Math 2.0

Several complex variables and analytic spaces Definition

Oka manifold

Builds on Complex manifoldHolomorphic mapCompact setOka–Weil theorem

Definition

Let YY be a complex manifold with a distance function dd that induces its topology. The manifold YY has the convex approximation property if the following holds for every n≥1n \ge 1. Let K⊆ℂnK \subseteq \C^n be a compact convex set, let UU be an open neighbourhood of KK, and let f ⁣:U→Yf \colon U \to Y be a holomorphic map. Then for every ε>0\varepsilon > 0 there is a holomorphic map F ⁣:ℂn→YF \colon \C^n \to Y with d(F(z),f(z))<εd(F(z), f(z)) < \varepsilon for all z∈Kz \in K.

The manifold YY is an Oka manifold if it has the convex approximation property.

Remarks

The property does not depend on the choice of dd, since any two such distances are uniformly equivalent on a compact neighbourhood of the compact set f(K)f(K). Example: Y=ℂmY = \C^m is Oka. A compact convex set is polynomially convex. So by the Oka–Weil theorem each component of ff is a uniform limit on KK of polynomials, and polynomial maps FF approximate ff on KK. Non-example: the unit disc D⊆ℂD \subseteq \C is not Oka. Take n=1n = 1, K={∣z∣≤1/2}K = \set{\abs{z} \le 1/2} and f(z)=zf(z) = z; any holomorphic F ⁣:ℂ→DF \colon \C \to D is bounded, hence constant by Liouville's theorem, and a constant cannot be closer than 1/21/2 to zz at both z=1/2z = 1/2 and z=−1/2z = -1/2. Forstnerič proved that the convex approximation property is equivalent to the full Oka principle for maps from Stein manifolds to YY, with approximation and interpolation; this justifies the definition.

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To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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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 …conceptHolomorphic map, concept, no evidence levelHolomorphic mapconceptPolynomially convex set, concept, no evidence levelPolynomially convex s…conceptPolynomials in several variables, concept, no evidence levelPolynomials in severa…conceptUniform convergence, concept, no evidence levelUniform convergenceconceptOka–Weil theorem, statement, no evidence levelOka–Weil theoremstatementOka manifold, definition, no evidence levelOka manifolddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Compact set concept
  2. Complex manifold concept
  3. Holomorphic function of several variables concept
  4. Holomorphic map concept
  5. Polynomially convex set concept
  6. Polynomials in several variables concept
  7. Uniform convergence concept
  8. Oka–Weil theorem statement
  9. Oka manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stein_manifold
About this page
TagM0SSFESZQY
URIm2://M0SSFESZQY
Kinddefinition
WritingW0 stub
MSC32Q56
DefinesOka manifold
UsesComplex manifold, Holomorphic map, Compact set, Oka–Weil theorem

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