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Multifaceted applications of rigid geometry to algebraic geometry

Research Project

Project/Area Number 23540015
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KATO Fumiharu  熊本大学, 自然科学研究科, 教授 (50294880)

Project Period (FY) 2011 – 2013
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2012: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2011: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywordsリジッド幾何学 / 非アルキメデス的幾何学 / 非アルキメデス的一意化 / 国際情報交流
Research Abstract

The outcomes of the present research project is two-fold:
(1) Construction high-dimensional non-archimedean orbifolds via rigid analytic uniformizations: This part of the research has mainly been conducted by the joint-work with Daniel Allcock, University of Texas in Austin. Based on the past researches in this field, we have succeeded to give an explicit and algebro-geometrically significant algebraic surface that comes from 2-adic non-archimedean uniformization process.
(2) Building up the solid foundations of rigid geometry: This has been promised in the initial research proposal of this project, meant to be one of the main things to do in the second and third years. This part of the research saw a very successful achievement; with Kazuhiro Fujiwara (Nagoya), we have finished a foundational book `Foundations of Rigid Geometry', containing more than 700 pages, which has been accepted to be published from European Mathematical Society, Publishing House.

Report

(4 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Research-status Report
  • 2011 Research-status Report
  • Research Products

    (31 results)

All 2014 2013 2012 2011 Other

All Journal Article (7 results) (of which Peer Reviewed: 7 results) Presentation (21 results) (of which Invited: 16 results) Book (3 results)

  • [Journal Article] A combinatorial Li-Yau inequality and rational points on curves2014

    • Author(s)
      Cornelissen, G.; Kato, F.; Kool, J.
    • Journal Title

      Math. Ann.

      Volume: 未定

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The densest lattices in PGL3(Q2)2013

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

      Adv. Math.

      Volume: 242 Pages: 102-115

    • DOI

      10.1016/j.aim.2013.04.003

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Densest lattices in PGL3Q22013

    • Author(s)
      Allcock, Daniel; Kato, Fumiharu
    • Journal Title

      Adv. Math

      Volume: 未定

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Un anneau de deformation universel et conducteur superieur2012

    • Author(s)
      Kato, F.
    • Journal Title

      Proc. Japan. Acad. Ser. A Math. Sci.

      Volume: 88 Issue: 2 Pages: 25-27

    • DOI

      10.3792/pjaa.88.25

    • Related Report
      2013 Final Research Report 2012 Research-status Report 2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Topological rings in rigidGeometry2011

    • Author(s)
      Fumiharu Kato
    • Journal Title

      LMS Lecture Notes Series

      Volume: 283 Pages: 103-144

    • DOI

      10.1017/cbo9780511667534.004

    • ISBN
      9780521149761, 9780511667534
    • Related Report
      2013 Final Research Report 2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] On Hausdorff completions of commutative rings in rigid geometry2011

    • Author(s)
      Kazuhiro Fujiwara
    • Journal Title

      Journal of Algebra

      Volume: 322 Issue: 1 Pages: 293-321

    • DOI

      10.1016/j.jalgebra.2011.02.001

    • Related Report
      2013 Final Research Report 2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] A combinatorial Li-Yau inequality and rational points on curves

    • Author(s)
      Cornelissen, G. ; Kato, F. ; Kool, J
    • Journal Title

      Math, Ann

      Volume: (掲載予定)

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Presentation] Comparison of several theories in non-archimedean geometry2014

    • Author(s)
      加藤文元
    • Organizer
      Kyushu Algebraic Number Theory
    • Place of Presentation
      九州大学数理学研究院/マス・フォア・インダストリ研究所3階大講義室1
    • Year and Date
      2014-02-05
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Combinatorial Li-Yau inequality and rational points on curves2014

    • Author(s)
      加藤文元
    • Organizer
      Arithmetic and Algebraic Geometry 2014
    • Place of Presentation
      東京大学大学院数理科学研究科大講義室
    • Year and Date
      2014-01-27
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Combinatorial Li-Yau inequality and rational points on curves2014

    • Author(s)
      加藤文元
    • Organizer
      p-adic cohomology and its applications
    • Place of Presentation
      東北大学大学院理学研究科数学教室 Kawai Hall
    • Year and Date
      2014-01-08
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Fake projective planes via p-adic uniformization2013

    • Author(s)
      加藤文元
    • Organizer
      Symposium "Combinatorics & arithmetic geometry"
    • Place of Presentation
      Universiteit Utrecht, Minnaertgebouw, room 205
    • Year and Date
      2013-05-08
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Yet another fake projective plane via p-adic uniformization2013

    • Author(s)
      加藤文元
    • Organizer
      九州代数的整数論2013
    • Place of Presentation
      九州大学伊都キャンパス伊都図書館3階大講義室1
    • Year and Date
      2013-02-11
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Yet another fake projective plane via p-adic uniformization2013

    • Author(s)
      加藤文元
    • Organizer
      Arithmetic and Algebraic Geometry 2013
    • Place of Presentation
      東京大学数理大講義室
    • Year and Date
      2013-01-28
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Accessory parameter in degenerations2012

