Non-Pluripolar Product in Pluripotential Theory

• Non-pluripolar product is defined as a truncation-based operator that discards mass on pluripolar sets while extending classical Bedford–Taylor theory to cases with unbounded potentials.
• Monotonicity and integration by parts principles show that increasing singularity in potentials reduces total non-pluripolar mass and ensures weak convergence.
• The framework extends to relative products, complex dynamics, and prescribed-singularity Monge–Ampère equations, supporting broad applications in complex geometry.

The non-pluripolar product is the truncation-based intersection product used in pluripotential theory to define wedge products of closed positive currents or Monge–Ampère measures when the underlying plurisubharmonic or quasi-plurisubharmonic potentials are unbounded. Its defining feature is that it discards mass on pluripolar sets while retaining a closed positive current that coincides with the Bedford–Taylor product in the bounded regime. On compact Kähler manifolds it is written $\langle T_1\wedge\cdots\wedge T_p\rangle$; on domains in $\mathbb C^n$ it appears as the non-pluripolar complex Monge–Ampère measure $\mathrm{NP}(dd^c u)^n$. The notion underlies modern intersection theory for singular currents, comparison and monotonicity theorems, integration by parts, prescribed-singularity Monge–Ampère equations, and several extensions to relative products, vector bundles, dynamics, and Hermitian geometry (Nyström, 2017, Do et al., 2024, Xia, 2022).

1. Definition and basic formalism

On a compact Kähler manifold $X$, if $T_j=\theta_j+dd^c u_j$ are closed positive $(1,1)$-currents with $u_j$ $\theta_j$-psh, the standard Boucksom–Eyssidieux–Guedj–Zeriahi construction defines the non-pluripolar product by truncating local potentials. In local coordinates one considers

$$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$

and the increasing limit is a closed positive current independent of the choice of local potentials. For a single potential $u$, the self-product is the non-pluripolar Monge–Ampère measure

$\mathbb C^n$0

The construction is symmetric, multilinear, and does not charge pluripolar sets (Dang et al., 10 Mar 2025, Nyström, 2017).

On a bounded domain $\mathbb C^n$1, the local definition is the analogous truncation formula

$\mathbb C^n$2

equivalently,

$\mathbb C^n$3

for every Borel set $\mathbb C^n$4. The truncation stabilizes on sets where $\mathbb C^n$5 is not too singular, and the resulting measure vanishes on pluripolar sets (Do et al., 2024, Do et al., 24 Jul 2025).

When the potentials are locally bounded, the non-pluripolar product coincides with the classical Bedford–Taylor wedge product. More generally, when the potentials have small unbounded locus on a compact Kähler manifold, the product agrees with the Bedford–Taylor product on the complement of a closed pluripolar set $\mathbb C^n$6, which is the standard reduction to bounded Bedford–Taylor theory. This bounded or small-unbounded-locus agreement is the baseline from which the unbounded theory is extended (Xia, 2019, Xia, 2022).

2. Singularity ordering, monotonicity, and full mass

A central organizing principle is the comparison of singularity types. For $\mathbb C^n$7-psh potentials $\mathbb C^n$8, $\mathbb C^n$9 is less singular than $\mathrm{NP}(dd^c u)^n$0 if $\mathrm{NP}(dd^c u)^n$1 for some constant $\mathrm{NP}(dd^c u)^n$2; they have the same singularity type if each is less singular than the other. In this ordering, making the potentials more singular decreases total non-pluripolar mass. The mixed monotonicity theorem states that if $\mathrm{NP}(dd^c u)^n$3 is less singular than $\mathrm{NP}(dd^c u)^n$4 for each $\mathrm{NP}(dd^c u)^n$5, then

$\mathrm{NP}(dd^c u)^n$6

For self-products, this becomes

$\mathrm{NP}(dd^c u)^n$7

whenever $\mathrm{NP}(dd^c u)^n$8 is less singular than $\mathrm{NP}(dd^c u)^n$9. These results remove the earlier small-unbounded-locus restriction and establish that singularity increase forces mass loss (Darvas et al., 2017, Nyström, 2017).

