# Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on
hermitian manifolds

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.

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate
complex Monge-Ampère equations on complex manifolds. It only relies on
compactness and envelopes properties of quasi-plurisubharmonic functions. In a
prequel [GL21a] we have shown how this method allows one to obtain new and
efficient proofs of several fundamental results in Kähler geometry. In [GL21b]
we have studied the behavior of Monge-Ampère volumes on hermitian manifolds.
We extend here the techniques of [GL21a] to the hermitian setting and use the
bounds established in [GL21b], producing new relative a priori estimates, as
well as several existence results for degenerate complex Monge-Ampère
equations on compact hermitian manifolds.

###### Key words and phrases:

Monge-Ampère equation, a priori estimates

###### 2010 Mathematics Subject Classification:

32W20, 32U05, 32Q15, 35A23

###### Contents

  1. 1 Preliminaries
    1. 1.1 Positivity properties and envelopes
      1. 1.1.1 Monge-Ampère measure
      2. 1.1.2 Positivity assumptions
      3. 1.1.3 Envelopes
    2. 1.2 Comparison and domination principles
      1. 1.2.1 Comparison principle
      2. 1.2.2 Domination principle
  2. 2 Uniform a priori estimates
    1. 2.1 Global $L^{\infty}$-bounds
      1. 2.1.1 Hermitian forms
      2. 2.1.2 Semi-positive forms
      3. 2.1.3 Stability estimate
    2. 2.2 Relative $L^{\infty}$-bounds
  3. 3 Existence of locally bounded solutions
    1. 3.1 Bounded solutions
      1. 3.1.1 The normalizing constant
      2. 3.1.2 Existence of solutions
    2. 3.2 Uniqueness of solutions
    3. 3.3 Locally bounded solutions
  4. 4 Geometric applications
    1. 4.1 Higher order regularity of solutions
    2. 4.2 Singular hermitian Calabi conjecture

## Introduction

Complex Monge-Ampère equations have been one of the most powerful tools in
Kähler geometry since Yau’s solution to the Calabi conjecture [Yau78]. In
recent years degenerate complex Monge-Ampère equations have been intensively
studied by many authors, in relation to the Minimal Model Program (see [GZ,
Don18, BBEGZ, BBJ21] and the references therein). The main analytical input
came here from pluripotential theory which allowed Kołodziej [Kol98] to
establish uniform a priori estimates in quite degenerate settings.

The study of complex Monge-Ampère equations on compact hermitian manifolds was
undertaken by Cherrier [Cher87] and Hanani [Han96]. It has gained considerable
interest in the last decade, after Tosatti and Weinkove solved an appropriate
version of Yau’s theorem in [TW10b], following important progress by Guan-Li
[GL10] and geometric motivation for constructing special hermitian metrics
(see e.g. [FLY12]). The smooth Gauduchon-Calabi-Yau conjecture has been solved
by Székélyhidi-Tosatti-Weinkove [STW17], while the pluripotential theory has
been partially extended by Dinew, Kołodziej, and Nguyen [DK12, KN15, Din16,
KN19], allowing these authors to establish various uniform a priori estimates
in a non Kähler setting.

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 thus applies
in the hermitian setting, providing several new results that we now describe
more precisely.

We let $X$ denote a compact complex manifold of complex dimension $n$,
equipped with a hermitian metric $\omega_{X}$. We fix $\omega$ a semi-positive
$(1,1)$-form and set

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

where $d=\partial+\overline{\partial}$,
$d^{c}=i(\overline{\partial}-\partial)$, and ${\rm PSH}(X,\omega)$ is the set
of $\omega$-plurisubharmonic functions: these are functions $u$ which are
locally the sum of a smooth and a plurisubharmonic function, and such that
$\omega+dd^{c}u\geq 0$ is a positive current.

When $\omega$ is closed, simple integration by parts reveal that
$v_{-}(\omega)=\int_{X}\omega^{n}$ is positive as soon as the differential
form $\omega$ is positive at some point. Bounding from below $v_{-}(\omega)$
is a much more delicate issue in general which we discuss at length in
[GL21b]. Our first main result is the following uniform a priori estimate when
$v_{-}(\omega)$ is positive (Theorem 2.2):

######  Theorem A.

Let $\omega$ be semi-positive with $v_{-}(\omega)>0$. Let $\mu$ be a
probability measure such that ${\rm PSH}(X,\omega)\subset L^{m}(\mu)$ for some
$m>n$. Any solution $\varphi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ to
$(\omega+dd^{c}\varphi)^{n}=c\mu$, where $c>0$, satisfies

${\rm Osc}_{X}(\varphi)\leq T$

for some uniform constant $T$ which depends on an upper bound on
$\frac{c}{v_{-}(\omega)}$ and

$A_{m}({\mu}):=\sup\left\\{\left(\int_{X}(-\psi)^{m}d\mu\right)^{\frac{1}{m}}\;:\;\psi\in{\rm
PSH}(X,\omega)\text{ with }\sup_{X}\psi=0\right\\}.$

This result covers the case when $\mu=fdV_{X}$ is absolutely continuous with
respect to Lebesgue measure, with density $f$ belonging to $L^{p}$, $p>1$, or
to an appropriate Orlicz class $L^{w}$ (for some convex weight $w$ with ”fast
growth” at infinity), thus partially extending the case of hermitian forms
treated by Dinew-Kołodziej [DK12] and Kołodziej-Nguyen [KN15, KN21].

We also provide a new and direct alternative proof of this a priori estimate
when $\omega$ is hermitian, relying only on the local resolution of the
classical Dirichlet problem for the complex Monge-Ampère equation, and
twisting the right hand side with an exponential (see Theorem 2.1).

As far as solutions to such equations are concerned, we obtain the following:

######  Theorem B.

Let $\omega$ be a semi-positive $(1,1)$ form which is either big or such that
$v_{-}(\omega)>0$. Fix $0\leq f\in L^{p}(dV_{X})$, where $p>1$ and
$\int_{X}fdV_{X}=1$. Then

  * •

there exists a unique constant $c(\omega,f)>0$ and a bounded $\omega$-psh
function $\varphi$ such that $(\omega+dd^{c}\varphi)^{n}=c(\omega,f)fdV_{X};$

  * •

for any $\lambda>0$ there exists a unique $\varphi_{\lambda}\in{\rm
PSH}(X,\omega)\cap L^{\infty}(X)$ such that

$(\omega+dd^{c}\varphi_{\lambda})^{n}=e^{\lambda\varphi_{\lambda}}fdV_{X}.$

By analogy with the Kähler setting, we say here that $\omega$ is big if there
exists an $\omega$-psh function with analytic singularities $\rho$ such that
$\omega+dd^{c}\rho\geq\delta\omega_{X}$ for some $\delta>0$. A celebrated
result of Demailly-Păun [DP04, Theorem 0.5] ensures, when $X$ is Kähler, that
the existence of $\rho$ is a consequence of the condition $v_{-}(\omega)>0$.
This result has been partially extended to the hermitian setting in [GL21b].

A slight refinement of our technique allows one to establish important
stability estimates (see Theorems 2.3 and 3.5). Similar results have been
obtained by quite different methods over the last decade (see [Blo11, DK12,
Szek18, KN19, LPT20]).

There are several geometric situations when one can not expect the Monge-
Ampère potential $\varphi$ to be globally bounded. These corresponds to
probability measures $\mu=fdV_{X}$ whose density $f$ belongs to an Orlicz
class $L^{w}$, for some convex weight $w$ with slow growth (see section 2.2).
Our next main result provides the following a priori estimate, which extends
to the hermitian setting a result proved by DiNezza-Lu [DnL17] in the context
of quasi-projective varieties:

######  Theorem C.

Let $\omega$ be a semi-positive $(1,1)$ form which is big and such that
$v_{-}(\omega)>0$. Let $\mu=fdV_{X}$ be a probability measure, where $0\leq
f=ge^{-A\psi}\in L^{w}$ with $g\in L^{p}(dV_{X})$ for some $p>1$, $A>0$ and
$\psi\in{\rm PSH}(X,\omega)$. Assume $\varphi$ is a bounded $\omega$-psh
function such that $(\omega+dd^{c}\varphi)^{n}=cfdV_{X}$ and
$\sup_{X}\varphi=0$. Then

$\alpha\psi-\beta\leq\varphi\leq 0$

for any $0<\alpha\leq 1$, where $\beta>0$ is a uniform constant that depends
on $p$, the weight $w$, and upper bounds for
$||g||_{L^{p}},\frac{A}{\alpha},\frac{c}{v_{-}(\omega)}$, and the Luxembourg
norm $\|f\|_{w}$.

The proof of DiNezza-Lu uses generalized Monge-Ampère capacities, hence relies
on Bedford-Taylor’s comparison principle. It has been shown by Chiose [Chi16]
that this comparison principle holds only under the restrictive condition
$dd^{c}\omega=dd^{c}\omega^{2}=0$. Our approach is completely different: we
use a quasi-psh envelope construction to replace $f$ by $e^{-2A\varphi}$ (a
similar idea has been recently used in [LN20, DDL19]) and then use Theorem A,
whose proof also heavily relies on quasi-psh envelopes.

We then move on to show the existence of solutions to such degenerate complex
Monge-Ampère equations. In this context we prove the following :

######  Theorem D.

Let $\omega$ be a semi-positive $(1,1)$ form which is big and such that
$v_{-}(\omega)>0$. Fix $\rho\in{\rm PSH}(X,\omega)$ with analytic
singularities along a divisor $E$, such that $\omega+dd^{c}\rho$ dominates a
hermitian form. Let $\mu=fdV_{X}$ be a probability measure, where $0\leq f$ is
smooth and positive in $X\setminus D$, and $f=e^{\psi^{+}-\psi^{-}}$ for some
quasi-plurisubharmonic functions $\psi^{\pm}$.

Then there exist $c>0$ and $\varphi\in{\rm PSH}(X,\omega)$ such that

  * •

$\varphi$ is smooth in the open set $X\setminus(D\cup E)$;

  * •

$\alpha(\psi^{-}+\rho)-\beta(\alpha)\leq\varphi$ in $X$ and
$\sup_{X}\varphi=0$;

  * •

$(\omega+dd^{c}\varphi)^{n}=cfdV_{X}$ in $X\setminus(D\cup E)$;

where $0<\alpha\leq 1$ is arbitrarily small, and $\beta(\alpha)$ is a uniform
constant.

This result can be seen as a generalization of the main result of [TW10b]. It
encompasses the case of smooth Monge-Ampère equations on mildly singular
compact hermitian varieties, as well as more degenerate settings, hermitian
analogues of the main results of [DnL17, Theorems 1 and 3]. It is obtained as
a combination of Theorems 3.7 and 4.2. When moreover $f\in L^{p}(dV_{X})$,
$p>1$, then $\varphi$ is globally bounded (one can take $\alpha=0$) and it
suffices to assume that $\omega$ is big (see Theorem 4.1).

We finally apply our results to solve a singular version of the hermitian
Calabi-Yau theorem. We work over a compact complex variety $V$ which has log
terminal singularities (see Section 4.2 for a precise definition). If the
first Bott-Chern class $c_{1}^{BC}(V)$ vanishes, one says that $V$ is a
$\mathbb{Q}$-Calabi-Yau variety. In that context we construct many Ricci-flat
hermitian metrics:

######  Theorem E.

Let $V$ be a $\mathbb{Q}$-Calabi-Yau variety and $\omega_{V}$ a hermitian
form. There exists a function $\varphi\in{\rm PSH}(V,\omega_{V})\cap
L^{\infty}(V)$ such that

  * •

$\varphi$ is smooth in $V_{reg}$;

  * •

$\omega_{V}+dd^{c}\varphi$ is a hermitian form and ${\rm
Ric}(\omega_{V}+dd^{c}\varphi)=0$ in $V_{reg}$.

We actually prove a singular hermitian analogue of Yau’s celebrated solution
to the Calabi conjecture, see Theorem 4.5. We expect many further geometric
implications of the present work, but leave this for a future project.

Comparison with other works. Yau’s proof of his famous $L^{\infty}$-a priori
estimate [Yau78] goes through a delicate Moser iteration process. A PDE proof
of the $L^{\infty}$-estimate has been recently provided by Guo-Phong-Tong
[GPT21] using an auxiliary Monge-Ampère equation, inspired by the recent
breakthrough by Chen-Cheng on constant scalar curvature metrics [CC21]. An
important generalization of Yau’s estimate has been provided by Kołodziej
[Kol98] using pluripotential techniques. This technique has been further
generalized in [EGZ09, EGZ08, DP10, BEGZ10] in order to deal with less
positive or collapsing classes. All these works require the underlying
manifold to be Kähler.

Błocki has provided a different approach [Blo05, Blo11] which is based on the
Alexandroff-Bakelman-Pucci maximum principle and a stability estimate due to
Cheng-Yau ($L^{2}$-case) and Kołodziej ($L^{p}$-case). Błocki’s method works
in the hermitian case and has been generalized to various settings by
Székélehydi, Tosatti and Weinkove [STW17, Szek18, TW18]. It requires the
reference form to be strictly positive, but applies to a large family of
equations (see also [TW15] for a slightly weaker $L^{\infty}$ estimate based
on Moser iteration process).

The first steps of pluripotential theory have been developed in the hermitian
setting by Dinew, Kołodziej and Nguyen [DK12, KN15, Din16, KN19]. The presence
of torsion requires the reference form to be positive in order to control
error terms in delicate integration by parts.

Our approach consists in showing that the volume of the sublevel sets
$(\varphi<-t)$ goes down to zero in finite time by directly measuring their
$\mu$-size. It relies on weak compactness of normalized
$\omega$-plurisubharmonic functions and basic properties of quasi-psh
envelopes, allowing us to deal with semi-positive forms.

######  Acknowledgements.

We thank D.Angella and V.Tosatti for useful discussions, as well as T.D.Tô and
C.-M.Pan for a careful reading of a first draft. 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\. Preliminaries

In the whole article we let $X$ denote a compact complex manifold of complex
dimension $n\geq 1$, equipped with a hermitian form $\omega_{X}$, $dV_{X}$ a
smooth probability measure, and $\omega$ a smooth semi-positive $(1,1)$-form
on $X$ such that $\int_{X}\omega^{n}>0$.

###  1.1. Positivity properties and envelopes

####  1.1.1. Monge-Ampère measure

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

######  Definition 1.1.

Quasi-plurisubharmonic functions
$\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ satisfying
$\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$ in the weak sense of currents
are called $\omega$-psh functions.

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

We refer the reader to [Dem, GZ, Din16] for basic properties of $\omega$-psh
functions. Recall that:

  * •

The set ${\rm PSH}(X,\omega)$ is a closed subset of $L^{1}(X)$, for the
$L^{1}$-topology.

  * •

${\rm PSH}(X,\omega)\subset L^{r}(X)$ for $r\geq 1$; the induced
$L^{r}$-topologies are equivalent;

  * •

the subset ${\rm PSH}_{A}(X,\omega):=\\{u\in{\rm
PSH}(X,\omega)\;:\;-A\leq\sup_{X}u\leq 0\\}$ is compact in $L^{r}(X)$ for any
$r\geq 1$ and all $A\geq 0$.

The complex Monge-Ampère measure $(\omega+dd^{c}u)^{n}=\omega_{u}^{n}$ is
well-defined for any $\omega$-psh function $u$ which is bounded, as follows
from Bedford-Taylor theory.

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 1.2]):

######  Lemma 1.2.

Let $\varphi,\psi$ be bounded $\omega$-psh functions such that
$\varphi\leq\psi$, then

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

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

We shall also need the following generalization of the inequality of
arithmetic and geometric means (see [N16, Lemma 1.9]):

######  Lemma 1.3.

Let $\varphi_{1},\ldots,\varphi_{n}$ be bounded $\omega$-psh functions such
that $(\omega+dd^{c}\varphi_{i})^{n}\geq f_{i}dV_{X}$. Then
$(\omega+dd^{c}\varphi_{1})\wedge\cdots\wedge(\omega+dd^{c}\varphi_{n})\geq(\Pi_{i=1}^{n}f_{i})^{\frac{1}{n}}dV_{X}.$

####  1.1.2. Positivity assumptions

On a few occasions we will need to assume slightly stronger positivity
properties of the form $\omega$:

######  Definition 1.4.

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

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

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 1.5.

We say that

  * •

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

  * •

$\omega$ is uniformly non-collapsing if $v_{-}(\omega)>0$, where

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

These positivity notions are studied at length in [GL21b]. It is shown there
that condition (B) implies non-collapsing, we further expect it implies
uniform non-collapsing (at least in the case if $X$ belongs to the Fujiki
class).

######  Definition 1.6.

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

If $V$ is a compact complex space endowed with a hermitian form $\omega_{V}$,
and $\pi:X\rightarrow V$ is a resolution of singularities, then
$\omega=\pi^{*}\omega_{V}$ is big. This follows from classical arguments (see
e.g. [FT09, Proposition 3.2]).

It is expected that $\omega$ is big if and only if $v_{-}(\omega)>0$. This is
a generalization of a conjecture of Demailly-Păun [DP04, Conjecture 0.8],
which has been addressed in [GL21b]: it is in particular shown in [GL21b,
Theorem 4.6] that if $v_{+}(\omega_{X})<+\infty$ and $v_{-}(\omega)>0$ then
$\omega$ is big, where

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

####  1.1.3. Envelopes

######  Definition 1.7.

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

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

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

The following has been established in [GL21b, Theorem 2.3]:

######  Theorem 1.8.

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
$\\{P_{\omega}(h)=h\\}$.

A useful consequence is the following (see [GL21b, Lemma 2.5]):

######  Lemma 1.9.

Fix $\lambda\geq 0$ and let $u,v$ be bounded $\omega$-psh functions. Fix two
smooth semi-positive $(1,1)$-forms $\omega_{1},\omega_{2}$ such that
$\omega_{1}\geq\omega,\omega_{2}\geq\omega$.

