---
title: Non-Pluripolar Product in Pluripotential Theory
url: https://www.emergentmind.com/topics/non-pluripolar-product-ff692703-bf66-4c81-8d93-4fb851f7ae0e
type: topic
---

# Non-Pluripolar Product in Pluripotential Theory

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 [1705.05796], [1907.06359], [2004.11111], [2106.10883].

## 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
\[
\langle(dd^c u)^n\rangle
=
\lim_{M\to\infty}
\mathbf1_{\{u>-M\}}\,(dd^c\max(u,-M))^n.
\]
The same construction defines the \(p\)-fold non-pluripolar product associated with an \(\omega\)-psh function \(\phi\) on \(\mathbf P^k\),
\[
\langle\phi\rangle^p
=
\lim_{j\to\infty}
\bigl(\omega+dd^c\max\{\phi,-j\}\bigr)^p\bigm|_{\{\phi>-j\}},
\]
which coincides with the Bedford–Taylor product when \(\phi\) is bounded [1811.03017], [1907.06359], [2407.00937].

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 [1703.01950], [2006.16803], [2106.10883].

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 \(\mathbf1_{\{u>-k\}}\) or \(\mathbf1_{\cap_j\{u_j>-k\}}\) 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 [2507.18116], [2407.00937].

## 2. Energy classes and extended Monge–Ampère currents

The non-pluripolar product interacts naturally with finite-energy classes. On \((\mathbf P^k,\omega)\), Guedj–Zeriahi’s hierarchy includes
\[
E(\mathbf P^k,\omega)
=
\{\phi\in \mathrm{PSH}(\mathbf P^k,\omega): \int\langle\phi\rangle^k=\int\omega^k\},
\]
and for \(p\ge 1\),
\[
E^p(\mathbf P^k,\omega)
=
\{\phi\in E(\mathbf P^k,\omega): E_p(\phi)<\infty\},
\]
with
\[
E_p(\phi)
:=
\frac1{p+1}\sum_{j=0}^p
\int_{\mathbf P^k}
|\phi|\,\langle\phi\rangle^j\wedge\omega^{k-j}.
\]
The normalized class
\[
E^1(\mathbf P^k,\omega)_0
=
\{\phi\in E^1(\mathbf P^k,\omega): \sup_{\mathbf P^k}\phi=0\}
\]
is the input class in Ahn–Nguyen’s equidistribution theorem [1811.03017].

Andersson, Witt Nyström, and Wulcan introduced a distinct but related finite-energy framework. For a domain \(\Omega\subset\mathbf C^n\), the class \(G(\Omega)\) consists of \(u\in\mathrm{PSH}(\Omega)\) such that \(\langle(dd^c u)^p\rangle\) is locally finite and \(u\in L^1_{\mathrm{loc}}(\langle(dd^c u)^p\rangle)\) for all \(p\le n-1\). On a compact Kähler manifold \((X,\omega)\), they define the global class
\[
G(X,\omega)
=
\{\varphi\in\mathrm{PSH}(X,\omega): \varphi\in L^1(\langle(dd^c\varphi+\omega)^j\rangle)\ \forall j\le n-1\}.
\]
The abstract states that \(G(X,\omega)\) includes \(\omega\)-psh functions with analytic singularities and the class \(\mathcal E(X,\omega)\) of \(\omega\)-psh functions of finite energy, although it is not convex itself [2106.10883].

A further refinement is the family of extended currents
\[
[dd^c u]^p
=
dd^c\bigl(u\,\langle(dd^c u)^{p-1}\rangle\bigr),
\]
defined for \(u\in G(\Omega)\), and globally
\[
[dd^c\varphi+\omega]^p
=
dd^c(\varphi+h)\wedge\langle dd^c(\varphi+h)\rangle^{p-1}
\quad (\text{locally } dd^c h=\omega).
\]
These currents dominate the ordinary non-pluripolar products:
\[
[dd^c u]^p\ge \langle(dd^c u)^p\rangle,
\qquad
[dd^c\varphi+\omega]^p\ge \langle(dd^c\varphi+\omega)^p\rangle.
\]
Their difference from the non-pluripolar product is encoded by the excess currents
\[
S_p(u)=[dd^c u]^p-\langle(dd^c u)^p\rangle,
\qquad
S_p(\varphi)=[dd^c\varphi+\omega]^p-\langle(dd^c\varphi+\omega)^p\rangle.
\]
Natural truncations recover these extended currents in the limit, and a mass formula expresses the loss of mass of \(\langle(dd^c\varphi+\omega)^n\rangle\) through the lower-order excess currents [2106.10883].

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 \([dd^c u]^p\) retain information about the escaping mass that the non-pluripolar product omits [2106.10883].

