# Complex Monge-Ampère equations on quasi-projective varieties

E. Di Nezza  Institut Mathématiques de Toulouse
Université Paul Sabatier
31062 Toulouse
France<EMAIL_ADDRESS>and  Chinh H. Lu  Chalmers
University of Technology
Mathematical Sciences
412 96 Gothenburg
Sweden<EMAIL_ADDRESS>
(Date:
The authors are partially supported by the french ANR project MACK)

###### Abstract.

We introduce generalized Monge-Ampère capacities and use these to study
complex Monge-Ampère equations whose right-hand side is smooth outside a
divisor. We prove, in many cases, that there exists a unique normalized
solution which is smooth outside the divisor.

##  1\. Introduction

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$ and let
$D$ be a divisor on $X$. Let $f$ be a non-negative function such that
$\int_{X}f\omega^{n}=\int_{X}\omega^{n}$. Consider the following complex
Monge-Ampère equation

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

When $f$ is smooth and positive on $X$, it follows from the seminal work of
Yau [37] that (1.1) admits a unique normalized smooth solution $\varphi$ such
that $\omega+dd^{c}\varphi$ is a Kähler form. Recall that this result solves
in particular the Calabi conjecture and allows to construct Ricci flat metrics
on $X$ whenever $c_{1}(X)=0$.

It is very natural to look for a similar result when $f$ is merely smooth and
positive on the complement of $D$, e.g. when studying Calabi’s conjecture on
quasi-projective manifolds (see e.g. [33, 34, 35] and [25]) for recent
developments). The study of conical Kähler-Einstein metrics (Kähler-Einstein
metrics in the complement of a divisor with a precise behavior near $D$) has
played a major role in the resolution of the Yau-Tian-Donaldson conjecture for
Fano manifolds (see [20],[21],[14, 15, 16],[32]).

However no systematic study of the regularity of solutions to such complex
Monge-Ampère equations has ever been done, this is the main goal of this
article. It follows from [24] that (1.1) has a unique (up to an additive
constant) solution in the finite energy class $\mathcal{E}(X,\omega)$. We say
that the solution is normalized if $\sup_{X}\varphi=0$. The problem thus boils
down to showing that such a normalized solution is smooth in $X\setminus D$
and understanding its asymptotic behavior along $D$.

As in the classical case of Yau [37] the main difficulty is in establishing a
priori $\mathcal{C}^{0}$ bounds. Since, in general the solution $\varphi$ is
unbounded, the idea is to bound $\varphi$ from below by some (singular)
$\omega$-psh function.

Our first main result shows that the solution $\varphi$ is smooth in
$X\setminus D$ when $f$ satisfies the mild condition $\mathcal{H}_{f}$:

$\displaystyle\ \ \quad f=e^{\psi^{+}-\psi^{-}},\ \psi^{\pm}\ {\rm are\ quasi\
plurisubharmonic\ on}\ X,\ \ \psi^{-}\in L^{\infty}_{\rm loc}(X\setminus D).$

Let us stress that $D$ is here an arbitrary divisor.

######  Theorem 1.

Assume that $0<f\in\mathcal{C}^{\infty}(X\setminus D)$ satisfies Condition
$\mathcal{H}_{f}$. Then the solution $\varphi$ is also smooth on $X\setminus
D$.

In Theorem 1, the density $f$ is only in $L^{1}(X)$ and there is no regularity
assumption on $D$. Hence we do not have any information about the behavior of
$\varphi$ near $D$. If we assume more regularity on $f$ and $D$, we will get
more precise $\mathcal{C}^{0}$-bounds.

Assume that $D=\sum_{j=1}^{N}D_{j}$ is a simple normal crossing divisor (snc
for short). For each $j=1,...,N$, let $L_{j}$ be the holomorphic line bundle
defined by $D_{j}$. Let $s_{j}$ be a holomorphic section of $L_{j}$ such that
$D_{j}=\\{s_{j}=0\\}$. Fix a hermitian metric $h_{j}$ on $L_{j}$ such that
$|s_{j}|:=|s_{j}|_{h_{j}}\leq 1/e$.

We say that $f$ satisfies Condition $\mathcal{S}(B,\alpha)$ for some constants
$B>0,\alpha>0$ if

$f\leq\frac{B}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{1+\alpha}}\ .$

######  Theorem 2.

Assume that $f\leq e^{-\phi}$ for some quasi-plurisubharmonic function $\phi$.
Then for each $a>0$ there exists $A>0$ depending on
$\int_{X}e^{-2\varphi/a}\omega^{n}$ such that

$\varphi\geq a\phi-A.$

More precisely, if $f$ satisfies Condition $\mathcal{S}(B,\alpha)$ for some
$\alpha>0$, $B>0$, then the following holds:

  * (a)

if $\alpha>1$ then $\varphi$ is continuous on $X$, $\varphi\geq-C$, with
$C=C(B,\alpha)$.

  * (b)

if $\alpha=1$ then there exists $A_{1},A_{2}>0$ depending on $B$ such that

$\varphi\geq\sum_{j=1}^{N}-A_{1}\left[\log(-\log|s_{j}|+A_{2})\right],$

  * (c)

if $\alpha\in(0,1)$ then for each $\beta\in(1-\alpha,1)$ and $a>0$ there
exists $A>0$ depending on $a,\alpha,\beta,B$ such that

$\varphi\geq\sum_{j=1}^{N}-a(-\log|s_{j}|)^{\beta}-A.$

Remark. It follows from Skoda’s theorem [30] that
$\int_{X}e^{-2\varphi/a}\omega^{n}$ is finite for all $a>0$, since
$\varphi\in\mathcal{E}(X,\omega)$ has zero Lelong number at all points [24].

When the behavior of $f$ near the divisor $D$ looks exactly like

$\frac{1}{\prod_{j=1}^{N}|s_{j}|^{2}|\log|s_{j}||^{1+\alpha}},\
\alpha\in(0,1]$

we show in Proposition 4.4 and Proposition 4.5 that $\varphi(x)$ converges to
$-\infty$ as $x$ approaches $D$ with precise rates. In particular there is no
bounded solution to (1.1).

When $f\in L^{p}(\omega^{n})$ for some $p>1$, it follows from the work of
Kołodziej [27] that the solution of (1.1) is actually uniformly bounded (and
even Hölder continuous) on the whole of $X$.

In our result, the density $f$ is merely in $L^{1}$. The first part of Theorem
2 says that when $\alpha>1$ the solution is continuous on $X$. Kołodziej’s
result [27, Theorem 2.5.2] also applies when $f$ satisfies
$\mathcal{S}(B,\alpha)$ for $\alpha>n$ but can not be applied to a density $f$
as above if $\alpha\leq n$.

Observe furthermore that $\alpha=1$ is a critical exponent as is easily seen
when $n=1$. In any dimension, when $f$ has singularities of Poincaré type,

$\frac{1/C}{\prod_{j=1}^{N}|s_{j}|^{2}|\log|s_{j}||^{2}}\leq
f(z)\leq\frac{C}{\prod_{j=1}^{N}|s_{j}|^{2}|\log|s_{j}||^{2}}$

along $D$ we show in Section 4.3 that the solution is locally uniformly
bounded on compact subsets of $X\setminus D$ and goes to $-\infty$ along $D$
with a certain rate. If moreover $f$ has a ”very precise” behavior near $D$ it
follows from the recent work of Auvray (see [1]) that $\varphi$ goes to
$-\infty$ along $D$ like $\sum_{j=1}^{N}-\log(-\log|s_{j}|)$. We stress that
this condition is very restrictive while in our result we only need a very
weak condition on the density. Recall also that in [33] the authors
constructed ”almost complete” Kähler Einstein metrics of negative Ricci
curvature on $X\setminus D$. In this case the $\mathcal{C}^{0}$ estimate
follows easily from the maximum principle.

In order to prove the $\mathcal{C}^{0}$-estimate we follow and generalize
Kołodziej’s approach. We introduce and study the $\psi$-Capacity of a Borel
subset $E\subset X$,

${\rm Cap}_{\psi}(E):=\sup\left\\{\int_{E}(\omega+dd^{c}u)^{n}\ \big{|}\
u\in{\rm PSH}(X,\omega),\ \psi-1\leq u\leq\psi\right\\}$

where $\psi\in{\rm PSH}(X,\omega)$ and here $(\omega+dd^{c}u)^{n}$ is the
nonpluripolar Monge-Ampère measure of $u$ (see Section 2 for the definition).
When $\psi$ is constant, $\psi\equiv C$, we recover the Monge-Ampère capacity,

${\rm Cap}_{\omega}={\rm Cap}_{C}.$

A similar notion has been studied in [13] in a local context. These
generalized capacities are interesting for themselves. In this paper we only
need some of their properties and refer the reader to [18] for a more
systematic study.

One of the advantages of the Kołodziej’s approach for the $\mathcal{C}^{0}$
estimates is that it also works in the case of semipositive and big classes as
shown in [4], [22] and [8]. Thus it is not surprising that our method is still
valid in this situation.

