Let $\tilde{X}_{0}$
be the strict transform of $X_{0}$ on $X^{\prime}$. There exist divisors $E,F$
on $X^{\prime}$ that satisfy the following:

  1. (1)

$E,F$ are effective $\mathbb{Q}$-divisors without common components,

  2. (2)

$E$ is $f$-exceptional,

  3. (3)

for each $k\in\mathbb{N}$, there exists an ample $\mathbb{Q}$-Cartier divisor
$H_{k}$ on $Z$ such that

$M^{\prime}+\frac{1}{k}E\sim_{\mathbb{Q}}g^{*}H_{k}+\frac{1}{k}F,$

  4. (4)

$\tilde{X}_{0}$ is not a component of $E\cup F$.

###### Proof.

There is a nef and big$/\Delta$ divisor $H^{\prime}$ on a smooth complex space
$Z$ such that $M^{\prime}\sim_{\mathbb{Q}}g^{*}H^{\prime}$. Then for any
$k\in\mathbb{N}$, $H^{\prime}\sim_{\mathbb{Q}}H_{k}+\frac{1}{k}B$ where
$H_{k}$ is ample and $B\geq 0$ (see [KM98, Proposition 2.6.1 (3)]). Let
$c=\operatorname{mult}_{X_{0}}(f_{*}g^{*}B)$, then

(4.1.2) $g^{*}B-c(\pi\circ f)^{*}(0)=g^{*}B-c(\tilde{X}_{0}+E_{f})$

does not contain $\tilde{X}_{0}$ in its support, where $E_{f}$ is some
$f$-exceptional divisor (here $0\in\Delta$ is viewed as a divisor and
$(\pi\circ f)^{*}(0)$ is the pull-back of $0$). However, $g^{*}B-c(\pi\circ
f)^{*}(0)$ may not be effective.

We have

$g^{*}B-c(\pi\circ f)^{*}(0)=g^{*}B-c(\tau\circ
g)^{*}(0)=g^{*}(B-c\tau^{*}(0)).$

Write

(4.1.3) $B-c\tau^{*}(0)=\Theta-E^{-},$

where $\Theta,E^{-}$ are $\mathbb{Q}$-Cartier effective divisors without
common components. But $g^{*}\Theta,g^{*}E^{-}$ may have common components.
Let

$\Lambda\coloneqq g^{*}\Theta\wedge g^{*}E^{-}$

be the effective $\mathbb{Q}$-Cartier divisor such that

$\operatorname{mult}_{P}\Lambda=\min\\{\operatorname{mult}_{P}g^{*}\Theta,\operatorname{mult}_{P}g^{*}E^{-}\\}$

for each prime divisor $P$. Then $g^{*}\Theta-\Lambda,g^{*}E^{-}-\Lambda$ are
effective divisors without common components.

By (4.1.3),

$g^{*}B-c(\pi\circ
f)^{*}(0)=g^{*}\Theta-g^{*}E^{-}=(g^{*}\Theta-\Lambda)-(g^{*}E^{-}-\Lambda).$

Moreover, the negative component (i.e. the component with negative
coefficients) of $g^{*}B-c(\pi\circ f)^{*}(0)$ is $f$-exceptional by (4.1.2).
Therefore, $g^{*}E^{-}-\Lambda$ is $f$-exceptional. By the choice of $c$ (see
(4.1.2)),

$\tilde{X}_{0}\not\subset\operatorname{Supp}((g^{*}\Theta-\Lambda)\cup(g^{*}E^{-}-\Lambda)).$

Let $E=g^{*}E^{-}-\Lambda$ and $F=g^{*}\Theta-\Lambda$, then

$M^{\prime}+\frac{1}{k}E\sim_{\mathbb{Q}}g^{*}H_{k}+\frac{1}{k}F.$

∎

###  4.2. Construction metrics for the abundant nef part

Under the above notation and assumptions, for the morphism $f:X^{\prime}\to
X$, let $\tilde{X}_{0}$ be the strict transform of $X_{0}$. Replacing $f$ by a
projective resolution, we can assume that $\tilde{X}_{0}$ is smooth. Let
$f_{0}:\tilde{X}_{0}\to X_{0}$ be the corresponding morphism. Set
$E_{0}\coloneqq E|_{\tilde{X}_{0}},F_{0}\coloneqq F|_{\tilde{X}_{0}}$ which
are well-defined $\mathbb{Q}$-Cartier divisors by Lemma 4.2 (4), and set
$\Upxi_{0}\coloneqq\Upxi|_{\tilde{X}_{0}}$ (see Lemma 4.1 for the definition
of $\Upxi$).

######  Lemma 4.3.

After shrinking $\Delta$ around $0$, there is a metric $h_{k}^{\prime}$ (non-
canonically constructed) for the $\mathbb{Q}$-Cartier divisor
$M^{\prime}+\frac{1}{k}E$ such that $h^{\prime}_{k}$ is a non-negative metric
defined by global sections as (3.2.1). Moreover,

  1. (1)

$h^{\prime}_{k}|_{\tilde{X}_{0}}\not\equiv\infty$,

  2. (2)

$\varphi_{k}^{\prime}\approx\frac{1}{k}\varphi_{F}$, where
$\varphi_{k}^{\prime},\varphi_{F}$ are the local weights of
$h_{k}^{\prime},\hbar_{F}$ respectively, and

  3. (3)

$\varphi_{k,0}^{\prime}\approx\frac{1}{k}\varphi_{F_{0}}$, where
$\varphi_{k,0}^{\prime},\varphi_{F_{0}}$ are the local weights of
$h^{\prime}_{k}|_{\tilde{X}_{0}},\hbar_{F_{0}}$ respectively.

###### Proof.

We have $M^{\prime}+\frac{1}{k}E\sim_{\mathbb{Q}}g^{*}H_{k}+\frac{1}{k}F$.
Because $H_{k}$ is ample, after shrinking $\Delta$ around $0$, $|mH_{k}|$
induces an embedding $Z\hookrightarrow\mathbb{P}_{\Delta}^{r}$ for some
$m\in\mathbb{N}$. Pulling back a Fubini-Study metric of
$\mathcal{O}_{\mathbb{P}_{\Delta}^{r}}(1)$, we have a positive and continuous
metric $h_{mH_{k}}$ for $mH_{k}$ which is defined by global sections. Let

$h_{H_{k}}\coloneqq(h_{mH_{k}})^{\frac{1}{m}}\quad\text{and}\quad\chi_{k}\coloneqq
g^{*}h_{H_{k}}.$

Then $\chi_{k}$ is defined by the global sections of $g^{*}H_{k}$, and it is a
smooth metric with non-negative curvature current for the semi-ample divisor
$g^{*}H_{k}$. Multiplying with the non-negative metric $\hbar_{F}$ for $F$
(see (3.2.2)), we get the desired metric

$h_{k}^{\prime}\coloneqq\chi_{k}\cdot\hbar_{F}^{\frac{1}{k}}$

for $g^{*}H_{k}+\frac{1}{k}F$, and thus for $M^{\prime}+\frac{1}{k}E$ (see
Remark 3.9). As $\tilde{X}_{0}\not\subset\operatorname{Supp}F$,
$h^{\prime}|_{\tilde{X}_{0}}\not\equiv\infty$. This is (1).

(2) follows as the local weight $\varphi_{\chi_{k}}$ for $\chi_{k}$ is a
continuous function. We can choose trivializations such that
$\varphi^{\prime}_{k}=\varphi_{\chi_{k}}+\frac{1}{k}\varphi_{F}$.

We have
$h_{k}^{\prime}|_{\tilde{X}_{0}}=\chi_{k}|_{\tilde{X}_{0}}\cdot(\hbar_{F}|_{\tilde{X}_{0}})^{\frac{1}{k}}$
and $\hbar_{F}|_{\tilde{X}_{0}}=\hbar_{F_{0}}$. As the local weight for
$\chi_{k}|_{\tilde{X}_{0}}$ is still a continuous function, (3) holds for the
same reason as (2). ∎

By Lemma 4.2, $E$ is $f$-exceptional, and thus

$M=f_{*}M^{\prime}=f_{*}(M^{\prime}+\frac{1}{k}E)\sim_{\mathbb{Q}}f_{*}(g^{*}H_{k}+\frac{1}{k}F).$

Because $g^{*}H_{k}$ and $\frac{1}{k}F$ admit metrics $\chi_{k}$ and
$\hbar_{F}^{\frac{1}{k}}$ which are both defined by global sections,
$h_{k}^{\prime}=\chi_{k}\cdot\hbar_{F}^{\frac{1}{k}}$ is also defined by
global sections by Proposition 3.11. Moreover, $h_{k}^{\prime}$ is a metric
for $M^{\prime}+\frac{1}{k}E$ (see Remark 3.9). Thus $h_{k}\coloneqq
f_{*}h_{k}^{\prime}$ is a metric for $M$ which is still defined by global
sections (see Section 3.3).

