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Several complex variables and analytic spaces Definition

Stein manifold

Builds on Complex manifoldHolomorphic function of several variablesCompact setHolomorphically convex hullSecond-countable space

Definition

Let XX be a complex manifold of complex dimension nn that is second countable, and let 𝒪(X)\mathcal{O}(X) be the algebra of holomorphic functions on XX. For a compact set K⊆XK \subseteq X, its holomorphically convex hull is K^={x∈X:∣f(x)∣≤sup⁡K∣f∣ for all f∈𝒪(X)}.\widehat{K} = \set{x \in X : \abs{f(x)} \le \sup_{K} \abs{f} \text{ for all } f \in \mathcal{O}(X)}. The manifold XX is a Stein manifold if:

  1. XX is holomorphically convex: K^\widehat{K} is compact for every compact K⊆XK \subseteq X.
  2. XX is holomorphically separable: for all x≠yx \ne y in XX there is f∈𝒪(X)f \in \mathcal{O}(X) with f(x)≠f(y)f(x) \ne f(y).

Remarks

Some authors add a third condition: at every point xx there are f1,…,fn∈𝒪(X)f_1, \dots, f_n \in \mathcal{O}(X) that form local coordinates near xx. This condition follows from the first two. Example: X=ℂnX = \C^n is Stein. The coordinate functions separate points. The hull K^\widehat{K} is closed, and ∣zj∣≤max⁡K∣zj∣\abs{z_j} \le \max_K \abs{z_j} on K^\widehat{K} for each jj. So K^\widehat{K} lies in a compact polydisc, and it is compact. Non-example: a compact connected complex manifold XX of positive dimension is not Stein. Every f∈𝒪(X)f \in \mathcal{O}(X) is constant, by the maximum principle, so clause 2 fails. A domain in ℂn\C^n is a Stein manifold if and only if it is a domain of holomorphy.

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Compact set, concept, no evidence levelCompact setconceptComplex manifold, concept, no evidence levelComplex manifoldconceptHolomorphic function of several variables, concept, no evidence levelHolomorphic function …conceptHolomorphically convex hull, concept, no evidence levelHolomorphically conve…conceptSecond-countable space, concept, no evidence levelSecond-countable spaceconceptStein manifold, definition, no evidence levelStein manifolddefinition
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  1. Compact set concept
  2. Complex manifold concept
  3. Holomorphic function of several variables concept
  4. Holomorphically convex hull concept
  5. Second-countable space concept
  6. Stein manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stein_manifold
About this page
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Kinddefinition
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MSC32Q28, 32E10
DefinesStein manifold
UsesComplex manifold, Holomorphic function of several variables, Compact set, Holomorphically convex hull, Second-countable space

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