# Hermitian null loci

Quang-Tuan Dang  The Abdus Salam International Centre for Theoretical Physics
(ICTP), Str. Costiera, 11, 34151 Trieste, TS, Italy<EMAIL_ADDRESS>
(Date: September 3, 2024)

###### Abstract.

We establish a transcendental generalization of Nakamaye’s theorem to compact
complex manifolds when the form is not assumed to be closed. We apply the
recent analytic technique developed by Collins–Tosatti to show that the non-
Hermitian locus of a nef and big $(1,1)$-form, which is not necessarily
closed, on a compact complex manifold equals the union of all positive-
dimensional analytic subvarieties where the restriction of the form is not big
(null locus). As an application, we can give an alternative proof of the
Nakai–Moishezon criterion of Buchdahl and Lamari for complex surfaces and
generalize this result in higher dimensions. This is also used for studying
degenerate complex Monge–Ampère equations on compact Hermitian manifolds.
Finally, we investigate finite time non-collapsing singularities of the
Chern–Ricci flow, partially answering a question raised by Tosatti and
Weinkove.

###### Key words and phrases:

Hermitian currents, null loci, Chern–Ricci flows, complex manifolds

###### 2020 Mathematics Subject Classification:

32W20, 53E30, 32U15

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Null loci and Nakai–Moishezon criterions
  4. 4 Singular Hermitian metrics and envelopes
  5. 5 Finite time singularities of the Chern–Ricci flow

##  1\. Introduction

Characterizing the augmented base locus of a nef and big line bundle on a
smooth algebraic variety over $\mathbb{C}$ was initially motivated by
applications to diophantine approximation problem [Nak15]. Nakamaye showed in
[Nak00] that the augmented base locus of a nef and big line bundle on a
projective manifold equals its null locus (which is the union of all
irreducible subvarieties where the restriction of the bundle is not big).
There are numerous generalizations of the latter result in
$\mathbb{R}$-divisors [ELM+09], in possibly singular varieties [Bir17, CL14,
BCL14], and in positive characteristics [CMM14]. We refer interested readers
to [Tos18b] for a comprehensive overview.

Our primary goal in this paper is to extend Nakamaye’s theorem to the context
of compact non-Kähler manifolds with non-closed differential forms. Let $X$ be
a compact complex manifold of dimension $n$, equipped with a Hermitian metric
$\omega_{X}$. We denote by ${\rm d^{c}}=\frac{1}{2i}(\partial-\bar{\partial})$
so that $\mathrm{dd^{c}}=i\partial\bar{\partial}$. Let $\theta$ be a smooth
real (1,1) form and let $[\theta]_{\partial\bar{\partial}}$ denote the
$\partial\bar{\partial}$-class of forms (resp. currents):
$[\theta]_{\partial\bar{\partial}}=\\{\theta+\mathrm{dd^{c}}\varphi:u\in\mathcal{C}^{\infty}(X,\mathbb{R})\;(\text{resp.}\,u\in
L^{1}(X))\\}.$ For simplicity, we omit the subscript $\partial\bar{\partial}$.
When $\theta$ is closed, the $\partial\bar{\partial}$-class $[\theta]$
coincides with its Bott–Chern cohomology class in $H^{1,1}_{\rm
BC}(X,\mathbb{R})$. A $\partial\bar{\partial}$-class $[\theta]$ is said to be
Hermitian (or positive) if it contains at least a positive definite form,
i.e., $\exists\,\varphi\in\mathcal{C}^{\infty}(X,\mathbb{R})$ such that
$\theta+\mathrm{dd^{c}}\varphi$ is a Hermitian metric. Although $\theta$ is
not closed in general, one can still meaningfully define the concept (in the
analytic sense) of nef $\partial\bar{\partial}$-classes; cf. Def. 2.1.
Similarly, a $\partial\bar{\partial}$-class $[\theta]$ is said to be big if it
contains a positive current dominating a Hermitian metric. It should be
emphasized that the previous concepts can be defined when the underlying
variety is possibly singular; cf. Sect. 2.3. By Demailly’s regularization
theorem, the Hermitian current in a big class can be taken to have (almost)
analytic singularities (in the sense of [DP04, Theorem 3.2]). In particular,
the Hermitian current is smooth in the complement of a Zariski closed subset
of $X$. The smallest possible Zariski closed subset of $X$ that one can take
in this way is the non-Hermitian locus of a big $\partial\bar{\partial}$-class
$[\theta]$, denoted by $\operatorname{E_{\textrm{{nH}}}}(\theta)$; cf. [Bou04,
Definition 3.16].

A nef and big $\partial\bar{\partial}$-class $[\theta]$, in general, may not
be positive, and another subset to measure its non-positivity is its null
locus, denoted by $\operatorname{{\rm Null}}(\theta)$, explicitly introduced
by Nakamaye (possibly investigated earlier). The latter subset is defined as
the union of all positive-dimensional irreducible analytic subvarieties
$V\subset X$ where the restriction $\theta|_{V}$ is not big. Clearly, the null
locus is a proper subset of $X$. Repeating line by line the proof of
Collins–Tosatti, one can prove that given a nef and big
$\partial\bar{\partial}$-class $[\theta]$, its non-Hermitian locus and its
null locus coincide. The proof does not require the closedness of the
representative form.

######  Theorem 1.1 ([CT15, Theorem 1.1]).

Let $X$ be a compact complex manifold. Let $\theta$ be a smooth real (1,1)
form whose $\partial\bar{\partial}$-class $[\theta]$ is nef and big. Then

$\operatorname{E_{\textrm{{nH}}}}(\theta)=\operatorname{{\rm Null}}(\theta).$

In the Kähler case (or the Fujiki class $\mathcal{C}$ in general), the bigness
of a closed nef form $\theta$ is equivalent to the positivity property of its
self-intersection number, i.e., $\int_{X}\theta^{n}>0$. Unlike the Kähler
case, the difficulty we face is that the volume of two forms in the same
$\partial\bar{\partial}$-class might be different in general. Let $X$ be a
compact complex manifold of dimension $n$, equipped with a Hermitian metric
$\omega_{X}$. A question raised by Guedj–Lu is whether

$v_{+}(X,\omega_{X})=\sup\left\\{\int_{X}(\omega_{X}+\mathrm{dd^{c}}\varphi)^{n}:\varphi\in\mathcal{C}^{\infty}(X),\omega_{X}+\mathrm{dd^{c}}\varphi>0\right\\}<+\infty;$
$v_{-}(X,\omega_{X})=\inf\left\\{\int_{X}(\omega_{X}+\mathrm{dd^{c}}\varphi)^{n}:\varphi\in\mathcal{C}^{\infty}(X),\omega_{X}+\mathrm{dd^{c}}\varphi>0\right\\}>0.$

We refer to [AGL23] for some classes of manifolds satisfying the condition
above. We say that a Hermitian metric $\omega_{X}$ satisfies the volume
condition ($H_{X}$) if:

($H_{X}$)
$v_{+}(X,\omega_{X})<+\infty\quad\text{and}\;v_{-}(V,\omega_{X}|_{V})>0$

for all positive-dimensional closed submanifold $V\subset X$.

######  Remark 1.2.

It has been shown in [AGL23, Theorem A] that for any closed submanifold
$V\subset X$,

$v_{+}(X,\omega_{X})<+\infty\Longrightarrow v_{+}(V,\omega_{X}|_{V})<+\infty.$

It is important to emphasize that the assumption ($H_{X}$) occurs in some
contexts of complex differential geometry. For instance, if $\omega_{X}$
satisfies the following condition $\mathrm{dd^{c}}\omega_{X}=0$, ${\rm
d}\omega_{X}\wedge{\rm d^{c}}\omega_{X}=0$ then by [Chi16b], it is equivalent
to the volume with respect to a form belonging to the
$\partial\bar{\partial}$-class $[\omega_{X}]$ is unchanged. That is, for any
closed submanifold $V\subset X$,

$\int_{V}(\omega_{X}|_{V}+\mathrm{dd^{c}}\varphi)^{\dim
V}=\int_{V}\omega_{X}|_{V}^{\dim
V},\forall\;{\omega_{X}}|_{V}+\mathrm{dd^{c}}\varphi>0.$

These metrics were introduced by Guan–Li [GL10], which are called the Guan–Li
metrics in this paper, although they were probably studied earlier; cf.
[FT11].

Given $\theta$ a smooth real (1,1) form on $X$, if $\theta$ is nef, we can
define

$\widehat{v}_{-}(X,\theta):=\inf_{\varepsilon>0}v_{-}(X,(1-\varepsilon)\theta+\varepsilon\omega_{X}).$

Although the form $(1-\varepsilon)\theta+\varepsilon\omega_{X}$ is not
positive, one could find by definition, for any $\varepsilon>0$, a smooth
function $\varphi_{\varepsilon}$ such that
$\theta+\mathrm{dd^{c}}\varphi_{\varepsilon}\geq-\varepsilon\omega_{X}$ so
that

$(1-\varepsilon)\theta+\varepsilon\omega_{X}+\mathrm{dd^{c}}(1-\varepsilon)\varphi_{\varepsilon}\geq\varepsilon^{2}\omega_{X}.$

Hence the quantity $v_{-}(X,(1-\varepsilon)\theta+\varepsilon\omega_{X})$ can
be understood as
$v_{-}(X,(1-\varepsilon)\theta+\varepsilon\omega_{X}+\mathrm{dd^{c}}(1-\varepsilon)\varphi_{\varepsilon})$.
Our main theorem is the following:

######  Theorem A.

Let $X$ be a compact complex manifold equipped with a Hermitian metric
$\omega_{X}$. Assume that $\omega_{X}$ satisfies the condition ($H_{X}$). Let
$\theta$ be a smooth real (1,1) form whose $\partial\bar{\partial}$ class is
nef with $\widehat{v}_{-}(X,\theta)>0$. Then $[\theta]$ is big and

(1.1) $\operatorname{E_{\textrm{{nH}}}}(\theta)=\operatorname{{\rm
Null}}(\theta)=\bigcup_{\widehat{v}_{-}(V,\theta|_{V})=0}V$

where the union is taken over all purely positive-dimensional irreducible
analytic subvarieties $V$ of $X$.

In particular, when $\theta$ is closed or merely satisfies the so-called
Guan–Li condition: $\mathrm{dd^{c}}\theta=0$, ${\rm d}\theta\wedge{\rm
d^{c}}\theta=0$ (equivalent to $\mathrm{dd^{c}}\theta^{k}=0$ for any
$k=1,\dots,n-1$), then we obtain
$\widehat{v}_{-}(V,\theta|_{V})=\int_{V}\theta^{\dim V}$ by [GL22, Lemma 4.4],
provided that $v_{+}(X,\omega_{X})<\infty$. This directly implies the result
of Collins–Tosatti in the case of compact complex manifolds belonging to the
Fujiki class $\mathcal{C}$.

A direct application of Theorem A is to establish the Nakai–Moishezon
criterion for some compact non-Kähler manifolds, which generalizes the
fundamental result of Demailly–Păun [DP04] for compact Kähler manifolds, and
Buchdahl [Buc00] and Lamari [Lam99] for compact non-Kähler surfaces. Our proof
relies essentially on the mass concentration result of Demailly–Păun, whose
proof was simplified by Chiose [Chi16a] and Popovici [Pop16]. Using Theorem A
allows one to avoid the inductive argument on dimension established in [DP04].
Inspired by the works of Collins–Tosatti [CT16], and Das–Hacon–Păun [DHP22],
we establish the Nakai–Moishezon criterion on a possibly singular compact
analytic normal variety, but the proof relies on the induction on dimension.

######  Theorem B.

Let $X$ be a compact analytic normal variety of pure dimension $n$ and
$\omega_{X}$ be a Hermitian metric on $X$ such that
$\mathrm{dd^{c}}\omega_{X}^{k}=0$ for any $k=1,\ldots,n-1$. Let $\theta$ be a
smooth real (1,1) form on $X$ such that

  1. (1)

$\mathrm{dd^{c}}\theta^{k}=0$, for any $k=1,\dots,n-1$;

  2. (2)

for any $k=1,\ldots,n$,

$\int_{X}\theta^{k}\wedge\omega_{X}^{n-k}>0;$

  3. (3)

for every irreducible analytic subvariety $V\subset X$ with $\dim V>0$,

$\int_{V}\theta^{k}\wedge\omega_{X}^{\dim V-k}>0,\,\;\forall\,k=1,\dots,\dim
V.$

Then the $\partial\bar{\partial}$-class $[\theta]$ is Hermitian.

We remark that the condition (2) can be absorbed in the (3) when $V=X$. In the
case of complex smooth surfaces, thanks to [Buc99, Proposition 5], we can
relax the condition $(3)$ to require only that $\int_{V}\theta>0$ for every
irreducible curve $V\subset X$ with strictly negative self-intersection
$V\cdot V<0$. We refer to Sect. 2.3 for the definition of differential forms,
Hermitian metrics, and integrals on singular analytic varieties.

The following application is to show the existence of solutions to degenerate
complex Monge–Ampère equations in the context of nef and big form. We extend
the result of Guedj–Lu [GL23, Theorem D] for semi-positive reference forms to
nef ones. We precisely obtain the following.

######  Theorem C.

Let $(X,\omega_{X})$ an $n$-dimensional compact complex manifold. Let $\theta$
be a smooth real (1,1) form whose $\partial\bar{\partial}$-class $[\theta]$ is
nef and big. Let $f\in\mathcal{C}^{\infty}(X,\mathbb{R})$. Then, there exists
a constant $c>0$ and an $\theta$-psh function $\varphi$ which is smooth on a
Zariski open set $U$ of $X$, $\sup_{X}\varphi=0$ and

$(\theta+\mathrm{dd^{c}}\varphi)^{n}=cf\omega_{X}^{n}$

on $U$. More precisely, $U$ is taken to be the complement
$X\backslash\operatorname{{\rm Null}}(\theta)$ of the null locus of $\theta$.

Extending the works of Berman [Ber19] to the non-Kähler case, we study the
partial regularity of $\theta$-plurisubharmonic envelope when the
$\partial\bar{\partial}$-class $[\theta]$ is assumed to be nef and big. Fixing
a function $f\in\mathcal{C}^{\infty}(X,\mathbb{R})$ and a Hermitian metric
$\omega_{X}$ on $X$, for $\beta>0$ , we consider the following complex
Monge–Ampère equation for a smooth solution $\varphi_{\beta}$:

(1.2)
$(\theta+2^{-\beta}\omega_{X}+\mathrm{dd^{c}}\varphi_{\beta})^{n}=e^{\beta(\varphi_{\beta}-f)}\omega_{X}^{n}.$

By the fundamental works of Cherrier [Che87] and Tosatti–Weinkove [TW10a], the
equation (1.2) admits a unique smooth solution. Indeed, since $\theta$ is nef
so its $\partial\bar{\partial}$-class $[\theta+2^{-\beta}\omega_{X}]$ is
Hermitian for any $\beta>0$. This shows the existence of smooth solutions and
the form
$\omega_{\beta}:=\theta+2^{-\beta}\omega_{X}+\mathrm{dd^{c}}\varphi_{\beta}$
defines a Hermitian metric in $[\theta+2^{-\beta}\omega_{X}]$. Since
$[\theta]$ is big, we can define the so-called $\theta$-plurisubharmonic
envelope $P_{\theta}(f)$ of $f$ as the largest $\theta$-plurisubharmonic
function lying below $f$. We study the regularity of the envelope outside the
null locus.

######  Theorem D.

Let $\theta$ be a smooth real (1,1) form on a $n$-dimensional compact complex
manifold $X$ whose $\partial\bar{\partial}$ class $[\theta]$ is nef and big.
Let $f\in\mathcal{C}^{\infty}(X,\mathbb{R})$ and let $P_{\theta}(f)$ denote
the corresponding $\theta$-plurisubharmonic envelope of $f$. Let
$\varphi_{\beta}$ be the unique smooth solution of the complex Monge–Ampère
equation (1.2). Then, as $\beta\to\infty$, the functions $\varphi_{\beta}$
converge to $P_{\theta}(f)$ in $\mathcal{C}^{1,\alpha}_{\rm
loc}(X\backslash\operatorname{{\rm Null}}(\theta))$ for any $\alpha\in]0,1[$.

The proof is an adaptation of [Ber19, Theorem 1.2] with slight changes. When
$\theta$ is Hermitian, a strong result was proved in [CZ19] that
$P_{\theta}(f)$ belongs to $\mathcal{C}^{1,1}(X)$.

Finally, we study finite time singularities of the Chern–Ricci flow on compact
complex manifolds, partially answering a question raised by Tosatti–Weinkove.
Let $\omega=\omega(t)$ be a solution of the Chern–Ricci flow on $X$, for
$t\in[0,T[$, starting at a Hermitian metric $\omega_{0}$,

(1.3) $\frac{\partial\omega}{\partial t}=-\operatorname{{\rm
Ric}}(\omega),\quad\omega(0)=\omega_{0}$

where $\operatorname{{\rm Ric}}(\omega)$ is the Chern–Ricci form (also called
first Chern form) of the Hermitian metric $\omega$. It is classical that
solving the Chern–Ricci flow boils down to solving a nonlinear parabolic
scalar equation in $\varphi=\varphi(t)$ of the form

$\frac{\partial\varphi}{\partial
t}=\log\frac{(\theta_{t}+\mathrm{dd^{c}}\varphi)^{n}}{\omega_{0}^{n}},\quad\theta_{t}+\mathrm{dd^{c}}\varphi>0,\;\varphi(0)=0$

where $\theta_{t}:=\omega_{0}-t\operatorname{{\rm Ric}}(\omega_{0})$. Tosatti
and Weinkove [TW15] characterized the maximal existence time $T$ of the flow
as

$T=\sup\\{t>0:\exists\,\psi\in\mathcal{C}^{\infty}(X)\;\text{with}\;\theta_{t}+\mathrm{dd^{c}}\psi>0\\}$

by this, we mean that $T$ is such that for all $t<T$, the
$\partial\bar{\partial}$-class $[\theta_{t}]$ is Hermitian. Suppose the flow
develops a finite time singularity $T<\infty$. We observe that the limiting
$\partial\bar{\partial}$ class $[\theta_{T}]$ is nef but not Hermitian. We
define the set of singularities of the flow, denoted by
$\operatorname{\textrm{{Sing}}}(\omega(t))$, to be the complement of the set
of all points $x\in X$ such that there is a neighborhood $U\ni x$ on which
$\omega(t)$ converge smoothly to a Hermitian metric $\omega_{T}$ as $t\to
T^{-}$. Furthermore, we can show that the singularity set of the Chern–Ricci
flow is equal to the set on which the Chern scalar curvature blows up. The
latter was studied by Gill–Smith [GS15], generalizing Zhang’s analog result
for Kähler–Ricci flows.

Tosatti and Weinkove [TW22, Question 6.1] posed the question of whether
singularities of the Chern–Ricci flow develop precisely along analytic
subvarieties of $X$? In the Kähler case, this question was initially raised by
Feldman–Ilmanen–Knopf and later affirmatively answered by Collins–Tosatti
[CT15, Theorem 1.5]. Gill–Smith [GS15, Theorem 1.2] gave an affirmative answer
for the case when $X$ is a complex surface and $\omega_{0}$ is Gauduchon. As
observed in [TW22, Secion 6], the answer is affirmative when the volume with
respect to $\omega(t)$ shrinks to zero as $t\to T$. We investigate non-
collapsing the case when $\widehat{v}_{-}(X,\theta_{T})>0$ and provide an
answer to this question on some underlying manifolds.

######  Theorem E.

Let $(X,\omega_{0})$ be a compact Hermitian manifold of dimension $n$. Let
$\omega(t)$ be a solution of the Chern–Ricci flow starting at $\omega_{0}$,
which develops a singularity at finite time $T$. Assume that the limiting form
$\theta_{T}=\omega_{0}-T\operatorname{{\rm Ric}}(\omega_{0})$ satisfies
$\widehat{v}_{-}(X,\theta_{T})>0$. Then if either

  * •

$n=2$ and $\omega_{0}$ is arbitrary,

  * •

or $n\geq 3$, $\mathrm{dd^{c}}\omega_{0}=0$, ${\rm d}\omega_{0}\wedge{\rm
d^{c}}\omega_{0}=0$,

then the metrics $\omega(t)$ develop singularities precisely along the
analytic set $\operatorname{{\rm Null}}(\theta_{T})$.

