# Continuity of Monge-Ampère potentials in big cohomology classes

Quang-Tuan Dang  Laboratoire de Mathématiques D’Orsay, Université Paris-
Saclay, CNRS, 91405 Orsay, France<EMAIL_ADDRESS>Institut de Mathematiques de Toulouse, Université de Toulouse; CNRS, 118 route
de Narbonne, 31400 Toulouse, France<EMAIL_ADDRESS>
(Date: August 29, 2024)

###### Abstract.

Extending Di Nezza-Lu’s approach [16] to the setting of big cohomology
classes, we prove that solutions of degenerate complex Monge-Ampère equations
on compact Kähler manifolds are continuous on a Zariski open set. This allows
us to show that singular Kähler-Einstein metrics on log canonical varieties of
general type have continuous potentials on the ample locus outside of the non-
klt part.

###### Key words and phrases:

Complex Monge-Ampère equation, big cohomology class

###### 2020 Mathematics Subject Classification:

32U15, 32Q15, 32W20

The author is partially supported by the French ANR project PARAPLUI

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Regularity of solutions

##  1\. Introduction

Finding canonical metrics on complex varieties is a fundamental problem of
complex geometry. As evidenced by recent developments in Kähler geometry in
connection with the Minimal Model Program, it is natural and necessary to
allow the varieties $Y$ in question to be singular. Working on a
desingularization $\pi:X\rightarrow Y$ is led to consider degenerate complex
Monge-Ampère equations of the form

(1.1) $(\theta+dd^{c}\varphi)^{n}=e^{\lambda\varphi}fdV,$

where $\theta$ is a smooth closed real $(1,1)$-form representing a big
cohomology class $\alpha$ on $X$, $\lambda\in\\{0,\pm 1\\}$, and $f$ is a
density of the form $f=e^{\psi_{+}-\psi_{-}}$, the functions
$\psi_{+},\psi_{-}$ being quasi-plurisubharmonic on $X$. The integrability
properties of $f$ depend on the singularities of $Y$.

Finding a Kähler-Einstein metric on a stable variety $Y$ boils down to solving
(1.1) on $X$ for $\lambda=1$ and $f\in L^{1-\delta}(X,dV)$ for all
$\delta\in(0,1)$. Building on the variational approach developed in [5],
Berman and Guenancia [6] have proved that the equation (1.1) admits a unique
finite energy solution $\varphi$, in the sense of [20, 5]. When $\\{\theta\\}$
is additionally nef, they established the smoothness of $\varphi$ on a Zariski
open set. As in the classical case of Yau [28], the main difficulty lies in
establishing an _a priori $\mathcal{C}^{0}$-estimate_. Unfortunately the
(normalized) solution $\varphi$ to (1.1) is in general unbounded, so a natural
idea is to try and bound such solution from below by a reference quasi-
plurisubharmonic function.

This motivates the following general question: for which densities $f\geq 0$
is the solution $\varphi$ locally bounded in some Zariski open subset of $X$?
The main result of this paper is the following:

######  Theorem 1.1.

Let $X$ be a compact Kähler manifold of dimension $n$ and fix a smooth closed
real $(1,1)$-form $\theta$ which represents a big cohomology class. Let
$\varphi\in\mathcal{E}(X,\theta)$ be the unique normalized solution to

(1.2) $(\theta+dd^{c}\varphi)^{n}=fdV,\;\sup_{X}\varphi=0.$

Assume that $f\leq e^{-\phi}$ for some quasi-plurisubharmonic function $\phi$
on $X$. Then $\varphi$ is continuous on ${\rm Amp}(\theta)\setminus
E_{1}(\phi)$, where ${\rm Amp}(\theta)$ is the ample locus of $\theta$ and
$E_{1}(\phi)=\\{x\in X:\nu(\phi,x)\geq 1\\}$, with $\nu(\phi,x)$ being the
Lelong number of $\phi$ at $x$.

Let us recall that $E_{1}(\phi)$ which is called _the Lelong super-level_ set
of $\phi$, is an analytic subset of $X$ by Siu’s result [24]. We refer the
reader to Section 2.3 for the definition of the _ample locus_ of a big
cohomology class.

Since $\mu=fdV$ is non-pluripolar, it is known [8, Section 3] that there
exists a unique normalized solution $\varphi\in\mathcal{E}(X,\theta)$, so the
point is to study its regularity. The idea of the proof is that we first use
Demailly’s equisingular approximation [12, 13] (see Theorem 2.1) to replace
$\phi$ by a quasi-psh function $\phi_{1}$ which has analytic singularities
with polar locus $Z$ contained in the set of points where the Lelong number of
$\phi$ is greater than or equal to $1$. We then adapt the approach of Di Nezza
and Lu [16] (see Theorem 3.2) to prove the continuity of $\varphi$ in the
complement of $E_{1}(\phi)$ in the ample locus of $\theta$. We also prove a
slightly stronger version of Theorem 1.1 valid for more singular densities
(see Theorem 3.1).

When the density $f$ is smooth in a Zariski open set (i.e. outside an analytic
subset), one expects the solution $\varphi$ to be smooth in a Zariski open set
(see [18, Question 21, 22]), but we are unable to prove this for the moment.
When $f$ belongs to $L^{p}(X)$ for some $p>1$, the Hölder continuity of
$\varphi$ on the ample locus was shown in [14]. The smoothness of $\varphi$
when $f$ is smooth is largely open.

Under the extra assumption that the class $\\{\theta\\}$ is nef, the
regularity properties for the solutions for the degenerate Monge-Ampère
equation (1.2) have been studied by many authors (see [8, 6, 16] and the
references therein). The strategy in these papers is that one first
establishes a relative uniform estimate which allows to adapt classical ideas
of Yau [28] and Siu [25] to obtain locally uniform estimates for the
Laplacian, and one finally uses Evans-Krylov’s general regularity theory to
conclude. In the above case functions in $\mathcal{E}(X,\theta)$ have zero
Lelong numbers (see [20, Corollary 1.8], [10, Theorem 1.1]). Using this
property Di Nezza and Lu [16] have generalized Kołodziej’s approach [22] to
establish a relative uniform estimate. Let us mention that in the general case
of a big class even the ”least singular” potential $V_{\theta}$ may have
positive Lelong numbers. To overcome this difficulty we exploit fine
properties of quasi-plurisubharmonic envelopes inspired by [11, 23].

Our approach allows us to deal with non-nef data. As an application we prove
that the unique singular Kähler-Einstein metric obtained in [6] is continuous
on some Zariski open subset. More precisely, we have the following:

######  Corollary 1.2.

Let $(Y,\Delta)$ be a projective log canonical pair of general type, i.e. the
canonical line bundle $K_{Y}+\Delta$ is big. Then there is a unique singular
Kähler-Einstein metric $\omega$ on $Y$ such that

$\displaystyle{\rm Ric}(\omega)=-\omega+[\Delta]$

in the weak sense of currents, and such that $\int_{Y}\omega^{n}={\rm
vol}(K_{Y}+\Delta)$. Furthermore $\omega$ has continuous potentials on ${\rm
Amp}(K_{Y}+\Delta)\cap(Y,\Delta)_{\rm reg}\setminus\lfloor\Delta\rfloor$.
Here, $(Y,\Delta)_{\rm reg}$ denotes the locus of points $p\in Y$ where the
pair $(Y,\Delta)$ is log smooth at $p$, i.e. $Y$ is smooth and $\Delta$ has
simple normal crossing (snc) support on a neighborhood of $p$, and
$\lfloor\Delta\rfloor$ denotes the integral part of $\Delta$, i.e. if
$\Delta=\sum d_{i}\Delta_{i}$, then $\lfloor\Delta\rfloor=\sum\lfloor
d_{i}\rfloor\Delta_{i}$.

