A new generalization of the Lelong number

Abstract

We will introduce a quantity which measures the singularity of a plurisubharmonic function φ relative to another plurisubharmonic function ψ, at a point a. We denote this quantity by ν_{a, ψ}(φ). It can be seen as a generalization of the classical Lelong number in a natural way: if ψ=(n−1)log| ⋅ −a|, where n is the dimension of the set where φ is defined, then ν_{a, ψ}(φ) coincides with the classical Lelong number of φ at the point a. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {z: ν_{z, ψ}(φ)≥c} where c>0, are in fact analytic sets, provided that the weightψ satisfies some additional conditions.