On the directional Lelong-Demailly numbers of positive currents

0.16 MB·

by  Mohamed Zaway

ON THE DIRECTIONAL LELONG-DEMAILLY NUMBERS OF POSITIVE
CURRENTS
MOHAMED ZAWAY,HAITHEMHAWARI,ANDNOUREDDINEGHILOUFI
4
1
0
2 Abstract. In this paper we study the existence of the directional Lelong-Demailly numbers

n of positive plurisubharmonic or plurisuperharmonic currents. We prove the independence of
a these numbers to the system of coordinates. Moreover these numbers will be given by locally
J integrable functions.

0
2 Sur les nombres de Lelong-Demailly directionnels des courants positifs.
R´esum´e. Danscetarticleon´etudiel’existencedesnombresdeLelong-Demaillydirectionnelsdes

] courants positifs plurisousharmoniques ou plurisurharmoniques. On montre l’ind´ependance de
V
cesnombresdusyst`emedescoordonn´ees. Depluscesnombresseront donn´espardesfonctions
C
localement int´egrables.
.
h
t 1. Introduction
a
m
The problem of the existence of Lelong numbers of positive currents on open subsets of Cn
[
is still open, it was considered firstly by Lelong in the 1950’th, who had solved it in the case

1 of positive closed currents, then in 1982’th this problem was solved by Skoda [5] in the case
v
of positive plurisubharmonic (psh) currents. Replacing the standard Ka¨hler form in the def-
0
inition of the Lelong number by a (1,1)−form relative to a psh function, Demailly [2] prove
2
9 the existence of so called generalized Lelong numbers or Lelong-Demailly numbers for positive
4 closed currents. Finally this problem was studied by the second author [3] in case of positive
.
1 plurisuperharmonic (prh) currents. In 1996, Alessandrini and Bassanelli [1] consider an anal-
0 ogous problem, the existence of the directional Lelong numbers of positive currents and they
4
solved it in plurisubharmoniccase. A generalized form of this problem appear then, (directional
1
: Lelong-Demailly numbers) and it is solved by Toujani [6] in the same case. A natural question
v
can be asked in this statement, is there a relationship between the two notions? Our goal is to
i
X
study the case of positive prh currents where we show in particular that there is no relationship
r between the two notions.
a
In the hole of this paper, we consider n, m, k ∈ N such that k ≤ n. We use (z,t) to indicate
an element of CN := Cn ×Cm. Let Ω := Ω ×Ω be an open subset of CN and B be an open
1 2
subset relatively compact in Ω . We consider two C2 positive psh (psh) functions ϕ and v on
2
Ω and Ω respectively such that logϕ is psh in the open subset {ϕ > 0}. We assume that the
1 2
two functions ϕ and v are semi-exhaustive (i.e. there exists R = R(ϕ) > 0 such that for every
c ∈]−∞,R[, one has {z ∈ Ω ; ϕ(z) < c} ⊂⊂ Ω ). For 0 < r < R and r < r < R, we set:
1 1 1 2
B (r) = {z ∈Ω; ϕ(z) < r} and B (r ,r ) = {z ∈ Ω; r ≤ ϕ(z) < r }.
ϕ ϕ 1 2 1 2
To simplify the notations, we set: β := ddcϕ(z), β := ddcv(t) and α := ddclogϕ(z). In the
ϕ v ϕ
classical case, we set ω = ddc|z|2 and ω = ddc|t|2 the Ka¨hler forms of Cn and Cm respectively,
z t
and ω = ω +ω the corresponding one of CN.
t z
Let T be a positive current of bidegree (k,k) on Ω. The purpose of this paper is to study the
directional Lelong-Demailly number of T with respect to B relatively to ϕ and v defined as the
2000 Mathematics Subject Classification. 32U25; 32U40; 32U05.
Key words and phrases. Lelong number,positive plurisubharmonic current,plurisubharmonic function.
