Non-Pluripolar Monge–Ampère Measure

• Non-pluripolar Monge–Ampère measure is the extension of the complex MA operator for singular plurisubharmonic potentials that discards contributions from pluripolar sets.
• It employs truncation techniques to ensure locality, monotonicity, and stability, underpinning the computation of mass invariants on compact Kähler and Hermitian manifolds.
• The operator’s framework facilitates solving complex Monge–Ampère equations with prescribed singularities, fostering advancements in relative full-mass and finite-energy theories.

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 (Nyström, 2017).

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 (Pang et al., 14 Jul 2026).

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

$(X,\omega)$0

in the sense of currents. The set of such functions is denoted $(X,\omega)$1. By the $(X,\omega)$2-lemma, every closed positive $(X,\omega)$3-current in $(X,\omega)$4 can be written as $(X,\omega)$5 for some $(X,\omega)$6. For positive closed $(X,\omega)$7-currents $(X,\omega)$8, the non-pluripolar product

$(X,\omega)$9

is a closed positive $(1,1)$0-current; when $(1,1)$1, it is a positive measure (Nyström, 2017).

In big cohomology classes, the canonical reference potential is

$(1,1)$2

For $(1,1)$3, the paper on moving prescribed singularities writes

$(1,1)$4

and regards the total mass $(1,1)$5 as the Monge–Ampère mass of $(1,1)$6 (Pang et al., 14 Jul 2026).

Local theories use analogous notation. On a bounded domain $(1,1)$7, the non-pluripolar complex Monge–Ampère measure of $(1,1)$8 is denoted either $(1,1)$9 or $\theta$0, depending on the source, and is defined by truncation. On compact Hermitian manifolds with a semipositive big $\theta$1-form $\theta$2, one writes

$\theta$3

with the explicit convention that for unbounded $\theta$4 this means the non-pluripolar Monge–Ampère measure, not the classical Bedford–Taylor product (Do et al., 2024, Alehyane et al., 4 Nov 2025).

A further Hermitian variant uses a smooth closed real $\theta$5-form $\theta$6 with $\theta$7 and a bounded $\theta$8-psh potential $\theta$9. There the operator is defined for $u\in \mathrm{PSH}(X,\theta)$0 by a truncation relative to $u\in \mathrm{PSH}(X,\theta)$1, and the resulting measure is written

$u\in \mathrm{PSH}(X,\theta)$2

The boundedness of $u\in \mathrm{PSH}(X,\theta)$3 is the device that replaces semipositivity of the smooth background form (Pang et al., 2024).

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 \mathrm{PSH}(X,\theta)$4,

$u\in \mathrm{PSH}(X,\theta)$5

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 (Pang et al., 14 Jul 2026).

On bounded domains, the corresponding local formula is

$u\in \mathrm{PSH}(X,\theta)$6

or equivalently, for every Borel set $u\in \mathrm{PSH}(X,\theta)$7,

$u\in \mathrm{PSH}(X,\theta)$8

A stabilization property holds on lower level sets: if $u\in \mathrm{PSH}(X,\theta)$9, then for all $$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$0,

$$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$1

This makes the operator local on strata where $$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$2 is already bounded below (Do et al., 2024, Do et al., 24 Jul 2025).

In the compact Hermitian setting with bounded $$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$3, the truncation becomes

$$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$4

This is the non-pluripolar Monge–Ampère measure associated to $$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$5 (Pang et al., 2024).

A crucial distinction emerges in the strictly pseudoconvex bounded-domain theory with background $$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$6. There one has both the full Cegrell-type measure $$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$7 and its non-pluripolar part

$$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$8

For $$\mathrm{MA}_\theta(u)=\langle (\theta+dd^c u)^n\rangle,$$9, the paper proves

$u$0

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 (Alehyane et al., 28 Sep 2025).

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 $u$1-psh functions agree on a plurifine open set $u$2, then

$u$3

The same principle is used in product form throughout the mixed theory (Nyström, 2017).

A related maximum principle states that for $u$4,

$u$5

In particular, if $u$6, then

$u$7

This inequality is a basic device in domination and comparison arguments (Dang et al., 10 Mar 2025).

A sharp contact-set identity was obtained for singular $u$8-psh functions under $u$9 obstacles. If $\int_X \mathrm{MA}_\theta(u)$0, $\int_X \mathrm{MA}_\theta(u)$1, and $\int_X \mathrm{MA}_\theta(u)$2, then

$\int_X \mathrm{MA}_\theta(u)$3

In particular,

$\int_X \mathrm{MA}_\theta(u)$4

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 (Nezza et al., 2019).

Capacity theory supplies the measure-theoretic control needed for singular products. On compact Kähler manifolds, capacities $\int_X \mathrm{MA}_\theta(u)$5 attached to positive-mass singularity types vanish exactly on pluripolar sets, and all such capacities are quantitatively comparable to the standard Monge–Ampère capacity $\int_X \mathrm{MA}_\theta(u)$6 via continuous control functions. This makes precise that the non-pluripolar theory sees the same negligible sets across different big classes and singularity types (Lu, 2020).

