Degenerations of Calabi-Yau manifolds and mirror symmetry

2021 Fiscal Year Final Research Report

• Project/Area Number: 16K05105
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Gakushuin University
• Principal Investigator: Hosono Shinobu 学習院大学, 理学部, 教授 (60212198)
• Project Period (FY): 2016-04-01 – 2022-03-31
• Keywords: カラビ・ヤウ多様体 / ミラー対称性 / 周期積分 / モジュライ空間 / 多変数超幾何微分方程式

Outline of Final Research Achievements

From the study of string theory in theoretical physics, certain special manifolds, called Calabi-Yau manifolds, were attracted attentions of many researchers, and around 1990, a mysterious symmetry called "mirror symmetry" was discovered in world of Calabi-Yau manifolds. Since the discovery of the mirror symmetry, extensive study toward mathematical understanding of the symmetry has been done. Now, we have two approaches to study the symmetry; one is categorical and the other is geometric method. In this research project, PI has made achievements in constructing explicit and interesting Calabi-Yau manifolds aiming to reveal mirror symmetry. These have been done by developing necessary methods to analyze behavior of some integral called period integral.

Free Research Field

複素幾何学

Academic Significance and Societal Importance of the Research Achievements

カラビ・ヤウ多様体のミラー対称性は、その発見から３０年近くが経過し、関連する数学の分野に多数の影響を与えてきました。しかしながら、対称性の数学的の完全な理解には至っておらず、現在も不思議な対称性と思われています。このような対称性が現れる興味深いカラビ・ヤウ多様体を構成することは、ミラー対称性を深く理解する上で大切な役割を果たします。また、具体的な例を構築すると同時に、それらを調べる手段・方法を整備して確立することは、ミラー対称性の解明に限らず、派生する様々な数学の問題への応用に寄与するもので、数学の大切な蓄積となります。