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Study on symplectic structures and contact structures

Research Project

Project/Area Number 09640095
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionHOKKAIDO UNIVERSITY (1998)
Ochanomizu University (1997)

Principal Investigator

ONO Kaoru  Grad.School of Sci., Hokkaido Univ.Prof., 大学院・理学研究科, 教授 (20204232)

Co-Investigator(Kenkyū-buntansha) TSUKADA Kazumi  Fac.of Sci., Ochanomizu Univ.Prof., 理学部, 教授 (30163760)
OHTA Hiroshi  Grad.School of Math., Nagoya Univ.Asso.Prof., 大学院・多元数理科学研究科, 助教授 (31400070)
FUKAYA Kenji  Grad.School of Sci., Kyoto Univ.Prof., 大学院・理学研究科, 教授 (30165261)
KANDA Yutaka  Grad.School of Sci., Hokkaido Univ.Inst., 大学院・理学研究科, 助手 (30280861)
IZUMIYA Shyuichi  Grad.School of Sci., Hokkaido Univ.Prof., 大学院・理学研究科, 教授 (80127422)
大場 清  お茶の水女子大学, 理学部, 助手 (80242337)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 1998: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1997: ¥2,000,000 (Direct Cost: ¥2,000,000)
KeywordsSymplectic structure / contact structure / J-holomorphic curves / Floer homology / Gromov-Witten invariant / Lagrangian submanifold / Seiberg-Witten invariant / symplectic 構造 / Floer ホモロジー / Gromov-Witten 不変量 / Seiberg-Witten 不変量 / Symplectic 構造 / ラグランジアン部分多様体
Research Abstract

Fukaya and Ono had introduced the notion of Kuranishi structure and constructed Floer homology for periodic Hamiltonian systems and Gromov-Witten invariants. We continue this direction and proceeded to investigation on Floer homology of Lagrangian intersections and ALPHA_*-category proposed by Fukaya. In order to deal with transversality for the defining equation of the moduli space of holomorphic maps, we need to use multi-valued perturbations and should work with rational coefficients. Hence we need to give coherent orientation for various moduli spaces. We showed that a spin structure on the lagrangian submanifold gives & natural orientation on moduli spaces. We also have another problem, namely there are examples of a pair of Lagrangian submanifolds so that the usual definition of the boundary operator for Floer complex does not give the boundary operator. We consider the boundary values of holomorphic disks systematically and defined a sort of obstruction classes for defining Floer homology for pairs of Lagrangian submanifolds. It is also shown that if such obstruction classes vanish, we can modify the definition of the boundary operator and define Floer homology. This is a result due to Fukaya, Kontsevich, Oh, Ohta and Ono.
Ohta and Ono studied topology of symplectically filling 4-manifold of links of simple singularities. Based on Taubes' theorem, we got a partial result in this direction. Inspired by this work, Kanda extended Taubes' result to certain non-compact symplectic manifolds.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (23 results)

All Other

All Publications (23 results)

  • [Publications] Kenji Fukaya: "Arnold conjecture and Gromov-Witten invariants" Topology. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kaoru Ono: "On Arnold's conjecture for symplectic fixed points" Banach Center Publications. 45. 13-24 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kenji Fukaya: "Anti-self-dual equations on 4-manifolds with degenerate metric" Geometric and Functional Analysis. 8. 446-528 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yutaka Kanda: "On Thurston-Bennequin invariant of Legendrian knots and non-exactness of the Bennequin inequality" Inventions Mathematicae. 133. 227-242 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shyuichi Izumiya: "Generic affine differential geometry of plane curves" Proceedings of the Edinburgh Mathematical Society. 41. 301-314 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kiyoshi Ohba: "Cutting and Pasting of Riemann surfaces with Abelian differentials, I" International Journal of Mathematics. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kenji Fukaya: "Arnold conjecture and Gromov-Witten invariants" Topology. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kaoru Ono: "On Arnold's conjecture for symplectic fixed points" Banach center Publications. 45. 13-24 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kenji Fukaya: "Anti-self-dual equations on 4-manifolds with degenerate metric" Geometric and Functional Analysis. 8. 446-528 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yutaka Kanda: "On Thurston-Bennequin invariant of Legendrian Knots and non-exactness of the Bennequin inequality" Inventiones Mathematicae. 133. 227-242 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shyuichi Izumiya: "Generic affiue differentialgeometry of plane curves" Proceedings of the Edinburgh Mathematical Society. 41. 301-314 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kiyoshi Oba: "Cutting and Pasting of Riemann surfaces with Abelian differentials, I" International Journal of Mathematics. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kenji Fukaya: "Arnold confecture and Gromov-Witten invariants" Topology (in press).

    • Related Report
      1998 Annual Research Report
  • [Publications] Kaoru Ono: "On Arnold's conjecture for symplectic fixed points" Banach Center Publications. 45. 13-24 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kenji Fukaya: "Anti-self-dual equations on 4-manifolds with-degenerate metric" Geometric and Functional Analysis. 8. 466-528 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Yutaka Kanda: "On Thurston-Bennequin invariant of Legendrian Knots and non-exactness of the Bennequin inequality" Inventiones Mathematical. 133. 227-242 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Shyuichi Izumiya: "Generic affine differential geometry of plane curves" Proceedings of the Edinburgh Mathematical Society. 41. 301-314 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kiyoshi Ohba: "Cutting and Pasting of Riemann surfaces with Abelian differentials I" International Journal of Mathematics. (in press).

    • Related Report
      1998 Annual Research Report
  • [Publications] F.Connolly, H-V, Le, K.Ono: "Almost complex structures which are compatible with Kahler or Symplectic" Ann Global Analysis and Geometry. 15. 325-334 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kaoru Ono: "On Arnold's conjecture for symplectic fixed points" Banach Center Publications,(to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] H.Ohta and K.Ono: "Symplectic 4-manifolds with b^+_2=1" Lecture Notes in Pure and Applied Mathematics. 184. 237-244 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kenji Fukaya: "The symplectic S-cobordism conjecture ; a summary" Lecture Notes in Pure and Applied Mathematics. 184. 209-219 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kazumi Tsukada: "Holomorphic Maps of Compact Generalized Hopf Manifolds" Geometriae Dedicata. 68. 61-71 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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