The same circle of results gives a global Bedford–Taylor-type continuity principle. If $X$0 converge in capacity to $X$1, and the total masses satisfy the semicontinuity condition

$X$2

then the non-pluripolar products converge weakly. This is the global substitute for Bedford–Taylor continuity in the singular setting, and the mass hypothesis is essential because truncations can lose mass to pluripolar sets (Darvas et al., 2017).

The relative full mass framework refines this monotonicity theory. Given a reference potential $X$3, the class $X$4 consists of potentials more singular than $X$5 with full non-pluripolar mass relative to $X$6: $X$7 The envelope $X$8 is the canonical representative of the singularity type, and if $X$9 with $T_j=\theta_j+dd^c u_j$0, then $T_j=\theta_j+dd^c u_j$1 is characterized by the equivalent envelope conditions recorded in Theorem 3.14 of (Darvas et al., 2017). A later refinement replaces $T_j=\theta_j+dd^c u_j$2-comparison by a capacity-controlled order $T_j=\theta_j+dd^c u_j$3, defined through

$T_j=\theta_j+dd^c u_j$4

for some $T_j=\theta_j+dd^c u_j$5 and $T_j=\theta_j+dd^c u_j$6. Under this stronger relation, mixed non-pluripolar masses still satisfy monotonicity, and equality of masses holds for potentials with the same singularity type in capacity (Dang et al., 10 Mar 2025).

A common misconception is that the non-pluripolar product is merely a singular version of the classical wedge product with no substantive mass deficit. The loss-of-mass results show otherwise: positive Lelong numbers can force the self-product $T_j=\theta_j+dd^c u_j$7 to fail to have full expected mass, and this failure can be quantified. In big nef classes, the deficit $T_j=\theta_j+dd^c u_j$8 admits lower bounds in terms of generic Lelong numbers along maximal analytic sets with positive generic Lelong number (Vu, 2021).

3. Integration by parts and weak stability

The integration by parts formula for non-pluripolar products is a decisive structural result. Let $T_j=\theta_j+dd^c u_j$9 be a compact Kähler manifold of dimension $(1,1)$0, $(1,1)$1 for $(1,1)$2, and set

$(1,1)$3

If $(1,1)$4 and $(1,1)$5, meaning each pair differs by a bounded function, then

$(1,1)$6

Here $(1,1)$7 is interpreted as the signed measure obtained from the difference of two non-pluripolar products, and the definition is independent of the chosen representatives inside the bounded singularity class. By polarization, the identity extends to fully mixed big classes (Xia, 2019).

This theorem genuinely extends the earlier BEGZ integration-by-parts statement, which required all potentials to have small unbounded locus. The new result works for arbitrary $(1,1)$8-psh potentials provided only that the differences $(1,1)$9 and $u_j$0 are bounded, equivalently that the relevant pairs have the same singularity type. A plausible implication is that many arguments previously restricted to Zariski-open bounded loci can now be formulated directly at the level of singular currents (Xia, 2019).

The proof uses Witt Nyström’s construction on $u_j$1. For $u_j$2, one introduces an auxiliary potential $u_j$3 with small unbounded locus. Its Monge–Ampère pushforward is computed explicitly: $u_j$4 and the normalized pushforwards converge in total variation to $u_j$5. This creates a bridge from the small-unbounded-locus theory on $u_j$6 to arbitrary unbounded potentials on $u_j$7 (Xia, 2019).

The same paper isolates two weak convergence principles that recur throughout the non-pluripolar theory: decreasing approximants that are uniformly bounded off a pluripolar set preserve weighted wedge products with bounded difference functions, and quasi-continuous bounded functions converging in capacity can be inserted into wedge products without destroying weak convergence. These are Bedford–Taylor-type stability statements adapted to the non-pluripolar framework (Xia, 2019).

