# Volumes of Bott–Chern classes

Sébastien Boucksom and Vincent Guedj and Chinh H. Lu  Sorbonne Université and
Université Paris Cité
CNRS
IMJ-PRG
F-75005 Paris
France<EMAIL_ADDRESS>Institut Universitaire de France &
Institut de Mathématiques de Toulouse
Université de Toulouse; CNRS, UPS
118 route de Narbonne, F-31400 Toulouse
France<EMAIL_ADDRESS>Univ Angers, CNRS, LAREMA, SFR
MATHSTIC
2 Bd de Lavoisier, 49000 Angers
France<EMAIL_ADDRESS>
###### Abstract.

We study the volumes of transcendental and possibly non-closed Bott–Chern
$(1,1)$-classes on an arbitrary compact complex manifold $X$. We show that the
latter belongs to the class $\mathcal{C}$ of Fujiki if and only if it has the
bounded mass property -i.e. its Monge-Ampère volumes have a uniform upper-
bound- and there exists a closed Bott–Chern class with positive volume. This
yields a positive answer to a conjecture of Demailly–Păun–Boucksom. To this
end we extend to the hermitian context the notion of non-pluripolar products
of currents, allowing for the latter to be merely quasi-closed and quasi-
positive. We establish a quasi-monotonicity property of Monge-Ampère masses,
and moreover show the existence of solutions to degenerate complex Monge-
Ampère equations in big classes, together with uniform a priori estimates.
This extends to the hermitian context fundamental results of
Boucksom–Eyssidieux–Guedj–Zeriahi.

###### 2010 Mathematics Subject Classification:

32W20, 32U05, 32Q15, 35A23

###### Contents

  1. 1 Quasi-closed and quasi-positive currents
  2. 2 Non-pluripolar products
  3. 3 The lower and upper volume of a $dd^{c}$-class
  4. 4 Monge-Ampère equations in big classes

## Introduction

Complex Monge-Ampère equations are powerful tools in complex geometry, as
demonstrated in foundational works ranging from Yau’s resolution of the Calabi
conjecture [Yau78] to the construction of singular canonical metrics by
Eyssidieux-Guedj-Zeriahi [EGZ09]. As illustrated in [BEGZ10], the Bedford-
Taylor pluripotential theory [BT76, BT87, Kol98] is highly robust, extendable
to the broad context of big cohomology classes. This extension has led to many
profound applications in Kähler geometry over the past decade.

Following Yau’s celebrated work, the Hermitian Monge-Ampère equation was
explored by Cherrier [Cher87], who solved it in some particular cases. The
general case solved by Tosatti and Weinkove [TW10] twenty years later
underscores the difficulty of translating techniques from the Kähler setting
to the Hermitian context. In recent years, Hermitian pluripotential theory has
become central to the advancement of hermitian geometry, as evidenced by
[DK12, KN19, GL22, GL23]. This discussion underlines the necessity of
extending the results of [BEGZ10] to the Hermitian context, which is the
primary objective of our paper.

We start by extending the non-pluripolar products of currents, allowing for
the latter to be merely quasi-closed and quasi-positive.

######  Theorem A.

Let $(X,\omega_{X})$ be a compact Hermitian manifold of dimension $n$ such
that $\overline{\mathrm{vol}}(\omega_{X})<+\infty$. For any $p=1,\dots,n$,
there exists a unique symmetric, multilinear pairing

$(T_{1},\dots,T_{p})\mapsto T_{1}\wedge\dots\wedge T_{p}$

defined on $p$-tuples of quasi-closed, quasi-positive currents and with values
in nonpluripolar $(p,p)$-currents, such that:

  * (i)

if the $T_{i}$’s have bounded potentials, then the pairing is the Bedford-
Taylor Monge-Ampère product;

  * (ii)

for any tuple of $(1,1)$-forms $\theta_{1},\dots,\theta_{p}$, the operator

$(\varphi_{1},\dots,\varphi_{p})\mapsto\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}),$

defined on all tuples of quasi-psh functions, is local in the plurifine
topology.

Moreover, $T_{1},\dots,T_{p}\geq 0\Longrightarrow T_{1}\wedge\dots\wedge
T_{p}\geq 0.$

A $(1,1)$-current $T$ is quasi-closed if $dT$ is a smooth form. It is quasi-
positive if $T\geq-A\omega_{X}$, for some constant $A$. When
$\theta=\theta_{1}=...=\theta_{n}$ and $u=u_{1}=...=u_{n}$, we simply write

$(\theta+dd^{c}u)^{n}=\bigwedge_{i}(\theta_{i}+dd^{c}u_{i}).$

Our proof is partially inspired by the work of Dinew-Kołodziej [DK12], which
is further elaborated in [GL22]. Due to the lack of local potentials of the
forms $\theta_{i}$, we locally add strictly plurisubharmonic functions to
employ Bedford-Taylor’s construction of the Monge-Ampère measure.
Specifically, we approximate the unbounded potentials through truncation and
then consider the Bedford-Taylor product

$\bigwedge_{i}(\theta_{i}+dd^{c}\max(\varphi_{i},-t)).$

However, this procedure results in currents of order zero with coefficients
that are signed measures instead of positive ones, unless $\theta_{i}\geq 0$.
The key point is that when restricted to the plurifine open set
$\cap_{i=1}^{p}(\varphi_{i}>-t)$ these signed measures are, in fact, positive.
To ensure our construction functions properly, we critically assume that $X$
has the bounded mass property, meaning it satisfies

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

The condition $\overline{\mathrm{vol}}(\omega_{X})<+\infty$ is independent of
the positive form $\omega_{X}$, and it is a bimeromorphic invariant [GL22,
Theorem A]. In particular $\overline{\mathrm{vol}}(\omega_{X})<+\infty$ if $X$
belongs to the class ${\mathcal{C}}$ of Fujiki. Several conditions have been
provided in [AGL23] to ensure the finiteness of
$\overline{\mathrm{vol}}(\omega_{X})$. To date, we are not aware of any
examples where $\overline{\mathrm{vol}}(\omega_{X})$ is infinite.

Our second main result establishes the following crucial quasi-monotonicity
property of Monge-Ampère masses, which extends [WN19, DDL23] to the hermitian
setting.

######  Theorem B.

Assume $\overline{\mathrm{vol}}(\omega_{X})<+\infty$, and fix $\theta$ a
smooth $(1,1)$-form such that $\pm\theta\leq C\omega_{X}$. If
$\varphi,\psi\in\operatorname{PSH}(\theta)$ satisfy $\varphi\leq\psi+O(1)$,
then

$\int_{X}(\theta+dd^{c}\varphi)^{n}\leq\int_{X}(\theta+dd^{c}\psi)^{n}+\Delta_{\theta},$

where
$\Delta_{\theta}:=\sup_{\varphi,\psi\in\operatorname{PSH}(C\omega_{X})\cap
L^{\infty}(X)}\left\\{\int_{X}(\theta+dd^{c}\varphi)^{n}-\int_{X}(\theta+dd^{c}\psi)^{n}\right\\}.$

Stokes’ theorem shows that $\Delta_{\theta}=0$ if $\theta$ is closed. In
particular, the total mass of $T^{n}$ is the same for all currents
$T\in\\{\theta\\}$ with minimal singularities. It is thus natural to set

$\operatorname{vol}(\\{\theta\\})=\int_{X}T_{\min}^{n}.$

This, together with [BEGZ10], shows that our volume notion is compatible with
the one introduced by Boucksom in [Bou02].

The situation is more subtle when $\theta$ is not closed, as positive currents
with minimal singularities do not necessarily have the same Monge-Ampère mass
(unless $dd^{c}\theta=0$ and $dd^{c}\theta^{2}=0$, see [GL22] and references
therein). When $\\{\theta\\}$ is big we thus introduce

$\underline{\mathrm{vol}}(\\{\theta\\}):=\underline{\mathrm{vol}}(T_{min})\;\;\text{
and
}\;\;\overline{\mathrm{vol}}(\\{\theta\\}):=\overline{\mathrm{vol}}(T_{min}),$

with

$\underline{\mathrm{vol}}(T):=\inf_{S}\int_{X}S^{n},\quad\overline{\mathrm{vol}}(T):=\sup_{S}\int_{X}S^{n},$

where $S$ ranges over all quasi-closed positive $(1,1)$-currents in the same
$dd^{c}$-class as $T$ and with equivalent singularities. By definition, the
upper and lower volumes of $T$ only depend on its $dd^{c}$-class and
singularity class, and satisfy

$0\leq\underline{\mathrm{vol}}(T)\leq\overline{\mathrm{vol}}(T)<\infty,$

as $X$ has the bounded mass property (see Lemma 2.4). Note also that

$T>0\Longrightarrow\overline{\mathrm{vol}}(T)>0.$

The analogous result for $\underline{\mathrm{vol}}(T)$ is unclear in general,
even when $T$ is smooth (see §3.2.3).

We establish in Section 3 key properties of these volumes, and notably show
that $\underline{\mathrm{vol}}$ is a continuous function (Theorem 3.20).
Indeed another motivation of our paper stems from the Grauert-Riemenschneider
conjecture [GR70], which asks whether the existence of a semi-positive
holomorphic line bundle $L\rightarrow X$ with $c_{1}(L)^{n}>0$ implies that
$X$ is Moishezon, i.e. bimeromorphically equivalent to a projective manifold.
This conjecture has been solved positively by Siu [Siu84] (see also [Dem85]).
One can relax the semi-positivity assumption and replace the condition
$c_{1}(L)^{n}>0$ by the positivity of the volume $\operatorname{vol}(L)>0$.
Recall that $\operatorname{vol}(L)$ measures the asymptotic growth of the
space of holomorphic sections,

$\operatorname{vol}(L)=\limsup_{k\rightarrow+\infty}\frac{n!}{k^{n}}h^{0}(X,kL).$

Demailly and Păun have further proposed a transcendental version of this
conjecture [DP04, Conjecture 0.8]: given a nef class $\alpha\in
H^{1,1}_{BC}(X,\mathbb{R})$ with $\alpha^{n}>0$, they conjectured that
$\alpha$ should contain a Kähler current; in particular $X$ should belong the
class ${\mathcal{C}}$. Recall that the Bott-Chern cohomology group
$H_{BC}^{1,1}(X,\mathbb{R})$ is the quotient of closed real smooth
$(1,1)$-forms, by the image of ${\mathcal{C}}^{\infty}(X,\mathbb{R})$ under
the $dd^{c}$-operator. The following answer to [DP04, Conjecture 0.8] has been
proposed in [GL22, Theorem C]:

$\alpha\text{ contains a K\"{a}hler current
}\Longleftrightarrow\overline{\mathrm{vol}}(\omega_{X})<+\infty.$

The existence of a class $\alpha\in H^{1,1}_{BC}(X,\mathbb{R})$ that is nef
and satisfies $\alpha^{n}>0$ is however not granted on an arbitrary manifold
in the Fujiki class (see [JM22]). It is thus natural to extend the conjecture
of Demailly-Păun, by removing the nef assumption, as was proposed by Boucksom
in [Bou02, Conjecture 4.1].

Theorem 3.20 applied to closed forms provides a characterization of the Fujiki
class, and yields the following answer to the Boucksom-Demailly-Păun
conjecture.

######  Theorem C.

Let $(X,\omega_{X})$ be a compact hermitian manifold. The following are
equivalent:

  1. (1)

$X$ belongs to the class ${\mathcal{C}}$ of Fujiki.

  2. (2)

$\overline{\mathrm{vol}}(\omega_{X})<+\infty$ and there exists a class
$\alpha\in H_{BC}^{1,1}(X,\mathbb{R})$ such that
$\operatorname{vol}(\alpha)>0$.

In the final Section 4 we extend some of the main results of [BEGZ10] by
solving complex Monge-Ampère equations associated to a (non-closed) big form
$\theta$, and by establishing uniform a priori estimates in this context,
elaborating on [GL23]. We say that a form $\theta$ is big if there exists a
$\theta$-psh function $\rho$ with analytic singularities such that
$\theta+dd^{c}\rho$ dominates a hermitian form. We fix such a big form, and we
no longer assume that $X$ has the bounded mass property. We then extend the
key result [BEGZ10, Theorem B] as follows.

######  Theorem D.

Assume $0\leq f\in L^{p}(X)$ with $p>1$ and $\|f\|_{p}>0$. Then there exists
$(\varphi,c)\in\operatorname{PSH}(X,\theta)\times(0,+\infty)$ such that
$\sup_{X}\varphi=0$,

$V_{\theta}-C\leq\varphi\leq V_{\theta}\;\;\text{ and
}\;\;(\theta+dd^{c}\varphi)^{n}=cf\omega_{X}^{n},$

where

  * •

the constant $c>0$ is uniquely determined by $f,X,\theta,\omega_{X}$, and

  * •

$C>0$ is a uniform constant that only depends on $(X,\omega_{X})$, $\theta$,
$p$ and $\|f\|_{p}$.

Here $V_{\theta}=\sup\\{u,u\in PSH(\theta)\;\text{ with }\;u\leq 0\\}$ is a
$\theta$-psh function with minimal singularities. As in [BEGZ10, Section 6],
we also show the existence of solutions to Monge-Ampère equations twisted by
an exponential (Theorem 4.5). However instead of deducing Theorem 4.5 from
Theorem D by a fixed point argument, we first prove Theorem 4.5 and then
deduce Theorem D by a deformation argument.

The uniqueness of the solution $\varphi$ is an open problem in the Hermitian
setting, even when $\theta$ is a Hermitian form (see [KN19] for a positive
result when $f$ is uniformly bounded away from zero). Showing higher
regularity of the solutions in the Zariski open set $\Omega=(\rho>-\infty)$,
-when $f$ is smooth- is an important open problem which is also largely open
in the Kähler case ([BEGZ10] treats the case when the class $\\{\theta\\}$ is
moreover nef).

######  Acknowledgment.

The authors are partially supported by the fondation Charles Defforey and the
Institut Universitaire de France. The third named author acknowledges partial
support from the PARAPLUI ANR-20-CE40-0019 project and the Centre Henri
Lebesgue ANR-11-LABX-0020-01.

##  1\. Quasi-closed and quasi-positive currents

Throughout the article, $X$ denotes a compact complex manifold, of complex
dimension $n$. We also pick a reference Hermitian $(1,1)$-form $\omega_{X}>0$.
The purpose of this section is to review some basic properties of quasi-psh
functions, quasi-closed quasi-positive $(1,1)$-currents, and their
$dd^{c}$-classes.

###  1.1. Quasi-closed currents and $dd^{c}$-classes

For each $p,q\in\mathbb{N}$, denote by $\Omega^{p,q}(X)$ the Fréchet space of
smooth $(p,q)$-forms on $X$.

######  Definition 1.1.

We define the _Bott–Chern space_ $\mathrm{BC}^{p,q}(X)$ of (possibly non-
closed) _$dd^{c}$ -classes of bidegree $(p,q)$_ as the cokernel of
$dd^{c}\colon\Omega^{p-1,q-1}(X)\to\Omega^{p,q}(X)$, i.e.

$\mathrm{BC}^{p,q}(X):=\frac{\Omega^{p,q}(X)}{dd^{c}\Omega^{p-1,q-1}(X)}.$

We denote by $\\{\theta\\}\in\mathrm{BC}^{p,q}(X)$ the $dd^{c}$-class of a
$(p,q)$-form $\theta$. When $p=q$, the real operator $dd^{c}$ induces a map
$dd^{c}\colon\Omega^{p-1,p-1}(X,\mathbb{R})\to\Omega^{p,p}(X,\mathbb{R})$
between spaces of real forms, and we denote by

$\mathrm{BC}^{p,p}(X,\mathbb{R})\subset\mathrm{BC}^{p,p}(X)$

its cokernel. Setting $\bar{d}\\{\theta\\}:=d\theta$ for any $(p,q)$-form
$\theta$ defines a linear map

$\bar{d}\colon\mathrm{BC}^{p,q}(X)\to\Omega^{p+1,q}(X)\oplus\Omega^{p,q+1}(X),$
(1.1)

whose kernel coincides with the usual _Bott–Chern cohomology space_

$H_{\mathrm{BC}}^{p,q}(X)=\frac{\Omega^{p,q}(X)\cap\ker
d}{dd^{c}\Omega^{p-1,q-1}(X)}\hookrightarrow\mathrm{BC}^{p,q}(X).$

When $X$ is Kähler (or even Fujiki), the latter coincides with the Dolbeault
space $H^{p,q}(X)$.

We endow the (infinite dimensional) complex vector space
$\mathrm{BC}^{p,q}(X)$ with the quotient topology. Thus
$\\{\theta_{j}\\}\to\\{\theta\\}$ in $\mathrm{BC}^{p,q}(X)$ iff
$\theta_{j}\to\theta$ smoothly for an appropriate choice of representatives.
Note that (1.1) is continuous in this topology.

######  Lemma 1.2.

The space $H_{\mathrm{BC}}^{p,q}(X)$ is finite dimensional, and
$\mathrm{BC}^{p,q}(X)$ is a (Hausdorff) Fréchet space.

###### Proof.

The first point is well-known, see for instance [Dem, Theorem VI.12.4]. It
equivalently says that $dd^{c}\Omega^{p-1,q-1}(X)$ has finite codimension in
the closed subspace $\Omega^{p,q}(X)\cap\ker d$ of the Fréchet space
$\Omega^{p,q}(X)$. Thus $dd^{c}\Omega^{p-1,q-1}(X)$ is closed in
$\Omega^{p,q}(X)\cap\ker d$, and hence in $\Omega^{p,q}(X)$, which proves the
second point. ∎

As is well-known, Bott–Chern cohomology can also be described in terms of
currents. This is more generally the case for $\mathrm{BC}^{p,q}(X)$. To this
end, it will be convenient to introduce the following terminology.

