In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the Siu semicontinuity theorem implies that the
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 comple ...
of a closed
positive current In mathematics, more particularly in complex geometry,
algebraic geometry and complex analysis, a positive current
is a positive (''n-p'',''n-p'')-form over an ''n''-dimensional complex manifold,
taking values in distributions.
For a formal defi ...
on a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
is
semicontinuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of Extended real number, extended real-valued Function (mathematics), functions that is weaker than Continuous function, continuity. An extended real-valued function f is ...
. More precisely, the points where the Lelong number is at least some constant form a complex
subvariety
A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms.
Plant taxonomy
Subvariety is ranked:
*below that of variety (''varietas'')
*above that of form (''forma'').
Subva ...
. This was conjectured by and proved by . generalized Siu's theorem to more general versions of the Lelong number.
References
*
*
*
*{{Citation , author1-link=Yum-Tong Siu , last1=Siu , first1=Yum-Tong , title=Analyticity of sets associated to Lelong numbers and the extension of closed positive currents , doi=10.1007/BF01389965 , mr=0352516 , year=1974 , journal=
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...
, issn=0020-9910 , volume=27 , issue=1–2 , pages=53–156
Complex manifolds
Theorems in complex analysis