1
2 M.ZAWAY,H.HAWARI,ANDN.GHILOUFI
limit,
ν(T,ϕ,B) := lim ν(T,ϕ,B,r)
r 0+
→
where ν(T,ϕ,B,.) is the function defined by
1
ν(T,ϕ,B,r) := T ∧βn k ∧βm.
rn k ϕ− v
− ZBϕ(r)×B
In classical case, we use
1
ν(T,B):= lim T ∧ωn k ∧ωm.
r→0+ r2(n−k) Z{|z|<r}×B z− t
These numbers are studied in the case of positive psh current by Alessandrini and Bassanelli [1]
and they showed that it’s independent to the system of coordinates in the classical case. The
general case has been studied by Toujani [6].
The main tool in this paper is the following formula:
Proposition 1. (Lelong-Jensen Formula) (See [6]) We assume that the current T is positive psh
or prh on Ω. Let(Tj)j N be a smooth regularizing sequence of T. Thenfor every0 <r1 < r2 < R
∈
and for all 1 ≤ p ≤n−k, 0≤ q <p, one has
1
T ∧α p q 1∧βn k p+q+1∧βm
q+1 j ϕ − − ϕ− − v
r2 ZBϕ(r2)×B 1
− T ∧αp q 1∧βn k p+q+1∧βm
q+1 j ϕ− − ϕ− − v
r1 ZBϕ(r1)×B
= T ∧α p∧β n k p∧β m
j ϕ ϕ − − v
ZBϕ(r1,r2)×B
r2 1 1
+ − ds ddcT ∧αp q 1∧βn k p+q ∧βm
Zr1 sq+1 r2q+1! ZBϕ(s)×B j ϕ− − ϕ− − v
r1 1 1
+ − ds ddcT ∧αp q 1∧βn k p+q ∧βm
q+1 q+1 j ϕ− − ϕ− − v
Z0 r1 r2 ! ZBϕ(s)×B
This equality still true when we replace T by T for q = p−1.
j
As a consequence of this formula, if T is a positive psh current on Ω then the function
ν(T,ϕ,B,.) is positive and increasing on ]0,R[ so its limit at 0, ν(T,ϕ,B), exists. This result is
true when m = 0 (Lelong-Demailly numbers).
This statement is so different for positive prh currents as the following examples shown:
Example 1. The current T := −log|z |2[z = 0] is an example of positive prh current of
0 2 1
bidimension (1,1) on the unit ball of C2 which has no Lelong number at 0, but it has (classical)
directionalLelongnumbers(inbothdirections)andthiscananswertothequestionaskedbefore.
Example 2. The current T := −log(|z |2 +|z |2)[z = 0] is positive prh of bidimension (2,2)
1 1 3 2
on the unit ball B of C3 which has no directional Lelong number with respect to the disc
B := D(0, 1) ⊂ {(0,0)}×C.
2
ON THE DIRECTIONAL LELONG-DEMAILLY NUMBERS 3
In fact, for r < √3 we have {|z |2+|z |2 < r2}×B ⊂⊂B. Thanks to Fubini-Tonelli theorem
2 1 2
we have
ν(T ,B,r)
1
−1 i i
= log(|z |2+|z |2) dz ∧dz ∧ dz ∧dz
π2r2 1 3 2 1 1 2 3 3
Z{|z1|<r}×B
1 i i
= −log(|z |2+|z |2) dz ∧dz dz ∧dz
π2r2 1 3 2 1 1 2 3 3
1 ZB Z|z21π|<r r i !
= −tlog(t2+|z |2)dtdθ dz ∧dz
π2r2 3 2 3 3
2 ZB(cid:18)Z10 Z0 r t(cid:19)3 i
= − r2log(r2+|z |2)+ dt dz ∧dz
πr2 2 3 t2+|z |2 2 3 3
ZB(cid:18) Z0 3 (cid:19)
2 1 r2 |z |2 i
= − r2log(r2+|z |2)+ − 3 log(r2+|z |2)−log(|z |2) dz ∧dz
πr2 2 3 2 2 3 3 2 3 3
ZB(cid:18) (cid:19)
= 2 21 −1r2log(r2+s2)+ r2 − s2 log(r(cid:2)2+s2)−log(s2) sds (cid:3)
r2 2 2 2
Z0 (cid:18) (cid:19)
−log(r2+ 1) r2log(r2+ 1) 1 (cid:2)logr 1 (cid:3)
≥ 4 + 4 + (− − ).