######  Definition 1.3.

We say that a current $T$ on $X$ is _quasi-closed_ if $dT$ is smooth.

In particular, any smooth form is quasi-closed.

######  Lemma 1.4.

A $(p,q)$-current $T$ on $X$ is quasi-closed iff it admits a decomposition

$T=\theta+dd^{c}U$

where $\theta$ is a smooth $(p,q)$-form and $U$ a $(p-1,q-1)$-current.
Furthermore, any other such decomposition is of the form

$T=(\theta+dd^{c}\gamma)+dd^{c}(U-\gamma)$

for a smooth $(p-1,q-1)$-form $\gamma$.

###### Proof.

While this result is likely well-known, we provide the simple argument for the
convenience of the reader. The existence of a current $U$ such that
$T-dd^{c}U$ is smooth implies that $dT=d(T-dd^{c}U))$ is smooth. Conversely,
assume $dT$ is smooth. We first claim that there exists a $(p+q-1)$-current
$S$ such that $T+dS$ is smooth. Indeed, since de Rham cohomology can be
computed using either forms or currents, the smooth $(p+q+1)$-form $dT$ is
exact, and we can thus find a $(p+q)$-form $\alpha$ such that $dT=d\alpha$.
The $(p+q)$-current $T-\alpha$ is then closed, and we can now find a
$(p+q-1)$-current $S$ such that $T-\alpha+dS$ is smooth, hence the claim.

Since $T+dS=T+(\partial+\overline{\partial})S$ is smooth, decomposing
according to type shows that

$T+\partial
S^{p-1,q}+\overline{\partial}S^{p,q-1},\quad\overline{\partial}S^{p-1,q}\quad\text{and}\quad\partial
S^{p,q-1}$

are smooth. Arguing as above, this time with Dolbeault cohomology, it follows
that

$S^{p-1,q}+{\bar{\partial}R}\quad\text{and}\quad S^{p,q-1}+\partial Q$

are smooth for some $(p-1,q-1)$-currents $R,Q$. Thus

$T+\partial
S^{p-1,q}+\overline{\partial}S^{p,q-1}=T+\partial\overline{\partial}(R-Q)=:\theta,$

is smooth, which yields the desired decomposition $T=\theta+dd^{c}U$.

Consider now another decomposition $T=\theta^{\prime}+dd^{c}U^{\prime}$. Then
$dd^{c}(U-U^{\prime})$ is a smooth $(p,q)$-form, which $dd^{c}$-exact as a
current, and hence as a form. Thus $dd^{c}(U-U^{\prime})=dd^{c}\gamma$ for a
smooth $(p-1,q-1)$-form $\gamma$, and the last point follows. ∎

As a direct consequence of Lemma 1.4, the inclusion of smooth forms into
quasi-closed currents induces a vector space isomorphism

$\mathrm{BC}^{p,q}(X)\simeq\frac{\\{T\text{ quasi-closed
}(p,q)\text{-current}\\}}{\\{dd^{c}U\mid U\,(p-1,q-1)\text{-current}\\}}.$
(1.2)

We denote by $\\{T\\}\in\mathrm{BC}^{p,q}(X)$ the $dd^{c}$-class of a quasi-
closed $(p,q)$-current $T$. Note that

$\bar{d}\\{T\\}=dT.$

When $T$ is closed, $\\{T\\}$ lies in the finite dimensional space
$H^{p,q}_{\mathrm{BC}}(X)$, and $T\mapsto\\{T\\}$ is further weakly continuous
on closed currents, since the weak quotient topology of
$H^{p,q}_{\mathrm{BC}}(X)$ is Hausdorff, and hence coincides with the strong
quotient topology.

However, $T\mapsto\\{T\\}$ is _not_ weakly continuous on quasi-closed
currents. Indeed, a weakly convergent sequence of quasi-closed currents
$T_{j}\to T$ can only satisfy $\\{T_{j}\\}\to\\{T\\}$ if
$\bar{d}\\{T_{j}\\}=dT_{j}$ converges smoothly to $\bar{d}\\{T\\}=dT$.

###  1.2. Quasi-psh functions

Recall that a _quasi-plurisubharmonic_ function (_quasi-psh_ for short) is a
function $\varphi\colon X\to\mathbb{R}\cup\\{-\infty\\}$ that is locally the
sum of a psh function and a smooth function. In particular, $\varphi$ is usc
and integrable. Quasi-psh functions are actually in $L^{p}(X)$ for any
$p\in[1,\infty)$, and the induced topologies are further all equivalent.

The _plurifine topology_ of $X$ is defined as the topology generated by all
quasi-psh functions on $X$. We shall say that an operator
$(\varphi_{1},\dots,\varphi_{p})\mapsto F(\varphi_{1},\dots,\varphi_{p})$,
defined on certain tuples of quasi-psh functions and with values in order $0$
currents, is _local in the plurifine topology_ if, for any plurifine open
$O\subset X$ and functions $\varphi_{i},\psi_{i}$, we have

$\varphi_{i}=\psi_{i}\text{ on }O\text{ for all
}i\Longrightarrow\operatorname{\mathbf{1}}_{O}F(\varphi_{1},\dots,\varphi_{p})=\operatorname{\mathbf{1}}_{O}F(\psi_{1},\dots,\psi_{p}).$

Any quasi-psh function $\varphi$ satisfies $\theta+dd^{c}\varphi\geq 0$ in the
sense of currents for some smooth $(1,1)$-form $\theta$. We then say that
$\varphi$ is _$\theta$ -psh_, and denote by

$\operatorname{PSH}(\theta)=\operatorname{PSH}(X,\theta)$

the set of such functions, which sits as a closed convex subspace of
$L^{1}(X)$.

When $\theta$ is closed, it admits a local potential near each point of $X$,
i.e. a smooth function $\rho$ such that $\theta=dd^{c}\rho$, and a function
$\varphi$ is then $\theta$-psh iff $\rho+\varphi$ is psh. The following well-
known observation allows to reduce many of the basic properties of
$\theta$-psh functions for a possibly non-closed $\theta$ to the closed case.

######  Lemma 1.5.

Pick a real $(1,1)$-form $\theta$, and $\varepsilon>0$. Then any point of $X$
admits a neighborhood $U\subset X$ with a closed $(1,1)$-form
$\theta^{\prime}$ such that
$-\varepsilon\omega_{X}\leq\theta-\theta^{\prime}\leq\varepsilon\omega_{X}$ on
$U$.

###### Proof.

For any $x\in X$, it suffices to pick local coordinates $(z_{1},\dots,z_{n})$
centered at $x$, and to define $\theta^{\prime}$ as the constant $(1,1)$-form
in these coordinates which coincides with $\theta$ at $x$. The result then
holds by continuity of $\theta$ at $x$. ∎

Given any $(1,1)$-form $\theta$ and any function $f\colon
X\to[-\infty,\infty]$, the _$\theta$ -psh envelope_ of $f$ is defined as the
pointwise supremum

$\operatorname{P}_{\theta}(f):=\sup\left\\{\varphi\mid\varphi\in\operatorname{PSH}(\theta),\,\varphi\leq
f\right\\}.$ (1.3)

Denote by $\operatorname{P}_{\theta}^{\star}(f)$ its usc regularization. Then
one of the following holds:

  * (a)

$\operatorname{P}_{\theta}^{\star}(f)\equiv-\infty$ (i.e. there exists no
$\theta$-psh function $\varphi$ such that $\varphi\leq f$);

  * (b)

$\operatorname{P}_{\theta}^{\star}(f)\equiv+\infty$;

  * (c)

$\operatorname{P}_{\theta}^{\star}(f)$ is the largest $\theta$-psh function
such that $\operatorname{P}_{\theta}^{\star}(f)\leq f$ outside a pluripolar
subset of $X$.

Note that if $f$ is usc and (c) holds then
$\operatorname{P}_{\theta}(\varphi)$ is already usc (and hence $\theta$-psh),
since $\operatorname{P}^{\star}_{\theta}(\varphi)$ is then a candidate in the
envelope.

######  Definition 1.6.

We say that a quasi-psh function $\varphi$ has _analytic singularities (with
smooth remainder)_ if $\varphi$ can be locally written as

$\varphi=\frac{1}{2m}\log\sum_{j=1}^{N}|f_{j}|^{2}+u$

for some $m\in\mathbb{Z}_{>0}$ and some choice of holomorphic functions
$f_{1},\dots,f_{N}$ and smooth function $u$. If we can further take $N=1$ then
we say that $\varphi$ has _divisorial singularities (with smooth remainder)_.

If $\varphi$ has analytic singularities, then it is smooth on the Zariski open
set $\\{\varphi>-\infty\\}$. If $\varphi$ further has divisorial
singularities, then the effective $\mathbb{Q}$-divisor
$E:=\tfrac{1}{m}\mathrm{div}(f_{1})$ is globally defined, and the
Lelong–Poincaré formula yields

$dd^{c}\varphi=\beta+[E]$

where $\beta$ is a closed smooth $(1,1)$-form.

As pointed out in [D+23, Remark 2.7], the notion of analytic singularities
with smooth remainder needs to be handled with care, as it depends on the
choice of local holomorphic functions $(f_{j})$, and not only on the ideal
sheaf they generate. At any rate, the following global result holds:

######  Lemma 1.7.

Assume $\varphi$ is a quasi-psh function with analytic singularities. Then
there exists a modification $\pi\colon Y\to X$, isomorphic over the Zariski
open set $\\{\varphi>-\infty\\}$, such that $\pi^{\star}\varphi$ has
divisorial singularities.

###### Proof.

By assumption, $X$ admits a finite open cover $(U_{\alpha})$ such that

$\varphi|_{U_{\alpha}}=\frac{1}{2m_{\alpha}}\log\sum_{j=1}^{N_{\alpha}}|f_{\alpha
j}|^{2}+u_{\alpha}$

with $f_{\alpha j}\in\mathcal{O}(U_{\alpha})$, $u_{\alpha}\in
C^{\infty}(U_{\alpha})$ and $m_{\alpha}\in\mathbb{Z}_{>0}$. Set
$m:=\prod_{\alpha}m_{\alpha}$, and denote by
$\mathcal{I}\subset\mathcal{O}_{X}$ the ideal sheaf of germs of holomorphic
functions $f$ such that $|f|\leq Ce^{m\varphi}$ locally. On $U_{\alpha}$ this
condition is equivalent to $|f|\leq C\max_{j}\left|f_{\alpha
j}^{m/m_{\alpha}}\right|$. By the well-known analytic characterization of
integral closure, this shows that $\mathcal{I}|_{U_{\alpha}}$ coincides with
the integral closure of the ideal sheaf
$\mathcal{I}_{\alpha}\subset\mathcal{O}_{U_{\alpha}}$ generated by $(f_{\alpha
j}^{m/m_{\alpha}})_{1\leq j\leq N_{\alpha}}$, and it follows that the ideal
sheaf $\mathcal{I}$ is coherent.

Pick a log resolution $\pi\colon Y\to X$ of $\mathcal{I}$, and denote by $D$
the effective divisor of $Y$ such that
$\mathcal{I}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-D)$. Then $\pi$ factors
through the normalization of the blowup of $\mathcal{I}$, which coincides on
$U_{\alpha}$ with the normalization of the blowup of $\mathcal{I}_{\alpha}$,
as the two ideal sheaves share the same integral closure. As a result, the
functions $(\pi^{\star}f_{\alpha j}^{m/m_{\alpha}})_{j}$ generate
$\mathcal{O}(-D)$ on $\pi^{-1}(U_{\alpha})$. Given a local equation $f_{D}$ of
$D$, we thus have $\pi^{\star}f_{\alpha j}^{m/m_{\alpha}}=f_{D}g_{\alpha j}$
for a family of local holomorphic functions $(g_{\alpha j})_{j}$ without
common zeroes. This shows

$\pi^{\star}\varphi=\frac{1}{2m_{\alpha}}\log\sum_{j}|f_{D}|^{2m_{\alpha}/m}|g_{\alpha
j}|^{2m_{\alpha}/m}+\pi^{\star}u_{\alpha}=\frac{1}{2m}\log|f_{D}|+v_{\alpha}$

with $v_{\alpha}:=u_{\alpha}+\frac{1}{2m_{\alpha}}\log\sum_{j}|g_{\alpha
j}|^{2m_{\alpha}/m}\in C^{\infty}$, which proves, as desired, that
$\pi^{\star}\varphi$ has divisorial singularities. ∎

The importance of functions with analytic singularities stems from the
following consequence of Demailly’s fundamental regularization theorem
[Dem92]:

######  Theorem 1.8 (Demailly).

Any $\varphi\in\operatorname{PSH}(\theta)$ can be written as the limit of a
decreasing sequence
$\varphi_{j}\in\operatorname{PSH}(\theta+\varepsilon_{j}\omega_{X})$ with
analytic singularities, where $\varepsilon_{j}>0$ decreasing to zero.

###  1.3. Quasi-positive currents

A $(1,1)$-current $T$ is said to be _quasi-positive_ if $T\geq\gamma$ for some
smooth $(1,1)$-form $\gamma$. As a consequence of Lemma 1.4, we then have:

######  Lemma 1.9.

For any $(1,1)$-current $T$, the following are equivalent:

  * (i)

$T$ is quasi-closed and quasi-positive;

  * (ii)

$T=\theta+dd^{c}\varphi$ for a smooth real $(1,1)$-form $\theta$ and a quasi-
psh function $\varphi$.

Furthermore, any other decomposition as in (ii) is of the form

$T=(\theta+dd^{c}\tau)+dd^{c}(\varphi-\tau)$

with $\tau\in C^{\infty}(X)$. As a result, the pull-back of a quasi-closed,
quasi-positive $(1,1)$-current $T$ by any surjective holomorphic map $f\colon
Y\to X$ can be defined by setting

$f^{\star}T:=f^{\star}\theta+dd^{c}f^{\star}\varphi$

for any decomposition $T=\theta+dd^{c}\varphi$ as above. In constrast, quasi-
closed, quasi-positive currents are in general not preserved under push-
forward.

######  Example 1.10.

Assume $n=2$, and let $\pi\colon Y\to X$ be the blowup of a point $p\in X$.
Then there exists a Hermitian form $\omega_{Y}$ on $Y$ such that
$\pi_{\star}\omega_{Y}$ is not quasi-closed. Indeed, since any smooth
$(1,1)$-form can be written as a difference of Hermitian forms, it would
otherwise follow that $d\pi_{\star}\theta=\pi_{\star}d\theta$ is smooth for
any smooth $(1,1)$-form $\theta$. However this fails for

$\theta=\chi dd^{c}(\pi^{*}f),$

where $\chi\in C^{\infty}_{c}(Y)$ is a cut-off function supported near a point
of the exceptional divisor, and $f\in C^{\infty}(X)$ coincides with
$\|z\|^{2}$ in a local chart centered at $p$. Indeed

$\pi_{\star}d\theta=d\chi\circ\pi^{-1}\wedge dd^{c}\|z\|^{2}$

is a differential form in $X\setminus\\{p\\}$ whose coefficients are not even
bounded near $p$.

Extending standard terminology for closed positive currents, we shall say that
a quasi-closed, quasi-positive current $T=\theta+dd^{c}\varphi$

  * •

has _bounded potentials_ on a given compact subset $K\subset X$ if $\varphi$
is bounded on $K$;

  * •

is _more singular than_ $T^{\prime}=\theta^{\prime}+dd^{c}\varphi^{\prime}$ if
$\varphi\leq\varphi^{\prime}+O(1)$, and that $T,T^{\prime}$ have _equivalent
singularities_ if $\varphi=\varphi^{\prime}+O(1)$;

  * •

has _analytic singularities_ if the quasi-psh function $\varphi$ has analytic
singularities (see Definition 1.6).

As consequence of Lemma 1.7, we have:

######  Lemma 1.11.

For any quasi-closed, quasi-positive current $T$ with analytic singularities,
there exists a modification $\pi\colon Y\to X$ such that

$\pi^{\star}T=\beta+[E]$

where $\beta$ is a smooth $(1,1)$-form and $E$ an effective
$\mathbb{Q}$-divisor.

Note further that $T$ is closed iff $\beta$ is.

###  1.4. Positivity for $dd^{c}$-classes

Recall that the space

$\mathrm{BC}^{1,1}(X,\mathbb{R})=\Omega^{1,1}(X,\mathbb{R})/dd^{c}C^{\infty}(X)$

of real, bidegree $(1,1)$ $dd^{c}$-classes is a Fréchet space (see Lemma 1.2),
containing the usual (finite dimensional) Bott–Chern cohomology space
$H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})$.

######  Definition 1.12.

We say that a class in $\mathrm{BC}^{1,1}(X,\mathbb{R})$ is:

  * •

_Hermitian_ if it can be represented by a Hermitian form;

  * •

_nef_ if it is a limit of Hermitian classes.

The set of Hermitian classes is an open convex cone in
$\mathrm{BC}^{1,1}(X,\mathbb{R})$, which we call the _Hermitian cone_. Its
closure, the _nef cone_ , is the set of all nef classes.

######  Lemma 1.13.

A class $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is nef iff it admits
a sequence of representatives $\theta_{j}$ such that
$\theta_{j}\geq-\varepsilon_{j}\omega_{X}$ with $\varepsilon_{j}\to 0$.