######  Lemma 4.4.

Under the above notation and assumptions, let $h_{k}=f_{*}h_{k}^{\prime}$ be
the metric for $M=f_{*}(M^{\prime}+\frac{1}{k}E)$. Then

  1. (1)

$h_{k}$ has non-negative curvature current,

  2. (2)

$h_{k}|_{X_{0}}\not\equiv\infty$,

  3. (3)

$f^{*}\varphi_{k}+\frac{1}{k}\varphi_{E}\approx\frac{1}{k}\varphi_{F}+\varphi_{\Upxi}$,
where $\varphi_{k},\varphi_{E},\varphi_{F},\varphi_{\Upxi}$ are local weights
for $h_{k},\hbar_{E},\hbar_{F},\hbar_{\Upxi}$ respectively, and

  4. (4)

$f_{0}^{*}\varphi_{k,0}+\frac{1}{k}\varphi_{E_{0}}\approx\frac{1}{k}\varphi_{F_{0}}+\varphi_{\Upxi_{0}}$,
where $\varphi_{k,0},\varphi_{E_{0}},\varphi_{F_{0}},\varphi_{\Upxi_{0}}$ are
local weights for
$h_{k}|_{X_{0}},\hbar_{E_{0}},\hbar_{F_{0}},\hbar_{\Upxi_{0}}$ respectively.

###### Proof.

As the metric defined by global sections always has non-negative curvature
current, we have (1).

For (2), by Lemma 4.3, $h^{\prime}_{k}|_{\tilde{X}_{0}}\not\equiv\infty$. By

$f^{*}M=f^{*}(f_{*}(M^{\prime}+\frac{1}{k}E))=(M^{\prime}+\frac{1}{k}E)+\Upxi-\frac{1}{k}E,$

Lemma 3.12 implies $f^{*}h_{k}=h_{k}^{\prime}\cdot\hbar_{\Upxi-\frac{1}{k}E}$,
which is the same as

(4.2.1)
$f^{*}h_{k}\cdot\hbar_{E}^{\frac{1}{k}}=h_{k}^{\prime}\cdot\hbar_{\Upxi}.$

As $E,\Upxi$ are $f$-exceptional divisors,
$h_{E}|_{\tilde{X}_{0}}\not\equiv\infty,h_{\Upxi}|_{\tilde{X}_{0}}\not\equiv\infty$.
Hence $f^{*}h_{k}|_{\tilde{X}_{0}}\not\equiv\infty$. This implies
$h_{k}|_{\tilde{X}_{0}}\not\equiv\infty$ as $f_{0}$ is a proper modification.

(3) follows from (4.2.1) and Lemma 4.3 (2).

For (4), restricting (4.2.1) to $\tilde{X}_{0}$, we have

$f^{*}h_{k}|_{\tilde{X}_{0}}\cdot\hbar_{E}^{\frac{1}{k}}|_{\tilde{X}_{0}}=h_{k}^{\prime}|_{\tilde{X}_{0}}\cdot\hbar_{\Upxi}|_{\tilde{X}_{0}}.$

Moreover,
$\hbar_{E}|_{\tilde{X}_{0}}=\hbar_{E_{0}},\hbar_{\Upxi}|_{\tilde{X}_{0}}=\hbar_{\Upxi_{0}}$.
By $f^{*}h_{k}|_{\tilde{X}_{0}}=f_{0}^{*}(h_{k}|_{X_{0}})$, we have

$f_{0}^{*}(h_{k}|_{X_{0}})\cdot\hbar_{E_{0}}^{\frac{1}{k}}=h_{k}^{\prime}|_{\tilde{X}_{0}}\cdot\hbar_{\Upxi_{0}}.$

The result follows from Lemma 4.3 (3). ∎

######  Remark 4.5.

In application, it is possible to work with $h_{\min}$, the metric with
minimal singularities for $M$ (which also makes sense on complex spaces under
suitable modifications). Comparing with $h_{k}$, we have
$\varphi_{\min}\preceq\varphi_{k}$ where $\varphi_{\min}$ is the local weight
for $h_{\min}$. In the following, we adopt the more direct approach to work
with $h_{k}$.

##  5\. Extension theorems and invariance of plurigenera

###  5.1. Multiplier ideal sheaves

Recall that for a complex manifold $X$ and a Cartier divisor $L$ with a metric
$h$. If the local weight of $h$ is $\varphi$, then the multiplier ideal sheaf
$\mathcal{I}(h)$ is the sheaf of germs of holomorphic functions $\alpha$ such
that $|\alpha|^{2}e^{-2\varphi}$ is locally integrable. Therefore,

$\begin{split}H^{0}(U,\mathcal{I}(h)\otimes\mathcal{O}_{X}(L))=\\{&s\in
H^{0}(U,\mathcal{O}_{X}(L))\mid|\theta(s)|^{2}e^{-2\varphi}\in L^{1}_{\rm
loc}(U),\\\ &\text{where~{}}\theta\text{~{}is a trivialization
of~{}}\mathcal{O}_{X}(L)\\}.\end{split}$

We have the following observation.

######  Proposition 5.1.

Let $X$ be a complex manifold. Let $L$ be a Cartier divisor with a metric $h$
and $E$ be a Cartier divisor with a metric $\hbar_{E}$. If
$s\in\mathscr{M}_{X}(X)$ is a global section of both $L$ and $L+E$ (i.e.
$\operatorname{div}(s)+L\geq 0,\operatorname{div}(s)+L+E\geq 0$), then $s\in
H^{0}(X,\mathcal{O}_{X}(L)\otimes\mathcal{I}(h))$ iff $s\in
H^{0}(X,\mathcal{O}_{X}(L+E)\otimes\mathcal{I}(h\hbar_{E}))$.

###### Proof.

Suppose that $\theta_{L}:\mathcal{O}(L)|_{U}\simeq\mathcal{O}_{U}$ is a
trivialization of $\mathcal{O}(L)$ on $U\subset X$. Shrinking $U$, let
$\sigma_{E}\in\mathscr{M}_{X}(U)$ be a local equation of $E$. There is a
trivialization $\theta_{E}:\mathcal{O}(E)|_{U}\simeq\mathcal{O}_{U}$ by
multiplying $\sigma_{E}$. Therefore, locally at a point $x\in U$,

$\int|\theta_{L}(s)|^{2}e^{-2\varphi_{h}}~{}dV_{X}=\int|\theta_{L}\theta_{E}(s)|^{2}e^{-2\varphi_{h}}e^{-2\log|\sigma_{E}|}~{}dV_{X},$

where $\varphi_{h}$ is the local weight of $h$ corresponding to $\theta_{L}$.
∎

###  5.2. Extension theorem for twisted pairs

We need a modification of [Tak07, Theorem 3.1] (see Lemma 5.2). In fact, we
only work in the setting of [Tak07, Theorem 3.1] instead of generalizing this
result (see Remark 5.3).

Let $\mu:W\to X$ be a log resolution and $\tau:W\to X\to\Delta$ be the
corresponding morphism. Let $N$ be a Cartier divisor on $W$ and
$\tilde{h}_{N}$ be a metric for $N$. Suppose that $X_{0}$ is a normal complex
subspace and $Y_{0}\subset\tau^{-1}(0)$ is the strict transform of $X_{0}$.
Taking a higher log resolution, we can assume that $Y_{0}$ is a smooth
divisor. Viewing $X_{0}$ as an effective Cartier divisor, we have
$\mu^{*}X_{0}=Y_{0}+\Theta$ with $\Theta\geq 0$ a $\mu$-exceptional divisor.
In particular, $\Theta_{0}\coloneqq\Theta|_{Y_{0}}\geq 0$. Restricting

$K_{W}+Y_{0}+\Theta\sim K_{W}$

to $Y_{0}$ and by the adjunction formula, we have

(5.2.1) $K_{Y_{0}}+\Theta_{0}\sim K_{W}|_{Y_{0}}.$

Therefore, a section of $\mathcal{O}(mK_{W}|_{Y_{0}})$ corresponds to a
section of $\mathcal{O}(mK_{Y_{0}}+m\Theta_{0})$, and its integrability with
respect to the metric $\hbar_{m\Theta_{0}}$ makes sense (see Remark 3.9).

We explain the natural isomorphism

$\tilde{\nu}:\mathcal{O}(K_{Y_{0}}+\Theta_{0})\simeq\mathcal{O}(K_{W}|_{Y_{0}})$

following [Tak07, Page 5-6]. The pull-back of the coordinate function $t$ on
$\Delta$ is regarded as a holomorphic function on $W$ and it is still denoted
by $t$. Suppose that $w_{1},\ldots,w_{n-1},y_{0}$ are local coordinates with
$Y_{0}=\\{y_{0}=0\\}$. Then $t=\xi y_{0}$ with $\Theta=\\{\xi=0\\}$. For a
sufficiently small open set $U\subset Y_{0}$, let

$\tau\omega_{Y_{0}}\coloneqq\tau dw_{1}\wedge\cdots\wedge dw_{n-1}\in
H^{0}(U,\mathcal{O}(K_{Y_{0}}+\Theta_{0})),$

where $\tau\in H^{0}(U,\mathcal{O}(\Theta_{0}))$ is a section. Let
$\tilde{\tau}$ be a local meromorphic extension of $\tau$ near $U$ such that
$\operatorname{div}(\tilde{\tau})+\Theta\geq 0$. Then
$\tilde{\tau}dw_{1}\wedge\cdots\wedge dw_{n-1}$ is a local extension of
$\tau\omega_{Y_{0}}$. By abuse of notation, $\omega_{Y_{0}}$ is still used to
denote a local extension of $\omega_{Y_{0}}$ to $W$. In fact, in what follows,
we always restrict the above forms to $Y_{0}$, hence the results are
independent of the choice of extensions.

