# smooth deformation limit of Moishezon manifolds is Moishezon

Mu-lin Li  School of Mathematics, Hunan University, China<EMAIL_ADDRESS>,
Sheng Rao  School of Mathematics and statistics, Wuhan University, Wuhan
430072, China<EMAIL_ADDRESS>,  Kai Wang  School of Mathematics and
statistics, Wuhan University, Wuhan 430072, China<EMAIL_ADDRESS>and
Meng-jiao Wang  School of Mathematics and statistics, Wuhan University, Wuhan
430072, China<EMAIL_ADDRESS>
###### Abstract.

We prove the conjecture that the deformation limit of Moishezon manifolds
under a smooth deformation over a unit disk in $\mathbb{C}$ is Moishezon.

The authors are partially supported by NSFC (Grant No. 11922115).

##  1\. introduction

Let $\pi:\mathcal{X}\rightarrow\Delta$ be a proper holomorphic submersion from
a complex manifold to a unit disc in $\mathbb{C}$, where the fibers
$X_{t}:=\pi^{-1}(t)$ are compact complex manifolds over $\Delta$. We call such
a morphism a _smooth family_ of compact complex manifolds. There are several
longstanding conjectures on the deformation limits of the compact complex
manifolds in Fujiki class $\mathcal{C}$, Kähler, Moishezon and projective
manifolds, respectively, within a smooth family. More precisely, if any
$X_{t}$ ($t\neq 0$) is in Fujiki class $\mathcal{C}$ (or Kähler), then $X_{0}$
is in Fujiki class $\mathcal{C}$; if any $X_{t}$ ($t\neq 0$) is Moishezon (or
projective), then $X_{0}$ is Moishezon. A compact complex manifold is
_Moishezon_ if it is bimeromorphic to a projective manifold. Equivalently, a
compact complex manifold is _Moishezon_ if and only if its algebraic dimension
is its complex dimension. The _algebraic dimension_ $a(X)$ of a compact
complex manifold $X$ is its transcendental degree of the field of meromorphic
functions on $X$ over $\mathbb{C}$.

According to Kodaira’s classification of surfaces (e.g., [Si83, Bu99, Lm99]),
every compact complex surface with the even first Betti number is Kähler. So
Ehresmann’s theorem [Em47] and the deformation invariance of Hodge numbers
imply that the limit surface $X_{0}$ of the compact complex surfaces in Fujiki
class $\mathcal{C}$ (or equivalently Kähler surfaces) is Kähler, while [Po13,
Corollary 1.3] proves the deformation limit of Moishezon (or equivalently
projective) surfaces to be projective.

As for the higher dimension case, Hironaka’s example [Hi62] shows that one can
not expect the central fiber to be necessarily projective or Kähler even if
all fibers $X_{t}$ are projective for $t\in\Delta^{*}$. D. Popovici [Po13]
proved that if all fibers $X_{t}$ are projective for $t\in\Delta^{*}$ and
$X_{0}$ satisfies the deformation invariance for Hodge number of type $(0,1)$
or admits a strongly Gauduchon metric, then $X_{0}$ is Moishezon. Recall that
a Hermitian metric $\omega$ on an $n$-dimensional compact complex manifold is
called a _strongly Gauduchon metric_ in [Po13] (or _Gauduchon metric_) if
$\partial\omega^{n-1}$ is $\bar{\partial}$-exact (or $\bar{\partial}$-closed).
There exists a Gauduchon metric on any compact complex manifold by [Ga77]. If
$X$ satisfies the $\partial\bar{\partial}$-lemma, then the Gauduchon metric
$\omega$ on $X$ is a strongly Gauduchon metric. D. Barlet [Ba15] showed that
if all fibers $X_{t}$ are Moishezon for $t\in\Delta^{*}$ and $X_{0}$ satisfies
either of the above two conditions, then $X_{0}$ is still Moishezon. The
second author–I-H. Tsai [RT21] prove that if there exist uncountably many
Moishezon fibers in the family, then any fiber with either of the above two
conditions is still Moishezon. J. Kollár [Ko22a] proposes a conjecture in the
flat family case that if all fibers $X_{t}$ are Moishezon for $t\in\Delta^{*}$
and $X_{0}$ is irreducible with rational singularities in a proper flat
morphism $f:{\cal X}\rightarrow\Delta$, then $X_{0}$ is Moishezon. A brief
progress review of this topic and a series of (other) stimulating open
questions can also be found in [Ko22a, Section 1].

As for the general case, we prove:

######  Theorem 1.1.

Assume that there exist uncountably many Moishezon fibers in a smooth family.
Then all fibers are Moishezon.

Theorem 1.1 is proposed as [RT21, Conjecture 1.6] and actually a direct
corollary of Theorem 1.2 and the second author–Tsai’s above result.

######  Theorem 1.2.

Assume that the fiber $X_{t}$ is Moishezon for each
$t\in\Delta^{*}:=\Delta\setminus\\{0\\}$. Then the central fiber $X_{0}$ is
Moishezon.

Now let us sketch our approach to prove Theorem 1.2. In what follows, we don’t
usually distinguish between $\Delta$ and $\Delta_{\epsilon}$ ($0<\epsilon\ll
1$) or similarly $\Delta^{*}$ and $\Delta_{\epsilon}^{*}$. Firstly, by the
theory of constructible sheaves [Di04], we construct a surjective map from a
constructible sheaf $\mathcal{G}^{\prime}$ via Bott–Chern and Kodaira–Spencer
theories to a local system ${\cal
F}:=\jmath_{*}\jmath^{-1}\mathcal{G}^{\prime}$ over $\Delta$, where
$\jmath:\Delta^{*}\rightarrow\Delta$ the inclusion. Here
$\mathcal{G}^{\prime}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}$ is
isomorphic to the smooth sheaf $\mathcal{G}$ over $\Delta$ with stalks
$H^{1,1}_{BC}(X_{t},\mathbb{C})$ of the Bott–Chern cohomology groups.
Secondly, we construct a constant section $\\{[Z_{t}]\\}_{t\in\Delta^{*}}$ of
$R^{2}\pi_{*}\mathbb{C}$ over $\Delta^{*}$ given by the fundamental classes of
$(n-1)$-cycles parameterized by a good filling $\mathcal{T}$. Thirdly, we
extend the section $\\{[Z_{t}]\\}_{t\in\Delta^{*}}$ to the whole disk as a
global section of
$\mathcal{F}_{sm}:=\mathcal{F}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}$
and obtain its preimage $s^{\prime}$ under the surjective morphism
$\mathcal{G}\rightarrow\mathcal{F}_{sm}$. Finally, we can control the volumes
of these cycles by the integration of the section $s^{\prime}$ wedged with a
smooth family of Gauduchon metrics over $X_{t}$ of $t\in\Delta^{*}$. Hence, we
obtain the semi-continuity of the algebraic dimension in the smooth family of
compact complex manifolds by the Barlet’s theory of relative codimension 1
cycle-space [Ba15, Lemma 2.2.3] (or Lemma 2.9) which is parameterized by the
good filling $\overline{\mathcal{T}}$, to prove the Theorem 1.2.

As several applications of Theorem 1.1 to a smooth family
$\pi:\mathcal{X}\rightarrow\Delta$ with uncountably many Moishezon fibers, we
prove that $\pi$ satisfies the deformation invariance of plurigenera and also
the property that the restriction of

$\mu:\mathcal{C}_{n-1}({\cal X}/\Delta)\rightarrow\Delta$

to each irreducible component of the relative $(n-1)$-cycle space
$\mathcal{C}_{n-1}({\cal X}/\Delta)$ is proper over $\Delta$. In particular,
if there are uncountably many projective fibers in $\pi$, $\pi$ is pseudo-
projective and thus the projective locus of $\pi$ is either at most a union of
countably many points or a Zariski dense open subset of $\Delta$.

Finally, using our approach, one can refine Barlet’s upper semi-continuity
[Ba15, Theorem 1.0.3] of algebraic dimensions under
$\partial\bar{\partial}$-lemma by removing the assumption on the deformation
invariance of Hodge number of type $(0,1)$ there. Theorem 1.3 can be regarded
as a generalization of Theorem 1.1. A. Fujiki-M. Pontecorvo’s example in
[FP10, Theorem 1.1] shows that this does not hold in general.

######  Theorem 1.3.

Assume that the fiber $X_{t}$ satisfies the $\partial\bar{\partial}$-lemma for
each $t\in\Delta^{*}$ and that $a:=\inf_{t\in\Delta^{*}}a(X_{t})$ of the
algebraic dimensions $a(X_{t})$ or (equivalently) there are uncountably many
fibers of algebraic dimension $a$. Then $a(X_{t})\geq a$ for all $t\in\Delta$.

