| Project/Area Number |
17K14193
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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 | Hokkaido University (2021-2022)
Kyoto Sangyo University (2017-2020) |
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Principal Investigator |
Kasuya Naohiko 北海道大学, 理学研究院, 准教授 (70757765)
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 複素曲面 / 接触構造 / トポロジー / 複素構造 / 4次元多様体 / 強擬凹境界 / 開複素多様体 / 開複素曲面 / 楕円曲線 / 対数変換 / ４次元トポロジー / ケーラーでない複素曲面 |
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| Outline of Final Research Achievements |
In this research program, I gave an affirmative answer to the problem "Is any closed co-oriented contact 3-manifold realizable as the boundary of a strongly pseudoconcave surface?" Moreover, I proved that a complex surface concavely filling the contact structure can be made Kaehler or non-Kaehler which ever you want. This is the main result that I obtained. Moreover, I showed that any two closed contact 3-manifolds can be connected by a complex cobordism, and such a cobordism can be taken to be Kaehler. In this case, however, the Kaehler structure is not compatible with the boundary contact structure, so the cobordism is not a Kaehler cobordism, in general. These results were summarized in a research paper written with Daniele Zuddas, which has been submitted a journal and under review now.
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| Academic Significance and Societal Importance of the Research Achievements |
強擬凸複素曲面は複素幾何・接触トポロジーの両面から盛んに研究されており、その境界の接触構造には「Stein filliable, 特にtightである」という強い制約がかかることが知られている。本研究課題では当初の目標であった「任意の3次元閉接触多様体は強擬凹複素曲面の境界として実現可能か？」という問いを肯定的に解決し、さらに接触多様体を充填する複素曲面はケーラーにも非ケーラーにもとれることを証明した。この成果は、強擬凹曲面は強擬凸の場合と異なり、境界接触構造に関して柔軟性を持っていることを示しており、複素曲面および接触構造の研究における「強擬凹」という新たな方向の重要性を示唆している。
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