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 Cⁿ is

${\displaystyle \liminf _{z\rightarrow x}{\frac {\phi (z)}{\log |z-x|}}.}$

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 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, MR 0095967
• Lelong, Pierre (1968), Fonctions plurisousharmoniques et formes différentielles positives, Paris: Gordon & Breach, MR 0243112
• 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, MR 2743817