Non-Pluripolar Product in Pluripotential Theory

• Non-Pluripolar Product is a truncation-based extension that redefines classical Bedford–Taylor wedge-products for singular plurisubharmonic potentials by eliminating pluripolar mass.
• It is constructed via potential truncation and convergence methods, ensuring closed, positive currents while preserving key properties such as monotonicity and compatibility with bounded cases.
• This concept underpins finite-energy classes, complex Monge–Ampère equations, dynamical systems, and singular Chern–Weil theory in modern pluripotential analysis.

The non-pluripolar product is a truncation-based extension of Bedford–Taylor wedge-products from locally bounded plurisubharmonic or quasi-plurisubharmonic potentials to singular ones. Given closed positive $(1,1)$-currents with unbounded local potentials, one first truncates the potentials from below, forms the classical Bedford–Taylor wedge on the truncated data, restricts to the region where the truncation has not altered the original potentials, and then lets the truncation level tend to infinity. The resulting current is closed, positive, and by construction puts no mass on pluripolar sets. In modern pluripotential theory it is the basic intersection operation underlying non-pluripolar Monge–Ampère measures, finite-energy classes, relative intersections with higher-bidegree currents, comparison and monotonicity theorems, prescribed-singularity Monge–Ampère equations, complex dynamics, and singular Chern–Weil theory (Darvas et al., 2017, Xia, 2019, Vu, 2020, Andersson et al., 2021).

1. Definition and core formalism

On a compact Kähler manifold $X$, let $T_j=\theta_j+dd^c u_j$ be closed positive $(1,1)$-currents, where $\theta_j$ is smooth and $u_j$ is quasi-plurisubharmonic. For each integer $k\ge 1$, set

$$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$

The non-pluripolar product is defined by

$$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$

Locally on domains in $\mathbf C^n$, this specializes to

$X$0

The same construction defines the $X$1-fold non-pluripolar product associated with an $X$2-psh function $X$3 on $X$4,

$X$5

which coincides with the Bedford–Taylor product when $X$6 is bounded (Ahn et al., 2018, Xia, 2019, Do et al., 2024).

Several structural properties are built into the definition. The product is a closed positive current of the expected bidegree; it puts no mass on pluripolar sets; it is compatible with classical Bedford–Taylor theory when the potentials are locally bounded; and it is plurifinely local, meaning that it depends only on the values of the potentials off pluripolar sets. In the local and global formulations appearing in the literature, symmetry, multilinearity, monotonicity in singularities, and continuity under decreasing regularization are recurrent basic properties (Nyström, 2017, Vu, 2020, Andersson et al., 2021).

A common misconception is that the non-pluripolar product is merely the classical wedge-product written with singular potentials. The truncation formulas show that this is not the case. The indicator $X$7 or $X$8 explicitly removes the part of the Bedford–Taylor mass carried by the singular locus, so the construction is designed to retain only the non-pluripolar component (Do et al., 24 Jul 2025, Do et al., 2024).

2. Energy classes and extended Monge–Ampère currents

The non-pluripolar product interacts naturally with finite-energy classes. On $X$9, Guedj–Zeriahi’s hierarchy includes

$T_j=\theta_j+dd^c u_j$0

and for $T_j=\theta_j+dd^c u_j$1,

$T_j=\theta_j+dd^c u_j$2

with

$T_j=\theta_j+dd^c u_j$3

The normalized class

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

is the input class in Ahn–Nguyen’s equidistribution theorem (Ahn et al., 2018).

Andersson, Witt Nyström, and Wulcan introduced a distinct but related finite-energy framework. For a domain $T_j=\theta_j+dd^c u_j$5, the class $T_j=\theta_j+dd^c u_j$6 consists of $T_j=\theta_j+dd^c u_j$7 such that $T_j=\theta_j+dd^c u_j$8 is locally finite and $T_j=\theta_j+dd^c u_j$9 for all $(1,1)$0. On a compact Kähler manifold $(1,1)$1, they define the global class

$(1,1)$2

The abstract states that $(1,1)$3 includes $(1,1)$4-psh functions with analytic singularities and the class $(1,1)$5 of $(1,1)$6-psh functions of finite energy, although it is not convex itself (Andersson et al., 2021).