128r2 2 r2 64 256
The result is obtained by tending r to 0. (cid:3)
Asthesecondexampleshows,weneedasuitableconditiontoprovetheexistenceofdirectional
Lelong-Demailly numbers; for this, as it was introduced in [3], we say that T (B, ϕ and v)
satisfy Condition (C) if the function s 7−→ s 1ν(ddcT,ϕ,B,s) is integrable in a neighborhood
−
of 0. Using the Lelong-Jensen formula, we can see that Condition (C) is satisfied for a positive
psh current.
Theorem 1. If the current T is positive prh and satisfies Condition (C) then the directional
Lelong-Demailly number ν(T,ϕ,B) exists.
Proof. For 0 < r < R, we consider
r sn k ν(ddcT,ϕ,B,s)
−
g(r) := ν(T,ϕ,B,r)+ −1 ds.
rn k s
Z0 (cid:18) − (cid:19)
Thanks to Condition (C), the function g is well defined. Moreover it is positive on ]0,R[
because the function ν(ddcT,B,.) is negative on ]0,R[.
AsthecurrentT is positiveprh,thenusingLelong-Jensen formula,wehave for0< r < r < R,
1 2
g(r )−g(r )
2 1
1 r2
= ν(T,ϕ,B,r )−ν(T,ϕ,B,r )+ sn k 1ν(ddcT,ϕ,B,s)ds
2 1 rn k − −
2− Z0
1 r1 r2 ν(ddcT,ϕ,B,s)
− sn k 1ν(ddcT,ϕ,B,s)ds− ds
r n k − − s
1 − Z0 Zr1
= T ∧αn k ∧βm ≥ 0.
ϕ− v
ZBϕ(r1,r2)×B
Hence the function g is positive and increasing on ]0,R[, thus its limit when r goes to 0 exists.
As Condition (C) is satisfied and (s/r)n k−1 is uniformly bounded in neighborhood of 0, then
−
r sn k ν(ddcT,ϕ,B,s)
−
lim −1 ds = 0.
r→0+Z0 (cid:18)rn−k (cid:19) s
Therefore
lim g(r) = lim ν(T,ϕ,B,r) = ν(T,ϕ,B).
r 0+ r 0+
→ → (cid:3)
4 M.ZAWAY,H.HAWARI,ANDN.GHILOUFI
Example 3. Letz = (z ,z )andT (z,t) := dlog(|z |2+|z |2)∧dclog(|z |2+|z |2). Thecurrent
1 2 2 1 2 1 2
T is positive prh of bidegree (1,1) on C3 and ddcT (z,t) = −[z = 0]. If B := D(0,1) ⊂ {0}×C
2 2
is the unit disc of C then
ν(ddcT ,B,r)= −[z = 0]∧ddc|t|2 = −1
2
Z z<r B
{| | }×
and
1
ν(T ,B,r) = T ∧ω ∧ω
2 r2 2 z t
1Z{|z|<r}×B 1
= idz ∧dz ∧idz ∧dz ∧idt∧dt = 1.
8π3r2 |z|2 1 1 2 2
Z{|z|<r}×B
This example proves that Condition (C) in Theorem 1 is not necessary for the existence of
directional Lelong-Demailly numbers.
2. Independence to the system of coordinates
In this part we prove the independence of Lelong-Demailly numbers of positive psh or prh
currents to the system of coordinates, this result is due to Alessandrini and Bassanelli [1] in the
classical case.
Proposition 2. Let T be a positive prh or psh current of bidegree (k,k) on Ω and p ∈ [2,+∞[.
For every r ∈]0,R(ϕ)], we have
ν(T,ϕp,B,rp)
(2.1) = pn k ν(T,ϕ,B,r)+ r ν(ddcT,ϕ,B,s) sn−k − s(n−k)p ds .
− Z0 s rn−k r(n−k)p! !
In particular, if the current T is positive prh, then ν(T,ϕ,B) exists if and only if ν(T,ϕp,B)
exists for some p ≥ 2.
Furthermore, in both cases (with the assumption that ν(T,ϕ,B) exists if T is prh) we have
p−1
ν(T,ϕp,B)= pn k ν(T,ϕ,B)+ ν(ddcT,ϕ,B) .