This theorem generalizes Gill–Smith’s result when the underlying manifold $X$
is a surface, but the initial metric $\omega_{0}$ is given to be arbitrary and
also partially answers the question of Tosatti–Weinkove. With assumptions in
this theorem, we have that the limiting form $\theta_{T}$ is big, so its null
locus $\operatorname{{\rm Null}}(\theta_{T})$ is a proper analytic subset of
$X$ (since $[\theta_{T}]$ is not Hermitian), and our theorem shows that no
singularities develop on its complement. We have seen from Theorem A that
$\operatorname{{\rm Null}}(\theta_{T})$ is precisely the union of all purely
positive-dimensional irreducible subvarieties $V$ of $X$ with
$v_{-}(V,\omega(t))$ shrinking to zero as $t\to T^{-}$, so singularities of
the flow develop at least there. We will see in Theorem 5.1 that the scalar
curvature of $\omega(t)$ is locally uniformly bounded on compact sets outside
singularities. Furthermore, it should be emphasized that in the setting of our
theorem, as $t\to T^{-}$, the metrics $\omega(t)$ converge weakly to a
positive current $\omega_{T}$ which is smooth on
$X\backslash\operatorname{{\rm Null}}(\theta_{T})$ and has minimal
singularities.

A crucial case of such finite-time singularities occurs when the
$\partial\bar{\partial}$-class $[\theta_{T}]$ contains a pullback of a
Hermitian metric via the blowdown of an exceptional divisor to a point.
Interesting results were obtained by Tosatti–Weinkove [TW13], Tô [Tô18], and
Nie [Nie17], in which they proved that the Chern–Ricci flow performs a
canonical surgical contraction.

This paper is structured as follows. In Sect. 2, we introduce notation and
definitions, and briefly review fundamental results extended to the non-Kähler
context. In Sect. 3, we establish an extension-type theorem for Hermitian
currents with analytic singularities, which enables us to prove Theorem A, and
we also prove Theorem B there. Sect. 4 is dedicated to the study of degenerate
complex Monge–Ampère equations, where we prove Theorems C and D. Finally, we
investigate finite time singularities of the Chern–Ricci flow and prove
Theorem E in Sect. 5.

### Notation

Unless otherwise mentioned, $X$ denotes a compact Hermitian manifold with a
reference Hermitian metric $\omega_{X}$. We interchangeably use the Hermitian
metric $\omega_{X}$ and its associated Riemannian metric $g$. We denote by
${\rm d}=\partial+\bar{\partial}$ and ${\rm
d^{c}}=\frac{i}{2}(\bar{\partial}-\partial)$ so that
$\mathrm{dd^{c}}=i\partial\bar{\partial}.$ Throughout the paper, we use $C$ to
represent a positive constant, which may vary from line to line and can be
uniformly controlled.

### Acknowledgement

We would like to express our gratitude to V. Tosatti for suggesting this
problem, his interest in this work, and his valuable comments. We are grateful
to T.-D. Tô for insightful discussions and C. H. Lu for carefully reading the
first draft. This work is partially supported by the ANR project PARAPLUI.

##  2\. Preliminaries

Let $X$ be a $n$-dimensional compact complex manifold. Fix $\omega_{X}$ a
reference Hermitian metric.

###  2.1. $\partial\bar{\partial}$-classes

It is well-known that the $\partial\bar{\partial}$-lemma does not hold on an
arbitrarily complex manifold, so it is better to work with the
$\partial\bar{\partial}$-class of differential forms. We denote by
$\mathcal{A}^{p,q}(X,\mathbb{C})$ (resp. $\mathcal{A}^{p,q}(X,\mathbb{R})$)
the space of smooth (resp. real) forms of bidgree $(p,q)$. We say that two
forms $\theta,\eta\in\mathcal{A}^{p,q}(X,\mathbb{C})$ are equivalent if they
are modulo to the $\partial\bar{\partial}$-exact ones, i.e., there exists a
smooth form $\psi\in\mathcal{A}^{p-1,q-1}(X,\mathbb{C})$ such that

$\theta-\eta=\mathrm{dd^{c}}\psi.$

Since then, we let $[\theta]_{\partial\bar{\partial}}$ denote the
$\partial\bar{\partial}$-class with smooth representative $\theta$. For
simplicity, we skip the subscript $\partial\bar{\partial}$, and unless we
emphasize the $\partial\bar{\partial}$ class $[\theta]$ is interpreted as the
set of forms $\theta+\mathrm{dd^{c}}\psi$ for all
$\psi\in\mathcal{C}^{\infty}(X)$.

We remark that a ${\rm d}$-exact form is not $\partial\bar{\partial}$-exact in
the general context of non-Kähler manifolds. A compact complex manifold $X$ is
said to be a $\partial\bar{\partial}$-manifold if it satisfies the exactness
property: for any $\rm d$-closed pure-type form $u$ on $X$, $u$ is $d$-exact
$\Longleftrightarrow$ $u$ is $\partial\bar{\partial}$-exact for any $d$-closed
form $u$ on $X$. In particular, a compact Kähler manifold is a
$\partial\bar{\partial}$-manifold.

######  Definition 2.1.

A smooth real (1,1) form $\theta$ is called nef if for any $\varepsilon>0$
there exists a form $\theta_{\varepsilon}\in[\theta]$ such that
$\theta_{\varepsilon}\geq-\varepsilon\omega_{X}$, i.e.,
$\exists\,\psi_{\varepsilon}\in\mathcal{C}^{\infty}(X,\mathbb{R})$ such that
$\theta+\mathrm{dd^{c}}\psi_{\varepsilon}\geq-\varepsilon\omega_{X}$. In this
case, the $\partial\bar{\partial}$-class $[\theta]$ is said to be nef.

If there exists a Hermitian metric $\omega\in[\theta]$, then we say that the
$\partial\bar{\partial}$-class $[\theta]$ is Hermitian/positive.

It should be emphasized that unlike in the Kähler case, even if two Hermitian
metrics $\omega$ and $\omega^{\prime}$ are in the same
$\partial\bar{\partial}$ class, i.e., there is
$\varphi\in\mathcal{C}^{\infty}(X)$ such that
$\omega^{\prime}=\omega+\mathrm{dd^{c}}\varphi$, the volume with respect to
$\omega$ does not equal the one with respect to $\omega^{\prime}$. It was
shown by Chiose [Chi16b, Theorem 0.1] that the preservation of Monge–Ampère
masses is equivalent to the condition introduced by Guan–Li [GL10]:
$\int_{X}\omega^{n}=\int_{X}\omega_{X}^{n}$ for all $\omega\in[\omega_{X}]$
Hermitian if and only if $\mathrm{dd^{c}}\omega_{X}=0$ and ${\rm
d}\omega_{X}\wedge{\rm d^{c}}\omega_{X}=0$. The latter condition is equivalent
to $\mathrm{dd^{c}}\omega_{X}^{k}=0$ for all $k=1,\dots,n-1$; see also
[Chi16b, Theorem 0.2] for more equivalent conditions.

We recall some quantities on Monge–Ampère volume bounds introduced by Guedj–Lu
[GL22].

######  Definition 2.2.

We define

$v_{+}(X,\omega_{X})=\sup\left\\{\int_{X}\omega^{n}:\forall\,\omega\in[\omega_{X}],\,\omega\,\text{Hermitian}\right\\}$

and

$v_{-}(X,\omega_{X})=\inf\left\\{\int_{X}\omega^{n}:\forall\,\omega\in[\omega_{X}],\,\omega\,\text{Hermitian}\right\\}.$

A question raised by Guedj–Lu [GL22] is whether $v_{+}(X,\omega_{X})<+\infty$
and/or $v_{-}(X,\omega_{X})>0$. We remark that these conditions somewhat
depend on the complex structure, but they are independent of the choice of a
Hermitian metric; see [GL22, Prop. 3.2]. Furthermore, they are bimeromorphic
invariants, and so is the condition ($H_{X}$).

######  Theorem 2.3 ([GL22, Thm. 3.7]).

Let $\pi:X\to Y$ be a bimeromorphic map between compact complex manifolds
equipped with Hermitian metrics $\omega_{X}$ and $\omega_{Y}$. Then
$\omega_{X}$ satisfies the condition ($H_{X}$) if and only if $\omega_{Y}$
satisfies the condition ($H_{X}$).

######  Definition 2.4.

If $\theta$ is nef, we can define

$\widehat{v}_{-}(X,\theta):=\inf_{\varepsilon>0}v_{-}(X,(1-\varepsilon)\theta+\varepsilon\omega_{X}).$

Although the form $(1-\varepsilon)\theta+\varepsilon\omega_{X}$ is not
positive, one could find by definition, for any $\varepsilon>0$, a smooth
function $\varphi_{\varepsilon}$ such that
$\theta+\mathrm{dd^{c}}\varphi_{\varepsilon}\geq-\varepsilon\omega_{X}$ so
that

$(1-\varepsilon)\theta+\varepsilon\omega_{X}+\mathrm{dd^{c}}(1-\varepsilon)\varphi_{\varepsilon}\geq\varepsilon^{2}\omega_{X}.$

Hence the quantity $v_{-}(X,(1-\varepsilon)\theta+\varepsilon\omega_{X})$ is
interpreted as
$v_{-}(X,(1-\varepsilon)\theta+\varepsilon\omega_{X}+\mathrm{dd^{c}}(1-\varepsilon)\varphi_{\varepsilon})$.
We expect that the definition above coincides with the one when $\theta$ is
closed. The necessary condition is that $\theta$ satisfies the Guan–Li
property.

######  Lemma 2.5 ([GL22, Lemma 4.4]).

If $v_{+}(X,\omega_{X})<\infty$ and $\theta$ is a smooth nef (1,1) form
satisfying $\mathrm{dd^{c}}\theta^{k}=0$ for any $k=1,\ldots,n-1$ then
$\widehat{v}_{-}(X,\theta)=\int_{X}\theta^{n}$. Moreover, for any positive-
dimensional closed submanifold $V\subset X$,
$\widehat{v}_{-}(V,\theta|_{V})=\int_{V}\theta^{\dim V}$.

The proof is identical to that of [GL22, Lemma 4.4], which we omit here.
Notice that the Guan–Li property is sufficient to guarantee the preservation
of Monge–Ampère masses.

###  2.2. Quasi-plurisubharmonic functions and Hermitian currents

Recall that an upper semi-continuous function
$\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is quasi-plurisubharmonic
(quasi-psh for short) if it is locally described as the sum of a
plurisubharmonic (psh) function and a smooth function.

######  Definition 2.6.

Let $\theta$ be a smooth (1,1) form on $X$. One says that

  * •

a quasi-psh function $\varphi$ is $\theta$-plurisubharmonic ($\theta$-psh for
short) if $\theta+\mathrm{dd^{c}}\varphi\geq 0$ in the sense of currents. We
let $\operatorname{{\rm PSH}}(X,\theta)$ denote the space of all $\theta$-psh
functions on $X$ which is not identically $-\infty$;

  * •

$T$ is a positive (1,1) current on $X$ in the $\partial\bar{\partial}$-class
$[\theta]$ if $T=\theta+\mathrm{dd^{c}}\varphi$ for some
$\varphi\in\operatorname{{\rm PSH}}(X,\theta)$, and vice versa.

For any positive (1,1) current $T$, we let $\operatorname{\textrm{{Sing}}}(T)$
denote the singular set of $T$, namely, if $T=\theta+\mathrm{dd^{c}}\varphi$
then $\operatorname{\textrm{{Sing}}}(T)=\\{\varphi=-\infty\\}$.

We refer the reader to nice references [Dem12, GZ17] for more details about
the basic properties of psh functions and positive currents.

The complex Monge–Ampère measure $(\theta+\mathrm{dd^{c}}u)^{n}$ is well-
defined for any $\theta$-psh function $u$, which is bounded on $X$, as follows
from Bedford–Taylor’s theory; cf. [DK12] or [GL22] for a brief recap. We
recall the following maximum principle.

######  Lemma 2.7 (Maximum principle).

Let $\varphi$, $\psi$ be bounded $\theta$-psh functions in $U\subseteq X$ such
that $\varphi\geq\psi$. Then

$\mathbf{1}_{\\{\varphi=\psi\\}}(\theta+\mathrm{dd^{c}}\varphi)^{n}\geq\mathbf{1}_{\\{\varphi=\psi\\}}(\theta+\mathrm{dd^{c}}\psi)^{n}.$

###### Proof.

The proof directly follows Bedford–Taylor’s maximum principle [GZ17, Theorem
3.27] in the local context. See also [GL22, Lemma 1.2] for a proof in the
global one. ∎

######  Definition 2.8.

Let $\mathcal{I}\subset\mathcal{O}_{X}$ be an analytic coherent ideal sheaf
and $c>0$. A quasi-psh function $\varphi$ on X is said to have almost analytic
singularities of type $(\mathcal{I},c)$ if the following are satisfied

  1. (i)

for any $x\in X$, there exists a neighborhood $U$ of $x$ such that on $U$ we
have

$\varphi=c\log\left(\sum_{j}|f_{j}|^{2}\right)+g$

where the $f_{j}$’s are local generators of a coherent sheaf $\mathcal{I}$,
and $g$ is a bounded function on $U$,

  2. (ii)

there is a (global) proper modification $\pi:X^{\prime}\to X$ of $X$, obtained
as a finite composition of blow-ups with smooth centers, and an effective
divisor $\sum_{\ell}a_{\ell}D_{\ell}$ with normal crossings such that one can
write $\varphi\circ\pi$ locally as

$\varphi\circ\pi=c\sum_{\ell}a_{\ell}\log|g_{\ell}|^{2}+h$

where $(g_{\ell}=0)$ are local equations of the divisors $D_{\ell}$ and $h$ is
a smooth function.

If a current $T=\theta+\mathrm{dd^{c}}\varphi$ with $\varphi$ a quasi-psh
function having almost analytic singularities (along subscheme
$V(\mathcal{I})$ induced by $\mathcal{I}$), then one says that $T$ has almost
analytic singularities.

Let us emphasize that the definition above is a bit lengthy but turns out to
be the most suitable one for this paper. We will see below the
$\partial\bar{\partial}$-class with appropriate positivity contains plenty of
currents with such singularities.

######  Definition 2.9.

We say that a real (1,1)-current $T$ is a Hermitian current if there exists a
Hermitian metric $\omega$ such that $T\geq\omega$.

The $\partial\bar{\partial}$-class $[\theta]$ is said to be big if it contains
a Hermitian current, i.e., there exists $\varphi\in\operatorname{{\rm
PSH}}(X,\theta)$ such that $T=\theta+\mathrm{dd^{c}}\varphi\geq\omega$ for
some Hermitian form $\omega$. In this case, the smooth representative $\theta$
is also called a big (1,1) form.

We introduce the notion of psh envelopes, understood as the least singularity
function lying below a prescribed one.

######  Definition 2.10.

Given a Borel function $f:X\to\mathbb{R}$, we define the $\theta$-psh envelope
of $f$ by

$P_{\theta}(f):=(\sup\\{u\in\operatorname{{\rm PSH}}(X,\theta):u\leq
f\\})^{*}$

where the ∗ means that we take the upper semi-continuous regularization.

The $\theta$-psh envelope $P_{\theta}(f)$ might be $\equiv-\infty$ when the
$\partial\bar{\partial}$-class $[\theta]$ contains no positive currents. When
$\theta$ is big, and $f$ is bounded from below, $P_{\theta}(f)$ is well-
defined as a $\theta$-psh function.

We recall the basic type of regularization due to Demailly; cf. [DP04, Lemma
2.1, Theorem 3.2].

######  Theorem 2.11 (Demailly’s regularization theorem).

Let $X$ be a compact complex manifold equipped with a Hermitian form $\omega$.
Let $T=\theta+\mathrm{dd^{c}}\varphi$ be a real (1,1) current; i.e., $\theta$
is a smooth real (1,1) form and $\varphi$ is quasi-psh. Suppose that
$T\geq\gamma$ for some smooth (1,1) form $\gamma$. Then, there exists a
decreasing sequence of quasi-psh functions $\varphi_{m}$ such that

  1. (1)

$T_{m}=\theta+\mathrm{dd^{c}}\varphi_{m}$ are currents with almost analytic
singularities;

  2. (2)

$T_{m}\to T$ weakly, and $\varphi_{m}$ decreases to $\varphi$ as $m\to\infty$;

  3. (3)

$T_{m}\geq\gamma-\varepsilon_{m}\omega$ for some sequence
$\varepsilon_{m}\searrow 0$.

By replacing $T$ (resp. $\gamma$) with $T-\theta$ (resp. $\gamma-\theta$) we
can work with $T=\mathrm{dd^{c}}\varphi$ a closed quasi-positive current. The
proof relies on the appropriate technique using an asymptotic Bergman kernel
procedure as we describe what happens locally. If $\varphi$ is a psh function
on the ball $B\subset\mathbb{C}^{n}$ one considers for any $m\geq 1$ the
Hilbert space $\mathcal{H}(B,m\varphi)$ of $L^{2}$ holomorphic function on $B$
such that $\int_{B}|f|^{2}e^{-m\varphi}dV<\infty$. We choose a Hilbert basic
$(f_{k}^{m})_{k\in\mathbb{N}}$ of $\mathcal{H}(B,m\varphi)$ and put
$\varphi_{m}:=\frac{1}{m}\log\left(\sum_{k\in\mathbb{N}}|f_{k}^{m}|^{2}\right)$.
One can check that $\varphi_{m}$ is well-defined, psh, with analytic
singularities and almost increasing. The difficult part is then to prove that
$\varphi_{m}$ converges to $\varphi$ as $m$ goes to infinity. To get the
global approximations, one glues the local ones by using a partition of unity
argument. In this gluing process, a small loss of positivity is established
due to the change of holomorphic coordinates. This is where the
$\varepsilon_{m}$’s come from. For each $m$, the corresponding modification
$\mu_{m}$ is then obtained by principalization of singularities of the
multiplier ideal sheaf $\mathcal{I}(m\varphi)$ so that
$\mu_{m}^{*}\mathcal{I}(m\varphi)$ is the principal ideal sheaf associated
with a normal crossing divisor, due to the fundamental result of Hironaka; cf.
[Wło09].

From the above regularization, we are able to approximate a positive current
by Hermitian currents with almost analytic singularities, with reasonable
control of singularities. In particular, a big $\partial\bar{\partial}$-class
$[\theta]$ contains various Hermitian currents with almost analytic
singularities.

We introduce the definition of the non-Hermitian locus of a big
$\partial\bar{\partial}$-class, analog to the one in [Bou04].

######  Definition 2.12 (non-Hermitian locus).

If $\theta$ is a big (1,1) form on $X$, we define its non-Hermitian locus to
be the set

$\operatorname{E_{\textrm{{nH}}}}(\theta)=\bigcap_{T\in[\theta]}\operatorname{\textrm{{Sing}}}(T)$

where the intersection is taken over all Hermitian currents with almost
analytic singularities in the $\partial\bar{\partial}$-class $[\theta]$.

We investigate the bigness in some particular context of fixed Hermitian
metric.

######  Theorem 2.13 (Guedj–Lu).

Let $X$ be a compact complex manifold equipped with a Hermitian metric
$\omega_{X}$ satisfying $v_{+}(\omega_{X})<+\infty$. Let $\theta$ be a nef
(1,1) form with $\widehat{v}_{-}(X,\theta)>0$. Then the
$\partial\bar{\partial}$-class $[\theta]$ is big, i.e., there exists a
Hermitian current $T$ in $[\theta]$ with almost analytic singularities.