4. Relative products and higher-dimensional extensions

The relative non-pluripolar product generalizes the usual product by incorporating an additional positive current $u_j$8 of higher bidegree. If $u_j$9 are closed positive $\theta_j$0-currents and $\theta_j$1 is a closed positive current of bidimension $\theta_j$2, the local construction truncates the $\theta_j$3-potentials and wedges with $\theta_j$4. In one notation this is written

$\theta_j$5

and in another

$\theta_j$6

The defining limit is taken on the region where all potentials stay above the truncation level, under a uniform local mass bound. When $\theta_j$7, the usual non-pluripolar product is recovered (Vu, 2020, Xia, 2022).

The relative theory preserves the essential formal properties of the classical product. It is symmetric in the $\theta_j$8-factors, local, compatible with restriction outside complete pluripolar sets, and satisfies an iteration or tower property. It also behaves functorially: in the vector-bundle formulation, it is compatible with proper push-forward and flat pull-back, and if the $\theta_j$9-currents have locally bounded potentials, it coincides with the classical Bedford–Taylor wedge product with $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$0 (Xia, 2022, Vu, 2020).

Monotonicity persists in the relative setting. If $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$1 is less singular than $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$2 in the same cohomology class, then

$$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$3

This leads to the notion of full mass intersection relative to $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$4. A necessary obstruction is given by Lelong numbers: if $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$5 are in Kähler classes and are of full mass intersection relative to $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$6, then an irreducible analytic subset along which all generic Lelong numbers are strictly positive must satisfy a codimension restriction; in the extremal case $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$7, such a set must be empty (Vu, 2020).

Density-current theory gives a geometric comparison. For currents in Kähler classes, $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$8 are of $$\mathbf{1}_{\cap_{j=1}^p\{v_j>-k\}}\, dd^c\max\{v_1,-k\}\wedge\cdots\wedge dd^c\max\{v_p,-k\},$$9-relative full mass intersection if and only if the relative non-pluripolar product agrees with the Dinh–Sibony product: $u$0 More generally, every density current dominates the pullback of the relative non-pluripolar product: $u$1 This identifies the relative non-pluripolar product as the minimal singular intersection object inside the density-current formalism (Vu, 2020).

Two later extensions broaden the scope of the theory. On certain compact Hermitian manifolds satisfying

$u$2

equivalently $u$3 for all $u$4, the relative non-pluripolar product is always well-defined and satisfies mass monotonicity under singularity domination (Li et al., 30 May 2025). For Griffiths positive singular Hermitian vector bundles, the theory is lifted to $u$5 and yields Segre operators

$u$6

with Chern operators defined by universal polynomials in the Segre operators. In the $u$7-good regime, this leads to Chern–Weil formulae expressed through b-divisors and the Riemann–Zariski space (Xia, 2022).

5. Prescribed-singularity Monge–Ampère equations

The non-pluripolar product is the correct operator for Monge–Ampère equations with very singular data. On compact Kähler manifolds, one studies

$u$8

where $u$9 is a positive non-pluripolar measure, $\mathbb C^n$00, and $\mathbb C^n$01 is a model potential, meaning $\mathbb C^n$02. Under the additional assumption that $\mathbb C^n$03 has small unbounded locus, the variational method of Berman–Boucksom–Guedj–Zeriahi gives existence and uniqueness in $\mathbb C^n$04. The model condition is necessary for arbitrary right-hand sides in $\mathbb C^n$05, $\mathbb C^n$06, and the theory yields applications to singular Kähler–Einstein metrics with prescribed singularity type (Darvas et al., 2017).

The local counterpart on a bounded domain $\mathbb C^n$07 encodes singularity type through the model envelope $\mathbb C^n$08. A negative psh function is model if and only if

$\mathbb C^n$09

and a central characterization is

$\mathbb C^n$10

The basic local Dirichlet-type problem is

$\mathbb C^n$11

for a positive Borel measure $\mathbb C^n$12 vanishing on pluripolar sets. If there exists a subsolution $\mathbb C^n$13 with $\mathbb C^n$14 and $\mathbb C^n$15, then the Perron envelope

$\mathbb C^n$16

solves the equation; under an additional subsolution in $\mathbb C^n$17, the solution is unique (Do et al., 2024).

