From Surf Wiki (app.surf) — the open knowledge base
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 (p,p) current u on a complex manifold has a Lelong number n(u,x) 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 \phi at a point x of Cn is : \liminf_{z\rightarrow x}\frac{\phi(z)}{\log |z-x|}.
For a point x of an analytic subset A of pure dimension k, the Lelong number ν(A,x) is the limit of the ratio of the areas of A ∩ B(r,x) and a ball of radius r in Ck as the radius tends to zero. (Here B(r,x) is a ball of radius r centered at x.) 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(A,x) is always an integer.
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about Lelong number — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report