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Relative pluripotential theory on compact Kähler manifolds

Tamás Darvas (University of Maryland)
Eleonora Di Nezza (Sorbonne Université)
Chinh H. Lu (Université d’Angers)

(To the memory of Jean–Pierre Demailly (1957-2022))

Abstract

Given a compact Kähler manifold, we survey the study of complex Monge–Ampère type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of Guedj–Zeriahi about the finite energy range of the complex Monge–Ampère operator.

Chapter 1 Main results

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$ , and let $\theta$ be another smooth Kähler form. We consider the complex Monge–Ampère equation

(\theta+dd^{c}u)^{n}=f\omega^{n}.

(1.1)

When $f$ is smooth and positive, Yau [Yau78], proved that this equation admits a unique smooth solution, resolving the famous Calabi conjecture. This landmark result is one of the cornerstones for studying canonical Kähler metrics.

One can consider conical/edge behaviour along a divisor for solutions of (1.1). This is relevant in birational geometry and in the study of K-stability. It has been explored in many works, including [Don11, JMR16, GP16].
When $f\geq 0$ , $f\in L^{p}(\omega^{n}),\ p>1,$ and $u$ is allowed to be bounded $\theta$ -plurisubharmonic, a solution to (1.1) exists by deep estimates of Kołodziej [Koł98, Koł03]. Furthermore, if one replaces $f\omega^{n}$ by a suitable Radon measure, it was shown in [GZ07] that solutions to (1.1) exist in a suitable finite energy space, mirroring results of Cegrell [Ceg98] in the local case.

When the cohomology class $\{\theta\}$ is allowed to be degenerate, solutions to (1.1) and related equations have been explored in [EGZ09, BEGZ10, BBGZ13].

There have been numerous other extensions and generalizations in recent years. In this paper, we survey recent work done in [DDL18a, DDL21], where we consider solutions of (1.1) that have a prescribed singularity profile. We also give a general answer to a question raised by Guedj–Zeriahi [GZ07].

To fix notation and terminology, we assume that $\theta$ is a closed smooth real $(1,1)$ -form such that $\{\theta\}$ is big, and let $\phi\in\textup{PSH}(X,\theta)$ . Let $f\geq 0$ , $f\in L^{p}(\omega^{n}),\ p>1$ .

We are looking for solutions $u\in\textup{PSH}(X,\theta)$ of (1.1) satisfying

\phi-C\leq u\leq\phi+C,\textup{ for some }C\in\mathbb{R}.

(1.2)

If $\varphi$ and $\varphi^{\prime}$ are two $\theta$ -plurisubharmonic functions on $X$ , then $\varphi^{\prime}$ is said to be less singular than $\varphi$ , i.e. $\varphi\preceq\varphi^{\prime}$ , if they satisfy $\varphi\leq\varphi^{\prime}+C$ for some $C\in\mathbb{R}$ . We say that $\varphi$ has the same singularity as $\varphi^{\prime}$ , i.e. $\varphi\simeq\varphi^{\prime}$ , if $\varphi\preceq\varphi^{\prime}$ and $\varphi^{\prime}\preceq\varphi$ . The latter condition is easily seen to yield an equivalence relation, whose equivalence classes are denoted by $[\varphi]$ , $\varphi\in{\rm PSH}(X,\theta)$ . Using this terminology (1.2) simply says that $[u]=[\phi]$ .

Typically $\phi$ is unbounded, hence so is $u$ , and the left-hand-side of (1.1) has to be interpreted as the non-pluripolar complex Monge-Ampère measure of $u$ , considered in [BEGZ10] (see (2.2) below):

\theta_{u}^{n}:=\langle(\theta+dd^{c}u)^{n}\rangle.

As it turns out, this problem is well posed only for potentials $\phi$ having “model” singularity type $[\phi]$ , that includes the case of analytic singularities, treated in [PS14].

We need to consider the following notion of envelope, only dependent on the singularity type $[\phi]$ :

P_{\theta}[\phi]=\sup\{v\in\textup{PSH}(X,\theta)\textup{ s.t. }v\leq 0,\ [\phi]=[v]\}.

Since $\phi-\sup_{X}\phi\leq P_{\theta}[\phi]$ , we have that $[\phi]\leq[P_{\theta}[\phi]]$ and typically equality does not happen. When $[\phi]=[P_{\theta}[\phi]]$ , we say that $\phi$ has model singularity type.

We state our first main result, initially proved in [DDL21]:

Theorem 1.1.

Suppose $\phi\in\textup{PSH}(X,\theta)$ and $[\phi]=[P_{\theta}[\phi]]$ . Let $f\in L^{p}(\omega^{n})$ , $p>1$ such that $f\geq 0$ and $\int_{X}f\omega^{n}=\int_{X}\theta_{\phi}^{n}>0$ . Then the following hold:
(i) There exists $u\in\textup{PSH}(X,\theta)$ , unique up to a constant, such that $[u]=[\phi]$ and

\theta_{u}^{n}=f\omega^{n}.

(1.3)

(ii) For any $\lambda>0$ there exists a unique $v\in\textup{PSH}(X,\theta)$ , such that $[v]=[\phi]$ and

\theta_{v}^{n}=e^{\lambda v}f\omega^{n}.

(1.4)

As it turns out, the condition $[\phi]=[P[\phi]]$ is very natural and necessary for well posedness in this context. Due to Theorem 5.22, if (1.3) has a unique solution for any choice of $f\in L^{\infty}$ , then the condition $[\phi]=[P[\phi]]$ must hold.

To prove this theorem, one needs to build up the variational approach of [BEGZ10] relative to the model singularity type $[\phi]$ , motivating the title of this survey.

When $K_{X}>0$ and $\lambda=1$ (or $K_{X}$ trivial and $\lambda=0$ ) solutions of (1.4) can be interpreted as Kähler–Einstein metrics with prescribed singularity.

Remark 1.2.

Potentials $\phi$ with analytic singularity type can be locally written as

\phi=\frac{c}{2}\log\big{(}\sum_{j}|f_{j}|^{2}\big{)}+g,

where $f_{j}$ are holomorphic, $c\in\mathbb{Q}_{+}$ and $g$ is bounded. Such potentials are always of model type ([RW14, Remark 4.6], [RS05], see also Proposition 5.24). In particular, discrete logarithmic singularity types are of model type, making connection with pluricomplex Green currents [CG09, PS14, RS05].

Our reader may wonder if there are other interesting enough potentials with model singularity type. We believe this to be the case:

$\bullet$ By Theorem 3.14 below, $P_{\theta}[\psi]=P_{\theta}[P_{\theta}[\psi]]$ for any $\psi\in\textup{PSH}(X,\theta)$ with $\int_{X}\theta_{\psi}^{n}>0$ . In particular, $P_{\theta}[\psi]$ is a model potential, giving an abundance of potentials with model type singularity.

$\bullet$ By Proposition 5.23 below, if $\psi\in\textup{PSH}(X,\theta)$ has small unbounded locus, and $\theta_{\psi}^{n}/\omega^{n}\in L^{p}(\omega^{n}),\ p>1$ with $\int_{X}\theta_{\psi}^{n}>0$ , then $\psi$ has model type singularity.

$\bullet$ Due to [RW14, DDL18], potentials with model type singularity naturally arise as degenerations along geodesic rays and in particular along test configurations.

As an application to Theorem 1.1 we prove the log-concavity property for non-pluripolar products, initially conjectured in [BEGZ10], proved in [DDL21].

Theorem 1.3.

Let $T_{1},...,T_{n}$ be positive closed $(1,1)$ -currents on a compact Kähler manifold $X$ . Then

\int_{X}\langle T_{1}\wedge\ldots\wedge T_{n}\rangle\geq\left(\int_{X}\langle T_{1}^{n}\rangle\right)^{\frac{1}{n}}\ldots\left(\int_{X}\langle T_{n}^{n}\rangle\right)^{\frac{1}{n}}.

In particular, $T\to\log\int_{X}\langle T^{n}\rangle$ is concave function on the space of positive $(1,1)$ -currents.

Next we consider the situation when the right hand side of (1.1) is a non-pluripolar measure (not necessarily of the type $f\omega^{n}$ ), and answer a question of Guedj–Zeriahi in our relative context.

For this we need to introduce relative finite energy classes. Let $\phi\in\textup{PSH}(X,\theta)$ such that $\phi=P[\phi]$ and $\int_{X}\theta_{\phi}^{n}>0$ .

By a foundational result of Witt Nyström [Wit19], if $[u]\leq[\phi]$ then $\int_{X}\theta_{u}^{n}\leq\int_{X}\theta_{\phi}^{n}$ (see Theorem 3.3 below). We say that $u$ has relative full mass with respect to $\phi$ (notation: $u\in\mathcal{E}(X,\theta,\phi)$ ) if this inequality is extremized, i.e., $[u]\leq[\phi]$ and $\int_{X}\theta_{u}^{n}=\int_{X}\theta_{\phi}^{n}$ .

It is argued in Theorem 5.17 that for any Radon measure $\mu$ not charging pluripolar sets such that $\mu(X)=\int_{X}\theta_{\phi}^{n}$ , there exists a unique solution $u\in\mathcal{E}(X,\theta,\phi))$ to the equation

\theta_{u}^{n}=\mu.

(1.5)

As pointed out in [GZ07], for potential theoretic reasons, it is natural to consider weighted subspaces of $\mathcal{E}(X,\theta,\phi)$ . A weight is a continuous increasing function $\chi:[0,\infty)\rightarrow[0,\infty)$ such that $\chi(0)=0$ and $\lim_{t\to\infty}\chi(t)=\infty$ .

Let us assume that the weight $\chi$ satisfies the following condition

\forall t\geq 0,\;\forall\lambda\geq 1,\;\chi(\lambda t)\leq\lambda^{M}\chi(t),

(1.6)

where $M\geq 1$ is a fixed constant. Let $\mathcal{E}_{\chi}(X,\theta,\phi)$ denote the set of all $u\in\mathcal{E}(X,\theta,\phi)$ such that

E_{\chi}(u,\phi):=\int_{X}\chi(|u-\phi|)\theta_{u}^{n}<\infty.

When $\phi=V_{\theta}$ , we denote $\mathcal{E}(X,\theta)=\mathcal{E}(X,\theta,V_{\theta})$ , $\mathcal{E}_{\chi}(X,\theta)=\mathcal{E}_{\chi}(X,\theta,V_{\theta}$ ) and $E_{\chi}(u)=E_{\chi}(u,V_{\theta})$ . Compared to [GZ07], we have changed the sign of the weight, but the weighted classes are the same. Also, notice that both low and high energy classes of [GZ07] satisfy the condition (1.6), allowing our treatment to be a bit more universal.

One may ask, under what condition is the solution $u\in\mathcal{E}(X,\theta,\phi)$ of (1.5) an element of $\mathcal{E}_{\chi}(X,\theta,\phi)$ . The theorem below provides precise answers to this question, containing perhaps the only novel result of this work:

Theorem 1.4.

Le $\mu$ be a Radon measure with $\mu(\{\phi=-\infty\})=0$ and $\int_{X}\theta_{\phi}^{n}=\mu(X)>0$ . For $\chi$ satisfying (1.6) the following conditions are equivalent:

(i)

There exists a constant $C>0$ such that, for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ with $\sup_{X}u=0$ , we have

$\int_{X}\chi(\phi-u)d\mu\leq CE_{\chi}(u,\phi)^{M/(M+1)}+C.$
(ii)

$\chi(|\phi-u|)\in L^{1}(\mu)$ , for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ .
(iii)

$\mu=\theta_{\varphi}^{n}$ , for some $\varphi\in\mathcal{E}_{\chi}(X,\theta,\phi)$ , with $\sup_{X}\varphi=0$ .

For slightly less general $\chi$ , the equivalence between (i) and (iii) was obtained in [DV21]. The condition $\mu(\{\phi=-\infty\})=0$ is necessary. Without it, the $\mu$ -integrability of $\chi(\phi-u)$ can not be discussed, as the values of this function are not defined on the set $\{\phi=-\infty\}$ . Of course this condition is vacuous if $\phi=0$ .

In the particular case when $\theta$ is Kähler and $\phi=0$ , the equivalence between (ii) and (iii) answers a question asked by Guedj–Zeriahi in the comments following [GZ07, Theorem 4.1]. In [GZ07, Theorem C] the authors obtain the above result for $\chi(t)=t^{p},\ p\geq 1$ .

Prerequisites.

An effort has been made to keep prerequisites at a minimum. However due to size constraints, such requirements on part of the reader are inevitable. We assume that our reader is familiar with the basics of Bedfor-Taylor theory [BT76, BT82], and finite energy pluripotential theory in the big case, as elaborated in [BEGZ10]. The recent book [GZ17] is a comprehensive source that can initiate novice readers into this subject.

Organization.

In Chapter 2 we recall terminology of finite energy pluripotential theory from [BEGZ10] and prove some preliminary results, touching [DDL18b, DT21] in the process. In Chapter 3 we prove the monotonicity theorem due to Witt Nyström [Wit19] and the authors [DDL18a] for non-pluripolar product masses. Here we also introduce and study the basic concepts of relative pluripotential theory.

In Chapter 4 we prove a result about comparison of capacities and prove an integrating by parts formula, making contact with [Xia19, Lu21, Vu21].

In Chapter 5, we finally solve the complex Monge–Ampère equations (1.3) and (1.4) using the variational method of [BBGZ13] as in [DDL18a]. Due to the availability of integration parts, the technical assumption of small unbounded locus from [DDL18a] can be avoided, making our treatment here more transparent, compared to the original arguments in [DDL18a, DDL21].

The proof of Theorem 1.4 is given in Chapter 6, containing the novel results of this survey.

Relation to other works.

Since [DDL18b, DDL18a, DDL18, DDL21] appeared, a number of works have taken up the topics of this survey, and developed it further. Due to size constraints we can not treat these here, but let us mention a few exciting directions.

The work [DDL21a] introduced a pseudo-metric on the space of singularity types and studied the stability of solutions to (1.3), as the singularity type is varied. More precise results have been recently obtained in [DV22]. Connections with Lelong numbers have been studied in [Vu20].

The works [DX21, DZ22, DXZ23] used relative pluripotential theory to study partial Bergman kernels, K-stability and the Ross–Witt Nyström correspondence.

The works [Xia19a, Tru19, Gup22] have explored the metric geometry of the relative finite energy classes surveyed in this work.

Finally, A. Trusiani started an elaborate study of Kahler–Einstein metrics with prescribed singularity type in the Fano case [Tru20, Tru23].

Acknowledgments.

The first named author was partially supported by an Alfred P. Sloan Fellowship and National Science Foundation grant DMS–1846942. The second author is supported by the project SiGMA ANR-22-ERCS-0004-02. The third named author is partially supported by Centre Henri Lebesgue ANR-11-LABX-0020-01. The second and third named authors are partially supported by the project PARAPLUI ANR-20-CE40-0019.

Chapter 2 Preliminaries

We recall results from pluripotential theory of big cohomology classes, particularly results about non-pluripolar complex Monge–Ampere measures. We borrow notation and terminology from [BEGZ10], and we refer to this work for further details.

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$ . Let $\theta$ be a smooth closed $(1,1)$ -form on $X$ such that $\{\theta\}$ is big, i.e., there exists $\psi\in{\rm PSH}(X,\theta)$ such that $\theta+dd^{c}\psi\geq\varepsilon\omega$ for some small constant $\varepsilon>0$ . Here, $d$ and $d^{c}$ are real differential operators defined as $d:=\partial+\bar{\partial},\,d^{c}:=\frac{i}{2\pi}\left(\bar{\partial}-\partial\right).$ A function $\varphi:X\rightarrow\mathbb{R}\cup\{-\infty\}$ is quasi-plurisubharmonic (qpsh) if it can be locally written as the sum of a plurisubharmonic function and a smooth function. $\varphi$ is called $\theta$ -plurisubharmonic ( $\theta$ -psh) if it is qpsh and $\theta+dd^{c}\varphi\geq 0$ in the sense of currents. We let ${\rm PSH}(X,\theta)$ denote the set of $\theta$ -psh functions that are not identically $-\infty$ .

A $\theta$ -psh function $\varphi$ is said to have analytic singularity type if there exists a constant $c>0$ such that locally on $X$ ,

\varphi=\frac{c}{2}\log\sum_{j=1}^{N}|f_{j}|^{2}+g,

where $g$ is bounded and $f_{1},\dots,f_{N}$ are local holomorphic functions. The ample locus ${\rm Amp}(\{\theta\})$ of $\{\theta\}$ is the set of points $x\in X$ such that there exists a Kähler current $T\in\{\theta\}$ with analytic singularity type and smooth in a neighbourhood of $x$ . The ample locus ${\rm Amp}(\{\theta\})$ is a Zariski open subset, and it is nonempty [Bou04].

Let $x\in X$ . Fixing a holomorphic chart $x\in U\subset X$ , the Lelong number $\nu(\varphi,x)$ of $\varphi\in\textup{PSH}(X,\theta)$ is defined as follows:

\nu(\varphi,x)=\sup\{\gamma\geq 0\textup{ s.t. }\varphi(z)\leq\gamma\log\|z-x\|+O(1)\textup{ on }U\}.

(2.1)

One can also associate to $\varphi$ a multiplier ideal sheaf $\mathcal{I}(\varphi)$ whose germs are holomorphic functions $f$ for which $|f|^{2}e^{-\varphi}$ is integrable.

If $\varphi$ and $\varphi^{\prime}$ are two $\theta$ -psh functions on $X$ , then $\varphi^{\prime}$ is said to be less singular than $\varphi$ , i.e. $\varphi\preceq\varphi^{\prime}$ , if they satisfy $\varphi\leq\varphi^{\prime}+C$ for some $C\in\mathbb{R}$ . We say that $\varphi$ has the same singularity as $\varphi^{\prime}$ , i.e. $\varphi\simeq\varphi^{\prime}$ , if $\varphi\preceq\varphi^{\prime}$ and $\varphi^{\prime}\preceq\varphi$ . The latter condition is easily seen to yield an equivalence relation, whose equivalence classes are denoted by $[\varphi]$ , $\varphi\in{\rm PSH}(X,\theta)$ .

A $\theta$ -psh function $\varphi$ is said to have minimal singularity type if it is less singular than any other $\theta$ -psh function. Such $\theta$ -psh functions with minimal singularity type always exist, one can consider for example

V_{\theta}:=\sup\left\{\varphi\,\,\theta\text{-psh},\varphi\leq 0\text{ on }X\right\}.

Trivially, a $\theta$ -psh function with minimal singularity type is locally bounded in ${\rm Amp}(\{\theta\})$ . It follows from [DT23, Theorem 1.1] that $V_{\theta}$ is $C^{1,\bar{1}}$ in the ample locus ${\rm Amp}(\{\theta\})$ .

Given $\theta^{1}+dd^{c}\varphi_{1},...,$ $\theta^{p}+dd^{c}\varphi_{p}$ positive $(1,1)$ -currents, where $\theta^{j}$ are closed smooth real $(1,1)$ -forms, following the construction of Bedford-Taylor [BT87] in the local setting, it has been shown in [BEGZ10] that the sequence of currents

{\bf 1}_{\bigcap_{j}\{\varphi_{j}>V_{\theta_{j}}-k\}}(\theta^{1}+dd^{c}\max(\varphi_{1},V_{\theta_{1}}-k))\wedge...\wedge(\theta^{p}+dd^{c}\max(\varphi_{p},V_{\theta_{p}}-k))

is non-decreasing in $k$ and converges weakly to the so called non-pluripolar product

\langle\theta^{1}_{\varphi_{1}}\wedge\ldots\wedge\theta^{p}_{\varphi_{p}}\rangle.

(2.2)

In the following, with a slight abuse of notation, we will denote the non-pluripolar product simply by $\theta^{1}_{\varphi_{1}}\wedge\ldots\wedge\theta^{p}_{\varphi_{p}}$ . When $p=n$ , the resulting positive $(n,n)$ -current is a Borel measure that does not charge pluripolar sets. Pluripolar sets are Borel measurable sets that are contained in some set $\{\psi=-\infty\}$ , where $\psi\in\textup{PSH}(X,\theta)$ .

For a $\theta$ -psh function $\varphi$ , the non-pluripolar complex Monge-Ampère measure of $\varphi$ is

\theta_{\varphi}^{n}:=\langle(\theta+dd^{c}\varphi)^{n}\rangle.

The volume of a big class $\{\theta\}$ is defined by

{\rm Vol}(\{\theta\}):=\int_{{\rm Amp}(\{\theta\})}\theta_{V_{\theta}}^{n}.

Alternatively, by [BEGZ10, Theorem 1.16], in the above expression one can replace $V_{\theta}$ with any $\theta$ -psh function with minimal singularity type. A $\theta$ -psh function $\varphi$ is said to have full Monge–Ampère mass if

\int_{X}\theta_{\varphi}^{n}={\rm Vol}(\{\theta\}),

and we then write $\varphi\in\mathcal{E}(X,\theta)$ .

An important property of the non-pluripolar product is that it is local with respect to the plurifine topology (see [BT87, Corollary 4.3],[BEGZ10, Section 1.2]). This topology is the coarsest such that all qpsh functions with values in $\mathbb{R}$ are continuous. For convenience we record the following version of this result for later use.

Lemma 2.1.

Fix closed smooth big $(1,1)$ -forms $\theta^{1},...,\theta^{n}$ . Assume that $\varphi_{j},\psi_{j},j=1,...,n$ are $\theta^{j}$ -psh functions such that $\varphi_{j}=\psi_{j}$ on $U$ an open set in the plurifine topology. Then

{\bf 1}_{U}\theta^{1}_{\varphi_{1}}\wedge...\wedge\theta^{n}_{\varphi_{n}}={\bf 1}_{U}\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{n}_{\psi_{n}}.

Lemma 2.1 will be referred to as the plurifine locality property. We will often work with sets of the form $\{u<v\}$ , where $u,v$ are quasi-psh functions. These are always open in the plurifine topology.

Convergence theorems.

The Monge–Ampère capacity of a Borel set $E\subset X$ is defined as

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

A function $u$ is called quasi-continuous if for each $\varepsilon>0$ , there exists an open set $U$ such that ${\rm Cap}_{\omega}(U)<\varepsilon$ and the restriction of $u$ on $X\setminus U$ is continuous.

A sequence of functions $u_{j}$ converges in capacity to $u$ if, for any $\delta>0$ ,

\lim_{j\to\infty}{\rm Cap}_{\omega}(\{x\in X\;:\;|u_{j}(x)-u(x)|>\delta\})=0.

We recall a classical convergence theorem from Bedford-Taylor theory. We refer to [GZ17, Theorem 4.26] for a proof of this result, which is a slight generalization of [Xin96, Theorem 1].

Proposition 2.2.

Let $U\subset\mathbb{C}^{n}$ be an open set. Suppose $\{f_{j}\}_{j}$ are uniformly bounded quasi-continuous functions which converge in capacity to another quasi-continuous function $f$ on $U$ . Let $\{u^{j}_{1}\}_{j},\{u^{j}_{2}\}_{j},\ldots,\{u^{j}_{n}\}_{j}$ be uniformly bounded plurisubharmonic functions on $\Omega$ , converging in capacity to $u_{1},u_{2},\ldots,u_{n}$ respectively. Then we have the following weak convergence of measures:

f_{j}i\partial\bar{\partial}u_{1}^{j}\wedge i\partial\bar{\partial}u_{2}^{j}\wedge\ldots\wedge i\partial\bar{\partial}u_{n}^{j}\to fi\partial\bar{\partial}u_{1}\wedge i\partial\bar{\partial}u_{2}\wedge\ldots\wedge i\partial\bar{\partial}u_{n}.

Definition 2.3.

A Borel set $E$ is called quasi-open if for each $\varepsilon>0$ , there exists an open set $U$ such that

{\rm Cap}_{\omega}((U\setminus E)\cup(E\setminus U))\leq\varepsilon.

Quasi-closed sets are defined similarly.

As mentioned above, we will often work with sets of the form $\{u<v\}$ where $u$ and $v$ are quasi-psh functions. In general these sets are not open but merely quasi-open.

If a sequence of positive measures $\mu_{j}$ converges weakly in $U$ to a positive measure $\mu$ then a elementary argument shows that if $E$ is open and $V$ is closed then

\liminf_{j\to\infty}\mu_{j}(E)\geq\mu(E),\qquad\limsup_{j\to\infty}\mu_{j}(V)\leq\mu(V).

Using the above facts one can easily argue the following result:

Lemma 2.4.

Assume $u_{j}$ is a sequence of uniformly bounded $\theta$ -psh functions in $U\subset X$ converging in capacity to a $\theta$ -psh function $u$ . Suppose $\{f_{j}\}_{j}$ are uniformly bounded non-negative quasi-continuous functions which converge in capacity to another quasi-continuous function $f\geq 0$ on $U$ .
If $E\subset U$ is a quasi-open set then

\liminf_{j\to\infty}\int_{E}f_{j}(\theta+dd^{c}u_{j})^{n}\geq\int_{E}f(\theta+dd^{c}u)^{n}.

If $V\subset U$ is a quasi-closed set then

\limsup_{j\to\infty}\int_{V}f_{j}(\theta+dd^{c}u_{j})^{n}\leq\int_{V}f(\theta+dd^{c}u)^{n}.

We will also need the following basic convergence result:

Lemma 2.5.

Assume that $\mu_{j}$ is a sequence of positive Borel measures converging weakly to $\mu$ . Assume that there exists a continuous function $f:[0,\infty)\rightarrow[0,\infty)$ with $f(0)=0$ such that, for any Borel set $E$ ,

\mu_{j}(E)+\mu(E)\leq f\left({\rm Cap}_{\omega}(E)\right).

Let $u_{j}$ be a sequence of uniformly bounded quasi-continuous functions which converges in capacity to a bounded quasi-continuous function $u$ . Then $u_{j}\mu_{j}\rightarrow u\mu$ in the sense of Radon measures on $X$ .

Proof.

Fixing $\varepsilon>0$ there exists a continuous function $v$ on $X$ such that

{\rm Cap}_{\omega}(\{x\in X\;:u(x)\neq v(x)\})<\varepsilon.

Let $A>0$ be a constant such that $|u_{j}|+|u|+|v|\leq A$ on $X$ . Fix $\delta>0$ . For $j>N$ large enough we have, by the assumption that $u_{j}$ converges in capacity to $u$ , that

{\rm Cap}_{\omega}(\{x\in X\;:\;|u_{j}(x)-u(x)|>\delta)<\varepsilon.

Fixing a continuous function $\chi$ and $j>N$ , it follows from the above that

	$\displaystyle\|\int_{X}(\chi u_{j}\mu_{j}-\chi ud\mu)\|\leq\int_{X}\|\chi(u_{j}-u)\|\mu_{j}+\|\int_{X}\chi u(\mu_{j}-\mu)\|$
	$\displaystyle\leq\delta\int_{X}\|\chi\|\mu_{j}+A\sup_{X}\|\chi\|\mu_{j}(\{\|u_{j}-u\|>\delta\})$
	$\displaystyle+\|\int_{X}\chi(u-v)(\mu_{j}-\mu)\|+\|\int_{X}\chi v(\mu_{j}-\mu)\|$
	$\displaystyle\leq\delta\int_{X}\|\chi\|\mu_{j}+A\sup_{X}\|\chi\|f(\varepsilon)+\|\int_{X}\chi(u-v)(\mu_{j}-\mu)\|+\|\int_{X}\chi v(\mu_{j}-\mu)\|$
	$\displaystyle\leq\delta\int_{X}\|\chi\|\mu_{j}+2A\sup_{X}\|\chi\|f(\varepsilon)+\|\int_{X}\chi v(\mu_{j}-\mu)\|.$

Since $v$ is continuous on $X$ the last term converges to $0$ as $j\to\infty$ . This completes the proof. ∎

The following lower-semicontinuity property of non-pluripolar products from [DDL18a] will be key in the sequel:

Theorem 2.6.

Let $\theta^{j},j\in\{1,\ldots,n\}$ be smooth closed real $(1,1)$ -forms on $X$ whose cohomology classes are big. Suppose that for all $j\in\{1,\ldots,n\}$ we have $u_{j},u_{j}^{k}\in\textup{PSH}(X,\theta^{j})$ such that $u^{k}_{j}\to u_{j}$ in capacity as $k\to\infty$ , and let $\chi_{k},\chi\geq 0$ be quasi-continuous and uniformly bounded such that $\chi_{k}\to\chi$ in capacity. Then

\liminf_{k\to\infty}\int_{X}\chi_{k}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}\geq\int_{X}\chi\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}.

(2.3)

If additionally,

\int_{X}\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}\geq\limsup_{k\rightarrow\infty}\int_{X}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}},

(2.4)

then $\chi_{k}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}\to\chi\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}$ in the weak sense of measures on $X$ .

Proof.

Set $\Omega:=\bigcap_{j=1}^{n}{\rm Amp}({\theta^{j}})$ and fix an open relatively compact subset $U$ of $\Omega$ . Then the functions $V_{\theta^{j}}$ are bounded on $U$ . We now use a classical idea in pluripotential theory. Fix $C>0,\varepsilon>0$ and consider

f_{j}^{k,C,\varepsilon}:=\frac{\max(u_{j}^{k}-V_{\theta^{j}}+C,0)}{\max(u_{j}^{k}-V_{\theta^{j}}+C,0)+\varepsilon},\ j=1,...,n,\ k\in\mathbb{N}^{*},

and

u_{j}^{k,C}:=\max(u_{j}^{k},V_{\theta^{j}}-C).

Observe that for $C,j$ fixed, the functions $u_{j}^{k,C}\geq V_{\theta^{j}}-C$ are uniformly bounded in $U$ (since $V_{\theta^{j}}$ is bounded in $U$ ) and converge in capacity to $u_{j}^{C}$ as $k\to\infty$ . Moreover, $f_{j}^{k,C,\varepsilon}=0$ if $u_{j}^{k}\leq V_{\theta^{j}}-C$ . By locality of the non-pluripolar product we can write

f^{k,C,\varepsilon}\chi_{k}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}=f^{k,C,\varepsilon}\chi_{k}\theta^{1}_{u^{k,C}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k,C}_{n}},

where $f^{k,C,\varepsilon}=f_{1}^{k,C,\varepsilon}\cdots f_{n}^{k,C,\varepsilon}$ . For each $C,\varepsilon$ fixed the functions $f^{k,C,\varepsilon}$ are quasi-continuous, uniformly bounded (with values in $[0,1]$ ) and converge in capacity to $f^{C,\varepsilon}:=f_{1}^{C,\varepsilon}\cdots f_{n}^{C,\varepsilon}$ , where $f_{j}^{C,\varepsilon}$ is defined by

f_{j}^{C,\varepsilon}:=\frac{\max(u_{j}-V_{\theta^{j}}+C,0)}{\max(u_{j}-V_{\theta^{j}}+C,0)+\varepsilon}.