Recall that the _ample locus_ of a big line bundle $L$, denoted by
$\textrm{Amp}(L)$ may be defined as the ample locus of its first Chern class
$\alpha=c_{1}(L)$ (see Definition 2.2).

If $K_{Y}+\Delta$ is additionally _nef_ , then the potential of $\omega$ was
already known to be smooth on $\textrm{Amp}(K_{Y}+\Delta)\cap(Y,\Delta)_{\rm
reg}\setminus\lfloor\Delta\rfloor$, as follows from the combined arguments of
Boucksom-Eyssidieux-Guedj-Zeriahi [8] (see also [4]) and Berman-Guenancia [6].

### Organization of the paper

The paper is organized as follows. In Section 2 we recall basic pluripotential
theory that will be needed later on. The proof of Theorem 1.1 is given in
Section 3.1, while Corollary 1.2 is proved in Section 3.2.

### Notations

In the whole article we fix

  * •

$X$ a $n$-dimensional compact Kähler manifold,

  * •

$dV$ a smooth volume form on $X$,

  * •

$\alpha\in H^{1,1}(X,\mathbb{R})$ a big cohomology class, and $\theta$ a
smooth representative of $\alpha$

  * •

a Kähler form $\omega$ so that $\omega\geq\theta$.

### Acknowledgements

I would like to express my gratitude to my advisors Vincent Guedj and Chinh H.
Lu for their help and various interesting discussions. I warmly thank the
referees for useful corrections, comments, and suggestions which improve the
presentation of this paper.

##  2\. Preliminaries

The purpose of this section is to recall some essential materials in
pluripotential theory which will be used later.

###  2.1. Quasi-psh functions

Recall that an upper semi-continuous function
$\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is called quasi-
plurisubharmonic (quasi-psh for short) if it is locally the sum of a smooth
and a plurisubharmonic (psh for short) function. We say that $\varphi$ is
$\theta$-plurisubharmonic ($\theta$-psh for short) if it is quasi-psh, and
$\theta+dd^{c}\varphi\geq 0$ in the sense of currents, where $d^{c}$ is
normalized so that $dd^{c}=\frac{i}{\pi}\partial\bar{\partial}$.

By the $dd^{c}$-lemma any closed positive $(1,1)$-current $T$ cohomologous to
$\theta$ can be written as $T=\theta+dd^{c}\varphi$ for some $\theta$-psh
function $\varphi$ which is furthermore unique up to an additive constant.

We let $\operatorname{{\rm PSH}}(X,\theta)$ denote the set of all $\theta$-psh
functions which are not identically $-\infty$. This set is endowed with the
$L^{1}(X)$-topology. By Hartog’s lemma $\varphi\mapsto\sup_{X}\varphi$ is
continuous in this weak topology. Since the set of closed positive currents in
a fixed cohomology class is compact (in the weak topology), it follows that
the set of $\varphi\in\operatorname{{\rm PSH}}(X,\theta)$, with
$\sup_{X}\varphi=0$ is compact.

Quasi-psh functions are in general singular, and a convenient way to measure
their singularities is the Lelong numbers. Let $x_{0}\in X$. Fixing a
holomorphic chart $x_{0}\in V_{x_{0}}\subset X$, the Lelong number
$\nu(\varphi,x_{0})$ of a quasi-psh function $\varphi$ at $x_{0}\in X$ is
defined as follows:

$\displaystyle\nu(\varphi,x_{0}):=\sup\\{\gamma\geq
0:\varphi(z)\leq\gamma\log\|z-x_{0}\|+O(1),\;\text{on}\;V_{x_{0}}\\}.$

We remark here that this definition does not depend on the choice of local
charts. In particular, if $\varphi=\log|f|$ in a neighborhood $V_{x_{0}}$ of
$x_{0}$, for some holomorphic function $f$, then $\nu(\varphi,x_{0})$ is equal
to the vanishing order
$\textrm{ord}_{x_{0}}(f):=\sup\\{k\in\mathbb{N}:D^{\gamma}f(x_{0})=0,\forall\,|\gamma|<k\\}$.
We can also define the _Lelong super-level sets_ , for $c>0$,

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

We also use the notation $E_{c}(T)$ for a closed positive $(1,1)$-current $T$.
A well-known result of Siu [24] asserts that the Lelong super-level sets
$E_{c}(\varphi)$ are analytic subsets of $X$. We refer the reader to [12,
Remark 3.2] for a simple proof.

###  2.2. Demailly’s equisingular approximation

We next recall the basic result on the approximation of psh functions by psh
functions with analytic singularities. For details about this, we refer the
reader to [12, 13].

Following Demailly [12], a closed positive $(1,1)$-current
$T=\theta+dd^{c}\varphi$ and its global potential $\varphi$ are said to have
_analytic singularities_ if there exists $c>0$ such that

$\displaystyle\varphi=c\log\left[\sum_{j=1}^{N}|f_{j}|^{2}\right]+v,$

locally on $X$, where $v$ is a smooth function and the $f_{j}$’s are
holomorphic functions.

Thanks to $dd^{c}$-Lemma, the problem of approximating a positive closed
$(1,1)$-current is reduced to approximating a quasi-psh function. The
following result of Demailly [12, 13] on the equisingular approximation of a
quasi-psh function by quasi-psh functions with analytic singularities is
crucial:

######  Theorem 2.1 (Demailly’s equisingular approximation).

Let $\varphi$ be a $\theta$-psh function on $X$. There exists a decreasing
sequence of quasi-psh functions $(\varphi_{m})$ such that

  1. (1)

$(\varphi_{m})$ converges pointwise and in $L^{1}(X)$ to $\varphi$ as
$m\to+\infty$,

  2. (2)

$\varphi_{m}$ has the same singularities as $1/2m$ times a logarithm of a sum
of squares of holomorphic functions,

  3. (3)

$\theta+dd^{c}\varphi_{m}\geq-\varepsilon_{m}\omega$, where
$\varepsilon_{m}>0$ decreases to 0 as $m\to+\infty$,

  4. (4)

$\int_{X}e^{2m(\varphi_{m}-\varphi)}dV<+\infty$;

  5. (5)

$\varphi_{m}$ is smooth outside the analytic subset $E_{1/m}(\varphi)$.

###### Proof.

We refer the reader to [13, Theorem 1.6, Lemma 1.10] for a proof. ∎

###  2.3. Big cohomology classes

A cohomology class $\alpha\in H^{1,1}(X,\mathbb{R})$ is big if it contains a
Kähler current, i.e. there is a positive closed current $T\in\alpha$ and
$\varepsilon>0$ such that $T\geq\varepsilon\omega$. Theorem 2.1 enables us in
particular to approximate a Kähler current $T$ inside its cohomology class by
Kähler currents $T_{m}$ with analytic singularities, with a very good control
of the singularities. A big class therefore contains plenty of Kähler currents
with analytic singularities.

######  Definition 2.2.

We let $\textrm{Amp}(\alpha)$ denote the ample locus of $\alpha$, i.e. the
Zariski open subset of all points $x\in X$ for which there exists a Kähler
current $T_{x}\in\alpha$ with analytic singularities such that $T_{x}$ is
smooth in a neighborhood of $x$.

It follows from the work of Boucksom [7, Theorem 3.17 (ii)] that one can find
a single Kähler current $T_{0}\in\alpha$ with analytic singularities such that

$\text{Amp}(\alpha)=X\backslash\textrm{Sing}(T_{0}).$

In particular $T_{0}$ is smooth in the ample locus $\textrm{Amp}(\alpha)$.