(i) If $(\omega_{1}+dd^{c}u)^{n}\leq e^{\lambda u}fdV_{X}$ and
$(\omega_{2}+dd^{c}v)^{n}\leq e^{\lambda v}gdV_{X}$, then

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

(ii) If $(\omega+dd^{c}u)^{n}\geq e^{\lambda u}fdV_{X}$ and
$(\omega+dd^{c}v)^{n}\geq e^{\lambda v}gdV_{X}$, then

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

###### Proof.

The second statement follows from [GL21b, Lemma 2.5]. We prove the first one.
Setting $\varphi:=P_{\omega}(\min(u,v))$, by Theorem 1.8 and Lemma 1.2 we have

$\displaystyle(\omega+dd^{c}\varphi)^{n}$ $\displaystyle\leq{\bf
1}_{\\{\varphi=u<v\\}}(\omega+dd^{c}\varphi)^{n}+{\bf
1}_{\\{\varphi=v\\}}(\omega+dd^{c}\varphi)^{n}$ $\displaystyle\leq{\bf
1}_{\\{\varphi=u<v\\}}(\omega_{1}+dd^{c}\varphi)^{n}+{\bf
1}_{\\{\varphi=v\\}}(\omega_{2}+dd^{c}\varphi)^{n}$ $\displaystyle\leq{\bf
1}_{\\{\varphi=u<v\\}}(\omega_{1}+dd^{c}u)^{n}+{\bf
1}_{\\{\varphi=v\\}}(\omega_{2}+dd^{c}v)^{n}$ $\displaystyle\leq{\bf
1}_{\\{\varphi=u<v\\}}e^{\lambda\varphi}\max(f,g)dV_{X}+{\bf
1}_{\\{\varphi=v\\}}e^{\lambda\varphi}\max(f,g)dV_{X}$ $\displaystyle\leq
e^{\lambda\varphi}\max(f,g)dV_{X}.$

∎

The following is a key tool to our new approach for uniform estimates:

######  Lemma 1.10.

Fix $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ a concave increasing
function such that $\chi^{\prime}(0)\geq 1$. Let $\varphi,\phi$ be bounded
$\omega$-psh functions with $\varphi\leq\phi$, and set
$\psi=\phi+\chi\circ(\varphi-\phi)$. Then

$(\omega+dd^{c}P_{\omega}(\psi))^{n}\leq
1_{\\{P_{\omega}(\psi)=\psi\\}}(\chi^{\prime}\circ(\varphi-\phi))^{n}(\omega+dd^{c}\varphi)^{n}.$

The proof is identical to that of [GL21a, Lemma 1.6], a consequence of Theorem
1.8 and Lemma 1.2.

###  1.2. Comparison and domination principles

####  1.2.1. Comparison principle

The comparison principle plays a central role in Kähler pluripotential theory.
A ”modified comparison principle” has been established by Kołodziej-Nguyen
[KN15, Theorem 0.2] when $\omega$ is hermitian. We extend the latter in this
section, assuming that $\omega$ is merely big: we fix

  * •

an $\omega$-plurisubharmonic function $\rho$ with analytic singularities such
that $\omega+dd^{c}\rho\geq\delta\omega_{X}$ for some $\delta>0$; we set
$\Omega:=\\{\rho>-\infty\\}$;

  * •

a constant $B_{1}>0$ such that for all $x\in\Omega$,

$-B_{1}\omega_{\rho}^{2}\leq dd^{c}\omega\leq B_{1}\omega_{\rho}^{2},\text{
and }-B_{1}\omega_{\rho}^{3}\leq d\omega\wedge d^{c}\omega\leq
B_{1}\omega_{\rho}^{3}.$

The existence of $B_{1}$ is clear since $-B\omega_{X}^{2}\leq dd^{c}\omega\leq
B\omega_{X}^{2}$ for some $B>0$, and $-B\omega_{X}^{3}\leq d\omega\wedge
d^{c}\omega\leq B\omega_{X}^{3}$.

######  Theorem 1.11.

Assume $\omega$ is big and let $\rho,B_{1}$ be as above. Let $u$ be a bounded
$\omega$-psh function, and set $m:=\inf_{X}(u-\rho)$. Then for $s>0$ small
enough we have

$(1-4B_{1}s(n-1)^{2})^{n}\int_{\\{u<\rho+m+s\\}}\omega_{\rho}^{n}\leq\int_{\\{u<\rho+m+s\\}}\omega_{u}^{n}.$

In particular, $\omega$ is non-collapsing.

###### Proof.

We set $\phi=\max(u,\rho+m+s)$ and $U:=\\{u<\phi\\}=\\{u<\rho+m+s\\}$. Observe
that $U$ is relatively compact in the open set $\Omega$. For each $k\geq 0$ we
set $T_{k}:=\omega_{u}^{k}\wedge\omega_{\phi}^{n-k}$ and $T_{l}=0$ for $l\leq
0$. Set $a=B_{1}s(n-1)^{2}$. We prove by induction on $k=0,1,...,n-1$ that

(1.1) $(1-4a)\int_{U}T_{k}\leq\int_{U}T_{k+1}.$

The conclusion follows since $\omega_{\phi}^{n}=\omega_{\rho}^{n}$ on the
plurifine open set $U=\\{u<\phi\\}$.

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

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

Noting that $X\setminus U=\\{u=\phi\\}$ we can write

$\int_{U}(T_{0}-T_{1})=\int_{U}(\omega_{\phi}^{n}-\omega_{u}\wedge\omega_{\phi}^{n-1})\leq\int_{X}(\omega_{\phi}^{n}-\omega_{u}\wedge\omega_{\phi}^{n-1})=\int_{X}dd^{c}(\phi-u)\wedge\omega_{\phi}^{n-1}.$

Observe that

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

Since $0\leq\phi-u\leq s$ and $U=\\{u<\phi\\}$, it follows from Stokes’
theorem that

$\displaystyle\int_{X}dd^{c}(\phi-u)\wedge\omega_{\phi}^{n-1}$
$\displaystyle=\int_{X}(\phi-u)dd^{c}\omega_{\phi}^{n-1}$ $\displaystyle\leq
sB_{1}(n-1)\int_{U}(\omega_{\rho}^{2}\wedge\omega_{\phi}^{n-2}+(n-2)\omega_{\rho}^{3}\wedge\omega_{\phi}^{n-3})$
$\displaystyle\leq sB_{1}(n-1)^{2}\int_{U}\omega_{\phi}^{n},$

using that $\omega_{\rho}^{k}\wedge\omega_{\phi}^{n-k}=\omega_{\phi}^{n}$ on
the plurifine open set $U$. We thus get $\int_{U}(T_{0}-T_{1})\leq
sB_{1}(n-1)^{2}\int_{U}T_{0}$, proving (1.1) for $k=0$.

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

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

The same arguments as above therefore show that

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

where the third inequality uses the induction hypothesis, while the fourth
follows from the upper bound $4a<1/8$. From this we obtain (1.1) for $k$.

We finally prove that $\omega$ is non-collapsing. If $u\in{\rm PSH}(X,\omega)$
is bounded and $\omega_{u}^{n}=0$ then the first statement of the proposition
implies, since $\omega_{\rho}^{n}\geq\delta^{n}\omega_{X}^{n}$, that
$\omega_{X}^{n}(u<\rho+m+s)=0$ for $s>0$ small enough. Since $u$ and $\rho$
are $\omega$-psh and $\omega_{X}>0$, this implies $u\geq\rho+m+s$,
contradicting the definition of $m$. ∎

####  1.2.2. Domination principle

Several versions of the domination principle have been established in [LPT20,
Proposition 2.2], [GL21b, Proposition 2.8]. We shall need the following
generalization, valid for mildly unbounded $\omega$-psh functions:

######  Proposition 1.12.

Fix $\rho$ an $\omega$-psh function with analytic singularities such that
$\omega+dd^{c}\rho\geq\delta\omega_{X}$, with $\delta>0$. Let $u,v$ be
$\omega$-psh functions such that, for all $\varepsilon>0$,

$\inf_{X}(\min(u,v)-\varepsilon\rho)>-\infty.$

If $\omega_{u}^{n}\leq c\omega_{v}^{n}$ on $\\{u<v\\}\cap\\{\rho>-\infty\\}$
for some $c\in[0,1)$, then $u\geq v$.

The condition $\inf_{X}(\min(u,v)-\varepsilon\rho)>-\infty$ can be
equivalently formulated as follows: for any $\varepsilon>0$ there exists
$C_{\varepsilon}>0$ such that $u,v\geq\varepsilon\rho-C_{\varepsilon}$. In
particular $u$ and $v$ are locally bounded in the Zariski open set
$\Omega=\\{\rho>-\infty\\}$, hence the Monge-Ampère measures $\omega_{u}^{n}$
and $\omega_{v}^{n}$ are well-defined in $\Omega$.

###### Proof.

We fix a constant $a>0$ so small that $\omega_{u}^{n}\leq
c_{1}\omega_{\phi}^{n}$ on $\\{u<v\\}$, where $0<c_{1}<1$ and
$\phi=(1-a)v+a\rho$. By adding a constant we can assume that $v\geq\rho$ so
that $\omega_{u}^{n}\leq c_{1}\omega_{\phi}^{n}$ on $\\{u<\phi\\}$.

We now fix $b>1$ large and consider $\varphi:=P_{\omega}(bu-(b-1)\phi)$. It
follows from Theorem 1.8 that $\varphi$ is bounded on $X$ and
$\omega_{\varphi}^{n}$ is supported on the contact set
$\mathcal{C}:=\\{\varphi=bu-(b-1)\phi\\}$. Since
$b^{-1}\varphi+(1-b^{-1})\phi\leq u$ with equality on the contact set, Lemma
1.2 yields

${\bf 1}_{\mathcal{C}}b^{-n}\omega_{\varphi}^{n}+{\bf
1}_{\mathcal{C}}(1-b^{-1})^{n}\omega_{\phi}^{n}\leq{\bf
1}_{\mathcal{C}}\omega_{u}^{n}.$

Thus for $b>1$ large enough $\omega_{\varphi}^{n}$ vanishes in
$\mathcal{C}\cap\\{u<\phi\\}=\mathcal{C}\cap\\{\varphi<\phi\\}$, hence on
$\\{\varphi<\max(\varphi,\phi)\\}$ since $\omega_{\varphi}^{n}$ is supported
on $\mathcal{C}$. The domination principle ensures that $\varphi\geq\phi$. We
infer $u\geq\phi$ since

$\varphi=P_{\omega}(bu-(b-1)\phi))\leq bu-(b-1)\phi.$

Thus $u\geq(1-a)v+a\rho$ and letting $a\to 0^{+}$ yields the conclusion. ∎

Here is a useful consequence of the domination principle.

######  Corollary 1.13.

Assume $\omega$ is non-collapsing and let $u,v$ be bounded $\omega$-psh
functions. If

$(\omega+dd^{c}u)^{n}\leq\tau(\omega+dd^{c}v)^{n}$

for some constant $\tau>0$, then $\tau\geq 1$.

In particular if $(\omega+dd^{c}u)^{n}=c\mu$ and
$(\omega+dd^{c}v)^{n}=c^{\prime}\mu$ for the same measure $\mu$, then
$c=c^{\prime}$.

###### Proof.

If $\tau<1$ the domination principle yields $u\geq v+C$ for any constant $C$,
a contradiction. ∎

The same result holds when $u,v$ are mildly singular:

######  Corollary 1.14.

Assume $\omega$ is big and fix $\rho$ an $\omega$-psh function with analytic
singularities such that $\omega+dd^{c}\rho\geq\delta\omega_{X}$, with
$\delta>0$. Let $u,v$ be $\omega$-psh functions such that for all
$\varepsilon>0$,

$\inf_{X}(\min(u,v)-\varepsilon\rho)>-\infty.$

  1. (1)

If $(\omega+dd^{c}u)^{n}=\tau(\omega+dd^{c}v)^{n}$ then $\tau=1$.

  2. (2)

If $e^{-\lambda v}(\omega+dd^{c}v)^{n}\geq e^{-\lambda u}(\omega+dd^{c}u)^{n}$
for some $\lambda>0$, then $v\leq u$.

###### Proof.

The proof of (1) is similar to that of Corollary 1.13, so we focus on (2)
whose proof follows again from Proposition 1.12: fix $\delta>0$ and observe
that in $\\{u<v-\delta\\}\cap\\{\rho>-\infty\\}$ we have

$(\omega+dd^{c}u)^{n}\leq e^{\lambda(u-v)}(\omega+dd^{c}v)^{n}\leq
e^{-\lambda\delta}(\omega+dd^{c}(v-\delta))^{n},$

with $c=e^{-\lambda\delta}<1$. Thus $v-\delta\leq u$ and the conclusion
follows by letting $\delta\rightarrow 0$. ∎

##  2\. Uniform a priori estimates

###  2.1. Global $L^{\infty}$-bounds

####  2.1.1. Hermitian forms

When $\omega$ is a hermitian form and $\mu$ is a smooth volume form, it has
been shown by Tosatti-Weinkove [TW10b] that there exists a unique $c>0$ and a
unique smooth sup-normalized $\omega$-psh function $\varphi$ such that

$(\omega+dd^{c}\varphi)^{n}=c\mu.$

This landmark result relies on several previous attempts to generalize Yau’s
result to the hermitian setting, notably by Cherrier [Cher87], Hanani [Han96]
and Guan-Li [GL10]. The key result that was missing and provided by [TW10b] is
an a priori $L^{\infty}$-estimate for the solution. An alternative a priori
estimate using pluripotential techniques has been provided by Dinew-Kołodziej
in [DK12], who treated the case when $\mu=fdV_{X}$ with $f\in L^{p}$, $p>1$.

######  Theorem 2.1 (Dinew-Kołodziej).

Assume $\omega$ is a hermitian form, $p>1$ and $f\in L^{p}(dV_{X})$ is such
that
$A^{-1}\leq\left(\int_{X}f^{\frac{1}{n}}dV_{X}\right)^{n}\leq\left(\int_{X}f^{p}dV_{X}\right)^{\frac{1}{p}}\leq
A$ for some $A>1$. If $c>0$ and $u\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$
are such that $(\omega+dd^{c}u)^{n}=cfdV_{X}$, then

$c+c^{-1}+{\rm Osc}_{X}(u)\leq T,$

where the constant $T$ only depends on $p,n,\omega$ and an upper bound for
$A$.

The proof by Dinew-Kołodziej is a non trivial extension of the pluripotential
approach developed by Kołodziej in the Kähler case [Kol98], bypassing extra
difficulties coming from the non closedness of $\omega$. We provide a direct
proof of this result here, that only relies on local resolutions of Monge-
Ampère equations.

###### Proof.

Step 1. Constructing a bounded subsolution. We claim that there exist uniform
constants $0<m=m(p,\omega)<M=M(p,\omega)$ such that for any $0\leq g\in L^{p}$
with $\int_{X}g^{p}dV_{X}\leq 1$, we can find $v\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ such that

$(\omega+dd^{c}v)^{n}\geq mgdV_{X}\;\;\text{ and }\;\;{\rm Osc}_{X}v\leq M.$

Consider indeed a finite double cover of $X$ by small ”balls”
$B_{j},B_{j}^{\prime}=\\{\rho_{j}<0\\}$, with $B_{j}\subset\subset
B_{j}^{\prime}$ which are bounded in a local holomorphic chart. Here
$\rho_{j}:X\rightarrow\mathbb{R}$ denotes a smooth function which is strictly
plurisubharmonic in a neighborhood of $B_{j}^{\prime}$. We solve
$(dd^{c}v_{j})^{n}=gdV_{X}$ in $B_{j}^{\prime}$ with $-1$ boundary values. It
follows from [Kol98] that the plurisubharmonic solution $v_{j}$ is uniformly
bounded in $B_{j}^{\prime}$. Considering $\max(v_{j},\lambda_{j}\rho_{j})$ we
can choose $\lambda_{j}>1$ and obtain a uniformly bounded function $w_{j}$
with the following properties:

  * •

$w_{j}$ coincides with $v_{j}$ in $B_{j}$ where it satisfies
$(dd^{c}w_{j})^{n}=gdV_{X}$;

  * •

$w_{j}$ is plurisubharmonic in $B_{j}^{\prime}$ and uniformly bounded;

  * •

$w_{j}$ coincides with $\lambda_{j}\rho_{j}$ in $X\setminus B_{j}^{\prime}$
and in a neighborhood of $\partial B_{j}^{\prime}$.

As $w_{j}$ is smooth where it is not plurisubharmonic, its curvature
$dd^{c}w_{j}$ is bounded below by $-\delta^{-1}\omega$ for some uniform
$\delta>0$. Thus $\delta w_{j}$ is $\omega$-psh and
$v:=\frac{\delta}{N}\sum_{j=1}^{N}w_{j}$ is the bounded subsolution we are
looking for, since in $B_{j}$ we obtain

$(\omega+dd^{c}v)^{n}\geq\frac{\delta^{n}}{N^{n}}(\omega+dd^{c}w_{j})^{n}\geq\frac{\delta^{n}}{N^{n}}(dd^{c}w_{j})^{n}=\frac{\delta^{n}}{N^{n}}gdV_{X}.$

Step 2. Uniform a priori bounds. We can normalize $u$ by $\sup_{X}u=0$. It
follows from Skoda’s uniform integrability (see [GZ, Theorem 8.11]) that one
can find $\varepsilon>0$, $p^{\prime}=p^{\prime}(\varepsilon,p)\in(1,p)$, and
$C=C({\varepsilon},p)>0$ independent of $u$ such that $g=e^{-\varepsilon
u}f\in L^{p^{\prime}}$ with

$||g||_{p^{\prime}}\leq||f||_{p}\cdot||e^{-\varepsilon\frac{p}{p-p^{\prime}}u}||_{\frac{p}{p-p^{\prime}}}\leq
C({\varepsilon},p)A.$

Let $v$ be the bounded subsolution provided by Step 1 for the density
$\frac{g}{||g||_{p^{\prime}}}$, i.e.

$(\omega+dd^{c}v)^{n}\geq
m\frac{g}{||g||_{p^{\prime}}}dV_{X}\geq\frac{m^{\prime}}{A}fdV_{X}=\frac{m^{\prime}}{Ac}(\omega+dd^{c}u)^{n},$

using that $u\leq 0$ hence $g\geq f$. It follows from Corollary 1.13 that
$c\geq m^{\prime}/A$. Note that the upper bound for $c$ follows easily from
Lemma 1.3.

We finally observe that the uniform bound on $v$ also provides a uniform bound
for $u$. Indeed since $v$ is bounded we obtain

$(\omega+dd^{c}v)^{n}\geq m^{\prime\prime}e^{-\varepsilon u}fdV_{X}\geq
e^{\varepsilon(v-u-C)}cfdV_{X},$

hence Corollary 1.14 ensures that $u\geq v-C$. ∎

####  2.1.2. Semi-positive forms

We now extend this key $L^{\infty}$-estimate to the case when the form
$\omega$ is not necessarily positive, assuming instead that $v_{-}(\omega)>0$:

######  Theorem 2.2.

Let $\omega$ be semi-positive with $v_{-}(\omega)>0$. Let $\mu$ be a
probability measure such that ${\rm PSH}(X,\omega)\subset L^{m}(\mu)$ for some
$m>n$. Any solution $\varphi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ to
$(\omega+dd^{c}\varphi)^{n}=c\mu$, where $c>0$, satisfies

${\rm Osc}_{X}(\varphi)\leq T$

for some uniform constant $T$ which depends on upper bounds for
$\frac{c}{v_{-}(\omega)}$ and

$A_{m}({\mu}):=\sup\left\\{\left(\int_{X}(-\psi)^{m}d\mu\right)^{\frac{1}{m}}\;:\;\psi\in{\rm
PSH}(X,\omega)\text{ with }\sup_{X}\psi=0\right\\}.$

Since any quasi-psh function belongs to $L^{r}(dV_{X})$ for all $r>1$, this
theorem applies to measures $\mu=fdV_{X}$, where $f\in L^{p}$ with $p>1$, as
follows from Hölder inequality. As in [Kol98, Theorem 2.5.2] our technique
also covers the case of more general densities. We refer the reader to [GL21a,
Section 2.2] for more details.

