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Slicing of Negative Plurisubharmonic Currents Arising From Analytic Subsets

Hedi Khedhiri
University of Monastir, Preparatory Institute for Engineering Studies, Research Laboratory LR18ES16,

Road Ibn Eljazzar Monastir 5019, Tunisia

Correspondence: khediri_h@yahoo.fr

Abstract. In this paper, first we focus on the slicing of negative plurisubharmonic currents whichare finite sums involving currents arising from analytic subsets with geometric complete intersection.Next, we provide significant results on the integrability across analytic subsets, of the coefficients ofsuch currents and of their slices.

1. Introduction
In this paper, we let Ω be a domain in Cn such that the unit polydisc ∆n satisfies ∆n b Ω and

ϕ be a plurisubharmonic function (psh for short), locally bounded on Ω. We let N , k , p and n benonzero fixed arbitrary natural numbers such that k ≤ p ≤ n and we consider the n-complex space
Cn with variables z such that

Cn = Ck × Cn−k , z = (z ′, z ′′), z ′ ∈ Ck , z ′′ ∈ Cn−k .

Assuming that ϕ depends only on the variable z ′ ∈ Ck and the support Sϕ of the associatedMong-Ampère measure
µϕ = (ddcϕ)kis ∆k , then, the slice (or the ϕ-slice) denoted 〈T, π, a〉ϕ of a current T ∈ D ′p,p(Ω), at point a ∈ Sϕ,is studied and well defined in [10], such that

< T, π, a >ϕ (Ψ) := lim
ε→0

1

µϕ(Bk(a, ε))

∫
Bk(a,ε)×Cn−k

T ∧ (ddc ϕ̃)k ∧Ψ. (1.1)
The definition (1.1) makes sense as well as the limit exists in C for any test form Ψ ∈ D(p−k,p−k)(Ω),where, ϕ̃ = ϕ ◦ π stands for the composite of ϕ with the projection map

π :
Cn −→ Ck

(z ′, z ′′) 7−→ z ′

Received: 6 Jun 2024.2020 Mathematics Subject Classification. 32U05, 31C10, 32U25, 32C30, 32U40.
Key words and phrases. plurisubharmonic function, plurisubharmonic current, slice, analytic set, complete intersection.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 2and Bk(a, ε) is the open ball in Ck centered at point a and of radius ε > 0.In the present work we are concerned by topics studied in [9–12]. Particularly, we will studytopics based on the definition (1.1) recently investigated in [11], for negative psh currents of smallsupport. A fundamental result was provided showing that for any psh function v non identicallyequals −∞, there exists a pluripolar subset E in Ck , such that the slice < v, π, a >ϕ of v at point
a is well defined and is expressed explicitly by v(a, .) as well as the point a lies outside E.As a consequence, an existence slicing result was deduced for negative psh currents having theirsupports contained in a strip. In particular, for positive or negative closed currents with supportcontained in a strip, it was established that the slices are well defined and are vanishing everywhereas soon as they are vanishing outside a pluripolar subset.In this paper, we shall develop the work in [11] by providing others slicing results for negativepsh currents which are arising from analytic subsets having suitable intersections.Among several currents, we are interested on those written as finite sums of the form∑

1≤j≤N
(log |fj |)

[
Xj
]
,

∑
1≤j≤N

vj [Xj ],
∑

1≤j≤N
U[Xj ] ∧ [Yj ], or ∑

1≤j≤N
[Xj ] ∧ U[Yj ],

where (fj)1≤j≤N is a finite family of holomorphic functions non identically vanishing on ∆n, (Xj)1≤j≤Nis a finite family of analytic subsets all of pure dimension p, (Yj)1≤j≤N is another finite family ofanalytic subsets such that each Yj forms a complete intersection with Xj in ∆n, (vj)1≤j≤N is afinite family of psh functions non identically −∞ on ∆n and U[Xj ] is the Lelong-Skoda potentialassociated to the current [Xj ], (1 ≤ j ≤ N).
The first result describes properly the slices of the current ∑

1≤j≤N
(log |fj |)

[
Xj
].

Theorem 1.1. Assume we have in ∆n, a finite family (Xj)1≤j≤N of analytic subsets all of the
same pure dimension q > k and a finite family (fj)1≤j≤N of holomorphic functions non identically
vanishing such that for all 1 ≤ j ≤ N the hypersurface Yj = {fj = 0} yields a complete intersection
near the origin with Xj . If σj : Xj ∩ π−1(a) −→ ∆n denotes the canonical injection, then, there
exist, a closed pluripolar subset E of ∆k , an integer m ∈ N and homogenous polynomials Qm,j on
Xj , 1 ≤ j ≤ N , such that for all a ∈ ∆k r E the slice 〈

∑
1≤j≤N

(log|fj |)[Xj ], π, a〉ϕ is defined such

that(1) If σj∗fj 6≡ 0 on Xj ∩ π−1(a), ∀ 1 ≤ j ≤ N, then

〈
∑

1≤j≤N
(log|fj |)[Xj ], π, a〉ϕ =

∑
1≤j≤N

(
log |σj∗fj |

) [
Xj ∩ π−1(a)

]
,

(2) If σj∗fj ≡ 0 and Qm,j |Xj∩π−1(a) 6≡ 0, ∀ 1 ≤ j ≤ N, then

〈
∑

1≤j≤N
(log|fj |)[Xj ], π, a〉ϕ =

∑
1≤j≤N

(
log |σj∗Qm,j |

) [
Xj ∩ π−1(a)

]
.

