Lelong number

Summary

The Lelong number of a plurisubharmonic function $u$ at a point $x$ measures how fast $u$ tends to $-\infty$ near $x$, compared with $\log \abs{z - x}$. The same name is used for the analogous density of a closed positive current at a point. For $u = \log \abs{f}$ with $f$ holomorphic, the Lelong number is the order of vanishing of $f$ at $x$. Lelong numbers are the basic local invariant of singularities in pluripotential theory.