With the information above we can apply Proposition 2.2 to get that

f^{k,C,\varepsilon}\chi_{k}\theta^{1}_{u^{k,C}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k,C}_{n}}\rightarrow f^{C,\varepsilon}\chi\theta^{1}_{u^{C}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{C}_{n}}\ \textrm{as}\ k\to\infty,

in the weak sense of measures on $U$ . In particular since $0\leq f^{k,C,\varepsilon}\leq 1$ we have that

	$\displaystyle\liminf_{k\to\infty}\int_{X}\chi_{k}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}$	$\displaystyle\geq$	$\displaystyle\liminf_{k\to\infty}\int_{U}f^{k,C,\varepsilon}\chi\theta^{1}_{u^{k,C}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k,C}_{n}}$
		$\displaystyle\geq$	$\displaystyle\int_{U}f^{C,\varepsilon}\chi\theta^{1}_{u^{C}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{C}_{n}}.$

Now, letting $\varepsilon\to 0$ and then $C\to\infty$ , by definition of the non-pluripolar product we obtain

\liminf_{k\to\infty}\int_{X}\chi_{k}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}\geq\int_{U}\chi\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}.

Finally, letting $U$ increase to $\Omega$ and noting that the complement of $\Omega$ is pluripolar, we conclude the proof of the first statement of the theorem.

To prove the last statement, we first notice that we actually have equality in (2.4) and the limsup is a lim, as one can just plug $\chi=1$ in (2.3).

Now let $B\in\mathbb{R}$ such that $\chi,\chi_{k}\leq B$ . By (2.3) we get that

\liminf_{k\to\infty}\int_{X}(B-\chi_{k})\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}\geq\int_{X}(B-\chi)\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}.

Flipping the signs and using (equality in) (2.4), we conclude the following inequality, finishing the proof:

\limsup_{k\to\infty}\int_{X}\chi_{k}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}\leq\int_{X}\chi\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}.

∎

Envelopes.

If $f$ is a function on $X$ , we define the envelope of $f$ in the class ${\rm PSH}(X,\theta)$ by

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

with the convention that $\sup\emptyset=-\infty$ . Observe that $P_{\theta}(f)\in{\rm PSH}(X,\theta)$ if and only if there exists some $u\in{\rm PSH}(X,\theta)$ lying below $f$ . Note also that $V_{\theta}=P_{\theta}(0)$ , and that $P_{\theta}(f+C)=P_{\theta}(f)+C$ for any constant $C$ .

In the particular case $f=\min(\psi,\phi)$ , we denote the envelope as $P_{\theta}(\psi,\phi):=P_{\theta}(\min(\psi,\phi))$ . We observe that $P_{\theta}(\psi,\phi)=P_{\theta}(P_{\theta}(\psi),P_{\theta}(\phi))$ , so w.l.o.g. we can assume $\psi,\phi$ are two $\theta$ -psh functions.

In our first technical result about envelopes, we show that the mass $\theta_{P_{\theta}(f)}^{n}$ is concentrated on the contact set $\{P_{\theta}(f)=f\}$ :

Theorem 2.7.

Assume $f$ is quasi-continuous on $X$ and $P_{\theta}(f)\in{\rm PSH}(X,\theta)$ . Then

\int_{\{P_{\theta}(f)<f\}}(\theta+dd^{c}P_{\theta}(f))^{n}=0.

Proof.

We can assume that $\theta\leq\omega$ . Since $P_{\theta}(f)\leq C$ is bounded from above, by replacing $f$ by $\min(f,C)$ we can assume that $f$ is bounded from above. Shifting $f$ by a constant we can also assume $f\leq 0$ .

Step 1. We assume that $f$ is bounded from below $f\geq-C_{0}$ . For each $j\geq 1$ , there is an open set $U_{j}\subset X$ such that ${\rm Cap}_{\omega}(U_{j})\leq 2^{-j-1}$ and the restriction of $f$ on $X\setminus U_{j}$ is continuous. By taking $\cup_{k\geq j}U_{k}$ we can assume that the sequence $U_{j}$ is decreasing. By the Tietze extension theorem, there is a function $f_{j}$ continuous on $X$ such that $f_{j}=f$ on $D_{j}:=X\setminus U_{j}$ , moreover $-C_{0}\leq f_{j}\leq 0$ . For each $j$ we define

g_{j}:=\sup_{k\geq j}f_{k}.

We observe that $g_{j}$ is lower-semicontinuous on $X$ , $g_{j}=f$ on $D_{j}$ , and $g_{j}$ decreases to some function $g$ on $X$ . Since the sequence $(D_{j})$ is increasing, it follows that $g_{j}=f$ on $D_{k}$ for all $k\leq j$ . Thus letting $j\to\infty$ gives $g=f$ on $D_{k}$ for all $k$ . This implies $g=f$ in $X$ except for a set of capacity equal to zero. Hence $g=f$ quasi-everywhere in $X$ , and $P_{\theta}(f)=P_{\theta}(g)$ . Since $g_{j}$ is lower-semicontinuous in $X$ , by the balayage method, [BT82, Corollary 9.2], we have

\int_{\Omega}(1-e^{P_{\theta}(g_{j})-g_{j}})(\theta+dd^{c}P_{\theta}(g_{j}))^{n}=0,

where $\Omega$ is the ample locus of $\{\theta\}$ . Fix an open set $G\Subset\Omega$ . Since $f_{j}$ is uniformly bounded on $X$ , we infer that $P_{\theta}(g_{j})-V_{\theta}$ is uniformly bounded, hence there is a constant $B>0$ such that $-B\leq P_{\theta}(g_{j})\leq 0$ in $G$ . It follows from the plurifine locality that, for all Borel set $E$ ,

{\bf 1}_{G\cap E}(\theta+dd^{c}P_{\theta}(g_{j}))^{n}\leq{\bf 1}_{G\cap E}(\omega+dd^{c}\max(P_{\theta}(g_{j}),-B))^{n}\leq B^{n}{\rm Cap}_{\omega}(E).

It follows from all the above that

	$\displaystyle\int_{G}$	$\displaystyle\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}$
		$\displaystyle=\int_{D_{j}\cap G}\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}+\int_{U_{j}\cap G}\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}$
		$\displaystyle=\int_{D_{j}\cap G}(1-e^{P_{\theta}(g_{j})-g_{j}})(\theta+dd^{c}P_{\theta}(g_{j}))^{n}+\int_{U_{j}\cap G}\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}$
		$\displaystyle\leq B^{n}\sup_{X}\|1-e^{P_{\theta}(g_{j})-f}\|{\rm Cap}_{\omega}(U_{j})\leq C2^{-j-1}.$

The functions $|1-e^{P_{\theta}(g_{j})-f}|$ are uniformly bounded and (by construction) converge in capacity to the quasi-continuous function $|1-e^{P_{\theta}(f)-f}|$ . It thus follows from Proposition 2.2 that

|1-e^{P_{\theta}(g_{j})-f}|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}\;\text{ weakly converges to }\;|1-e^{P_{\theta}(f)-f}|(\theta+dd^{c}P_{\theta}(f))^{n},

hence

	$\displaystyle\liminf_{j\to\infty}(C2^{-j-1})$	$\displaystyle\geq\liminf_{j\to\infty}\int_{G}\|1-e^{P(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}$
		$\displaystyle\geq\int_{G}\|1-e^{P_{\theta}(f)-f}\|(\theta+dd^{c}P_{\theta}(f))^{n}\geq 0.$

We then infer that

\int_{G}|1-e^{P_{\theta}(f)-f}|(\theta+dd^{c}P_{\theta}(f))^{n}=0.

Letting $G$ increase to $\Omega$ , we can conclude that $(\theta+dd^{c}P_{\theta}(f))^{n}$ is concentrated on the contact set $\{P_{\theta}(f)=f\}$ .

Step 2. For the general case we approximate $f$ by $f_{j}:=\max(f,-j)$ . By the first step, we have

\int_{\{P_{\theta}(f_{j})<f_{j}\}}(\theta+dd^{c}P_{\theta}(f_{j}))^{n}=0.

Fix $C>0$ and an open set $G\Subset\Omega$ . Consider the quasi-open set $U:=G\cap\{P_{\theta}(f)>V_{\theta}-C\}$ . Since $P_{\theta}(f_{j})\geq P_{\theta}(f)$ , thanks to the plurifine property we have

{\bf 1}_{U}(\theta+dd^{c}\max(P_{\theta}(f_{j}),V_{\theta}-C))^{n}={\bf 1}_{U}(\theta+dd^{c}P_{\theta}(f_{j}))^{n},

hence

\int_{U}(1-e^{P_{\theta}(f_{j})-f_{j})})(\theta+dd^{c}\max(P_{\theta}(f_{j}),V_{\theta}-C))^{n}=0.

Observe that $P_{\theta}(f_{j})$ decreases to $P_{\theta}(f)$ , hence it converges in capacity. Letting $j\to\infty$ and using Lemma 2.4 we obtain

\int_{U}(1-e^{P_{\theta}(f)-f)})(\theta+dd^{c}\max(P_{\theta}(f),V_{\theta}-C))^{n}=0,

hence

\int_{U}(1-e^{P_{\theta}(f)-f)})(\theta+dd^{c}P_{\theta}(f))^{n}=0.

From this, letting $C\to\infty$ and $U\nearrow\Omega$ we obtain the result. ∎

In order to prove a regularity result for $\theta_{V_{\theta}}^{n}$ , we will need two preliminaries lemmas. We recall that $C^{1,\overline{1}}(X)$ denotes the space of continuous function with bounded distributional Laplacian w.r.t. $\omega$ .

Lemma 2.8.

If $f_{1},f_{2}\in C^{1,\bar{1}}(X)$ , then $P_{\omega}(f_{1},f_{2})\in C^{1,\bar{1}}(X)$ and for $i=1,2$ the functions $f_{i}$ and $P_{\omega}(f_{1},f_{2})$ are equal up to second order at almost every point on the set $\{P_{\omega}(f_{1},f_{2})=f_{i}\}$ . In particular, the measures

{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{1}\}}\omega_{f_{1}}^{n},\quad{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{2}\}}\omega_{f_{2}}^{n},\quad(j=1,2),

are positive and

	$\displaystyle{\omega}_{P_{\omega}(f_{1},f_{2})}^{n}$	$\displaystyle=$	$\displaystyle{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{1}\}}\omega_{f_{1}}^{n}+{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{2}\}}\omega_{f_{2}}^{n}-{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{1}=f_{2}\}}\omega_{f_{1}}^{n}$
		$\displaystyle=$	$\displaystyle{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{1}\}}\omega_{f_{1}}^{n}+{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{2}\}}\omega_{f_{2}}^{n}-{\bf 1}_{\{P_{\omega}(f_{1},f_{2})=f_{1}=f_{2}\}}\omega_{f_{2}}^{n}.$

The $C^{1,\bar{1}}$ regularity of the envelope $P_{\omega}(f_{1})$ is due to Berman [Ber19]. For the $C^{1,\bar{1}}$ regularity of the envelope $P_{\theta}(f_{1})$ in the case $\{\theta\}$ is big we refer to [DT23]. The $C^{1,\bar{1}}$ regularity of the rooftop envelope $P_{\omega}(f_{1},f_{2})$ was proved in [DR16, Theorem 2.5(ii)]. For a detailed presentation of these results, we refer to [Dar19, Appendix A.1]. For Hessian estimates which give the optimal $C^{1,1}$ regularity, see [Tos18, Theorem 3.1]. Using the $C^{1,\bar{1}}$ estimates, one can reason the same way as in the short argument of [Dar17, Proposition 2.2] to conclude (2.8).

Lemma 2.9.

Let $\varphi,\psi\in{\rm PSH}(X,\theta)$ . Then

\theta_{\max(\varphi,\psi)}^{n}\geq{\bf 1}_{\{\psi\leq\varphi\}}\theta_{\varphi}^{n}+{\bf 1}_{\{\varphi<\psi\}}\theta_{\psi}^{n}.

(2.6)

In particular, if $\varphi\leq\psi$ then ${\bf 1}_{\{\varphi=\psi\}}\theta_{\varphi}^{n}\leq{\bf 1}_{\{\varphi=\psi\}}\theta_{\psi}^{n}.$

Proof.

Let $\psi_{k}:=\max(\psi,V_{\theta}-k)$ and $\varphi_{k}:=\max(\varphi,V_{\theta}-k)$ .

By the locality of the Monge–Ampère measure with respect to the plurifine topology it follows that

{\bf 1}_{\{\psi_{k}>\varphi_{k}\}}\theta_{\max(\psi_{k},\varphi_{k})}^{n}={\bf 1}_{\{\psi_{k}>\varphi_{k}\}}\theta_{\psi_{k}}^{n}.

and

{\bf 1}_{\{\psi_{k}<\varphi_{k}\}}\theta_{\max(\psi_{k},\varphi_{k})}^{n}={\bf 1}_{\{\psi_{k}<\varphi_{k}\}}\theta_{\varphi_{k}}^{n}

holds in the ample locus of $\{\theta\}$ where all the functions above are locally bounded. As the non-pluripolar products are extended trivially over $X$ , we see that the above inequality holds over $X$ in the sense of measures. Considering $\max(\psi_{k},\varphi_{k}+t)$ and letting $t\searrow 0$ we obtain

\theta_{\max(\varphi_{k},\psi_{k})}^{n}\geq{\bf 1}_{\{\psi_{k}\leq\varphi_{k}\}}\theta_{\varphi_{k}}^{n}+{\bf 1}_{\{\varphi_{k}<\psi_{k}\}}\theta_{\psi_{k}}^{n}.

Multiplying with ${\bf 1}_{\{\min(\varphi,\psi)>V_{\theta}-k\}}$ , and using plurifine locality we arrive at

{\bf 1}_{\{\min(\varphi,\psi)>V_{\theta}-k\}}\theta_{\max(\varphi,\psi)}^{n}\geq{\bf 1}_{\{\min(\varphi,\psi)>V_{\theta}-k\}\cap\{\psi\leq\varphi\}}\theta_{\varphi}^{n}+{\bf 1}_{\{\min(\varphi,\psi)>V_{\theta}-k\}\cap\{\varphi<\psi\}}\theta_{\psi}^{n}.

Letting $k\to\infty$ , (2.6) follows. ∎

We are now ready to prove a regularity result about $\theta_{V_{\theta}}^{n}$ , following [DT21].

Theorem 2.10.

If $\phi\in{\rm PSH}(X,\theta)$ and $\phi\leq 0$ , then

{\bf 1}_{\{\phi=0\}}(\theta+dd^{c}\phi)^{n}={\bf 1}_{\{\phi=0\}}\theta^{n}.

In particular, one has

(\theta+dd^{c}V_{\theta})^{n}={\bf 1}_{\{V_{\theta}=0\}}\theta^{n}.

Proof.

It suffices to prove the first statement as the second follows from this and Theorem 2.7. Without loss of generality, we can assume $\theta\leq\omega$ . Note that $\phi$ is a $\omega$ -psh function as well. We proceed in two steps.

Step 1. We want to prove in this step that

{\bf 1}_{\{\phi=0\}}(\omega+dd^{c}\phi)^{n}={\bf 1}_{\{\phi=0\}}\omega^{n}.

Let $f_{j}$ be a sequence of smooth $\omega$ -psh functions on $X$ decreasing to $\phi$ . Set $\phi_{j}=P_{\omega}(f_{j},0)$ and observe that $\phi\leq\phi_{j}$ . Using

\{\phi=0\}\subseteq\{\phi_{j}=0\}.

(2.7)

and Lemma 2.8 we then get

\omega_{\phi_{j}}^{n}={\bf 1}_{\{\phi_{j}=0\}}\omega^{n}\geq{\bf 1}_{\{\phi=0\}}\omega^{n}.

(2.8)

The functions $\max(\phi_{j},-1)$ are uniformly bounded and decrease to $\max(\phi,-1)$ , hence

(\omega+dd^{c}\max(\phi_{j},-1))^{n}\to(\omega+dd^{c}\max(\phi,-1))^{n}

weakly. Since $\{\phi=0\}=X\setminus\{\phi<0\}$ is a closed subset of $X$ , we get

\limsup_{j\to\infty}\int_{\{\phi=0\}}(\omega+dd^{c}\max(\phi_{j},-1))^{n}\leq\int_{\{\phi=0\}}(\omega+dd^{c}\max(\phi,-1))^{n}.

Observe also that $\{\phi=0\}\subset\{\phi_{j}>-1\}$ and $\{\phi=0\}\subset\{\phi>-1\}$ . By plurifine locality we thus have

\limsup_{j\to\infty}\int_{\{\phi=0\}}(\omega+dd^{c}\phi_{j})^{n}\leq\int_{\{\phi=0\}}(\omega+dd^{c}\phi)^{n}.

(2.9)

By (2.8) we then get

\int_{\{\phi=0\}}\omega^{n}\leq\int_{\{\phi=0\}}(\omega+dd^{c}\phi)^{n}.

Then, by Lemma 2.9 we get the identity

{\bf 1}_{\{\phi=0\}}\omega_{\phi}^{n}={\bf 1}_{\{\phi=0\}}\omega^{n}.

(2.10)

Step 2. Now, fix $A>0$ such that $\theta+A\omega$ is a Kähler form. Since $\phi$ is $\theta$ -psh function, it follows that $\phi$ is $(\theta+t\omega)$ -psh, for $t\geq 0.$ Let $g\in C^{0}(X,\mathbb{R})$ and consider the function

Q(t):=\int_{\{\phi=0\}}g(\theta+t\omega+dd^{c}\phi)^{n}-\int_{\{\phi=0\}}g(\theta+t\omega)^{n}

defined for $t\geq 0.$ Then by multilinearity of the non-pluripolar product and the multilinearity of the product of forms, it is clear that $Q(t)$ is a polynomial in $t$ . Thanks to (2.10) we can infer that for any $t>A$

{\bf 1}_{\{\phi=0\}}(\theta+t\omega+dd^{c}\phi)^{n}={\bf 1}_{\{\phi=0\}}(\theta+t\omega)^{n}.

This implies that the polynomial $Q(t)$ is identically zero for $t>A,$ hence $Q(t)\equiv 0$ . It then follows that $Q(0)=0.$ Since $g\in C^{0}(X,\mathbb{R})$ is arbitrary we have the desired equality between measures. ∎

As it turns out, (2.9) follows from a more general result. This was proved in [DDL21a, Proposition 4.6], and the proof we give below fixes an imprecision in the original argument.

Lemma 2.11.

Suppose that $u_{j},u\in\textup{PSH}(X,\theta)$ with $u_{j},u\leq$ 0. If $\|u_{j}-u\|_{L^{1}}\to 0$ then

\limsup_{j\rightarrow+\infty}\int_{\{u_{j}=0\}}\theta_{u_{j}}^{n}\leq\int_{\{u=0\}}\theta_{u}^{n}.

Proof.

Let $v_{k}=\textup{usc}(\sup_{j\geq k}u_{j})$ . It is well known that $v_{k}\searrow u$ . Also $u_{k}\leq v_{k}\leq 0$ , so $\{u_{k}=0\}\subset\{v_{k}=0\}$ . As a result, due to Lemma 2.9, we have ${\bf 1}_{\{u_{k}=0\}}\theta_{u_{k}}^{n}\leq{\bf 1}_{\{u_{k}=0\}}\theta_{v_{k}}^{n}.$

Let $b,c>0$ . Using plurifine locality and the above we get that

\int_{\{u_{j}=0\}}\theta_{u_{j}}^{n}\leq\int_{\{u_{j}=0\}}\theta_{v_{j}}^{n}\leq\int_{\{v_{j}=0\}}\theta_{v_{j}}^{n}=\int_{\{v_{j}=0\}}\theta_{\max(v_{j},V_{\theta}-c)}^{n}\leq\int_{X}e^{bv_{j}}\theta_{\max(v_{j},V_{\theta}-c)}^{n}.

Since $e^{bv_{j}},e^{bu}$ are uniformly bounded and non-negative function, taking the limit Theorem 2.6 gives that

\limsup_{j\rightarrow+\infty}\int_{\{u_{j}=0\}}\theta_{u_{j}}^{n}\leq\int_{X}e^{bu}\theta_{\max(u,V_{\theta}-c)}^{n}.

Now letting $b\to\infty$ , and subsequently $c\to\infty$ , the conclusion follows. ∎

In our investigation of relative pluripotential theory, the following envelope construction will be essential: given $\phi\in\textup{PSH}(X,\theta)$ we consider

\textup{PSH}(X,\theta)\ni\psi\to\ P_{\theta}[\psi](\phi)\in\textup{PSH}(X,\theta).

This was introduced by Ross and Witt Nyström [RW14] in their construction of geodesic rays, building on ideas of Rashkovskii and Sigurdsson [RS05] in the local setting. Starting from the “rooftop envelope” $P_{\theta}(\psi,\phi)$ we introduce

P_{\theta}[\psi](\phi):=\Big{(}\lim_{C\to\infty}P_{\theta}(\psi+C,\phi)\Big{)}^{*}.

It is easy to see that $P_{\theta}[\psi](\phi)$ only depends on the singularity type of $\psi$ . When $\phi=V_{\theta}$ , we will simply write $P_{\theta}[\psi]:=P_{\theta}[\psi](V_{\theta})$ , and we refer to this potential as the envelope of the singularity type $[\psi]$ . We note the following simple concavity result about the operator $P_{\theta}[\cdot](\phi)$ .

Lemma 2.12.

The operator $P_{\theta}[\cdot](\phi)$ is concave: if $u,v,\phi\in{\rm PSH}(X,\theta)$ and $t\in(0,1)$ then

P_{\theta}[tu+(1-t)v](\phi)\geq tP_{\theta}[u](\phi)+(1-t)P_{\theta}[v](\phi).

Proof.

Assume $u,v\in{\rm PSH}(X,\theta)$ and $t\in(0,1)$ . Then, for all $C>0$ ,

P_{\theta}(\min(tu+(1-t)v+C,\phi))\geq tP_{\theta}(\min(u+C,\phi))+(1-t)P_{\theta}(\min(v+C,\phi)),

because the right-hand side is a $\theta$ -psh function lying below $\min(tu+(1-t)v+C,0)$ . Letting $C\nearrow\infty$ we arrive at the result. ∎

Chapter 3 The basics of relative pluripotential theory

3.1 Monotonicity of non-pluripolar product masses

In what follows, unless otherwise stated, we work with $\theta$ a smooth closed real $(1,1)$ -form whose cohomology class is big. We start with the following result, saying that potentials with the same singularity type have also the same global mass.

Lemma 3.1.

Let $u,v\in{\rm PSH}(X,\theta)$ . If $u$ and $v$ have the same singularity type, then $\int_{X}\theta_{u}^{n}=\int_{X}\theta_{v}^{n}$ .

The above result is due to Witt Nyström [Wit19]. A different proof has been given in [LN22] using the monotonicity of the energy functional. We give below a direct proof using a standard approximation process. Another different proof has been recently given in [Vu21], where generalized non-pluripolar products of positive currents are studied.

Proof.

Step 1. Assume that $\theta$ is a Kähler form.

We first prove the following claim: if there exists a constant $C>0$ such that $u=v$ on the open set $U:=\{\min(u,v)<-C\}$ then $\int_{X}\theta_{u}^{n}=\int_{X}\theta_{v}^{n}$ .

Fix $t>C$ . Since $u=v$ on $U$ , we have $\max(u,-t)=\max(v,-t)$ on $U$ . Since $U$ is open, we have

{\bf 1}_{U}(\theta+dd^{c}\max(u,-t))^{n}={\bf 1}_{U}(\theta+dd^{c}\max(v,-t))^{n}.

We also have $\{u\leq-t\}=\{v\leq-t\}\subset U$ . Indeed, if $u(x)\leq-t$ then $x\in U$ because $-t<-C$ . But on $U$ we have $u=v$ , hence $v(x)=u(x)\leq-t$ .

By plurifine locality we thus have

	$\displaystyle\theta_{\max(u,-t)}^{n}$	$\displaystyle={\bf 1}_{\{u>-t\}}\theta_{\max(u,-t)}^{n}+{\bf 1}_{\{u\leq-t\}}\theta_{\max(u,-t)}^{n}$
		$\displaystyle={\bf 1}_{\{u>-t\}}\theta_{u}^{n}+{\bf 1}_{\{v\leq-t\}}\theta_{\max(v,-t)}^{n},$

and

\theta_{\max(v,-t)}^{n}={\bf 1}_{\{v>-t\}}\theta_{v}^{n}+{\bf 1}_{\{v\leq-t\}}\theta_{\max(v,-t)}^{n}.

Integrating over $X$ , since $\int_{X}\theta_{\max(u,-t)}^{n}=\int_{X}\theta_{\max(v,-t)}^{n}={\rm Vol}({\theta})$ (recall that $\max(u,-t)$ and $\max(u,-t)$ are bounded functions), we get

\int_{\{u>-t\}}\theta_{u}^{n}=\int_{\{v>-t\}}\theta_{v}^{n}.

Letting $t\to\infty$ , the claim follows.

Now we prove the general case when $\theta$ is Kähler. Since $u$ and $v$ have the same singularity type, we can assume that $v\leq u\leq v+B\leq 0$ , for some positive constant $B$ . For each $a\in(0,1)$ we set $v_{a}:=av$ , $u_{a}:=\max(u,v_{a})$ and $C:=Ba(1-a)^{-1}$ . To use the claim above we need to check that $u_{a}=v_{a}$ on the open set $U_{a}:=\{\min(u_{a},v_{a})<-C\}$ . Observe that $\min(u_{a},v_{a})=v_{a}$ because $v_{a}\leq u_{a}$ . If $x\in U_{a}$ then $av(x)<-C$ hence $(1-a)v(x)<-B$ , which implies (recall that $v+B\geq u$ )

av(x)\geq v(x)+B\geq u(x).

We infer that $v_{a}(x)\geq u(x)$ , hence $u_{a}(x)=v_{a}(x)$ . We can thus apply the claim above to get $\int_{X}\theta_{u_{a}}^{n}=\int_{X}\theta_{v_{a}}^{n}$ . Since non-pluripolar products are multilinear, see [BEGZ10, Proposition 1.4], we have that

\int_{X}\theta_{v_{a}}^{n}=a^{n}\int_{X}\theta_{v}^{n}+\sum_{k=0}^{n-1}a^{k}(1-a)^{n-k}\int_{X}\theta_{v}^{k}\wedge\theta^{n-k}\to\int_{X}\theta_{v}^{n}

as $a\nearrow 1$ . Since $u_{a}\searrow u$ as $a\nearrow 1$ , by Theorem 2.6 we have

\liminf_{a\to 1^{-}}\int_{X}\theta_{u_{a}}^{n}\geq\int_{X}\theta_{u}^{n}.

We thus have $\int_{X}\theta_{u}^{n}\leq\int_{X}\theta_{v}^{n}$ . By symmetry we get equality, finishing the proof of Step 1.

Step 2. We treat the general case when $\{\theta\}$ is merely big. We use an idea from [DT21]. Fix $s>0$ so large that $\theta+s\omega$ is Kähler. For $t>s$ we have, by the first step,

\int_{X}(\theta+t\omega+dd^{c}u)^{n}=\int_{X}(\theta+t\omega+dd^{c}v)^{n}.

Since non-pluripolar products are multilinear ([BEGZ10, Proposition 1.4]) we have for all $t>s$ ,

\sum_{k=0}^{n}\binom{n}{k}\int_{X}\theta_{u}^{k}\wedge\omega^{n-k}t^{n-k}=\sum_{k=0}^{n}\binom{n}{k}\int_{X}\theta_{v}^{k}\wedge\omega^{n-k}t^{n-k}.

We thus obtain an equality between two polynomials for all $t\geq 0$ , and identifying the coefficients we infer the desired equality. ∎

We continue with the following immediate generalization of the above result:

Proposition 3.2.

Let $\theta^{j},j\in\{1,\ldots,n\}$ be smooth closed real $(1,1)$ -forms on $X$ whose cohomology classes are pseudoeffective. Let $u_{j},v_{j}\in\textup{PSH}(X,\theta^{j})$ such that $u_{j}$ has the same singularity type as $v_{j},\ j\in\{1,\ldots,n\}$ . Then

\int_{X}\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}=\int_{X}\theta^{1}_{v_{1}}\wedge\ldots\wedge\theta^{n}_{v_{n}}.

The proof of this result uses the ideas from [BEGZ10, Corollary 2.15].

Proof.

First we note that we can assume that the classes $\{\theta^{j}\}$ are in fact big. Indeed, if this is not the case we can just replace each $\theta^{j}$ with $\theta^{j}+\varepsilon\omega$ , and using the multi-linearity of the non-pluripolar product ([BEGZ10, Proposition 1.4]) we can let $\varepsilon\to 0$ at the end of our argument to conclude the statement for pseudoeffective classes.

For each $t\in\Delta=\{t=(t_{1},...,t_{n})\in\mathbb{R}^{n}\ |\ t_{j}>0\}$ consider $u_{t}:=\sum_{j}t_{j}u_{j}$ , $v_{t}:=\sum_{j}t_{j}v_{j}$ and $\theta^{t}:=\sum_{j}t_{j}\theta^{j}$ . Clearly, $\{\theta^{t}\}$ is big, and $u_{t}$ has the same singularity type as $v_{t}$ . Hence it follows from Lemma 3.1 that $\int_{X}(\theta^{t}_{u_{t}})^{n}=\int_{X}(\theta^{t}_{v_{t}})^{n}$ for all $t\in\Delta$ . On the other hand, using multi-linearity of the non-pluripolar product again ([BEGZ10, Proposition 1.4]), we see that both $t\to\int_{X}(\theta^{t}_{u_{t}})^{n}$ and $t\to\int_{X}(\theta^{t}_{v_{t}})^{n}$ are homogeneous polynomials of degree $n$ . Our last identity forces all the coefficients of these polynomials to be equal, giving the statement of our result. ∎

Using the above result we argue the monotonicity of non-pluripolar products, conjectured in [BEGZ10] and proved in [DDL18a] (see [Vu21, Theorem 1.1] for a more general result):

Theorem 3.3.

Let $\theta^{j},j\in\{1,\ldots,n\}$ be smooth closed real $(1,1)$ -forms on $X$ whose cohomology classes are pseudoeffective. Let $u_{j},v_{j}\in\textup{PSH}(X,\theta^{j})$ be such that $u_{j}$ is less singular than $v_{j}$ for all $j\in\{1,\ldots,n\}$ . Then

\int_{X}\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}\geq\int_{X}\theta^{1}_{v_{1}}\wedge\ldots\wedge\theta^{n}_{v_{n}}.

Proof.

By the same reason as in Proposition 3.2, we can assume that the classes $\{\theta^{j}\}$ are in fact big. For each $t>0$ we set $v_{j}^{t}:=\max(u_{j}-t,v_{j})$ for $j=1,...,n$ . Observe that the $v_{j}^{t}$ converge decreasingly to $v_{j}$ as $t\to\infty$ . In particular, by [GZ05, Proposition 3.7] the convergence holds in capacity. As $v_{j}^{t}$ and $u_{j}$ have the same singularity type, it follows from Proposition 3.2 that

\int_{X}\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}=\int_{X}\theta^{1}_{v_{1}^{t}}\wedge\ldots\wedge\theta^{n}_{v_{n}^{t}}.

Letting $t\to\infty$ , the first part of Theorem 2.6 allows to conclude the argument. ∎

Remark 3.4.

Condition (2.4) in Theorem 2.6 is automatically satisfied if $u^{k}_{j}\nearrow u_{j}$ a.e. as $k\to\infty$ . Indeed, in this case $u_{j}^{k}\to u_{j}$ in capacity (see [GZ17, Proposition 4.25]), and by Theorem 3.3 we have $\int_{X}\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}\geq\limsup_{k}\int_{X}\theta^{1}_{u^{k}_{1}}\wedge\ldots\wedge\theta^{n}_{u^{k}_{n}}$ .

