Relative Non-Pluripolar Product

• Relative non-pluripolar product is a singular intersection current defined by truncating closed positive (1,1)-currents relative to an auxiliary higher-degree current.
• It preserves non-pluripolar mass by discarding singular, polar contributions through bounded approximations and a plurifine localization approach.
• The product exhibits symmetry, homogeneity, and a tower property, linking with density currents and energy theory under the relative full mass condition.

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 (Vu, 2020, Vu, 2020).

1. Definition and ambient framework

Let $X$ be a compact Kähler manifold of dimension $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$0. The basic data are closed positive $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$1-currents $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$2, a closed positive $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$3-current $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$4, and the dimensional constraint

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$5

The output is a closed positive $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$6-current

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$7

Locally one writes

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$8

or globally, after fixing a smooth representative $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$9,

$T_1,\dots,T_m$0

with $T_1,\dots,T_m$1 $T_1,\dots,T_m$2-psh. The truncations are

$T_1,\dots,T_m$3

The relative non-pluripolar product is then defined by

$T_1,\dots,T_m$4

Equivalently, on the truncation region,

$T_1,\dots,T_m$5

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_1,\dots,T_m$6-variables is discarded (Vu, 2020).

This construction is relative in a precise sense. It is not the naive wedge

$T_1,\dots,T_m$7

formed formally in the presence of singularities. Rather, one first regularizes the $T_1,\dots,T_m$8’s by bounded truncations, wedges these bounded approximants against $T_1,\dots,T_m$9, and only then passes to the limit on the non-polar region. When $(1,1)$0, or equivalently when $(1,1)$1 is the current of integration along $(1,1)$2, one recovers the usual non-pluripolar product: $(1,1)$3 (Vu, 2020).

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)$4-entries, and homogeneous. It is also local in the plurifine topology, and it ignores the part of the auxiliary current $(1,1)$5 carried by the polar sets of the $(1,1)$6’s. More precisely, if

$(1,1)$7

then

$(1,1)$8

If $(1,1)$9 is a locally complete pluripolar set and $T$0 has no mass on $T$1, then the relative product also has no mass on $T$2 (Vu, 2020).

The product admits an iteration or tower property. If

$T$3

then

$T$4

This allows one to absorb some of the $T$5-currents into the reference current and is central in inductive arguments (Vu, 2020).

A notable subtlety is additivity. In general the relative non-pluripolar product is not additive in the $T$6-variables. However, if $T$7 has no mass on the polar sets $T$8, then one has

$T$9

(Vu, 2020).

The basic cohomological monotonicity statement is that if $(p,p)$0 is cohomologous to $(p,p)$1 and less singular than $(p,p)$2 for every $(p,p)$3, then

$(p,p)$4

This leads to the notion of $(p,p)$5-relative full mass intersection: if the $(p,p)$6 are chosen with minimal singularities in their classes, then $(p,p)$7 are of $(p,p)$8-relative full mass intersection when equality holds. Equivalently,

$(p,p)$9

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 (Vu, 2020).

A truncation criterion makes the full-mass condition concrete. If

$m+p\le n=\dim X$0

then full mass relative to $m+p\le n=\dim X$1 is equivalent to the vanishing of the deep-singularity contribution: $m+p\le n=\dim X$2 (Vu, 2020).

3. Density currents and the Dinh–Sibony comparison

A second intersection theory enters through density currents. Given several positive currents $m+p\le n=\dim X$3, one considers the tensor product current on $m+p\le n=\dim X$4 and tangent currents along the diagonal $m+p\le n=\dim X$5. A density current associated to $m+p\le n=\dim X$6 is such a tangent current, and the Dinh–Sibony product

$m+p\le n=\dim X$7

is defined when the density current is unique and pulled back from the base. For bounded-potential $m+p\le n=\dim X$8-currents this agrees with the classical wedge product, and similarly

$m+p\le n=\dim X$9

in the bounded-potential regime (Vu, 2020).

The main comparison theorem states that if the cohomology class of each $T$0 is Kähler, then

$T$1

if and only if

$T$2

Thus the relative non-pluripolar product and the Dinh–Sibony product coincide exactly under the relative full-mass hypothesis (Vu, 2020).

Even without full mass, there is a universal domination statement. If $T$3 is a density current associated to $T$4, then

$T$5

where $T$6 is the normal-bundle projection from the diagonal in $T$7. 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 $T$8-dimension (Vu, 2020).

