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Foundations of rigid geometry from birational viewpoint

Research Project

Project/Area Number 15H03607
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNagoya University

Principal Investigator

FUJIWARA KAZUHIRO  名古屋大学, 多元数理科学研究科, 教授 (00229064)

Co-Investigator(Kenkyū-buntansha) HESSELHOLT LARS  名古屋大学, 多元数理科学研究科, 教授 (10436991)
加藤 文元  東京工業大学, 理学院, 教授 (50294880)
Research Collaborator OHKUBO SHUN  名古屋大学, 大学院多元数理科学研究科, 助教 (20755160)
KATO FUMIHARU   (50294880)
KONDO SHIGEYUKI  名古屋大学, 大学院多元数理科学研究科, 教授 (50186847)
SAITO TAKESHI  東京大学, 大学院数理科学研究科, 教授 (70201506)
SAITO SHUJI  東京大学, 大学院数理科学研究科, 教授 (50153804)
TAKAHASHI RYO  名古屋大学, 大学院多元数理科学研究科, 准教授 (40447719)
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥16,120,000 (Direct Cost: ¥12,400,000、Indirect Cost: ¥3,720,000)
Fiscal Year 2018: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2017: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2016: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2015: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Keywordsリジッド幾何学 / 数論幾何学 / 代数幾何学 / 整数論 / 可換環論 / モチーフ理論 / パーフェクトイド空間 / 代数学 / パーフェクトイド / 非アルキメデス的バナッハ環
Outline of Final Research Achievements

Aimed for establishing foundational aspects of rigid geometry, "Foundations of rigid geometry I" (joint with F. Kato at titech) was published from EMS (2018). A new aspect, i.e., spectral theory of filtered rings, is added during the project. Also, perfectoid spaces, which have close connection to rigid geometry, are put into our plan. After P. Scholze's introduction, many applications of perfectoid spaces are known up to now. The principal investigator has tried to understand phenomena behind cohomological purity conjectures from perfectoid viewpoint, and has given a new proof of the absolute purity conjecture.

Academic Significance and Societal Importance of the Research Achievements

リジッド幾何学とは非アルキメデス解析をベースとした解析幾何学である. 非アルキメデス解析は通常現れる実数に基づいた解析と異なる側面があり, 特に代数的な視点が多く現れる. またリジッド幾何的対象が整数論に現れることが多いため, その研究は整数の持つ隠れた幾何学的性質や代数的構造の理解に役立つ. 研究代表者は以前より「リジッド幾何学とは形式幾何学の双有理幾何学である」との視点に基づいた研究を行っており, 国際的にも独自性が高いものと考えている. 確固とした基礎理論は科学にとって重要であり, この研究期間中にもヨーロッパ数学会出版局からリジッド幾何学の基礎付けを著書として発表している.

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • Research Products

    (11 results)

All 2018 2017 2016 2015

All Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results) Presentation (6 results) (of which Int'l Joint Research: 6 results,  Invited: 6 results) Book (1 results) Funded Workshop (1 results)

  • [Journal Article] Topological Hochschild homology and the Hasse-Weil zeta function2018

    • Author(s)
      Hesselholt, Lars
    • Journal Title

      Contemp. Math.

      Volume: 708 Pages: 157-180

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A fake projective plane via 2-adic uniformization with torsion2017

    • Author(s)
      Daniel Allcock and Fumiharu Kato
    • Journal Title

      Tohoku Math. J.

      Volume: 69

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A combinatorial Li-Tau inequality and rational points on curves2015

    • Author(s)
      Gunther Cornelissen, Fumiharu Kato, Janne Kool
    • Journal Title

      Math. Ann.

      Volume: 361 Issue: 1-2 Pages: 211-258

    • DOI

      10.1007/s00208-014-1067-x

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Presentation] Higer algebra and arithmetic2018

    • Author(s)
      Hesselholt, Lars
    • Organizer
      Rothschild lecture, Isaac Newton Institute
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Cohomological purity from perfectoid viewpoint2018

    • Author(s)
      Kazuhiro Fujiwara
    • Organizer
      Motives in Tokyo
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Henselian rigid geometry2016

    • Author(s)
      Fumiharu Kato
    • Organizer
      Conference “Algebraic Geometry in East Asia 2016
    • Place of Presentation
      University of Tokyo
    • Year and Date
      2016-01-19
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Proper dominant descent in rigid geometry2015

    • Author(s)
      Kazuhiro Fujiwara
    • Organizer
      Non-archimedean analytic geometry: theory and practice
    • Place of Presentation
      Papeete, French Polynesia
    • Year and Date
      2015-08-25
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Zariski Main Theorem for henselian rigid space2015

    • Author(s)
      Fumiharu Kato
    • Organizer
      Non-archimedean analytic geometry: theory and practice
    • Place of Presentation
      Papeete, French Polynesia
    • Year and Date
      2015-08-25
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-archimedean geometry ― past and present2015

    • Author(s)
      Fumiharu Kato
    • Organizer
      A Symposium on the History of Functional Analysis
    • Place of Presentation
      School of Mathematics, Northwest University, Xian, China
    • Year and Date
      2015-05-09
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Book] Foundations of Rigid Geometry I2018

    • Author(s)
      Kazuhiro Fujiwara, Fumiharu Kato
    • Total Pages
      863
    • Publisher
      EMS publishing house
    • ISBN
      9783037191354
    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] Motives in Tokyo2017

    • Place of Presentation
      日本, 東京, 東京大学大学院数理科学研究科
    • Year and Date
      2017-02-20
    • Related Report
      2016 Annual Research Report

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Published: 2015-04-16   Modified: 2020-03-30  

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