Given $\varphi,\psi\in\operatorname{{\rm PSH}}(X,\theta)$, we say that
$\varphi$ is less singular than $\psi$, and denote by $\psi\preceq\varphi$, if
there exists a constant $C$ such that $\psi\leq\varphi+C$ on $X$. We say that
$\varphi,\psi$ have the same singularity type, and denote by
$\varphi\simeq\psi$ if $\varphi\preceq\psi$ and $\psi\preceq\varphi$.

######  Definition 2.3.

A $\theta$-psh function is said to have minimal singularities if it is less
singular than any $\theta$-psh function.

Such a function is not unique in general, only its class of singularities is.
Following Demailly, one defines the extremal function

$\displaystyle V_{\theta}:=\sup\\{\varphi\in\operatorname{{\rm
PSH}}(X,\theta):\varphi\leq 0\\}.$

It is a $\theta$-psh function with minimal singularities. By the analysis
above $V_{\theta}$ is locally bounded on the ample locus ${\rm Amp}(\alpha)$.
Of course we have $V_{\theta}\equiv 0$ if $\theta$ is semi-positive.

###  2.4. Non-pluripolar Monge-Ampère operator

Let $\varphi_{1},\cdots,\varphi_{n}\in\operatorname{{\rm PSH}}(X,\theta)$ with
minimal singularities. Then they are locally bounded on the ample locus ${\rm
Amp}(\alpha)$. Following the construction of Bedford-Taylor [1, 2] in the
local setting, it has been shown in [8, Section 1.2] that the product

$(\theta+dd^{c}\varphi_{1})\wedge\cdots\wedge(\theta+dd^{c}\varphi_{n})$

is well-defined as a positive Radon measure on ${\rm Amp}(\alpha)$ and it has
finite total mass. One can then extend it trivially on the whole $X$.

In particular, if $\varphi_{1}=\cdots=\varphi_{n}=\varphi$ then this procedure
defines the (non-pluripolar) Monge-Ampère measure of a function
$\varphi\in\operatorname{{\rm PSH}}(X,\theta)$ with minimal singularities. For
a general $\varphi\in\operatorname{{\rm PSH}}(X,\theta)$, its canonical
approximants $\varphi^{j}:=\max(\varphi,V_{\theta}-j)$, $j>0$ have minimal
singularities. One can show that the sequence of Borel positive measures ${\bf
1}_{\\{\varphi>V_{\theta}-j\\}}(\theta+dd^{c}\varphi^{j})$ is increasing in
$j$. Its (strong) limit

$\operatorname{{\rm
MA}}_{\theta}(\varphi)=(\theta+dd^{c}\varphi)^{n}:=\lim_{j\to+\infty}\nearrow{\bf
1}_{\\{\varphi>V_{\theta}-j\\}}(\theta+dd^{c}\varphi^{j})^{n}$

is the _non-pluripolar Monge-Ampère measure_ of $\varphi$. The volume of a big
class $\alpha=\\{\theta\\}$ is given by the total mass of the non-pluripolar
Monge-Ampère measure of $V_{\theta}$, i.e.

${\rm\operatorname{{\rm Vol}}}(\alpha):=\int_{X}\operatorname{{\rm
MA}}_{\theta}(V_{\theta}).$

We say that $\varphi\in\operatorname{{\rm PSH}}(X,\theta)$ has full Monge-
Ampère mass if $\int_{X}\operatorname{{\rm
MA}}_{\theta}(\varphi)=\operatorname{{\rm Vol}}(\alpha)$. We let

$\displaystyle\mathcal{E}(X,\theta):=\left\\{\varphi\in\operatorname{{\rm
PSH}}(X,\theta):\int_{X}\operatorname{{\rm
MA}}_{\theta}(\varphi)=\operatorname{{\rm Vol}}(\alpha)\right\\}$

denote the set of $\theta$-psh functions with full Monge-Ampère mass. Note
that $\theta$-psh functions with minimal singularities have full Monge-Ampère
mass (see [8, Theorem 1.16] for more details), but the converse is not true.

We recall here the plurifine locality of the non-pluripolar product, which
will be used several times in this paper.

######  Lemma 2.4.

Assume that $\varphi$, $\psi$ are $\theta$-psh function such that
$\varphi=\psi$ on an open set $U$ in the plurifine topology. Then

$\mathbf{1}_{U}\operatorname{{\rm
MA}}_{\theta}(\varphi)=\mathbf{1}_{U}\operatorname{{\rm MA}}_{\theta}(\psi).$

We stress in particular that sets of the form $\\{u<v\\}$, where $u$, $v$ are
quasi-psh functions, are open in the plurifine topology.

###### Proof.

The proof for locally bounded functions can be found in [3, Corollary 4.3] or
[8, Section 1.2]. For the general case we write $\varphi$ (resp. $\psi$) as
the decreasing limits of its canonical approximants
$\varphi^{t}:=\max(\varphi,V_{\theta}-t)$ (resp.
$\psi^{t}:=\max(\psi,V_{\theta}-t)$). We observe that $\varphi^{t}$ (resp.
$\psi^{t}$) is locally bounded on the ample locus ${\rm Amp}(\theta)$. By the
result for locally bounded functions we have

$\displaystyle\mathbf{1}_{U\cap(\varphi>V_{\theta}-t)}\operatorname{{\rm
MA}}_{\theta}(\varphi^{t})=\mathbf{1}_{U_{\cap}(\psi>V_{\theta}-t)}\operatorname{{\rm
MA}}_{\theta}(\psi^{t}).$

Letting $t\to+\infty$, we conclude the proof. ∎

###  2.5. Capacities

####  2.5.1. The Monge-Ampère capacity

For the convenience of the reader we recall here a few facts contained in
[21]. Let $K$ be a Borel subset of $X$. The Monge-Ampère capacity is

$\displaystyle\operatorname{{\rm
Cap}}_{\omega}(K):=\sup\left\\{\int_{K}(\omega+dd^{c}u)^{n}:u\in\operatorname{{\rm
PSH}}(X,\omega),\,-1\leq u\leq 0\right\\}.$

######  Lemma 2.5.

Let $\nu=gdV$ be a Radon positive measure with $0\leq g\in L^{p}(dV)$ for some
$p>1$. Then there exists $B>0$ depending on $n$, $p$ $\omega$, $dV$ and
$\|g\|_{L^{p}(dV)}$ such that, for all Borel subsets $K$ of $X$,

$\displaystyle\nu(K)\leq B\cdot\operatorname{{\rm Cap}}(K)^{4}.$

###### Proof.

This result was proved by Kolodziej; see [22, Section 2.5]. We refer the
readers to [19, Proposition 3.1] for an alternative proof. ∎

####  2.5.2. The generalized capacity

We present here a generalization of this notion introduced by Di Nezza and Lu
[16, 15] (see also [9, Section 4.1]).

######  Definition 2.6.

Let $\psi\in\operatorname{{\rm PSH}}(X,\theta)$. We define the _$\psi$
-relative capacity_ of a Borel subset $K\subset X$ by

$\displaystyle\operatorname{{\rm
Cap}}_{\theta,\psi}(K):=\sup\left\\{\int_{K}\operatorname{{\rm
MA}}_{\theta}(u):u\in\operatorname{{\rm PSH}}(X,\theta),\,\psi-1\leq
u\leq\psi\right\\}.$

Note that when $\theta$ is Kähler, a related notion of capacity has been
studied in [15, 16]. The (generalized) Monge-Ampère capacity plays a vital
role in establishing uniform estimates for complex Monge-Ampère equation (see
e.g. [19, 8, 15, 16] and the references therein). We shall use the
$\psi$-capacity $\operatorname{{\rm Cap}}_{\theta,\psi}$ in the proof of
Theorem 3.2.

The following results are important for the sequel.