Actually, a compact complex manifold satisfying the
$\partial\bar{\partial}$-lemma has many nice properties, such as the
deformation invariance of all-type Hodge numbers and Bott–Chern numbers and
also Hodge decomposition. Based on this and the approach to prove Theorem 1.2,
we can obtain a ($\Delta$-proper) good filling for $\pi$ which is the closure
of a ($\Delta^{*}$-proper) good filling for the restricted family
$\tilde{\pi}$ over $\Delta^{*}$. Notice that the general fiber of
$\tilde{\pi}$ is of algebraic dimension $a$ in the setting of Theorem 1.3.

Throughout this paper, denote by $\pi:\mathcal{X}\rightarrow\Delta$ a smooth
family of $n$-dimensional compact complex manifolds over the unit disk in
$\mathbb{C}$ and $f:\mathcal{X}\rightarrow B$ one over arbitrary dimensional
base. All complex manifolds are assumed to be connected. The notation
$h^{i}(X,L)$ for any non-negative integer $i$ denotes
$\dim_{\mathbb{C}}H^{i}(X,L)$ for a holomorphic line bundle $L$ over a compact
complex manifold $X$. The _general_ points of a complex space always refer to
the ones outside a thin subset of $\Delta$.

Acknowledgement: The authors would like to express their gratitude to
Professor D. Barlet for his careful reading and suggesting us consider Theorem
1.3, Professor Kang Zuo for several useful discussions on Hodge theory, and
Professors Xiao-Lei Liu, Wei Xia, Song Yang, Xiangdong Yang, Dr. Jian Chen,
Tianzhi Hu for their useful comments and suggestion. The second author also
wants to thank Professors Kefeng Liu and Fangyang Zheng for their constant
encouragement and help on deformation theory and complex geometry.

##  2\. Preliminaries: constructible sheaves and Barlet’s cycle spaces

To start with, we introduce the preliminaries on constructible sheaves and
Barlet’s theory of relative codimension 1 cycle-space, to be used in this
paper.

###  2.1. Constructible sheaves

A nice reference on constructible sheaves is [Di04].

Let $X$ be a complex manifold and $E$ a smooth complex vector bundle of rank
$r$ over $X$. A _flat structure_ in $E$ is given by an open cover
$\\{U_{i},s_{i}\\}_{i\in I}$ with local frame fields such that the transition
functions $g_{{i}{j}}$ are all constant matrices in $GL(r;\mathbb{C})$ where
$s_{j}=s_{i}g_{{i}{j}}$. A vector bundle $E$ with a flat structure is said to
be _flat_.

Denote by $\text{mod}(\mathbb{C}_{X})$ the abelian category of sheaves of
$\mathbb{C}_{X}$-module over $X$, where $\mathbb{C}_{X}$ is the constant sheaf
on $X$. A _$\mathbb{C}_{X}$ -local system_ on $X$ is a sheaf
$\mathcal{L}\in\text{mod}(\mathbb{C}_{X})$ which is locally constant, i.e.,
there is an open cover $\\{U_{i}\\}$ of $X$ and a family of
$\mathbb{C}$-vector spaces $V_{i}$ such that
$\mathcal{L}|_{U_{i}}\cong\mathcal{V}_{i}$, the constant sheaf on $U_{i}$
associated to $V_{i}$. When $X$ is connected, the family $(V_{i})$ can be
replaced by a single $\mathbb{C}$-vector space. A constant local system (i.e.,
one in which the covering can be chosen as $\\{X\\}$) is also called a
_trivial local system_.

######  Remark 2.1.

There is a bijective correspondence between local systems on $X$ up to
isomorphism and flat bundles on $X$ up to isomorphism which are frequently
used in the context.

Let $\mathcal{P}=(X_{j})_{j\in J}$ be a locally finite partition of $X$ into
non-empty, connected and locally closed subsets, which are called the _strata_
of $\mathcal{P}$. The partition $\mathcal{P}$ is called _admissible_ if it
satisfies the following conditions:

  1. (1)

The frontier condition, i.e., each frontier $\partial
X_{j}=\overline{X}_{j}\setminus X_{j}$ is a union of strata in $\mathcal{P}$;

  2. (2)

Constructibility, i.e., for all $j\in J$ the spaces $\overline{X}_{j}$ and
$\partial X_{j}$ are closed complex analytic subspaces in $X$.

A sheaf ${\cal F}\in\text{mod}(\mathbb{C}_{X})$ is _weakly constructible_ if
there is an admissible partition $\mathcal{P}=(X_{j})_{j\in J}$ such that the
restriction ${\cal F}|_{X_{j}}$ is a $\mathbb{C}$-local system for all $j\in
J$. A sheaf ${\cal F}\in\text{mod}(\mathbb{C}_{X})$ is _constructible_ if it
is weakly constructible and all its stalks ${\cal F}_{x}$ for $x\in X$ are
finite $\mathbb{C}$-modules.

The following two lemmas about constructible sheaves are used in process of
proving the Theorem 1.2.

######  Lemma 2.2.

Let $\mathcal{M}$ be a constructible sheaf on the unit disk $\Delta$ with
respect to the partition $\\{0\\}\cup\Delta^{*}$. Then

$H^{1}(\Delta,\mathcal{M})=0.$

###### Proof.

Let $r:\Delta\rightarrow[0,1)$ be the proper $\mathbb{R}$-analytic function
defined by $r(z)=x^{2}+y^{2}$ for $z=x+iy$. Then [Di04, Corollary 4.3.11]
gives

$H^{1}(\Delta,\mathcal{M})\cong H^{1}(r^{-1}(0),\mathcal{M})=0.$

∎

######  Lemma 2.3 ([Di04, Remark 2.5.3]).

Let $\mathcal{M}$ and $\mathcal{N}$ be locally constant sheaves on
$\Delta^{*}$ such that their restrictions to a dense Zariski open subset of
$\Delta^{*}$ are isomorphic. Then $\mathcal{M}$ and $\mathcal{N}$ are
isomorphic.

###  2.2. Barlet’s cycle spaces

We refer to [BM19] for a systematic study on Barlet’s theory of cycle spaces.

####  2.2.1. Absolute case

In this subsection, let $X$ always be a compact complex manifold of dimension
$n$. For any nonnegative integer $k\leq n$, a _$k$ -cycle_ in $X$ is a formal
linear combination

$Z:=\sum_{i\in I}k_{i}Z_{i}$

where $I$ is a finite set, $k_{i}$ are nonzero positive integers and
$(Z_{i})_{i\in I}$ are pairwise distinct complex $k$-dimensional irreducible
analytic subsets of $X$. By Poincaré duality there exists an isomorphism

$\gamma:H_{2k}(X,\mathbb{Z})\rightarrow H^{2n-2k}(X,\mathbb{Z})$

associating the homology class $[Z]\in H_{2k}(X,\mathbb{Z})$ to a cohomology
class $[\eta_{Z}]\in H^{2n-2k}(X,\mathbb{Z})$ of a $d$-closed $(2n-2k)$-form
$\eta_{Z}$ such that for any $d$-closed $2k$-form $\alpha$, there holds

$\int_{Z}\alpha=\int_{X}\alpha\wedge\eta_{Z}.$

Then we call the class $[\eta_{Z}]$ the _fundamental class_ of $Z$. If $X$ is
projective or Moishezon, then the fundamental class $[\eta_{Z}]$ of a
$k$-cycle $Z$ belongs in the Dolbeault cohomology $H^{n-k,n-k}(X,\mathbb{C})$.

The _support_ $|Z|$ of the cycle $Z$ is the analytic subset

$|Z|:=\bigcup_{i\in I}Z_{i}.$

A cycle $Z$ is called _compact_ if $|Z|$ is compact. Let $\mathcal{C}_{k}(X)$
be the set of all compact $k$-cycles of $X$ and

$\mathcal{C}(X)=\bigcup_{0\leq{k}\leq{n}}\mathcal{C}_{k}(X).$

For any compact irreducible analytic subset $\mathcal{A}$ of
$\mathcal{C}_{n-1}(X)$ there exists a meromorphic map

$\mathcal{K}_{\mathcal{A}}:X\dashrightarrow\mathcal{C}_{a-1}(\mathcal{A})$

from the generic point $x$ of $X$ to the set

$\\{T\in\mathcal{C}_{a-1}(\mathcal{A}):\ \text{there exists}\ t\in|T|\ s.t.,\
x\in|t|\\},$

where $a\geq 1$ is the complex dimension of $\mathcal{A}$. The map
$\mathcal{K}_{\mathcal{A}}$ is said to be the _Kodaira map_ of $\mathcal{A}$,
introduced by F. Campana in [Ca80]. The algebraic dimension of $X$ is
$\max_{\Gamma}\dim_{\mathbb{C}}\mathcal{K}_{\Gamma}(X)$, where $\Gamma$ is any
irreducible component in $\mathcal{C}_{n-1}(X)$.