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

The measure and its total mass must be distinguished. For a smooth $\int_X \mathrm{MA}_\theta(u)$7-psh potential $\int_X \mathrm{MA}_\theta(u)$8, one has

$\int_X \mathrm{MA}_\theta(u)$9

For singular $u:X\to[-\infty,\infty)$0, the total mass may be strictly smaller, and this mass loss is one of the main invariants of singularity (Nyström, 2017).

The foundational monotonicity theorem states that if $u:X\to[-\infty,\infty)$1 and $u:X\to[-\infty,\infty)$2 is less singular than $u:X\to[-\infty,\infty)$3, meaning

$u:X\to[-\infty,\infty)$4

then

$u:X\to[-\infty,\infty)$5

Thus stronger singularities decrease the total non-pluripolar Monge–Ampère mass. A global comparison principle follows: $u:X\to[-\infty,\infty)$6 The result was conjectured by Boucksom–Eyssidieux–Guedj–Zeriahi and first known under a small-unbounded-locus hypothesis; the general case removed that restriction (Nyström, 2017).

The mixed-product version asserts that if $u:X\to[-\infty,\infty)$7 is less singular than $u:X\to[-\infty,\infty)$8 in each pseudoeffective class $u:X\to[-\infty,\infty)$9, then

$\theta$0

This underlies the relative pluripotential theory of prescribed singularities and the definition of classes $\theta$1 by equality of total non-pluripolar mass relative to a reference potential $\theta$2 (Darvas et al., 2017).

A later refinement introduced the weaker notion of being less singular in capacity. If there exist $\theta$3 and $\theta$4 such that

$\theta$5

then the same mass monotonicity conclusion holds. In particular, having the same singularity type in capacity implies equality of total masses: $\theta$6 The same work gives a relative full-mass characterization: $\theta$7 iff $\theta$8 and $\theta$9 for some $(X,\omega)$00 and $(X,\omega)$01 (Dang et al., 10 Mar 2025).

5. Capacity, convergence, and stability

Weak convergence of non-pluripolar Monge–Ampère measures is delicate because weak $(X,\omega)$02 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 $(X,\omega)$03 with positive Monge–Ampère mass are comparable to $(X,\omega)$04 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 (Lu, 2020).

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 $(X,\omega)$05, $(X,\omega)$06 are normalized positive-mass model potentials, and $(X,\omega)$07 in total variation with

$(X,\omega)$08

then for the normalized solutions $(X,\omega)$09 of

$(X,\omega)$10

one has

$(X,\omega)$11

This identifies capacity convergence as the intrinsic topology of moving prescribed singularities detected by the non-pluripolar operator (Pang et al., 14 Jul 2026).

A compact Hermitian analogue gives a direct weak-convergence criterion. If $(X,\omega)$12 and

$(X,\omega)$13

for a positive non-pluripolar Radon measure $(X,\omega)$14, while $(X,\omega)$15 in $(X,\omega)$16, then $(X,\omega)$17, the sequence converges in capacity, and

$(X,\omega)$18

This is a genuinely Hermitian weak-convergence theorem for non-pluripolar Monge–Ampère measures (Pang et al., 2024).

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 $(X,\omega)$19 with small unbounded locus and

$(X,\omega)$20

existence and uniqueness hold in the singularity class $(X,\omega)$21 for equations

$(X,\omega)$22

with $(X,\omega)$23 a non-pluripolar measure of matching mass. This inaugurates a relative full-mass and relative finite-energy theory $(X,\omega)$24, $(X,\omega)$25 (Darvas et al., 2017).

On compact Hermitian manifolds with semipositive big $(X,\omega)$26, the same operator underlies equations

$(X,\omega)$27

for positive Radon measures $(X,\omega)$28 vanishing on pluripolar sets, with solutions in a relative full-mass class $(X,\omega)$29 defined by envelope conditions rather than fixed cohomological mass (Alehyane et al., 4 Nov 2025).

Local theories on bounded hyperconvex domains similarly formulate prescribed-singularity equations via the non-pluripolar operator: $(X,\omega)$30 or

$(X,\omega)$31

Existence and uniqueness then rest on envelope methods, maximum inequalities, and singularity classes $(X,\omega)$32 or $(X,\omega)$33 (Do et al., 24 Jul 2025, Do et al., 2024).

A complementary bounded-domain theory with background $(X,\omega)$34 shows exactly where non-pluripolarity stops. For $(X,\omega)$35,

$(X,\omega)$36

but the full Cegrell-type measure may have an extra term on $(X,\omega)$37. 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 (Alehyane et al., 28 Sep 2025).

Two broader developments sharpen the scope of the subject. First, the non-pluripolar product is bimeromorphically invariant: $(X,\omega)$38 so in top degree the non-pluripolar Monge–Ampère measure is a birationally meaningful object in big classes (Nezza, 2013). Second, an extension of the theory introduced positive closed currents

$(X,\omega)$39

dominating the non-pluripolar current $(X,\omega)$40, with defect currents

$(X,\omega)$41

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 (Andersson et al., 2021).