Mirror symmetry of Calabi-Yau and Fano manifolds form the viewpoint of moduli theory

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K17817
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra; Geometry
• Research Institution: Keio University (2021-2022); Kyoto University (2017-2020)
• Principal Investigator: KANAZAWA Atsushi 慶應義塾大学, 総合政策学部(藤沢), 准教授 (40784492)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: Calabi-Yau多様体 / ミラー対称性 / モジュライ空間 / Landau-Ginzburg模型 / 安定性条件 / Weil-Petersson幾何 / SYZミラー対称性 / Fano多様体

Outline of Final Research Achievements

We study mirror symmetry of Calabi-Yau and Fano manifolds from the viewpoint of moduli theory. More concretely, we consider the problem that how Calabi-Yau manifolds, Fano manifolds and Landau-Ginzburg models are related under degenerations of complex and Kahler structures. Moreover, inspired by mirror symmetry, we also build some foundations of differential geometric structures of Kahler moduli spaces of Calabi-Yau manifolds.

Free Research Field

複素幾何, シンプレクティック幾何, 数理物理

Academic Significance and Societal Importance of the Research Achievements

ミラー対称性は異なる分野に潜む共通の本質を抽出し, 様々な分野を有機的に結び付けることが期待される. 本研究においても興味深い現象が発見され, 今後の発展のための具体例の蓄積がなされた. より技術的な側面に関しては, (1)Calabi-Yau多様体の退化においてミラー対称性がどのように振る舞うかに関して理解が大きく進展した. (2)Calabi-Yau多様体の複素構造とKahler構造のモジュライ空間の構造の理解が深まった.