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Stein manifold
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(Definition)
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Stein manifold is a generalization of the concept of the domain of holomorphy to manifolds. Furthermore, Stein manifolds are the generalizations of Riemann surfaces in higher dimensions. Every noncompact Riemann surface is a Stein manifold by a theorem of Behnke and Stein. Note that every domain of holomorphy in ${\mathbb{C}}^n$ is a Stein manifold. It is not hard to see that every closed complex submanifold of a Stein manifold is Stein.
Theorem 1 (Remmert, Narasimhan, Bishop) If $M$ is a Stein manifold of dimension $n$ . There exists a proper holomorphic embedding of $M$ into ${\mathbb{C}}^{2n+1}$ .
Note that no compact complex manifold can be Stein since compact complex manifolds have no holomorphic functions. On the other hand, every compact complex manifold is holomorphically convex.
- 1
- Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"Stein manifold" is owned by jirka.
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Cross-references: compact, embedding, complex submanifold, closed, theorem, Riemann surfaces, manifolds, domain of holomorphy, coordinate system, holomorphic, function, holomorphically convex, properties, dimension, complex, complex manifold
There are 4 references to this entry.
This is version 4 of Stein manifold, born on 2005-02-22, modified 2008-03-31.
Object id is 6799, canonical name is SteinManifold.
Accessed 3220 times total.
Classification:
| AMS MSC: | 32E10 (Several complex variables and analytic spaces :: Holomorphic convexity :: Stein spaces, Stein manifolds) |
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Pending Errata and Addenda
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