## 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 \(\varphi_1,\varphi_2\in\mathrm{PSH}(X,\theta)\) and \(\varphi_1\le \varphi_2\), then
\[
\int_X \langle(\theta+dd^c\varphi_1)^n\rangle
\ge
\int_X \langle(\theta+dd^c\varphi_2)^n\rangle.
\]
Witt Nyström proved this in full generality, removing the earlier small-unbounded-locus restriction [1703.01950]. Darvas–Di Nezza–Lu established the corresponding mixed statement: if \(u_j\) is less singular than \(v_j\) for each \(j\), then
\[
\int_X\langle T_1\wedge\cdots\wedge T_n\rangle
\ge
\int_X\langle T_1'\wedge\cdots\wedge T_n'\rangle
\]
for \(T_j=\theta_j+dd^c u_j\) and \(T_j'=\theta_j+dd^c v_j\) [1705.05796].

Monotonicity yields comparison principles. Witt Nyström obtained the Bedford–Taylor-type inequality
\[
\int_{\{\varphi<\psi\}}
\langle(\theta+dd^c\psi)^n\rangle
\le
\int_{\{\varphi<\psi\}}
\langle(\theta+dd^c\varphi)^n\rangle,
\]
while local Xing-type comparison principles adapted to \(\langle(dd^c\cdot)^n\rangle\) were proved for bounded domains in \(\mathbf C^n\) [1703.01950], [2407.00937].

The notion of full mass packages the equality case. If \(T_j\) lie in classes \(\alpha_j\), one has
\[
\bigl\|\langle T_1\wedge\cdots\wedge T_m\rangle\bigr\|
\le
\|\alpha_1\wedge\cdots\wedge\alpha_m\|,
\]
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
\[
\bigl\|\alpha^m-\{\langle T^m\rangle\}\bigr\|
\]
in terms of generic Lelong numbers along maximal analytic strata [2101.05483].

Recent work makes the relation between full mass and singularities sharper. If \(\alpha\) is a big class on a compact Kähler manifold and \(D\) is an effective divisor, then the full-mass condition along \(D\),
\[
\langle\alpha^{\,n-1}\rangle\wedge\{D\}
=
\langle\alpha^{\,n-1}\rangle\big|_{X|D},
\]
forces
\[
\nu\bigl(\langle\alpha^{\,n-1}\rangle,x\bigr)=0
\quad\text{for every }x\in\mathrm{Supp}(D).
\]
In particular, on projective manifolds the Lelong numbers of \(\langle\alpha^{\,n-1}\rangle\) are zero at every point [2508.14669]. 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 [2503.07534].

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 \((p,p)\)-current \(T\) is fixed. If \(T_j=\theta_j+dd^c u_j\), one defines
\[
\bigl\langle T_1\wedge\cdots\wedge T_m\dot\wedge T\bigr\rangle
=
\lim_{k\to\infty}
\mathbf1_{\cap_j\{u_j>-k\}}
\,(T_{1,k}\wedge\cdots\wedge T_{m,k})\wedge T.
\]
When \(T=[X]\), the current of integration along \(X\), this recovers the absolute non-pluripolar product. Relative products relative to divisors or subvarieties are obtained by taking \(T=[D]\) or a corresponding integration current [2004.11111], [2006.16803].

The relative theory has its own monotonicity and full-mass notions. If \(T_j'\) is less singular than \(T_j\) in the same cohomology class, then
\[
\bigl\{\langle T_1\wedge\cdots\wedge T_m\dot\wedge T\rangle\bigr\}
\le
\bigl\{\langle T_1'\wedge\cdots\wedge T_m\dot\wedge T\rangle\bigr\}
\]
in the compact Kähler setting. Vu also proved a Lelong obstruction: if the generic Lelong numbers of \(T_1,\dots,T_m\) and of \(T\) along an irreducible analytic set \(V\) are all positive, then relative full-mass intersection forces
\[
\dim V < n-p-m.
\]
Weighted classes of currents of relative full-mass intersection are convex in each cohomology class [2004.11111].

A parallel comparison with the Dinh–Sibony theory of density currents is now part of the standard picture. If \(R_\infty\) is a tangent density current of \(T_1\otimes\cdots\otimes T_m\otimes T\) along the diagonal, then
\[
\pi^*\bigl\langle T_1\wedge\cdots\wedge T_m\dot\wedge T\bigr\rangle
\le
R_\infty.
\]
Under Kähler-class hypotheses, the relative product equals the Dinh–Sibony product if and only if the currents have \(T\)-relative full-mass intersection [2006.16803].

The theory extends beyond the Kähler case. On compact Hermitian manifolds carrying a Hermitian form \(\omega\) satisfying
\[
\partial\bar\partial\,\omega=0,
\qquad
\partial\omega\wedge\bar\partial\omega=0,
\]
Li–Su proved that the relative non-pluripolar product is always well-defined and that monotonicity survives at the level of masses:
\[
\bigl\|\langle T_1\wedge\cdots\wedge T_m\dot\wedge T\rangle\bigr\|
\le
\bigl\|\langle T_1'\wedge\cdots\wedge T_m'\dot\wedge T\rangle\bigr\|
\]
whenever each \(T_j'\) is less singular than \(T_j\) [2505.24702].

