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Wild motivic integration and application to singularities

Research Project

Project/Area Number 18H01112
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionOsaka University (2020-2022)
Tohoku University (2018-2019)

Principal Investigator

Yasuda Takehiko  大阪大学, 大学院理学研究科, 教授 (30507166)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥17,030,000 (Direct Cost: ¥13,100,000、Indirect Cost: ¥3,930,000)
Fiscal Year 2022: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2021: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2020: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2019: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
KeywordsDeligne-Mumfordスタック / 野性商特異点 / モジュライ空間 / モチーフ積分 / 局所エタール基本群 / スタック / Batyrev-Manin予想 / 特異点 / 代数的スタック / 野性分岐 / 有理点 / 弦モチーフ / v関数 / 野性的商特異点 / マッカイ対応 / トーサー / 商特異点 / 量公式 / 野性的Deligne-Mumfordスタック
Outline of Final Research Achievements

In this research project, we have constructed and applied new theories in multiple areas of algebraic geometry involving singularities, such as wild Deligne-Mumford stacks, motivic integration, and wild quotient singularities. Key achievements include the generalization of motivic integration to wild Deligne-Mumford stacks of equal characteristic, the proof of the wild McKay correspondence, the construction of the moduli space of formal torsors, the finiteness of the local etale fundamental group of two-dimensional KLT singularities, and the formulation of the Batyrev-Manin conjecture for Deligne-Mumford stacks. These research results were presented at several international conferences.

Academic Significance and Societal Importance of the Research Achievements

本研究課題は、代数幾何学の特異点に関わる複数の重要なトピックを対象として、新たな理論構築とその応用を行った。研究成果は、代数幾何学や整数論分野内での知見の拡大につながった。特に、野性的Deligne-Mumfordスタックやモチーフ積分、野性商特異点などの理解を深めることで、それらを用いる先端的な研究の進展が期待される。研究成果はarXiv等のインターネット上で公開され、誰でも見ることができる。また、多くの国際研究集会で発表し、より多くの人に研究成果が伝わるように努めた。多くの国際共同研究を通して、日本の数学研究の水準向上や国際競争力強化に貢献できたと思う。

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • Research Products

    (36 results)

All 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (6 results) Journal Article (1 results) (of which Peer Reviewed: 1 results) Presentation (27 results) (of which Int'l Joint Research: 19 results,  Invited: 27 results) Funded Workshop (2 results)

  • [Int'l Joint Research] KU Leuven(ベルギー)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] University of Basel(スイス)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] UNAM(メキシコ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] EPFL(スイス)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] The University of Florence(イタリア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] The University of Florence(イタリア)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Pathological Quotient Singularities in Characteristic Three Which Are Not Log Canonical2021

    • Author(s)
      Yamamoto Takahiro
    • Journal Title

      Michigan Mathematical Journal

      Volume: Advance Publication Issue: 4 Pages: 1-14

    • DOI

      10.1307/mmj/1600308172

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Presentation] Stringy invariants and finiteness of local fundamental groups2023

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Non-Archimedean methods in arithmetic and geometry
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Another height function for Deligne-Mumford stacks via twisted sectors2022

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Geometry of Arithmetic Statistics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Orbifold pseudo-effective cones2022

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Rationality, Moduli Spaces, and Related Topics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Batyrev-Manin conjecture for Deligne-Mumford stacks2022

    • Author(s)
      Takehiko Yasuda
    • Organizer
      PRIMA Congress 2022
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the isomorphism problem of projective schemes2021

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Algebraic Geometry in East Asia
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Local fundamental groups of log terminal singularities via stringy motives2021

    • Author(s)
      TakehikoYasuda
    • Organizer
      Recent Developments in Algebraic Geometry, Arithmetic and Dynamics Part 1
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The isomorphism problem for projective schemes2021

    • Author(s)
      Takehiko Yasuda
    • Organizer
      MSJ-KMS Joint Meeting 2021
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Introduction to algebraic stacks (with emphasis on application to singularities)2021

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Interactions of New Trends in Algebraic Geometry and Singularities
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Local etale fundamental groups of log terminal surface singularities2021

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Algebraic Number Theory and Related Topics
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Quotient singularities via stringy motives2021

    • Author(s)
      Takehiko Yasuda
    • Organizer
      2020 OIST Workshop “Quantum Math, Singularities and Applications”
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] モチーフ積分とMcKay対応2020

    • Author(s)
      安田健彦
    • Organizer
      東京工業大学談話会
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] 位数$p^2$の巡回群による商特異点2020

    • Author(s)
      安田健彦
    • Organizer
      特異点セミナー)
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] The motivic McKay correspondence in arbitrary characteristics2020

    • Author(s)
      Takehiko Yasuda
    • Organizer
      The McKay correspondence, mutation and related topics
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Motivic integration over DM stacks and its applications2020

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Arithmetic Geometry seminar at the University of Munster
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Motivic integration over DM stacks and its applications2020

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Online Algebraic Geometry Seminar, Osaka University
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] The wild McKay correspondence of arbitrary finite groups2019

    • Author(s)
      Takehiko Yasuda
    • Organizer
      月曜特異点セミナー
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 野性McKay対応概説 -数論的視点と最新成果-2019

    • Author(s)
      Takehiko Yasuda
    • Organizer
      第27回整数論サマースクール
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] The wild McKay correspondence for an arbitrary finite group2019

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Interaction Between Algebraic Geometry and QFT
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The wild McKay correspondence for an arbitrary finite group2019

    • Author(s)
      Takehiko Yasuda
    • Organizer
      RIMS & OIST Workshop: On the Problem of Resolution of Singularities and its Vicinity
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Motivic integration and the wild McKay correspondence2019

    • Author(s)
      Takehiko Yasuda
    • Organizer
      a lecture series at EPFL
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] モチーフ積分とマッカイ対応2019

    • Author(s)
      Takehiko Yasuda
    • Organizer
      第24回代数学若手研究会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Computational aspects of the McKay correspondence2019

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Higher Dimensional Arithmetic Geometry
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A moduli problem in the wild McKay correspondence2018

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Categorical and Analytic Invariants in Algebraic Geometry Ⅵ
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 特異点と有理点2018

    • Author(s)
      Takehiko Yasuda
    • Organizer
      熊本大学談話会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Motivic mass formulae for formal torsors2018

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Arithmetic Geometry : l-adic and p-adic aspects
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Cohen-Macaulayness of klt singularities in positive characteristics2018

    • Author(s)
      Takehiko Yasuda
    • Organizer
      代数学シンポジウム
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Quotient singularities by the cyclic group of order p2018

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Modern Algebraic Geometry
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Singularities and Arithmetics2020

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] 高次元代数多様体の有理点2019

    • Related Report
      2019 Annual Research Report

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Published: 2018-04-23   Modified: 2024-01-30  

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