    • Author(s)
      加藤文元
    • Organizer
      Symposium on Arithmetic & Geometry
    • Place of Presentation
      九州大学伊都キャンパス伊都図書館3階小講義室2
    • Year and Date
      2012-06-02
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] アクセサリー・パラメーターと退化2012

    • Author(s)
      加藤文元
    • Organizer
      アクセサリーパラメーター研究会
    • Place of Presentation
      熊本大学理学部3号館4階共同研究室
    • Year and Date
      2012-03-15
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] アクセサリー・パラメーターと退化2012

    • Author(s)
      加藤文元
    • Organizer
      アクセサリー・パラメーター研究会
    • Place of Presentation
      熊本大学 理学部3号館4階 共同研究室
    • Related Report
      2011 Research-status Report
  • [Presentation] Adic formal algebraic spaces of finite ideal type and rigid GAGA on algebraic spaces2011

    • Author(s)
      加藤文元
    • Organizer
      2011 Taiwan-Japan Workshop on Arithmetic Algebraic Geometry and Related Topics
    • Place of Presentation
      Institute of Mathematics, Academia Sinica
    • Year and Date
      2011-11-18
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] The densest lattices in PGL3Q22011

    • Author(s)
      加藤文元
    • Organizer
      南九州代数系集会
    • Place of Presentation
      鹿児島大学郡元キャンパス理学部1号館101号室
    • Year and Date
      2011-08-28
    • Related Report
      2013 Final Research Report
  • [Presentation] On lattices in PGL3(Q2)2011

    • Author(s)
      加藤文元
    • Organizer
      九州大学代数学セミナー(招待講演)
    • Place of Presentation
      九州大学 伊都キャンパス 伊都図書館3階 小講義室 2
    • Related Report
      2011 Research-status Report
  • [Presentation] The densest lattices in PGL3(Q2)2011

    • Author(s)
      加藤文元
    • Organizer
      南九州代数系集会
    • Place of Presentation
      鹿児島大学 郡元キャンパス 理学部1号館101号室
    • Related Report
      2011 Research-status Report
  • [Presentation] Adic formal algebraic spaces of finite ideal type and rigid GAGA on algebraic spaces2011

    • Author(s)
      加藤文元
    • Organizer
      2011 Taiwan-Japan Workshop on Arithmetic Algebraic Geometry and Related Topics(招待講演)
    • Place of Presentation
      Institute of Mathematics, Academia Sinica(台湾・台北市)
    • Related Report
      2011 Research-status Report
  • [Presentation] Fake projective planes via p-adic uniformisation

    • Author(s)
      Fumiharu Kato
    • Organizer
      Symposium "Combinatorics & arithmetic geometry"
    • Place of Presentation
      Universiteit Utrecht, Minnaertgebouw, room 205
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Combinatorial Li-Yau inequality and rational points on curves

    • Author(s)
      Fumiharu Kato
    • Organizer
      p-adic cohomology and its applications, 2014
    • Place of Presentation
      東北大学大学院理学研究科数学教室 Kawai Hall
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Combinatorial Li-Yau inequality and rational points on curves

    • Author(s)
      Fumiharu Kato
    • Organizer
      Arithmetic and Algebraic Geometry 2014
    • Place of Presentation
      東京大学大学院数理科学研究科大講義室
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Comparison of several theories in non-archimedean geometry

    • Author(s)
      Fumiharu Kato
    • Organizer
      Kyushu Algebraic Number Theory 2014
    • Place of Presentation
      九州大学数理学研究院/マス・フォア・インダストリ研究所3階大講義室1
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Yet another fake projective plane via p-adic uniformization

    • Author(s)
      加藤文元
    • Organizer
      九州大数的整数論(KANT)2013
    • Place of Presentation
      九州大学 伊都キャンパス 伊都図書館3階 大講義室 1
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Yet another fake projective plane via p-adic uniformization

    • Author(s)
      加藤文元
    • Organizer
      Arithmetic and Algebraic Geometry 2013
    • Place of Presentation
      東京大学 数理大講義室
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Accessory parameter in degenerations

    • Author(s)
      加藤文元
    • Organizer
      Symposium on ARITHMETIC & GEOMETRY
    • Place of Presentation
      九州大学 伊都キャンパス 伊都図書館3階 小講義室 2
    • Related Report
      2012 Research-status Report
    • Invited
  • [Book] Foundations of Rigid Geometry I2014

    • Author(s)
      Fujiwara, K.; Kato, F.
    • Total Pages
      728
    • Publisher
      European Mathematical Society Publishing House
    • Related Report
      2013 Annual Research Report
  • [Book] リジッド幾何学入門2013

    • Author(s)
      加藤文元
    • Total Pages
      304
    • Publisher
      岩波書店
    • Related Report
      2012 Research-status Report
  • [Book] Foundations of Rigid Geometry I, to appear from European Mathematical Society Monograph Series

    • Author(s)
      Fujiwara, K. ; Kato, F
    • Total Pages
      728
    • Related Report
      2013 Final Research Report

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Published: 2011-08-05   Modified: 2023-03-16  

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