A new generalization of the Lelong number

Journal article, 2013

We will introduce a quantity which measures the singularity of a plurisubharmonic function phi relative to another plurisubharmonic function psi, at a point a. We denote this quantity by nu (a,psi) (phi). It can be seen as a generalization of the classical Lelong number in a natural way: if psi=(n-1)log| a <...aEuro parts per thousand a'a|, where n is the dimension of the set where phi is defined, then nu (a,psi) (phi) coincides with the classical Lelong number of phi 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:nu (z,psi) (phi)a parts per thousand yenc} where c > 0, are in fact analytic sets, provided that the weight psi satisfies some additional conditions.