######  Lemma 2.7.

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

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

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

###### Proof.

The proof is almost the same as the one of [16, Lemma 2.6] in the Kähler case,
i.e. $\theta=\omega$ is Kähler. For the reader’s convenience, we give the
proof here. The right-continuity is straightforward . For the second
statement, we first assume that $\psi\leq V_{\theta}$. Fix
$u\in\operatorname{{\rm PSH}}(X,\theta)$ such that $\psi-1\leq u\leq\psi$. The
generalized comparison principle ([8, Corollary 2.3]) yields

$\displaystyle\int_{\\{\varphi<\psi-t\\}}\operatorname{{\rm
MA}}_{\theta}(u)\leq\int_{\\{\varphi<u-t+1\\}}\operatorname{{\rm
MA}}_{\theta}(u)$
$\displaystyle\leq\int_{\\{\varphi<u-t+1\\}}\operatorname{{\rm
MA}}_{\theta}(\varphi)$
$\displaystyle\leq\int_{\\{\varphi<V_{\theta}-t+1\\}}\operatorname{{\rm
MA}}_{\theta}(\varphi).$

The last term goes to $0$ as $t\to+\infty$ as
$\varphi\in\mathcal{E}(X,\theta)$. This finishes the proof. ∎

######  Lemma 2.8.

Let $\psi$ be a quasi-psh function such that
$\theta+dd^{c}\psi\geq\delta\omega$ for some $\delta\in(0,1)$. Then for any
Borel set $K\subset X$,

$\displaystyle\operatorname{{\rm
Cap}}_{\omega}(K)\leq\frac{1}{\delta^{n}}\operatorname{{\rm
Cap}}_{\theta,\psi}(E).$

###### Proof.

Let $u$ be a $\omega$-psh function such that $-1\leq u\leq 0$. We then have
that $\varphi:={\psi+\delta u}$ is a candidate defining $\operatorname{{\rm
Cap}}_{\theta,\psi}$. It follows that

$\displaystyle\delta^{n}\int_{K}(\omega+dd^{c}u)^{n}$
$\displaystyle\leq\int_{K}(\theta+dd^{c}\psi+dd^{c}(\delta
u))^{n}\leq\operatorname{{\rm Cap}}_{\theta,\psi}(K),$

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

Generalizing [22, Ineq. (2.3.2)], we have the following result which is a
simple consequence of the generalized comparison principle ([8, Corollary
2.3]).

######  Lemma 2.9.

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

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

###### Proof.

Let $u$ be a $\theta$-psh function such that $\psi-1\leq u\leq\psi$. We then
have

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

Since $s^{n}\operatorname{{\rm MA}}_{\theta}(u)\leq\operatorname{{\rm
MA}}_{\theta}(su+(1-s)\psi)$ and $\varphi$ has full Monge-Ampère mass, it
follows from the generalized comparison principle ([8, Corollary 2.3]) that

$\displaystyle s^{n}\int_{\\{\varphi<\psi-t-s\\}}\operatorname{{\rm
MA}}_{\theta}(u)$ $\displaystyle\leq\int_{\\{\varphi<\psi-
t-s\\}}\operatorname{{\rm MA}}_{\theta}(su+(1-s)\psi)$
$\displaystyle\leq\int_{\\{\varphi<su+(1-s)\psi-t\\}}\operatorname{{\rm
MA}}_{\theta}(su+(1-s)\psi)$
$\displaystyle\leq\int_{\\{\varphi<su+(1-s)\psi-t\\}}\operatorname{{\rm
MA}}_{\theta}(\varphi)\leq\int_{\\{\varphi<\psi-t\\}}\operatorname{{\rm
MA}}_{\theta}(\varphi).$

Since $u$ was taken arbitrarily as a candidate in the definition of
$\operatorname{{\rm Cap}}_{\theta,\psi}$, the proof therefore finishes. ∎

###  2.6. Quasi-psh envelopes

For a Borel function $h$, we let $P_{\theta}(h)$ denote the largest
$\theta$-psh function lying below $h$:

$P_{\theta}(h):=\left(\sup\\{\varphi\in\operatorname{{\rm
PSH}}(X,\theta):\varphi\leq h\;\text{on}\;X\\}\right)^{*}.$

######  Proposition 2.10.

Fix $\varphi\in\mathcal{E}(X,\theta)$. Then for any $b>0$,
$P_{\omega}(b\varphi-bV_{\theta})$ is a $\omega$-psh function with full Monge-
Ampère mass.

The proof given below is inspired by [11, Lemma 4.3].

###### Proof.

We first show that the function $P_{\omega}(b\varphi-
bV_{\theta})\not\equiv-\infty$ for all $b>0$.

For each $j\in\mathbb{N}$ we set $\varphi_{j}:=\max(\varphi,V_{\theta}-j)$ and
$\psi_{j}:=P_{\omega}(b\varphi_{j}-bV_{\theta})$. We observe that $(\psi_{j})$
is a decreasing sequence of $\omega$-psh functions, and $\psi_{j}\geq-jb$ for
each $j$. Therefore the proof would follow if we could show that
$\lim_{j}\psi_{j}$ is not identically $-\infty$. We let for each $j$,
$D_{j}:=\\{\psi_{j}=b\varphi_{j}-bV_{\theta}\\}$ denote the contact set.
Observe that the sets $D_{j}$ are non-empty for $j$ large enough. Fix $t>0$.
We see that

$\\{\psi_{j}\leq-t\\}\cap D_{j}=\\{\varphi_{j}\leq
V_{\theta}-t/b\\}\subset\\{\varphi\leq V_{\theta}-t/b\\}.$

Set $\tilde{\omega}:=\left(\frac{1}{b}+1\right)\omega$. By Lemma 2.11 below
and plurifine locality we have for $j>t/b$,

(2.1) $\displaystyle\int_{\\{\psi_{j}\leq-t\\}}(\omega+dd^{c}\psi_{j})^{n}$
$\displaystyle=\int_{\\{\psi_{j}\leq-t\\}}{\bf
1}_{D_{j}}(\omega+dd^{c}\psi_{j})^{n}$ $\displaystyle\leq
b^{n}\int_{\\{\psi_{j}\leq-t\\}}{\bf
1}_{D_{j}}(\tilde{\omega}+dd^{c}\varphi_{j})^{n}$ $\displaystyle\leq
b^{n}\int_{\\{\varphi\leq
V_{\theta}-t/b\\}}(\tilde{\omega}+dd^{c}{\varphi_{j}})^{n}$
$\displaystyle=b^{n}\left(\int_{X}(\tilde{\omega}+dd^{c}{\varphi_{j}})^{n}-\int_{\\{\varphi>V_{\theta}-t/b\\}}(\tilde{\omega}+dd^{c}\varphi)^{n}\right),$

since $\varphi_{j}=\varphi$ on $\\{\varphi>V_{\theta}-t/b\\}$ for $j>t/b$.
Suppose by contradiction that $\sup_{X}\psi_{j}\rightarrow-\infty$ as
$j\to+\infty$. It thus follows that $\\{\psi_{j}\leq-t\\}=X$ for $j$ large
enough, $t$ being fixed. Hence, for $j>0$ large enough, (2.1) becomes

$\int_{X}\omega^{n}\leq
b^{n}\left(\int_{X}(\tilde{\omega}+dd^{c}\varphi_{j})^{n}-\int_{\\{\varphi>V_{\theta}-t/b\\}}(\tilde{\omega}+dd^{c}\varphi)^{n}\right).$

Letting $j\rightarrow+\infty$, we obtain

(2.2) $\int_{X}\omega^{n}\leq
b^{n}\left(\int_{X}(\tilde{\omega}+dd^{c}\varphi)^{n}-\int_{\\{\varphi>V_{\theta}-t/b\\}}(\tilde{\omega}+dd^{c}\varphi)^{n}\right),$

where we have used that

$(\tilde{\omega}+dd^{c}\varphi_{j})^{n}=\sum_{k=0}^{n}\binom{n}{k}(\tilde{\omega}-\theta)^{k}\wedge(\theta+dd^{c}\varphi_{j})^{n-k}\to(\tilde{\omega}+dd^{c}\varphi)^{n}$

in the weak sense of measures on $X$, thanks to [9, Theorem 2.3, Remark 2.5].
Finally, letting $t\rightarrow+\infty$, in (2.2) we obtain a contradiction.
Consequently, $\psi_{j}$ decreases to a $\omega$-psh function, we infer that
$P_{\omega}(b\varphi-bV_{\theta})$ is a $\omega$-psh function for any $b>0$.