−
p(n−k)
(cid:18) (cid:19)
Proof. Let ǫ > 0. If we replace ϕ by ϕ = ϕ+ǫ and ψ by ψ = (ϕ+ǫ)p in the Lelong-Jensen
ǫ ǫ
formula, we obtain for every 0 <r < ǫ < r = r < R(ϕ),
1 2
ν(T,ϕ ,B,r)
ǫ
1 r
= T ∧αn k ∧βm− sn k 1ν(ddcT,ϕ ,B,s)ds
(2.2) ϕ−ǫ v rn k − − ǫ
ZBϕǫ(ǫ,r)×B − Zǫ r ν(ddcT,ϕ ,B,s)
ǫ
+ ds
s
Zǫ
Using ddcT instead of T, Equality (2.2) becomes
ν(ddcT,ψ ,B,rp) = ddcT ∧αn k 1∧βm
ǫ ψ−ǫ − v
ZBψǫ(ǫp,rp)×B
(2.3)
= pn k 1 ddcT ∧αn k 1∧βm
− − ϕ−ǫ − v
= pn k 1νZ(Bdϕdǫc(Tǫ,r,)ϕ×B,B,r)
− − ǫ
ON THE DIRECTIONAL LELONG-DEMAILLY NUMBERS 5
Let now
ν(T,ψ ,B,rp)
ǫ
rp sn k ν(ddcT,ψ ,B,s)
= T ∧αn k ∧βm+ 1− − ǫ ds
ψ−ǫ v rp(n k) s
ZBψǫ(ǫp,rp)×B Zǫp (cid:18) − (cid:19)
r sp(n k) ν(ddcT,ψ ,B,sp)
= T ∧αn k ∧βm+p 1− − ǫ ds
ZBψǫ(ǫp,rp)×B ψ−ǫ v Zǫ rp(n−k)! s
r sp(n k) ν(ddcT,ϕ ,B,s)
= pn k T ∧αn k ∧βm+pn k 1− − ǫ ds
(2.4) − ZBϕǫ(ǫ,r)×B ϕ−ǫ r v sn k− Zνǫ(dd cT,ϕr,pB(n,−sk))! s
= pn k ν(T,ϕ ,B,r)− 1− − ǫ ds
− ǫ rn k s
(cid:26) Zǫ (cid:18) − (cid:19) (cid:27)
r sp(n k) ν(ddcT,ϕ ,B,s)
+pn k 1− − ǫ ds
−
Zǫ rp(n−k)! s
r ν(ddcT,ϕ ,B,s) sn k sp(n k)
= pn k ν(T,ϕ ,B,r)+ ǫ − − − ds .
− ǫ s rn k rp(n k)
Zǫ − − ! !
here we have used successively Equality (2.2) with ψ instead of ϕ , then the change of vari-
ǫ ǫ
able s 7→ sp, next Equality (2.3) and finally equality (2.2).
When ǫ → 0, Equality (2.4) gives Equality (2.1).
Thanks to Equality (2.1), for every r ∈]0,R(ϕ)[, we have
1 r sn k 1 sp(n k) 1
ν(T,ϕ,B,r)− ν(T,ϕp,B,rp) = − ν(ddcT,ϕ,B,s) − − − − − ds.
pn k rn k rp(n k)
− Z0 − − !
For a technical reason, we distinguish the two cases:
• First case ddcT ≥ 0: The current ddcT is positive closed, so ν(ddcT,ϕ,B,.) is a positive
increasing function. Hence
p−1 r sn k 1 sp(n k) 1
ν(ddcT,ϕ,B) ≤ ν(ddcT,ϕ,B,s) − − − − − ds
p(n−k) rn k rp(n k)
Z0 − − !
p−1
≤ ν(ddcT,ϕ,B,r) .
p(n−k)
It follows that
p−1 1
− ν(ddcT,ϕ,B,r) ≤ ν(T,ϕ,B,r)− ν(T,ϕp,B,rp)
p(n−k) pn k
−
p−1
≤ − ν(ddcT,ϕ,B).
p(n−k)
• Second case ddcT ≤ 0: The current ddcT is negative and closed, so ν(ddcT,ϕ,B,.) is a
negative decreasing function. A similar computation proves that we have
p−1 1
− ν(ddcT,ϕ,B) ≤ ν(T,ϕ,B,r)− ν(T,ϕp,B,rp)
p(n−k) pn k
−
p−1
≤ − ν(ddcT,ϕ,B,r).
p(n−k)
In both cases, the two terms of the right hand and the left hand have the same limit when
r → 0+. This completes the proof of the proposition. (cid:3)
Remark 1. If the current T is positive psh (resp. prh satisfying Condition (C)) then
ν(T,ϕp,B) = pn kν(T,ϕ,B).