Since for all $u\in{\rm PSH}(X,\theta)$ we have $P(u,V_{\theta}+C)\nearrow P_{\theta}[u]$ as $C\to\infty$ , we conclude that

\int_{X}\theta_{u}^{n}=\int_{X}\theta_{P_{\theta}[u]}^{n}.

Similarly, for $u_{j}\in\textup{PSH}(X,\theta^{j})$ the same ideas allow to conclude that

\int_{X}\theta^{1}_{u_{1}}\wedge\ldots\wedge\theta^{n}_{u_{n}}=\int_{X}\theta^{1}_{P_{\theta}[u_{1}]}\wedge\ldots\wedge\theta^{n}_{P_{\theta}[u_{n}]}.

3.2 Model potentials and relative full mass classes

Rooftop envelopes revisited.

We survey some results from [DDL18a]. We first start with the following simple observation.

Lemma 3.5.

Assume $u,v\in{\rm PSH}(X,\theta)$ and $u$ is more singular than $v$ . Then

\sup_{X}(u-P_{\theta}[v])=\sup_{X}u.

Proof.

Since $P_{\theta}[v]\leq 0$ , we have $\sup_{X}(u-P_{\theta}[v])\geq\sup_{X}u$ . By the assumption that $u$ is more singular than $v$ , we have $u-\sup_{X}u\leq\min(v+C,0)$ for some constant $C$ , hence $u-\sup_{X}u\leq P_{\theta}[v]$ . This proves the other inequality, finishing the proof. ∎

We next prove the following result about the complex Monge–Ampère measure of $P_{\theta}(u,v)$ from [GLZ19, Lemma 4.1] (see [Dar17] for a particular result in the Kähler case).

Theorem 3.6.

Suppose $\varphi,\psi,P_{\theta}(\varphi,\psi)\in{\rm PSH}(X,\theta)$ . Then

\theta_{P_{\theta}(\varphi,\psi)}^{n}\leq{\bf 1}_{\{P_{\theta}(\varphi,\psi)=\varphi\}}\theta_{\varphi}^{n}+{\bf 1}_{\{P_{\theta}(\varphi,\psi)=\psi\}}\theta_{\psi}^{n}.

In particular,

\theta_{P_{\theta}[\psi](\varphi)}^{n}\leq{\bf 1}_{\{P_{\theta}[\psi](\varphi)=\varphi\}}\theta_{\varphi}^{n}\quad\text{and}\quad\theta_{P_{\theta}[\psi]}^{n}\leq{\bf 1}_{\{P_{\theta}[\psi]=0\}}\theta^{n}.

Proof.

Since the function $\min(\varphi,\psi)$ is quasi-continuous on $X$ , by Theorem 2.7, the Monge–Ampère measure $\theta_{P_{\theta}(\varphi,\psi)}^{n}$ is concentrated on the contact set $\{P_{\theta}(\varphi,\psi)=\min(\varphi,\psi)\}$ . By Lemma 2.9 we have

{\bf 1}_{\{P_{\theta}(\varphi,\psi)=\varphi\}}\theta_{P_{\theta}(\varphi,\psi)}^{n}\leq{\bf 1}_{\{P_{\theta}(\varphi,\psi)=\varphi\}}\theta_{\varphi}^{n}\quad\text{and}\quad{\bf 1}_{\{P_{\theta}(\varphi,\psi)=\psi\}}\theta_{P_{\theta}(\varphi,\psi)}^{n}\leq{\bf 1}_{\{P_{\theta}(\varphi,\psi)=\psi\}}\theta_{\psi}^{n}.

From this the first inequality follows. Using this, for each $t>0$ we have

	$\displaystyle\theta_{P_{\theta}(\psi+t,\varphi)}^{n}$	$\displaystyle\leq{\bf 1}_{\{P_{\theta}(\psi+t,\varphi)=\psi+t\}}\theta_{\psi}^{n}+{\bf 1}_{\{P_{\theta}(\psi+t,\varphi)=\varphi\}}\theta_{\varphi}^{n}$
		$\displaystyle\leq{\bf 1}_{\{\psi+t\leq\varphi\}}\theta_{\psi}^{n}+{\bf 1}_{\{P_{\theta}(\psi+t,\varphi)=\varphi\}}\theta_{\varphi}^{n}.$

Since $\theta_{\psi}^{n}$ vanishes on the pluripolar set $\{\psi=-\infty\}$ , we have $\lim_{t\to\infty}{\bf 1}_{\{\psi\leq\varphi-t\}}\theta_{\psi}^{n}=0$ . Observe also that $P_{\theta}(\psi+t,\varphi)\nearrow P_{\theta}[\psi](\varphi)$ as $t\nearrow\infty$ , hence Theorem 2.6 and Remark 3.4 ensure that $\theta_{P_{\theta}(\psi+t,\varphi)}^{n}$ converges weakly to $\theta_{P_{\theta}[\psi](\varphi)}^{n}$ . We also have that $\{P_{\theta}(\psi+t,\varphi)=\varphi\}\subset\{P_{\theta}[\psi](\varphi)=\varphi\}$ . Letting $t\to\infty$ thus gives

\theta_{P_{\theta}[\psi](\varphi)}^{n}\leq{\bf 1}_{\{P_{\theta}[\psi](\varphi)=\varphi\}}\theta_{\varphi}^{n}.

In particular, when $\varphi=V_{\theta}$ , this yields

\theta_{P_{\theta}[\psi]}^{n}=\theta_{P_{\theta}[\psi](V_{\theta})}^{n}\leq{\bf 1}_{\{P_{\theta}[\psi](V_{\theta})=V_{\theta}\}}\theta_{V_{\theta}}^{n}\leq{\bf 1}_{\{P_{\theta}[\psi]=0\}}\theta^{n},

where the last inequality follows from Theorem 2.10 since $\{P_{\theta}[\psi]=0\}\subset\{V_{\theta}=0\}$ . ∎

The domination principle and uniqueness.

Definition 3.7.

Given a potential $\phi\in{\rm PSH}(X,\theta)$ , the relative full mass class $\mathcal{E}(X,\theta,\phi)$ is the set of all $\theta$ -psh functions $u$ such that $u$ is more singular than $\phi$ and $\int_{X}\theta_{u}^{n}=\int_{X}\theta_{\phi}^{n}$ .

Remark 3.8.

If $u,v\in\mathcal{E}(X,\theta,\phi)$ then $\max(u,v)\in\mathcal{E}(X,\theta,\phi)$ . Indeed, since both $u$ and $v$ are more singular than $\phi$ , $\max(u,v)$ is also more singular than $\phi$ . By Theorem 3.3,

\int_{X}\theta_{u}^{n}\leq\int_{X}\theta_{\max(u,v)}^{n}\leq\int_{X}\theta_{\phi}^{n}=\int_{X}\theta_{u}^{n},

hence all the inequalities become equalities.

Definition 3.9.

A model potential is a $\theta$ -psh function $\phi$ such that $P_{\theta}[\phi]=\phi$ and $\int_{X}\theta_{\phi}^{n}>0$ .

Let us prove an important technical result from [DDL21a].

Theorem 3.10.

Assume that $u,v\in{\rm PSH}(X,\theta)$ , $u\leq v$ , $\int_{X}\theta_{u}^{n}>0$ and $b>1$ is such that

b^{n}\int_{X}\theta_{u}^{n}>(b^{n}-1)\int_{X}\theta_{v}^{n}.

(3.1)

Then $P_{\theta}(bu+(1-b)v)\in{\rm PSH}(X,\theta)$ .

To be clear, $P_{\theta}(bu+(1-b)v):=\textup{usc}(\sup\{h\in\textup{PSH}(X,\theta)\textup{ such that }h+(b-1)v\leq bu\})$ , and the set of candidates for this supremum is typically empty. Observe that, if $u$ and $v$ have the same positive mass then (3.1) is trivially satisfied.

Proof.

Set $\phi:=P_{\theta}[v]$ . It suffices to prove that $P_{\theta}(bu-(b-1)\phi)\in{\rm PSH}(X,\theta)$ since $v\leq P_{\theta}[v]$ , hence $bu+(1-b)v\geq bu+(1-b)P_{\theta}[v]$ . By Remark 3.4 we have $\int_{X}\theta_{v}^{n}=\int_{X}\theta_{\phi}^{n}$ .

For $j\in\mathbb{N}$ we set $u_{j}:=\max(u,\phi-j)$ and $\varphi_{j}:=P(bu_{j}+(1-b)\phi)$ . Observe that $\varphi_{j}$ is a decreasing sequence of $\theta$ -psh functions, having the same singularity type as $\phi$ . The proof is finished if we can show that $\varphi:=\lim_{j}\varphi_{j}$ is not identically $-\infty$ . Assume by contradiction that $\sup_{X}\varphi_{j}\to-\infty$ . Set $\psi_{j}:=b^{-1}\varphi_{j}+(1-b^{-1})\phi$ and $D_{j}:=\{b^{-1}\varphi_{j}+(1-b^{-1})\phi=u_{j}\}$ . Note that $\psi_{j}\leq u_{j}$ with equality on the contact set $D_{j}$ . Lemma 2.9 thus ensures that

{\bf 1}_{D_{j}}b^{-n}\theta_{\varphi_{j}}^{n}\leq{\bf 1}_{D_{j}}\theta_{\psi_{j}}^{n}\leq{\bf 1}_{D_{j}}\theta_{u_{j}}^{n}.

(3.2)

Also, observe that by Theorem 2.7, the Monge–Ampère measure $\theta_{\varphi_{j}}^{n}$ is supported on $\{\varphi_{j}=bu_{j}+(1-b)\phi\}=D_{j}$ .

Fix $j>k>0$ . We note that $u_{j}=u$ on $\{u>\phi-k\}$ and, since $u_{j}$ has the same singularity type as $\phi$ , by Lemma 3.1 and the plurifine locality we have

\int_{\{u\leq\phi-k\}}\theta_{u_{j}}^{n}=\int_{X}\theta_{u_{j}}^{n}-\int_{\{u>\phi-k\}}\theta_{u_{j}}^{n}=\int_{X}\theta_{\phi}^{n}-\int_{\{u>\phi-k\}}\theta_{u}^{n}.

(3.3)

Since $\{u_{j}\leq\phi-k\}=\{u\leq\phi-k\}$ , Theorem 2.7 and (3.2) we obtain

		$\displaystyle\theta_{\varphi_{j}}^{n}(\{\varphi_{j}\leq\phi-bk\})\leq b^{n}{\bf 1}_{D_{j}}\theta_{u_{j}}^{n}(\{\varphi_{j}\leq\phi-bk\})\leq b^{n}\theta_{u_{j}}^{n}(\{bu_{j}+(1-b)\phi\leq\phi-bk\})$
		$\displaystyle=b^{n}\theta_{u_{j}}^{n}(\{u_{j}\leq\phi-k\})\leq b^{n}\theta_{u_{j}}^{n}(\{u\leq\phi-k\})\leq b^{n}\bigg{(}\int_{X}\theta_{\phi}^{n}-\int_{\{u>\phi-k\}}\theta_{u}^{n}\bigg{)},$		(3.4)

where in the last inequality we have used (3.3). Since $\phi=P_{\theta}[v]$ , by Lemma 3.5 we have $\sup_{X}(\varphi_{j}-\phi)=\sup_{X}\varphi_{j}\to-\infty$ . From this we see that $\{\varphi_{j}\leq\phi-bk\}=X$ for $j\geq j_{0}$ large enough, $k$ being fixed. Thus,

\int_{X}\theta_{\phi}^{n}=\int_{X}\theta_{\varphi_{j_{0}}}^{n}\leq b^{n}\bigg{(}\int_{X}\theta_{\phi}^{n}-\int_{\{u>\phi-k\}}\theta_{u}^{n}\bigg{)}

where the first identity follows from the fact that $\varphi_{j_{0}}$ and $\phi$ have the same singularity type. Letting $j_{0}\to\infty$ in (3.2), and then $k\to\infty$ gives

\int_{X}\theta_{\phi}^{n}\leq b^{n}\left(\int_{X}\theta_{\phi}^{n}-\int_{X}\theta_{u}^{n}\right),

contradicting (3.1). Consequently, $\varphi_{j}$ decreases to a function in ${\rm PSH}(X,\theta)$ , finishing the proof. ∎

In some instances the conclusion of the above result can be strengthened:

Lemma 3.11.

Assume $u,v\in{\rm PSH}(X,\theta)$ and $u\leq v$ . Assume also that, for all $b>1$ , $P_{\theta}(bu-(b-1)v)\in{\rm PSH}(X,\theta)$ . Then, for all $b>1$ ,

\int_{X}(\theta+dd^{c}P_{\theta}(bu-(b-1)v))^{n}=\int_{X}\theta_{v}^{n}.

Proof.

Fix $t>b$ and observe that $bu-(b-1)v=t^{-1}b(tu-(t-1)v)+(1-t^{-1}b)v$ , where $t^{-1}b<1$ . Hence

\varphi:=P_{\theta}(bu-(b-1)v)\geq t^{-1}bP_{\theta}(tu-(t-1)v)+(1-t^{-1}b)v,

and by Lemma 2.12,

P_{\theta}[\varphi]\geq t^{-1}bP_{\theta}[P_{\theta}(tu-(t-1)v)]+(1-t^{-1}b)P_{\theta}[v].

Set $\psi_{t}:=P_{\theta}[P_{\theta}(tu-(t-1)v)]$ . Then $\sup_{X}\psi_{t}=0$ and $\psi_{t}\in{\rm PSH}(X,A\omega)$ for some fixed constant $A>0$ . It follows from [GZ05, Proposition 2.7] that the functions $\psi_{t}$ stay in a compact set of $L^{1}(X)$ . Letting $t\to\infty$ we thus have

P_{\theta}[\varphi]\geq P_{\theta}[v].

Since $u\leq v$ , we also have $\varphi\leq v$ . Combining this and the above inequality we see that $P_{\theta}[\varphi]=P_{\theta}[v]$ , and using Remark 3.4 we arrive at the result. ∎

Next we prove the domination principle of relative pluripotential theory, extending a result of Dinew from the Kähler case [BL12].

Theorem 3.12.

Assume $\phi$ is a model potential and $u,v\in\mathcal{E}(X,\theta,\phi)$ . Then

\theta_{u}^{n}(\{u<v\})=0\Longrightarrow u\geq v.

Proof.

Since $\max(u,v)\in\mathcal{E}(X,\theta,\phi)$ , we can assume that $u\leq v\leq 0$ . By Lemma 2.9 we have

\theta_{v}^{n}\geq{\bf 1}_{\{u=v\}}\theta_{v}^{n}\geq{\bf 1}_{\{u=v\}}\theta_{u}^{n}=\theta_{u}^{n},

where the last identity follows from the fact that $\theta_{u}^{n}(u<v)=0$ . Comparing the total mass we thus have $\theta_{u}^{n}=\theta_{v}^{n}$ and the measure is concentrated on $\{u=v\}$ .

Fix $b>1$ and set $\varphi_{b}:=P_{\theta}(bu-(b-1)v)$ . Since the masses of $u$ and $v$ are equal the assumption in Theorem 3.10 is trivially satisfied, hence $\varphi_{b}\in{\rm PSH}(X,\theta)$ . By Lemma 3.11 above, we have $\int_{X}\theta_{\varphi_{b}}^{n}=\int_{X}\theta_{\phi}^{n}>0$ .

Set $\psi_{b}:=b^{-1}\varphi_{b}+(1-b^{-1})v$ and $D_{b}:=\{b^{-1}\varphi_{b}+(1-b^{-1})v=u\}=\{\varphi_{b}=bu-(b-1)v\}$ . Note that $\psi_{b}\leq u$ with equality on the contact set $D_{b}$ . Lemma 2.9 and Theorem 2.7 thus ensure that

b^{-n}\theta_{\varphi_{b}}^{n}\leq b^{-n}\theta_{\varphi_{b}}^{n}+{\bf 1}_{D_{b}}(1-b^{-1})^{n}\theta_{v}^{n}={\bf 1}_{D_{b}}b^{-n}\theta_{\varphi_{b}}^{n}+{\bf 1}_{D_{b}}(1-b^{-1})^{n}\theta_{v}^{n}\leq{\bf 1}_{D_{b}}\theta_{\psi_{b}}^{n}\leq{\bf 1}_{D_{b}}\theta_{u}^{n}.

We then infer

b^{-n}\theta_{\varphi_{b}}^{n}\leq\theta_{u}^{n},

hence $\theta_{\varphi_{b}}^{n}$ is concentrated on $\{\varphi_{b}=bu-(b-1)v=u\}$ . Since $\varphi_{b}\leq u$ , it thus follows from Lemma 2.9 that

\theta_{\varphi_{b}}^{n}={\bf 1}_{\{\varphi_{b}=u\}}\theta_{\varphi_{b}}^{n}\leq\theta_{u}^{n}.

Comparing the total mass, we obtain $\theta_{\varphi_{b}}^{n}=\theta_{u}^{n}$ , hence

\int_{X}e^{\varphi_{b}}\theta_{u}^{n}=\int_{X}e^{\varphi_{b}}\theta_{\varphi_{b}}^{n}=\int_{X}e^{u}\theta_{u}^{n}>0.

Note also that, since $u\leq v$ , $\varphi_{b}$ is decreasing in $b$ . The above thus implies that $\varphi:=\lim_{b\to\infty}\varphi_{b}$ is not identically $-\infty$ . Moreover, for any $x\in\{u<v\}$ , we have $\varphi_{b}(x)\leq bu(x)-(b-1)v(x)\leq v(x)+b(u(x)-v(x))$ . Letting $b\to\infty$ , we see that $\varphi(x)=-\infty$ . This implies that $\{u<v\}$ is pluripolar, hence empty. It then follows that $u\geq v$ as desired. ∎

The following result states that the non-pluripolar Monge–Ampère measure determines the potential within a relative full mass class. This extends a result of Dinew from the Kähler case [Din09].

Theorem 3.13.

Assume $\phi$ is a model potential and $u,v\in\mathcal{E}(X,\theta,\phi)$ . Then

\theta_{u}^{n}=\theta_{v}^{n}\Longrightarrow u-v\;\text{is constant}.

Proof.

We normalize $u,v$ by $\sup_{X}u=\sup_{X}v=0$ . The goal is then to prove that $u=v$ . Then $\max(u,v)\in\mathcal{E}(X,\theta,\phi)$ and by Lemma 2.9, $\theta_{\max(u,v)}^{n}\geq\mu:=\theta_{u}^{n}$ . Comparing the total mass we see that $\theta_{\max(u,v)}^{n}=\mu$ . Thus, replacing $v$ with $\max(u,v)$ , we can assume that $u\leq v$ .

Fix $b>1$ and set $\varphi_{b}:=P_{\theta}(bu-(b-1)v)$ . Since the masses of $u$ and $v$ are equal, Theorem 3.10 ensures that $\varphi_{b}\in{\rm PSH}(X,\theta)$ . By Lemma 3.11, we have $\int_{X}\theta_{\varphi_{b}}^{n}=\int_{X}\theta_{\phi}^{n}>0$ . Since $\varphi_{b}\leq u$ , we infer that $\varphi_{b}\in\mathcal{E}(X,\theta,\phi)$ .

Set $\psi_{b}:=b^{-1}\varphi_{b}+(1-b^{-1})v$ and $D_{b}:=\{b^{-1}\varphi_{b}+(1-b^{-1})v=u\}$ . Note that $\psi_{b}\leq u$ with equality on the contact set $D_{b}$ . Lemma 2.9 and Theorem 2.7 thus ensure that

{\bf 1}_{D_{b}}\theta_{\psi_{b}}^{n}\leq{\bf 1}_{D_{b}}\theta_{u}^{n}={\bf 1}_{D_{b}}\mu,

hence

b^{-n}\theta_{\varphi_{b}}^{n}+{\bf 1}_{D_{b}}(1-b^{-1})^{n}\theta_{v}^{n}={\bf 1}_{D_{b}}b^{-n}\theta_{\varphi_{b}}^{n}+{\bf 1}_{D_{b}}(1-b^{-1})^{n}\theta_{v}^{n}\leq{\bf 1}_{D_{b}}\theta_{\psi_{b}}^{n}\leq{\bf 1}_{D_{b}}\theta_{u}^{n}.

It thus follows that $\theta_{\varphi_{b}}^{n}=f\mu$ for some $f\in L^{1}(\mu)$ whose support is contained in $D_{b}$ . The mixed Monge–Ampère inequalities, [BEGZ10, Proposition 1.11] yield

\theta_{\varphi_{b}}^{j}\wedge\theta_{v}^{n-j}\geq f^{j/n}\mu,

hence

$\displaystyle\theta_{\psi_{b}}^{n}$	$\displaystyle=\sum_{j=0}^{n}\binom{n}{j}(b^{-1}\theta_{\varphi_{b}})^{j}\wedge\left((1-b^{-1})\theta_{v}\right)^{n-j}$
	$\displaystyle\geq\sum_{j=0}^{n}\binom{n}{j}b^{-j}(1-b^{-1})^{n-j}f^{\frac{j}{n}}\mu$
	$\displaystyle=(b^{-1}f^{1/n}+1-b^{-1})^{n}\mu.$	(3.5)

Since ${\bf 1}_{D_{b}}\mu\geq{\bf 1}_{D_{b}}\theta_{\psi_{b}}^{n}$ , it follows from the above that $f^{1/n}\leq 1$ on $D_{b}$ . Since $f$ is supported in $D_{b}$ we get $f\leq 1$ , and so $\theta_{\varphi_{b}}^{n}\leq\mu$ . Comparing the total masses (noting that $\varphi_{b}\in\mathcal{E}(X,\theta,\phi)$ ) gives $f=1$ . Hence $\theta_{\varphi_{b}}^{n}=\mu$ , and from (3.2) we also get $\theta_{\psi_{b}}^{n}\geq\mu$ . Since $\psi_{b}\leq u$ , we then infer, via Theorem 3.3, that $\theta_{\varphi_{b}}^{n}=\theta_{\psi_{b}}^{n}=\mu$ is concentrated on $\{\varphi_{b}=bu-(b-1)v\}$ . Now, $\{\psi_{b}<u\}=\{\varphi_{b}<bu-(b-1)v\}$ , hence

\theta_{\psi_{b}}^{n}(\psi_{b}<u)=0.

Invoking the domination principle (Theorem 3.12) we obtain $\psi_{b}=u$ , hence $\varphi_{b}=bu-(b-1)v$ . Since $u\leq v\leq 0$ and $\sup_{X}u=0$ , it follows that there exists $x\in X$ with $u(x)=v(x)=0$ . We then infer $\sup_{X}\varphi_{b}=0$ , hence the functions $\varphi_{b}=v+b(u-v)$ decrease to some $\varphi\in{\rm PSH}(X,\theta)$ which is $-\infty$ in $\{u<v\}$ . It follows that $\{u<v\}$ is a pluripolar set, hence $u\geq v$ almost everywhere. Since $u,v$ are quasi-psh functions we can conclude that $u\geq v$ everywhere. ∎

Maximality of model potentials.

Based on our previous findings, one wonders if the following set of potentials has a maximal element:

F_{\phi}:=\bigg{\{}v\in{\rm PSH}(X,\theta)\;:\;\phi\leq v\leq 0\ \textrm{and}\ \int_{X}\theta_{v}^{n}=\int_{X}\theta_{\phi}^{n}\bigg{\}}.

In other words, does there exist a least singular potential that is less singular than $\phi$ but has the same full mass as $\phi$ . As shown in the following result, if $\int_{X}\theta_{\phi}^{n}>0$ , this is indeed the case, moreover this maximal potential is equal to $P_{\theta}[\phi]$ .

Theorem 3.14.

Assume that $\phi\in{\rm PSH}(X,\theta)$ and $\int_{X}\theta_{\phi}^{n}>0$ . Then

P_{\theta}[\phi]=\sup_{v\in F_{\phi}}v.

In particular, $P_{\theta}[\phi]=P_{\theta}[P_{\theta}[\phi]]$ .

As shown in Remark 3.4, $P_{\theta}[\phi]\in F_{\phi}$ , hence by Theorem 3.14, $P_{\theta}[\phi]$ is the maximal element of $F_{\phi}$ .

Proof.

Let $v\in F_{\phi}$ . By Theorem 3.6 we have

\displaystyle\theta_{P_{\theta}[\phi]}^{n}(\{P_{\theta}[\phi]<v\})

\displaystyle\leq{\bf 1}_{\{P_{\theta}[\phi]=0\}}\theta^{n}(\{P_{\theta}[\phi]<v\})\leq{\bf 1}_{\{P_{\theta}[\phi]=0\}}\theta^{n}(\{P_{\theta}[\phi]<0\})=0.

As $\int_{X}\theta_{\phi}^{n}=\int_{X}\theta_{v}^{n}$ , by Remark 3.4 we have

\int_{X}\theta^{n}_{P_{\theta}[\phi]}=\int_{X}\theta_{\phi}^{n}=\int_{X}\theta_{v}^{n}=\int_{X}\theta^{n}_{P_{\theta}[v]}>0.

Consequently, $P_{\theta}[\phi],v\in\mathcal{E}(X,\theta,P_{\theta}[v])$ and Theorem 3.12 now ensures that $P_{\theta}[\phi]\geq v$ , hence $P_{\theta}[\phi]\geq\sup_{v\in F_{\phi}}v$ . As $P_{\theta}[\phi]\in F_{\phi}$ , it follows that $P_{\theta}[\phi]=\sup_{v\in F_{\phi}}v$ .

For the last statement notice that $P_{\theta}[\phi]=\sup_{v\in F_{\phi}}v\geq\sup_{v\in F_{P_{\theta}[\phi]}}v=P_{\theta}[P_{\theta}[\phi]]$ , since $F_{\phi}\supset F_{P_{\theta}[\phi]}$ . The reverse inequality is trivial. ∎

As a consequence of this last result, we obtain the following characterization of membership in $\mathcal{E}(X,\theta,\phi)$ .

Theorem 3.15.

Suppose $\phi\in\textup{PSH}(X,\theta)$ with $\int_{X}\theta_{\phi}^{n}>0$ and $\phi\leq 0$ . The following are equivalent:
(i) $u\in\mathcal{E}(X,\theta,\phi)$ .
(ii) $\phi$ is less singular than $u$ , and $P_{\theta}[u](\phi)=\phi$ .
(iii) $\phi$ is less singular than $u$ , and $P_{\theta}[u]=P_{\theta}[\phi]$ .

As a consequence of the equivalence between (i) and (iii), we see that the potential $P_{\theta}[u]$ stays the same for all $u\in\mathcal{E}(X,\theta,\phi)$ , i.e., it is an invariant of this class. In particular, since $\mathcal{E}(X,\theta,\phi)\subset\mathcal{E}(X,\theta,P_{\theta}[\phi])$ , by the last statement of Theorem 3.14, it seems natural to only consider potentials $\phi$ that are in the image of the operator $\psi\to P_{\theta}[\psi]$ , when studying classes of relative full mass $\mathcal{E}(X,\theta,\phi)$ . What is more, in the next section it will be clear that considering such $\phi$ is not just more natural, but also necessary when trying to solve complex Monge–Ampère equations with prescribed singularity.

Proof.

Assume that (i) holds. By Theorem 3.6 it follows that $P_{\theta}[u](\phi)\geq\phi$ a.e. with respect to $\theta^{n}_{P_{\theta}[u](\phi)}$ . Theorem 3.12 gives $P_{\theta}[u](\phi)=\phi$ , hence (ii) holds.

Suppose (ii) holds. We can assume that $u\leq\phi\leq 0$ . Then $P_{\theta}[u]\geq P_{\theta}[u](\phi)=\phi$ . By the last statement of the previous theorem, this implies that

P_{\theta}[u]=P_{\theta}[P_{\theta}[u]]\geq P_{\theta}[\phi].

As the reverse inequality is trivial, (iii) follows.

Lastly, assume that (iii) holds. By Remark 3.4 it follows that

\int_{X}\theta_{u}^{n}=\int_{X}\theta_{P_{\theta}[u]}^{n}=\int_{X}\theta_{P_{\theta}[\phi]}^{n}=\int_{X}\theta_{\phi}^{n},

hence (i) holds. ∎

Corollary 3.16.

Suppose $\phi\in\textup{PSH}(X,\theta)$ such that $\int_{X}\theta_{\phi}^{n}>0$ . Then $\mathcal{E}(X,\theta,\phi)$ is convex. Moreover, given $\psi_{1},\ldots,\psi_{n}\in\mathcal{E}(X,\theta,\phi)$ we have

\int_{X}\theta_{\psi_{1}}^{s_{1}}\wedge\ldots\wedge\theta_{\psi_{n}}^{s_{n}}=\int_{X}\theta_{\phi}^{n},

(3.6)

where $s_{j}\geq 0$ are integers such that $\sum_{j=1}^{n}s_{j}=n$ .

Proof.

Let $u,v\in\mathcal{E}(X,\theta,\phi)$ and fix $t\in(0,1)$ . It follows from Theorem 3.15 that $P_{\theta}[v](\phi)=P_{\theta}[u](\phi)=\phi$ . By Lemma 2.12,

P_{\theta}[tv+(1-t)u](\phi)\geq tP_{\theta}[v](\phi)+(1-t)P_{\theta}[u](\phi)=\phi.

As the reverse inequality is trivial, another application of Theorem 3.15 gives $tv+(1-t)u\in\mathcal{E}(X,\theta,\phi)$ .

We prove the last statement. Since $\mathcal{E}(X,\theta,\phi)$ is convex, given $\psi_{1},\ldots,\psi_{n}\in\mathcal{E}(X,\theta,\phi)$ we know that any convex combination $\psi:=\sum_{j=1}^{n}s_{j}\psi_{j}$ with $0\leq s_{j}\leq 1$ and $\sum_{j}s_{j}=n$ , belongs to $\mathcal{E}(X,\theta,\phi)$ . Hence

\int_{X}\bigg{(}\sum_{j}s_{j}\theta_{\psi_{j}}\bigg{)}^{n}=\int_{X}\theta_{\psi}^{n}=\int_{X}\theta_{\phi}^{n}=\int_{X}\bigg{(}\sum_{j}s_{j}\theta_{\phi}\bigg{)}^{n}.

We have an identity of two homogeneous polynomials of degree $n$ . All the coefficients of these polynomials have to be equal, giving (3.6). ∎

Lemma 3.17.

Assume $\phi\in{\rm PSH}(X,\theta)$ is a model potential. If $b>1$ and $u,v\in\mathcal{E}(X,\theta,\phi)$ then $P_{\theta}(bu-(b-1)v)\in\mathcal{E}(X,\theta,\phi)$ .

Proof.

The proof is similar to that of [LN22, Corollary 3.20]. Since $u,v\in\mathcal{E}(X,\theta,\phi)$ , both $u$ and $v$ are more singular than $\phi$ , we can assume that $\max(u,v)\leq\phi$ . It follows from Theorem 3.10 (and the comment below it) that $P_{\theta}(bu-(b-1)\phi)\in{\rm PSH}(X,\theta)$ for all $b>1$ . Lemma 3.11 ensures that $P_{\theta}(bu-(b-1)\phi)\in\mathcal{E}(X,\theta,\phi)$ for all $b>1$ . Set $\psi:=P_{\theta}(bu-(b-1)v)$ and $\varphi:=P_{\theta}(bu-(b-1)\phi)$ . Then $\psi\geq\varphi$ and $\varphi\in\mathcal{E}(X,\theta,\phi)$ . We also have

b^{-1}\psi+(1-b^{-1})v\leq u,

which, by Lemma 2.12, gives

b^{-1}P_{\theta}[\psi]+(1-b^{-1})P_{\theta}[v]\leq P_{\theta}[b^{-1}\psi+(1-b^{-1})v]\leq P_{\theta}[u]\leq P_{\theta}[\phi]=\phi.

