A new generalization of the Lelong number

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