Let $\theta$ be a smooth semipositive form on $X$ such that
$\int_{X}\theta^{n}>0$. Let $f$ be a non-negative function such that
$\int_{X}f\omega^{n}=\int_{X}\theta^{n}$. Consider the following degenerate
complex Monge-Ampère equation

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

It follows from [6] that (1.2) admits a unique normalized solution
$\varphi\in\mathcal{E}(X,\theta)$. As in the Kähler case, it is interesting to
investigate the regularity properties of $\varphi$ if we know that the density
$f$ is smooth, strictly positive outside a divisor $D$ and verifies Condition
$\mathcal{H}_{f}$. We can not expect $\varphi$ to be smooth on $X\setminus D$
since $\theta$ may be zero somewhere there. Our result below shows that the
solution is smooth on $X\setminus(D\cup E)$, where $E$ is an effective simple
normal crossing divisor on $X$ such that $\\{\theta\\}-c_{1}(E)$ is ample.

######  Theorem 3.

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$ and $D$
be an arbitrary divisor on $X$. Let $E$ be an effective snc divisor on $X$,
and $\theta$ be a smooth semipositive form on $X$ such that
$\int_{X}\theta^{n}>0$ and $\\{\theta\\}-c_{1}(E)$ is ample. Assume that
$0<f\in\mathcal{C}^{\infty}(X\setminus D)$ satisfies Condition
$\mathcal{H}_{f}$. Let $\varphi$ be the unique normalized solution to equation
(1.2). Then $\varphi$ is smooth on $X\setminus(D\cup E)$.

Remark. The condition we impose on $\\{\theta\\}$ is natural in studying
Kähler Einstein metrics on singular varieties (see [9]).

Let us say some words about the organization of the paper. In Section 2, we
introduce the generalized $\psi$-Capacity, and establish their basic
properties. The proof of Theorem 1 will be given in Section 3. We provide some
volume-capacity estimates in Section 4.1. We then use these to prove Theorem 2
and discuss about the asymptotic behavior of solutions near the divisor in
Section 4.2. Finally we consider the case of semipositive and big classes in
Section 5.

Acknowledgments. It is our pleasure to thank our advisors Vincent Guedj and
Ahmed Zeriahi for providing constant help, many suggestions and
encouragements. We also thank Robert Berman and Bo Berndtsson for very useful
comments. We are indebted to Sébastien Boucksom and Henri Guenancia for a very
careful reading of a preliminary version of this paper, for their suggestions
which improve the presentation of the paper.

##  2\. Preliminaries

Let $(X,\omega)$ be a compact Kähler manifold. We first recall basic facts
about finite energy classes of $\omega$-psh functions on $X$. The reader can
find more details about these in [24].

###  2.1. Finite energy classes

######  Definition 2.1.

We let ${\rm PSH}(X,\omega)$ denote the class of $\omega$-plurisubharmonic
functions ($\omega$-psh for short) on $X$, i.e. the class of functions
$\varphi$ such that locally $\varphi=\rho+u$, where $\rho$ is a local
potential of $\omega$ and $u$ is a plurisubharmonic function.

Let $\varphi$ be some (unbounded) $\omega$-psh function on $X$ and consider
$\varphi_{j}:=\max(\varphi,-j)$ the canonical approximation by bounded
$\omega$-psh functions. It follows from [24] that

${{\bf 1}}_{\\{\varphi_{j}>-j\\}}(\omega+dd^{c}\varphi_{j})^{n}$

is a non-decreasing sequence of Borel measures. We denote by
$(\omega+dd^{c}\varphi)^{n}$ (or $\mathrm{MA}\,(\varphi)$ for short if
$\omega$ is fixed and no confusion can occur) this limit:

$\mathrm{MA}\,(\varphi)=(\omega+dd^{c}\varphi)^{n}=\lim_{j\to+\infty}{{\bf
1}}_{\\{\varphi_{j}>-j\\}}(\omega+dd^{c}\varphi_{j})^{n}.$

It was shown in [24] that the Monge-Ampère measure $\mathrm{MA}\,(\varphi)$
puts no mass on pluripolar sets. This is the non-pluripolar part of the Monge-
Ampère of $\varphi$. Note that its total mass $\mathrm{MA}\,(\varphi)(X)$ can
take value in $\left[0,\int_{X}\omega^{n}\right]$.

######  Definition 2.2.

We let $\mathcal{E}(X,\omega)$ denote the class of $\omega$-psh function
having full Monge-Ampère mass:

$\mathcal{E}(X,\omega):=\left\\{\varphi\in{\rm PSH}(X,\omega)\ \big{|}\
\int_{X}\mathrm{MA}\,(\varphi)=\int_{X}\omega^{n}\right\\}.$

Let us stress that $\omega$-psh functions with full Monge-Ampère mass have
mild singularities. Indeed, it was shown in [24, Corollary 1.8] that

$\nu(\varphi,x)=0,\forall\varphi\in\mathcal{E}(X,\omega),\ x\in X.$

We also recall that, for every $\varphi\in\mathcal{E}(X,\omega)$ and
$\psi\in{\rm PSH}(X,\omega)$, the _generalized comparison principle_ holds
(see [8, Corollary 2.3]), namely

$\int_{\\{\varphi<\psi\\}}(\omega+dd^{c}\psi)^{n}\leq\int_{\\{\varphi<\psi\\}}(\omega+dd^{c}\varphi)^{n}.$

Let $\chi:\mathbb{R}^{-}\to\mathbb{R}^{-}$ be an increasing function such that
$\chi(0)=0$ and $\chi(-\infty)=-\infty$.

######  Definition 2.3.

Let $\mathcal{E}_{\chi}(X,\omega)$ denote the set of $\omega$-psh functions
with finite $\chi$-energy,

$\mathcal{E}_{\chi}(X,\omega):=\\{\varphi\in\mathcal{E}(X,\omega)\;\,\
\big{|}\ \;\,\chi(-|\varphi|)\in L^{1}(\mathrm{MA}\,(\varphi))\\}.$

For $p>0$, we use the notation

${\mathcal{E}}^{p}(X,\omega):=\mathcal{E}_{\chi}(X,\omega),\text{ when
}\chi(t)=-(-t)^{p}.$

###  2.2. The $\psi$-Capacity

######  Definition 2.4.

Let $\psi\in{\rm PSH}(X,\omega)$. We define the $\psi$-Capacity of a Borel
subset $E\subset X$ by

${\rm Cap}_{\psi}(E):=\sup\left\\{\int_{E}\mathrm{MA}\,(u)\ |\ u\in{\rm
PSH}(X,\omega),\ \psi-1\leq u\leq\psi\right\\}.$

Then the Monge-Ampère capacity corresponds to $\psi\equiv{\rm constant}$ (see
[3], [28], [23]). We list below some basic properties of the $\psi$-Capacity.

######  Proposition 2.5.

  * (i)

If $E_{1}\subset E_{2}\subset X$ then ${\rm Cap}_{\psi}(E_{1})\leq{\rm
Cap}_{\psi}(E_{2})$ .

  * (ii)

If $E_{1},E_{2},...$ are Borel subsets of $X$ then

${\rm
Cap}_{\psi}\left(\bigcup_{j=1}^{\infty}E_{j}\right)\leq\sum_{j=1}^{+\infty}{\rm
Cap}_{\psi}(E_{j}).$

  * (iii)

If $E_{1}\subset E_{2}\subset...$ are Borel subsets of $X$ then

${\rm
Cap}_{\psi}\left(\bigcup_{j=1}^{\infty}E_{j}\right)=\lim_{j\to+\infty}{\rm
Cap}_{\psi}(E_{j}).$

The following results are elementary and important for the sequel. We stress
that these results still hold in the case when $\omega$ is merely semipositive
and big rather than Kähler.

######  Lemma 2.6.

Let $\psi\in{\rm PSH}(X,\omega)$ and $\varphi\in\mathcal{E}(X,\omega)$. Then
the function

$H(t):={\rm Cap}_{\psi}(\\{\varphi<\psi-t\\}),\ t\in\mathbb{R},$

is right-continuous and $H(t)\to 0$ as $t\to+\infty$.

###### Proof.

The right-continuity of $H$ follows from (iii) of Proposition 2.5. Let us
prove the second statement. We can assume that $\psi\leq 0$ on $X$. Fix
$v\in{\rm PSH}(X,\omega)$ such that $\psi-1\leq v\leq\psi$. We apply the
comparison principle to obtain

$\displaystyle\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(v)\leq\int_{\\{\varphi<v-t+1\\}}\mathrm{MA}\,(v)\leq\int_{\\{\varphi<-t+1\\}}\mathrm{MA}\,(\varphi).$

The last term goes to zero as $t$ goes to $+\infty$ since
$\varphi\in\mathcal{E}(X,\omega)$. ∎

######  Lemma 2.7.

Let $(X,\omega)$ be a compact Kähler manifold and $\psi\in{\rm
PSH}(X,\omega/2)$. Then we have

${\rm Cap}_{\omega/2}(E)\leq{\rm Cap}_{\psi}(E),$

where ${\rm Cap}_{\omega/2}$ is the Monge-Ampère Capacity with respect to the
Kähler metric $\omega/2$ introduced in [28] and studied in [23], and ${\rm
Cap}_{\psi}$ is the generalized $\psi$-Capacity with respect to the Kähler
metric $\omega$.

