1998 Fiscal Year Final Research Report Summary
Study on symplectic structures and contact structures
| Project/Area Number |
09640095
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | HOKKAIDO UNIVERSITY (1998)
Ochanomizu University (1997) |
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Principal Investigator |
ONO Kaoru Grad.School of Sci., Hokkaido Univ.Prof., 大学院・理学研究科, 教授 (20204232)
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| 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)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Symplectic structure / contact structure / J-holomorphic curves / Floer homology / Gromov-Witten invariant / Lagrangian submanifold / Seiberg-Witten invariant |
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| 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.
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Research Products
(12 results)
