# Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes

Vincent Guedj & Chinh H. Lu  Institut de Mathématiques de Toulouse
Université de Toulouse
118 route de Narbonne
31400 Toulouse, France
<EMAIL_ADDRESS>https://www.math.univ-toulouse.fr/ guedj/
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405,
Orsay, France<EMAIL_ADDRESS>https://www.imo.universite-paris-saclay.fr/ lu/

###### Abstract.

In [GL21a] we have developed a new approach to $L^{\infty}$-a priori estimates
for degenerate complex Monge-Ampère equations, when the reference form is
closed. This simplifying assumption was used to ensure the constancy of the
volumes of Monge-Ampère measures.

We study here the way these volumes stay away from zero and infinity when the
reference form is no longer closed. We establish a transcendental version of
the Grauert-Riemenschneider conjecture, partially answering conjectures of
Demailly-Păun [DP04] and Boucksom-Demailly-Păun-Peternell [BDPP13].

Our approach relies on a fine use of quasi-plurisubharmonic envelopes. The
results obtained here will be used in [GL21b] for solving degenerate complex
Monge-Ampère equations on compact Hermitian varieties.

###### Key words and phrases:

Monge-Ampère equation, a priori estimates

###### 2010 Mathematics Subject Classification:

32W20, 32U05, 32Q15, 35A23

###### Contents

  1. 1 Non collapsing forms
    1. 1.1 Positivity properties
    2. 1.2 Comparison principle
  2. 2 Envelopes
    1. 2.1 Basic properties
    2. 2.2 Locally vs globally pluripolar sets
    3. 2.3 Domination principle
  3. 3 Bounds on Monge-Ampère masses
    1. 3.1 Global bounds
    2. 3.2 Bimeromophic invariance
  4. 4 Weak transcendental Morse inequalities
    1. 4.1 Nef and big forms
    2. 4.2 Demailly-Păun conjecture
    3. 4.3 Transcendental holomorphic Morse inequalities

## Introduction

The study of complex Monge-Ampère equations on compact Hermitian (non Kähler)
manifolds has gained considerable interest in the last decade, after Tosatti
and Weinkove established an appropriate version of Yau’s theorem in [TW10].
The smooth Gauduchon-Calabi-Yau conjecture has been further solved by
Székelyhidi-Tosatti-Weinkove [STW17], while the pluripotential theory has been
partially extended by Dinew, Kołodziej, and Nguyen [DK12, KN15, Din16, KN19].

As in Yau’s original proof [Yau78], the method of [TW10] consists in
establishing a priori estimates along a continuity path, and the most delicate
estimate turns out again to be the a priori $L^{\infty}$-estimate. The fact
that the reference form is not closed introduces several new difficulties:
there are many extra terms to handle when using Stokes theorem, and it becomes
non trivial to get uniform bounds on the total Monge-Ampère volumes involved
in the estimates.

In [GL21a] we have developed a new approach for establishing uniform a priori
estimates, restricting to the context of Kähler manifolds for simplicity.
While the pluripotential approach consists in measuring the Monge-Ampère
capacity of sublevel sets $(\varphi<-t)$, we directly measure the volume of
the latter, avoiding delicate integration by parts. Our approach applies in
the Hermitian setting, once certain Monge-Ampère volumes are under control.
Understanding the behavior of these volumes is the main focus of this article,
while [GL21b] is concerned with the resolution of degenerate complex Monge-
Ampère equations.

We let $X$ denote a compact complex manifold of complex dimension $n$,
equipped with a Hermitian metric $\omega_{X}$. The first difficulty we face is
to decide whether

$v_{+}(\omega_{X}):=\sup\left\\{\int_{X}(\omega_{X}+dd^{c}\varphi)^{n}\;:\;\varphi\in{\rm
PSH}(X,\omega_{X})\cap L^{\infty}(X)\right\\}$

is finite. Here $d=\partial+\overline{\partial}$,
$d^{c}=i(\partial-\overline{\partial})$, and ${\rm PSH}(X,\omega_{X})$ is the
set of $\omega_{X}$-plurisubharmonic functions: these are functions
$u:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ which are locally given as the sum
of a smooth and a plurisubharmonic function, and such that
$\omega_{X}+dd^{c}u\geq 0$ is a positive current. The complex Monge-Ampère
measure $(\omega_{X}+dd^{c}u)^{n}$ is well-defined by [BT82].

Building of works of Chiose [Chi16] and Guan-Li [GL10] we provide several
results which ensure that the condition $v_{+}(\omega_{X})<+\infty$ is
satisfied:

  * •

for any compact complex manifold $X$ of dimension $n\leq 2$;

  * •

for any threefold which admits a pluriclosed metric
$dd^{c}\tilde{\omega}_{X}=0$;

  * •

as soon as there exists a metric $\tilde{\omega}_{X}$ such that
$dd^{c}\tilde{\omega}_{X}=0$ and $dd^{c}\tilde{\omega}_{X}^{2}=0$;

  * •

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

The Fujiki class is the class of compact complex manifolds that are
bimeromorphically equivalent to Kähler manifolds.

We also need to bound the Monge-Ampère volumes from below. Given $\omega$ a
semi-positive form, we introduce several positivity properties:

  * •

we say $\omega$ is non-collapsing if there is no bounded
$\omega$-plurisubharmonic function $u$ such that $(\omega+dd^{c}u)^{n}\equiv
0$;

  * •

$\omega$ satisfies condition $(B)$ if there exists a constant $B>0$ such that

$-B\omega^{2}\leq dd^{c}\omega\leq B\omega^{2}\;\;\text{ and
}\;\;-B\omega^{3}\leq d\omega\wedge d^{c}\omega\leq B\omega^{3};$

  * •

we say $\omega$ is uniformly non-collapsing if

$v_{-}(\omega):=\inf\left\\{\int_{X}(\omega+dd^{c}u)^{n}\;:\;u\in{\rm
PSH}(X,\omega)\cap{L}^{\infty}(X)\right\\}>0.$

The non-collapsing condition is the minimal positivity condition one should
require. We show in Proposition 2.8 that it implies the domination principle,
a useful extension of the classical maximum principle. We provide a simple
example showing that having positive volume $\int_{X}\omega^{n}>0$ does not
prevent from being collapsing (see Example 3.5).

After providing a simplified proof of Kołodziej-Nguyen modified comparison
principle (see [KN15, Theorem 0.5] and Theorem 1.5), we show that condition
(B) implies non-collapsing. The former condition is e.g. satisfied by any form
$\omega$ which is the pull-back of a Hermitian form on a singular Hermitian
variety.

When $\omega$ is closed, simple integration by parts reveal that
$v_{-}(\omega)=\int_{X}\omega^{n}$ is positive as soon as $\omega$ is positive
at some point. Bounding from below $v_{-}(\omega)$ is a much more delicate
issue in general. We show in Proposition 3.4 that $\omega$ is uniformly non-
collapsing if one restricts to $\omega$-psh functions that are uniformly
bounded by a fixed constant $M$:

$v_{M}^{-}(\omega):=\inf\left\\{\int_{X}(\omega+dd^{c}u)^{n}\;:\;u\in{\rm
PSH}(X,\omega)\text{ with }-M\leq u\leq 0\right\\}>0.$

For non uniformly bounded functions we show the following:

######  Theorem A.

The condition $v_{+}(\omega_{X})<+\infty$ is independent of the choice of
$\omega_{X}$; it is moreover invariant under bimeromorphic change of
coordinates.

The condition $v_{-}(\omega_{X})>0$ is also independent of the choice of
$\omega_{X}$ and invariant under bimeromorphic change of coordinates.

In particular these conditions both hold true if $X$ belongs to the Fujiki
class.

We are not aware of a single example of a compact complex manifold such that
$v_{+}(\omega_{X})=+\infty$ or $v_{-}(\omega_{X})=0$. This is an important
open problem.

The proof of Theorem A relies on a fine use of quasi-plurisubharmonic
envelopes. These envelopes have been systematically studied in [GLZ19] in the
Kähler framework. Adapting and generalizing [GLZ19] to this Hermitian setting,
we prove in Section 2 the following:

######  Theorem B.

Given a Lebesgue measurable function $h:X\rightarrow\mathbb{R}$, we define the
$\omega$-plurisubharmonic envelope of $h$ by
$P_{\omega}(h):=\left(\sup\\{u\in{\rm PSH}(X,\omega)\;:\;u\leq
h\\}\right)^{*},$ where the star means that we take the upper semi-continuous
regularization. If $h$ is bounded below, quasi-lower-semi-continuous, and
$P_{\omega}(h)<+\infty$, then

  * •

$P_{\omega}(h)$ is a bounded $\omega$-plurisubharmonic function;

  * •

$P_{\omega}(h)\leq h$ in $X\setminus P$, where $P$ is pluripolar;

  * •

$(\omega+dd^{c}P_{\omega}(h))^{n}$ is concentrated on the contact set
$\\{P_{\omega}(h)=h\\}$.

An influential conjecture of Grauert-Riemenschneider [GR70] asked 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 in [Siu84] (with complements by [Siu85] and Demailly
[Dem85]).

Demailly and Păun have proposed a transcendental version of this conjecture
(see [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, i.e. a positive closed
$(1,1)$-current which dominates a Hermitian form. 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.

This influential conjecture has been further reinforced by Boucksom-Demailly-
Păun-Peternell who proposed a weak transcendental form of Demailly’s
holomorphic Morse inequalities [BDPP13, Conjecture 10.1]. This stronger
conjecture has been solved recently by Witt-Nyström when $X$ is projective
[WN19].

Building on works of Chiose [Chi13], Xiao [Xiao15] and Popovici [Pop16] we
obtain the following answer to the qualitative part of these conjectures:

######  Theorem C.

Let $\alpha,\beta\in H^{1,1}_{BC}(X,\mathbb{C})$ be nef classes such that
$\alpha^{n}>n\alpha^{n-1}\cdot\beta.$ The following properties are equivalent:

  1. (1)

$\alpha-\beta$ contains a Kähler current;

  2. (2)

$v_{+}(\omega_{X})<+\infty$;

  3. (3)

$X$ belongs to the Fujiki class.

A consequence of our analysis is that the conjectures of Demailly-Păun and
Boucksom-Demailly-Păun-Peternell can be extended to non closed forms, making
sense outside the Fujiki class. Progresses in the theory of complex Monge-
Ampère equations on compact hermitian manifolds have indeed shown that it is
useful to consider $dd^{c}$-perturbations of non closed nef forms. It is
therefore natural to try and consider an extension of Theorem C. These are the
contents of Theorem 4.6 (when $\beta=0$) and Theorem 4.15 (when $\beta\neq
0$).

######  Acknowledgements.

We thank Daniele Angella for several useful discussions. This work has
benefited from State aid managed by the ANR under the ”PIA” program bearing
the reference ANR-11-LABX-0040 (research project HERMETIC). The authors are
also partially supported by the ANR project PARAPLUI.

##  1\. Non collapsing forms

In the whole article we let $X$ denote a compact complex manifold of complex
dimension $n\geq 1$, and we fix $\omega$ a smooth semi-positive $(1,1)$-form
on $X$.

###  1.1. Positivity properties

####  1.1.1. Monge-Ampère operators

A function is quasi-plurisubharmonic (quasi-psh for short) if it is locally
given as the sum of a smooth and a psh function.

Given an open set $U\subset X$, quasi-psh functions
$\varphi:U\rightarrow\mathbb{R}\cup\\{-\infty\\}$ satisfying
$\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$ in the weak sense of currents
are called $\omega$-psh functions on $U$. Constant functions are $\omega$-psh
functions since $\omega$ is semi-positive. A ${\mathcal{C}}^{2}$-smooth
function $u\in\mathcal{C}^{2}(X)$ has bounded Hessian, hence $\varepsilon u$
is $\omega$-psh on $X$ if $0<\varepsilon$ is small enough and $\omega$ is
positive (i.e. Hermitian).

######  Definition 1.1.

We let ${\rm PSH}(X,\omega)$ denote the set of all $\omega$-plurisubharmonic
functions which are not identically $-\infty$.

The set ${\rm PSH}(X,\omega)$ is a closed subset of $L^{1}(X)$, for the
$L^{1}$-topology. We refer the reader to [Dem, GZ, Din16] for basic properties
of $\omega$-psh functions.

The complex Monge-Ampère measure $(\omega+dd^{c}u)^{n}$ is well-defined for
any $\omega$-psh function $u$ which is bounded, as follows from Bedford-Taylor
theory: if $\beta=dd^{c}\rho$ is a Kähler form that dominates $\omega$ in a
local chart, the function $u$ is $\beta$-psh hence the positive currents
$(\beta+dd^{c}u)^{j}$ are well defined for $0\leq j\leq n$; one thus sets

$(\omega+dd^{c}u)^{n}:=\sum_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)(-1)^{n-j}(\beta+dd^{c}u)^{j}\wedge(\beta-\omega)^{n-j}.$

We refer to [DK12] for an adaptation of [BT82] to the Hermitian context.

