Compactifications of Mumford-Tate domains and log geometry

• Project/Area Number: 16K05093
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Hitotsubashi University
• Principal Investigator: NAKAYAMA Chikara 一橋大学, 大学院経済学研究科, 教授 (70272664)
• Project Period (FY): 2016-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: ホッジ理論 / 代数幾何

Outline of Final Research Achievements

We construct various compactifications of the Mumford-Tate domain (variants of the period domains with the actions of algebraic groups) as moduli spaces of mixed log Hodge structures endowed with the action of algebraic groups. In this process, the compactification of nilpotent orbits, the compactification by SL(2)-orbits and the compactification by Borel-Serre orbits are constructed, and the fundamental diagram including these spaces is established.

Academic Significance and Societal Importance of the Research Achievements

ある数学的対象全部の集合に自然な空間構造を入れたものをモジュライ空間といい、モジュライ空間を調べやすくするために無限遠点を追加してコンパクト化することが重要である。log 幾何は各種モジュライ空間のコンパクト化を構成する広く一般的な枠組みを提供する。本研究では log 幾何を応用し、従来構成されていなかった、混合マンフォード-テイト領域のコンパクト化に成功した。

Research Products

• [Int'l Joint Research] シカゴ大学(米国)
• [Int'l Joint Research] シカゴ大学(米国)
• [Int'l Joint Research] シカゴ大学(米国)
• [Int'l Joint Research] シカゴ大学(米国)
• [Int'l Joint Research] シカゴ大学(米国)
• [Journal Article] Logarithmic abelian varieties, Part VII: Moduli (2021)
  • Author(s): Takeshi Kajiwara, Kazuya Kato, Chikara Nakayama
  • Journal Title: Yokohama Mathematical Journal Volume: 67 Pages: 9-48
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On log motives (2020)
  • Author(s): Tetsushi, Ito ; Kazuya, Kato ; Chikara, Nakayama ; Sampei, Usui
  • Journal Title: Tunisian Journal of Mathematics Volume: 2 Issue: 4 Pages: 733-789
  • DOI: 10.2140/tunis.2020.2.733
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Taro fujisawa, Chikara Nakayama (2020)
  • Author(s): Geometric polarized log Hodge structures with a base of log rank one
  • Journal Title: Kodai Mathematical Journal Volume: 43 Pages: 57-83
  • Peer Reviewed
• [Journal Article] Logarithmic abelian varieties, Part VI: Projective models (2019)
  • Author(s): Takeshi Kajiwara, Kazuya Kato, Chikara Nakayama
  • Journal Title: Yokohama Mathematical Journal Volume: 65
  • Peer Reviewed / Int'l Joint Research
• [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
  • NAID: 120006633341
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram (2018)
  • Author(s): Kazuya Kato, Chikara Nakayama, Sampei Usui
  • Journal Title: Kyoto Journal of Mathematics Volume: 印刷中
  • NAID: 120006732768
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Logarithmic \'etale cohomology, II (2017)
  • Author(s): Chikara Nakayama
  • Journal Title: Advances in Mathematics Volume: 314 Pages: 663-725
  • DOI: 10.1016/j.aim.2017.05.006
  • Peer Reviewed
• [Journal Article] Log abelian varieties (Survey) (2017)
  • Author(s): Chikara Nakayama
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B64
  • Peer Reviewed
• [Journal Article] Categorical characterization of strict morphisms of fs log schemes (2017)
  • Author(s): Yuichiro Hoshi and Chikara Nakayama
  • Journal Title: Mathematical Journal of Okayama University Volume: 59
  • NAID: 120005898804
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram (2017)
  • Author(s): Kazuya Kato, Chikara Nakayama, and Sampei Usui
  • Journal Title: Kyoto Journal of Mathematics Volume: 印刷中
  • NAID: 120006732768
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Extended period domains, algebraic groups, and higher Albanese manifolds (2017)
  • Author(s): Kazuya Kato, Chikara Nakayama, and Sampei Usui
  • Journal Title: S. Zucker's volume Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Log Hodge theory (2021)
  • Author(s): Chikara Nakayama
  • Organizer: 代数セミナー
  • Invited
• [Presentation] Geometric polarized log Hodge structures over the base of log rank one (2018)
  • Author(s): Chikara Nakayama
  • Organizer: ワークショップ「ホッジ理論と代数幾何学」
  • Invited
• [Presentation] Log motives and the Hodge realization (2018)
  • Author(s): Chikara Nakayama
  • Organizer: Workshop: Log geometry, degenerations and related topics.
  • Invited
• [Presentation] Log mixed Hodge 理論における無限遠点の捉え方 (1) ― Log higher Albanese manifolds ― (2017)
  • Author(s): Chikara Nakayama
  • Organizer: ワークショップ「ホッジ理論と代数幾何学」
  • Invited
• [Book] Hodge Theory and L^2-analysis (2017)
  • Author(s): Editor: Lizhen Ji
  • Total Pages: 597
  • Publisher: Higher education press
  • ISBN: 9787040477771