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

A major analytic difficulty in the singular setting is whether one can move \(dd^c\) 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
\[
u=\varphi_1-\varphi_2,\qquad v=\psi_1-\psi_2,
\]
with \([\varphi_1]=[\varphi_2]\) and \([\psi_1]=[\psi_2]\), then
\[
\int_X u\,dd^c v\wedge\theta_{\gamma_1}\wedge\cdots\wedge\theta_{\gamma_{n-1}}
=
\int_X v\,dd^c u\wedge\theta_{\gamma_1}\wedge\cdots\wedge\theta_{\gamma_{n-1}}.
\]
No small-unbounded-locus assumption is required on \(\varphi_j,\psi_j\). The proof uses Witt Nyström’s approximation \(\Phi_N[\varphi]\) on \(X\times\mathbf P^N\), two weak-convergence lemmas, and classical Bedford–Taylor integration by parts on the approximating space [1907.06359].

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
\[
(\omega+dd^c u)^n=f\,\omega^n
\]
in the non-pluripolar sense, and similarly for right-hand sides of the form \(e^{\lambda u}f\,\omega^n\) [1705.05796].

Local analogues are now available on bounded hyperconvex domains. One existence theorem assumes a nonnegative Borel measure \(\mu\) vanishing on pluripolar sets, a model plurisubharmonic function \(\phi\), and a measurable non-decreasing \(F:\mathbf R\times\Omega\to[0,+\infty)\) with a subsolution \(\psi\in \mathrm{PSH}^-(\Omega)\), \(P[\psi]=0\), satisfying
\[
NP[\psi]\ge F(\phi,\cdot)\,d\mu.
\]
Then there is a unique \(u\in\mathrm{PSH}^-(\Omega)\) with \(P[u]=\phi\) solving
\[
\langle(dd^c u)^n\rangle = F(u(z),z)\,d\mu,
\qquad
P[u]=\phi,
\]
and \(u\) lies in the local energy class \(N(\phi)\) [2507.18116].

A related Dirichlet-type problem on bounded domains in \(\mathbf C^n\) uses Perron envelopes. If \(\mu\) vanishes on pluripolar sets and there exists a subsolution \(v\in\mathrm{PSH}^-(\Omega)\), \(P[v]=0\), with \(\langle(dd^c v)^n\rangle\ge \mu\), then the upper envelope of the subsolution family solves
\[
\langle(dd^c u)^n\rangle=\mu,
\qquad
P[u]=0.
\]
In hyperconvex domains, every finite non-pluripolar Radon measure admits a unique such solution [2407.00937].

## 6. Dynamical, geometric, and vector-bundle applications

In complex dynamics, Ahn–Nguyen studied the pull-back of non-pluripolar products under holomorphic endomorphisms \(f:\mathbf P^k\to\mathbf P^k\) of algebraic degree \(d\ge 2\). If \(\phi\in E^1(\mathbf P^k,\omega)_0\), \(T\) is the Green \((1,1)\)-current, and \(1\le p\le k\), then the normalized pull-backs converge exponentially fast:
\[
\bigl|\langle d^{-pn}f^{*n}\langle\phi\rangle^p-T^p,\psi\rangle\bigr|
\le C\,\lambda^n,
\qquad
\lambda=\frac{d_{p-1}}{d^p}<1.
\]
Here \(T^p=(\omega+dd^c G)^p\) is well-defined because the Green potential \(G\) is Hölder continuous. In particular, when \(p=1\), one recovers exponential convergence of \(d^{-n}f^{*n}(\omega+dd^c\phi)\) to \(T\) [1811.03017].

In complex geometry, non-pluripolar products now enter singular Chern–Weil theory for vector bundles. For a Griffiths positive singular Hermitian metric \(h_E\) on a holomorphic vector bundle \(E\), the induced singular metric on \(\mathcal O_{\mathbf P(E^\vee)}(1)\) yields Segre currents
\[
s_i(\hat E)\cap T
=
(-1)^i\,p_*\bigl(c_1(\mathcal O(1))^{r+i}\cap p^*T\bigr),
\]
where the symbol \(\cap\) denotes the relative non-pluripolar product. Chern currents are then obtained from the universal polynomials expressing Chern classes in terms of Segre classes [2210.15342].

This framework supports a notion of \(\mathcal I\)-good singularities for vector bundles, defined by requiring that the induced metric on \(\mathcal O_{\mathbf P(E^\vee)}(1)\) be \(\mathcal I\)-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 [2210.15342].

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.

Source: https://www.emergentmind.com/topics/non-pluripolar-product-ff692703-bf66-4c81-8d93-4fb851f7ae0e