Then

(5.2.2)
$\tilde{\nu}:\tau\omega_{Y_{0}}\mapsto(\tilde{\tau}\omega_{Y_{0}}\wedge
dt)|_{Y_{0}}.$

Note that this is well-defined because $dt$ is independent of the choices of
the open set $U$. In order to write $(\omega_{Y_{0}}\wedge dt)|_{Y_{0}}$ in
terms of $(dw_{1}\wedge\cdots\wedge dw_{n-1}\wedge dy_{0})|_{Y_{0}}$, note
that

$dt=\xi dy_{0}+y_{0}d\xi$

and $dw_{1}\wedge\cdots\wedge dw_{n-1}\wedge d\xi=0$. Thus

(5.2.3)
$\tilde{\nu}:\tau\omega_{Y_{0}}\mapsto(\tilde{\tau}\xi\omega_{Y_{0}}\wedge
dy_{0})|_{Y_{0}}.$

Because $\Theta=\\{\xi=0\\}$ and $\operatorname{div}(\tilde{\tau})+\Theta\geq
0$, $\tilde{\tau}\xi$ is holomorphic. Hence
$(\tilde{\tau}\xi\omega_{Y_{0}}\wedge dy_{0})|_{Y_{0}}\in
H^{0}(U,\mathcal{O}(K_{W}|_{Y_{0}}))$. From such local expressions,
$\tilde{\nu}$ is seen to be an isomorphism. Moreover, the above discussion
also gives a natural isomorphism

(5.2.4)
$\mathcal{O}(mK_{Y_{0}}+m\Theta_{0})\xrightarrow{\sim}\mathcal{O}(mK_{W}|_{Y_{0}}).$

######  Lemma 5.2.

Under the above notation and assumptions. If $\tilde{h}_{N}|_{Y_{0}}$ is well-
defined, then for each $m\in\mathbb{N}$, as long as
$\tilde{h}_{N}\hbar_{m\Theta}$ has non-negative curvature current, any section
of

$H^{0}(Y_{0},\mathcal{O}(mK_{W}|_{Y_{0}}+N|_{Y_{0}})\otimes\mathcal{I}(\tilde{h}_{N}|_{Y_{0}}\hbar_{m\Theta_{0}}))$

extends over $W$.

###### Proof.

The argument is identically as that for [Pău07, Theorem 1] (a simplified and
twisted version of [Siu02]). Therefore, we just sketch the argument.

We show that the naturally defined morphisms $p,\nu,q$ (defined below) give
the following commutative diagram

(5.2.5)
$\textstyle{\mathcal{O}_{W}(mK_{W}+N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{\mathcal{O}_{Y_{0}}(mK_{Y_{0}}+m\Theta_{0}+N|_{Y_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\mathcal{O}_{Y_{0}}(mK_{W}|_{Y_{0}}+N|_{Y_{0}}).}$

It suffices to define the morphism when $N=0$. First, $p$ is defined through

$\mathcal{O}_{W}(mK_{W})\xrightarrow{\sim}\mathcal{O}_{W}(mK_{W}+mY_{0}+m\Theta_{0})\to\mathcal{O}_{Y_{0}}(mK_{Y_{0}}+m\Theta_{0}).$

Suppose that locally
$\mathcal{O}_{Y_{0}}(K_{Y_{0}})|_{U}=\omega_{Y_{0}}\mathcal{O}_{U}$, then the
above is given in local expressions by

$\alpha(\omega_{Y_{0}}\wedge dy_{0})^{\otimes
m}\mapsto\frac{\alpha}{t^{m}}(\omega_{Y_{0}}\wedge dy_{0})^{\otimes
m}\mapsto\frac{\alpha_{0}y_{0}^{m}}{t^{m}}\omega_{Y_{0}}^{\otimes m},$

where $\alpha$ is a local section of $\mathcal{O}_{W}$ and
$\alpha_{0}=\alpha|_{Y_{0}}$.

The $\nu$ is given as (5.2.4) and the $q$ is the natural restriction map. To
show $q=\nu\circ p$, set $\xi_{0}\coloneqq\xi|_{Y_{0}}$. As $t=\xi y_{0}$, we
have

$\frac{\alpha_{0}y_{0}^{m}}{t^{m}}\omega_{Y_{0}}^{\otimes
m}=\frac{\alpha_{0}}{\xi_{0}^{m}}\omega_{Y_{0}}^{\otimes m}.$

Thus

$p:\alpha(\omega_{Y_{0}}\wedge dy_{0})^{\otimes
m}\mapsto\frac{\alpha_{0}}{\xi_{0}^{m}}\omega_{Y_{0}}^{\otimes m}.$

By $dt=\xi dy_{0}+y_{0}d\xi$, we have

(5.2.6) $\omega_{Y_{0}}\wedge dy_{0}=\frac{1}{\xi}\omega_{Y_{0}}\wedge dt.$

Thus

$q:\alpha(\omega_{Y_{0}}\wedge dy_{0})^{\otimes
m}\mapsto\alpha_{0}(\frac{1}{\xi_{0}}(\omega_{Y_{0}}\wedge
dt)|_{Y_{0}})^{\otimes m}.$

By (5.2.2),
$\nu:\mathcal{O}(mK_{Y_{0}}+m\Theta_{0})\xrightarrow{\sim}\mathcal{O}(mK_{W}|_{Y_{0}})$
is given by

$\nu:\tau_{0}\omega_{Y_{0}}^{\otimes
m}\mapsto\tilde{\tau}_{0}(\omega_{Y_{0}}\wedge dt)^{\otimes m}|_{Y_{0}},$

where $\tau_{0}\in H^{0}(U,\mathcal{O}(m\Theta_{0}))$ and $\tilde{\tau}_{0}$
is a meromorphic extension of $\tau_{0}$. Take
$\tau_{0}=\frac{\alpha_{0}}{\xi_{0}^{m}}$, we have $q=\nu\circ p$.

We claim that any section of

$H^{0}(Y_{0},\mathcal{O}(mK_{Y_{0}}+(N|_{Y_{0}}+m\Theta_{0}))\otimes\mathcal{I}(\tilde{h}_{N}|_{Y_{0}}h_{m\Theta_{0}}))$

extends over $W$ (i.e. such section has preimage in
$H^{0}(W,\mathcal{O}(mK_{W}+N))$). To show this, the argument of [Pău07,
Theorem 1] goes through with the version of the Ohsawa-Takegoshi extension
theorem in [Pău07, Theorem 2.1] replaced by [Tak07, Lemma 3.6].

Now, as $\nu$ induces the isomorphism

$\begin{split}&H^{0}(Y_{0},\mathcal{O}(mK_{Y_{0}}+(N|_{Y_{0}}+m\Theta_{0}))\otimes\mathcal{I}(\tilde{h}_{N}|_{Y_{0}}\hbar_{m\Theta_{0}}))\\\
\xrightarrow{\sim}&H^{0}(Y_{0},\mathcal{O}(mK_{W}|_{Y_{0}}+(N|_{Y_{0}}))\otimes\mathcal{I}(\tilde{h}_{N}|_{Y_{0}}\hbar_{m\Theta_{0}})),\end{split}$

the commutativity of the diagram (5.2.5) gives the desired result. ∎

######  Remark 5.3.

When $N=0$, Lemma 5.2 is weaker than [Tak07, Theorem 3.1] because

$H^{0}(Y_{0},\mathcal{O}(mK_{W}|_{Y_{0}})\otimes\mathcal{I}(h_{m\Theta_{0}}))\subset
H^{0}(Y_{0},\mathcal{O}(mK_{W}|_{Y_{0}})).$

It is likely that under the assumption of non-negative curvature current of
$\tilde{h}_{N}$, any section of

$H^{0}(Y_{0},\mathcal{O}(mK_{Y_{0}}+m\Theta_{0}+N|_{Y_{0}})\otimes\mathcal{I}(\tilde{h}_{N}|_{Y_{0}}))$

extends over $W$.

###  5.3. The construction of $\mathcal{G}_{m}(h)$

Let $X$ be a complex space with canonical singularities. Note that the
canonical divisor $K_{X}$ is a Weil divisor which may not be Cartier. There
exists $\ell\in\mathbb{N}$ such that $\ell K_{X}$ is Cartier. Let $\mu:W\to X$
be a resolution. Let $w_{i},i=1\ldots,n$ be local coordinates on $W$ and
$z_{i},i=1\ldots,n$ be local coordinates on $X_{\rm reg}$. There is a
multiple-valued meromorphic function $J(\mu)$ such that

$(d\mu^{*}z_{1}\wedge\cdots\wedge
d\mu^{*}z_{n})^{\otimes\ell}=J(\mu)^{\ell}(dw_{1}\wedge\cdots\wedge
dw_{n})^{\otimes\ell},$

where $J(\mu)^{\ell}$ is a (single-valued) meromorphic function. In what
follows, over possibly singular locus, we are only concerned with $|J(\mu)|$,
hence the multi-valuedness of $J(\mu)$ will not cause any problem.

If we write $E=\mu^{*}K_{X}-K_{W}$ with $E$ the $\mu$-exceptional
$\mathbb{Q}$-Cartier divisor, then $J(\mu)$ is the local equation of $-E$ (up
to multiply a nowhere vanishing function). As $X$ has canonical singularities,
$E\leq 0$ and $J(\mu)$ is a multiple-valued holomorphic function.