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 (Vu, 2020).

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$9-admissible dsh functions. If $T_j$0 with $T_j$1 carrying no mass on the pole sets of the quasi-psh pieces, then one can define

$T_j$2

and more generally, after setting

$T_j$3

also

$T_j$4

This extends the relative product from positive currents to differential expressions needed in energy estimates (Vu, 2020).

The central analytic identity is the integration by parts formula. If $T_j$5 has bidegree $T_j$6 and $T_j$7 are bounded $T_j$8-admissible dsh functions, then

$T_j$9

A weighted version with $X$0 is also available: $X$1 These identities are the technical engine behind relative energy monotonicity and convexity results (Vu, 2020).

Given fixed reference currents $X$2, the relative joint energy is defined by

$X$3

and weighted classes $X$4 are those for which this expression is finite. In the diagonal case one gets the relative weighted full-mass classes $X$5 (Vu, 2020, Vu, 2020).

The principal structural consequence is convexity. The class of currents with finite relative energy is convex, and more generally if each diagonal $X$6-tuple $X$7 is of full mass intersection relative to $X$8 with weight $X$9, then the mixed tuple $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$00 is also of full mass intersection relative to $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$01 with weight $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$02. In particular,

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$03

is convex. When $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$04, these constructions reduce to the ordinary non-pluripolar product and the usual BEGZ/Guedj–Zeriahi weighted energy classes (Vu, 2020, Vu, 2020).

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

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$05

then one always has the cohomological inequality

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$06

Moreover, if equality holds, then for every $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$07,

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$08

Thus relative full mass forces the singularities of the absolute positive product $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$09 and the auxiliary current $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$10 not to overlap pointwise (Nguyen et al., 20 Aug 2025).

This becomes especially geometric when $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$11 is the current of integration along an effective divisor $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$12. For a big class $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$13,

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$14

If the restricted volume class has full mass, namely

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$15

then

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$16

On projective manifolds, the paper further obtains

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$17

for every big class $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$18 (Nguyen et al., 20 Aug 2025).

The relative product also quantifies loss of mass in ordinary non-pluripolar self-products. In the big nef setting, if $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$19 is a closed positive current and

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$20

then $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$21 is pseudoeffective and

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$22

where the sum is over irreducible analytic subsets of suitable dimension. Applied incrementally with $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$23, this yields quantitative lower bounds for the mass defect of self-products: $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$24 In this analysis the relative non-pluripolar product is the device that isolates the cohomological loss created by adding one more singular factor (Vu, 2021).

6. Extensions, adjacent theories, and scope

The compact Kähler theory has been extended to a class of compact Hermitian manifolds. If $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$25 is compact Hermitian and

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$26

equivalently $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$27 for every $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$28, then the relative non-pluripolar product is always well-defined for any closed positive $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$29-currents $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$30 and any closed positive $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$31-current $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$32 with $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$33. In this Hermitian setting one has monotonicity of total masses,

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$34

when $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$35 is less singular than $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$36, although the stronger Kähler cohomology-class monotonicity is not available in general (Li et al., 30 May 2025).

A closely related local theory exists on bounded domains and bounded hyperconvex domains in $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$37, but there the relative aspect is encoded differently. The papers on prescribed singularity do not introduce an explicit operator of the form

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$38

nor a relative Monge–Ampère operator $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$39 in this truncation form. Instead they work with the ordinary non-pluripolar Monge–Ampère measure

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$40

together with a prescribed singularity condition

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$41

or classes such as $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$42 and $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$43. This suggests that, locally, the relative aspect can be built into the admissible singularity class rather than into a separate wedge operator (Do et al., 24 Jul 2025, Do et al., 2024).

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 $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$44, Xia passes to the induced singular metric on $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$45 over $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$46 and defines Segre operators by

$$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$47

where the product upstairs is the scalar relative non-pluripolar product and $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$48 is a closed dsh current. This projectivized construction yields Chern currents, functorial pullback and pushforward properties, and Chern–Weil type formulae in the $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$49-good setting (Xia, 2022).

The present scope remains selective. The modern theory gives a robust relative intersection product for singular $$\big\langle T_1\wedge\cdots\wedge T_m \dot\wedge T\big\rangle,$$50-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.