###### Proof.

The proof is very similar to that of [GL21a, Theorem 2.1]. Shifting by an
additive constant we can assume $\sup_{X}\varphi=0$. Set

$T_{\max}:=\sup\\{t>0\;:\;\mu(\varphi<-t)>0\\}.$

Our goal is to establish a precise bound on $T_{max}$. By definition,
$-T_{max}\leq\varphi$ almost everywhere with respect to $\mu$, hence
everywhere by the domination principle, providing the desired a priori bound
${\rm Osc}_{X}(\varphi)\leq T_{max}$.

We let $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ denote a concave
increasing function such that $\chi(0)=0$ and $\chi^{\prime}(0)=1$. We set
$\psi=\chi\circ\varphi$, $u=P(\psi)$ and observe that

$\omega+dd^{c}\psi=\chi^{\prime}\circ\varphi\omega_{\varphi}+[1-\chi^{\prime}\circ\varphi]\omega+\chi^{\prime\prime}\circ\varphi
d\varphi\wedge d^{c}\varphi\leq\chi^{\prime}\circ\varphi\omega_{\varphi}.$

It follows from Lemma 1.10 that

$\frac{1}{c}(\omega+dd^{c}u)^{n}\leq
1_{\\{u=\psi\\}}(\chi^{\prime}\circ\varphi)^{n}\mu.$

Controlling the energy of $u$. We fix $\varepsilon>0$ so small that
$n<n+3\varepsilon=m.$ The concavity of $\chi$ and the normalization
$\chi(0)=0$ yields $|\chi(t)|\leq|t|\chi^{\prime}(t)$. Since
$u=\chi\circ\varphi$ on the support of $(\omega+dd^{c}u)^{n}$, Hölder’s
inequality yields

$\displaystyle\int_{X}(-u)^{\varepsilon}\frac{(\omega+dd^{c}u)^{n}}{c}$
$\displaystyle\leq$
$\displaystyle\int_{X}(-\chi\circ\varphi)^{\varepsilon}(\chi^{\prime}\circ\varphi)^{n}d\mu\leq\int_{X}(-\varphi)^{\varepsilon}(\chi^{\prime}\circ\varphi)^{n+\varepsilon}d\mu$
$\displaystyle\leq$
$\displaystyle\left(\int_{X}(-\varphi)^{n+2\varepsilon}d\mu\right)^{\frac{\varepsilon}{n+2\varepsilon}}\left(\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu\right)^{\frac{n+\varepsilon}{n+2\varepsilon}}$
$\displaystyle\leq$ $\displaystyle
A_{m}(\mu)^{\varepsilon}\left(\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu\right)^{\frac{n+\varepsilon}{n+2\varepsilon}}$

using that $\varphi$ belongs to the set of $\omega$-psh functions $v$
normalized by $\sup_{X}v=0$ which is compact in $L^{n+2\varepsilon}(\mu)$.

Controlling the norms $||u||_{L^{m}}$. We are going to choose below the weight
$\chi$ in such a way that
$\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu=B\leq 2$ is a finite
constant under control. This provides a uniform lower bound on $\sup_{X}u$ as
we now explain: indeed

$\displaystyle 0\leq(-\sup_{X}u)^{\varepsilon}\,\frac{v_{-}(\omega)}{c}$
$\displaystyle\leq$
$\displaystyle(-\sup_{X}u)^{\varepsilon}\int_{X}\frac{(\omega+dd^{c}u)^{n}}{c}$
$\displaystyle\leq$
$\displaystyle\int_{X}(-u)^{\varepsilon}\frac{(\omega+dd^{c}u)^{n}}{c}$
$\displaystyle\leq$ $\displaystyle 2A_{m}(\mu)^{\varepsilon}$

yields

$-\left(\frac{2c}{v_{-}({\omega})}\right)^{1/\varepsilon}A_{m}(\mu)\leq\sup_{X}u\leq
0.$

We infer that $u$ belongs to a compact set of $\omega$-psh functions, hence

$||u||_{L^{m}(\mu)}\leq\left[1+\left(\frac{2c}{v_{-}({\omega})}\right)^{1/\varepsilon}\right]A_{m}({\mu})=:\tilde{A}^{m}.$

From $u\leq\chi\circ\varphi\leq 0$ we get
$||\chi\circ\varphi||_{L^{m}}\leq||u||_{L^{m}}\leq\tilde{A}^{m}$. Using
Chebyshev inequality we thus obtain

(2.1) ${\mu}(\varphi<-t)\leq\frac{\tilde{A}}{|\chi|^{m}(-t)}.$

Choice of $\chi$. Recall that if $g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$
is increasing with $g(0)=1$, then

$\int_{X}g\circ(-\varphi)d\mu=\mu(X)+\int_{0}^{T_{\max}}g^{\prime}(t){\mu}(\varphi<-t)dt.$

Setting $g(t)=[\chi^{\prime}(-t)]^{n+2\varepsilon}$ we define $\chi$ by
imposing $\chi(0)=0$, $\chi^{\prime}(0)=1$, and

$g^{\prime}(t)=\frac{1}{(1+t)^{2}{\mu}(\varphi<-t)}\;\text{ if}\;t<T_{max}.$

This choice guarantees that $\chi$ is concave increasing with
$\chi^{\prime}\geq 1$ on $\varphi(X)$, and

$\int_{X}(\chi^{\prime}\circ\varphi)^{n+2\varepsilon}d\mu\leq\mu(X)+\int_{0}^{+\infty}\frac{dt}{(1+t)^{2}}\leq
2.$

A slight technical point is that one should first work on a compact
subinterval $[0,T^{\prime}]$ with $0<T^{\prime}<T_{max}$ and then let
$T^{\prime}$ tend to $T_{max}$. We refer the reader to the proof of [GL21a,
Theorem 2.1] for more details on this twist.

Conclusion. We set $h(t)=-\chi(-t)$ and work with the positive counterpart of
$\chi$. Note that $h(0)=0$ and
$h^{\prime}(t)=[g(t)]^{\frac{1}{n+2\varepsilon}}$ is positive increasing,
hence $h$ is convex. Observe also that $g(t)\geq g(0)=1$ hence
$h^{\prime}(t)=[g(t)]^{\frac{1}{n+2\varepsilon}}\geq 1$ yields

(2.2) $h(1)=\int_{0}^{1}h^{\prime}(s)ds\geq 1.$

Together with (2.1) our choice of $\chi$ yields, for all $t\in[0,T_{max})$,

$\frac{1}{(1+t)^{2}g^{\prime}(t)}={\mu}(\varphi<-t)\leq\frac{\tilde{A}}{h^{m}(t)}.$

This reads

$h^{m}(t)\leq\tilde{A}(1+t)^{2}g^{\prime}(t)=(n+2\varepsilon)\tilde{A}(1+t)^{2}h^{\prime\prime}(t)(h^{\prime})^{n+2\varepsilon-1}(t).$

Multiplying by $h^{\prime}$, integrating between $0$ and $t$, we infer that
for all $t\in[0,T_{max})$,

$\displaystyle\frac{h^{m+1}(t)}{m+1}$ $\displaystyle\leq$
$\displaystyle(n+2\varepsilon)\tilde{A}\int_{0}^{t}(1+s)^{2}h^{\prime\prime}(s)(h^{\prime})^{n+2\varepsilon}(s)$
$\displaystyle\leq$
$\displaystyle\frac{(n+2\varepsilon)\tilde{A}(t+1)^{2}}{n+2\varepsilon+1}\left((h^{\prime})^{n+2\varepsilon+1}(t)-1\right)$
$\displaystyle\leq$
$\displaystyle\tilde{A}(1+t)^{2}(h^{\prime})^{n+2\varepsilon+1}(t).$

Recall that we have set $m=n+3\varepsilon$ so that

$\alpha:=m+1=(n+2\varepsilon+1)+\varepsilon>\beta:=n+2\varepsilon+1>2.$

The previous inequality then reads ${(1+t)^{-\frac{2}{\beta}}}\leq
C{h^{\prime}(t)}{h(t)^{-\frac{\alpha}{\beta}}},$ for some uniform constant $C$
depending on $n,m,\tilde{A}$. Since $\alpha>\beta>2$ and $h(1)\geq 1$ (by
(2.2)), integrating the above inequality between $1$ and $T_{max}$ we obtain
$T_{max}\leq C^{\prime},$ for some uniform constant $C^{\prime}$ depending on
$C,\alpha,\beta$. ∎

####  2.1.3. Stability estimate

Adapting similarly the proof of [GL21a, Theorem 2.4], we also obtain:

######  Theorem 2.3.

Let $\omega,\mu$ be as in Theorem 2.2. Let $\varphi\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ be such that $\sup_{X}\varphi=0$ and
$(\omega+dd^{c}\varphi)^{n}=c\mu$ for some $c>0$. Then

$\sup_{X}(\phi-\varphi)_{+}\leq
T\left(\int_{X}(\phi-\varphi)_{+}d\mu\right)^{\gamma},$

for any $\phi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$, where
$\gamma=\gamma(n,m)>0$ and $T$ is a uniform constant which depends on upper
bounds for $\frac{c}{v_{-}(\omega)}$, $||\phi||_{L^{\infty}}$, and

$A_{m}({\mu}):=\sup\left\\{\left(\int_{X}(-\psi)^{m}d\mu\right)^{\frac{1}{m}}\;:\;\psi\in{\rm
PSH}(X,\omega)\text{ with }\sup_{X}\psi=0\right\\}.$

If $\phi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ also satisfies
$(\omega+dd^{c}\phi)^{n}=c^{\prime}\mu^{\prime}$ for some $c^{\prime}>0$ and
$\mu^{\prime}$ having similar properties as that of $\mu$, the above result
yields an $L^{1}-L^{\infty}$-stability estimate,

$||\phi-\varphi||_{L^{\infty}(X)}\leq
M||\phi-\varphi||_{L^{1}(\mu+\mu^{\prime})}^{\gamma}.$

Thus if a sequence of such solutions $\varphi_{j}$ converges in $L^{1}(\mu)$,
this should allow one to conclude that it actually uniformly converges. We
shall use this strong information on several occasions in the sequel.

###  2.2. Relative $L^{\infty}$-bounds

We consider in this section the degenerate complex Monge-Ampère equation

(2.3) $(\omega+dd^{c}\varphi)^{n}=cfdV_{X},$

where $0\leq f\in L^{1}(X)$. It follows from abstract measure theoretic
arguments that $f$ belongs to an Orlicz class $L^{w}(dV_{X})$ for some convex
increasing weight $w:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
$\frac{w(t)}{t}\rightarrow+\infty$ as $t\rightarrow+\infty$. The Luxemburg
norm

$||f||_{w}:=\inf\left\\{r>0\;:\;\int_{X}w\left(\frac{f(x)}{r}\right)dx<1\right\\}$

is finite if and only if $\int_{X}w\circ f\,dV_{X}<+\infty$.

If the weight $w$ grows fast enough at infinity (e.g. $w(t)=t^{p}$ with $p>1$,
or $w(t)=t(\log t)^{m}$ with $m>n$), then one can expect the solution to be
uniformly bounded (see [Kol98, Theorem 2.5.2]). We assume here this is not the
case.

When $f\leq e^{-\psi}$ for some quasi-psh function $\psi$, and $X$ is Kähler,
it has been shown by DiNezza-Lu [DnL17, Theorem 2] that the normalized
solution $\varphi$ to (2.3) is locally bounded in the complement of the
pluripolar set $\\{\psi=-\infty\\}$. We extend this important result here to
the hermitian setting:

######  Theorem 2.4.

Assume $v_{-}(\omega)>0$ and fix $A>0$. Fix $\mu=fdV_{X}$ a probability
measure with $f\in L^{w}(dV_{X})$ and assume $f\leq ge^{-A\psi}$, where $g\in
L^{p}(dV_{X})$ for some $p>1$ and $\psi\in{\rm PSH}(X,\omega)$. If $\varphi$
is a bounded $\omega$-psh function such that
$(\omega+dd^{c}\varphi)^{n}=cfdV_{X}$ and $\sup_{X}\varphi=0$, then

$\psi-\beta\leq\varphi\leq 0$

where $\beta>0$ depends on $p$, $w$ and upper bounds for
$c,v_{-}(\omega)^{-1},||g||_{L^{p}},\|f\|_{w},A$.

Proceeding by approximation, we are going to use in the sequel this lower
bound to ensure that some approximating solutions $\varphi_{j}$ are uniformly
bounded from below by a fixed function $\psi-\beta$. In particular any cluster
point $\varphi$ of the $\varphi_{j}$’s will be locally bounded in
$X\setminus\\{\psi=-\infty\\}$. Replacing $A$ and $\psi$ by $A/\alpha$ and
$\alpha\psi$, with $0<\alpha\leq 1$, and increasing $\beta$ in the statement,
one obtains

$\alpha\psi-C_{\alpha}\leq\varphi\leq 0.$

An interesting consequence of this improved a priori estimate is that it
produces limiting functions $\varphi$ having zero Lelong number at all points.

The proof of DiNezza-Lu makes use of a theory of generalized Monge-Ampère
capacities further developed in [DnL15]. It seems quite delicate to extend
their arguments to the hermitian setting, as they rely on very precise
integration by parts which produce several extra terms when $\omega$ is not
closed. Our new approach allows us to bypass these difficulties.

###### Proof.

Uniform integrability of $\varphi$. Set $\chi_{1}(t):=-(w^{*})^{-1}(-t)$,
where $w^{*}$ denotes the Legendre transform of $w$. Thus
$\chi_{1}:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ is a convex increasing,
such that $\chi_{1}(-\infty)=-\infty$ and

$\int_{X}(-\chi_{1}\circ\varphi)(\omega+dd^{c}\varphi)^{n}\leq c\int_{X}w\circ
fdV_{X}+c\int_{X}(-\varphi)dV_{X}\leq C_{0},$

as follows from the additive version of Hölder-Young inequality and the
compactness of $\sup$-normalized $\omega$-psh functions. We set, for $t\leq
0$,

$\chi(t)=-\int_{t}^{0}\sqrt{(-\chi_{1})(s)}ds\;\;\text{ and
}\;\;\chi_{2}(t)=-\sqrt{(-\chi_{1})}\circ\chi^{-1}(t).$

The weight $\chi$ is concave increasing with $\chi(-\infty)=-\infty$ and
$\chi^{\prime}(-\infty)=+\infty$, while $\chi_{2}$ is convex increasing with
$\chi_{2}(-\infty)=-\infty$. Set
$\Phi=P_{\omega}(\chi\circ\varphi)\leq\chi\circ\varphi\leq 0$. It follows from
Lemma 1.10 that

$\displaystyle\int_{X}(-\chi_{2})\circ\Phi\,(\omega+dd^{c}\Phi)^{n}$
$\displaystyle\leq$
$\displaystyle\int_{X}(-\chi_{2}\circ\chi\circ\varphi)\,\chi^{\prime}\circ\varphi\,(\omega+dd^{c}\varphi)^{n}$
$\displaystyle=$
$\displaystyle\int_{X}(-\chi_{1}\circ\varphi)(\omega+dd^{c}\varphi)^{n}\leq
C_{0}.$

Since $v_{-}(\omega)>0$ this provides a uniform bound on $\sup_{X}\Phi$
depending on $C_{0}$, $v_{-}(\omega)$ and $\chi_{2}$, so that $\Phi$ belongs
to a compact family of $\omega$-psh functions. It follows from Skoda’s uniform
integrability result that there exist uniform constants $\alpha,C>0$ such that

$\int_{X}\exp(-\alpha\chi\circ\varphi)dV_{X}\leq\int_{X}\exp(-\alpha\Phi)dV_{X}\leq
C<+\infty.$

Finally since $\chi(t)$ grows faster than $t$ at infinity, we infer that for
any $\lambda>0$,

(2.4) $\int_{X}\exp(-\lambda\varphi)dV_{X}\leq M_{\lambda}$

for a uniform constant $M_{\lambda}$ which depends on upper bounds for
$||f||_{w}$ and $v_{-}(\omega)^{-1}$.

Quasi-psh envelope. Consider $u:=P_{\omega}(2\varphi-\psi)$. One can show that
$\sup_{X}u\leq C_{0}$ is uniformly bounded from above by generalizing [GZ,
Theorem 9.17.1] to the hermitian setting. The previous argument also shows
that $\sup_{X}P_{\omega}(2\varphi)$ is uniformly bounded from below, hence the
same holds for $P_{\omega}(2\varphi-\psi)$ as we can assume without loss of
generality that $\psi\leq 0$.

Since $\varphi$ is bounded, we have $u=P_{\omega}(2\varphi-\max(\psi,-t))$ for
$t>0$ large enough. We can thus assume that $\psi$ is also bounded. Since
$2\varphi-\psi$ is bounded and quasi continuous, it follows from Theorem 1.8
that the Monge-Ampère measure $(\omega+dd^{c}u)^{n}$ is concentrated on the
contact set ${\mathcal{C}}=\\{u=2\varphi-\psi\\}$. We claim that

$(\omega+dd^{c}u)^{n}\leq 1_{{\mathcal{C}}}2^{n}(\omega+dd^{c}\varphi)^{n}.$

Indeed consider $v=u+\psi$. This is a $(2\omega)$-psh function such that
$v\leq 2\varphi$; it follows therefore from Lemma 1.2 that

$1_{\\{v=2\varphi\\}}(2\omega+dd^{c}v)^{n}\leq
1_{\\{v=2\varphi\\}}(2\omega+dd^{c}(2\varphi))^{n}$

The claim follows since $\\{v=2\varphi\\}={\mathcal{C}}$ and
$(\omega+dd^{c}u)^{n}\leq(2\omega+dd^{c}v)^{n}$. Here the boundedness of
$\psi$ ensures that the Monge-Ampère measure $(\omega+dd^{c}v)^{n}$ is well-
defined. Now $(\omega+dd^{c}\varphi)^{n}=cfdV_{X}\leq cge^{-A\psi}dV_{X}$,
with

$1_{{\mathcal{C}}}ge^{-A\psi}=1_{{\mathcal{C}}}ge^{Au}e^{-2A\varphi}\leq
e^{A\sup_{X}u}ge^{-2A\varphi}.$

Using Hölder inequality and (2.4), we infer that
$g^{\prime}=ge^{-2A\varphi}\in L^{p^{\prime}}$, for some $p^{\prime}>1$
arbitrarily close to $p$, and with an upper bound on
$||g^{\prime}||_{L^{p^{\prime}}}$ that only depends on $w$ and upper bounds
for $A,||g||_{L^{p}}$, and $v_{-}(\omega)^{-1}$.

