Lelong number


In mathematics, the Lelong number is an invariant of a point of a complex analytic variety that in some sense measures the local density at that point. It was introduced by. More generally a closed positive current u on a complex manifold has a Lelong number n for each point x of the manifold. Similarly a plurisubharmonic function also has a Lelong number at a point.

Definitions

The Lelong number of a plurisubharmonic function φ at a point x of Cn is
For a point x of an analytic subset A of pure dimension k, the Lelong number ν is the limit of the ratio of the areas of A ∩ B and a ball of radius r in Ck as the radius tends to zero. In other words the Lelong number is a sort of measure of the local density of A near x. If x is not in the subvariety A the Lelong number is 0, and if x is a regular point the Lelong number is 1. It can be proved that the Lelong number ν is always an integer.