Let $L$ be a Cartier divisor. As $X$ has canonical singularities, for any
$m\in\mathbb{N}$ ($mK_{X}$ may not be Cartier), there is a natural pull-back

$\mu^{*}\mathcal{O}_{X}(mK_{X})\to\mathcal{O}_{W}(mK_{W})$

which induces the natural map

$\mu^{*}\mathcal{O}_{X}(mK_{X}+L)\to\mathcal{O}_{W}(mK_{W}+\mu^{*}L).$

To be precise, suppose that on an open set $U\subset X_{\operatorname{reg}}$,
we have

$s\in H^{0}(U,\mathcal{O}_{X}(mK_{X}+L))$

such that

$s=\alpha(dz_{1}\wedge\cdots\wedge dz_{n})^{\otimes m}$

with $\alpha\in H^{0}(U,\mathcal{O}(L))$. Let

$d\mu^{*}z_{1}\wedge\cdots\wedge d\mu^{*}z_{n}=J(\mu)dw_{1}\wedge\cdots\wedge
dw_{n}.$

As $U$ is smooth, $J(\mu)$ is a single-valued holomorphic function on $U$.
Then

(5.3.1) $\mu^{*}s=\mu^{*}\alpha\cdot J(\mu)^{m}(dw_{1}\wedge\cdots\wedge
w_{n})^{\otimes m},$

where $\mu^{*}\alpha\cdot J(\mu)^{m}\in
H^{0}(\mu^{-1}(U),\mathcal{O}_{W}(\mu^{*}L))$. As $E\leq 0$,

$\mu^{*}\mathcal{O}_{X}(mK_{X}+L)\hookrightarrow\mathcal{O}_{W}(mK_{W}+\mu^{*}L),$

and thus $\mu^{*}\alpha\cdot J(\mu)^{m}(dw_{1}\wedge\cdots\wedge
w_{n})^{\otimes m}$ extends over $W$.

Let $X$ be a normal complex space and $L$ be a Cartier divisor with a metric
$h$. Assume that $\varphi$ is a local weight of $h$ under some trivialization.
The following type of ideal sheaves generalizes multiplier ideal sheaves.

######  Definition 5.4 (Definition of $\mathcal{G}_{m}(h)$).

Under the above notation and assumptions. Let $\mathcal{G}_{m}(h)$ be the
sheaf of germs of holomorphic functions such that for any $x\in X$,

$\begin{split}\mathcal{G}_{m}(h)_{x}=\\{&\alpha\in\mathcal{O}_{X,x}\mid|\mu^{*}\alpha|^{2}|J(\mu)|^{2m}e^{-2\mu^{*}\varphi}\in
L_{\rm loc}^{1}(\mu^{-1}(U)),\\\ &\text{where~{}}\mu:W\to X\text{~{}is a
resolution and~{}}U\text{~{}is a neighborhood of~{}}x\\}.\end{split}$

######  Remark 5.5.

(1) Because of (3.1.2), the above definition is independent of the choice of
trivializations. However, $\alpha$ may depend on $\mu$. (2) For technical
reasons (see Lemma 5.12), we require $\mu$ to be a (global) resolution of $X$
instead of a neighborhood of $x$.

When $X$ is smooth, we certainly have
$\mathcal{G}_{m}(h)\supset\mathcal{I}(h)$. Furthermore, if $m=1$, then
$\mathcal{G}_{1}(h)=\mathcal{I}(h)$ by the change-of-variables formula (see
(5.4.2)). The following example shows that the inclusion
$\mathcal{G}_{m}(h)\supset\mathcal{I}(h)$ may be strict.

######  Example 5.6.

Let $x,y$ be the coordinates of $\mathbb{C}^{2}$. Let $h=e^{-\varphi}$ with
$\varphi\coloneqq\frac{1}{2}\log(|x|^{4}+|y|^{4})$ be the metric for the
trivial divisor. Then $1\not\in\mathcal{I}(h)$. We claim that
$1\in\mathcal{G}_{2}(h)$. In fact, let $\mu:W\to\mathbb{C}^{2}$ be the blow-up
of the origin. Then
$W=\\{(x,y)\times[u:v]\in\mathbb{C}^{2}\times\mathbb{P}^{1}\mid xv=yu\\}$. It
is covered by two pieces $U_{1}=\\{(x,y,z)\in\mathbb{C}^{2}\mid xz=y)\\}$ and
$U_{2}=\\{(x,y,w)\in\mathbb{C}^{2}\mid x=yw)\\}$. By the symmetry of
$\varphi$, it suffices to consider the $L_{\rm loc}^{1}$-property on one
piece, say $U_{1}$. Choose $x,z$ as local coordinates on $U_{1}$. By
$dy=xdz+zdx$, we have

$xdx\wedge dz=d\mu^{*}x\wedge d\mu^{*}y.$

Thus $J(\mu)=x$. Then locally on $U_{1}$,

$\begin{split}&\int 1\cdot|J(\mu)|^{4}e^{-2\mu^{*}\varphi}~{}dV_{W}\\\
=&\int\frac{|x|^{4}}{|x|^{4}+|y|^{4}}~{}dV_{W}=\int\frac{1}{1+|z|^{4}}~{}dV_{W}<\infty.\end{split}$

It is natural to ask the following question:

######  Question 5.7.

Is $\mathcal{G}_{m}(h)$ a coherent sheaf?

To study the above question, it seems that the $L^{2}$-extension theorem also
needs to be generalized in the bimeromorphic setting.

###  5.4. Proof of Theorem 1.1, Theorem 1.2 and Corollary 1.3

Let $\pi:X\to\Delta$ be a projective contraction from a
$\mathbb{Q}$-Gorenstein complex space $X$ to the disc $\Delta$. Let $L$ be a
Cartier divisor on $X$. Assume that $X_{0}\subset X$ is the fiber over
$0\in\Delta$, and it is a normal complex subspace. As $X_{0}$ is a Cartier
divisor, by adjunction formula,

$(K_{X}+X_{0})|_{X_{0}}\sim_{\mathbb{Q}}K_{X_{0}}.$

Thus $X_{0}$ is also $\mathbb{Q}$-Gorenstein, and
$(mK_{X}+L)|_{X_{0}}\sim_{\mathbb{Q}}mK_{X_{0}}+L|_{X_{0}}$ as $X_{0}$ is
linearly equivalent to $0$ on $X$. Note that we do not assume that
$mK_{X_{0}}$ is Cartier. We explain the meaning of extending sections from
$X_{0}$ to $X$.

######  Lemma 5.8.

Under the above notation and assumptions, there exists a natural map

$H^{0}(X,\mathcal{O}(mK_{X}+L))\to
H^{0}(X_{0},\mathcal{O}(mK_{X_{0}}+L|_{X_{0}})).$

###### Proof.

Let $V_{0}\subset X_{0}$ be the smooth locus of $X_{0}$. As $X_{0}$ is a
Cartier divisor, there exists a neighborhood $V\supset V_{0}$ such that
$V\subset X$ is a smooth open variety. Let $\tilde{V}_{0}=X_{0}\cap{V}$, then
$\tilde{V}_{0}\supset V_{0}$.

Let $j:\tilde{V}_{0}\to V$ be the closed embedding. Because $V$ is smooth, we
have

$\mathcal{O}_{V}(mK_{X}+L)\to j_{*}\mathcal{O}_{\tilde{V}_{0}}(mK_{X}+L),$

and
$\mathcal{O}_{\tilde{V}_{0}}(mK_{X}+L)\simeq\mathcal{O}(mK_{\tilde{V}_{0}}+L|_{\tilde{V}_{0}})$
as $\tilde{V}_{0}$ is linearly equivalent to $0$ on $V$. As
$\operatorname{codim}_{X_{0}}(X_{0}\backslash V_{0})\geq 2$ and
$\tilde{V}_{0}\supset V_{0}$,

$H^{0}(\tilde{V}_{0},\mathcal{O}(mK_{\tilde{V}_{0}}+L|_{\tilde{V}_{0}}))\simeq
H^{0}(X_{0},\mathcal{O}(mK_{X_{0}}+L|_{X_{0}})).$

Therefore, there exist natural maps

$\begin{split}&H^{0}(X,\mathcal{O}(mK_{X}+L))\to
H^{0}(V,\mathcal{O}_{V}(mK_{X}+L))\\\
\to&H^{0}(\tilde{V}_{0},\mathcal{O}(mK_{\tilde{V}_{0}}+L|_{\tilde{V}_{0}}))\to
H^{0}(X_{0},\mathcal{O}(mK_{X_{0}}+L|_{X_{0}})).\end{split}$

∎

Assume that $X$ has canonical singularities. Let $\mu:W\to X$ be a log
resolution of $(X,X_{0})$. Let $Y_{0}$ be the strict transform of $X_{0}$.
Suppose that $X_{0}$ has canonical singularities. Let
$\mu_{0}\coloneqq\mu|_{Y_{0}}:Y_{0}\to X_{0}$ and
$E_{0}=\mu_{0}^{*}K_{X_{0}}-K_{Y_{0}}$. The following is the key extension
lemma.

######  Lemma 5.9.

Under the above notation and assumptions. Let $L$ be a Cartier divisor on $X$
with a non-negative metric $h$. Suppose that $h|_{X_{0}}$ is well-defined.
Then for each $m\in\mathbb{N}$, any section $s\in
H^{0}(X_{0},\mathcal{O}_{X_{0}}(mK_{X_{0}}+L|_{X_{0}}))$ such that

$\mu_{0}^{*}s\in
H^{0}\left(Y_{0},\mathcal{O}(mK_{Y_{0}}+\mu_{0}^{*}(L|_{X_{0}}))\otimes\mathcal{I}(\mu_{0}^{*}(h|_{X_{0}}))\right)$

extends over $X$ (i.e. $s$ has a preimage in $H^{0}(X,\mathcal{O}(mK_{X}+L))$
under the natural map in Lemma 5.8).

