---
title: Relative Non-Pluripolar Product
url: https://www.emergentmind.com/topics/relative-non-pluripolar-product
type: topic
---

# Relative Non-Pluripolar Product

Searching arXiv for recent and foundational papers on relative non-pluripolar products.
The relative non-pluripolar product is a singular intersection current of closed positive \((1,1)\)-currents taken relative to a fixed closed positive current of higher bidegree. In the compact Kähler setting, it is denoted
\[
\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,
\]
where \(T_1,\dots,T_m\) are closed positive \((1,1)\)-currents, \(T\) is a closed positive \((p,p)\)-current, and \(m+p\le n=\dim X\). Introduced by Duc-Viet Vu as a relative extension of the BEGZ non-pluripolar product, it keeps the ambient current \(T\) in the Bedford–Taylor truncation scheme and discards pluripolar contributions coming from the singularities of the \(T_j\)’s [2004.11111, 2006.16803].

## 1. Definition and ambient framework

Let \(X\) be a compact Kähler manifold of dimension \(n\). The basic data are closed positive \((1,1)\)-currents \(T_1,\dots,T_m\), a closed positive \((p,p)\)-current \(T\), and the dimensional constraint
\[
m+p\le n.
\]
The output is a closed positive \((m+p,m+p)\)-current
\[
\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle.
\]

Locally one writes
\[
T_j=dd^c u_j,
\]
or globally, after fixing a smooth representative \(\theta_j\in \{T_j\}\),
\[
T_j=\theta_j+dd^c u_j,
\]
with \(u_j\) \(\theta_j\)-psh. The truncations are
\[
u_{j,k}:=\max\{u_j,-k\},
\qquad
T_{j,k}:=\theta_j+dd^c u_{j,k}.
\]
The relative non-pluripolar product is then defined by
\[
\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle
=
\lim_{k\to\infty}
\mathbf 1_{\cap_{j=1}^m\{u_j>-k\}}
\bigwedge_{j=1}^m T_{j,k}\wedge T.
\]
Equivalently, on the truncation region,
\[
\mathbf 1_{\cap_{j=1}^m\{u_j>-k\}}
\bigwedge_{j=1}^m T_{j,k}\wedge T
=
\mathbf 1_{\cap_{j=1}^m\{u_j>-k\}}
\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle.
\]
The indicator functions record the defining feature of the construction: only the plurifinely relevant part where all potentials are finite is retained, and pluripolar concentration in the \(T_j\)-variables is discarded [2006.16803].

This construction is relative in a precise sense. It is not the naive wedge
\[
T_1\wedge\cdots\wedge T_m\wedge T
\]
formed formally in the presence of singularities. Rather, one first regularizes the \(T_j\)’s by bounded truncations, wedges these bounded approximants against \(T\), and only then passes to the limit on the non-polar region. When \(T\equiv 1\), or equivalently when \(T\) is the current of integration along \(X\), one recovers the usual non-pluripolar product:
\[
\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge 1\big\rangle
=
\big\langle T_1\wedge\cdots\wedge T_m\big\rangle
\]
[2004.11111].

## 2. Structural properties and relative full mass

On compact Kähler manifolds the relative non-pluripolar product is always well-defined. It is closed, positive, symmetric in the \((1,1)\)-entries, and homogeneous. It is also local in the plurifine topology, and it ignores the part of the auxiliary current \(T\) carried by the polar sets of the \(T_j\)’s. More precisely, if
\[
T' := \mathbf 1_{X\setminus \bigcup_{j=1}^m I_{T_j}}\,T,
\]
then
\[
\big\langle \bigwedge_{j=1}^m T_j \dot\wedge T\big\rangle
=
\big\langle \bigwedge_{j=1}^m T_j \dot\wedge T'\big\rangle.
\]
If \(A\) is a locally complete pluripolar set and \(T\) has no mass on \(A\), then the relative product also has no mass on \(A\) [2005.13241].

The product admits an iteration or tower property. If
\[
R:=\Big\langle \bigwedge_{j=l+1}^m T_j \dot\wedge T\Big\rangle,
\]
then
\[
\Big\langle \bigwedge_{j=1}^m T_j \dot\wedge T\Big\rangle
=
\Big\langle \bigwedge_{j=1}^l T_j \dot\wedge R\Big\rangle.
\]
This allows one to absorb some of the \((1,1)\)-currents into the reference current and is central in inductive arguments [2005.13241].