We stress that the above result insures ${\rm Cap}_{\psi}(E)>0$ for any Borel
subset $E$ which is not pluripolar.

###### Proof.

Let $u\in{\rm PSH}(X,\omega/2)$ be such that $-1\leq u\leq 0$. Then
$\varphi:=\psi+u$ is a candidate defining ${\rm Cap}_{\psi}$. Using the
definition of the Monge-Ampère meausure it is not difficult to see that

$\int_{E}(\omega/2+dd^{c}u)^{n}\leq\int_{E}(\omega+dd^{c}\varphi)^{n}\leq{\rm
Cap}_{\psi}(E),$

and taking the supremum over all $u$ we get the result. ∎

The following result generalizes Lemma 2.3 in [22].

######  Proposition 2.8.

Let $\varphi\in\mathcal{E}(X,\omega)$, $\psi\in{\rm PSH}(X,\omega)$. Then for
all $t>0$ and $0\leq s\leq 1$ we have

$s^{n}{\rm Cap}_{\psi}(\\{\varphi<\psi-
t-s\\})\leq\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi).$

###### Proof.

Let $u\in{\rm PSH}(X,\omega)$ such that $\psi-1\leq u\leq\psi$. Observe the
following trivial inclusion

$\\{\varphi<\psi-
t-s\\}\subset\\{\varphi<su+(1-s)\psi-t\\}\subset\left\\{\varphi<\psi-t\right\\}.$

It thus follows from the generalized comparison principle (see [8, Corollary
2.3]) that

$\displaystyle s^{n}\int_{\\{\varphi<\psi-t-s\\}}\mathrm{MA}\,(u)$
$\displaystyle\leq$ $\displaystyle\int_{\\{\varphi<\psi-
t-s\\}}\mathrm{MA}\,(su+(1-s)\psi)$ $\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<su+(1-s)\psi-t\\}}\mathrm{MA}\,(su+(1-s)\psi)$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi).$

By taking the supremum over all candidates $u$ we get the result. ∎

##  3\. Smooth solution in a general case

In this section we prove Theorem 1. The most difficult part is the
$\mathcal{C}^{0}$ estimate which is followed by a simple observation: if
$\varphi\in\mathcal{E}(X,\omega)$, $\sup_{X}\varphi=0$ is such that
$\mathrm{MA}\,(\varphi)\leq e^{-\phi}\omega^{n}$, for some quasi-psh function
$\phi$, then $\varphi$ is bounded from below by $a\phi-A$, for some positive
constants $a,A$.

###  3.1. Uniform estimate

In this subsection we assume that $0\leq f\in L^{1}(X)$ is such that
$\int_{X}f\omega^{n}=\int_{X}\omega^{n}$. Let
$\varphi\in\mathcal{E}(X,\omega)$ be the unique normalized solution to

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

Here we normalize $\varphi$ such that $\sup_{X}\varphi=0$. We prove the
following $\mathcal{C}^{0}$ estimate:

######  Theorem 3.1.

Assume that $f\leq e^{-\phi}$ for some quasi-plurisubharmonic function $\phi$.
Let $\varphi\in\mathcal{E}(X,\omega)$ be the unique normalized solution to
(3.1). Then for any $a>0$, there exists $A>0$ depending on
$\int_{X}e^{-2\varphi/a}\omega^{n}$ such that

$\varphi\geq a\phi-A.$

Moreover, if $\phi$ is bounded in a compact subset $K\subset X$ then $\varphi$
is continuous on $K$.

###### Proof.

We can assume that $\phi\leq 0$. Observe that it is enough to prove Theorem
3.1 for $a>0$ small enough. Fix $a>0$ such that $\psi:=a\phi$ belongs to ${\rm
PSH}(X,\omega/2)$. It follows from Lemma 2.7 that ${\rm Cap}_{\omega}\leq
2^{n}{\rm Cap}_{\omega/2}\leq 2^{n}{\rm Cap}_{\psi}$. Fix $s\in[0,1]$, $t>0$
and apply Proposition 2.8 to get

(3.2) $s^{n}{\rm Cap}_{\psi}(\varphi<\psi-
t-s)\leq\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi).$

By assumption on $f$ we have

$\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi)\leq\int_{\\{\varphi<\psi-t\\}}e^{-\varphi/a}e^{\psi/a}\mathrm{MA}\,(\varphi)\leq\int_{\\{\varphi<\psi-t\\}}e^{-\varphi/a}\omega^{n}.$

It follows from [23] that

${\rm Vol}_{\omega}\leq\exp{\left(\frac{-C_{1}}{{\rm
Cap}_{\omega}^{1/n}}\right)}.$

Thus using Hölder inequality we get from (3.2) that

$s^{n}{\rm Cap}_{\psi}(\varphi<\psi-t-s)\leq C_{2}\left({\rm
Cap}_{\omega}(\varphi<\psi-t)\right)^{2}\leq C_{3}\left({\rm
Cap}_{\psi}(\varphi<\psi-t)\right)^{2},$

where $C_{3}$ depends only on $\int_{X}e^{-2\varphi/a}\omega^{n}$. Now,
consider the following function

$H(t)=\left[{\rm Cap}_{\psi}(\\{\varphi<\psi-t\\})\right]^{1/n},\ t>0.$

By the arguments above we get

$sH(t+s)\leq C_{4}H(t)^{2},\ \forall t>0,\forall s\in[0,1],$

where $C_{4}>0$ depends only on $\int_{X}e^{-2\varphi/a}\omega^{n}$. It
follows from Lemma 2.6 that $H$ is right-continuous and $H(+\infty)=0$. Thus
by [22, Lemma 2.4] we get $\varphi\geq\psi-C_{5}$, where $C_{5}$ only depends
on $\int_{X}e^{-2\varphi/a}\omega^{n}$.

Now, assume that $\phi$ is bounded on a compact subset $K\subset X$. Set
$\psi:=a\phi$ as above. Let us prove that $\varphi$ is continuous on $K$. For
convenience, we normalize $\varphi$ so that $\sup_{X}\varphi=-1$. Let
$0\geq\varphi_{j}$ be a sequence of continuous $\omega$-psh functions on $X$
decreasing to $\varphi$. Fix $\lambda\in(0,1)$. For each $j\in\mathbb{N}$ set

$\psi_{j}:=\lambda\varphi_{j}+(1-\lambda)\psi-(1-\lambda)A-2.$

Then $\psi_{j}$ belongs to ${\rm PSH}(X,\frac{1+\lambda}{2}\omega)$ and
$\psi_{j}\leq\varphi_{j}-2$. Set

$H_{j}(t):=\left[{\rm
Cap}_{\psi_{j}}\left(\\{\varphi<\psi_{j}-t\\}\right)\right]^{1/n},t>0.$

We can argue as above and use Proposition 2.8 to get

$sH_{j}(t+s)\leq C_{1}H_{j}(t)^{2},\ \forall t>0,\ \forall s\in[0,1],$

where $C_{1}>0$ depends on $\int_{X}e^{-2\varphi/(1-\lambda)a}$. Let
$\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ be an increasing convex weight
such that $\chi(0)=0$, $\chi(-\infty)=-\infty$ and
$\varphi\in\mathcal{E}_{\chi}(X,\omega)$. By the comparison principle we also
get

$\displaystyle{\rm Cap}_{\psi_{j}}(\varphi<\psi_{j})$ $\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<\psi_{j}+1\\}}\mathrm{MA}\,(\varphi)\leq\int_{\\{\varphi<\varphi_{j}-1\\}}\mathrm{MA}\,(\varphi)$
$\displaystyle\leq$
$\displaystyle\frac{1}{-\chi(-1)}\int_{X}(-\chi\circ(\varphi-\varphi_{j}))f\omega^{n}.$

The latter converges to $0$ as $j\to+\infty$, since $\varphi_{j}$ decreases to
$\varphi$. Thus for $j$ big enough we have $H_{j}(0)\leq 1/(2C_{1})$. It then
follows from [22, Remark 2.5] that $H_{j}(t)=0$ if $t\geq t_{\infty}$ where
$t_{\infty}\leq C_{2}H_{j}(0)$ and $C_{2}$ depends on $C_{1}$. We then get

$\varphi\geq\lambda\varphi_{j}+(1-\lambda)\psi-C_{2}H_{j}(0).$

Now, letting $j\to+\infty$, we get

$\lim_{j\to+\infty}\inf_{K}(\varphi-\varphi_{j})\geq(\lambda-1)\sup_{K}|\psi|.$

Finally, letting $\lambda\to 1$ we get the continuity of $\varphi$ on $K$. ∎

###  3.2. Laplacian estimate

The following a priori estimate generalizes [29].

