A new generalization of the Lelong number

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 $\{z:{\nu }_{z,g}(f)\ge c>0\}$, are in fact analytic sets, under certain conditions on the weight g.

29 pages