−
The current T defined on C3 by
2
T (z,t) := dlog(|z |2+|z |2)∧dclog(|z |2+|z |2)
2 1 2 1 2
6 M.ZAWAY,H.HAWARI,ANDN.GHILOUFI
shows that this equality is not true if Condition (C) is not satisfied; so the second equality in
Proposition 2 is sharp.
Theorem 2. Let T be a positive psh or prh current of bidegree (k,k) on Ω and ϕ, ψ two psh
functions such that
logψ(z)
liminf ≥ ℓ.
ϕ(z) 0 logϕ(z)
→
Assume that T, ψ and B satisfy Condition (C). Then ν(T,ψ,B) ≥ ℓn kν(T,ϕ,B).
−
In particular, if logψ(z) ∼ ℓlogϕ(z) when ϕ(z) ∈V (0) then ν(T,ψ,B) = ℓn kν(T,ϕ,B).
−
This Theorem is known with ”comparison theorem”; a such theorem was proved by Demailly
in case of positive closed currents. This result allows us to prove the independenceof directional
Lelong-Demailly numbers with a changement of the system of coordinates.
Proof. Replacing ψ by ψp and ℓ by (p−1)ℓ where p ≥ 2 large enough, we can assume that
logψ(z)
liminf > ℓ ≥ 2.
ϕ(z) 0 logϕ(z)
→
Hence we have
ψ(z)
lim = 0.
ϕ(z) 0 ϕ(z)ℓ
→
So the function Ψ = ψ + ǫϕℓ ∼ ǫϕℓ for every ǫ > 0. As ℓ ≥ 2, Ψ is C2. Thanks to
ǫ ǫ
ϕ(z) V(0)
∈
dominated convergence theorem,
1 1
lim T ∧βn k ∧βm = T ∧βn k ∧βm
ǫ→0rn−k Z{Ψǫ<r}×B Ψ−ǫ v rn−k Z{ψ<r}×B ψ− v
for every r ∈]0,R(ψ)[.
If the current T is positive prh then the function g defined by
ǫ
r sn k ν(ddcT,Ψ ,B,s)
− ǫ
g (r)= ν(T,Ψ ,B,r)+ −1 ds ≥ 0
ǫ ǫ rn k s
Z0 (cid:18) − (cid:19)
is increasing, thus
1 r sn k ν(ddcT,Ψ ,B,s)
g (r) = T ∧βn k ∧βm+ − −1 ǫ ds
ǫ rn k Ψ−ǫ v rn k s
− Z1{Ψǫ<r}×B Z0 (cid:18) − (cid:19)
≥ lim T ∧βn k ∧βm
ρ→0ρn1−k Z{Ψǫ<ρ}×B Ψ−ǫ v
= lim T ∧βn k ∧βm
ρ→0ρn1−k Z{ǫϕℓ<ρ}×B Ψ−ǫ v
≥ lim T ∧βn k ∧βm = ν(T,ϕℓ,B).
ρ→0ρn−k Z{ǫϕℓ<ρ}×B ǫϕ−ℓ v
Where we use the fact that Ψ ∼ ǫϕℓ and T ∧(ddc(ψ+ǫϕℓ))n k ≥ T ∧(ǫddc(ϕℓ))n k.
ǫ − −
Thesameresultcanbeobtainedinthecaseofpositivepshcurrentbyusingg (r) =ν(T,Ψ ,B,r).
ǫ ǫ
If ǫ → 0, we obtain ν(T,ψ,B,r) ≥ ν(T,ϕℓ,B). Thus, if r → 0, ν(T,ψ,B) ≥ ν(T,ϕℓ,B).
