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Remmert-Stein theorem
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(Theorem)
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For a complex analytic subvariety  and  a regular point, let  denote the complex dimension of  near the point 
The condition that
 for all regular  is the same as saying that all the irreducible components of  are of dimension strictly greater than  To show that the restriction on
the dimension of  is “sharp,” consider the following example where the dimension of  equals the dimension of  . Let
 be our coordinates and let  be defined by
 in
 where  is defined by  The closure of  in
 cannot possibly be analytic. To see this look for example at
 If
 is analytic then  ought to be a zero dimensional complex analytic set and thus a set of isolated points, but it has a limit point  by Picard's theorem.
Finally note that there are various generalizations of this theorem where the set  need not be a variety, as long as it is of small enough dimension. Alternatively, if  is of finite volume, we can weaken the restrictions on  even further.
- 1
- Klaus Fritzsche, Hans Grauert. From Holomorphic Functions to Complex Manifolds, Springer-Verlag, New York, New York, 2002.
- 2
- Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.
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"Remmert-Stein theorem" is owned by jirka. [ full author list (2) ]
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Cross-references: even, volume, finite, variety, Picard's theorem, limit point, isolated points, zero dimensional, analytic, coordinates, restriction, strictly, irreducible components, regular, analytic variety, closure, point, near, dimension, complex, regular point, complex analytic subvariety
There are 2 references to this entry.
This is version 6 of Remmert-Stein theorem, born on 2005-02-22, modified 2008-02-04.
Object id is 6805, canonical name is RemmertSteinTheorem.
Accessed 1830 times total.
Classification:
| AMS MSC: | 32A60 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Zero sets of holomorphic functions) | | | 32C25 (Several complex variables and analytic spaces :: Analytic spaces :: Analytic subsets and submanifolds) |
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Pending Errata and Addenda
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