It remains to show that $P_{\omega}(b\varphi-bV_{\theta})$ has full Monge-
Ampère mass. Observe that $P_{\omega}(b\varphi-
bV_{\theta})\geq\frac{b}{A}P_{\omega}(A\varphi-AV_{\theta})$ for $A>b$. Using
monotonicity of mass (see e.g. [27, Theorem 1.2]), we obtain

$\displaystyle\int_{X}\left(\omega+dd^{c}P_{\omega}(b\varphi-
bV_{\theta})\right)^{n}\geq\left(1-\frac{b}{A}\right)^{n}\int_{X}\omega^{n}+\left(\frac{b}{A}\right)^{n}\int_{X}\left(\omega+dd^{c}P_{\omega}(A\varphi-
AV_{\theta})\right)^{n}.$

Letting $A\to+\infty$ we thus finish the proof. ∎

######  Lemma 2.11.

Fix $b>0$, $\varphi$ and $P_{\omega}(b\varphi-
bV_{\theta})\in\operatorname{{\rm PSH}}(X,\omega)$. Then the measure
$(\omega+dd^{c}{P_{\omega}(b\varphi-bV_{\theta})})^{n}$ is supported on the
contact set $D:=\\{P_{\omega}(b\varphi-bV_{\theta})=b\varphi-bV_{\theta}\\}$,
and

${\bf 1}_{D}(\omega+dd^{c}{P_{\omega}(b\varphi-bV_{\theta})})^{n}\leq
b^{n}\mathbf{1}_{D}\left(\left(1+\frac{1}{b}\right)\omega+dd^{c}\varphi\right)^{n}.$

###### Proof.

We refer the readers to [11, Lemma 4.4] for a proof of the first statement.

For the second one, set $u=\frac{1}{b}P_{\omega}(b\varphi-
bV_{\theta})+V_{\theta}$. Then $u$ is a
$\tilde{\omega}:=\left(\frac{1}{b}+1\right)\omega$-psh function, and
$u\leq\varphi$. It follows from [21, Corollary 10.8] that

(2.3)
$\displaystyle\mathbf{1}_{\\{u=\varphi\\}}(\tilde{\omega}+dd^{c}u)^{n}\leq\mathbf{1}_{\\{u=\varphi\\}}(\tilde{\omega}+dd^{c}\varphi)^{n}.$

Furthermore, the measure $(\omega+dd^{c}{P_{\omega}(b\varphi-
bV_{\theta})})^{n}$ is supported on the contact set $D=\\{b\varphi-
bV_{\theta}=P_{\omega}(b\varphi-bV_{\theta})\\}=\\{u=\varphi\\}$, hence

(2.4) $\displaystyle\dfrac{1}{b^{n}}(\omega+dd^{c}{P_{\omega}(b\varphi-
bV_{\theta})})^{n}$
$\displaystyle=\mathbf{1}_{\\{u=\varphi\\}}\dfrac{1}{b^{n}}(\omega+dd^{c}{P_{\omega}(b\varphi-
bV_{\theta})})^{n}$
$\displaystyle\leq\mathbf{1}_{\\{u=\varphi\\}}(\tilde{\omega}+dd^{c}u)^{n}.$

Combining (2.3) and (2.4) we obtain the desired estimate. ∎

##  3\. Regularity of solutions

###  3.1. Proof of the Main Theorem

In this section we prove Theorem 1.1. The key ingredient is an adaptation of
Di Nezza-Lu’s approach [16] (see also [15]).

Given a non-negative Radon measure $\mu$ whose total mass is
$\operatorname{{\rm Vol}}(\alpha)$, we consider the Monge-Ampère equation

(3.1) $\operatorname{{\rm MA}}_{\theta}(\varphi)=\mu.$

The systematic study of such equations in big cohomology classes has been
initiated in [8]. It has been shown there that (3.1) admits a unique
normalized solution $\varphi\in\mathcal{E}(X,\theta)$ if and only if $\mu$ is
a non pluripolar measure on $X$.

Our goal is to prove the following:

######  Theorem 3.1.

Assume $\nu=gdV$ to be a Radon measure, with $0\leq g\in L^{p}(dV)$ for some
$p>1$. Let $\mu$ be a non-pluripolar measure such that
$\mu(X)=\operatorname{{\rm Vol}}(\alpha)$. Assume that $\mu=fd\nu$, with
$f\leq e^{-\phi}$ for some quasi-psh function $\phi$ on $X$. Let
$\varphi\in\mathcal{E}(X,\theta)$ be the unique normalized solution to (3.1).
Then $\varphi$ is continuous on $\textrm{Amp}(\alpha)\setminus E_{1/q}(\phi)$,
where $q$ denotes the conjugate exponent of $p$.

Note that Theorem 1.1 in the introduction is a particular case of Theorem 3.1.
We first establish this result under an extra assumption. More precisely, we
have the following theorem, which is closely similar to [16, Theorem 3.1] in
the case that $\theta$ is Kähler.

######  Theorem 3.2.

Let $\varphi\in\mathcal{E}(X,\theta)$ be normalized by $\sup_{X}\varphi=0$.
Assume that $\operatorname{{\rm MA}}_{\theta}(\varphi)\leq e^{-\phi}gdV$, for
some quasi-psh function $\phi$ on $X$, and $0\leq g\in L^{p}(dV)$, with $p>1$.
Assume that $\phi$ is locally bounded on an open set $U\subset{\rm
Amp}(\alpha)$. Then $\varphi$ is continuous on $U$.

###### Proof of Theorem 3.2.

We fix a $\theta$-psh function $\rho_{0}$ on $X$ such that

$\theta+dd^{c}\rho_{0}\geq 2\delta_{0}\omega,$

for some small constant $\delta_{0}>0$. Replacing $\rho_{0}$ by
$\rho_{0}-\sup_{X}\rho_{0}$, we can always assume that $\rho_{0}\leq
V_{\theta}$. Moreover, we can choose $\rho_{0}$ such that it is smooth in the
ample locus, with analytic singularities thanks to [7, Theorem 3.17 (ii)].

We will divide the proof in three steps.

Step 1. We prove that $\varphi$ is locally bounded on $U$.

We pick $a>0$ so small that $a\phi$ belongs to $\operatorname{{\rm
PSH}}(X,\delta_{0}\omega)$. Set $\psi:=\rho_{0}+a\phi$. We thus have
$\theta+dd^{c}\psi\geq\delta_{0}\omega$, and $\psi\leq V_{\theta}+a\phi$. We
claim that

(3.2) $\varphi\geq\psi-A,$

for $A>0$ depending only $\delta_{0}$, $p$, $dV$, $\|g\|_{L^{p}(dV)}$, and
$\int_{X}e^{-2P_{\omega}(a^{-1}\varphi-a^{-1}V_{\theta})}gdV$.

We remark that by Proposition 2.10, for any $b>0$, $P_{\omega}(b\varphi-
bV_{\theta})$ is a $\omega$-psh function with full Monge-Ampère mass, hence it
has zero Lelong numbers (see [20, Corollary 1.8]). Therefore, Skoda’s
integrability theorem [26] ensures that $e^{-P_{\omega}(b\varphi-
bV_{\theta})}$ belongs to $L^{q}(dV)$ for all $q<+\infty$. In particular,
$\int_{X}e^{-2P_{\omega}(b\varphi-bV_{\theta})}gdV$ is finite for any $b>0$.