###### Proof.

Let $L_{0}=L|_{X_{0}},h_{0}=h|_{X_{0}}$. By assumption (see Definition 3.3),
$\mu^{*}h$ has non-negative curvature current. Let $\mu^{*}X_{0}=Y_{0}+\Theta$
and $\Theta_{0}=\Theta|_{Y_{0}}$. Then $\mu^{*}h\hbar_{m\Theta}$ has non-
negative curvature current by $\Theta\geq 0$.

By Proposition 5.1 and (5.2.1),

$\begin{split}&H^{0}(Y_{0},\mathcal{O}(mK_{Y_{0}}+\mu_{0}^{*}L_{0})\otimes\mathcal{I}(\mu_{0}^{*}h_{0}))\\\
\subset&H^{0}(Y_{0},\mathcal{O}(mK_{Y_{0}}+m\Theta_{0}+\mu_{0}^{*}L_{0})\otimes\mathcal{I}(\mu_{0}^{*}h_{0}\hbar_{m\Theta_{0}}))\\\
\simeq&H^{0}(Y_{0},\mathcal{O}(mK_{W}|_{Y_{0}}+\mu_{0}^{*}L_{0})\otimes\mathcal{I}(\mu_{0}^{*}h_{0}\hbar_{m\Theta_{0}})).\end{split}$

By Lemma 5.2, $\mu_{0}^{*}s$ extends over $W$. That is, there exists
$\tilde{\omega}\in H^{0}(W,\mathcal{O}(mK_{W}+\mu^{*}L))$ such that
$\tilde{\omega}|_{Y_{0}}=\mu_{0}^{*}s$.

We have the following diagram

(5.4.1)
$\textstyle{H^{0}(W,\mathcal{O}(mK_{W}+\mu^{*}L))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{H^{0}(Y_{0},\mathcal{O}(mK_{W}|_{Y_{0}}+\mu_{0}^{*}L_{0}))}$$\textstyle{H^{0}(X,\mathcal{O}(mK_{X}+L))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{b}$$\scriptstyle{d}$$\textstyle{H^{0}(X_{0},\mathcal{O}(mK_{X_{0}}+L_{0})),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$

where $a$ is the restriction map, $d$ comes from Lemma 5.8, $b$ comes from
that $X$ has canonical singularities, and $c$ is an inclusion which comes from
(5.2.1) and $X_{0}$ has canonical singularities. The desired extension follows
from the diagram chasing once we know that the diagram is commutative.

To check the commutativity of the diagram, we follow the notation in the
discussion before Lemma 5.2. Besides, it is enough to work locally in the
smooth loci of $X_{0},Y_{0}$ and $W$. Then

$b:\mathcal{O}_{X}(mK_{X}+L)\to\mathcal{O}_{W}(m\mu^{*}K_{X}+\mu^{*}L)\to\mathcal{O}(mK_{W}+\mu^{*}L)$

is given by

$\beta(\omega_{X_{0}}\wedge dt)^{\otimes
m}\mapsto\mu^{*}\beta(\mu^{*}\omega_{X_{0}}\wedge dt)^{\otimes
m}\mapsto\mu^{*}\beta(J(\mu_{0})\omega_{Y_{0}}\wedge dt)^{\otimes m},$

where $\beta$ is a local section of $L$, $\omega_{X_{0}}$ and $\omega_{Y_{0}}$
are local generators of $\mathcal{O}(K_{X_{0}})$ and $\mathcal{O}(K_{Y_{0}})$
respectively, and $\mu^{*}\omega_{X_{0}}=J(\mu_{0})\omega_{Y_{0}}$. Note that
the $t$ in each term is the corresponding pull-back of the coordinate $t$ on
$\Delta$.

The map

$\begin{split}c:\mathcal{O}(mK_{X_{0}}+L_{0})\to\mathcal{O}(m\mu_{0}^{*}K_{X_{0}}+\mu_{0}^{*}L_{0})&\hookrightarrow\mathcal{O}(mK_{Y_{0}}+\mu_{0}^{*}L_{0})\\\
&\to\mathcal{O}(mK_{W}|_{Y_{0}}+\mu_{0}^{*}L_{0})\end{split}$

is given by

$\beta_{0}\omega_{X_{0}}^{\otimes
m}\mapsto\mu_{0}^{*}\beta_{0}(\mu_{0}^{*}\omega_{X_{0}})^{\otimes
m}\mapsto\mu_{0}^{*}\beta_{0}(J(\mu_{0})\omega_{Y_{0}})^{\otimes
m}\mapsto\mu_{0}^{*}\beta_{0}J(\mu_{0})^{m}(\omega_{Y_{0}}\wedge
dt|_{Y_{0}})^{\otimes m},$

where $\beta_{0}$ is a local section of $L_{0}$ and
$\mu_{0}^{*}\omega_{X_{0}}=J(\mu_{0})\omega_{Y_{0}}$. The last map is (5.2.4).

For the same local section $\beta$ of $L$, the map $a$ is given by

$a:\mu^{*}\beta(\omega_{Y_{0}}\wedge dt)^{\otimes
m}\mapsto(\mu^{*}\beta|_{Y_{0}})(\omega_{Y_{0}}\wedge dt|_{Y_{0}})^{\otimes
m}.$

(Strictly speaking, locally around $Y_{0}$, a section of
$\mathcal{O}(mK_{W}+\mu^{*}L)$ should be represented by
$\mu^{*}\beta(\omega_{Y_{0}}\wedge dy_{0})^{\otimes m}$. Note
$\omega_{Y_{0}}\wedge dy_{0}=\frac{1}{\xi}\omega_{Y_{0}}\wedge dt$ by
(5.2.6).)

Finally, the map $d$ is given by

$d:\beta(\omega_{X_{0}}\wedge dt)^{\otimes
m}\mapsto\beta|_{Y_{0}}\omega_{X_{0}}^{\otimes m}.$

Thus the diagram (5.4.1) commutes. ∎

######  Remark 5.10.

Lemma 5.9 is enough to show the desired extension result Theorem 1.2. The
reason is that for any section of
$H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}})))$, the integrability
requirement is always satisfied when taking the resolution $W=X^{\prime}$ (see
(5.4.7), the key is that after taking a resolution, we have an extra weight
$\varphi_{-mB_{0}}$).

######  Lemma 5.11.

Let $X$ be a $\mathbb{Q}$-Gorenstein complex space. Let $g:V\to X$ and $u:W\to
V$ be resolutions. Set $f=g\circ u:W\to X$. Let $L$ be a Cartier divisor on
$X$ with a metric $h$. For any $m\in\mathbb{N}$ and $s\in
H^{0}(X,\mathcal{O}(mK_{X}+L))$, if $g^{*}s\in
H^{0}(V,\mathcal{O}(mK_{V}+g^{*}L)\otimes\mathcal{I}(g^{*}h))$, then
$f^{*}s\in H^{0}(W,\mathcal{O}(mK_{W}+f^{*}L)\otimes\mathcal{I}(f^{*}h))$.

###### Proof.

By $V$ smooth and $g^{*}s\in H^{0}(V,\mathcal{O}(mK_{V}+g^{*}L))$, we have
$f^{*}s\in H^{0}(W,\mathcal{O}(mK_{W}+f^{*}L))$. Hence we only need to check
the integrability.

Let $v_{i},i=1,\ldots,n$ and $w_{i},i=1,\ldots,n$ be local coordinates of $V$
and $W$ respectively. Assume that $g^{*}s=\sigma(dv_{1}\wedge\cdots\wedge
dv_{n})^{\otimes m}$ with $\sigma$ a local section of $g^{*}L$, then locally
we have

$\int\|\sigma\|^{2}_{g^{*}h}~{}dV_{V}<\infty.$

By the change-of-variables formula, this is the same as

(5.4.2) $\int\|u^{*}\sigma\|^{2}_{u^{*}g^{*}h}|J(u)|^{2}~{}dV_{W}<\infty,$

where $J(u)$ is the local equation of the $u$-exceptional divisor
$K_{W}-u^{*}K_{V}$.

On the other hand,

$f^{*}s=u^{*}(\sigma(dv_{1}\wedge\cdots\wedge dv_{n})^{\otimes
m})=(u^{*}\sigma)J(u)^{2m}(dw_{1}\wedge\cdots\wedge dw_{n})^{\otimes m}.$

Thus $f^{*}s\in H^{0}(W,\mathcal{O}(mK_{W}+f^{*}L)\otimes\mathcal{I}(f^{*}h))$
means that locally

$\int\|u^{*}\sigma\|^{2}_{u^{*}g^{*}h}|J(u)|^{2m}~{}dV_{W}<\infty.$

As $m\geq 1$ and $J(u)$ is holomorphic, the claim follows. ∎

######  Lemma 5.12.

Let $X$ be a compact complex space with canonical singularities. Let $L$ be a
Cartier divisor on $X$ with a metric $h$. For some $m\in\mathbb{N}$, $s\in
H^{0}(X,\mathcal{O}_{X}(mK_{X}+L)\otimes\mathcal{G}_{m}(h))$ if and only if
there exists a resolution $\mu:W\to X$ such that

$\mu^{*}s\in
H^{0}(W,\mathcal{O}_{W}(mK_{W}+\mu^{*}L)\otimes\mathcal{I}(\mu^{*}h)).$

###### Proof.

The sufficient part follows from the definition. In the following, we show the
necessary part.

Let $\varphi$ be the local weight for $h$. By $X$ compact, there are open sets
$U_{j},1\leq j\leq k$ and projective resolutions $\mu_{j}:W_{j}\to X,1\leq
j\leq k$ such that $s\in H^{0}(U_{j},\mathcal{O}_{X}(mK_{X}+L))$ and

(5.4.3) $|\mu^{*}_{j}\alpha|^{2}|J(\mu_{j})|^{2m}e^{-2\mu_{j}^{*}\varphi}\in
L^{1}_{\rm loc}(\mu_{j}^{-1}(U_{j})),$

where $s=\alpha\cdot(\omega_{X})^{\otimes m}$ with $\omega_{X}$ a local
generator for $\mathcal{O}(K_{X})$ and $\alpha$ a local section of $L$ (see
Remark 5.5 (2)). More precisely, we have

$\mu_{j}^{*}s=(\mu_{j}^{*}\alpha)\cdot J(\mu_{j})^{m}(\omega_{W_{j}})^{\otimes
m},$

where $(\mu_{j}^{*}\alpha)\cdot J(\mu_{j})^{m}$ is first defined on the smooth
locus, but it extends as a local section of $\mu_{j}^{*}L$ on $W_{j}$ as $X$
has canonical singularities.

Hence, (5.4.3) is the same as

(5.4.4) $\mu_{j}^{*}s\in
H^{0}(\mu^{-1}_{j}(U_{j}),\mathcal{O}_{W_{j}}(mK_{W_{j}}+\mu_{j}^{*}L)\otimes\mathcal{I}(\mu_{j}^{*}h)).$

Let $\mu:W\to X$ be a resolution such that $\mu$ factors through each
$\mu_{j},1\leq j\leq k$. We claim that $\mu$ satisfies the desired property.
By construction, the natural morphism
$\nu_{j}:\mu^{-1}(U_{j})\to\mu^{-1}_{j}(U_{j})$ is a resolution. By (5.4.4)
and Lemma 5.11,

$\nu_{j}^{*}(\mu_{j}^{*}s)\in
H^{0}(\mu^{-1}(U_{j}),\mathcal{O}_{W}(mK_{W}+\mu^{*}L)\otimes\mathcal{I}(\mu^{*}h)).$

As $\nu_{j}^{*}(\mu_{j}^{*}s)=(\mu^{*}s)|_{\mu^{-1}(U_{j})}$, the claim
follows. ∎

######  Lemma 5.13.

Let $X$ be a complex space, and $X_{0}\subset X$ be a reduced irreducible
compact complex subspace of codimension $1$. Suppose that
$\nu_{0}:\tilde{X}_{0}\to X_{0}$ is a proper modification. Then there exist
smooth complex spaces $Y_{0}\subset Y$ and resolutions $\mu:Y\to
X,\tau_{0}:Y_{0}\to\tilde{X}_{0}$ such that
$\mu|_{Y_{0}}=\nu_{0}\circ\tau_{0}$.

###### Proof.

By Hironaka’s Chow lemma (see [Pet94a, Chapter VII, Theorem 2.8, Corollary
2.9]), there exist a blow-up $\sigma_{0}:Z_{0}\to X_{0}$ of an ideal sheaf
$\mathfrak{I}_{0}$ on $X_{0}$ and a proper modification
$\mu_{0}:Z_{0}\to\tilde{X}_{0}$ such that $\sigma_{0}=\nu_{0}\circ\mu_{0}$.
Replacing $\nu_{0}$ by $\sigma_{0}$, we can assume that $\nu_{0}$ is the blow-
up of the ideal sheaf $\mathfrak{I}_{0}$ on $X_{0}$. Let $\mathfrak{I}$ be the
kernel of $\mathcal{O}_{X}\to\mathcal{O}_{X_{0}}/\mathfrak{I}_{0}$. Let
$X^{\prime}=\operatorname{Proj}_{X}\oplus_{i=0}^{\infty}\mathfrak{I}^{i}\to X$
be the blow-up of $\mathfrak{I}$. Then there exist embeddings
$\tilde{X}_{0}\subset X_{0}\times_{X}X^{\prime}\subset X^{\prime}$. Let $Y\to
X^{\prime}$ be a log resolution of $(X^{\prime},\tilde{X}_{0})$ with $Y_{0}$
the strict transform of $\tilde{X}_{0}$. Then the corresponding morphisms
satisfy the claim. ∎

###### Proof of Theorem 1.1.

By Lemma 5.12, for a fixed $s\in
H^{0}(X_{0},\mathcal{O}_{X_{0}}(mK_{X_{0}}+L|_{X_{0}})\otimes\mathcal{G}_{m}(h|_{X_{0}}))$,
there is a resolution $\nu_{0}:\tilde{X}_{0}\to X_{0}$ such that
$\nu_{0}^{*}s\in
H^{0}(\tilde{X}_{0},\mathcal{O}_{\tilde{X}_{0}}(mK_{\tilde{X}_{0}}+\nu_{0}^{*}(L|_{X_{0}}))\otimes\mathcal{I}(\nu_{0}^{*}(h|_{X_{0}})))$.
By Lemma 5.13, there are smooth complex spaces $Y_{0}\subset Y$ and
resolutions $\mu:Y\to X,\tau_{0}:Y_{0}\to\tilde{X}_{0}$ such that
$\mu_{0}\coloneqq\mu|_{Y_{0}}=v_{0}\circ\tau_{0}$. By Lemma 5.11,

$\mu_{0}^{*}s\in
H^{0}(Y_{0},\mathcal{O}_{Y_{0}}(mK_{Y_{0}}+\mu_{0}^{*}(L|_{X_{0}}))\otimes\mathcal{I}(\mu_{0}^{*}(h|_{X_{0}}))).$

Then the claim follows from Lemma 5.9. ∎

######  Remark 5.14.

[Tak07, Theorem 1] does not assume that $X_{0}$ has canonical singularities.
This is because the $m$-genus in [Tak07] is defined by using its smooth model
which does not coincide with our definition (take $L=0$) when the
singularities are worse than the canonical singularities.

Using this result, we show the extension theorem for g-pairs with abundant nef
part.

######  Remark 5.15.

In Theorem 1.2, $mK_{X_{0}}$ is not assumed to be Cartier. Instead, it is just
a Weil divisor on $X_{0}$. A priori, $K_{X_{0}}$ could be even not
$\mathbb{Q}$-Cartier. But if there exists $m$ such that $mM|_{X_{0}}$ is
Cartier, then $K_{X_{0}}$ is $\mathbb{Q}$-Cartier as $K_{X_{0}}+M_{0}$ is
$\mathbb{Q}$-Cartier (this is included in the definition of g-canonical
singularities).

###### Proof of Theorem 1.2.

As $\pi_{*}\mathcal{O}_{X}(mK_{X}+mM)$ is a coherent sheaf on $\Delta$, by
Cartan’s Theorem A, it suffices to show the extension after shrinking
$\Delta$. Hence, by [Kaw99a, Main Theorem], we can assume that $X$ has
canonical singularities.

Let $L=mM$ and $M^{\prime}$ be the nef$/\Delta$ $\mathbb{Q}$-divisor on
$X^{\prime}$ such that $f_{*}M^{\prime}=M$ and satisfying Definition 2.3. To
apply Theorem 1.1, it suffices to construct a metric $h$ for $L$ satisfying
conditions of Theorem 1.1 and show

$H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}})))=H^{0}(X_{0},\mathcal{O}_{X_{0}}(mK_{X_{0}}+L|_{X_{0}})\otimes\mathcal{G}_{m}(h|_{X_{0}})).$

By the adjunction formula (see (2.3.1)), we have a g-lc pair
$(X_{0},B_{0}+M_{0})$ such that $K_{X_{0}}+B_{0}+M_{0}=K_{X_{0}}+M|_{X_{0}}$.
As $(X_{0},B_{0}+M_{0})$ has g-canonical singularities, $B_{0}=0$ and thus
$M_{0}=M|_{X_{0}}$. Then

$K_{\tilde{X}_{0}}+D_{0}+M^{\prime}_{0}=f_{0}^{*}(K_{X_{0}}+M_{0}),$

where $M_{0}^{\prime}\coloneqq M^{\prime}|_{\tilde{X}_{0}}$ and $D_{0}\leq 0$
is an $f_{0}$-exceptional divisor. Besides,

$K_{\tilde{X}_{0}}+B_{0}=f_{0}^{*}K_{X_{0}},$

where $B_{0}$ is an $f_{0}$-exceptional divisor. Because $(X_{0},M_{0})$ has
g-canonical singularities and $K_{X_{0}}$ is $\mathbb{Q}$-Cartier, $X_{0}$
also has canonical singularities, and thus $B_{0}\leq 0$. By Lemma 4.1,
$M^{\prime}+\Upxi=f^{*}M$ with $\Upxi\geq 0$. We have

$M_{0}^{\prime}+\Upxi_{0}=f_{0}^{*}(M_{0}),$

where $\Upxi_{0}\coloneqq\Upxi|_{X_{0}}$. Combining the above equations, we
have

(5.4.5) $D_{0}=B_{0}+\Upxi_{0}.$

Shrinking $\Delta$ further, let $h_{k}$ be a metric for $M$ as in Lemma 4.4
and set $h_{k,0}=h_{k}|_{X_{0}}$. For a fixed $m\in\mathbb{N}$, we claim that
there exists $k\gg 1$ such that

$H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}})))=H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}}))\otimes\mathcal{G}_{m}(h_{k,0}^{m})).$

Then the theorem follows from Theorem 1.1.