Since $(\omega+dd^{c}u)^{n}\leq 2^{n}ce^{AC_{0}}g^{\prime}dV_{X}$, we can
invoke Theorem 2.2 to ensure that the oscillation of $u$ is uniformly bounded.
The desired lower bound follows since $2\varphi=(2\varphi-\psi)+\psi\geq
u+\psi$. ∎

##  3\. Existence of locally bounded solutions

We now show the existence of solutions to degenerate complex Monge-Ampère
equations under various positivity assumptions on $\omega$. We first establish
a general existence result for bounded solutions in section 3.1, then treat
the case of solutions that are merely locally bounded in a Zariski open set in
section 3.3. Higher regularity of these solutions will be studied in section
4.1.

###  3.1. Bounded solutions

####  3.1.1. The normalizing constant

Fix $p>1$ and $0\leq f\in L^{p}(X,dV_{X})$ with $\int_{X}fdV>0$. It follows
from [KN15, Theorem 0.1] and [N16, Corollary 0.2] that for each
$\varepsilon>0$, there exists a unique constant
$c_{\varepsilon}=c(\omega+\varepsilon\omega_{X},f)>0$ such that one can find
$u\in{\rm PSH}(X,\omega+\varepsilon\omega_{X})\cap L^{\infty}(X)$ satisfying

$(\omega+\varepsilon\omega_{X}+dd^{c}u)^{n}=c_{\varepsilon}fdV.$

Observe that $\varepsilon\mapsto c_{\varepsilon}$ is increasing. Indeed assume
$\varepsilon>\varepsilon^{\prime}$ and
$u_{\varepsilon},u_{\varepsilon^{\prime}}$ are bounded solutions to the
corresponding equations. Then

$\displaystyle(\omega+\varepsilon\omega_{X}+dd^{c}u_{\varepsilon^{\prime}})^{n}$
$\displaystyle\geq$
$\displaystyle(\omega+\varepsilon^{\prime}\omega_{X}+dd^{c}u_{\varepsilon^{\prime}})^{n}$
$\displaystyle=$ $\displaystyle
c_{\varepsilon^{\prime}}fdV=\frac{c_{\varepsilon^{\prime}}}{c_{\varepsilon}}(\omega+\varepsilon\omega_{X}+dd^{c}u_{\varepsilon})^{n}.$

It follows therefore from Corollary 1.13 that $c_{\varepsilon}\geq
c_{\varepsilon^{\prime}}$. We can thus consider:

######  Definition 3.1.

We set $c(\omega,f):=\lim_{\varepsilon\to 0}c_{\varepsilon}.$

This definition is independent of the choice of the hermitian metric
$\omega_{X}$:

######  Proposition 3.2.

The constant $c(\omega,f)$ does not depend on $\omega_{X}$. Moreover

$c(\omega,f)=\inf\\{c(\omega^{\prime},f)\;:\;\omega^{\prime}\text{ hermitian
form such that }\omega^{\prime}>\omega\\}.$

###### Proof.

Fix $\omega_{1},\omega_{2}$ two hermitian metrics on $X$. Then
$A^{-1}\omega_{1}\leq\omega_{2}\leq A\omega_{1}$ for some positive constant
$A$. It follows that, for all $\varepsilon>0$,

$(\omega+A^{-1}\varepsilon\omega_{1})\leq(\omega+\varepsilon\omega_{2})\leq(\omega+A\varepsilon\omega_{1}).$

Corollary 1.13 ensures that

$c(\omega+A^{-1}\varepsilon\omega_{1},f)\leq
c(\omega+\varepsilon\omega_{2},f)\leq c(\omega+A\varepsilon\omega_{1},f).$

Letting $\varepsilon\to 0$ yields the conclusion. The last assertion follows
similarly. ∎

####  3.1.2. Existence of solutions

######  Lemma 3.3.

Fix $p>1$ and assume either $\omega$ is big or $v_{-}(\omega)>0$. Then there
exists $0<c=c(p,\omega)$ such that for any $f\in L^{p}(dV_{X})$ with
$\int_{X}f^{p}dV_{X}\leq 1$, we can find $u\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ such that

$(\omega+dd^{c}u)^{n}\geq cfdV_{X}\;\;\text{and}\;\;-1\leq u\leq 0.$

###### Proof.

We first assume $\omega$ is big. We fix $\rho\in{\rm PSH}(X,\omega)$ with
analytic singularities such that $\omega+dd^{c}\rho\geq\delta\omega_{X}$ for
some $\delta>0$, with $\sup_{X}\rho=-1$. Since $\rho$ belongs to $L^{r}$ for
any $r>1$, Hölder inequality ensures that $|\rho|^{2n}f\in L^{q}(dV_{X})$ for
some $q>1$. It follows from the first step of the proof of Theorem 2.1 that
there exist a uniform $c_{1}>0$ and $v\in{\rm PSH}(X,\omega_{X})\cap
L^{\infty}(X)$ such that $\sup_{X}v=-1$ and

$(\omega_{X}+dd^{c}v)^{n}\geq c_{1}|\rho|^{2n}fdV_{X}.$

Observe that the function $\delta v+\rho$ is $\omega$-psh with
$\omega+dd^{c}(\delta v+\rho)\geq\delta(\omega_{X}+dd^{c}v)$. Set $u:=-(\delta
v+\rho)^{-1}=\chi\circ(\delta v+\rho)$ with $\chi(t)=-t^{-1}$ convex
increasing on $\mathbb{R}^{-}$. Our normalizations ensure $-1\leq u\leq 0$ and
a direct computation yields

$\omega+dd^{c}u\geq\delta\chi^{\prime}\circ(\delta
v+\rho)(\omega_{X}+dd^{c}v)=\frac{\delta}{(\delta
v+\rho)^{2}}(\omega_{X}+dd^{c}v).$

We infer $\omega_{u}^{n}\geq|\delta
v+\rho|^{-2n}\delta^{n}c_{1}|\rho|^{2n}fdV_{X}$. Since $v\leq-1$ is bounded
and $\rho\leq-1$, it follows that $\omega_{u}^{n}\geq afdV_{X}$ for some
uniform constant $a>0$.

We now assume $v_{-}(\omega)>0$. Set $\omega_{j}=\omega+2^{-j}\omega_{X}$.
Rescaling $\omega_{X}$ if necessary we can assume that
$\omega\leq\omega_{j}\leq\omega_{X}$. Using [KN15, Theorem 0.1] we solve

$(\omega_{j}+dd^{c}u_{j})^{n}=c_{j}(1+f)dV_{X},\;\sup_{X}u_{j}=0$

for any fixed $j>0$, where $u_{j}\in{\rm PSH}(X,\omega_{j})\cap
L^{\infty}(X)$. Fix $a>0$ such that $\omega_{X}^{n}\geq adV_{X}$, and observe
that $(\omega_{X}+dd^{c}u_{j})^{n}\geq(\omega_{j}+dd^{c}u_{j})^{n}\geq
c_{j}dV_{X},$ hence

$(\omega_{X}+dd^{c}u_{j})\wedge\omega_{X}^{n-1}\geq
c_{j}^{\frac{1}{n}}a^{\frac{n-1}{n}}dV_{X},$

as follows from Lemma 1.3.

We infer that $c_{j}$ is uniformly bounded from above. Since
$||(1+f)||_{p}\leq 2$ and $v_{-}(\omega_{j})\geq v_{-}(\omega)>0$, it follows
from Theorem 2.2 that $u_{j}$ is uniformly bounded.

Extracting and relabelling we can assume that $u_{j}\rightarrow u\in{\rm
PSH}(X,\omega)$. Theorem 2.2 yields $-T\leq u\leq 0$ for some uniform constant
$T>0$. Set

$\Phi_{j}:=\left(\sup_{\ell\geq j}u_{\ell}\right)^{*}.$

By Lemma 1.9, we have

$(\omega+2^{-j}\omega_{X}+dd^{c}\Phi_{j})^{n}\geq v_{-}(\omega)fdV_{X}.$

Since $\Phi_{j}\searrow u\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$, we obtain
$(\omega+dd^{c}u)^{n}\geq v_{-}(\omega)fdV_{X}$.

If $0\leq T\leq 1$ we are done, otherwise we consider $\tilde{u}=u/T\in{\rm
PSH}(X,\omega)$ and observe that $-1\leq\tilde{u}\leq 0$ with
$(\omega+dd^{c}\tilde{u})^{n}\geq T^{-n}v_{-}(\omega)fdV_{X}$. ∎

######  Theorem 3.4.

Let $\mu=fdV_{X}$ be a probability measure, where $0\leq f\in L^{p}(dV_{X})$
for some $p>1$. Assume either $\omega$ is big or $v_{-}(\omega)>0$.

  1. (1)

For any $\lambda>0$ there exists a unique $\varphi_{\lambda}\in{\rm
PSH}(X,\omega)\cap L^{\infty}(X)$ such that

$(\omega+dd^{c}\varphi_{\lambda})^{n}=e^{\lambda\varphi_{\lambda}}fdV_{X}.$

Moreover $||\varphi_{\lambda}||_{\infty}\leq C$, where $C$ depends on upper
bounds on $\|f\|_{p}$, $\|f\|_{1/n}^{-1},\lambda^{-1}$.

  2. (2)

One has $c(\omega,f)>0$ and there exists $\varphi\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ such that

$(\omega+dd^{c}\varphi)^{n}=c(\omega,f)\mu.$

Here $c(\omega,f)$ denotes the normalizing constant from Definition 3.1. It is
possible to study Hölder regularity of $\varphi$ following [DDGKPZ], [LPT20],
[KN19], but we leave it for a future project.

###### Proof.

We first prove (1). For simplicity we assume $\lambda=1$. By [N16, Theorem
0.1], for each $\varepsilon>0$ there exists $\varphi_{\varepsilon}\in{\rm
PSH}(X,\omega+\varepsilon\omega_{X})\cap L^{\infty}(X)$ such that

$(\omega+\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon})^{n}=e^{\varphi_{\varepsilon}}\mu.$

The bounded subsolution $u$ obtained in Lemma 3.3 satisfies

$(\omega+\varepsilon\omega_{X}+dd^{c}u)^{n}\geq(\omega+dd^{c}u)^{n}\geq
e^{u+\ln c}\mu,$

since $u\leq 0$. The domination principle now yields $u+\ln
c\leq\varphi_{\varepsilon}$, for all $\varepsilon>0$.

It follows again from the domination principle that $\varphi_{\varepsilon}$
decreases to some function $\varphi\in{\rm PSH}(X,\omega)$ which is bounded
below by the previous estimate, and bounded above by $\varphi_{1}$, hence it
is uniformly bounded.

The uniqueness is a simple consequence of Corollary 1.14.

We now prove (2). It follows from [KN15, KN19, Theorem 0.1] that there exists
$c_{j}>0$ and $\varphi_{j}\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ such that
$\sup_{X}\varphi_{j}=-1$ and

$(\omega_{j}+dd^{c}\varphi_{j})^{n}=c_{j}\mu,$

where $\omega_{j}:=\omega+2^{-j}\omega_{X}$. By Lemma 3.3 we have a lower
bound $c_{j}\geq a$, while the domination principle ensures that $j\mapsto
c_{j}$ is decreasing, so $c_{j}\rightarrow c(\omega,f)\in[a,c_{1}]$.

Fix $\gamma>0$ so small that $g_{j}:=e^{-\gamma\varphi_{j}}f$ is uniformly
bounded in $L^{q}(X)$ for some $q>1$. This is possible thanks to the Skoda
integrability theorem [Sko72] and Hölder inequality. By the previous step the
solutions $u_{j}\in{\rm PSH}(X,\omega)$ to

$(\omega+dd^{c}u_{j})^{n}=e^{\gamma u_{j}}g_{j}dV_{X}$

are uniformly bounded. Since $u_{j}$ satisfies

$(\omega+2^{-j}\omega_{X}+dd^{c}u_{j})^{n}\geq e^{\gamma(u_{j}-\gamma^{-1}\ln
c_{1})}c_{1}g_{j}dV_{X},$

while $\varphi_{j}$ satisfies

$(\omega+2^{-j}\omega_{X}+dd^{c}\varphi_{j})^{n}\leq
e^{\gamma\varphi_{j}}c_{1}g_{j}dV_{X},$

the domination principle ensures that $\varphi_{j}\geq u_{j}-\gamma^{-1}\ln
c_{1}\geq-C$.

Extracting a subsequence we can assume that $\varphi_{j}$ converges in $L^{1}$
and a.e. to $\varphi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$. Set

$\Phi_{j}^{+}:=\left(\sup_{\ell\geq j}\varphi_{\ell}\right)^{*}\;\;\text{ and
}\;\;\Phi_{j}^{-}:=P_{\omega}\left(\inf_{\ell\geq j}\varphi_{\ell}\right).$

The function $\Phi_{j}^{+}$ (resp. $\Phi_{j}^{-}$) is $\omega_{j}$-psh (resp.
$\omega$-psh) and uniformly bounded on $X$. It follows from Lemma 1.9 that

$(\omega_{j}+dd^{c}\Phi_{j}^{+})^{n}\geq c_{j}^{-}fdV_{X}\;\;\text{ with
}\;\;c_{j}^{-}:=\inf_{\ell\geq j}c_{\ell}.$

Since $\Phi_{j}^{+}$ decreases to $\varphi$, while $c_{j}^{-}$ increases to
$c$, it follows from the continuity of the complex Monge-Ampère along
monotonic sequences (see [BT76, BT82]) that

$(\omega+dd^{c}\varphi)^{n}\geq cfdV_{X}\;\;\text{ on }\;\;X.$

Writing
$(\omega_{j}+dd^{c}\varphi_{j})^{n}=e^{\varphi_{j}-\varphi_{j}}c_{j}fdV_{X}$,
it follows from Lemma 1.9 that

$(\omega+dd^{c}\Phi_{j}^{-})^{n}\leq e^{\Phi_{j}^{-}-\inf_{\ell\geq
j}\varphi_{\ell}}c_{j}^{+}fdV_{X}\;\;\text{ with
}\;\;c_{j}^{+}:=\sup_{\ell\geq j}c_{\ell}.$

The functions $\Phi_{j}^{-}$ increase to some $\varphi^{\prime}\in{\rm
PSH}(X,\omega)\cap L^{\infty}(X)$ which satisfies

$(\omega+dd^{c}\varphi^{\prime})^{n}\leq
e^{\varphi^{\prime}-\varphi}cfdV_{X}\;\text{on}\;X.$

The domination principle then yields $\varphi^{\prime}\geq\varphi$ but we also
have by construction that $\varphi^{\prime}\leq\varphi$, hence
$\varphi=\varphi^{\prime}$ is a solution. ∎

###  3.2. Uniqueness of solutions

We now study the dependence of the solution with respect to the density,
dealing as well with the case when the RHS depends on the solution in a convex
way.

######  Theorem 3.5.

Assume $\omega$ is either big or such that $v_{-}(\omega)>0$. Fix
$f_{1},f_{2}\in L^{p}(dV_{X})$ with $p>1$ and
$A^{-1}\leq\left(\int_{X}f_{i}^{\frac{1}{n}}dV_{X}\right)^{n}\leq\left(\int_{X}f_{i}^{p}dV_{X}\right)^{\frac{1}{p}}\leq
A$, for some constant $A>1$. Assume $\varphi_{1},\varphi_{2}\in{\rm
PSH}(X,\omega)\cap L^{\infty}(X)$ satisfy

$(\omega+dd^{c}\varphi_{i})^{n}=e^{\lambda\varphi_{i}}f_{i}dV_{X}.$

  1. (1)

If $\lambda>0$, then $||\varphi_{1}-\varphi_{2}||_{\infty}\leq
T||f_{1}-f_{2}||^{\frac{1}{n}},$ where $T$ is a constant which depends on
$n,p$ and upper bounds for $A$, $\lambda^{-1},\lambda$.

  2. (2)

If $\lambda=0$ and $f_{1}\geq c_{0}>0$, then the same estimate holds with $T$
depending on $n,p$ and upper bounds for $A,c_{0}^{-1}$.

In particular, there is at most one bounded $\omega$-psh solution $\varphi$ to
the equation $(\omega+dd^{c}\varphi)^{n}=e^{\lambda\varphi}f\,dV_{X}$ in case
$\lambda>0$, or when $\lambda=0$ and $f\geq c_{0}>0$.

When $\omega$ is hermitian such a stability estimate has been provided by
Kołodziej-Nguyen for $\lambda=0$ [KN19]. We adapt some arguments of Lu-Phung-
Tô [LPT20], [GLZ18], who treated the case $\lambda>0$, and obtained refined
estimates in the case $\lambda=0$.

###### Proof.

(1) We first consider the simpler case when $\lambda>0$. Rescaling we can
assume without loss of generality that $\lambda=1$. We first observe that by
Theorem 3.4 $\varphi_{1},\varphi_{2}$ are uniformly bounded. Since
$\varphi_{1},\varphi_{2}$ play the same role, it suffices to show that
$\varphi_{1}-\varphi_{2}\leq T\|f_{1}-f_{2}\|_{p}^{1/n}$. Using Theorem 3.4 we
solve

$(\omega+dd^{c}u)^{n}=c\left(\frac{|f_{1}-f_{2}|}{\|f_{1}-f_{2}\|_{p}^{1/n}}+1\right)dV_{X},$

with $\sup_{X}u=0$ and $|u|,c,c^{-1}$ uniformly bounded. We can assume that

$\varepsilon:=e^{(\sup_{X}u-\log c)/n}\|f_{1}-f_{2}\|_{p}^{1/n}$

is small enough. We next consider

$\phi:=(1-\varepsilon)\varphi_{1}+\varepsilon u-C\varepsilon,$

and proceed exactly the same as in [LPT20, Theorem 1.1] to prove that

$(\omega+dd^{c}\phi)^{n}\geq e^{\phi}f_{2}dV_{X}.$

The domination principle thus gives $\phi\leq\varphi_{2}$, yielding the
desired estimate.

(2) We now take care of the case $\lambda=0$. As usual we use $C$ to denote
various uniform constants. By Theorem 3.4 we have that
$\sup_{X}(|\varphi_{1}|+|\varphi_{2}|)\leq C$.

Step 1. We first assume that $0<f_{1},f_{2}$ are smooth, and

(3.1) $2^{-\varepsilon}f_{1}\leq f_{2}\leq 2^{\varepsilon}f_{1},$

for some small constant $\varepsilon>0$. Our goal is to show that
$|\varphi_{1}-\varphi_{2}|\leq C\varepsilon$.