The proof of the claim above follows the approach of Di Nezza and Lu [16, 15].
To see this, fix $s\in[0,1]$, $t>0$. Set $d\nu=gdV$, $b=a^{-1}$. Using Lemma
2.9 and the assumption on $\operatorname{{\rm MA}}_{\theta}(\varphi)$ we have

(3.3) $\displaystyle s^{n}\operatorname{{\rm
Cap}}_{\theta,\psi}(\\{\varphi<\psi-t-s\\})$
$\displaystyle\leq\int_{\\{\varphi<\psi-t\\}}\operatorname{{\rm
MA}}_{\theta}(\varphi)$
$\displaystyle\leq\int_{\\{\varphi<\psi-t\\}}e^{b(\psi-\varphi)}e^{-\phi}d\nu$
$\displaystyle\leq\int_{\\{\varphi<\psi-t\\}}e^{-(b\varphi-bV_{\theta})}d\nu$
$\displaystyle\leq\int_{\\{\varphi<\psi-t\\}}e^{-P_{\omega}(b\varphi-
bV_{\theta})}d\nu,$

where we have used that $b\psi\leq bV_{\theta}+\phi$ in the third inequality.
Using Hölder’s inequality we have

(3.4) $\displaystyle\int_{\\{\varphi<\psi-t\\}}e^{-P_{\omega}(b\varphi-
bV_{\theta})}d\nu\leq\left(\nu({\\{\varphi<\psi-t\\}})\right)^{1/2}\left(\int_{X}e^{-2P_{\omega}(b\varphi-
bV_{\theta})}d\nu\right)^{1/2}.$

By Lemma 2.5, one can find a constant $B>0$ depending on $n$, $p$, $\omega$,
$dV$, and $\|g\|_{L^{p}(dV)}$ such that

$\displaystyle\nu(\cdot)^{1/2}\leq B(\operatorname{{\rm
Cap}}_{\omega}(\cdot))^{2}.$

Since $\theta+dd^{c}\psi\geq\delta_{0}\omega$ it follows from Lemma 2.8 that
$\operatorname{{\rm Cap}}_{\omega}\leq\delta_{0}^{-n}\operatorname{{\rm
Cap}}_{\theta,\psi}$, hence

(3.5) $\displaystyle\nu(\cdot)^{1/2}\leq B\delta_{0}^{-2n}\operatorname{{\rm
Cap}}_{\theta,\psi}(\cdot)^{2}.$

By (3.3), (3.4) and (3.5) we thus get

(3.6) $s^{n}\operatorname{{\rm Cap}}_{\theta,\psi}(\\{\varphi<\psi-s-t\\})\leq
C\operatorname{{\rm Cap}}_{\theta,\psi}(\\{\varphi<\psi-t\\})^{2},$

where $C$ depends on $\omega$, $\delta_{0}$, $n$, $p$, $\|g\|_{L^{p}(dV)}$,
and $\int_{X}e^{-2P_{\omega}(b\varphi-bV_{\theta})}d\nu$. Set

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

By the estimate (3.6) we get

$\displaystyle sH(t+s)\leq C^{1/n}H(t)^{2}.$

It follows from Lemma 2.7 that the function $H$ is right-continuous and
$H(+\infty)=0$. We can thus apply [19, Lemma 2.4] which yields $H(t_{0}+2)=0$,
where $t_{0}>0$ is such that

$\displaystyle H(t_{0})<\dfrac{1}{2C^{1/n}}.$

Therefore, for $A=t_{0}+2$ we have $\varphi\geq\psi-A$ on $X\backslash P$ for
some Borel subset $P$ such that $\operatorname{{\rm Cap}}_{\theta,\psi}(P)=0$.
By Lemma 2.8 we have $\operatorname{{\rm Cap}}_{\omega}(P)=0$ so $P$ is a
pluripolar set. Hence $\varphi\geq\psi-A$ everywhere.

Using Hölder’s inequality it follows from (3.3) (take $s=1$) that

$\displaystyle H(t)^{n}$
$\displaystyle\leq\left(\int_{X}e^{-2P_{\omega}(b\varphi-
bV_{\theta}))}d\nu\right)^{1/2}\left(\int_{\\{\varphi<\psi-t+1\\}}d\nu\right)^{1/2}$
$\displaystyle\leq\left(\int_{X}e^{-2P_{\omega}(b\varphi-
bV_{\theta})}gdV\right)^{1/2}\left(\dfrac{1}{t-1}\int_{X}|\psi-\varphi|gdV\right)^{1/2}.$

The last integral is bounded by a uniform constant: using Hölder’s inequality
again we have
$\int_{X}|\psi-\varphi|gdV\leq\|g\|_{L^{p}(dV)}\left(\|\psi\|_{L^{q}(dV)}+\|\varphi\|_{L^{q}(dV)}\right)$
with $q=p/(p-1)$. Since $\varphi$ belongs to the compact set of $\theta$-psh
functions normalized by $\sup_{X}\varphi=0$, its $L^{q}$ norm is bounded by an
absolute constant only depending on $\theta,dV$ and $p$. Consequently, we can
choose $t_{0}>0$ to be only dependent on $dV$, $p$, $\|g\|_{L^{p}(dV)}$, and
an upper bound for $\int_{X}e^{-2P_{\omega}(b\varphi-bV_{\theta})}d\nu$.

Step 2. There exists a sequence of functions $\varphi_{j}\in\operatorname{{\rm
PSH}}(X,\theta)\cap\mathcal{C}^{0}(\textrm{Amp}(\alpha))$ which decreases
towards $\varphi$.

For convenience, we normalize $\varphi$ so that $\sup_{X}\varphi=-1$. Let
$0\geq h_{j}$ be a sequence of smooth functions decreasing to $\varphi$. Then
the sequence of $\theta$-psh functions $\varphi_{j}:=P_{\theta}(h_{j})$
decreases to $\varphi$ as $j\to+\infty$. Indeed, since the operator
$P_{\theta}$ is monotone, the sequence $\varphi_{j}$ is decreasing to a
$\theta$-psh function $u$ and since $\varphi_{j}\geq\varphi$ for all $j$ we
have $u\geq\varphi$. Moreover, $u(x)\leq\varphi_{j}(x)\leq
h_{j}(x),\forall\;x\in X$, for all $j$, hence $u(x)\leq\varphi(x)$, as
claimed. Furthermore, we have that $\varphi_{j}$ is continuous in
$\textrm{Amp}(\alpha)$ for each $j$. In fact, [4, Ineq. (1.2)] gives an upper
bound on the Monge-Ampère measure of $\varphi_{j}$:

$\operatorname{{\rm
MA}}_{\theta}(\varphi_{j})\leq\mathbf{1}_{D}\operatorname{{\rm
MA}}_{\theta}(h_{j}),\quad D=\\{\varphi_{j}=h_{j}\\}.$

The equality also holds following the work of Di Nezza and Trapani [17]. In
particular $\operatorname{{\rm MA}}_{\theta}(\varphi_{j})$ has an
$L^{\infty}$-density, hence it follows from [14, Theorem D] that $\varphi_{j}$
is Hölder continuous in $\textrm{Amp}(\alpha)$, as claimed.

Step 3. We show that the decreasing convergence $\varphi_{j}\to\varphi$ is
locally uniform on $U$ and finish the proof.