In particular if logψ(z) ∼ ℓlogϕ(z), for ǫ > 0 (with ℓ(1−ǫ) ≥ 2), there exists η > 0 such
that if |ϕ(z)| < η, then ϕ(z)ℓ(1+ǫ) ≤ ψ(z) ≤ ϕ(z)ℓ(1 ǫ). So we can apply the previous inequality
−
to obtain
ℓn k(1+ǫ)n kν(T,ϕ,B) ≤ ν(T,ψ,B) ≤ ℓn k(1−ǫ)n kν(T,ϕ,B).
− − − −
When ǫ → 0, we obtain ν(T,ψ,B) = ℓn kν(T,ϕ,B). (cid:3)
−
ON THE DIRECTIONAL LELONG-DEMAILLY NUMBERS 7
3. Main result
The main result of this paper is the following theorem:
Theorem 3. Let T be a positive prh current of bidegree (k,k), 0 ≤ k ≤ n on Ω. We assume
that T, B satisfy Condition (C). Then there exists an open subset V ⊂ B and a function
0 0
f ∈L1 (V) such that
loc
1
ν(T,B) = lim T ∧ωn k ∧ωm = f(t)ωm
r→0r2(n−k) Z{|z|<r}×B z− t ZB t
for every open ball B ⊂⊂ V.
The analogous of this result in the case of positive psh currents was proved by Alessandrini
and Bassanelli [1]. For k = 0 or k = n the proof is simple, indeed
• If k = 0 then h := T is a positive prh function and
H(z) := h(z,t)ωm
t
ZB
is also prh, so ν(H,z) =H(z). Therefore
1 1
ν(h,B) = lim h(z,t)ωn ∧ωm = lim H(z)ωn
r→0r2n Z{|z|<r}×B z t r→0r2n Z|z|<r z
= h(0,t)ωm.
t
ZB
• If k = n, one has
ν(T,B)= lim T ∧ωm.
t
r→0Z{|z|<r}×B
Let Θ = 1lΩrYT, where Y = {0}×Cm∩Ω. The subsetY is analytic in Ω of codimension
k = n. Since ||T|| is finite for every compact subsetK ⊂ Ω, the current Θ has a locally
K
finite mass in neighborhood of every point of Y. As T is a positive prh current, then T
and Θ are C−flat currents. So T = 1l T +Θ, where Θ is the trivial extension of Θ by
Y
0 across Y. Thanks to the support theorem, there exists a positive prh function f on Y
such that 1l T = f[Y]. Hence T = f[Y]+Θeand it folelows that
Y
T ∧ωm = f(t)ωem+ Θ∧ωm
t t t
Z{|z|<r}×B ZB Z{|z|<r}×B
for every open ball B of Ω and r > 0 small enough. But e
2
Θ∧ωm ≤ Θ∧ωm = ||Θ||({|z| < r}×B)
t
Z{|z|<r}×B Z{|z|<r}×B
and e e e
lim||Θ||({|z| < r}×B)= ||Θ||(B) = 0
r 0
→
which gives the result.
e e
3.1. Preliminary lemmas. To prove the main result in case 1≤ k ≤ n−1, we need a special
transformation of coordinates (introduced by Siu [4]) given by w = w(z,t) = (w ,....,w )
1 m+n
such that (z,w ) = (z ,....,z ,w ,....,w ) form a system of coordinates of CN for every 1 ≤
I 1 n i1 im
i < i < ..... < i ≤ m+n. We denote by ω := ddc|w |2.
1 2 m I I
The first lemma is a version of Lelong-Jensen formula adapted to ω :
I
8 M.ZAWAY,H.HAWARI,ANDN.GHILOUFI
Lemma 1. (See [1]) Let ψ be a smooth (k,k)-form on Ω. For any increasing m−multiindex I
and every 1 ≤ p ≤ n−k, 0 ≤ q < p, 0 < r,c, such that {(z,t) ∈ Ω; |z| < r, |w | < c} ⊂⊂ Ω,
I
we have
ψ∧θ p∧ω n k p∧ωm
z z − − I
Z{|z1|<r, |wI|<c}
= ψ∧θ p q 1∧ωn k p+q+1∧ωm
r2(q+1) z − − z− − I
r Z{1|z|<r, |wI|<c}
− 2sds ddcψ∧θ p q 1∧ω n k p+q ∧ω m
z − − z − − I
s2(q+1)
Z0r 1 Z{|z|<s, |wI|<c}
+ 2sds ddcψ∧θp q 1∧ωn k p+q ∧ωm
r2(q+1) z− − z− − I
Z0 Z{|z|<s, |wI|<c}
where θ := ddclog|z|2.