###### Proof.

Assume the existence of representatives $\theta_{j}=\theta+dd^{c}\varphi_{j}$
such that $\theta_{j}\geq-\varepsilon_{j}\omega_{X}$. Then
$\theta^{\prime}_{j}:=\theta_{j}+2\varepsilon_{j}\omega_{X}>0$, and
$\theta^{\prime}_{j}-dd^{c}\varphi_{j}=\theta+2\varepsilon_{j}\omega_{X}\to\theta$
smoothly, which shows that $\\{\theta\\}=\lim_{j}\\{\theta_{j}\\}$ is nef.
Conversely, assume $\\{\theta\\}$ is nef. Then
$\\{\theta\\}=\lim_{j}\\{\omega_{j}\\}$ for a sequence of Hermitian forms
$\omega_{j}$, and hence $\omega_{j}+dd^{c}\tau_{j}\to\theta$ for some smooth
functions $\tau_{j}$. In particular,
$\theta-(\omega_{j}+dd^{c}\tau_{j})\geq-\varepsilon_{j}\omega_{X}$ with
$\varepsilon_{j}\to 0$, and hence $\theta-
dd^{c}\tau_{j}\geq-\varepsilon_{j}\omega_{X}$. ∎

######  Definition 1.14.

We say that a class $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is:

  * •

_pseudo-effective_ (_psef_ for short) if it can be represented by a positive
$(1,1)$-current $T=\theta+dd^{c}\varphi\geq 0$;

  * •

_big_ if it can be represented by a current $T=\theta+dd^{c}\varphi$ which is
_strictly positive_ , i.e. $T\geq\varepsilon\omega_{X}$ for $0<\varepsilon\ll
1$.

As a consequence of Demailly’s regularization (Theorem 1.8 above), any big
class can be represented by a strictly positive current with analytic
singularities.

######  Lemma 1.15.

The psef cone of $\mathrm{BC}^{1,1}(X,\mathbb{R})$ is a closed convex cone,
whose interior coincides with the big cone.

In particular, the Hermitian cone is contained in the big cone, and the nef
cone is contained in the psef cone.

###### Proof.

It is straightforward to check that the big cone coincides with the interior
of the psef cone. To see that the latter is closed, assume
$\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is the limit of a sequence of
psef classes $\\{\theta_{j}\\}$. For an appropriate choice of representatives,
we then have $\theta_{j}\to\theta$ smoothly, and hence $\theta_{j}\leq
C\omega_{X}$ for a uniform constant $C>0$. Since $\\{\theta_{j}\\}$ is psef,
we can shoose
$\varphi_{j}\in\operatorname{PSH}(\theta_{j})\subset\operatorname{PSH}(C\omega_{X})$,
normalized by $\sup_{X}\varphi_{j}=0$. By weak compactness of normalized
$C\omega_{X}$-psh functions, we may assume, after passing to a subsequence,
that $\varphi_{j}$ converges weakly to a $C\omega_{X}$-psh function $\varphi$.
Since $\theta_{j}+dd^{c}\varphi_{j}\geq 0$, we infer $\theta+dd^{c}\varphi\geq
0$, which shows that $\\{\theta\\}$ is psef. ∎

######  Remark 1.16.

When $X$ is Kähler, the big cone of $H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})$,
i.e. its intersection with the big cone of $\mathrm{BC}^{1,1}(X,\mathbb{R})$,
coincides with the interior of the psef cone of
$H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})$. However, this fails in general in the
non-Kähler case: if $X$ is for instance a Hopf surface, then
$H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})$ is one-dimensional, and its psef cone
has nonempty interior since $X$ carries a nonzero closed positive
$(1,1)$-current; however, $X$ does not admit any closed strictly positive
current, and the big cone of $H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})$ is thus
empty.

For later use, we also record the following useful characterization of psef
classes, which follows from the Hahn–Banach theorem (see [Lam99, Lemma 3.3]):

######  Lemma 1.17.

A class $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is psef iff
$\int_{X}\theta\wedge\gamma\geq 0$ for every $dd^{c}$-closed, positive
$(n-1,n-1)$-form $\gamma>0$.

A quasi-closed positive current $T$ has _minimal singularities_ (within its
$dd^{c}$-class) if it is less singular than any other positive current in the
psef class $\\{T\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$. As observed by
Demailly, such currents exist in any psef class $\\{\theta\\}$. Indeed, the
upper envelope

$V_{\theta}:=\operatorname{P}_{\theta}(0)=\sup\left\\{\varphi\in\operatorname{PSH}(\theta)\mid\varphi\leq
0\text{ on }X\right\\}$

yields a positive current $T=\theta+dd^{c}V_{\theta}$ with minimal
singularities. Currents with minimal singularities in a psef class are however
usually far from unique.

###  1.5. The bounded mass property

Following [GL22] we introduce the following:

######  Definition 1.18.

We define the _upper volume_ of $\omega_{X}$ as

$\overline{\mathrm{vol}}(\omega_{X}):=\sup_{\varphi}\int_{X}(\omega_{X}+dd^{c}\varphi)^{n}\in[0,+\infty],$

where $\varphi$ ranges over all smooth $\omega_{X}$-psh functions.

The above supremum is unchanged if $\varphi$ ranges instead over all bounded
$\omega_{X}$-psh functions, since any such $\varphi$ is a decreasing limit of
smooth $\omega_{X}$-psh functions $\varphi_{j}$ (by Theorem 1.8), and

$\int_{X}(\omega+dd^{c}\varphi_{j})^{n}\to\int_{X}(\omega+dd^{c}\varphi)^{n}$

by continuity of Bedford–Taylor products along decreasing sequences.

Any other Hermitian form $\omega^{\prime}_{X}$ satisfies
$C^{-1}\omega_{X}\leq\omega^{\prime}_{X}\leq C\omega_{X}$ for some $C>0$,
which implies
$C^{-1}\operatorname{PSH}(\omega_{X})\subset\operatorname{PSH}(\omega^{\prime}_{X})\subset
C\operatorname{PSH}(\omega_{X})$, and hence

$C^{-n}\overline{\mathrm{vol}}(\omega_{X})\leq\overline{\mathrm{vol}}(\omega^{\prime}_{X})\leq
C^{n}\overline{\mathrm{vol}}(\omega_{X}).$

We may thus introduce:

######  Definition 1.19.

Let $X$ be compact complex manifold. We say that $X$ has the _bounded mass
property_ if $\overline{\mathrm{vol}}(\omega_{X})$ is finite for some (hence
any) Hermitian form $\omega_{X}$.

The following properties are straightforward:

  * •

the bounded mass property holds whenever $dd^{c}\omega_{X}^{p}=0$ for all $p$
(which, as is well-known, follows from the case $p=1,2$);

  * •

it is inherited by any holomorphic image of $X$.

In particular, the bounded mass property holds whenever $n\leq 2$ (using a
Gauduchon metric $\omega_{X}$), and also when $X$ is a Fujiki manifold (i.e.
bimeromorphic to a compact Kähler manifold).

By [AGL23, Theorem A] and [GL22, Theorem 3.7], we further have:

######  Proposition 1.20.

The bounded mass property is inherited by complex submanifolds, and is
bimeromorphically invariant.

In dimension $n=3$, the bounded mass property is known to hold in the
following cases (see [AGL23]):

  * •

there exists a Hermitian form $\omega_{X}$ with $dd^{c}\omega_{X}\geq 0$;

  * •

for ‘most’ complex nilmanifolds;

  * •

for the twistor space of a compact antiselfdual $4$-manifold.

At the time of this writing, we do not know any example where the bounded mass
property fails, and it is an intriguing problem whether it always holds. For
instance:

######  Example 1.21.

Does the Hopf manifold $X=(\mathbb{C}^{n}\setminus\\{0\\})/2^{\mathbb{Z}}$ of
dimension $n\geq 3$ have the bounded mass property? See [AGL23, §3.4.2] for a
discussion.

##  2\. Non-pluripolar products

As before, $X$ denotes a compact complex manifold, of complex dimension $n$,
endowed with a reference Hermitian form $\omega_{X}$. We show that the bounded
mass property yields a notion of non-pluripolar products of quasi-closed,
quasi-positive currents, and extend to this context some basic properties of
non-pluripolar masses.

###  2.1. Products of currents with bounded potentials

For any $p=1,\dots,n$, the classical work of Bedford–Taylor [BT87] extends the
operator

$(\varphi_{1},\dots,\varphi_{p})\mapsto dd^{c}\varphi_{1}\wedge\dots\wedge
dd^{c}\varphi_{p}$

from tuples of (locally defined) smooth psh functions to bounded ones, with
values in closed positive $(p,p)$-currents that are _nonpluripolar_ , in the
sense that they put no mass on pluripolar sets. This operator is symmetric and
multilinear, continuous along decreasing sequences, and further has the key
property of being local in the plurifine topology (see §1.2).

Since any quasi-psh function is locally the sum of a psh function and a smooth
function, these results readily extend to tuples of bounded quasi-psh
functions $\varphi_{1},\dots,\varphi_{p}$, the closed $(p,p)$-current
$dd^{c}\varphi_{1}\wedge\dots\wedge dd^{c}\varphi_{p}$ being now merely of
order $0$ (i.e. with signed measures as local coefficients), and still
nonpluripolar.

More generally, given arbitrary $(1,1)$-forms $\theta_{1},\dots,\theta_{p}$,
we define an operator

$(\varphi_{1},\dots,\varphi_{p})\mapsto\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i})$
(2.1)

on tuples of bounded quasi-psh functions by forcing multilinearity, i.e.

$\bigwedge_{i=1}^{p}(\theta_{i}+dd^{c}\varphi_{i}):=\sum_{I\subset\\{1,\dots,p\\}}\bigwedge_{i\in
I}\theta_{i}\wedge\bigwedge_{i\notin I}dd^{c}\varphi_{i}.$ (2.2)

This formula shows that the operator (2.1) is still local in the plurifine
topology. As the notation suggests, one easily checks that (2.2) only depends
on the currents $T_{i}=\theta_{i}+dd^{c}\varphi_{i}$, and yields a symmetric,
multilinear pairing

$(T_{1},\dots,T_{p})\mapsto T_{1}\wedge\dots\wedge T_{p}$ (2.3)

from tuples of quasi-closed, quasi-positive currents with bounded potentials
to nonpluripolar $(p,p)$-currents. We next extend this as follows:

######  Proposition 2.1.

For any compact subset $K\subset X$, there exists a unique symmetric,
multilinear pairing

$(T_{1},\dots,T_{p})\mapsto\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge
T_{p}$

defined on tuples of quasi-closed, quasi-positive currents with bounded
potentials on $K$ and with values in nonpluripolar $(p,p)$-currents, such
that:

  * (i)

when the $T_{i}$’s have bounded potentials, the pairing is compatible with
(2.3);

  * (ii)

for any tuple of $(1,1)$-forms $\theta_{1},\dots,\theta_{p}$, the operator

$(\varphi_{1},\dots,\varphi_{p})\mapsto\operatorname{\mathbf{1}}_{K}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}),$

acting on tuples of quasi-psh functions bounded on $K$ is local in the
plurifine topology.

Moreover, the construction is compatible with restriction, i.e.

$\operatorname{\mathbf{1}}_{L}T_{1}\wedge\dots\wedge
T_{p}=\operatorname{\mathbf{1}}_{L}\left(\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge
T_{p}\right)$ (2.4)

for any compact subset $L\subset K$, and we further have

$T_{1},\dots,T_{p}\geq
0\Longrightarrow\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge T_{p}\geq
0.$ (2.5)

We emphasize that the notation
$\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge T_{p}$ above should be
taken as a whole, i.e. we do not try to make sense of $T_{1}\wedge\dots\wedge
T_{p}$ itself (see however Theorem 2.2 below).

###### Proof.

Pick $(1,1)$-forms $\theta_{1},\dots,\theta_{p}$ and quasi-psh functions
$\varphi_{1},\dots,\varphi_{p}$ that are bounded on $K$. The latter is then
contained in the plurifine open subset

$O:=\bigcap_{i}\\{\varphi_{i}>-t_{0}\\}$

for some $t_{0}$. For any $t\geq t_{0}$ and $i=1,\dots,p$, the bounded quasi-
psh function $\max\\{\varphi_{i},-t\\}$ coincides with $\varphi_{i}$ on $O$.
By (i), (ii), we thus necessarily have

$\operatorname{\mathbf{1}}_{K}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i})=\operatorname{\mathbf{1}}_{K}\bigwedge_{i}(\theta_{i}+dd^{c}\max\\{\varphi_{i},-t\\}),$
(2.6)

for $t$ large enough, where the right-hand side was defined above (in the
sense of Bedford–Taylor). This proves uniqueness. To get existence, we use
(2.6) as a definition, and first check that it only depends on the quasi-
closed, quasi-positive currents $T_{i}=\theta_{i}+dd^{c}\varphi_{i}$,
$i=1,\dots,p$. Any other decomposition of $T_{i}$ is of the form

$T_{i}=(\theta_{i}+dd^{c}\tau_{i})+dd^{c}(\varphi_{i}-\tau_{i})$

with $\tau_{i}\in C^{\infty}(X)$ (see Lemma 1.4). For $t$ large enough, the
bounded quasi-psh functions

$\max\\{\varphi_{i}-\tau_{i},-t\\},\quad\max\\{\varphi_{i},-t\\}-\tau_{i}$

are both equal to $\varphi_{i}-\tau_{i}$ on the plurifine open $O$. By
plurifine locality, this yields

$\operatorname{\mathbf{1}}_{K}\bigwedge_{i}\left(\left(\theta_{i}+dd^{c}\tau_{i}\right)+dd^{c}\max\\{\varphi_{i}-\tau_{i},-t\\}\right)=\operatorname{\mathbf{1}}_{K}\bigwedge_{i}\left(\left(\theta_{i}+dd^{c}\tau_{i}\right)+dd^{c}\left(\max\\{\varphi_{i},-t\\}-\tau_{i}\right)\right),$

which coincides in turn with
$\operatorname{\mathbf{1}}_{K}\bigwedge_{i}\left(\theta_{i}+dd^{c}\max\\{\varphi_{i},-t\\}\right)$.
By (2.6) we infer

$\operatorname{\mathbf{1}}_{K}\bigwedge_{i}\left((\theta_{i}+dd^{c}\tau_{i})+dd^{c}(\varphi_{i}-\tau_{i})\right)=\operatorname{\mathbf{1}}_{K}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}),$

which proves, as desired, that this current only depends on the $T_{i}$’s.
Similar considerations shows that
$\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge T_{p}$ is a multilinear
function of the $T_{i}$. Symmetry is trivial, and (2.4) holds by uniqueness.

The proof of (2.5) is slightly more involved. Assume $T_{i}\geq 0$ for all
$i$. Since

$\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge T_{p}\geq 0$

is a local property, it is enough to argue in a small open neighborhood
$\Omega$ of a given point of $X$, on which we can thus find a Kähler form
$\omega=dd^{c}\rho$ with $\rho$ smooth and bounded. Pick $\varepsilon>0$. By
Lemma 1.5, each point of $\Omega$ admits an open neighborhood $U\subset\Omega$
(depending on $\varepsilon$) and closed $(1,1)$-forms $\theta_{i}^{\prime}$ on
$U$ such that
$-\varepsilon\omega\leq\theta_{i}-\theta^{\prime}_{i}\leq\varepsilon\omega$ on
$U$. After shrinking $U$, we further assume
$\theta^{\prime}_{i}=dd^{c}\tau_{i}$ with $\tau_{i}$ smooth and bounded on
$U$. Then

$dd^{c}(\tau_{i}+\varepsilon\rho+\varphi_{i})=\theta_{i}^{\prime}+\varepsilon\omega+dd^{c}\varphi_{i}\geq\theta_{i}+dd^{c}\varphi_{i}\geq
0,$

and hence $\tau_{i}+\varepsilon\rho+\varphi_{i}$ is psh on $U$. For each
$t>0$, $\max\\{\tau_{i}+\varepsilon\rho+\varphi_{i},-t\\}$ is thus psh and
bounded on $U$, and it further coincides with the bounded quasi-psh function
$\tau_{i}+\varepsilon\rho+\max\\{\varphi_{i},-t\\}$ on the plurifine open
$O\cap U$ for $t$ large enough. By plurifine locality of Bedford–Taylor
products, we get for each $I\subset\\{1,\dots,p\\}$

$\operatorname{\mathbf{1}}_{K\cap U}\bigwedge_{i\in
I}\left(\theta^{\prime}_{i}+\varepsilon\omega+dd^{c}\varphi_{i}\right)=\operatorname{\mathbf{1}}_{K\cap
U}\bigwedge_{i\in
I}dd^{c}\left(\tau_{i}+\varepsilon\rho+\max\\{\varphi_{i},-t\\}\right)$
$=\operatorname{\mathbf{1}}_{K\cap U}\bigwedge_{i\in
I}dd^{c}\max\\{\tau_{i}+\varepsilon\rho+\varphi_{i},-t\\}\geq 0.$

Since $\theta_{i}+\varepsilon\omega\geq\theta^{\prime}_{i}$ on $U$, we infer

$\operatorname{\mathbf{1}}_{K\cap
U}\bigwedge_{i=1}^{p}(\theta_{i}+2\varepsilon\omega+dd^{c}\varphi_{i})=\sum_{I\subset\\{1,\dots,p\\}}\operatorname{\mathbf{1}}_{K\cap
U}\bigwedge_{i\in
I}(\theta^{\prime}_{i}+\varepsilon\omega+dd^{c}\varphi_{i})\wedge\bigwedge_{i\notin
I}(\theta_{i}+\varepsilon\omega-\theta^{\prime}_{i})\geq 0.$

As this holds for $U$ ranging over an open cover of $\Omega$, we infer

$\operatorname{\mathbf{1}}_{K\cap\Omega}\bigwedge_{i=1}^{p}(\theta_{i}+2\varepsilon\omega+dd^{c}\varphi_{i})\geq
0.$

This holds for any $\varepsilon>0$, and this current converges weakly to
$\operatorname{\mathbf{1}}_{K\cap\Omega}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i})$
as $\varepsilon\to 0$, as one easily sees using multilinearity. We conclude
$\operatorname{\mathbf{1}}_{K}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i})\geq
0$ on $\Omega$, which finishes the proof of (2.5). ∎

###  2.2. Non-pluripolar products

In this Section and 2.3 _we assume that the manifold $X$ has the bounded mass
property_ (see §1.5). Generalizing [BEGZ10, §1] (which dealt with closed
currents on a Kähler manifold), we then show:

######  Theorem 2.2.

For any $p=1,\dots,n$, there exists a unique symmetric, multilinear pairing

$(T_{1},\dots,T_{p})\mapsto T_{1}\wedge\dots\wedge T_{p}$

defined on $p$-tuples of quasi-closed, quasi-positive currents and with values
in nonpluripolar $(p,p)$-currents, such that:

  * (i)

if the $T_{i}$’s have bounded potentials, then the pairing is compatible with
(2.3);

  * (ii)

for any tuple of $(1,1)$-forms $\theta_{1},\dots,\theta_{p}$, the operator

$(\varphi_{1},\dots,\varphi_{p})\mapsto\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}),$

defined on all tuples of quasi-psh functions, is local in the plurifine
topology.