A further refinement is the family of extended currents

$(1,1)$7

defined for $(1,1)$8, and globally

$(1,1)$9

These currents dominate the ordinary non-pluripolar products: $\theta_j$0 Their difference from the non-pluripolar product is encoded by the excess currents

$\theta_j$1

Natural truncations recover these extended currents in the limit, and a mass formula expresses the loss of mass of $\theta_j$2 through the lower-order excess currents (Andersson et al., 2021).

This enlargement of the formalism is significant because it separates two phenomena that coincide in the locally bounded setting: the classical Monge–Ampère current obtained from regularizations, and the non-pluripolar part obtained after discarding pluripolar mass. The extended currents $\theta_j$3 retain information about the escaping mass that the non-pluripolar product omits (Andersson et al., 2021).

3. Monotonicity, comparison, and the role of singularities

A central theorem is monotonicity of total non-pluripolar mass with respect to singularities. On a compact Kähler manifold, if $\theta_j$4 and $\theta_j$5, then

$\theta_j$6

Witt Nyström proved this in full generality, removing the earlier small-unbounded-locus restriction (Nyström, 2017). Darvas–Di Nezza–Lu established the corresponding mixed statement: if $\theta_j$7 is less singular than $\theta_j$8 for each $\theta_j$9, then

$u_j$0

for $u_j$1 and $u_j$2 (Darvas et al., 2017).

Monotonicity yields comparison principles. Witt Nyström obtained the Bedford–Taylor-type inequality

$u_j$3

while local Xing-type comparison principles adapted to $u_j$4 were proved for bounded domains in $u_j$5 (Nyström, 2017, Do et al., 2024).

The notion of full mass packages the equality case. If $u_j$6 lie in classes $u_j$7, one has

$u_j$8

and the currents are said to be of full-mass intersection when equality holds. Vu and subsequent work show that positive Lelong numbers obstruct this regime. In the big and nef setting, Vu gave a quantitative lower bound for the mass defect

$u_j$9

in terms of generic Lelong numbers along maximal analytic strata (Vu, 2021).

Recent work makes the relation between full mass and singularities sharper. If $k\ge 1$0 is a big class on a compact Kähler manifold and $k\ge 1$1 is an effective divisor, then the full-mass condition along $k\ge 1$2,

$k\ge 1$3

forces

$k\ge 1$4

In particular, on projective manifolds the Lelong numbers of $k\ge 1$5 are zero at every point (Nguyen et al., 20 Aug 2025). Dang–Do–Pham compare singularities of closed positive currents whose non-pluripolar complex Monge–Ampère masses are equal and provide a short alternative proof of monotonicity, generalizing results of Witt Nyström, Darvas–Di Nezza–Lu, Lu–Nguyên, and Vu (Dang et al., 10 Mar 2025).

These results clarify a frequent source of confusion. The non-pluripolar product is not merely a way to define singular intersections; it is also a device that records loss of mass caused by singularities. Positive Lelong numbers, failure of full mass, and the appearance of excess currents are different manifestations of the same obstruction.

4. Relative non-pluripolar products and density currents

Vu introduced a relative version in which an additional closed positive $k\ge 1$6-current $k\ge 1$7 is fixed. If $k\ge 1$8, one defines

$k\ge 1$9

When $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$0, the current of integration along $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$1, this recovers the absolute non-pluripolar product. Relative products relative to divisors or subvarieties are obtained by taking $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$2 or a corresponding integration current (Vu, 2020, Vu, 2020).

The relative theory has its own monotonicity and full-mass notions. If $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$3 is less singular than $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$4 in the same cohomology class, then

$$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$5

in the compact Kähler setting. Vu also proved a Lelong obstruction: if the generic Lelong numbers of $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$6 and of $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$7 along an irreducible analytic set $$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$8 are all positive, then relative full-mass intersection forces

$$u_{j,k}:=\max\{u_j,-k\}, \qquad T_{j,k}:=\theta_j+dd^c u_{j,k}.$$9

Weighted classes of currents of relative full-mass intersection are convex in each cohomology class (Vu, 2020).

A parallel comparison with the Dinh–Sibony theory of density currents is now part of the standard picture. If $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$0 is a tangent density current of $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$1 along the diagonal, then

$$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$2

Under Kähler-class hypotheses, the relative product equals the Dinh–Sibony product if and only if the currents have $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$3-relative full-mass intersection (Vu, 2020).

