Study of geometric structures via holomorphic curves

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03313
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Rikkyo University
• Principal Investigator: Nishinou Takeo 立教大学, 理学部, 准教授 (50420394)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 変形理論 / 正則曲線

Outline of Final Research Achievements

I conducted research on holomorphic curves on complex manifolds and related objects. In particular, I developed a method to determine the obstructions to deforming singular curves on complex surfaces through local calculations. As an application of this method, I proved a correspondence between holomorphic curves on Abelian surfaces and tropical curves on real 2-dimensional tori, which was a long-standing problem. On the other hand, by investigating gauge theory on 2-dimensional complex tori, I proved that as a limit of Hermitian-Yang-Mills connections on complex tori, a Lagrangian submanifold on the mirror torus naturally correspond, and I partially proved the mirror symmetry conjecture related to D-branes.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

以前知られていた手法では扱いが難しい対象について, 新しい手法を開発することにより研究を可能にした。具体的には, 計算が難しい障害がある場合の変形理論について, 障害の計算を局所的な計算に帰着させることにより, 長年未解決であった問題の解決に役立てた。また, これも扱いが難しい, 横断正則性が成り立たない状況でのゲージ理論について, 新たな手法を開発することで研究を進め, ミラー対称性予想の一部を証明した。