A notable subtlety is additivity. In general the relative non-pluripolar product is not additive in the \((1,1)\)-variables. However, if \(T\) has no mass on the polar sets \(I_{T_1}\cup I_{T_1'}\), then one has
\[
\big\langle  (T_1+T'_1) \wedge \bigwedge_{j=2}^m T_j \dot\wedge T\big\rangle
=
\langle T_1 \wedge \bigwedge_{j=2}^m T_j \dot\wedge T\rangle
+
\langle T'_1 \wedge \bigwedge_{j=2}^m T_j \dot\wedge T\rangle
\]
[2005.13241].

The basic cohomological monotonicity statement is that if \(T_j'\) is cohomologous to \(T_j\) and less singular than \(T_j\) for every \(j\), then
\[
\Big\{ \big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle \Big\}
\le
\Big\{ \big\langle T_1'\wedge\cdots\wedge T_m' \dot\wedge T\big\rangle \Big\}.
\]
This leads to the notion of \(T\)-relative full mass intersection: if the \(T_j'\) are chosen with minimal singularities in their classes, then \(T_1,\dots,T_m\) are of \(T\)-relative full mass intersection when equality holds. Equivalently,
\[
\Big\{ \big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle \Big\}
=
\bigwedge_{j=1}^m \{T_j\}\wedge \{T\}.
\]
In the Kähler-class setting this is the relative analogue of the full Monge–Ampère mass condition, and it is the central hypothesis in later comparison theorems [2006.16803].

A truncation criterion makes the full-mass condition concrete. If
\[
T_j=\theta_j+dd^c u_j,\qquad
T_{j,k}=\theta_j+dd^c\max\{u_j,-k\},
\]
then full mass relative to \(T\) is equivalent to the vanishing of the deep-singularity contribution:
\[
\int_{\cup_{j=1}^m\{u_j\le -k\}}
\bigwedge_{j=1}^m T_{j,k}\wedge T\wedge \omega^{n-m-p}
\to 0
\]
[2004.11111].

## 3. Density currents and the Dinh–Sibony comparison

A second intersection theory enters through density currents. Given several positive currents \(T_1,\dots,T_m\), one considers the tensor product current on \(X^m\) and tangent currents along the diagonal \(\Delta_m\). A density current associated to \(T_1,\dots,T_m\) is such a tangent current, and the Dinh–Sibony product
\[
T_1\curlywedge\cdots\curlywedge T_m
\]
is defined when the density current is unique and pulled back from the base. For bounded-potential \((1,1)\)-currents this agrees with the classical wedge product, and similarly
\[
T_1\curlywedge\cdots\curlywedge T_m\curlywedge T
=
T_1\wedge\cdots\wedge T_m\wedge T
\]
in the bounded-potential regime [2006.16803].

The main comparison theorem states that if the cohomology class of each \(T_j\) is Kähler, then
\[
T_1,\dots,T_m \text{ are of \(T\)-relative full mass intersection}
\]
if and only if
\[
\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle
=
T_1\curlywedge\cdots\curlywedge T_m\curlywedge T.
\]
Thus the relative non-pluripolar product and the Dinh–Sibony product coincide exactly under the relative full-mass hypothesis [2006.16803].

Even without full mass, there is a universal domination statement. If \(R_\infty\) is a density current associated to \(T_1,\dots,T_m,T\), then
\[
\pi_{m+1}^*
\big\langle \wedge_{j=1}^m T_j \dot\wedge T\big\rangle
\le
R_\infty,
\]
where \(\pi_{m+1}\) is the normal-bundle projection from the diagonal in \(X^{m+1}\). This inequality is the basic bridge between pluripotential-theoretic intersection and density-current intersection. The proof uses exact truncation identities, bounded-potential lemmas along the diagonal, and the fact that under full mass the relevant density class has minimal \(h\)-dimension [2006.16803].

The Kähler assumption is essential in the equivalence theorem. It is explicitly stated that the result is false in general for arbitrary pseudoeffective classes, because minimal-singularity currents in non-nef classes may have positive Lelong numbers somewhere, obstructing identification with density currents [2006.16803].