Since $P_{\theta}[v]=\phi$ , we can thus conclude that $P_{\theta}[\psi]\leq\phi$ . On the other hand $\phi=P_{\theta}[\varphi]\leq P_{\theta}[\psi]$ . Thus $P_{\theta}[\psi]=\phi$ and $\psi\in\mathcal{E}(X,\theta,\phi)$ . ∎

Lemma 3.18.

Assume that $u,v,w\in\textup{PSH}(X,\theta)$ are such that $\int_{X}\theta_{u}^{n}+\int_{X}\theta_{v}^{n}>\int_{X}\theta_{w}^{n}$ and $\max(u,v)\leq w$ . Then $P_{\theta}(u,v)\in{\rm PSH}(X,\theta)$ .

Proof.

We can assume without loss of generality that $u,v,w\leq 0$ . Replacing $w$ with $P_{\theta}[\varepsilon V_{\theta}+(1-\varepsilon)w]$ for small enough $\varepsilon>0$ we can also assume that $w$ is a model potential.

For $j\geq 0$ we set $u_{j}:=\max(u,w-j),v_{j}:=\max(v,w-j)$ , $h_{j}:=P(u_{j},v_{j})$ . Observe that $u_{j},v_{j},h_{j}$ have the same singularity type as $w$ . Fix $s>0$ big enough, such that for all $j>s$ , we have

\int_{\{u>w-s\}}\theta_{u_{j}}^{n}+\int_{\{v>w-s\}}\theta_{v_{j}}^{n}=\int_{\{u>w-s\}}\theta_{u}^{n}+\int_{\{v>w-s\}}\theta_{v}^{n}>\int_{X}\theta_{w}^{n},

where in the equality above we used Lemma 2.1. Observe that such $s$ exists thanks to the assumption.
It follows from Theorem 3.6 and the above estimate, that for $j>s$ ,

	$\displaystyle\int_{\{h_{j}\leq w-s\}}\theta_{h_{j}}^{n}$	$\displaystyle\leq\int_{\{u_{j}\leq w-s\}}\theta_{u_{j}}^{n}+\int_{\{v_{j}\leq w-s\}}\theta_{v_{j}}^{n}$
		$\displaystyle=2\int_{X}\theta_{w}^{n}-\int_{\{u>w-s\}}\theta_{u}^{n}-\int_{\{v>w-s\}}\theta_{v}^{n}<\int_{X}\theta_{w}^{n},$

where in the identity above we used the fact that $\{u_{j}\leq w-s\}=\{u\leq w-s\}$ and that $u_{j},v_{j}$ and $w$ have the same singularity type. Since $u_{j},v_{j}$ decrease to $u,v$ respectively, it follows that $h_{j}\searrow P(u,v)$ . We now rule out the possibility that $P(u,v)\equiv-\infty$ . Indeed, suppose $\sup_{X}h_{j}$ decreases to $-\infty$ . From Lemma 3.5 we obtain that $\sup_{X}h_{j}=\sup_{X}(h_{j}-w)\searrow-\infty$ . But then, for $j$ large enough the set $\{h_{j}\leq w-s\}$ coincides with $X$ , contradicting our last integral estimate, since each $h_{j}$ has the same singularity type as $w$ . ∎

Corollary 3.19.

If $\phi\in{\rm PSH}(X,\theta)$ be a model potential and $u,v\in\mathcal{E}(X,\theta,\phi)$ then $P_{\theta}(u,v)\in\mathcal{E}(X,\theta,\phi)$ .

Proof.

Fixing $b>1$ , from Lemma 3.17 we infer that $u_{b}:=P_{\theta}(bu-(b-1)\phi)$ and $v_{b}:=P_{\theta}(bv-(b-1)\phi)$ belong to $\mathcal{E}(X,\theta,\phi)$ . Lemma 3.18 then ensures that $P_{\theta}(u_{b},v_{b})\in{\rm PSH}(X,\theta)$ . We also have

P_{\theta}(u,v)\geq b^{-1}P_{\theta}(u_{b},v_{b})+(1-b^{-1})\phi.

Comparing the total mass via Theorem 3.3 and letting $b\to\infty$ , we obtain the result. ∎

Plainly speaking, by the next lemma, the fixed point set of the map $\psi\to P[\psi]$ is stable under the operation $(\psi,\phi)\to P(\psi,\phi)$ .

Lemma 3.20.

Suppose $u_{0},u_{1}\in\textup{PSH}(X,\theta)$ are such that $P(u_{0},u_{1})\in\textup{PSH}(X,\theta)$ , and $P[u_{0}]=u_{0}$ and $P[u_{1}]=u_{1}$ . Then $P[P(u_{0},u_{1})]=P(u_{0},u_{1}).$

Proof.

As $P(u_{0},u_{1})\leq\min(u_{0},u_{1})\leq 0$ and $P[P(u_{0},u_{1})]\leq\min(P[u_{0}],P[u_{1}])$ , it follows that

P(u_{0},u_{1})\leq P[P(u_{0},u_{1})]\leq P(P[u_{0}],P[u_{1}])=P(u_{0},u_{1}).

This shows that all the inequalities above are in fact equalities. ∎

Proposition 3.21.

Let $\phi,\psi\in\textup{PSH}(X,\theta)$ be such that $\phi=P[\phi]$ , $\psi=P[\psi]$ , and $P(\phi,\psi)\in{\rm PSH}(X,\theta)$ . If $u\in\mathcal{E}(X,\theta,\phi)$ , $v\in\mathcal{E}(X,\theta,\psi)$ and $\int_{X}\theta_{P(\phi,\psi)}^{n}>0$ then

P(u,v)\in\mathcal{E}(X,\theta,P(\phi,\psi)).

Proof.

We can assume that $u\leq\phi$ and $v\leq\psi$ .

Step 1. We first prove that $P(u,\psi)\in\mathcal{E}(X,\theta,P(\phi,\psi))$ . By assumption we have

\int_{X}\theta_{u}^{n}+\int_{X}\theta_{P(\phi,\psi)}^{n}>\int_{X}\theta_{\phi}^{n},

and Lemma 3.18 gives $P(u,\psi)=P(u,P(\phi,\psi))\in{\rm PSH}(X,\theta)$ . Also, it follows from Theorem 3.10 that $u_{b}:=P_{\theta}(bu-(b-1)\phi)\in{\rm PSH}(X,\theta)$ for all $b>1$ . By definition, for $1<b<t$ we have

\phi\geq u_{b}\geq bt^{-1}u_{t}+(1-bt^{-1})\phi.

Comparing the total mass via Theorem 3.3 and letting $t\to\infty$ we see that $u_{b}\in\mathcal{E}(X,\theta,\phi)$ . As above, Lemma 3.18 ensures that $P(u_{b},\psi)\in{\rm PSH}(X,\theta)$ . On the other hand we also have

\phi\geq u\geq b^{-1}u_{b}+(1-b^{-1})\phi,

therefore $P(\phi,\psi)\geq P(u,\psi)\geq b^{-1}P(u_{b},\psi)+(1-b^{-1})P(\phi,\psi)$ . Comparing the total mass via Theorem 3.3 and letting $b\to\infty$ we arrive at $\int_{X}\theta_{P(\phi,\psi)}^{n}\geq\int_{X}\theta_{P(u,\psi)}^{n}\geq\int_{X}\theta_{P(\phi,\psi)}^{n}$ , hence the conclusion.

Step 2. We prove that $P(u,v)\in{\rm PSH}(X,\theta)$ . It follows from Theorem 3.3, the assumption $v\in\mathcal{E}(X,\theta,\psi)$ , and the first step that

\int_{X}\theta_{P(u,\psi)}^{n}+\int_{X}\theta_{v}^{n}=\int_{X}\theta_{P(\phi,\psi)}^{n}+\int_{X}\theta_{\psi}^{n}>\int_{X}\theta_{\psi}^{n}.

Since $\max(P(u,\psi),v)\leq\psi$ , Lemma 3.18 can be applied giving $P(u,v)=P(P(u,\psi),v)\in{\rm PSH}(X,\theta)$ .

Step 3. We conclude the proof. It follows from Theorem 3.10 that $v_{b}:=P_{\theta}(bv-(b-1)\psi)\in{\rm PSH}(X,\theta)$ for all $b>1$ . For $1<b<t$ we have

\psi\geq v_{b}\geq bt^{-1}v_{t}+(1-bt^{-1})\psi.

Comparing the total mass via Theorem 3.3 and letting $t\to\infty$ we see that $v_{b}\in\mathcal{E}(X,\theta,\psi)$ . By the second step we have that $P(u,v_{b})\in{\rm PSH}(X,\theta)$ . On the other hand we also have

\psi\geq v\geq b^{-1}v_{b}+(1-b^{-1})\psi,

therefore $P(u,\psi)\geq P(u,v)\geq b^{-1}P(u,v_{b})+(1-b^{-1})P(u,\psi)$ . Comparing the total mass via Theorem 3.3 and letting $b\to\infty$ we arrive at $\int_{X}\theta_{P(u,\psi)}^{n}\geq\int_{X}\theta_{P(u,v)}^{n}\geq\int_{X}\theta_{P(u,\psi)}^{n}$ . Combining this and the first step we arrive at the conclusion. ∎

Comparison principle.

We note the partial comparison principle for functions s of relative full mass, generalizing a result of Dinew from [Din09]:

Proposition 3.22.

Suppose $\psi_{k}\in\textup{PSH}(X,\theta^{k}),k=1,\ldots,j\leq n$ and $\phi\in{\rm PSH}(X,\theta)$ is a model potential. If $u,v\in\mathcal{E}(X,\theta,\phi)$ then

\int_{\{u<v\}}\theta_{v}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}\leq\int_{\{u<v\}}\theta_{u}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}.

Proof.

The proof follows the argument of [BEGZ10, Proposition 2.2] with a vital ingredient from Theorem 3.3.

Since $\max(u,v)\in\mathcal{E}(X,\theta,\phi)$ , we have, by Theorem 3.15, $P_{\theta}[\max(u,v)]=P_{\theta}[u]=P_{\theta}[v]=\phi$ . It thus follows from Theorem 3.3 and Remark 3.4 that

$\displaystyle\int_{X}\theta_{\phi}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}$	$\displaystyle=$	$\displaystyle\int_{X}\theta_{v}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}$
	$\displaystyle\leq$	$\displaystyle\int_{X}\theta_{\max(u,v)}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}$
	$\displaystyle=$	$\displaystyle\int_{X}\theta_{\phi}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}.$

Hence the inequality above is in fact equality. By locality of the non-pluripolar product we then can write:

$\displaystyle\int_{X}\theta_{\max(u,v)}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}$	$\displaystyle\geq$	$\displaystyle\int_{\{u>v\}}\theta_{u}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}+\int_{\{v>u\}}\theta_{v}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}$
	$\displaystyle=$	$\displaystyle\int_{X}\theta_{u}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}-\int_{\{u\leq v\}}\theta_{u}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}$
		$\displaystyle+\int_{\{v>u\}}\theta_{v}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}$
	$\displaystyle=$	$\displaystyle\int_{X}\theta_{\max(u,v)}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}-\int_{\{u\leq v\}}\theta_{u}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}$
		$\displaystyle+\int_{\{v>u\}}\theta_{v}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge...\wedge\theta^{j}_{\psi_{j}}.$

We thus get

\displaystyle\int_{\{u<v\}}\theta_{v}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}\leq\int_{\{u\leq v\}}\theta_{u}^{n-j}\wedge\theta^{1}_{\psi_{1}}\wedge\ldots\wedge\theta^{j}_{\psi_{j}}.

Replacing $u$ with $u+\varepsilon$ in the above inequality, and letting $\varepsilon\searrow 0$ , by the monotone convergence theorem we arrive at the result. ∎

The above result yields the following important consequence, generalizing [BEGZ10, Corollary 2.3].:

Corollary 3.23.

Suppose $\phi\in\textup{PSH}(X,\theta)$ is a model potential and assume that $u,v\in\mathcal{E}(X,\theta,\phi)$ . Then

\int_{\{u<v\}}\theta_{v}^{n}\leq\int_{\{u<v\}}\theta_{u}^{n}\quad\textup{ and }\quad\int_{\{u\leq v\}}\theta_{v}^{n}\leq\int_{\{u\leq v\}}\theta_{u}^{n}.

The second inequality follows from the first inequality applied to $u$ and $v+\varepsilon$ and $\varepsilon\searrow 0$ .

Chapter 4 Generalized Monge–Ampère capacities and integration by parts

4.1 Generalized Monge–Ampère capacities

In this section we prove a comparison of Monge–Ampère capacities which will be used in the proof of the integration by parts formula in the next section. We first start with a version of the Chern-Levine-Nirenberg inequality.

Lemma 4.1.

Let $u,v,\psi\in{\rm PSH}(X,\omega)$ . Assume that $v\leq u\leq v+B$ for some positive constant $B$ . Then

\int_{X}\psi\omega_{u}^{n}\geq\int_{X}\psi\omega_{v}^{n}-nB\int_{X}\omega^{n}.

Proof.

By the monotone convergence theorem we can assume that $\psi$ is bounded. By subtracting a constant we can assume that $u\leq 0$ . We first prove the lemma under the assumption that $u=v$ on the open set

U:=\{\min(u,v)=v<-C\},

for some positive constant $C$ .
We approximate $u$ and $v$ by $u^{t}:=\max(u,-t)$ and $v^{t}:=\max(v,-t)$ . For $t>0$ we apply the integration by parts formula for bounded $\omega$ -psh functions, which is a consequence of Stokes theorem, to get

\int_{X}\psi(\omega_{u^{t}}^{n}-\omega_{v^{t}}^{n})=\int_{X}(u^{t}-v^{t})dd^{c}\psi\wedge S^{t},

where $S^{t}:=\sum_{k=0}^{n-1}\omega_{u^{t}}^{k}\wedge\omega_{v^{t}}^{n-1-k}$ . Note that $\int_{X}\omega\wedge S^{t}=n\int_{X}\omega^{n}$ as there are $n$ terms in the sum and each of them is equal to $\int_{X}\omega^{n}$ by Stokes’ theorem. Since $u^{t}\geq v^{t}$ we can continue the above estimate and obtain

\int_{X}\psi(\omega_{u^{t}}^{n}-\omega_{v^{t}}^{n})=\int_{X}(u^{t}-v^{t})(\omega_{\psi}\wedge S^{t}-\omega\wedge S^{t})\geq-Bn\int_{X}\omega^{n}.

For $t>B+C$ we have that $u^{t}=v^{t}$ on the open set $U$ which contains $\{u\leq-t\}=\{v\leq-t\}$ . It thus follows that ${\bf 1}_{U}\omega_{u^{t}}^{n}={\bf 1}_{U}\omega_{v^{t}}^{n}$ . Thus, for $t>B+C$ we have

\displaystyle\int_{\{v>-t\}}\psi(\omega_{u}^{n}-\omega_{v}^{n})=\int_{\{v>-t\}}\psi(\omega_{u_{t}}^{n}-\omega_{v_{t}}^{n})

\displaystyle=

\displaystyle\int_{X}\psi(\omega_{u^{t}}^{n}-\omega_{v^{t}}^{n})\geq-Bn\int_{X}\omega^{n}.

Letting $t\to\infty$ we finish the first step.

We now treat the general case. By approximating $\psi$ from above by smooth $\omega$ -psh functions, see [Dem94], [BK07], we can assume that $\psi$ is smooth (in fact, we only need the continuity of $\psi$ ). We fix $a\in(0,1)$ and set $v_{a}:=av$ , $u_{a}:=\max(u,v_{a})$ . Setting $C:=a(1-a)^{-1}B$ we have that $u_{a}=v_{a}$ on $U=\{\min(u_{a},v_{a})=v_{a}<-C\}$ (see arguments in Step 1 of the proof of Lemma 3.1). We can thus apply the first step to get

\int_{X}\psi\omega_{u_{a}}^{n}\geq\int_{X}\psi\omega_{v_{a}}^{n}-nB\int_{X}\omega^{n}.

Observe that $v_{a}\searrow v$ and $u_{a}\searrow u$ as $a\nearrow 1$ . Also, by the multilinearity of non-pluripolar products, we have

\lim_{a\to 1^{-}}\int_{X}(\omega+dd^{c}v_{a})^{n}=\lim_{a\to 1^{-}}\int_{X}((1-a)\omega+a\omega_{v})^{n}=\int_{X}(\omega+dd^{c}v)^{n}.

Recalling that $v\leq u\leq 0$ , we have $u\leq u_{a}=\max(u,av)\leq\max(u,au)=au$ . By Theorem 3.3 we thus have

\int_{X}\omega_{u}^{n}\leq\int_{X}\omega_{u_{a}}^{n}\leq\int_{X}(\omega+add^{c}u)^{n}.

Using the multilinearity of non-pluripolar products, we then have

\int_{X}(\omega+dd^{c}u)^{n}\leq\lim_{a\to 1^{-}}\int_{X}(\omega+dd^{c}u_{a})^{n}\leq\lim_{a\to 1^{-}}\int_{X}(\omega+add^{c}u)^{n}=\int_{X}(\omega+dd^{c}u)^{n}.

Hence the above inequalities are equalities. It then follows from Theorem 2.6 that the positive measures $\omega_{u_{a}}^{n},\omega_{v_{a}}^{n}$ converge respectively to $\omega_{u}^{n},\omega_{v}^{n}$ in the weak sense of Radon measures as $a\nearrow 1$ . Since $\psi$ is continuous on $X$ we thus obtain

\int_{X}\psi\omega_{u}^{n}\geq\int_{X}\psi\omega_{v}^{n}-nB\int_{X}\omega^{n}.

∎

We next use the Chern-Levine-Nirenberg inequality to compare Monge–Ampère capacities.

Definition 4.2.

Given $\phi\in{\rm PSH}(X,\theta)$ and $E\subset X$ a Borel subset we define

{\rm Cap}_{\theta,\phi}(E):=\sup\left\{\int_{E}\theta_{u}^{n}\;:\;u\in{\rm PSH}(X,\theta),\;\phi-1\leq u\leq\phi\right\}.

Note that in the Kähler case a related notion of capacity has been studied in [DL17, DL15]. In the case when $\phi=V_{\theta}$ we recover the Monge–Ampère capacity used in [BEGZ10, Section 4.1].

Lemma 4.3.

The relative Monge–Ampère capacity ${\rm Cap}_{\theta,\phi}$ is inner regular, i.e.

{\rm Cap}_{\theta,\phi}(E)=\sup\{{\rm Cap}_{\theta,\phi}(K)\;:\;K\subset E,\;K\;\textrm{is compact}\}.

Proof.

By definition ${\rm Cap}_{\theta,\phi}(E)\geq{\rm Cap}_{\theta,\phi}(K)$ for any compact set $K\subset E$ . Fix $\varepsilon>0$ . There exists $u\in{\rm PSH}(X,\theta)$ such that $\phi-1\leq u\leq\phi$ and

\int_{E}\theta_{u}^{n}\geq{\rm Cap}_{\theta,\phi}(E)-\varepsilon.

Since $\theta_{u}^{n}$ is an inner regular Borel measure it follows that there exists a compact set $K\subset E$ such that $\int_{K}\theta_{u}^{n}\geq\int_{E}\theta_{u}^{n}-\varepsilon\geq{\rm Cap}_{\theta,\phi}(E)-2\varepsilon$ . Hence ${\rm Cap}_{\theta,\phi}(K)\geq{\rm Cap}_{\theta,\phi}(E)-2\varepsilon$ . Taking the supremum over all the compact set $K\subset E$ , we arrive at the conclusion. ∎

Proposition 4.4.

Assume $\phi\in{\rm PSH}(X,\omega)$ is a model potential. There exists a constant $C>0$ depending on $X,\omega,n$ such that, for all Borel set $E$ , we have

{\rm Cap}_{\omega,\phi}(E)\leq C{\rm Cap}_{\omega}(E)^{1/n}.

Proof.

By inner regularity of the capacities, we can assume that $E=K$ is compact. We can also assume that $K$ is non-pluripolar, otherwise the inequality is trivial. Let $V_{K}^{*}$ be the global extremal function which it is defined as

V_{K}:=\sup\{u\in{\rm PSH}(X,\omega),\;\;u\leq 0\;{\rm on}\;K\}.

We recall that $V_{K}^{*}\geq 0$ and that let $M_{K}:=\sup_{X}V_{K}^{*}<\infty$ if and only if $K$ is non-pluripolar. By [GZ05, Theorem 5.2], $\omega_{V_{K}^{*}}^{n}$ is concentrated on $K$ . If $M_{K}<1$ then

\int_{X}\omega^{n}=\int_{X}\omega_{V_{K}^{*}}^{n}=\int_{K}\omega_{V_{K}^{*}}^{n}=\int_{K}\omega_{V_{K^{*}}-1}^{n}\leq{\rm Cap}_{\omega}(K),

while ${\rm Cap}_{\omega,\phi}(K)\leq\int_{X}\omega^{n}$ . Hence for $C\geq(\int_{X}\omega^{n})^{1-1/n}$ , the desired inequality holds.

Assume now that $M_{K}\geq 1$ . Then $v:=M_{K}^{-1}V_{K}^{*}-1$ is $\omega$ -psh and takes values in $[-1,0]$ , hence

M_{K}^{-n}\int_{X}\omega^{n}=M_{K}^{-n}\int_{X}\omega_{V_{K}^{*}}^{n}=M_{K}^{-n}\int_{K}\omega_{V_{K}^{*}}^{n}\leq\int_{K}\omega_{v}^{n}\leq{\rm Cap}_{\omega}(K).

Set $\psi:=V_{K}^{*}-M_{K}$ and let $u$ be a $\omega$ -psh function such that $\phi-1\leq u\leq\phi$ . Since $\psi=-M_{K}$ on $K$ modulo a pluripolar set, it follows from Lemma 4.1 that

M_{K}\int_{K}\omega_{u}^{n}\leq\int_{X}(-\psi)\omega_{u}^{n}\leq\int_{X}(-\psi)\omega_{\phi}^{n}+n\int_{X}\omega^{n}\leq\int_{X}(-\psi)\omega^{n}+n\int_{X}\omega^{n}\leq A,

for a constant $A>0$ depending on $X,\omega,n$ . We have used above the fact that, $\omega_{\phi}^{n}\leq{\bf 1}_{\{\phi=0\}}\omega^{n}$ since $\phi$ is a model potential (see Theorem 3.6), hence $\int_{X}(-\psi)\omega_{\phi}^{n}\leq\int_{X}(-\psi)\omega^{n}$ . Taking the supremum over all such $u$ yields

{\rm Cap}_{\omega,\phi}(K)\leq AM_{K}^{-1}\leq C{\rm Cap}_{\omega}(K)^{1/n},

where the last inequality follows from [GZ05, Theorem 7.1]. ∎

Lemma 4.5.

Fix $\varphi,\psi\in{\rm PSH(X,\theta)}$ such that $\psi\leq\varphi$ and $\int_{X}\theta_{\varphi}^{n}=\int_{X}\theta_{\psi}^{n}$ . Then there exists a continuous function $g:[0,\infty)\rightarrow[0,\infty)$ with $g(0)=0$ such that, for all Borel sets $E$ ,

{\rm Cap}_{\theta,\psi}(E)\leq g\left({\rm Cap}_{\theta,\varphi}(E)\right).

Our proof uses an idea from [GLZ19].

Proof.

We can assume that $\varphi\leq 0$ . We claim that if $v\in{\rm PSH}(X,\theta)$ with $\varphi-t\leq v\leq\varphi$ (for $t\geq 0$ ) then for any Borel set $E$ we have

\int_{E}\theta_{v}^{n}\leq\max(t,1)^{n}{\rm Cap}_{\theta,\varphi}(E).

If $t\in[0,1]$ then $v$ is a candidate defining the capacity ${\rm Cap}_{\theta,\varphi}$ , hence the desired inequality holds. For $t>1$ , the function $v_{t}:=t^{-1}v+(1-t^{-1})\varphi$ is $\theta$ -psh and $\varphi-1\leq v_{t}\leq\varphi$ . Since non-pluripolar products are multilinear, we thus have

t^{-n}\int_{E}\theta_{v}^{n}\leq\int_{E}\theta_{v_{t}}^{n}\leq{\rm Cap}_{\theta,\varphi}(E),

yielding the claim.

Let $u$ be a $\theta$ -psh function such that $\psi-1\leq u\leq\psi$ . Fix $t>1$ and set $u_{t}:=\max(u,\varphi-2t)$ , $E_{t}:=E\cap\{u>\varphi-2t\}$ , $F_{t}:=E\cap\{u\leq\varphi-2t\}$ . Observe that $\varphi-2t\leq u_{t}\leq\varphi$ . By plurifine locality and the claim we have that

\int_{E_{t}}\theta_{u}^{n}=\int_{E_{t}}\theta_{u_{t}}^{n}\leq(2t)^{n}{\rm Cap}_{\theta,\varphi}(E_{t})\leq(2t)^{n}{\rm Cap}_{\theta,\varphi}(E).

On the other hand, using the inclusions

F_{t}\subset\left\{\psi-1\leq\frac{u+\varphi}{2}-t\right\}\subset\{\psi-1\leq\varphi-t\}\subset\{\psi-1\leq-t\}

and the comparison principle, Corollary 3.23, we infer

	$\displaystyle\int_{F_{t}}\theta_{u}^{n}$	$\displaystyle\leq\int_{\{\psi-1\leq\frac{u+\varphi}{2}-t\}}\theta_{u}^{n}\leq 2^{n}\int_{\{\psi-1\leq\frac{u+\varphi}{2}-t\}}\theta_{\frac{u+\varphi}{2}}^{n}$
		$\displaystyle\leq 2^{n}\int_{\{\psi\leq\varphi-t+1\}}\theta_{\psi}^{n}\leq 2^{n}\int_{\{\psi\leq-t+1\}}\theta_{\psi}^{n}.$

Taking the supremum over all candidates $u$ we obtain

{\rm Cap}_{\theta,\psi}(E)\leq(2t)^{n}{\rm Cap}_{\theta,\varphi}(E)+2^{n}\int_{\{\psi\leq-t+1\}}\theta_{\psi}^{n}.

Set $t:=({\rm Cap}_{\theta,\varphi}(E))^{-1/2n}$ . If $t>1$ we get ${\rm Cap}_{\theta,\psi}(E)\leq g\left({\rm Cap}_{\theta,\varphi}(E)\right)$ , where $g$ is defined on $[0,\infty)$ by

g(s):=(2^{n}+{\rm Vol}(\{\theta\}))s^{1/2}+2^{n}\int_{\{\psi\leq-s^{-1/(2n)}+1\}}\theta_{\psi}^{n}.

Observe that $g(0)=0$ since $\theta_{\psi}^{n}$ does not charge the pluripolar set $\{\psi=-\infty\}$ .

If $t\leq 1$ then $s:={\rm Cap}_{\theta,\varphi}(E)\geq 1$ , and by the choice of $g$ above we have

{\rm Cap}_{\theta,\psi}(E)\leq{\rm Vol}(\{\theta\})\leq g\left({\rm Cap}_{\theta,\varphi}(E)\right),

finishing the proof. ∎

Theorem 4.6.

Assume that $\psi\in{\rm PSH}(X,\theta)$ . Then there exists a continuous function $f:[0,\infty)\rightarrow[0,\infty)$ with $f(0)=0$ such that, for any Borel set $E$ ,

{\rm Cap}_{\theta,\psi}(E)\leq f\left({\rm Cap}_{\omega}(E)\right).

Proof.

We can assume that $\theta\leq\omega$ . Since ${\rm PSH}(X,\theta)\subseteq{\rm PSH}(X,\omega)$ , for any Borel set $E$ we have ${\rm Cap}_{\theta,\psi}(E)\leq{\rm Cap}_{\omega,\psi}(E)$ . Also, by Proposition 4.4 and Lemma 4.5 we get

{\rm Cap}_{\theta,\psi}(E)\leq{\rm Cap}_{\omega,\psi}(E)\leq g\left({\rm Cap}_{\omega,P_{\omega}[\psi]}(E)\right)\leq g\left(C{\rm Cap}_{\omega}(E)^{1/n}\right).

∎

4.2 Integration by parts

The integration by parts formula was recently obtained [Xia19] using Witt Nyström’s construction. We give an argument here following [Lu21] (see [Vu21] for a more general result). We first start with the following key lemma.

Lemma 4.7.

Let $\varphi_{1},\varphi_{2},\psi_{1},\psi_{2}\in{\rm PSH}(X,\theta)$ be such that $[\varphi_{1}]=[\varphi_{2}]$ and $[\psi_{1}]=[\psi_{2}]$ . Then

\int_{X}(\varphi_{1}-\varphi_{2})\left(\theta_{\psi_{1}}^{n}-\theta_{\psi_{2}}^{n}\right)=\int_{X}(\psi_{1}-\psi_{2})(S_{1}-S_{2}),

where $S_{j}:=\sum_{k=0}^{n-1}\theta_{\varphi_{j}}\wedge\theta_{\psi_{1}}^{k}\wedge\theta_{\psi_{2}}^{n-k-1}$ , $j=1,2$ .

Proof.

It follows from Proposition 3.2 that

\int_{X}(\theta_{\psi_{1}}^{n}-\theta_{\psi_{2}}^{n})=\int_{X}(S_{1}-S_{2})=0.

By adding a constant we can assume that $\varphi_{1},\varphi_{2},\psi_{1},\psi_{2}$ are negative and that $\varphi_{1}\leq\varphi_{2}$ and $\psi_{1}\leq\psi_{2}$ . Let $B>0$ be a constant such that

\varphi_{2}\leq\varphi_{1}+B,\quad\psi_{2}\leq\psi_{1}+B.

Step 1. We assume $\theta$ is Kähler and $\psi_{1},\psi_{2},\varphi_{1},\varphi_{2}$ are $\lambda\theta$ -psh for some $\lambda\in(0,1)$ . Observe that the last condition ensures that their Monge–Ampère mass is strictly positive. Step 1.1. We also assume that there exists $C>0$ such that $\psi_{1}=\psi_{2}$ on $U:=\{\min(\psi_{1},\psi_{2})<-C\}$ and $\varphi_{1}=\varphi_{2}$ on $V:=\{\min(\varphi_{1},\varphi_{2})<-C\}$ .

For a function $u\in{\rm PSH}(X,\theta)$ we consider its canonical approximant $u^{t}:=\max(u,-t)$ , $t>0$ . It follows from Stokes’ theorem that

\int_{X}(\varphi_{1}^{t}-\varphi_{2}^{t})\left(\theta_{\psi_{1}^{t}}^{n}-\theta_{\psi_{2}^{t}}^{n}\right)=\int_{X}(\psi_{1}^{t}-\psi_{2}^{t})(S_{1}^{t}-S_{2}^{t}),

where $S_{j}^{t}:=\sum_{k=0}^{n-1}\theta_{\varphi_{j}^{t}}\wedge\theta_{\psi_{1}^{t}}^{k}\wedge\theta_{\psi_{2}^{t}}^{n-k-1}$ , $j=1,2$ . We now consider the limit as $t\to\infty$ in the above equality.

