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2000 Fiscal Year Final Research Report Summary

Logarithmic deformations of complex projective hypersurfaces with ordinary singularities and their period maps

Research Project

Project/Area Number 11640086
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 Unvi., Faculty of Science, Professor, 理学部, 教授 (80027375)

Co-Investigator(Kenkyū-buntansha) OHMOTO Toru  Kagoshima Univ., Faculty of Sciences, Associate Professor, 理学部, 助教授 (20264400)
YOKURA Shoji  Kagoshima Univ., Faculty of Sciences, Professor, 理学部, 教授 (60182680)
MIYAJIMA Kimio  Kagoshima Univ., Faculty of Sciences, Professor, 理学部, 教授 (40107850)
NAKASHIMA Masaharu  Kagoshima Univ., Faculty of Sciences, Professor, 理学部, 教授 (40041230)
AIKOU Tadashi  Kagoshima Univ., Faculty of Sciences, Associate Professor, 理学部, 助教授 (00192831)
Project Period (FY) 1999 – 2000
KeywordsOrdinary singularity / Normal crossing variety / Logarithmic deformation / Infinitesimal mixed Torelli problem / Kodaira-Spencer map / Chech de Rham cohomology / Cubic hyper-resolution / Rigid singularity
Research Abstract

1. We have formulated the infinitesimal mixed Torelli problem for a locally trivial analytic family of complex projective surfaces with ordinary singularities, parametrized by a manifold, relativizing the notion of cubic hyper-resolution due to V.Navarro Aznar, F.Guillen et al., and have gave cohomological sufficient conditions for this problem to be affirmatively solved. Furthermore we have constructed a few examples for which these sufficient conditions are satisfied.
2. We have also considered the infinitesimal mixed Torelli problem for complex projective threefolds of so-called type (n, r_1, r_2, r_3, r_4). In this procedure we have found a certain weakly normal, non-isolated singularity which is a degenerate one of an ordinary triple point, and is described as (xy)^2+(yz)^2+(zx)^2+wxyz=0 by use of affine coordinates. It has turned out that singularity is a cone over the Steiner surface which is a rational surface with ordinary singularities in P^3 (C). The normalization of it is a cone over P^2 (C) embedded in P^5 (C) by the Veronese map of degree 2, a rational isolated singularity of multiplicity 4, and is rigid under deformation.
3. Besides the above results, we have obtained a formula which gives the Euler number of the non-singular normalization of a complex hypersurface with ordinary singularities in P^4 (C), generalizing the classical one for a complex hypersurface with ordinary singularities in P^3 (C) due to Enriques. This work is in preparation to be published.

  • Research Products

    (25 results)

All Other

All Publications (25 results)

  • [Publications] S.Tsuboi: "Infinitesdimal locally trivial deformation spaces of algebraic surfaces with ordinary singularities"Proc.Japan Acad.. 75A,No.7. 99-102 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Tsuboi: "Infinitesimal Parameter Spaces of Locally Trivial Deformations of Compact Complex Surfaces with Ordinary Singularities"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
  • [Publications] S.Tsuboi and F.Guillen: "Simultaneous Cubic Hyper-resolutins of Locally Trivial Analytic Families of Complex Projective Varieties and Cohomological Descent"The Reports of the Faculty of Science Kagoshima University. 33. 1-33 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Tsuboi: "A Certain Degenerate Ordinary Singularity of Dimension Three"Proceedings of the Eighth International Conference on Finite or Infinite Dimensional Complex Analysis. (in press). 6 printed pages

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Miyajima: "CR construction of the flat deformations of normal isolated singularities"J.Alg.Geom.. 8(3). 315-327 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Ohmoto and S.Yokura: "Product formula of the Milnor class"Bull.Polish Academy of Sciences. 48(4). 388-401 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Tsuboi: "Infinitesimal locally trivial deformation spaces of compact complex surfaces with ordinary singularities."Proc.Japan Acad.. 75A, Ser.A, No.7. 99-102 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Tsuboi: "Infinitesimal mixed Torelli problem for algebraic surfaces with ordinary singularities"Proceedings of the symposium on Hodge theory and algebraic geometry, supported by Grant-in-Aid for Sciencific Research (A)(1). No.11304001. 86-96 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [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(欧文)」より
  • [Publications] S.Tsuboi and F.Guillen: "Simultaneous Cubic Hyper-resolutins of Locally Trivial Analytic Families of Complex Projective Varieties and Cohomological Descent."The Reports of the Faculty of Science, Kagoshima University.. No.33. 1-33 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Tsuboi: "A Certain Degenerate Ordinary Singularity of Dimension Three"Proceedings of the Eighth International Conference on Finite or Infinite Dimensional Complex Analysis, Shandon University. (in press). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Miyajima: "CR construction of the flat deformations of normal isolated singularities"J.Alg.Geom.. 8(3). 403-470 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Miyajima: "A note on the closed rengeness of vector bundle-valued tangential Cauchy-Riemann complex, Analysis and Geometry in Several Complex Variables"Proceedings of the 40th Taniguchi Symposium, ed. G.Komatsu and M.Kuranishi, Trends in Mathematics, Birkhauser. 193-208 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Miyajima: "CR geometry/analysis and deformation of isolated singularities"J.Korean Math.Soc.. 37(2). 193-223 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Yokura: "On a Verdier-type Riemann-Roch for Chern-Shwartz-MacPherson class"Topology and Its Application. 94. 315-327 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Yokura: "On characteristic classes of complete intersections, Algebraic Geometry-Hirzebruch 70"Contemporary Mathemthematics. 241. 349-369 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Aikou: "Conformal flatness of complex Finsler structures"Publ.Math.Debrecen. 54/1-2. 165-179 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Aikou: "Some remarks on Finsler vector bundles"Publ.Math.Debrecen. 57/2-4. 367-373 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ailou: "Some remarks on the conformal equivalence of complex Finsler structure s"in Finslerian Geometries : A Metting of Minds (edited by P.L.Antonelli), Kluwer Academic Publishers. 35-52 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ohmoto: "On Characteristic Classes of Fibers/Images of Mappings, "Singularity theory and its application", "Kokyuroku""RIMS, Kyoto University. 1122. 134-139 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ohmoto, and Shoji Yokura: "Product formula of the Milnor class"Bull.Polish Academy of Sciences. vol.48, No.4. 388-401 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Nakashima: "Explicit A-stable Runge-Kutta methods for Linear Stiff-Equations, 3rd IMACS/IEEE International Multiconference on"Circuits, Systems Communications and Computers.(Military Institution of Univ, Piraeus, GREECE, 4-8 July 1999), Abstracts. 634-638 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Nakashima: "Explicit A-stable Runge-Kutta methods for Linear Stiff-Equations, 3rd IMACS/IEEE International Multiconference on"Circuits, Systems, Communications and Computers.(Military Institution of Univ, Piraeus, GREECE, 4-8 July 1999). Recent Advances in Information Science and Technology (edited by P.DN, E.Mastorakis). 231-236 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Nakashima: "Explicit A-stable Rational Runge-Kutta methods for Parabolic Differential equations, Pararllel and Distributed Processing Techniques and Applications International Conference, (Las Vegas, Nevada, USA June 26-29, 2000)"Proceedings of the International Conference on Pararllel and Distributed Processing Techniques. 2847-2853 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Nakashima: "Explicit A-stable Rational Runge-Kutta methods for parabolic differential equations (II), International Symposium on Applied Mathematics (Dalian, China Aug 14, 2000-Aug 18, 2000)"Proceedings of International Symposium on Applied Mathematics. 31-31 (2000)

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      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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