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2002 Fiscal Year Final Research Report Summary

A topological and analytical study on three dimensional singular complex projective hypersurfaces

Research Project

Project/Area Number 13640083
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKagoshima University

Principal Investigator

TSUBOI Shoji  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (80027375)

Co-Investigator(Kenkyū-buntansha) OHMOTO Toru  Kagoshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (20264400)
YOKURA Shoji  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (60182680)
MIYAJIMA Kimio  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (40107850)
NAKASHIMA Masaharu  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (40041230)
OBITSU Kunio  Kagoshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (00325763)
Project Period (FY) 2001 – 2002
KeywordsSingular hypersurfaces / Ordinary singularities / Normalizations / Chem numbers / Isolated rational singularities / Mixed Hodge structures / Rational integrals of the 2^<nd> kind / Poincare residue map
Research Abstract

(1) Let Y be an algebraic hypersurface with ordinary singularities, i.e., ordinary n-ple points, ordinary cuspidal points and stationary points, in the 4-dimensional complex projective space, and let X be its normalization. For such Y and X, we have proved numerical formulas which describe the Chem numbers c_1(X)^3, c_1(X)c_2(X), c_3(X) of X in terms of numerical characteristics of Y and its singular locus. As an application, we have derived a numerical formula which gives the Euler-Poincare characteristic of X with coefficients in the sheaf of holomorphic vector fields on X.
(2) We have given an example of hypersurfaces which have ordinary n-ple points (2≦n≦4), ordinary cuspidal points and degenerate ordinary triple points only as singularities, and whose normalizations have isolated rational quadruple points only as singularities. From Schlessinger's criterion, it follows that these isolated singular points are rigid under small deformations.
(3) For a (n+1)-dimensional complex algebraic manifold X, embedded in a projective space, and its non-singular hyperplane section Y which is sufficiently ample, we have proved the following:
(a) F^kH^p(X-Y,C)【similar or equal】I^p _k(X,(p+1)Y), F^kH^p(X,C)_0【similar or equal】I^p _k(X,(p+1)Y)_0 (0≦k≦p, 0≦p≦n+1),
(b) F^kGr^<w[q]> _qH^p(X-Y,C)【similar or equal】I^p _k(X,(p+1)Y)_0, F^kGr^<w[q]> _<q+1>H^p(X-Y,C)【similar or equal】 I^p _k(X,(p+1)Y)/I^p _k(X,(p+1)Y)_0 (0≦k≦p, 0≦p≦n+1),
(c) F^kH^n(Y,C)_0 【similar or equal】Res(I^<n+1> _<k+1>(X,(n+2)Y))【symmetry】r*(I^n _k(X,(n+1)Y')_0,
where H^p(X,C)_0 and H^p(Y,C)_0 denote the p-th cohomology of X and Y, respectively, and I^p _k(X,(p+1)Y) denotes the De Rham cohomology of closed rational differential forms which has poles of order p-k+1 at most along Y, I^p _k(X,(p+1)Y)_0 the subspace of I^p _k(X,(p+1)Y) generated by closed rational differential forms of the second kind, and Y' a sufficiently ample hyperplane section of X, intersecting with X transversely.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] S.Tsuboi: "On certain hypersurfaces with non-isolated singularities in P^4(C)"Pro.Japan Aca.. 79A, No.1. 1-4 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities"Proceedings of the Japan-Korea joint Workshop in Math.2001, Department of Mathematics, Yamaguchi Univ.. 115-124 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Yokura(with L.Ernstrom): "On bivariant Chern-Schwartz-MacPherson classes with values in Chow groups"Selecta Mathematica. Vol.8,No.1. 1-25 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Yokura: "Bivariant theories constructible functions and Grothendieck transformations"Topology and Its Application. 123. 283-296 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Yokura: "Remarks on Ginzburg's Bivariant Chern classes"Proc.Amer.Math.Soc.. 130. 3645-3471 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Obitsu: "The asymptotic behavior of Eisenstein series and a comparison of The Weil-Petersson and the Zograf-Takhtajan metrics"Publ.RIMS.Kyoto Univ.. 37. 459-478 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shoji Tsuboi: "On certain hypersurfaces with non-isolated singularities in P^4c"Proc. Japan Acad.. 79A-1. 1-4 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shoji Yokura, (With L. Emstrom): "On bivariant Chem-Schwartz-MacPherson classes with values in Chow groups"Selecta Mathematica. Vol. 8, No. 1. 1-25 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shoji Yokura: "Bivariant theories of constructible functions and Grothendieck transformations"Topology and its Applications. Vol. 123. 283-296 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shoji Yokura: "Remarks on Ginzburg's bivariant Chem classes"Proc. Amer. Math. Soc.. Vol. 130. 3465-3471 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kunio Obitsu: "The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics"Publ. RIMS. Kyoto Univ.. 37. 459-478 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kimio Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities, in Proceedings of the Japan-Korea Joint Workshop in Mathematics "Complex Analysis and Related Topics""Department of Mathematics, Yamaguchi University. 115-124 (2002)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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