The mixed Monge-Ampère measures
$(\omega+dd^{c}u)^{j}\wedge(\omega+dd^{c}v)^{n-j}$ are also well defined for
any $0\leq j\leq n$, and any bounded $\omega$-psh functions $u,v$. We recall
the following classical inequality (see [GL21a, Lemma 1.3]):

######  Lemma 1.2.

Let $\varphi,\psi$ be bounded $\omega$-psh functions in $U\subset X$ such that
$\varphi\leq\psi$. Then

${\bf
1}_{\\{\psi=\varphi\\}}(\omega+dd^{c}\varphi)^{j}\wedge(\omega+dd^{c}\psi)^{n-j}\leq{\bf
1}_{\\{\psi=\varphi\\}}(\omega+dd^{c}\psi)^{n},$

for all $1\leq j\leq n$.

####  1.1.2. Condition (B) and non-collapsing

We always assume in this article that $\int_{X}\omega^{n}>0$. On a few
occasions we will need to assume positivity properties that are possibly
slightly stronger:

######  Definition 1.3.

We say $\omega$ satisfies condition (B) if there exists $B\geq 0$ such that

$-B\omega^{2}\leq dd^{c}\omega\leq B\omega^{2}\;\;\text{ and
}\;\;-B\omega^{3}\leq d\omega\wedge d^{c}\omega\leq B\omega^{3}.$

Here are three different contexts where this condition is satisfied:

  * •

any Hermitian metric $\omega>0$ satisfies condition (B);

  * •

if $\pi:X\rightarrow Y$ is a desingularization of a singular compact complex
variety $Y$ and $\omega_{Y}$ is a Hermitian metric, then
$\omega=\pi^{*}\omega_{Y}$ satisfies condition (B);

  * •

if $\omega$ is semi-positive and closed, then it satisfies condition (B).

Combining these one obtains further settings where condition (B) is satisfied.

######  Definition 1.4.

We say $\omega$ is non-collapsing if for any bounded $\omega$-psh function,
the complex Monge-Ampère measure $(\omega+dd^{c}u)^{n}$ has positive mass:
$\int_{X}\omega_{u}^{n}>0$.

We shall see in Corollary 1.6 below that condition (B) implies non-collapsing.

###  1.2. Comparison principle

The comparison principle plays a central role in Kähler pluripotential theory.
Its proof breaks down in the Hermitian setting, as it heavily relies on the
closedness of the reference form $\omega$ through the preservation of Monge-
Ampère masses. In that context the following ”modified comparison principle”
has been established by Kołodziej-Nguyen [KN15, Theorem 0.2]:

######  Theorem 1.5.

Assume $\omega$ satisfies condition $(B)$ and let $u,v$ be bounded
$\omega$-psh functions. For $\lambda\in(0,1)$ we set
$m_{\lambda}=\inf_{X}\\{u-(1-\lambda)v\\}$. Then

$\left(1-\frac{4B(n-1)^{2}s}{\lambda^{3}}\right)^{n}\int_{\\{u<(1-\lambda)v+m_{\lambda}+s\\}}\omega^{n}_{(1-\lambda)v}\leq\int_{\\{u<(1-\lambda)v+m_{\lambda}+s\\}}\omega^{n}_{u}.$

for all $0<s<\frac{\lambda^{3}}{32B(n-1)^{2}}$.

The proof by Kołodziej-Nguyen relies on the main result of [DK12], together
with extra fine estimates. We propose here a simplified proof.

###### Proof.

Set $\phi:=\max(u,(1-\lambda)v+m_{\lambda}+s)$,
$U_{\lambda,s}:=\\{u<(1-\lambda)v+m_{\lambda}+s\\}$. For $0\leq k\leq n$ we
set $T_{k}:=\omega_{u}^{k}\wedge\omega_{\phi}^{n-k}$, and $T_{l}=0$ if $l<0$.
Set $a=Bs\lambda^{-3}(n-1)^{2}$. We are going to prove by induction on
$k=0,1,...,n-1$ that

(1.1) $(1-4a)\int_{U_{\lambda,s}}T_{k}\leq\int_{U_{\lambda,s}}T_{k+1}.$

The conclusion follows since $(\omega_{\phi})^{n}=(\omega_{(1-\lambda)v})^{n}$
in the plurifine open set $U_{\lambda,s}$.

We first prove (1.1) for $k=0$. Since $u\leq\phi$, Lemma 1.2 ensures that

${\bf 1}_{\\{u=\phi\\}}\omega_{\phi}^{n}\geq{\bf
1}_{\\{u=\phi\\}}\omega_{u}\wedge\omega_{\phi}^{n-1}.$

Observing that $U_{\lambda,s}=\\{u<\phi\\}$ we infer

$\int_{X}dd^{c}(\phi-u)\wedge\omega_{\phi}^{n-1}=\int_{X}(\omega_{\phi}^{n}-\omega_{u}\wedge\omega_{\phi}^{n-1})\geq\int_{U_{\lambda,s}}\omega_{\phi}^{n}-\int_{U_{\lambda,s}}\omega_{u}\wedge\omega_{\phi}^{n-1}.$

A direct computation shows that

$\displaystyle dd^{c}\omega_{\phi}^{n-1}$ $\displaystyle=$
$\displaystyle(n-1)dd^{c}\omega\wedge\omega_{\phi}^{n-2}+(n-1)(n-2)d\omega\wedge
d^{c}\omega\wedge\omega_{\phi}^{n-3}$ $\displaystyle\leq$
$\displaystyle(n-1)B\omega^{2}\wedge\omega_{\phi}^{n-2}+(n-1)(n-2)B\omega^{3}\wedge\omega_{\phi}^{n-3},$

since $\omega$ satisfies condition (B). As $\phi-u\geq 0$, it follows from
Stokes theorem that

$\int_{X}dd^{c}(\phi-u)\wedge\omega_{\phi}^{n-1}\leq(n-1)B\left\\{\int_{X}(\phi-u)\omega^{2}\wedge\omega_{\phi}^{n-2}+(n-2)\int_{X}(\phi-u)\omega^{3}\wedge\omega_{\phi}^{n-3}\right\\}.$

Observe that

  * •

$\lambda\omega\leq\omega_{(1-\lambda)v}$ hence
$\omega^{j}\wedge\omega_{\phi}^{n-j}\leq\lambda^{-j}(\omega_{(1-\lambda)v})^{j}\wedge\omega_{\phi}^{n-j}$,

  * •

$(\omega_{(1-\lambda)v})^{j}\wedge\omega_{\phi}^{n-j}=\omega_{\phi}^{n}$ in
the plurifine open set $U_{\lambda,s}$,

  * •

and $0\leq\phi-u\leq s$ and $\phi-u=0$ on $X\setminus U_{\lambda,s}$,

to conclude that $\int_{X}(\phi-u)\omega^{j}\wedge\omega_{\phi}^{n-j}\leq
s\lambda^{-j}\int_{U_{\lambda,s}}\omega_{\phi}^{n}$, for $j=2,3$, hence

$\int_{U_{\lambda,s}}\omega_{\phi}^{n}-\int_{U_{\lambda,s}}\omega_{u}\wedge\omega_{\phi}^{n-1}\leq\int_{X}dd^{c}(\phi-u)\wedge\omega_{\phi}^{n-1}\leq\frac{Bs(n-1)^{2}}{\lambda^{3}}\int_{U_{\lambda,s}}\omega_{\phi}^{n},$

since $\lambda^{-2}\leq\lambda^{-3}$. This yields (1.1) for $k=0$.

We asume now that (1.1) holds for all $j\leq k-1$, and we check that it still
holds for $k$. Observe that

$\displaystyle dd^{c}\left(\omega_{u}^{k}\wedge\omega_{\phi}^{n-[k+1]}\right)$
$\displaystyle=$ $\displaystyle
kdd^{c}\omega\wedge\omega_{u}^{k-1}\wedge\omega_{\phi}^{n-[k+1]}+(n-[k+1])dd^{c}\omega\wedge\omega_{u}^{k}\wedge\omega_{\phi}^{n-[k+2]}$
$\displaystyle+$ $\displaystyle 2k(n-[k+1])d\omega\wedge
d^{c}\omega\wedge\omega_{u}^{k-1}\wedge\omega_{\phi}^{n-[k+2]}+k(k-1)d\omega\wedge
d^{c}\omega\wedge\omega_{u}^{k-2}\wedge\omega_{\phi}^{n-[k+1]}$
$\displaystyle+$ $\displaystyle(n-[k+1])[n-(k+2)]d\omega\wedge
d^{c}\omega\wedge\omega_{u}^{k}\wedge\omega_{\phi}^{n-[k+3]}.$

The same arguments as above therefore show that

$\displaystyle\int_{U_{\lambda,s}}(T_{k}-T_{k+1})$
$\displaystyle\leq\int_{X}(T_{k}-T_{k+1})=\int_{X}(\phi-u)dd^{c}(\omega_{u}^{k}\wedge\omega_{\phi}^{n-[k+1]})$
$\displaystyle\leq\frac{Bs}{\lambda^{3}}\int_{U_{\lambda,s}}\left(k(k-1)T_{k-2}+2k[n-k]T_{k-1}+(n-[k+1])^{2}T_{k}\right)$
$\displaystyle\leq
a\left(\frac{1}{(1-4a)^{2}}+\frac{1}{1-4a}+1\right)\int_{U_{\lambda,s}}T_{k}\leq
4a\int_{U_{\lambda,s}}T_{k},$

where in the third inequality above we have used the induction hypothesis,
while the fourth inequality follows from the upper bound $4a<1/8$. From this
we obtain (1.1) for $k$, finishing the proof. ∎

######  Corollary 1.6.

If $\omega$ satisfies condition $(B)$ then $\omega$ is non-collapsing.

###### Proof.

It follows from Theorem 1.5 that the domination principle holds (see [LPT,
Proposition 2.2]). The latter implies in particular that if $u,v$ are
$\omega$-psh and bounded, then $e^{-v}(\omega+dd^{c}v)^{n}\geq
e^{-u}(\omega+dd^{c}u)^{n}\Longrightarrow v\leq u$ (see [LPT, Proposition
2.3]). There can thus be no bounded $\omega$-psh function $u$ such that
$(\omega+dd^{c}u)^{n}=0$. Otherwise the previous inequality applied with a
constant function $v=A$ yields $u\geq A$ for any $A$, a contradiction. ∎

##  2\. Envelopes

We consider here envelopes of $\omega$-psh functions, extending some results
of [GLZ19] that have been established for Kähler manifolds.

###  2.1. Basic properties

######  Definition 2.1.

A Borel set $E\subset X$ is (locally) pluripolar if it is locally contained in
the $-\infty$ locus of some psh function: for each $x\in X$, there exists an
open neighborhood $U$ of $x$ and $u\in{\rm PSH}(U)$ such that $E\cap
U\subset\\{u=-\infty\\}$.

######  Definition 2.2.

Given a Lebesgue measurable function $h:X\rightarrow\mathbb{R}$, we define the
$\omega$-psh envelope of $h$ by

$P_{\omega}(h):=\left(\sup\\{u\in{\rm PSH}(X,\omega)\;:\;u\leq
h\,\,\text{quasi-everywhere in}\,\,X\\}\right)^{*},$

where the star means that we take the upper semi-continuous regularization,
while quasi-everywhere means outside a locally pluripolar set.

When $\omega$ is Hermitian and $h$ is ${\mathcal{C}}^{1,1}$-smooth, then so is
$P_{\omega}(h)$ (see [Ber19, CZ19, CM20]) and one can show that

(2.1) $(\omega+dd^{c}P_{\omega}(h))^{n}={\bf
1}_{\\{P_{\omega}(h)=h\\}}(\omega+dd^{c}h)^{n}.$

For less regular obstacle $h$ we have the following:

######  Theorem 2.3.

If $h$ is bounded from below, quasi-l.s.c., and $P_{\omega}(h)<+\infty$, then

  * •

$P_{\omega}(h)$ is a bounded $\omega$-plurisubharmonic function;

  * •

$P_{\omega}(h)\leq h$ in $X\setminus P$, where $P$ is pluripolar;

  * •

$(\omega+dd^{c}P_{\omega}(h))^{n}$ is concentrated on the contact set
$\\{P_{\omega}(h)=h\\}$.