####  2.2.2. Relative case

Let

$f:\mathcal{X}\rightarrow B$

be a smooth family of $n$-dimensional compact complex manifolds with
$\dim_{\mathbb{C}}B=s$. There is a _relative $k$-cycle space_
$\mathcal{C}_{k}({\cal X}/B)\subset\mathcal{C}_{k}({\cal X})$ such that the
image $f(|Z|)$ for any $Z\in\mathcal{C}_{k}({\cal X}/B)$ is a point in $B$ for
$0\leq k\leq n$. And the set $\mathcal{C}_{k}({\cal X}/B)$ is a closed
analytic subset of $\mathcal{C}_{k}({\cal X})$. Denote by

$\mathcal{C}({\cal X}/B):=\bigcup_{0\leq k\leq n}\mathcal{C}_{k}({\cal X}/B)$

the _relative Barlet space_ of ${\cal X}$ over $B$. By the definition of
relative cycle space, there is a canonical morphism

$\mu:\mathcal{C}({\cal X}/B)\rightarrow B.$

Bishop’s sequence theorem of analytic sets plays an important role in proving
the properness of the morphism $\mu$.

######  Proposition 2.4 ([Bi64, Theorem 1]).

Let $(Z_{m})_{m\in\mathbb{N}}$ be a sequence of pure $n$-dimensional analytic
subsets in an open subset $U$ of $\mathbb{C}^{n}$, converging to a set $Z$ of
$U$ in the sense of Hausdorff in $U$. If the volume of $Z_{m}$ is finite and
bounded by some constant, then $Z$ is an analytic subset of $U$ of pure
dimension $n$.

Recall that a real smooth $(2n-2)$-form $\omega$ on an $n$-dimensional compact
complex manifold $X$ is called a _strongly Gauduchon form_ if it is
$d$-closed, and the $(n-1,n-1)$-part of $\omega$ is positive. A smooth
$B$-relative $(2n-2)$-form on ${\cal X}$ is called a _relative strongly
Gauduchon form_ for the smooth family $f$ if it induces a strongly Gauduchon
form on each fiber of $f$.

######  Remark 2.5.

[Po13, Proposition 4.2] tells us that a compact complex manifold is strongly
Gauduchon if and only if it has a strongly Gauduchon form. And Barlet [Ba15]
observes that if every fiber of a smooth family of compact complex manifolds
is a strongly Gauduchon manifold, then there exists a relative strongly
Gauduchon form for the smooth family. In fact, it is a consequence of the
deformation openness of the strongly Gauduchon manifold and the partition of
unity of the base.

######  Lemma 2.6 ([Ba15, Proposition 3.0.6]).

Let $f:\mathcal{X}\rightarrow B$ admit a relative strongly Gauduchon form.
Then the restriction of

$\mu:\mathcal{C}_{n-1}({\cal X}/B)\rightarrow B$

to each irreducible component of $\mathcal{C}_{n-1}({\cal X}/B)$ is proper
over $B$.

Let $\mathcal{T}$ be an irreducible $B$-proper analytic subset of
$\mathcal{C}_{n-1}({\cal X}/B)$ and set $\dim_{\mathbb{C}}\mathcal{T}=t$. If
the analytic set $\mathcal{T}$ satisfies the conditions

  1. (1)

the second projection from the graph
$\mathcal{G}_{\mathcal{T}}\subset\mathcal{T}\times_{B}{\cal X}$ which is the
tautological family parameterized by $\mathcal{T}$ to ${\cal X}$ is
surjective;

  2. (2)

the general member is irreducible for any tautological family parameterized by
$\mathcal{T}$,

then the set $\mathcal{T}$ is said to be a good filling for the family $f$.

######  Remark 2.7.

In the proof of [Ba15, Theorem 1.0.2], Barlet shows the existence of the good
filling for a smooth family $\mathcal{X}\rightarrow B$ if the restriction of
the morphism $\mu:\mathcal{C}_{n-1}({\cal X}/B)\rightarrow B$ to each
irreducible component of $\mathcal{C}_{n-1}({\cal X}/B)$ is proper over $B$.

Barlet [Ba15] introduces the relative Kodaira map
$\mathcal{K}_{\mathcal{T}/B}$. For each good filling $\mathcal{T}$ we can
define a proper $B$-modification
$\alpha_{\mathcal{T}}:\mathcal{X}_{\mathcal{T}}\rightarrow{\cal X}$ and a
holomorphic $B$-map

$\mathcal{K}_{\mathcal{T}/B}:\mathcal{X}_{\mathcal{T}}\rightarrow\mathcal{C}_{t-s-1}(\mathcal{T}/B),$

obtained by composition of the “fiber-map” of
$\mathcal{G}_{\mathcal{T}}\rightarrow{\cal X}$ and the $B$-relative direct-
image map of relative $(t-s-1)$-cycles via the $B$-map
$\mathcal{G}_{\mathcal{T}}\rightarrow\mathcal{T}$.

######  Remark 2.8.

[Ba15, Lemma 2.2.2] shows that if $\mathcal{T}$ is a ($B$-proper) good filling
for $f$, then the map $\mathcal{K}_{\mathcal{T}/B}$ is proper and the image
$\mathcal{I}_{\mathcal{T}}$ of $\mathcal{K}_{\mathcal{T}/B}$ is proper over
$B$. And the morphism

$\tau:\mathcal{I}_{\mathcal{T}}\rightarrow B$

satisfies the property that the complex dimension of the fiber
$\mathcal{I}_{\mathcal{T}}(b):=\tau^{-1}(b)$ of $\tau$ equals to the algebraic
dimension of the fiber $X_{b}$ of $f$ for any general $b\in B$.

We present [Ba15, Lemma 2.2.3] of Barlet in the special case of
$1$-dimensional base.

######  Lemma 2.9.

Let $\pi:{\cal X}\rightarrow\Delta$ be a smooth family of $n$-dimensional
compact complex manifolds. And assume that:

  1. (1)

the irreducible analytic subset $\mathcal{T}$ of $\mathcal{C}_{n-1}({\cal
X}/\Delta)$ is a ($\Delta$-proper) good filling for $\pi$;

  2. (2)

there exists a Zariski dense subset $D_{\mathcal{T}}\subset\Delta$ such that

$\inf_{q\in
D_{\mathcal{T}}}[\dim_{\mathbb{C}}\mathcal{I}_{\mathcal{T}}(q)]=d.$

Then the inequality $a(X_{o})\geq d$ holds for each $o\in\Delta$.

##  3\. Deformation limit of Moishezon manifolds

In this section, we use constructible sheaves and Barlet’s theory of relative
codimension 1 cycle-space to prove Theorem 1.2 and then Theorem 1.1.

###  3.1. Construction via Bott–Chern and Kodaira–Spencer theories

Let $\pi:{\cal X}\to\Delta$ be a smooth family of $n$-dimensional compact
complex manifolds in this subsection. Denote by ${\cal E}_{\Delta}$ the
structure sheaf of smooth functions on $\Delta$. Let

${\cal A}^{p,q}_{{\cal X}/\Delta}=\wedge^{p}{\cal A}_{{\cal
X}/\Delta}^{1,0}\otimes\wedge^{q}{\cal A}_{{\cal X}/\Delta}^{0,1}$

be the ${\cal E}_{\Delta}$-sheaf of smooth relative $(p,q)$-differential forms
on ${\cal X}$, where

${\cal A}_{{\cal X}/\Delta}^{1,0}:={\cal A}_{{\cal X}}^{1,0}/\pi^{*}{\cal
A}_{\Delta}^{1,0},\quad{\cal A}_{{\cal X}/\Delta}^{0,1}:={\cal A}_{{\cal
X}}^{0,1}/\pi^{*}{\cal A}_{\Delta}^{0,1}.$

Kodaira–Spencer [KS60, (27)] defined two differential operators

$\overline{\partial}:{\cal A}^{p,q}_{{\cal X}/\Delta}\to{\cal
A}^{p,q+1}_{{\cal X}/\Delta},\quad\partial:{\cal A}^{p,q}_{{\cal
X}/\Delta}\to{\cal A}^{p+1,q}_{{\cal X}/\Delta}$

such that the restrictions satisfy
$\imath_{t}^{*}\circ\overline{\partial}=\overline{\partial}_{t}\circ\imath_{t}^{*}$
and $\imath_{t}^{*}\circ\partial=\partial_{t}\circ\imath_{t}^{*}$, where

$\imath_{t}:X_{t}\hookrightarrow\mathcal{X}$

is the natural holomorphic embedding, $\overline{\partial}_{t}$ and
$\partial_{t}$ are the ordinary differential operators on $X_{t}$.