Moreover,

$T_{1},\dots,T_{p}\geq 0\Longrightarrow T_{1}\wedge\dots\wedge T_{p}\geq 0.$

The current $T_{1}\wedge\dots\wedge T_{p}$ is called the _non-pluripolar
product_ of the $T_{i}$’s. When $\theta_{i}=\theta$ and $\varphi_{i}=\varphi$,
we write

$(\theta+dd^{c}\varphi)^{p}:=\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}).$

The key input in the proof is the following:

######  Lemma 2.3.

Let $T_{1},\dots,T_{p}$ be quasi-closed, quasi-positive currents, and pick a
compact subset $K\subset X$ on which the $T_{i}$’s have bounded potentials.
Then the total variation of the current
$\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge T_{p}$ defined in
Proposition 2.1 is bounded by a constant independent of $K$.

More precisely, write $T_{i}=\theta_{i}+dd^{c}\varphi_{i}$ with $\theta_{i}$
smooth and $\varphi_{i}$ quasi-psh, and assume

$\pm\theta_{i}\leq C\omega_{X}\quad\text{and}\quad
dd^{c}\varphi_{i}\geq-C\omega_{X}$

for some $C>0$ and for all $i$. Then the total variation of
$\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge T_{p}$ is bounded by a
constant only depending on $\omega_{X}$ and $C$.

###### Proof.

By multilinearity, we may assume without loss of generality that
$\theta_{i}=\omega_{X}$ and $T_{i}\geq 0$, i.e.
$\varphi_{i}\in\operatorname{PSH}(\omega_{X})$. By (2.5), we then have
$\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge T_{p}\geq 0$, and

$\int\operatorname{\mathbf{1}}_{K}T_{1}\wedge\dots\wedge
T_{p}\wedge\omega_{X}^{n-p}\leq
n^{n}\int\operatorname{\mathbf{1}}_{K}(\omega_{X}+dd^{c}\psi)^{n}$

with $\psi:=\tfrac{1}{n}\sum_{i=1}^{p}\varphi_{i}$, by multilinearity. By
construction, the right-hand side is unchanged when we replace $\psi$ with the
bounded $\omega_{X}$-function $\max\\{\psi,-t\\}$ for $t\gg 1$ (see
Proposition 2.1), and

$\int\operatorname{\mathbf{1}}_{K}(\omega_{X}+dd^{c}\psi)^{n}=\int\operatorname{\mathbf{1}}_{K}(\omega_{X}+dd^{c}\max\\{\psi,-t\\})^{n}$
$\leq\int_{X}(\omega_{X}+dd^{c}\max\\{\psi,-t\\})^{n}\leq\overline{\mathrm{vol}}(\omega_{X})<\infty,$

see §1.5. ∎

###### Proof of Theorem 2.2.

Write $T_{i}=\theta_{i}+dd^{c}\varphi_{i}$ with $\theta_{i}$ smooth and
$\varphi_{i}$ quasi-psh. For each $m\in\mathbb{N}$, the $T_{i}$’s have bounded
potentials on the compact subset $K_{m}:=\bigcap_{i}\\{\varphi_{i}\geq-m\\}$,
and $(K_{m})_{m}$ is an increasing sequence such that

$X\setminus\bigcup_{m}K_{m}=\bigcup_{i}\\{\varphi_{i}=-\infty\\}$

is pluripolar. For each $m$, Proposition 2.1 yields a nonpluripolar
$(p,p)$-current

$\Theta_{m}:=\operatorname{\mathbf{1}}_{K_{m}}T_{1}\wedge\dots\wedge T_{p},$

which further satisfies
$\Theta_{m}=\operatorname{\mathbf{1}}_{K_{m}}\Theta_{m+1}$. By Lemma 2.3, the
total variation of $\Theta_{m}$ is further bounded independently of $m$, and
it follows that there exists a unique nonpluripolar $(p,p)$-current $\Theta$
on $X$ such that $\operatorname{\mathbf{1}}_{K_{m}}\Theta=\Theta_{m}$ for all
$m$. By (i), (ii), this current $\Theta$ necessarily coincides with
$T_{1}\wedge\dots\wedge T_{p}$, and the rest now follows from Proposition 2.1.
∎

We further record the following consequence of Lemma 2.3:

######  Lemma 2.4.

For all quasi-closed positive currents $T_{1},\dots,T_{n}\geq 0$, we have

$\int_{X}T_{1}\wedge\dots\wedge T_{n}\leq M$

for a constant $M>0$ only depending on the psef classes
$\\{T_{i}\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$.

###  2.3. Monotonicity of non-pluripolar masses

Arguing as in [DDL18, Theorem 2.3], we establish the following semicontinuity
property of non-pluripolar Monge–Ampère masses.

######  Proposition 2.5.

For any $(1,1)$-forms $\theta_{1},\dots,\theta_{n}$,

$\int_{X}(\theta_{1}+dd^{c}\varphi_{1})\wedge\dots\wedge(\theta_{n}+dd^{c}\varphi_{n})$

is a lower semicontinuous function of the tuple
$\varphi_{i}\in\operatorname{PSH}(\theta_{i})$, $i=1,\dots,n$, with respect to
convergence in capacity.

We refer to [GZ, Chapter 4] for the notion of convergence in capacity. Suffice
it to say here that any monotonically convergent sequence of $C\omega_{X}$-psh
functions converges in capacity. The proof of Proposition 2.5 relies on the
following continuity property of Monge–Ampère integrals, which is a direct
consequence of the corresponding local statement, see [GZ, Theorem 4.25].

######  Lemma 2.6.

Pick $C>0$, and assume $\varphi_{i}^{k}\in\operatorname{PSH}(C\omega_{X})$ is
uniformly bounded and converges in capacity to
$\varphi_{i}\in\operatorname{PSH}(C\omega_{X})$ as $k\to\infty$, for
$i=1,\dots,n$. Let also $(f_{k})$ be a bounded sequence of quasi-continuous
functions, converging in capacity to a quasi-continuous function $f$. Then

$\lim_{k}\int_{X}f_{k}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}^{k})=\int_{X}f\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}).$

Recall that a function $f$ is _quasi-continuous_ if is continuous outside open
subsets of arbitrarily small capacity. Any quasi-psh function is quasi-
continuous, and we thus have:

######  Example 2.7.

Pick finitely many quasi-psh functions $\varphi_{i}$, and consider the
plurifine open set $O:=\\{\varphi>0\\}$ with $\varphi:=\min_{i}\varphi_{i}$.
For each $\varepsilon>0$, the function $\chi_{\varepsilon}\colon X\to[0,1]$
defined by

$\chi_{\varepsilon}:=\frac{\max\\{\varphi,0\\}}{\max\\{\varphi,0\\}+\varepsilon}$

is then quasi-continuous, and $\chi_{\varepsilon}$ increases pointwise to
$\operatorname{\mathbf{1}}_{O}$ as $\varepsilon\searrow 0$.

###### Proof of Proposition 2.5.

For each $k$ and $t,\varepsilon>0$, consider as in Example 2.7 the quasi-
continuous approximations

$\chi_{k,\varepsilon,t}:=\frac{\max\\{\min_{i}\varphi_{i}^{k}+t,0\\}}{\max\\{\min_{i}\varphi_{i}^{k}+t,0\\}+\varepsilon},\quad\chi_{\varepsilon,t}:=\frac{\max\\{\min_{i}\varphi_{i}+t,0\\}}{\max\\{\min_{i}\varphi_{i}+t,0\\}+\varepsilon}$

of the characteristic functions of the plurifine open sets

$O_{k,t}:=\bigcap_{i}\\{\varphi_{i}^{k}>-t\\},\quad
O_{t}:=\bigcap_{i}\\{\varphi_{i}>-t\\}.$

Since $\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}^{k})\geq 0$ and
$\chi_{k,\varepsilon,t}$ vanishes outside $O_{k,t}$, plurifine locality of
nonpluripolar products yields

$\int_{X}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}^{k})\geq\int_{X}\chi_{k,\varepsilon,t}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}^{k})=\int_{X}\chi_{k,\varepsilon,t}\,\bigwedge_{i}(\theta_{i}+dd^{c}\max\\{\varphi_{i}^{k},-t\\}).$

Now $(\chi_{k,\varepsilon,t})_{k}$ is a bounded sequence of quasi-continuous
fonctions converging in capacity to the quasi-continuous function
$\chi_{\varepsilon,t}$, while $(\max\\{\varphi_{i}^{k},-t\\})_{k}$ is a
bounded sequence of $C\omega_{X}$-psh functions converging in capacity to
$\max\\{\varphi_{i},-t\\}$. By Lemma 2.6, we thus have

$\lim_{k}\int_{X}\chi_{k,\varepsilon,t}\,\bigwedge_{i}(\theta_{i}+dd^{c}\max\\{\varphi_{i}^{k},-t\\})=\int_{X}\chi_{\varepsilon,t}\bigwedge_{i}(\theta_{i}+dd^{c}\max\\{\varphi_{i},-t\\})$
$=\int_{X}\chi_{\varepsilon,t}\,\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}),$

where the last equality holds by plurifine locality again. We thus get

$\liminf_{k}\int_{X}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}^{k})\geq\int_{X}\chi_{\varepsilon,t}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}),$

and the result follows since

$\lim_{t\to\infty}\lim_{\varepsilon\to
0}\int_{X}\chi_{\varepsilon,t}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i})=\lim_{t\to\infty}\int_{O_{t}}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i})=\int_{X}\bigwedge_{i}(\theta_{i}+dd^{c}\varphi_{i}).$

∎

Finally, we provide a version, in the present non-Kähler setting, of the
monotonicity of nonpluripolar masses with respect to singularities. To state
the result, pick any $(1,1)$-form $\theta$, let $C\geq 0$ be the smallest
constant such that $\pm\theta\leq C\omega_{X}$, and set

$\Delta_{\theta}:=\sup_{\varphi,\psi\in\operatorname{PSH}(C\omega_{X})\cap
L^{\infty}}\left\\{\int_{X}(\theta+dd^{c}\varphi)^{n}-\int_{X}(\theta+dd^{c}\psi)^{n}\right\\}.$
(2.7)

By Stokes’ theorem, $\Delta_{\theta}=0$ if $\theta$ is closed. In general,
Lemma 2.4 yields $\Delta_{\theta}\in[0,+\infty)$ and

$\Delta_{\theta+\varepsilon\omega_{X}}=\Delta_{\theta}+O(\varepsilon)$ (2.8)

for $\varepsilon\in[0,1]$.

######  Theorem 2.8.

Assume $\varphi,\psi\in\operatorname{PSH}(\theta)$ satisfy
$\varphi\leq\psi+O(1)$. Then

$\int_{X}(\theta+dd^{c}\varphi)^{n}\leq\int_{X}(\theta+dd^{c}\psi)^{n}+\Delta_{\theta}.$

###### Proof.

The proof goes along the same lines as [DDL23, Lemma 3.1]. Assume first
$\varphi=\psi$ outside some compact subset $K\subset X$ on which
$\varphi,\psi$ are bounded. Pick $t\gg 1$ such that $\varphi,\psi>-t$ on $K$,
and consider the bounded $C\omega_{X}$-psh functions

$\varphi_{t}:=\max\\{\varphi,-t\\},\quad\psi_{t}:=\max\\{\psi,-t\\}.$

By plurifine locality, we then have

$\int_{K}(\theta+dd^{c}\varphi)^{n}=\int_{K}(\theta+dd^{c}\varphi_{t})^{n},\quad\int_{K}(\theta+dd^{c}\psi)^{n}=\int_{K}(\theta+dd^{c}\psi_{t})^{n},$

while $\varphi=\psi,\varphi_{t}=\psi_{t}$ on the open set $X\setminus K$
similarly yields

$\int_{X\setminus K}(\theta+dd^{c}\varphi)^{n}=\int_{X\setminus
K}(\theta+dd^{c}\psi)^{n},\quad\int_{X\setminus
K}(\theta+dd^{c}\varphi_{t})^{n}=\int_{X\setminus
K}(\theta+dd^{c}\psi_{t})^{n}.$

This implies

$\int_{X}(\theta+dd^{c}\varphi)^{n}-\int_{X}(\theta+dd^{c}\psi)^{n}=\int_{X}(\theta+dd^{c}\varphi_{t})^{n}-\int_{X}(\theta+dd^{c}\psi_{t})^{n},$

and the result follows since the right-hand side is bounded by
$\Delta_{\theta}$. Returning to the general case, set

$\varphi_{\varepsilon,t}:=\max\\{(1+\varepsilon)\varphi,\psi-t\\}\in\operatorname{PSH}(\theta+C\varepsilon\omega_{X}),$

where $\varepsilon,t>0$ are fixed for the moment. Since
$\varphi\leq\psi+O(1)$, we have $\varphi_{\varepsilon,t}=\psi-t$ outside the
compact set $\\{\varphi\geq-s\\}$ for $s\gg 1$, and the first part of the
proof thus yields

$\int_{X}(\theta+C\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon,t})^{n}\leq\int_{X}(\theta+C\varepsilon\omega_{X}+dd^{c}\psi)^{n}+\Delta_{\theta+C\varepsilon\omega_{X}}.$

For $0<\varepsilon\leq\delta$, we infer

$\int_{X}(\theta+C\delta\omega_{X}+dd^{c}\varphi_{\varepsilon,t})^{n}=\int_{X}(\theta+C\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon,t})^{n}+O(\delta)$
$\leq\int_{X}(\theta+C\varepsilon\omega_{X}+dd^{c}\psi)^{n}+\Delta_{\theta}+O(\delta),$

using Lemma 2.3 and (2.8). Since
$\varphi_{t,\varepsilon}\in\operatorname{PSH}(\theta+C\delta\omega_{X})$
increases to $\max\\{\varphi,\psi-t\\}$ as $\varepsilon\to 0$, Proposition 2.5
now yields

$\int_{X}(\theta+C\delta\omega+dd^{c}\max\\{\varphi,\psi-t\\})^{n}\leq\int_{X}(\theta+dd^{c}\psi)^{n}+\Delta_{\theta}+O(\delta).$

Letting $\delta\to 0$, we get

$\int_{X}(\theta+dd^{c}\max\\{\varphi,\psi-t\\})^{n}\leq\int_{X}(\theta+dd^{c}\psi)^{n}+\Delta_{\theta}.$

Since $\max\\{\varphi,\psi-t\\}$ decreases to $\varphi$ as $t\to\infty$, the
conclusion now follows from Proposition 2.5. ∎

###  2.4. Monge–Ampère measures of quasi-psh envelopes

In this section, we fix a big $dd^{c}$-class
$\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$. We also pick
$\rho\in\operatorname{PSH}(\theta)$ with analytic singularities such that
$\theta+dd^{c}\rho\geq\omega_{X}$ (after perhaps scaling $\omega_{X}$), and
consider the Zariski open subset

$\Omega:=\\{\rho>-\infty\\}.$

######  Theorem 2.9.

Pick a quasi-continuous function $f\colon X\to[-\infty,\infty]$, and assume
that the usc regularization $\operatorname{P}_{\theta}^{\star}(f)$ of its
$\theta$-psh envelope (1.3) is $\theta$-psh. Then

$\int_{\\{\operatorname{P}_{\theta}^{\star}(f)<f\\}}(\theta+dd^{c}\operatorname{P}_{\theta}^{\star}(f))^{n}=0.$

###### Proof.

Replacing $f$ with $\min\\{f,\sup_{X}\operatorname{P}_{\theta}^{\star}(f)\\}$,
we may assume without loss that $f$ is bounded above. Assume further that $f$
is bounded. When $f$ is lsc, the result follows from a standard balayage
argument. If $f$ is merely bounded, we can find a decreasing sequence
$(f_{j})$ of bounded lsc functions that decreases pointwise to $f$ outside a
pluripolar set (see [GLZ19, Lemma 2.4] and [DDL23, Theorem 2.7]). Using that
$u_{j}:=\operatorname{P}^{\star}_{\theta}(f_{j})$ (resp.
$u:=\operatorname{P}_{\theta}^{\star}(f)$) is the largest $\theta$-psh
function dominated by $f_{j}$ (resp. $f$) outside a pluripolar set, it is easy
to see that $u_{j}$ decreases pointwise to $u$. By the previous step of the
proof, we further have

$\int_{X}\frac{\max\\{f_{j}-u_{j},0\\}}{\max\\{f_{j}-u_{j},0\\}+\varepsilon}(\theta+dd^{c}u_{j})^{n}=0$

for each $\varepsilon>0$. Letting $j\to+\infty$, and arguing as in Proposition
2.5, we obtain

$\int_{X}\frac{\max\\{f-u,0\\}}{\max\\{f-u,0\\}+\varepsilon}(\theta+dd^{c}u)^{n}=0,$

and letting $\varepsilon\to 0$ yields the result in the bounded case. If $f$
is not bounded, we approximate it by $f_{j}=\min(f,-j)$, and proceed as above.
∎

######  Remark 2.10.

When $\operatorname{P}_{\theta}^{\star}(f)$ is locally bounded in $\Omega$,
the measure $(\theta+dd^{c}\operatorname{P}_{\theta}^{\star}(f))^{n}$ is
understood in the classical sense of Bedford–Taylor in $\Omega$, and can thus
be viewed as a nonpluripolar measure on $X$ without assuming the bounded mass
property. The same remark applies for the results below.