Recall that a function $h$ is quasi-lower-semicontinuous (quasi-l.s.c.) if for
any $\varepsilon>0$, there exists an open set $G$ of capacity smaller than
$\varepsilon$ such that $h$ is continuous in $X\setminus G$. Quasi-psh
functions are quasi-continuous (see [BT82]), as well as differences of the
latter.

###### Proof.

The proof is an adaptation of [GLZ19, Proposition 2.2, Lemma 2.3, Proposition
2.5], which deal with the case when $\omega$ is Kähler.

Since $P_{\omega}(h)$ is bounded from above, up to replacing $h$ with
$\min(h,C)$ with $C>\sup_{X}P_{\omega}(h)$ we can assume that $h$ is bounded.

Step 1: $h$ is smooth, $\omega$ is Hermitian. Building on Berman’s work
[Ber19], it was shown by Chu-Zhou in [CZ19] that the smooth solutions
$\varphi_{\beta}$ to

$(\omega+dd^{c}\varphi_{\beta})^{n}=e^{\beta(\varphi_{\beta}-h)}\omega^{n}$

converge uniformly to $P_{\omega}(h)$ along with uniform
${\mathcal{C}}^{1,1}$-estimates. As a consequence, the measures
$(\omega+dd^{c}\varphi_{\beta})^{n}$ converge weakly to
$(\omega+dd^{c}P_{\omega}(h))^{n}$. For each fixed $\varepsilon>0$, we have
the inclusions of open sets
$\\{P_{\omega}(h)<h-2\varepsilon\\}\subset\\{\varphi_{\beta}<h-\varepsilon\\}$
for $\beta$ large enough, yielding

$\displaystyle\int_{\\{P_{\omega}(h)<h-2\varepsilon\\}}(\omega+dd^{c}P_{\omega}(h))^{n}$
$\displaystyle\leq\liminf_{\beta\to+\infty}\int_{\\{P_{\omega}(h)<h-2\varepsilon\\}}(\omega+dd^{c}\varphi_{\beta})^{n}$
$\displaystyle\leq\liminf_{\beta\to+\infty}\int_{\\{P_{\omega}(h)<h-2\varepsilon\\}}e^{-\beta\varepsilon}\omega^{n}=0.$

Step 2: $h$ is lower semi-continuous, $\omega$ is Hermitian. If $h$ is
continuous, we can approximate it uniformly by smooth functions $h_{j}$.
Letting $u_{j}:=P(h_{j})$ the previous step ensures that

$\int_{X}(h_{j}-u_{j})(\omega+dd^{c}u_{j})^{n}=0.$

As $h_{j}\to h$ uniformly we also have that $u_{j}\to u:=P(h)$ uniformly and
the desired property follows from Bedford-Taylor’s convergence theorem.

When $h$ is merely lower semi-continuous, we let $(h_{j})$ denote a sequence
of continuous functions which increase pointwise to $h$ and set
$u_{j}=P(h_{j})$. Then $u_{j}\nearrow u$ a.e. on $X$ for some bounded function
$u\in{\rm PSH}(X,\omega)$. Since $u_{j}\leq h_{j}\leq h$ quasi-everywhere on
$X$ we infer $u\leq h$ quasi-everywhere on $X$, hence $u\leq P(h)$. For each
$k<j$, the second step ensures that

$\int_{\\{u<h_{k}\\}}(\omega+dd^{c}u_{j})^{n}\leq\int_{\\{u_{j}<h_{j}\\}}(\omega+dd^{c}u_{j})^{n}=0.$

Since $\\{u<h_{k}\\}$ is open, letting $j\to+\infty$ and then $k\to+\infty$ we
arrive at

$\int_{\\{u<h\\}}(\omega+dd^{c}u)^{n}=0.$

We also have that $P(h)\leq h$ quasi-everywhere on $X$, hence

$\int_{\\{u<P(h)\\}}(\omega+dd^{c}u)^{n}=0,$

and [LPT, Proposition 2.2] then ensures that $u=P(h)$.

Step 3: $h$ is quasi-l.s.c., $\omega$ is Hermitian. By [GLZ19, Lemma 2.4] we
can find a decreasing sequence $(h_{j})$ of lsc functions such that
$h_{j}\searrow h$ q.e. on $X$ and $h_{j}\to h$ in capacity. Then
$u_{j}:=P(h_{j})\searrow u:=P(h)$. By Step 2 we know that for all $j>k$,

$\int_{\\{u_{k}<h\\}}(\omega+dd^{c}u_{j})^{n}\leq\int_{\\{u_{j}<h_{j}\\}}(\omega+dd^{c}u_{j})^{n}=0.$

Since $\\{u_{k}<h\\}$ is quasi-open and the functions $u_{j}$ are uniformly
bounded, letting $j\to+\infty$ we obtain

$\int_{\\{u_{k}<h\\}}(\omega+dd^{c}u)^{n}=0.$

Letting $k\to+\infty$ yields the desired result.

Step 4: the general case. We approximate $\omega\geq 0$ by the Hermitian forms
$\omega_{j}=\omega+j^{-1}\omega_{X}>0$. Observe that $j\mapsto
u_{j}=P_{\omega_{j}}(h)$ decreases to $u=P_{\omega}(h)$ as $j$ increases to
$+\infty$. For $0<k<j$, the previous step ensures that

$\int_{\\{u_{k}<h\\}}(\omega+j^{-j}\omega_{X}+dd^{c}u_{j})^{n}=0.$

As the set $\\{u_{k}<h\\}$ is quasi-open and $u_{j}$ is uniformly bounded we
can let $j\to+\infty$ and use Bedford-Taylor’s convergence theorem to get

$\int_{\\{u_{k}<h\\}}(\omega+dd^{c}u)^{n}=0,$

We finally let $k\to+\infty$ to conclude. ∎

For later use we extend the latter result to a setting where $P_{\omega}(f)$
is not necessarily globally bounded:

######  Corollary 2.4.

If $f$ is quasi-lower-semicontinuous and $P_{\omega}(f)$ is locally bounded in
a non-empty open set $U\subset X$ then $(\omega+dd^{c}P_{\omega}(f))^{n}$ is a
well-defined positive Borel measure in $U$ which vanishes in
$U\cap\\{P_{\omega}(f)<f\\}$.

###### Proof.

Let $(f_{j})$ be a sequence of l.s.c. functions decreasing to $f$ quasi-
everywhere. Then $u_{j}:=P_{\omega}(f_{j})$ is a bounded $\omega$-psh function
such that $(\omega+dd^{c}u_{j})^{n}=0$ on $\\{u_{j}<f_{j}\\}$. Since $u_{j}$
decreases to $u:=P_{\omega}(f)$, Bedford-Taylor’s convergence theorem ensures
that $\omega_{u_{j}}^{n}\to\omega_{u}^{n}$ in $U$.

Fix $U^{\prime}$ a relatively compact open set $U^{\prime}\Subset U$. For each
$k$ fixed the set $\\{u_{k}<f\\}$ is quasi open and the functions $u_{j},u$
are uniformly bounded in $U^{\prime}$, hence

$\liminf_{j\to+\infty}\int_{\\{u_{k}<f\\}\cap
U^{\prime}}\omega_{u_{j}}^{n}\geq\int_{\\{u_{k}<f\\}\cap
U^{\prime}}\omega_{u}^{n},$

which implies, after letting $k\to+\infty$, that $\omega_{u}^{n}$ vanishes in
$U^{\prime}\cap\\{u<f\\}$. We finally let $U^{\prime}$ increase to $U$ to
conclude. ∎

We shall use later on the following :

######  Lemma 2.5.

Let $u,v$ be bounded $\omega$-psh functions. Then

  1. (1)

$(\omega+dd^{c}P(\min(u,v)))^{n}\leq(\omega+dd^{c}u)^{n}+(\omega+dd^{c}v)^{n}$;

  2. (2)

if $(\omega+dd^{c}u)^{n}=fdV_{X}$ and $(\omega+dd^{c}v)^{n}=gdV_{X}$, then

$(\omega+dd^{c}P(\min(u,v)))^{n}\leq\max(f,g)dV_{X},$

while

$(\omega+dd^{c}\max(u,v)))^{n}\geq\min(f,g)dV_{X}.$

###### Proof.

We set $w=P(\min(u,v))$. Since $\min(u,v)$ is quasi-continuous, it follows
from Theorem 2.3 that the Monge-Ampère measure $\omega_{w}^{n}$ has support in

$\\{P(\min(u,v))=\min(u,v)\\}=\\{P(\min(u,v))=u<v\\}\cup\\{P(\min(u,v))=v\\}.$

Thus

(2.2) $\omega_{w}^{n}\leq{\bf 1}_{\\{w=u<v\\}}\omega_{w}^{n}+{\bf
1}_{\\{w=v\\}}\omega_{w}^{n}.$

Since $w=P(\min(u,v))\leq u$ and $w=P(\min(u,v))\leq u$, Lemma 1.2 yields

${\bf 1}_{\\{w=u\\}}\omega_{w}^{n}\leq{\bf
1}_{\\{w=u\\}}\omega_{u}^{n}\leq\omega_{u}^{n}$

as well as ${\bf 1}_{\\{w=v\\}}\omega_{w}^{n}\leq\omega_{v}^{n}$. Together
with (2.2) we infer $\omega_{w}^{n}\leq\omega_{u}^{n}+\omega_{v}^{n}$ as
claimed.

When $(\omega+dd^{c}u)^{n}=fdV_{X}$ and $(\omega+dd^{c}v)^{n}=gdV_{X}$, we
obtain

${\bf 1}_{\\{w=u<v\\}}\omega_{w}^{n}\leq{\bf 1}_{\\{w=u<v\\}}fdV_{X}\leq{\bf
1}_{\\{w=u<v\\}}\max(f,g)dV_{X}$

and ${\bf 1}_{\\{w=v\\}}\omega_{w}^{n}\leq{\bf 1}_{\\{w=v\\}}gdV_{X}\leq{\bf
1}_{\\{w=u<v\\}}\max(f,g)dV_{X}$, hence

$\omega_{w}^{n}\leq\left\\{{\bf 1}_{\\{w=u<v\\}}+{\bf
1}_{\\{w=v\\}}\right\\}\max(f,g)dV_{X}\leq\max(f,g)dV_{X}.$

The last item follows from

$(\omega+dd^{c}\max(\varphi,\psi))^{n}\geq{\bf 1}_{\\{u\leq
v\\}}\omega_{u}^{n}+{\bf 1}_{\\{v>u\\}}\omega_{v}^{n}\geq\min(f,g)dV_{X}.$

∎

###  2.2. Locally vs globally pluripolar sets

A classical result of Josefson asserts that a locally pluripolat set $E$ in
$\mathbb{C}^{n}$ is globally pluripolar, i.e. there exists a psh function
$u\in{\rm PSH}(\mathbb{C}^{n})$ such that $E\subset\\{u=-\infty\\}$. This
result has been extended to compact Kähler manifolds in [GZ05], and to the
Hermitian setting in [Vu19]: if $E\subset X$ is locally pluripolar and
$\omega_{X}$ is a Hermitian form, one can find $u\in{\rm PSH}(X,\omega_{X})$
such that $E\subset\\{u=-\infty\\}$.

We further extend this result to the case of non-collapsing forms:

######  Lemma 2.6.

If $E$ is (locally) pluripolar and $\omega\geq 0$ is non-collapsing then
$E\subset\\{u=-\infty\\}$ for some $u\in{\rm PSH}(X,\omega)$.

The proof is a consequence of Theorem 2.3 and analogous results established on
Kähler manifolds.

###### Proof.

As in [GZ05, Theorem 5.2] it is enough to check that
$V_{E,\omega}^{*}\equiv+\infty$, where

$V_{E,\omega}(x)=\sup\\{\varphi(x)\;:\;\varphi\in{\rm PSH}(X,\omega)\text{ and
}\varphi\leq 0\;\text{quasi-everywhere on }E\\}.$

Here quasi-everywhere means outside a locally pluripolar set. If it is not the
case then $V_{E,\omega}^{*}$ is a bounded $\omega$-psh function on $X$. We can
assume that $E\subset U\Subset V\Subset V^{\prime}$ is contained in a
holomorphic chart $V^{\prime}$. By Josefson’s theorem (see [GZ, Theorem 4.4])
we can find $u\in L^{1}_{{\rm loc}}(V^{\prime})$ a psh function in
$V^{\prime}$ such that $E\subset\\{u=-\infty\\}$. Let $u_{j}$ be a sequence of
smooth psh functions in a neighborhood of $V$ such that $u_{j}\searrow u$. Fix
$N\in\mathbb{N}$ and for $j$ large enough we set

$K_{j,N}:=\\{x\in
V\;:\;u_{j}(x)\leq-N\\},\;\varphi_{j,N}:=V_{K_{j,N},\omega}^{*},$

and note that $\varphi_{j,N}\searrow\varphi_{N}\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ as $j\to+\infty$. We also have that $E\subset\cup_{j\geq
1}K_{j,N}$, hence $0\leq\varphi_{N}\leq V_{E,\omega}^{*}$. We can thus find
$j_{N}$ so large that $\varphi_{j,N}\leq\sup_{X}V_{E,\omega}^{*}+1$ for all
$j\geq j_{N}$.

