---
title: Non-Pluripolar Monge–Ampère Measure
url: https://www.emergentmind.com/topics/non-pluripolar-monge-ampere-measure
type: topic
---

# Non-Pluripolar Monge–Ampère Measure

The non-pluripolar Monge–Ampère measure is the extension of the complex Monge–Ampère operator to singular plurisubharmonic or \(\theta\)-plurisubharmonic potentials obtained by discarding the part of the classical wedge product that would concentrate on pluripolar sets. On a compact Kähler manifold \((X,\omega)\) with a smooth closed real \((1,1)\)-form \(\theta\), for \(u\in \mathrm{PSH}(X,\theta)\) one writes
\[
\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,
\]
and this is the non-pluripolar Monge–Ampère measure of \(u\). Its total mass
\[
\int_X \mathrm{MA}_\theta(u)
\]
is a central invariant of singularity type, with monotonicity and full-mass phenomena governing much of modern pluripotential theory [1703.01950].

The theory now spans compact Kähler manifolds, compact Hermitian manifolds, and bounded hyperconvex or strictly pseudoconvex domains, with parallel but not identical truncation procedures. A recurring theme is that the non-pluripolar operator is the correct singular Monge–Ampère object when one wants a measure that never charges pluripolar sets, while broader Cegrell-type extensions may deliberately retain a pluripolar part and thereby solve equations with right-hand sides outside the non-pluripolar range [2607.12797].

## 1. Definition and geometric frameworks

In the compact Kähler setting, a function
\[
u:X\to[-\infty,\infty)
\]
is \(\theta\)-plurisubharmonic if it is upper semicontinuous, locally integrable, and
\[
\theta+dd^c u\ge 0
\]
in the sense of currents. The set of such functions is denoted \(\mathrm{PSH}(X,\theta)\). By the \(dd^c\)-lemma, every closed positive \((1,1)\)-current in \([\theta]\) can be written as \(\theta+dd^c u\) for some \(u\in \mathrm{PSH}(X,\theta)\). For positive closed \((1,1)\)-currents \(T_j=\theta_j+dd^c u_j\), the non-pluripolar product
\[
\langle T_1\wedge\cdots\wedge T_p\rangle
\]
is a closed positive \((p,p)\)-current; when \(p=n\), it is a positive measure [1703.01950].

In big cohomology classes, the canonical reference potential is
\[
V_\theta:=\sup\{u\in\operatorname{PSH}(X,\theta):u\le 0\}.
\]
For \(u\in\operatorname{PSH}(X,\theta)\), the paper on moving prescribed singularities writes
\[
\theta_u^n=\big\langle(\theta+dd^c u)^n\big\rangle
\]
and regards the total mass \(\int_X\theta_u^n\) as the Monge–Ampère mass of \(u\) [2607.12797].

Local theories use analogous notation. On a bounded domain \(\Omega\subset\mathbb C^n\), the non-pluripolar complex Monge–Ampère measure of \(u\in \mathrm{PSH}(\Omega)\) is denoted either \(NP(dd^c u)^n\) or \(NP(u)\), depending on the source, and is defined by truncation. On compact Hermitian manifolds with a semipositive big \((1,1)\)-form \(\theta\), one writes
\[
\theta_u^n:=(\theta+dd^c u)^n
\]
with the explicit convention that for unbounded \(u\) this means the non-pluripolar Monge–Ampère measure, not the classical Bedford–Taylor product [2407.00937] [2511.02339].

A further Hermitian variant uses a smooth closed real \((1,1)\)-form \(\beta\) with \(\int_X\beta^n>0\) and a bounded \(\beta\)-psh potential \(\rho\). There the operator is defined for \(u\in \mathrm{PSH}(X,\beta)\) by a truncation relative to \(\rho\), and the resulting measure is written
\[
\beta_u^n=(\beta+dd^c u)^n.
\]
The boundedness of \(\rho\) is the device that replaces semipositivity of the smooth background form [2412.11547].

## 2. Truncation procedures and the “non-pluripolar” condition

The decisive feature of the non-pluripolar construction is that it removes mass on pluripolar sets. In the compact Kähler big-class formalism, for \(u\in\operatorname{PSH}(X,\theta)\),
\[
\theta_u^n
=
\big\langle(\theta+dd^cu)^n\big\rangle
:=
\lim_{k\to+\infty}
\mathbf 1_{\{u>V_\theta-k\}}
\,\theta_{\max\{u,V_\theta-k\}}^n.
\]
This is the global non-pluripolar product of Boucksom–Eyssidieux–Guedj–Zeriahi, and it is local in the plurifine topology, multilinear, and non-charging on pluripolar sets [2607.12797].

