# Difference between revisions of "Stein manifold"

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2) for any two different points $ x, y \in M $ | 2) for any two different points $ x, y \in M $ | ||

there is a function $ f \in {\mathcal O} ( M) $ | there is a function $ f \in {\mathcal O} ( M) $ | ||

− | such that $ f( x) \neq f( y) $( | + | such that $ f( x) \neq f( y) $ (holomorphic separability); |

− | holomorphic separability); | ||

3) in a neighbourhood of any point there is a holomorphic chart whose coordinate functions belong to $ {\mathcal O} ( M) $. | 3) in a neighbourhood of any point there is a holomorphic chart whose coordinate functions belong to $ {\mathcal O} ( M) $. | ||

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The class of Stein manifolds was introduced by K. Stein [[#References|[1]]] as a natural generalization of the notion of a [[Domain of holomorphy|domain of holomorphy]] in $ \mathbf C ^ {n} $. | The class of Stein manifolds was introduced by K. Stein [[#References|[1]]] as a natural generalization of the notion of a [[Domain of holomorphy|domain of holomorphy]] in $ \mathbf C ^ {n} $. | ||

Any closed analytic submanifold in $ \mathbf C ^ {n} $ | Any closed analytic submanifold in $ \mathbf C ^ {n} $ | ||

− | is a Stein manifold; conversely, any $ n $- | + | is a Stein manifold; conversely, any $ n $-dimensional Stein manifold has a proper holomorphic imbedding in $ \mathbf C ^ {2n} $ (cf. [[Proper morphism|Proper morphism]]). Any non-compact Riemann surface is a Stein manifold. The direct generalization of a Stein manifold is a [[Stein space|Stein space]]. |

− | dimensional Stein manifold has a proper holomorphic imbedding in $ \mathbf C ^ {2n} $( | ||

− | cf. [[Proper morphism|Proper morphism]]). Any non-compact Riemann surface is a Stein manifold. The direct generalization of a Stein manifold is a [[Stein space|Stein space]]. | ||

See also the references to [[Stein space|Stein space]]. | See also the references to [[Stein space|Stein space]]. |

## Latest revision as of 01:32, 19 January 2022

*holomorphically-complete manifold*

A paracompact complex analytic manifold $ M $ with the following properties:

1) for any compact set $ K \subset M $ the set

$$ \left \{ { x \in X } : {| f( x) | \leq \sup _ {z \in K } | f( z) | ( f \in {\mathcal O} ( M)) } \right \} , $$

where $ {\mathcal O} ( M) $ is the algebra of holomorphic functions on $ M $, is compact (holomorphic convexity);

2) for any two different points $ x, y \in M $ there is a function $ f \in {\mathcal O} ( M) $ such that $ f( x) \neq f( y) $ (holomorphic separability);

3) in a neighbourhood of any point there is a holomorphic chart whose coordinate functions belong to $ {\mathcal O} ( M) $.

The requirement of holomorphic convexity can be replaced by the following one: For any sequence $ \{ {x _ {n} } : {n = 1, 2 , . . . } \} \subset M $ without limit points there is a function $ f \in {\mathcal O} ( M) $ such that $ \sup _ {n} | f( x _ {n} ) | = \infty $.

The class of Stein manifolds was introduced by K. Stein [1] as a natural generalization of the notion of a domain of holomorphy in $ \mathbf C ^ {n} $. Any closed analytic submanifold in $ \mathbf C ^ {n} $ is a Stein manifold; conversely, any $ n $-dimensional Stein manifold has a proper holomorphic imbedding in $ \mathbf C ^ {2n} $ (cf. Proper morphism). Any non-compact Riemann surface is a Stein manifold. The direct generalization of a Stein manifold is a Stein space.

See also the references to Stein space.

#### References

[1] | K. Stein, "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem" Math. Ann. , 123 (1951) pp. 201–222 |

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Stein manifold.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Stein_manifold&oldid=51897