###### Proof.

This was proved in [GL22, Theorem 4.6], which was inspired by the one of
Chiose [Chi16a] and Popovici [Pop16] relying on the characterization result of
Lamari [Lam99]. For the reader’s convenience, we give the proof here.
According to the Hahn–Banach theorem as in [Lam99, Lemme 3.3], it thus
suffices to show that there exists a constant $\delta_{0}>0$ such that

$\int_{X}\theta\wedge\eta^{n-1}\geq\delta_{0}\int_{X}\omega_{X}\wedge\eta^{n-1}$

for all Gauduchon metrics $\eta$. Suppose by contradiction that for each
$\varepsilon>0$ there exists Gauduchon metrics $\eta_{\varepsilon}$ such that

$\int_{X}\theta\wedge\eta_{\varepsilon}^{n-1}\leq\varepsilon\int_{X}\omega_{X}\wedge\eta_{\varepsilon}^{n-1}.$

We can normalize $\eta_{\varepsilon}$ so that
$\int_{X}\omega_{X}\wedge\eta_{\varepsilon}^{n-1}=1$. We fix a function
$\psi_{\varepsilon}\in\mathcal{C}^{\infty}(X)$ such that
$\theta_{\varepsilon}:=\theta+\varepsilon\omega_{X}+\mathrm{dd^{c}}\psi_{\varepsilon}$
is a Hermitian metric. By the main result of [TW10a] there exist
$c_{\varepsilon}>0$ and $u_{\varepsilon}\in\operatorname{{\rm
PSH}}(X,\theta_{\varepsilon})\cap\mathcal{C}^{\infty}(X)$ such that
$\sup_{X}u_{\varepsilon}=0$ and

$(\theta_{\varepsilon}+\mathrm{dd^{c}}u_{\varepsilon})^{n}=c_{\varepsilon}\omega_{X}\wedge\eta_{\varepsilon}^{n-1}.$

By normalization we have

$c_{\varepsilon}=\int_{X}(\theta_{\varepsilon}+\mathrm{dd^{c}}u_{\varepsilon})^{n}\geq\widehat{v}_{-}(X,\theta)>0.$

We apply [GL22, Lemma 4.13] which reformulates the one in [Pop16, Lemma 3.1]
to obtain

(2.1)
$\int_{X}(\theta_{\varepsilon}+\mathrm{dd^{c}}u_{\varepsilon})\wedge\eta^{n-1}_{\varepsilon}\times\int_{X}(\theta_{\varepsilon}+\mathrm{dd^{c}}u_{\varepsilon})^{n-1}\wedge\omega_{X}\geq\frac{c_{\varepsilon}}{n}.$

The first term on the left-hand side can be written as
$\int_{X}(\theta+\varepsilon\omega_{X})\wedge\eta_{\varepsilon}^{n-1}$ since
$\eta_{\varepsilon}$ is Gauduchon and by assumption,

$\int_{X}(\theta+\varepsilon\omega_{X})\wedge\eta_{\varepsilon}^{n-1}\leq
2\varepsilon.$

For the second term, it follows by the assumption that

(2.2)
$\int_{X}(\theta_{\varepsilon}+\mathrm{dd^{c}}u_{\varepsilon})^{n-1}\wedge\omega_{X}\leq
Mv_{+}(\omega_{X})$

which is bounded from above. Therefore we obtain

$2\varepsilon Mv_{+}(\omega_{X})\geq\frac{\widehat{v}_{-}(X,\theta)}{n}>0$

which is a contradiction as $\varepsilon\to 0$. The positive current can be
taken to have almost analytic singularities thanks to Theorem 2.11. ∎

######  Theorem 2.14.

Let $\theta$ be a big (1,1)-form. Then there exists a Hermitian current
$T_{0}$ with almost analytic singularities such that
$\operatorname{E_{\textrm{{nH}}}}(\theta)=\operatorname{\textrm{{Sing}}}(T_{0})$.
In particular, the non-Hermitian locus of a big class $[\theta]$ is an
analytic subvariety of $X$.

###### Proof.

The proof is quite close to that of [Bou04, Theorem 3.17]. We denote by
$\mathcal{S}$ the set of all Hermitian currents with almost analytic
singularities in the class $[\theta]$. Fix
$T_{j}=\theta+\mathrm{dd^{c}}\varphi_{j}\in\mathcal{S}$ for $j=1,2$. First, we
claim that there exists a Hermitian current with almost analytic singularities
$T$ such that
$\operatorname{\textrm{{Sing}}}(T)\subset\operatorname{\textrm{{Sing}}}(T_{1})\cap\operatorname{\textrm{{Sing}}}(T_{2})$.
Indeed, if we set $\varphi_{3}=\max(\varphi_{1},\varphi_{2})$ then
$T_{3}=\theta+\mathrm{dd^{c}}\phi_{3}$ is a Hermitian current such that
$(\varphi_{3}=-\infty)\subset\operatorname{\textrm{{Sing}}}(T_{1})\cap\operatorname{\textrm{{Sing}}}(T_{2})$.
By Theorem 2.11, we can find a Hermitian current
$T=\theta+\mathrm{dd^{c}}\varphi\in\mathcal{S}$ with almost analytic
singularities such that $\varphi\geq\varphi_{3}$ so
$\operatorname{\textrm{{Sing}}}(T)\subset\operatorname{\textrm{{Sing}}}(T_{1})\cap\operatorname{\textrm{{Sing}}}(T_{2})$.
This proves our claim.

Thanks to the claim above and Theorem 2.11 again, we can construct a sequence
of currents $T_{m}=\theta+\mathrm{dd^{c}}\varphi_{m}\in\mathcal{S}$ such that
$\operatorname{\textrm{{Sing}}}(T_{m})$ is a decreasing sequence of anlytic
subsets. By the strong Noetherian property,
$\operatorname{\textrm{{Sing}}}(T_{m})$ must be stationary for $m\geq m_{0}$.
Therefore, we obtain
$\operatorname{E_{\textrm{{nH}}}}(\theta)=\operatorname{\textrm{{Sing}}}(T_{0})$
for $T_{0}:=T_{m_{0}}$, as desired.

Moreover, if $\operatorname{E_{\textrm{{nH}}}}(\theta)=\varnothing$ then
$\operatorname{\textrm{{Sing}}}(T_{0})=\varnothing$, it means that $T_{0}$ is
smooth on $X$, so $T_{0}$ is in fact a Hermitian form. ∎

To summarize, we have the following.

######  Theorem 2.15.

Let $X$ be a compact complex manifold equipped with a positive form
$\omega_{X}$ satisfying $v_{+}(\omega_{X})<+\infty$. Let $\theta$ be a nef
(1,1) form with $\widehat{v}_{-}(X,\theta)>0$. Then there exists a Hermitian
current $T$ in the $\partial\bar{\partial}$-class $[\theta]$ with almost
analytic singularities such that

$\operatorname{\textrm{{Sing}}}(T)=\operatorname{E_{\textrm{{nH}}}}(\theta).$

We recall the following lemma, which implies that the non-Hermitian locus of a
big form has no isolated points.

######  Lemma 2.16.

If $\theta$ is a big (1,1)-form, then the analytic set
$\operatorname{E_{\textrm{{nH}}}}(\theta)$ has no isolated points.

###### Proof.

We refer to [CT15, Lemma 3.1]. The closeness is not involved. ∎

###  2.3. Extension on complex analytic spaces

We give here a short recap of the extension of the pluripotential theory to
compact complex spaces. We refer the interested readers to [Dem85] for more
details.

Let $X$ be a reduced complex analytic space of pure dimension $n$. We will let
$X_{\rm reg}$ denote the locus of regular points of $X$, and $X_{\rm
sing}:=X\backslash X_{\rm reg}$ the locus of singular ones. We see that
$X_{\rm reg}$ is a complex manifold of dimension $n$ and $X_{\rm sing}$ is an
analytic subset of $X$ of complex codimension $\geq$ 1\. If
$X=\bigcup_{\alpha}U_{\alpha}$ is an open covering, we consider local
embeddings $j_{\alpha}:U_{\alpha}\hookrightarrow\mathbb{C}^{N}$ onto analytic
subsets of $\mathbb{C}^{N}$ for some $N\geq 1$. By these local embeddings, one
can define the sheaves of smooth $(p,q)$-forms $\mathcal{A}^{p,q}$, and the
space of currents is, by definition, the dual of the space of differential
forms as in the smooth case. The differential operators ${\rm d}$, $\partial$,
$\bar{\partial}$, ${\rm d^{c}}$, and $\mathrm{dd^{c}}$ satisfy the usual rules
and are then well defined by duality. We can also define the concepts of
Hermitian $\partial\bar{\partial}$-classes, nef
$\partial\bar{\partial}$-classes, big $\partial\bar{\partial}$-classes, etc.

We next move on to the notion of psh functions. Similarly, by local embeddings
above, we say that a function $u:X\to\mathbb{R}\cup\\{-\infty\\}$ is psh on
$X$ if it is locally the restrictions of a psh function defined an open set
$\Omega_{\alpha}$ of $\mathbb{C}^{N}$ containing $j_{\alpha}(U_{\alpha})$.

Another way to define a psh function is introduced by Demailly: a function $u$
is weakly psh on $X$ if it is locally bounded from above on $X$ and its
restriction to the complex manifold $X_{\rm reg}$ is psh. One can extend it to
$X$ by

$u^{*}(x):=\limsup_{X_{\rm reg}\ni y\to x}u(y).$

The function $u^{*}$ is upper semi-continuous, locally integrable on $X$, and
$\mathrm{dd^{c}}u^{*}$ is a positive current. Since $X$ is locally integrable,
thanks to [Dem85, Théorème 1.10], $u^{*}$ is psh on $X$. In the same way, one
can define the notion of quasi-psh functions with almost analytic
singularities, Hermitian currents, non-Hermitian loci, etc.

We end this section by discussing integration over analytic (irreducible)
subvarieties of a smooth complex manifold $X$. If $V$ is an analytic
subvariety of $X$ of pure dimension $k>0$, for any smooth $(k,k)$ form
$\alpha$ on $X$, then the integral

$\int_{V}\alpha:=\int_{V_{\rm reg}}\alpha$

is finite thanks to the fundamental result of Lelong; see e.g. [GH78, page
32]. In the same way, we can define the quantities $v_{-}(V,\omega_{V})$,
$v_{+}(V,\omega_{V})$ and $\widehat{v}_{-}(V,\theta)$. Furthermore, for smooth
real $(k-1,k-1)$-form $\beta$, we have by [GH78, page 33],

$\int_{V}\mathrm{dd^{c}}\beta=0.$

In particular, we can extend Lemma 2.5 to the singular context, namely, for
any nef (1,1) form $\theta$ with $\mathrm{dd^{c}}\theta^{\ell}=0$ for all
$1\leq\ell\leq k$, then

$\widehat{v}_{-}(V,\theta|_{V})=\int_{V}\theta^{k}.$

##  3\. Null loci and Nakai–Moishezon criterions

###  3.1. Extension theorem

We define the null locus of a big form, following the algebraic version in
[Nak00].

######  Definition 3.1 (Null locus).

Let $\theta$ be a big (1,1) form. We define the null locus of $\theta$,
denoted by $\operatorname{{\rm Null}}(\theta)$, to be the set

$\operatorname{{\rm Null}}(\theta)=\bigcup_{\theta|_{V}\,\text{not big}}V,$

where the union is taken over all purely positive-dimensional irreducible
analytic subvarieties $V\subset X$.

We describe the null locus with the following example.

######  Example 3.2.

Let $Y$ be a compact complex manifold of dimension $n$ equipped with a
positive (1,1) form $\omega_{Y}$. Let $\pi:X\to Y$ be a blow-up of a point
$y\in Y$ with the exceptional divisor
$E=\pi^{-1}(y)\simeq\mathbb{C}\mathbb{P}^{n-1}$. The form
$\theta=\pi^{*}\omega_{Y}$ is semi-positive, so it is nef. It was shown (from
e.g. [DP04, Lemma 3.5]) that there exists a smooth Hermitian metric $h$ on the
line bundle $\mathcal{O}_{X}(E)$ with curvature $\Theta_{h}(E)$ such that
$\pi^{*}\omega_{Y}-\varepsilon\Theta_{h}(E)$ is positive definite on $X$ for
$\varepsilon$ sufficiently small. In particular, let $s$ be a section of
$\mathcal{O}_{X}(E)$ vanishing along $E$ of order 1. Then for any
$\varepsilon$ small, we have that
$\pi^{*}\omega_{Y}+\varepsilon\mathrm{dd^{c}}\log|s|^{2}_{h}$ is a Hermitian
current on $X$ that belongs to the $\partial\bar{\partial}$-class $[\theta]$.
One obviously sees that the latter current has almost analytic singularities
precisely along $E$. Since $\theta$ is positive on $X\backslash E$, we also
observe that

$E=\bigcup_{\theta|_{V}\;\text{not big}}V,$

where the union is taken over all positive dimensional irreducible analytic
subvarieties $V\subset X$.

The above example motivates us to prove our main theorem in this section.

######  Theorem 3.3.

Let $X$ be a compact complex manifold of dimension $n$. Let $\theta$ be a nef
and big (1,1) form. Then

$\operatorname{E_{\textrm{{nH}}}}(\theta)=\operatorname{{\rm Null}}(\theta).$

The key ingredient in the proof of the theorem above is an extension-type
result for Hermitian current with almost analytic singularities in a nef and
big $\partial\bar{\partial}$-class (or merely big in the case of surfaces) on
complex subvarieties. We mimic the main technical idea of [CT15, Theorem 3.2]
in which they used Hironaka’s resolutions of singularities and Richberg’s
gluing technique in order to glue psh functions with analytic singularities.

######  Theorem 3.4.

Let $X$ be a compact complex manifold of dimension $n$. Let $\theta$ be a
smooth real (1,1) form on $X$, whose $\partial\bar{\partial}$-class is nef and
big. Let $E=V\cup\cup_{i}Y_{i}$ be an analytic closed subset of $X$ where $V$
and $Y_{i}$ are irreducible components of $E$, and $V$ is a compact complex
subvariety of purely positive dimension. Let $S=\theta+\mathrm{dd^{c}}\psi$ be
a Hermitian current with almost analytic singularities precisely along $E$.
Let $T=\theta|_{V}+\mathrm{dd^{c}}\varphi$ be a Hermitian current in the class
$[\theta|_{V}]$ on $V$ with almost analytic singularities. Then there exists a
Hermitian current with almost analytic singularities $\widetilde{T}$ on $X$ in
the $\partial\bar{\partial}$-class $[\theta]$ such that $\widetilde{T}$ is
smooth in a neighborhood of the generic point of $V$.

We have the following lemma, which is crucial in Richberg’s gluing procedure.

######  Lemma 3.5.

Let $M\subset\mathbb{C}^{N}$ be a complex manifold equipped with a positive
definite (1,1) form $\omega$ and let $E\subset M$ be an analytic subvariety.
Assume $\theta$ is a continuous (1,1) form. Let $W_{j}^{\prime}\Subset
W_{j}\subset M$ be open subsets for $j=1,2$. Let
$\varphi_{j}\in\operatorname{{\rm PSH}}(W_{j},\theta)$ such that
$\varphi_{1}|_{E\cap W_{1}\cap W_{2}}=\varphi_{2}|_{E\cap W_{1}\cap W_{2}}$
and $\varphi_{1}-\varphi_{2}$ is continuous on $W_{1}\cap W_{2}$. Then for any
$\varepsilon>0$, there exists an open neighborhood $W$ of $E$ and
$\varphi\in\operatorname{{\rm PSH}}(M,\theta+\varepsilon\omega)$ such that

  1. (1)

$E\cap(\overline{W_{1}^{\prime}}\cup\overline{W_{2}^{\prime}})\subset E\cap
W$, $W\Subset W_{1}\cup W_{2}$,

  2. (2)

$\varphi|_{E\cap W\cap W_{j}}=\varphi_{j}|_{E\cap W\cap W_{j}}$, for $j=1,2$,

  3. (3)

$\varphi|_{W\backslash\overline{W_{2}}}=\varphi_{1}|_{W\backslash\overline{W_{2}}}$,
and
$\varphi|_{W\backslash\overline{W_{1}}}=\varphi_{2}|_{W\backslash\overline{W_{1}}}$.

Moreover, if $\varphi_{j}$ (for $j=1,2$) has almost analytic singularities of
the same type, one also obtains $\varphi$ to have almost analytic
singularities.

###### Proof.

The proof is quite similar to that of [Ric68, Satz 3.1], we provide the proof
for reader’s convenience. We consider two open sets $U_{j}$ in $M$ for $j=1,2$
such that $W_{j}^{\prime}\Subset U_{j}\Subset W_{j}$. We set
$K:=E\cap\overline{W_{1}^{\prime}}\cup\overline{W_{2}^{\prime}}\cap\overline{U_{1}}\cap\overline{U_{2}}$
and $K_{j}:=K\cap\partial U_{j}$. We see that $K$, $K_{1}$, and $K_{2}$ are
compact subsets of $W_{1}\cap W_{2}$. Since $K_{1}\cap K_{2}=\varnothing$, we
can pick $\gamma_{1}$ a smooth nonnegative cutoff function being identically 1
in a neighborhood of $K_{2}$ in $M$ and $\gamma_{2}$ a smooth nonnegative
cutoff function being identically 1 in a neighborhood of $K_{1}$ in $M$ such
that the supports of $\gamma_{1}$ and $\gamma_{2}$ are disjoint. Next, we
choose $\eta>0$ so small that
$\theta+\mathrm{dd^{c}}(\varphi_{j}-\eta\gamma_{j})\geq-\varepsilon\omega$. We
define

$\varphi_{0}:=\max(\varphi_{1}-\eta\gamma_{1},\varphi_{2}-\eta\gamma_{2}).$

Since $K\subset E\cap W_{1}\cap W_{2}$, $\varphi_{1}|_{E\cap W_{1}\cap
W_{2}}=\varphi_{2}|_{E\cap W_{1}\cap W_{2}}$, and $\varphi_{1}-\varphi_{2}$ is
continuous on $W_{1}\cap W_{2}$ there exists an open neighborhood of $K_{2}$
such that $\varphi_{1}-\varphi_{2}<\eta$ there, hence

$\varphi_{1}-\eta\gamma_{1}=\varphi_{1}-\eta<\varphi_{2}=\varphi_{2}-\eta\gamma_{2}.$

Similarly, there exists an open neighborhood of $K_{1}$ on which
$\varphi_{2}-\eta\gamma_{2}<\varphi_{1}-\eta\gamma_{1}$. Therefore, there
exists an open neighborhood $W_{0}\Subset W_{1}\cap W_{2}$ of $K$ such that

$\varphi_{0}|_{W_{0}\backslash\overline{W_{2}}}=\varphi_{1}|_{W_{0}\backslash\overline{W_{2}}},\quad\varphi_{0}|_{W_{0}\backslash\overline{W_{1}}}=\varphi_{2}|_{W_{0}\backslash\overline{W_{1}}}.$

From the construction of $W_{0}$ and $\varphi_{0}$, we can define $W$ and
$\varphi_{0}$ as follows. We set

$W:=W_{0}\cup(U_{1}\backslash\overline{U_{2}})\cup(U_{2}\backslash\overline{U_{1}})$

which is a neighborhood of $K$ in $W_{1}\cap W_{2}$. We define the function
$\varphi$ on $W$ such that

$\varphi|_{W_{0}}=\varphi_{0}|_{W_{0}},\;\varphi|_{U_{1}\backslash\overline{U_{2}}}=\varphi_{1}|_{U_{1}\backslash\overline{U_{2}}},\;\varphi|_{U_{2}\backslash\overline{U_{1}}}=\varphi_{2}|_{U_{2}\backslash\overline{U_{1}}}.$

The last statement follows from the construction above. ∎

###### Proof of Theorem 3.4.

The proof is quite close to that of [CT15, Theorem 3.2], which does not seem
to require the closeness of the reference form $\theta$. For the sake of
completeness, we include the proof here.