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 3The second result generalizes Theorem 1.1 to psh functions. It provides the existence and theexpression of the slices of a negative psh current of the form ∑
1≤j≤N

vj [Xj ].
Theorem 1.2. Assume we have in ∆n, a finite family (Xj)1≤j≤N of analytic subsets all of pure
dimension p and a finite family (vj)1≤j≤N of negative psh functions such that for all 1 ≤ j ≤ N the
set of singular points of vj is contained in a hypersurface Yj that yields a complete intersection
with Xj . Then, there is a pluripolar subset E of ∆k , such that for all a ∈ ∆k r E, the current∑

1≤j≤N vj [Xj ] admits a slice expressed by

〈
∑

1≤j≤N
vj .
[
Xj
]
, π, a〉ϕ =

∑
1≤j≤N

vj |Xj∩π−1(a).
[
Xj ∩ π−1(a)

]
.

Next, we let N(x) = − 1
(n−1)(4π)n

1
|x |2n−2 be the Newton kernel in Cn and we let η be a positivesmooth function with compact support in Ω such that 0 ≤ η ≤ 1, η ≡ 1 on a neighborhood of ∆n.For a given analytic subset X in ∆n, we take the Lelong-Skoda potential U[X] associated to thecurrent [X], as the (n − p − 1, n − p − 1)-negative current defined by the following integral

U[X](z) =

∫
ξ∈Cn

η(ξ)N(z − ξ)[X](ξ) ∧
(
ddc |z − ξ|2

)n−1
.

We prove the third result of this paper on the existence and the expression of the slices of thewedge product current ∑
1≤j≤N

U[Xj ] ∧ [Yj ] introduced and studied in [9].
Theorem 1.3. Assume we have in ∆n, finite families (Xj)1≤j≤N and (Yl)1≤l≤N of analytic subsets
such that for all 1 ≤ j ≤ N,, Xj is of pure dimension p and Yj is of pure dimension q and Xj
yields a complete intersection with Yj . Then, there exists a subset E contained in a countable
union of analytic subsets of ∆k of dimensions ≤ k − 1 such that for all a ∈ ∆k r E, the slice of∑
1≤j≤N

U[Xj ] ∧ [Yj ] at point a is given by

〈
∑

1≤j≤N
U[Xj ] ∧ [Yj ], π, a〉ϕ =

∑
1≤j≤N

j∗aU[Xj ] ∧
[
Yj ∩ π−1(a)

]
.

In the sequel, we show that the above results enjoy interesting properties like the integrability ofcoefficients of the above currents and their slices. For instance, the following Theorem 1.4 providesan important integrability property for a given psh function v not identically −∞ on ∆n. In fact, itdescribes the integrability of the function exp(−v) across an analytic subset X . Consequently, weget the following beautiful result in terms of slices, about the integrability of exp(−v) across theintersection π−1(a) ∩X .
Theorem 1.4. Assume we have a finite family (vj)1≤j≤N of psh functions in ∆n such that for all
1 ≤ j ≤ N the set of singular points of vj is contained in a hypersurface X that yields a complete
intersection with another analytic subset Y in ∆n. Then, there exist a constant α > 0 and a

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 4

pluripolar subset E of ∆k , such that for any point a ∈ ∆krE, the function
∑

1≤j≤N
exp (−αvj)|Y ∩π−1(a)

lies in L1
loc

(
Y ∩ π−1(a)

)
.

Finally, as an other application we establish the following result valid for analytic subsets Xand Y yielding a complete intersection in Ω.
Theorem 1.5. Assume we have analytic subsets X and Y of pure dimensions p and q respectively,
yielding a complete intersection in ∆n. Then, there exist a constant δ > 0 and a pluripolar subset
E in Ck , such that for all a ∈ ∆k rE, the coefficients of the current j∗a (U[X])∧

[
Y ∩ π−1(a)

]
lie in

L1+δ
loc (Y ∩ π−1(a)).

2. Preliminaries
Let Ω be a domain in Cn, we use the standard notations for the operators d = ∂ + ∂̄ and

dc = i(∂̄ − ∂); the operator ddc is then defined by ddc = 2i∂∂. The space D ′p,p(Ω) of (n− p, n−
p)-currents (of bidimension (p, p)) on Ω, is the dual of the space Dp,p(Ω) of smooth compactlysupported (p, p)-forms on Ω. We say that a current T ∈ D ′p,p(Ω) is positive, if for all smooth
(1, 0)-forms α1, . . . , αp on Ω, the product

T ∧ iα1 ∧ ᾱ1 ∧ . . . ∧ iαp ∧ ᾱp

is a positive measure. The current T is said to be closed if dT = 0 and psh if ddcT ≥ 0.In particular, if p = n, then T is a psh function u on Ω, that is a distribution such that∑
1≤j,k≤n

