2006 Fiscal Year Final Research Report Summary
Study on Floer cohomology, mirror symmetry conjecture and singularity
| Project/Area Number |
15340020
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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) |
KOBAYASHI Ryoichi Nagoya University, Graduate School of Mathematics, Professor, 大学院多元数理科学研究科, 教授 (20162034)
SATO Hajime Nagoya University, Graduate School of Mathematics, Professor, 大学院多元数理科学研究科, 教授 (30011612)
TSUCHIYA Akihiro Nagoya University, Graduate School of Mathematics, Professor, 大学院多元数理科学研究科, 教授 (90022673)
KANNO Hiroaki Nagoya University, Graduate School of Mathematics, Professor, 大学院多元数理科学研究科, 教授 (90211870)
FUKAYA Kenji Kyoto University, Department of Mathematics, Professor, 大学院理学研究科, 教授 (30165261)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Symplectic geometry / Floer cohomology / Lagrangian submanifold / A_∞ algebra / mirror symmetry / deformation theory / obstruction theory / quantum cohomology |
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| Research Abstract |
We revised our paper on obstruction theory and deformation theory of Lagrangian Floer cohomology. This is a joint work with K. Fukaya, Y-G. Oh and K. Ono. In March 2006, we wrote up the manuscript (more than 800 pages) of Chapter 1,2,3,4,5,6,7 and 9 and made the preprint version which were distributed in the world. We established the transversality argument on the spaces with Kuranishi structure. This will play very important role to derive homotopy algebraic structure from various moduli spaces. During the revision, we define the potential function in the context of our-filtered A_∞ algebra and show that it coincides with the super potential of the Landau-Ginzburg model in physics literature. This was predicted by physists Hori and Vafa. In 2006, we have finished to write up the remaining chapters, Chapter 8 and Chapter 10. In Chapter 8, we study the case of semi-positive Lagrangian submanifolds. In this case, we can establish our results over Z or Z_2 coefficients. We apply the results to, for example, the Arnold-Givental conjecture. In Chapter 10, we investigate how the moduli space of pseudo holomorphic discs changes under Lagrangian surgery. We also describe the analytic detail. Moreover, we describe the action of cycles of the ambient symplectic manifold to the filtered A_∞ algebra in terms of L_∞ homomorphism. And we discovered a new relation between the torsion part of our Floer cohomology and Hofer distance of Hamilton isotopy.
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