Math 2.0

Several complex variables and analytic spaces Analytic spaces Theorem

Oka's normalization theorem

Builds on Reduced complex spaceNormal complex spaceNormalizationCoherent analytic sheafNowhere dense setComplex analytic variety

Statement

Let XX be a reduced complex space and let N⊆XN \subseteq X be the set of points of XX that are not normal. Then NN is a nowhere dense analytic subset of XX, and there are a normal complex space X~\tilde{X} and a holomorphic map ν ⁣:X~→X\nu \colon \tilde{X} \to X such that:

  1. ν\nu is finite, that is, closed with finite fibres, and surjective;
  2. ν−1(N)\nu^{-1}(N) is nowhere dense in X~\tilde{X};
  3. ν\nu restricts to a biholomorphic map X~∖ν−1(N)→X∖N\tilde{X} \setminus \nu^{-1}(N) \to X \setminus N.

The pair (X~,ν)(\tilde{X}, \nu) is unique up to a unique isomorphism over XX; it is the normalization of XX.

Remarks

The key step, due to Oka, is that the sheaf 𝒪^X\hat{\mathcal{O}}_X of weakly holomorphic functions is a coherent analytic sheaf of 𝒪X\mathcal{O}_X-modules. A weakly holomorphic function is one that is holomorphic on the regular points and locally bounded. Then X~\tilde{X} is the analytic spectrum of the finite 𝒪X\mathcal{O}_X-algebra 𝒪^X\hat{\mathcal{O}}_X.

Example: for the cusp X={z12=z23}⊆ℂ2X = \set{z_1^2 = z_2^3} \subseteq \C^2, the map ν ⁣:ℂ→X\nu \colon \C \to X, t↦(t3,t2)t \mapsto (t^3, t^2), is the normalization. It is bijective and finite, N={0}N = \set{0}, and off t=0t = 0 its inverse (z1,z2)↦z1/z2(z_1, z_2) \mapsto z_1 / z_2 is holomorphic. For the cross X={z1z2=0}X = \set{z_1 z_2 = 0}, the normalization is the disjoint union of two copies of ℂ\C, mapped onto the two axes; here ν\nu is not injective over 00.

Going deeper

For a student

Normalization reduces many questions about reduced spaces to normal spaces, where the Riemann extension theorem and local irreducibility are available. For curves the normalization is smooth, so it is a resolution of singularities. It is the analytic counterpart of the normalization of an integral scheme by integral closure of its rings.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Coherent analytic sheaf, concept, no evidence levelCoherent analytic she…conceptComplex analytic variety, concept, no evidence levelComplex analytic vari…conceptNormal complex space, concept, no evidence levelNormal complex spaceconceptNormalization, concept, no evidence levelNormalizationconceptNowhere dense set, concept, no evidence levelNowhere dense setconceptReduced complex space, concept, no evidence levelReduced complex spaceconceptOka's normalization theorem, statement, no evidence levelOka's normalization t…statement
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. Coherent analytic sheaf concept
  2. Complex analytic variety concept
  3. Normal complex space concept
  4. Normalization concept
  5. Nowhere dense set concept
  6. Reduced complex space concept
  7. Oka's normalization theorem statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_scheme
About this page
TagBP0ZYHGD0S
URIm2://BP0ZYHGD0S
Kindstatement, theorem
WritingW0 stub
MSC32C20, 32C35
StatusThis copy of the site has no build data. Run m2 status BP0ZYHGD0S.
AboutNormal complex space, Normalization
UsesReduced complex space, Normal complex space, Normalization, Coherent analytic sheaf, Nowhere dense set, Complex analytic variety

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