Let $m=\inf_{X}(\varphi_{1}-\varphi_{2})$ and
$M=\sup_{X}(\varphi_{1}-\varphi_{2})$. To simplify the notation we set
$\mu=f_{1}^{p}dV_{X}$. Recall that by Lemma 3.3 there exists a constant
$c=c(\omega,p)>0$ such that if $0\leq f\in L^{p}(X,dV_{X})$ with
$\|f\|_{p}\leq 1$ then we can find $u\in{\rm PSH}(X,\omega)$ such that
$(\omega+dd^{c}u)^{n}\geq 4cfdV_{X}$ with $-1\leq u\leq 0$. We define

$t_{0}:=\sup\\{t\in[m,M]\;:\;\mu(\varphi_{1}<\varphi_{2}+t)\leq c^{p}\\},$
$T_{0}:=\inf\\{t\in[m,M]\;:\;\mu(\varphi_{1}>\varphi_{2}+t)\leq c^{p}\\}.$

Step 1.1. We claim that $t_{0}\leq m+C\varepsilon$ and $T_{0}\geq
M-C\varepsilon$.

Assume $t\leq t_{0}$ and $\mu(u<v+t)\leq c^{p}$. Set

$\hat{f}:=\frac{1}{2c}{\bf
1}_{\\{\varphi_{1}<\varphi_{2}+t\\}}f_{1}+\frac{1}{2\|f_{1}\|_{p}}{\bf
1}_{\\{\varphi_{1}\geq\varphi_{2}+t\\}}f_{1}.$

It follows from Lemma 3.3 that $c(\omega,\hat{f})\geq 4c$, since

$\displaystyle\int_{X}\hat{f}^{p}dV_{X}$
$\displaystyle\leq\frac{1}{(2c)^{p}}\int_{\\{\varphi_{1}<\varphi_{2}+t\\}}f^{p}_{1}dV+2^{-p}\leq
1.$

Using Theorem 3.4 we find $\phi\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ such
that $\sup_{X}(\phi-\varphi_{2})=0$ and

$(\omega+dd^{c}\phi)^{n}=c(\omega,\hat{f})\hat{f}dV_{X}.$

Since $\hat{f}\geq C^{-1}f_{1}$, it follows that $4c\leq c(\omega,\hat{f})\leq
C$, hence $\phi$ is uniformly bounded.

We next consider $\psi:=(1-\varepsilon)\varphi_{2}+\varepsilon\phi+t$. Then
$\\{\varphi_{1}<\psi\\}\subset\\{\varphi_{1}<\varphi_{2}+t\\}$ and on the
latter set we have

$\omega_{\psi}^{n}\geq((1-\varepsilon)2^{-\varepsilon/n}+2^{1/n}\varepsilon)^{n}f_{1}dV\geq(1+\gamma)f_{1}dV=(1+\gamma)\omega_{\varphi_{1}}^{n},$

for some $\gamma>0$. The domination principle ensures that
$\varphi_{1}\geq\psi\geq\varphi_{2}+t-C\varepsilon$, hence
$t\leq\varphi_{1}-\varphi_{2}+C\varepsilon$. Since this inequality is true for
all $x\in X$, we can take $x$ such that
$\varphi_{1}(x)-\varphi_{2}(x)=\inf_{X}(\varphi_{1}-\varphi_{2})$ and obtain

(3.2) $t_{0}\leq m+C\varepsilon.$

Using a similar argument we obtain

(3.3) $T_{0}\geq M-C\varepsilon.$

Step 1.2. Fixing $t_{0}<a<b<T_{0}$, we claim that

(3.4) $I_{a,b}:=\int_{E_{a,b}}d(\varphi_{1}-\varphi_{2})\wedge
d^{c}(\varphi_{1}-\varphi_{2})\wedge\omega_{\varphi_{1}}^{n-1}\leq
C(b-a)(\varepsilon+b-t_{0}),$

and

(3.5) $I_{a,b}\int_{E_{a,b}}\omega_{\varphi_{1}}\wedge\omega_{X}^{n-1}\geq
C^{-1}\left(\int_{E_{a,b}}|\partial(\varphi_{1}-\varphi_{2})|dV_{X}\right)^{2},$

where $E_{a,b}:=\\{\varphi_{2}+a<\varphi_{1}<\varphi_{2}+b\\}$.

To prove (3.4) we set $\varphi:=\max(\varphi_{1},\varphi_{2}+a)$,
$\psi:=\max(\varphi_{1},\varphi_{2}+b)$, and consider

$J:=\int_{X}(\varphi-\psi)(\omega_{\psi}^{n}-\omega_{\varphi}^{n})=\int_{X}(\varphi-\psi)dd^{c}(\psi-\varphi)\wedge
T,$

where
$T=\sum_{j=0}^{n-1}\omega_{\varphi}^{j}\wedge\omega_{\psi}^{n-1-j}\geq\omega_{\varphi}^{n-1}$.
Integrating by parts we obtain

$\displaystyle J$ $\displaystyle=\int_{X}d(\varphi-\psi)\wedge
d^{c}(\varphi-\psi)\wedge T-\frac{1}{2}\int_{X}(\varphi-\psi)^{2}dd^{c}T$
$\displaystyle\geq\int_{E_{a,b}}d(\varphi_{1}-\varphi_{2})\wedge
d^{c}(\varphi_{1}-\varphi_{2})\wedge\omega_{\varphi_{1}}^{n-1}-C(b-a)^{2}.$

We have used here $0\leq\psi-\varphi\leq b-a$,
$\omega_{\varphi_{1}}^{n-1}=\omega_{\varphi}^{n-1}$ on
$\\{\varphi_{1}>\varphi_{2}+a\\}$, and

$dd^{c}T\geq-C(\omega^{2}\wedge(\omega_{\varphi_{1}}+\omega_{\varphi_{2}})^{n-2}+\omega^{3}\wedge(\omega_{\varphi_{1}}+\omega_{\varphi_{2}})^{n-3}),$

the integral over $X$ of the latter form being uniformly bounded. We have also
used that $\partial\psi=0$ a.e. on
$\\{\varphi_{1}\geq\varphi_{2}+b\\}\cup\\{\varphi_{1}\leq\varphi_{2}+a\\}$.

On the other hand, $(\varphi-\psi)(\omega_{\varphi}^{n}-\omega_{\psi}^{n})=0$
on $\\{\varphi_{1}<\varphi_{2}+a\\}\cup\\{\varphi_{1}\geq\varphi_{2}+b\\}$,
and

$\left|\int_{E_{a,b}}(\varphi-\psi)(\omega_{\varphi}^{n}-\omega_{\psi}^{n})\right|\leq(b-a)\int_{X}|f-g|dV\leq
C(b-a)\varepsilon.$

Now

$\displaystyle\left|\int_{\\{\varphi_{1}=\varphi_{2}+a\\}}(\varphi-\psi)(\omega_{\psi}^{n}-\omega_{\varphi}^{n})\right|$
$\displaystyle=(b-a)\left|\int_{\\{\varphi_{1}=\varphi_{2}+a\\}}(\omega_{\varphi}^{n}-\omega_{\varphi_{2}}^{n})\right|$
$\displaystyle=(b-a)\left|\int_{\\{\varphi_{1}\leq\varphi_{2}+a\\}}(\omega_{\varphi}^{n}-\omega_{\varphi_{2}}^{n})\right|$
$\displaystyle\leq
C(b-a)\varepsilon+(b-a)\left|\int_{\\{\varphi_{1}\leq\varphi_{2}+a\\}}(\omega_{\varphi}^{n}-\omega_{\varphi_{1}}^{n})\right|$
$\displaystyle=C(b-a)\varepsilon+(b-a)\left|\int_{X}(\omega_{\varphi}^{n}-\omega_{\varphi_{1}}^{n})\right|$
$\displaystyle\leq C(b-a)\varepsilon+(b-a)C\sup_{X}(\varphi-\varphi_{1})$
$\displaystyle\leq C(b-a)(\varepsilon+a-t_{0}).$

These inequalities are justified as follows:

  * •

in the second line we used the identity
$\omega_{\varphi}^{n}=\omega_{\varphi_{2}}^{n}$ on
$\\{\varphi_{1}<\varphi_{2}+a\\}$.

  * •

in the third line we used
$\left|\int_{\\{\varphi_{1}\leq\varphi_{2}+a\\}}(\omega_{\varphi_{1}}^{n}-\omega_{\varphi_{2}}^{n})\right|\leq\int_{X}|f-g|dV_{X}\leq
C\varepsilon.$

  * •

in the fourth line we used the identity
$\omega_{\varphi}^{n}=\omega_{\varphi_{1}}^{n}$ on
$\\{\varphi_{1}>\varphi_{2}+a\\}$.

  * •

the fifth line consists in the following integration by part,

$\left|\int_{X}(\omega_{\varphi}^{n}-\omega_{\varphi_{1}}^{n})\right|=\left|\int_{X}(\varphi-\varphi_{1})dd^{c}\sum_{j=0}^{n-1}\omega_{\varphi}^{j}\wedge\omega_{\varphi_{1}}^{n-1-j}\right|\leq
C\sup_{X}(\varphi-\varphi_{1}).$

Altogether this yields (3.4), as

$I_{a,b}\leq C(b-a)(\varepsilon+a-t_{0}+b-a)\leq
C_{1}(b-a)(b-t_{0}+\varepsilon).$

We next prove (3.5). The following pointwise inequality is a reformulation of
[Pop16, Lemma 3.1]: if $\alpha\geq 0$ is a $(1,1)$-form and
$\omega_{1},\omega_{2}$ are Hermitian forms then

$\frac{\alpha\wedge\omega_{1}^{n-1}}{\omega_{2}^{n}}\frac{\omega_{1}\wedge\omega_{2}^{n-1}}{\omega_{2}^{n}}\geq\frac{1}{n}\frac{\omega_{1}^{n}}{\omega_{2}^{n}}\frac{\alpha\wedge\omega_{2}^{n-1}}{\omega_{2}^{n}}.$

Applying this for $\alpha=d(\varphi_{1}-\varphi_{2})\wedge
d^{c}(\varphi_{1}-\varphi_{2})$, $\omega_{1}=\omega_{\varphi_{1}}$,
$\omega_{2}=\omega_{X}$ we obtain

$\displaystyle\frac{d(\varphi_{1}-\varphi_{2})\wedge
d^{c}(\varphi_{1}-\varphi_{2})\wedge\omega_{\varphi_{1}}^{n-1}}{\omega_{X}^{n}}\frac{\omega_{\varphi_{1}}\wedge\omega_{X}^{n-1}}{\omega_{X}^{n}}$
$\displaystyle\geq\frac{1}{n}\frac{\omega_{\varphi_{1}}^{n}}{\omega_{X}^{n}}|\partial(\varphi_{1}-\varphi_{2})|^{2}$
$\displaystyle\geq\frac{c_{0}}{nC}|\partial(\varphi_{1}-\varphi_{2})|^{2}.$

Applying Cauchy-Schwarz inequality we obtain (3.5). Let us stress that this is
the only place where the extra assumption $f_{1}\geq c_{0}$ is used.

Step 1.3. We finally set $t_{1}=t_{0}+\varepsilon$,
$t_{k}:=t_{0}+2^{k-1}(t_{1}-t_{0})$, for $k\geq 2$, so that
$t_{k+1}-t_{k}=2^{k-1}\varepsilon$. Fix $k$ such that $t_{0}\leq
t_{k}<t_{k+1}\leq T_{0}$. Then

$\mu(\varphi_{1}<\varphi_{2}+t_{k})\geq
c^{p}\;\text{and}\;\mu(\varphi_{1}>\varphi_{2}+t_{k+1})\geq c^{p},$

hence Lemma 3.6 below (with $h=\varphi_{1}-\varphi_{2}-t_{k}$ and
$\delta=t_{k+1}-t_{k}$) yields

$\int_{\\{\varphi_{2}+t_{k}<\varphi_{1}\leq\varphi_{2}+t_{k+1}\\}}|\partial(\varphi_{1}-\varphi_{2})|dV_{X}\geq
C^{-1}(t_{k+1}-t_{k})=C^{-1}2^{k-1}\varepsilon.$

Combining this with (3.4) and (3.5) for $a=t_{k}$ and $b=t_{k+1}$ we obtain

$\displaystyle
C2^{2k}\varepsilon^{2}\int_{E_{a,b}}\omega_{\varphi_{1}}\wedge\omega_{X}^{n-1}\geq
I_{a,b}\int_{E_{a,b}}\omega_{\varphi_{1}}\wedge\omega_{X}^{n-1}\geq
C^{-1}2^{2k}\varepsilon^{2},$

hence

$\displaystyle\int_{\\{\varphi_{2}+t_{k}<\varphi_{1}\leq\varphi_{2}+t_{k+1}\\}}\omega_{\varphi_{1}}\wedge\omega_{X}^{n-1}\geq
C^{-1}.$

We take $N$ such that $t_{N}<T_{0}\leq t_{N+1}$. Summing up for $k$ from $1$
to $N$ we obtain

$\int_{\\{\varphi_{2}+t_{1}<\varphi_{1}\leq\varphi_{2}+t_{N}\\}}\omega_{\varphi_{1}}\wedge\omega_{X}^{n-1}\geq
NC^{-1}.$

Since the integral $\int_{X}\omega_{\varphi_{1}}\wedge\omega_{X}^{n-1}$ is
uniformly bounded, this yields a uniform upper bound for $N$. Using (3.2) and
(3.3) we thus obtain the desired bound

$M-m=M-T_{0}+T_{0}-t_{N+1}+\sum_{k=1}^{N+1}(t_{k}-t_{k-1})+t_{0}-m\leq
C\varepsilon.$

Step 2. To remove the assumption (3.1) on $f_{1},f_{2}$ from Step 1 one can
proceed as in [LPT20] to which we refer the reader (this requires the case
$\lambda=1$ which has been treated independently above).

We finally remove the smoothness assumption. Let $f_{1,j},f_{2,j}$ be
sequences of smooth positive densities converging to $f_{1},f_{2}$
respectively in $L^{p}(X,dV_{X})$. Let also $\varphi_{1,j}$, $\varphi_{2,j}$
be smooth functions decreasing to $\varphi_{1},\varphi_{2}$ respectively.
Using [Cher87] we solve, for $i=1,2$,

$(\omega_{j}+dd^{c}u_{i,j})^{n}=e^{u_{i,j}-\varphi_{i,j}}f_{i,j}dV_{X},$

where $\omega_{j}=\omega+2^{-j}\omega_{X}$. Theorem 3.4 ensures that the
functions $u_{i,j}$ are uniformly bounded. Note that the constants
$c(p,\omega_{j})$ from Lemma 3.3 satisfy $c(p,\omega_{j})\geq c(p,\omega)>0$.
By Step 1 we have

$|u_{1,j}-u_{2,j}|\leq T\|g_{1,j}-g_{2,j}\|_{p}^{1/n},$

where $g_{i,j}=e^{u_{i,j}-\varphi_{i,j}}f_{i,j}$. As $j\to+\infty$, arguing as
in the proof of Theorem 3.4 we can show that $u_{i,j}$ converge to $u_{i}$
solving

$(\omega+dd^{c}u_{i})^{n}=e^{u_{i}-\varphi_{i}}f_{i}dV_{X}.$

The domination principle ensures that $u_{i}=\varphi_{i}$, hence
$\|g_{1,j}-g_{2,j}\|_{p}\to\|f_{1}-f_{2}\|_{p}$ finishing the proof. ∎

We have used the following elementary result (see [KN19, Lemma 2.6]):

######  Lemma 3.6.

Let $h$ be a real-valued function in $W^{1,1}(X)$ such that

${\rm Vol}(h\leq 0)\geq\gamma\;\;\text{and}\;\;{\rm
Vol}(h\geq\delta)\geq\gamma$

where $\delta>0$ and $\gamma>0$ are constants. Then

$\int_{\\{0<h<\delta\\}}|\partial h|dV_{X}\geq
C^{-1}\gamma^{\frac{2n-1}{2n}}\delta.$

Here $W^{1,1}(X)$ denotes the set of $L^{1}$ functions whose gradient belongs
to $L^{1}$.

###  3.3. Locally bounded solutions

We assume in this section that $v_{-}(\omega)>0$ and $\omega$ is big. We fix
$\rho$ an $\omega$-psh function such that

  * •

$\rho$ has analytic singularities and $\sup_{X}\rho=0$;

  * •

$\omega+dd^{c}\rho\geq\delta\omega_{X}$ is a hermitian current;

Given $\psi$ a quasi-plurisubharmonic function on $X$ and $c>0$, we set

$E_{c}(\psi):=\\{x\in X\;:\;\nu(\psi,x)\geq c\\},$

where $\nu(\psi,x)$ denotes the Lelong number of $\psi$ at $x$. A celebrated
theorem of Siu ensures that for any $c>0$, the set $E_{c}(\psi)$ is a closed
analytic subset of $X$.

######  Theorem 3.7.

Assume $\omega$ is a big form and $v_{-}(\omega)>0$. Fix $p>1$ and
$q=\frac{p}{p-1}$. Let $\mu=fdV_{X}$ be a probability measure, where
$f=ge^{-\psi}$ with $0\leq g\in L^{p}(dV_{X})$, and $\psi$ is quasi-
plurisubharmonic. Assume $f$ belongs to a fixed Orlicz class $L^{w}$. Then
there exist a unique constant $c>0$ and a function $\varphi\in{\rm
PSH}(X,\omega)$ such that

  * •

$\varphi$ is locally bounded and continuous in
$\Omega:=X\setminus\\{\rho=-\infty\\}\cup E_{q^{-1}}(\psi)$;

  * •

$(\omega+dd^{c}\varphi)^{n}=cfdV_{X}\;\;\text{ in }\;\;\Omega;$

  * •

for any $\alpha>0$, $\alpha(A^{-1}\psi+\rho)-\beta\leq\varphi\leq 0;$

where $A>0$ is a constant such that $dd^{c}\psi\geq-A\omega_{\rho}$ and
$\beta>0$ depends on $p,w$, upper bounds for $\alpha^{-1}$,
$||g||_{L^{p}},||f||_{w}$, $v_{-}(\omega)^{-1}$.