We divide the proof into several steps, which we briefly organize as follows.
First, we lift our problem to a modification $\widetilde{X}$ of $X$ and
$\widetilde{V}$ of $V$; precisely, we use the resolution of singularities,
allowing us to reduce the singularities of $T$ to a simple normal crossing
divisor on the strict transform $\widetilde{V}$. Next, we construct local
extensions on a finite covering of $\widetilde{V}$ in $\widetilde{X}$. We then
use the technique of Richberg to glue the local potentials to construct a
local extension of Hermitian current in a neighborhood of $\widetilde{V}$. The
crucial point is that local potentials have the same singularity type on any
non-empty intersection. Finally, after possibly reducing the Lelong numbers of
the global Hermitian current $S$, we can glue the Hermitian current on the
neighborhood to $S$ to obtain the desired one.

Step 1: Lifting to the desingularization. Let
$\mathcal{I}_{T}\subset\mathcal{O}_{V}$ be the coherent sheaf of ideals
defining the singularities of $T$ on $V$. It follows from Hironaka’s theorem
that we can find a principlization of the sheaf of ideals $\mathcal{I}_{T}$
(cf. [Hir77, Wło09]), that is, a finite sequence

$p^{\prime}:V^{\prime}=V_{r}\xrightarrow{p_{r}}V_{r-1}\rightarrow\cdots\rightarrow
V_{1}\xrightarrow{p_{1}}V_{0}=V$

of blow-ups $p_{i}$ with smooth centers $C_{i-1}\subset V_{i-1}$. Next,
regarding the first center $C_{0}$ as a submanifold of $X$, we blow it up
inside $X$ and iterate $r$ times to obtain a sequence

$\pi:\widetilde{X}=X_{r}\xrightarrow{\pi_{r}}X_{r-1}\xrightarrow{\pi_{r-1}}\cdots\xrightarrow{\pi_{2}}X_{1}\xrightarrow{\pi_{1}}X_{0}=X$

of blow-ups $\pi_{i}:X_{i}\to X_{i-1}$ with smooth centers $C_{i-1}\subset
X_{i-1}$. We observe that if $\widetilde{V}$ is the strict transform of $V$
under $\pi$, then $\widetilde{V}\to V^{\prime}$ is a proper finite
bimeromorphic morphism, so it is indeed an isomorphism since $V^{\prime}$ is
smooth (due to Zariski’s main theorem). We see, moreover, that the exceptional
divisor $E_{i}$ of $\pi_{i}$ has simple normal crossings with
$\widetilde{V}_{i}$ and $E_{i}\cap\widetilde{V}_{i}$ coincides with the
exceptional divisor of $p_{i}$. Finally, on $\widetilde{X}$, we get a divisor
$D=\sum D_{\ell}$ which has simple normal crossings with the strict transform
$\widetilde{V}$ of $V$ via birational morphism $\pi$ and

$(\pi|_{\widetilde{V}})^{*}\mathcal{I}_{T}=\mathcal{O}_{\widetilde{V}}\left(-\sum_{\ell}a_{\ell}D_{\ell}|_{\widetilde{V}}\right),\quad
a_{\ell}>0.$

Note that since the analytic subvariety cutting out by $\mathcal{I}_{T}$ has
codimension at least 2 in $X$, it follows that all divisors $D_{\ell}$ are
$\pi$-exceptional divisors.

By assumptions, we find a $\varepsilon>0$ so small that

$T=\theta|_{V}+\mathrm{dd^{c}}\varphi\geq 3\varepsilon\omega|_{V}$

in the sense of currents on $V$, where $\omega$ denotes a Hermitian metric on
$X$. It follows from [DP04, Lemma 3.5] that there is a small $\delta>0$ such
that

$\pi^{*}T+\delta\mathrm{dd^{c}}\log|s_{D}|^{2}_{h_{D}}\geq
3\widetilde{\omega}|_{\widetilde{V}}$

for some Hermitian metric $\widetilde{\omega}$ on $\widetilde{X}$, where
$s_{D}$ is a holomorphic section defining $D=\sum_{\ell}D_{\ell}$ and $h_{D}$
is a smooth Hermitian metric on the line bundle
$\mathcal{O}_{\widetilde{X}}(D)$. We set
$\widetilde{\varphi}:=\varphi\circ\pi+\delta\log|s_{D}|^{2}_{h_{D}}$, defining
a function on $\widetilde{V}$ such that

$\pi^{*}\theta|_{\widetilde{V}}+\mathrm{dd^{c}}(\varphi\circ\pi)+\mathrm{dd^{c}}(\delta\log|s_{D}|^{2}_{h_{D}})_{\widetilde{V}}\geq
3\varepsilon\widetilde{\omega}|_{\widetilde{V}}.$

In the next step, we will explain how to extend $\widetilde{\varphi}$ to a
small neighborhood of $\widetilde{V}$.

Step 2: Gluing technique of Richberg. Since $\widetilde{V}$ is a closed
submanifold of $\widetilde{X}$ and $\sum_{\ell}D_{\ell}|_{\widetilde{V}}$ has
simple normal crossing support, we can find a finite open covering
$\\{W_{j}\\}_{j=1}^{N}$ of $\widetilde{V}$ such that on each local chart
$W_{j}$ there exist local coordinates $z_{j}=(z_{j}^{1},\ldots z_{j}^{n})$
with $\widetilde{V}\cap W_{j}=\\{z_{j}^{1}=\cdots=z_{j}^{n-k}=0\\}$, where
$k=\dim\widetilde{V}$, and with $\cup_{\ell}D_{\ell}\cap
W_{j}=\\{z^{i_{1}}_{j}\cdots z^{i_{p}}_{j}=0\\}$, $n-k+1\leq
i_{1},\ldots,i_{p}\leq n$. We denote by
$z^{\prime}_{j}=(z^{1}_{j},\ldots,z^{n-k}_{j})$ and
$z^{\prime\prime}_{j}=(z^{n-k+1}_{j},\ldots,z^{n}_{j})$. We define a function
$\varphi_{j}$ on each $W_{j}$ by

$\varphi_{j}(z_{j}^{\prime},z_{j}^{\prime\prime})=(\varphi\circ\pi)(0,z^{\prime\prime}_{j})+A|z^{\prime}_{j}|^{2}+\delta\log|s_{D}|^{2}_{h_{D}}$

for some $A>0$ being chosen so sufficiently large that

$\pi^{*}\theta+\mathrm{dd^{c}}\varphi_{j}\geq
2\varepsilon\widetilde{\omega}\;\text{on}\;W_{j}$

holds for all $j$ (we can shrink the $W_{j}^{\prime}s$ if necessary). Since
$\varphi$ is smooth at the generic point of $V$, one sees immediately that all
functions $\varphi_{j}$ are smooth in the neighborhood of the generic point of
$\widetilde{V}\cap W_{j}$.

We are going to apply Lemma 3.5 to glue the functions $\varphi_{j}$ together
to obtain a function $\varphi^{\prime}$ defined in a neighborhood $W$ of
$\widetilde{V}$ in $\widetilde{X}$ such that
$\varphi^{\prime}|_{\widetilde{V}}=\widetilde{\varphi}$ and

$\pi^{*}\theta+\mathrm{dd^{c}}\varphi^{\prime}\geq\varepsilon\widetilde{\omega}.$

This could be done if $\varphi_{i}-\varphi_{j}$ would be continuous on any
overlap $W_{i}\cap W_{j}$, $\forall\,1\leq i,j\leq N$. Assuming this, we
finish this step by induction of the number of open sets $\\{W_{j}\\}_{1\leq
j\leq N}$ covering $\widetilde{V}$. We take open sets $W_{j}^{\prime}$ in
$\widetilde{X}$ such that $W_{j}^{\prime}\Subset W_{j}$ and
$\widetilde{V}\subset\cup_{j=1}^{N}W^{\prime}_{j}$. If $N=1$ we pick
$W=W_{1}^{\prime}$ and $\varphi:=\varphi_{1}$. Otherwise, we start with two
open sets $W_{1}$ and $W_{2}$ in the covering such that $W_{1}\cap
W_{2}\cap\widetilde{V}\neq\varnothing$. From Lemma 3.5, we can find a
neighborhood $W_{1,2}$ of $(W_{1}^{\prime}\cup
W^{\prime}_{2})\cap\widetilde{V}$ in $\widetilde{X}$ and define a function
$\varphi_{1,2}$ on $W_{1,2}$ such that

$\varphi|_{\widetilde{V}\cap W_{1,2}\cap
W_{j}}=\varphi_{j}|_{\widetilde{V}\cap W_{1,2}\cap W_{j}},\quad j=1,2$

and $\pi^{*}\theta+\mathrm{dd^{c}}\varphi_{1,2}\geq
2\varepsilon(1-2^{-N})\widetilde{\omega}$. We apply the inductive assumption
with $\widetilde{V}\subset W_{1,2}\cup\bigcup_{j=3}^{n}W_{j}$ and functions
$\varphi_{1,2}$, $\varphi_{j}$ to procedure an open neighborhood $W$ of
$\widetilde{V}$ in $\widetilde{X}$ and a function $\varphi^{\prime}$ defined
on $W$ with having almost analytic singularities, which satisfies
$\varphi^{\prime}|_{\widetilde{V}}=\widetilde{\varphi}$ and
$\pi^{*}\theta+\mathrm{dd^{c}}\varphi^{\prime}\geq\varepsilon\widetilde{\omega}$
holds on $W$.

It remains to show that for any two open sets $W_{1}$, $W_{2}$, with
$W_{1}\cap W_{2}\cap\widetilde{V}\neq\varnothing$, the difference
$\varphi_{1}-\varphi_{2}$ is continuous (smooth) on $W_{1}\cap W_{2}$. We are
done if none of $D_{\ell}$ intersects $W_{1}\cap W_{2}$. Otherwise, up to
relabelling local coordinates, we can write $D_{\ell}\cap
W_{1}=\\{z_{1}^{\ell}=0\\}$ and $D_{\ell}\cap W_{2}=\\{z_{2}^{\ell}=0\\}$. We
then see that on $W_{1}\cap W_{2}$, $z_{2}^{\ell}=f^{\ell}\cdot z^{\ell}_{1}$
for some non-vanishing holomorphic function $f^{\ell}$.

It follows from the construction that $\varphi_{j}$ has almost analytic
singularities along $\cup_{\ell}D_{\ell}\cap W_{j}$, for $j=1,2$. The sheaf of
ideals defining the singularities of $\varphi_{j}$ corresponds to an effective
divisor $\sum_{\ell}a_{j}^{\ell}D_{\ell}$. We have on each $W_{j}$,

$\varphi_{j}=c_{j}\left(\sum_{r=1}^{p}a_{j}^{i_{r}}\log|z_{j}^{i_{r}}|^{2}\right)+h_{j}$

for some positive constants $c_{j}$, $a_{j}^{i_{r}}$, for some smooth
functions $h_{j}$ on $W_{j}$, and $\cup_{\ell}D_{\ell}\cap
W_{j}=\\{\prod_{r=1}^{p}|z_{j}^{i_{r}}|^{2a_{j}^{i_{r}}}=0\\}$. We remark that
the term $\delta\log|s_{D}|^{2}_{h_{D}}$ is absorbed in the first one of the
expression. Since $\varphi_{1}=\varphi_{2}$ on $\widetilde{V}\cap W_{1}\cap
W_{2}$ we must have $c_{1}=c_{2}$ and $a^{i_{r}}_{1}=a_{2}^{i_{r}}$ there. The
relation $z_{2}^{\ell}=f^{\ell}\cdot z^{\ell}_{1}$ implies that
$\varphi_{1}-\varphi_{2}$ is smooth on $W_{1}\cap W_{2}$. Furthermore, the
function $\max(\varphi_{1},\varphi_{2})$ defined on $W_{1}\cap W_{2}$ has
almost analytic singularities of the same type as $\varphi_{1}$ and
$\varphi_{2}$, and coincide with $\widetilde{\varphi}$ on $\widetilde{V}\cap
W_{1}\cap W_{2}$.

Step 3: Reducing Lelong numbers. From Step 2, we obtain a $\pi^{*}\theta$-psh
function $\varphi^{\prime}$ defined on $W$ (a neighborhood of $\widetilde{V}$
in $\widetilde{X}$) which has almost analytic singularities and
$\pi^{*}\theta+\mathrm{dd^{c}}\varphi^{\prime}\geq\varepsilon\widetilde{\omega}$
there. One moreover has that $\varphi^{\prime}\equiv-\infty$ on all the
components $D_{\ell}$ intersecting $W$, and $\varphi^{\prime}$ has the
strictly positive generic Lelong numbers along each such component. Let
$\widetilde{Y}_{i}$ be the strict transform of $Y_{i}$ in $\widetilde{X}$. Up
to shrinking $W$ slightly, we may assume that $\varphi^{\prime}$ is defined on
an open neighborhood $\widetilde{W}$ of $W$. By [DP04, Lemma 2.1, Theorem
3.2], there exists a globally defined $C_{0}\widetilde{\omega}$-psh function
$\psi$ on $\widetilde{X}$ for some $C_{0}>0$, with almost analytic
singularities along $\cup_{i}\widetilde{Y}_{i}$. We then define
$\varphi^{\prime\prime}:=\varphi^{\prime}+\frac{\varepsilon}{2C_{0}}\psi$,
which has almost analytic singularities along
$(\cup_{\ell}D_{\ell}\cup\cup_{i}\widetilde{Y}_{i})\cap\widetilde{W}$, and
satisfies
$\pi^{*}\theta+\mathrm{dd^{c}}\varphi^{\prime\prime}\geq\frac{\varepsilon}{2}\widetilde{\omega}$
on $\widetilde{W}$. Note that the generic Lelong number of
$\varphi^{\prime\prime}$ along
$(\cup_{\ell}D_{\ell}\cup\cup_{i}\widetilde{Y}_{i})\cap\widetilde{W}$ is
always strictly positive. We explain below how to reduce its generic Lelong
number, for which we are able to extend $\varphi^{\prime\prime}$ in the whole
$\widetilde{X}$.

We work now with the function $\psi$ in the assumption. One sees immediately
that the function $\psi\circ\pi$ has singularities along
$\cup_{\ell}D_{\ell}\cup\cup_{i}\widetilde{Y}_{i}\cup\widetilde{V}$. We now
set $\widetilde{\psi}:=\psi\circ\pi+\delta\log|s_{D}|^{2}_{h_{D}}$ where
$s_{D}$ and $h_{D}$ are defined in Step 1, $\delta>0$ small enough. By [DP04,
Lemma 3.5] again, we have

$\pi^{*}\theta+\mathrm{dd^{c}}\widetilde{\psi}\geq\gamma\widetilde{\omega}$

for some small constant $\gamma>0$. Since the form $\pi^{*}\theta$ is nef, for
any small $\lambda>0$ there exists a smooth function $\rho_{\lambda}$ such
that
$\pi^{*}\theta+\mathrm{dd^{c}}\rho_{\lambda}\geq-\lambda\pi^{*}\omega\geq-\lambda\widetilde{\omega}$
on $\widetilde{X}$. Up to adding a positive constant, we can assume that
$\rho_{\lambda}>0$. Therefore, we have

$\pi^{*}\theta+\mathrm{dd^{c}}\left(\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}\right)\geq\lambda^{2}\gamma\widetilde{\omega}.$

Let us stress that this is the only place where the nefness of $\theta$ is
used.

Fix any point
$x\in(\cup_{\ell}D_{\ell}\cup\cup_{i}\widetilde{Y}_{i})\cap\partial W$, we
choose a local chart $U$ centered at $x$ on which $\varphi^{\prime\prime}$ is
also defined, $U\cap\widetilde{V}=\varnothing$, with local coordinates
$(z^{1},\ldots,z^{n})$. Thanks to the fundamental theorem of Hironaka [Hir77],
there is a principalization $q:\widetilde{U}\to U$ of the sheaf of ideals of
the singularities of $\varphi^{\prime\prime}$ and $\widetilde{\psi}$ on $U$
such that this sheaf of a simple normal crossing divisor. Namely, on
$\widetilde{U}$, we have

$\begin{split}q^{*}\varphi^{\prime\prime}=c\left(\sum_{p}a^{i_{p}}\log|z^{i_{p}}|^{2}\right)+g\\\
q^{*}\widetilde{\psi}=d\left(\sum_{p}b^{i_{p}}\log|z^{i_{p}}|^{2}\right)+h\end{split}$

where $c$, $d$, $a^{i_{p}}$, $b^{i_{p}}$ are positive constants and $g$, $h$
are smooth functions on $\widetilde{U}$. Therefore, we can choose $\lambda>0$
so small that
$\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}>\varphi^{\prime\prime}$ on
a possibly smaller open neighborhood of $x$. Since $x$ was taken arbitrarily,
we can find an open neighborhood of
$(\cup_{\ell}D_{\ell}\cup\cup_{i}\widetilde{Y}_{i})\cap\partial W$ so that
$\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}>\varphi^{\prime\prime}$
there.

Step 4: Completing the proof. From the previous step, there exist an open
neighborhood $U$ of $\cup_{\ell}D_{\ell}\cup\cup_{i}\widetilde{Y}_{i}$ and
$\lambda\in]0,1[$ so small such that
$\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}>\varphi^{\prime\prime}$ on
a neighborhood of $\partial W\cap U$. Since on $\partial W\backslash U$ we can
find a constant $B>0$ so large that

$\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}>\varphi^{\prime\prime}-B$

hence
$\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}>\varphi^{\prime\prime}-B$
in a neighborhood of $\partial W$. Therefore, we can define a function $\Phi$
on $\widetilde{X}$ by

$\Phi=\begin{cases}\max(\varphi^{\prime\prime},\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}+B)&\text{on}\;\widetilde{W}\\\
\lambda\widetilde{\psi}+(1-\lambda)\rho_{\lambda}+B&\text{on}\;\widetilde{X}\backslash\widetilde{W}.\end{cases}$

One easily sees that
$T^{\prime}=\pi^{*}\theta+\mathrm{dd^{c}}{\Phi}\geq\varepsilon^{\prime}\widetilde{\omega}$
for some $0<\varepsilon^{\prime}<\min(\varepsilon,\lambda\gamma)$. We easily
see that $\Phi=\varphi^{\prime\prime}$ in an open neighborhood of the generic
point of $\widetilde{V}$ because
$\widetilde{\psi}|_{\widetilde{V}}\equiv-\infty$. Therefore,
$\Phi|_{\widetilde{V}}$ is smooth in an open neighborhood of the generic point
of $\widetilde{V}$. Pushing forward $T^{\prime}$ on $X$, we obtain a Hermitian
current $\widetilde{T}=\pi_{*}T^{\prime}$ in the
$\partial\bar{\partial}$-class $[\theta]$ which is an extension of $T$ such
that the restriction $\widetilde{T}|_{V}$ is smooth in an open neighborhood of
the generic point of ${V}$. The poof is complete. ∎

###### Proof of Theorem 3.3.

We first show that $\operatorname{{\rm
Null}}(\theta)\subset\operatorname{E_{\textrm{{nH}}}}(\theta)$. Let $x\in X$
and $V$ an irreducible analytic subavriety of $X$ of positive dimension such
that $\theta|_{V}$ is not big. Suppose by contradiction that $x$ does not
belong to $\operatorname{E_{\textrm{{nH}}}}(\theta)$, so there is a Hermitian
current $T\in[\theta]$ with almost analytic singularities and smooth in an
open neighborhood of $x$. We write $T=\theta+\mathrm{dd^{c}}\varphi$ for some
$\theta$-psh function $\varphi$. However,
$\theta|_{V}+\mathrm{dd^{c}}\varphi|_{V}$ defines a Hermitian current with
almost analytic singularities on $V$ because $\varphi|_{V}\not\equiv-\infty$,
a contradiction.

For the inverse implication $\operatorname{{\rm
Null}}(\theta)\supset\operatorname{E_{\textrm{{nH}}}}(\theta)$, suppose by
contradiction that there exists a point
$x\in\operatorname{E_{\textrm{{nH}}}}(\theta)\backslash\operatorname{{\rm
Null}}(\theta)$. Let $V$ be an irreducible component of the closed analytic
set $\operatorname{E_{\textrm{{nH}}}}(\theta)$ containing $x$. Thanks to Lemma
2.16, the pure dimension of $V$ is strictly positive.

