Canonical Kahler metrics and Moduli spaces

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13389
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kyoto University
• Principal Investigator: Odaka Yuji 京都大学, 理学研究科, 准教授 (30700356)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: モジュライ空間 / K3曲面 / コンパクト化 / Ricci-平坦計量 / K安定性 / トロピカル幾何学 / 非アルキメデス幾何学 / 局所対称空間

Outline of Final Research Achievements

Firstly, in a joint work with Yoshiki Oshima, we largely developped the understanding of our non-algebro-geometric compactification of moduli spaces which is related to compact limits of canonical metrics, in the case of abelian varieties, by using theories of symmetric spaces. Later, we further developped the theory also in the case of type II degeneration of K3 surfaces, not only for the case of type III degenerations which was where our primary interest lie then. （This was inspired by the works of Honda-Sun-Zhang, Oshima, Alexeev-Brunyate-Engel etc. ) Further, I obtained various results in the direction to develop K-moduli theories for Calabi-Yau varieties (in the most generalized sense). I also obtained other results with my PhD students.

Free Research Field

代数幾何学、moduli空間

Academic Significance and Societal Importance of the Research Achievements

広義Calabi-Yau多様体、そのmoduli空間や標準計量の挙動は、代数幾何学や微分幾何はもちろん、幾何学においても数理物理学においても、時折数論幾何学においても調べられる古典的で重要な対象である。大島氏との仕事や、この数年間に考えた弱K-モジュライ理論の発展は、これらの理論の新しい礎につながっていくことが期待される。