Study of harmonic bundles and related objects
| Project/Area Number |
15K04843
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| 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 / ホロノミックD加群 / Riemann-Hilbert対応 / 差分加群 / q-差分加群 / ワイルド調和束 / ヒッグス束 / ストークス構造 / フーリエ変換 / Kontsevich complex / 調和バンドル / 小林-Hitchin対応 / Riemann・Hilbert対応 / リーマン・ヒルベルト対応 / enhanced ind-sheaf / ディラク型特異点 / GKZ超幾何系 / Hitchin-WKB-問題 / asymptotic decoupling / limiting configuration |
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| 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.
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| 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。
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Report
(6 results)
Research Products
(42 results)