Assume first that $V$ is smooth. Since $\theta|_{V}$ is big, we can find a
Hermitian current $T=\theta|_{V}+\mathrm{dd^{c}}\varphi$ with almost analytic
singularities (Theorm 2.11). We decompose

$\operatorname{E_{\textrm{{nH}}}}(\theta)=V\cup\bigcup_{i\in I}Y_{i}$

for disjoint irreducible components $V$, $Y_{i}$ of
$\operatorname{E_{\textrm{{nH}}}}(\theta)$ which are all purely positive
dimensional as follows from Lemma 2.16. We can thus apply Theorem 3.4 with
$E=\operatorname{E_{\textrm{{nH}}}}(\theta)$, with the current
$S=\theta+\mathrm{dd^{c}}\psi$ in Theorem 2.14 to obtain a Hermitian current
$\widetilde{T}=\theta+\mathrm{dd^{c}}\Phi$ in the
$\partial\bar{\partial}$-class $[\theta]$ on $X$ satisfying that
$\widetilde{T}|_{V}=T$ and $T$ is smooth in a neighborhood of the generic
point of $V$. In particular, we must have
$V\nsubseteq\operatorname{\textrm{{Sing}}}(\widetilde{T})$, which is a
contradiction since
$V\subset\operatorname{E_{\textrm{{nH}}}}(\theta)\subset\operatorname{\textrm{{Sing}}}(\widetilde{T})$.

If $V$ is singular, we take an embedded desingularization of $V$, that is,
$\pi:\widetilde{X}\to X$, a finite composition of blow-ups with smooth
centers. The strict transform $\widetilde{V}$ is smooth and is in simple
normal crossings with the exceptional divisor. Since $\pi$ is bimeromorphic,
$\pi^{*}\theta$ is nef and big on $\widetilde{X}$, has the same properties on
$\widetilde{V}$. Then there exists a Hermitian current $S\in[\pi^{*}\theta]$
with almost analytic singularities along an analytic subset
$\operatorname{E_{\textrm{{nH}}}}(\pi^{*}\theta)$. Thanks to [Tos18b,
Proposition 2.5],

$\widetilde{V}\subset\operatorname{E_{\textrm{{nH}}}}(\pi^{*}\theta)=\textrm{Exc}(\pi)\cup\pi^{-1}(\operatorname{E_{\textrm{{nH}}}}(\theta)).$

We can apply Theorem 3.4 with $S$ and a Hermitian current
$T=\pi^{*}\theta|_{\widetilde{V}}+\mathrm{dd^{c}}\varphi$ to obtain a
Hermitian current
$\widetilde{T}=\pi^{*}\theta+\mathrm{dd^{c}}\widetilde{\Phi}$ in the
$\partial\bar{\partial}$-class $[\pi^{*}\theta]$ such that
$\widetilde{T}|_{\widetilde{V}}$ is smooth in a neighborhood of the generic
point of $\widetilde{V}$. Hence, the direct image $\pi_{*}\widetilde{T}$ is
smooth near the generic point of $V$, which is again a contradiction. ∎

The following example tells us that the nef condition in Theorem 3.3 is
essential even in the algebraic version of Nakamaye’s theorem; we would like
to thank V. Tosatti for pointing out this example.

######  Example 3.6.

Let $\pi_{1}:X_{1}\to\mathbb{C}\mathbb{P}^{2}$ be a blow-up at one point, with
exceptional divisor $F_{1}$. Let $\pi_{2}:X\to X_{1}$ be the blow-up of a
point $y\in F_{1}$ with exceptional divisor $F_{2}$. The exceptional divisor
$E$ of $\pi=\pi_{2}\circ\pi_{1}$ has two components $E=E_{1}+E_{2}$ where
$E_{1}$ is the strict transform of $F_{1}$ under $\pi_{2}$ while
$E_{2}=F_{2}$. We have that $E_{1}^{2}=-2$ , $E_{2}^{2}=-1$ and $E_{1}\cdot
E_{2}=1$. We take $L=\pi^{*}H+E_{1}$ where $H$ is a hyperplane bundle on
$\mathbb{C}\mathbb{P}^{2}$. It is clear that $L$ is big, and its positive part
in the divisorial Zariski decomposition is $\pi^{*}H$, which is nef.
Nakamaye’s theorem yields

$\mathrm{E_{nH}}(L)=\mathrm{E_{nH}}(\pi^{*}H)=E_{1}\cup E_{2}.$

However, since the restriction $L|_{E_{2}}$ is a line bundle on a (rational)
curve with degree $E_{1}\cdot E_{2}=1$, it is ample. This implies
$\operatorname{E_{\textrm{{nH}}}}(L)\neq\operatorname{{\rm Null}}(L)$.

###  3.2. Proof of Theorem A

Under some extra assumptions on the manifold, we can determine the null locus
as follows.

######  Theorem 3.7.

Let $X$ be a $n$-dimensional compact complex manifold equipped with a
Hermitian metric $\omega_{X}$. Assume that $\omega_{X}$ satisfies the volume
condition ($H_{X}$). Let $\theta$ be a nef (1,1) form. Then $\theta$ is big if
and only if $\widehat{v}_{-}(X,\theta)>0$ and moreover

(3.1) $\operatorname{E_{\textrm{{nH}}}}(\theta)=\operatorname{{\rm
Null}}(\theta)=\bigcup_{\widehat{v}_{-}(V,\theta|_{V})=0}V$

where the union is taken over all purely positive-dimensional irreducible
analytic subvarieties of $V\subset X$.

###### Proof.

We denote by $N$ the set of the right-hand side of (3.1). The first statement
is known in [GL22, Theorem 4.6]. It suffices to show that
$\operatorname{E_{\textrm{{nH}}}}(\theta)=N$.

We first show that $N\subset\operatorname{E_{\textrm{{nH}}}}(\theta)$. Let
$x\in X$ and $V$ an irreducible analytic subavriety of $X$ of dimension $k>0$
such that $\widehat{v}_{-}(V,\theta|_{V})=0$. Suppose by contradiction that
$x$ does not belong to $\operatorname{E_{\textrm{{nH}}}}(\theta)$, so there is
a Hermitian current $T\in[\theta]$ with almost analytic singularities and
smooth in an open neighborhood of $x$. We write
$T=\theta+\mathrm{dd^{c}}\varphi$ for some $\theta$-psh function $\varphi$.
Set $\psi=\varphi|_{V}$.

Suppose first that $V$ is smooth. Assume that
$\theta|_{V}+\mathrm{dd^{c}}\psi\geq\omega_{V}$ for some Hermitian metric
$\omega_{V}$, and $\sup_{V}\psi=0$. Fix $\varepsilon>0$ and
$u\in\operatorname{{\rm
PSH}}(V,\theta|_{V}+\varepsilon\omega_{V})\cap\mathcal{C}^{\infty}(V)$. Set
$v:=P_{\omega_{V}}(u-\psi)$. We note that $u-\psi$ is merely well-defined
outside a pluripolar set and $v+\psi\leq u$ almost everywhere, hence
everywhere. It follows from [GZ17, Theorem 9.17] that $v$ is bounded from
above. By the classical pluipotential theory (see e.g. [GL22, Theorem B]), $v$
is a bounded $\omega$-psh function and $(\omega_{V}+\mathrm{dd^{c}}v)^{k}$ is
supported on the contact set

$D=\\{v=u-\psi\\}\subset\\{\psi>-\infty\\}$

where the product $(\cdot)^{k}$ is understood as the Monge–Ampère product in
the sense of Bedford–Taylor; cf. [GL22, Theorem 2.3]. By the maximum principle
(Lemma 2.7), we have

$\mathbf{1}_{D\cap\\{\psi>-\infty\\}}(\theta|_{V}+\varepsilon\omega_{V}+\mathrm{dd^{c}}(v+\psi))^{n}\leq\mathbf{1}_{D\cap\\{\psi>-\infty\\}}(\theta|_{V}+\varepsilon\omega_{V}+\mathrm{dd^{c}}u)^{n}.$

Hence,

$\int_{V}(\omega_{V}+\mathrm{dd^{c}}v)^{n}=\int_{D}(\omega_{V}+\mathrm{dd^{c}}v)^{n}\leq\int_{V}(\theta|_{V}+\varepsilon\omega_{V}+\mathrm{dd^{c}}u)^{n}$

since $D\cap\\{\psi>-\infty\\}=D$. Therefore,

$0<v_{-}(V,\omega_{V})\leq v_{-}(V,\theta|_{V}+\varepsilon\omega_{V})$

by assumption. Taking the infimum for $\varepsilon>0$ gives a contradiction.

If $V$ is singular, we consider $\pi:X^{\prime}\to X$ an embedded resolution
of singularities obtained by a finite composition of blow-ups with smooth
centers and assume $V^{\prime}$ is the strict transform of $V$. We set
$\psi^{\prime}=(\varphi\circ\pi)|_{V^{\prime}}+\delta\log|s|^{2}_{h}$ where
$s$ is a section defining the exceptional divisor $\textrm{Exc}(\pi)$, and
$\delta>0$ small enough. Then
$\pi^{*}\theta|_{V^{\prime}}+\mathrm{dd^{c}}\psi^{\prime}\geq\omega_{V^{\prime}}$
for some Hermitian metric $\omega_{V^{\prime}}$. We apply the argument above
with $\psi^{\prime}$ and $\omega_{V^{\prime}}$ to obtain

$0<v_{-}(V^{\prime},\omega_{V^{\prime}})\leq
v_{-}(V^{\prime},\pi^{*}\theta+\varepsilon\omega_{V^{\prime}})$

a contradiction.

Conversely, suppose by contradiction that there exists a point
$x\in\operatorname{E_{\textrm{{nH}}}}(\theta)\backslash N$. Let $V$ be an
irreducible component of the closed analytic set
$\operatorname{E_{\textrm{{nH}}}}(\theta)$ containing $x$. Thanks to Lemma
2.16, we have that the pure dimension of $V$ is strictly positive, denoted by
$k$.

Assume first that $V$ is smooth. We see that
$\widehat{v}_{-}(V,\theta|_{V})>0$ because $x\notin N$. Then, by the first
statement, we can find a Hermitian current
$T=\theta|_{V}+\mathrm{dd^{c}}\varphi$ with almost analytic singularities
(thanks to Demailly’s regularization theorem). We decompose

$\operatorname{E_{\textrm{{nH}}}}(\theta)=V\cup\bigcup_{i}Y_{i}$

for disjoint irreducible components of
$\operatorname{E_{\textrm{{nH}}}}(\theta)$ which are all purely positive
dimension by Lemma 2.16. We apply Theorem 3.4 with
$E=\operatorname{E_{\textrm{{nH}}}}(\theta)$, with the current
$S=\theta+\mathrm{dd^{c}}\psi$ in Theorem 2.14 to obtain a Hermitian current
$\widetilde{T}=\theta+\mathrm{dd^{c}}\Phi$ in the
$\partial\bar{\partial}$-class $[\theta]$ on $X$ satisfying that
$\widetilde{T}|_{V}=T$ and $T$ is smooth in a neighborhood of the generic
point of $V$. In particular, we must have
$V\nsubseteq\operatorname{\textrm{{Sing}}}(\widetilde{T})$, which is a
contradiction because

$V\subset\operatorname{E_{\textrm{{nH}}}}(\theta)\subset\operatorname{\textrm{{Sing}}}(\widetilde{T}).$

If $V$ is not smooth, we argue the same as in the proof of Theorem 3.3 to
obtain again a contradiction; see also [CT15, page 1180]. ∎

We end this section with some examples of the manifold $X$ equipped with a
Hermitian metric $\omega_{X}$ satisfying the volume assumption ($H_{X}$).

######  Example 3.8.

Obviously, if $X$ is Kähler or belongs to the Fujiki class $\mathcal{C}$, then
it always admits a metric $\omega_{X}$ satisfying the condition ($H_{X}$).

When $\dim_{\mathbb{C}}X=2$ or 3 and $\omega_{X}$ is pluriclosed, i.e.,
$\mathrm{dd^{c}}\omega_{X}=0$ , it has been shown in [AGL23, Theorem 3.3] that
the condition ($H_{X}$) is satisfied.

If $\omega_{X}$ satisfies the curvature condition given by Guan–Li, namely

$\mathrm{dd^{c}}\omega_{X}=0,\;{\rm d}\omega_{X}\wedge{\rm d^{c}}\omega_{X}=0$

which is equivalent to $\mathrm{dd^{c}}\omega_{X}^{k}=0$ for $k=1,\dots,n-1$,
then we have that

$\int_{V}(\omega_{X}+\mathrm{dd^{c}}\varphi)^{\dim V}=\int_{V}\omega_{X}^{\dim
V}$

for any compact submanifold (without boundary) of $X$. We can precisely choose
$X$ to be the product of a complex surface and a Kähler manifold. Namely,
$X=M\times N$ where $M$ is a complex surface and $N$ is Kähler, and
$\omega_{X}=p_{1}^{*}\omega_{M}+p_{2}^{*}\omega_{N}$ where $\omega_{M}$ is a
$\mathrm{dd^{c}}$-closed metric (it always exists thanks to Gauduchon’s
theorem) and $\omega_{N}$ is Kähler metric.

######  Example 3.9.

It was shown in [GL22, Theorem 3.7] that the condition ($H_{X}$) is
bimeromorphic invariant. Therefore, we can find various compact complex
manifolds with such conditions. For instance, we assume that a compact complex
manifold $X$ admits a Hermitian metric $\omega_{X}$ satisfying the condition
($H_{X}$) and $\pi:X^{\prime}\to X$ a finite composition of blow-ups of $X$
with smooth centers, then $X^{\prime}$ carries a Hermitian metric
$\omega_{X^{\prime}}$ with the same property.

###  3.3. Nakai–Moishezon’s criterion on some compact Hermitian manifolds

As a direct application of Theorem A, we discuss the following Nakai–Moishezon
criterion for some compact non-Kähler manifolds.

######  Theorem 3.10.

Let $X$ be a $n$-dimensional compact complex manifold equipped with a
Hermitian metric $\omega_{X}$. Assume that $\omega_{X}$ satisfies the volume
condition ($H_{X}$). Let $\theta$ be a nef (1,1) form. Then the
$\partial\bar{\partial}$-class $[\theta]$ is Hermitian if and only if
$\widehat{v}_{-}(V,\theta|_{V})>0$ for all irreducible analytic subvarieties
$V\subset X$.

###### Proof.

If the $\partial\bar{\partial}$ class $[\theta]$ is Hermitian we can find a
Hermitian metric $\omega$ such that $\omega=\theta+\mathrm{dd^{c}}\varphi$ for
$\varphi\in\mathcal{C}^{\infty}(X)$, and we have $\omega\geq\delta\omega_{X}$
for some $\delta>0$. If $V\subset X$ is a closed smooth manifold, then by
[GL22, Lemma 4.4],

$\widehat{v}_{-}(V,\theta|_{V})=v_{-}(V,\omega|_{V})\geq\delta^{n}v_{-}(V,\omega_{X}|_{V})>0.$

If $V$ is not smooth, then thanks to Hironaka’s theorem [Hir77, Wło09], we can
take $\pi:X^{\prime}\to X$ be an embedded resolution of singularities of $V$,
obtained by composition of blow up with smooth centers. Let $V^{\prime}$ be
the strict transform of $V$, which is now a closed smooth submanifold of
$X^{\prime}$. Thanks to [GL22, Theorem 3.7] the condition ($H_{X}$) is
preserved. That theorem also yields

$v_{-}(V^{\prime},\omega^{\prime})\geq\delta^{n}v_{-}(V^{\prime},\pi^{*}\omega_{X})>0.$

Conversely, with $V=X$ we observe that $\widehat{v}_{-}(X,\theta)>0$, hence it
follows from Theorem 2.15 that there exist a Hermitian current $T$ in the
$\partial\bar{\partial}$ class $[\theta]$ with almost analytic singularities
along $\operatorname{E_{\textrm{{nH}}}}(\theta)$ . We apply Theorem 3.7 to get
that the non-Hermitian (null) locus

$\operatorname{E_{\textrm{{nH}}}}(\theta)=\operatorname{{\rm
Null}}(\theta)=\bigcup_{\widehat{v}_{-}(V,\theta|_{V})=0}V$

is empty. Therefore, the Hermitian current $T$ is, indeed, a Hermitian metric.
∎

Consequently, we can give an alternative proof of the Nakai–Moishezon
criterion for compact non-Kähler surfaces [Buc00, Lam99]. We extend this
result in higher dimensions, and also show that it holds on singular compact
varieties. We are grateful to V. Tosatti for pointing out this to us.

###### Proof of Theorem B.

We first treat the case when $X$ is smooth. Obviously, the metric $\omega_{X}$
satisfies the condition ($H_{X}$). Thanks to Theorem 3.10, it suffices to show
that $\theta$ is nef. Indeed, if $\theta$ were nef, we would have that
$\widehat{v}_{-}(V,\theta|_{V})=\int_{V}\theta^{\dim V}$ by (1). We choose a
number $t>0$ so large that the $\partial\bar{\partial}$-class
$[\theta+t\omega]$ is Hermitian, and we denote by $t_{0}$ the minimum of such
$t$. It thus remains to prove that $t_{0}\leq 0$. Suppose by contradiction
that $t_{0}>0$. We see that $[\theta+t_{0}\omega]$ is automatically nef, but
not Hermitian. Since $t_{0}>0$, the conditions (1) and (2) yield

$\widehat{v}_{-}(X,\theta+t_{0}\omega_{X})=\int_{X}(\theta+t_{0}\omega_{X})^{n}>0$

By Theorem 2.13, the $\partial\bar{\partial}$-class $[\theta+t_{0}\omega_{X}]$
is big. On the other hand, the condition (3) and Theorem 3.10 imply that the
$\partial\bar{\partial}$-class $[\theta+t_{0}\omega_{X}]$ is Hermitian, a
contradiction. Therefore, we must have $t_{0}\leq 0$ so $\theta$ is nef.

If $X$ is singular, thanks to Hironaka’s theorem, we can take
$\pi:X^{\prime}\to X$ to be an embedded resolution of singularities of $X$,
obtained by finite compositions of blow-ups with smooth centers, with
exceptional divisor $D$. On $X^{\prime}$,
$\omega_{X^{\prime}}:=\pi^{*}\omega_{X}+\mathrm{dd^{c}}\delta\log|s_{D}|_{h}$
is a Hermitian metric for $\delta>0$ sufficiently small, which satisfies
$\mathrm{dd^{c}}\omega_{X^{\prime}}^{k}=0$ for any $1\leq k\leq n-1$. We apply
the previous argument with $\pi^{*}\theta$ and $\omega_{X^{\prime}}$ we can
find a smooth function $\psi$ on $X^{\prime}$ such that
$\Theta=\pi^{*}\theta+\mathrm{dd^{c}}\psi$ is a Hermitian metric on
$X^{\prime}$. The direct image $\pi_{*}\Theta$ is a Hermitian current with
singularities along $X_{\rm sing}$. In order to remove the singularities of
$\pi_{*}\Theta$ we apply the induction technique on dimension used in [CT16,
page 94] or [DHP22, pages 19–23] (probably used earlier in [DP04]), so we only
give the main ideas. Suppose by induction that the theorem holds for any
compact normal analytic variety of pure dimension smaller than $n$ and that
for any analytic subset $Y\subset X$, there exists an open neighborhood
$W_{Y}$ of $Y$ and a smooth function $\rho_{Y}$ on $W_{Y}$ such that
$\theta|_{W_{Y}}+\mathrm{dd^{c}}\rho_{Y}\geq\varepsilon\omega_{X}|_{W_{Y}}$
for $\varepsilon>0$. The latter can be obtained from the same arguments in the
proof of Theorem 3.4 or from the aforementioned references. On the
desingularization $p:Y^{\prime}\to Y$, we can find a Hermitian metric
$p^{*}\theta|_{Y^{\prime}}+\mathrm{dd^{c}}\psi_{Y}$, its push forward is a
Hermitian current with singularities along $Z\subset Y$ ($p:Y\backslash
p^{-1}(Z)\to Y\backslash Z$ is biholomorphic). By induction hypothesis, we can
find a smooth function $\rho_{Z}$ and a neighborhood $W_{Z}$ of $Z$ on which
$\theta_{W_{Z}}+\mathrm{dd^{c}}\rho_{Z}\geq\delta\omega_{X}|_{Z}$ with
$\delta>0$. Hence, using the regularized maximum function; cf. [Dem12, page
43], we can glue $\pi_{*}\psi_{Y}$ with $\rho_{Z}$ to get the desired one.
Finally, we apply with $\pi_{*}\psi$ and $\rho_{X_{\rm sing}}$ to conclude. ∎

##  4\. Singular Hermitian metrics and envelopes

###  4.1. Singular Hermitian metrics

This section is devoted to the study of degenerations of Hermitian metrics on
compact complex manifolds. We first treat the case of smooth densities,
proving the following.

