Non-abelian Hodge decomposition on non-Kahler complex manifolds
Project/Area Number |
15K17533
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Research Category |
Grant-in-Aid for Young Scientists (B)
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Allocation Type | Multi-year Fund |
Research Field |
Geometry
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Research Institution | Osaka University (2016-2019) Tokyo Institute of Technology (2015) |
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 |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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Keywords | 平坦束 / Higgs束 / 非可換ホッジ理論 / ヒッグス束 / 佐々木多様体 / 調和計量 / Variation of MHS / Sasakian多様体 / mixed Hodge構造 / ケーラー多様体 / 基本群 / ループ群 / ループホッジ理論 / 複素幾何 / ホッジ理論 |
Outline of Final Research Achievements |
It is known that on compact Kahler manifolds, there is a correspondence between flat vector bundles and Higgs bundles so called non-abelian Hodge decomposition.But the non-abelian Hodge decomposition on non-kahler manifolds is not known. In this research, I was constructing the theory of non-abelian Hodge decomposition on non-kahler manifolds by using Twistor structure theory (a generalization of Hodge theory) and Bott-CHern cohomology theory. An important application of this research is to prove that the non-abelian Hodge decomposition exists on compact Sasakian manifolds (an important class of non-kohler manifold). More precisely, I proved that on compact Sasakian manifolds there exists a correspondence between semi-simple flat vector bundles and poly-stable basic Higgs bundles.
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Academic Significance and Societal Importance of the Research Achievements |
ケーラー多様体上で非可換ホッジ分解は多様体の分類(Moduli)に関する理論や基本群の表現に関する理論など多様な応用がある。しかしケーラーという仮定は幾何学においては非常に限定的であり、これらの理論が非ケーラー多様体に拡張できることが望まれている。 ケーラー多様体は偶数次元の空間しか扱えないが、佐々木多様体で非可換ホッジ分解が得られたことにより奇数次元上でも理論が展開できるようになり、現実の物理理論とリンクするような幾何学理論が構築されることが期待される。
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Report
(6 results)
Research Products
(23 results)