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 Karl Stein (1951). 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. 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 Karl Stein (1951). 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. Suppose X {displaystyle X} is a complex manifold of complex dimension n {displaystyle n} and let O ( X ) {displaystyle {mathcal {O}}(X)} denote the ring of holomorphic functions on X . {displaystyle X.} We call X {displaystyle X} a Stein manifold if the following conditions hold: 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 H 1 ( X , O X ∗ ) = 0 {displaystyle H^{1}(X,{mathcal {O}}_{X}^{*})=0} . The exponential sheaf sequence leads to the following exact sequence: Now Cartan's theorem B shows that H 1 ( X , O X ) = H 2 ( X , O X ) = 0 {displaystyle H^{1}(X,{mathcal {O}}_{X})=H^{2}(X,{mathcal {O}}_{X})=0} , therefore H 2 ( X , Z ) = 0 {displaystyle H^{2}(X,mathbb {Z} )=0} . This is related to the solution of the second Cousin problem. 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.