2002 Fiscal Year Final Research Report Summary
Floer homology, string theory and contact geometry
| Project/Area Number |
12640066
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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 | Nagoya University |
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Principal Investigator |
OHTA Hiroshi Nagoya University, Graduate School of Mathematics, Associated Professor, 大学院・多元数理科学研究科, 助教授 (50223839)
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| Co-Investigator(Kenkyū-buntansha) |
TSUCHIYA Akihiro Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (90022673)
SATO Hajime Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (30011612)
KOBAYASHI Ryoichi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20162034)
FUKAYA Kenji Kyoto University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (30165261)
MINAMI Kazuhiko Nagoya University, Graduate School of Mathematics, Associated Professor, 大学院・多元数理科学研究科, 助教授 (40271530)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Symplectic geometry / Floer homology / Mirror symmetry Conjecture / A_∞ algebra / Lagrangian submanifold / Simple singularity / pseudo holomorphic map / Monopole equations |
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| Research Abstract |
1. The link of an isolated singularity in an algebraic surface carries a natural contact structure. In general, there are various topological types of the symplectic fillings of the link. We proved that the diffeomorphism types of the minimal fillings of the link of so-called simple singularity are unique, by using the monopole equations and theory of pseudo-holomorphic curves. Moreover, we completely determined the diffeomorphism types of the minimal fillings of the links of simple elliptic singularities. By product, we also proved that symplectic 4-manifold which contains a positive rational curve with, a (2, 3)-cusp point must be rational surface. These results were obtained by joint works with K.Ono, one of our investigators.
2. We constructed a filtered A_∞ algebra associated to a Lagrangian submanifold L. This gives a quantum deformation of the classical rational de Rham complex of L as A_∞ algebras. Based on our A_∞ algebra, we constructed an obstruction and deformation theories in Floer cohomology for Lagrangian intersections. From the point of view of minor symmetry conjecture, we established the mathematical foundations of the symplectic side. We also applied our theory to some concrete problems, for example, Arnold conjecture, Arnold-Givental conjecture and Maslov index conjecture, in symplectic geometry. These results were obtained by joint works with K.Fukaya, Y-G Oh and K.Ono. To establish a theory for higher genus cases is a problem for further research in the future.
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