stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein. This characterization is not, however, the original definition: it follows as a consequence of the embedding theorem for Stein manifolds. It does show that for the case of dimension 1, a connected Riemann surface is a Stein manifold if and only if it is not compact. Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" holomorphic functions taking values in the complex numbers. See for example Cartan's theorems A and B, relating to sheaf cohomology. The initial impetus was to have a description of the properties of the domain of definition of the (maximal) analytic continuation of an analytic function. In the GAGA set of analogies, Stein manifolds correspond to affine varieties. Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory".

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