######  Proposition 2.11.

Assume $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is big. If
$u,v\in\operatorname{PSH}(\theta)$ then

$(\theta+dd^{c}\max\\{u,v\\})^{n}\geq\operatorname{\mathbf{1}}_{\\{u\geq
v\\}}(\theta+dd^{c}u)^{n}+\operatorname{\mathbf{1}}_{\\{v>u\\}}(\theta+dd^{c}v)^{n}.$

In particular, if $u\leq v$, then

$\operatorname{\mathbf{1}}_{\\{u=v\\}}(\theta+dd^{c}u)^{n}\leq\operatorname{\mathbf{1}}_{\\{u=v\\}}(\theta+dd^{c}v)^{n}.$

###### Proof.

If $u$ and $v$ have minimal singularities, then they are locally bounded in
$\Omega$. By [GL22, Lemma 1.2], the result thus holds on $\Omega$, and hence
on $X$ since all measures involved are nonpluripolar. In the general case, we
approximate $u,v$ by the $\theta$-psh functions with minimal singularities

$u_{t}:=\max\\{u,V_{\theta}-t\\},\quad v_{t}:=\max\\{v,V_{\theta}-t\\}.$

The previous step yields

$(\theta+dd^{c}\max\\{u_{t},v_{t}\\})^{n}\geq\operatorname{\mathbf{1}}_{\\{v_{t}\leq
u_{t}\\}}(\theta+dd^{c}u_{t})^{n}+\operatorname{\mathbf{1}}_{\\{u_{t}<v_{t}\\}}(\theta+dd^{c}v_{t})^{n}.$

Multiplying with
$\operatorname{\mathbf{1}}_{\\{\min\\{u,v\\}>V_{\theta}-t\\}}$, we get

$\displaystyle\operatorname{\mathbf{1}}_{\\{\min\\{u,v\\}>V_{\theta}-t\\}}(\theta+dd^{c}\max\\{u,v\\})^{n}$
$\displaystyle\leq\operatorname{\mathbf{1}}_{\\{\min\\{u,v\\}>V_{\theta}-t\\}}\operatorname{\mathbf{1}}_{\\{v\leq
u\\}}(\theta+dd^{c}u)^{n}$
$\displaystyle+\operatorname{\mathbf{1}}_{\\{\min\\{u,v\\}>V_{\theta}-t\\}}\operatorname{\mathbf{1}}_{\\{u<v\\}}(\theta+dd^{c}v)^{n}.$

We finally let $t\to+\infty$ to obtain the result. ∎

As a consequence we infer the following minimum principle.

######  Corollary 2.12.

Pick $u,v\in\operatorname{PSH}(\theta)$ such that there exists
$w\in\operatorname{PSH}(\theta)$ with $w\leq u,v$, and hence
$\operatorname{P}_{\theta}(\min\\{u,v\\})\in\operatorname{PSH}(\theta)$. Then

$(\theta+dd^{c}\operatorname{P}_{\theta}(\min\\{u,v\\}))^{n}\leq\operatorname{\mathbf{1}}_{\\{\operatorname{P}_{\theta}(\min\\{u,v\\})=v\leq
u\\}}(\theta+dd^{c}v)^{n}+\operatorname{\mathbf{1}}_{\\{\operatorname{P}_{\theta}(\min\\{u,v\\})=u<v\\}}(\theta+dd^{c}u)^{n}.$

##  3\. The lower and upper volume of a $dd^{c}$-class

As above, $X$ denotes an $n$-dimensional compact complex manifold, assumed
throughout this section to have the bounded mass property. We introduce and
study the notion of lower and upper volume of a $dd^{c}$-class.

###  3.1. The lower and upper volume of a current

Extending [GL22, Definition 3.1], we introduce:

######  Definition 3.1.

The _lower volume_ and _upper volume_ of a quasi-closed positive
$(1,1)$-current $T\geq 0$ are defined as

$\underline{\mathrm{vol}}(T):=\inf_{S}\int_{X}S^{n},\quad\overline{\mathrm{vol}}(T):=\sup_{S}\int_{X}S^{n},$

where $S$ ranges over all quasi-closed positive $(1,1)$-currents in the same
$dd^{c}$-class as $T$ and with equivalent singularities.

Writing $T=\theta+dd^{c}\varphi$ with $\theta$ smooth and
$\varphi\in\operatorname{PSH}(\theta)$, we equivalently have

$\underline{\mathrm{vol}}(T)=\inf_{\psi}\int_{X}(\theta+dd^{c}\psi)^{n},\quad\overline{\mathrm{vol}}(T)=\sup_{\psi}\int_{X}(\theta+dd^{c}\psi)^{n}$
(3.1)

where $\psi$ ranges over all $\theta$-psh functions such that
$\psi=\varphi+O(1)$.

By definition, the upper and lower volumes of $T$ only depend on its
$dd^{c}$-class and singularity class, and satisfy

$0\leq\underline{\mathrm{vol}}(T)\leq\overline{\mathrm{vol}}(T)<\infty,$

see Lemma 2.4. Note also that

$T>0\Longrightarrow\overline{\mathrm{vol}}(T)>0.$ (3.2)

The analogous result for $\underline{\mathrm{vol}}(T)$ is unclear in general,
even when $T$ is smooth (see §3.2.3). As a consequence of Theorem 2.8, we
have:

######  Example 3.2.

If $T$ is closed, then
$\underline{\mathrm{vol}}(T)=\overline{\mathrm{vol}}(T)=\int_{X}T^{n}$.

We first collect a few simple properties:

######  Proposition 3.3.

Pick a quasi-closed current $T\geq 0$. Then:

  * (i)

for all $t\in\mathbb{R}_{>0}$ we have

$\underline{\mathrm{vol}}(tT)=t^{n}\underline{\mathrm{vol}}(T),\quad\overline{\mathrm{vol}}(tT)=t^{n}\overline{\mathrm{vol}}(T);$

  * (ii)

for any quasi-closed current $T^{\prime}\geq 0$ and $\varepsilon\in[0,1]$, we
have

$\underline{\mathrm{vol}}(T+\varepsilon
T^{\prime})\leq\underline{\mathrm{vol}}(T)+C\varepsilon$ (3.3)

for a constant $C>0$ only depending on $\\{T\\},\\{T^{\prime}\\}$;

  * (iii)

for any effective $\mathbb{R}$-divisor $E$ we have

$\underline{\mathrm{vol}}(T+[E])=\underline{\mathrm{vol}}(T),\quad\overline{\mathrm{vol}}(T+[E])=\overline{\mathrm{vol}}(T);$

  * (iv)

for any modification $\pi\colon Y\to X$ we have

$\underline{\mathrm{vol}}(\pi^{\star}T)=\underline{\mathrm{vol}}(T),\quad\overline{\mathrm{vol}}(\pi^{\star}T)=\overline{\mathrm{vol}}(T).$

###### Proof.

(i) is obvious from the definition. Pick $T^{\prime}\geq 0$ quasi-closed and
$0\leq S\in\\{T\\}$ with singularities equivalent to $T$. Then $0\leq
S+\varepsilon T^{\prime}\in\\{T+\varepsilon T^{\prime}\\}$ has singularities
equivalent to $T+\varepsilon T^{\prime}$, and hence $\int_{X}(S+\varepsilon
T^{\prime})^{n}\geq\underline{\mathrm{vol}}(T+\varepsilon T^{\prime})$. For
$j=0,\dots,n$, $\int_{X}S^{j}\wedge T^{\prime n-j}$ is further bounded in
terms of $\\{S\\}=\\{T\\}$ and $\\{T^{\prime}\\}$, by Lemma 2.4. Thus

$\int_{X}S^{n}\geq\int_{X}(S+\varepsilon
T^{\prime})^{n}-C\varepsilon\geq\underline{\mathrm{vol}}(T+\varepsilon
T^{\prime})-C\varepsilon$

for a constant $C>0$ only depending on $\\{T\\},\\{T^{\prime}\\}$. Taking the
infimum over $S$ yields (ii).

Next pick an effective $\mathbb{R}$-divisor $E$, and note that any $0\leq
S\in\\{T\\}$ satisfies $S+[E]\in\\{T+[E]\\}$ and
$\int_{X}S^{n}=\int_{X}(S+[E])^{n}$. To prove (iii), it is thus enough to show
that $S\mapsto S+[E]$ sets a 1–1 correspondence between positive currents in
$\\{T\\}$ with the same singularities as $T$ and positive currents in
$\\{T+[E]\\}$ with the same singularities as $T+[E]$. One direction is clear.
Conversely, pick $0\leq S^{\prime}\in\\{T+[E]\\}$ with the same singularities
as $T+[E]$. Write

$T=\theta+dd^{c}\psi,\quad[E]=\theta_{E}+dd^{c}\psi_{E},\quad
S^{\prime}=\theta+\theta_{E}+dd^{c}\varphi^{\prime}$

with $\theta,\theta_{E}$ smooth and $\psi\in\operatorname{PSH}(\theta)$,
$\psi_{E}\in\operatorname{PSH}(\theta_{E})$,
$\varphi^{\prime}\in\operatorname{PSH}(\theta+\theta_{E})$. By assumption we
have $\varphi^{\prime}=\psi+\psi_{E}+O(1)$. In particular,
$\varphi^{\prime}-\psi_{E}$ is $\theta$-psh on
$X\setminus\operatorname{supp}E$ and bounded above, and hence uniquely extends
to a $\theta$-psh function $\varphi$ on $X$, which satisfies
$\varphi=\psi+O(1)$. This shows that $S:=\theta+dd^{c}\varphi\geq 0$ lies in
$\\{T\\}$, has the same singularities as $T$, and satisfies
$S^{\prime}=S+[E]$, which concludes the proof of (iii).

Finally, pick a modification $\pi\colon Y\to X$. Since $\pi$ has connected
fibers, each positive current in $\\{\pi^{\star}T\\}$ is of the form
$\pi^{\star}S$ with $0\leq S\in\\{T\\}$, and $S$ has the same singularities as
$T$ iff $\pi^{\star}S$ has the same singularities as $\pi^{\star}T$. Since
$\int_{X}S^{n}=\int_{Y}(\pi^{\star}S)^{n}$, we get (iv). ∎

######  Remark 3.4.

The analogue of (3.3) for $\overline{\mathrm{vol}}$ holds when $T>0$ (with a
constant depending on $T$), but is unclear in general.

We shall in fact mostly consider the upper and lower volumes of currents with
analytic singularities (see §1.3), which basically reduces to the smooth case
thanks to the following consequence of Proposition 3.3 (iii),(iv).

######  Lemma 3.5.

Assume $T\geq 0$ is quasi-closed with analytic singularities, and pick a
modification $\pi\colon Y\to X$ such that $\pi^{\star}T=\beta+[E]$ with
$\beta\geq 0$ smooth and $E$ an effective $\mathbb{Q}$-divisor. Then
$\underline{\mathrm{vol}}(T)=\underline{\mathrm{vol}}(\beta)$ and
$\overline{\mathrm{vol}}(T)=\overline{\mathrm{vol}}(\beta)$.

Building on [GL22], we infer the following monotonicity property.

######  Proposition 3.6.

Let $T_{1},T_{2}\geq 0$ be quasi-closed positive currents with analytic
singularities. If $T_{1}\leq T_{2}$, then
$\underline{\mathrm{vol}}(T_{1})\leq\underline{\mathrm{vol}}(T_{2})$ and
$\overline{\mathrm{vol}}(T_{1})\leq\overline{\mathrm{vol}}(T_{2})$.

###### Proof.

After passing to a joint resolution of singularities of $T_{1}$ and $T_{2}$ we
may assume $T_{i}=\beta_{i}+[E_{i}]$ with $\beta_{i}\geq 0$ smooth and $E_{i}$
an effective $\mathbb{Q}$-divisor, and we then have
$\underline{\mathrm{vol}}(T_{i})=\underline{\mathrm{vol}}(\beta_{i})$ (see
Lemma 3.5). Then $T_{1}\leq T_{2}$ implies
$\beta_{1}+[E_{1}]\leq\beta_{2}+[E_{2}]$, and hence $\beta_{1}\leq\beta_{2}$.
By [GL22, Proposition 3.2] we get
$\underline{\mathrm{vol}}(\beta_{1})\leq\underline{\mathrm{vol}}(\beta_{2})$
and
$\overline{\mathrm{vol}}(\beta_{1})\leq\overline{\mathrm{vol}}(\beta_{2})$,
which concludes the proof. ∎

While we are not able at the moment to extend this result to arbitrary quasi-
closed positive currents, quasi-psh envelopes enable us the following related
monotonicity result in the general case.

######  Proposition 3.7.

Fix smooth real $(1,1)$-forms $\theta_{1},\theta_{2}$ and
$\varphi_{1}\in\operatorname{PSH}(\theta_{1})$,
$\varphi_{2}\in\operatorname{PSH}(\theta_{2})$ with
$\varphi_{1}\leq\varphi_{2}$ and $\theta_{1}\leq\theta_{2}$. Then the positive
currents $T_{1}=\theta_{1}+dd^{c}\varphi_{1}$,
$T_{2}=\theta_{2}+dd^{c}\varphi_{2}$ satisfy

$\underline{\mathrm{vol}}(T_{1})\leq\underline{\mathrm{vol}}(T_{2}),\quad\overline{\mathrm{vol}}(T_{1})\leq\overline{\mathrm{vol}}(T_{2}).$

In particular, the upper and lower volumes of any quasi-closed positive
current $T$, which only depend on its $dd^{c}$-class and singularity class,
are further monotone increasing functions of the singularity class.

###### Proof.

Pick $\psi_{2}\in\operatorname{PSH}(\theta_{2})$ such that
$\psi_{2}=\varphi_{2}+O(1)$, and set

$\varphi_{t}:=\operatorname{P}_{\theta_{1}}(\min\\{\varphi_{1}+t,\psi_{2}\\})$

for $t>0$. Using $\varphi_{1}\leq\varphi_{2}=\psi_{2}+O(1)$ we get
$\varphi_{t}\in\operatorname{PSH}(\theta_{1})$,
$\varphi_{t}=\varphi_{1}+O(1)$, and hence
$\underline{\mathrm{vol}}(T_{1})\leq\int_{X}(\theta_{1}+dd^{c}\varphi_{t})^{n}$.
By Theorem 2.9, we further have

$\displaystyle(\theta_{1}+dd^{c}\varphi_{t})^{n}$
$\displaystyle\leq\operatorname{\mathbf{1}}_{\\{\varphi_{t}=\varphi_{1}+t\\}}(\theta_{1}+dd^{c}\varphi_{1})^{n}+\operatorname{\mathbf{1}}_{\\{\varphi_{t}=\psi_{2}\\}}(\theta_{2}+dd^{c}v_{t})^{n}$
$\displaystyle\leq\operatorname{\mathbf{1}}_{\\{\varphi_{1}+t\leq\psi_{2}\\}}(\theta_{1}+dd^{c}\varphi_{1})^{n}+\operatorname{\mathbf{1}}_{\\{\varphi_{t}=\psi_{2}\\}}(\theta_{2}+dd^{c}\psi_{2})^{n}$
$\displaystyle\leq\operatorname{\mathbf{1}}_{\\{\varphi_{1}\leq\psi_{2}-t\\}}(\theta_{1}+dd^{c}\varphi_{1})^{n}+(\theta_{2}+dd^{c}\psi_{2})^{n}.$

Integrating over $X$ and letting $t\to+\infty$, we obtain
$\underline{\mathrm{vol}}(T_{1})\leq\int_{X}(\theta_{2}+dd^{c}\psi_{2})^{n}$.
Taking the infimum over $\psi_{2}$ we obtain the first inequality.

Next pick $\psi_{1}\in\operatorname{PSH}(\theta_{1})$ such that
$\psi_{1}=\varphi_{1}+O(1)$ is bounded, and set

$\psi_{t}:=\max\\{\psi_{1},\varphi_{2}-t\\}$

for $t>0$. Then $\psi_{t}\in\operatorname{PSH}(\theta_{2})$,
$\psi_{t}=\varphi_{2}+O(1)$, and $\psi_{t}\searrow\psi_{1}$ as $t\to+\infty$.
By lower-semicontinuity of non-pluripolar masses (see Proposition 2.5), we
infer

$\int_{X}(\theta_{2}+dd^{c}\psi_{1})^{n}\leq\liminf_{t\to+\infty}\int_{X}(\theta_{2}+dd^{c}\psi_{t})^{n}\leq\overline{\mathrm{vol}}(T_{2})$

Taking supremum over all $\psi_{1}$ concludes the proof. ∎

###  3.2. The lower and upper volume of a $dd^{c}$-class

####  3.2.1. The case of a big $dd^{c}$-class

Recall from §1.4 that a $dd^{c}$-class
$\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is big if it contains a
strictly positive current, which can further be chosen to have analytic
singularities (see Theorem 1.8).