###### Proof.

Reduction to analytic singularities. We let $q$ denote the conjugate exponent
of $p$, set $r=\frac{2p}{p+1}$, and note that $1<r<p$. If the Lelong numbers
of $\psi$ are all less than $\frac{1}{q}$, it follows from Hölder inequality
that $f\in L^{r}(dV_{X})$, since

$\int_{X}f^{r}dV_{X}=\int_{X}g^{r}e^{-r\psi}dV_{X}\leq\left(\int_{X}g^{p}dV_{X}\right)^{\frac{r}{p}}\cdot\left(\int_{X}e^{-\frac{pr}{p-r}\psi}dV_{X}\right)^{\frac{p-r}{p}},$

where the last integral is finite by Skoda’s integrability theorem [GZ,
Theorem 8.11] if $\frac{pr}{p-r}\nu(\psi,x)<2$ for all $x\in X$, which is
equivalent to $\nu(\psi,x)<\frac{1}{q}$.

It is thus natural to expect that the solution $\varphi$ will be locally
bounded in the complement of the closed analytic set $E_{q^{-1}}(\psi)$. It
follows from Demailly’s equisingular approximation technique (see [Dem15])
that there exists a sequence $(\psi_{m})$ of quasi-psh functions on $X$ such
that

  * •

$\psi_{m}\geq\psi$ and $\psi_{m}\rightarrow\psi$ (pointwise and in $L^{1}$);

  * •

$\psi_{m}$ has analytic singularities, it is smooth in $X\setminus
E_{m^{-1}}(\psi)$;

  * •

$dd^{c}\psi_{m}\geq-K\omega_{X}$, for some uniform constant $K>0$;

  * •

$\int_{X}e^{2m(\psi_{m}-\psi)}dV_{X}<+\infty$ for all $m$.

We choose $m=[q]$, set $g_{m}:=ge^{\psi_{m}-\psi}$, and observe that

$\displaystyle\int_{X}g_{m}^{r}$ $\displaystyle\leq$
$\displaystyle\left(\int_{X}e^{2m(\psi_{m}-\psi)}dV_{X}\right)^{\frac{1}{2m}}\cdot\left(\int_{X}g_{m}^{\frac{2mr}{2m-r}}dV_{X}\right)^{\frac{2m-r}{2m}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{X}e^{2m(\psi_{m}-\psi)}dV_{X}\right)^{\frac{1}{2m}}\cdot\left(\int_{X}g_{m}^{p}dV_{X}\right)^{\frac{2m-r}{2m}}<+\infty$

if we choose $r^{-1}=p^{-1}+(2m)^{-1}<1$ so that $\frac{2mr}{2m-r}=p$. By
replacing $\psi$ by $\psi_{[q]}\geq\psi$ and $g$ by $g_{m}\in L^{r}$ in the
sequel, we can thus assume that $\psi$ has analytic singularities and is
smooth in $X\setminus E_{q^{-1}}(\psi)$.

Bounded approximation. The densities $f_{j}:=ge^{-\max(\psi,-j)}$ belong to
$L^{p}$ and increase towards $f$. By Theorem 3.4 there exists
$\varphi_{j}\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ such that

$(\omega+dd^{c}\varphi_{j})^{n}=c_{j}f_{j}dV_{X},$

where $c_{j}>0$ and $\sup_{X}\varphi_{j}=0$.

Observe that $dV_{X}\geq\delta_{1}\omega_{X}^{n}$ for some $\delta_{1}>0$. It
follows therefore from the arithmetico-geometric means inequality that

$c_{j}^{1/n}f_{j}^{1/n}\delta_{1}^{1/n}\omega_{X}^{n}\leq(\omega+dd^{c}\varphi_{j})\wedge\omega_{X}^{n-1}.$

As $j$ increases to $+\infty$, we have
$\int_{X}f_{j}^{1/n}\omega_{X}^{n}\rightarrow\int_{X}f^{1/n}\omega_{X}>0$,
while

$\int_{X}(\omega+dd^{c}\varphi_{j})\wedge\omega_{X}^{n-1}\leq\int_{X}\omega\wedge\omega_{X}^{n-1}+B\int_{X}(-\varphi_{j})\omega_{X}^{n},$

using that $dd^{c}\omega_{X}^{n-1}\leq B\omega_{X}^{n}$ and $\varphi_{j}\leq
0$. Thus the positive constants $c_{j}$ are uniformly bounded from above (the
functions $\varphi_{j}$ are $\omega$-psh and normalized, hence they are
relatively compact in $L^{1}(X)$).

They are also bounded away from zero, since

$c_{j}\int_{X}f_{j}dV_{X}=\int_{X}(\omega+dd^{c}\varphi_{j})^{n}\geq
v_{-}(\omega),$

and $\int_{X}f_{j}dV_{X}\rightarrow\int_{X}fdV_{X}=1$.

$L^{\infty}$-estimates. Fix $A>0$ large enough so that
$\omega+dd^{c}\psi\geq-A\omega_{\rho}$. Since $c\leq c_{j}\leq C$, $f_{j}\leq
f$ where $f$ belongs to a fixed Orlicz class $L^{w}$, $f_{j}\leq
ge^{-A\tilde{\psi}}$ where $\tilde{\psi}=\frac{\psi}{A}+\rho\in{\rm
PSH}(X,\omega)$, it follows from Theorem 2.4 that

$\alpha(A^{-1}\psi+\rho)-\beta\leq\varphi_{j}\leq 0.$

Thus the functions $\varphi_{j}$ are locally uniformly bounded in
$\Omega:=X\setminus\\{\rho+\psi=-\infty\\}$.

We can thus extract a subsequence such that $c_{j}\longrightarrow c>0$ and

  * •

$\varphi_{j}\rightarrow\varphi$ a.e. and in $L^{1}$, with $\varphi\in{\rm
PSH}(X,\omega)$;

  * •

$\sup_{X}\varphi=0$, as follows from Hartogs lemma (see [GZ, Theorem 1.46]);

  * •

$\alpha(A^{-1}\psi+\rho)-\beta\leq\varphi\leq 0$ in $\Omega$,

hence the Monge-Ampère measure $(\omega+dd^{c}\varphi)^{n}$ is well-defined in
$\Omega$.

Convergence of the approximants. Set

$\Phi_{j}^{+}:=\left(\sup_{\ell\geq j}\varphi_{\ell}\right)^{*}\;\;\text{ and
}\;\;\Phi_{j}^{-}:=P_{\omega}\left(\inf_{\ell\geq j}\varphi_{\ell}\right).$

The function $\Phi_{j}^{+}$ (resp. $\Phi_{j}^{-}$) is $\omega_{j}$-psh (resp.
$\omega$-psh) and locally bounded in $\Omega$. Note that $\Phi_{j}^{+}$
decreases to $\varphi$, while $\Phi_{j}^{-}$ increases to some $\omega$-psh
function $\Phi^{\prime}\leq\varphi$. We are going to show that
$\Phi^{\prime}=\varphi$.

We also set $f_{j}^{+}:=\sup_{\ell\geq
j}f_{\varepsilon_{\ell}},\;f_{j}^{-}:=\inf_{\ell\geq
j}f_{\varepsilon_{\ell}},\;c_{j}^{+}=\sup_{\ell\geq
j}c_{\varepsilon_{\ell}},\;c_{j}^{-}=\inf_{\ell\geq j}c_{\varepsilon_{\ell}}.$
It follows from Lemma 1.9 that

$(\omega+dd^{c}\Phi_{j}^{+})^{n}\geq c_{j}^{-}f_{j}^{-}dV_{X}.$

Since $\Phi_{j}^{+}$ decreases to $\varphi$, while $c_{j}^{-}f_{j}^{-}$
increases to $cf$, it follows from the continuity of the complex Monge-Ampère
along monotonic sequences that

$(\omega+dd^{c}\varphi)^{n}\geq cfdV_{X}\;\;\text{ in }\;\;\Omega.$

It follows from Lemma 1.9 that

$(\omega+dd^{c}\Phi_{j}^{+})^{n}\geq c_{j}^{-}f_{j}^{-}dV_{X}.$

Since $\Phi_{j}^{+}$ decreases to $\varphi$, while $c_{j}^{-}f_{j}^{-}$
increases to $cf$, it follows from the continuity of the complex Monge-Ampère
along monotonic sequences that

$(\omega+dd^{c}\varphi)^{n}\geq cfdV_{X}\;\;\text{ in }\;\;\Omega.$

It also follows from Lemma 1.9 that

$(\omega+dd^{c}\Phi_{j}^{-})^{n}\leq e^{\Phi_{j}^{-}-\inf_{\ell\geq
j}\varphi_{j}}c_{j}^{+}f_{j}^{+}dV_{X},$

so $\Phi_{j}^{-}$ increase to some $\Phi^{\prime}\in{\rm PSH}(X,\omega)$ such
that

$(\omega+dd^{c}\Phi^{\prime})^{n}\leq e^{\Phi^{\prime}-\varphi}cfdV_{X}\leq
e^{\Phi^{\prime}-\varphi}(\omega+dd^{c}\varphi)^{n}$

in $\Omega$. It thus follows from Proposition 1.12, that
$\Phi^{\prime}\geq\varphi$, hence $\Phi^{\prime}=\varphi$ since by
construction $\Phi^{\prime}\leq\varphi$. Altogether this shows that $\varphi$
solves the desired equation.

The uniqueness of the constant $c>0$ follows from Corollary 1.14.

Continuity of the solutions. We finally prove that any solution $\phi\in{\rm
PSH}(X,\omega)$ satisfying $\inf_{X}(\phi-\varepsilon\rho)>-\infty$, for all
$\varepsilon>0$, is continuous in $\Omega$. We can assume without loss of
generality that $\phi\leq-1$. Note that $\phi$ has zero Lelong number
everywhere on $X$. Using Demailly’s regularization we can thus find $\phi_{j}$
smooth functions such that $\omega+dd^{c}\phi_{j}\geq-2^{-j}\omega_{X}$, and
$\phi_{j}\searrow\phi$. Fix $\varepsilon>0$ and solve

$(\omega+dd^{c}u_{j,\varepsilon})^{n}=e^{u_{j,\varepsilon}}(g_{j,\varepsilon}e^{-2\varepsilon^{-1}\phi}+h)dV_{X},$

where $h=\omega^{n}/dV_{X}$, $g_{j,\varepsilon}=\varepsilon^{-n}{\bf
1}_{\\{\phi<\phi_{j}-\varepsilon\\}}g$. Observe that
$g_{j,\varepsilon}e^{-\varepsilon^{-1}\phi}\to 0$ in $L^{p}(X)$ as
$j\to+\infty$. It follows from Theorem 3.5 that the functions
$u_{j,\varepsilon}$ uniformly converge to $0$ as $j\to+\infty$. Consider

$v_{j,\varepsilon}:=(1-\varepsilon)\phi_{j}+\frac{\varepsilon}{2}(\psi+A\rho+u_{j,\varepsilon})-C\varepsilon,$

where $A>0$ is a constant ensuring that $\omega+dd^{c}(\psi+A\rho)\geq 0$, and
$C>0$ is a constant chosen below. A direct computation shows that

$(\omega+dd^{c}v_{j,\varepsilon})^{n}\geq 2^{-n}e^{u_{j,\varepsilon}}{\bf
1}_{\\{\phi<\phi_{j}-\varepsilon\\}}ge^{-\varepsilon^{-1}\phi}dV_{X}.$

Recall that, by Theorem 2.4,
$\phi_{j}\geq\phi\geq\frac{1}{2}(\psi+A\rho)-C_{1}$, for some positive
constant $C_{1}$. On the set $\\{\phi<v_{j,\varepsilon}\\}$ we have

$\phi<\phi_{j}-\varepsilon\left(\phi_{j}-\frac{1}{2}(\psi+A\rho)\right)+\frac{\varepsilon}{2}\sup_{X}|u_{j,\varepsilon}|-C\varepsilon<\phi_{j}-\varepsilon,$

if we choose $C>1+\sup_{X}|u_{j,\varepsilon}|+C_{1}$. On this set we also have

$2^{-n}e^{u_{j,\varepsilon}}ge^{-\varepsilon^{-1}\phi}\geq
2^{-n}e^{-\sup_{X}|u_{j,\varepsilon}|}e^{C-\psi}\geq 2ge^{-\psi},$

if we choose $C>2^{n}+\sup_{X}|u_{j,\varepsilon}|+\ln(2)$. Thus

$(\omega+dd^{c}\phi)^{n}\leq
2^{-1}(\omega+dd^{c}v_{j,\varepsilon})^{n}=2^{-1}(\omega+dd^{c}\max(\phi,v_{j,\varepsilon}))^{n}$

on the set $\\{\phi<v_{j,\varepsilon}\\}$. Thus
$\phi\geq\max(\phi,v_{j,\varepsilon})\geq v_{j,\varepsilon}$ by Proposition
1.12. Fixing a compact set $K\Subset\Omega$ and letting $j\to+\infty$ we
obtain

$\liminf_{j\to+\infty}\inf_{K}(\phi-\phi_{j})\geq-C^{\prime}\varepsilon.$

Since $\phi_{j}\to\phi$ uniformly on $K$ as $\varepsilon\to 0$, we conclude
that $\phi\in{\mathcal{C}}^{0}(\Omega)$. ∎

######  Remark 3.8.

This relative $L^{\infty}$-bound requires one to assume both that $\omega$ is
big and $v_{-}(\omega)>0$. It is natural to expect that it suffices to assume
that $\omega$ is big, however such an estimate is an open problem even in the
simplest case when $\omega$ is a hermitian form ! On the other hand these two
conditions are expected to be equivalent; such is the case e.g. when $X$
belongs to the Fujiki class (see [GL21b]).

##  4\. Geometric applications

In this section we apply the previous analysis on compact complex varieties
$V$ with mild singularities. We refer the reader to [EGZ09, Section 5] for an
introduction to complex analysis on singular varieties. Roughly speaking one
can consider local embeddings
$j_{\alpha}V_{\alpha}\hookrightarrow\mathbb{C}^{N}$ and consider objects
(quasi-psh functions, forms, etc) that are restrictions of global ones.

One can also consider $\pi:X\rightarrow V$ a resolution of singularities and
pull-back these objects to $X$ in order to study the corresponding equations
in a smooth environment. The draw-back is a loss of positivity along some
divisor $E=\pi^{-1}(V_{sing})$ which lies above the singular locus of $V$.
Considering a hermitian form $\omega_{V}$ on $V$, we are thus lead to work
with $\omega=\pi^{*}\omega_{V}$, which is semi-positive and big but no longer
hermitian. We fix a $\omega$-psh function $\rho$ with analytic singularities
along a divisor $E$ such that $\omega+dd^{c}\rho\geq 3\delta\omega_{X}$, for
some $\delta>0$.

###  4.1. Higher order regularity of solutions

In this section we establish higher order regularity of the solution under
appropriate assumptions on the density.

######  Theorem 4.1.

Assume $\omega$ is big or $v_{-}(\omega)>0$. Assume
$f=e^{\psi^{+}-\psi^{-}}\in L^{p}(dV_{X})$, $p>1$, and $f$ is smooth and
positive in $X\setminus D$, with $\psi^{\pm}$ quasi-plurisubharmonic
functions. Then there exist $c>0$ and $\varphi\in{\rm PSH}(X,\omega)$ such
that $\sup_{X}\varphi=0$ and

  * •

$\varphi$ is smooth in the open set $X\setminus(D\cup E)$;

  * •

$-T\leq\varphi\leq 0$ in $X$ for some uniform $T>0$;

  * •

$(\omega+dd^{c}\varphi)^{n}=cfdV_{X}$ on $X$.

We also establish higher regularity under less restrictive assumptions on the
density $f$, but we then need to assume that $\omega$ is both big and
satisfies $v_{-}(\omega)>0$, in order to have a good relative
$L^{\infty}$-bound:

######  Theorem 4.2.

Assume $\omega$ is big and $v_{-}(\omega)>0$. Assume
$f=e^{\psi^{+}-\psi^{-}}\in L^{1}(dV_{X})$ is smooth and positive in
$X\setminus D$, with $\psi^{\pm}$ quasi-plurisubharmonic functions. Then there
exist $c>0$ and $\varphi\in{\rm PSH}(X,\omega)$ such that $\sup_{X}\varphi=0$
and

  * •

$\varphi$ is smooth in the open set $X\setminus(D\cup E)$;

  * •

for any $\alpha>0$ there is $\beta(\alpha)>0$ with
$\alpha(\delta\psi^{-}+\rho)-\beta(\alpha)\leq\varphi\leq 0$ in $X$;

  * •

$(\omega+dd^{c}\varphi)^{n}=cfdV_{X}$ in $X\setminus(D\cup E)$.

One can keep track of the dependence of the constants on the data, as was done
in Theorem 3.7. This result can be seen as a generalization of the main result
of [TW10b] which dealt with the case when $\omega>0$ is hermitian on $X$ and
$f$ is globally smooth and positive.

###### Proof.

We prove Theorem 4.1 and Theorem 4.2 at once.

Smooth approximation. We assume without loss of generality that
$\psi^{\pm}\leq 0$. For a Borel function $g$ we let
$g_{\varepsilon}:=\rho_{\varepsilon}(g)$ denote the Demailly regularization of
$g$ (see [Dem94, (3.1)]):

$\rho_{\varepsilon}(g)(x):=\frac{1}{\varepsilon^{2n}}\int_{\zeta\in
T_{x}X}g({\rm
exph}_{x}(\zeta))\chi\left(\frac{|\zeta|^{2}}{\varepsilon^{2}}\right)d\lambda(\zeta).$

Since $\psi^{-}$ is quasi-psh, the corresponding regularization
$\psi^{-}_{\varepsilon}$ satisfies
$\psi^{-}\leq\psi^{-}_{\varepsilon}+A\varepsilon^{2}$, while
$\psi^{+}_{\varepsilon}\leq 0$. Moreover the functions
$\psi_{\varepsilon}^{\pm}$ are quasi-psh with
$dd^{c}\psi_{\varepsilon}^{\pm}\geq-K^{\pm}\omega_{X}$ for uniform constants
$K^{\pm}\geq 0$. In particular

$K^{-}\omega+dd^{c}\left(\delta\psi_{\varepsilon}^{-}+K^{-}\rho\right)\geq-K^{-}\delta\omega_{X}+K^{-}(\omega+dd^{c}\rho)\geq
0,$

so $\alpha\psi_{\varepsilon}^{-}+\rho\in{\rm PSH}(X,\omega)$ for all
$0<\alpha<\frac{\delta}{K^{-}}$. Up to replacing $\delta$ with
$\frac{\delta}{K^{-}}$, we assume in the sequel that $K^{-}=1$.