Let $\rho$ be a smooth psh function in $V$ such that $dd^{c}\rho\geq\omega$.
The Chern-Levine-Nirenberg inequality (see [GZ, Theorem 3.14]) ensures that,
for $j\geq j_{N}$,

$\displaystyle\int_{K_{j,N}}(\omega+dd^{c}\varphi_{j,N})^{n}$
$\displaystyle\leq\int_{K_{j,N}}(dd^{c}(\varphi_{j,N}+\rho))^{n}$
$\displaystyle\leq\frac{1}{N}\int_{V}|\varphi_{j,N}|(dd^{c}(\varphi_{j,N}+\rho))^{n}$
$\displaystyle\leq\frac{C}{N},$

for some uniform constant $C>0$. The function which is zero on ${K_{j,N}}$ and
$+\infty$ elsewhere is lower semi-continuous on $X$ since $K_{j,N}$ is
compact. It thus follows from Theorem 2.3 that

$\int_{X}(\omega+dd^{c}\varphi_{j,N})^{n}=\int_{K_{j,N}}(\omega+dd^{c}\varphi_{j,N})^{n}\leq\frac{C^{\prime}}{N}.$

Letting $j\to+\infty$ we obtain $\int_{X}(\omega+dd^{c}\varphi_{N})^{n}\leq
C^{\prime}/N$. Now $\varphi_{N}\nearrow\varphi$ as $N\rightarrow+\infty$, for
some $\varphi\in{\rm PSH}(X,\omega)$ which is bounded since
$0\leq\varphi_{N}\leq V_{E,\omega}^{*}$. We thus obtain
$\int_{X}(\omega+dd^{c}\varphi)^{n}=0$, yielding a contradiction since
$\omega$ is non-collapsing and $\varphi$ is bounded. ∎

Since locally pluripolar sets are ${\rm PSH}(X,\omega)$-pluripolar, arguing as
in the proof of [GLZ19, Proposition 2.2], one finally obtains:

######  Corollary 2.7.

Let $f$ be a Borel function such that $P_{\omega}(f)\in{\rm PSH}(X,\omega)$.
Then

$P_{\omega}(f)=\left(\sup\\{u\in{\rm PSH}(X,\omega)\;:\;u\leq
f\,\,\text{in}\,\,X\\}\right)^{*}.$

###  2.3. Domination principle

We now establish the following generalization of the domination principle:

######  Proposition 2.8.

Assume $\omega$ is non-collapsing and fix $c\in[0,1)$. If $u,v$ are bounded
$\omega$-psh functions such that $\omega_{u}^{n}\leq c\omega_{v}^{n}$ on
$\\{u<v\\}$, then $u\geq v$.

The usual domination principle corresponds to the case $c=0$ (see [LPT,
Proposition 2.2]).

###### Proof.

Fixing $a>0$ arbitrarily small, we are going to prove that $u\geq v-a$ on $X$.
Assume by contradiction that $E=\\{u<v-a\\}$ is not empty. Since $u,v$ are
quasi-psh, the set $E$ has positive Lebesgue measure. For $b>1$ we set

$u_{b}:=P_{\omega}(bu-(b-1)v).$

It follows from Theorem 2.3 that $(\omega+dd^{c}u_{b})^{n}$ is concentrated on
the set

$D:=\\{u_{b}=bu-(b-1)v\\}.$

Note also that $b^{-1}u_{b}+(1-b^{-1})v\leq u$ with equality on $D$. Therefore

${\bf 1}_{D}(\omega+dd^{c}(b^{-1}u_{b}+(1-b^{-1})v))^{n}\leq{\bf
1}_{D}\omega_{u}^{n},$

as follows from Lemma 1.2, hence

${\bf 1}_{D}b^{-n}(\omega+dd^{c}u_{b})^{n}+{\bf
1}_{D}(1-b^{-1})^{n}(\omega+dd^{c}v)^{n}\leq{\bf 1}_{D}\omega_{u}^{n}.$

We choose $b$ so large that $(1-b^{-1})^{n}>c$. Multiplying the above
inequality by ${\bf 1}_{\\{u<v\\}}$ and noting that $\omega_{u}^{n}\leq
c\omega_{v}^{n}$ on $\\{u<v\\}$, we obtain

${\bf 1}_{D\cap\\{u<v\\}}(\omega+dd^{c}u_{b})^{n}=0.$

Since $u_{b}$ is bounded and $\omega$ is non-collapsing, we know that
$\omega_{u_{b}}^{n}(D)=\omega_{u_{b}}^{n}(X)>0$. We infer that the set
$D\cap\\{u\geq v\\}$ is not empty, and on this set we have

$u_{b}=bu-(b-1)v\geq u\geq-C,$

since $u$ is bounded. It thus follows that $\sup_{X}u_{b}$ is uniformly
bounded from below. As $b\rightarrow+\infty$ the functions
$u_{b}-\sup_{X}u_{b}$ converge to a function $u_{\infty}$ which is $-\infty$
on $E$, but not identically $-\infty$ hence it belongs to ${\rm
PSH}(X,\omega)$. This implies that the set $E$ has Lebesgue measure $0$, a
contradiction. ∎

Here is a direct consequence of the domination principle:

######  Corollary 2.9.

Assume $\omega$ is non-collapsing and let $u,v$ be bounded $\omega$-psh
functions. Then for all $\varepsilon>0$,

$e^{-\varepsilon v}(\omega+dd^{c}v)^{n}\geq e^{-\varepsilon
u}(\omega+dd^{c}u)^{n}\Longrightarrow v\leq u.$

###### Proof.

Fix $a>0$. On the set $\\{u<v-a\\}$ we have $\omega_{u}^{n}\leq
e^{-\varepsilon a}\omega_{v}^{n}$. Proposition 2.8 thus gives $u\geq v-a$.
This is true for all $a>0$, hence $u\geq v$. ∎

##  3\. Bounds on Monge-Ampère masses

In the sequel we fix a Hermitian form $\omega_{X}$ on $X$.

###  3.1. Global bounds

Since the semi-positive $(1,1)$-form $\omega$ is not necessarily closed, the
mass of the complex Monge-Ampère measures $(\omega+dd^{c}u)^{n}$ is (in
general) not constantly equal to $V_{\omega}:=\int_{X}\omega^{n}>0$.

######  Definition 3.1.

For $1\leq j\leq n$ we consider

$v_{-,j}(\omega):=\inf\left\\{\int_{X}(\omega+dd^{c}u)^{j}\wedge\omega^{n-j},\;u\in{\rm
PSH}(X,\omega)\cap{L}^{\infty}(X)\right\\}$

and

$v_{+,j}(\omega):=\sup\left\\{\int_{X}(\omega+dd^{c}u)^{j}\wedge\omega^{n-j},\;u\in{\rm
PSH}(X,\omega)\cap{L}^{\infty}(X)\right\\}.$

We set $v_{-}(\omega):=v_{-,n}(\omega)$ and $v_{+}(\omega)=v_{+,n}(\omega)$.
When $\omega>0$ is Hermitian, the supremum and infimum in the definition of
$v_{+,j}(\omega)$ and $v_{-,j}(\omega)$ can be taken over ${\rm
PSH}(X,\omega)\cap C^{\infty}(X)$ as follows from Demailly’s approximation
[Dem92] and Bedford-Taylor’s convergence theorem [BT76, BT82].

It is an interesting open problem to determine when $v_{-}(\omega_{X})$ is
positive and/or $v_{+}(\omega_{X})$ is finite. These conditions may depend on
the complex structure, but they are independent of the choice of Hermitian
metric.

####  3.1.1. Monotonicity and invariance properties

######  Proposition 3.2.

Let $0\leq\omega_{1}\leq\omega_{2}$ be semi-positive $(1,1)$-forms. Then

(3.1) $v_{-}(\omega_{1})\leq v_{-}(\omega_{2})\;\;\text{ and
}\;\;v_{+}(\omega_{1})\leq v_{+}(\omega_{2}).$

Moreover

1) $v_{+}(\omega_{X})<+\infty\Longleftrightarrow
v_{+}(\omega_{X}^{\prime})<+\infty$ for any other Hermitian metric
$\omega_{X}^{\prime}$.

2) $0<v_{-}(\omega_{X})\Longleftrightarrow 0<v_{-}(\omega_{X}^{\prime})$ for
any other Hermitian metric $\omega_{X}^{\prime}$.

###### Proof.

Since any $\omega_{1}$-psh function $u$ is also $\omega_{2}$-psh, we obtain

$\int_{X}(\omega_{1}+dd^{c}u)^{n}\leq\int_{X}(\omega_{2}+dd^{c}u)^{n}\leq
v_{+}(\omega_{2}).$

which shows that $v_{+}(\omega_{1})\leq v_{+}(\omega_{2})$. We now bound
$v_{-}(\omega_{2})$ from below. Let $v$ be a bounded $\omega_{2}$-psh function
and let $u=P_{\omega_{1}}(v)$ denote its $\omega_{1}$-psh envelope. Then $u$
is a bounded $\omega_{2}$-psh function and $u\leq v$ on $X$. Lemma 1.2 and
Theorem 2.3 thus ensure that

$(\omega_{1}+dd^{c}u)^{n}\leq{\bf
1}_{\\{u=v\\}}(\omega_{2}+dd^{c}u)^{n}\leq{\bf
1}_{\\{u=v\\}}(\omega_{2}+dd^{c}v)^{n}.$

We therefore obtain $v_{-}(\omega_{1})\leq v_{-}(\omega_{2})$. This proves
(3.1).

Let now $\omega,\omega^{\prime}$ be two Hermitian metrics (we simplify
notations). Observe that $v_{\pm}(A\omega)=A^{n}v_{\pm}(\omega)$ for all
$A>0$. Since $A^{-1}\omega^{\prime}\leq\omega\leq A\omega$ for an appropriate
choice of the constant $A>1$, items 1) and 2) follow from (3.1). ∎

We now establish bounds on the mixed Monge-Ampère quantities:

######  Proposition 3.3.

  1. (1)

One always has $v_{+,1}(\omega)<+\infty$.

  2. (2)

If $\omega$ is Hermitian then $0<v_{-,1}(\omega)$.

  3. (3)

If $dd^{c}\omega^{n-2}=0$ then $v_{+,2}(\omega)<+\infty$.

  4. (4)

If $dd^{c}\omega=0$ and $dd^{c}\omega^{2}=0$ then
$v_{-,j}(\omega)=v_{+,j}(\omega)=V_{\omega}\in\mathbb{R}_{+}^{*}$.

  5. (5)

For all $0\leq\ell\leq j\leq n$ one has $v_{+,\ell}(\omega)\leq
2^{j}v_{+,j}(\omega)$.

  6. (6)

$v_{+,n-1}(\omega)<+\infty$ if and only if $v_{+,n}(\omega)<+\infty$.

A Hermitian metric such that $dd^{c}(\omega^{n-2})=0$ is called Astheno-
Kähler. These metrics play an important role in the study of harmonic maps
(see [JY93]). A Hermitian metric satisfying $dd^{c}\omega=0$ is called SKT or
pluriclosed in the literature. When $n=3$ the Astheno-Kähler and the
pluriclosed condition coincide, and the third item is due to Chiose [Chi16,
Question 0.8]. Examples of compact complex manifolds admitting a pluriclosed
metric can be found in [FPS04, Ot20].

Condition (4) has been introduced by Guan-Li in [GL10]. It has been shown by
Chiose [Chi16] that it is equivalent to the invariance of Monge-Ampère masses:
$\int_{X}(\omega+dd^{c}u)^{n}=\int_{X}\omega^{n}$ for all smooth $\omega$-psh
functions if and only if $dd^{c}\omega^{j}=0$ for all $j=1,2$. Note that any
compact complex surface admits a Gauduchon metric $dd^{c}\omega=0$ [Gaud77],
which also satisfies $dd^{c}\omega^{2}=0$ for bidegree reasons.

###### Proof.

One can assume without loss of generality that $\omega\leq\tilde{\omega}$,
where $\tilde{\omega}$ is a Gauduchon metric. It follows that for any
$\varphi\in{\rm PSH}(X,\omega)$,

$\int_{X}(\omega+dd^{c}\varphi)\wedge\omega^{n-1}\leq\int_{X}(\omega+dd^{c}\varphi)\wedge\tilde{\omega}^{n-1}=\int_{X}\omega\wedge\tilde{\omega}^{n-1},$

hence $v_{+,1}(\omega)\leq\int_{X}\omega\wedge\tilde{\omega}^{n-1}<+\infty$.