Consider the following complex (e.g., [YY23, Proposition 2.1])

${\cal L}^{\bullet}(1,1):0\longrightarrow{\cal A}^{0,0}_{{\cal
X}/\Delta}\stackrel{{\scriptstyle\partial\bar{\partial}}}{{\longrightarrow}}{\cal
A}_{{\cal X}/\Delta}^{1,1}\stackrel{{\scriptstyle d}}{{\longrightarrow}}{\cal
A}_{{\cal X}/\Delta}^{1,2}\oplus{\cal A}_{{\cal
X}/\Delta}^{2,1}\longrightarrow\cdots.$

Put

$\mathcal{G}:=R^{1}\pi_{*}{\cal L}^{\bullet}(1,1).$

We need Lemma 3.1 to show the base change and get $\mathcal{G}_{t}\cong
H^{1,1}_{BC}(X_{t},\mathbb{C})$.

######  Lemma 3.1.

There holds $\Gamma(X_{t},{\cal A}_{{\cal X}/\Delta}^{p,q})=\Gamma(X_{t},{\cal
A}^{p,q}_{X_{t}})$ for any $t\in\Delta$.

###### Proof.

By definition we have $\Gamma(X_{t},{\cal A}_{{\cal
X}/\Delta}^{p,q})\subseteq\Gamma(X_{t},{\cal A}^{p,q}_{X_{t}})$. The converse
follows from [RYY19, Lemma 2.1] and we give a different proof here. As
Ehresmann’s theorem implies that ${\cal X}$ is diffeomorphic to
$X_{t}\times\Delta$, there exists a smooth projection

$\pi_{1}:{\cal X}\cong X_{t}\times\Delta\to X_{t},$

such that

(3.1) $\pi_{1}\circ\imath_{t}=\text{Id}_{X_{t}}$

(e.g., [Vo02, Remark 9.4]). Let $\alpha\in\Gamma(X_{t},{\cal
A}^{p,q}_{X_{t}})$. Then $\pi_{1}^{*}\alpha\in\Gamma({\cal X},{\cal
A}^{p+q}_{{\cal X}/\Delta})$. Let

$P^{p,q}:\Gamma({\cal X},{\cal A}^{p+q}_{{\cal X}/\Delta})\to\Gamma({\cal
X},{\cal A}^{p,q}_{{\cal X}/\Delta})$

and

$P_{t}^{p,q}:\Gamma(X_{t},{\cal A}^{p+q}_{X_{t}})\to\Gamma(X_{t},{\cal
A}^{p,q}_{X_{t}})$

be two projections, and then the smooth form
$P^{p,q}\pi_{1}^{*}\alpha\in\Gamma({\cal X},{\cal A}^{p,q}_{{\cal
X}/\Delta})$. Obviously, (3.1) gives

$(P^{p,q}\pi_{1}^{*}\alpha)|_{X_{t}}=P_{t}^{p,q}((\pi_{1}^{*}\alpha)|_{X_{t}})=P_{t}^{p,q}(\alpha)=\alpha$

and then $\Gamma(X_{t},{\cal A}^{p,q}_{X_{t}})\subseteq\Gamma(X_{t},{\cal
A}^{p,q}_{{\cal X}/\Delta})$. ∎

Let

$\mathcal{D}^{\bullet}(p):0\longrightarrow{\cal A}^{p,0}_{{\cal
X}/\Delta}\stackrel{{\scriptstyle\bar{\partial}}}{{\longrightarrow}}{\cal
A}_{{\cal
X}/\Delta}^{p,1}\stackrel{{\scriptstyle\bar{\partial}}}{{\longrightarrow}}{\cal
A}_{{\cal
X}/\Delta}^{p,2}\stackrel{{\scriptstyle\bar{\partial}}}{{\longrightarrow}}\cdots$

be the Dolbeault complex. Put

(3.2) ${\cal F}^{2}:=R^{0}\pi_{*}\mathcal{D}^{\bullet}(2).$

The morphism $\rho^{\bullet}:{\cal
L}^{\bullet}(1,1)\to\mathcal{D}^{\bullet}(1)$ in the abelian category of
complexes of sheaves of $\text{mod}(\mathcal{E}_{\Delta})$

$\begin{CD}0@>{}>{}>{\cal A}^{0,0}_{{\cal
X}/\Delta}@>{{}^{\partial\bar{\partial}}}>{}>{\cal A}_{{\cal
X}/\Delta}^{1,1}@>{{}^{d}}>{}>{\cal A}_{{\cal X}/\Delta}^{1,2}\oplus{\cal
A}_{{\cal X}/\Delta}^{2,1}@>{}>{}>\cdots\\\
@V{{}^{\partial}}V{}V@V{{}^{Id}}V{}V@V{}V{}V\\\
0@>{}>{}>\mathcal{A}^{1,0}_{{\cal
X}/\Delta}@>{{}^{\bar{\partial}}}>{}>\mathcal{A}_{{\cal
X}/\Delta}^{1,1}@>{{}^{\bar{\partial}}}>{}>\mathcal{A}_{{\cal
X}/\Delta}^{1,2}@>{}>{}>\cdots\end{CD}$

induces a morphism

(3.3) $\rho:\mathcal{G}\to R^{1}\pi_{*}\mathcal{D}^{\bullet}(1)$

in $\text{mod}(\mathcal{E}_{\Delta})$. We use the Kodaira–Spencer’s proof of
upper semi-continuity to construct a smooth complex vector bundle which
contains $\mathcal{G}$ as a torsion free subsheaf after shrinking the base.

In what follows, we don’t usually distinguish between $\Delta$ and
$\Delta_{\epsilon}$ ($0<\epsilon\ll 1$) or similarly $\Delta^{*}$ and
$\Delta_{\epsilon}^{*}$.

######  Lemma 3.2.

The sheaf $\mathcal{G}=R^{1}\pi_{*}{\cal L}^{\bullet}(1,1)$ is a torsion free
${\cal E}_{\Delta}$-subsheaf of a locally free sheaf.

###### Proof.

Fixing a Hermitian metric on a compact complex manifold, one can define the
Bott–Chern Laplacian operator

$\Delta_{BC}:=(\partial\overline{\partial})(\partial\overline{\partial})^{*}+(\partial\overline{\partial})^{*}(\partial\overline{\partial})+(\overline{\partial}^{*}\partial)(\overline{\partial}^{*}\partial)^{*}+(\overline{\partial}^{*}\partial)^{*}(\overline{\partial}^{*}\partial)+\overline{\partial}^{*}\overline{\partial}+\partial^{*}\partial,$

which is a self-adjoint elliptic differential operator.

As in [KS60, Part I], let $\Delta_{BC,t}$ be the family of Bott–Chern
Laplacian elliptic operators which act on $\Gamma(X_{t},{\cal
A}^{1,1}_{X_{t}})$ for any $t\in\Delta$ and $\lambda_{h}(t)$ the corresponding
eigenvalues with

$0\leq\lambda_{1}(t)\leq\lambda_{2}(t)\leq\cdots\leq\lambda_{h}(t)\leq\cdots.$

And the function $t\mapsto\lambda_{h}(t)$ is continuous over $\Delta$. Suppose
$0=\lambda_{1}(0)=\cdots=\lambda_{h_{0}}(0)<\lambda_{h_{0}+1}(0)$. Let
$D:=D_{\epsilon}$ be an $\epsilon$-disc in $\mathbb{C}$ with
$0<\epsilon<\lambda_{h_{0}+1}(0)$. Therefore, the boundary $C$ of $D$ does not
meet any value of $\lambda_{h}(0)$ for $h\geq 1$. Denote by
$\mathbf{F}_{t}(C)$ the linear subspace of eigenfuntions corresponding to the
eigenvalues $\lambda_{h}(t)\in D$. Then by [KS60, Proposition 2, Lemma 7],
after shrinking $\Delta$ small enough, one knows that

$J:=\bigcup_{t\in\Delta}\mathbf{F}_{t}(C)$

forms a smooth complex vector bundle over $\Delta$. Let $\mathcal{J}$ be the
associated smooth locally free sheaf of $J$. The definition of Bott–Chern
cohomology implies $\mathcal{G}\subset\mathcal{J}$ as a subsheaf. So the sheaf
$\mathcal{G}$ is a torsion free ${\cal E}_{\Delta}$-sheaf. ∎

Because $J$ is a smooth complex vector bundle over $\Delta$, the choice of the
eigenfunctions as coordinates associates a local system $\mathfrak{J}$ to $J$.