We fix $0<\varepsilon\leq 1$ and set
$\omega_{\varepsilon}:=\omega+\varepsilon\omega_{X}$. It follows from [TW10b]
that there exist unique constants $c_{\varepsilon}>0$ and smooth
$\omega_{\varepsilon}$-psh functions $\varphi_{\varepsilon}$ such that

$(\omega+\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon})^{n}=c_{\varepsilon}e^{\psi^{+}_{\varepsilon}-\psi^{-}_{\varepsilon}}dV_{X}.$

with $\sup_{X}\varphi_{\varepsilon}=0$. Note that by Jensen inequality we have

$e^{\psi^{+}_{\varepsilon}-\psi^{-}_{\varepsilon}}\leq f_{\varepsilon}.$

Since $f_{\varepsilon}$ converges to $f$ in $L^{1}(X,dV_{X})$, extracting a
subsequence we can assume that
$e^{\psi^{+}_{\varepsilon}-\psi^{-}_{\varepsilon}}\leq F$, for some $F\in
L^{1}(dV_{X})$ which belongs to an Orlicz class $L^{w}$, for some convex
increasing weight $w:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
$\frac{w(t)}{t}\rightarrow+\infty$ as $t\rightarrow+\infty$.

${\mathcal{C}}^{0}$-estimates. This is the only place where the proofs of
Theorem 4.1 and Theorem 4.2 slightly differ, and also the reason why the
assumptions on $\omega$ look slightly stronger in the second case.

Arguing as in the proof of Theorem 3.7, one shows that the constants
$c_{\varepsilon}$ are uniformly bounded away from zero and $+\infty$. If $f$
belongs to $L^{p}(dV_{X})$ for some $p>1$, it follows from Theorem 2.2 (if
$v_{-}(\omega)>0$) and Theorem 3.4 (if $\omega$ is big) that
$-T\leq\varphi_{\varepsilon}\leq 0$, for some uniform constant $T$ independent
of $\varepsilon>0$.

If we rather assume that $f\in L^{1}(dV_{X})$ with $\omega$ big and
$v_{-}(\omega)>0$, note that

$(\omega+\varepsilon\omega_{X}+dd^{c}\varphi_{\varepsilon})^{n}\leq
Ce^{-A(\delta\psi^{-}_{\varepsilon}+\rho)}dV_{X},$

with $A=\delta^{-1}$. It then follows from Theorem 2.4 that for any $\alpha>0$
there exists $\beta(\alpha)$ such that

(4.1)
$\alpha(\delta\psi^{-}_{\varepsilon}+\rho)-\beta(\alpha)\leq\varphi_{\varepsilon}\leq
0.$

Thus $(\varphi_{\varepsilon})$ is locally uniformly bounded in
$\Omega:=X\setminus(E\cup D)$.

${\mathcal{C}}^{2}$-estimates. In the sequel we fix $\alpha=1$ and we
establish uniform bounds on $\Delta_{\omega_{X}}\varphi_{\varepsilon}$ on
compact subsets of $\Omega$. We follow the computations of [TW10a, Proof of
Theorem 2.1] and [Tô18] with a twist in order to deal with unbounded
functions. We use $C$ to denote various uniform constants which may be
different. Consider

$H:=\log{\rm Tr}_{\omega_{X}}(\tilde{\omega})-\gamma(u)$

where

$\tilde{\omega}=\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon},\;\;u=(\varphi_{\varepsilon}-\rho-2\delta\psi^{-}_{\varepsilon}+\beta+1)>1,$

and $\gamma:\mathbb{R}\rightarrow\mathbb{R}$ is a smooth concave increasing
function such that $\gamma(+\infty)=+\infty$. We are going to show that $H$ is
uniformly bounded from above for an appropriate choice of $\gamma$. Since $u$
is uniformly bounded on compact subsets of $\Omega$, this will yield for each
$\varepsilon>0$ and $K\subset\subset\Omega$ a uniform bound

$\Delta_{\omega_{X}}(\varphi_{\varepsilon})={\rm
Tr}_{\omega_{X}}(\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon})-{\rm
Tr}_{\omega_{X}}(\omega_{\varepsilon})\leq C_{K}.$

We let ${g}$ denote the Riemannian metric associated to $\omega_{X}$ and
$\tilde{g}$ the one associated to
$\tilde{\omega}=\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon}$. To simplify
notations we will omit the subscript $\varepsilon$ in the sequel. Since
$\psi^{\pm}$ are quasi-psh, up to multiplying $\omega_{X}$ with a large
constant, we can assume that $\omega_{X}+dd^{c}\psi^{\pm}\geq 0$.

Since $\rho\rightarrow-\infty$ on $E$ the maximum of $H$ is attained at some
point $x_{0}\in X\setminus E$. We use special coordinates at this point, as
defined by Guan-Li in [GL10]:

$g_{i\bar{j}}=\delta_{ij},\;\;\frac{\partial g_{i\bar{i}}}{\partial
z_{j}}=0\;\;\text{and}\;\;\tilde{g}_{i\bar{j}}\;\text{is diagonal}.$

To achieve this we use a linear change of coordinates so that
$g_{i\bar{j}}=\delta_{ij}$ and $\tilde{g}_{i\bar{j}}$ is diagonal at $x_{0}$,
and we then make a change of coordinates as in [GL10, (2.19)]. We first
compute

$\displaystyle\Delta_{\tilde{\omega}}{\rm Tr}_{\omega_{X}}(\tilde{\omega})$
$\displaystyle=\sum_{i,j,k,l}\tilde{g}^{i\bar{j}}\partial_{i}\partial_{\bar{j}}(g^{k\bar{l}}\tilde{g}_{k\bar{l}})$
$\displaystyle=\sum_{i,k}\tilde{g}^{i\bar{i}}\tilde{g}_{k\bar{k}i\bar{i}}-2\Re\left(\sum_{i,j,k}\tilde{g}^{i\bar{i}}g_{j\bar{k}\bar{i}}\tilde{g}_{k\bar{j}i}\right)+\sum_{i,j,k}\tilde{g}^{i\bar{i}}g_{j\bar{k}i}g_{k\bar{j}\bar{i}}\tilde{g}_{k\bar{k}}$
$\displaystyle+\sum_{i,j,k}\tilde{g}^{i\bar{i}}g_{k\bar{j}i}g_{j\bar{k}\bar{i}}\tilde{g}_{k\bar{k}}-\sum_{i,k}\tilde{g}^{i\bar{i}}g_{k\bar{k}\bar{i}i}\tilde{g}_{k\bar{k}}$
$\displaystyle\geq\sum_{i,k}\tilde{g}^{i\bar{i}}\tilde{g}_{k\bar{k}i\bar{i}}-2\Re\left(\sum_{i,j,k}\tilde{g}^{i\bar{i}}g_{j\bar{k}\bar{i}}\tilde{g}_{k\bar{j}i}\right)-C{\rm
Tr}_{\omega_{X}}(\tilde{\omega}){\rm Tr}_{\tilde{\omega}}(\omega_{X}).$

Using this and

${\rm Tr}_{\omega_{X}}{\rm
Ric}(\tilde{\omega})=\sum_{i,k}\tilde{g}^{i\bar{i}}\left(-\tilde{g}_{i\bar{i}k\bar{k}}+\sum_{j}\tilde{g}^{j\bar{j}}\tilde{g}_{i\bar{j}k}\tilde{g}_{j\bar{i}\bar{k}}\right)$

we obtain

(4.2) $\displaystyle\Delta_{\tilde{\omega}}{\rm
Tr}_{\omega_{X}}(\tilde{\omega})$
$\displaystyle\geq\sum_{i,j,k}\tilde{g}^{i\bar{i}}\tilde{g}^{j\bar{j}}\tilde{g}_{i\bar{j}k}\tilde{g}_{j\bar{i}\bar{k}}-{\rm
Tr}_{\omega_{X}}{\rm Ric}(\tilde{\omega})-C{\rm
Tr}_{\tilde{\omega}}(\omega_{X})$ $\displaystyle-C{\rm
Tr}_{\omega_{X}}(\tilde{\omega}){\rm
Tr}_{\tilde{\omega}}(\omega_{X})-2\Re\left(\sum_{i,j,k}\tilde{g}^{i\bar{i}}g_{j\bar{k}\bar{i}}\tilde{g}_{k\bar{j}i}\right),$

noting that $|\tilde{g}_{i\bar{i}k\bar{k}}-\tilde{g}_{k\bar{k}i\bar{i}}|\leq
C$. Here ${\rm Ric}(\tilde{\omega})$ is the Chern-Ricci form of
$\tilde{\omega}$. From ${\rm Tr}_{\omega_{X}}(\tilde{\omega}){\rm
Tr}_{\tilde{\omega}}(\omega_{X})\geq n$, (4.2) and

${\rm Ric}(\tilde{\omega})={\rm Ric}(\omega_{X})-dd^{c}(\psi^{+}-\psi^{-})\leq
C\omega_{X}+dd^{c}\psi^{-},$

we obtain

(4.3) $\displaystyle\Delta_{\tilde{\omega}}{\rm
Tr}_{\omega_{X}}(\tilde{\omega})$
$\displaystyle\geq\sum_{i,j,k}\tilde{g}^{i\bar{i}}\tilde{g}^{j\bar{j}}\tilde{g}_{i\bar{j}k}\tilde{g}_{j\bar{i}\bar{k}}-{\rm
Tr}_{\omega_{X}}(\omega_{X}+dd^{c}\psi^{-})-C{\rm
Tr}_{\tilde{\omega}}(\omega_{X})$
$\displaystyle-2\Re\left(\sum_{i,j,k}\tilde{g}^{i\bar{i}}g_{j\bar{k}\bar{i}}\tilde{g}_{k\bar{j}i}\right)-C{\rm
Tr}_{\omega_{X}}(\tilde{\omega}){\rm Tr}_{\tilde{\omega}}(\omega_{X}).$

Our special choice of coordinates at $x_{0}$ ensures that
$g_{j\bar{j}\bar{i}}=0$. Using Cauchy-Schwarz inequality and
$|\tilde{g}_{k\bar{j}i}-\tilde{g}_{i\bar{j}k}|\leq C$, we therefore obtain

$\displaystyle\left|2\Re\left(\sum_{i,j,k}\tilde{g}^{i\bar{i}}g_{j\bar{k}\bar{i}}\tilde{g}_{k\bar{j}i}\right)\right|$
$\displaystyle\leq\left|2\Re\left(\sum_{i}\sum_{j\neq
k}\tilde{g}^{i\bar{i}}g_{j\bar{k}\bar{i}}\tilde{g}_{i\bar{j}k}\right)\right|+C{\rm
Tr}_{\tilde{\omega}}(\omega_{X})$ $\displaystyle\leq\sum_{i}\sum_{j\neq
k}\left(\tilde{g}^{i\bar{i}}\tilde{g}^{j\bar{j}}\tilde{g}_{i\bar{j}k}\tilde{g}_{j\bar{i}\bar{k}}+\tilde{g}^{i\bar{i}}\tilde{g}_{j\bar{j}}g_{j\bar{k}\bar{i}}g_{k\bar{j}i}\right)+C{\rm
Tr}_{\tilde{\omega}}(\omega_{X})$ $\displaystyle\leq\sum_{i}\sum_{j\neq
k}\tilde{g}^{i\bar{i}}\tilde{g}^{j\bar{j}}\tilde{g}_{i\bar{j}k}\tilde{g}_{j\bar{i}\bar{k}}+C{\rm
Tr}_{\omega_{X}}(\tilde{\omega}){\rm Tr}_{\tilde{\omega}}(\omega_{X})+C{\rm
Tr}_{\tilde{\omega}}(\omega_{X}).$

Together with (4.3) this yields

(4.4) $\Delta_{\tilde{\omega}}{\rm Tr}_{\omega_{X}}(\tilde{\omega})\geq
I-C{\rm Tr}_{\omega_{X}}(\omega_{X}+dd^{c}\psi^{-})-C{\rm
Tr}_{\omega_{X}}(\tilde{\omega}){\rm Tr}_{\tilde{\omega}}(\omega_{X})-C{\rm
Tr}_{\tilde{\omega}}(\omega_{X})$

with
$I:=\sum_{i,j}\tilde{g}^{i\bar{i}}\tilde{g}^{j\bar{j}}\tilde{g}_{i\bar{j}j}\tilde{g}_{j\bar{i}\bar{j}}$.
We next compute

$\displaystyle|\partial{\rm
Tr}_{\omega_{X}}(\tilde{\omega})|_{\tilde{\omega}}^{2}$
$\displaystyle=\sum_{i,j,k}\tilde{g}^{i\bar{i}}\tilde{g}_{j\bar{j}i}\tilde{g}_{k\bar{k}\bar{i}}$
$\displaystyle=\sum_{i,j,k}\tilde{g}^{i\bar{i}}(T_{ij\bar{j}}+\tilde{g}_{i\bar{j}j})(\overline{T}_{ik\bar{k}}+\tilde{g}_{k\bar{i}\bar{k}})$
$\displaystyle=\sum_{i,j,k}\tilde{g}^{i\bar{i}}\tilde{g}_{i\bar{j}j}\tilde{g}_{k\bar{i}\bar{k}}+\sum_{i,j,k}\tilde{g}^{i\bar{i}}T_{ij\bar{j}}\overline{T}_{ik\bar{k}}+2\Re\left(\tilde{g}^{i\bar{i}}T_{ij\bar{j}}\tilde{g}_{k\bar{i}\bar{k}}\right).$

where $T_{ij\bar{j}}=\tilde{g}_{j\bar{j}i}-\tilde{g}_{i\bar{j}j}$ is the
torsion term corresponding to $\omega+\varepsilon\omega_{X}$ which is under
control: $|T_{ij\bar{j}}|\leq C$. We bound the first term by Cauchy-Schwarz
inequality

$\displaystyle\sum_{i,j,k}\tilde{g}^{i\bar{i}}\tilde{g}_{i\bar{j}j}\tilde{g}_{k\bar{i}\bar{k}}$
$\displaystyle=\sum_{i}\tilde{g}^{i\bar{i}}\left|\sum_{j}\tilde{g}_{i\bar{j}j}\right|^{2}$
$\displaystyle\leq\left(\sum_{i}\tilde{g}^{i\bar{i}}\tilde{g}^{j\bar{j}}\tilde{g}_{i\bar{j}j}\tilde{g}_{j\bar{i}\bar{j}}\right)\left(\sum_{j}\tilde{g}_{j\bar{j}}\right)=I{\rm
Tr}_{\omega_{X}}(\tilde{\omega}).$

We thus get

(4.5) $\frac{|\partial{\rm
Tr}_{\omega_{X}}(\tilde{\omega})|^{2}_{\tilde{\omega}}}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}\leq\frac{I}{{\rm
Tr}_{\omega_{X}}(\tilde{\omega})}+C\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}+\frac{2}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}\Re\left(\sum\tilde{g}^{i\overline{i}}T_{ij\overline{j}}\tilde{g}_{k\bar{i}\bar{k}}\right),$

Since $\partial_{\overline{i}}H=0$ at the point $x_{0}$, we obtain by
differentiating $H$ once

$\tilde{g}_{k\overline{k}\overline{i}}={\rm
Tr}_{\omega_{X}}(\tilde{\omega})\gamma^{\prime}(u)u_{\overline{i}}.$

Cauchy-Schwarz inequality yields

$\left|\frac{2}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}\Re\left(\sum\tilde{g}^{i\overline{i}}T_{ij\overline{j}}\tilde{g}_{k\overline{k}\overline{i}}\right)\right|\leq
C\frac{\gamma^{\prime}(u)^{2}}{(-\gamma^{\prime\prime}(u))}\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}+(-\gamma^{\prime\prime}(u))|\partial
u|_{\tilde{\omega}}^{2}.$

Noting that $|\tilde{g}_{k\bar{k}\bar{i}}-\tilde{g}_{k\bar{i}\bar{k}}|\leq C$
we infer

$\left|\frac{2}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}\Re\left(\sum\tilde{g}^{i\overline{i}}T_{ij\overline{j}}\tilde{g}_{k\overline{i}\overline{k}}\right)\right|\leq
C\left(\frac{\gamma^{\prime}(u)^{2}}{(-\gamma^{\prime\prime}(u))}+1\right)\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}+(-\gamma^{\prime\prime}(u))|\partial
u|_{\tilde{\omega}}^{2}.$

Since $0\geq\Delta_{\tilde{\omega}}H$ at $x_{0}$, it follows from (4.4), (4.5)
that

(4.6) $\displaystyle 0$
$\displaystyle\geq\Delta_{\tilde{\omega}}H=\frac{\Delta_{\tilde{\omega}}{\rm
Tr}_{\omega_{X}}(\tilde{\omega})}{{\rm
Tr}_{\omega_{X}}(\tilde{\omega})}-\frac{|\partial{\rm
Tr}_{\omega_{X}}(\tilde{\omega})|^{2}_{\tilde{\omega}}}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}-\gamma^{\prime}(u)\Delta_{\tilde{\omega}}(u)-\gamma^{\prime\prime}(u)|\partial
u|_{\tilde{\omega}}^{2}$ $\displaystyle\geq-\frac{C{\rm
Tr}_{\omega_{X}}(\omega_{X}+dd^{c}\psi^{-})}{{\rm
Tr}_{\omega_{X}}(\tilde{\omega})}-\gamma^{\prime}(u)(n-\delta{\rm
Tr}_{\tilde{\omega}}(3\omega_{X}+2dd^{c}\psi^{-}))$
$\displaystyle-C\left(\frac{\gamma^{\prime}(u)^{2}}{(-\gamma^{\prime\prime}(u))}+1\right)\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}-C\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{{\rm Tr}_{\omega_{X}}(\tilde{\omega})}-C{\rm
Tr}_{\tilde{\omega}}(\omega_{X}).$

We now choose the function $\gamma$ so as to obtain a simplified information.
We set

$\gamma(u):=\frac{C+1}{\min(\delta,1)}u+\ln(u).$

Since $u\geq 1$, we observe that

$\frac{C+1}{\min(\delta,1)}\leq\gamma^{\prime}(u)\leq
1+\frac{C+1}{\min(\delta,1)}\;\;\;\text{ and
}\;\;\;\frac{\gamma^{\prime}(u)^{2}}{|\gamma^{\prime\prime}(u)|}+1\leq
C_{1}u^{2}.$

By incorporating this into (4.6) we obtain

$\displaystyle 0$ $\displaystyle\geq-\frac{C{\rm
Tr}_{\omega_{X}}(\omega_{X}+dd^{c}\psi^{-})}{{\rm
Tr}_{\omega_{X}}(\tilde{\omega})}-C_{2}+(C+1)({\rm
Tr}_{\tilde{\omega}}(\omega_{X})+{\rm
Tr}_{\tilde{\omega}}(\omega_{X}+dd^{c}\psi^{-}))$ $\displaystyle-
C_{2}(u^{2}+1)\frac{{\rm Tr}_{\tilde{\omega}}(\omega_{X})}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}-C\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{{\rm Tr}_{\omega_{X}}(\tilde{\omega})}-C{\rm
Tr}_{\tilde{\omega}}(\omega_{X}).$

Using ${\rm Tr}_{\omega_{X}}(\omega_{X}+dd^{c}\psi^{-})\leq{\rm
Tr}_{\tilde{\omega}}(\omega_{X}+dd^{c}\psi^{-}){\rm
Tr}_{\omega_{X}}(\tilde{\omega})$ we thus arrive at

(4.7) $0\geq{\rm Tr}_{\tilde{\omega}}(\omega_{X})-C_{2}(u^{2}+1)\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}-C\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{{\rm
Tr}_{\omega_{X}}(\tilde{\omega})}-C_{2}.$

At the point $x_{0}$ we have the following alternative:

  * •

if ${\rm Tr}_{\omega_{X}}(\tilde{\omega})^{2}\geq 4C_{2}(u^{2}+1)+(4C)^{2}$
then

$C_{2}(u^{2}+1)\frac{{\rm Tr}_{\tilde{\omega}}(\omega_{X})}{({\rm
Tr}_{\omega_{X}}(\tilde{\omega}))^{2}}\leq\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{4}\;\;\text{and}\;\;C\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{{\rm
Tr}_{\omega_{X}}(\tilde{\omega})}\leq\frac{{\rm
Tr}_{\tilde{\omega}}(\omega_{X})}{4},$

hence from (4.7) we get ${\rm Tr}_{\tilde{\omega}}(\omega_{X})\leq 2C_{2}$.
Now

${\rm Tr}_{\omega_{X}}(\tilde{\omega})\leq
n\frac{\tilde{\omega}^{n}}{\omega_{X}^{n}}\left({\rm
Tr}_{\tilde{\omega}}(\omega_{X})\right)^{n-1}\leq
n(2C_{2})^{n-1}ce^{\psi^{+}-\psi^{-}}$

yields ${\rm Tr}_{\omega_{X}}(\tilde{\omega})\leq Ce^{\psi^{+}-\psi^{-}}$. It
follows therefore from (4.1) that

$\displaystyle H(x_{0})$
$\displaystyle\leq\log(2C_{2})-\psi^{-}-\frac{C+1}{\min(\delta,1)}(\varphi-\rho-2\delta\psi^{-})$
$\displaystyle\leq\log(2C_{2})-(C+1)(\varphi-\rho-\delta\psi^{-})\leq C_{3}.$

  * •

If ${\rm Tr}_{\omega_{X}}(\tilde{\omega})^{2}\leq 4C_{2}(u^{2}+1)+(4C)^{2}$
then

$H(x_{0})\leq\log\sqrt{4C_{2}(u^{2}+1)+(4C)^{2}}-\gamma(u)\leq C_{4}.$

Thus $H(x_{0})$ is uniformly bounded from above, yielding the desired
estimate.