######  Theorem 3.2.

Let $\mu$ be a positive measure on $X$ of the form
$\mu=e^{\psi^{+}-\psi^{-}}\omega^{n}$ where $\psi^{+},\psi^{-}$ are smooth on
$X$. Let $\varphi\in\mathcal{C}^{\infty}(X)$ be such that $\sup_{X}\varphi=0$
and

$(\omega+dd^{c}\varphi)^{n}=e^{\psi^{+}-\psi^{-}}\omega^{n}.$

Assume given a constant $C>0$ such that

$dd^{c}\psi^{\pm}\geq-C\omega,\ \ \sup_{X}\psi^{+}\leq C.$

Assume also that the holomorphic bisectional curvature of $\omega$ is bounded
from below by $-C$. Then there exists $A>0$ depending on $C$ and
$\int_{X}e^{-2(4C+1)\varphi}\omega^{n}$ such that

$0\leq n+\Delta_{\omega}\varphi\leq Ae^{-2\psi^{-}}.$

We follow the lines in Appendix B of [7]. We recall the following result:

######  Lemma 3.3.

Let $\alpha,\beta$ be positive $(1,1)$-forms. Then

$n\left(\frac{\alpha^{n}}{\beta^{n}}\right)^{\frac{1}{n}}\leq{\rm
tr}_{\beta}(\alpha)\leq
n\left(\frac{\alpha^{n}}{\beta^{n}}\right)\cdot\left({\rm
tr}_{\alpha}(\beta)\right)^{n-1}.$

###### Proof of Theorem 3.2.

Set $\omega_{\varphi}:=\omega+dd^{c}\varphi$. Since the holomorphic
bisectional curvature of $\omega$ is bounded from below by $-C$, it follows
from Lemma 2.2 in [10] that

(3.3) $\Delta_{\omega_{\varphi}}\log{\rm
tr}_{\omega}(\omega_{\varphi})\geq\frac{{\rm
tr}_{\omega}(dd^{c}\psi^{+}-dd^{c}\psi^{-})}{{\rm
tr}_{\omega}(\omega_{\varphi})}-C{\rm tr}_{\omega_{\varphi}}(\omega).$

Since $dd^{c}\psi^{+}\geq-C\omega$, using the trivial inequality $n\leq{\rm
tr}_{\omega}(\omega_{\varphi}){\rm tr}_{\omega_{\varphi}}(\omega)$ we thus get
from (3.3) that

(3.4) $\displaystyle\Delta_{\omega_{\varphi}}\log{\rm
tr}_{\omega}(\omega_{\varphi})$ $\displaystyle\geq$ $\displaystyle-\frac{{\rm
tr}_{\omega}(C\omega+dd^{c}\psi^{-})}{{\rm
tr}_{\omega}(\omega_{\varphi})}-C{\rm tr}_{\omega_{\varphi}}(\omega)$
$\displaystyle\geq$ $\displaystyle-2C{\rm
tr}_{\omega_{\varphi}}(\omega)-\frac{\Delta\psi^{-}}{{\rm
tr}_{\omega}(\omega_{\varphi})}\,.$

By assumption we have $0\leq C\omega+dd^{c}\psi^{-}\leq{\rm
tr}_{\omega_{\varphi}}(C\omega+dd^{c}\psi^{-})\omega_{\varphi}$. Applying
${\rm tr}_{\omega}$ to the previous inequality yields

$Cn+\Delta\psi^{-}\leq(C{\rm
tr}_{\omega_{\varphi}}(\omega)+\Delta_{\omega_{\varphi}}\psi^{-}){\rm
tr}_{\omega}(\omega_{\varphi}),$

and hence

$-\Delta\psi^{-}\geq-(C{\rm
tr}_{\omega_{\varphi}}(\omega)+\Delta_{\omega_{\varphi}}\psi^{-}){\rm
tr}_{\omega}(\omega_{\varphi}).$

Thus, plugging this into (3.4) we obtain

(3.5) $\displaystyle\Delta_{\omega_{\varphi}}\log{\rm
tr}_{\omega}(\omega_{\varphi})\geq-3C{\rm
tr}_{\omega_{\varphi}}(\omega)-\Delta_{\omega_{\varphi}}\psi^{-}.$

We want now to apply the maximum principle to the function

$H:=\log{\rm tr}_{\omega}(\omega_{\varphi})+2\psi^{-}-(1+4C)\varphi,$

Let $x_{0}\in X$ be such that $H$ achieves its maximum on $X$ at $x_{0}$. Then
at $x_{0}$ we get

$\displaystyle 0\geq\Delta_{\omega_{\varphi}}H$ $\displaystyle\geq$
$\displaystyle{\rm tr}_{\omega_{\varphi}}(\omega)-n(1+4C).$

Furthermore, by Lemma 3.3 we get

${\rm tr}_{\omega}(\omega_{\varphi})(x_{0})\leq
ne^{\psi^{+}-\psi^{-}}(x_{0})\left({\rm
tr}_{\omega_{\varphi}}(\omega)\right)^{n-1}(x_{0})\leq
A_{1}e^{\psi^{+}-\psi^{-}}(x_{0}),$

and hence, since $\sup_{X}\psi^{+}\leq C$,

$\log{\rm tr}_{\omega}(\omega_{\varphi})(x_{0})\leq\log
A_{1}+\psi^{+}(x_{0})-\psi^{-}(x_{0})\leq A_{2}-\psi^{-}(x_{0})\,.$

It follows that

$H(x)\leq H(x_{0})\leq A_{3}+\psi^{-}(x_{0})-(1+4C)\varphi(x_{0}).$

By assumption and the $\mathcal{C}^{0}$ estimate in Theorem 3.1 we have
$\varphi\geq a\psi^{-}-A_{4}$, where $a=1/(4C+1)$ and $A_{4}$ depends on $C$
and $\int_{X}e^{-2\varphi/a}\omega^{n}$. Thus

$\log{\rm tr}_{\omega}(\omega_{\varphi})\leq A_{5}-2\psi^{-}.$

We finally infer as desired

${\rm tr}_{\omega}(\omega_{\varphi})\leq A_{6}e^{-2\psi^{-}}.$

∎

We are now ready to prove Theorem 1.

###  3.3. Proof of Theorem 1

Let $\varphi\in\mathcal{E}(X,\omega)$ be the unique normalized solution to

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

By assumption we can write $\log f=\psi^{+}-\psi^{-}$, where $\psi^{\pm}$ are
quasi psh functions on $X$, $\psi^{-}$ is locally bounded on $X\setminus D$,
and there is a uniform constant $C>0$ such that

$dd^{c}\psi^{\pm}\geq-C\omega,\ \sup_{X}\psi^{+}\leq C.$

We now approximate $\psi^{\pm}$ by using Demailly’s regularization operator
$\rho_{\varepsilon}$. We recall the construction: if $u$ is a quasi-psh
function on $X$ and $\varepsilon>0$ we set

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

Here $\chi\in\mathcal{C}^{\infty}(\mathbb{R})$ is a cut-off function supported
in $[-1,1]$, $\int_{\mathbb{R}}\chi(t)dt=1$, and

${\rm exph}:TX\rightarrow X,\ \ \zeta\mapsto{\rm exph}_{z}(\zeta)$

is the formal holomorphic part of the Taylor expansion of the exponential map
defined by the metric $\omega$. For more details, see [17]. Observe that by
Jensen’s inequality, $\rho_{\varepsilon}(e^{u})\geq
e^{\rho_{\varepsilon}(u)}$. Applying this smoothing regularization to
$\psi^{\pm}$ we get, for $\varepsilon>0$ small enough,

$dd^{c}\rho_{\varepsilon}(\psi^{\pm})\geq-C_{1}\omega,\ \
e^{\rho_{\varepsilon}(\psi^{+}-\psi^{-})}\leq
e^{-\rho_{\varepsilon}(\psi^{-})+C_{1}},$

where $C_{1}$ depends on $C$ and the Lelong numbers of the currents
$C\omega+dd^{c}\psi^{\pm}$. Now, for each $\varepsilon>0$, let
$\varphi_{\varepsilon}\in\mathcal{C}^{\infty}(X)$ be the unique normalized
solution to