###  3.2. Proof of Theorems 1.2 and 1.1

Throughout this subsection, let $\pi:{\cal X}\to\Delta$ be a smooth family of
$n$-dimensional compact complex manifolds with all fibers $X_{t}$ for $t\neq
0$ being Moishezon.

######  Lemma 3.3.

The sheaf

$\mathcal{G}^{\prime}:=\mathfrak{J}\cap\mathcal{G}$

is a constructible sheaf with respect to the partition
$\mathcal{P}:=\\{0\\}\cup\Delta^{*}$ and the sheaf $\mathcal{G}$ is isomorphic
to $\mathcal{G}^{\prime}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}$.

###### Proof.

Since the fiber $X_{t}$ is Moishezon for any $t\in\Delta^{*}$, the Bott–Chern
number $h_{BC}^{1,1}(X_{t},\mathbb{C})$ is invariant for any $t\in\Delta^{*}$
and thus $\mathcal{G}|_{\Delta^{*}}$ is a smooth complex vector bundle over
$\Delta^{*}$. Then $\mathcal{G}^{\prime}|_{\Delta^{*}}$ is a local system
corresponding to $\mathcal{G}|_{\Delta^{*}}$ and then $\mathcal{G}^{\prime}$
is a constructible sheaf with respect to the partition $\mathcal{P}$.
Obviously,
$\mathcal{G}^{\prime}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}\subset\mathcal{G}$.
Moreover, $\mathcal{G}_{t}\cong\mathcal{G}^{\prime}_{t}$ for any $t\in\Delta$
and thus
$\mathcal{G}\subset\mathcal{G}^{\prime}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}$.
So
$\mathcal{G}^{\prime}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}\cong\mathcal{G}$.
∎

Now we come to:

###### Proof of Theorem 1.2.

The proof is divided into four steps.

Step $1$: construct a surjective map from $\mathcal{G}^{\prime}$ to a local
system ${\cal F}:=\jmath_{*}\jmath^{-1}\mathcal{G}^{\prime}$, where
$\jmath:\Delta^{*}\rightarrow\Delta$ the inclusion.

The sheaf $\mathcal{G}^{\prime}$ is a constructible sheaf on $\Delta$ with
respect to the partition $\\{0\\}\cup\Delta^{*}$ by Lemma 3.3. And [Dm12,
(13.6) of Chapter IV] and Lemma 3.1 give $\mathcal{G}_{t}\cong
H_{BC}^{1,1}(X_{t},\mathbb{C})$ for any $t\in\Delta$. [Di04, Corollary 2.4.3]
gives the exact sequence in $\text{mod}(\mathbb{C}_{\mathcal{X}})$

$0\to\mathcal{H}^{0}_{\\{0\\}}(\mathcal{G}^{\prime})\to\mathcal{G}^{\prime}\to\jmath_{*}\jmath^{-1}\mathcal{G}^{\prime}\to\mathcal{H}^{1}_{\\{0\\}}(\mathcal{G}^{\prime})\to
0,$

where $\mathcal{H}^{i}_{\\{0\\}}(\mathcal{G}^{\prime})$ $(i=0,1)$ are the
local cohomology sheaves with supports in $\\{0\\}$. Put

${\cal F}:=\jmath_{*}\jmath^{-1}\mathcal{G}^{\prime}.$

Then it is a constant sheaf, since $\mathcal{G}^{\prime}|_{\Delta^{*}}$ is a
constant sheaf. Because the canonical morphism
$\mathcal{G}^{\prime}|_{\Delta^{*}}\to R^{2}\pi_{*}\mathbb{C}|_{\Delta^{*}}$
is injective, the induced morphism ${\cal
F}=\jmath_{*}\jmath^{-1}\mathcal{G}^{\prime}\to R^{2}\pi_{*}\mathbb{C}$ is
also injective. Let $\Delta_{\epsilon}:=\\{z\in\mathbb{C}:|z|<\epsilon\\}$ be
a small disc. [Di04, Remark 4.3.12, Theorem 3.3.10] and Lemma 2.2 imply

$H_{\\{0\\}}^{1}(\Delta_{\epsilon},\mathcal{G}^{\prime})\cong
H_{c}^{1}(\Delta_{\epsilon},\mathcal{G}^{\prime})\cong
H^{1}(\Delta_{\epsilon},(\mathcal{G}^{\prime})^{\vee})^{\vee}=0,$

where $H_{c}^{1}(\Delta_{\epsilon},\mathcal{G}^{\prime})$ is the cohomology
with compact support. Thus,
$\mathcal{H}^{1}_{\\{0\\}}(\mathcal{G}^{\prime})=0$ since $\epsilon$ can be
chosen small enough. Then we have the exact sequence

(3.4)
$0\to\mathcal{H}^{0}_{\\{0\\}}(\mathcal{G}^{\prime})\to\mathcal{G}^{\prime}\to{\cal
F}\to 0.$

Step $2$: construct a section of $R^{2}\pi_{*}\mathbb{C}$ over $\Delta^{*}$
given by the fundamental classes of $(n-1)$-cycles parameterized by a good
filling.

Since all fibers $X_{t}$ are Moishezon for $t\in\Delta^{*}$, there exists a
strongly Gauduchon form on each fiber $X_{t}$ for $t\in\Delta^{*}$. So there
is a relative strongly Gauduchon form for the family

$\tilde{\pi}:=\pi|_{{\cal X}^{*}}:{\cal
X}^{*}:=\pi^{-1}(\Delta^{*})\rightarrow\Delta^{*}$

from Remark 2.5. Then the restriction of the holomorphic morphism

$\mu:\mathcal{C}_{n-1}({{\cal X}}^{*}/\Delta^{*})\to\Delta^{*}$

to each irreducible component of $\mathcal{C}_{n-1}({{\cal
X}}^{*}/\Delta^{*})$ is proper by Lemma 2.6. Thanks to the proof of [Ba15,
Theorem 1.0.2], we can obtain a ($\Delta^{*}$-proper) good filling
$\mathcal{T}$ for the family $\tilde{\pi}$ such that
$\dim_{\mathbb{C}}\mathcal{I}_{\mathcal{T}}(t)$ is the algebraic dimension
$a(X_{t})$ of $X_{t}$ for the general $t\in\Delta^{*}$.

Denote by $\alpha$ the fundamental class of the relative $(n-1)$-cycles in the
family which are parameterized by the good filling $\mathcal{T}$. By the Hodge
decomposition theorem, the fundamental class $\alpha\in
H^{1,1}(X_{t},\mathbb{C})\cap H^{2}(X_{t},\mathbb{Z})$ for any
$t\in\Delta^{*}$. In fact, $[Z_{t}]$ is the integration current on the fiber
$X_{t}$ of a member $Z_{t}$ of the family $\mathcal{T}$ in the fiber $X_{t}$
for $t\in\Delta^{*}$. Then there is a commutative diagram

---
$\textstyle{\mathcal{C}_{n-1}({{\cal
X}}^{*}/\Delta^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \ \ p}$$\textstyle{R^{2}{\tilde{\pi}}_{*}\mathbb{Z}_{{{\cal
X}}^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta^{*}}$

where $p$ is the map from $\mathcal{C}_{n-1}({{\cal X}}^{*}/\Delta^{*})$ to
$R^{2}{\tilde{\pi}}_{*}\mathbb{Z}_{{{\cal X}}^{*}}$. Since the map $p$ is
continuous and $R^{2}{\tilde{\pi}}_{*}\mathbb{Z}_{{\cal X}^{*}}$ is discrete,
the cohomology classes induced by the analytic cycles $\\{[Z_{t}]\\}\in
H^{2}(X_{t},\mathbb{Z})$ remain the same $\alpha$ for all $t\in\Delta^{*}$.
Then we can obtain a family $\\{[Z_{t}]\\}_{t\in\Delta^{*}}$ of flat classes
in $H^{2}(X_{t},\mathbb{Z})$ and then regard $\\{[Z_{t}]\\}_{t\in\Delta^{*}}$
as a section of $R^{2}\pi_{*}\mathbb{C}$ over $\Delta^{*}$.

Step $3$: extend the section $\\{[Z_{t}]\\}_{t\in\Delta^{*}}$ over the whole
disk to become of $\mathcal{F}_{sm}$ and get its preimage $s^{\prime}$ of
$\mathcal{G}$.

