• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

F-singularities and singularities in birational geometry in characteristic zero

Research Project

Project/Area Number 26400039
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionThe University of Tokyo

Principal Investigator

Takagi Shunsuke  東京大学, 大学院数理科学研究科, 准教授 (40380670)

Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
KeywordsF特異点 / 可換環論 / 特異点論 / 局所コホモロジー / 代数幾何学 / 標準特異点 / 対数的端末特異点 / 一般超平面切断 / F正則特異点 / 乗数イデアル / 判定イデアル
Outline of Final Research Achievements

``F-singularities" are a generic term used to refer to singularities defined in terms of Frobenius maps, and there are four basic classes of F-singularities, F-regular, F-pure, F-rational and F-injective singularities. F-singularities are expected to correspond to the singularities in birational geometry in characteristic zero, and many researchers have studied this correspondence. In this research project, we obtained the following two results related to this correspondence: (1) When X is a numerically Q-Gorenstein variety over an algebraically closed field of characteristic zero, we proved that the multiplier ideals on X (in the sense of de Fernex-Hacon) coincide, after reduction to characteristic p>>0, with the test ideals. (2) We introduced a new class of F-singularities, F-nilpotent singularities, and gave a Hodge theoretic interpretation of (3-dimensional) F-nilpotent normal isolated singularities.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (31 results)

All 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (5 results) Journal Article (10 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 10 results,  Acknowledgement Compliant: 8 results) Presentation (15 results) (of which Int'l Joint Research: 9 results,  Invited: 15 results) Remarks (1 results)

  • [Int'l Joint Research] ユタ大学(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] 北京大学(中国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] University of Utah/University of Michigan/University of Kansas(米国)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Universita di Genova(Italy)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Tata Institute of Fundamental Research(India)

    • Related Report
      2015 Research-status Report
  • [Journal Article] F-singularities: applications of characteristic p methods to singularity theory2018

    • Author(s)
      Shunsuke Takagi, Kei-ichi Watanabe
    • Journal Title

      Sugaku Expositions

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report 2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] F-singularities: applications of characteristic p methods to singularity theory2018

    • Author(s)
      Shunsuke Takagi and Kei-ichi Watanabe
    • Journal Title

      Sugaku Expositions

      Volume: 印刷中

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A Gorenstein criterion for strongly F-regular and log terminal singularities2017

    • Author(s)
      Anurag Singh, Shunsuke Takagi, Matteo Varbaro
    • Journal Title

      International Mathematics Research Notices

      Volume: 印刷中 Pages: rnw195-rnw195

    • DOI

      10.1093/imrn/rnw195

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Nilpotence of Frobenius action and the Hodge filtration on local cohomology2017

    • Author(s)
      Vasudevan Srinivas, Shunsuke Takagi
    • Journal Title

      Advances in Mathematics

      Volume: 305 Pages: 456-478

    • DOI

      10.1016/j.aim.2016.09.029

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] The weak ordinarity conjecture and F-singularities2017

    • Author(s)
      Bhargav Bhatt, Karl Schwede, Shunsuke Takagi
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Surfaces of globally F-regular and F-split type2016

    • Author(s)
      Yoshinori Gongyo, Shunsuke Takagi
    • Journal Title

      Mathematische Annalen

      Volume: 364 Issue: 3-4 Pages: 841-855

    • DOI

      10.1007/s00208-015-1238-4

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The weak ordinarity conjecture and F-singularities2016

    • Author(s)
      Bhargav Bhatt, Karl Schwede, Shunsuke Takagi
    • Journal Title

      Advanced Studies in Pure Mathematics

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] On the relationship between depth and cohomological dimension2016

    • Author(s)
      Hailong Dao, Shunsuke Takagi
    • Journal Title

      Compositio Mathematica

      Volume: 152 Issue: 4 Pages: 876-888

    • DOI

      10.1112/s0010437x15007678

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Comparing multiplier ideals to test ideals on numerically Q-Gorenstein varieties2015

    • Author(s)
      Tommaso de Fernex, Roi Docampo, Shunsuke Takagi and Kevin Tucker
    • Journal Title

      Bull. London Math. Soc.