$(\omega+dd^{c}\varphi_{\varepsilon})^{n}=c_{\varepsilon}e^{\rho_{\varepsilon}(\psi^{+})-\rho_{\varepsilon}(\psi^{-})}\omega^{n}=f_{\varepsilon}\omega^{n},$

where $c_{\varepsilon}>0$ is a normalization constant. Since
$e^{\rho_{\varepsilon}(\log f)}$ converges point-wise to $f$ on $X$ and since
$e^{\rho_{\varepsilon}(\log f)}\leq\rho_{\varepsilon}(e^{\log f})$, by the
General Lebesgue Dominated Convergence Theorem we see that
$e^{\rho_{\varepsilon}(\log f)}$ converges to $f$ in $L^{1}(X)$ as
$\varepsilon\to 0$. This implies that $c_{\varepsilon}$ converges to $1$ as
$\varepsilon\to 0$. Then we can assume that $c_{\varepsilon}\leq 2$. Thus we
get the following uniform control

$f_{\varepsilon}\leq e^{-\rho_{\varepsilon}(\psi^{-})+C_{2}}.$

By Lemma 3.4 below we know that $\varphi_{\varepsilon}$ converges to $\varphi$
in $L^{1}(X)$. Thus the set

$\mathcal{U}:=\\{\varphi_{\varepsilon}\ \ \big{|}\ \
\varepsilon>0\\}\cup\\{\varphi\\}$

is compact in $L^{1}(X)$. Then it follows from the uniform Skoda integrability
theorem (Lemma 3.5 below) that for any $A>0$ we have

$\sup_{\varepsilon>0}\int_{X}e^{-A\varphi_{\varepsilon}}\omega^{n}<+\infty.$

Thus, we can apply Theorem 3.2 to find $C_{3}>0$ under control such that

$\Delta_{\omega}\varphi_{\varepsilon}\leq C_{3}e^{-2\psi^{-}}.$

Fix a compact $K\Subset X\setminus D$, $k\geq 2$ and $\beta\in(0,1)$. Now
since $0<f\in\mathcal{C}^{\infty}(X\setminus D)$ we have uniform controls on
the derivatives of all orders of $\log f_{\varepsilon}$ on $K$. Using the
standard Evans-Krylov method and Schauder estimates we then obtain

$\|\varphi_{\varepsilon}\|_{\mathcal{C}^{k,\beta}(K)}\leq C_{K,k,\beta}.$

This explains the smoothness of $\varphi$ on $X\setminus D$.

######  Lemma 3.4.

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$. Let $(f_{j})$
be a sequence of non-negative functions on $X$ such that
$\int_{X}f_{j}\omega^{n}=\int_{X}\omega^{n}$. Assume that $f_{j}$ converges in
$L^{1}(X)$ and point-wise to $f$. For each $j$, let
$\varphi_{j}\in\mathcal{E}(X,\omega)$ be the unique normalized solution to
$\mathrm{MA}\,(\varphi_{j})=f_{j}\omega^{n}$. Then $\varphi_{j}$ converges in
$L^{1}(X)$ to $\varphi\in\mathcal{E}(X,\omega)$ the unique normalized solution
to $\mathrm{MA}\,(\varphi)=f\omega^{n}$.

###### Proof.

We can assume that $\varphi_{j}$ converges in $L^{1}(X)$ to $\psi\in{\rm
PSH}(X,\omega)$. It follows from the Hartogs lemma that $\sup_{X}\psi=0$. For
each $j\in\mathbb{N}$ set

$\psi_{j}:=\left(\sup_{k\geq j}\varphi_{j}\right)^{*}\ {\rm and}\
u_{j}:=\max(\psi_{j},\varphi-1).$

Then we see that $\psi_{j}\downarrow\psi$ and $u_{j}\downarrow
u:=\max(\psi,\varphi-1)\in\mathcal{E}(X,\omega)$. We also have that
$\sup_{X}u=0$. It follows from the comparison principle that

$\mathrm{MA}\,(u_{j})\geq\min\left(f,\inf_{k\geq
j}f_{k}\right)\omega^{n}=g_{j}\omega^{n}.$

By the continuity of the Monge-Ampère operator along decreasing sequences in
$\mathcal{E}(X,\omega)$ we get

$\mathrm{MA}\,(u)=\lim_{j\to+\infty}\mathrm{MA}\,(u_{j})\geq\lim_{j\to+\infty}g_{j}\omega^{n}=f\omega^{n}.$

Then the equality holds since they have the same total mass. Finally, by the
uniqueness result in the class $\mathcal{E}(X,\omega)$ (see [19]) we deduce
that $u=\varphi$, which implies that $\psi=\varphi$. The proof is thus
complete. ∎

By [24], functions in $\mathcal{E}(X,\omega)$ have zero Lelong number at every
point on $X$. Thus the following lemma is a direct consequence of the uniform
Skoda integrability theorem due to Zeriahi [38]:

######  Lemma 3.5.

Let $\mathcal{U}$ be a compact family of functions in $\mathcal{E}(X,\omega)$.
Then for each $C_{1}>0$ there exists $C_{2}$ depending on $C_{1}$ and
$\mathcal{U}$ such that

$\int_{X}e^{-C_{1}\phi}\omega^{n}\leq C_{2},\ \forall\phi\in\mathcal{U}.$

##  4\. Asymptotic behavior near the divisor

In Theorem 3.1 we have given a very general $\mathcal{C}^{0}$ estimate. We
only assumed that the density $f$ is bounded by $e^{-\phi}$ for some quasi
plurisubharmonic function $\phi$, and there is no regularity assumption on
$D$. It is therefore natural to investigate the asymptotic behavior of the
solution near $D$ when we have more information about $D$ and about the
behavior of $f$ near $D$.

Let $X$ be a compact Kähler manifold of dimension $n$ and let $\omega$ be a
Kähler form on $X$. Let $D=\sum_{j=1}^{N}D_{j}$ be a simple normal crossing
divisor on $X$. Here ”simple normal crossing” means that around each
intersection point of $k$ components $D_{j_{1}},...,D_{j_{k}}$ ($k\leq N$), we
can find complex coordinates $z_{1},...,z_{n}$ such that for each $l=1,...,k$
the hypersurface $D_{j_{l}}$ is locally given by $z_{l}=0$. For each $j$, let
$L_{j}$ be the holomorphic line bundle defined by $D_{j}$. Let $s_{j}$ be a
holomorphic section of $L_{j}$ defining $D_{j}$, i.e $D_{j}=\\{s_{j}=0\\}$. We
fix a hermitian metric $h_{j}$ on $L_{j}$ such that
$|s_{j}|:=|s_{j}|_{h_{j}}\leq 1/e$.

We say that $f$ satisfies Condition $\mathcal{S}(B,\alpha)$ for some
$B>0,\alpha>0$ if

(4.1) $f\leq\frac{B}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{1+\alpha}}.$

###  4.1. Volume-capacity domination

######  Lemma 4.1.

Assume that $f$ satisfies (4.1) for some $B>0,\alpha>0$. Then for each
$0<\gamma<\alpha$ we can find $A>0$ which only depends on
$B,\alpha,\gamma,\omega$ such that

${\rm Vol}_{f}(E):=\int_{E}f\omega^{n}\leq A\,{\rm
Cap}_{\omega}(E)^{\gamma},\;\forall E\subset X,$

where ${\rm Cap}_{\omega}$ is the Monge-Ampère capacity introduced in [28],
[23].

Before giving the proof of the lemma, let us recall the definition and basic
facts about Cegrell’s classes. We refer the reader to [11, 12] for more
details.

Let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. The class
$\mathcal{E}_{0}(\Omega)$ consists of bounded psh functions which vanish on
the boundary and have finite total mass.

We say that $u\in\mathcal{E}^{p}(\Omega),p>0$ if there exists a sequence
$(u_{j})\subset\mathcal{E}_{0}(\Omega)$ decreasing to $u$ such that

$\sup_{j}\int_{\Omega}(-u_{j})^{p}(dd^{c}u_{j})^{n}<+\infty.$

A function $u$ belongs to $\mathcal{F}(\Omega)$ if there exists a sequence
$(u_{j})\subset\mathcal{E}_{0}(\Omega)$ decreasing to $u$ such that

$\sup_{j}\int_{\Omega}(dd^{c}u_{j})^{n}<+\infty.$

We recall the local Monge-Ampère capacity introduced in [3]: for any Borel
subset $E\subset\Omega$, we define

${\rm Cap}_{\rm BT}(E,\Omega):=\sup\left\\{\int_{E}(dd^{c}u)^{n}\ \ \big{|}\ \
u\in{\rm PSH}(\Omega)\ ,\ -1\leq u\leq 0\right\\}.$

The relative extremal function of $E$ with respect to $\Omega$ is

$u_{E,\Omega}:=\sup\left\\{u\in{\rm PSH}(\Omega)\ \ \big{|}\ \ u\leq 0\ {\rm
on}\ \Omega\ ,\ u\leq-1\ {\rm on}\ E\right\\}.$

###### Proof of Lemma 4.1.

It follows from [28] that ${\rm Cap}_{\omega}$ is comparable to the local
capacity ${\rm Cap}_{\rm BT}(\cdot,\Omega)$, where $\Omega$ is an open subset
contained in a local chart. By considering $E$ a small subset contained in a
local chart we reduce the problem to showing that

(4.2) ${\rm Vol}_{g}(E)\leq A_{1}{\rm Cap}_{\rm
BT}(E,\mathbb{D}^{n})^{\alpha},\forall
E\Subset\mathbb{D}^{n}_{\delta}\Subset\mathbb{D}^{n},$

where $\mathbb{D}^{n}$ is the unit polydisk in $\mathbb{C}^{n}$, $\delta>0$
small enough and fixed, and

$g(z)=g(z_{1},...,z_{n}):=\frac{1}{\prod_{j=1}^{k}|z_{j}|^{2}(1-\log|z_{j}|)^{1+\alpha}},k\leq
n.$

We prove (4.2) by induction using the ideas in [2]. We start with the case
$n=1$.

Set $E_{r}:=E\cap\partial\mathbb{D}_{r}$, for any $r\in[0,t]$. Define now
$\tilde{E}:=\\{r\in[0,t]\,|\,E_{r}\neq\emptyset\\}$ and denote by
$l(\tilde{E})$ the length of $\tilde{E}$. Since the function
$r\mapsto\frac{1}{r(1-\log r)^{1+\alpha}}$ is non-increasing when $r$ is
small, we obtain

$\displaystyle\int_{E}g(z)dV(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\int_{\tilde{E}}\frac{drd\theta}{r(1-\log
r)^{1+\alpha}}$ $\displaystyle\leq$ $\displaystyle
2\pi\int_{0}^{\ell({\tilde{E}})}\frac{dr}{r(1-\log r)^{1+\alpha}}$
$\displaystyle\leq$ $\displaystyle\frac{C_{1}}{(-\log l(\tilde{E}))^{\alpha}}$
$\displaystyle\leq$ $\displaystyle C_{2}\,\left[{\rm Cap}_{\rm
BT}(E,\mathbb{D})\right]^{\alpha},$

where the last inequality follows from [26, p.1336].

Assume that the result holds for $n-1$. Let us prove it for $n$. Without loss
of generality we can assume that $E$ is compact in $\mathbb{D}^{n}$. We can
also assume that $k=n$ (if $k<n$ the situation is much easier). Set
$h=h_{E,\mathbb{D}^{n}}^{*}$ the relative extremal function of $E$. Consider

$g_{n}(w):=\frac{1}{|w|^{2}(1-\log|w|)^{1+\alpha}}\ ,\
g_{n-1}(z):=\frac{1}{\prod_{j=1}^{n-1}|z_{j}|^{2}(1-\log|z_{j}|)^{1+\alpha}}.$

For each $w\in\mathbb{D}$ set

$E_{w}=\\{z\in\mathbb{D}^{n-1}\ |\ h(z,w)\leq-1\\}\ \ {\rm and}\ \
h_{w}=h(\cdot,w).$

By induction hypothesis we get

${\rm Vol}_{g}(E)=\int_{\mathbb{D}}{\rm
Vol}_{g_{n-1}}(E_{w})g_{n}(w)dV_{2}(w)\leq A_{1}\int_{\mathbb{D}}\left[{\rm
Cap}_{\rm BT}(E_{w},\mathbb{D}^{n-1})\right]^{\gamma}g_{n}(w)dV(w).$

Fix now $w\in\mathbb{D}$ and denote by $u=h_{E_{w},\mathbb{D}}^{*}$ the
relative extremal function of $E_{w}$. Since $h\in\mathcal{F}(\mathbb{D}^{n})$
it follows from [2, Theorem 3.1] that
$h_{w}\in\mathcal{E}^{1}(\mathbb{D}^{n-1})$. We also have $h_{w}\leq u$ and
$h_{w}=-1$ on $E_{w}.$ Using integration by parts we get

$\displaystyle{\rm Cap}_{\rm
BT}(E_{w},\mathbb{D}^{n-1})\leq\int_{\mathbb{D}^{n-1}}(-h_{w})(dd^{c}u)^{n-1}\leq\int_{\mathbb{D}^{n-1}}(-h_{w})(dd^{c}h_{w})^{n-1}=:-\varphi(w).$

By [2, Theorem 3.1] we know that $\varphi\in\mathcal{F}(\mathbb{D})$.
Moreover, we also have $\varphi\geq-A_{0}$ for some universal constant $A_{0}$
(here $A_{0}$ depends on $\delta$). Indeed, let $v$ be the relative extremal
function of $\mathbb{D}_{\delta}^{n}$ with respect to $\mathbb{D}^{n}$. Since
$h\geq v$, it is easy to see that for each $w\in\mathbb{D}$, $h_{w}\geq
v_{w}$. From this we get a uniform lower bound for $\varphi$. Since $E$ is
compact in $\mathbb{D}^{n}$ we also get

$\mu=\int_{\mathbb{D}}dd^{c}\varphi=\int_{\mathbb{D}^{n}}(dd^{c}h)^{n}={\rm
Cap}_{\rm BT}(E,\mathbb{D}^{n}).$

Thus, using the previous part (when $n=1$) we obtain

$\displaystyle{\rm Vol}_{g}(E)$ $\displaystyle\leq$ $\displaystyle
A_{1}\int_{\mathbb{D}}(-\varphi(w))^{\gamma}g_{n}(w)dV_{2}(w)$
$\displaystyle=$ $\displaystyle A_{2}\int_{0}^{A_{0}}t^{\gamma-1}{\rm
Vol}_{g_{n}}(\varphi<-t)dt$ $\displaystyle\leq$ $\displaystyle
A_{3}\int_{0}^{A_{0}}t^{\gamma-\beta_{1}-1}\mu^{\beta_{1}}dt$ $\displaystyle=$
$\displaystyle A_{4}\left[{\rm Cap}_{\rm
BT}(E,\mathbb{D}^{n})\right]^{\beta_{1}}.$

Here, we choose $\beta_{1}<\gamma$ so that the integrals converge. In the
above we have used the fact that

${\rm Cap}_{\rm BT}(v<-t)\leq\frac{1}{t}\int_{\mathbb{D}}dd^{c}v,\ \forall
v\in\mathcal{F}(\mathbb{D}),\ \forall t>0.$

Since $\beta_{1}$ can be chosen arbitrarily near $\gamma$ (and the constant
$A_{4}$ will increase), the result follows. ∎

When $\alpha=1$ we get the following estimate.

######  Lemma 4.2.

Let $\mu=f\omega^{n}$,
$f=\frac{1}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{2}}$. Then there exists
$A>0$ such that for every Borel subset $E\subset X$ we have

(4.3) $\mu(E)\leq A\cdot\left[\eta+(-\log\eta)^{n}{\rm
Cap}_{\omega}(E)\right],\ \forall\eta\in(0,1/e).$

###### Proof.

We only give a sketch of the proof since it is essentially a copy of the proof
of Lemma 4.1 with a small change. We also use the same notation as there.
Without loss of generality we can assume that
$E\Subset\mathbb{D}^{n}_{\delta}\Subset\mathbb{D}^{n}$ for some small fixed
$\delta$. The function $\varphi$ belongs to $\mathcal{F}(\mathbb{D})$. The
same arguments as in Lemma 4.1 show that $\varphi$ is also bounded from below
by $-A_{1}$ for some universal constant $A_{1}>0$. In the final step we get

$\displaystyle{\rm Vol}_{g}(E)$ $\displaystyle\leq$ $\displaystyle
A_{2}\int_{\mathbb{D}}\left(\eta+(-\log\eta)^{n-1}(-\varphi(w))\right)g_{n}(w)dV_{2}(w)$
$\displaystyle=$ $\displaystyle
A_{3}\eta+A_{2}(-\log\eta)^{n-1}\int_{0}^{A_{1}}{\rm
Vol}_{g_{n}}(\varphi<-t)dt$ $\displaystyle\leq$ $\displaystyle
A_{3}\eta+A_{4}\eta^{2}(-\log\eta)^{n-1}+A_{5}(-\log\eta)^{n-1}\int_{\eta^{2}}^{A_{1}}{\rm
Cap}_{\omega}(\varphi<-t)dt$ $\displaystyle\leq$ $\displaystyle
A_{6}\eta+A_{5}(-\log\eta)^{n-1}\int_{\eta^{2}}^{A_{1}}\frac{1}{t}\left[\int_{\mathbb{D}}dd^{c}\varphi\right]dt$
$\displaystyle\leq$ $\displaystyle
A_{6}\eta+A_{7}(-\log\eta)^{n}\int_{\mathbb{D}}dd^{c}\varphi.$

∎

######  Lemma 4.3.

Let $\varphi\in\mathcal{E}(X,\omega)$ be such that $\sup_{X}\varphi=0$ and
$\mu=\mathrm{MA}\,(\varphi)$ satisfies (4.3) for some $A>0$. Then there exists
$C,c>0$ depending on $A$ such that

${\rm Cap}_{\omega}(\varphi<-t)\leq Ce^{-ct},\ \forall t>0,$

In particular, if $\beta<c$ then $\int_{X}e^{-\beta\varphi}d\mu\leq
C^{\prime}$, with $C^{\prime}=C(\beta,A)>0$.