For a section $s\in
H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}})))$, assume that
$\theta:\mathcal{O}(L)|_{U}\simeq\mathcal{O}_{U}$ is a trivialization and on
$U_{\rm reg}$, $s$ can be written as $\alpha(dx_{1}\wedge\cdots\wedge
dx_{n-1})^{\otimes m}$. By Lemma 5.12, it suffices to show

(5.4.6)
$\int\|f_{0}^{*}\alpha\|^{2}_{f_{0}^{*}(h^{m}_{k,0})}|J(f_{0})|^{2m}~{}dV_{{\tilde{X}_{0}}}<\infty.$

(5.4.6) holds for $k\gg 1$ by Lemma 4.4 (4) and (5.4.5). In fact, first note
that $J(f_{0})$ is the local equation of $-B_{0}$, hence
$|J(f_{0})|=e^{-\varphi_{B_{0}}}$, where $\varphi_{B_{0}}$ is the local weight
for $\hbar_{B_{0}}$. Thus

$\|f_{0}^{*}\alpha\|^{2}_{f_{0}^{*}(h_{k,0}^{m})}|J(f_{0})|^{2m}=\|f_{0}^{*}(\theta(\alpha))\|^{2}\cdot
e^{-2mf_{0}^{*}\varphi_{k,0}}\cdot e^{-2m\varphi_{B_{0}}},$

where $\varphi_{k,0}$ is the local weight for $h_{k,0}$. By Lemma 4.4 (4),

$f_{0}^{*}\varphi_{k,0}+\frac{1}{k}\varphi_{E_{0}}\approx\frac{1}{k}\varphi_{F_{0}}+\varphi_{\Upxi_{0}}.$

By (5.4.5), $mB_{0}=mD_{0}-m\Upxi_{0}$. Thus

(5.4.7) $\begin{split}-mf_{0}^{*}\varphi_{k,0}-m\varphi_{B_{0}}&\approx
m(-\frac{1}{k}\varphi_{F_{0}}-\varphi_{\Upxi_{0}}+\frac{1}{k}\varphi_{E_{0}})-m\varphi_{D_{0}}+m\varphi_{\Upxi_{0}}\\\
&\approx
m(-\frac{1}{k}\varphi_{F_{0}}+\frac{1}{k}\varphi_{E_{0}}-\varphi_{D_{0}}).\end{split}$

Note $-D_{0}\geq 0$, and thus for a fixed $m$, we can take $k\gg 1$ such that
the integrability of (5.4.6) holds. In fact, it is enough to choose $k$ such
that

$\frac{2m}{k}\nu(\Theta_{\hbar_{F_{0}}}(F_{0}),y)<1\text{~{}for
all~{}}y\in\tilde{X}_{0},$

where $\nu(\Theta_{h_{F_{0}}}(F_{0}),y)$ is the Lelong number of the curvature
current at $y$. ∎

The following remark explains the crucial point of Theorem 1.2.

######  Remark 5.16.

Even in the smooth case, for the metric $h=h_{k}$ constructed in the proof of
Theorem 1.2, we may have

$H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}})))\subsetneqq
H^{0}(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+M|_{X_{0}}))\otimes\mathcal{I}(h|_{X_{0}})).$

Hence [Pău07, Theorem 1] does not apply. In fact, under the notation of the
proof of Theorem 1.2, assuming that $X_{0}$ is smooth, by the change-of-
variables formula, we have

$\int\|\alpha\|_{h_{0}^{m}}^{2}~{}dV_{{X_{0}}}=\int\|f_{0}^{*}\alpha\|_{f_{0}^{*}h_{0}^{m}}^{2}|J(f_{0})|^{2}~{}dV_{{\tilde{X}_{0}}},$

where $J(f_{0})$ is the local equation of
$K_{\tilde{X}_{0}}-f_{0}^{*}K_{X_{0}}=-B_{0}\geq 0$ (c.f. (5.4.6)). If
$\tilde{\theta}:\mathcal{O}(f_{0}^{*}(L|_{X_{0}}))|_{f_{0}^{-1}(U)}\simeq\mathcal{O}_{f_{0}^{-1}(U)}$
is a trivialization, then it becomes

$\int|\tilde{\theta}(f_{0}^{*}\alpha)|^{2}e^{-2mf_{0}^{*}\varphi_{k,0}}e^{-2\varphi_{B_{0}}}~{}dV_{{\tilde{X}_{0}}}.$

However (c.f. (5.4.7)),

$\begin{split}-mf_{0}^{*}\varphi_{k,0}-\varphi_{B_{0}}&\approx
m(-\frac{1}{k}\varphi_{F_{0}}-\varphi_{\Upxi_{0}}+\frac{1}{k}\varphi_{E_{0}})-\varphi_{D_{0}}+\varphi_{\Upxi_{0}}\\\
&\approx
m(-\frac{1}{k}\varphi_{F_{0}}+\frac{1}{k}\varphi_{E_{0}})-\varphi_{D_{0}}-(m-1)\varphi_{\Upxi_{0}}.\end{split}$

Note $\Upxi_{0}\geq 0$, for $m\gg 1$, we do not have the integrability.

The new extension theorem works in this setting because the integrability
requirement is for

$\int\|f_{0}^{*}\alpha\|^{2}_{f_{0}^{*}(h_{k,0}^{m})}|J(f_{0})|^{2m}~{}dV_{{\tilde{X}_{0}}}.$

The extra $|J(f_{0})|^{2m}$ makes the integral finite.

###### Proof of Corollary 1.3.

If $\mathcal{F}$ is a sheaf of $\mathcal{O}_{\Delta}$-module and $t\in\Delta$
is a point, then
$\mathcal{F}\otimes\mathbb{C}(t)=\mathcal{F}_{t}/m_{t}\mathcal{F}_{t}$, where
$m_{t}\subset\mathcal{O}_{\Delta,t}$ is the maximal ideal corresponding to $t$
and $\mathbb{C}(t)\coloneqq\mathcal{O}_{\Delta,t}/m_{t}$. The following
argument is similar to [Tak07, Proof of Theorem 1.1].

Let $L=mM$ and $L_{t}=L|_{X_{t}}$. By the same argument as Lemma 5.8, there is
a natural map

$\pi_{*}\mathcal{O}_{X}(mK_{X}+L)\to
H^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t}))$

which is surjective by Theorem 1.2. For $\alpha\in m_{t}$ and
$\sigma\in(\pi_{*}\mathcal{O}_{X}(mK_{X}+L))_{t}$,
$(\alpha\otimes\sigma)|_{X_{t}}=0$, thus the above map induces the surjective
map

(5.4.8) $\pi_{*}\mathcal{O}_{X}(mK_{X}+L)\otimes\mathbb{C}(t)\to
H^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t})).$

We show that this map is also injective. Let $U=X-{\rm Sing}X_{t}$. We claim
that the following natural maps give a short exact sequence

(5.4.9)
$\begin{split}0&\to\mathcal{O}_{U}(mK_{X}+(m-1)X_{t}+L)\to\mathcal{O}_{U}(mK_{X}+mX_{t}+L)\\\
&\to\iota_{*}\mathcal{O}_{U\cap X_{t}}(mK_{X_{t}}+L_{t})\to 0,\end{split}$

where $\iota:U\cap X_{t}\to U$. It is enough to check the exactness on stalks.
Let $z\in U$. If $z\not\in X_{t}$, then $\iota_{*}\mathcal{O}_{U\cap
X_{t}}(mK_{X_{t}}+L_{t})_{z}=0$ and
$\mathcal{O}_{U}(mK_{X}+(m-1)X_{t}+L)_{z}\simeq\mathcal{O}_{U}(mK_{X}+mX_{t}+L)_{z}$
as we can locally invert the defining equation of $X_{t}$. If $z\in U\cap
X_{t}$, then by the choice of $U$, $z$ is a smooth point of $X_{t}$. As
$X_{t}$ is Cartier, there is a smooth open set of $U$ which contains $z$. Then
the exactness follows. Let $j:U\hookrightarrow X$. Pushing forward (5.4.9), we
have

$\begin{split}0&\to\mathcal{O}_{X}(mK_{X}+(m-1)X_{t}+L)\to\mathcal{O}_{X}(mK_{X}+mX_{t}+L)\\\
&\to j_{*}\iota_{*}\mathcal{O}_{U\cap X_{t}}(mK_{X_{t}}+L_{t}).\end{split}$

As $\operatorname{codim}_{X_{t}}(X_{t}-U\cap X_{t})\geq 2$, the natural map

$\eta_{*}\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t})\to
j_{*}\iota_{*}\mathcal{O}_{U\cap X_{t}}(mK_{X_{t}}+L_{t})$

is an isomorphism, where $\eta:X_{t}\to X$. In conclusion, there is an exact
sequence

$0\to\mathcal{O}_{X}(mK_{X}+(m-1)X_{t}+L)\rightarrow\mathcal{O}_{X}(mK_{X}+mX_{t}+L)\to\eta_{*}\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t}).$

Pushing forward by $\pi$ and taking the stalk at $t$, we have

$\begin{split}0&\to\pi_{*}\mathcal{O}_{X}(mK_{X}+(m-1)X_{t}+L)_{t}\rightarrow\pi_{*}\mathcal{O}_{X}(mK_{X}+mX_{t}+L)_{t}\\\
&\to H^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t})).\end{split}$

Let $\mathcal{F}\coloneqq\pi_{*}\mathcal{O}_{X}(mK_{X}+(m-1)X_{t}+L)$ and
$\mathcal{G}\coloneqq\pi_{*}\mathcal{O}_{X}(mK_{X}+mX_{t}+L)$, then

$\begin{split}&\operatorname{Ker}(\pi_{*}\mathcal{O}_{X}(mK_{X}+mX_{t}+L)\otimes\mathbb{C}(t)\to
H^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t})))\\\
\simeq&(\mathcal{F}_{t}+m_{t}\mathcal{G}_{t})/m\mathcal{G}_{t}.\end{split}$

We claim that $(\mathcal{F}_{t}+m_{t}\mathcal{G}_{t})/m_{t}\mathcal{G}_{t}=0$.
In fact, as $m_{t}=(z-t)\mathcal{O}_{\Delta,t}$, $\pi^{*}(z-t)$ is the
defining equation of $X_{t}$, we have $\mathcal{F}_{t}\subset
m_{t}\mathcal{G}_{t}$. Because $X_{t}\sim 0$, we have
$\mathcal{O}_{X}(mK_{X}+(m-1)X_{t}+L)\simeq\mathcal{O}_{X}(mK_{X}+L)$. Thus
(5.4.8) is an isomorphism.

