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Research on moduli of the boundary structure of isolated singularities

Research Project

Project/Area Number 12640080
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

MIYAJIMA Kimio  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (40107850)

Co-Investigator(Kenkyū-buntansha) AIKOU Tadashi  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (00192831)
YOKURA Shoji  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (60182680)
TSUOBI Shoji  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (80027375)
OBITSU Kunio  Kagoshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (00325763)
OHMOTO Toru  Kagoshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (20264400)
黒川 隆英  鹿児島大学, 理学部, 教授 (20124852)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2000: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsCR structure / isolated singularity / moduli space / deformation / hypersurface singularity / cone singularity / quasi-homogeneous singularity / quotient singularity / CR構造 / モジュライ空間
Research Abstract

Under the guiding principle that the stably embeddable deformation of CR structures is the boundary analogue of the deformation of normal isolated singularities, we investigate description of stably embeddable deformation of CR structures on a link of normal isolated singularities. In particular, we concentrated on establishing the method describing it in the following three cases; (i) complete intersection singularities, (ii) quasi-homogeneous singularities and (iii) quotient singularities. The main results are as follows, (i) Complete intersection singularities: Deformation of singularities is controlled by means of CR functions on its link, and we can deduce so-called tha Kas-Schlessinger theorem, (ii) Quasi-homogeneous singularities: A grading induced by the S^1-action associ ating with the quasi-homogenety is introduced in the deformation space. We realized that the grading controlls the grading of deformations of defining equations of the singularities. Furthermore, if the singularities are cone singularities, the grading controlls deformation of resolution of singularities. These graded arguments provide the CR-version of the the orems of H. Pinkham and J. Wahl on deformation of quasi-homogenous singularities, (iii) Higher dimensional quotient singularities: Based on the sphere analysis, our construction of semi-universal family of stably embeddable deformation of CR structures provides the Sch lessingers rigidity theorem, (iv) Cyclic quotient surface singularities: (these singularities are the origin of several interesting geometries, e.g. twistor space, hyper Kaler manifold and quiver): In the case of the degree <__- 4, we obtained the CR description of the semi-universal deformation of the singularity and also description of the simultaneous resolution; in the case of degree >__- 5, we obtained an algorithm constructing the semi-universal deformation.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (69 results)

All Other

All Publications (69 results)

