Stein manifold について

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Stein manifold
In the theory of several complex variables and complex manifolds in mathematics, 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.
== Definition ==
A complex manifold

X

of complex dimension

n

is called a Stein manifold if the following conditions hold:
*

X

is holomorphically convex, i.e. for every compact subset

K \subset X

, the so-called ''holomorphic convex hull'',
::

\bar K = \,

:is again a ''compact'' subset of

X

. Here

\mathcal O(X)

denotes the ring of holomorphic functions on

X

.
*

X

is holomorphically separable, i.e. if

x \neq y

are two points in

X

, then there is a holomorphic function
::

f \in \mathcal O(X)

:such that

f(x) \neq f(y).

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