On bounded domains, the corresponding local formula is
\[
NP(dd^c u)^n
=
\lim_{j\to\infty}
\mathbf 1_{\{u>-j\}}
(dd^c\max\{u,-j\})^n,
\]
or equivalently, for every Borel set \(E\subset \Omega\),
\[
\int_E NP(dd^c u)^n
=
\lim_{j\to\infty}
\int_{E\cap\{u>-j\}}
(dd^c\max\{u,-j\})^n.
\]
A stabilization property holds on lower level sets: if \(E\subset\{u>-k\}\), then for all \(j\ge k\),
\[
\int_E(dd^c\max\{u,-j\})^n
=
\int_E(dd^c\max\{u,-k\})^n.
\]
This makes the operator local on strata where \(u\) is already bounded below [2407.00937] [2507.18116].

In the compact Hermitian setting with bounded \(\rho\), the truncation becomes
\[
u_k:=\max\{u,\rho-k\},\qquad
\beta_u^n
=
\lim_{k\to\infty}
\mathbf 1_{\{u>\rho-k\}}
(\beta+dd^c u_k)^n.
\]
This is the non-pluripolar Monge–Ampère measure associated to \(u\in \mathrm{PSH}(X,\beta)\) [2412.11547].

A crucial distinction emerges in the strictly pseudoconvex bounded-domain theory with background \(\omega\). There one has both the full Cegrell-type measure \((\omega+dd^c u)^n\) and its non-pluripolar part
\[
NP(\omega+dd^c u)^n
=
\lim_{t\to+\infty}
\mathbf 1_{\{u>-t\}}
(\omega+dd^c\max(u,-t))^n.
\]
For \(u\in \mathcal E(\Omega,\omega)\), the paper proves
\[
NP(\omega+dd^c u)^n
=
\mathbf 1_{\{u>-\infty\}}(\omega+dd^c u)^n,
\]
so the non-pluripolar measure is exactly the restriction of the full measure to the non-polar locus. This sharply separates non-pluripolar theory from full Cegrell-type theories that may retain a pluripolar component [2509.23944].

## 3. Structural properties: locality, contact sets, and comparison

The non-pluripolar Monge–Ampère measure inherits strong locality properties. In the compact Kähler setting, if two \(\theta\)-psh functions agree on a plurifine open set \(O\), then
\[
\mathbf 1_O\,\mathrm{MA}_\theta(\varphi)
=
\mathbf 1_O\,\mathrm{MA}_\theta(\psi).
\]
The same principle is used in product form throughout the mixed theory [1703.01950].

A related maximum principle states that for \(u,v\in\mathrm{PSH}(X,\theta)\),
\[
\theta_{\max(u,v)}^n
\ge
\mathbf 1_{\{v\le u\}}\theta_u^n
+
\mathbf 1_{\{u<v\}}\theta_v^n.
\]
In particular, if \(u\le v\), then
\[
\mathbf 1_{\{u=v\}}\theta_u^n
\le
\mathbf 1_{\{u=v\}}\theta_v^n.
\]
This inequality is a basic device in domination and comparison arguments [2503.07534].

A sharp contact-set identity was obtained for singular \(\theta\)-psh functions under \(C^{1,1}\) obstacles. If \(f\in C^{1,1}(X)\), \(\varphi\in \mathrm{PSH}(X,\theta)\), and \(\varphi\le f\), then
\[
\mathbf 1_{\{\varphi=f\}}\theta_\varphi^n
=
\mathbf 1_{\{\varphi=f\}}\theta_f^n.
\]
In particular,
\[
\theta_{P_\theta(f)}^n
=
\mathbf 1_{\{P_\theta(f)=f\}}\theta_f^n,
\qquad
\theta_{P_\theta[\varphi](f)}^n
=
\mathbf 1_{\{P_\theta[\varphi](f)=f\}}\theta_f^n.
\]
These formulas identify the Monge–Ampère measure of an envelope with the obstacle measure on the contact set and are a global singular analogue of the obstacle-problem principle [1912.12720].

Capacity theory supplies the measure-theoretic control needed for singular products. On compact Kähler manifolds, capacities \(\mathrm{Cap}_{\theta,\psi}\) attached to positive-mass singularity types vanish exactly on pluripolar sets, and all such capacities are quantitatively comparable to the standard Monge–Ampère capacity \(\mathrm{Cap}_\omega\) via continuous control functions. This makes precise that the non-pluripolar theory sees the same negligible sets across different big classes and singularity types [2005.04264].

## 4. Total mass, full-mass classes, and singularity comparison

The measure and its total mass must be distinguished. For a smooth \(\theta\)-psh potential \(u\), one has
\[
\mathrm{MA}_\theta(u)=(\theta+dd^c u)^n
\quad\text{and}\quad
\int_X\mathrm{MA}_\theta(u)=\int_X\theta^n.
\]
For singular \(u\), the total mass may be strictly smaller, and this mass loss is one of the main invariants of singularity [1703.01950].

The foundational monotonicity theorem states that if \(\varphi,\psi\in\mathrm{PSH}(X,\theta)\) and \(\varphi\) is less singular than \(\psi\), meaning
\[
\psi\le \varphi+O(1),
\]
then
\[
\int_X \mathrm{MA}_\theta(\varphi)\ge \int_X \mathrm{MA}_\theta(\psi).
\]
Thus stronger singularities decrease the total non-pluripolar Monge–Ampère mass. A global comparison principle follows:
\[
\int_{\{\varphi<\psi\}}\mathrm{MA}_\theta(\psi)
\le
\int_{\{\varphi<\psi\}}\mathrm{MA}_\theta(\varphi).
\]
The result was conjectured by Boucksom–Eyssidieux–Guedj–Zeriahi and first known under a small-unbounded-locus hypothesis; the general case removed that restriction [1703.01950].

The mixed-product version asserts that if \(u_j\) is less singular than \(v_j\) in each pseudoeffective class \(\{\theta_j\}\), then
\[
\int_X \langle (\theta_1+dd^c u_1)\wedge\cdots\wedge(\theta_n+dd^c u_n)\rangle
\ge
\int_X \langle (\theta_1+dd^c v_1)\wedge\cdots\wedge(\theta_n+dd^c v_n)\rangle.
\]
This underlies the relative pluripotential theory of prescribed singularities and the definition of classes \(\mathcal E(X,\theta,\phi)\) by equality of total non-pluripolar mass relative to a reference potential \(\phi\) [1705.05796].

A later refinement introduced the weaker notion of being less singular in capacity. If there exist \(h\in\mathcal E(X,\omega)\) and \(c\ge 0\) such that
\[
\lim_{k\to+\infty}
k^n\operatorname{Cap}_\omega(\{\varphi<\psi+ch-k\})=0,
\]
then the same mass monotonicity conclusion holds. In particular, having the same singularity type in capacity implies equality of total masses:
\[
\int_X\theta_\varphi^n=\int_X\theta_\psi^n.
\]
The same work gives a relative full-mass characterization: \(u\in\mathcal E(X,\theta,\phi)\) iff \(u\le \phi\) and \(u\ge \phi+ch\) for some \(c>0\) and \(h\in\mathcal E(X,\omega)\) [2503.07534].

## 5. Capacity, convergence, and stability

Weak convergence of non-pluripolar Monge–Ampère measures is delicate because weak \(L^1\) convergence of potentials alone is generally too weak. On compact Kähler manifolds, capacities attached to singularity types provide the right topology. A comparison theorem shows that all capacities \(\mathrm{Cap}_{\theta,\psi}\) with positive Monge–Ampère mass are comparable to \(\mathrm{Cap}_\omega\) through continuous control functions. This is used to prove an alternative direct proof of the integration by parts formula for non-pluripolar products, a key ingredient in variational methods [2005.04264].

In moving big classes with prescribed model singularities of positive mass, total variation convergence of the right-hand side non-pluripolar positive Radon measures is the exact “no-mass-loss” condition behind stability. If \(\theta_j\to \theta\), \(\phi_j\to\phi\) are normalized positive-mass model potentials, and \(\mu_j\to\mu\) in total variation with
\[
\mu_j(X)=\int_X\theta_{j,\phi_j}^n,\qquad
\mu(X)=\int_X\theta_\phi^n>0,
\]
then for the normalized solutions \(u_j,u\) of
\[
\theta_{j,u_j}^n=\mu_j,\qquad
\theta_u^n=\mu,
\]
one has
\[
\phi_j\to\phi \text{ in }\operatorname{Cap}_\omega
\quad\Longleftrightarrow\quad
u_j\to u \text{ in }\operatorname{Cap}_\omega.
\]
This identifies capacity convergence as the intrinsic topology of moving prescribed singularities detected by the non-pluripolar operator [2607.12797].

