Study of harmonic bundles and related objects

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04843
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Mochizuki Takuro 京都大学, 数理解析研究所, 教授 (10315971)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 調和束 / モノポール / ツイスターD加群 / Kontsevich複体 / 漸近挙動 / Dirac型特異点 / Kobayashi-Hitchin対応

Outline of Final Research Achievements

We applied the knowledge and results in the study of harmonic bundles to solve new problems and to study new subjects. We studied the natural deformation family of
harmonic bundles on a compact Riemann surface. We particularly established the Hitchin-WKB problem, and determined the limiting configuration in the rank 2 case.
We also clarified the relation between the Kontsevich complexes and the mixed twistor D-modules associated to algebraic functions, which allows us to deduce many known results for Kontsevich complexes from a general theory of mixed twistor D-modules. Moreover, we started to study monopoles and difference modules. We established simple characterizations of Dirac type singularity of monopoles. We also established the Kobayashi-Hitchin correspondence for holomorphic bundles on non-compact Kahler manifolds with infinite volume.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

調和束に関する以前の研究で得られていた知見を、新しい問題に適用することで興味深い進展が得られ、さらに以前の研究結果をより汎用性の高いものにすることができました。また、モノポールと差分加群の間の新しい対応を追求することで、Dirac型特異点の特徴付けや体積無限大のケーラー多様体上のKobayashi-Hitchin対応などの基礎的な意義を持つ成果が得られました。