###### Proof.

Fix $s,t>1$. By standard application of the comparison principle we get

$\displaystyle{\rm Cap}_{\omega}(\varphi<-t-s)$ $\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<-t\\}}\left(\omega+\frac{1}{s}dd^{c}\varphi\right)^{n}$
$\displaystyle\leq$
$\displaystyle\frac{1}{s^{n}}\int_{\\{\varphi<-t\\}}\sum_{k=0}^{n}C^{k}_{n}(s-1)^{k}\omega^{k}\wedge\omega_{\varphi}^{n-k}$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<-t\\}}\omega^{n}+\frac{2^{n}}{s}\int_{\\{\varphi<-t\\}}\mathrm{MA}\,(\varphi),$

where the last inequality follows from the partial comparison principle (see
[19, Theorem 2.3]). It follows from [23] that

$\int_{\\{\varphi<-t\\}}\omega^{n}\leq C_{1}e^{-at},a>0.$

Choose $s:=2^{n}Ae$ and fix $\varepsilon<\min(1,a,1/s)$. Set

$F(t):=\frac{e^{\varepsilon t}}{t^{n}}{\rm Cap}_{\omega}(\varphi<-t),\ t\geq
1.$

Now, if we choose $\eta=e^{-t}$ in (4.3) and plug (4.3) into (4.1) we get

$F(t^{2}+s)\leq C_{2}+bF(t),$

where $b=2^{n}Ae^{\varepsilon s}/s<1$. This yields $\sup_{t\geq 1}F(t)\leq
C_{3}$, for some $C_{3}>0$ depending on $A$. We finally get

${\rm Cap}_{\omega}(\varphi<-t)\leq Ce^{-ct},c<\varepsilon.$

The last statement easily follows since it follows from [4, Lemma 2.3] that

$\int_{\\{\varphi<-t\\}}\mathrm{MA}\,(\varphi)\leq t^{n}{\rm
Cap}_{\omega}(\varphi<-t),\ \ \forall t\geq 1.$

∎

###  4.2. Proof of Theorem 2

Assume in this section that $f$ satisfies Condition $\mathcal{S}(B,\alpha)$
for some $B>0,\alpha>0$. The first part of Theorem 2 was proved in Theorem
3.1. We divide the remaining parts into three cases depending on the value of
$\alpha$.

####  4.2.1. The case when $\alpha>1$

The continuity of $\varphi$ and the $\mathcal{C}^{0}$ estimate follow directly
from Lemma 4.1 and Kołodziej’s classical result (see [27]).

####  4.2.2. The case when $0<\alpha<1$

Fix $\beta\in(1-\alpha,1)$ and set $\delta=\alpha+\beta-1$, and

$u_{\beta}:=\sum_{j=1}^{N}-a(-\log|s_{j}|)^{\beta},$

where $a>0$ is small enough so that $u_{\beta}\in{\rm PSH}(X,\omega)$. By
Theorem 3.1 we have

$\varphi\geq\sum_{j=1}^{N}\log|s_{j}|-C_{0},$

for some positive constant $C_{0}$ depending on $B$. By simple computations we
obtain

$\mathrm{MA}\,(\varphi)\leq\frac{C_{1}f_{1-\beta}\omega^{n}}{(-\varphi)^{\delta}},$

for some positive constant $C_{1}$ depending on $C_{0}$. Here for each $r>0$,
we set

$f_{r}:=\frac{1}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{1+r}}.$

We also get

$\mathrm{MA}\,(u_{\beta}-C_{2})\geq\frac{C_{1}f_{1-\beta}\omega^{n}}{(-u_{\beta}+C_{2})^{\delta}},$

where $C_{2}>0$ depends on $C_{1},\delta$. The comparison principle yields
that $\varphi\geq u_{\beta}-C_{2}$.

####  4.2.3. The case when $\alpha=1$

Consider the model function

$\psi:=-A_{1}\sum_{j=1}^{N}\log(-\log|s_{j}|+A_{2}),$

where $A_{1}>0$ is big and $A_{2}$ is chosen so that $\psi$ is $\omega/2$-psh
on $X$. It follows from Lemma 4.2 and Lemma 4.3 that
$\int_{X}e^{-c\varphi}f\omega^{n}<C_{1}$ for some small constant $c>0$
depending on $B$. Here $C_{1}$ depends on $c$ and $B$. Thus, for $t>0$, $p>1$,
by Hölder inequality we get

$\displaystyle\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi)$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<\psi-t\\}}e^{-c\varphi/p}e^{c\psi/p}f\omega^{n}$
$\displaystyle\leq$
$\displaystyle\left(\int_{X}e^{-c\varphi}f\omega^{n}\right)^{1/p}\left(\int_{\\{\varphi<\psi-t\\}}e^{c\psi/(p-1)}f\omega^{n}\right)^{1-1/p}$
$\displaystyle\leq$ $\displaystyle C_{2}\left({\rm
Cap}_{\psi}(\varphi<\psi-t)\right)^{\gamma},$

where $\gamma<A_{1}c/p+(p-1)/p$ and $C_{2}>0$ is a universal constant. The
last inequality follows from the volume-capacity domination (Lemma 4.1) and
from Lemma 2.7. Now if $A_{1}c>1$ we can choose $\gamma>1$ and the result
follows as in Theorem 3.1.

###  4.3. Regularity near the divisor $D$

In this subsection we will discuss about the behavior of the solution to
equation (1.1) near the divisor $D$. We prove the following result when
$\alpha<1$.

######  Proposition 4.4.

Consider $f=\frac{h}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{1+\alpha}}$,
where $1/B\leq h\leq B$ on $X$ and $\alpha\in(0,1)$. Assume that $f$ is
normalized so that $\int_{X}f\omega^{n}=\int_{X}\omega^{n}$. Let
$\varphi\in\mathcal{E}(X,\omega)$ be the unique normalized solution of (1.1).
Then for each $0<p<1-\alpha$ and each $1-\alpha<q<1$, we have

$-a_{1}(-\log|s|)^{q}-A_{1}\leq\varphi\leq-a_{2}(-\log|s|)^{p}+A_{2},$

where $a_{1},A_{1}>0$ depend on $B,\alpha,q$ while $a_{2},A_{2}>0$ depend on
$B,\alpha,p$. In particular, the solution $\varphi$ goes to $-\infty$ on $D$.

###### Proof.

One inequality has been proved in Theorem 2. Let us prove the upper bound. We
normalize $\varphi$ such that $\sup_{X}\varphi=-1$. To simplify the notation
we denote, for each $r>0$,

$f_{r}:=\frac{1}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{1+r}}.$

Fix $p\in(0,1-\alpha)$ set $\delta:=(1-\alpha-p)/p>0$. Consider
$u_{p}:=-\sum_{j=1}^{N}a_{2}(-\log|s_{j}|)^{p}$, where $a_{2}>0$ is small so
that $u_{p}$ is $\omega$-psh on $X$. Then we can find $C_{3}>0$ such that

$\mathrm{MA}\,(u_{p})\leq\frac{C_{3}f\omega^{n}}{(-u_{p})^{\delta}},$

while since $\varphi\leq 0$, for some $A_{2}>0$ big enough (for instance
$A_{2}^{\delta}=C_{3}$) we have

$\mathrm{MA}\,(\varphi-
A_{2})\geq\frac{C_{3}f\omega^{n}}{(-\varphi+A_{2})^{\delta}}.$

The comparison principle then yields the desired upper bound. ∎

By the same way we obtain a similar upper bound when $\alpha=1$.

######  Proposition 4.5.

Assume that $f$ is normalized so that $\int_{X}f\omega^{n}=\int_{X}\omega^{n}$
and

$f\geq\frac{1}{B\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{2}}.$

Let $\varphi\in\mathcal{E}(X,\omega)$ be the unique normalized solution of
(1.1). Then for any $p\in(0,1)$ there exist $a,A>0$ depending on $B,p$ such
that

$\varphi\leq-a\sum_{j}\left[\log(-\log|s_{j}|)\right]^{p}+A.$

In particular, $\varphi$ is not bounded and goes to $-\infty$ on $D$.

###### Proof.

The proof uses the same arguments as in Proposition 4.4. ∎

##  5\. The case of semipositive and big classes

In this section we prove Theorem 3. For convenience let us recall the setting.
We assume that $(X,\omega)$ is a compact Kähler manifold of dimension $n$ and
$D$ is an arbitrary divisor on $X$. Let $E=\sum_{j=1}^{M}a_{j}E_{j}$ be an
effective snc divisor on $X$. Let $\theta$ be a smooth semipositive form on
$X$ such that $\int_{X}\theta^{n}>0$ and $\\{\theta\\}-c_{1}(E)$ is ample.
Consider the following degenerate complex Monge-Ampère equation

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

where $0\leq f\in L^{1}(X,\omega^{n})$ satisfies the compatibility condition
$\int_{X}f\omega^{n}=\int_{X}\theta^{n}$.