If $\omega$ is Hermitian one can similarly bound from below $\omega$ by a
Gauduchon form and conclude that $v_{-,1}(\omega)>0$.

We claim that $\int_{X}(\omega+dd^{c}\varphi)^{2}\wedge\omega^{n-2}\leq M$ is
uniformly bounded from above when $\varphi\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ is normalized and $dd^{c}\omega^{n-2}=0$. Indeed

$\displaystyle\int_{X}(\omega+dd^{c}\varphi)^{2}\wedge\omega^{n-2}$
$\displaystyle=$ $\displaystyle\int_{X}\omega^{n}+2\int_{X}\omega^{n-1}\wedge
dd^{c}\varphi+\int_{X}\omega^{n-2}\wedge(dd^{c}\varphi)^{2}$ $\displaystyle=$
$\displaystyle\int_{X}\omega^{n}+2\int_{X}\varphi
dd^{c}\omega^{n-1}-\int_{X}\varphi dd^{c}\omega^{n-2}\wedge dd^{c}\varphi.$

The latter integral vanishes since $dd^{c}\omega^{n-2}=0$. The second one is
uniformly bounded since the functions $\varphi$ belong to a compact subset of
$L^{1}(X)$. Altogether this shows that $v_{+,2}(\omega)<+\infty$ if
$dd^{c}(\omega^{n-2})=0$.

Since $dd^{c}(\omega^{2})=2d\omega\wedge d^{c}\omega+2\omega\wedge
dd^{c}\omega$, the Guan-Li condition is equivalent to $dd^{c}\omega=0$ and
$d\omega\wedge d^{c}\omega=0$. For ${u\in{\rm
PSH}(X,\omega)\cap{\mathcal{C}}^{\infty}(X)}$ we use the binomial expansion of
the Monge-Ampère measure $(\omega+dd^{c}u)^{n}$ to obtain

$\int_{X}(\omega+dd^{c}u)^{n}=\int_{X}\omega^{n}+n\int_{X}\omega^{n-1}\wedge
dd^{c}u+\cdots+n\int_{X}\omega\wedge(dd^{c}u)^{n-1}+\int_{X}(dd^{c}u)^{n}.$

Observe that $dd^{c}\\{du\wedge
d^{c}u\wedge(dd^{c}u)^{n-2-j}\\}=-(dd^{c}u)^{n-j},$ while
$\int_{X}(dd^{c}u)^{n}=0$ by Stokes theorem, hence

$\displaystyle dd^{c}\left\\{\omega^{j}\wedge du\wedge
d^{c}u\wedge(dd^{c}u)^{n-2-j}\right\\}=-\omega^{j}\wedge(dd^{c}u)^{n-j}$
$\displaystyle+$ $\displaystyle j\omega^{j-1}\wedge dd^{c}\omega\wedge
du\wedge d^{c}u\wedge(dd^{c}u)^{n-2-j}$ $\displaystyle+$ $\displaystyle
j(j-1)\omega^{j-2}\wedge d\omega\wedge d^{c}\omega\wedge du\wedge
d^{c}u\wedge(dd^{c}u)^{n-2-j}.$

If $dd^{c}\omega=0$ and $d\omega\wedge d^{c}\omega=0$ we infer from Stokes
theorem $\int_{X}\omega^{j}\wedge(dd^{c}u)^{n-j}=0$, hence
$\int_{X}(\omega+dd^{c}u)^{n}=\int_{X}\omega^{n}$ for all $u\in{\rm
PSH}(X,\omega)\cap{\mathcal{C}}^{\infty}(X)$, showing that
$v_{-}(\omega)=v_{+}(\omega)=V_{\omega}$ is both finite and positive.
Expanding similarly the mixed Monge-Ampère measure
$(\omega+dd^{c}u)^{j}\wedge\omega^{n-j}$ one obtains 4).

Observe that for any $\varphi\in{\rm PSH}(X,\omega)\cap L^{\infty}$ and
$0\leq\ell\leq j\leq n$ one has

(3.2)
$\int_{X}(\omega+dd^{c}\varphi)^{\ell}\wedge\omega^{n-\ell}\leq\int_{X}(2\omega+dd^{c}\varphi)^{j}\wedge\omega^{n-j}\leq
2^{j}v_{+,j}(\omega).$

In particular $v_{+,n-1}(\omega)\leq 2^{n}v_{+,n}(\omega)$ hence
$v_{+,n}(\omega)<+\infty\Rightarrow v_{+,n-1}(\omega)<+\infty$. We finally
show conversely that $v_{+,n-1}(\omega)<+\infty\Rightarrow
v_{+,n}(\omega)<+\infty$ by proving

$v_{+,n}(\omega)\leq 2^{2n-2}v_{+,n-1}(\omega).$

Observe indeed that

$\displaystyle 0$ $\displaystyle=\int_{X}(\omega+dd^{c}\varphi-\omega)^{n}$
$\displaystyle=\int_{X}(\omega+dd^{c}\varphi)^{n}+\sum_{k=1}^{n}(-1)^{k}\binom{n}{k}(\omega+dd^{c}\varphi)^{n-k}\wedge\omega^{k}$
$\displaystyle\geq\int_{X}(\omega+dd^{c}\varphi)^{n}-\sum_{1\leq 2k+1\leq
n}\binom{n}{2k+1}(\omega+dd^{c}\varphi)^{n-2k-1}\wedge\omega^{2k+1}.$

Using (3.2) we thus get

$v_{+,n}(\omega)\leq\sum_{1\leq 2k+1\leq
n}\binom{n}{2k+1}2^{n-1}v_{+,n-1}(\omega)=2^{2n-2}v_{+,n-1}(\omega).$

∎

####  3.1.2. Uniformly bounded functions

Restricting to uniformly bounded $\omega$-psh functions, it is natural to
consider

$v^{-}_{M}(\omega):=\inf\left\\{\int_{X}(\omega+dd^{c}u)^{n}\;:\;u\in{\rm
PSH}(X,\omega)\text{ with }-M\leq u\leq 0\right\\}$

where $M\in\mathbb{R}^{+}$, and

$v^{+}_{M}(\omega):=\sup\left\\{\int_{X}(\omega+dd^{c}u)^{n}\;:\;u\in{\rm
PSH}(X,\omega)\text{ with }-M\leq u\leq 0\right\\}.$

These quantities are always under control as we now explain:

######  Proposition 3.4.

Assume $\omega$ is non-collapsing. For any $M\in\mathbb{R}^{+}$, one has

$0<v^{-}_{M}(\omega)\leq v^{+}_{M}(\omega)<+\infty.$

###### Proof.

The finiteness of $v^{+}_{M}(\omega)$ follows easily from integration by
parts, it is e.g. a simple consequence of [DK12, Theorem 3.5].

In order to show that $v^{-}_{M}(\omega)$ is positive we argue by
contradiction. Assume there exists $u_{j}\in{\rm PSH}(X,\omega)$ such that
$-M\leq u_{j}\leq 0$ and $\int_{X}(\omega+dd^{c}u_{j})^{n}\leq 2^{-j}$. For
$j\in\mathbb{N}$ fixed, the sequence

$k\mapsto v_{j,k}:=P_{\omega}(\min(u_{j},u_{j+1},\ldots,u_{j+k}))$

decreases towards a $\omega$-psh function $w_{j}$ such that $-M\leq w_{j}\leq
0$. It follows therefore from Lemma 2.5 that

$\int_{X}(\omega+dd^{c}w_{j})^{n}=\lim_{k\rightarrow+\infty}\int_{X}(\omega+dd^{c}v_{j,k})^{n}\leq\sum_{\ell=0}^{+\infty}\int_{X}(\omega+dd^{c}v_{j+\ell})^{n}\leq
2^{-j+1}.$

Thus the sequence $j\mapsto w_{j}$ increases to a bounded $\omega$-psh
function $w$ such that $(\omega+dd^{c}w)^{n}=0$, which yields a contradiction.
∎

######  Example 3.5.

We provide here an example of a semi-positive form $\omega$ such that
$\int_{X}\omega^{n}>0$ but $\omega$ is collapsing, in particular
$v_{-}(\omega)=0$. Let $X=Y\times Z$ where $Y,Z$ are two compact complex
manifolds of dimension $m\geq 1$, $p\geq 1$ respectively, and ${\rm
dim}X=n=p+m$. Fix a smooth function $u$ on $Y$ such that
$\omega_{Y}+dd^{c}u<0$ is negative in a small open set $U\subset Y$. Let
$0\leq\rho\leq 1$ be a cut-off function on $Y$ supported in $U$. The smooth
$(1,1)$-form $\omega$ defined by

$\omega=\rho\circ\pi_{1}(\pi_{1}^{*}\omega_{Y}+\pi_{2}^{*}\omega_{Z}).$

is semipositive on $X$ and satisfies $\omega(y,z)=0$ for $y\notin U$.

Set now $\phi:=P_{\omega}(u\circ\pi_{1})$ and let
$\mathcal{C}:=\\{\phi=u\circ\pi_{1}\\}$ denote the contact set. The Monge-
Ampère measure $(\omega+dd^{c}\phi)^{n}$ is concentrated on $\mathcal{C}$.
Arguing as in [Ber09, Proposition 3.1] one can show that
$\mathcal{C}\subset\\{x\in X,\;\omega+dd^{c}u\circ\pi_{1}(x)\geq 0\\}$. Since
$\omega+dd^{c}(u\circ\pi_{1})<0$ is negative in $U\times Z$, it follows that
$\mathcal{C}\subset X\setminus(U\times Z)$. Now $\omega=0$ outside $U\times
Z$, hence

$(\omega+dd^{c}\phi)^{n}\leq{\bf 1}_{\mathcal{C}}(dd^{c}u\circ\pi_{1})^{n}=0,$

because $u\circ\pi_{1}$ depends only on $y$. It thus follows that
$(\omega+dd^{c}\phi)^{n}=0$ on $X$.

###  3.2. Bimeromophic invariance

######  Lemma 3.6.

Let $f:X\rightarrow Y$ be a proper holomorphic map between compact complex
manifolds of dimension $n$, equipped with Hermitian forms
$\omega_{X},\omega_{Y}$. Then

  * •

$v_{+}(\omega_{X})<+\infty\Longrightarrow v_{+}(\omega_{Y})<+\infty$;

  * •

$v_{-}(\omega_{Y})>0\Longrightarrow v_{-}(\omega_{X})>0$ if $f$ has connected
fibers.

It follows from Zariski’s main theorem that $f$ has connected fibers if it is
bimeromorphic.

###### Proof.

Up to rescaling, we can assume that $f^{*}\omega_{Y}\leq\omega_{X}$. Fix
$\varphi\in{\rm PSH}(Y,\omega_{Y})\cap L^{\infty}(Y)$. Then $\varphi\circ
f\in{\rm PSH}(X,\omega_{X})\cap L^{\infty}(X)$ with

$\int_{Y}(\omega_{Y}+dd^{c}\varphi)^{n}=\int_{X}(f^{*}\omega_{Y}+dd^{c}\varphi\circ
f)^{n}\leq\int_{X}(\omega_{X}+dd^{c}\varphi\circ f)^{n}\leq
v_{+}(\omega_{X}),$

thus $v_{+}(\omega_{Y})\leq v_{+}(\omega_{X})$ and the first assertion is
proved.

Consider now $\psi\in{\rm PSH}(X,\omega_{X})\cap L^{\infty}(X)$ and set
$u=P_{f^{*}\omega_{Y}}(\psi)$. The function $u$ is $f^{*}\omega_{Y}$, hence
plurisubharmonic on the fibers of $f$. If the latter are connected we obtain
that $u$ is constant on them, i.e. $u=\varphi\circ f$ for some function
$\varphi\in{\rm PSH}(Y,\omega_{Y})\cap L^{\infty}(Y)$. Since
$(f^{*}\omega_{Y}+dd^{c}u)^{n}\leq{\bf
1}_{\\{u=\psi\\}}(f^{*}\omega_{Y}+dd^{c}\psi)^{n}$, we infer

$v_{-}(\omega_{Y})\leq\int_{Y}(\omega_{Y}+dd^{c}\varphi)^{n}=\int_{X}(f^{*}\omega_{Y}+dd^{c}u)^{n}\leq\int_{X}(\omega_{X}+dd^{c}\psi)^{n}$

so that $v_{-}(\omega_{Y})\leq v_{-}(\omega_{X})$, proving the second
assertion. ∎

We conversely show that the properties $v_{+}(\omega_{X})<+\infty$ and
$v_{-}(\omega_{X})>0$ are invariant under blow ups and blow downs with smooth
centers:

######  Theorem 3.7.

Let $X$ and $Y$ be compact complex manifolds which are bimeromorphically
equivalent. Then

  * •

$v_{+}(\omega_{X})<+\infty$ if and only if $v_{+}(\omega_{Y})<+\infty$;

  * •

$v_{-}(\omega_{X})>0$ if and only if $v_{-}(\omega_{Y})>0$.

