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Concept# Stein manifold

Summary

In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.
Suppose is a complex manifold of complex dimension and let denote the ring of holomorphic functions on We call a Stein manifold if the following conditions hold:
is holomorphically convex, i.e. for every compact subset , the so-called holomorphically convex hull,
is also a compact subset of .
is holomorphically separable, i.e. if are two points in , then there exists such that
Let X be a connected, non-compact Riemann surface. A deep theorem of Heinrich Behnke and Stein (1948) asserts that X is a Stein manifold.
Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so . The exponential sheaf sequence leads to the following exact sequence:
Now Cartan's theorem B shows that , therefore .
This is related to the solution of the second Cousin problem.
The standard complex space is a Stein manifold.
Every domain of holomorphy in is a Stein manifold.
It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.
The embedding theorem for Stein manifolds states the following: Every Stein manifold of complex dimension can be embedded into by a biholomorphic proper map.
These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the ambient space (because the embedding is biholomorphic).
Every Stein manifold of (complex) dimension n has the homotopy type of an n-dimensional CW-complex.
In one complex dimension the Stein condition can be simplified: a connected Riemann surface is a Stein manifold if and only if it is not compact.

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