∂2u

∂zj∂z̄k
idzj ∧ dzk

is a positive current of bidimension (n − 1, n − 1), the derivatives are taken here in the sense ofdistributions.We denote Psh(Ω) the set of psh functions on Ω and L∞loc(Ω)∩ Psh(Ω) the subset of elementsin Psh(Ω) which are locally bounded. The Kähler form on Cn is denoted β(t) = ddc |t|2. It canbe decomposed into the sum β′(t ′) + β′′(t ′′) where β′ and β′′ are Kähler forms on Ck and Cn−krespectively. For a point a ∈ Ck , we denote ja the map defined by
ja :
Cn−k −→ {a} × Cn−k

z ′′ 7−→ (a, z ′)

3. Proof of Theorem 1.1: slicing of the current (log |f |)[X]

First of all, we require the following well known propositions:
Proposition 3.1. Let X be an analytic subset of ∆n and let m be its complex dimension. Then, the
following statements hold:

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 5(1) if m < k , then π(X) is contained in a countable union of analytic subsets of ∆k of dimen-
sions ≤ m.(2) If m ≥ k , then the set

Z = {a ∈ ∆k / dimC(X ∩ π−1(X)) ≥ m − k + 1}

is contained in a countable union of analytic subsets of ∆k of dimensions ≤ k − 1.

As a consequence of the expansion of holomorphic functions in power series we have:
Proposition 3.2. Let f 6≡ 0 be a non vanishing holomorphic function on ∆n such that for all
z ′′ ∈ ∆n−k , f (0, z ′′) = 0. Then, there exist a natural number m ≥ 1 and holomorphic functions
aµ(z ′′) on ∆n−k , such that, for all (z ′; z ′′) ∈ ∆k × ∆n−k , we have f (z ′, z ′′) =

∑∞
j=mQj(z

′; z ′′)

where Qm(z ′, z ′′) =
∑
|µ|=m z

′µaµ(z ′′) and |µ| = µ1 + · · ·+ µk .

Now, given a psh function v on ∆n such that v is locally integrable on the analytic subset X ,then, for all test form Ψ, for all fixed ε > 0 and all a ∈ Sϕ, we shall justify the well definition andthe finiteness of the quantity ∫
Bk(a,ε)×Cn−k

v [X] ∧ (ddc ϕ̃)k ∧Ψ.

Proposition 3.3. Let X ⊂ ∆n be analytic of pure dimension p > k , let v ∈ Psh(∆n) ∩ L1
loc(X) be

negative and let ϕ ∈ Psh(∆k)∩L∞loc(∆k) be given such that Sϕ = ∆k . Then, for any point a ∈ ∆k ,
for any fixed ε > 0 and for any positive test form Ψ ∈ D(p−k,p−k)(∆n), we have

1

µϕ(Bk(a, ε))

∫
Bk(a,ε)×Cn−k

−v [X] ∧ (ddc ϕ̃)k ∧Ψ <∞.

Proof. The result is local. We may take a = 0, without loss of generality we may choose the testform Ψ such that Ψ = ψ(z)β′′n−k ∈ D(n−p,n−p)(∆n−k) where ψ is a positive test function on ∆n−k .Let denote
Γε :=

1

µϕ(Bk(a, ε))

∫
Bk(a,ε)×Cn−k

−v [X] ∧ (ddc ϕ̃)k ∧Ψ. (3.2)
There exists a neighborhood U = U ′ × U ′′ ⊂ ∆k × ∆p−k × ∆n−p of 0 and a coordinate system

(t, ζ, z ′′) such that the projection
πX :

X ∩ U → ∆p = ∆k × ∆p−k(
(t, ζ), z ′′

)
7→ (t, ζ)is a ramified covering of X . Let Z be the ramification locus of πX and set

XZ = X ∩ ((U ′ r Z)× U ′′) ⊂ Xreg.

The restriction of
πXZ : XZ → U ′ r Z

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 6is then a covering with a finite sheet number. The expression of Γε given by (3.2) will be transformedas
Γε =

1

µϕ(Bk(0, ε))

∫
XZ∩Bk(0,ε)×Cn−k

πXZ
∗(− v(ddcϕ)k ∧Ψ

)
. (3.3)

We may choose ε > 0 small enough, so that
Uε := XZ ∩ Bk(0, ε)× Cn−kBk(0, ε)× ∆p−k × {0}Cn−p .Then, the expression given by (3.3) can be written as

Γε =
1

µϕ(Bk(0, ε))

∫
Bk(0,ε)

w(t)(ddcϕ)k , (3.4)
where

w(t) =

∫
∆p−k

π∗XZ (−vψ)(t, ζ)dλp−k(ζ).

Furthermore, following [6] we have
||w(ddcϕ)k ||Bk(0,ε) ≤ ||w ||L1(Bk(0,2ε))||ϕ||kL∞(Bk(0,2ε)), (3.5)

the inequality (3.5) implies that w is a locally integrable function with respect to the positivemeasure µϕ and hence the integral given by (3.4) is finite. �

Now we are ready to give the proof of Theorem 1.1.
Proof. The result is local. Without loss of generalities we may suppose N = 1. Furthermore, forsimplicity, we may consider the slice at point a = 0. Put T = log |f |[Y ] and

Γε(T ) =
1

µϕ(Bk(0, ε))

∫
Bk(0,ε)×Cn−k

T ∧ (ddc ϕ̃)k ∧Ψ, (3.6)
where Ψ is a test form such that Ψ = h(z)β′′n−k ∈ D(n−q,n−q)(∆n−k) where h is a positive smoothfunction with compact support. We may find a neighborhood U = U ′×U ′′ of 0 in ∆k×∆q−k×∆n−qand a coordinate system (t, ζ, z ′′) such that the projection

πY :
Y ∩ U → ∆q(

(t, ζ), z ′′
)
7→ (t, ζ)defines a ramified covering of Y . Let Z be the ramification locus of π and YZ = Y ∩((U ′rZ)×U ′′) ⊂

Yreg . The restriction of πYZ : YZ → U ′ r Z is then a covering with a finite sheet number. Equality
(3.6) can be written as

Γε(T ) =
1

µϕ(Bk(0, ε))

∫
YZ∩Bk(0,ε)×Cn−k

π∗
(

log |f |(ddcϕ)k ∧Ψ
)
. (3.7)

We have to find the limit, as ε→ 0, of (6.20). For ε > 0 small enough, the set Uε := YZ∩Bk(0, ε)×
Cn−k can be viewed as Bk(0, ε) × ∆q−k × {0}Cn−q . The integral in the right hand side of (6.20)can be written such that

Γε(T ) =
1

µϕ(Bk(0, ε))

∫
Bk(0,ε)

wh(t)(ddcϕ)k , (3.8)

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 7where
wh(t) =

∫
∆q−k log |f (t, ζ, 0)|h(ζ, 0)dλq−k(ζ)

=
∫
YZ∩π−1(0) log |i∗

YZ∩π−1(0)
f (t, ζ, z ′′)|i∗

YZ∩π−1(0)
h(ζ, z ′′),and iYZ∩π−1(0) : YZ ∩π−1(0)→ ∆n is the canonical injection. Let E be the closed pluripolar subsetof ∆k defined by

E = {a ∈ ∆k : w(a) = −∞ or w 6∈ L1
loc(µϕ) near a}.

By Theorem 1.1 in [11], for all a ∈ ∆k r E, we have:(1) if (iYZ∩π−1(0))∗f 6≡ 0, then, as ε→ 0, the limit of the integral given by (3.8) will be
limε→0 Γε = wh(0)

=
∫

∆q−k log |f (0, ζ, 0)|h(ζ, 0)dλq−k(ζ)

=
∫
Y ∩π−1(0)(iYZ∩π−1(0))∗[log |f | ∧Ψ]

= 〈
(
iYZ∩π−1(0)

)∗
(log |f |)

[
Y ∩ π−1(0)

]
, (iYZ∩π−1(0))∗(Ψ)〉ϕ.(2) If (iYZ∩π−1(0))∗f ≡ 0, then by Proposition 3.2 applied with the function (iYZ∩π−1(0))∗f on

∆q = ∆k × ∆q−k , there exist an integer m ∈ N, a non vanishing homogenous polynomial
Qm(t, ζ) and smooth functions (t, ζ) 7→ R(t, ζ) ∈ C∞(∆k × ∆q−k) such that

f (t, ζ) = Qm(t, ζ) + tR(t, ζ) on Bk(0, ε)× ∆p−k .

Thanks to Taylor’s formula, the rest R(t, ζ) can be chosen such that |R(t, ζ)| ≤ 1. Thiswith the triangle inequality legitimate the following inequalities
log ||Qm| − |tR|| ≤ log ||f (t, ζ)|| = log |(iYZ∩π−1(0))∗f (t, w, z ′′)|

≤ log(|Qm|+ |tR|).Since (t, ζ) ∈ Bk(0, ε) × ∆p−k and |R(t, ζ)| ≤ 1, then, as ε → 0, |t| → 0 and |tR| → 0.Hence, we get
limε→0 Γε = wh(0)

=
∫

∆q−k log |Qm(0, ζ)|h(ζ, z ′′)dλq−k(ζ)

=
∫
Y ∩π−1(0)(iYZ∩π−1(0))∗[log |Qm(0, ζ| ∧Ψ]

= 〈(log |(iYZ∩π−1(0))∗Qm|)
[
Y ∩ π−1(0)

]
, (iYZ∩π−1(0))∗(Ψ)〉ϕ.The proof is completed. �

The following example illustrates Theorem 1.1.
Example 3.4. In C3 we let f (z) = z2

1 + z2
2 − z3, Y = {z1 = −z3}, k = 1 and ϕ(z ′) = |z ′|2 = |z1|2.It is clear that the subsets X = {f = 0} and Y yield a complete intersection near 0. Hence,according to [9], we deduce that log |f | ∈ L1
loc(Y ). By Theorem 1.1, we have

〈log |f |.
[
Y
]
, π, 0〉 = log |z2

2 − z3|.
[
z1 = z3 = 0

]
.

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 84. Proof of Theorem 1.2: slicing of the current v [X]

Proof. We use notations and arguments as in the proof of Theorem 1.1 and we may suppose N = 1.We have
Γε(v [X]) =

1

µϕ(Bk(0, ε))

∫
Bk(0,ε)×Cn−k

v [X] ∧ (ddc ϕ̃)k ∧Ψ. (4.9)
Let Xreg be the set of regular points of X. Assume that 0 ∈ Xreg and put

Z1 = {a ∈ ∆k : dimC(X ∩ π−1(a)) > p − k}.

