In mathematics, the Lelong number is an
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (hydr ...
''u'' on a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
has a Lelong number ''n''(''u'',''x'') for each point ''x'' of the manifold. Similarly a
plurisubharmonic function In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic ...
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
:
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
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*{{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
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, location=Providence, R.I. , series=IAS/Park City Math. Ser. , isbn= 978-0-8218-4908-8 , mr=2743817 , year=2010 , volume=17 , chapter=Three variations on a theme in complex analytic geometry , chapter-url=https://books.google.com/books?id=wwgEP4frWvAC&pg=PA183 , pages=183–294
Complex manifolds