Deformations and classification of log Calabi-Yau varieties

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14509
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kobe University
• Principal Investigator: Sano Taro 神戸大学, 理学研究科, 准教授 (10773195)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: Fano多様体 / Calabi-Yau多様体

Outline of Final Research Achievements

In joint work with Hashimoto, we constructed examples of non-Kahler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers. I also constructed such examples in any dimension >3. We also computed their algebraic dimensions. Moreover, I proved the birational boundedness of K3 surfaces and Abelian surfaces which can appear as boundaries of 3-fold plt CY pairs.
In joint work with Tasin, we proved that most of Fano weighted hypersurfaces (of index 1) are K-stable. As an application, in joint work with Liu and Tasin, we constructed infinitely many families of Sasaki-Einstein metrics on odd-dimensional spheres.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

ケーラーでないカラビヤウ多様体の例の構成は代数幾何的手法に基づいて複素幾何的に興味深い例の構成に成功しており、広く興味深いと思われる。また、双有理有界性を証明したK3曲面やAbel曲面は長年研究がなされてきた対象であり、学術的に一定の価値がある。
また、Liu氏、Tasin氏との共同研究では代数的な手法を使って微分幾何学の長年の予想を解決した研究として、学術的価値は高いと思われる。