# 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 Lelong (1957). 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 φ at a point *x* of **C**^{n} is

- [math]\displaystyle{ \liminf_{z\rightarrow x}\frac{\phi(z)}{\log |z-x|}. }[/math]

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 **C**^{k} 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

- Lelong, Pierre (1957), "Intégration sur un ensemble analytique complexe",
*Bulletin de la Société Mathématique de France***85**: 239–262, ISSN 0037-9484, http://www.numdam.org/item?id=BSMF_1957__85__239_0 - Lelong, Pierre (1968),
*Fonctions plurisousharmoniques et formes différentielles positives*, Paris: Gordon & Breach, https://books.google.com/books/about/Fonctions_plurisousharmoniques_et_formes.html?id=cy_vAAAAMAAJ - Varolin, Dror (2010), "Three variations on a theme in complex analytic geometry", in McNeal, Jeffery; Mustaţă, Mircea,
*Analytic and algebraic geometry*, IAS/Park City Math. Ser.,**17**, Providence, R.I.: American Mathematical Society, pp. 183–294, ISBN 978-0-8218-4908-8, https://books.google.com/books?id=wwgEP4frWvAC&pg=PA183

Original source: https://en.wikipedia.org/wiki/Lelong number.
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