For each $j=1,...,M$ let $K_{j}$ be the holomorphic line bundle defined by
$E_{j}$. Let $\sigma_{j}$ be a holomorphic section of $K_{j}$ that vanish on
$E_{j}$. We fix hermitian metric $h_{j}$ on $K_{j}$ such that
$|\sigma_{j}|\leq 1/e$. Since $\\{\theta\\}-c_{1}(E)$ is ample, we can assume
that

$\theta+dd^{c}\phi=\omega_{0}+[E],$

where $\omega_{0}$ is a Kähler form on $X$ and

$\phi:=\sum_{j=1}^{M}a_{j}\log|\sigma_{j}|.$

By rescaling $\omega$ we can also assume that $\omega_{0}\geq\omega$. Recall
that $f$ satisfies Condition $\mathcal{H}_{f}$ on $X$, i.e. there is a
constant $C>0$ such that

(5.2) $f=e^{\psi^{+}-\psi^{-}},\ \ dd^{c}\psi^{\pm}\geq-C\omega,\
\sup_{X}\psi^{+}\leq C,\ \ \psi^{-}\in L^{\infty}_{\rm loc}(X\setminus D).$

###  5.1. Uniform estimate

The following $\mathcal{C}^{0}$-lower bound can be proved in the same ways as
we have done in Theorem 3.1:

######  Theorem 5.1.

Assume that $D,E$ and $\theta$ are as above and $f$ satisfies (5.2). Let
$\varphi$ be the unique normalized solution to equation (5.1). Then $\varphi$
is uniformly bounded away from $D\cup E$. More precisely, for any $a>0$ there
exists $A>0$ depending on $C$ and $\int_{X}e^{-2\varphi/a}\omega^{n}$ such
that

$\varphi\geq a\psi^{-}+\phi-A.$

###### Proof.

It suffices to prove the result for small $a>0$. Fix $a>0$ very small so that

$\psi:=a\psi^{-}+\frac{1}{2}\phi\in{\rm PSH}(X,\theta/2).$

It follows from Proposition 3.1 in [22] that

${\rm Vol}_{\omega}\leq C_{1}\exp\left(\frac{-C_{2}}{\left[{\rm
Cap}_{\theta/2}\right]^{1/n}}\right),$

for some universal constants $C_{1},C_{2}>0$. Now, the same proof of Lemma 2.7
yields

${\rm Cap}_{\theta/2}\leq{\rm Cap}_{\psi},$

where ${\rm Cap}_{\psi}$ is the generalized capacity defined by the form
$\theta$ and $\psi$:

${\rm Cap}_{\psi}(E):=\sup\left\\{\int_{E}(\theta+dd^{c}u)^{n}\ \ \big{|}\ \
u\in{\rm PSH}(X,\theta),\ \psi-1\leq u\leq\psi\right\\}.$

Then we can repeat the arguments in the proof of Theorem 3.1 to get the
result. ∎

###  5.2. Laplacian estimate

We now prove a $C^{2}$ a priori estimate in the semipositive and big case.
Even when $f$ is smooth on $X$, $\varphi$ is only smooth in the ample locus of
$\theta$. To get rid of this, we replace $\theta$ by $\theta+t\omega$, $t>0$.
In principle, the $\mathcal{C}^{2}$ estimate will depends heavily on $t>0$ and
we will have serious problem when $t\downarrow 0$. But, fortunately, the so-
called Tsuji’s trick (see [36]) allows us to get around this difficulty. In
the sequel, we follow essentially the ideas in [8].

######  Theorem 5.2.

Let $f=e^{\psi^{+}-\psi^{-}}$ where $\psi^{+},\psi^{-}$ are smooth on $X$. Fix
$t\in(0,1)$. Let $\varphi\in\mathcal{C}^{\infty}(X)$ be the unique normalized
solution to

$(\theta+t\omega+dd^{c}\varphi)^{n}=e^{\psi^{+}-\psi^{-}}\omega^{n}.$

Assume given a constant $C>0$ such that

$dd^{c}\psi^{\pm}\geq-C\omega,\ \ \sup_{X}\psi^{+}\leq C.$

Assume also that the holomorphic bisectional curvature of $\omega$ is bounded
from below by $-C$. Then there exists $A>0$ depending on $C$ and
$\int_{X}e^{-2(4C+1)\varphi}\omega^{n}$ such that

$\Delta_{\omega}\varphi\leq Ae^{-2\psi^{-}-(4C+1)\phi}.$

###### Proof.

Ignoring the dependence on $t$, we denote
$\omega_{\varphi}:=\theta+t\omega+dd^{c}\varphi$. Consider the following
function

$H:=\log{\rm tr}_{\omega}(\omega_{\varphi})+2\psi^{-}-(4C+1)(\varphi-\phi),$

Since $\phi$ goes to $-\infty$ on $E$, we see that $H$ attains its maximum on
$X\setminus E$ at some point $x_{0}\in X\setminus E$. From now on we carry all
computations on $X\setminus E$. We can argue as in Theorem 3.2 to obtain

(5.3) $\displaystyle\Delta_{\omega_{\varphi}}\log{\rm
tr}_{\omega}(\omega_{\varphi})\geq-3C{\rm
tr}_{\omega_{\varphi}}(\omega)-\Delta_{\omega_{\varphi}}\psi^{-}.$

Since $\omega_{0}+t\omega\geq\omega$ we get

(5.4) $\Delta_{\omega_{\varphi}}(\varphi-\phi)\leq{\rm
tr}_{\omega_{\varphi}}(\omega_{\varphi}-\omega_{0}-t\omega)\leq n-{\rm
tr}_{\omega_{\varphi}}(\omega).$

Therefore, from (5.3) and (5.4) we deduce that on $X\setminus E$

$\displaystyle\Delta_{\omega_{\varphi}}H$ $\displaystyle\geq$
$\displaystyle{\rm tr}_{\omega_{\varphi}}(\omega)-n(4C+1).$

We now apply the maximum principle to the function $H$ at $x_{0}$:

$0\geq\Delta_{\omega_{\varphi}}H(x_{0})\geq{\rm
tr}_{\omega_{\varphi}}(\omega)(x_{0})-n(4C+1).$

Furthermore, by Lemma 3.3 we get

${\rm tr}_{\omega}(\omega_{\varphi})(x_{0})\leq
ne^{\psi^{+}-\psi^{-}}(x_{0})\left({\rm
tr}_{\omega_{\varphi}}(\omega)\right)^{n-1}(x_{0})\leq
A_{1}e^{\psi^{+}-\psi^{-}}(x_{0}),$

and hence, since $\sup_{X}\psi^{+}\leq C$,

$\log{\rm tr}_{\omega}(\omega_{\varphi})(x_{0})\leq\log
A_{1}+\psi^{+}(x_{0})-\psi^{-}(x_{0})\leq A_{2}-\psi^{-}(x_{0})\,.$

It follows that

$H(x)\leq H(x_{0})\leq A_{2}+\psi^{-}(x_{0})-(4C+1)(\varphi-\phi)(x_{0}).$

By assumption and the $\mathcal{C}^{0}$ estimate in Theorem 5.1 we have

$\varphi\geq\frac{1}{4C+1}\psi^{-}+\phi-A_{3},$

where $A_{3}$ depends on $C$ and $\int_{X}e^{-2(4C+1)\varphi}\omega^{n}$. Thus

$\log{\rm tr}_{\omega}(\omega_{\varphi})\leq
A_{4}-2\psi^{-}+(4C+1)(\varphi-\phi).$

We finally get

${\rm tr}_{\omega}(\omega_{\varphi})\leq A_{5}e^{-2\psi^{-}-(4C+1)\phi}.$

∎

###### Proof of Theorem 3.

We proceed as in Section 3.3. We also borrow the notations there. Let
$\rho_{\varepsilon}(\psi^{\pm})$ be the Demailly’s smoothing regularization of
$\psi^{\pm}$. For each $\varepsilon>0$ let $\varphi_{\varepsilon}$ be the
unique smooth function such that $\sup_{X}\varphi_{\varepsilon}=0$ and

$(\theta+\varepsilon\omega+dd^{c}\varphi_{\varepsilon})^{n}=c_{\varepsilon}e^{\rho_{\varepsilon}(\psi^{+})-\rho_{\varepsilon}(\psi^{-})}\omega^{n},$

where $c_{\varepsilon}$ is a normalization constant. As in Section 3.3 we have
a uniform control on the right-hand side:

$c_{\varepsilon}e^{\rho_{\varepsilon}(\psi^{+})-\rho_{\varepsilon}(\psi^{-})}\leq
e^{C-\psi^{-}_{\varepsilon}}.$

Now, we can copy the arguments in Section 3.3 since our uniform estimate and
laplacian estimate do not depend on $\varepsilon$. The proof is thus complete.
∎

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