As $\\{[Z_{t}]\\}_{t\in\Delta^{*}}$ gives a constant section of
$R^{2}{\pi}_{*}{\mathbb{C}}$ over $\Delta^{*}$, one can canonically extend it
across the origin $0$, and denotes by $s$ the extended section over $\Delta$.
Then we have the sequence of smooth sheaves

$\mathcal{E}_{\Delta}\stackrel{{\scriptstyle
s}}{{\longrightarrow}}R^{2}\pi_{\ast}\mathbb{C}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}(R^{2}\pi_{\ast}\mathbb{C}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta})/(\mathcal{F}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta})\longrightarrow
0,$

where the first morphism is induced by tensor product with the section $s$.
Since $\\{[Z_{t}]\\}_{t\in\Delta^{*}}\in\Gamma(\Delta^{*},\mathcal{F})$ and
$(R^{2}\pi_{\ast}\mathbb{C}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta})/(\mathcal{F}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta})$
is a smooth complex vector bundle, $\phi\circ{s}$ is zero on the whole disc
$\Delta$. Thus,

$s=s\otimes_{\mathbb{C}_{\Delta}}1\in\Gamma(\Delta,\mathcal{F}_{sm}),$

where

(3.5)
$\mathcal{F}_{sm}:=\mathcal{F}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}$

is the associated smooth complex vector bundle to $\mathcal{F}$ on $\Delta$.

Lemma 3.3 implies that
$\mathcal{G}\cong\mathcal{G}^{\prime}\otimes_{\mathbb{C}_{\Delta}}\mathcal{E}_{\Delta}$.
By the exact sequence (3.4), the morphism

$\psi:\mathcal{G}\rightarrow\mathcal{F}_{sm}$

is surjective. So the germ $s_{0}$ of the section $s$ at ${0}$ has a preimage
$s^{\prime}_{0}$ in $\mathcal{G}_{0}$, which is the germ of a section
$s^{\prime}\in\Gamma(\Delta_{\epsilon},\mathcal{G})$ for small $\epsilon$.
Since $\psi(s^{\prime}_{0})=s_{0}$ means $\psi(s^{\prime})_{0}=s_{0}$, there
is an open neighborhood (still denoted by $\Delta_{\epsilon}$) of $0$ such
that $\psi(s^{\prime})|_{\Delta_{\epsilon}}=s|_{\Delta_{\epsilon}}$. As $\psi$
is an isomorphism over $\Delta^{*}$,

$s^{\prime}_{t}=s_{t}={\\{[Z_{t}]\\}}$

for $t\in\Delta_{\epsilon}\setminus\\{0\\}$ possibly after shrinking the base.

Step $4$: control the volumes of the cycles and prove the central fiber to be
Moishezon.

Without loss of generality, we denote by $\Delta$ the $\epsilon$-disk
$\Delta_{\epsilon}$. Since Lemma 3.2 tells us that $\mathcal{G}$ is a subsheaf
of the smooth locally free sheaf $\mathcal{J}$, one has
$s^{\prime}\in\Gamma(\Delta,\mathcal{G})\subset\Gamma(\Delta,\mathcal{J})$.
Then there is a $d$-closed smooth $(1,1)$-form $\Omega$ on ${\cal X}$ such
that $[\Omega|_{X_{t}}]=s^{\prime}_{t}$ in $H^{1,1}_{BC}(X_{t},\mathbb{C})$
for any $t\in\Delta$. By [Ga77] we obtain a smooth family of Gauduchon metrics
$\omega_{t}$ of positive definite $(1,1)$-forms on the fibers $X_{t}$ for all
$t\in\Delta$. Therefore, there exists a positive constant number $C$ such that

$\int_{X_{t}}{\\{[Z_{t}]\\}\wedge\omega_{t}^{n-1}}=\int_{X_{t}}{s^{\prime}_{t}\wedge\omega_{t}^{n-1}}=\int_{X_{t}}{\Omega|_{X_{t}}\wedge\omega_{t}^{n-1}}\leq
C,$

for any $t\in\Delta\setminus\\{0\\}.$

Denote by $\overline{\mathcal{T}}$ the closure of $\mathcal{T}$ in
$\mathcal{C}_{n-1}(\mathcal{X}/\Delta)$. By Proposition 2.4,
$\overline{\mathcal{T}}$ is an irreducible component of
$\mathcal{C}_{n-1}(\mathcal{X}/\Delta)$ which is proper over $\Delta$. Lemma
2.9 implies the semi-continuity of the dimensions of the fibers of the
projection $\mathcal{I}_{\overline{\mathcal{T}}}\rightarrow\Delta$, and thus
for any $t\in\Delta$, $a(X_{t})\geq n$, or equivalently, $a(X_{t})=n$. Hence,
the central fiber $X_{0}$ is Moishezon. ∎

######  Remark 3.4.

For a smooth family with general projective fibers, we might also use the same
method as above to control the mass of a family of positive forms in [Po13]
and thus obtain an integral Kähler current on the central fiber along the
approach in [Po13], to prove the central fiber to be Moishezon. In this way,
we just need to consider a sequence of points in $\Delta^{*}$ which converges
to the origin and control the mass of positive forms over this sequence of
points. Then the proof would rely much on [Po08].

###### Proof of Theorem 1.1.

The second author–I-H. Tsai [RT21] prove that if there exist uncountably many
Moishezon fibers in a smooth family over the unit disk in $\mathbb{C}$, then
any fiber satisfying the deformation invariance for Hodge number of type
$(0,1)$ or admits a strongly Gauduchon metric is still Moishezon. Moreover,
Grauert’s upper semi-continuity theorem tells us that a general fiber in a
smooth family indeed satisfies the deformation invariance for Hodge number of
any type (e.g., [LRT24, Corollary 2.10]). So any general fiber is Moishezon
and Theorem 1.2 thus concludes the proof. ∎

##  4\. Applications to Moishezon families

We give several applications of Theorem 1.1 to deformation invariance of
plurigenera, structure of family, projective locus and cycle space theory.

By Theorem 1.1 and the bimeromorphic embedding proved in [RT21, RT22], a
smooth family with uncountably many Moishezon fibers is Moishezon, i.e.,
bimeromorphic to a projective family over the unit disk.

###  4.1. Deformation invariance of plurigenera

The first application to generically fiberwise Moishezon family is deformation
invariance of plurigenera. For any positive integer $m$, the $m$-genus
$P_{m}(X)$ of a compact complex manifold $X$ is defined by

$P_{m}(X):=\dim_{\mathbb{C}}H^{0}(X,K_{X}^{\otimes{m}}),$

where $K_{X}$ is the canonical line bundle of $X$. It is a bimeromorphic
invariant and the _$m$ -genus of an arbitrary compact complex variety_ is thus
defined as the $m$-genus of its arbitrary non-singular model. Similarly, a
compact complex variety $M$ is _of general type_ if the Kodaira–Iitaka
dimension of the canonical bundle of its arbitrary non-singular model is just
the complex dimension of $M$.

######  Proposition 4.1 ([RT22, Main Theorem 1.2]).

Let $\pi:\mathcal{X}\rightarrow\Delta$ be a family such that any of the the
following conditions holds:

  1. (1)

the family $\pi$ is holomorphic and any fiber $X_{t}$ for $t\in\Delta$ is
Moishezon.

  2. (2)

each (not necessarily projective) compact variety $X_{t}$ at $t\in\Delta$ has
only canonical singularities and uncountable fibers therein being of general
type.

Then the _$m$ -genus_ $P_{m}(X_{t})$ for each positive integer $m$ is
independent of $t\in\Delta$.

######  Corollary 4.2.

Let $\pi:{\cal X}\rightarrow\Delta$ be a smooth family with uncountably many
Moishezon fibers. Then the $m$-genus $P_{m}(X_{t})$ is independent of
$t\in\Delta$ for any positive integer $m$.

###### Proof.

This is a direct corollary of Theorem 1.1 and Proposition 4.1. ∎

######  Remark 4.3.

The (possibly incomplete) introduction on the progress of the deformation
invariance of plurigenera can be found in [RT22, Introduction].

###  4.2. Pseudo-projective families and projective loci

[RT22, Definition 4.28] introduces a new notion that a family
$\pi:\mathcal{X}\rightarrow\Delta$ is called _pseudo-projective_ if there
exists a bimeromorphic morphism

$\tau:\mathcal{Y}\rightarrow{\cal X},$

over $\Delta$ where $\rho:\mathcal{Y}\rightarrow\Delta$ is a projective family
with $\mathcal{Y}$ being a complex manifold, such that $\tau$ induces a
biholomorphism from $\rho^{-1}(U)$ to $\pi^{-1}(U)$ for some dense Zariski
open subset $U$ of $\Delta$.

