Nombre de Lelong directionnel d'un courant positif plurisousharmonique

Abstract

Let T be a positive psh current of bidegree ( k , k ) on a neighborhood Ω of 0 in C N = C n × C m ( n = N − m ⩾ k ), let L = { 0 } × C m and B a Borel subset of L such that B ⊂ ⊂ Ω . We denote ( z , t ) ∈ C n × C m and consider two C 2 positive semi-exhaustive psh functions on Ω, ( z , t ) ↦ φ ( z ) et ( z , t ) ↦ v ( t ) such that log φ is also psh on the open set { φ > 0 } . We prove here that T admits a directional Lelong number along L with respect to the functions φ and v. If m = 0 and φ ( z ) = | z | 2 , we get the classical Lelong number of T at 0. If m = 0 and T is a d-closed positive current, we get the number introduced by J.-P. Demailly. If φ ( z ) = | z | 2 and v ( t ) = | t | 2 , we get the number introduced by Alessandrini–Bassanelli. The method first consists in proving a Lelong–Jensen type formula. Finally we prove a theorem on the existence of a positive psh function f on L such that the Lelong number of T is given by f. This theorem generalizes a result proved by Alessandrini–Bassanelli with φ ( z ) = | z | 2 and v ( t ) = | t | 2 . To cite this article: M. Toujani, C. R. Acad. Sci. Paris, Ser. I 343 (2006).