Global open submanifolds of compact complex manifolds

2019 Fiscal Year Final Research Report

• Project/Area Number: 19K21024
• Project/Area Number (Other): 18H05834 (2018)
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund (2019); Single-year Grants (2018)
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Osaka City University
• Principal Investigator: Koike Takayuki 大阪市立大学, 大学院理学研究科, 講師 (30784706)
• Project Period (FY): 2018-08-24 – 2020-03-31
• Keywords: 上田理論 / レビ平坦超曲面

Outline of Final Research Achievements

We studied geometric complex analysis mainly on a domain whose boundaries are Levi-flat hypersurface in a K3 surface and a blow-up of the projective plane at nine points.
As a joint work with Takato Uehara at Okayama University, our projects have gotten some progress into a Kahler geometrical aspects of such an open submanifolds. Related to this, we also studied some types of new geometrical constructions of K3 surfaces which corresponds to the degeneration of K3 surfaces of type III. At the same time, my research project on a deformation of the blow-up of the projective plane at nine points, which comes from the change of the choice of nine points configurations, have developed. As a result, we found a new sufficient condition for the nine points configurations so that the blow-up admits Levi-flat hypersurfaces.

Free Research Field

複素幾何学

Academic Significance and Societal Importance of the Research Achievements

多変数関数論の歴史を遡ると必ず登場する楕円曲線及び楕円積分に関する理論は, 様々な現代数学の源流と呼ぶに相応しいものである. 事実, その代数学的・幾何学的・解析学的性質の解明やそれらの関連についての考察は, 現代にまで通用する様々なアイディアを導き出している. 本研究で主な役割を担うK3曲面はその自然な一般化といえ, 具体例ではある一方で, 数学内外の非常に広範な範囲に及ぶ一般性を秘めた対象である. またヒルベルトの第14問題にも関連する具体例である射影曲面の9点爆発もまた別の文脈から非常に重要な具体例であり, これらに関する本研究は意義深いと言える.