######  Theorem 4.1.

Let $X$ be a $n$-dimensional compact complex manifold equipped with a
Hermitian metric $\omega_{X}$. Let $\theta$ be a smooth real (1,1) form whose
$\partial\bar{\partial}$-class is nef and big. Let
$f\in\mathcal{C}^{\infty}(X,\mathbb{R})$. Then, there exists a constant $c>0$
and an $\theta$-psh function $\varphi$ which is smooth on a Zariski open $U$
such that $\sup_{X}\varphi=0$ and

$(\theta+\mathrm{dd^{c}}\varphi)^{n}=cf\omega_{X}^{n}$

on $U$. More precisely, $U$ can be taken to be the complement of the null
locus of $\theta$.

###### Proof.

Since the class $[\theta]$ is nef we have for any $t>0$, there exists a smooth
function $u_{t}$ such that $\theta+t\omega_{X}+\mathrm{dd^{c}}u_{t}$ is a
positive Hermitian form. Set $\theta_{t}:=\theta+t\omega_{X}$. Since $\theta$
is nef, the $\partial\bar{\partial}$-class $[\theta_{t}]$ is Hermitian. Hence
the main result of [TW10a] ensures that there exists a constant $c_{t}>0$ and
a smooth $\theta_{t}$-psh function $\varphi_{t}$ normalized by
$\sup_{X}\varphi_{t}=0$ such that

$(\theta_{t}+\mathrm{dd^{c}}\varphi_{t})^{n}=c_{t}f\omega_{X}^{n},$

where $c_{t}$ equals $\int_{X}\omega_{t}^{\prime n}/\int_{X}f\omega_{X}^{n}$
with $\omega_{t}^{\prime}:=\theta_{t}+\mathrm{dd^{c}}\varphi_{t}$. We observe
that $\omega_{t}^{\prime}$ is the (unique) Hermitian metric in the
$\partial\bar{\partial}$-class $[\theta_{t}]$ with prescribed Chern–Ricci
form. We will show that as $t\to 0^{+}$, $\omega_{t}^{\prime}$ converges to
some metric $\omega_{0}=\theta+\mathrm{dd^{c}}\varphi$ on
$X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$ in
$\mathcal{C}^{\infty}_{\rm
loc}(X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta))$. We divide the
proof into several steps.

Step 1: $\mathcal{C}^{0}$-estimates. Thanks to Theorem 2.14, we can find a
$\theta$-psh function $\rho$ with almost analytic singularities precisely
along $\operatorname{E_{\textrm{{nH}}}}(\theta)$, normalized by
$\sup_{X}\rho=0$ such that

$\theta+\mathrm{dd^{c}}\rho\geq 2\delta_{0}\omega_{X}.$

In particular, $\rho$ is smooth on the complement of the analytic subset
$\operatorname{E_{\textrm{{nH}}}}(\theta)$ of $X$. For $t$ sufficiently small,
we have

$\theta_{t}+\mathrm{dd^{c}}\rho\geq\delta_{0}\omega_{X}.$

We set $v_{t}=P_{\delta_{0}\omega_{X}}(\varphi_{t}-\rho)$. Since the set
$E=\\{\rho>-1\\}$ is not empty, it is non-pluripolar. We have that the global
extremal function $V_{E,\delta_{0}\omega_{X}}$ of $E$ is not identically
$+\infty$ (cf. [GZ17, Theorem 9.17]). On $E$, $v_{t}\leq\varphi_{t}-\rho\leq
1$, hence $v_{t}-1\leq V^{*}_{E,\delta_{0}\omega_{X}}$ on $X$. Therefore
$\sup_{X}v_{t}<+\infty$ independent of $t$. Since $\varphi_{t}-\rho$ is
bounded from below and lower semi-continuous, it follows from [GL22, Theorem
2.3] that $v_{t}$ is a bounded $(\delta_{0}\omega_{X})$-psh function and
$(\delta_{0}\omega_{X}+\mathrm{dd^{c}}v_{t})^{n}$ is concentrated on the
contact set

$D_{t}:=\\{v_{t}=\varphi_{t}-\rho\\}\subset\\{\rho>-\infty\\}.$

Note that $D_{t}$ is not empty and $\sup_{X}v_{t}$ is uniformly (in $t$)
bounded from below since $\rho\leq 0$. Since $v_{t}+\rho\leq\varphi_{t}$
outside a pluripolar set, hence on $X$. It follows from the maximum principle
(Lemma 2.7) that

$\mathbf{1}_{D_{t}}(\theta_{t}+\mathrm{dd^{c}}(v_{t}+\rho))^{n}\leq\mathbf{1}_{D_{t}}(\theta_{t}+\mathrm{dd^{c}}\varphi_{t})^{n}.$

To summarize, we have

$\displaystyle(\delta_{0}\omega_{X}+\mathrm{dd^{c}}v_{t})^{n}$
$\displaystyle=\mathbf{1}_{D_{t}}(\delta_{0}\omega_{X}+\mathrm{dd^{c}}v_{t})^{n}$
$\displaystyle\leq\mathbf{1}_{D_{t}}(\theta_{t}+\mathrm{dd^{c}}(v_{t}+\rho))^{n}$
$\displaystyle\leq\mathbf{1}_{D_{t}}(\theta_{t}+\mathrm{dd^{c}}\varphi_{t})^{n}$
$\displaystyle\leq c_{t}f\omega_{X}^{n}.$

We show that the constant $c_{t}$ can be controlled, not being dependent on
$t$. Let $G$ be the Gaduchon factor of $\omega_{X}$, i.e.,
$G\in\mathcal{C}^{\infty}(X)$ such that
$\mathrm{dd^{c}}(e^{G}\omega_{X}^{n-1})=0$. By the arithmetic-geometric
inequality, we have

$c_{t}^{\frac{1}{n}}f^{\frac{1}{n}}e^{\frac{(n-1)G}{n}}\omega_{X}^{n}\leq(\theta_{t}+\mathrm{dd^{c}}\varphi_{t})\wedge
e^{G}\omega_{X}^{n-1}.$

Integrating both sides on $X$ we get a upper bound on $c_{t}$ depending on a
lower bound for $\int_{X}f^{\frac{1}{n}}e^{\frac{(n-1)G}{n}}\omega_{X}^{n}$
and $\int_{X}\omega_{X}^{n}$.

We have shown that $(\delta_{0}\omega_{X}+\mathrm{dd^{c}}v_{t})^{n}\leq
c_{t}f\omega_{X}^{n}$ with $c_{t}>0$ uniformly bounded from above. By
Dinew–Kolodziej–Nguyen’s uniform estimate (see also [GL23, Theorem 2.2]), we
see that $v_{t}\geq-C_{0}$ for a uniform constant $C>0$ only depending on $n$,
$p$, $\delta_{0}$, $\omega_{X}$, and $\|f\|_{p}$ with $p>1$. Here, we have
used the fact that $\sup_{X}v_{t}$ is uniformly bounded, hence $v_{t}$ belongs
to a compact set of quasi-psh functions. Therefore, we have

$\varphi_{t}\geq\rho-C_{0}.$

For a lower bound of $c_{t}$, by [GL23, Lemma 2.1] we can find a bounded
$\omega_{X}$-psh function $u$ and $c=c(n,\omega_{X},\|f\|_{p})>0$, with $p>1$
such that $(\omega_{X}+\mathrm{dd^{c}}u)^{n}\geq cf\omega_{X}^{n}$. We have

$c_{t}(\theta_{t}+\mathrm{dd^{c}}(\rho+\delta_{0}u))^{n}\geq\delta_{0}^{n}c_{t}(\omega_{X}+\mathrm{dd^{c}}u)^{n}\geq
c\delta_{0}^{n}(\theta_{t}+\mathrm{dd^{c}}\varphi_{t})^{n}$

hence $c_{t}\geq c\delta_{0}^{n}>0$ by the domination principle [GL23,
Corollary 1.14].

Step 2: Laplacian estimates. In the sequel, we establish uniform bounds on
$\Delta_{\omega_{X}}\varphi_{t}$ on compact subsets of
$X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$. The difficulty in this
part is dealing with the torsion term corresponding $\omega_{X}$. We follow
the trick due to Phong–Sturm [PS10]. We use $C$ for positive constants whose
values may change from line to line.

We set $u:=\varphi-\rho+C_{0}+1$ so that $e^{-u}\in[0,1]$, and
$\omega^{\prime}:=\theta_{t}+\mathrm{dd^{c}}\varphi_{t}$ (we omit the
subscript $t$ for simplicity). We denote by $g$ (resp. $g^{\prime}$) the
Riemann metric associated to $\omega_{X}$ (resp. $\omega^{\prime}$). We also
denote by $\nabla$ and $T$ the Chern connection and torsion of $g$. Consider

$Q:=\log\operatorname{tr}_{g}g^{\prime}-Au+e^{-u}$

with $A>0$ to be determined hereafter. We are going to show that $Q$ is
uniformly bounded from above. If this would be done, then for each $t$ and any
compact subset $K\subset\subset
X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$ we obtain a uniform bound
for $\Delta_{\omega_{X}}\varphi_{t}$ on $K$.

Since $\rho\to-\infty$ near $\operatorname{E_{\textrm{{nH}}}}(\theta)$ hence
$Q$ attains its maximum at some point $x_{0}\in
X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$. We assume that
$\operatorname{tr}_{g}g^{\prime}(x_{0})\geq 1$; otherwise we are done. We
compute as in [TW15, Sect. 9] at the point $x_{0}$,

(4.1)
$\begin{split}\Delta_{g^{\prime}}\log\operatorname{tr}_{g}g^{\prime}&\geq\frac{2}{(\operatorname{tr}_{g}g^{\prime})^{2}}\textrm{Re}\left(g^{\prime\bar{q}k}T^{i}_{ik}\partial_{\bar{q}}\operatorname{tr}_{g}g^{\prime}\right)+\frac{\Delta_{g}\log
c_{t}f}{\operatorname{tr}_{g}g^{\prime}}\\\
&-C\operatorname{tr}_{g^{\prime}}g-C\end{split}$

with $C>0$ only depending on $\omega_{X}$ and its torsion terms and Riemann
curvature. Notice that we used $\operatorname{tr}_{g}g^{\prime}(x_{0})\geq 1$.
We observe that $|\Delta_{g}\log f|\leq B$ for a uniform constant $B>0$. At
the maximum point $x_{0}$ of $Q$ we have $\partial_{\bar{q}}Q=0$, hence

$\frac{1}{\operatorname{tr}_{g}g^{\prime}}\partial_{\bar{q}}\operatorname{tr}_{g}g^{\prime}-(A+e^{-u})\partial_{\bar{q}}u=0.$

The Cauchy–Schwarz inequality yields

(4.2)
$\begin{split}\left|\frac{2}{(\operatorname{tr}_{g}g^{\prime})^{2}}\textrm{Re}\left(g^{\prime\bar{q}k}T^{i}_{ik}\partial_{\bar{q}}\operatorname{tr}_{g}g^{\prime}\right)\right|&\leq\left|\frac{2}{\operatorname{tr}_{g}g^{\prime}}\textrm{Re}\left((A+e^{-u})\partial_{\bar{q}}ug^{\prime\bar{q}k}T^{i}_{ik}\right)\right|\\\
&\leq e^{-u}|\partial
u|^{2}_{g^{\prime}}+C(A+1)^{2}e^{u}\frac{\operatorname{tr}_{g^{\prime}}g}{(\operatorname{tr}_{g}g^{\prime})^{2}}\end{split}$

with $C>0$ under control. If at $x_{0}$,
$(\operatorname{tr}_{g}g^{\prime})^{2}\leq C(A+1)^{2}e^{u}$ then

$Q\leq C+\frac{1}{2}u-Au+e^{-u}\leq C+1$

we are done. Otherwise, from (4.1) together with (4.2), we have at $x_{0}$

$\begin{split}0\geq\Delta_{g^{\prime}}Q&\geq-e^{-u}|\partial
u|^{2}_{g^{\prime}}-(B+C+1)\operatorname{tr}_{g^{\prime}}g-C\\\
&\quad-(A+e^{-u})\operatorname{tr}_{\omega^{\prime}}(\omega^{\prime}-(\theta_{t}+\mathrm{dd^{c}}\rho))+e^{-u}|\partial
u|^{2}_{g^{\prime}}\\\
&\geq(A\delta_{0}-B-C-1)\operatorname{tr}_{\omega^{\prime}}\omega_{X}-(A+1)n-C\end{split}$

where we used $\theta_{t}+\mathrm{dd^{c}}\rho\geq\delta_{0}\omega_{X}$.
Choosing $A=(B+C+2)\delta_{0}^{-1}$, at $x_{0}$, we obtain

$\operatorname{tr}_{\omega^{\prime}}\omega_{X}\leq C$

Hence, at $x_{0}$

(4.3) $\operatorname{tr}_{\omega_{X}}\omega^{\prime}\leq n\frac{\omega^{\prime
n}}{\omega_{X}^{n}}(\operatorname{tr}_{\omega^{\prime}}\omega_{X})^{n-1}\leq
nC^{n-1}c_{t}f$

which is uniformly bounded from above since $c_{t}$ is so. It therefore
follows that

$\operatorname{tr}_{\omega_{X}}\omega^{\prime}\leq
Ce^{-A\rho}\quad\text{on}\;X.$

Step 3: Higher order estimates and conclusion. For each compact subset
$K\subset\subset X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$, we
obtain uniform bounds on $\|\Delta_{\omega_{X}}\varphi_{t}\|_{L^{\infty}(K)}$
as follows from the previous step. We apply a complex version of the
Evans–Krylov–Trudinger estimate (see [TW10b, Sect. 4] in the non-Kähler
context) and Schauder’s estimates to obtain for any $\alpha\in]0,1[$,
$j\in\mathbb{N}$,

$\sup_{t}\|\varphi_{t}\|_{\mathcal{C}^{j,\alpha}(K)}\leq
C_{j,\alpha}<+\infty.$

This guarantees that the family $(\varphi_{t})_{t>0}$ is relatively compact in
$\mathcal{C}_{\rm
loc}^{\infty}(X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta))$. Up to
exacting and relabelling, we can pass the limit to get

$c=\lim_{t\to 0}c_{t}>0,\quad\varphi=\lim_{t\to 0}\varphi_{t}$

satisfying that on $X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$,
$\varphi$ is smooth and
$(\theta+\mathrm{dd^{c}}\varphi)^{n}=cf\omega_{X}^{n}$. By Hartogs’ lemma, we
obtain $\sup_{X}\varphi=0$. ∎

The same result still holds when the density $f$ is mildly singular.

######  Theorem 4.2.

Let $\theta$ be a smooth real (1,1) form whose $\partial\bar{\partial}$-class
is nef and big. Assume $f=e^{\psi^{+}-\psi^{-}}\in L^{p}(X,\omega_{X})$,
$p>1$, and $f$ is smooth and positive in the complement of a divisor
$X\backslash D$, with $\psi^{\pm}$ quasi-psh functions on $X$. Then, there
exists a constant $c>0$ and $\varphi\in\operatorname{{\rm PSH}}(X,\theta)$
such that $\sup_{X}\varphi=0$ and on
$X\backslash(D\cup\operatorname{E_{\textrm{{nH}}}}(\theta))$, $\varphi$ is
smooth and

$(\theta+\mathrm{dd^{c}}\varphi)^{n}=cf\omega_{X}^{n}.$

###### Proof.

The proof is almost identical to the previous one, so we only give the main
idea. Without loss of generality, we assume that $\sup_{X}\psi^{\pm}\leq-1$.
For a quasi-psh function $\psi$, the Demailly regularization
$\psi_{t}:=\rho_{t}(\psi)$ of $\psi$, with $t>0$ so small, is defined by

$\rho_{t}(\psi):=\frac{1}{t^{2n}}\int_{\zeta\in T_{x}X}\psi(\operatorname{{\rm
exph}}_{x}(\zeta))\chi\left(\frac{|\zeta|^{2}}{t^{2}}\right)dV_{\omega_{X}}$

where $\chi$ is a cut-off function normalized by $\int_{\mathbb{R}}\chi(t)=1$
and $\operatorname{{\rm exph}}$ is the formal holomorphic part of the Taylor
expansion of the exponential map of the Chern connection on the tangent space
$TX$ associated with $\omega_{X}$ (see [Dem94, (3.1)]). We have that the
corresponding regularization $\psi_{t}^{\pm}$ is smooth on $X$ and satisfies
$\mathrm{dd^{c}}\psi_{t}^{\pm}\geq-A\omega_{X}$ where $A>0$ is a uniform
constant only depending on the Lelong numbers of $\psi^{\pm}$. Up to
rescaling, we assume in the sequel that $A=1$. We observe that
$f_{t}:=e^{\psi^{+}_{t}-\psi^{-}_{t}}$ converges to $f$ in $L^{p}$ by the
dominated convergence theorem. Up extracting subsequence, we may assume that
$f_{t}\leq F$ for some $F\in L^{p}(X,\omega_{X})$.

As explained in the previous part, it suffices to obtain uniform estimates and
Laplacian estimates of the potential $\varphi_{t}$ solution of

$(\theta+t\omega_{X}+\mathrm{dd^{c}}\varphi_{t})^{n}=c_{t}f_{t}\omega_{X}^{n},\;\sup_{X}\varphi_{t}=0.$

We use the same trick as above with minor modifications. The same argument
yields that $c_{t}$ has a uniform upper bound only depending on $\omega_{X}$,
$n$ and $\int_{X}f^{\frac{1}{n}}\omega_{X}^{n}$. From this we have

$(\theta_{t}+\mathrm{dd^{c}}\varphi_{t})^{n}\leq F\omega_{X}^{n}$

with $F\in L^{p}(X)$, $p>1$. It follows that $C_{0}\geq\varphi\geq\rho-C_{0}$
for a constant $C_{0}>0$ depending on $n$, $\omega_{X}$, $p$, and an upper
bound for $\|F\|_{p}$.

For the Laplacian estimate, we consider

$Q:=\log\operatorname{tr}_{\omega_{X}}\omega^{\prime}-Au+e^{-u}$

where $A>0$ sufficiently large and

$\omega^{\prime}:=\theta+t\omega_{X}+\mathrm{dd^{c}}\varphi_{t},\;\;u:=\varphi_{t}-\rho-2\delta\psi^{-}+C_{0}+1,\;\delta\in]0,\delta_{0}[.$

By the same arguments as in Step 2 of the proof of Theorem 4.1 (see also
[GL23, Theorem 4.2]), one obtains

$\Delta_{\omega_{X}}\varphi_{t}\leq Ce^{-A(\rho+2\delta\psi^{-})}.$

The conclusion of the proof is then roughly the same as in the smooth case. ∎

###  4.2. Regularity of envelopes

Our main theorem in this section is the following.

######  Theorem 4.3.

Let $X$ be a $n$-dimensional compact complex manifold equipped with a
Hermitian metric $\omega_{X}$. Let $\theta$ be a smooth real (1,1) form whose
$\partial\bar{\partial}$-class is nef and big. Let
$f\in\mathcal{C}^{\infty}(X,\mathbb{R})$. Then
$P_{\theta}(f)\in\mathcal{C}^{1,\alpha}(X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta))$
with $\alpha\in]0,1[$.

