| Project/Area Number |
17K05223
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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 | Tokyo Metropolitan University |
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Principal Investigator |
Sakai Takashi 東京都立大学, 理学研究科, 教授 (30381445)
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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 |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 対称空間 / 等質空間 / 対称対 / 対称三対 / 対蹠集合 / 旗多様体 / Lagrange部分多様体 / Floerホモロジー / s多様体 / 部分多様体 / 二重調和写像 |
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| Outline of Final Research Achievements |
In this research project, we studied structures of the intersection of two real forms in a complex flag manifold. We showed that the intersection is an antipodal set of the complex flag manifold when two real forms intersect transversely. As an application of the antipodal structure of the intersection, we calculated the Z_2-Floer homology of a pair of real forms in a complex flag manifold equipped with a Kahler-Einstein metric.
We introduced the notion of generalized s-manifolds as a generalization of symmetric spaces. We gave a method of constructing generalized s-manifolds using Γ-symmetric pairs, and studied their maximal antipodal sets and antipodal numbers. As a generalization of symmetric R-spaces, we gave natural Γ-symmetric structures on R-spaces, and described their maximal antipodal sets explicitly.
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| Academic Significance and Societal Importance of the Research Achievements |
対称対およびルート系の理論の一般化が，複素旗多様体内の実形の交叉の研究や，二重調和部分多様体の研究など幾何学の研究において有用であることがわかった．本研究課題において得られた結果および技術は今後の研究に大いに役立つものと期待される．
本研究課題において，対称空間の一般化概念として一般化されたs多様体を導入した．これは非可換群やLie群などこれまでにない対称性を持つ空間であり，今後更なる研究の進展が期待される．また，本研究課題において得られたR空間上のΓ対称空間の構造は対称R空間の自然な拡張であると言え，学術的に意義のあるものである．
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