A new generalization of the Lelong number

Licentiate thesis, 2009

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 $ \nu_{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 $\nu_{a,\psi}(\varphi)$ 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: \nu_{z,\psi}(\varphi) \geq c \}$ where $c>0$, are in fact analytic sets, provided that the \textit{weight} $\psi$ satisfies some additional conditions.