Abstract Let T be a positive plurisubharmonic (psh for short) current of bidegree ( k , k ) on a neighborhood Ω of 0 in C N = C n × C m ( n = N − m ⩾ k ), B be a Borel subset of L : = { 0 } × C m such that B ⋐ Ω . Taking ( z , t ) ∈ C n × C m , we define a C 2 positive semi-exhaustive psh function on Ω, ( z , t ) ↦ φ ( z ) , such that log ⁡ φ is also psh on the open set { φ > 0 } and consider ( z , t ) ↦ v ( t ) a continuous semi-exhaustive psh function on Ω. This paper aims to prove that T admits a generalized directional Lelong number along L with respect to the functions φ 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.

中文翻译：

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正多次谐波电流的广义定向勒隆数

摘要 设 T 为 CN = C n × C m ( n = N − m ⩾ k ) 中邻域 Ω 上的双度 ( k , k ) 正多次谐波（简称 psh）电流，B 为L : = { 0 } × C m 使得 B ⋐ Ω 。取 ( z , t ) ∈ C n × C m ，我们在 Ω 上定义一个 C 2 正半穷举 psh 函数， ( z , t ) ↦ φ ( z ) ，使得 log ⁡ φ 在开集上也是 psh { φ > 0 } 并考虑 ( z , t ) ↦ v ( t ) Ω 上的连续半穷举 psh 函数。本文旨在证明T在函数φ和v上承认沿L的广义有向Lelong数。此外，我们给出了L上存在正的psh函数f的定理，使得T的Lelong数为由 f 给出。该定理概括了 Alessandrini-Bassanelli 和 Toujani 研究的结果。