| Project/Area Number |
17K14181
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Osaka City University (2021)
Gakushuin University (2017-2020) |
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Principal Investigator |
Kawai Kotaro 大阪市立大学, 数学研究所, 特別研究員 (60728343)
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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)
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| Keywords | special holonomy / G2多様体 / Spin(7)多様体 / モジュライ空間 / Hessian計量 / formality / dHYM接続 / dDT接続 / G2-dDT接続 / multi-moment map / 変形理論 / Cayley equality / associator equality / coassociative部分多様体 / associative部分多様体 / Cayley部分多様体 |
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| Outline of Final Research Achievements |
I found obstructions for manifolds to be formal. In particular, I showed that the Massey product vanishes in many cases. I also found obstructions for the existence of a certain class of Riemannian manifolds. I introduced the notion of homogeneous pairs, and unraveled the geometric structure of moduli spaces of several kind of geometric structures.
I studied dHYM, dDT connections, which can be considered to be mirrors of calibrated submanifolds. I first established the deformation theory and show the mirror version of Cayley, associator equality. This implies various facts such as the property that dDT connections minimize the mirror volume. In addition, I showed the short-term existence and uniqueness of the negative gradient flow of the mirror volume, and gave some characterizations of dDT connections.
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| Academic Significance and Societal Importance of the Research Achievements |
多様体がformalになる障害を与えることで、formalでない多様体を発見するのをより容易にした。同様に、ある条件をみたさないリーマン多様体の発見もより容易にした。homogeneous pairは、多くの幾何構造のモジュライ空間を含む概念であるので、そのようなモジュライ空間の性質を統一的に扱えるようになる。
また部分多様体のミラーが、部分多様体の理論やゲージ理論と多くの類似点を持つことがわかった。更なる類似が成り立つことが期待され、大きな発展が見込まれる。逆に、このミラー側の研究を発展させることで、部分多様体やゲージ理論の研究への応用も期待できる。
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