      Volume: 47 Issue: 2 Pages: 359-369

    • DOI

      10.1112/blms/bdv006

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] The weak ordinarity conjecture and F-singularities2015

    • Author(s)
      Bhargav Bhatt, Karl Schwede and Shunsuke Takagi
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 印刷中

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] A Gorenstein criterion for strongly Fregular and log terminal singularities2016

    • Author(s)
      高木 俊輔
    • Organizer
      都の西北代数幾何学シンポジウム
    • Place of Presentation
      早稲田大学
    • Year and Date
      2016-11-16
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] F特異点と3次元準射影多様体の一般の超平面切断2016

    • Author(s)
      高木 俊輔
    • Organizer
      広島大学理学部数学科談話会
    • Place of Presentation
      広島大学
    • Year and Date
      2016-10-25
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] F-singularities and singularities in birational geometry2016

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Differential forms in algebraic geometry
    • Place of Presentation
      University of Freiburg
    • Year and Date
      2016-09-19
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] General hyperplane sections of canonical 3-folds in positive characteristic2016

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Higher Dimensional Birational Geometry and Characteristic p > 0
    • Place of Presentation
      Centre International de Rencontres Mathematiques
    • Year and Date
      2016-09-15
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A Gorenstein criterion for strongly Fregular rings2016

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Commutative Algebra and its Interactions with Algebraic Geometry
    • Place of Presentation
      University of Michigan
    • Year and Date
      2016-07-10
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A Gorenstein criterion for strongly F-regular and log terminal singularities2016

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Beijing Algebraic Geometry Colloquium 5th meeting
    • Place of Presentation
      Beijing International Center for Mathematical Research
    • Year and Date
      2016-04-09
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the tautness of F-singularities2015

    • Author(s)
      Shunsuke Takagi
    • Organizer
      International Conference on Singularity Theory -In Honor of Henry Laufer’s 70th Birthday
    • Place of Presentation
      Tsinghua Sanya International Mathematics Forum
    • Year and Date
      2015-12-14
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] F-singularities and weak ordinarity conjecture2015

    • Author(s)
      Shunsuke Takagi
    • Organizer
      The 10th Anniversary Tokyo-Seoul Conference in Mathematics -Algebraic/Arithmetic/Complex Geometry
    • Place of Presentation
      東京大学
    • Year and Date
      2015-12-03
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Vanishing of local cohomology and local Picard groups2015

    • Author(s)
      高木 俊輔
    • Organizer
      第3回杜の都代数幾何学研究集会@福岡
    • Place of Presentation
      福岡大学
    • Year and Date
      2015-11-25
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Frobenius action on local cohomology and the Hodge filtration2015

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Workshop on Algebraic Geometry
    • Place of Presentation
      Fudan University
    • Year and Date
      2015-10-30
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Vanishing of local cohomology and local Picard groups2015

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Higher Dimensional Algebraic Geometry, 2015
    • Place of Presentation
      Taiwan National University
    • Year and Date
      2015-08-22
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Frobenius action on local cohomology and the Hodge filtration2015

    • Author(s)
      Shunsuke Takagi
    • Organizer
      2015 Summer Research Institute on Algebraic Geometry
    • Place of Presentation
      University of Utah
    • Year and Date
      2015-07-17
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Frobenius action on local cohomology and the Hodge filtration2014

    • Author(s)
      高木俊輔
    • Organizer
      代数多様体とその周辺
    • Place of Presentation
      琉球大学
    • Year and Date
      2014-10-01
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] F特異点論の最近の進展について2014

    • Author(s)
      高木俊輔
    • Organizer
      第59回代数学シンポジウム
    • Place of Presentation
      東京大学
    • Year and Date
      2014-09-08
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Nilpotency of Frobenius and divisor class groups2014

    • Author(s)
      Shunsuke Takagi
    • Organizer
      FRG Special Month in Ann Arbor
    • Place of Presentation
      ミシガン大学
    • Year and Date
      2014-06-09
    • Related Report
      2014 Research-status Report
    • Invited
  • [Remarks] Shunsuke Takagi

    • URL

      http://www.ms.u-tokyo.ac.jp/~stakagi/

    • Related Report
      2016 Annual Research Report 2015 Research-status Report 2014 Research-status Report

URL: 

Published: 2014-04-04   Modified: 2022-02-16  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi