Math 2.0

Several complex variables and analytic spaces Analytic spaces Theorem

Hironaka's resolution for complex spaces

Builds on Reduced complex spaceModification of a complex spaceComplex manifoldRegular and singular points of an analytic setProper mapNowhere dense setResolution of singularitiesBlowing upOka's normalization theorem

Statement

Let XX be a reduced complex space whose topology has a countable basis. Let XregX_{\mathrm{reg}} and XsingX_{\mathrm{sing}} be its sets of regular and singular points. Then there are a complex manifold X~\tilde{X} and a proper surjective holomorphic map π ⁣:X~→X\pi \colon \tilde{X} \to X such that:

  1. π\pi restricts to a biholomorphic map π−1(Xreg)→Xreg\pi^{-1}(X_{\mathrm{reg}}) \to X_{\mathrm{reg}};
  2. π−1(Xsing)\pi^{-1}(X_{\mathrm{sing}}) is nowhere dense in X~\tilde{X}.

In particular π\pi is a modification of XX, called a resolution of singularities.

Remarks

Hironaka proved resolution of singularities for algebraic varieties over fields of characteristic 00; the method was then carried over to complex spaces. The resolution is built from a sequence of blow-ups along smooth centres contained in the singular locus, chosen by a local invariant that drops at each step. For a non-compact space infinitely many blow-ups may be needed, but they form a locally finite family, and the canonical choice of centres lets local resolutions glue. The map can also be chosen so that π−1(Xsing)\pi^{-1}(X_{\mathrm{sing}}) is a hypersurface with normal crossings.

Example: for a reduced curve, the normalization is a resolution; for the cusp {z12=z23}\set{z_1^2 = z_2^3} it is t↦(t3,t2)t \mapsto (t^3, t^2) from ℂ\C. For the cone {z1z2=z32}\set{z_1 z_2 = z_3^2} in ℂ3\C^3, one blow-up of the vertex gives a resolution whose exceptional set is a projective line.

Going deeper

For a student

Resolution is used to define invariants of singularities and to prove local estimates such as the Łojasiewicz inequality. It also extends integration and Hodge theory to singular spaces, and reduces questions on plurisubharmonic functions and multiplier ideals to normal crossing situations. The resolution is not unique: blowing up a point of X~\tilde{X} over XsingX_{\mathrm{sing}} gives another one.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

Show the path as a graph
Algebraic variety, concept, no evidence levelAlgebraic varietyconceptBirational map, concept, no evidence levelBirational mapconceptBlowing up, concept, no evidence levelBlowing upconceptCoherent analytic sheaf, concept, no evidence levelCoherent analytic she…conceptComplex analytic variety, concept, no evidence levelComplex analytic vari…conceptComplex manifold, concept, no evidence levelComplex manifoldconceptModification of a complex space, concept, no evidence levelModification of a com…conceptNormal complex space, concept, no evidence levelNormal complex spaceconceptNormalization, concept, no evidence levelNormalizationconceptNowhere dense set, concept, no evidence levelNowhere dense setconceptProper map, concept, no evidence levelProper mapconceptProper morphism, concept, no evidence levelProper morphismconceptReduced complex space, concept, no evidence levelReduced complex spaceconceptRegular and singular points of an analytic set, concept, no evidence levelRegular and singular …conceptSingular point of an algebraic variety, concept, no evidence levelSingular point of an …conceptResolution of singularities, statement, no evidence levelResolution of singula…statementOka's normalization theorem, statement, no evidence levelOka's normalization t…statementHironaka's resolution for complex spaces, statement, no evidence levelHironaka's resolution…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.
  • no evidence level
  1. Algebraic variety concept
  2. Birational map concept
  3. Blowing up concept
  4. Coherent analytic sheaf concept
  5. Complex analytic variety concept
  6. Complex manifold concept
  7. Modification of a complex space concept
  8. Normal complex space concept
  9. Normalization concept
  10. Nowhere dense set concept
  11. Proper map concept
  12. Proper morphism concept
  13. Reduced complex space concept
  14. Regular and singular points of an analytic set concept
  15. Singular point of an algebraic variety concept
  16. Resolution of singularities statement
  17. Oka's normalization theorem statement
  18. Hironaka's resolution for complex spaces statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Resolution_of_singularities
About this page
TagGWJWP187QE
URIm2://GWJWP187QE
Kindstatement, theorem
WritingW0 stub
MSC32S45, 14E15
StatusThis copy of the site has no build data. Run m2 status GWJWP187QE.
AboutModification of a complex space
UsesReduced complex space, Modification of a complex space, Complex manifold, Regular and singular points of an analytic set, Proper map, Nowhere dense set, Resolution of singularities, Blowing up, Oka's normalization theorem

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