Stein manifold
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 formula_1 is a complex manifold of complex dimension formula_2 and let formula_3 denote the ring of holomorphic functions on formula_4 We call formula_1 a Stein manifold if the following conditions hold: 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 formula_15. The exponential sheaf sequence leads to the following exact sequence: Now Cartan's theorem B shows that formula_17, therefore formula_18. This is related to the solution of the second Cousin problem. Properties and examples of Stein manifolds. 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). 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". Relation to smooth manifolds. Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n>2, and when n=2 the same holds provided the 2-handles are attached with certain framings (framing less than the Thurston–Bennequin framing). Every closed smooth  4-manifold is a union of two Stein 4-manifolds glued along their common boundary.