Then, by Proposition 3.1, Z1 is contained in a countable union of analytic subsets of ∆k of dimen-sions ≤ k − 1. As the dimension m of the set Xsing of singular points, satisfies m ≤ p − 1, then,by Proposition 3.1, there exists a set Z2 contained in a countable union of analytic subsets of ∆kof dimensions ≤ k − 1, such that for all a ∈ ∆k rZ2, Xsing ∩π−1(a) is an analytic subset of ∆k ofdimension m − k (otherwise is empty). Put Z = Z1 ∪ Z2 and denote π̄ := π|Xreg . The expression
(4.9) can be transformed to

Γε(v [X]) =
1

µϕ(Bk(a, ε))

∫
Xregr(π̄−1(Z))∩(Bk(a,ε)×Cn−k)

π̄∗[v(ddcϕ)k ∧Ψ]. (4.10)
In a local chart of coordinates (z1, . . . , zk , w1, . . . , wn−k), such that

Xreg r (π̄−1(Z)) ∩ (Bk(a, ε)× Cn−k) = Bk(a, ε)× Cp−k × {0}Cn−p ,

when ε > 0 is small enough, the integral (4.10), can be written as
Γε =

1

µϕ(Bk(a, ε))

∫
∆pr(π̄−1(Z))∩(Bk(a,ε)×Cn−k)

v(z ′, w)(ddcϕ)k ∧Ψ(z ′, w). (4.11)
According to [11], since z ′ 7→ v(z ′, w) is locally integrable on Xreg r (π̄−1(Z), then when ε→ 0,we get the limit of (4.11), as follows

limε→0 Γε(v [X]) =
∫

∆p−k v(a, .)ψ(a, w)β′′p−k

=
∫
Xreg\(π̄−1(Z))∩π−1{a} v(a, .)ψ(a, w)β′′p−k

= 〈v|X∩π−1(a).
[
X ∩ π−1(a)

]
, (iX∩π−1(a))∗Ψ〉.This finishes the proof. �

We illustrate Theorem 1.2 by the following example.
Example 4.1. In C4 = C× C3, we take

v(z) = log(|z1|3 + |z3|5), Y = {z4 = z2
1 z

4
2} , k = 1 and ϕ(z ′) = |z ′|2 = |z1|2.

The subset X = {z1 = z3 = 0} that contains {v = −∞}, yields a complete intersection with thesubset Y near 0. By Theorem 1.2, up to a constant, we have
〈v [Y ], π, 0〉 = log |z3|

[
z1 = z2 = z4 = 0

]
.

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 9For the next section, we will be concerned with points a ∈ ∆k at which the slice 〈U[X]∧ [Y ], π, a〉of the current U[X] ∧ [Y ], is well defined and how it can be expressed.Remember that the wedge product U[X] ∧ [Y ] was defined in [10] as a current on Y , as wellas, the analytic subsets X and Y yield a complete intersection in Ω. Actually a necessary andsufficient condition was established, by expressing that the current U[X] has integrable coefficientswith respect to the trace measure of the current [Y ], if and only if X and Y yield a completeintersection in Ω. The condition of complete intersection is optimal for the definition of this wedgeproduct in the weak sense of currents. Recall that, X and Y yield a complete intersection, if for anyirreducible components Xj of X and Yk of Y , we have Codim(Xj ∩ Yk) = CodimXj + CodimYk .

5. Proof of Theorem 1.3: slicing of the current U[X] ∧ [Y ]

In order to avoid complications, we assume here that the function ϕ is smooth and hence themeasure µϕ can be considered as the Lebesgue measure on Ck , with density a smooth functiondenoted µϕ(z ′).
Proof. Let Ψ ∈ D(p+q−n+1−k,p+q−n+1−k)(∆n) be a test form. We have to prove the existence of apluripolar subset E of ∆k such that for all a 6∈ E, we have

lim
ε→0

Γε =

∫
∆n
J∗aU[X] ∧

[
Y ∩ π−1(a)

]
∧ j∗a (Ψ)

where
Γε =

1

µϕ(Bk(a, ε))

∫
Bk(a,ε)×Cn−k

U[X] ∧ [Y ] ∧ (ddc ϕ̃)k ∧Ψ. (5.12)
Since the potential U[X] involved in (5.12), is a finite linear combination of forms with coefficients
BI,J(s, z) given by the following expression

BI,J(s, z) =

∫
ξ∈Cn

η(ξ)N(z − ξ)[X](ξ) ∧ βp−s(ξ) ∧ 2s i s
2

dξI ∧ dξJ (|I| = |J| = s). (5.13)
We need to find limε→0 Γε(BI,J(s, z)), where Γε(BI,J(s, z)) is given by

Γε(BI,J(s, z)) :=
1

µϕ(Bk(a, ε))

∫
Bk(a,ε)×Cn−k

BI,J(s, z) ∧ [Y ] ∧ (ddc ϕ̃)k ∧Ψ. (5.14)
Since the current U[X]∧ [Y ] is well defined then, the functions z 7→ BI,J(s, z) are locally integrableon Y , and hence Γε(BI,J(s, z)) given by (5.14), can be written as

Γε := Γε(BI,J(s, z))

= 1
µϕ(Bk(a,ε))

∫
(ξ,z)∈Vε η(ξ)N(z − ξ)

[
X × Y

]
(ξ, z) ∧ (ddcϕ)k ∧ F (ξ, z),

where
F (ξ, z) = βp−s(ξ) ∧ 2s i s

2

dξI ∧ dξJ ∧Ψ(z) and Vε = Cn ×
[
Bk(a, ε)× Cn−k

]
.