###### Proof.

The proof follows the PDE method of R. Berman [Ber19]. Up to replacing
$\theta$ with $\theta_{f}=\theta+\mathrm{dd^{c}}f$, we may assume that $f=0$.
We consider, for $\beta>1$ fixed positive numbers, the following complex
Monge–Ampère equations for a smooth function

$(\theta+2^{-\beta}\omega_{X}+\mathrm{dd^{c}}\varphi_{\beta})^{n}=e^{\beta\varphi_{\beta}}\omega_{X}^{n}.$

Note that the $\partial\bar{\partial}$-class $[\theta+2^{-\beta}\omega_{X}]$
is Hermitian for $\beta>0$. By the fundamental results of Cherrier [Che87] and
Tosatti–Weinkove [TW10a] there exists indeed a unique smooth solution
$\varphi_{\beta}$ to the previous equation.

$\mathcal{C}^{0}$-estimates. Using the domination principle, we have

$\left(1-\frac{1}{\beta}\right)P_{\theta}(0)+\frac{1}{\beta}\rho-\frac{C\log\beta}{\beta}\leq\varphi_{\beta}\leq\frac{C}{\beta}$

with $C>0$ under control. In particular, $\varphi_{\beta}$ converges to
$P_{\theta}(0)$ in $\mathcal{C}^{0}_{\rm
loc}(X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta))$, as
$\beta\to+\infty$.

Indeed, we assume that $\theta+2^{-\beta}\omega_{X}\leq A\omega_{X}$ for a
uniform constant $A>0$. We have

$(A\omega_{X}+\mathrm{dd^{c}}\varphi_{\beta})\geq e^{\beta(\varphi-n\log
A/\beta)}A^{n}\omega_{X}^{n}.$

It follows from the domination principle that $\varphi_{\beta}\leq n\log
A/\beta$. Conversely, fixing an $\theta$-psh function $u\geq\rho$, we consider

$v_{\beta}:=\left(1-\frac{1}{\beta}\right)u+\frac{1}{\beta}\rho-\frac{n\log(\delta_{0}^{-1}\beta)}{\beta}$

where $\rho$ is an $\theta$-psh function such that $\sup_{X}\rho\leq 0$ and
$\theta+\mathrm{dd^{c}}\rho\geq 2\delta_{0}\omega_{X}$ for small
$\delta_{0}>0$. We have

$(\theta+2^{-\beta}\omega_{X}+\mathrm{dd^{c}}v_{\beta})^{n}\geq(\delta_{0}/\beta)^{n}\omega_{X}^{n}\geq
e^{\beta v}\omega_{X}^{n}.$

It follows from the domination principle [GL23, Corollary 1.14] that
$\varphi_{\beta}\geq v_{\beta}$. Taking the supremum for all candidates
$u\geq\rho$, we obtain

$\varphi_{\beta}\geq\left(1-\frac{1}{\beta}\right)P_{\theta}(0)+\frac{1}{\beta}\rho-\frac{C\log\beta}{\beta}.$

In particular, since $P_{\theta}(0)$ dominates all $\theta$-psh function $u$
with $\sup_{X}u\leq 0$ and $\beta>1$, there exists a uniform constant
$C_{0}>0$ independent of $\beta$ such that

$\varphi_{\beta}\geq\rho-C_{0}.$

Laplacien estimates. Fixing $\beta>1$ and setting
$\omega^{\prime}:=\theta+2^{-\beta}\omega_{X}+\mathrm{dd^{c}}\varphi_{\beta}$,
we let $g$ denotes the Riemannian metric associated to $\omega_{X}$ and
$g^{\prime}$ the one associated to $\omega^{\prime}$. Setting
$u:=\varphi_{\beta}-\rho-C_{0}+1$, we consider

$Q:=\log\operatorname{tr}_{g}g^{\prime}-Bu+e^{-u}$

for $B>0$ to be chosen hereafter. Since $u\to-\infty$ near
$\operatorname{E_{\textrm{{nH}}}}(\theta)$ hence $Q$ attains its maximum at
some point $x_{0}\in X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$. At
this point, we have $\partial_{\bar{q}}Q=0$. Therefore, the same arguments
yields (cf. (4.1), (4.2)), at $x_{0}$

$\begin{split}0\geq-C(A+1)^{2}e^{u}\frac{\operatorname{tr}_{g^{\prime}}g}{(\operatorname{tr}_{g}g^{\prime})^{2}}+\frac{\Delta_{g}\beta\varphi_{\beta}}{\operatorname{tr}_{g}g^{\prime}}+(B\delta_{0}-C)\operatorname{tr}_{g}g^{\prime}-C\end{split}$

If at $x_{0}$, $(\operatorname{tr}_{g}g^{\prime})^{2}\leq C(A+1)^{2}e^{u}$ we
are done. Otherwise, since
$\Delta_{g}\varphi_{\beta}=\operatorname{tr}_{g}g^{\prime}-\operatorname{tr}_{g}(\theta+2^{-\beta}\omega_{X})$,
it thus follows that

$0\geq(B\delta_{0}-C-1)\operatorname{tr}_{g^{\prime}}g-C-(n+1)B+\beta-\beta\frac{\sup_{X}\operatorname{tr}_{g}(\theta+2^{-\beta}\omega_{X})}{\operatorname{tr}_{g}g^{\prime}}.$

Choosing $B=(C+1)\delta_{0}^{-1}$ gives

$(C+(n+1)B)\operatorname{tr}_{g}g^{\prime}+\beta\sup_{X}\operatorname{tr}_{g}(\theta+2^{-\beta}\omega_{X})\geq\beta\operatorname{tr}_{g}g^{\prime}$

at $x_{0}$. Hence for $\beta>C+(n+1)B=:\beta_{0}$,

$\operatorname{tr}_{g}g^{\prime}(x_{0})\leq\frac{1}{1-\beta_{0}/\beta}M,\;\text{with}\;M=\sup_{X}\operatorname{tr}_{g}(\theta+\omega_{X})$

It follows that for $\beta>\beta_{0}$,

(4.4)
$\operatorname{tr}_{\omega_{X}}(\theta+2^{-\beta}\omega_{X}+\mathrm{dd^{c}}\varphi_{\beta})\leq\exp\left(\frac{1}{1-\beta_{0}/\beta}M\right)\delta_{0}^{-n}e^{B\sup_{X}\varphi_{\beta}}e^{-B\rho}.$

We have known that $\sup_{X}\varphi_{\beta}$ is uniformly bounded in terms of
$A>0$. Therefore, for $\beta>\beta_{0}$, for any compact subset $K\subset
X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$,

$-C_{K}\leq\Delta_{\omega_{X}}\varphi_{\beta}\leq C_{K}.$

After passing a subsequence, the basic compactness result yields that
$\varphi_{\beta}$ converges to a function $\varphi$ in
$\mathcal{C}^{1,\alpha}$ for any fixed $\alpha\in]0,1[$. It follows from the
$\mathcal{C}^{0}$-estimates that $\varphi=P_{\theta}(0)$. This completes the
proof. ∎

######  Remark 4.4.

We can relax the assumption of $f$ by merely assuming that
$f\in\mathcal{C}^{0}(X)$, $\sup_{X}|f|\leq C$, and $\mathrm{dd^{c}}f\leq
C\omega_{X}$ for a fixed constant $C>0$. Indeed, we approximate $f$ uniformly
by smooth functions $f_{j}$ with $\mathrm{dd^{c}}f_{j}\leq(C+1)\omega_{X}$.
Let $\varphi_{\beta,j}$ be the solutions of the complex Monge–Ampère equations
obtained by replacing $\theta_{f}$ with
$\theta_{f_{j}}=\theta+\mathrm{dd^{c}}f_{j}$. We still have that
$M_{j}:=\sup_{X}\operatorname{tr}_{\omega_{X}}(\theta_{f_{j}}+\omega_{X})\leq
M<\infty$ and $\sup_{X}\varphi_{\beta,j}\leq C$ independent of $j$,
$\beta\geq\beta_{0}$. Hence, we obtain the precise Laplacian estimate as in
(4.4)

$\operatorname{tr}_{\omega_{X}}(\theta_{f_{j}}+2^{-\beta}\omega_{X}+\mathrm{dd^{c}}\varphi_{\beta,j})\leq\exp\left(\frac{1}{1-\beta_{0}/\beta}M\right)\delta_{0}^{-n}e^{B\sup_{X}\varphi_{\beta,j}}e^{-B\rho}.$

This shows that $\Delta_{\omega_{X}}P_{\theta}(f_{j})$ is locally bounded on
$X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta)$ as $\beta\to+\infty$.
Letting $j\to+\infty$, since $P_{\theta}(f_{j})$ uniformly converges to
$P_{\theta}(f)$ we arrive at the conclusion.

##  5\. Finite time singularities of the Chern–Ricci flow

Let $(X,\omega_{0})$ be a compact Hermitian manifold of dimension $n$. We
consider a solution $\omega=\omega(t)$ of the Chern–Ricci flow starting from
$\omega_{0}$ is given by a smooth family of Hermitian metrics satisfying

(5.1) $\begin{cases}\frac{\partial\omega}{\partial t}=-\operatorname{{\rm
Ric}}(\omega),\;0\leq t<T\\\ \omega(0)=\omega_{0}.\end{cases}$

One observes that just like the Kähler–Ricci flow, the Chern–Ricci flow is
equivalent to a parabolic equation for a scalar function. Namely, solving a
solution to the Chern–Ricci flow starting at $\omega_{0}$ boils down to
solving the parabolic complex Monge–Ampère equation, for $\varphi=\varphi(t)$
with $t\in[0,T[$,

(5.2) $\frac{\partial\varphi}{\partial
t}=\log\frac{(\theta_{t}+\mathrm{dd^{c}}\varphi)^{n}}{\omega_{0}^{n}},\quad\theta_{t}+\mathrm{dd^{c}}\varphi>0,\quad\varphi(0)=0$

where $\theta_{t}:=\omega_{0}-t\operatorname{{\rm Ric}}(\omega_{0})$. Since
$X$ is compact, by the standard parabolic theory, the Chern–Ricci flow always
admits a unique solution on some time interval $[0,T[$ for some
$0<T\leq\infty$; see e.g. [Tos18a, Theorem 3.5]. As follows from [TW15], the
maximal existence time $T$ can be characterized by

$T=\sup\\{t>0:\exists\,\psi\in\mathcal{C}^{\infty}(X)\,\text{with}\,\theta_{t}+\mathrm{dd^{c}}\psi>0\\}.$

When $\omega_{0}$ satisfies the Guan–Li condition
$\mathrm{dd^{c}}\omega_{0}^{k}=0$ for all $1\leq k\leq n-1$, we can check that
this condition is preserved by the Chern–Ricci flow, also does $\theta_{t}$.
Thanks to the Nakai–Moishezon criterion (Theorem B), we obtain a geometric
characterization of the maximal existence time

$T=\sup\left\\{t>0:\int_{V}\theta_{t}^{k}\wedge\omega_{0}^{\dim
V-k}>0,\forall\,\text{irreducible subvarieties}\,V\subset X\right\\}.$

This extends the result [TW15, Theorem 1.3] to higher dimensions.

In this section, we study finite time singularities of the Chern–Ricci flow,
proving theorem E. Suppose that $T<\infty$, so the flow develops singularities
at a finite time. Following [CT15] (see also [TW22]), the set on which
singularities of the Chern–Ricci flow develop on $M$ (depending on the initial
metric $\omega_{0}$) is defined by

(5.3)
$\begin{split}\operatorname{\textrm{{Sing}}}(\omega(t))=X\backslash\\{x\in
X|U\ni x,\exists\,\omega_{T}\,&\text{Hermitian metric on}\,U\\\
&\text{s.t.}\,\omega(t)\xrightarrow{t\to
T^{-}}\omega_{T}\,\text{in}\,\mathcal{C}^{\infty}_{\rm loc}(U)\\}.\end{split}$

Following Z. Zhang [Zha10], we define the singularity formation set of the
flow $\Sigma$ (which depends on the initial metric $\omega_{0}$) to be the
complement of the set of points $x\in X$ such that there exists an open
neighborhood $U$ of $x$ a constant $C>0$ with
$|\textrm{Rm}(t)|_{\omega(t)}\leq C$ on $[0,T[\times U$, where
$\textrm{Rm}(t)$ denotes the curvature tensor of $\omega(t)$. Also, we define
the set $\Sigma^{\prime}$ to be the complement of the set of points $x\in X$
such that there exists an open neighborhood $U$ of $x$ a constant $C>0$ with
$|R(t)|_{\omega(t)}\leq C$ on $[0,T[\times U$, where $R(t)$ is the scalar
curvature of $\omega(t)$.

It was shown by Gill–Smith [GS15] that the singularity set of the flow is
equal to the region on which the Chern scalar curvature blows up. The latter
set is the singularity formation locus of the flow due to [EMT11]. We give two
equivalent definitions of the singularity set
$\operatorname{\textrm{{Sing}}}(\omega(t))$.

######  Theorem 5.1.

Assume that the Chern–Ricci flow (5.1) exists on the maximal time interval
$[0,T[$ with $T<\infty$. Then

$\operatorname{\textrm{{Sing}}}(\omega(t))=\Sigma=\Sigma^{\prime}.$

###### Proof.

This was proved by M. Gill and K. Smith. However, it is unclear how the
derivative $\partial_{i}Q_{1}$ in (3.9) of [GS15] vanishes at the maximum
point of $Q_{2}$. We give the proof here, mimicking their estimates with minor
modifications.

Since a uniform bound on the curvature tensor implies an upper bound on the
scalar curvature, we have $\Sigma^{\prime}\subset\Sigma$. If the metrics
$\omega(t)$ converges smoothly to a Hermitian metric $\omega_{U}$ on
$[0,T[\times U$ for some open subset $U$, then we have that
$|\textrm{Rm}(t)|_{\omega(t)}\leq C$, hence the inclusion
$\operatorname{\textrm{{Sing}}}(\omega(t))\subset\Sigma$. It remains to show
that if for any $x\in X$ with an open neighborhood $U$ of $x$ and $C>0$
satisfying $|R(t)|\leq C$ on $[0,T[\times U$, then on a possibly smaller open
neighborhood $U^{\prime}$ of $x$ we have that $\omega(t)$ converges smoothly
to a Hermitian metric $\omega_{U^{\prime}}$ as $t\to T^{-}$. We remark that
most of the estimates from the proof of Theorem 5.3 are local and hence could
be applied again here.

Fixing $x\in K\subset\subset U$ a compact subset, we will establish uniform
estimates on $\omega(t)$ in $K$. It follows from the flow equation that

$\frac{\partial\varphi}{\partial
t}=\log\frac{\omega(t)^{n}}{\omega_{0}^{n}},\quad\frac{\partial}{\partial
t}\left(\frac{\partial\varphi}{\partial t}\right)=-R\leq C$

on $[0,T[\times U$. Integrating in time implies that
$|\partial_{t}\varphi|\leq C$ and $|\varphi|\leq C$ on $[0,T[\times U$.
Therefore the quantity $H=t\partial_{t}\varphi-\varphi-nt$ is uniformly
bounded on $[0,T[\times U$, also is $\partial_{t}H=-tR-n$, and

$\left(\frac{\partial}{\partial
t}-\Delta_{\omega}\right)H=-\operatorname{tr}_{\omega}\omega_{0}.$

We set

$Q:=\log\operatorname{tr}_{\omega_{0}}\omega+AH+e^{H}$

for $A>0$ to be determined hereafter. The quantity $Q$ is achieved its maximum
at some point $(t_{0},x_{0})\in[0,T^{\prime}]\times K$ with $T^{\prime}<T$. It
follows from [TW15, Proposition 3.1] (also [TW15, (4.2)]) that

(5.4) $\left(\frac{\partial}{\partial
t}-\Delta_{\omega}\right)\log\operatorname{tr}_{\omega_{0}}\omega\leq\frac{2}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}\textrm{Re}(g^{\bar{q}k}(T_{0})^{p}_{kp}\partial_{\bar{q}}\operatorname{tr}_{\omega_{0}}\omega)+C\operatorname{tr}_{\omega}\omega_{0},$

where $(T_{0})^{p}_{kp}$ denote the torsion terms corresponding to
$\omega_{0}$. At the maximum point $(t_{0},x_{0})$ of $Q$ we have
$\partial_{\bar{q}}Q=0$ hence

$\frac{1}{\operatorname{tr}_{\omega_{0}}\omega}\partial_{\bar{q}}\operatorname{tr}_{\omega_{0}}\omega+A\partial_{\bar{q}}H+e^{H}\partial_{\bar{q}}H=0.$

Therefore, the Cauchy–Schwarz inequality yields

(5.5)
$\begin{split}\left|\frac{2}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}\textrm{Re}(g^{\bar{q}k}(T_{0})^{p}_{kp}\partial_{\bar{q}}\operatorname{tr}_{\omega_{0}}\omega)\right|&\leq\left|\frac{2}{\operatorname{tr}_{\omega_{0}}\omega}\textrm{Re}((A+e^{H})g^{\bar{q}k}(T_{0})^{p}_{kp}\partial_{\bar{q}}H\right|\\\
&\leq e^{H}|\partial
H|^{2}_{\omega}+C(A+1)^{2}e^{-H}\frac{\operatorname{tr}_{\omega}\omega_{0}}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}\end{split}$

for uniform $C>0$ only depending on the torsion term. From (5.4) and (5.5), we
have at $(t_{0},x_{0})$,

$\begin{split}0\leq\left(\frac{\partial}{\partial
t}-\Delta_{\omega}\right)Q&\leq
C(A+1)^{2}e^{-H}\frac{\operatorname{tr}_{\omega}\omega_{0}}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}+C\operatorname{tr}_{\omega}\omega_{0}-(A+e^{H})\operatorname{tr}_{\omega}\omega_{0}.\end{split}$

If $(\operatorname{tr}_{\omega}\omega_{0})^{2}\leq C(A+1)^{2}e^{-H}$ at
$(t_{0},x_{0})$, then

$Q\leq C-\frac{1}{2}H+AH+e^{H}\leq C_{1}$

we are done. Otherwise, we choose $A=C+1$, then at $(t_{0},x_{0})$,

$\operatorname{tr}_{\omega}\omega_{0}\leq C.$

On the other hand, since $\omega^{n}/\omega_{0}^{n}=\partial_{t}\varphi\leq
C$, hence

$\operatorname{tr}_{\omega_{0}}\omega\leq
C(\operatorname{tr}_{\omega}\omega_{0})^{n-1}\frac{\omega^{n}}{\omega_{0}^{n}}\leq
C.$

This implies that $Q$, and thus $\operatorname{tr}_{\omega_{0}}\omega$ is
bounded from above on $[0,T[\times K$ since $T^{\prime}$ was taken
arbitrarily. Finally, the local higher-order estimates for $\omega$ on
$[0,T[\times U^{\prime}$ with $U^{\prime}\subset U$ follow by Sherman–Weinkove
[SW13]. The proof is complete. ∎

Consequently, the scalar curvature blows up at a finite time singularity,
generalizing Zhang’s analogous result [Zha10] in the Kähler case.

######  Corollary 5.2 ([GS15, Theorem 1.1]).

If the Chern–Ricci flow develops a finite time singularity, then the
singularity formation set $\Sigma$ is nonempty, and furthermore,

$\limsup_{t\to T}\left(\sup_{X}R(t)\right)=+\infty.$

###### Proof.

If $\Sigma=\varnothing$ then by Theorem 5.1, replacing $K$ with $X$, we would
have that $\omega(t)$ converges in $\mathcal{C}^{\infty}$ to a Hermitian
metric $\omega_{T}$ on $X$, a contradiction. The blow-up of the supremum of
the scalar curvature again follows from Theorem 5.1. ∎

We next compare the singularity formation set of the flow
$\operatorname{\textrm{{Sing}}}(\omega(t))$ with the null locus of the
limiting form $\theta_{T}=\omega_{0}-T\operatorname{{\rm Ric}}(\omega_{0})$ in
some specific contexts. We see that the $\partial\bar{\partial}$ class
$[\theta_{T}]$ is nef, but not Hermitian. Since the definition of
$\widehat{v}_{-}(X,\theta_{T})$ does not depend on the choice of reference
Hermitian metric $\omega_{X}$ and for any $\varepsilon>0$,

$(1-\varepsilon)\theta_{T}+\varepsilon\omega_{0}=\theta_{(1-\varepsilon){T}}$

so we have

$\widehat{v}_{-}(V,\theta_{T}|_{V})=\inf_{t\to T^{-}}v_{-}(V,\omega(t)|_{V})$

for all irreducible analytic subvarieties $V\subset X$. We define the set-
theoretic union of all such subvarieties for which the infimum volume shrinks
to zero as $t\to T^{-}$,

(5.6) $N(\omega(t)):=\bigcup_{v_{-}(V,\omega(t)|_{V})\xrightarrow{t\to T}0}V.$

In particular, the set $N(\omega(t))$ is a closed analytic subset of $X$.