Fix $\lambda\in(0,1)$. For any $j\in\mathbb{N}$, set

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

where $A>0$ is the uniform constant so that $\varphi\geq\psi-A$ (see (3.2)).
If we pick $\varepsilon_{j}\leq\frac{\lambda}{2(1-\lambda)}\delta_{0}$ for
$j>0$ big enough, then
$\theta+dd^{c}\psi_{j}\geq\frac{\lambda}{2}\delta_{0}\omega$. We observe by
definitions that $\varphi_{j}\leq V_{\theta}$, hence
$\psi_{j}\leq\lambda\psi+(1-\lambda)V_{\theta}\leq V_{\theta}+\lambda a\phi$.
Set

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

For any $s\in[0,1]$, $t>0$, we can argue as above to obtain

$\displaystyle sH_{j}(t+s)\leq C^{1/n}H_{j}(t)^{2},$

for $C>0$ only depending on $p$, $dV$, $\|g\|_{L^{p}(dV)}$, $\delta_{0}$,
$\lambda$, and $\int_{X}e^{-2P_{\omega}(c\varphi-cV_{\theta})}gdV$, with
$c=(\lambda a)^{-1}$. Let $\chi$ be an increasing convex weight such that
$\chi(0)=0$, $\chi(-\infty)=-\infty$, and $\varphi$ has finite $\chi$-energy
(see [8, Proposition 2.11]). Since $\varphi_{j}\geq\varphi\geq\psi-A$ we have
$\psi_{j}\leq\varphi_{j}-2\lambda$. It follows from Lemma 2.9 (take
$s=\lambda$, $\psi=\psi_{j}+\lambda$) that

$\displaystyle\lambda^{n}\operatorname{{\rm
Cap}}_{\theta,\psi_{j}}\\{\varphi<\psi_{j}\\}$
$\displaystyle\leq\int_{\\{\varphi<\psi_{j}+\lambda\\}}\operatorname{{\rm
MA}}_{\theta}(\varphi)\leq\int_{\\{\varphi<\varphi_{j}-\lambda\\}}\operatorname{{\rm
MA}}_{\theta}(\varphi)$
$\displaystyle\leq\dfrac{1}{-\chi(-\lambda)}\int_{X}(-\chi\circ(\varphi-\varphi_{j}))\operatorname{{\rm
MA}}_{\theta}(\varphi).$

The latter converges to $0$ as $j\rightarrow+\infty$ since $\varphi_{j}$
decreases to $\varphi$, namely $H_{j}(0)$ goes to $0$ as $j\to+\infty$. We
thus take $j>0$ so big that $H_{j}(0)\leq 1/(2C^{1/n})$. It thus follows from
[19, Remark 2.5] that $H_{j}(t)=0$ if $t\geq t_{0}$ where $t_{0}\leq
4C^{1/n}H_{j}(0)$. We then have

(3.7)
$\displaystyle\varphi\geq\lambda\psi+(1-\lambda)\varphi_{j}-(A+2)\lambda-4C^{1/n}H_{j}(0).$

We have $H_{j}(0)\to 0$ as $j\to\infty$. Letting $j\rightarrow+\infty$, we
thus obtain

(3.8)
$\displaystyle\lim_{j\rightarrow+\infty}\inf_{K}(\varphi-\varphi_{j})\geq-\lambda(\sup_{K}|\psi|+A+2),$

for any compact subset $K$ of $U$. We observe by definition that
$\psi=\rho_{0}+a\phi$ is bounded on $K$. Finally, letting $\lambda\rightarrow
0$ in (3.8) we obtain that the convergence $\varphi_{j}\to\varphi$ is locally
uniform on $U$, hence $\varphi$ is continuous on $U$. ∎

###### Proof of Theorem 3.1.

By rescaling, we may assume without loss of generality that $\phi$ is a
$\omega$-psh function. By Theorem 2.1, we can find a quasi-psh function
${\phi}_{m}$ with analytic singularities such that
$\omega+dd^{c}{\phi}_{m}\geq-\varepsilon_{m}\omega$ for $\varepsilon_{m}>0$
decreasing to $0$ as $m\to+\infty$, and

$\int_{X}e^{2m({\phi}_{m}-{\phi})}dV<+\infty.$

Note that ${\phi}_{m}$ is smooth outside the analytic set
$E_{1/m}(\phi)\subset X$. We see that

$\displaystyle\operatorname{{\rm MA}}_{\theta}(\varphi)\leq
e^{-\phi_{m}}e^{({\phi}_{m}-{\phi})}gdV.$

Set now $g^{\prime}=e^{({\phi}_{m}-{\phi})}g$. We choose $m=[q]$, where $[q]$
denotes the integer part of $q$. We have $2m>q$, hence there is a constant
$p^{\prime}>1$ such that $\frac{1}{p^{\prime}}=\frac{1}{p}+\frac{1}{2m}$. It
follows from Hölder’s inequality that $g^{\prime}\in L^{p^{\prime}}(dV)$.
Observe that $\phi_{m}$ is smooth in the complement $X\setminus E_{1/m}(\phi)$
of the analytic set $E_{1/m}(\phi)\subset X$, in particular it is locally
bounded on $\textrm{Amp}(\alpha)\setminus
E_{1/m}(\phi)\supset\textrm{Amp}(\alpha)\setminus E_{1/q}(\phi)$. We can thus
apply Theorem 3.2 to complete the proof.

In particular if $p=\infty$ then we can choose $m=1$ to conclude the proof of
Theorem 1.1. ∎

###  3.2. Kähler-Einstein metrics on log canonical pairs of general type

####  3.2.1. Log canonical singularities

A _pair_ $(Y,\Delta)$ is by definition a connected complex normal projective
variety $Y$ and an effective Weil $\mathbb{Q}$-divisor $\Delta$. We will say
that the pair $(Y,\Delta)$ has _log canonical singularities_ if $K_{Y}+\Delta$
is $\mathbb{Q}$-Cartier, and if for some (or equivalently any) log resolution
$\pi:X\to Y$ of $(Y,\Delta)$, we have

$K_{X}=\pi^{*}(K_{Y}+\Delta)+\sum_{i}a_{i}E_{i},$

where $E_{i}$ are either exceptional divisors or components of the strict
transform of $\Delta$, and the coefficients $a_{i}\in\mathbb{Q}$ satisfy the
inequality $a_{i}\geq-1$ for all $j$. The divisor $\sum_{i}E_{i}$ has simple
normal crossing support. We denote the singular set of $Y$ by $Y_{\rm sing}$
and let $Y_{\rm reg}:=Y\setminus Y_{\rm sing}$.

Let $m$ be a positive integer such that $m(K_{Y}+\Delta)$ is Cartier. If we
choose $\sigma$ a local generator of $m(K_{Y}+\Delta)$ defined on an open
subset $U$ of $Y$, then $(i^{mn^{2}}\sigma\wedge\bar{\sigma})^{1/m}$ defines a
smooth volume form on $U\cap(Y_{\rm reg}\setminus\text{Supp}(\Delta))$. If
$f_{i}$ is a local equation of $E_{i}$ around a point $\pi^{-1}(U)$, then we
can see that

(3.9)
$\pi^{*}\left(i^{mn^{2}}\sigma\wedge\bar{\sigma}\right)^{1/m}=\prod_{i}|f_{i}|^{2a_{i}}dV$

locally on $\pi^{-1}(U)$ for some local volume form $dV$.