As the second application, one has:

######  Corollary 4.4.

Let $\pi:{\cal X}\rightarrow\Delta$ be a smooth family with uncountably many
projective fibers. Then $\pi$ is pseudo-projective.

###### Proof.

The proof is implicitly contained in [RT22, Lemma 4.25, Remark 4.26]. In fact,
all fibers are Moishezon by Theorem 1.1 and according to the argument in
Corollary 4.5 below or [RT21, RT22], it is not difficult to find a (global)
line bundle $\mathcal{L}$ on $\mathcal{X}$ such that its restriction to any
general fiber $X_{t_{0}}$ is very ample and
$h^{0}(X_{t},\mathcal{L}|_{X_{t}})$ is constant for any $t\in\Delta$
sufficiently close to $t_{0}$. Then by (the proof of) [RT22, Theorem 1.8], the
global line bundle $\mathcal{L}$ induces a bimeromorphic map $\Phi$ from
$\mathcal{X}$ to a subvariety $\mathcal{W}$ of $\mathbb{P}^{m}\times\Delta$
over $\Delta$ for some positive integer $m$, which implies that $\Phi$ is
biholomorphic on $\mathcal{X}\setminus{\mathcal{S}}$ for some proper
subvariety $\mathcal{S}$ of $\mathcal{X}$. So $R:=\pi(\mathcal{S})$ is a
proper analytic subset of $\Delta$ since the restriction
$\Phi_{t_{0}}:X_{t_{0}}\rightarrow W_{t_{0}}$ to the fiber of $\mathcal{W}$
over $t_{0}$ is biholomorphic. So $\Phi$ is biholomorphic at least on
$\mathcal{X}\setminus{\mathcal{S}}$ by the definition of $\mathcal{S}$, and
thus, $\mathcal{W}$ is smooth outside ${\pi_{\mathcal{W}}^{-1}(R)}$ for
$\pi_{\mathcal{W}}:\mathcal{W}\rightarrow\Delta$ since $\mathcal{X}$ is
smooth. Conversely, $\mathcal{W}$ is bimeromorphic to $\mathcal{X}$ and
$\pi_{\mathcal{W}}:\mathcal{W}\rightarrow\Delta$ is a projective morphism. By
Remmert’s elimination of indeterminacy [Re57] and Hironaka’s Chow lemma [Hi75,
Corllary 1 and 2], one modifies $\mathcal{W}$ as a complex manifold
$\mathcal{Y}$ such that the composition $\mathcal{Y}\rightarrow{\mathcal{W}}$
over $\Delta$ is a bimeromorphic morphism with
$\mathcal{Y}\setminus\pi_{\mathcal{Y}}^{-1}(R)\cong\mathcal{W}\setminus{\pi_{\mathcal{W}}^{-1}(R)}$
as
$\mathcal{W}\setminus{\pi_{\mathcal{W}}^{-1}(R)}\cong\mathcal{X}\setminus{\mathcal{S}}$.
Hence, $\pi_{\mathcal{Y}}:\mathcal{Y}\rightarrow\Delta$ is the desired
projective morphism and the morphism $\pi$ is pseudo-projective. ∎

######  Corollary 4.5.

Let $\pi:\mathcal{X}\rightarrow\Delta$ be a smooth family. Define the
_projective locus_ of $\pi$ as

$\mathcal{P}(\pi):=\\{t\in\Delta:\text{$X_{t}$ is projective}\\}.$

Then $\mathcal{P}(\pi)$ is either at most a union of countably many points or
a Zariski dense open subset of $\Delta$.

It seems not difficult to generalize Corollary 4.5 to a proper morphism of
complex analytic spaces as in [Ko22b, Section 5], where a complete reference
list on projective loci is given.

###### Proof.

This follows from Corollary 4.4 directly. For reader’s convenience, we give a
more elementary proof here, also to provide more details of proving Corollary
4.4. Assume that there are uncountably many projective fibers in $\pi$.

Step $1$: construct a global line bundle ${\cal L}$ over ${{\cal X}}$ such
that ${\cal L}|_{X_{t_{0}}}$ is ample for some $t_{0}\in\Delta$.

By the exponential sequence, one has the commutative diagram of exact
sequences

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathcal{X},\mathcal{O}^{*}_{\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(\mathcal{X},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H^{2}(\mathcal{X},\mathcal{O}_{\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}({X_{t}},\mathcal{O}_{X_{t}}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(X_{t},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(X_{t},\mathcal{O}_{{X_{t}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

for any $t\in\Delta$. Applying the Leray spectral sequence to $\pi$ for the
sheaf ${\cal O}_{{{\cal X}}}$, one obtains the exact sequence

$\textstyle{H^{1}(\Delta,R^{1}\pi_{*}{\cal O}_{{{\cal
X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}({{\cal
X}},{\cal O}_{{{\cal
X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\Delta,R^{2}\pi_{*}{\cal
O}_{{{\cal
X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since $\Delta$ is a holomorphic domain, the cohomology
$H^{1}(\Delta,R^{1}\pi_{*}{\cal O}_{{{\cal X}}})$ vanishes and thus

$H^{2}({{\cal X}},{\cal O}_{{{\cal X}}})\cong H^{0}(\Delta,R^{2}\pi_{*}{\cal
O}_{{{\cal X}}}).$

Let $c:=c_{1}(A)$ where ${A}$ is an ample line bundle on some fiber $X_{t}$
and denote by $c\in H^{2}(X_{t},\mathbb{Z})$ the element in $H^{2}({{\cal
X}},\mathbb{Z})$. The composed morphism

$H^{2}({{\cal X}},\mathbb{Z})\subset H^{2}({{\cal X}},\mathbb{C})\to
H^{0}(\Delta,R^{2}\pi_{*}{\cal O}_{{{\cal X}}})$

induces a section $s_{c}\in H^{0}(\Delta,R^{2}\pi_{*}{\cal O}_{{{\cal X}}})$.
Since any fiber $X_{t}$ is Moishezon for $t\in\Delta$, the $(0,2)$-type Hodge
number $h^{0,2}(X_{t})$ is constant on $\Delta$. So $R^{2}\pi_{*}{{\cal
O}_{{\cal X}}}$ is locally free over $\Delta$ by Grauert’s continuity theorem
and the map

$R^{2}\pi_{*}\mathcal{O}_{\mathcal{X}}(t):=(R^{2}\pi_{*}\mathcal{O}_{\mathcal{X}})_{t}\otimes\mathbb{C}(t)\rightarrow
H^{2}(X_{t},\mathcal{O}_{{X_{t}}})$

is bijective for each $t\in\Delta$ by Grauert’s base change theorem. Thus, the
zero locus $Z_{c}$ of $s_{c}$ is an analytic subset of $\Delta$. Since there
is an uncountable subset $T\subseteq\Delta$ such that all the fibers $X_{t}$
for $t\in T$ are projective, one has

$\bigcup_{c}Z_{c}\supseteq T,$

where the union takes over all integral classes $c_{1}(L_{t})\in
H^{2}(X,\mathbb{Z})$ of an ample line bundle $L_{t}$ on any fiber $X_{t}$
$(t\in T)$. Here $X$ is the underlying differentiable manifold of the family’s
fibers. Since a countable union of proper analytic subsets is still countable,
there exits an analytic $Z_{c}=\Delta$, and there thus exists a global line
bundle ${\cal L}$ over ${{\cal X}}$ such that ${\cal L}|_{X_{t_{0}}}$ is an
ample line bundle for some $t_{0}\in\Delta$ by Nakai–Moishezon criterion of
ampleness.

Step $2$: apply the Zariski openness of the ampleness of a line bundle over
the total space due to Grothendieck [Gr61, THÉORÈME 2.1] to conclude the
proof. For reader’s convenience, we give a proof here.

Let $m$ be a sufficiently large integer such that ${\cal L}^{\otimes
m}|_{X_{t_{0}}}$ is very ample and

$h^{i}(X_{t_{0}},{\cal L}^{\otimes m}|_{X_{t_{0}}})=0$

for $i>0$. Denote by ${\cal F}$ the direct image $\pi_{*}{\cal L}^{\otimes m}$
of ${\cal L}^{\otimes m}$ under $\pi$.