###### Proof.

A celebrated result of Hironaka ensures that any bimeromorphic map between
compact complex manifolds is a finite composition of blow ups and blow downs
with smooth centers. We can thus assume that $f:X\rightarrow Y$ is the blow up
of $Y$ along a smooth center.

We fix $\psi$ a quasi-plurisubharmonic function such that
$\pi^{*}\omega_{Y}+dd^{c}\psi\geq\delta\omega_{X}$. The existence of $\psi$
follows from a classical argument in complex geometry (see [BL70], [FT09,
Proposition 3.2]). By Demailly’s approximation theorem we can further assume
that $\psi$ has analytic singularities. Up to scaling we can assume without
loss of generality that $\delta=1$, and we set $\Omega=\\{x\in
X\;:\;\psi(x)>-\infty\\}$.

We already know by Lemma 3.6 that $v_{+}(\omega_{X})<+\infty\Longrightarrow
v_{+}(\omega_{Y})<+\infty$. Assume conversely that
$v_{+}(\omega_{Y})<+\infty$. For any $\varphi\in{\rm PSH}(X,\omega_{X})\cap
L^{\infty}(X)$,

$\displaystyle\int_{X}(\omega_{X}+dd^{c}\varphi)^{n}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}(\pi^{*}\omega_{Y}+dd^{c}(\psi+\varphi))^{n}$
$\displaystyle\leq$
$\displaystyle\liminf_{j\rightarrow+\infty}\int_{\pi(\Omega)}(\pi^{*}\omega_{Y}+dd^{c}(\max[\psi+\varphi,-j]))^{n}.$

The function $u_{j}=\max[\psi+\varphi,-j]$ is $\pi^{*}\omega_{Y}$-psh and
bounded in $\Omega$. It is constant on the fibers of $\pi$, hence
$u_{j}=v_{j}\circ\pi$ with $v_{j}\in{\rm PSH}(\pi(\Omega),\omega_{Y})\cap
L^{\infty}(\Omega)$. As $v_{j}$ is bounded, it extends trivially through the
analytic set $\pi(\partial\Omega)$ as a bounded $\omega_{Y}$-psh function.
Thus

$\int_{\pi(\Omega)}(\pi^{*}\omega_{Y}+dd^{c}u_{j})^{n}=\int_{Y}(\omega_{Y}+dd^{c}v_{j})^{n}\leq
v_{+}(\omega_{Y})$

yields $v_{+}(\omega_{X})\leq v_{+}(\omega_{Y})<+\infty$.

We now assume that $v_{-}(\omega_{X})>0$. Pick $v\in{\rm
PSH}(Y,\omega_{Y})\cap L^{\infty}(Y)$ and set
$u=P_{\omega_{X}}(v\circ\pi-\psi)$. Observe that $u\in{\rm
PSH}(X,\omega_{X})\cap L^{\infty}(X)$ and recall that
$(\omega_{X}+dd^{c}u)^{n}$ is concentrated on the contact set
${\mathcal{C}}=\\{u+\psi=v\circ\pi\\}$ (see Theorem 2.3). Since $u+\psi$ and
$v\circ\pi$ are both $\pi^{*}\omega_{Y}$-psh, locally bounded in $\Omega$,
with $u+\psi\leq v\circ\pi$, it follows from Lemma 1.2 that

$1_{\mathcal{C}}(\pi^{*}\omega_{Y}+dd^{c}(u+\psi))^{n}\leq
1_{\mathcal{C}}(\pi^{*}\omega_{Y}+dd^{c}v\circ\pi)^{n}\leq(\pi^{*}\omega_{Y}+dd^{c}v\circ\pi)^{n}.$

Now $\pi^{*}\omega_{Y}+dd^{c}(u+\psi)\geq\omega_{X}+dd^{c}u$ and
$(\omega_{X}+dd^{c}u)^{n}$ is concentrated on ${\mathcal{C}}$ so

$1_{\mathcal{C}}(\pi^{*}\omega_{Y}+dd^{c}(u+\psi))^{n}\geq(\omega_{X}+dd^{c}u)^{n}.$

We infer

$\displaystyle v_{-}(\omega_{X})\leq\int_{X}(\omega_{X}+dd^{c}u)^{n}$
$\displaystyle\leq$
$\displaystyle\int_{\mathcal{C}}(\pi^{*}\omega_{Y}+dd^{c}(u+\psi))^{n}$
$\displaystyle\leq$
$\displaystyle\int_{X}(\pi^{*}\omega_{Y}+dd^{c}v\circ\pi)^{n}=\int_{Y}(\omega_{Y}+dd^{c}v)^{n},$

showing that $v_{-}(\omega_{Y})\geq v_{-}(\omega_{X})>0$. The reverse
implication $v_{-}(\omega_{Y})>0\Longrightarrow v_{-}(\omega_{X})>0$ follows
from Lemma 3.6. ∎

Recall that a compact complex manifold $X$ belongs to the Fujiki class
${\mathcal{C}}$ if there exists a holomorphic bimeromophic map
$\pi:Y\rightarrow X$, where $Y$ is compact Kähler. Since
$v_{+}(\omega_{X})=v_{-}(\omega_{X})=\int_{X}\omega_{X}^{n}\in\mathbb{R}_{+}^{*}$
when $\omega_{X}$ is a Kähler form, we obtain the following:

######  Corollary 3.8.

If $X$ belongs to the Fujiki class $\mathcal{C}$ then

$0<v_{-}(\omega_{X})\leq v_{+}(\omega_{X})<+\infty.$

##  4\. Weak transcendental Morse inequalities

###  4.1. Nef and big forms

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. This is a
finite dimensional vector space as $X$ is compact.

Nefness and bigness are fundamental positivity properties of holomorphic line
bundles in complex algebraic geometry (see [Laz]). Their transcendental
counterparts have been defined and studied by Demailly (see [Dem]):

######  Definition 4.1.

  * •

A cohomology class $\alpha\in H_{BC}^{1,1}(X,\mathbb{R})$ is nef if for any
$\varepsilon>0$, one can find a smooth closed real $(1,1)$-form
$\theta_{\varepsilon}\in\alpha$ such that
$\theta_{\varepsilon}\geq-\varepsilon\omega_{X}$.

  * •

A Hermitian current on $X$ is a positive current $T$ of bidegree $(1,1)$ which
dominates a Hermitian form, i.e. there exists $\delta>0$ such that
$T\geq\delta\omega_{X}$.

  * •

A cohomology class $\alpha\in H_{BC}^{1,1}(X,\mathbb{R})$ is big if it can be
represented by a closed Hermitian current (a Kähler current).

It follows from an approximation result of Demailly [Dem92] that one can
weakly approximate a Hermitian current by Hermitian currents with analytic
singularities. In particular a big cohomology class can be represented by a
Kähler current with analytic singularities.

By analogy with the above setting, we propose the following definitions:

######  Definition 4.2.

Let $\omega$ be a smooth real $(1,1)$ form on $X$.

  * •

We say that $\omega$ is nef if for any $\varepsilon>0$ there exists a smooth
quasi-plurisubharmonic function $\varphi_{\varepsilon}$ such that
$\omega+dd^{c}\varphi_{\varepsilon}\geq-\varepsilon\omega_{X}$.

  * •

We say that $\omega$ is big if there exists a $\omega$-psh function $\rho$
with analytic singularities such that $\omega+dd^{c}\rho$ dominates a
Hermitian form.

Note that ${\rm PSH}(X,\omega)$ is non empty in both cases: indeed
$\rho\in{\rm PSH}(X,\omega)$ in the latter case, while one can extract
$\varphi_{\varepsilon_{j}}\rightarrow\varphi\in{\rm PSH}(X,\omega)$ in the
former, normalizing the potentials $\varphi_{\varepsilon_{j}}$ by imposing
$\sup_{X}\varphi_{\varepsilon_{j}}=0$.

When $X$ is a compact Kähler manifold and $\alpha\in
H^{1,1}_{BC}(X,\mathbb{R})$ is nef with $\alpha^{n}>0$, a celebrated result of
Demailly-Păun [DP04, Theorem 0.5] ensures the existence of a Kähler current
representing $\alpha$. This result is the key step in establishing a
transcendental Nakai-Moishezon criterion (see [DP04, Main theorem]).

We study in the sequel a possible extension of this result to the Hermitian
setting. We thus need to extend the definition of $v_{-}$ to nef forms:

######  Definition 4.3.

If $\omega$ is a nef $(1,1)$-form, we set

$\hat{v}_{-}(\omega):=\inf_{\varepsilon>0}v_{-}(\omega+\varepsilon\omega_{X}).$

Although the form $\omega+\varepsilon\omega_{X}$ needs not be semi-positive,
one can find by definition a semi-positive form
$\omega+\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon}$ cohomologous to
$\omega+\varepsilon\omega_{X}$, and it is understood here that
$v_{-}(\omega+\varepsilon\omega_{X}):=v_{-}(\omega+\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon})$.
By (3.1), the definition of $\hat{v}_{-}(\omega)$ is independent of the choice
of the Hermitian form $\omega_{X}$.

It is natural to expect that this definition is consistent with the previous
one when $\omega$ is semi-positive, and that $\hat{v}_{-}(\omega)=\alpha^{n}$
when $\omega$ is a closed form representing a nef class $\alpha\in
H^{1,1}_{BC}(X,\mathbb{R})$:

######  Lemma 4.4.

If $\omega$ is semi-positive then $\hat{v}_{-}(\omega)=v_{-}(\omega)$. If
$v_{+}(\omega_{X})<+\infty$ and $\omega$ is a closed form representing a nef
class in $H^{1,1}_{BC}(X,\mathbb{R})$, then $\hat{v}_{-}(\omega)=\alpha^{n}$.

When $X$ is Kähler, it is classical that any nef class $\alpha\in
H_{BC}^{1,1}(X,\mathbb{R})$ satisfies $\alpha^{n}\geq 0$. This inequality is
no longer obvious on an arbitrary hermitian manifold (we thank J.-P.Demailly
for emphasizing this issue) but, as a consequence of the above lemma, it
remains true when $v_{+}(\omega_{X})<+\infty$.

###### Proof.

Assume first that $\omega$ is semi-positive and set
$\omega_{\varepsilon}:=\omega+\varepsilon\omega_{X}$, for
$\varepsilon\in(0,1)$. Proposition 3.2 ensures that $v_{-}(\omega)\leq
v_{-}(\omega_{\varepsilon})$, hence $v_{-}(\omega)\leq\hat{v}_{-}(\omega)$. On
the other hand, for any $u\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ we have

$\displaystyle\int_{X}(\omega+dd^{c}u)^{n}$
$\displaystyle=\int_{X}(\omega_{\varepsilon}+dd^{c}u-\varepsilon\omega_{X})^{n}$
$\displaystyle\geq\int_{X}(\omega_{\varepsilon}+dd^{c}u)^{n}-C\varepsilon$
$\displaystyle\geq\hat{v}_{-}(\omega)-C\varepsilon,$

where $C$ is a constant depending on $u$, but it is harmless as we will let
$\varepsilon\to 0$ while keeping $u$ fixed. Doing so we obtain
$\int_{X}(\omega+dd^{c}u)^{n}\geq\hat{v}_{-}(\omega)$, and taking infimum over
such $u$ we obtain $v_{-}(\omega)\geq\hat{v}_{-}(\omega)$, proving the first
statement.

