Floer theory, mirror symmetry conjecture and applications to symplectic geometry

2014 Fiscal Year Final Research Report

• Project/Area Number: 23340015
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: OHTA Hiroshi 名古屋大学, 多元数理科学研究科, 教授 (50223839)
• Co-Investigator(Renkei-kenkyūsha): FUKAYA Kenji 京都大学, 大学院理学研究科, 教授 (30165261) ; ONO Kaoru 京都大学, 数理解析研究所, 教授 (20204232) ; KANNO Hiroaki 名古屋大学, 多元数理科学研究科, 教授 (90211870)
• Project Period (FY): 2011-04-01 – 2015-03-31
• Keywords: シンプレクティック幾何 / フレアー理論 / ミラー対称性 / ラグランジュ部分多様体 / 倉西構造 / 仮想基本類 / トーリック多様体 / フロベニウス多様体構造

Outline of Final Research Achievements

We had established the fundamental theory of Lagrangian intersection Floer theory based on the filtered A infinity algebra in the last decade. In this research period, we studied its applications, especially to the mirror symmetry conjecture. We proved an ring isomorphism between the big quantum cohomology of any compact smooth toric manifold and the Jacobian ring of the potential function which we introduced from the context in the Lagrangian Floer theory. Moreover, we studied the correspondence of the Frobenius manifold structures between them. We went further to study of equivalence at (derived) categorical level, so called homological mirror symmetry conjecture. We showed generation criteria for the Fukaya category of compact symplectic manifold and obtained a generator for compact toric manifold. We used the virtual fundamental chain technique to obtain these results. For this purpose we also provided the technical details of the theory of Kuranishi structure and brushed it up.

Free Research Field

幾何学