Define

$A_{i}=\\{t\in\Delta:\ h^{i}(X_{t},{\cal L}^{\otimes m}|_{X_{t}})\geq 1\\}.$

By Grauert’s upper semi-continuity theorem, we know that $A_{i}$ is a proper
analytic subset of $\Delta$ for each $i\geq 1$. Then

$U^{\prime}:=\Delta\setminus\bigcup_{i\geq 1}A_{i}$

is a Zariski open set in $\Delta$ which contains $t_{0}$. By Riemann–Roch
theorem, the dimension $h^{0}(X_{s},{\cal L}^{\otimes m}|_{X_{s}})$ is
invariant for $s\in U^{\prime}$. The sheaf ${\cal F}$ is locally free on a
neighborhood of $t_{0}$ and

${\cal F}({t_{0}})\cong H^{0}(X_{t_{0}},{\cal L}^{\otimes m}|_{X_{t_{0}}}).$

From the local freeness of $\mathcal{F}$, we have the exact sequence

$0\to{\cal F}\otimes\mathcal{I}_{t_{0}}\to{\cal F}\to{\cal F}\otimes{\cal
O}_{t_{0}}\to 0,$

where $\mathcal{I}_{t_{0}}$ is the ideal sheaf over $t_{0}$. Thus, the
morphism $H^{0}(\Delta,{\cal F})\to{\cal F}(t_{0})$ is surjective, since
$\Delta$ is a holomorphic domain.

Set $N=\dim_{\mathbb{C}}H^{0}(X_{t_{0}},{\cal L}^{\otimes m}|_{X_{t_{0}}})$.
Suppose that $\sigma_{1},\cdots,\sigma_{N}\in H^{0}({{\cal X}},{\cal
L}^{\otimes m})\cong H^{0}(\Delta,{\cal F})$ are global sections such that
$\\{\sigma_{i}\\}$ generate ${\cal F}(t_{0})$, i.e., the vector space
$H^{0}(X_{t_{0}},{\cal L}^{\otimes m}|_{X_{t_{0}}})$ is spanned by the
sections $\sigma_{i}|_{X_{t_{0}}}$. Let

$\phi\colon{{\cal
X}}\dashrightarrow\widetilde{\mathbb{P}}:=\mathbb{P}^{N}\times{\Delta}$

be the meromorphic map induced by $\\{\sigma_{i}\\}$. The restriction
$\phi_{s}\colon X_{s}\dashrightarrow\widetilde{\mathbb{P}}_{s}$ is defined by
the complete linear system $|{\cal L}^{\otimes m}|_{X_{s}}|$ for each $s\in
U^{\prime}$. Thus $\phi_{t_{0}}$ is an embedding.

Denote by $Z_{1}$ the common zeros of $\\{\sigma_{i}\\}$ and let $Z_{2}$ be
the zero loci of

$\left\\{d\bigg{(}\frac{\sigma_{j_{1}}}{\sigma_{i}}\bigg{)}\wedge\cdots\wedge
d\bigg{(}\frac{\sigma_{j_{n}}}{\sigma_{i}}\bigg{)}\right\\},$

where $1\leq i,j_{k}\leq N,j_{k}\neq i$ and $1\leq k\leq n$. These are two
analytic subsets of ${{\cal X}}$ and let $Z=Z_{1}\cup Z_{2}$. Since $\pi$ is
proper, the image $\pi(Z)$ is an analytic subset of $\Delta$. But
$t_{0}\notin\pi(Z)$, and $\pi(Z)$ is thus a discrete set of points in
$\Delta$. Let $U:=\Delta\setminus\pi(Z)$. Then $\phi_{s}$ is a holomorphic
morphism and locally embedding for any $s\in U$. Thus ${\cal L}^{\otimes
m}|_{X_{s}}\cong\phi_{s}^{*}({\cal O}_{\widetilde{\mathbb{P}}_{s}}(1))$ are
ample for all $s\in U$. ∎

###  4.3. Barlet’s theory of relative codimension 1 cycle-space

As the final (and direct) application of Theorem 1.1, one has:

######  Corollary 4.6.

Let $\pi:{\cal X}\rightarrow\Delta$ be a smooth family with uncountably many
Moishezon fibers. Then the restriction of

$\mu:\mathcal{C}_{n-1}({\cal X}/\Delta)\rightarrow\Delta$

to each irreducible component of $\mathcal{C}_{n-1}({\cal X}/\Delta)$ is
proper over $\Delta$.

###### Proof.

It follows from Theorem 1.1 and [Ba15, Proposition 3.0.6] (or Lemma 2.6 here).
∎

## References

  * [Ba15] D. Barlet, Two semi-continuity results for the algebraic dimension of compact complex manifolds, J. Math. Sci. Univ. Tokyo 22 (2015), no. 1, 39-54.
  * [BM19] D. Barlet, J. Magnússon, Complex analytic cycles I, basic results on complex geometry and foundations for the study of cycles, Grundlehren Math. Wiss., 356 [Fundamental Principles of Mathematical Sciences] Springer, Cham; Société Mathématique de France, Paris, 2019.
  * [Bi64] E. Bishop, Conditions for the analyticity of certain sets, Michigan Math. J. 11 (1964), 289-304.
  * [Bu99] N. Buchdahl, On compact Kähler surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, vii, xi, 287-302.
  * [Ca80] F. Campana, Application de l’espace des cycles à la classification biméromorphe des espaces analytiques kählériens compacts, Revue de l’Inst. E. Cartan no.2 (Nancy), pp. 1-162.
  * [Dm12] J.-P. Demailly, Complex analytic and differential geometry, J.-P. Demailly’s CADG e-book.
  * [Di04] A. Dimca, Sheaves in topology, Universitext. Springer-Verlag, Berlin, 2004.
  * [Em47] C. Ehresmann, Sur les espaces fibres differentiables, C. R. A cad. Sci. Paris, 224, 1611-1612 (1947).
  * [FP10] A. Fujiki, M. Pontecorvo, Non-upper-semicontinuity of algebraic dimension for families of compact complex manifolds, Math. Ann. 348 (2010), no. 3, 593¨C599.
  * [Ga77] P. Gauduchon, Le théoréme de l’excentricité nulle, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 5, A387-A390.
  * [Gr61] A. Grothendieck, Techniques de construction en géométrie analytique. VIII. Rapport sur les théorèmes de finitude de Grauert et Remmert, Séminaire Henri Cartan, Volume 13 (1960-1961) no. 2, Talk no. 15, p.1-10.
  * [Hi62] H. Hironaka, An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures, Ann. of Math. (2) 75 (1962), 190-208.
  * [Hi75] H. Hironaka, Flattening theorem in complex-analytic geometry, Amer. J. Math. 97 (1975), pp. 503-547.
  * [KS60] K. Kodaira, D. C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2) 71 (1960), 43-76.
  * [Ko22a] J. Kollár, Moishezon morphisms, Pure Appl. Math. Q. 18 (2022), no.4, 1661-1687.
  * [Ko22b] J. Kollár,  Seshadri’s criterion and openness of projectivity, Proc. Indian Acad. Sci. Math. Sci. 132 (2022), no.2, Paper No. 40.
  * [Lm99] A. Lamari, Courants kählériens et surfaces compactes. (French) [Kähler currents and compact surfaces], Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, vii, x, 263-285.
  * [LRT24] Yi Li, Sheng Rao, I-Hsun Tsai, A deformation property when uncountably many fibers are Moishezon or of general type, preprint, 2024.
  * [Po08] D. Popovici, Regularization of currents with mass control and singular Morse inequalities, J. Differential Geom. 80 (2008), no. 2, 281-326.
  * [Po13] D. Popovici, Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics, Invent. Math.194 (2013), no. 3, 515-534.
  * [RT21] Sheng Rao, I-Hsun Tsai, Deformation limit and bimeromorphic embedding of Moishezon manifolds, Commun. Contemp. Math. 23 (2021), no. 8, Paper No. 2050087.
  * [RT22] Sheng Rao, I-Hsun Tsai, Invariance of plurigenera and Chow-type lemma, Asian J. Math. 26 (2022), no. 4, 507-554.
  * [RYY19] Sheng Rao, Song Yang, Xiangdong Yang, Dolbeault cohomologies of blowing up complex manifolds, J. Math. Pures Appl. 130 (2019), 68-92.
  * [Re57] R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann. 133 (1957), pp. 328-370.
  * [Si83] Y.-T. Siu, Every K3 surface is Kähler, Invent. Math. 73 (1983), no. 1, 139-150.
  * [Vo02] C. Voisin, Hodge theory and complex algebraic geometry. I. Translated from the French original by Leila Schneps. Cambridge Studies in Advanced Mathematics, $76$. Cambridge University Press, Cambridge, 2002.
  * [YY23] Song Yang, Xiangdong Yang, Bott–Chern hypercohomology and bimeromorphic invariants, Complex Manifolds 10 (2023), no. 1, Paper No. 20220148.