  • [Publications] K.Miyajima: "CR geometry/analysis and deformation of isolated singularities"J. Korean Math. Soc.. 37. 193-223 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 宮嶋公夫: "強凝凸CR多様体と正規孤立特異点の変形"数学. 53. 172-184 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities""Complex Analysis and Related Topics", Proc. of the Japan-Korea Joint Workshop in Mathematics 2001. 115-124 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities"in " Selected Topics in Cauchy-Riemann Geometry", (Ed.by S.Dragomir). (印刷中). 24 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"Sugaku Expositions, A.M.S.. (印刷中). 19 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Tsuboi: "Infinitesimal Parameter Spaces of Locally Trivial Deformations of Compact Complex Surfaces with Ordinary Sungularities"Finite or Infinite Dimensional Complex Analysis (Marcel Dekker, Inc.). 523-532 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Tsuboi, Guillen: "Simultaneous cubic hyper-resolutions of locally trivial analytic families of c Complex projective varieties and cohomological descent"The Rep. Of Fac. of Sci., Kagoshima Univ.. 33. 1-33 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Tsuboi: "A certain degeneratoe ordinary singularity of dimension three"Finite or Infinite Dimensional Complex Analysis, Shandon Sci. and Tech. Press. 223-228 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Tsuboi: "The Euler number of the non-singular normalization of an algebraic threefold with Ordinary singularities"Proc. of the International Symp. on Singularity Theory and its Applications. 113-119 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.-P.Brasselet, S.Yokura: "Remarks on bivariant constructible functions"Advanced Studies in Pure Mathematics. 29. 53-77 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] L.Ernstrom, S.Yokura: "On bivariant Chern-Schwartz-MacPherson classes with values in Chow groups"Selecta Mathematica. 7(印刷中). 25 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Yokura: "Bivariant theories of constructible functions and Grothendieck transformations"Topology and Its Applications. (印刷中). 14 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Yokura: "Verdier-Riemann-Roch for Chern class and Milnor class"Asian J. Mathematics. (印刷中). 23 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Yokura: "Remarks on Ginzburg's bivariant Chern classes"Proc. of A.M.S.. (印刷中). 8 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Aikou: "Some remarks on Finsler vector bundles"Publ. Math. Debrecen. 57. 1-13 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Aikou: "Some remarks on the conformal equivalence of complex Finsler structures""Finslerian Geometries : A Meeting of Minds" (ed ited by P.L.Antonelli). 23 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Aikou: "Applications of Bott connection to Finsler geometry""Steps in Differential Geometry", The Institute of Mathematics, Univ. of Debrecen. 1-13 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Aikou: "Differential geometry of Kaehler fibrations and its application to Finsler geometry"Far East Journal Mathematical Science. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Ohmoto, S.Yokura: "Product formula of the Milnor class"Bull. Polish Academy of Sciences. 48. 388-401 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kurokawa: "On relation between Bessel potential spaces and Riesz potential spaces"Potential Analysis. 12. 299-323 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Obitsu: "The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics"Proceedings of the Second ISSAC Congress. 2. 1037-1048 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [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
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Miyajima,K.: "CR geometry/analysis and deformation of isolated singularities"J. Korean Math. Soc.. 37. 193-223 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Miyajima,K.: "Strongly pseudoconvex CR manifolds and deformations of normal isolated singularities (in Japanese)"Sugaku. 53. 172-184 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tsuboi,S. and Guillen: "Simultaneous cubic hyper-resolutions of locally trivial analytic families of complex projective varieties and cohomological descent"Finite or Infinite Dimensional Complex Analysis )Marcel Dekker, Inc.). 523-532 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tsuboi,S. and Guillen: "Simultaneous cubic hyper-resolutions of locally trivial analytic families of complex projective varieties and cohomological descent"The Rep. of the Fac. of Sci., Kagoshima Univ.. 33. 1-33 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tsuboi,S.: "A certain degenerate ordinary singularity of dimension three"Finite or infinite Dimensional Complex Analysis, Shandon Science and Technology Press. 223-228 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tsuboi,S.: "The Euler number of the non-singular normalization of an algebraic threefold with ordinary singulaities"Proceeings of the International Symposium on Singularity Theory and its Applicationsy. 113-119 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Brasselet,J.-P. and Yokura.S.: "Remarks on bivariant constructible functions"Advanced Studies in Pure Mathematics. 29. 53-77 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yokura,S.: "An Application of bivarriant theory to Milnor classes"Topology and Its Applications. 115. 43-61 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ernstrom,L. and Yokura,S.: "On bivariant Chern-Schwartz-MacPherson classes with values in Chow groups"Selecta Mathematica. 25

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yokura,S.: "Bivariant theories of constructible functions and Grothendieck transformations"Topology and Its Applications. 14

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yokura,S.: "Verdier-Riemann-Roch for Chern class and Milnor class"Asian J. Mathematics. 23

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yokura,S.: "Remarks on Ginzburg's bivariant Chern Classes"Proc. Amer. Math. Soc.. 8

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Aikou,T.: "Some remarks on Finsler vector bundles"Publ. Math. Debrecen. 57. 367-373 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Aikou,T.: "Some remarks on the conformal equivalence of complex Finsler structures"Finslerian Geometries: A Meeting of Minds )ed. P.L.Antonelli). 35-52 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Aikou,T.: "Applications of Bott connection to Finsler geometry"in "Steps in Differential Geometry", The Institute of Mathematics and Informations, University of Debrecen. 1-13 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Aikou,T.: "Differential geometry of Kahler filiations and its application to Finsler geometry"Far East Journal Mathematical Science. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ohmoto,T. and Yokura,S.: "Product formula of the Milnor class"Bull. Polish Academy of Sciences. 48. 388-401 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kurokawa,T.: "On relations between Bessel potential spaces and Riesz potential spaces"Potentical Analysys. 12. 299-323 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Obitsu,K.: "The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the ZografTakhtaj an metrics"Proceedings of the Second ISSAC Congress. Vol. 2. 1037-1048 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Obitsu,K.: "The asymptotic behavior of Eisensteiu series and a comparison of the Weil-Petersson and the Zograf-Takhtaj an metrics"Publ. RIMS. Kyoto Univ.. 37. 459-478 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Miyajima,K.: "Reformation theory of CR structures on a boundary of normal isolated singularities"Comlex Analysis and Related Topics, Proceedings of the Japan-Kerea Joint Workshop in Mathematics 2001, Yamaguchi University. 115-124 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Miyajima,K.: "CR description of the formal deformations of quasi-homogeneous singularities"Selected Topics in Cauchy-Riemann Geometry )Ed. by S. Dragomir), Quaderni di Matematica, Series ed. by Dipartimento di Matematica, Seconda Universita di Napoli, Caserta. 24

