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. Résumé. Danscetarticleonétudiel’existencedesnombresdeLelong-Demaillydirectionnelsdes ] courants positifs plurisousharmoniques ou plurisurharmoniques. On montre l’indépendance de V cesnombresdusystèmedescoordonnées. Depluscesnombresseront donnéspardesfonctions C localement intégrables. . 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| 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 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 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| 0 small enough. But e 2 Θ∧ωm ≤ Θ∧ωm = ||Θ||({|z| < r}×B) t Z{|z| 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 | 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| 0fsuch that {|z| < r}×B ⊂ U one has r sds lim −ddcT ∧θp 1∧ωn k p∧ωm j→+∞Z0 s2 Z{|z|

On the directional Lelong-Demailly numbers of positive currents

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