A compact Hermitian analogue gives a direct weak-convergence criterion. If \(u_j\in\mathcal E(X,\beta+dd^c\rho)\) and
\[
(\beta+dd^c\rho+dd^c u_j)^n\le \mu
\]
for a positive non-pluripolar Radon measure \(\mu\), while \(u_j\to u\) in \(L^1(X)\), then \(u\in\mathcal E(X,\beta+dd^c\rho)\), the sequence converges in capacity, and
\[
(\beta+dd^c u_j)^n\rightharpoonup (\beta+dd^c u)^n.
\]
This is a genuinely Hermitian weak-convergence theorem for non-pluripolar Monge–Ampère measures [2412.11547].

## 6. Equations, range questions, and broader developments

The non-pluripolar Monge–Ampère measure is the basic weak operator in modern complex Monge–Ampère equations with prescribed singularities. On compact Kähler manifolds, for a model potential \(\phi=P_\theta[\phi]\) with small unbounded locus and
\[
\int_X \theta_\phi^n>0,
\]
existence and uniqueness hold in the singularity class \([\phi]\) for equations
\[
\theta_u^n=\mu
\quad\text{or}\quad
\theta_u^n=e^{\lambda u}\mu,
\]
with \(\mu\) a non-pluripolar measure of matching mass. This inaugurates a relative full-mass and relative finite-energy theory \(\mathcal E(X,\theta,\phi)\), \(\mathcal E^1(X,\theta,\phi)\) [1705.05796].

On compact Hermitian manifolds with semipositive big \(\theta\), the same operator underlies equations
\[
(\theta+dd^c u)^n=e^{\lambda u}\mu
\quad\text{or}\quad
(\theta+dd^c u)^n=c\mu,
\]
for positive Radon measures \(\mu\) vanishing on pluripolar sets, with solutions in a relative full-mass class \(\mathcal E(X,\theta,\phi)\) defined by envelope conditions rather than fixed cohomological mass [2511.02339].

Local theories on bounded hyperconvex domains similarly formulate prescribed-singularity equations via the non-pluripolar operator:
\[
NP(u)=F(u,\cdot)\,d\mu,\qquad P[u]=\phi,
\]
or
\[
NP(dd^c u)^n=\mu,\qquad P[u]=0.
\]
Existence and uniqueness then rest on envelope methods, maximum inequalities, and singularity classes \(\mathcal N_{NP}(\phi)\) or \(\mathcal M_{NP}\) [2507.18116] [2407.00937].

A complementary bounded-domain theory with background \(\omega\) shows exactly where non-pluripolarity stops. For \(u\in \mathcal E(\Omega,\omega)\),
\[
NP(\omega+dd^c u)^n=\mathbf 1_{\{u>-\infty\}}(\omega+dd^c u)^n,
\]
but the full Cegrell-type measure may have an extra term on \(\{u=-\infty\}\). This permits solvability for measures with pluripolar mass, which lie beyond the range of the non-pluripolar operator alone. The distinction is conceptual: the non-pluripolar measure is the ambient singular Monge–Ampère measure with the pluripolar part removed [2509.23944].

Two broader developments sharpen the scope of the subject. First, the non-pluripolar product is bimeromorphically invariant:
\[
f_*\langle T_1\wedge\cdots\wedge T_p\rangle
=
\langle f_*T_1\wedge\cdots\wedge f_*T_p\rangle,
\]
so in top degree the non-pluripolar Monge–Ampère measure is a birationally meaningful object in big classes [1311.7301]. Second, an extension of the theory introduced positive closed currents
\[
[dd^c u]^p
\]
dominating the non-pluripolar current \(\langle dd^c u\rangle^p\), with defect currents
\[
S_p(u):=[dd^c u]^p-\langle dd^c u\rangle^p,
\]
and a mass formula expressing precisely the loss of mass of the non-pluripolar Monge–Ampère measure. This suggests that the classical non-pluripolar operator is the “mass-preserving away from pluripolar sets” part of a larger singular intersection theory [2106.10883].

Source: https://www.emergentmind.com/topics/non-pluripolar-monge-ampere-measure