If $\theta_{T}$ is big, we can find a $\theta_{T}$-psh function $\rho$ with
almost analytic singularities along
$\operatorname{E_{\textrm{{nH}}}}(\theta_{T})$ such that

$\theta_{T}+\mathrm{dd^{c}}\rho\geq 2\delta_{0}\omega_{0}$

for some $\delta_{0}>0$. Up to subtracting a positive constant, we can assume
that $\sup_{X}\rho\leq 0$. We have

$\theta_{t}+\mathrm{dd^{c}}\rho=\frac{1}{T}\left((T-t)(\omega_{0}+\mathrm{dd^{c}}\rho)+t(\theta_{T}+\mathrm{dd^{c}}\rho)\right)\geq\delta_{0}\omega_{0}$

if $t\in[T-\varepsilon,T[$ for some $\varepsilon>0$.

We are interested in the behavior of the metrics $\omega(t)$ as $t\to T^{-}$,
proving the following

######  Theorem 5.3.

Assume that the limiting form $\theta_{T}$ is big. Then there exists a
positive (1,1) current $\omega_{T}$ on $X$ in the $\partial\bar{\partial}$
class $[\theta_{T}]$, which is smooth precisely away from
$\operatorname{E_{\textrm{{nH}}}}(\theta_{T})$ and has minimal singularities,
such that as $t\to T^{-}$ the metrics $\omega(t)$ converge to $\omega_{T}$ in
both current and $\mathcal{C}_{\rm
loc}^{\infty}(X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta_{T}))$
senses. In particular,
$\operatorname{\textrm{{Sing}}}(\omega(t))\subset\operatorname{E_{\textrm{{nH}}}}(\theta_{T})$.

######  Lemma 5.4.

There is a uniform constant $C_{0}>0$ such that on $[0,T[\times X$ we have

  1. (i)

$C_{0}\geq\varphi\geq\rho-C_{0}$;

  2. (ii)

$C_{0}\geq\partial_{t}{\varphi}\geq C_{0}\rho-C_{0}$.

###### Proof.

The proof follows almost verbatim from the Kähler case [CT15, Sect. 4]. We
also refer to [TW13, Sect. 2] or [Dan23, Sect. 5] for a proof. ∎

######  Lemma 5.5.

There is a constant $C>0$ such that on $[0,T[\times X$ we have

$\operatorname{tr}_{\omega_{0}}\omega\leq Ce^{-C\rho}.$

###### Proof.

The proof relies on the trick of Phong–Sturm [PS10], we repeat the estimates
derived in the proof of [TW15, Lemma 4.1] with minor modifications. Set
$\psi=\varphi-\rho+C_{0}\geq 0$, so $e^{-\psi}\in]0,1]$. We compute the
evolution of

$Q=\log\operatorname{tr}_{\omega_{0}}\omega-A\psi+e^{-\psi}$

for $A>0$ to be determined hereafter. It suffices to show that $Q$ is
uniformly bounded from above. We observe that $Q\leq C$ on
$[0,T-\varepsilon]\times X$ for a uniform $C>0$. Fixing
$T-\varepsilon<T^{\prime}<T$, suppose that $Q$ attains its maximum at some
point $(t_{0},x_{0})\in[0,T^{\prime}]\times X$ with
$t\in]T-\varepsilon,T^{\prime}]$. In what follows, we compute at this point.
At the maximum point $(t_{0},x_{0})$ of $Q$ we have $\partial_{i}Q=0$ hence

$\frac{1}{\operatorname{tr}_{\omega_{0}}\omega}\partial_{i}\operatorname{tr}_{\omega_{0}}\omega-A\partial_{i}\psi-e^{-\psi}\partial_{i}\psi=0.$

Therefore, the Cauchy–Schwarz inequality yields

(5.7)
$\begin{split}\left|\frac{2}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}\textrm{Re}(g^{\bar{q}k}(T_{0})^{p}_{kp}\partial_{\bar{q}}\operatorname{tr}_{\omega_{0}}\omega)\right|&\leq\left|\frac{2}{\operatorname{tr}_{\omega_{0}}\omega}\textrm{Re}((A+e^{-\psi})g^{\bar{q}k}(T_{0})^{p}_{kp}\partial_{\bar{q}}\psi\right|\\\
&\leq
e^{-\psi}|\partial\psi|^{2}_{\omega}+C(A+1)^{2}e^{\psi}\frac{\operatorname{tr}_{\omega}\omega_{0}}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}\end{split}$

for uniform $C>0$ only depending on the torsion term. It thus follows from
(5.4) and (5.7) that, at $(t_{0},x_{0})$,

(5.8) $\begin{split}0\leq\left(\frac{\partial}{\partial
t}-\Delta_{\omega}\right)Q&\leq
C(A+1)^{2}e^{\psi}\frac{\operatorname{tr}_{\omega}\omega_{0}}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}+C\operatorname{tr}_{\omega}\omega_{0}\\\
&\quad-(A+e^{-\psi})\partial_{t}{\varphi}+(A+e^{-\psi})\operatorname{tr}_{\omega}(\omega-(\alpha_{t}+\mathrm{dd^{c}}\rho))\\\
&\leq
C(A+1)^{2}e^{\psi}\frac{\operatorname{tr}_{\omega}\omega_{0}}{(\operatorname{tr}_{\omega_{0}}\omega)^{2}}+(C-A\delta_{0})\operatorname{tr}_{\omega}\omega_{0}+(A+1)\log\frac{\omega_{0}^{n}}{\omega^{n}}\end{split}$

where we have used $\theta_{t}+\mathrm{dd^{c}}\rho\geq\delta_{0}\omega_{0}$.
If at $(t_{0},x_{0})$, $(\operatorname{tr}_{\omega_{0}}\omega)^{2}\leq
e^{\psi}C(A+1)^{2}$ then at the same point we obtain

$Q\leq C+\frac{1}{2}\psi-A\psi+e^{-\psi}\leq C+1$

since $\psi\geq 0$, we are done. Otherwise, we choose
$A=\delta_{0}^{-1}(C+2)$. Hence, from (5.8) one gets at the maximum of $Q$,

$\operatorname{tr}_{\omega}\omega_{0}\leq
C\log\frac{\omega_{0}^{n}}{\omega^{n}}+C,$

so

$\operatorname{tr}_{\omega_{0}}\omega\leq
n(\operatorname{tr}_{\omega}\omega_{0})^{n-1}\frac{\omega^{n}}{\omega_{0}^{n}}\leq
C\frac{\omega^{n}}{\omega_{0}^{n}}\left(\log\frac{\omega_{0}^{n}}{\omega^{n}}\right)^{n-1}+C\leq
C^{\prime}$

since $\omega^{n}/\omega_{0}^{n}\leq C_{0}$ by Lemma 5.4 and $y\mapsto y|\log
y|^{n-1}$ is bounded from above as $y\to 0$. Thanks to Lemma 5.4 $(iii)$, $Q$
is bounded from above at its maximum, finishing the proof. ∎

###### Proof of Theorem 5.3.

We apply Lemmas 5.4 and 5.5 to get that for any compact set $K\subset
X\backslash\operatorname{E_{\textrm{{nH}}}}(\theta_{T})$, $k\in\mathbb{N}$,
there is a constant $C_{K,k}>0$ such that

$\|\varphi(t)\|_{\mathcal{C}^{k}(K)}\leq C_{K,k}$

using the complex parabolic Evans–Krylov theory together with Schauder’s
estimate; see also Sherman–Weinkove’s local curvature estimates [SW13]. This
implies that for some constant $C_{K}>0$,

$\frac{\partial}{\partial t}(\varphi(t)+C_{K}t)\geq 0$

hence $\varphi(t)+C_{K}t$ is increasing in $t$ as well as being bounded from
above. This yields that $\varphi(t)$ has a limit as $t\to T^{-}$, denoted by
$\varphi_{T}$. An elementary argument yields that $\varphi(t)$ converges in
$\mathcal{C}^{\infty}$ on compact subsets of
$X\backslash\operatorname{E_{\textrm{{nH}}}}(\alpha)$. By weak compactness
$\varphi_{T}$ extends to a $\theta_{T}$-psh function on $X$ and $\varphi(t)$
converges to $\varphi_{T}$ in $L^{1}(X)$. The current $\omega_{T}$ can be
expressed as $\omega_{T}=\theta_{T}+\mathrm{dd^{c}}\varphi_{T}$.

Next, we show that $\varphi_{T}$ has minimal singularities, that is, for any
$\theta_{t}$-psh function $u$ there is a constant $C>0$ such that

$\varphi\geq u-C\quad\text{on}\,[0,T[\times X.$

Thanks to Demailly’s regularization theorem (Theorem 2.11), we can find
functions with logarithmic singularities $u_{m}$ decreasing to $u$ such that

$\theta_{T}+\mathrm{dd^{c}}u_{m}\geq-\varepsilon_{m}\omega_{0},\quad\varepsilon_{m}\searrow
0.$

We set

$G=(T-t)\partial_{t}\varphi+\varphi+nt-\varepsilon_{m}(t\partial_{t}\varphi-\varphi-
nt)-u_{m}.$

On $X\backslash\operatorname{\textrm{{Sing}}}(u_{m})$, $G$ is smooth and

$\left(\frac{\partial}{\partial
t}-\Delta_{\omega}\right)G=\operatorname{tr}_{\omega}(\theta_{T}+\varepsilon_{m}\omega_{0}+\mathrm{dd^{c}}u_{m})\geq
0.$

Since $\sup_{X}u_{m}\leq C$ uniformly, so $G$ is achieved its minimum in
$X\backslash\operatorname{\textrm{{Sing}}}(u_{m})$. Hence the minimum
principle implies that $G\geq-C$ independent of $m$, so

$(T-t-\varepsilon_{m})\partial_{t}\varphi+(1+\varepsilon_{m})\varphi(t)\geq
u_{m}-C\geq u-C.$

We take $m\to+\infty$ and use the fact that $\partial_{t}\varphi\leq C_{0}$ to
conclude. ∎

######  Remark 5.6.

The current $\omega_{T}$ has minimal singularities but is not a Hermitian
current. Indeed, we write $\omega_{T}=\theta_{T}+\mathrm{dd^{c}}\varphi_{T}$.
If $\omega_{T}$ were Hermitian, we would have that the current
$\theta_{T}+\mathrm{dd^{c}}(1-\varepsilon)\varphi_{T}$ is Hermitian for
$\varepsilon>0$ sufficiently small. By minimality, we have
$\varphi_{T}+C\geq(1-\varepsilon)\varphi_{T}$ for a constant $C>0$ depending
on $\varphi_{T}$, hence $\varphi_{T}\geq-C\varepsilon^{-1}$ is bounded. This
implies that $\operatorname{\textrm{{Sing}}}(\varphi_{T})=\varnothing$, hence
the $\partial\bar{\partial}$-class $[\theta_{T}]$ is Hermitian, a
contradiction.

######  Theorem 5.7.

Let $(X^{n},\omega_{0})$ be a compact Hermitian manifold. Let $\omega(t)$ be a
solution of the Chern–Ricci flow starting at $\omega_{0}$, which develops a
singularity at finite time $T$. Assume that the limiting form
$\theta_{T}=\omega_{0}-T\operatorname{{\rm Ric}}(\omega_{0})$ satisfies
$\widehat{v}_{-}(X,\theta_{T})>0$. Then if either

  * •

$n=2$,

  * •

or $n\geq 3$, $\mathrm{dd^{c}}\omega_{0}=0$, ${\rm d}\omega_{0}\wedge{\rm
d^{c}}\omega_{0}=0$,

then the metrics $\omega(t)$ develop singularities precisely along
$N(\omega(t))=\operatorname{E_{\textrm{{nH}}}}(\theta_{T})$.

Recall here that the set $N(\omega(t))$ is defined by (5.6). We will see in
the proof that $N(\omega(t))$ is the set-theoretic union of all such
subvarieties for which the volume shrinks to zero as $t\to T^{-}$. This,
therefore, partially answers the question of Tosatti–Weinkove [TW22, Question
6.1].

###### Proof.

By assumption, the metric $\omega_{0}$ always satisfies the condition
($H_{X}$). Since $\theta_{T}$ is nef and $\widehat{v}_{-}(X,\theta_{T})>0$, so
$\theta_{T}$ is big. Thanks to Theorems 5.3 and 3.7, we have that

$\operatorname{\textrm{{Sing}}}(\omega(t))\subset\operatorname{E_{\textrm{{nH}}}}(\theta_{T})=N(\omega(t)).$

It suffices to show that the metrics $\omega(t)$ develop singularities along
$N(\omega(t))$. Suppose by contradiction that there is $x\in N(\omega(t))$
with a neighborhood $U\ni x$ on which as $t\to T^{-}$, the metrics $\omega(t)$
converges smoothly to a Hermitian metric $\omega_{U}$ and there is a e
positive-dimension irreducible subvariety $V\subset X$ containing $x$ and
$v_{-}(V,\omega(t)|_{V})\to 0$ as $t\to T^{-}$. We divide it into two cases.

If $n=2$, we mus have $\dim V=1$ because $\widehat{v}_{-}(X,\theta_{T})>0$.
Then

$v_{-}(V,\omega(t)|_{V})=\int_{V}\omega(t)\geq\int_{V\cap
U}\omega(t)\xrightarrow{t\to T}\int_{V\cap U}\omega_{U}>0,$

a contradiction.

If $n\geq 3$ and $\mathrm{dd^{c}}\omega_{0}=0$ and ${\rm
d}\omega_{0}\wedge{\rm d^{c}}\omega_{0}=0$ then
$\mathrm{dd^{c}}\omega_{0}^{k}=0$, $\forall\,1\leq k\leq n-1$. The latter
condition is preserved along the Chern–Ricci flow, i.e.,
$\mathrm{dd^{c}}\omega(t)^{k}=0$ for all $1\leq k\leq n-1$. Then we have

$v_{-}(V,\omega(t)|_{V})=\int_{V}\omega(t)^{\dim V}$

for every irreducible subvariety $V\subset X$, thanks to Lemma 2.5. On the
other hand,

$\int_{V}\omega(t)^{\dim V}\geq\int_{V\cap U}\omega(t)^{\dim
V}\xrightarrow{t\to T}\int_{V\cap U}\omega_{U}^{\dim V}>0,$

a contradiction. ∎

In the setup above, we show that the Chern–Ricci flow performs a canonical
surgical contraction:

######  Theorem 5.8.

Let $X^{n}$ be compact Hermitian manifolds equipped with a Hermitian metric
$\omega_{0}$ with $\mathrm{dd^{c}}\omega_{0}=0$ and ${\rm
d}\omega_{0}\wedge{\rm d^{c}}\omega_{0}=0$. Assume that the solution
$\omega(t)$ of the Chern–Ricci flow (5.1) starting at $\omega_{0}$ exists on
$[0,T[$ with $T<\infty$ and the limiting form
$\theta_{T}=\omega_{0}-T\operatorname{{\rm Ric}}(\omega_{0})$ satisfies
$\int_{X}\theta_{T}^{n}>0$. Assume there exists a map $\pi:X\to Y$ blowing
down disjoint irreducible exceptional divisors $E_{i}$ on $X$ to points
$y_{i}\in Y$ for $i=1,\dots,p$, with $Y$ a compact Hermitian manifold, such
that there exist a function $f\in\mathcal{C}^{\infty}(X,\mathbb{R})$ and a
smooth real (1,1)-form $\eta$ on $Y$ with

($\ast$) $\theta_{T}+\mathrm{dd^{c}}f=\pi^{*}\eta.$

Then we have the following

  1. (i)

As $t\to T^{-}$, the metrics $\omega(t)$ converges to a smooth Hermitian
metric $\omega_{T}$ on $X^{\circ}:=X\backslash\bigcup_{i=1}^{p}E_{i}$ in
$\mathcal{C}^{\infty}_{\rm loc}(X^{\circ})$. Using $\pi$, $\omega_{T}$ can be
regarded as a Hermitian metric on
$Y^{\circ}:=Y\backslash\\{y_{1},\dots,y_{p}\\}$.

  2. (ii)

Let $d_{\omega_{T}}$ be the distance function induced by $\omega_{T}$ on
$Y^{\circ}$. Then there exists a unique metric $d_{T}$ on $Y$ extending
$d_{\omega_{T}}$ to be zero on $\\{y_{1},\dots,y_{p}\\}$ such that $(Y,d_{T})$
is a compact metric space homeomorphic to $Y$.

  3. (iii)

The metric space $(X,\omega(t))\rightarrow(Y,d_{T})$ as $t\to T^{-}$ in the
Gromov–Hausdoff sense.

  4. (iv)

A smooth maximal solution $g(t)$ of the Chern–Ricci flow on $Y$ starting at
$\omega_{T}$ exists for $t\in]T,T_{Y}[$ with $T<T_{Y}\leq\infty$, and $g(t)$
converges to $\omega_{T}$ in $\mathcal{C}^{\infty}_{\rm loc}(Y^{\circ})$ as
$t\to T^{+}$.

  5. (v)

The metric space $(Y,g(t))\rightarrow(Y,d_{T})$ as $t\to T^{+}$ in the
Gromov–Hausdorff sense.

###### Proof.

We observe that $\mathrm{dd^{c}}\omega_{0}^{k}=0$ for all $1\leq k\leq n-1$,
this property is preserved along the Chern–Ricci flow. Since $\theta_{T}$ is
nef and $\int_{X}\theta_{T}^{n}>0$ we have that $\theta_{T}$ is big. By the
previous analysis, together with assumptions, we know that

$\operatorname{E_{\textrm{{nH}}}}(\theta_{T})=\operatorname{{\rm
Null}}(\theta_{T})=\bigcup_{1\leq i\leq p}E_{i}.$

The $(i)$ follows immediately from Theorem 5.3.

We show that in the condition ($\ast$ ‣ 5.8) the form $\eta$ can be replaced
with a Hermitian metric $\omega_{Y}$ on $Y$. Indeed, it is obvious that $\eta$
is nef and for any irreducible subvariety $V\subset Y$ with $k=\dim V>0$, we
have that

$\int_{V}\eta^{k}=\int_{\pi^{-1}(V)}(\pi^{*}\eta)^{k}=\int_{\pi^{-1}(V)}\theta_{T}^{k}>0$

Thanks to Nakai–Moishezon’s criterion (Theorem 3.10) the
$\partial\bar{\partial}$-class $[\eta]$ is Hermitian on $Y$. We thus obtain
$(ii)$ and $(iii)$ as follows from [TW13, Theorem 1.3].

We have seen that $\omega_{T}$ is a positive current with minimal
singularities, so it has bounded potentials. The push forward
$\pi_{*}\omega_{T}$ is therefore a positive current on $Y$ with bounded
potentials. In this case, we have
$\pi_{*}\omega_{T}=\omega_{Y}+\mathrm{dd^{c}}\psi_{T}$ where $\psi_{T}$ is a
bounded $\omega_{Y}$-psh function, which is smooth in $Y^{\circ}$. With this
condition, the $(iv)$ and $(v)$ was proved by Nie [Nie17] and Tô [Tô18]
independently. ∎

The major problems are whether the metric space $(Y,d_{T})$ is the metric
completion of $(Y^{\circ},d_{\omega_{T}})$, and we can remove the condition
($\ast$ ‣ 5.8). Those may require new techniques.

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