Geometry of log abelian varieties and its application

• Project/Area Number: 15K04811
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yokohama National University
• Principal Investigator: Kajiwara Takeshi 横浜国立大学, 大学院工学研究院, 教授 (00250663)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 対数構造 / 退化多様体 / アーベル多様体

Outline of Final Research Achievements

We study moduli space of abelian varieties, and our purpose is to find suitable degenerating objects in logarithmic geometry. In our research, we have studied foundation on polarizations of logarithmic abelian varieties, and also local moduli spaces of them and GAGF. We show cubic theorem on torsors with relevant multiplicative groups on logarithmic abelian varieties, and existence of projective models of them.

Academic Significance and Societal Importance of the Research Achievements

対数アーベル多様体は、アーベル多様体の退化と考えられる多様体では、群構造と完備性という、代数幾何における扱いやすい条件が両立しない点を補う、新しい空間であり、対数代数空間の例である。アーベル多様体の幾何を対数アーベル多様体へ拡張することで、退化アーベル多様体のさまざまな様相が統一的にとらえらる。本研究では、代数幾何の基本的な不変量や射影性の概念を定式化し、また、代数幾何と形式幾何との対応を確立した。これにより、本理論の基礎づけ、応用に貢献している。

Research Products

• [Journal Article] Log abelian varieties, Part VI:local moduli and GAGF (2020)
  • Author(s): Takeshi Kajiwara, Kazuya Kato, and Chikara Nakayama
  • Journal Title: Yokomama Mathematical Journal Volume: 65 Pages: 53-75
  • Peer Reviewed / Open Access
• [Journal Article] Logarithmic abelian varieties, part V: projective models (2018)
  • Author(s): Takeshi Kajiwara, Kazuya Kato, Chikara Nakayama
  • Journal Title: Yokohama Mathematical Journal Volume: 64 Pages: 21-82
  • NAID: 120006633341
  • Peer Reviewed / Open Access