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Miyajima,K.: "Strongly pseudoconvex CR manifolds and deformations of normal isolated singularities"Sugaku Exposition, A.M.S. )(in press)). 19

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 宮嶋公夫: "強擬凸CR多様体と正規孤立特異点の変形"数学. 53(2). 172-184 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities""Complex Analysis and Related Topics", Proc. of the Japan-Korea Joint Workshop in Mathematics 2001. 115-124 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities"in "Selected Topics in Cauchy-Riemann Geometry", (Ed. by S.Dragomir). (印刷中). 24 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"Sugaku Expositions, A.M.S.. (印刷中). 19 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Tsuboi: "A Certain Degenerate Ordinary Singularity of Dimension Three"Finite or infinite Dimensinal Complex Analysis, Shandon Science and Technology Press. 223-228 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Tsuboi: "The Euler number of the non-singular normalization of an algebraic threefold with ordinary singulaities"Proceeings of the International Symposium on Singularity Theory and its Applications. 113-119 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Yokura: "An application of bivariant theory to Milnor classes"Topology and Its Applications. 115. 43-61 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] L.Ernstrom, S.Yokora: "On bivariant Chem-Schwartz-MacPherson classes with values in Chow groups"Selecta Mathematica. 7(印刷中). 25 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Yokura: "Bivariant theories of constructible functions and Grothendieck transformations"Topology and Its Applications. (印刷中). 14 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Yokura: "Verdier-Riemann-Roch for Chem class and Milnor class"Asian J. Mathematics. (印刷中). 23 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Yokura: "Remarks on Ginzburg's bivariant Chern classes"Proc. of A.M.S.. (印刷中). 8 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Aikou: "Applications of Bott connection to Finsler geometry""Steps in Differential Geometry", The Institute of Mathematics, Univ. of Debrecen. 1-13 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Obitsu: "Differential geometry of Kaehler fibrations and its application to Finsler geometry"Far East Journal Mathematical Science. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [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. 49-478 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Miyajima: "CR geometry/analysis and deformation of isolated singularities"J.Korean Math.Soc.. 37(2). 193-223 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Tsuboi: "Infinitesimal Parameter Spaces of Locally Trivial Deformations of Compact Complex Surfaces with Ordinary Sungularities"Finite or Infinite Dim.Complex Anal. (Marcel Dekker, Inc.). 523-532 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Tsuboi and Guillen: "Simultaneous Cubic Hyper-resolutins of Locally Trivial Analytic Families of Complex Projective Varieties and Cohomological Descent"The Rep.of the Fac.of Sci., Kagoshima University. 33. 1-33 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Tsuboi: "A Certain Degenerate Ordinary Singularity of Dimension Three (印刷中)"Proc.of the 8th International Conference on Finite or Infinite Dimensional Complex Analysis, Shandon University. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Ohmoto and S.Yokura: "Product formula of the Milnor class"Bull.Polish Academy of Sciences. 48(4). 388-401 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Aikou: "Some remarks on Finsler vector bundles"Publ.Math.Debrecen. 57/2-4. 367-373 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Aikou: "Some remarks on the conformal equivalence of complex Finsler structures"Finslerian Geometries : A Metting of Minds (edited by P.L.Antonelli),. 35-52 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kurokawa: "On relations between Bessel potential spaces and Riesz potential spaces"Potential Analysis. 12. 299-323 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] J.-P.Brasselet and S.Yokura: "Remarks on bivariant constructible functions"Advanced Studies in Pure Mathematics. 29. 53-77 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Yokura: "An application of bivariant theory to Milnor classes (印刷中)"Topology and Its Applications. (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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