Fix $t>C$ . By assumption we have $\varphi_{1}^{t}=\varphi_{2}^{t}$ on $V$ and $\{\varphi_{1}\leq-t\}=\{\varphi_{2}\leq-t\}\subset V$ . The same properties for $\psi_{1},\psi_{2}$ also hold: $\psi_{1}^{t}=\psi_{2}^{t}$ in the open set $U$ and $\{\psi_{1}\leq-t\}=\{\psi_{2}\leq-t\}\subset U$ . It thus follows that

{\bf 1}_{U}\theta_{\psi_{1}^{t}}^{n}={\bf 1}_{U}\theta_{\psi_{2}^{t}}^{n}\quad\text{and}\quad{\bf 1}_{V}S_{1}^{t}={\bf 1}_{V}S_{2}^{t},

hence multiplying with the characteristic functions ${\bf 1}_{\{\psi_{1}\leq-t\}}$ and ${\bf 1}_{\{\varphi_{1}\leq-t\}}$ respectively gives

{\bf 1}_{\{\psi_{1}\leq-t\}}\theta_{\psi_{1}^{t}}^{n}={\bf 1}_{\{\psi_{1}\leq-t\}}\theta_{\psi_{2}^{t}}^{n}\quad\text{and}\quad{\bf 1}_{\{\varphi_{1}\leq-t\}}S_{1}^{t}={\bf 1}_{\{\varphi_{1}\leq-t\}}S_{2}^{t}.

By plurifine locality of the non-pluripolar product we thus have

	$\displaystyle\int_{X}(\varphi_{1}^{t}-\varphi_{2}^{t})\left(\theta_{\psi_{1}^{t}}^{n}-\theta_{\psi_{2}^{t}}^{n}\right)$	$\displaystyle=\int_{\{\psi_{1}>-t\}\cap\{\varphi_{1}>-t\}}(\varphi_{1}^{t}-\varphi_{2}^{t})(\theta_{\psi_{1}^{t}}^{n}-\theta_{\psi_{2}^{t}}^{n})$
		$\displaystyle=\int_{\{\psi_{1}>-t\}\cap\{\varphi_{1}>-t\}}(\varphi_{1}-\varphi_{2})(\theta_{\psi_{1}}^{n}-\theta_{\psi_{2}}^{n}),$

and

\int_{X}(\psi_{1}^{t}-\psi_{2}^{t})(S_{1}^{t}-S_{2}^{t})=\int_{\{\psi_{1}>-t\}\cap\{\varphi_{1}>-t\}}(\psi_{1}-\psi_{2})(S_{1}-S_{2}).

Since $\varphi_{1}-\varphi_{2}$ and $\psi_{1}-\psi_{2}$ are bounded, using the dominated convergence theorem we finish Step 1.1.

Step 1.2. We remove the assumption made in Step 1.1.
Recall that $\psi_{1},\psi_{2},\varphi_{1},\varphi_{2}$ are $\lambda\theta$ -psh for some $\lambda\in(0,1)$ . For each $\varepsilon\in(0,1-\lambda)$ we define

\psi_{2,\varepsilon}:=\max(\psi_{1},(1+\varepsilon)\psi_{2})\ ;\ \varphi_{2,\varepsilon}:=\max(\varphi_{1},(1+\varepsilon)\varphi_{2}).

Since $\theta$ is assumed to be Kähler, and $\varepsilon\in(0,1-\lambda)$ , the functions $(1+\varepsilon)\psi_{2}$ and $(1+\varepsilon)\varphi_{2}$ are still $\theta$ -psh. Also, $\psi_{1}\leq\psi_{2,\varepsilon}\leq(1+\varepsilon)(\psi_{1}+B)$ and $\varphi_{1}\leq\varphi_{2,\varepsilon}\leq(1+\varepsilon)(\varphi_{1}+B)$ . These are $\theta$ -psh functions satisfying the assumptions in Step 1.1 with $C=B+B\varepsilon^{-1}$ . Indeed, if $\varphi_{1}(x)<-C$ then

(1+\varepsilon)\varphi_{2}(x)=\varphi_{2}(x)+\varepsilon\varphi_{2}(x)\leq\varphi_{1}(x)+B+\varepsilon(B-C)=\varphi_{1}(x).

It then follows that $\varphi_{2,\varepsilon}=\varphi_{1}$ on $V=\{\varphi_{1}=\min(\varphi_{1},\varphi_{2,\varepsilon})<-C\}$ . Similarly we have that $\psi_{2,\varepsilon}=\psi_{1}$ on $U=\{\psi_{1}=\min(\psi_{1},\psi_{2,\varepsilon})<-C\}$ , with the same $C$ .
We can thus apply Step 1.1 to $\psi_{1}$ , $\psi_{2,\varepsilon}$ , $\varphi_{1}$ , $\varphi_{2,\varepsilon}$ to obtain

\int_{X}(\varphi_{1}-\varphi_{2,\varepsilon})\left(\theta_{\psi_{1}}^{n}-\theta_{\psi_{2,\varepsilon}}^{n}\right)=\int_{X}\left(\psi_{1}-\psi_{2,\varepsilon}\right)(S_{1,\varepsilon}-S_{2,\varepsilon}),

where $S_{1,\varepsilon}:=\sum_{k=0}^{n-1}\theta_{\varphi_{1}}\wedge\theta_{\psi_{1}}^{k}\wedge\theta_{\psi_{2,\varepsilon}}^{n-k-1}$ and $S_{2,\varepsilon}:=\sum_{k=0}^{n-1}\theta_{\varphi_{2,\varepsilon}}\wedge\theta_{\psi_{1}}^{k}\wedge\theta_{\psi_{2,\varepsilon}}^{n-k-1}$ .
By Theorem 4.6 there exists a continuous function $f:[0,\infty)\rightarrow[0,\infty)$ with $f(0)=0$ such that for every Borel set $E$ ,

{\rm Cap}_{\theta,\psi}(E)\leq f({\rm Cap}_{\theta}(E)),

where

\psi:=\frac{\varphi_{1}+\varphi_{2}+\psi_{1}+\psi_{2}}{5}-B.

Note that $\psi$ is $\theta$ -psh and $\int_{X}\theta_{\psi}^{n}>0$ . Indeed, recalling that in this step $\theta$ is Kähler, we have

\int_{X}\theta_{\psi}^{n}=5^{-n}\int_{X}(\theta+\theta_{\varphi_{1}}+\theta_{\varphi_{2}}+\theta_{\psi_{1}}+\theta_{\psi_{2}})^{n}\geq 5^{-n}\theta^{n}>0.

Since we have assumed that $\psi_{2}-B\leq\psi_{1}\leq\psi_{2}\leq 0$ and $\varphi_{2}-B\leq\varphi_{1}\leq\varphi_{2}\leq 0$ , we get

\varphi_{2}-B\leq\varphi_{1}\leq\varphi_{2,\varepsilon}=\max(\varphi_{1},(1+\varepsilon)\varphi_{2})\leq\max(\varphi_{1},\varphi_{2})=\varphi_{2},

and

\psi_{2}-B\leq\psi_{1}\leq\psi_{2,\varepsilon}=\max(\psi_{1},(1+\varepsilon\psi_{2}))\leq\max(\psi_{1},\psi_{2})=\psi_{2}.

In particular $\psi_{1}\simeq\psi_{2}\simeq\psi_{2,\varepsilon}$ and $\varphi_{1}\simeq\varphi_{2}\simeq\varphi_{2,\varepsilon}$ .
Set $u_{\varepsilon}:=\frac{1}{5}(\varphi_{1}+\varphi_{2,\varepsilon}+\psi_{2,\varepsilon}+\psi_{1})$ . Using the above inequalities we get

	$\displaystyle\psi$	$\displaystyle\leq\frac{\varphi_{1}+\psi_{1}+\varphi_{2}+\psi_{2}}{5}-\frac{2B}{5}$
		$\displaystyle\leq\frac{\varphi_{1}+\psi_{1}+\varphi_{2,\varepsilon}+\psi_{2,\varepsilon}}{5}=u_{\varepsilon}$
		$\displaystyle\leq\frac{\varphi_{1}+\psi_{1}+\varphi_{2}+\psi_{2}}{5}=\psi+B.$

Also observe that there exists a positive constant $C^{\prime}$ such that

S_{j,\varepsilon}\leq C^{\prime}(5\theta+dd^{c}(\varphi_{1}+\varphi_{2,\varepsilon}+\psi_{2,\varepsilon}+\psi_{1}))^{n}=C^{\prime}5^{n}\theta_{u_{\varepsilon}}^{n}

We then obtain that for any Borel set $E$ and any $\varepsilon\in(0,1-\lambda),\;j=1,2$ ,

\int_{E}S_{j,\varepsilon}\leq C^{\prime}5^{n}B^{n}{\rm Cap}_{\theta,\psi}(E)\leq C^{\prime\prime}f({\rm Cap}_{\theta}(E)),

where the first inequality follows from the arguments at the beginning of the proof of Lemma 4.5, and the second follows from Theorem 4.6.

Since $\psi_{2,\varepsilon}$ and $\varphi_{2,\varepsilon}$ are increasing to $\psi_{2}$ and $\varphi_{2}$ respectively, for each $j\in\{1,2\}$ we also have that $S_{j,\varepsilon}\to S_{j}$ and $\theta_{\psi_{2,\varepsilon}}^{n}\to\theta_{\psi_{2}}^{n}$ as $\varepsilon\to 0$ in the weak sense of Radon measures (see Theorem 2.6 and Remark 3.4). By the above, these measures are uniformly dominated by ${\rm Cap}_{\theta}$ .
Note also that $\varphi_{1}-\varphi_{2,\varepsilon},\varphi_{1}-\varphi_{2},\psi_{2,\varepsilon}-\psi_{1},\psi_{2}-\psi_{1}$ are uniformly bounded and quasi-continuous (because difference of quasi-psh functions). Moreover, $\psi_{2,\varepsilon}-\psi_{1}\to\psi_{2}-\psi_{1}$ , and $\varphi_{1}-\varphi_{2,\varepsilon}\to\varphi_{1}-\varphi_{2}$ in capacity as $\varepsilon\to 0$ . It thus follows from Lemma 2.5 that

\lim_{\varepsilon\to 0}\int_{X}(\varphi_{1}-\varphi_{2,\varepsilon})\left(\theta_{\psi_{1}}^{n}-\theta_{\psi_{2,\varepsilon}}^{n}\right)=\int_{X}(\varphi_{1}-\varphi_{2})\left(\theta_{\psi_{1}}^{n}-\theta_{\psi_{2}}^{n}\right)

and

\lim_{\varepsilon\to 0}\int_{X}\left(\psi_{1}-\psi_{2,\varepsilon}\right)(S_{1,\varepsilon}-S_{2,\varepsilon})=\int_{X}\left(\psi_{1}-\psi_{2}\right)(S_{1}-S_{2}),

finishing the proof of Step 1.2.

Step 2. We merely assume that $\{\theta\}$ is big. We can assume that $-\omega\leq\theta\leq\omega$ . For $s>2$ we consider $\theta_{s}:=\theta+s\omega$ , which is Kähler, and we observe that $\varphi_{1},\varphi_{2},\psi_{1},\psi_{2}$ are $\lambda\theta_{s}$ -psh for some $\lambda\in(0,1)$ . We can thus apply the first step to get

\int_{X}u\left((\theta_{s}+dd^{c}\psi_{1})^{n}-(\theta_{s}+dd^{c}\psi_{2})^{n}\right)=\int_{X}vT_{s},

where $u=\varphi_{1}-\varphi_{2}$ , $v=\psi_{1}-\psi_{2}$ and

	$\displaystyle T_{s}=$	$\displaystyle\sum_{k=0}^{n-1}(\theta_{s}+dd^{c}\varphi_{1})\wedge(\theta_{s}+dd^{c}\psi_{1})^{k}\wedge(\theta_{s}+dd^{c}\psi_{2})^{n-k-1}$
	$\displaystyle-$	$\displaystyle\sum_{k=0}^{n-1}(\theta_{s}+dd^{c}\varphi_{2})\wedge(\theta_{s}+dd^{c}\psi_{1})^{k}\wedge(\theta_{s}+dd^{c}\psi_{2})^{n-k-1}.$

We thus obtain an equality between two polynomials in $s$ . Identifying the coefficients we arrive at the conclusion. ∎

Next we prove the integration by parts formula, extending the one in [BEGZ10] which only applies to the case of potentials with small unbounded locus.

Theorem 4.8.

Let $u,v\in L^{\infty}(X)$ be differences of quasi-psh functions, and $\phi_{j}\in\textup{PSH}(X,\theta^{j})$ , $j\in\{1,\ldots,n-1\}$ with $\{\theta^{j}\}$ big. Then

\int_{X}udd^{c}v\wedge\theta^{1}_{\phi_{1}}\wedge\ldots\wedge\theta^{n-1}_{\phi_{n-1}}=\int_{X}vdd^{c}u\wedge\theta^{1}_{\phi_{1}}\wedge\ldots\wedge\theta^{n-1}_{\phi_{n-1}}.

Proof.

We first assume that $\theta$ is Kähler, $u=\varphi_{1}-\varphi_{2}$ and $v=\psi_{1}-\psi_{2}$ where $\psi_{1},\psi_{2},\varphi_{1},\varphi_{2}$ are $\theta$ -psh. Fix $\phi\in{\rm PSH}(X,\theta)$ and for each $s\in[0,1]$ , $j=1,2$ , we set $\psi_{j,s}:=s\psi_{j}+(1-s)\phi$ . Note that $\psi_{1,s}\simeq\psi_{2,s}$ . It follows from Lemma 4.7 that for any $s\in[0,1]$ ,

\int_{X}u\left(\theta_{\psi_{1,s}}^{n}-\theta_{\psi_{2,s}}^{n}\right)=\int_{X}(\psi_{1,s}-\psi_{2,s})T_{s}=\int_{X}svT_{s},

where

T_{s}:=\sum_{k=0}^{n-1}\theta_{\varphi_{1}}\wedge\theta_{\psi_{1,s}}^{k}\wedge\theta_{\psi_{2,s}}^{n-k-1}-\sum_{k=0}^{n-1}\theta_{\varphi_{2}}\wedge\theta_{\psi_{1,s}}^{k}\wedge\theta_{\psi_{2,s}}^{n-k-1}.

We thus have an identity between two polynomials in $s$ . We then compute the first derivative in $s=0$ and we find that for $j=1,2$

\frac{\partial}{\partial s}(s\theta_{\psi_{j}}+(1-s)\theta_{\phi})^{n}|_{s=0}=n\theta_{\psi_{j}}\wedge\theta_{\phi}^{n-1}-n\theta_{\phi}^{n}

and

\frac{\partial}{\partial s}(sT_{s})|_{s=0}=T_{0}.

Noticing that $T_{0}=ndd^{c}u\wedge\theta_{\phi}^{n-1}$ , we obtain

\int_{X}udd^{c}v\wedge\theta_{\phi}^{n-1}=\int_{X}vdd^{c}u\wedge\theta_{\phi}^{n-1}.

For the general case, i.e. $\{\theta\}$ is merely big, we can write $u=\varphi_{1}-\varphi_{2}$ and $v=\psi_{1}-\psi_{2}$ , where $\psi_{1},\psi_{2},\varphi_{1},\varphi_{2}$ are $A\omega$ -psh, for some $A>0$ large enough. We apply the first step with $\theta$ replaced by $\theta+t\omega$ , for $t>A$ to get

\int_{X}udd^{c}v\wedge(t\omega+\theta_{\phi})^{n-1}=\int_{X}vdd^{c}u\wedge(t\omega+\theta_{\phi})^{n-1}.

Identifying the coefficients of these two polynomials in $t$ we obtain

\int_{X}udd^{c}v\wedge\theta_{\phi}^{n-1}=\int_{X}vdd^{c}u\wedge\theta_{\phi}^{n-1}.

We now consider $\theta=s_{1}\theta_{1}+....+s_{n-1}\theta_{n-1}$ , $\phi:=s_{1}\phi_{1}+...+s_{n-1}\phi_{n-1}$ with $s_{1},...,s_{n-1}\in[0,1]$ and $\sum_{j=1}^{n-1}s_{j}=1$ . We obtain an identity between two polynomials in $(s_{1},...,s_{n-1})$ , and identifying the coefficients we arrive at the result. ∎

Chapter 5 Complex Monge–Ampère equations with prescribed singularity type

Let $\theta$ be a smooth closed real $(1,1)$ -form on $X$ such that $\{\theta\}$ is big and $\phi\in\textup{PSH}(X,\theta)$ . By $\mathrm{PSH}(X,\theta,\phi)$ we denote the set of $\theta$ -psh functions that are more singular than $\phi$ . We say that $v\in\textup{PSH}(X,\theta,\phi)$ has relatively minimal singularity type if $v$ has the same singularity type as $\phi$ . Clearly, $\mathcal{E}(X,\theta,\phi)\subset\textup{PSH}(X,\theta,\phi)$ .

Let $\mu$ be a non-pluripolar positive measure on $X$ such that $\mu(X)=\int_{X}\theta_{\phi}^{n}>0$ . Our aim is to study existence and uniqueness of solutions to the following equation of complex Monge–Ampère type:

\theta_{u}^{n}=\mu,\ \ \ u\in\mathcal{E}(X,\theta,\phi).

(5.1)

It is not hard to see that this equation does not have a solution for arbitrary $\phi$ . Indeed, suppose for the moment that $\theta=\omega$ , and choose $\phi\in\mathcal{E}(X,\omega):=\mathcal{E}(X,\omega,0)$ unbounded. It is clear that $\mathcal{E}(X,\omega,\phi)\subsetneq\mathcal{E}(X,\omega,0)$ . By [BEGZ10, Theorem A], the (trivial) equation $\omega_{u}^{n}=\omega^{n},\ u\in\mathcal{E}(X,\omega,0)$ is only solved by potentials $u$ that are constant over $X$ , hence we cannot have $u\notin\mathcal{E}(X,\omega,\phi)$ .

This simple example suggests that we need to be more selective in our choice of $\phi$ , to make (5.1) well posed. As it turns out, the natural choice is to take $\phi$ such that $P_{\theta}[\phi]=\phi$ (see Theorem 5.22 for concrete evidence). Therefore, for the rest of this section we ask that $\phi$ additionally satisfies:

\phi=P_{\theta}[\phi].

(5.2)

We recall that such a potential $\phi$ is model, and $[\phi]$ is a model type singularity. As $V_{\theta}=P_{\theta}[V_{\theta}]$ , one can think of such $\phi$ as generalizations of $V_{\theta}$ , the potential with minimal singularity from [BEGZ10].

One wonders if maybe model type potentials (those that satisfy (5.2)) always have small unbounded locus. Sadly, this is not the case, as we now point out. Suppose $\theta$ is a Kähler form, and $\{x_{j}\}_{j}\subset X$ is a dense countable subset. Also let $v_{j}\in\textup{PSH}(X,\theta)$ such that $v_{j}<0$ , $\int_{X}(-v_{j})\theta^{n}=1$ , and $v_{j}$ has a positive Lelong number at $x_{j}$ . Then $u=\sum_{j}\frac{1}{2^{j}}v_{j}\in\textup{PSH}(X,\theta)$ has positive Lelong numbers at all $x_{j}$ . As we have argued in Lemma 5.1 below, the Lelong numbers of $P_{\theta}[u]$ are the same as those of $u$ , hence the model type potential $P_{\theta}[u]$ cannot have small unbounded locus.

Lemma 5.1.

Suppose that $u\in\textup{PSH}(X,\theta)$ . Then $\mathcal{I}(tu)=\mathcal{I}(tP_{\theta}[u])$ for all $t>0$ . In particular, all the Lelong numbers of $u$ and $P_{\theta}[u]$ are the same.

Recall that $\mathcal{I}(v)$ is the multiplier ideal sheaf of a quasi-psh function $v$ , whose germs are defined by

\mathcal{I}(v,x):=\left\{f\in{\mathcal{O}}_{x}\ :\ \int_{U}|f|^{2}e^{-v}\omega^{n}<+\infty\quad{\rm for\ some\ open\ subset\ }U\subset X,x\in U\right\}.

We use the valuative criteria of integrability of Boucksom–Favre–Jonsson. For a more elementary argument see the proof of [DDL18b, Theorem 1.1].

Proof.

Since $P_{\theta}(u+C,V_{\theta})\nearrow P_{\theta}[u]$ a.e., as $C\nearrow\infty$ , from the Guan–Zhou openess theorem [GZ15] it follows that $\mathcal{I}(tP_{\theta}(u+C,V_{\theta}))=\mathcal{I}(tP_{\theta}[u])$ for $C>0$ big enough. However $[P_{\theta}(u+C,V_{\theta})]=[u]$ for all $C>0$ , so we get that $\mathcal{I}(tu)=\mathcal{I}(tP_{\theta}[u])$ . The conclusion about Lelong numbers follows from the equivalence between (i) (applied with the proper modification $\pi=id$ ) and (ii) in [BFJ08, Theorem A]. ∎

5.1 The relative finite energy class

To develop the variational approach to (5.1), we study the relative version of the Monge–Ampère energy, and its bounded locus $\mathcal{E}^{1}(X,\theta,\phi)$ . For $u\in\mathcal{E}(X,\theta,\phi)$ with relatively minimal singularity type, we define the Monge–Ampère energy of $u$ relative to $\phi$ as

\mathrm{I}_{\phi}(u):=\frac{1}{n+1}\sum_{k=0}^{n}\int_{X}(u-\phi)\theta_{u}^{k}\wedge\theta_{\phi}^{n-k}.

Before we study this energy closely, we note the following inequality:

Lemma 5.2.

Assume $u,v\in\mathcal{E}(X,\theta,\phi)$ have the same singularity type. Then

\int_{X}(u-v)\theta_{u}^{k}\wedge\theta_{v}^{n-k}\leq\int_{X}(u-v)\theta_{u}^{k-1}\wedge\theta_{v}^{n-k+1}.

Proof.

Adding a constant to $u$ does not affect the inequality (by Theorem 3.3), so we can assume that $v\leq u$ . By the partial comparison principle (Proposition 3.22), we have

	$\displaystyle\int_{X}(u-v)\theta_{u}^{k}\wedge\theta_{v}^{n-k}$	$\displaystyle=\int_{0}^{\infty}\theta_{u}^{k}\wedge\theta_{v}^{n-k}(u>v+t)dt$
		$\displaystyle\leq\int_{0}^{\infty}\theta_{u}^{k-1}\wedge\theta_{v}^{n-k+1}(u>v+t)dt$
		$\displaystyle=\int_{X}(u-v)\theta_{u}^{k-1}\wedge\theta_{v}^{n-k+1}.$

∎

In the next theorem we collect basic properties of the Monge–Ampère energy:

Theorem 5.3.

Suppose $u,v\in\mathcal{E}(X,\theta,\phi)$ have relatively minimal singularity type. Then:
(i) $\mathrm{I}_{\phi}(u)-\mathrm{I}_{\phi}(v)=\frac{1}{n+1}\sum_{k=0}^{n}\int_{X}(u-v)\theta_{u}^{k}\wedge\theta_{v}^{n-k}.$ In particular $\mathrm{I}_{\phi}(u)\leq\mathrm{I}_{\phi}(v)$ if $u\leq v$ .
(ii) If $u\leq\phi$ then, $\int_{X}(u-\phi)\theta_{u}^{n}\leq I_{\phi}(u)\leq\frac{1}{n+1}\int_{X}(u-\phi)\theta_{u}^{n}.$
(iii) $\mathrm{I}_{\phi}$ is concave along affine curves. Also, the following estimates hold:

\int_{X}(u-v)\theta_{u}^{n}\leq I_{\phi}(u)-I_{\phi}(v)\leq\int_{X}(u-v)\theta_{v}^{n}.

Proof.

Using Theorem 4.8, it is possible to repeat the arguments of [BEGZ10, Proposition 2.8], almost word for word. As a courtesy, we present a detailed proof.

We compute the derivative of $f(t):=I_{\phi}(u_{t}),t\in[0,1]$ , where $u_{t}:=tu+(1-t)v$ . By the multi-linearity property of the non-pluripolar product we see that $f(t)$ is a polynomial in $t$ . Using integration by parts (Theorem 4.8), one can check the following formula:

$\displaystyle f^{\prime}(t)$	$\displaystyle=$	$\displaystyle\frac{1}{n+1}\bigg{(}\sum_{k=0}^{n}\int_{X}(u-v)\theta_{u_{t}}^{k}\wedge\theta_{\phi}^{n-k}+\sum_{k=1}^{n}\int_{X}k(u_{t}-\phi)dd^{c}(u-v)\wedge\theta_{u_{t}}^{k-1}\wedge\theta_{\phi}^{n-k}\bigg{)}$
	$\displaystyle=$	$\displaystyle\frac{1}{n+1}\bigg{(}\sum_{k=0}^{n}\int_{X}(u-v)\theta_{u_{t}}^{k}\wedge\theta_{\phi}^{n-k}+\sum_{k=1}^{n}\int_{X}k(u-v)(\theta_{u_{t}}-\theta_{\phi})\wedge\theta_{u_{t}}^{k-1}\wedge\theta_{\phi}^{n-k}\bigg{)}$
	$\displaystyle=$	$\displaystyle\int_{X}(u-v)\theta_{u_{t}}^{n}.$

Computing one more derivative, we arrive at

f^{\prime\prime}(t)=n\int_{X}(u-v)dd^{c}(u-v)\wedge\theta_{u_{t}}^{n-1}=n\int_{X}(u-v)(\theta_{u}-\theta_{v})\wedge\theta_{u_{t}}^{n-1}\leq 0,

where the inequality follows from Lemma 5.2.

Now, the function $t\mapsto f^{\prime}(t)$ is continuous on $[0,1]$ , thanks to convergence property of the Monge–Ampère operator (see Theorem 2.6). It thus follows that

I_{\phi}(u_{1})-I_{\phi}(u_{0})=\int_{0}^{1}f^{\prime}(t)dt=\int_{0}^{1}\int_{X}(u-v)\theta_{u_{t}}^{n}dt.

Using the multi-linearity of the non-pluripolar product again, we get that

	$\displaystyle\int_{0}^{1}\int_{X}(u-v)\theta_{u_{t}}^{n}dt$	$\displaystyle=$	$\displaystyle\sum_{k=0}^{n}\left(\int_{0}^{1}\binom{n}{k}t^{k}(1-t)^{n-k}dt\right)\int_{X}(u-v)\theta_{u}^{k}\wedge\theta_{v}^{n-k}$
		$\displaystyle=$	$\displaystyle\frac{1}{n+1}\sum_{k=0}^{n}\int_{X}(u-v)\theta_{u}^{k}\wedge\theta_{v}^{n-k}.$

This verifies (i), and another application of Lemma 5.2 finishes the proof of (iii).

For (ii) we observe that since $u-\phi\leq$ each term $\int_{X}(u-\phi)\theta_{u}^{k}\wedge\theta_{\phi}^{n-k}$ is negative, hence $I_{\phi}(u)\leq\int_{X}(u-\phi)\theta_{u}^{n}$ . The left-hand side inequality of (ii) follows from Lemma 5.2. ∎

Lemma 5.4.

Suppose $u_{j},u\in\mathcal{E}(X,\theta,\phi)$ have relatively minimal singularity type such that $u_{j}\searrow u$ . Then ${\rm I}_{\phi}(u_{j})\searrow\mathrm{I}_{\phi}(u)$ .

Proof.

From Theorem 5.3 it follows that

|{\rm I}_{\phi}(u_{j})-{\rm I}_{\phi}(u)|={\rm I}_{\phi}(u_{j})-{\rm I}_{\phi}(u)\leq\int_{X}(u_{j}-u)\theta_{u}^{n}.

An application of the dominated convergence theorem finishes the argument. ∎

We can now define the Monge–Ampère energy for arbitrary $u\in\mathrm{PSH}(X,\theta,\phi)$ using a familiar formula:

{\rm I}_{\phi}(u):=\inf\{{\rm I}_{\phi}(v)\ |\ v\in\mathcal{E}(X,\theta,\phi),\;v\ \textrm{has relatively minimal singularity type, and }u\leq v\}.

Lemma 5.5.

If $u\in{\rm PSH}(X,\theta,\phi)$ then ${\rm I}_{\phi}(u)=\lim_{t\to\infty}{\rm I}_{\phi}(\max(u,\phi-t))$ .

Proof.

It follows from the above definition that ${\rm I}_{\phi}(u)\leq\lim_{t\to\infty}{\rm I}_{\phi}(\max(u,\phi-t))$ . Assume now that $v\in{\rm PSH}(X,\theta,\phi)$ is such that $u\leq v$ , and $v$ has the same singularity type as $\phi$ (i.e. $v$ is a candidate in the definition of $I_{\phi}(u)$ ). Then for $t$ large enough we have $\max(u,\phi-t)\leq v$ , hence the other inequality follows from monotonicity of ${\rm I}_{\phi}$ . ∎

Let $\mathcal{E}^{1}(X,\theta,\phi)$ be the set of all $u\in{\rm PSH}(X,\theta,\phi)$ such that ${\rm I}_{\phi}(u)$ is finite. As a result of Lemma 5.5 and Theorem 5.3 $(i)$ we observe that ${\rm I}_{\phi}$ is non-decreasing in ${\rm PSH}(X,\theta,\phi)$ . Consequently, $\mathcal{E}^{1}(X,\theta,\phi)$ is stable under the max operation, moreover we have the following familiar characterization of $\mathcal{E}^{1}(X,\theta,\phi)$ :

Lemma 5.6.

Let $u\in\mathrm{PSH}(X,\theta,\phi)$ . Then $u\in\mathcal{E}^{1}(X,\theta,\phi)$ if and only if $u\in\mathcal{E}(X,\theta,\phi)$ and $\int_{X}(u-\phi)\theta_{u}^{n}>-\infty$ .

Proof.

Let $u\in\mathcal{E}^{1}(X,\theta,\phi)$ . We can assume that $u\leq\phi$ . For each $C>0$ we set $u^{C}:=\max(u,\phi-C)$ .

If $I_{\phi}(u)>-\infty$ (say $I_{\phi}(u)>-A$ ), then by the monotonicity property we have $I_{\phi}(u^{C})\geq I_{\phi}(u)$ . Since $u^{C}\leq\phi$ , an application of Theorem 5.3 $(ii)$ gives that $\int_{X}(u^{C}-\phi)\theta_{u^{C}}^{n}\geq-A,\ \forall C$ , for some $A>0$ . From this we obtain that

\int_{\{u\leq\phi-C\}}\theta_{u^{C}}^{n}\leq\int_{\{u\leq\phi-C\}}\frac{\phi-u}{C}\,\theta_{u^{C}}^{n}\leq\int_{X}\frac{\phi-u^{C}}{C}\,\theta_{u^{C}}^{n}\leq\frac{A}{C}\to 0,

as $C\to\infty$ . By plurifine locality and the above, we have

\int_{\{u>\phi-C\}}\theta_{u}^{n}=\int_{\{u>\phi-C\}}\theta_{u^{C}}^{n}=\int_{X}\theta_{u^{C}}^{n}-\int_{\{u\leq\phi-C\}}\theta_{u^{C}}^{n}=\int_{X}\theta_{\phi}^{n}-\int_{\{u\leq\phi-C\}}\theta_{u^{C}}^{n}\to\int_{X}\theta_{\phi}^{n}

as $C\to\infty$ , showing that $u\in\mathcal{E}(X,\theta,\phi)$ . Moreover,

\int_{X}(u^{C}-\phi)\theta_{u^{C}}^{n}\leq\int_{\{u>\phi-C\}}(u-\phi)\theta_{u}^{n}.