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 10Let [X × Y ]reg be the set of regular points of X × Y. We may assume that 0 ∈ [X × Y ]reg . Put
Z1 = {a ∈ ∆k : dimC((X × Y ) ∩ (π−1(a)× π−1(a)) > p + q − k}

= {a ∈ ∆k : dimC(X ∩ π−1(a))× (Y ∩ π−1(a)) > p + q − k}.

Following Proposition 3.1, Z1 is contained in a countable union of analytic subsets of ∆k ofdimensions ≤ k − 1. As the dimension m of the subset (X × Y )sing of singular points, satisfies
m ≤ p + q − 1, then, again by Proposition 3.1, there exists a subset Z2 contained in a countableunion of analytic subsets of ∆k of dimensions ≤ k − 1, such that for any point a ∈ ∆k rZ2, the set
(X × Y )sing ∩ (π−1(a) × π−1(a) is analytic in ∆k of dimension m − k (otherwise is empty). Put
Z = Z1 ∪ Z2 and denote π̄ := π|(X×Y )reg , we have

Γε =

∫
(X×Y )regr(π̄−1(Z))∩Vε

π̄∗
[
η(ξ)N(z − ξ)(ddcϕ)k ∧ F (ξ, z)

]
. (5.15)

We can find local coordinates
(ξ, z) = (ξ, (z ′, w)) =

(
ξ, (z1, . . . , zk , w1, . . . , wn−k)

)
,

so that, for all ε > 0 small enough,
(X × Y )reg r (π̄−1(Z)) ∩ Vε = Cp × {0}Cn−p × Bk(a, ε)× Cq−k × {0}Cn−q .

Since µϕ(Bk(a, ε)) ∼ ω2kε
2kµϕ(a) as ε → 0. Then, by an application of Fubini’s theorem andby the change of variables (ξ, z ′) ↔ (ξ, z

′−a
ε ), when ε > 0 is small enough, the equality (5.15)can be transformed to the following

Γε =
1

ω2kµϕ(a)

∫
V

µϕ(a + εt)η(ξ)N((a + εt, w)− ξ)F (ξ, (a + εt, w))dν(t, w), (5.16)
where V = Cp+q−k ×Bk(0, 1) and dν(t, w) = dλk(t)⊗ dλp+q−k(w). By letting ε→ 0 in (5.16),we get

limε→0 Γε =

∫
Cp+q−k

η(ξ)N((a, w)− ξ)F (ξ, (a, w))dλp+q−k(w)

=

∫
(X×Y )reg∩(π−1(a)×π−1(a))

η(ξ)N((a, w)− ξ) ∧ F (ξ, (a, w))

=

∫
(X∩π−1(a))reg×(Y ∩π−1(a))reg

η(ξ)N((a, w)− ξ) ∧ F (ξ, (a, w))

=

∫
Cn−k

BI,J(s, (a, w)) ∧ [Y ∩ π−1(a)] ∧Ψ(a, z ′′)

= 〈J∗a(BI,J(s, z)) ∧ [Y ∩ π−1(a)], j∗a (Ψ)〉.The last equality holds true since we have
dim

(
(X × Y )sing ∩ (π−1(a)× π−1(a))

)
< p + q − k.

This achieves the proof of Theorem 1.3. �

Let illustrate Theorem 1.3 with the following example.

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 11

Example 5.1. Consider in C5 = C × C4, X = {z2 = z3 = z4 = 0} and Y = {z4 = z2
2 z

2
3} take

k = 1 and ϕ(z ′) = |z ′|2 = |z1|2. Since X is smooth, then a potential of [X] is
U =

log(|z2|2 + |z3|2 + |z4|2)ddc log(|z2|2 + |z3|2 + |z4|2)

|z2|2 + |z3|2 + |z4|2
.

As X and Y yield a complete intersection, then by [9], the current U ∧ [Y ] is well defined. Thecurrent U ∧ [Y ] is given by the following
U ∧ [Y ] = i∗Y

[
log(|z2|2 + |z3|2 + |z2z3|4)ddc log(|z2|2 + |z3|2 + |z2z3|4)

|z2|2 + |z3|2 + |z2z3|4

]
.

We have
X ∩ π−1(0) = {z1 = z2 = z3 = z4 = 0} and Y ∩ π−1(0) = {z1 = z2

4 − z2z3 = 0}.

It is clear that X∩π−1(0) and Y ∩π−1(0) yield a complete intersection in {0}×C4. If σ = iY ∩π−1(0),then,
〈U ∧ [Y ], π, 0〉 = σ∗

[
log(|z2|2 + |z3|2 + |z2z3|4)ddc log(|z2|2 + |z3|2 + |z2z3|4)

|z2|2 + |z3|2 + |z2z3|4

]
.

6. Proofs of Theorem 1.4 and Theorem 1.5 with applications
This section aims to give the proof of Theorem 1.4 and the proof of Theorem 1.5. We may suppose

N = 1. Indeed, for the proof of Theorem 1.5, we use the fact that a finite union of pluripolar subsetsis pluripolar. In addition, Theorem 1.4 is an immediate application of the following Theorem 6.1.In fact, if for all 1 ≤ j ≤ N , there exists αj > 0 for which Theorem 6.1 holds for the psh function
vj , then for α = min1≤j≤N (αj), Theorem 1.4 can be deduced from Theorem 6.1 in terms of slicesand it works with the psh function v =

∑
1≤j≤N exp(−αvj).