z
The following proposition is the crucial point in the proof of the main result:
Proposition 3. Let T be a positive current on Ω satisfying Condition (C) and (Tj)j N its
∈
regularizing sequence. Thenthere exist0< r ,csuchthatforanyincreasing m−multiindexI and
0
every0 ≤ p ≤n−k, 0 < r ≤ r wehave {(z,t) ∈Ω; |z| <r, |w |< c} ⊂ {{|z| < 2r}×B } ⊂⊂ Ω
0 I 0
and
sup T ∧θp∧ωn k p∧ωm < +∞.
j z z− − I
j Z{|z|<r, |wI|<c}
Proof. For p = 0, the result is clear.
So let 1 ≤ p < n−k. Let r > 0 such that {|z| < 2r }×B ⊂⊂ Ω. There exists c > 0, small
0 0 0
enough, such that for any increasing m−multiindex I,
{(z,t) ∈ Ω; |z| < r , |w | <c} ⊂{|z| < 2r }×B .
0 I 0 0
Then for every 0< r ≤ r one has
0
{(z,t) ∈ Ω; |z| < r, |w |< c} ⊂ {|z| < 2r}×B .
I 0
If we choose q = p−1 in previous lemma, we obtain
T ∧θ p∧ωn k p∧ω m
j z z− − I
Z1{|z|<r, |wI|<c}
≤ T ∧ωn k ∧ωm
r2p j z− I
Zr{|z|1<r, |wI|<c}
− 2sds ddcT ∧ωn k 1∧ωm
s2p j z− − I
1Z0 Z{|z|<s, |wI|<c}
≤ T ∧ωn k ∧ωm
r2p j z− I
Zr{|z|<r, |wI|<c}
+ 2sds −ddcT ∧θp∧ωn k 1 p∧ωm
j z z− − − I
1Z0 Z{|z|<s, |wI|<c}
≤ T ∧ωn k ∧ωm
r2p j z− I
Z{|z|<r, |wI|<c}
+r2 −ddcT ∧θp∧ωn k 1 p∧ωm.
j z z− − − I
Z{|z|<r, |wI|<c}
Furthermore, for r outside a set at most countable, we have
lim T ∧ωn k ∧ωm = T ∧ωn k ∧ωm.
j z− I z− I
j→+∞Z{|z|<r, |wI|<c} Z{|z|<r, |wI|<c}
Since −ddcT is a positive closed current, then thanks to Alessandrini-Bassanelli [1], we have
sup −ddcT ∧θp∧ωn k 1 p∧ωm < +∞.
j z z− − − I
j Z{|z|<r, |wI|<c}
ON THE DIRECTIONAL LELONG-DEMAILLY NUMBERS 9
In the case p = n − k, it is easy to see that there exist A > 0 and a closed C form φ on
∞
{|z| < 2r }×B such that ωm ≤ A(ω ∧φ+ωm) in neighborhood of {|z| < r , |w |< c}. So
0 0 I z t 0 I
r 1
sup 2sds −ddcT ∧ωn k 1∧ωm
s2(n k) j z− − I
j Z0 r −A Z{|z|<s, |wI|<c}
≤ sup 2sds −ddcT ∧ωn k 1∧(ω ∧φ+ωm)
s2(n k) j z− − z t
j Z0r A− Z{|z|<s, |wI|<c}
≤ sup 2sds −ddcT ∧ωn k 1∧(ω ∧φ+ωm)
s2(n k) j z− − z t
j Z0 − Z{|z|<2s}×B0
r r ν(−ddcT,B,2s)
≤ 2n kA 2sds −ddcT ∧θn k ∧φ+ ds .
− z− s
Z0 Z z<2s B0 Z0 !
{| | }×
Thanks to Condition (C), the last integral is finite. (cid:3)
Remark 2. (See [1]) Let U := ∩ {(z,t) ∈ Ω; |z| < r , |w | < c}, then thanks to the last
I 0 I
proposition, there exist a subsequence of (Tj)j N, noted in the same way, and some currents
T(0),T(1),...,T(n k) defined on U such that ∈
−
lim T^∧θp = T(p)
j z
j +
→ ∞(cid:16) (cid:17)
weakly on U.