######  Definition 3.8.

The _lower volume_ and _upper volume_ of a big $dd^{c}$-class
$\\{\theta\\}\in\mathrm{BC}^{1,1}(X)$ are respectively defined as

$\underline{\mathrm{vol}}(\\{\theta\\}):=\sup_{0\leq
T\in\\{\theta\\}}\underline{\mathrm{vol}}(T),\quad\overline{\mathrm{vol}}(\\{\theta\\}):=\sup_{0\leq
T\in\\{\theta\\}}\overline{\mathrm{vol}}(T).$ (3.4)

where $T$ ranges over all positive currents in $\\{\theta\\}$.

Note that

$0\leq\underline{\mathrm{vol}}(\\{\theta\\})\leq\overline{\mathrm{vol}}(\\{\theta\\})=\sup_{\varphi\in\operatorname{PSH}(\theta)}\int_{X}(\theta+dd^{c}\varphi)^{n}<\infty,$

see (3.1) and Lemma 2.4. By (3.2), we have

$\\{\theta\\}\text{ big
}\Longrightarrow\overline{\mathrm{vol}}(\\{\theta\\})>0,$

see §3.2.3 for a discussion of the analogous property for
$\underline{\mathrm{vol}}$.

######  Lemma 3.9.

For any positive current $T_{\min}\in\\{\theta\\}$ with minimal singularities,
we have

$\underline{\mathrm{vol}}(\\{\theta\\})=\underline{\mathrm{vol}}(T_{\min}),\quad\overline{\mathrm{vol}}(\\{\theta\\})=\overline{\mathrm{vol}}(T_{\min}).$

If $\theta$ is further closed, then

$\underline{\mathrm{vol}}(\\{\theta\\})=\overline{\mathrm{vol}}(\\{\theta\\})=\int_{X}T_{\min}^{n}.$

The last point ensures compatibility of Definition 3.8 with [Bou02, BEGZ10]
for usual (closed) Bott–Chern classes $\\{\theta\\}\in
H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})$.

###### Proof.

By Proposition 3.7, the upper and lower volume of $T\in\\{\theta\\}$ are both
monotone increasing functions of the singularity class. This proves the first
point, and the second one now follows from Theorem 2.8 (see Example 3.2). ∎

######  Lemma 3.10.

For any big class $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$, we have

$\underline{\mathrm{vol}}(\\{\theta\\})=\sup_{T}\underline{\mathrm{vol}}(T),\quad\overline{\mathrm{vol}}(\\{\theta\\})=\sup_{T}\overline{\mathrm{vol}}(T)$

where $T$ ranges over all positive currents in $\\{\theta\\}$ with analytic
singularities.

###### Proof.

Pick $\varphi_{\min}\in\operatorname{PSH}(\theta)$ with minimal singularities,
and $\rho\in\operatorname{PSH}(\theta)$ with analytic singularities such that
$\theta+dd^{c}\rho\geq\omega_{X}$ (after perhaps scaling $\omega_{X}$). By
Theorem 1.8, there exists a sequence
$\varphi_{j}\in\operatorname{PSH}(\theta+\varepsilon_{j}\omega_{X})$ with
analytic singularities decreasing to $\varphi_{\min}$, where
$\varepsilon_{j}\searrow 0_{+}$. Set
$\psi_{j}:=(1+\varepsilon_{j})^{-1}(\varphi_{j}+\varepsilon_{j}\rho)$ and
$T_{j}:=\theta+dd^{c}\psi_{j}$. Then $0\leq T_{j}\in\\{\theta\\}$ has analytic
singularities, and

$\theta+\varepsilon_{j}\omega_{X}+dd^{c}\varphi_{j}\leq(1+\varepsilon_{j})T_{j}$

By Proposition 3.6 and Proposition 3.7, this yields

$\underline{\mathrm{vol}}(\\{\theta\\})=\underline{\mathrm{vol}}(\theta+dd^{c}\varphi_{\min})\leq\underline{\mathrm{vol}}(\theta+\varepsilon_{j}\omega_{X}+dd^{c}\varphi_{j})\leq(1+\varepsilon_{j})^{n}\underline{\mathrm{vol}}(T_{j}).$

Thus
$\limsup_{j}\underline{\mathrm{vol}}(T_{j})\geq\underline{\mathrm{vol}}(\\{\theta\\})$,
and similarly for $\overline{\mathrm{vol}}$. ∎

######  Proposition 3.11.

For all big classes
$\\{\theta\\},\\{\theta^{\prime}\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$, we
have

$\theta\leq\theta^{\prime}\Longrightarrow\underline{\mathrm{vol}}(\\{\theta\\})\leq\underline{\mathrm{vol}}(\\{\theta^{\prime}\\}),\quad\overline{\mathrm{vol}}(\\{\theta\\})\leq\overline{\mathrm{vol}}(\\{\theta^{\prime}\\}).$

Furthermore, the upper and lower volume functions are continuous on the big
cone of $\mathrm{BC}^{1,1}(X,\mathbb{R})$.

###### Proof.

The first point follows from Proposition 3.7. Pick a convergence sequence of
big classes $\\{\theta_{j}\\}\to\\{\theta\\}$, so that $\theta_{j}\to\theta$
smoothly for some choice of representatives. Then
$-\varepsilon_{j}\omega_{X}\leq\theta_{j}-\theta\leq\varepsilon_{j}\omega_{X}$
with $\varepsilon_{j}\to 0$. By the first point, it therefore suffices to show

$\underline{\mathrm{vol}}(\\{\theta\pm\varepsilon\omega_{X}\\})\to\underline{\mathrm{vol}}(\\{\theta\\}),\quad\overline{\mathrm{vol}}(\\{\theta\pm\varepsilon\omega_{X}\\})\to\overline{\mathrm{vol}}(\\{\theta\\})$

as $\varepsilon\to 0$. To see this, pick a strictly positive current with
analytic singularities $S\in\\{\theta\\}$. We may assume $S\geq\omega_{X}$.
For any $0<\varepsilon<1$ and $T\in\\{\theta+\varepsilon\omega_{X}\\}$ with
analytic singularities,

$T_{\varepsilon}:=(1-\varepsilon)(T-\varepsilon\omega_{X})+\varepsilon
S=(1-\varepsilon)T+\varepsilon(S-\omega_{X})+\varepsilon^{2}\omega_{X}$

is then a positive current with analytic singularities in $\\{\theta\\}$.
Since $T_{\varepsilon}\geq(1-\varepsilon)T$, Proposition 3.6 yields
$\underline{\mathrm{vol}}(\\{\theta\\})\geq\underline{\mathrm{vol}}(T_{\varepsilon})\geq(1-\varepsilon)^{n}\underline{\mathrm{vol}}(T)$.
Taking the supremum over $T$ we get

$\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})\geq\underline{\mathrm{vol}}(\\{\theta\\})\geq(1-\varepsilon)^{n}\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\}),$

by Lemma 3.10, and hence
$\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})\to\underline{\mathrm{vol}}(\\{\theta\\})$.
To prove
$\underline{\mathrm{vol}}(\\{\theta-\varepsilon\omega_{X}\\})\to\underline{\mathrm{vol}}(\\{\theta\\})$
we pick a positive current with analytic singularities $T\in\\{\theta\\}$.
Then

$S_{\varepsilon}:=(1-\varepsilon)T+\varepsilon
S-\varepsilon\omega_{X}\geq(1-\varepsilon)T\geq 0$

is a positive current with analytic singularities in
$\\{\theta-\varepsilon\omega_{X}\\}$. By Proposition 3.6, we infer
$\underline{\mathrm{vol}}(\\{\theta-\varepsilon\omega_{X}\\})\geq(1-\varepsilon)^{n}\underline{\mathrm{vol}}(T)$.
Taking the supremum over $T$ yields

$\underline{\mathrm{vol}}(\\{\theta\\})\geq\underline{\mathrm{vol}}(\\{\theta-\varepsilon\omega_{X}\\})\geq(1-\varepsilon)^{n}\underline{\mathrm{vol}}(\\{\theta\\}),$

thus
$\underline{\mathrm{vol}}(\\{\theta-\varepsilon\omega_{X}\\})\to\underline{\mathrm{vol}}(\\{\theta\\})$.
The proof of
$\overline{\mathrm{vol}}(\\{\theta\pm\varepsilon\omega_{X}\\})\to\overline{\mathrm{vol}}(\\{\theta\\})$
is similar. ∎

We conclude this section with the following Fujita-type approximation result
(compare [Bou02, Theorem 1.4]).

######  Proposition 3.12.

For any big class $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ we have

$\underline{\mathrm{vol}}(\\{\theta\\})=\sup_{Y}\underline{\mathrm{vol}}(\omega_{Y}),\quad\overline{\mathrm{vol}}(\\{\theta\\})=\sup_{Y}\overline{\mathrm{vol}}(\omega_{Y})$

where the suprema range over all modifications $\pi\colon Y\to X$ and
decompositions

$\\{\pi^{\star}\theta\\}=\\{\omega_{Y}\\}+\\{E\\}$

with $\omega_{Y}>0$ smooth and $E$ an effective $\mathbb{Q}$-divisor.

###### Proof.

Pick a modification $\pi\colon Y\to X$ and a decomposition

$\\{\pi^{\star}\theta\\}=\\{\omega_{Y}\\}+\\{E\\}$

with $\omega_{Y}>0$ smooth and $E$ an effective $\mathbb{Q}$-divisor. Then
Proposition 3.14 yields

$\underline{\mathrm{vol}}(\omega_{Y})=\underline{\mathrm{vol}}(\\{\omega_{Y}\\})\leq\underline{\mathrm{vol}}(\\{\pi^{\star}\theta\\})=\underline{\mathrm{vol}}(\\{\theta\\}),$

where the last equality holds by Proposition 3.3 (iv).

Conversely, pick $\varepsilon>0$. By the first part of the proof, we can find
$0\leq T\in\\{\theta\\}$ with analytic singularities such that
$\operatorname{vol}(\\{\theta\\})<\underline{\mathrm{vol}}(T)+\varepsilon$ and
$T\geq\omega_{X}$, after rescaling $\omega_{X}$. Pick a modification
$\pi\colon Y\to X$ such that $\pi^{\star}T$ has divisorial singularities, i.e.
$\pi^{\star}T=\beta+[E]$ with $\beta$ smooth and $E$ an effective
$\mathbb{Q}$-divisor. Then $T\geq\omega_{X}$ implies
$\pi^{\star}T\geq\pi^{\star}\omega_{X}$, and hence
$\beta\geq\pi^{\star}\omega_{X}$.

After passing to a higher modification, we may assume that $\pi$ is obtained
as a finite sequence of blowups with smooth centers, so that there exists a
$\pi$-exceptional $\mathbb{Q}$-divisor $D\geq 0$ on $Y$ such that $-D$ is
$\pi$-ample. We can then find a smooth closed $(1,1)$-form
$\theta_{D}\in\\{D\\}$ such that $\pi^{\star}\omega_{X}-c\theta_{D}>0$ for all
$c>0$ small enough. Then

$\\{\pi^{\star}\theta\\}=\\{\pi^{\star}T\\}=\\{\omega_{c}\\}+\\{E+cD\\},$

with $\omega_{c}:=\beta-c\theta_{D}\geq\pi^{\star}\omega_{X}-c\theta_{D}>0$
for $0<c\ll 1$. By Lemma 3.9, we further have
$\underline{\mathrm{vol}}(\omega_{c})=\underline{\mathrm{vol}}(\\{\omega_{c}\\})$.
By Proposition 3.11, this converges to
$\underline{\mathrm{vol}}(\\{\beta\\})=\underline{\mathrm{vol}}(\beta)=\underline{\mathrm{vol}}(T)$
as $c\to 0$, and hence
$\underline{\mathrm{vol}}(\\{\theta\\})<\underline{\mathrm{vol}}(\omega_{c})+\varepsilon$
for $c$ small enough. The result for $\underline{\mathrm{vol}}$ follows. The
same argument applies for the upper volume $\overline{\mathrm{vol}}$,
finishing the proof. ∎

####  3.2.2. General $dd^{c}$-classes

Assume now that $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is psef. Then
$\\{\theta+\varepsilon\omega_{X}\\}$ is big for each $\varepsilon>0$, and its
lower and upper volumes are both monotone increasing functions of
$\varepsilon$ (see Proposition 3.11). We may thus introduce:

######  Definition 3.13.

The _lower volume_ and _upper volume_ of an arbitrary class
$\\{\theta\\}\in\mathrm{BC}^{1,1}(X)$ are respectively defined by setting

$\underline{\mathrm{vol}}(\\{\theta\\}):=\inf_{\varepsilon>0}\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})=\lim_{\varepsilon\to
0}\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\}),$ (3.5)
$\overline{\mathrm{vol}}(\\{\theta\\}):=\inf_{\varepsilon>0}\overline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})=\lim_{\varepsilon\to
0}\overline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})$ (3.6)

if $\\{\theta\\}$ is psef, and
$\underline{\mathrm{vol}}(\\{\theta\\})=\overline{\mathrm{vol}}(\\{\theta\\}):=0$
otherwise.

Proposition 3.11 implies that the definition is independent of the choice of
$\omega_{X}$, and is compatible with the big case.

######  Proposition 3.14.

Pick $\\{\theta\\}\in\mathrm{BC}^{1,1}(X)$. Then:

  * (i)

for any modification $\pi\colon Y\to X$, we have

$\underline{\mathrm{vol}}(\\{\pi^{\star}\theta\\})=\underline{\mathrm{vol}}(\\{\theta\\}),\quad\overline{\mathrm{vol}}(\\{\pi^{\star}\theta\\})=\overline{\mathrm{vol}}(\\{\theta\\});$

  * (ii)

for any $\\{\theta^{\prime}\\}\in\mathrm{BC}^{1,1}(X)$ such that
$\\{\theta^{\prime}\\}-\\{\theta\\}$ is psef, we have

$\underline{\mathrm{vol}}(\\{\theta^{\prime}\\}\geq\underline{\mathrm{vol}}(\\{\theta\\}),\quad\overline{\mathrm{vol}}(\\{\theta^{\prime}\\}\geq\overline{\mathrm{vol}}(\\{\theta\\});$

  * (iii)

if $\theta\geq 0$ then
$\underline{\mathrm{vol}}(\\{\theta\\})=\underline{\mathrm{vol}}(\theta)$.

###### Proof.

To prove (i), we may replace $\theta$ with $\theta+\varepsilon\omega_{X}$ and
assume that $\\{\theta\\}$ and $\\{\pi^{\star}\theta\\}$ are big. The result
is then a direct consequence of Proposition 3.3 (iv).

To prove (ii), we may assume that $\\{\theta\\}$ is psef, as the result is
otherwise trivial. By (3.5), we may further replace $\theta,\theta^{\prime}$
with $\theta+2\varepsilon\omega_{X},\theta^{\prime}+\varepsilon\omega_{X}$
with $\varepsilon>0$, and hence assume that
$\\{\theta\\},\\{\theta^{\prime}\\}$ and $\\{\theta^{\prime}-\theta\\}$ are
big. Pick a strictly positive current $S\in\\{\theta^{\prime}-\theta\\}$ with
analytic singularities. For each $0\leq T\in\\{\theta\\}$ with analytic
singularities, $0\leq T+S\in\\{\theta^{\prime}\\}$ has analytic singularities
as well. By Proposition 3.6, we infer
$\underline{\mathrm{vol}}(\\{\theta^{\prime}\\})\geq\underline{\mathrm{vol}}(T+S)\geq\underline{\mathrm{vol}}(T)$,
and taking the supremum over $T$ yields
$\underline{\mathrm{vol}}(\\{\theta^{\prime}\\})\geq\underline{\mathrm{vol}}(\\{\theta\\})$.
The same argument applies to $\overline{\mathrm{vol}}$, proving (ii).

Next finally $\theta\geq 0$. When $\\{\theta\\}$ is big, the equality
$\underline{\mathrm{vol}}(\\{\theta\\})=\underline{\mathrm{vol}}(\theta)$
follows from Lemma 3.9. In the general case, we thus have
$\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})=\underline{\mathrm{vol}}(\theta+\varepsilon\omega_{X})$
for $\varepsilon>0$. Now
$\underline{\mathrm{vol}}(\theta)\leq\underline{\mathrm{vol}}(\theta+\varepsilon\omega_{X})\leq\underline{\mathrm{vol}}(\theta)+O(\varepsilon)$
by Proposition 3.6 and (3.3), hence
$\underline{\mathrm{vol}}(\theta+\varepsilon\omega_{X})\to\underline{\mathrm{vol}}(\theta)$,
and (iii) follows. ∎

######  Remark 3.15.

When $\\{\theta\\}$ is further assumed to be big, the analogue of (iii) also
holds for $\overline{\mathrm{vol}}$, by Lemma 3.9. However, we do not know
whether this extends to the psef case, as $\lim_{\varepsilon\to
0}\overline{\mathrm{vol}}(\theta+\varepsilon\omega_{X})=\overline{\mathrm{vol}}(\theta)$
is then unclear.