Letting $C\to\infty$ we see that $\int_{X}(u-\phi)\theta_{u}^{n}>-A$ .

To prove the converse statement, assume that $u\in\mathcal{E}(X,\theta,\phi)$ and $\int_{X}(u-\phi)\theta_{u}^{n}>-\infty$ . For each $C>0$ the measures $\theta_{u}^{n}$ and $\theta_{u^{C}}^{n}$ have the same total mass and they coincide on $\{u>\phi-C\}$ . It follows that $\int_{\{u\leq\phi-C\}}\theta_{u^{C}}^{n}=\int_{\{u\leq\phi-C\}}\theta_{u}^{n}$ , and from this we deduce that

\displaystyle\int_{X}(u^{C}-\phi)\theta_{u^{C}}^{n}

\displaystyle=

\displaystyle-\int_{\{u\leq\phi-C\}}C\theta_{u}^{n}+\int_{\{u>\phi-C\}}(u-\phi)\theta_{u}^{n}\geq\int_{X}(u-\phi)\theta_{u}^{n}>-A.

It thus follows from Theorem 5.3 $(ii)$ that $I_{\phi}(u^{C})$ is uniformly bounded. Finally, it follows from Lemma 5.5 that $I_{\phi}(u^{C})\searrow I_{\phi}(u)$ as $C\to\infty$ , finishing the proof. ∎

We finish this section with a series of standard results listing various properties of the class $\mathcal{E}^{1}(X,\theta,\phi)$ .

Lemma 5.7.

Assume that $u_{j},u\in\mathcal{E}^{1}(X,\theta,\phi)$ such that $u_{j}\searrow u$ . Then ${\rm I}_{\phi}(u_{j})$ decreases to ${\rm I}_{\phi}(u)$ .

Proof.

Without loss of generality we can assume that $u_{j}\leq\phi$ for all $j$ . For each $C>0$ we set $u_{j}^{C}:=\max(u_{j},\phi-C)$ and $u^{C}:=\max(u,\phi-C)$ . Note that $u_{j}^{C},u^{C}$ have the same singularity type as $\phi$ . Then Lemma 5.4 insures that $\lim_{j}I_{\phi}(u_{j}^{C})=I_{\phi}(u^{C})$ . Monotonicity of $I_{\phi}$ gives now that $I_{\phi}(u)\leq\lim_{j}I_{\phi}(u_{j})\leq\lim_{j}I_{\phi}(u_{j}^{C})=I_{\phi}(u^{C})$ . Letting $C\to\infty$ , the result follows from Lemma 5.5. ∎

Lemma 5.8.

Assume that $\{u_{j}\}_{j}\subset\mathcal{E}^{1}(X,\theta,\phi)$ is decreasing, such that ${\rm I}_{\phi}(u_{j})$ is uniformly bounded. Then the limit $u:=\lim_{j}u_{j}$ belongs to $\mathcal{E}^{1}(X,\theta,\phi)$ and ${\rm I}_{\phi}(u_{j})$ decreases to ${\rm I}_{\phi}(u)$ .

Proof.

We can assume that $u_{j}\leq\phi$ for all $j$ . As all the terms in the definition of $I_{\phi}(u_{j})$ are negative, we notice that $-C\leq(n+1){\rm I}_{\phi}(u_{j})\leq\int_{X}(u_{j}-\phi)\theta_{\phi}^{n}\leq 0$ for some $C>0$ .

If $u\equiv-\infty$ , then $\sup_{X}u_{j}=\sup_{X}(u_{j}-\phi)\to-\infty$ (Lemma 3.5). This implies that $\int_{X}(u_{j}-\phi)\theta_{\phi}^{n}\leq\sup_{X}(u_{j}-\phi)\int_{X}\theta_{\phi}^{n}\to-\infty$ , a contradiction. Hence $u\in\textup{PSH}(X,\theta)$ .

By continuity along decreasing sequences (Lemma 5.7) we have

\lim_{j\rightarrow\infty}{\rm I}_{\phi}(\max(u_{j},\phi-C))={\rm I}_{\phi}(\max(u,\phi-C)).

It follows that ${\rm I}_{\phi}(\max(u,\phi-C))$ is uniformly bounded. Lemma 5.5 then insures that ${\rm I}_{\phi}(u)$ is finite, i.e., $u\in\mathcal{E}^{1}(X,\theta,\phi)$ . ∎

Corollary 5.9.

${\rm I}_{\phi}$ is concave along affine curves in ${\rm PSH}(X,\theta,\phi)$ . In particular, the set $\mathcal{E}^{1}(X,\theta,\phi)$ is convex.

Proof.

Let $u,v\in{\rm PSH}(X,\theta,\phi)$ and $u_{t}:=tu+(1-t)v,t\in(0,1)$ . If one of $u,v$ is not in $\mathcal{E}^{1}(X,\theta,\phi)$ then the conclusion is obvious. So, we can assume that both $u$ and $v$ belong to $\mathcal{E}^{1}(X,\theta,\phi)$ . For each $C>0$ we set $u_{t}^{C}:=t\max(u,\phi-C)+(1-t)\max(v,\phi-C)$ . By Theorem 5.3 $(iii)$ , $t\to{\rm I}_{\phi}(u_{t}^{C})$ is concave. Since $u_{t}^{C}$ decreases to $u_{t}$ as $C\to\infty$ , Lemma 5.8 gives the conclusion. ∎

Lemma 5.10.

Suppose $u,v\in\mathcal{E}^{1}(X,\theta,\phi)$ have the same singularity type. Then

\int_{X}(u-v)\theta_{u}^{n}\leq{\rm I}_{\phi}(u)-{\rm I}_{\phi}(v)\leq\int_{X}(u-v)\theta_{v}^{n}.

With a bit of extra work, one can also get rid of the assumption that $u$ and $v$ have the same singularity type [DDL18, Proposition 2.5].

Proof.

First, note that these estimates hold for $u^{C}:=\max(u,\phi-C),v^{C}:=\max(v,\phi-C)$ , by Theorem 5.3 $(iii)$ . It is easy to see that $u^{C}-v^{C}$ is uniformly bounded and converges in capacity to $u-v$ . Putting these last two facts together, Theorem 2.6 gives that

\displaystyle\bigg{|}\int_{X}(u^{C}-v^{C})\theta_{v^{C}}^{n}-\int_{X}(u-v)\theta_{v}^{n}\bigg{|}\to 0,\ \ \ \bigg{|}\int_{X}(u^{C}-v^{C})\theta_{u^{C}}^{n}-\int_{X}(u-v)\theta_{u}^{n}\bigg{|}\to 0.

The result follows from Lemma 5.5. ∎

5.2 Monge–Ampère equations with prescribed singularity type

Throughout this section $\phi$ is a $\theta$ -psh function satisfying $\phi=P_{\theta}[\phi]$ , and $\int_{X}\theta_{\phi}>0$ . We additionally assume the normalization

\int_{X}\theta_{\phi}^{n}=1.

This can always be achieved by rescaling our big class $\{\theta\}$ . In this section we consider the following complex Monge–Ampère equation in our prescribed setting:

\theta_{u}^{n}=e^{\lambda u}\mu,\ u\in\mathcal{E}(X,\theta,\phi).

(5.3)

where $\lambda\geq 0$ , $\mu$ is a positive non-pluripolar probability measure on $X$ .

When $\lambda>0$ , we adapt the variational method in [BBGZ13] to our needs. When $\lambda=0$ , it is also possible to use the variational method to solve the equation, but it requires a detailed study of the relative Monge–Ampère capacities, as carried out in [DDL18a]. In this survey we provide a simpler approach by using the solutions for the case $\lambda>0$ and letting $\lambda\to 0^{+}$ .

Proposition 5.11.

${\rm I}_{\phi}:\mathcal{E}^{1}(X,\theta,\phi)\to\mathbb{R}$ is upper semicontinuous with respect to the weak $L^{1}$ topology of potentials.

Proof.

Assume that $\{u_{j}\}_{j}$ is a sequence in $\mathcal{E}^{1}(X,\theta,\phi)$ $L^{1}$ -converging to $u\in\mathcal{E}^{1}(X,\theta,\phi)$ . We can assume that $u_{j}\leq 0$ for all $j$ . For each $k,\ell\in\mathbb{N}$ we set $v_{k,\ell}:=\max(u_{k},...,u_{k+\ell})$ . As $\mathcal{E}^{1}(X,\theta,\phi)$ is stable under the max operation, we have that $v_{k,\ell}\in\mathcal{E}^{1}(X,\theta,\phi)$ .

Moreover $v_{k,\ell}\nearrow\varphi_{k}:=\textup{usc}\left(\sup_{j\geq k}u_{j}\right)$ , hence by the monotonicity property we get ${\rm I}_{\phi}(\varphi_{k})\geq{\rm I}_{\phi}(v_{k,\ell})\geq{\rm I}_{\phi}(u_{k})>-\infty$ . As a result, $\varphi_{k}\in\mathcal{E}^{1}(X,\theta,\phi)$ . By Hartogs’ lemma $\varphi_{k}\searrow u$ as $k\to\infty$ . By Lemma 5.7 it follows that ${\rm I}_{\phi}(\varphi_{k})$ decreases to ${\rm I}_{\phi}(u)$ . Thus, using the monotonicity of ${\rm I}_{\phi}$ we get ${\rm I}_{\phi}(u)=\lim_{k\rightarrow\infty}{\rm I}_{\phi}(\varphi_{k})\geq\limsup_{k\rightarrow\infty}{\rm I}_{\phi}(u_{k}),$ finishing the proof. ∎

Next we describe the first order variation of $I_{\phi}$ , shadowing a result from [BB10]:

Proposition 5.12.

Let $u\in\mathcal{E}^{1}(X,\theta,\phi)$ and $\chi$ be a continuous function on $X$ . For each $t\in\mathbb{R}$ set $u_{t}:=P_{\theta}(u+t\chi)$ . Then $u_{t}\in\mathcal{E}^{1}(X,\theta,\phi)$ , $t\mapsto{\rm I}_{\phi}(u_{t})$ is differentiable, and its derivative is given by

\frac{d}{dt}{\rm I}_{\phi}(u_{t})=\int_{X}\chi\theta_{u_{t}}^{n},\ t\in\mathbb{R}.

Proof.

For $t\geq 0$ the potential $u+t\inf_{X}\chi$ is a candidate in each envelope, hence $u+t\inf_{X}\chi\leq u_{t}$ . Monotonicity of $I_{\phi}$ now implies that $u_{t}\in\mathcal{E}^{1}(X,\theta,\phi)$ . A similar argument implies that $u_{t}\in\mathcal{E}^{1}(X,\theta,\phi)$ for $t\leq 0$ .

Let $t\in\mathbb{R}$ and $s>0$ . As the singularity type of each $u_{t}$ is the same of that of $u$ , we can apply Lemma 5.10 and conclude:

\int_{X}(u_{t+s}-u_{t})\theta_{u_{t+s}}^{n}\leq{\rm I}_{\phi}(u_{t+s})-{\rm I}_{\phi}(u_{t})\leq\int_{X}(u_{t+s}-u_{t})\theta_{u_{t}}^{n}.

It follows from Theorem 2.7 that $\theta_{u_{t}}^{n}$ is supported on $\{u_{t}=u+t\chi\}$ . We thus have

\int_{X}(u_{t+s}-u_{t})\theta_{u_{t}}^{n}=\int_{X}(u_{t+s}-u-t\chi)\theta_{u_{t}}^{n}\leq\int_{X}s\chi\theta_{u_{t}}^{n},

since $u_{t+s}\leq u+(t+s)\chi$ . Similarly we have

\int_{X}(u_{t+s}-u_{t})\theta_{u_{t+s}}^{n}=\int_{X}(u+(t+s)\chi-u_{t})\theta_{u_{t+s}}^{n}\geq\int_{X}s\chi\theta_{u_{t+s}}^{n}.

Since $u_{t+s}$ converges uniformly to $u_{t}$ as $s\to 0$ , by Theorem 2.6 it follows that $\theta_{u_{t+s}}^{n}$ converges weakly to $\theta_{u_{t}}^{n}$ . As $\chi$ is continuous, dividing by $s>0$ and letting $s\to 0^{+}$ we see that the right derivative of ${\rm I}_{\phi}(u_{t})$ at $t$ is $\int_{X}\chi\theta_{u_{t}}^{n}$ . The same argument applies for the left derivative. ∎

The case $\lambda>0$ .

It suffices to treat the case $\lambda=1$ as the other cases can be done similarly.

We introduce the following functional on $\mathcal{E}^{1}(X,\theta,\phi)$ :

F(u):=F_{\mu}(u):={\rm I}_{\phi}(u)-L_{\mu}(u),\ u\in\mathcal{E}^{1}(X,\theta,\phi),

where $L_{\mu}(u):=\int_{X}e^{u}d\mu$ .

Theorem 5.13.

Assume that $u\in\mathcal{E}^{1}(X,\theta,\phi)$ maximizes $F$ on $\mathcal{E}^{1}(X,\theta,\phi)$ . Then $u$ solves the equation (5.3).

Proof.

Let $\chi$ be an arbitrary continuous function on $X$ and set $u_{t}:=P_{\theta}(u+t\chi)$ . It follows from Proposition 5.12 that $u_{t}\in\mathcal{E}^{1}(X,\theta,\phi)$ for all $t\in\mathbb{R}$ , that the function

g(t):={\rm I}_{\phi}(u_{t})-L_{\mu}(u+t\chi)

is differentiable on $\mathbb{R}$ , and its derivative is given by

g^{\prime}(t)=\int_{X}\chi\theta_{u_{t}}^{n}-\int_{X}\chi e^{(u+t\chi)}d\mu.

Moreover, as $u_{t}\leq u+t\chi$ , we have

g(t)\leq{\rm I}_{\phi}(u_{t})-L_{\mu}(u_{t})=F_{\mu}(u_{t})\leq\sup_{\mathcal{E}^{1}(X,\theta,\phi)}F_{\mu}=F(u)=g(0).

This means that $g$ attains a maximum at $0$ , hence $g^{\prime}(0)=0$ . Since $\chi\in C^{0}(X)$ is arbitrary it follows that $\theta_{u}^{n}=e^{\lambda u}\mu$ . ∎

Having computed the first order variation of the Monge–Ampère energy, we establish the following existence and uniqueness result.

Theorem 5.14.

Assume that $\mu$ is a positive non-pluripolar measure on $X$ . Then there exists a unique $u\in\mathcal{E}^{1}(X,\theta,\phi)$ such that

\theta_{u}^{n}=e^{u}\mu.

(5.4)

Proof.

We use the variational method. Let $\{u_{j}\}_{j}$ be a sequence in $\mathcal{E}^{1}(X,\theta,\phi)$ such that $\lim_{j}F(u_{j})=\sup_{\mathcal{E}^{1}(X,\theta,\phi)}F>-\infty$ . We claim that $\sup_{X}u_{j}$ is uniformly bounded from above. Indeed, assume that it were not the case. Then by relabeling the sequence we can assume that $\sup_{X}u_{j}$ increase to $\infty$ . By the compactness property [GZ05, Proposition 2.7] it follows that the sequence $\psi_{j}:=u_{j}-\sup_{X}u_{j}$ converges in $L^{1}(X,\omega^{n})$ to some $\psi\in{\rm PSH}(X,\theta)$ such that $\sup_{X}\psi=0$ . In particular, $\int_{X}e^{\psi}d\mu>0$ . It thus follows that

\int_{X}e^{u_{j}}d\mu=e^{\sup_{X}u_{j}}\int_{X}e^{\psi_{j}}d\mu\geq c\ e^{\sup_{X}u_{j}}

(5.5)

for some $c>0$ . Note also that $\psi_{j}\leq\phi$ since $\psi_{j}\in\mathcal{E}(X,\theta,\phi)$ and $\psi_{j}\leq 0$ and $\phi$ is the maximal function with these properties (see Theorem 3.14). It then follows that

{\rm I}_{\phi}(u_{j})={\rm I}_{\phi}(\psi_{j})+\sup_{X}u_{j}\leq\sup_{X}u_{j}.

(5.6)

From (5.5) and (5.6) we arrive at

\lim_{j\to\infty}F(u_{j})\leq\lim_{j\to\infty}\left(\sup_{X}u_{j}-ce^{\sup_{X}u_{j}}\right)=-\infty,

which is a contradiction. Thus $\sup_{X}u_{j}$ is bounded from above as claimed. Since $F(u_{j})\leq{\rm I}_{\phi}(u_{j})\leq\sup_{X}u_{j}$ it follows that ${\rm I}_{\phi}(u_{j})$ and hence $\sup_{X}u_{j}$ is also bounded from below. It follows again from [GZ05, Proposition 2.7] that a subsequence of $u_{j}$ (still denoted by $u_{j}$ ) converges in $L^{1}(X,\omega^{n})$ to some $u\in{\rm PSH}(X,\theta)$ . Since ${\rm I}_{\phi}$ is upper semicontinuous we have

\limsup_{j\to\infty}{\rm I}_{\phi}(u_{j})\leq{\rm I}_{\phi}(u),

hence $u\in\mathcal{E}^{1}(X,\theta,\phi)$ . Moreover, by continuity of $\varphi\mapsto\int_{X}e^{\varphi}d\mu$ we get that $F(u)\geq\sup_{\mathcal{E}^{1}(X,\theta,\phi)}F$ . Hence $u$ maximizes $F$ on $\mathcal{E}^{1}(X,\theta,\phi)$ . Now Theorem 5.13 shows that $u$ solves the desired complex Monge–Ampère equation. The next lemma address the uniqueness question. ∎

Lemma 5.15.

Assume that $u\in\mathcal{E}(X,\theta,\phi)$ is a solution of (5.4) and $v\in\mathcal{E}(X,\theta,\phi)$ is a subsolution, i.e., $\theta_{v}^{n}\geq e^{v}\mu.$ Then $u\geq v$ on $X$ .

Proof.

By the comparison principle for the class $\mathcal{E}(X,\theta,\phi)$ (Corollary 3.23) we have

\int_{\{u<v\}}\theta_{v}^{n}\leq\int_{\{u<v\}}\theta_{u}^{n}.

As $u$ is a solution and $v$ is a subsolution to (5.4) we then have

\int_{\{u<v\}}e^{v}d\mu\leq\int_{\{u<v\}}\theta_{v}^{n}\leq\int_{\{u<v\}}\theta_{u}^{n}=\int_{\{u<v\}}e^{u}d\mu\leq\int_{\{u<v\}}e^{v}d\mu.

It follows that all inequalities above are equalities, hence $\mu(\{u<v\})=0$ . Since $\mu=e^{u}\theta_{u}^{n}$ , it follows that $\theta_{u}^{n}(\{u<v\})=0$ . By the domination principle (Theorem 3.12) we get that $u\geq v$ everywhere on $X$ . ∎

The case $\lambda=0$ .

It immediately follows from Lemma 2.9 that subsolutions are preserved under taking maximums. In addition to this, the $L^{1}$ -limit of subsolutions is also a subsolution:

Lemma 5.16.

Let $(u_{j})$ be a sequence of $\theta$ -psh functions such that $\theta_{u_{j}}^{n}\geq f_{j}\mu$ , where $f_{j}\in L^{1}(X,\mu)$ and $\mu$ is a positive non-pluripolar Borel measure on $X$ . Assume that $f_{j}$ converge in $L^{1}(X,\mu)$ to $f\in L^{1}(X,\mu)$ , and $u_{j}$ converge in $L^{1}(X,\omega^{n})$ to $u\in{\rm PSH}(X,\theta)$ . Then $\theta_{u}^{n}\geq f\mu$ .

Proof.

By extracting a subsequence if necessary, we can assume that $f_{j}$ converge $\mu$ -a.e. to $f$ . For each $k$ we set $v_{k}:=\textup{usc}(\sup_{j\geq k}u_{j})$ . Then $v_{k}$ decreases pointwise to $u$ and Lemma 2.9 together with Theorem 2.6 gives

\theta_{v_{k}}^{n}\geq\left(\inf_{j\geq k}f_{j}\right)\mu.

Indeed $\max(u_{k},\ldots,u_{k+l})\nearrow v_{k}$ a.e., as $l\to\infty$ , making Theorem 2.6 applicable due to Remark 3.4.

To explain our notation below, for $t>0$ and a function $g$ we set $g^{t}:=\max(g,V_{\theta}-t)$ .

Note that $\{u>V_{\theta}-t\}\subset\{v_{k}>V_{\theta}-t\}$ . Multiplying both sides of the above estimate with ${\bf 1}_{\{u>V_{\theta}-t\}}$ , $t>0$ , and using the locality of the complex Monge–Ampère operator with respect to the plurifine topology we arrive at

\theta_{v_{k}^{t}}^{n}\geq{\bf 1}_{\{u>V_{\theta}-t\}}\left(\inf_{j\geq k}f_{j}\right)\mu.

Note that for $t>0$ fixed, $v_{k}^{t}$ decreases to $u^{t}$ , all having minimal singularity type (in particular they all have full mass); also the sequence $\left(\inf_{j\geq k}f_{j}\right)$ is increasing to $f$ . Letting $k\to\infty$ and using Theorem 2.6 we obtain

\theta_{u^{t}}^{n}\geq{\bf 1}_{\{u>V_{\theta}-t\}}f\mu,\ t>0.

Again, multiplying both sides with ${\bf 1}_{\{u>V_{\theta}-t\}}$ , $t>0$ , and using the locality of the complex Monge–Ampère operator with respect to the plurifine topology we arrive at

{\bf 1}_{\{u>V_{\theta}-t\}}\theta_{u}^{n}\geq{\bf 1}_{\{u>V_{\theta}-t\}}f\mu.

Finally, letting $t\to\infty$ we obtain the result. ∎

Theorem 5.17.

Assume that $\mu$ is a non-pluripolar positive measure such that $\mu(X)=1$ . Then there exists a unique $u\in\mathcal{E}(X,\theta,\phi)$ such that $\theta_{u}^{n}=\mu$ and $\sup_{X}u=0$ .

Proof.

The uniqueness follows from Theorem 3.13. For each $j>0$ , using Theorem 5.14 we solve

\theta_{v_{j}}^{n}=e^{j^{-1}v_{j}}\mu,\quad v_{j}\in\mathcal{E}(X,\theta,\phi).

We set $u_{j}:=v_{j}-\sup_{X}v_{j}$ , so that $\sup_{X}u_{j}=0$ . Up to extracting a subsequence, we can assume that $u_{j}\to u\in{\rm PSH}(X,\theta)$ in $L^{1}$ and almost everywhere. Since $u_{j}\leq\phi$ , we have $u\leq\phi$ . Observe also that $j^{-1}u_{j}$ converge in capacity to $0$ . Indeed, for any fixed $\varepsilon>0$ we have

{\rm Cap}_{\omega}(j^{-1}u_{j}<-\varepsilon)={\rm Cap}_{\omega}(u_{j}<-\varepsilon j)\to 0

as follows from [GZ05, Proposition 3.6].

Hence the functions $e^{j^{-1}u_{j}}$ converge in capacity to $1$ . It thus follows from Theorem 2.6 that

\lim_{j\to\infty}\int_{X}e^{j^{-1}u_{j}}d\mu=\mu(X)=1.

Since $\theta_{u_{j}}^{n}=e^{j^{-1}\sup_{X}v_{j}}e^{j^{-1}u_{j}}\mu$ , it follows from the above that $j^{-1}\sup_{X}v_{j}\to 0$ as $j\to\infty$ , hence $e^{j^{-1}v_{j}}$ converges to $1$ in $L^{1}(\mu)$ and almost everywhere. It thus follows from Lemma 5.16 that $\theta_{u}^{n}\geq\mu$ . Since $u\leq\phi$ , Theorem 3.3 ensures that $\int_{X}\theta_{u}^{n}\leq\int_{X}\theta_{\phi}^{n}=1=\mu(X)$ . Hence $\theta_{u}^{n}=\mu$ as desired. ∎

Log concavity of non-pluripolar masses.

We give a proof for the following result from [DDL21], initially conjectured in [BEGZ10, Conjecture 1.23]:

Theorem 5.18.

Let $T_{1},...,T_{n}$ be positive $(1,1)$ -currents on a compact Kähler manifold $X$ . Then

\int_{X}\langle T_{1}\wedge...\wedge T_{n}\rangle\geq\bigg{(}\int_{X}\langle T_{1}^{n}\rangle\bigg{)}^{\frac{1}{n}}...\bigg{(}\int_{X}\langle T_{n}^{n}\rangle\bigg{)}^{\frac{1}{n}}.

Proof.

We can assume that the classes of $T_{j}$ are big and their total masses are non-zero. Otherwise the right-hand side of the inequality to be proved is zero. Consider smooth closed real $(1,1)$ -forms $\theta^{j}$ , and $u_{j}\in{\rm PSH}(X,\theta^{j})$ such that $T_{j}=\theta^{j}_{u_{j}}$ .

We can assume that $\int_{X}\omega^{n}=1$ . For each $j=1,...,n$ , Theorem 5.17 ensures that there exists a normalizing constant $c_{j}>0$ and $\varphi_{j}\in\mathcal{E}(X,\theta^{j},P_{\theta^{j}}[u_{j}])$ such that $\big{(}\theta^{j}_{\varphi_{j}}\big{)}^{n}=c_{j}\omega^{n}$ .

Thus we have

c_{j}=\int_{X}\big{(}\theta^{j}_{\varphi_{j}}\big{)}^{n}=\int_{X}\big{(}\theta^{j}_{P_{\theta^{j}}[u_{j}]}\big{)}^{n}=\int_{X}\big{(}\theta^{j}_{u_{j}}\big{)}^{n}=\int_{X}\langle T_{j}^{n}\rangle.

Remark 3.4 and Theorem 3.15 then gives

\int_{X}\theta^{1}_{\varphi_{1}}\wedge...\wedge\theta^{n}_{\varphi_{n}}=\int_{X}\theta^{1}_{P_{\theta^{1}}[u_{1}]}\wedge...\wedge\theta^{n}_{P_{\theta^{n}}[u_{n}]}=\int_{X}\theta^{1}_{u_{1}}\wedge...\wedge\theta^{n}_{u_{n}}=\int_{X}\langle T_{1}\wedge\ldots\wedge T_{n}\rangle.

An application of the mixed Monge–Ampère inequalities ([BEGZ10, Proposition 1.11]) gives that $\theta^{1}_{\varphi_{1}}\wedge\ldots\wedge\theta^{n}_{\varphi_{n}}\geq c_{1}^{1/n}\ldots c_{n}^{1/n}\omega^{n}$ . The result follows from integrating this estimate. ∎

5.3 Relative boundedness of solutions

Recall that we are working with $\phi\in{\rm PSH}(X,\theta)$ such that $P_{\theta}[\phi]=\phi$ , and $\int_{X}\theta_{\phi}^{n}>0$ . Let $f\in L^{p}(\omega^{n})$ with $f\geq 0$ . In the previous section we have shown that the equation

\theta_{u}^{n}=f\omega^{n},\ \ u\in\mathcal{E}(X,\theta,\phi)

has a unique solution. In this section we will show that this solution has the same singularity type as $\phi$ . This generalizes [BEGZ10, Theorem B], that treats the particular case of solutions with minimal singularity type in a big class. Analogous results will be obtained for equations of the type (5.4) as well.

Our arguments follow the one in [GL21] which relies on quasi-psh envelopes. The original argument in [DDL18a, DDL21] was given using a Kolodziej type estimate, inspired from [Koł98].

In our study we make use of the following lemma multiple times:

Lemma 5.19.

Let $v\in\mathcal{E}(X,\theta,\phi)$ , $v\leq\phi$ , and $\gamma:\mathbb{R}^{+}\cup\{\infty\}\mapsto\mathbb{R}^{+}\cup\{\infty\}$ denote a concave continuous increasing function with $\gamma^{\prime}\leq 1$ . Then $\chi:=-\gamma(\phi-v)+\phi\in\textup{PSH}(X,\theta,\phi)$ and $\chi$ satisfies

\theta_{\chi}^{n}\geq(\gamma^{\prime}(\phi-v))^{n}\theta_{v}^{n}.

(5.7)

To be precise, in the above statement the function $\chi$ is equal to $-\infty$ on the pluripolar set $\{\phi=-\infty\}$ and $\chi=-\gamma(\phi-v)+\phi$ otherwise.

Proof.

Notice that there exists $\gamma_{k}:\mathbb{R}^{+}\cup\{\infty\}\mapsto\mathbb{R}^{+}\cup\{\infty\}$ smooth on $[0,\infty)$ , such that $\gamma_{k}\nearrow\gamma$ pointwise and $\gamma^{\prime}_{k}\leq 1$ . As a result, it will be enough to prove the theorem for $\gamma$ smooth.

First we want to show that $\chi=-\gamma(\phi-v)+\phi\in\textup{PSH}(X,\theta,\phi)$ . Observe that $\chi\leq\phi$ since $-\gamma\leq 0$ . In order to prove that $\chi$ is $\theta$ -psh, we will show that it is the decreasing limit of $\theta$ -psh functions.

Since the problem is local, we can work in a local chart $V\subset X$ where we write $\theta=dd^{c}g$ in $V$ . After slightly shrinking $V$ , the mollifications $v_{j}:=\rho_{1/j}*(v+g)-g$ and $\phi_{j}:=\rho_{1/j}*(\phi+g)-g$ are smooth approximants of $v$ and $\phi$ on $V$ . Moreover, they are $\theta$ -psh on $V$ , $v_{j}\searrow v$ , $\phi_{j}\searrow\phi$ , and $v_{j}\leq\phi_{j}$ . Let $\chi_{j}:=-\gamma(\phi_{j}-v_{j})+\phi_{j}$ . Then for any $j$ , $\chi_{j}$ is a smooth $\theta$ -psh function in $V$ since

	$\displaystyle\theta+dd^{c}\chi_{j}$	$\displaystyle=$	$\displaystyle\gamma^{\prime}(\phi_{j}-v_{j})\theta_{v_{j}}+(1-\gamma^{\prime}(\phi_{j}-v_{j}))\theta_{\phi_{j}}-\gamma^{\prime\prime}(\phi_{j}-v_{j})d(\phi_{j}-v_{j})\wedge d^{c}(\phi_{j}-v_{j})$		(5.8)
		$\displaystyle\geq$	$\displaystyle\gamma^{\prime}(\phi_{j}-v_{j})\theta_{v_{j}},$		(5.8)

where we used that $\gamma$ is concave and that $\gamma^{\prime}\leq 1$ .

We claim that $\chi_{j}$ is decreasing in $j$ . Indeed, for $s\in\mathbb{R}$ fixed, the function $t\mapsto-\gamma(t-s)+t$ is increasing in $[s,+\infty)$ because $1-\gamma^{\prime}(t-s)\geq 0$ . Using also that $\gamma$ is increasing we find that for $j\leq k$ we have

\chi_{j}=-\gamma(\phi_{j}-v_{j})+\phi_{j}\geq-\gamma(\phi_{j}-v_{k})+\phi_{j}\geq-\gamma(\phi_{k}-v_{k})+\phi_{k}=\chi_{k}.

