Stein manifold

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holomorphically-complete manifold

A paracompact complex analytic manifold with the following properties:

1) for any compact set the set

where is the algebra of holomorphic functions on , is compact (holomorphic convexity);

2) for any two different points there is a function such that (holomorphic separability);

3) in a neighbourhood of any point there is a holomorphic chart whose coordinate functions belong to .

The requirement of holomorphic convexity can be replaced by the following one: For any sequence without limit points there is a function such that .

The class of Stein manifolds was introduced by K. Stein [1] as a natural generalization of the notion of a domain of holomorphy in . Any closed analytic submanifold in is a Stein manifold; conversely, any -dimensional Stein manifold has a proper holomorphic imbedding in (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.


[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. A.L. Onishchik (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098