######  Proposition 3.16.

The lower and upper volume functions coincide on the subspace

$H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})\subset\mathrm{BC}^{1,1}(X,\mathbb{R})$

of usual (closed) Bott–Chern classes. If $\\{\theta\\}\in
H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})$ is further nef, then

$\underline{\mathrm{vol}}(\\{\theta\\})=\overline{\mathrm{vol}}(\\{\theta\\})=\int_{X}\theta^{n}.$

We simply denote by

$\operatorname{vol}\colon
H^{1,1}_{\mathrm{BC}}(X,\mathbb{R})\to\mathbb{R}_{\geq 0}$

the common restriction of $\underline{\mathrm{vol}},\overline{\mathrm{vol}}$.

###### Proof.

Assume $\theta$ is closed. If $\\{\theta\\}$ is not psef, then
$\underline{\mathrm{vol}}(\\{\theta\\})=\overline{\mathrm{vol}}(\\{\theta\\})=0$
by definition. Assume that $\\{\theta\\}$ is psef, and pick $0\leq
T_{\varepsilon}\in\\{\theta+\varepsilon\omega_{X}\\}$ with minimal
singularities for each $\varepsilon>0$. Since $\theta$ is closed, we have
$\Delta_{\theta}=0$ (see (2.7)), and hence
$\Delta_{\theta+\varepsilon\omega_{X}}=O(\varepsilon)$. By Theorem 2.8, we
infer
$\overline{\mathrm{vol}}(T_{\varepsilon})\leq\underline{\mathrm{vol}}(T_{\varepsilon})+O(\varepsilon)$,
i.e.

$\overline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})\leq\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})+O(\varepsilon),$

by Lemma 3.9. Letting $\varepsilon\to 0$ proves
$\underline{\mathrm{vol}}(\\{\theta\\})=\overline{\mathrm{vol}}(\\{\theta\\})$.

Assume next $\\{\theta\\}$ is further nef. Rescaling $\omega_{X}$ we can
assume $\theta\leq\omega_{X}$. For each smooth function
$\varphi\in\operatorname{PSH}(\theta+\varepsilon\omega_{X})\subset\operatorname{PSH}(X,2\omega_{X})$,
the bounded mass property and Stokes’ theorem yield

$\displaystyle\int_{X}(\theta+\varepsilon\omega_{X}+dd^{c}\varphi)^{n}$
$\displaystyle=\int_{X}(\theta+dd^{c}\varphi)^{n}+\sum_{k=1}^{n}\binom{n}{k}\varepsilon^{k}\int_{X}(\theta+dd^{c}\varphi)^{n-k}\wedge\omega_{X}^{k}$
$\displaystyle=\int_{X}\theta^{n}+\sum_{k=1}^{n}\binom{n}{k}\varepsilon^{k}\int_{X}(\theta-2\omega_{X}+2\omega_{X}+dd^{c}\varphi)^{n-k}\wedge\omega_{X}^{k}$
$\displaystyle=\int_{X}\theta^{n}+O(\varepsilon).$

Taking the infimum over all $\varphi$ yields
$\underline{\mathrm{vol}}(\\{\theta+\varepsilon\omega_{X}\\})=\int_{X}\theta^{n}+O(\varepsilon),$
hence $\underline{\mathrm{vol}}(\\{\theta\\})=\int_{X}\theta^{n}$. ∎

####  3.2.3. The positive volume property

As a consequence of Proposition 3.14 we have:

######  Corollary 3.17.

We either have $\underline{\mathrm{vol}}\equiv 0$ on
$\mathrm{BC}^{1,1}(X,\mathbb{R})$, or $\underline{\mathrm{vol}}>0$ on the big
cone.

###### Proof.

Assume there exists $\\{\theta_{0}\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ with
$\underline{\mathrm{vol}}(\\{\theta_{0}\\})>0$. For any big class
$\\{\theta\\}$, $t\\{\theta\\}-\\{\theta_{0}\\}$ is big for $t\gg 1$, and
hence
$t^{n}\underline{\mathrm{vol}}(\\{\theta\\})\geq\underline{\mathrm{vol}}(\\{\theta_{0}\\})>0$,
by Proposition 3.14. ∎

######  Definition 3.18.

We say that $X$ has the _positive volume property_ if
$\underline{\mathrm{vol}}>0$ on the big cone of
$\mathrm{BC}^{1,1}(X,\mathbb{R})$.

By Proposition 3.14 and Corollary 3.17, the positive volume property holds iff
$\underline{\mathrm{vol}}(\omega)>0$ for some Hermitian form $\omega>0$ on
$X$.

Like the bounded mass property, the positive volume property is
bimeromorphically invariant (by Proposition 3.14, or [GL22, Theorem 3.7]). It
always holds in dimension $n=2$; when $n=3$ it holds for Vaisman manifolds
[AGL23, Proposition 3.10], as well as for some nilmanifolds [AGL23, Theorem
4.4] and solvmanifolds [AGL23, Theorem 4.7]. No counterexample is known at the
moment of this writing.

Recall that the bounded mass property is inherited by any submanifold. Here we
establish a ‘dual’ result for the positive volume property:

######  Proposition 3.19.

Assume $\pi\colon X\to Y$ is a holomorphic submersion onto a compact complex
manifold. If $X$ has the positive volume property, then so does $Y$.

###### Proof.

Set $d:=\dim Y$, and pick a hermitian form $\beta_{X}$ on $X$. We claim that
there exists a semipositive $(1,1)$-form $\theta\geq 0$ on $X$ that is
(strictly) positive along the fibers of $\pi$, and satisfies
$\theta^{n-d+1}=0$. Indeed, pick a $\mathcal{C}^{\infty}$-projection $p\colon
T_{X}\to T_{X/Y}$ onto the holomorphic subbundle $T_{X/Y}=\ker
d\pi\hookrightarrow T_{X}$. The claim then holds with
$\theta(v,w):=\beta_{X}(p(v),p(w))$. Given a Hermitian form $\omega_{Y}>0$ on
$Y$, we then have $\omega_{X}:=\pi^{\star}\omega_{Y}+\theta>0$. For any smooth
$\varphi\in\operatorname{PSH}(\omega_{Y})$, we further have

$(\omega_{X}+dd^{c}\pi^{\star}\varphi)^{n}=(\pi^{\star}(\omega_{Y}+dd^{c}\varphi)+\theta)^{n}={n\choose
d}\pi^{\star}(\omega_{Y}+dd^{c}\varphi)^{d}\wedge\theta^{n-d},$

and hence

$0<\underline{\mathrm{vol}}(\omega_{X})\leq{n\choose
d}\int_{Y}f(\omega_{Y}+dd^{c}\varphi)^{d}\leq
C\int_{Y}(\omega_{Y}+dd^{c}\varphi)^{d},$

with $0<f:=\pi_{\star}\theta^{n-d}\in C^{\infty}(Y)$ (see for instance [Dem,
Section 2.15]). We infer $\underline{\mathrm{vol}}(\omega_{Y}))>0$, and the
result follows. ∎

###  3.3. Characterization of the Fujiki class

Generalizing [Bou02, Corollary 7.11] (which dealt with closed classes on a
Kähler manifold), we next show:

######  Theorem 3.20.

The lower volume function
$\underline{\mathrm{vol}}\colon\mathrm{BC}^{1,1}(X,\mathbb{R})\to\mathbb{R}_{\geq
0}$ is continuous. In particular, it vanishes outside of the big cone.

In other words, the last point states that
$\underline{\mathrm{vol}}(\\{\theta\\})>0$ implies $\\{\theta\\}$ is big. This
extends [GL22, Theorem 4.6], which dealt with nef classes. Note that the
analogue of Theorem 3.20 fails for the upper volume:

######  Example 3.21.

By [GL22, Example 3.5], any product $X=C\times C^{\prime}$ of two curves
carries a semipositive $(1,1)$-form $\theta$ such that
$\int_{X}\theta^{2}>\underline{\mathrm{vol}}(\\{\theta\\})=0$. Since $X$ has
the positive volume property (in fact $X$ is Kähler, hence
$\underline{\mathrm{vol}}(\omega_{X})>0$) $\\{\theta\\}$ is therefore not big,
while $\overline{\mathrm{vol}}(\\{\theta\\})\geq\int_{X}\theta^{2}>0$.

###### Proof of Theorem 3.20.

The second point formally follows from the first one, since any $\\{\theta\\}$
on the boundary of the psef cone satisfies
$\underline{\mathrm{vol}}(\\{\theta-\varepsilon\omega_{X}\\})=0$ for
$\varepsilon>0$ (see Definition 3.13. However, we actually first show the
former, by adapting (as in [GL22, Theorem 4.6]) a strategy due to Chiose
[Chi13, Theorem 3.1].

Assume thus that $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ satisfies
$\underline{\mathrm{vol}}(\\{\theta\\})>0$. By Definition 3.13, $\\{\theta\\}$
is necessarily psef, and we need to show that it is big, i.e. that
$\\{\theta-\delta\omega_{X}\\}$ is psef for some $\delta>0$. By Hahn–Banach
(see Lemma 1.17), this is equivalent to showing
$\int_{X}\theta\wedge\gamma\geq\delta$ for each $dd^{c}$-closed
$(n-1,n-1)$-form $\gamma>0$, normalized so that the volume form

$\mu:=\omega_{X}\wedge\gamma$

has mass $1$. Applying Proposition 3.12 to the big classes
$\\{\theta+\varepsilon\omega_{X}\\}$, $\varepsilon>0$, yields a sequence of
modifications $\pi_{j}\colon Y_{j}\to X$ and decompositions

$\pi_{j}^{\star}\\{\theta+\varepsilon_{j}\omega_{X}\\}=\\{\omega_{j}\\}+\\{E_{j}\\}$

where $\omega_{j}$ a Hermitian form on $Y_{j}$, $E_{j}$ an effective
$\mathbb{Q}$-divisor, $\varepsilon_{j}\to 0$ and
$\underline{\mathrm{vol}}(\omega_{j})\to\underline{\mathrm{vol}}(\\{\theta\\})$.

By [GL23, Theorem 4.1], for each $j$ there exists a positive current $0\leq
S_{j}\in\\{\omega_{j}\\}$ with bounded potentials, smooth outside the
exceptional locus of $\pi_{j}$, such that

$S_{j}^{n}=c_{j}\pi_{j}^{*}\mu$

for some constant $c_{j}>0$. Note that
$c_{j}=\int_{X}S_{j}^{n}\geq\underline{\mathrm{vol}}(\omega_{j})$, and hence

$\liminf_{j}c_{j}\geq\underline{\mathrm{vol}}(\\{\theta\\}).$ (3.7)

Denote by $T_{j}\in\\{\theta+\varepsilon_{j}\omega_{X}\\}$ the unique positive
current such that $\pi_{j}^{\star}T_{j}=S_{j}+[E_{j}]$. Then $T_{j}$ is smooth
on a nonempty Zariski open set $\Omega_{j}\subset X$, and satisfies
$T_{j}^{n}=c_{j}\mu$.

By Lemma 2.4 we have $\int_{X}T_{j}^{n-1}\wedge\omega_{X}\leq M$ for a
constant $M>0$ only depending on $\\{\theta\\}$ and $\\{\omega_{X}\\}$.
Introduce the set

$A_{j}:=\left\\{x\in\Omega_{j}\mid T_{j}^{n-1}\wedge\omega_{X}\geq 2M\mu\text{
at }x\right\\},$

and note that

$\mu(A_{j})\leq\frac{1}{2M}\int_{A_{j}}T_{j}^{n-1}\wedge\omega_{X}\leq\frac{1}{2},$

and hence $\mu(\Omega_{j}\setminus A_{j})\geq\tfrac{1}{2}$. At each point of
$\Omega_{j}\setminus A_{j}$ we have

$T_{j}^{n-1}\wedge\omega_{X}\leq 2M\mu=\frac{2M}{c_{j}}T_{j}^{n},$

which implies $T_{j}\geq\frac{c_{j}}{2nM}\omega_{X}$, and hence
$T_{j}\wedge\gamma\geq\frac{c_{j}}{2nM}\mu$. Since $\gamma$ is
$dd^{c}$-closed, we infer

$\int_{X}(\theta+\varepsilon_{j}\omega_{X})\wedge\gamma=\int_{X}T_{j}\wedge\gamma\geq\frac{c_{j}}{2nM}\mu(\Omega_{j}\setminus
A_{j})\geq\frac{c_{j}}{4nM},$

and (3.7) yields the desired estimate

$\int_{X}\theta\wedge\gamma\geq\delta:=\frac{\underline{\mathrm{vol}}(\\{\theta\\})}{4nM}>0.$

We now show that the lower volume function is continuous on
$\mathrm{BC}^{1,1}(X,\mathbb{R})$. By Proposition 3.11, it is continuous on
the big cone, and it is identically zero outside the psef cone. Assume
$\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$ is psef but not big. By the
first part of the proof, $\underline{\mathrm{vol}}(\\{\theta\\})=0$, and we
thus need to show $\underline{\mathrm{vol}}(\\{\theta_{j}\\})\to 0$ for any
smoothly convergent sequence $\theta_{j}\to\theta$. We have
$\theta_{j}\leq\theta+\varepsilon_{j}\omega_{X}$ with $\varepsilon_{j}\to 0$,
and hence
$\underline{\mathrm{vol}}(\\{\theta_{j}\\})\leq\underline{\mathrm{vol}}(\\{\theta+\varepsilon_{j}\omega_{X}\\})$,
by Proposition 3.14. By definition,
$\underline{\mathrm{vol}}(\\{\theta+\varepsilon_{j}\omega_{X}\\})$ converges
to $\underline{\mathrm{vol}}(\\{\theta\\})=0$, and we are done. ∎

######  Remark 3.22.

The above proof yields, more precisely, that $\\{\theta\\}$ is ‘more psef’
than $\delta\\{\omega_{X}\\}$ for
$\delta:=\underline{\mathrm{vol}}(\\{\theta\\})/4nM$ where

$M:=\sup\left\\{\int_{X}T^{n-1}\wedge\omega_{X}\mid 0\leq
T\in\\{\theta\\}\right\\}.$

A simple consequence is the following interesting characterization of the
Fujiki class, which establishes a transcendental version of the
Grauert–Riemenschneider conjecture:

######  Corollary 3.23.

Let $X$ be a compact complex manifold $X$ of dimension $n$. The following
properties are equivalent:

  * (i)

$X$ is a Fujiki manifold, i.e. bimeromorphic to a compact Kähler manifold;

  * (ii)

$X$ has the bounded mass property and it admits a closed positive
$(1,1)$-current $T$ such that $\int_{X}T^{n}>0$.

###### Proof.

The implication (i)$\Rightarrow$ (ii) follows from Proposition 1.20. The
existence of a closed positive $(1,1)$-current $T=\theta+dd^{c}\varphi$ such
that $\int_{X}T^{n}>0$ is equivalent to $\operatorname{vol}(\\{\theta\\})>0$,
by definition and Example 3.2. By Theorem 3.20, this implies that
$\\{\theta\\}$ is big, and it then well-known (see for instance Proposition
3.12) that $X$ admits a Kähler modification, and hence is Fujiki. ∎

##  4\. Monge-Ampère equations in big classes

In this section, we no longer asssume that $X$ has the bounded mass property.
We pick a big class $\\{\theta\\}\in\mathrm{BC}^{1,1}(X,\mathbb{R})$, fix a
$\theta$-psh function $\rho\leq 0$ with analytic singularities such that
$\theta+dd^{c}\rho\geq\omega_{X}$ (after rescaling $\omega_{X}$), and denote
by $\Omega\subset X$ the Zariski open set where $\rho$ is smooth. We also fix
a (smooth, positive) volume form $dV$ on $X$, and denote by

$\|f\|_{p}:=\left(\int_{X}|f|^{p}dV\right)^{1/p}$

the $L^{p}$-norm of a measurable function $f$. We are going to solve complex
Monge–Ampère equations in $\Omega$, extending [BEGZ10, Theorem B] to the
Hermitian setting.

###  4.1. The domination principle

We start by establishing a useful local maximum principle.

######  Lemma 4.1.

Let $U\Subset\Omega$ be an open subset that is biholomorphic to a ball in
$\mathbb{C}^{n}$. Let $u,v$ be bounded $\theta$-psh functions in $U$ such that
$\liminf_{z\to\partial U}(u-v)(z)\geq 0$. If
$(\theta+dd^{c}u)^{n}\leq(\theta+dd^{c}v)^{n}$ in $\\{u<v\\}$, then $u\geq v$
in $U$.