Theorem 6.1. Assume we have a psh function v on ∆n such that its set of singular points is
contained in a hypersurface X forming a complete intersection with the analytic subset Y of ∆n.
Then, there exists α > 0, such that the function exp (−αv)|Y lies in L1

loc(Y ).

Proof. The result is local, so it’s enough to prove it near a singular point z0 = 0 ∈ {v = −∞} ⊂ X .We may suppose z0 is a regular point of X ∩Y . Let q be the dimension of Y at z0 (1 ≤ q ≤ n−1),since X and Y yield a complete intersection near z0, then we may find a neighborhood V (z0) of
z0 and local coordinates (z1, . . . , zn) such that

K ∩X ∩ Y ⊂ K ∩ Y ∩ {zq = zq+1 = · · · = zn = 0}, K = V (z0). (6.17)
Let (vδ)δ>0 be a decreasing sequence of continuous psh functions such that limδ→0 vδ = v pointwise.We choose a fixed δ0 > 0 small enough, such that

vδ0
(z)− 1 ≤ v(z) ≤ vδ0

(z), ∀ z ∈ K rX ∩ Y. (6.18)

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 12According to a classical result due to H. J. Bremermann and P. Lelong [4, 14], there exists (fj)j asequence of holomorphic functions on ∆n such that the function vδ0
is the regularized supremumlimit on K of (1

j log |fj |
)
j
. Which means that

vδ0
(z) =

[
limj→∞ sup 1

j log |fj(z)|
]∗

= limε→0 supζ∈B(z,ε)[limj→∞
1
j log |fj(ζ)], z ∈ K.

(6.19)
For the compact K, in view of (6.18) and (6.19), we can find a natural number N that depends on
K, and analytic functions on ∆n, f1, . . . , fN , such that the function v and the function vN definedby

vN(z) = max
1≤j≤N

1

j
log |fj(z)|, (6.20)

satisfy the following inequalities
vN − 1 ≤ v ≤ vN ≤ 0 on K rX ∩ Y . (6.21)

We may suppose the compact K is sufficiently small so that
K ∩X ∩ Y ⊂ K ∩ Y ∩ {fj = 0, j = 1, . . . , N}.

By the Weierstrass preparation theorem (see [6]), for each indice 1 ≤ j ≤ N , one can write fj suchthat
fj(z) = hj(z)Pj(z

′, zn), (6.22)
where hj is an invertible holomorphic function on ∆n and Pj(z ′, zn) is a Weierstrass polynomial in
zn, of the form

Pj(z
′, zn) = z

mj
n + a1,j(z

′)z
mj−1
n + · · ·+ aν,j(z

′)z
mj−ν
n + · · ·+ amj ,j,(z

′), aν,j(0) = 0,

with mj ≥ 1 is the vanishing order of fj at z0 = 0 and (aν,j,(z
′))1≤µ≤mj are holomorphic coefficientson the polydisc ∆n−1 in Cn−1. Furthermore, for all 1 ≤ j ≤ N and all 1 ≤ ν ≤ mj , there is apositive constant C1 such that for all z ′ ∈ ∆n−1, the following inequality holds

|aν,j(z ′)| ≤ C1||z ′||ν . (6.23)
Therefore, by the triangle inequality, (6.23) provides the existence of a constant C2 = C3(K) > 0such that for all 1 ≤ j ≤ N and for all z = (z ′, zn) ∈ K, the following inequality holds

|Pj(z ′, zn)| ≤ C2||z ′||mj . (6.24)
On the other hand, since {fj = 0, j = 1, . . . , N} = {Pj = 0, j = 1, . . . , N} and since the polynomial
Pj vanishes at z0 = 0, then there is a constant C3 > 0, such that for any point z = (z ′, zn) ∈ Ksufficiently close to 0, the following inequality holds

|zn| ≤ C3||z ′||. (6.25)

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 13Hence, using the triangle inequality with (6.25) and reasoning by induction on the degree mj of
Pj , 1 ≤ j ≤ N , we can find a constant C4 = C4(K) > 0, such that for all 1 ≤ j ≤ N and for all
z = (z ′, zn) ∈ K, the polynomial Pj(z ′, zn) satisfies the following inequality

C4||z ′||mj ≤ |Pj(z ′, zn)|. (6.26)
In addition, since for all 1 ≤ j ≤ N , the holomorphic function hj is invertible and N depends onlyon K, then we can find a constant C5 = C5(K) > 0 such that for all z ∈ K, the function defined by

v ′N(z) = max
1≤j≤N

1

j
log |hj(z)|

satisfies the following inequality
− C5 ≤ v ′N(z) ≤ 0. (6.27)If we denote βN = min1≤j≤N

mj
j and wN(z) = wN(z ′, zn) = βN log ||z ′||, then following (6.22) thefunction vN defined by (6.20) is such that

vN(z) = max1≤j≤N
1
j log |Pj(z ′, z,n )|+ max1≤j≤N

1
j log |hj,δ>0(z)|

≤ βN log ||z ′||+ max1≤j≤N
1
j log |hj(z)|

= wN(z) + v ′N(z) ≤ 0.

