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''holomorphically-complete manifold''
 
''holomorphically-complete manifold''
  
A paracompact complex [[Analytic manifold|analytic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876301.png" /> with the following properties:
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A paracompact complex [[Analytic manifold|analytic manifold]] $  M $
 +
with the following properties:
  
1) for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876302.png" /> the set
+
1) for any compact set $  K \subset  M $
 +
the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876303.png" /></td> </tr></table>
+
$$
 +
\left \{ {
 +
x \in X } : {| f( x) | \leq  \sup _ {z \in K }  | f( z) |  ( f \in {\mathcal O} ( M)) } \right \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876304.png" /> is the algebra of holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876305.png" />, is compact (holomorphic convexity);
+
where $  {\mathcal O} ( M) $
 +
is the algebra of holomorphic functions on $  M $,  
 +
is compact (holomorphic convexity);
  
2) for any two different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876306.png" /> there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876307.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876308.png" /> (holomorphic separability);
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s0876309.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s08763010.png" /> without limit points there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s08763011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s08763012.png" />.
+
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 [[#References|[1]]] as a natural generalization of the notion of a [[Domain of holomorphy|domain of holomorphy]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s08763013.png" />. Any closed analytic submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s08763014.png" /> is a Stein manifold; conversely, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s08763015.png" />-dimensional Stein manifold has a proper holomorphic imbedding in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087630/s08763016.png" /> (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]].
+
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} $
 +
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]].
  
 
See also the references to [[Stein space|Stein space]].
 
See also the references to [[Stein space|Stein space]].

Revision as of 08:23, 6 June 2020


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
How to Cite This Entry:
Stein manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stein_manifold&oldid=18400
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article