###### Proof.

Fix $\varepsilon>0$ and set $v_{\varepsilon}=\max\\{u,v-\varepsilon\\}$. Then
$u=v_{\varepsilon}$ near the boundary of $U$, and the Bedford–Taylor maximum
principle (compare Proposition 2.11) thus yields

$(\theta+dd^{c}v_{\varepsilon})^{n}\geq\operatorname{\mathbf{1}}_{\\{v\geq
u+\varepsilon\\}}(\theta+dd^{c}v)^{n}+\operatorname{\mathbf{1}}_{\\{u>v-\varepsilon\\}}(\theta+dd^{c}u)^{n}$

on $U$. Since $(\theta+dd^{c}v)^{n}\geq(\theta+dd^{c}u)^{n}$ on $\\{v\geq
u+\varepsilon\\}\subset\\{v>u\\}$, we infer

$(\theta+dd^{c}u)^{n}\leq(\theta+dd^{c}v)^{n}.$

Let $\varphi:=\operatorname{P}^{\star}(u-v_{\varepsilon})$ denote the largest
psh function on $U$ lying below $u-v_{\varepsilon}$ outside a pluripolar set,
and let $h$ be a defining psh function for $U$: $h=0$ on $\partial U$ and
$h<0$ in $U$. Since $u-v_{\varepsilon}\leq 0$ with equality near $\partial U$,
we can find a large constant $A>0$ such that $Ah\leq u-v_{\varepsilon}$. Thus
$Ah\leq\varphi$, hence $\varphi$ vanishes on $\partial U$. Since
$\varphi+v_{\varepsilon}\leq u$, and the two functions are $\theta$-psh, by
the maximum principle we have

${\bf 1}_{D}(\theta+dd^{c}\varphi+dd^{c}v_{\varepsilon})^{n}\leq{\bf
1}_{D}(\theta+dd^{c}u)^{n},$

where $D=\\{\varphi+v_{\varepsilon}=u\\}$ is the contact set. Using

$(\theta+dd^{c}u)^{n}\leq(\theta+dd^{c}v_{\varepsilon})^{n}\leq(\theta+dd^{c}\varphi+dd^{c}v_{\varepsilon})^{n},$

we obtain ${\bf 1}_{D}(dd^{c}\varphi)^{n}=0$. Since $(dd^{c}\varphi)^{n}$ is
supported on $D$, we infer that $(dd^{c}\varphi)^{n}=0$, and the
Bedford–Taylor comparison principle ensures that $\varphi=0$. Thus $u\geq
v_{\varepsilon}=\max\\{u,v-\varepsilon\\}$ outside a pluripolar set, and hence
everywhere, and letting $\varepsilon\to 0$ we conclude that $u\geq v$. ∎

######  Lemma 4.2.

Let $u,v$ be $\theta$-psh functions on $X$ such that $\rho\leq\min\\{u,v\\}$
and $v\leq u+C$ for some constant $C$. If

${\bf 1}_{\\{u<v\\}}(\theta+dd^{c}u)^{n}\leq c{\bf
1}_{\\{u<v\\}}(\theta+dd^{c}v)^{n},$

for some $c\in[0,1)$, then $u\geq v$.

This domination principle contains a great deal of information. While the
statement is similar to [GL23, Proposition 1.12], the proof we provide here is
very different.

###### Proof.

Fix $\varepsilon\in(0,1)$ such that $(1-\varepsilon)^{n}>c$ and set
$v_{\varepsilon}:=(1-\varepsilon)v+\varepsilon\rho$. Since $\rho$ satisfies
$\theta+dd^{c}\rho\geq\omega_{X}$, we have

$\theta+dd^{c}\lambda\rho=\lambda(\theta+dd^{c}\rho)+(1-\lambda)\theta\geq\omega_{X}+(1-\lambda)\theta.$

Thus, when $|\lambda-1|$ is sufficiently small we have
$\theta+dd^{c}\lambda\rho\geq\omega_{X}/2$. Replacing $\rho$ with
$\lambda\rho-1$ for some $\lambda>1$ we can assume

$\lim_{x\to\partial\Omega}(u-\rho)=+\infty\;\text{and}\;\rho-v\leq-1.$

We are going to show that $u\geq v_{\varepsilon}$; the result then follows by
letting $\varepsilon\to 0$. Assume by contradiction that

$\inf_{\Omega}(u-v_{\varepsilon})=m<0.$

Let $(x_{j})$ be a sequence in $\Omega$ converging to $x_{0}\in\Omega$ such
that $\lim_{j}(u-v_{\varepsilon})(x_{j})=m$. Take a holomorphic coordinate
chart $(B,z)$ around $x_{0}$ so that $B=\\{|z|<1\\}\Subset\Omega$, and define
$\eta=-a|z|^{2}+a,$ where the constant $a>0$ is so small that
$\frac{\omega_{X}}{2}+dd^{c}\eta\geq 0$. Setting

$\varphi:=v_{\varepsilon}+\varepsilon\eta+m,$

we get $\liminf_{x\to\partial B}(u-\varphi)\geq 0$,

$\theta+dd^{c}\varphi=(1-\varepsilon)(\theta+dd^{c}v)+\varepsilon(\theta+dd^{c}\rho+dd^{c}\eta)\geq
0$

and

$(\theta+dd^{c}u)^{n}\leq
c(\theta+dd^{c}v)^{n}\leq(1-\varepsilon)^{n}(\theta+dd^{c}v)^{n}\leq(\theta+dd^{c}\varphi)^{n}\;\text{on}\;\\{u<\varphi\\}$

because

$\\{u<\varphi\\}\subset\\{u<v+\varepsilon(\rho-v+\eta)+m\\}\subset\\{u<v+m\\}\subset\\{u<v\\},$

if $a$ is small enough. Lemma 4.1 yields $u\geq\varphi$, and evaluating at
$x_{j}$ we obtain

$u(x_{j})-v_{\varepsilon}(x_{j})\geq\varepsilon\eta(x_{j})+m,$

which yields a contradiction as $j\to+\infty$, since $\eta(x_{0})>0$. ∎

As a simple consequence, we get:

######  Corollary 4.3.

Assume $u,v\in\operatorname{PSH}(\theta)$ are such that $u-v$ is bounded,
$\min\\{u,v\\}\geq\rho$, and

$(\theta+dd^{c}u)^{n}\leq c(\theta+dd^{c}v)^{n},$

for some $c\geq 0$. Then $c\geq 1$.

###  4.2. Solving complex Monge–Ampère equations

A crucial step in solving complex Monge–Ampère equations is to establish
uniform a priori estimates. Following [GL23] we construct suitable
subsolutions to bound the solutions via the domination principle.

######  Lemma 4.4.

Let $\eta\in\operatorname{PSH}(X,\theta)$ such that $0\geq\eta\geq\rho$. Let
$g\geq 0$ be a measurable function on $X$ such that $\|g\|_{q}\leq 1$, for
some exponent $q>1$. Then there exists $v\in\operatorname{PSH}(X,\theta)$ with
$\eta\leq v\leq\eta+1$ such that

$(\theta+dd^{c}v)^{n}\geq mgdV$

on $\Omega$, where $m$ is a positive constant only depending on $q$,
$(X,\omega_{X})$, and $\theta$.

###### Proof.

Set $h:=g(2-\rho)^{2n}$. By Hölder’s inequality, we have $\|h\|_{r}\leq C$ for
some $r\in(1,q)$ and a uniform constant $C>0$, and [GL23, Lemma 3.3] thus
yields $u\in\operatorname{PSH}(X,\omega_{X})$ such that $-1\leq u\leq 0$, and

$(\omega_{X}+dd^{c}u)^{n}\geq mhdV$

with $m>0$ uniformly bounded away from $0$. Set $u_{1}:=\rho+u$. Since we
assume $\theta+dd^{c}\rho\geq\omega_{X}$, we get
$\theta+dd^{c}u_{1}\geq\omega_{X}+dd^{c}u\geq 0$, thus $u_{1}$ is $\theta$-psh
and satisfies $(\theta+dd^{c}u_{1})^{n}\geq mhdV$.

Consider now $v:=\eta+\frac{1}{1+\eta-u_{1}}$. Observe that $1\leq 1+\eta-
u_{1}=1-u+\eta-\rho\leq 2-\rho$ and

$\displaystyle\theta+dd^{c}v$ $\displaystyle=$
$\displaystyle\theta+dd^{c}\eta-\frac{dd^{c}(\eta-u_{1})}{(1+\eta-
u_{1})^{2}}+\frac{2d(\eta-u_{1})\wedge d^{c}(\eta-u_{1})}{(1+\eta-u_{1})^{3}}$
$\displaystyle\geq$ $\displaystyle\left[1-\frac{1}{(1+\eta-
u_{1})^{2}}\right](\theta+dd^{c}\eta)+\frac{\theta+dd^{c}u_{1}}{(1+\eta-
u_{1})^{2}}$ $\displaystyle\geq$
$\displaystyle\frac{\theta+dd^{c}u_{1}}{(2-\rho)^{2}}.$

The function $v$ is thus $\theta$-psh, with $\eta\leq v\leq\eta+1$ and

$(\theta+dd^{c}v)^{n}\geq\frac{(\theta+dd^{c}u_{1})^{n}}{(2-\rho)^{2n}}=mgdV.$

∎

We now show that one can solve Monge-Ampère equations with respect to a big
form $\theta$. We first start with the equation twisted by an exponential.

######  Theorem 4.5.

Assume $0\leq f\in L^{p}(X)$ with $p>1$ and $\|f\|_{p}>0$. For each
$\lambda>0$, we can then find a unique $\varphi_{\lambda}\in PSH(X,\theta)$
such that $\varphi_{\lambda}=V_{\theta}+O(1)$ and

$\left(\theta+dd^{c}\varphi_{\lambda}\right)^{n}=e^{\lambda\varphi_{\lambda}}fdV$

on $\Omega$. Furthermore, $\sup_{X}|\varphi_{\lambda}-V_{\theta}|\leq C$ for a
constant $C>0$ only depending on $(X,\omega_{X})$, $\theta$, $p$, $\|f\|_{p}$
and $\lambda$.

###### Proof.

Uniqueness follows from the domination principle. To simplify the notation, we
assume that $\lambda=1$. To prove the existence, we approximate $V_{\theta}$
by a quasi-decreasing sequence $(\psi_{j})$ of $\theta$-psh functions with
analytic singularities: we first apply Demailly’s approximation (Theorem 1.8)
to get a decreasing sequence
$\tilde{\psi}_{j}\in\operatorname{PSH}(X,\theta+2^{-j}\omega)$ converging to
$V_{\theta}$, and we then take

$\psi_{j}=(1-2^{-j})\tilde{\psi}_{j}+2^{-j}\rho.$

Observe that $\psi_{j}\geq(1-2^{-j})V_{\theta}+2^{-j}\rho\geq\rho,$ thus

$\operatorname{P}_{\theta}^{\star}\left(\inf_{k\geq
j}\psi_{k}\right)\geq(1-2^{-j})V_{\theta}+2^{-j}\rho\nearrow
V_{\theta}\;\text{as}\;j\to+\infty.$

Resolving the singularities of $\psi_{j}$ and arguing as in the proof of
Theorem 3.20, one deduces from [GL23, Theorem 3.4] the existence of
$u_{j}\in\operatorname{PSH}(X,\theta)$ such that $u_{j}=\psi_{j}+O(1)$ and

$(\theta+dd^{c}u_{j})^{n}=e^{u_{j}}fdV.$

By Lemma 4.4 there exists $v_{j}\in\operatorname{PSH}(X,\theta)$ with
$\psi_{j}-1\leq v_{j}\leq\psi_{j}\leq 0$ such that

$(\theta+dd^{c}v_{j})^{n}\geq mfdV\geq e^{v_{j}+\log m}fdV.$

The domination principle thus yields a uniform constant $C^{\prime}$ such that

$\psi_{j}-C^{\prime}\leq v_{j}-C^{\prime}+1\leq u_{j}.$

Extracting a subsequence, we can assume that $u_{j}\to
u\in\operatorname{PSH}(X,\theta)$ almost everywhere on $X$. The above
estimates yield uniform bounds $V_{\theta}-C^{\prime}\leq u\leq V_{\theta}$.

We next consider the sequences

$\varphi_{j}=\operatorname{P}_{\theta}^{\star}\left(\inf_{k\geq
j}u_{k}\right)\;\;\text{ and }\;\;\tau_{j}=\mathrm{sup}^{\star}_{k\geq
j}u_{k}.$

These functions are locally bounded in $\Omega$ since

$\min\\{\varphi_{j},\tau_{j}\\}\geq(1-2^{-j})V_{\theta}+2^{-j}\rho-C^{\prime}\geq\rho-C^{\prime}.$

Note that $\tau_{j}$ decreases pointwise to $u$, while $\varphi_{j}$ increases
(pointwise outside a pluripolar set) to some function
$\varphi\in\operatorname{PSH}(X,\theta)$. The maximum principle moreover
yields, in $\Omega$,

$(\theta+dd^{c}\varphi_{j})^{n}\leq e^{\varphi_{j}}fdV,$

and

$(\theta+dd^{c}\tau_{j})^{n}\geq e^{\tau_{j}}fdV.$

Letting $j\to+\infty$, we obtain $\varphi\leq u$, $\varphi=u+O(1)$, and

$(\theta+dd^{c}\varphi)^{n}\leq e^{\varphi}fdV,\;\;\text{ while
}\;\;(\theta+dd^{c}u)^{n}\geq e^{u}fdV.$

The domination principle ensures $\varphi=u$, hence
$(\theta+dd^{c}u)^{n}=e^{u}fdV.$

∎

######  Theorem 4.6.

Assume $0\leq f\in L^{p}(X)$ with $p>1$ and $\|f\|_{p}>0$. Then there exists
$(\varphi,c)\in\operatorname{PSH}(X,\theta)\times(0,+\infty)$ such that
$V_{\theta}-C\leq\varphi\leq V_{\theta}$ and

$(\theta+dd^{c}\varphi)^{n}=cfdV$

on $\Omega$, where

  * •

the constant $c>0$ is uniquely determined by $f,X,\theta$, and

  * •

$C>0$ is a uniform constant that only depends on $(X,\omega_{X})$, $\theta$,
$p$ and $\|f\|_{p}$.

When $X$ is Kähler and $\theta$ is closed, this is [BEGZ10, Theorem B]. When
$\theta>0$ is a Hermitian form, this result is a combination of [TW10] and
[DK12, KN15]. When the class $\\{\theta\\}\in BC^{1,1}(X)$ is nef, one can
further show the existence of a solution which is smooth in $\Omega$ (see
[GL23, Theorem B] and [Dang24, Theorem C]). The uniqueness of $\varphi$ up to
an additive constant is largely open (see however [KN19, Theorem A]).

###### Proof.

The uniqueness of the constant $c$ follows from the domination principle (see
Corollary 4.3). For each $j\in\mathbb{N}^{*}$ we use Theorem 4.5 to find
$u_{j}\in\operatorname{PSH}(X,\theta)$ such that $u_{j}-V_{\theta}$ is bounded
and

$(\theta+dd^{c}u_{j})^{n}=e^{j^{-1}u_{j}}fdV.$

Setting $v_{j}:=u_{j}-\sup_{X}u_{j}$ and $f_{j}:=e^{j^{-1}v_{j}}f$, the above
equation becomes

$(\theta+dd^{c}v_{j})^{n}=c_{j}f_{j}dV,\;\text{ where
}\;c_{j}=e^{j^{-1}\sup_{X}u_{j}}.$

Extracting and relabelling we can assume that $v_{j}\to
v\in\operatorname{PSH}(X,\theta)$ in $L^{1}$ and almost everywhere, hence
$e^{j^{-1}v_{j}}\to 1$ in $L^{1}(X,dV)$.

Pick a Gauduchon metric $\omega_{G}$ on $X$ and write $\omega_{G}^{n}=hdV$.
The mixed Monge-Ampère inequality gives

$(\theta+dd^{c}u_{j})\wedge\omega_{G}^{n-1}\geq
e^{j^{-1}v_{j}/n}c_{j}^{1/n}f^{1/n}h^{1-1/n}dV.$

Integrating this inequality over $X$ yields an upper bound for $c_{j}$.

Fix a small constant $\varepsilon>0$ such that $e^{-\varepsilon v_{j}}f$ is
uniformly in $L^{q}(X,dV)$ for some $1<q<p$. This can be done thanks to the
uniform version of Skoda’s integrability theorem (see [GZ], [Zer01]). Using
Lemma 4.4, we can find $\eta_{j}\in\operatorname{PSH}(X,\theta)$ with
$V_{\theta}-1\leq\eta_{j}\leq V_{\theta}$ such that

$(\theta+dd^{c}\eta_{j})^{n}\geq mfe^{-\varepsilon v_{j}}dV\geq
mfe^{\varepsilon(\eta_{j}-v_{j})}dV.$

The domination principle yields $m\leq c_{j}$ and $v_{j}\geq\eta_{j}+\log
m\geq V_{\theta}-C$, for a uniform constant $C>0$. Extracting a subsequence,
we can assume $c_{j}\to c>0$.

Consider the following sequences

$\varphi_{j}=\operatorname{P}_{\theta}^{\star}\left(\inf_{k\geq
j}v_{k}\right),\;\tau_{j}=\mathrm{sup}^{\star}_{k\geq j}v_{k}.$

Then
$\min\\{\varphi_{j},\tau_{j}\\}\geq(1-2^{-j})V_{\theta}+2^{-j}\rho\geq\rho$.
In particular, these functions are locally bounded in $\Omega$. By the maximum
principle and the minimum principle, we have

$(\theta+dd^{c}\varphi_{j})^{n}\leq e^{\varepsilon(\varphi_{j}-\inf_{k\geq
j}v_{k})}\sup_{k\geq j}(c_{k}f_{k})dV,$

and

$(\theta+dd^{c}\tau_{j})^{n}\geq e^{\varepsilon(\tau_{j}-\sup_{k\geq
j}v_{k})}\inf_{k\geq j}(c_{k}f_{k})dV,$

in $\Omega$. Letting $j\to+\infty$, we obtain $V_{\theta}-C\leq\varphi\leq
v\leq V_{\theta}$, while

$(\theta+dd^{c}\varphi)^{n}\leq e^{\varepsilon(\varphi-v)}cfdV\;\;\text{ and
}\;\;(\theta+dd^{c}v)^{n}\geq e^{\varepsilon(v-v)}cfdV.$

The domination principle thus gives $\varphi=v$ and
$(\theta+dd^{c}v)^{n}=cfdV$, finishing the proof. ∎

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