Stein manifold

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.

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