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See also: capitalized

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In mathematics, the '''Lelong number''' is an [[invariant (mathematics)|invariant]] of a point of a [[complex analytic variety]] that in some sense measures the local density at that point. It was introduced by {{harvs|txt|last=Lelong|authorlink=Pierre Lelong|year=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'''<sup>''n''</sup> is
:<math> \liminf_{z\rightarrow x}\frac{\phi(z)}{\log |z-x|^2}.</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'''<sup>''k''</sup> 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. Thie proved that the Lelong number ν(''A'',''x'') is always an integer.

==References==
*{{Citation | last1=Lelong | first1=Pierre | title=Intégration sur un ensemble analytique complexe | url=http://www.numdam.org/item?id=BSMF_1957__85__239_0 | id={{MR|0095967}} | year=1957 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=85 | pages=239–262}}
*{{Citation | last1=Lelong | first1=Pierre | title=Fonctions plurisousharmoniques et formes différentielles positives | url=http://books.google.com/books/about/Fonctions_plurisousharmoniques_et_formes.html?id=cy_vAAAAMAAJ | publisher=Gordon & Breach | location=Paris | id={{MR|0243112}} | year=1968}}
*{{Citation | last1=Varolin | first1=Dror | editor1-last=McNeal | editor1-first=Jeffery | editor2-last=Mustaţă | editor2-first=Mircea | title=Analytic and algebraic geometry | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=IAS/Park City Math. Ser. | isbn= 978-0-8218-4908-8 | id={{MR|2743817}} | year=2010 | volume=17 | chapter=Three variations on a theme in complex analytic geometry | chapterurl=http://books.google.com/books?id=wwgEP4frWvAC&pg=PA183 | pages=183–294}}

[[Category:Complex manifolds]]

Arithmetico-geometric sequence - Revision history

143.105.128.62: /* See also */ capitalized