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. Definition 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 Non-compact Riemann surfaces are Stein manifolds 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. Properties and examples of Stein manifolds 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. This can be proved using a version of the Runge theorem for Riemann surfaces, due to Behnke and Stein. Every Stein manifold is holomorphically spreadable, i.e. for every point , there are holomorphic functions defined on all of which form a local coordinate system when