In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a closed, complex submanifold of the vector space of n complex dimensions. More intrinsically it can be defined as a complex manifold admitting a proper holomorphic embedding into C n {\displaystyle \mathbb {C} ^{n}} for some n {\displaystyle n} . 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.
Definition 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 two conditions hold:
X {\displaystyle X} is holomorphically convex, i.e. for every compact subset K ⊂ X {\displaystyle K\subset X} , the so-called holomorphic hull (or envelope of holomorphy) of K {\displaystyle K} , K ^ := { z ∈ X | | f ( z ) | ≤ sup w ∈ K | f ( w ) | ∀ f ∈ O ( X ) } , {\displaystyle {\widehat {K}}:=\left\{z\in X\,\left|\,|f(z)|\leq \sup _{w\in K}|f(w)|\ \forall f\in {\mathcal {O}}(X)\right.\right\},}
is also a compact subset of X {\displaystyle X} .
X {\displaystyle X} is holomorphically separable, i.e. if x ≠ y {\displaystyle x\neq y} are two distinct points in X {\displaystyle X} , then there exists f ∈ O ( X ) {\displaystyle f\in {\mathcal {O}}(X)} such that f ( x ) ≠ f ( y ) . {\displaystyle f(x)\neq f(y).}
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, and in particular every holomorphic line bundle, on this Riemann surface X is trivial. In particular, every line bundle is trivial. This is related to the solution of the second Cousin problem.
Properties and examples of Stein manifolds The standard complex space C n {\displaystyle \mathbb {C} ^{n}} is a Stein manifold. Every domain of holomorphy in C n {\displaystyle \mathbb {C} ^{n}} is a Stein manifold. Every Fatou–Bieberbach domain in C n {\displaystyle \mathbb {C} ^{n}} is a Stein manifold. 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 X {\displaystyle X} of complex dimension n {\displaystyle n} can be embedded into C 2 n + 1 {\displaystyle \mathbb {C} ^{2n+1}} 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).
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