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2005 Fiscal Year Final Research Report Summary

Residues on Singular Varieties

Research Project

Project/Area Number 15340016
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNiigata University (2005)
Hokkaido University (2003-2004)

Principal Investigator

SUWA Tatsuo  Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (40109418)

Co-Investigator(Kenkyū-buntansha) ITO Toshikazu  Ryukoku University, Faculty of Economy, Professor, 経済学部, 教授 (60110178)
OHMOTO Toru  Hokkaido University, Graduate School of Science, Assoc.Professor, 大学院・理学研究科, 助教授 (20264400)
OKA Mutsuo  Tokyo University of Science, Faculty of Science, Professor, 理学部, 教授 (40011697)
TAJIMA Shinichi  Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (70155076)
YOKURA Shoji  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (60182680)
Project Period (FY) 2003 – 2005
KeywordsSingular varieties / Localization of characteristic classes / Residues / Chern classes / Local Euler obstructions / Holomorphic foliations / Complex dynamical systems / Intersection theory
Research Abstract

The head investigator and the others did research on residues on singular varieties. More specifically :
1.We developed a theory of residues of Chern classes of vector bundles on singular varieties with respect to collections of sections. We also gave explicit expressions (analytic, algebraic and topological) of the residues at an isolated singular point.
2.We introduced the notion of multiplicity for functions on singular varieties and proved a generalization of a formula of Iversen for holomorphic maps onto a Riemann surface.
3.In the case of the top Chern class, the Thom class contains local informations and produces analytic., algebraic and topological invariants through the Bochner-Martinelli kernel. We found the "intermediate Thom classes" for other Chern classes.
4.In a collaboration with J.-P.Brasselet and J.Seade, we proved a "proportionality theorem" for the local Euler obstruction of 1-forms on singular varieties.
5.In a collaboration with F.Bracci, we prove the existence of parabolic curves at a fixed point of a holomorphic self-map of a singular complex surface, as an application of our residue theory. For this, we developed the intersection theory of curves in singular surfaces, using the Grothendieck residues on singular varieties.
6.Besides the above, Ito obtained important results on the Poincare-Hopf type theorems, Ohmoto on the characteristic classes of algebraic stacks, Oka on the fundamental group of the complement of algebraic curves, Tajima on Milnor and Tjurina numbers, Yokura
on motivic characteristic classes, respectively.

  • Research Products

    (13 results)

All 2006 2005

All Journal Article (12 results) Book (1 results)

  • [Journal Article] First order local invariants of apparent contours2006

    • Author(s)
      大本 亨
    • Journal Title

      Topology 45

      Pages: 27-45

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] First order local invariants of apparent contours2006

    • Author(s)
      T.Ohmoto
    • Journal Title

      Topology 45

      Pages: 27-45

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Residues of Chern classes on singular varieties2005

    • Author(s)
      諏訪 立雄
    • Journal Title

      Seminaire et Congres, Soc.Math.France 10

      Pages: 265-285

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A Poincare-Hopf-type theorem for holomorphic one-forms2005

    • Author(s)
      伊藤 敏和
    • Journal Title

      Topology 44

      Pages: 73-84

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On Fermat curves and maximal nodal curves2005

    • Author(s)
      岡 睦雄
    • Journal Title

      Michigan Math.J. 53

      Pages: 459-477

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Unimodal singularities and differential operators2005

    • Author(s)
      田島 慎一
    • Journal Title

      Seminaire et Congres, Soc.Math.France 10

      Pages: 191-208

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Bivariant Chern classes for morphisms with nonsingular target varieties2005

    • Author(s)
      與倉 昭治
    • Journal Title

      Cent.Eur.J.Math. 3

      Pages: 614-626

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Residues of Chern classes on singular varieties2005

    • Author(s)
      T.Suwa
    • Journal Title

      Seminaire et Congres, Soc.Math.France 10

      Pages: 265-285

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A Poincare-Hopf-type theorem for holomorphic one-forms2005

    • Author(s)
      T.Ito
    • Journal Title

      Topology 44

      Pages: 73-84

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On Fermat curves and maximal nodal curves2005

    • Author(s)
      M.Oka
    • Journal Title

      Michigan Math.J. 53

      Pages: 459-477

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Unimodal singularities and differential operators2005

    • Author(s)
      S.Tajima
    • Journal Title

      Seminaire et Congres, Soc.Math.France 10

      Pages: 191-208

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Bivariant Chern classes for morphisms with nonsingular target varieties2005

    • Author(s)
      S.Yokura
    • Journal Title

      Cent.Eur.J.Math. 3

      Pages: 614-626

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] Singularites Franco-Japonaise2005

    • Author(s)
      諏訪 立雄
    • Total Pages
      460
    • Publisher
      Soc.Math.France
    • Description
      「研究成果報告書概要(和文)」より

URL: 

Published: 2007-12-13  

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