The previous construction leads to the following _adapted measure_ which is
introduced in [19, Sect. 6]:

######  Definition 3.3.

Let $(Y,\Delta)$ be a pair and let $h$ be a smooth hermitian metric on the
$\mathbb{Q}$-line bundle $\mathcal{O}_{Y}(K_{Y}+\Delta)$. The corresponding
adapted measure $\mu_{Y,h}$ on $Y_{\rm reg}$ is locally defined by choosing a
nowhere zero section $\sigma$ of $\mathcal{O}_{Y}(m(K_{Y}+\Delta))$ over a
small open set $U$ and setting

$\mu_{Y,h}:=\frac{(i^{mn^{2}}\sigma\wedge\bar{\sigma})^{1/m}}{|\sigma|_{h^{m}}^{2/m}}.$

The point of the definition is that the measure $\mu_{Y,h}$ does not depend on
the choice of $\sigma$. This measure can be extended by zero across $Y_{\rm
sing}\cup\textrm{Supp}(\Delta)$. Remark that the restriction $\sigma|_{Y_{\rm
reg}}$ can be viewed as a meromorphic form with a pole of order $md_{i}$ on
$\Delta_{i}$ where $\Delta=\sum_{i}d_{i}\Delta_{i}$ is the decomposition of
$\Delta$ into prime divisors (see [19, Sect. 6.3]). The Lelong-Poincaré
formula yields

$-dd^{c}\log\mu_{Y,h}=[\Delta]-i\Theta_{h}(K_{Y}+\Delta)$

on $Y_{reg}$, where $[\Delta]$ is the integration current on $\Delta$.

####  3.2.2. Kähler-Einstein metrics

Let $(Y,\Delta)$ be a log canonical pair, with $n=\dim_{\mathbb{C}}Y$. Assume
that $K_{Y}+\Delta$ is a big $\mathbb{Q}$-line bundle.

We next recall the notion of (negatively curved) Kähler-Einstein metric
attached to a pair $(Y,\Delta)$. There are various equivalent definitions for
such an object (e.g. in [19, 8, 6]), we choose here the following definition
in the sense of [6, Sect. 3].

######  Definition 3.4.

We say that a closed positive current $\omega_{KE}\in c_{1}(K_{Y}+\Delta)$ on
$Y$ is a _(singular) Kähler-Einstein_ metric (_KE_ for short) with negative
curvature for $(Y,\Delta)$ if

  1. (1)

The non-pluripolar product $\omega_{KE}^{n}$ defines a (locally) absolutely
continuous measure on $Y_{\rm reg}$ with respect to $dz\wedge d\bar{z}$ and
$\log(\omega_{KE}^{n}/dz\wedge d\bar{z})\in L^{1}_{\rm loc}(Y_{\rm reg})$,
where $z=(z_{i})$ are local holomorphic coordinates;

  2. (2)

${\rm Ric}(\omega_{KE})=-\omega_{KE}+[\Delta]$ on $Y_{\rm reg}$;

  3. (3)

$\int_{Y_{\rm reg}}\omega^{n}_{KE}={\rm vol}(K_{Y}+\Delta)$.

The condition (1) allows us to define (on $Y_{\rm reg}$) the Ricci curvature
of $\omega_{KE}$ by setting
$\textrm{Ric}(\omega_{KE}):=-\log(\omega_{KE}^{n})$. Another way of thinking
of this is to interpret the positive measure $\omega_{KE}^{n}|_{Y_{\rm reg}}$
as a singular metric on $-K_{Y_{\rm reg}}$ whose curvature is then
$\textrm{Ric}(\omega_{KE})$ by definition.

Let $h$ be a smooth hermitian metric on $K_{Y}+\Delta$ with curvature $\eta$.
Finding a singular Kähler-Einstein metric is equivalent to solving the
following Monge-Ampère equation for an $\eta$-psh $\phi$ with full Monge-
Ampère mass

(3.10) $(\eta+dd^{c}\phi)^{n}=e^{\phi+c}\mu_{Y,h},$

for some $c\in\mathbb{R}$. Indeed, we set $\omega:=\eta+dd^{c}\phi$. Since
$\phi$ is locally integrable $\omega$ satisfies the condition (1) (see Eq.
(3.9)). We also have

${\rm Ric}(\omega)=-dd^{c}\log\omega^{n}=-dd^{c}\phi-
dd^{c}\log\mu_{Y,h}=-dd^{c}\phi-\eta+[\Delta]$

on $Y_{\rm reg}$. Condition (3) is clearly satisfied.

We now prove Corollary 1.2 in the introduction. Assume the initial pair
$(Y,\Delta)$ is lc of general type, i.e. the canonical bundle $K_{Y}+\Delta$
is big. We consider a log resolution $\pi:(X,D)\rightarrow(Y,\Delta)$ of the
pair, with $K_{X}+D=\pi^{*}(K_{Y}+\Delta)$. Here, $D=\sum a_{i}D_{i}$ is a
$\mathbb{R}$-divisor with simple normal crossing support (snc for short) on
$X$, consisting of $\pi$-exceptional divisors with coefficients in
$(-\infty,1]$, and of the strict transforms of the components of $\Delta$ with
coefficients in $(0,1]$. The (singular) Kähler-Einstein metric $\omega_{KE}$
for $(X,D)$, or equivalently the pull-back of the (singular) KE metric for
$(Y,\Delta)$ by $\pi$ can be written as $\omega_{KE}=\theta+dd^{c}\varphi$
where $\theta=\pi^{*}\eta$ is a smooth representative of $c_{1}(K_{X}+D)$ and
$\varphi$ is a $\theta$-psh function (with full Monge-Ampère mass) solving the
following Monge–Ampère equation

(3.11) $\operatorname{{\rm
MA}}_{\theta}(\varphi)=\frac{e^{\varphi}dV}{\prod_{i}|s_{i}|^{2a_{i}}}.$

Here $s_{i}$ are sections cutting out $D_{i}$ above, $|\cdot|_{i}$ are smooth
hermitian metrics on associated line bundles $\mathcal{O}_{X}(D_{i})$, and
$dV$ is the smooth volume form with prescribed Ricci curvature. We let $D_{\rm
nklt}$ denote the non-klt part of $D$, i.e. $D_{\rm
nklt}:=\cup_{a_{i}=1}D_{i}$. We remark that $\varphi$ goes to $-\infty$ near
$D_{\rm nklt}$ as $|s|^{-2}$ is not integrable.

As a consequence of Theorem 3.2 we have the following

######  Corollary 3.5.

There is a unique solution $\varphi\in\mathcal{E}(X,\theta)$ to the equation
(3.11) which is continuous on $\textrm{Amp}(\theta)\setminus D_{\rm nklt}$.

###### Proof.

The existence of a unique solution $\varphi\in\mathcal{E}(X,\theta)$ to the
Monge-Ampère equation (3.11) follows from [6, Theorem 4.2]. It remains to
prove the continuity of $\varphi$. It is convenient to differentiate the ”klt
part” of $D$ from its ”non-klt part”, so we set
$g=\prod_{a_{i}<1}|s_{i}|^{-2a_{i}}\in L^{p}$ for some $p>1$, and
$\phi=\sum_{a_{i}=1}2\log|s_{i}|$. We see that $\phi$ is smooth outside of the
non-klt locus $D_{\rm nklt}$, in particular it is locally bounded on
$\textrm{Amp}(\theta)\setminus D_{\rm nklt}$. Since $\varphi$ is bounded from
above, we can therefore apply Theorem 3.2 to complete the proof. ∎

With the notations above, since $\\{\theta\\}$ is a pull-back by $\pi$ of a
big class, we have

$\textrm{Amp}(\theta)=\pi^{-1}(\textrm{Amp}(K_{Y}+\Delta))\setminus\textrm{Exc}(\pi)$

which projects onto $\textrm{Amp}(K_{Y}+\Delta)\cap(Y,\Delta)_{\rm reg}$. We
next observe that the projection of the non-klt locus $D_{\rm nklt}$ is
contained in $(Y,\Delta)_{\rm sing}\cup\lfloor\Delta\rfloor$, with
$\lfloor\Delta\rfloor=\sum_{d_{i}=1}\Delta_{i}$. The proof of Corollary 1.2
thus follows.

## References

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