Assume now that $\omega$ is closed and $\\{\omega\\}\in
H^{1,1}_{BC}(X,\mathbb{R})$ is nef. We can also assume that
$-\omega_{X}\leq\omega\leq\omega_{X}$. We pick $\varphi\in{\rm
PSH}(X,\omega+\varepsilon\omega_{X})\cap C^{\infty}(X)$ and observe that ${\rm
PSH}(X,\omega+\varepsilon\omega_{X})\subset{\rm PSH}(X,2\omega_{X})$ for
$0<\varepsilon\leq 1$, hence

$\int_{X}(\omega+\varepsilon\omega_{X}+dd^{c}\varphi)^{n}=\int_{X}(\omega+dd^{c}\varphi)^{n}+\sum_{j=1}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\varepsilon^{j}\int_{X}\omega_{X}^{j}\wedge(\omega+dd^{c}\varphi)^{n-j}.$

Writing
$\omega+dd^{c}\varphi=(2\omega_{X}+dd^{c}\varphi)-(2\omega_{X}-\omega)$,
expanding $(\omega+dd^{c}\varphi)^{n-j}$ accordingly and using $0\leq
2\omega_{X}-\omega\leq 3\omega_{X}$, we obtain that
$\left|\int_{X}\omega_{X}^{j}\wedge(\omega+dd^{c}\varphi)^{n-j}\right|$ is
bounded from above by a finite sum of terms
$\int_{X}\omega_{X}^{\ell}\wedge(\omega_{X}+dd^{c}\varphi)^{n-\ell}$, each of
which is bounded from above by $3^{n}v_{+}(\omega_{X})$. Since
$\int_{X}(\omega+dd^{c}\varphi)^{n}=\alpha^{n}$, we end up with

$\alpha^{n}-C\varepsilon
v_{+}(\omega_{X})\leq\int_{X}(\omega+\varepsilon\omega_{X}+dd^{c}\varphi)^{n}\leq\alpha^{n}+C\varepsilon
v_{+}(\omega_{X}),$

using that $\varepsilon^{j}\leq\varepsilon$ for all $1\leq j\leq n$. We infer
$\hat{v}_{-}(\omega)=\alpha^{n}$. ∎

###  4.2. Demailly-Păun conjecture

####  4.2.1. Hermitian currents

The following is a natural generalization of [DP04, Conjecture 0.8]:

######  Question 4.5.

Let $X$ be a compact complex manifold. Let $\omega$ be a nef $(1,1)$-form such
that $\hat{v}_{-}(\omega)>0$. Does there exist a $\omega$-psh function
$\varphi$ with analytic singularities such that the current
$\omega+dd^{c}\varphi$ dominates a Hermitian form ?

We provide a partial answer to Question 4.5 following some ideas of Chiose
[Chi13]:

######  Theorem 4.6.

Let $\omega$ be a nef $(1,1)$-form.

  * •

If $\hat{v}_{-}(\omega)>0$ and $v_{+}(\omega_{X})<+\infty$ then $\omega$ is
big.

  * •

Conversely if $\omega$ is big and $v_{-}(\omega_{X})>0$ then
$\hat{v}_{-}(\omega)>0$.

###### Proof.

We assume without loss of generality that $\omega\leq\omega_{X}/2$.

We first assume that $\hat{v}_{-}(\omega)>0$, $v_{+}(\omega_{X})<+\infty$, and
we prove that $\omega$ is big. An application of Hahn-Banach theorem as in
[Lam99, Lemma 3.3] shows that the existence of a Hermitian current
$\omega+dd^{c}\psi\geq\delta\omega_{X}$ is equivalent to the inequalities

$\int_{X}\omega\wedge\theta^{n-1}\geq\delta\int_{X}\omega_{X}\wedge\theta^{n-1},$

for all Gauduchon metrics $\theta$. Assume by contradiction that there exists
a sequence of Gauduchon metrics $\theta_{j}$ such that

$\int_{X}\omega\wedge\theta_{j}^{n-1}\leq\frac{1}{j}\int_{X}\omega_{X}\wedge\theta_{j}^{n-1}.$

We can normalize the latter so that
$\int_{X}\omega_{X}\wedge\theta_{j}^{n-1}=1$.

Set $\omega_{j}=\omega+\frac{1}{j}\omega_{X}$ and note that
$\omega_{j}\leq\omega_{X}$ for $j\geq 2$. Since $\omega$ is nef, one can find
$\psi_{j}\in{\mathcal{C}}^{\infty}(X,\mathbb{R})$ such that
$\omega_{j}+dd^{c}\psi_{j}$ is a Hermitian form, hence the main result of
[TW10] ensures that there exist constants $C_{j}>0$ and $\varphi_{j}\in{\rm
PSH}(X,\omega_{j})\cap{\mathcal{C}}^{\infty}(X)$ such that
$\sup_{X}\varphi_{j}=0$ and

$(\omega_{j}+dd^{c}\varphi_{j})^{n}=C_{j}\omega_{X}\wedge\theta_{j}^{n-1}.$

It follows from Proposition 3.2 that

$C_{j}=\int_{X}(\omega_{j}+dd^{c}\varphi_{j})^{n}\geq
v_{-}(\omega_{j})\geq\hat{v}_{-}(\omega)>0,$

while by assumption
$\int_{X}(\omega_{j}+dd^{c}\varphi_{j})^{n-1}\wedge\omega_{X}\leq
M:=v_{+,n-1}(\omega_{X})$ is bounded from above.

We set $\alpha_{j}:=\omega_{j}+dd^{c}\varphi_{j}$ and consider

$E:=\\{x\in X,\;\omega_{X}\wedge\alpha_{j}^{n-1}\geq
2M\omega_{X}\wedge\theta_{j}^{n-1}\\}.$

This set has small $\omega_{X}\wedge\theta_{j}^{n-1}$ measure since

$\int_{E}\omega_{X}\wedge\theta_{j}^{n-1}\leq\frac{1}{2M}\int_{E}\omega_{X}\wedge\alpha_{j}^{n-1}\leq\frac{1}{2},$

thus $\int_{X\setminus E}\omega_{X}\wedge\theta_{j}^{n-1}\geq\frac{1}{2}$,
thanks to the normalization $\int_{X}\omega_{X}\wedge\theta_{j}^{n-1}=1$.

We can compare $\omega_{X}$ and $\alpha_{j}$ in $X\setminus E$ since

$\omega_{X}\wedge\alpha_{j}^{n-1}\leq
2M\omega_{X}\wedge\theta_{j}^{n-1}=\frac{2M}{C_{j}}\alpha_{j}^{n}\leq\frac{2M}{\hat{v}_{-}(\omega)}\alpha_{j}^{n}.$

Thus $\alpha_{j}\geq\frac{\hat{v}_{-}(\omega)}{2nM}\omega_{X}$ in $X\setminus
E$ and we infer

$\int_{X\setminus
E}\alpha_{j}\wedge\theta_{j}^{n-1}\geq\frac{\hat{v}_{-}(\omega)}{2nM}\int_{X\setminus
E}\omega_{X}\wedge\theta_{j}^{n-1}\geq\frac{\hat{v}_{-}(\omega)}{4nM}>0,$

which contradicts

$\displaystyle\int_{X}\alpha_{j}\wedge\theta_{j}^{n-1}$ $\displaystyle=$
$\displaystyle\int_{X}\omega\wedge\theta_{j}^{n-1}+\frac{1}{j}\int_{X}\omega_{X}\wedge\theta_{j}^{n-1}+\int_{X}dd^{c}\varphi_{j}\wedge\theta_{j}^{n-1}$
$\displaystyle\leq$
$\displaystyle\frac{2}{j}\int_{X}\omega_{X}\wedge\theta_{j}^{n-1}=\frac{2}{j}\rightarrow
0,$

where $\int_{X}dd^{c}\varphi_{j}\wedge\theta_{j}^{n-1}=0$ follows from the
Gauduchon property of $\theta_{j}$.

We next assume that $\omega$ is big, $v_{-}(\omega_{X})>0$, and we prove that
$\hat{v}_{-}(\omega)>0$ by an argument similar to that of Theorem 3.7. Fix a
$\omega$-psh function $\psi$ with analytic singularities such that
$\omega+dd^{c}\psi\geq\delta\omega_{X}$ for some $\delta>0$. We can assume
that $\delta=1$ and $\sup_{X}\psi=0$. We prove that
$v_{-}(\omega+\varepsilon\omega_{X})\geq v_{-}(\omega_{X})$ for all
$\varepsilon>0$. Fix $\varepsilon>0$, $u\in{\rm
PSH}(X,\omega+\varepsilon\omega_{X})\cap L^{\infty}(X)$, and set
$v=P_{\omega_{X}}(u-\psi)$. The open set $G=\\{\psi>-1\\}$ is not empty hence
it is non-pluripolar. On $G$ we have $u\leq u-\psi\leq u+1\leq\sup_{X}u+1$. It
follows that $v$ is a bounded $\omega_{X}$-psh function and
$(\omega_{X}+dd^{c}v)^{n}$ is supported on the contact set
$\mathcal{C}=\\{v=u-\psi\\}\subset\\{\psi>-\infty\\}$. Since $v+\psi\leq u$
with equality on $\\{\psi>-\infty\\}\cap\mathcal{C}$, Lemma 1.2 ensures that

${\bf
1}_{\\{\psi>-\infty\\}\cap\mathcal{C}}(\omega+\varepsilon\omega_{X}+dd^{c}(v+\psi))^{n}\leq{\bf
1}_{\\{\psi>-\infty\\}\cap\mathcal{C}}(\omega+\varepsilon\omega_{X}+dd^{c}u)^{n}.$

Using $\omega+dd^{c}\psi\geq\omega_{X}$ and the fact that
$(\omega_{X}+dd^{c}v)^{n}(\psi=-\infty)=0$ since $v$ is bounded, we thus
arrive at

$\int_{X}(\omega_{X}+dd^{c}v)^{n}\leq\int_{X}(\omega+\varepsilon\omega_{X}+dd^{c}u)^{n}.$

We thus get $v_{-}(\omega+\varepsilon\omega_{X})\geq v_{-}(\omega_{X})>0$, for
all $\varepsilon>0$, hence $\hat{v}_{-}(\omega)>0$. ∎

This result provides in particular the following answer to Question 4.5:

######  Corollary 4.7.

The answer to Question 4.5 is positive if

  * •

either $n=2$ ($X$ is any compact surface);

  * •

or $n=3$ and $X$ admits a pluriclosed metric;

  * •

or $n$ is arbitrary and $X$ belongs to the Fujiki class;

  * •

orelse $n$ is arbitrary and $X$ admits a Guan-Li metric.

Let us stress that the $2$-dimensional setting is due to Buchdahl [Buch99] and
Lamari [Lam99]. The three dimensional case follows from Proposition 3.3.

####  4.2.2. Transcendental Grauert-Riemenschneider conjecture

Let $L\rightarrow X$ be a semi-positive holomorphic line bundle with
$c_{1}(L)^{n}>0$. An influential conjecture of Grauert-Riemenschneider [GR70]
asked whether the existence of such a line bundle implies that $X$ is
Moishezon (i.e. bimeromorphically equivalent to a projective manifold).

This conjecture has been solved positively by Siu in [Siu84] (see also
[Dem85]). Demailly and Păun have proposed a transcendental version of this
conjecture:

######  Conjecture 4.8.

[DP04, Conjecture 0.8] Let $X$ be a compact complex manifold of dimension $n$.
Assume that $X$ posseses a nef class $\alpha\in H_{BC}^{1,1}(X,\mathbb{R})$
such that $\alpha^{n}>0$. Then $X$ belongs to the Fujiki class.

As a direct consequence of Theorem 4.6, Lemma 4.4, and Corollary 3.8, we
obtain the following answer to the transcendental Grauert-Riemenschneider
conjecture:

######  Theorem 4.9.

Let $X$ be a compact $n$-dimensional complex manifold. Let $\alpha\in
H_{BC}^{1,1}(X,\mathbb{R})$ be a nef class such that $\alpha^{n}>0$. The
following are equivalent:

  * •

$\alpha$ contains a Kähler current

  * •

$v_{+}(\omega_{X})<+\infty$.

Since a Kähler current with analytic singularities can be desingularized after
finitely many blow-ups producing a Kähler form, we obtain:

######  Corollary 4.10.

Let $\alpha\in H_{BC}^{1,1}(X,\mathbb{R})$ be a nef class such that
$\alpha^{n}>0$. Then $X$ belongs to the Fujiki class if and only if
$v_{+}(\omega_{X})<+\infty$.

###  4.3. Transcendental holomorphic Morse inequalities

The following conjecture has been proposed by Boucksom-Demailly-Păun-
Peternell, as a transcendental counterpart to the holomorphic Morse
inequalities for integral classes due to Demailly:

######  Conjecture 4.11.

[BDPP13, Conjecture 10.1.ii] Let $X$ be a compact $n$-dimensional complex
manifold. Let $\alpha,\beta\in H^{1,1}_{BC}(X,\mathbb{C})$ be nef classes such
that $\alpha^{n}>n\alpha^{n-1}\cdot\beta.$

Then $\alpha-\beta$ contains a Kähler current and ${\rm
Vol}(\alpha-\beta)\geq\alpha^{n}-n\alpha^{n-1}\cdot\beta$.

