Abstract
Let T be a positive plurisubharmonic (psh for short) current of bidegree (k, k) on a neighborhood Omega of 0 in C-N = C-n x C-m (n = N - m >= k), B be a Borel subset of L := {0} x C-m such that B subset of Omega. Taking (z, t) is an element of C-n x C-m, we define a C-2 positive semiexhaustive psh function on Omega, (z, t) bar right arrow phi(z) , such that log phi, is also psh on the open set {phi > 0} and consider (z, t) bar right arrow v(t) a continuous semi-exhaustive psh function on Omega. This paper aims to prove that T admits a generalized directional Lelong number along L with respect to the functions phi and v. Moreover, we give 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 results studied by Alessandrini-Bassanelli and Toujani. (C) 2019 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.