Math 2.0

Several complex variables and analytic spaces Definition

Stein space

Builds on Complex analytic spaceHolomorphically convex hullCompact setSecond-countable spaceHausdorff space

Definition

Let (X,π’ͺX)(X, \mathcal{O}_X) be a complex analytic space whose underlying topological space is Hausdorff and second countable. Let π’ͺ(X)\mathcal{O}(X) be the algebra of global holomorphic functions on XX, and let f(x)βˆˆβ„‚f(x) \in \C denote the value of f∈π’ͺ(X)f \in \mathcal{O}(X) at x∈Xx \in X. For a compact set KβŠ†XK \subseteq X, its holomorphically convex hull is K^={x∈X:∣f(x)βˆ£β‰€sup⁑K∣f∣ forΒ allΒ f∈π’ͺ(X)}\widehat{K} = \set{x \in X : \abs{f(x)} \le \sup_K \abs{f} \text{ for all } f \in \mathcal{O}(X)}. The space XX is a Stein space if:

  1. XX is holomorphically convex: K^\widehat{K} is compact for every compact KβŠ†XK \subseteq X.
  2. XX is holomorphically separable: for all xβ‰ yx \ne y in XX there is f∈π’ͺ(X)f \in \mathcal{O}(X) with f(x)β‰ f(y)f(x) \ne f(y).

Remarks

A Stein space whose underlying space is a complex manifold is a Stein manifold. Example: every closed complex subspace XX of β„‚N\C^N is Stein. The coordinate functions restricted to XX separate points, and K^\widehat{K} is a closed subset of XX contained in a closed ball of β„‚N\C^N, hence compact. Non-example: the complex projective line β„™1\mathbb{P}^1 is a complex space whose global holomorphic functions are constant, so clause 2 fails. An equivalent form: XX is Stein if and only if Hq(X,β„±)=0H^q(X, \mathcal{F}) = 0 for every coherent analytic sheaf β„±\mathcal{F} on XX and every qβ‰₯1q \ge 1. The direction from Stein to vanishing is Cartan's theorem B.

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Compact set, concept, no evidence levelCompact setconceptComplex analytic space, concept, no evidence levelComplex analytic spaceconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptHolomorphically convex hull, concept, no evidence levelHolomorphically conve…conceptSecond-countable space, concept, no evidence levelSecond-countable spaceconceptStein space, definition, no evidence levelStein spacedefinition
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  1. Compact set concept
  2. Complex analytic space concept
  3. Hausdorff space concept
  4. Holomorphically convex hull concept
  5. Second-countable space concept
  6. Stein space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stein_manifold
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Kinddefinition
WritingW0 stub
MSC32E10, 32C15
DefinesStein space
UsesComplex analytic space, Holomorphically convex hull, Compact set, Second-countable space, Hausdorff space

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