Note that this contains [DP04, Conjecture 0.8] as a particular case
($\beta=0$). This conjecture has recently been established by Witt Nyström
[WN19] when $X$ is projective. Building on works of Xiao [Xiao15] and Popovici
[Pop16] we propose the following characterization which answers the
qualitative part:

######  Theorem 4.12.

Let $\alpha,\beta\in H^{1,1}_{BC}(X,\mathbb{C})$ be nef classes such that
$\alpha^{n}>n\alpha^{n-1}\cdot\beta.$ The following are equivalent:

  * •

$\alpha-\beta$ contains a Kähler current;

  * •

$v_{+}(\omega_{X})<+\infty$.

###### Proof.

If $\alpha-\beta$ contains a Kähler current, then $X$ belongs to the Fujiki
class and we have already observed that $v_{+}(\omega_{X})<+\infty$ (see
Corollary 3.8).

We now assume that $v_{+}(\omega_{X})<+\infty$. Let $\omega$ and
$\omega^{\prime}$ be smooth closed real $(1,1)$-forms representing $\alpha$
and $\beta$ respectively. We can assume without los of generality that
$\omega\leq\frac{\omega_{X}}{2}$ and
$\omega^{\prime}\leq\frac{\omega_{X}}{2}$. For each $\varepsilon>0$ we fix
smooth functions $\varphi_{\varepsilon}\in{\rm
PSH}(X,\omega+\varepsilon\omega_{X})$ and $\psi_{\varepsilon}\in{\rm
PSH}(X,\omega^{\prime}+\varepsilon\omega_{X})$ such that
$\omega_{\varepsilon}:=\omega+\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon}$
and
$\omega_{\varepsilon}^{\prime}=\omega^{\prime}+\varepsilon\omega_{X}+dd^{c}\psi_{\varepsilon}$
are hermitian forms.

Assume by contradiction that $\alpha-\beta$ does not contain any Kähler
current. It follows from Hahn-Banach theorem as in [Lam99, Lemma 3.3] that
there exist Gauduchon metrics $\eta_{\varepsilon}$ such that

(4.1)
$\int_{X}(\omega_{\varepsilon}-\omega_{\varepsilon}^{\prime})\wedge\eta_{\varepsilon}^{n-1}\leq\varepsilon\int_{X}\omega_{\varepsilon}^{\prime}\wedge\eta_{\varepsilon}^{n-1}.$

We normalize $\eta_{\varepsilon}$ so that
$\int_{X}\omega_{\varepsilon}^{\prime}\wedge\eta_{\varepsilon}^{n-1}=1$.

Using [TW10] we can find unique constants $c_{\varepsilon}>0$ and normalized
functions $u_{\varepsilon}\in{\rm
PSH}(X,\omega_{\varepsilon})\cap{\mathcal{C}}^{\infty}(X)$ such that

$(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n}=c_{\varepsilon}\omega_{\varepsilon}^{\prime}\wedge\eta_{\varepsilon}^{n-1},\;\sup_{X}u_{\varepsilon}=0.$

Our normalization for $\eta_{\varepsilon}$ yields
$c_{\varepsilon}=\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n}.$
Applying Lemma 4.13 below with
$\theta_{1}=\omega_{\varepsilon}+dd^{c}u_{\varepsilon}$,
$\theta_{2}=c_{\varepsilon}\omega_{\varepsilon}^{\prime}$ and
$\theta_{3}=\eta_{\varepsilon}$, and recalling that
$\theta_{1}^{n}=\theta_{2}\wedge\theta_{3}^{n-1}$ with
$\int_{X}\theta_{1}^{n}=\int_{X}\theta_{2}\wedge\theta_{3}^{n-1}=c_{\varepsilon}$,
we obtain

$\left(\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})\wedge\eta_{\varepsilon}^{n-1}\right)\left(\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n-1}\wedge\omega_{\varepsilon}^{\prime}\right)\geq\frac{c_{\varepsilon}}{n}.$

Now
$\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})\wedge\eta_{\varepsilon}^{n-1}=\int_{X}\omega_{\varepsilon}\wedge\eta_{\varepsilon}^{n-1}$
because $\eta_{\varepsilon}$ is a Gauduchon metric, while (4.1) yields
$\int_{X}\omega_{\varepsilon}\wedge\eta_{\varepsilon}^{n-1}\leq(1+\varepsilon)\int_{X}\omega^{\prime}_{\varepsilon}\wedge\eta_{\varepsilon}^{n-1}=(1+\varepsilon)$,
hence

$(1+\varepsilon)\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n-1}\wedge\omega_{\varepsilon}^{\prime}\geq\frac{1}{n}\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n}.$

We finally claim that, as $\varepsilon\to 0$,

$\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n}\to\alpha^{n}\;\text{and}\;\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n-1}\wedge\omega_{\varepsilon}^{\prime}\to\alpha^{n-1}\cdot\beta,$

which yields the contradiction $n\alpha^{n-1}\cdot\beta\geq\alpha^{n}$.

We first explain why
$\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n}\to\alpha^{n}$.
Stokes theorem yields

$\displaystyle\alpha^{n}=\int_{X}(\omega+dd^{c}(u_{\varepsilon}+\varphi_{\varepsilon}))^{n}=\int_{X}(\omega+\varepsilon\omega_{X}+dd^{c}(u_{\varepsilon}+\varphi_{\varepsilon})-\varepsilon\omega_{X})^{n}$
$\displaystyle=$
$\displaystyle\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n}+\sum_{j=0}^{n-1}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\varepsilon^{n-j}(-1)^{n-j}\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{j}\wedge\omega_{X}^{n-j}.$

Since $\omega\leq\frac{\omega_{X}}{2}$, the function
$v_{\varepsilon}=u_{\varepsilon}+\varphi_{\varepsilon}$ is $\omega_{X}$-psh
for $0<\varepsilon\leq\frac{1}{2}$, hence

$0\leq\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{j}\wedge\omega_{X}^{n-j}\leq\int_{X}(\omega_{X}+dd^{c}v_{\varepsilon})^{j}\wedge\omega_{X}^{n-j}\leq
2^{n}v_{+}(\omega_{X}),$

as follows from Proposition 3.3. We infer

$\left|\alpha^{n}-\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n}\right|\leq\sum_{j=0}^{n-1}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\varepsilon^{n-j}2^{n}v_{+}(\omega_{X})\leq
4^{n}\varepsilon\,v_{+}(\omega_{X}).$

The conclusion thus follows by letting $\varepsilon\to 0$.

We similarly can check that

$\left|\alpha^{n-1}\cdot\beta-\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n-1}\wedge\omega_{\varepsilon}^{\prime}\right|\leq
2\cdot 6^{n}\varepsilon\,v_{+}(\omega_{X}).$

Using Stokes theorem again we indeed obtain that

$\displaystyle\alpha^{n-1}\cdot\beta$
$\displaystyle=\int_{X}(\omega+dd^{c}\varphi_{\varepsilon}+dd^{c}u_{\varepsilon})^{n-1}\wedge(\omega^{\prime}+dd^{c}\psi_{\varepsilon})$
$\displaystyle=\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon}-\varepsilon\omega_{X})^{n-1}\wedge(\omega_{\varepsilon}^{\prime}-\varepsilon\omega_{X})$
$\displaystyle=\int_{X}(\omega_{\varepsilon}+dd^{c}u_{\varepsilon})^{n-1}\wedge\omega^{\prime}_{\varepsilon}+O(\varepsilon).$

Each term
$\int_{X}(\omega_{X}+dd^{c}v_{\varepsilon})^{\ell}\wedge(\omega_{X}+dd^{c}\psi_{\varepsilon})^{p}\wedge\omega_{X}^{q}$,
with $\ell+p+q=n$, is bounded from above by $3^{n}v_{+}(\omega_{X})$, as one
can check by observing that the function
$\frac{v_{\varepsilon}+\psi_{\varepsilon}}{3}$ is $\omega_{X}$-psh with

$\int_{X}(\omega_{X}+dd^{c}v_{\varepsilon})^{\ell}\wedge(\omega_{X}+dd^{c}\psi_{\varepsilon})^{p}\wedge\omega_{X}^{q}\leq
3^{n}\int_{X}\left(\omega_{X}+dd^{c}\frac{v_{\varepsilon}+\psi_{\varepsilon}}{3}\right)^{n}.$

∎

We have used in the previous proof the following observation of Popovici:

######  Lemma 4.13.

Let $\theta_{1},\theta_{2},\theta_{3}$ be hermitian forms on $X$. Then

$\left(\int_{X}\theta_{1}\wedge\theta_{3}^{n-1}\right)\left(\int_{X}\theta_{1}^{n-1}\wedge\theta_{2}\right)\geq\frac{1}{n}\left(\int_{X}\sqrt{\frac{\theta_{2}\wedge\theta_{3}^{n-1}}{\theta_{1}^{n}}}\theta_{1}^{n}\right)^{2}.$

In particular if $\theta_{1}^{n}=\theta_{2}\wedge\theta_{3}^{n-1}$, then

$\left(\int_{X}\theta_{1}\wedge\theta_{3}^{n-1}\right)\left(\int_{X}\theta_{1}^{n-1}\wedge\theta_{2}\right)\geq\frac{1}{n}\left(\int_{X}\theta_{1}^{n}\right)^{2}.$

We provide the proof as a courtesy to the reader.

###### Proof.

It follows from Cauchy-Schwarz inequality that

$\left(\int_{X}\theta_{1}\wedge\theta_{3}^{n-1}\right)\left(\int_{X}\theta_{1}^{n-1}\wedge\theta_{2}\right)\geq\left(\int_{X}\sqrt{\frac{\theta_{1}\wedge\theta_{3}^{n-1}}{\theta_{1}^{n}}\frac{\theta_{1}^{n-1}\wedge\theta_{2}}{\theta_{1}^{n}}}\theta_{1}^{n}\right)^{2}.$

The elementary pointwise estimate

$Tr_{\theta_{3}}(\theta_{1})Tr_{\theta_{1}}(\theta_{2})\geq
Tr_{\theta_{3}}(\theta_{2}).$

is [Pop16, Lemma 3.1]. Multiplying by $\frac{\theta_{3}^{n}}{\theta_{1}^{n}}$
it can be reformulated as

(4.3)
$\frac{\theta_{1}\wedge\theta_{3}^{n-1}}{\theta_{1}^{n}}\cdot\frac{\theta_{2}\wedge\theta_{1}^{n-1}}{\theta_{1}^{n}}\geq\frac{1}{n}\frac{\theta_{2}\wedge\theta_{3}^{n-1}}{\theta_{1}^{n}}.$

The first inequality follows. Moreover when
$\theta_{1}^{n}=\theta_{2}\wedge\theta_{3}^{n-1}$, we infer

$\int_{X}\sqrt{\frac{\theta_{2}\wedge\theta_{3}^{n-1}}{\theta_{1}^{n}}}\theta_{1}^{n}=\int_{X}\theta_{1}^{n}.$

∎

Motivated by possible extensions of the conjectures of Demailly-Păun and
Boucksom-Demailly-Păun-Peternell, we introduce the following:

######  Definition 4.14.

Given $\omega_{1},\ldots,\omega_{n}$ hermitian forms we consider

$v_{-}(\omega_{1},\ldots,\omega_{n}):=\inf\left\\{\int_{X}(\omega_{1}+dd^{c}\varphi_{1})\wedge\cdots\wedge(\omega_{n}+dd^{c}\varphi_{n}),\;\varphi_{j}\in{\mathcal{P}}(\omega_{j})\right\\},$

and

$v_{+}(\omega_{1},\ldots,\omega_{n}):=\sup\left\\{\int_{X}(\omega_{1}+dd^{c}\varphi_{1})\wedge\cdots\wedge(\omega_{n}+dd^{c}\varphi_{n}),\;\varphi_{j}\in{\mathcal{P}}(\omega_{j})\right\\},$

where ${\mathcal{P}}(\omega_{j}):={\rm PSH}(X,\omega_{j})\cap L^{\infty}(X)$.
If the $\omega_{j}$’s are merely nef we set

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

and

$\hat{v}_{+}(\omega_{1},\ldots,\omega_{n}):=\inf_{\varepsilon>0}v_{+}(\omega_{1}+\varepsilon\omega_{X},\ldots,\omega_{n}+\varepsilon\omega_{X}).$

A straightforward generalization of Theorem 4.12 along the lines of Theorem
4.6 is the following:

######  Theorem 4.15.

Let $X$ be a compact $n$-dimensional complex manifold such that
$v_{+}(\omega_{X})<+\infty$. Let $\omega,\omega^{\prime}$ be nef
$(1,1)$-forms. If
$\hat{v}_{-}(\omega)>n\hat{v}_{+}(\omega,\ldots,\omega,\omega^{\prime})$ then
the form $\omega-\omega^{\prime}$ is big.

We leave the technical details to the reader.

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