## 4. Relative energy, integration by parts, and convexity

The relative non-pluripolar product supports an energy theory parallel to the BEGZ finite-energy formalism. A key step is the extension of the relative calculus to bounded \(T\)-admissible dsh functions. If \(v=\varphi_1-\varphi_2\) with \(T\) carrying no mass on the pole sets of the quasi-psh pieces, then one can define
\[
\langle dd^c v \dot\wedge T\rangle,
\qquad
\langle dv\wedge d^c v \dot\wedge T\rangle,
\]
and more generally, after setting
\[
R=\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\rangle,
\]
also
\[
\langle dd^c v\wedge T_1\wedge\cdots\wedge T_m \dot\wedge T\rangle
:=
\langle dd^c v\dot\wedge R\rangle.
\]
This extends the relative product from positive currents to differential expressions needed in energy estimates [2005.13241].

The central analytic identity is the integration by parts formula. If \(T\) has bidegree \((n-1,n-1)\) and \(v,w\) are bounded \(T\)-admissible dsh functions, then
\[
\int_X w \langle  dd^c v \dot\wedge T\rangle
=
\int_X v \langle dd^c w\dot\wedge T\rangle
=
- \int_X \langle dw \wedge d^c v \dot\wedge T \rangle.
\]
A weighted version with \(\chi\in C^3(\mathbb R)\) is also available:
\[
\int_X \chi(w) \langle  dd^c v \dot\wedge T\rangle
=
\int_X v \chi''(w) \langle dw \wedge d^c w \dot\wedge T\rangle
+
\int_X v \chi'(w) \langle dd^c w \dot\wedge T\rangle.
\]
These identities are the technical engine behind relative energy monotonicity and convexity results [2005.13241].

Given fixed reference currents \(P_j\in \alpha_j\), the relative joint energy is defined by
\[
E_{\xi,\mathbf{P}}(T_1,\ldots,T_m;T)
:=
\sum_{J}
\int_X -\xi
\left\langle
\bigwedge_{j \in J} T_j
\wedge
\bigwedge_{j \not \in J} P_j
\dot{\wedge} T
\right\rangle,
\]
and weighted classes \(\mathcal E_{\chi,\mathbf P}(T)\) are those for which this expression is finite. In the diagonal case one gets the relative weighted full-mass classes \(\mathcal E_{\chi,m}(\beta,T)\) [2005.13241, 2004.11111].

The principal structural consequence is convexity. The class of currents with finite relative energy is convex, and more generally if each diagonal \(m\)-tuple \((T_j,\dots,T_j)\) is of full mass intersection relative to \(T\) with weight \(\chi\), then the mixed tuple \(T_1,\dots,T_m\) is also of full mass intersection relative to \(T\) with weight \(\chi\). In particular,
\[
\mathcal E_{\chi,m}(\beta,T)
\]
is convex. When \(T\equiv 1\), these constructions reduce to the ordinary non-pluripolar product and the usual BEGZ/Guedj–Zeriahi weighted energy classes [2004.11111, 2005.13241].

## 5. Lelong numbers, mass loss, and restricted positive intersections

A central theme in later work is that relative full mass is obstructed by singularities measured through Lelong numbers. If
\[
R:=\langle T_1\wedge\cdots\wedge T_{n-1}\rangle,
\]
then one always has the cohomological inequality
\[
\big\{\langle \wedge_{j=1}^{n-1}T_j\ \dot{\wedge}\ T\rangle\big\}
\le
\{R\}\wedge\{T\}.
\]
Moreover, if equality holds, then for every \(x\in X\),
\[
\nu(R,x)\cdot \nu(T,x)=0.
\]
Thus relative full mass forces the singularities of the absolute positive product \(R\) and the auxiliary current \(T\) not to overlap pointwise [2508.14669].

This becomes especially geometric when \(T=[D]\) is the current of integration along an effective divisor \(D\). For a big class \(\alpha\),
\[
\langle \alpha^{n-1}\rangle|_{X|D}
=
\big\{\langle T_{\min,\alpha}^{\,n-1}\ \dot{\wedge}\ [D]\rangle\big\}.
\]
If the restricted volume class has full mass, namely
\[
\langle \alpha^{n-1}\rangle|_{X|D}
=
\langle \alpha^{n-1}\rangle\wedge \{D\},
\]
then
\[
\nu(\langle \alpha^{n-1}\rangle,x)=0
\qquad\text{for every }x\in \operatorname{Supp}(D).
\]
On projective manifolds, the paper further obtains
\[
\nu(\langle \alpha^{n-1}\rangle,x)=0
\qquad \forall x\in X
\]
for every big class \(\alpha\) [2508.14669].

