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Residues on Singular Varieties

Research Project

Project/Area Number 18340015
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

SUWA Tatsuo  Niigata University, Institute of Science and Technology, Professor (40109418)

Co-Investigator(Kenkyū-buntansha) OHMOTO Toru  Hokkaido University, Graduate School of Science, Assoc. Professor (20264400)
OKA Mutsuo  Tokyo University of Science, Faculty of Science, Professor (40011697)
SAITO Kyoji  Kyoto University, RIMS, Professor (20012445)
TAJIMA Shinichi  Niigata University, Institute of Science and Technology, Professor (70155076)
YOKURA Shoji  Kagoshima University, Faculty of Science, Professor (60182680)
Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥6,120,000 (Direct Cost: ¥5,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2007: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2006: ¥3,000,000 (Direct Cost: ¥3,000,000)
Keywordssingular varieties / localization of characteristic classes / residues / Chern classes / Atiyah classes / holomorphic foliations / complex dynamical systems / intersection theory / 局所Euler障害
Research Abstract

The head investigator and the others did research on residues on singular varieties and related subjects. More specifically:
1. We developed a theory of localization of Chern classes by frames of vector bundles on singular varieties. In the previous year, we gave explicit expressions (analytic, algebraic and topological)of the residues at an isolated singularity. In this research, we gave an expression in the case the singularity is not isolated.
2. As an application of 1 above, we developed an analytic intersection theory on singular varieties. This clarifies global intersections, local intersections and the relation between the two. In the global case, the localization theory of Chern classes is very effective and in the local case, Grothendieck residues on singular varieties play an essential role. The two situations are related by the residue theorem.
3. As a summary of collaboration with J.-P. Brasselet and J. Seade, we almost finished writing a book on the characteristic classes of singular varieties utilizing indices and residues of vector fields. This also includes a new simple proof of the Proportionality Theorem, which describes a fundamental property of the local Euler obstruction of singular varieties.
4. In the collaboration with M. Abate, F. Bracci and F. Tovena, we started to construct a localization theory of Atiyah classes of holomorphic vector bundles. This theory is expected to be very interesting and have many applications.
5. Besides the above, Ohmoto obtained important results on the characteristic classes of varieties with group actions, Oka on the fundamental group of the complement of algebraic curves, Saito on Lie algebras and singularities, Tajima on Milnor and Tjurina numbers, Yokura on motivic characteristic classes, respectively.

Report

(3 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • Research Products

    (20 results)

All 2008 2007 2006 Other

All Journal Article (15 results) (of which Peer Reviewed: 6 results) Presentation (3 results) Book (2 results)

  • [Journal Article] Residue Theoretical Approach to Intersection Theory2008

    • Author(s)
      T. Suwa
    • Journal Title

      Contemporary Math., Amer. Math. Soc. (in press)

      Pages: 55-55

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On the uniqueness of bivariant Chern class and bivariant Riemann-Roch transformations2007

    • Author(s)
      S. Yokura, J.-P. Brasselet and J. Schurmann
    • Journal Title

      Advances in Math. 210

      Pages: 797-812

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On the uniqueness of bivariant Chern class and bivariant Riemann-Roch transformations2007

    • Author(s)
      S, Yokura, J.-P, Brasselet, J, Schurmann
    • Journal Title

      Advances in Math 210

      Pages: 797-812

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the uniqueness of bivariant Chern class and bivariant Riemann-Roch transformations2007

    • Author(s)
      S. Yokura, J. -P. Brasselet and J. Schurmann
    • Journal Title

      Advances in Math. 210

      Pages: 797-812

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Introduction to Complex Analytic Geometry2007

    • Author(s)
      T.Suwa
    • Journal Title

      Proceedings of the Trieste Singularity Singularity Summer School and Workshop, World Scientific

      Pages: 161-221

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Proportionality of indices of 1-forms on singular varieties2007

    • Author(s)
      T.Suwa, J.-P.Brasselet, J.Seade
    • Journal Title

      Advanced Studies in Pure Mathematicss 46

      Pages: 49-65

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On weighted-degrees for algebraic local cohomologies associated with semiquasihomogeneous singularities2007

    • Author(s)
      S.Tajima, Y.Nakamura
    • Journal Title

      Advanced Studies in Pure Mathematicss 46

      Pages: 105-117

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Equivariant Chern classes of singular algebraic varieties with group actions2006

    • Author(s)
      T. Ohmoto
    • Journal Title

      Math. Proc. Cambridge Phil. Soc. 140

      Pages: 115-134

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Equivariant Chern classes of singular algebraic varieties with group actions2006

    • Author(s)
      T, Ohmoto
    • Journal Title

      Math. Proc. Cambridge Phil. Soc 140

      Pages: 115-134

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Equivariant Chern classes of singular algebraic varieties with grou actions2006

    • Author(s)
      T.Ohmoto
    • Journal Title

      Math. Proc. Cambridge Philos. Soc. 140

      Pages: 115-134

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Fundamental groups of the complements of certain plane non-tame toru sextics2006

    • Author(s)
      M.Oka, C.Eyral
    • Journal Title

      Topology Appl. 11

      Pages: 1705-1721

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Classes de Hirzebruch et classes de Chern motiviques2006

    • Author(s)
      S.Yokura, J.-P.Brasselet, J.Schurmann
    • Journal Title

      C. R. Math. Acad. Sci. Paris 342

      Pages: 325-328

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Residue Theoretical Approach to Intersection Theory

    • Author(s)
      T, Suwa
    • Journal Title

      Contemporary Math., Amer. Math. Soc (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Residue Theoretical Approach to Intersection Theory

    • Author(s)
      T. Suwa
    • Journal Title

      Contemporary Math., Amer. Math. Soc. (in press)

      Pages: 55-55

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Generating functions of orbifold Chern classes I: Symmetric Products

    • Author(s)
      T.Ohmoto
    • Journal Title

      Math. Proc. Cambridge Phil. Soc. (in press)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Localization of Atiyah classes2007

    • Author(s)
      T. Suwa
    • Organizer
      Niigata Workshop on Complex Geometry and Singularities
    • Place of Presentation
      クロスパル新潟
    • Year and Date
      2007-08-20
    • Related Report
      2007 Annual Research Report
  • [Presentation] Residue Theoretical Approach to Intersection Theory2006

    • Author(s)
      T. Suwa
    • Organizer
      Real and Complex Singularities
    • Place of Presentation
      Sao Carlos, Brazil
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Residue Theoretical Approach to Intersection Theory2006

    • Author(s)
      T, Suwa
    • Organizer
      Real and Complex Singularities
    • Place of Presentation
      Sao Carlos, Brazil(4 consecutive invited lectures)
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] Singularities in Geometry and Topology2007

    • Author(s)
      J. -P. Brasselet and T. Suwa
    • Total Pages
      329
    • Publisher
      Mathematical Society of Japan
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Book] Singularities in Geometry and Topology2007

    • Author(s)
      T.Suwa, J.-P.Brasselet
    • Total Pages
      329
    • Publisher
      Mathematical Society of Japan
    • Related Report
      2006 Annual Research Report

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Published: 2006-04-01   Modified: 2016-04-21  

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