Study on the formal groups of low-dimensional Calabi-Yau varieties

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03200
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Hokkaido University of Education
• Principal Investigator: GOTO Yasuhiro 北海道教育大学, 教育学部, 教授 (40312425)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 数論的代数幾何 / 形式群 / カラビ・ヤウ多様体 / デルサルト型多様体 / ファイブレーション / 国際研究者交流

Outline of Final Research Achievements

The formal groups of Calabi-Yau threeforlds with K3 or elliptic fibrations are considered. The relationship between the formal group of a threefold and those of its fibers were investigated, and in various special cases, the relationship of the formal groups was formulated explicitly. For threefolds of Delsarte type or of Borcea-Voisin type, the height of the formal groups was computed and the range of distribution of the values was determined. Similar calculations were done also on Calabi-Yau fourfolds of Delsarte type. In addition, an international workshop on Calabi-Yau varieties was organized every year.

Free Research Field

数物系科学

Academic Significance and Societal Importance of the Research Achievements

本研究は、各種ファイブレーションを持つ３次元カラビ・ヤウ多様体について、本体とファイバー間の形式群の関係性を調べた点で意義がある。これは、楕円曲線の形式群の変形を通して楕円型K3曲面の形式群を調べたArtinの研究の３次元化に向けた試みである。また、形式群の高さの計算を進め、３次元カラビ・ヤウ多様体の場合にその値分布を示すデータを更新し、４次元カラビ・ヤウ多様体においては高さの数値的データを得たことが新しい。