Arithmetic and Geometry over Calabi-Yau Varieties

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540053
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Hosei University
• Principal Investigator: KATSURA Toshiyuki 法政大学, 理工学部, 教授 (40108444)
• Co-Investigator(Renkei-kenkyūsha): KONDO Shigeyuki 名古屋大学, 大学院多元数理科学研究科, 教授 (50186847) ; SHIMADA Ichiro 広島大学, 大学院理学研究科, 教授 (10235616)
• Research Collaborator: GEER Gerard Van Der Universiteit van Amsterdam, Korteweg-de Vries Instituut, Professor
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: カラビ・ヤウ多様体 / K3曲面 / アーベル曲面 / 正標数 / 有理曲線 / 楕円曲線 / ネロン・セヴェリ群 / 直線配置

Outline of Final Research Achievements

Calabi-Yau varieties are very important varieties both in mathematics and in physics (theory of elementary particles). In this research, I examined the structure of Calabi-Yau varieties, in particular, K3 surfaces (two-dimensional Calabi-Yau varieties) in characteristic p >0. In fact, on the superspecial K3 surface in characteristic 5, there exist 96 smooth rational curves, and they are divided into 6 groups. Any two curves in the same group don't intersect each other, and if we choose two groups among six, the curves in the two groups make a beautiful configuration. I had also obtain several results on the structure of Chern class maps of abelian surfaces, geometric invariants of algebraic varieties in positive characteristic, and the structure of lines on a certain Fermat hypersurface in the 3-dimensional projective space. I announced in total 4 papers on these results.

Free Research Field

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