Note that $\pi_{*}\mathcal{O}_{X}(mK_{X}+L)$ is a coherent sheaf. By the upper
semi-continuity of

$\dim_{\mathbb{C}}(\pi_{*}\mathcal{O}_{X}(mK_{X}+L)\otimes\mathbb{C}(t))$

and the isomorphism (5.4.8),
$h^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t}))$ is upper semi-continuous.
Fix a $t_{0}\in\Delta$, any section of
$H^{0}(X_{t_{0}},\mathcal{O}_{X_{t_{0}}}(mK_{X_{t_{0}}}+L_{t_{0}}))$ extends
over $X$ by Theorem 1.2. Hence

$h^{0}(X_{t_{0}},\mathcal{O}_{X_{t_{0}}}(mK_{X_{t_{0}}}+L_{t_{0}}))\leq
h^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t}))$

for a general $t\in\Delta$. Thus,
$h^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}+L_{t}))$ must be a constant for
all $t\in\Delta$. ∎

###  5.5. Further discussions

The following example shows that nefness of $M$ along does not guarantee the
invariance of plurigenera.

######  Example 5.17.

Let $A/\mathbb{C}$ be an abelian variety and $A^{\vee}$ be its dual abelian
variety. If $\operatorname{Pic}^{0}(A)$ is the identity component of the
Picard variety of $A$, then $\operatorname{Pic}^{0}(A)=A^{\vee}$. Let
$\mathcal{P}$ be the Poincaré bundle on $A\times A^{\vee}$. Suppose that $0\in
A^{\vee}$ corresponds to $\mathcal{P}_{0}\simeq\mathcal{O}_{A}$. Let
$0\in\Delta\subset A^{\vee}$ be a disc containing $0$. Let $X=A\times\Delta$
and $\mathcal{P}_{\Delta}\coloneqq\mathcal{P}|_{X}$ be a line bundle on $X$.
$\mathcal{P}_{\Delta}$ is nef over $\Delta$ as $\mathcal{P}_{t}$ is
numerically trivial for each $t\in\Delta$. Moreover,
$\mathcal{O}(K_{X})=\mathcal{O}_{X}$. Note that for an abelian variety, a line
bundle $\mathcal{L}\in\operatorname{Pic}^{0}(A)$ has global sections if and
only if $\mathcal{L}\simeq\mathcal{O}_{A}$ (see [Mum70, Page 76, (vii)]).
Therefore, for each $m\in\mathbb{N}$,

$h^{0}(X_{X_{t}},\mathcal{O}_{X_{t}}(mK_{X_{t}})\otimes\mathcal{P}_{\Delta}^{\otimes
m}|_{X_{t}})=\begin{cases}1,&\text{ if~{} }t=0,\\\ 0,&\text{ if~{}
}t\in\Delta-\\{0\\}.\end{cases}$

Next, recall that for a non-smooth family of varieties, we have

######  Theorem 5.18 ([Tak07, Theorem 1.1]).

Let $\pi:X\to C$ be a proper surjective algebraic morphism with connected
fibers from a complex variety $X$ to a smooth curve $C$. Assume that every
fiber $X_{t}=\pi^{-1}(t)$ has only canonical singularities. Then
$h^{0}(X_{t},\mathcal{O}_{X_{t}}(mK_{X_{t}}))$ is independent of $t\in C$ for
any positive integer $m$.

Such result has been established by [Kaw99a, Theorem 6] under the additional
assumption that each fiber is of general type. On the other hand, for klt
singularities, local sections of fibers may not lift to global sections (see
[Kaw99b, Example 4.3]). For g-pairs, the additional nef part introduces the
singularities even if each $X_{t}$ is smooth.

The above discussions show that both assumptions on the abundant nef parts and
the g-canonical singularities are indispensable for Theorem 1.2 and Corollary
1.3.

## References

  * [Amb05] Florin Ambro.  The moduli b-divisor of an lc-trivial fibration.  Compos. Math., 141(2):385–403, 2005.
  * [BCHM10] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan.  Existence of minimal models for varieties of log general type.  J. Amer. Math. Soc., 23(2):405–468, 2010.
  * [Bir21] Caucher Birkar.  Generalised pairs in birational geometry.  EMS Surv. Math. Sci., 8(1-2):5–24, 2021.
  * [BP12] Bo Berndtsson and Mihai Păun.  Quantitative extensions of pluricanonical forms and closed positive currents.  Nagoya Math. J., 205:25 – 65, 2012.
  * [BZ16] Caucher Birkar and De-Qi Zhang.  Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs.  Publ. Math. Inst. Hautes Études Sci., 123:283–331, 2016.
  * [Dem12] Jean-Pierre Demailly.  Complex analytic and differential geometry.  2012\.
  * [DHP13] Jean-Pierre Demailly, Christopher D. Hacon, and Mihai Păun.  Extension theorems, non-vanishing and the existence of good minimal models.  Acta Math., 210(2):203–259, 2013.
  * [DPS00] Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider.  Pseudo-effective line bundles on compact Kähler manifolds.  Internat. J. Math., 12:689–741, 2000.
  * [FS20] Stefano Filipazzi and Roberto Svaldi.  Invariance of plurigenera and boundedness for generalized pairs.  Mat. Contemp., 47:114–150, 2020.
  * [GR56] Hans Grauert and Reinhold Remmert.  Plurisubharmonische funktionen in komplexen räumen.  Math. Z., 65:175–194, 1956.
  * [HMX13] Christopher D. Hacon, James McKernan, and Chenyang Xu.  On the birational automorphisms of varieties of general type.  Ann. of Math. (2), 177(3):1077–1111, 2013.
  * [HMX18] Christopher D. Hacon, James McKernan, and Chenyang Xu.  Boundedness of moduli of varieties of general type.  J. Eur. Math. Soc. 20, 20:865–901, 2018.
  * [Kaw85] Yujiro Kawamata.  Pluricanonical systems on minimal algebraic varieties.  Invent. Math., 79:567–588, 1985.
  * [Kaw98] Yujiro Kawamata.  Subadjunction of log canonical divisors, II.  Amer. J. Math., 120:893–899, 1998.
  * [Kaw99a] Yujiro Kawamata.  Deformations of canonical singularities.  J. Amer. Math. Soc., 12(1):85–92, 1999.
  * [Kaw99b] Yujiro Kawamata.  On the extension problem of pluricanonical forms.  In Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), volume 241 of Contemp. Math., pages 193–207. Amer. Math. Soc., Providence, RI, 1999.
  * [KM98] János Kollár and Shigefumi Mori.  Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 1998.
  * [Li20] Zhan Li.  Boundedness of the base varieties of certain fibrations.  arXiv:2002.06565, 2020.
  * [Li22] Zhan Li.  On finiteness of log canonical models.  Internat. J. Math., 33(2):Paper No. 2250012, 27, 2022.
  * [Mum70] David Mumford.  Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics.  Oxford University Press, London, 1970.
  * [Nar61] Raghavan Narasimhan.  The Levi problem for complex spaces.  Math. Ann., 142:355–365, 1961.
  * [Pău07] Mihai Păun.  Siu’s invariance of plurigenera: a one-tower proof.  J. Differ. Geom., 76:485–493, 2007.
  * [Pet94a] Thomas Peternell.  Modifications.  In Several Complex variables VII: sheaf-theoretical methods in complex analysis, pages 285–317. Springer-Verlag, 1994.
  * [Pet94b] Thomas Peternell.  Pseudoconvexity, the Levi problem and vanishing theorems.  In Several complex variables VII: sheaf-theoretical methods in complex analysis, pages 221–257. Springer-Verlag, 1994.
  * [RT20] Sheng Rao and I-Hsun Tsai.  Invariance of plurigenera and Chow-type lemma.  arXiv:2011.03306, 2020.
  * [Siu98] Yum-Tong Siu.  Invariance of plurigenera.  Invent. math., 134:661–673, 1998.
  * [Siu02] Yum-Tong Siu.  Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type.  In Complex geometry (Göttingen, 2000), pages 223–277. Springer, Berlin, 2002.
  * [Tak07] Shigeharu Takayama.  On the invariance and the lower semi-continuity of plurigenera of algebraic varieties.  J. Algebraic Geom., 16(1):1–18, 2007.
  * [Uen75] Kenji Ueno.  Classification theory of algebraic varieties and compact complex spaces.  Lecture Notes in Mathematics, Vol. 439. Springer-Verlag, 1975.
  * [Wło09] Jarosław Włodarczyk.  Resolution of singularities of analytic spaces.  In Proceedings of Gökova Geometry-Topology Conference 2008, pages 31–63. Gökova Geometry/Topology Conference (GGT), Gökova, 2009.