The relative product also quantifies loss of mass in ordinary non-pluripolar self-products. In the big nef setting, if \(T\in\alpha\) is a closed positive current and
\[
\gamma:= \{P\}\wedge \{T\}- \{\langle P \dot{\wedge} T\rangle \},
\]
then \(\gamma\) is pseudoeffective and
\[
\| \gamma\| \ge \sum_{V} \nu(P,V)\nu(T,V) (V),
\]
where the sum is over irreducible analytic subsets of suitable dimension. Applied incrementally with \(P=\langle T^{n-l-1}\rangle\), this yields quantitative lower bounds for the mass defect of self-products:
\[
\big\| \alpha^m - \{\langle T^m \rangle \}  \big\|
\ge
C \bigg(\sum_{V \in \mathcal V}\nu(T,V)^{n- \dim V} (V) \bigg)^{2^m}.
\]
In this analysis the relative non-pluripolar product is the device that isolates the cohomological loss created by adding one more singular factor [2101.05483].

## 6. Extensions, adjacent theories, and scope

The compact Kähler theory has been extended to a class of compact Hermitian manifolds. If \((X,\omega)\) is compact Hermitian and
\[
\partial\bar\partial \omega = 0,
\qquad
\partial\omega\wedge \bar\partial\omega=0,
\]
equivalently \(\partial\bar\partial\omega^k=0\) for every \(k\ge 1\), then the relative non-pluripolar product is always well-defined for any closed positive \((1,1)\)-currents \(T_1,\dots,T_m\) and any closed positive \((p,p)\)-current \(T\) with \(m+p\le n\). In this Hermitian setting one has monotonicity of total masses,
\[
\big\|\langle T_1\wedge \cdots \wedge T_m \dot\wedge T\rangle\big\|
\le
\big\|\langle T'_1\wedge \cdots \wedge T'_m \dot\wedge T\rangle\big\|,
\]
when \(T'_j\) is less singular than \(T_j\), although the stronger Kähler cohomology-class monotonicity is not available in general [2505.24702].

A closely related local theory exists on bounded domains and bounded hyperconvex domains in \(\mathbb C^n\), but there the relative aspect is encoded differently. The papers on prescribed singularity do not introduce an explicit operator of the form
\[
\mathbf 1_{\{u>\phi-k\}}
(dd^c\max(u,\phi-k))^m\wedge T,
\]
nor a relative Monge–Ampère operator \(\operatorname{MA}_\phi(u)\) in this truncation form. Instead they work with the ordinary non-pluripolar Monge–Ampère measure
\[
NP(u)=\lim_{M\to\infty}\mathbf 1_{\{u>-M\}}(dd^c\max\{u,-M\})^n
\]
together with a prescribed singularity condition
\[
P[u]=\phi
\]
or classes such as \(N(\phi)\) and \(MNP(H)\). This suggests that, locally, the relative aspect can be built into the admissible singularity class rather than into a separate wedge operator [2507.18116, 2407.00937].

The scalar relative product has also been used to define non-pluripolar intersection theory on vector bundles. For a Griffiths positive singular Hermitian vector bundle \(\hat E\), Xia passes to the induced singular metric on \(\widehat{\mathcal O(1)}\) over \(\mathbb P E^\vee\) and defines Segre operators by
\[
s_i(\hat E)\cap T
=
(-1)^i p_*\big(c_1(\widehat{\mathcal O(1)})^{r+i}\cap p^*T\big),
\]
where the product upstairs is the scalar relative non-pluripolar product and \(T\) is a closed dsh current. This projectivized construction yields Chern currents, functorial pullback and pushforward properties, and Chern–Weil type formulae in the \(\mathcal I\)-good setting [2210.15342].

The present scope remains selective. The modern theory gives a robust relative intersection product for singular \((1,1)\)-currents against a fixed positive current, a comparison with density currents, an energy calculus, and strong singularity-theoretic consequences. At the same time, adjacent local prescribed-singularity theories and projectivized vector-bundle theories show that “relative” can mean either relative to an auxiliary current, relative to a reference singularity type, or relative to a projective-bundle morphism. The shared principle is the same: singular intersections are constructed by truncation, controlled by plurifine locality, and interpreted through the loss or preservation of non-pluripolar mass.

Source: https://www.emergentmind.com/topics/relative-non-pluripolar-product