Study of microlocal category and homological mirror symmetry

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13405
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kyoto University (2022); Osaka University (2020-2021); The University of Tokyo (2018-2019)
• Principal Investigator: Kuwagaki Tatsuki 京都大学, 理学研究科, 准教授 (60814621)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 深谷圏 / シンプレクティック幾何 / 超局所層理論 / ミラー対称性 / RH対応 / WKB解析

Outline of Final Research Achievements

Homological mirror symmetry is a mysterious duality between symplectic geometry and algebraic geometry. Traditionally, the symplectic side is studied via Floer theory, which is a geometric analytic theory. Recent advances of algebraic analysis enables us to study the symplectic side through sheaf theory. In this study, we used the technology to prove homological mirror symmetry of toric varieties. Also, we also studied the technology itself, and found a relationship between WKB analysis and microlocal sheaf theory.

Free Research Field

Geometry

Academic Significance and Societal Importance of the Research Achievements

トーリック多様体のミラー対称性を層理論を用いた証明は、（私自身のものとは限らない）さまざまな派生研究（偏屈圏のミラー対称性、トーリック退化・因子のミラー対称性など）を生み、学術的意義は大きかったと考える。また、超局所層理論をもちいたシンプレクティック幾何の研究は、量子化とFloer理論の関係を明確にする上での端緒となると考えられ、今後の発展への期待が大きい。