It follows that $\chi_{j}$ decreases to some $\theta$ -psh function in $V$ which has to be $\chi$ . Indeed, the pointwise convergence $\chi_{j}\to\chi$ is seen to hold on the set $\{\phi>-\infty\}$ . Since we have $\chi_{j}\leq\phi_{j}\to-\infty$ on $\{\phi>-\infty\}$ , pointwise convergence holds everywhere on $X$ . It follows that $\chi$ is $\theta$ -psh on $X$ .

Next we now show that

\displaystyle\theta+dd^{c}\chi\geq\gamma^{\prime}(\phi-v)\theta_{v}.

(5.9)

For $t>0$ , let $\phi^{t}:=\max(\phi,V_{\theta}-t)$ , $v^{t}:=\max(v,V_{\theta}-t)$ and $\chi^{t}:=-\gamma({\phi^{t}-v_{t}})+\phi^{t}$ . Note that all of these potentials are locally bounded on $\textup{Amp}(\{\theta\})$ . As before, after slightly shrinking $V$ , the mollifications $v^{t}_{j}$ and $\phi^{t}_{j}$ are smooth approximants of $v^{t}$ and $\phi^{t}$ on $V$ . The same computations give that $\chi^{t}_{j}:=-\gamma({\phi^{t}_{j}-v^{t}_{j}})+\phi^{t}_{j}$ satisfies (5.8). By Proposition 2.2, letting $j\rightarrow\infty$ we obtain

\displaystyle\theta+dd^{c}\chi^{t}\geq\gamma^{\prime}(\phi^{t}-v^{t})(\theta+dd^{c}v^{t})\quad{\rm in}\;V.

Let $U_{t}=\{v>V_{\theta}-t\}=\{\phi>V_{\theta}-t\}\cap\{v>V_{\theta}-t\}$ . Using plurifine locality, we get that

\displaystyle{\bf 1}_{U_{t}}(\theta+dd^{c}\chi)={\bf 1}_{U_{t}}(\theta+dd^{c}\chi_{t})\geq{\bf 1}_{U_{t}}\gamma^{\prime}(\phi-v)(\theta+dd^{c}v)\quad{\rm in}\;V.

Letting $t\to\infty$ , we arrive at (5.9).

Unfortunately (5.9) does not directly imply (5.7) since $\gamma^{\prime}(\phi-v)\theta_{v}$ is not a closed form. However the above approximation process allows to conclude on the chart $V$ considered above.
Since $\chi^{t}_{j}$ and $v^{t}_{j}$ are smooth, from (5.9) we get that

\displaystyle(\theta+dd^{c}\chi^{t}_{j})^{n}\geq(\gamma^{\prime}(\phi^{t}_{j}-v^{t}_{j}))^{n}(\theta+dd^{c}v^{t}_{j})^{n}\quad{\rm in}\;V.

We now conclude similarly. By Proposition 2.2, letting $j\rightarrow\infty$ we obtain

\displaystyle(\theta+dd^{c}\chi^{t})^{n}\geq(\gamma^{\prime}(\phi^{t}-v^{t}))^{n}(\theta+dd^{c}v^{t})^{n}\quad{\rm in}\;V.

Let $U_{t}=\{\phi>V_{\theta}-t\}\cap\{v>V_{\theta}-t\}$ . Using plurifine locality, we get that

\displaystyle{\bf 1}_{U_{t}}(\theta+dd^{c}\chi)^{n}\geq{\bf 1}_{U_{t}}(\gamma^{\prime}(\phi-v))^{n}(\theta+dd^{c}v)^{n}\quad{\rm in}\;V.

Letting $t\to\infty$ , we arrive at the conclusion. ∎

Theorem 5.20.

Let $u\in\mathcal{E}(X,\theta,\phi)$ with $\sup_{X}u=0$ . If $\theta_{u}^{n}=f\omega^{n}$ for some $f\in L^{p}(\omega^{n}),\ p>1,$ then $u$ has the same singularity type as $\phi$ . More precisely:

\phi-C\Big{(}\|f\|_{L^{p}},p,\omega,\theta,\int_{X}\theta_{\phi}^{n}\Big{)}\leq u\leq\phi.

Proof.

To simplify the notation, we set $\mu=f\omega^{n}$ .

A priori estimate. We assume at the moment that $u-\phi$ is bounded, hence

T_{\max}:=\sup\{t>0\;:\;\mu(u<\phi-t)>0\}<\infty.

Our goal is to establish a uniform bound on $T_{\max}$ . Since $f\in L^{p}(\omega^{n})$ and ${\rm PSH}(X,\omega)\subset L^{q}(\omega^{n})$ for any $q>0$ , by Hölder inequality, ${\rm PSH}(X,\omega)\subset L^{r}(\mu)$ , for any $r>0$ and the quantity

A_{r}(\mu):=\sup\left\{\int_{X}(-h)^{r}d\mu\;:\;h\in{\rm PSH}(X,\omega),\;\sup_{X}h=0\right\}

is finite and it depends on an upper bound for $\|f\|_{p}$ .

By definition, $u\geq\phi-T_{\max}$ almost everywhere with respect to $\mu=\theta_{u}^{n}$ , hence everywhere by the domination principle (Theorem 3.12), providing the desired a priori bound. In particular $|u-\phi|\leq T_{\max}$ .

We let $\chi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ denote a convex increasing function such that $\chi(0)=0$ and $\chi^{\prime}(0)=1$ (hence $\chi^{\prime}(t)\geq 1$ ). Let $\gamma:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ denote the inverse function of $\chi$ , which is concave and increasing. We set $\psi=-\chi(\phi-u)+\phi$ , $v=P_{\theta}(\psi)$ , and observe that

\varphi:=-\gamma(\phi-v)+\phi\leq-\gamma(\phi-\psi)+\phi=-\gamma(\chi(\phi-u))+\phi\leq u

with equality on the contact set $\{v=\psi\}$ . Since $u$ has the same singularity type as $\phi$ , so does $v$ . In particular, by Proposition 3.2

\int_{X}\theta_{v}^{n}=\int_{X}\theta_{\phi}^{n}=\int_{X}\theta_{u}^{n}.

Observe that $\gamma^{\prime}\leq 1$ since $\chi^{\prime}(\gamma(t))\gamma^{\prime}(t)=1$ and that $\phi-u=\gamma(\phi-v)$ on $\{v=\psi\}$ . Lemmas 5.19 and 2.9 thus give

{\bf 1}_{\{v=\psi\}}(\chi^{\prime}(\phi-u))^{-n}\theta_{v}^{n}={\bf 1}_{\{v=\psi\}}(\gamma^{\prime}(\phi-v))^{n}\theta_{v}^{n}\leq{\bf 1}_{\{v=\psi\}}\theta_{\varphi}^{n}\leq{\bf 1}_{\{v=\psi\}}\theta_{u}^{n}\leq\theta_{u}^{n},

hence ${\bf 1}_{\{v=\psi\}}\theta_{v}^{n}\leq(\chi^{\prime}(\phi-u))^{n}\mu$ . By Theorem 2.7, we can infer $\theta_{v}^{n}\leq(\chi^{\prime}(\phi-u))^{n}\mu$ .

Step 1: Controlling the energy of $v$ . The convexity of $\chi$ and the normalization $\chi(0)=0$ yields $\chi(t)\leq t\chi^{\prime}(t)$ . Since $v=-\chi(\phi-u)+\phi$ on $\{v=\psi\}$ , the above inequality, the convexity of $\chi$ and Hölder inequality (with $p=n+2$ and $q=(n+2)/(n+1)$ ) yields

	$\displaystyle\int_{X}(\phi-v)\theta_{v}^{n}$	$\displaystyle=\int_{X}\chi(\phi-u)\theta_{v}^{n}\leq\int_{X}\chi(\phi-u)(\chi^{\prime}(\phi-u))^{n}d\mu$
		$\displaystyle\leq\int_{X}(\phi-u)(\chi^{\prime}(\phi-u))^{n+1}d\mu$
		$\displaystyle\leq\left(\int_{X}(\phi-u)^{n+2}d\mu\right)^{\frac{1}{n+2}}\left(\int_{X}(\chi^{\prime}(\phi-u))^{n+2}d\mu\right)^{\frac{n+1}{n+2}}$
		$\displaystyle\leq A_{n+2}(\mu)^{\frac{1}{n+2}}.\left(\int_{X}(\chi^{\prime}(\phi-u))^{n+2}d\mu\right)^{\frac{n+1}{n+2}}.$

Step 2: Controlling the norms $||u||_{L^{m}}$ . To simplify the notation we set $m=n+3$ . We are going to choose below the weight $\chi$ in such a way that $\int_{X}(\chi^{\prime}(\phi-u))^{n+2}d\mu\leq 2$ . This provides a uniform lower bound on $\sup_{X}v$ as we now explain. Indeed,

0\leq(-\sup_{X}v)\mu(X)=-\sup_{X}(v-\phi)\int_{X}\theta_{v}^{n}\leq\int_{X}(\phi-v)\theta_{v}^{n}\leq 2A_{n+2}(\mu)^{\frac{1}{n+2}}=C_{1},

where the first identity follows from Lemma 3.5. This yields $\sup_{X}v\geq-C_{1}\mu(X)^{-1}$ . We infer that $v$ belongs to a compact set of $\theta$ -psh functions, hence its norm $||v||_{L^{m}(\mu)}$ is under control. Since $v\leq\psi$ it follows that $\phi-v\geq\chi(\phi-u)\geq 0$ and hence

	$\displaystyle\int_{X}(\chi(\phi-u))^{m}d\mu$	$\displaystyle\leq\int_{X}(\phi-v)^{m}d\mu$
		$\displaystyle\leq\int_{X}\|v-\sup_{X}v+\sup_{X}v\|^{m}d\mu$
		$\displaystyle\leq 2^{m-1}\int_{X}(\|v-\sup_{X}v\|^{m}+\|\sup_{X}v\|^{m})d\mu$
		$\displaystyle\leq 2^{m-1}A_{m}(\mu)+2^{m-1}\|\sup_{X}v\|^{m}\mu(X)\leq C_{2}.$

Chebyshev’s inequality thus yields

{\mu}(u<\phi-t)\leq\frac{1}{(\chi(t))^{m}}\int_{X}(\chi(\phi-u))^{m}d\mu\leq\frac{C_{2}}{(\chi(t))^{m}}.

(5.10)

Step 3: Choice of $\chi$ . If $g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is strictly increasing with $g(0)=1$ , Lebesgue’s formula gives

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

(5.11)

Fix $0<T_{0}<T_{\max}$ . Setting $g(t)=(\chi^{\prime}(t))^{n+2}$ we define $\chi$ by imposing $\chi(0)=0$ , $\chi^{\prime}(0)=1$ , and

g^{\prime}(t)=\begin{cases}\dfrac{1}{(1+t)^{2}{\mu}(u<\phi-t)},\;\text{if}\;t\leq T_{0}\\ \;\\ \frac{1}{(1+t)^{2}}\;\;\;\;\text{ if}\;t>T_{0}\end{cases}.

This choice guarantees that $\chi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is convex increasing with $\chi^{\prime}\geq 1$ , and by (5.11) that

\int_{X}(\chi^{\prime}(\phi-u))^{n+2}d\mu\leq\mu(X)+\int_{0}^{\infty}\frac{dt}{(1+t)^{2}}=2.

Step 4: Conclusion. Observe that $g(t)\geq g(0)=1$ , hence $\chi^{\prime}(t)=(g(t))^{\frac{1}{n+2}}\geq 1$ . This yields

\chi(t)=\int_{0}^{t}\chi^{\prime}(s)ds\geq t.

(5.12)

In particular $\chi(t)\geq t$ . Together with (5.10), our choice of $\chi$ yields, for all $t\in[0,T_{0}]$ ,

\frac{1}{(1+t)^{2}g^{\prime}(t)}={\mu}(u<\phi-t)\leq\frac{C_{2}}{(\chi(t))^{m}}.

This reads

(\chi(t))^{m}\leq C_{2}(1+t)^{2}g^{\prime}(t)=(n+2)C_{2}(1+t)^{2}\chi^{\prime\prime}(t)(\chi^{\prime}(t))^{n+1},\quad\forall t\in[0,T_{0}].

Multiplying by $\chi^{\prime}\geq 1$ , integrating between $0$ and $t$ , we get that for all $t\in[0,T_{0}]$

$\displaystyle\frac{(\chi(t))^{m+1}}{m+1}$	$\displaystyle\leq$	$\displaystyle(n+2)C_{2}\int_{0}^{t}(1+s)^{2}\chi^{\prime\prime}(s)(\chi^{\prime}(s))^{n+2}\ ds$
	$\displaystyle\leq$	$\displaystyle(n+2)C_{2}\int_{0}^{t}[(1+s)^{2}\chi^{\prime\prime}(s)(\chi^{\prime}(s))^{n+2}+2(1+s)((\chi^{\prime}(s))^{n+3}-1)]\ ds$
	$\displaystyle=$	$\displaystyle\frac{(n+2)C_{2}(1+t)^{2}}{n+3}(1+s)^{2}\left((\chi^{\prime}(s))^{n+3}-1\right)\big{\|}_{s=0}^{s=t}$
	$\displaystyle\leq$	$\displaystyle\frac{(n+2)C_{2}(1+t)^{2}}{n+3}\left((\chi^{\prime}(t))^{n+3}-1\right)$
	$\displaystyle\leq$	$\displaystyle C_{3}(1+t)^{2}(\chi^{\prime}(t))^{n+3}.$

Recall that we choose $m=n+3$ so that $\alpha:=m+1>\beta:=n+3>2.$ The previous inequality then reads

(1+t)^{-\frac{2}{\beta}}\leq C_{4}{\chi^{\prime}(t)}{\chi(t)^{-\frac{\alpha}{\beta}}}.

Since $\alpha>\beta>2$ and $\chi(1)\geq 1$ (by (5.12)), integrating the above inequality between $1$ and $T_{0}$ we obtain

\frac{\beta-2}{\beta}(1+t)^{\frac{\beta-2}{\beta}}\big{|}_{1}^{T_{0}}\leq C_{4}\frac{\beta-\alpha}{\beta}\chi(t)^{\frac{-\alpha+\beta}{\beta}}\big{|}_{1}^{T_{0}}\leq C_{4}\frac{\alpha-\beta}{\beta}\chi(1)^{\frac{-\alpha+\beta}{\beta}}\leq C_{5}.

It then follows from the above that $T_{0}\leq C_{6},$ for some uniform constant $C_{6}>0$ . Since $T_{0}$ was chosen arbitrarily in $(0,T_{\max})$ the uniform estimate for $T_{\max}$ follows.

Relative boundedness of $u$ .

To finish the proof we finally prove that $u-\phi$ is bounded. We fix $1<q<p$ and a constant $\varepsilon>0$ so small that $e^{-\varepsilon h}f\in L^{q}(X)$ for all $h\in{\rm PSH}(X,\omega)$ . For each $j$ we solve

u_{j}\in\mathcal{E}(X,\theta,\phi),\;\theta_{u_{j}}^{n}={\bf 1}_{\{u>\phi-j\}}e^{\varepsilon(u_{j}-\max(u,\phi-j)}\mu.

Observe that for any fixed $j$ , $\max(u,\phi-j)$ is a subsolution of the above equation because

\theta_{\max(u,\phi-j)}^{n}\geq{\bf 1}_{\{u>\phi-j\}}\theta_{u}^{n}={\bf 1}_{\{u>\phi-j\}}e^{\varepsilon(\max(u,\phi-j)-\max(u,\phi-j))}\mu.

Lemma 5.15 then gives $u_{j}\geq\max(u,\phi-j)$ , hence $\sup_{X}u_{j}\geq 0$ and $u_{j}-\phi$ is bounded. If $j<k$ , then

\theta_{u_{k}}^{n}={\bf 1}_{\{u>\phi-k\}}{\bf 1}_{\{u>\phi-k\}}e^{\varepsilon(u_{k}-\max(u,\phi-k)}\mu\geq{\bf 1}_{\{u>\phi-j\}}e^{\varepsilon(u_{k}-\max(u,\phi-j)}\mu.

Invoking again Lemma 5.15 we obtain $u_{j}\geq u_{k}$ . The measures

\mu_{j}:={\bf 1}_{\{u>\phi-j\}}e^{\varepsilon(u_{j}-\max(u,\phi-j)}\mu=f_{j}\omega^{n}

have densities $f_{j}\leq e^{\varepsilon u_{1}}e^{-\varepsilon\max(u,\phi-j)}f\leq e^{\varepsilon\sup_{X}u_{1}}e^{-\varepsilon\max(u,\phi-j)}f$ which are uniformly bounded in $L^{q}(X)$ . By the above a priori estimate, we have a uniform bound $u_{j}\geq\phi-C$ , hence $v:=\lim_{j\to\infty}u_{j}$ satisfies $v\geq\phi-C$ and $\theta_{v}^{n}=e^{\varepsilon(v-u)}\mu$ . Since $\theta_{v}^{n}=e^{\varepsilon(u-u)}\mu$ , Lemma 5.15 ensures $u=v$ . Thus $u-\phi$ is bounded as desired. This finishes the proof. ∎

Corollary 5.21.

If $\lambda>0$ and $u\in\mathcal{E}(X,\theta,\phi)$ , $\theta_{u}^{n}=e^{\lambda u}f\omega^{n}$ for some $f\in L^{p}(\omega^{n}),\ p>1,$ then $u$ has the same singularity type as $\phi$ .

Proof.

Since $u$ is bounded from above on $X$ and $\lambda>0$ it follows that $e^{\lambda u}f\in L^{p}(X,\omega^{n})$ , $p>1$ . The result follows from Theorem 5.20. ∎

5.4 Naturality of model type singularities and examples

One may still wonder if our choice of model potentials is a natural one in our pursuit of complex Monge–Ampère equations with prescribed singularity. We address these doubts in the next well posedness result.

Theorem 5.22.

Suppose that $\psi\in\textup{PSH}(X,\theta)$ and the equation

\theta_{u}^{n}=f\omega^{n}

has a solution $u\in\textup{PSH}(X,\theta)$ with the same singularity type as $\psi$ , for all $f\in L^{\infty}(X),\ f\geq 0$ satisfying $\int_{X}\theta_{\psi}^{n}=\int_{X}f\omega^{n}>0$ . Then $\psi$ has model type singularity.

Proof.

Suppose that $[\psi]$ is not of model type. Then $P_{\theta}[\psi]$ is strictly less singular than $\psi$ . On the other hand $\mathcal{E}(X,\theta,\psi)\subset\mathcal{E}(X,\theta,P_{\theta}[\psi])$ as $\int_{X}\theta_{\psi}^{n}=\int_{X}\theta_{P_{\theta}[\psi]}^{n}$ .

By Theorem 3.6, there exists $g\in L^{\infty}$ such that $\theta^{n}_{P_{\theta}[\psi]}=g\omega^{n}.$ By uniqueness (Theorem 3.13), $P_{\theta}[\psi]$ is the only solution of this last equation inside $\mathcal{E}(X,\theta,P_{\theta}[\psi])$ .

Since $\mathcal{E}(X,\theta,\psi)\subset\mathcal{E}(X,\theta,P_{\theta}[\psi])$ , but $P_{\theta}[\psi]\notin\mathcal{E}(X,\theta,\psi)$ , we get that $\theta_{u}^{n}=g\omega^{n}$ cannot have any solution that has the same singularity type as $\psi$ . ∎

Next we point out a simple way to construct potential with model singularity types:

Proposition 5.23.

Suppose that $\psi\in\textup{PSH}(X,\theta)$ and $\theta_{\psi}^{n}=f\omega^{n}$ for some $f\in L^{p}(\omega^{n}),\ p>1$ with $\int_{X}f\omega^{n}>0$ . Then $\psi$ has model type singularity.

Proof.

We first observe that $\psi\in\mathcal{E}(X,\theta,P_{\theta}[\psi])$ . Since $\theta_{\psi}^{n}$ has $L^{p}$ density with $p>1$ , it thus follows from Theorem 5.20 that $\psi-P_{\theta}[\psi]$ is bounded on $X$ , hence $[\psi]=[P_{\theta}[\psi]]$ , implying that $\psi$ has model singularity type. ∎

As noticed in [RW14] (see [RS05] for the local case), all analytic singularity types are of model type.

Proposition 5.24.

Suppose $\psi\in\textup{PSH}(X,\theta)$ has analytic singularity type, i.e. $\psi$ can be locally written as $\frac{c}{2}\log\big{(}\sum_{j}|f_{j}|^{2}\big{)}+g$ , where $f_{j}$ are holomorphic, $c>0$ and $g$ is bounded. Then $[\psi]$ is of model type.

We give an argument that is different from the elementary one given in [DDL18a, Section 4.5]. Following [RW14], we use resolution of singularities, and get a slightly more general result (with $g$ being bounded instead of smooth).

Proof.

Let $\mathcal{I}$ be the coherent sheaf of holomorphic functions $f$ satisfying $|f|\leq Ae^{\frac{2\psi}{c}}$ for some $A>0$ and $\pi:Y\to X$ be a resolution of singularities of $\mathcal{I}$ , as in [Dem12, Remark 5.9]. We obtain that the singularity type of $\psi\circ\pi$ is modelled by an snc divisor $D=\sum_{j}\alpha_{j}D_{j}\subset Y$ , where $\alpha_{j}>0$ . That is to say, the Lelong numbers of $\psi\circ\pi$ along each $D_{j}$ is $\alpha_{j}$ , i.e., $\psi\circ\pi$ has $\alpha_{j}\log|z|$ type poles along each $D_{j}$ .

By Lemma 5.1 we know that the Lelong numbers of $P_{\pi^{*}\theta}[\psi\circ\pi]$ and $\psi\circ\pi$ are the same. In particular, we obtain that $[P_{\pi^{*}\theta}[\psi\circ\pi]]=[\psi\circ\pi]$ . Since any $\pi^{*}\theta$ -psh function can be written as $u\circ\pi$ , for some $\theta$ -psh function $u$ , it follows that for any $C>0$ , $P_{\pi^{*}\theta}(\psi\circ\pi+C,0)=P_{\theta}(\psi+C,0)\circ\pi$ . This means that $P_{\pi^{*}\theta}[\psi\circ\pi]=P_{\theta}[\psi]\circ\pi$ . Combining the above and pushing forward to $X$ we obtain that $[P_{\theta}[\psi]]=[\psi]$ , as desired. ∎

Chapter 6 The finite energy range of the complex Monge–Ampère operator

In this chapter we give a characterization of the Borel measures $\mu$ that are equal to the complex Monge–Ampere measure of some $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ , with $\chi$ having polynomial growth, and $\phi$ a model potential ( $\phi=P_{\theta}[\phi]$ ) with $\int_{X}\theta_{\phi}^{n}>0$ .

Before we can achieve this we need to develop more potential theory. A weight is a continuous increasing function $\chi:[0,\infty)\rightarrow[0,\infty)$ such that $\chi(0)=0$ and $\chi(\infty)=\infty$ . Denote by $\chi^{-1}$ its inverse function, i.e. such that $\chi(\chi^{-1}(t))=t$ for all $t\geq 0$ .

Throughout this section we assume that the weight $\chi$ satisfies the following condition

\forall t\geq 0,\;\forall\lambda\geq 1,\;\chi(\lambda t)\leq\lambda^{M}\chi(t),

(6.1)

where $M\geq 1$ is a fixed constant. Observe that from (6.1) it follows that

\forall t\geq 0,\;\forall\gamma<1,\;\chi(\gamma t)\geq\gamma^{M}\chi(t).

(6.2)

We fix $\phi$ a model potential and we let $\mathcal{E}_{\chi}(X,\theta,\phi)$ denote the set of all $u\in\mathcal{E}(X,\theta,\phi)$ such that

E_{\chi}(u,\phi):=\int_{X}\chi(|u-\phi|)\theta_{u}^{n}<\infty.

When $\phi=V_{\theta}$ , we denote $\mathcal{E}(X,\theta)=\mathcal{E}(X,\theta,V_{\theta})$ , $\mathcal{E}_{\chi}(X,\theta)=\mathcal{E}_{\chi}(X,\theta,V_{\theta}$ ) and $E_{\chi}(u)=E_{\chi}(u,V_{\theta})$ . Compared to [GZ07], we have changed the sign of the weight, but the weighted classes are the same. The following simple observation shows that the class $\mathcal{E}(X,\theta,\phi)$ is stable under adding a constant.

Lemma 6.1.

If $u$ belongs to $\mathcal{E}_{\chi}(X,\theta,\phi)$ then so does $u+C$ for any constant $C$ .

Proof.

Since $\chi$ satisfies (6.1), for $t>0$ and $s>0$ we have

\chi(t+s)=\chi(2(t+s)/2)\leq 2^{M}\chi((t+s)/2)\leq 2^{M}\chi(\max(t,s))\leq 2^{M}\max(\chi(t),\chi(s)),

where the last two inequalities follow from the fact that $\chi$ is increasing. Then

	$\displaystyle\int_{X}\chi(\|u+C-\phi\|)\theta_{u}^{n}$	$\displaystyle\leq\int_{X}\chi(\|u-\phi\|+\|C\|)\theta_{u}^{n}$
		$\displaystyle\leq 2^{M}\int_{X}\max(\chi(\|u-\phi\|),\chi(\|C\|))\theta_{u}^{n}$
		$\displaystyle\leq 2^{M}\max\left(\int_{X}\chi(\|u-\phi\|)\theta_{u}^{n},\ \chi(\|C\|)\int_{X}\theta_{u}^{n}\right)<\infty.$

∎

Next we prove a few of technical results that will be often used.

Lemma 6.2.

There exists a uniform constant $C>0$ such that for all $u\in{\rm PSH}(X,\theta,\phi)$ normalized with $\sup_{X}u=0$ we have

\int_{X}\chi(\phi-u)\theta_{\phi}^{n}\leq C.

Proof.

We recall that by Theorem 3.6, $\theta_{\phi}^{n}\leq{\bf 1}_{\{\phi=0\}}\theta^{n}\leq A\omega^{n}$ , for some $A>0$ . Also, thanks to (6.1), if $\phi-u\geq 1$ we have $\chi(\phi-u)\leq(\phi-u)^{M}\chi(1)$ ; otherwise (since $\chi$ is increasing) we have $\chi(\phi-u)\leq\chi(1)$ . In both cases we can infer that

\int_{X}\chi(\phi-u)\theta_{\phi}^{n}\leq C^{\prime}\int_{X}(|\phi|^{M}+|u|^{M}+1)\omega^{n}.

The conclusion then follows from the fact that $\int_{X}|h|^{M}\omega^{n}$ is uniformly bounded for $h\in{\rm PSH}(X,\theta)$ with $\sup_{X}h=0$ . ∎

Lemma 6.3.

Let $u\in\mathcal{E}(X,\theta,\phi)$ with $\sup_{X}u=0$ . Then, for any $j\in\{1,...,n\}$

\int_{X}\chi(\phi-u)\theta_{u}^{j}\wedge\theta_{\phi}^{n-j}\leq\int_{X}\chi(\phi-u)\theta_{u}^{n}.

Proof.

By Lemma 3.5 we have that $u\leq\phi\leq 0$ . By the partial comparison principle (Proposition 3.22) we have that for any $j\in\{1,...,n\}$

	$\displaystyle\int_{X}\chi(\phi-u)\theta_{u}^{j}\wedge\theta_{\phi}^{n-j}$	$\displaystyle=\int_{0}^{\infty}\theta_{u}^{j}\wedge\theta_{\phi}^{n-j}(u<\phi-\chi^{-1}(t))dt$
		$\displaystyle\leq\int_{0}^{\infty}\theta_{u}^{n}(u<\phi-\chi^{-1}(t))dt=\int_{X}\chi(\phi-u)\theta_{u}^{n}.$

∎

Lemmas 6.4, 6.5 below are essentially known by [GZ07], but we repeat the simple proof for the reader’s convenience. Similar simplifications can also be found in [Gup22].

Lemma 6.4.

If $u,v\in\mathcal{E}(X,\theta,\phi)$ and $u,v\leq 0$ , then

\int_{X}\chi(\phi-u)\theta_{v}^{n}\leq 2^{n+M}E_{\chi}(u,\phi)+E_{\chi}(v,\phi).

Proof.

By Lemma 3.5 we have that $u\leq\phi\leq 0$ . By the comparison principle, Corollary 3.23, we have

	$\displaystyle\int_{X}\chi(\phi-u)\theta_{v}^{n}$	$\displaystyle=\int_{0}^{\infty}\theta_{v}^{n}(u<\phi-\chi^{-1}(t))dt$
		$\displaystyle\leq\int_{0}^{\infty}\theta_{v}^{n}(2u<v+\phi-\chi^{-1}(t))dt+\int_{0}^{\infty}\theta_{v}^{n}(v<\phi-\chi^{-1}(t))dt$
		$\displaystyle\leq 2^{n}\int_{0}^{\infty}\left(\theta+dd^{c}{\frac{v+\phi}{2}}\right)^{n}\left(u<\frac{v+\phi-\chi^{-1}(t)}{2}\right)dt+E_{\chi}(v,\phi)$
		$\displaystyle\leq 2^{n}\int_{0}^{\infty}\theta_{u}^{n}(2u<2\phi-\chi^{-1}(t))dt+E_{\chi}(v,\phi)$
		$\displaystyle=2^{n}\int_{X}\chi(2\phi-2u)\theta_{u}^{n}+E_{\chi}(v,\phi)$
		$\displaystyle\leq 2^{n+M}E_{\chi}(u,\phi)+E_{\chi}(v,\phi),$

where in the second line we have used the trivial inclusion

\{2u\geq v+\phi-\chi^{-1}(t)\}\cap\{v\geq\phi-\chi^{-1}(t)\}\subseteq\{u\geq\phi-\chi^{-1}(t)\},

in the third one we have used

\theta_{v}^{n}\leq 2^{n}\left(\theta+dd^{c}\frac{v+\phi}{2}\right)^{n},

and in the last line we have used (6.1) with $\lambda=2$ . ∎

Lemma 6.5.

For all $u,v\in\mathcal{E}(X,\theta,\phi)$ with $u\leq v\leq 0$ we have

\int_{X}\chi(\phi-v)\,\theta_{v}^{n}\leq\int_{X}\chi(\phi-u)\,\theta_{v}^{n}\leq 2^{n+M}E_{\chi}(u,\phi).

Proof.

By the comparison principle, Corollary 3.23, and the fact that $u\leq v\leq\phi$ , we have

	$\displaystyle\int_{X}\chi(\phi-u)\,\theta_{v}^{n}$	$\displaystyle=\int_{0}^{\infty}\theta_{v}^{n}(u<\phi-\chi^{-1}(t))dt$
		$\displaystyle\leq\int_{0}^{\infty}\theta_{v}^{n}(2u<v+\phi-\chi^{-1}(t))dt$
		$\displaystyle\leq 2^{n}\int_{0}^{\infty}\theta_{\frac{v+\phi}{2}}^{n}\left(u<\frac{v+\phi-\chi^{-1}(t)}{2}\right)dt$
		$\displaystyle\leq 2^{n}\int_{0}^{\infty}\theta_{u}^{n}(2u<2\phi-\chi^{-1}(t))dt$
		$\displaystyle=2^{n}\int_{X}\chi(2\phi-2u)\theta_{u}^{n}\leq 2^{n+M}E_{\chi}(u,\phi).$

∎

Proposition 6.6.

If $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ and $u\leq v$ then $v\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . Moreover, the class $\mathcal{E}_{\chi}(X,\theta,\phi)$ is convex.

Proof.