Higher order estimates. With uniform bounds on
$||\Delta_{\omega_{X}}\varphi_{\varepsilon}||_{L^{\infty}(K)}$ in hands, we
can use a complex version of Evans-Krylov-Trudinger estimate (see [TW10a,
Section 4] in this context) and eventually differentiate the equation to
obtain -using Schauder estimates- uniform bounds, for each
$K\subset\subset\Omega$, $0<\beta<1$, $j\geq 0$,

$\sup_{\varepsilon>0}||\varphi_{\varepsilon}||_{{\mathcal{C}}^{j,\beta}(K)}=C_{j,\beta}(K)<+\infty,$

which guarantee that $\varphi_{\varepsilon}$ is relatively compact in
${\mathcal{C}}^{\infty}(\Omega)$.

We now extract a subsequence $\varepsilon_{j}\rightarrow 0$ such that

  * •

$c_{\varepsilon_{j}}\longrightarrow c>0$;

  * •

$\varphi_{\varepsilon_{j}}\rightarrow\varphi$ in $L^{1}$ with $\varphi\in{\rm
PSH}(X,\omega)$ and $\sup_{X}\varphi=0$ (Hartogs lemma);

  * •

$\varphi\in{\mathcal{C}}^{\infty}(\Omega)$ with
$(\omega+dd^{c}\varphi)^{n}=cfdV_{X}$ in $\Omega$;

  * •

$\varphi\geq\alpha(\delta\psi^{-}+\rho)-\beta$ in $X$,

where $\alpha>0$ is arbitrarily small, as follows from (4.1). When $f\in
L^{p}(dV_{X})$, $p>1$, the solution $\varphi$ is even uniformly bounded on
$X$. ∎

######  Remark 4.3.

By comparison with the Kähler setting, it is not clear how to make sense of
the Monge-Ampère measure $(\omega+dd^{c}\varphi)^{n}$ across the singularity
divisor $D\cup E$ if $\varphi$ is unbounded, and it is delicate to establish
uniqueness of the solution, even if the solution is globally bounded (see
however Theorem 3.5).

###  4.2. Singular hermitian Calabi conjecture

Let $V$ be a compact complex variety with log-terminal singularities, i.e. $V$
is a normal complex space such that the canonical bundle $K_{V}$ is
$\mathbb{Q}$-Cartier and for some (equivalently any) resolution of
singularities $\pi:X\rightarrow V$, we have

$K_{X}=\pi^{*}K_{V}+\sum_{i}a_{i}E_{i},$

where the $E_{i}$’s are exceptional divisors with simple normal crossings, and
the rational coefficients $a_{i}$ (the discrepancies) satisfy $a_{i}>-1$.

Given $\phi$ a smooth metric of $K_{V}$ and $\sigma$ a non vanishing local
holomorphic section of $K_{V}$, we consider the ”adapted volume form”

$\mu_{\phi}:=\left(\frac{i^{rn^{2}}\sigma\wedge\overline{\sigma}}{|\sigma|^{2}_{r\phi}}\right)^{\frac{1}{r}}.$

This measure is independent of the choice of $\sigma$, and it has finite mass
on $V$, since the singularities are log-terminal. Given $\omega_{V}$ a
hermitian form on $V$, there exists a unique metric $\phi=\phi(\omega_{V})$ of
$K_{V}$ such that

$\omega_{V}^{n}=\mu_{\phi}.$

######  Definition 4.4.

The Ricci curvature form of $\omega_{V}$ is ${\rm
Ric}(\omega_{V}):=-dd^{c}\phi$.

Recall that the Bott-Chern space $H_{BC}^{1,1}(V,\mathbb{R})$ is the space of
closed real $(1,1)$-forms modulo the image of $dd^{c}$ acting on real
functions. The form ${\rm Ric}(\omega_{V})$ determines a class $c_{1}^{BC}(V)$
which maps to the usual Chern class $c_{1}(V)$ under the natural surjection
$H_{BC}^{1,1}(V,\mathbb{R})\rightarrow H^{1,1}(V,\mathbb{R})$.

By analogy with the Calabi conjecture from Kähler geometry, it is natural to
wonder conversely, whether any representative $\eta\in c_{1}^{BC}(V)$ can be
realised as the Ricci curvature form of a hermitian metric $\omega_{V}$. We
provide a positive answer, as a consequence of Theorem 3.4 and Theorem 4.1:

######  Theorem 4.5.

Let $V$ be a compact hermitian variety with log terminal singularities
equipped with a hermitian form $\omega_{V}$. For every smooth closed real
$(1,1)$-form $\eta$ in $c_{1}^{BC}(V)$, there exists a function
$\varphi\in{\rm PSH}(V,\omega_{V})$ such that

  * •

$\varphi$ is globally bounded on $V$ and smooth in $V_{reg}$;

  * •

$\omega_{V}+dd^{c}\varphi$ is a hermitian form and ${\rm
Ric}(\omega_{V}+dd^{c}\varphi)=\eta$ in $V_{reg}$.

In particular if $c_{1}^{BC}(V)=0$, any hermitian form $\omega_{V}$ is
”$dd^{c}$-cohomologous” to a Ricci flat hermitian current. Understanding the
asymptotic behavior of these singular Ricci flat currents near the
singularities of $V$ is, as in the Kähler case, an important open problem.

###### Proof.

It is classical that solving the (singular) Calabi conjecture is equivalent to
solving a complex Monge-Ampère equation. We let $\pi:X\rightarrow V$ denote a
log resolution of singularities and observe that

$\pi^{*}\mu_{\phi}=fdV,\;\;\text{ where
}\;\;f=\prod_{i=1}^{k}|s_{i}|^{2a_{i}}$

has poles (corresponding to $a_{i}<0$) or zeroes (corresponding to $a_{i}>0$)
along the exceptional divisors $E_{i}=(s_{i}=0)$ and $dV$ is a smooth volume
form on $X$.

We set $\psi^{+}=\sum_{a_{i}>0}2a_{i}\log|s_{i}|$,
$\psi^{-}=\sum_{a_{i}<0}2|a_{i}|\log|s_{i}|$, and fix $\phi$ a smooth metric
of $K_{V}$ such that $\eta=-dd^{c}\phi$. Finding $\omega_{V}+dd^{c}\varphi$
such that ${\rm Ric}(\omega_{V}+dd^{c}\varphi)=\eta$ is thus equivalent to
solving the Monge-Ampère equation
$(\omega_{V}+dd^{c}\varphi)^{n}=c\mu_{\phi}$. Passing to the resolution this
boils down to solve

$(\omega+dd^{c}\tilde{\varphi})^{n}=ce^{\psi^{+}-\psi^{-}}dV$

on $X$, where $\omega=\pi^{*}\omega_{V}$ and
$\tilde{\varphi}=\varphi\circ\pi\in{\rm PSH}(X,\omega)$.

Since $\omega$ is semi-positive and big, and since $\psi^{\pm}$ are quasi-
plurisubharmonic functions which are smooth in $X^{0}=\pi^{-1}(V_{reg})$, it
follows from Theorem 3.4 and Theorem 4.1 that there exists a solution
$\tilde{\varphi}$ with all the required properties. The function
$\varphi=\pi_{*}\tilde{\varphi}$ is the potential we were looking for. ∎

## References

  * [BT76] E. Bedford, B.A. Taylor, _The Dirichlet problem for a complex Monge-Ampère equation_. Invent. Math. 37 (1976), no. 1, 1–44.
  * [BT82] E. Bedford, B. A. Taylor, _A new capacity for plurisubharmonic functions_. Acta Math. 149 (1982), no. 1-2, 1–40.
  * [BBEGZ] R.J.Berman, S.Boucksom, P.Eyssidieux, V.Guedj, A.Zeriahi, _Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties_ , J. Reine Ang. Math. 751 (2019), 27–89.
  * [BBJ21] R. J. Berman, S.Boucksom, M.Jonsson, _A variational approach to the Yau-Tian-Donaldson conjecture_ , J. Amer. Math. Soc. (2021).
  * [Blo05] Z. Błocki, _On uniform estimate in Calabi-Yau theorem._ Sci. China Ser. A 48 (2005), suppl., 244–247.
  * [Blo11] Z. Błocki, _On uniform estimate in Calabi-Yau theorem II._ Sci. China Math. 54 (2011), no. 7, 1375–1377.
  * [BEGZ10] S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, _Monge-Ampère equations in big cohomology classes_ , Acta Math. 205 (2010), no. 2, 199–262.
  * [Cher87] P.Cherrier, _Équations de Monge-Ampère sur les variétés hermitiennes compactes_ , Bull. Sci. Math. (2) 111 (1987), no. 4, 343–385.
  * [Chi16] I.Chiose, _On the invariance of the total Monge-Ampère volume of Hermitian metrics_ , Preprint arXiv:1609.05945.
  * [CC21] X.X. Chen, J. Cheng, _On the constant scalar curvature Kähler metrics._ J. Amer. Math. Soc. (2021).
  * [Dem94] J.P. Demailly, Regularization of closed positive currents of type $(1,1)$ by the flow of a Chern connection, Contributions to complex analysis and analytic geometry, 105–126, Friedr. Vieweg, Braunschweig (1994).
  * [Dem15] J.P. Demailly, On the cohomology of pseudoeffective line bundles. Complex geometry and dynamics, Abel Symp., vol. 10, Springer, Cham, 2015, pp. 51–99.
  * [Dem] J.P. Demailly, _Analytic methods in algebraic geometry_ , Surveys of Modern Mathematics, 1. International Press; Higher Education Press, Beijing, 2012. viii+231 pp.
  * [DDGKPZ] J.P. Demailly, S. Dinew, V. Guedj, H.H. Pham, S. Kołodziej, A. Zeriahi, Hölder continuous solutions to Monge-Ampère equations. J. Eur. Math. Soc. 16 (2014), 619–647.
  * [DP10] J.P. Demailly, N. Pali, _Degenerate complex Monge-Ampère equations over compact Kähler manifolds._ Internat. J. Math. 21 (2010), no. 3, 357–405.
  * [DP04] J.P. Demailly, M. Paun, _Numerical characterization of the Kähler cone of a compact Kähler manifold._ Ann. of Math. (2) 159 (2004), no. 3, 1247–1274.
  * [Din16] S. Dinew, _Pluripotential theory on compact Hermitian manifolds_ , Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), no. 1, 91–139.
  * [DK12] S. Dinew, S.Kołodziej, _Pluripotential estimates on compact Hermitian manifolds._ Advances in geometric analysis, 69-86, Adv. Lect. Math. (ALM), 21, Int. Press, 2012.
  * [DnL15] E. Di Nezza, C. H. Lu, _Generalized Monge-Ampère capacities_. Int. Math. Res. Not. IMRN 2015, no. 16, 7287–7322.
  * [DnL17] E. Di Nezza, C. H. Lu, _Complex Monge-Ampère equations on quasi-projective varieties_ , J. Reine Angew. Math. 727 (2017), 145–167.
  * [DDL19] T. Darvas, E. Di Nezza, C.H. Lu, The metric geometry of singularity types, J. Reine Angew. Math. 771 (2021), 137–170.
  * [Don18] S. Donaldson, _Some recent developments in Kähler geometry and exceptional holonomy_ , Proc. Int. Cong. Math., Rio de Janeiro 2018. Vol. I. 425–451, World Sci. Publ., NJ, 2018.
  * [EGZ08] P. Eyssidieux, V. Guedj, A. Zeriahi, _A priori $L^{\infty}$-estimates for degenerate complex Monge-Ampère equations_, I.M.R.N., Vol. 2008, Article ID rnn070, 8 pages.
  * [EGZ09] P. Eyssidieux, V. Guedj, A. Zeriahi, _Singular Kähler-Einstein metrics_ , J. Amer. Math. Soc. 22 (2009), no. 3, 607–639.
  * [FT09] A. Fino, A. Tomassini, _Blow-ups and resolutions of strong Kähler with torsion metrics._ Adv. Math. 221 (2009), no. 3, 914–935.
  * [FLY12] J. Fu, J. Li, S.-T. Yau, _Balanced metrics on non-Kähler Calabi-Yau threefolds._ J. Differential Geom. 90 (2012), no. 1, 81–129.
  * [GL10] B. Guan, Q. Li, _Complex Monge-Ampère equations and totally real submanifolds._ Adv. Math. 225 (2010), no. 3, 1185–1223.
  * [GL21a] V. Guedj, C. H. Lu, _Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds_ , Preprint arxiv:2106.04273 (2021).
  * [GL21b] V. Guedj, C. H. Lu, _Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes_ , Preprint arxiv:2106.04272 (2021).
  * [GLZ18] V. Guedj, C.H. Lu, A. Zeriahi, Stability of solutions to complex Monge-Ampère flows. Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2819–2836.
  * [GZ] V. Guedj, A. Zeriahi, _Degenerate complex Monge-Ampère equations_ , EMS Tracts in Mathematics, vol. 26, European Mathematical Society (EMS), Zürich, 2017.
  * [GPT21] B. Guo, D.H. Phong, F. Tong, _On $L^{\infty}$-estimates for complex Monge-Ampère equations._ Preprint arXiv:2106.02224.
  * [Han96] A. Hanani, _Équations du type de Monge-Ampère sur les variétés hermitiennes compactes_ , J. Funct. Anal. 137 (1996), no. 1, 49–75.
  * [Kol98] S. Kołodziej, _The complex Monge-Ampère equation_ , Acta Math. 180 (1998), 69–117.
  * [KN15] S. Kołodziej, N.C. Nguyen, _Weak solutions to the complex Monge-Ampère equation on compact Hermitian manifolds,_ Contemp. Math. 644 (2015), 141–158.
  * [KN19] S. Kołodziej, N.C.Nguyen, _Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds_ , Adv. Math. 346 (2019), 264–304.
  * [KN21] S. Kołodziej, N.C.Nguyen, _Continuous solutions to Monge-Ampère equations on Hermitian manifolds for measures dominated by capacity_. Calc. Var. P.D.E. 60 (2021), no. 3.
  * [LN20] C. H. Lu, V.D. Nguyen, Complex Hessian equations with prescribed singularity on compact Kähler manifolds, to appear in Annali della Scuola Normale Superiore di Pisa.
  * [LPT20] C. H. Lu, T.T. Phung, T.D. Tô, _Stability and Hölder regularity of solutions to complex Monge-Ampère equations on compact hermitian manifolds._ To appear in Ann. Inst. Fourier.
  * [N16] N.C. Nguyen, _The complex Monge-Ampère type equation on compact Hermitian manifolds and applications._ Adv. Math. 286 (2016), 240–285.
  * [Pop16] D. Popovici, _Sufficient bigness criterion for differences of two nef classes_ , Math. Ann. 364 (2016), 649–655.
  * [Sko72] H. Skoda, _Sous-ensembles analytiques d’ordre fini ou infini dans $\mathbb{C}^{n}$_, Bull. Soc. Math. France, 100 (1972), 353–408.
  * [Szek18] G. Székélyhidi, _Fully non-linear elliptic equations on compact Hermitian manifolds._ J. Differential Geom. 109 (2018), no. 2, 337–378.
  * [STW17] G. Székélyhidi, V. Tosatti, B. Weinkove, _Gauduchon metrics with prescribed volume form_ , Acta Math. 219 (2017), no. 1, 181–211.
  * [Tô18] T.D. Tô, Regularizing properties of complex Monge-Ampère flows II: Hermitian manifolds. Math. Ann. 372, 699–741 (2018).
  * [TW10a] V. Tosatti, B. Weinkove, _Estimates for the Complex Monge-Ampère Equation on Hermitian and Balanced Manifolds._ Asian J. Math. Vol. 14, No. 1, pp. 19–40, March 2010.
  * [TW10b] V. Tosatti, B. Weinkove, _The complex Monge-Ampère equation on compact Hermitian manifolds._ J. Amer. Math. Soc. 23 (2010), no. 4, 1187–1195.
  * [TW15] V. Tosatti, B. Weinkove, _On the evolution of a Hermitian metric by its Chern-Ricci form_ , J. Differential Geom. 99 (2015), no. 1, 125–163.
  * [TW18] V. Tosatti, B. Weinkove, _The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation._ Nonlinear analysis in geometry and applied mathematics. Part 2, 147–158, Int. Press, Somerville, MA, 2018.
  * [Yau78] S. T. Yau, _On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I._ Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.