Geometry of non-Kaehler open complex manifolds and 4-dimensional topology

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K14193
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hokkaido University (2021-2022); Kyoto Sangyo University (2017-2020)
• Principal Investigator: Kasuya Naohiko 北海道大学, 理学研究院, 准教授 (70757765)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 複素曲面 / 接触構造 / トポロジー

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.

Free Research Field

微分位相幾何学

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。