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Construction and evolution of log Hodge theory and applications of the fundamental diagram to geometry

Research Project

Project/Area Number 17K05200
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionOsaka University

Principal Investigator

USUI Sampei  大阪大学, その他部局等, 名誉教授 (90117002)

Co-Investigator(Kenkyū-buntansha) 中山 能力  一橋大学, 大学院経済学研究科, 教授 (70272664)
Project Period (FY) 2017-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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)
Keywordsホッジ理論 / log幾何 / 分類空間 / コンパクト化 / 冪零軌道 / SL(2)軌道 / Borel--Serre軌道 / Log ホッジ理論 / 混合ホッジ構造の分類空間 / Mumford--Tate領域 / 混合ホッジ構造 / 群作用付 / log混合ホッジ構造 / 基本図式 / log実解析関数 / log無限回可微分関数 / 対数幾何 / 対数混合ホッジ構造 / ホモトピー / 志村多様体 / 対数的モチーフ / 対数的混合ホッジ構造 / トロピカル コンパクト化 / 対数構造 / 代数幾何学 / モジュライ / モチーフ
Outline of Final Research Achievements

The joint researches of Kato--Nakayama--Usui continue. In part IV, we constructed fundamental diagram which relates various (partial) compactificatios of moduli of mixed Hodge structures. Part V generalized IV for mixed Hodge structures as tensor functors with group actions. In part VI, we defined log real analytic functions and log C infinity differentiable functions, studied their calculus, and understood SL(2)-orbit theorem geometrically.
The following are their applications: Compactification of higher Albanese varieties. Category of log (mixed) motives. Generalization of Goresky--Tai homomorphism from cohomology of reductive Borel--Serre compactification to that of toroidal compactification. Simplification of the formulation and the description of Deligne--Beilinson cohomology.

Academic Significance and Societal Importance of the Research Achievements

Log構造の良さ:Log構造を使って、比と偏角の空間を導入しそれらをホモトピーの立場から捉える。無限遠点での極限というかわりに境界点に立ちそこを中心とした座標を使って見渡せる。退化するホッジ構造族に対して、通常では失われる情報を、log構造を使って微細構造を捉えそれを研究できる。退化で一見減った情報がlog構造を使って回復できる。ミラー対称性との関係が見えてくるようだ。

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (25 results)

All 2023 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (4 results) Journal Article (5 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 5 results,  Open Access: 4 results) Presentation (16 results) (of which Int'l Joint Research: 4 results,  Invited: 16 results)

  • [Int'l Joint Research] シカゴ大学(米国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] シカゴ大学(米国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] シカゴ大学(米国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] シカゴ大学(米国)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Deligne--Beilinson cohomology and log Hodge theory2023

    • Author(s)
      Ito, T., Kato, K., Nakayama, C., Usui, S.
    • Journal Title

      Proc. Japan Acad.

      Volume: 99(A)

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A description of a result of Deligne by log higher Albanese map2020

    • Author(s)
      Usui, Sampei
    • Journal Title

      Journal of Singularities

      Volume: 21

    • DOI

      10.5427/jsing.2020.21q

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] On log motives2020

    • 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

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram2018

    • Author(s)
      Kato Kazuya, Nakayama Chikara, Usui Sampei
    • Journal Title

      Kyoto J. Math.

      Volume: 58 Issue: 2 Pages: 289-426

    • DOI

      10.1215/21562261-2017-0024

    • NAID

      120006732768

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Extended period domains, algebraic groups, and higher Albanese manifolds2017

    • Author(s)
      Kato Kazuya, Nakayama Chikara, Usui Sampei
    • Journal Title

      “Hodge Theory and L^2-analysis”, Advanced Lectures in Math.

      Volume: 39

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 退化する楕円曲線の積分周期について2022

    • Author(s)
      臼井三平
    • Organizer
      代数幾何学ミニワークショップ
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Relationship of various extensions of period domains with group actions2021

    • Author(s)
      臼井三平
    • Organizer
      阪大オンライン代数幾何学セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Degenerating Mixed Hodge structures with group action2021

    • Author(s)
      臼井三平
    • Organizer
      代数幾何学ミニワークショップ
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] 代数群のホッジ表現のモジュライのコンパクト化2020

    • Author(s)
      臼井三平
    • Organizer
      代数幾何学ミニワークショップ
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Relation among various compactifications of moduli of mixed Hodge structures with group action2019

    • Author(s)
      Usui, Sampei
    • Organizer
      The Fifth International Conference on History of Modern Mathematics, Xi’an
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Borel-Serre type compactification of moduli spaces of mixed Hodge structures2019

    • Author(s)
      臼井三平
    • Organizer
      代数幾何学ミニワークショップ
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Views through fundamental diagram of classifying spaces of degenerating Hodge structures2019

    • Author(s)
      臼井三平
    • Organizer
      代数幾何学セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Geometric polarized log Hodge structures over the base of log rank one2018

    • Author(s)
      中山能力
    • Organizer
      ワークショップ「ホッジ理論と代数幾何学」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Log Hodge structures, log abelian varieties, and log Drinfeld modules2018

    • Author(s)
      加藤和也
    • Organizer
      ワークショップ「ホッジ理論と代数幾何学」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Log motives and the Hodge realization2018

    • Author(s)
      Nakayama Chikara
    • Organizer
      Log geometry, degenerations and related topics, 2018/02/19--20, 神戸大学(兵庫県)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Log mixed Hodge structures with group action2018

    • Author(s)
      Usui Sampei
    • Organizer
      Log geometry, degenerations and related topics, 2018/02/19--20, 神戸大学(兵庫県)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Description of limits of polylogs by log higher Albanese map2018

    • Author(s)
      臼井三平
    • Organizer
      代数幾何学ミニワークショップ、2018/01/06--07, 兵庫県多可郡多可町 八千代プラザ
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Log higher Albanese manifolds (joint with K. Kato, C. Nakayama)2017

    • Author(s)
      Usui Sampei
    • Organizer
      Geometric and Algebraic Singularity Theory, 2017/09/10--16, Bedlewo, Poland
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Log Hodge 理論における無限遠点の捉え方 (1) Log higher Albanese manifolds2017

    • Author(s)
      中山能力
    • Organizer
      ワークショップ「ホッジ理論と代数幾何学」、2017/08/29--30、 東京電機大学 東京千住キャンパス
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Log Hodge 理論における無限遠点の捉え方 (2) Examples2017

    • Author(s)
      臼井三平
    • Organizer
      ワークショップ「ホッジ理論と代数幾何学」、2017/08/29--30、 東京電機大学 東京千住キャンパス
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Log mixed Hodge theory and geometry2017

    • Author(s)
      臼井三平
    • Organizer
      東京電機大学数学講演会、2017年5月22日、東京電機大学 東京千住キャンパス
    • Related Report
      2017 Research-status Report
    • Invited

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Published: 2017-04-28   Modified: 2024-01-30  

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