A new generalization of the Lelong number

TL;DR

This paper introduces a generalized Lelong number to measure singularities of plurisubharmonic functions relative to another, proving that its upper level sets are analytic under certain conditions.

Contribution

It extends the classical Lelong number by defining a new quantity that captures singularities relative to a weight function, with proven analyticity properties.

Findings

The generalized Lelong number $ u_{a,g}(f)$ effectively measures singularities.

Upper level sets of $ u_{a,g}(f)$ are analytic sets under specific conditions.

The new framework broadens understanding of singularities in complex analysis.

Abstract

We introduce a quantity which measures the singularity of a plurisubharmonic function f relative to another plurisubharmonic function g, at a point a. This quantity, which we denote by $\nu_{a,g}(f)$, can be seen as a generalization of the classical Lelong number, in a natural way. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form $\{z: \nu_{z,g}(f) \geq c > 0 \}$, are in fact analytic sets, under certain conditions on the weight g.