On bounded hyperconvex domains, the local theory extends to nonlinear right-hand sides: $\mathbb C^n$18 where $\mathbb C^n$19 is model, $\mathbb C^n$20 vanishes on pluripolar sets, and $\mathbb C^n$21 is continuous and nondecreasing in $\mathbb C^n$22 and locally $\mathbb C^n$23-integrable in $\mathbb C^n$24. If there exists $\mathbb C^n$25 with

$\mathbb C^n$26

then there exists a unique $\mathbb C^n$27 solving the equation. The proof combines an auxiliary fixed-point construction in the finite-mass case with domain exhaustion and monotone limits in the general case (Do et al., 24 Jul 2025).

Comparison principles, maximum constructions, and stability lemmas are built directly around the non-pluripolar operator in these local theories. Supremums of subsolutions remain subsolutions, monotone limits preserve lower bounds for the non-pluripolar measure, and Xing-type comparison inequalities imply uniqueness statements of the form $\mathbb C^n$28 under the stated hypotheses (Do et al., 2024, Do et al., 24 Jul 2025).

6. Dynamics, singular mass, and Lelong-number phenomena

In holomorphic dynamics, non-pluripolar products provide singular intersection currents whose normalized pull-backs converge to canonical Green currents. On $\mathbb C^n$29, if $\mathbb C^n$30 and $\mathbb C^n$31 is a holomorphic endomorphism of algebraic degree $\mathbb C^n$32, then for every $\mathbb C^n$33,

$\mathbb C^n$34

exponentially fast, where $\mathbb C^n$35 is the Green current. The proof uses the pull-back identity

$\mathbb C^n$36

and truncation of $\mathbb C^n$37 by $\mathbb C^n$38 (Ahn et al., 2018).

On a compact Kähler manifold with a surjective holomorphic endomorphism $\mathbb C^n$39 having simple action on cohomology, if $\mathbb C^n$40 are closed positive $\mathbb C^n$41-currents, then the normalized pull-backs of their non-pluripolar product satisfy

$\mathbb C^n$42

where $\mathbb C^n$43 is the main dynamical Green current and

$\mathbb C^n$44

The absence of pluripolar mass is essential in removing the singular contribution of the truncated exceptional sets (Ahn et al., 2023).

Another major development is the explicit separation between the non-pluripolar product and the singular mass that it ignores. For $\mathbb C^n$45, the current

$\mathbb C^n$46

dominates the non-pluripolar product: $\mathbb C^n$47 Globally, analogous currents $\mathbb C^n$48 are defined on $\mathbb C^n$49, and the mass formula

$\mathbb C^n$50

describes precisely the loss of mass of the non-pluripolar Monge–Ampère measure. This shows that the non-pluripolar product is intentionally blind to part of the singular mass; it is not a full singular intersection current in the sense of retaining every concentrated contribution (Andersson et al., 2021).

Lelong-number obstructions sharpen this picture. For a big class $\mathbb C^n$51, the class $\mathbb C^n$52 is defined by the non-pluripolar product of a current with minimal singularities. If

$\mathbb C^n$53

then

$\mathbb C^n$54

and if equality holds, then

$\mathbb C^n$55

Applied to divisorial restricted volumes, this implies that if

$\mathbb C^n$56

then $\mathbb C^n$57 for every $\mathbb C^n$58. In particular, on projective manifolds, the Lelong numbers of $\mathbb C^n$59 vanish at every point (Nguyen et al., 20 Aug 2025).

These developments collectively show that the non-pluripolar product occupies a precise position between classical Bedford–Taylor theory and more singular intersection formalisms. It preserves positivity, locality, and comparison in regimes where ordinary wedges fail, but it also deliberately excludes pluripolar mass. This suggests that its proper role is not to encode all singular intersection data, but to isolate the analytically stable part of Monge–Ampère and mixed-intersection theory that survives under severe singularities (Nyström, 2017, Andersson et al., 2021).