# Stein manifold

*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.

#### 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. A.L. Onishchik (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Stein_manifold&oldid=18400