In view of Lemma 6.1 we can assume that $v\leq 0$ . Lemma 6.5 then yields

E_{\chi}(v,\phi)=\int_{X}\chi(\phi-v)\,\theta_{v}^{n}\leq 2^{n+M}E_{\chi}(u,\phi).

We next prove that the class $\mathcal{E}_{\chi}(X,\theta,\phi)$ is convex. Assume $u$ and $v$ are in $\mathcal{E}_{\chi}(X,\theta,\phi)$ . It follows from Corollary 3.19 that $w:=P_{\theta}(u,v)$ belongs to $\mathcal{E}(X,\theta,\phi)$ . From Theorem 3.6 we also have

\int_{X}\chi(\phi-w)\theta_{w}^{n}\leq\int_{X}\chi(\phi-u)\theta_{u}^{n}+\int_{X}\chi(\phi-v)\theta_{v}^{n}<\infty,

hence $w\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . For $t\in[0,1]$ , since $w\leq tu+(1-t)v$ , it follows from Lemma 6.5 that $tu+(1-t)v\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . ∎

Lemma 6.7.

Assume $(u_{j})$ is a sequence in $\mathcal{E}_{\chi}(X,\theta,\phi)$ converging in $L^{1}$ to $u\in{\rm PSH}(X,\theta,\phi)$ . If $\sup_{j}E_{\chi}(u_{j},\phi)<\infty$ , then $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ .

Proof.

We can assume $u_{j}\leq 0$ for all $j$ . Define $v_{k}:=(\sup_{j\geq k}u_{j})^{*}\in{\rm PSH}(X,\theta,\phi)$ . Then $v_{k}\searrow u$ and $u$ is more singular than $\phi$ . By Theorem 3.3 we then have $\int_{X}\theta_{u}^{n}\leq\int_{X}\theta_{\phi}^{n}$ .
Since $0\geq v_{k}\geq u_{k}$ , we have $\chi(\phi-v_{k})\leq\chi(\phi-u_{k})$ . Lemma 6.5 then ensures that $E_{\chi}(v_{k},\phi)\leq 2^{n+M}E_{\chi}(u_{k},\phi)$ and the latter quantity is uniformly bounded by assumption.

Fix $t>0$ and consider $v_{k,t}:=\max(v_{k},\phi-t)$ . Note that $v_{k,t}\searrow\max(u,\phi-t)$ as $k\to\infty$ and that by Lemma 6.5 we have $E_{\chi}(v_{k,t},\phi)\leq 2^{n+M}E_{\chi}(v_{k},\phi)$ . This means that $E_{\chi}(v_{k,t},\phi)$ has a uniform upper bound. Moreover, the functions $\chi(\phi-v_{k,t})$ are quasi-continuous and uniformly bounded on $X$ . It thus follows from Theorem 2.6 that

\liminf_{k\to\infty}\int_{X}\chi(\phi-v_{k,t})(\theta+dd^{c}v_{k,t})^{n}\geq\int_{X}\chi(\phi-\max(u,\phi-t))(\theta+dd^{c}\max(u,\phi-t))^{n}.

The above estimate then gives a uniform upper bound for $E_{\chi}(\max(u,\phi-t),\phi)$ . We can thus infer that there exists a constant $C>0$ such that

C\geq\int_{X}\chi(\phi-\max(u,\phi-t))\theta_{\max(u,\phi-t)}^{n}\geq\chi(t)\int_{\{u\leq\phi-t\}}\theta_{\max(u,\phi-t)}^{n}.

Letting $t\to\infty$ yields that $\int_{\{u\leq\phi-t\}}\theta_{\max(u,\phi-t)}^{n}\rightarrow 0$ (since $\chi(t)\rightarrow\infty$ ). Hence, using the plurifine property of the Monge–Ampère operator and the fact that $u$ is more singular than $\phi$ , we get

\int_{X}\theta_{\phi}^{n}=\lim_{t\to\infty}\int_{X}\theta_{\max(u,\phi-t)}^{n}=\lim_{t\to\infty}\bigg{(}\int_{\{u\leq\phi-t\}}\theta_{\max(u,\phi-t)}^{n}+\int_{\{u>\phi-t\}}\theta_{u}^{n}\bigg{)}\leq\int_{X}\theta_{u}^{n}\leq\int_{X}\theta_{\phi}^{n}.

This means that $u\in\mathcal{E}(X,\theta,\phi)$ . Using plurifine locality, we also obtain

\int_{\{u>\phi-t\}}\chi(\phi-u)\theta_{u}^{n}\leq C,

and letting $t\to\infty$ , we see that $\int_{X}\chi(\phi-u)\theta_{u}^{n}\leq C$ . Hence $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . ∎

Lemma 6.8.

Let $\mu$ be a positive Borel measure on $X$ . Assume that $\mu(\{\phi=-\infty\})=0$ and $\chi(\phi-u)\in L^{1}(\mu)$ for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ and fix a constant $A>0$ . Then there exists a constant $C=C(A)>0$ such that, for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ with $\sup_{X}u=0$ and $E_{\chi}(u,\phi)\leq A$ we have

\int_{X}\chi(\phi-u)d\mu\leq C.

Observe that the condition $\mu(\{\phi=-\infty\})=0$ is necessary. Without it, $\mu$ -integrability of $\chi(\phi-u)$ can not be discussed, as the values of this function are not defined on the set $\{\phi=-\infty\}$ .

Proof.

Assume by contradiction that there exists a sequence $(u_{j})\subset\mathcal{E}_{\chi}(X,\theta,\phi)$ satisfying $\sup_{X}u_{j}=0$ and $E_{\chi}(u_{j},\phi)\leq A$ such that

\int_{X}\chi(\phi-u_{j})\,d\mu\geq 4^{jM},

where $M$ is the exponent appearing in (6.1).
Define $v_{k}:=P_{\theta}(\min_{1\leq j\leq k}(2^{-j}u_{j}+(1-2^{-j})\phi))\leq\phi$ . Observe that $v_{k}$ is decreasing and setting

v:=\lim_{k}v_{k},\quad\psi:=\sum_{j\geq 1}2^{-j}u_{j},

then, since $u_{j}\leq\phi$ , we can check that $\psi\leq 2^{-j}u_{j}+(1-2^{-j})\phi\leq\phi$ . It then follows that $v\geq\psi\in{\rm PSH}(X,\theta,\phi)$ . We fix $k\geq 2$ and for $1\leq l\leq k$ we set

D_{1}:=\left\{v_{k}=2^{-1}u_{1}+2^{-1}\phi\right\}

and

D_{l}:=\left\{v_{k}=2^{-l}u_{l}+(1-2^{-l})\phi<\min_{1\leq p\leq l-1}(2^{-p}u_{p}+(1-2^{-p})\phi)\right\}\quad{\rm for}\;\ell\geq 2

Note that the sets $D_{l}$ are pairwise disjoint and

\bigcup_{k\geq l\geq 1}D_{l}=\{v_{k}=\min_{1\leq j\leq k}(2^{-j}u_{j}+(1-2^{-j})\phi)\}.

It follows from Theorem 2.7 and Lemma 2.9 that

\int_{X}\chi(\phi-v_{k})\theta_{v_{k}}^{n}\leq\sum_{l=1}^{k}\int_{D_{l}}\chi(2^{-l}(\phi-u_{l}))(2^{-l}\theta_{u_{l}}+(1-2^{-l})\theta_{\phi})^{n}.

We note that

(2^{-l}\theta_{u_{l}}+(1-2^{-l})\theta_{\phi})^{n}\leq\theta_{\phi}^{n}+2^{-l}\sum_{p=1}^{n}\binom{n}{p}\theta_{u_{l}}^{p}\wedge\theta_{\phi}^{n-p}.

(6.3)

Since the sets $D_{l}$ are pairwise disjoint and $\chi$ is increasing, it then follows that

$\displaystyle\int_{X}\chi(\phi-v_{k})\theta_{v_{k}}^{n}$	$\displaystyle\leq$	$\displaystyle\sum_{l=1}^{k}\int_{D_{l}}\chi(2^{-l}(\phi-u_{l}))(2^{-l}\theta_{u_{l}}+(1-2^{-l})\theta_{\phi})^{n}$
	$\displaystyle\leq$	$\displaystyle\sum_{l=1}^{k}\int_{D_{l}}\chi(2^{-l}(\phi-u_{l}))\theta_{\phi}^{n}+2^{n}\sum_{l=1}^{k}2^{-l}\int_{X}\chi(\phi-u_{l})\theta_{u_{l}}^{n}$
	$\displaystyle\leq$	$\displaystyle\int_{X}\chi(\phi-\psi)\theta_{\phi}^{n}+2^{n}\sum_{l=1}^{k}2^{-l}\int_{X}\chi(\phi-u_{l})\theta_{u_{l}}^{n}$
	$\displaystyle\leq$	$\displaystyle C_{2}+2^{n}A.$

The second inequality follows from (6.3) together with Lemma 6.3. The third inequality follows from the fact $0\leq 2^{-l}(\phi-u_{l})\leq\sum_{l\geq 1}2^{-l}(\phi-u_{l})=\phi-\psi$ . The last line follows from Lemma 6.2.
It then follows from Lemma 6.7 that $v$ , which is the decreasing limit of $v_{k}$ , belongs to $\mathcal{E}_{\chi}(X,\theta,\phi)$ . On the other hand, since $v\leq 2^{-k}u_{k}+(1-2^{-k})\phi$ for any $k\geq 1$ , and thanks to (6.2) we have

\int_{X}\chi(\phi-v)d\mu\geq\int_{X}\chi(2^{-k}(\phi-u_{k}))d\mu\geq 2^{-kM}\int_{X}\chi(\phi-u_{k})d\mu\geq 2^{kM}\to\infty,

as $k\to\infty$ , contradicting the assumption. ∎

Lemma 6.9.

Assume $\mu$ is a positive Borel measure satisfying $\mu(\{\phi=-\infty\})=0$ and $\chi(\phi-u)\in L^{1}(\mu)$ for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . Then there exists a constant $C>0$ such that, for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ with $\sup_{X}u=0$ , we have

\int_{X}\chi(\phi-u)d\mu\leq C(E_{\chi}(u,\phi)+1).

Proof.

We prove the lemma by contradiction, assuming there exists a sequence $(u_{j})\subset\mathcal{E}_{\chi}(X,\theta,\phi)$ with $\sup_{X}u_{j}=0$ , but

\int_{X}\chi(\phi-u_{j})d\mu\geq j(E_{\chi}(u_{j},\phi)+1).

Set $\varepsilon_{j}:=(E_{\chi}(u_{j},\phi)+1)^{-1}<1$ and

\psi_{j}:=P_{\theta}(-\chi^{-1}(\varepsilon_{j}\chi(\phi-u_{j}))+\phi).

Since $\theta_{\psi_{j}}^{n}$ is supported on $\{\psi_{j}=-\chi^{-1}(\varepsilon_{j}\chi(\phi-u_{j}))+\phi\}$ (Theorem 2.7), we have

\int_{X}\chi(\phi-\psi_{j})\theta_{\psi_{j}}^{n}\leq\varepsilon_{j}\int_{X}\chi(\phi-u_{j})\theta_{\psi_{j}}^{n}\leq C_{1}.

In the last inequality we have used the fact that $u_{j}\leq\psi_{j}$ and that $\chi$ is increasing and Lemma 6.5. By Lemma 6.8 we thus have

\int_{X}\chi(\phi-\psi_{j})d\mu\leq C_{2}.

But since $\phi-\psi_{j}\geq\chi^{-1}(\varepsilon_{j}\chi(\phi-u_{j}))$ , we have

\int_{X}\chi(\phi-\psi_{j})d\mu\geq\int_{X}\varepsilon_{j}\chi(\phi-u_{j})d\mu\geq j,

which is a contradiction. ∎

Lemma 6.10.

There exists a constant $C>0$ such that, for all $u,v\in\mathcal{E}_{\chi}(X,\theta,\phi)$ with $\sup_{X}v=0$ and $u\leq 0$ , we have

\int_{X}\chi(\phi-v)\theta_{u}^{n}\leq C(1+E_{\chi}(u,\phi))E_{\chi}(v,\phi)^{M/(M+1)}+C.

(6.4)

The above result is due to Duc-Thai Do and Duc-Viet Vu [DV21]. Their proof uses generalized non-pluripolar products. We give below a less technical argument that relies only on the comparison principle.

Proof.

We observe that for all $t>1$ , $v+(t-1)\phi,tu\in\mathcal{E}_{\chi}(X,t\theta,t\phi)$ and $t\phi-(v+(t-1)\phi)=\phi-v$ (where equality holds outside a puripolar set). Fixing $t>1$ , by Lemma 6.4 (regarding $v+(t-1)\phi$ and $tu$ as elements of $\mathcal{E}(X,t\theta,t\phi)$ ) we have

$\displaystyle\int_{X}\chi(\phi-v)(t\theta+dd^{c}tu)^{n}$	$\displaystyle\leq$	$\displaystyle 2^{n+M}\int_{X}\chi(\phi-v)(t\theta+dd^{c}(v+(t-1)\phi))^{n}$
		$\displaystyle+\int_{X}\chi(t\phi-tu)(t\theta+dd^{c}tu)^{n}$
	$\displaystyle\leq$	$\displaystyle 2^{n+M}t^{n}\int_{X}\chi(\phi-v)\theta_{\phi}^{n}+t^{n-1}2^{2n+M}E_{\chi}(v,\phi)+t^{n+M}E_{\chi}(u,\phi),$

where the last inequality follows from Lemma 6.3 after observing that

(t\theta+dd^{c}(v+(t-1)\phi))^{n}=((t-1)\theta_{\phi}+\theta_{v})^{n}\leq t^{n}\theta_{\phi}^{n}+t^{n-1}2^{n}\sum_{j=1}^{n}\theta_{v}^{j}\wedge\theta_{\phi}^{n-j}.

Dividing by $t^{n}$ and using Lemma 6.2 we thus get

\int_{X}\chi(\phi-v)\theta_{u}^{n}\leq C_{1}+2^{2n+M}t^{-1}E_{\chi}(v,\phi)+t^{M}E_{\chi}(u,\phi).

Choosing $t=(1+E_{\chi}(v,\phi))^{{1}/{(M+1)}}$ , we finish the proof. ∎

Proposition 6.11.

Assume $\varphi\in\mathcal{E}(X,\theta,\phi)$ with $\sup_{X}\varphi=0$ satisfies $\theta_{\varphi}^{n}\leq A\theta_{u}^{n}$ and $[u]=[\phi]$ for some $A>0$ and $u\in{\rm PSH}(X,\theta)$ . Then there exists $\alpha>0$ such that

\int_{X}e^{\alpha(\phi-\varphi)}\theta_{\varphi}^{n}<\infty.

Proof.

We can assume that $\sup_{X}\varphi=0$ and $\phi-1\leq u\leq\phi$ . We argue as in [DL17, Lemma 4.3]. Fix $t>0$ and $s>1$ . Observe that away from the pluripolar set $\{\phi=-\infty\}$ we have that $\{\varphi<\phi-t-s\}\subseteq\{\varphi<\phi-t+s(u-\phi)\}\subseteq\{\varphi<\phi-t\}$ . By the assumption and the partial comparison principle (Proposition 3.22) we have

	$\displaystyle\int_{\{\varphi<\phi-t-s\}}\theta_{\varphi}^{n}$	$\displaystyle\leq A\int_{\{s^{-1}\varphi+(1-s^{-1})\phi<u-s^{-1}t\}}\theta_{u}^{n}$
		$\displaystyle\leq A\int_{\{s^{-1}\varphi+(1-s^{-1})\phi<u-s^{-1}t\}}(s^{-1}\theta_{\varphi}+(1-s^{-1})\theta_{\phi})^{n}$
		$\displaystyle\leq A\int_{\{\varphi<\phi-t\}}(s^{-1}\theta_{\varphi}+(1-s^{-1})\theta_{\phi})^{n}$
		$\displaystyle\leq A\int_{\{\varphi<\phi-t\}}\theta_{\phi}^{n}+A\sum_{k=1}^{n}s^{-k}\binom{n}{k}\int_{\{\varphi<\phi-t\}}\theta_{\varphi}^{k}\wedge\theta_{\phi}^{n-k}$
		$\displaystyle\leq AC_{0}\int_{\{\varphi<\phi-t\}}\omega^{n}+2^{n}As^{-1}\int_{\{\varphi<\phi-t\}}\theta_{\varphi}^{n}$
		$\displaystyle\leq C_{2}e^{-at}\int_{\{\varphi<-t\}}e^{a\|\varphi\|}\omega^{n}+2^{n}As^{-1}\int_{\{\varphi<\phi-t\}}\theta_{\varphi}^{n}$
		$\displaystyle\leq C_{3}e^{-at}+2^{n}As^{-1}\int_{\{\varphi<\phi-t\}}\theta_{\varphi}^{n}.$

In the fifth inequality we used Theorem 3.6. The last inequality holds for a choice of $a>0$ very small, so that the uniform Skoda integrability theorem (see [Sko72], [GZ17]) ensures

\int_{X}e^{-a\varphi}\omega^{n}\leq C_{1}

is uniformly bounded.

We fix $s$ so large that $2^{n}As^{-1}<(2e)^{-1}$ . Then, for $a=1/s$ we have $2^{n}As^{-1}e^{as}<1/2$ . Setting

F(t):=e^{at}\int_{\{\varphi<\phi-t\}}\theta_{\varphi}^{n},\;t>0,

we then have

F(t+s)\leq C_{4}+\frac{F(t)}{2},

from which we obtain, by induction on $k\in\mathbb{N}$ ,

F(t+ks)\leq C_{4}\sum_{j=1}^{k}2^{1-j}+2^{-k}F(t).

Any $t\geq 0$ can be written as $t=t_{0}+ks$ for some $t_{0}\in[0,s]$ and $k\in\mathbb{N}$ . The above estimate thus gives a uniform bound $F(t)\leq 2C_{4}+\sup_{[0,s]}F\leq C_{5}$ . Now, for $\alpha=a/2$ , we get

	$\displaystyle\int_{X}e^{\alpha(\phi-\varphi)}\theta_{\varphi}^{n}$	$\displaystyle=\int_{0}^{\infty}\alpha e^{\alpha t}\theta_{\varphi}^{n}(\{\varphi<\phi-t\})dt=\int_{0}^{\infty}\frac{a}{2}e^{-{\frac{a}{2}}t}e^{at}\theta_{\varphi}^{n}(\{\varphi<\phi-t\})dt,$
		$\displaystyle=\int_{0}^{\infty}\frac{a}{2}e^{-\frac{a}{2}t}F(t)dt\leq\frac{aC_{5}}{2}\int_{0}^{\infty}e^{-\frac{a}{2}t}dt<\infty.$

∎

Lemma 6.12.

If $E\subset X$ is a pluripolar set then there exists $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ such that $E\subset\{u=-\infty\}$ .

Proof.

If $\eta(t)=t^{M},t\geq 0$ then the inclusion $\mathcal{E}_{\eta}(X,\theta,\phi)\subset\mathcal{E}_{\chi}(X,\theta,\phi)$ holds because $\chi(t)\leq\chi(1)t^{M}$ for all $t\geq 1$ . We can thus assume that $\chi=\eta$ is convex. We claim that there exists $h\in\mathcal{E}^{1}(X,\theta,\phi)$ such that $E\subset\{h=-\infty\}$ . Indeed, it follows from [BBGZ13, Corollary 2.11] that $E\subset\{\psi=-\infty\}$ for some $\psi\in\mathcal{E}^{1}(X,\theta)$ . We can assume $\sup_{X}\psi=-1$ , which implies $\psi\leq V_{\theta}$ . Since $P_{\theta}(V_{\theta},\phi)=\phi$ , Proposition 3.21 ensures that the function $h:=P_{\theta}(\psi,\phi)$ belongs to $\mathcal{E}(X,\theta,\phi)$ . By Theorem 3.6 we have

\int_{X}(\phi-h)\theta_{h}^{n}\leq\int_{\{h=\psi\}}(\phi-\psi)\theta_{\psi}^{n}\leq\int_{X}(V_{\theta}-\psi)\theta_{\psi}^{n}<\infty.

We thus have that $h\in\mathcal{E}^{1}(X,\theta,\phi)$ and $E\subset\{\psi=-\infty\}\subset\{h=-\infty\}$ , proving the claim.

Now, let $\gamma:=\chi^{-1}$ be the inverse of $\chi$ (so $\gamma(t)=t^{1/M}$ and $\gamma^{\prime}(t)\leq 1$ if $t\geq 1$ ), which is a concave weight. By adding a constant, we can assume that $h\leq-1$ . Consider $u:=-\gamma(\phi-h)+\phi\in{\rm PSH}(X,\theta,\phi)$ , as in Lemma 5.19 (see also the discussion following the statement). We observe also that $E\subset\{u=-\infty\}$ and $u\geq h$ , hence $u\in\mathcal{E}(X,\theta,\phi)$ . By Lemma 6.5,

\int_{X}\chi(\phi-u)\theta_{u}^{n}=\int_{X}(\phi-h)\theta_{u}^{n}\leq 2^{n}E_{1}(h,\phi)<\infty.

We thus have that $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ , finishing the proof. ∎

Proposition 6.13.

Assume $\mu$ is a positive Radon measure satisfying $\int_{X}\theta_{\phi}^{n}=\mu(X)>0$ . Assume also that $\mu(\{\phi=-\infty\})=0$ and

\int_{X}\chi(\phi-\varphi)d\mu\leq aE_{\chi}(\varphi,\phi)+C,\quad\varphi\in\mathcal{E}_{\chi}(X,\theta,\phi),\;\sup_{X}\varphi=0,

(6.5)

for some constants $a\in(0,1)$ , $C>0$ . Then $\mu=(\theta+dd^{c}u)^{n}$ for some $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ .

Proof.

We prove the proposition by an argument going back to Cegrell [Ceg98]. We first claim that $\mu$ vanishes on pluripolar sets. Indeed, fix such a Borel set $E$ . It follows from Lemma 6.12 that $E\subset\{h=-\infty\}$ for some $h\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . We can assume that $h\leq\phi$ .

Since $\int_{E}\chi(\phi-h)d\mu\leq\int_{X}\chi(\phi-h)d\mu<\infty$ , we get that $\mu(E\cap\{\phi\neq-\infty\})=0$ . Since $\mu(\{\phi=-\infty\})=0$ , it follows that $\mu(E)=0$ .

We next claim that we can write

\mu=f\nu,\;\nu=(\theta+dd^{c}\phi_{0})^{n},

for some $\phi_{0}\in{\rm PSH}(X,\theta,\phi)$ such that $\phi-1\leq\phi_{0}\leq\phi$ , and some $0\leq f\in L^{1}(X,\nu)$ . To see this, we first find $\psi\in\mathcal{E}(X,\theta,\phi)$ such that $(\theta+dd^{c}\psi)^{n}=\mu$ and $\sup_{X}\psi=0$ (in particular $\psi\leq\phi$ ). The existence of $\psi$ follows from Theorem 5.17.

We set $\phi_{0}:=e^{\psi-\phi}+\phi$ as in Lemma 5.19 (see also the discussion following the statement). Then $\phi_{0}\in{\rm PSH}(X,\theta,\phi)$ , $[\phi_{0}]=[\phi]$ , and

\displaystyle(\theta+dd^{c}\phi_{0})^{n}\geq e^{n(\psi-\phi)}(\theta+dd^{c}\psi)^{n}=e^{n(\psi-\phi)}\mu.

(6.6)

From (6.6) it follows that $\mu$ is absolutely continuous with respect to $(\theta+dd^{c}\phi_{0})^{n}$ , proving the second claim.

Now, for each $j>1$ , let $\varphi_{j}\in\mathcal{E}(X,\theta,\phi)$ be the unique solution of

(\theta+dd^{c}\varphi_{j})^{n}=c_{j}\min(f,j)(\theta+dd^{c}\phi_{0})^{n},\;\sup_{X}\varphi_{j}=0,

where $c_{j}$ is a normalization constant to have equality between the total masses of the left and right-hand side. For $j$ large enough we have $c_{j}a<1$ , since $c_{j}\to 1$ and by assumption $a<1$ . We can thus assume that $c_{j}a<\lambda<1$ . It follows from Proposition 6.11 that

\int_{X}e^{\alpha_{j}(\phi-\varphi_{j})}(\theta+dd^{c}\varphi_{j})^{n}<\infty,

for some $\alpha_{j}>0$ . In particular, since for any $t>0$ $\chi(t)\leq Ce^{\alpha_{j}t}$ for some $C>0$ , we infer that $E_{\chi}(\varphi_{j},\phi)$ is finite. We claim that this bound is uniform in $j$ . Indeed, since $\theta_{\varphi_{j}}^{n}\leq c_{j}f\theta_{\phi_{0}}^{n}$ , we have

E_{\chi}(\varphi_{j},\phi)\leq\int_{X}\chi(\phi-\varphi_{j})c_{j}d\mu\leq ac_{j}E_{\chi}(\varphi_{j},\phi)+C

gives $E_{\chi}(\varphi_{j},\phi)\leq C(1-\lambda)^{-1}$ . Extracting a subsequence we can assume that $\varphi_{j}\to\varphi$ in $L^{1}$ . It follows from Lemma 6.7 that $\varphi\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . It then follows from Lemma 5.16 that $(\theta+dd^{c}\varphi)^{n}\geq\mu$ . Comparing the total mass, we obtain the equality, finishing the proof. ∎

We are now ready to prove our main result of this section.

Theorem 6.14.

Assume $\mu(\{\phi=-\infty\})=0$ and $\chi(|\phi-u|)\in L^{1}(\mu)$ , for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . Then $\mu=(\theta+dd^{c}\varphi)^{n}$ for some $\varphi\in\mathcal{E}_{\chi}(X,\theta,\phi)$ .

Proof.

Let $C$ be the constant in Lemma 6.9. Let $v\in\mathcal{E}(X,\theta,\phi)$ be the unique solution to

(\theta+dd^{c}v)^{n}=(4C)^{-1}\mu+b\omega^{n},\;\sup_{X}v=-1.

Here $b>0$ is a constant so that $\int_{X}(4C)^{-1}\mu+b\omega^{n}=\int_{X}\theta_{\phi}^{n}$ . The existence of $v$ follows from Theorem 5.17. By Lemma 6.9 there exists a constant $C_{1}>0$ such that, for all $\varphi\in\mathcal{E}_{\chi}(X,\theta,\phi)$ with $\sup_{X}\varphi=0$ , we have

\int_{X}\chi(\phi-\varphi)\theta_{v}^{n}\leq\frac{1}{4}E_{\chi}(\varphi,\phi)+C_{1}.

By Proposition 6.13, $v\in\mathcal{E}_{\chi}(X,\theta,\phi)$ . Since $\mu\leq 4C(\theta+dd^{c}v)^{n}$ , it follows from Lemma 6.10 that $\mu$ satisfies

\int_{X}\chi(\phi-u)\mu\leq 4C\int_{X}\chi(\phi-u)\theta_{v}^{n}\leq C^{\prime}(1+E_{\chi}(v,\phi))E_{\chi}(u,\phi)^{M/(M+1)}+C^{\prime}.

As a result, $\mu$ also satisfies (6.5). We can thus use Lemma 6.13 to complete the proof. ∎

We summarize the findings of this chapter in the theorem below.

Theorem 6.15.

Fix a Radon measure $\mu$ with $\mu(\{\phi=-\infty\})=0$ and $\int_{X}\theta_{\phi}^{n}=\mu(X)>0$ . Then the following are equivalent.

(i)

There exists a constant $C>0$ such that, for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ with $\sup_{X}u=0$ , we have

$\int_{X}\chi(\phi-u)d\mu\leq CE_{\chi}(u,\phi)^{M/(M+1)}+C.$
(ii)

$\chi(|\phi-u|)\in L^{1}(\mu)$ , for all $u\in\mathcal{E}_{\chi}(X,\theta,\phi)$ .
(iii)

$\mu=(\theta+dd^{c}\varphi)^{n}$ for some $\varphi\in\mathcal{E}_{\chi}(X,\theta,\phi)$ , with $\sup_{X}\varphi=0$ .

Proof.

(i) $\Longrightarrow$ (ii) is immediate. The implication (ii) $\Longrightarrow$ (iii) is Theorem 6.14. The implication (iii) $\Longrightarrow$ (i) is Lemma 6.10. ∎

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	$\displaystyle\|\int_{X}(\chi u_{j}\mu_{j}-\chi ud\mu)\|\leq\int_{X}\|\chi(u_{j}-u)\|\mu_{j}+\|\int_{X}\chi u(\mu_{j}-\mu)\|$
	$\displaystyle\leq\delta\int_{X}\|\chi\|\mu_{j}+A\sup_{X}\|\chi\|\mu_{j}(\{\|u_{j}-u\|>\delta\})$
	$\displaystyle+\|\int_{X}\chi(u-v)(\mu_{j}-\mu)\|+\|\int_{X}\chi v(\mu_{j}-\mu)\|$
	$\displaystyle\leq\delta\int_{X}\|\chi\|\mu_{j}+A\sup_{X}\|\chi\|f(\varepsilon)+\|\int_{X}\chi(u-v)(\mu_{j}-\mu)\|+\|\int_{X}\chi v(\mu_{j}-\mu)\|$
	$\displaystyle\leq\delta\int_{X}\|\chi\|\mu_{j}+2A\sup_{X}\|\chi\|f(\varepsilon)+\|\int_{X}\chi v(\mu_{j}-\mu)\|.$

	$\displaystyle\int_{G}$	$\displaystyle\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}$
		$\displaystyle=\int_{D_{j}\cap G}\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}+\int_{U_{j}\cap G}\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}$
		$\displaystyle=\int_{D_{j}\cap G}(1-e^{P_{\theta}(g_{j})-g_{j}})(\theta+dd^{c}P_{\theta}(g_{j}))^{n}+\int_{U_{j}\cap G}\|1-e^{P_{\theta}(g_{j})-f}\|(\theta+dd^{c}P_{\theta}(g_{j}))^{n}$
		$\displaystyle\leq B^{n}\sup_{X}\|1-e^{P_{\theta}(g_{j})-f}\|{\rm Cap}_{\omega}(U_{j})\leq C2^{-j-1}.$

	$\displaystyle\int_{X}(\chi(\phi-u))^{m}d\mu$	$\displaystyle\leq\int_{X}(\phi-v)^{m}d\mu$
		$\displaystyle\leq\int_{X}\|v-\sup_{X}v+\sup_{X}v\|^{m}d\mu$
		$\displaystyle\leq 2^{m-1}\int_{X}(\|v-\sup_{X}v\|^{m}+\|\sup_{X}v\|^{m})d\mu$
		$\displaystyle\leq 2^{m-1}A_{m}(\mu)+2^{m-1}\|\sup_{X}v\|^{m}\mu(X)\leq C_{2}.$

	$\displaystyle\int_{X}\chi(\|u+C-\phi\|)\theta_{u}^{n}$	$\displaystyle\leq\int_{X}\chi(\|u-\phi\|+\|C\|)\theta_{u}^{n}$
		$\displaystyle\leq 2^{M}\int_{X}\max(\chi(\|u-\phi\|),\chi(\|C\|))\theta_{u}^{n}$
		$\displaystyle\leq 2^{M}\max\left(\int_{X}\chi(\|u-\phi\|)\theta_{u}^{n},\ \chi(\|C\|)\int_{X}\theta_{u}^{n}\right)<\infty.$