The theory extends beyond the Kähler case. On compact Hermitian manifolds carrying a Hermitian form $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$4 satisfying

$$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$5

Li–Su proved that the relative non-pluripolar product is always well-defined and that monotonicity survives at the level of masses: $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$6 whenever each $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$7 is less singular than $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$8 (Li et al., 30 May 2025).

5. Integration by parts and Monge–Ampère equations with prescribed singularities

A major analytic difficulty in the singular setting is whether one can move $$\langle T_1\wedge\cdots\wedge T_m\rangle = \lim_{k\to\infty} \mathbf1_{\cap_j\{u_j>-k\}} \,T_{1,k}\wedge\cdots\wedge T_{m,k}.$$9 across non-pluripolar products without assuming small unbounded locus. The integration by parts formula proved by Xing-type methods and Witt Nyström approximation answers this on compact Kähler manifolds. If

$\mathbf C^n$0

with $\mathbf C^n$1 and $\mathbf C^n$2, then

$\mathbf C^n$3

No small-unbounded-locus assumption is required on $\mathbf C^n$4. The proof uses Witt Nyström’s approximation $\mathbf C^n$5 on $\mathbf C^n$6, two weak-convergence lemmas, and classical Bedford–Taylor integration by parts on the approximating space (Xia, 2019).

This identity is indispensable in variational treatments of complex Monge–Ampère equations. Darvas–Di Nezza–Lu used relative energy functionals and the monotonicity of non-pluripolar products to solve equations with prescribed singularity type on compact Kähler manifolds, under the assumption of small unbounded locus. They obtained existence and uniqueness for

$\mathbf C^n$7

in the non-pluripolar sense, and similarly for right-hand sides of the form $\mathbf C^n$8 (Darvas et al., 2017).

Local analogues are now available on bounded hyperconvex domains. One existence theorem assumes a nonnegative Borel measure $\mathbf C^n$9 vanishing on pluripolar sets, a model plurisubharmonic function $X$00, and a measurable non-decreasing $X$01 with a subsolution $X$02, $X$03, satisfying

$X$04

Then there is a unique $X$05 with $X$06 solving

$X$07

and $X$08 lies in the local energy class $X$09 (Do et al., 24 Jul 2025).

A related Dirichlet-type problem on bounded domains in $X$10 uses Perron envelopes. If $X$11 vanishes on pluripolar sets and there exists a subsolution $X$12, $X$13, with $X$14, then the upper envelope of the subsolution family solves

$X$15

In hyperconvex domains, every finite non-pluripolar Radon measure admits a unique such solution (Do et al., 2024).

6. Dynamical, geometric, and vector-bundle applications

In complex dynamics, Ahn–Nguyen studied the pull-back of non-pluripolar products under holomorphic endomorphisms $X$16 of algebraic degree $X$17. If $X$18, $X$19 is the Green $X$20-current, and $X$21, then the normalized pull-backs converge exponentially fast: $X$22 Here $X$23 is well-defined because the Green potential $X$24 is Hölder continuous. In particular, when $X$25, one recovers exponential convergence of $X$26 to $X$27 (Ahn et al., 2018).

In complex geometry, non-pluripolar products now enter singular Chern–Weil theory for vector bundles. For a Griffiths positive singular Hermitian metric $X$28 on a holomorphic vector bundle $X$29, the induced singular metric on $X$30 yields Segre currents

$X$31

where the symbol $X$32 denotes the relative non-pluripolar product. Chern currents are then obtained from the universal polynomials expressing Chern classes in terms of Segre classes (Xia, 2022).

This framework supports a notion of $X$33-good singularities for vector bundles, defined by requiring that the induced metric on $X$34 be $X$35-good in the line-bundle sense. On projective manifolds, the associated non-pluripolar Chern and Segre currents can be reinterpreted on the Riemann–Zariski space through b-divisors, leading to Chern–Weil type formulae for singular metrics (Xia, 2022).

Taken together, these developments show that the non-pluripolar product has become the standard intersection mechanism whenever singular positivity is present but pluripolar mass must be discarded. This suggests a unifying role across pluripotential theory, dynamics, and singular complex geometry: the same truncation-and-locality principle governs Monge–Ampère operators, relative intersections, dynamical equilibrium currents, and singular characteristic classes.