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.