(6.28)
Hence, in view of (6.21), (6.24), (6.26) and (6.28), we can find constants C6 = C6(K) > 0 and
C7 = C7(K) > 0 such that for all z ∈ K r Y ∩ X , the functions v and wN satisfy the followinginequalities

C6wN(z)− C7 ≤ v(z) ≤ wN(z) ≤ 0. (6.29)It is clear that (βN)N is a positive and decreasing sequence. Hence it has a limit β ≥ 0 as N → +∞.In view of (6.17) and (6.29) we have β > 0. Indeed, if not, the function v will be bounded near 0and then the point z0 is not a singular point of v . Therefore, by letting N → +∞ in (6.29), we canfind constants β > 0 and C8 = C8(K) > 0 such that for all z = (z ′, zn) ∈ K r Y ∩X , the function
v satisfies the following inequality

exp (−v(z)) ≤ C8||z ′||−β. (6.30)
Taking α > 0 so that αβ ∈ (0, 2q − 2), hence z 7→ 1

||z ||αβ ∈ L
1
loc(Cq−1), and investigating (6.17)with (6.30), we get the following∫

z∈K\X∩Y
exp(−αv(z)) ≤ C8

∫
z∈∆q−1

dλq−1(z)

||z ||αβ <∞. (6.31)
Consequently, our proof is achieved thanks to (6.31). �

Remark 6.2. Theorem 6.1 generalizes a result in [9] showing that, if Y = {f = 0} is a hyper-surface in Ω given by a holomorphic function not identically vanishing and if Y yields a completeintersection with another analytic subset X ⊂ Ω, then there exists δ > 0 such that the coefficientsof the current log |f |[X] lies in L1+δ
loc (X). Moreover, there exists α > 0, such that the coefficients

https://doi.org/10.28924/ada/ma.4.23


Eur. J. Math. Anal. 10.28924/ada/ma.4.23 14of the current |f |−α[X] lie in L1
loc(X). If we replace the hypersurface Y = {f = 0} by a positive

(1, 1)-closed current of the form T = ddcv for some psh function v , then we provide Theorem 1.4.Note that we invite interested readers to proceed defining and investigating quaternionic hyper-surfaces and quaternionic analytic sets in the space HC recently studied in [13] and find similarresults as given in this paper.
In the direction of Theorem 1.3 and proceeding as in the proof of Theorem 3.3 in [9], we finallygive the proof of Theorem 1.5.

Proof. We consider the analytic subsets X ∩ π−1(a), Y ∩ π−1(a) and
E = {a ∈ ∆k : dimC(X ∩ π−1(a))× (Y ∩ π−1(a)) > p + q − k}.

By Proposition 3.1, E is contained in a countable union of analytic subsets of ∆k of dimension
≤ k−1. Therefore the set E is pluripolar. We work as in [9] ( proof of Theorem 3.3) around a point
a 6∈ E. �

Competing interests
The author(s) declare(s) that there is no conflict of interest regarding the publication of thispaper.

References
[1] E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982) 1–41.[2] H. Ben Messaoud, H. El Mir, Operateur de Monge-Ampre et formule de tranchage pour un courant positifs fermé,C.R.A.S. Paris, T316 I (1993), 1173–1176.[3] H. Ben Messaoud, H. El Mir, Tranchage et prolongement des courants positifs fermés, Math. Ann. 307 (1997),473–487.[4] H.-J. Bremermann, On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions,Math. Ann. 131 (1956), 76–86.[5] C.O. Kiselman, Sur la définition de l’opérateur de Monge-Ampère complexe, Analyse Complexe, Proceedings ofjournées Fermat (SMF), Toulouse, May, 1983.[6] J.P. Demailly, Complex analytic and differential geometry, e-book, 2007.[7] H. Federer, Geometric measure theory, Springer, 1969.[8] R. Harvey, B. Shiffman, A characterization of Holomorphic chains, Ann. Math. 28 (1974), 553–587.[9] H. Khedhiri, Wedge product of currents, Lobachevskii J. Math. 31 (2010), 224–231.[10] H. Khedhiri, Slicing of currents associated to a plurisubharmonic function, Punj Univ. J. Math. 47 (2015), 21–34.[11] H. Khedhiri, ϕ-Slicing results for negative plurisubharmonic currents, Uzbek J. Math. 68 (2024), 102-112.[12] H. Khedhiri, On construction of positive closed currents with prescribed Lelong numbers, J. Sib. Fed. Univ. Math.Phys. 13 (2020), 331-341.[13] H. Khedhiri, T. Mkademi, Foundational aspects of a new matrix holomorphic structure, Arab J. Math. Sci. (2024).[14] P. Lelong, Intégration sur un ensemble analytique complexe, Bull. Soc. Math France. 853 (1957), 239–262.

https://doi.org/10.28924/ada/ma.4.23

	1. Introduction
	2. Preliminaries
	3. Proof of Theorem 1.1: slicing of the current (log|f|)[X]
	4. Proof of Theorem 1.2: slicing of the current v[X]
	5. Proof of Theorem 1.3: slicing of the current U[X][Y]
	6. Proofs of Theorem 1.4 and Theorem 1.5 with applications
	Competing interests
	References