Lemma 2. With the same notations as in Proposition 3, if we denote by
^
(ddcT)(p−q−1) := lim ddcTj ∧θzp−q−1
j +
→ ∞
then we have
r sds
−(ddcT)(p q 1)∧ωn k p+q ∧ωm
s2(q+1) − − z− − I
(3.1) Z0 rZ{|zs|d<ss, |wI|<c}
≤ lim −ddcT ∧θp q 1∧ωn k p+q ∧ωm < +∞.
j→+∞Z0 s2(q+1) Z{|z|<s, |wI|<c} j z− − z− − I
Lemma 3. Let Rp := log|z|ddcT ∧θp. There exist S(0),.....,S(n k 1) positive prh currents on
j j z − −
U such that
lim Rp = S(p).
j
j +
→ ∞
Furthermore, for every open ball B and r > 0fsuch that {|z| < r}×B ⊂ U one has
r sds
lim −ddcT ∧θp 1∧ωn k p∧ωm
j→+∞Z0 s2 Z{|z|<s}×B j z− z− − t
= −logr (ddcT)(p 1)∧ωn k p∧ωm+ S(p 1)∧ωn k p∧ωm.
− z− − t − z− − t
Z{|z|<r}×B Z{|z|<r}×B
Proof. The proof of this lemma is similar to lemma 3.3 in [1]. (cid:3)
3.2. Proof of the main result.
Lemma 4. If T(1) and S(0) are as in previous lemma, then
ν(T,B)= ν(T(1),B)−2ν(S(0),B).
10 M.ZAWAY,H.HAWARI,ANDN.GHILOUFI
Proof. Thanks to Lelong-Jensen formula, we have
(3.2) T ∧θ ∧ωn k 1∧ωm
j z z− − t
Z{|z|<r}×B
1
(3.3) = T ∧ωn k ∧ωm
r2 j z− t
Z{|z|<r}×B
r 1
(3.4) + 2sds −ddcT ∧ωn k 1∧ω m
s2 j z− − t
Z0 Z{|z|<s}×B
r 1
(3.5) − 2sds −ddcT ∧ωn k 1∧ωm
r2 j z− − t
Z0 Z{|z|<s}×B
It is easy to see that
lim (3.2)= T(1)∧ωn k 1∧ωm,
z− − t
j→+∞ Z{|z|<r}×B
1
lim (3.3) = T ∧ωn k ∧ωm
j→+∞ r2 Z{|z|<r}×B z− t
for r outside a subset which is at most countable.
Set G the function defined by
j
G (s):= −ddcT ∧ωn k 1∧ωm.
j j z− − t
Z{|z|<s}×B
Thefunction G is positive and increasing, so thanks to dominated convergence theorem, we get
j
r 1
lim (3.5) = 2sds −ddcT ∧ωn k 1∧ωm.
j→+∞ Z0 r2 Z{|z|<s}×B z− − t
Thanks to Lemma 3, one has
lim (3.4) =−2logr ddcT ∧ωn k 1∧ωm+ 2S(0) ∧ωn k 1∧ωm.
z− − t z− − t
j→+∞ Z{|z|<r}×B Z{|z|<r}×B
Hence
1
(3.6) A := T(1)−2S(0) ∧ωn k 1∧ωm
r2(n k 1) z− − t
− − Z{|z|<r}×B (cid:16) (cid:17)
1
(3.7) = T ∧ωn k ∧ωm
r2(n k) z− t
− Z{|z|<r}×B
2logr
(3.8) − ddcT ∧ωn k 1∧ωm
r2(n k 1) z− − t
− − Z{|z|<r}×B
1 r
(3.9) − 2sds −ddcT ∧ωn k 1∧ωm.
r2(n k) z− − t
− Z0 Z z<s B
{| | }×
If we set
G(s) := −ddcT ∧ω n k 1∧ωm,
z − − t
Z{|z|<s}×B
then by Lemma 2 with p = n−k and q = p−1, the function s 7−→ s1 2(n k)G(s) is integrable
− −
on [0,r]; it follows that
1 r r sG(s)
sG(s)ds ≤ ds < +∞.
r2(n k) s2(n k)
− Z0 Z0 −