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Research of Chow group using deformation

Research Project

Project/Area Number 16K13748
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionHiroshima University

Principal Investigator

Kimura Shun-ichi  広島大学, 理学研究科, 教授 (10284150)

Co-Investigator(Kenkyū-buntansha) 島田 伊知朗  広島大学, 理学研究科, 教授 (10235616)
石井 亮  名古屋大学, 多元数理科学研究科, 教授 (10252420)
高橋 宣能  広島大学, 理学研究科, 准教授 (60301298)
Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywordsモチーフ / モチビックゼータ / 有限次元性 / Trace Formula / Chow群 / 代数幾何 / Bloc 予想 / Bloch 予想 / 代数曲面 / 有理同値の変形 / 代数学 / 幾何学
Outline of Final Research Achievements

The biggest news related to this research is the proof of Finite Dimensionality of motives of algebraic varieties. Unfortunately for us, the final step is done by Joseph Ayoub, but undoubtedly our research contributed to this big result.
We also have two new mathematics spinned out of our research: (1) Infinitesimal Motivic Chow Series and (2) Algebraicity of Lagarias Q function in the context of Collatz conjecture for the polynomial rings over finite fields.

Academic Significance and Societal Importance of the Research Achievements

数学的対象を、そのモチビックゼータというプリズムを通して見る、という視点の重要さについて本研究は明らかにしたと思う。代数多様体のChow群をMumford は幾何的視点から「無限次元的」と表現したが、代数的な視点からモチビックゼータをとることで「有限次元的」な振る舞いをすることを本研究は予測し、またその有限次元性を仮定すると多くの大予想がそこから従うことも確かめた。本研究期間中に有限次元性予想は Ayoub に証明され、その意味では一つの大きな役割を終えたが、モチビックゼータの有限次元性や代数性などの視点は新しい分野で応用の窓口を広げている。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (25 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (2 results) Journal Article (11 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 11 results,  Open Access: 5 results,  Acknowledgement Compliant: 3 results) Presentation (10 results) (of which Int'l Joint Research: 9 results,  Invited: 10 results) Book (1 results) Funded Workshop (1 results)

  • [Int'l Joint Research] 中国科学技術大学(中国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] 中国科学技術大学/首都師範大学(中国)

    • Related Report
      2016 Research-status Report
  • [Journal Article] On an Enriques surface associated with a quartic Hessian surface2019

    • Author(s)
      Ichiro Shimada
    • Journal Title

      Canad. J. Math.

      Volume: 71 Issue: 1 Pages: 213-246

    • DOI

      10.4153/cjm-2018-022-7

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] QUANDLES ASSOCIATED TO GALOIS COVERS OF ARITHMETIC SCHEMES2019

    • Author(s)
      Nobuyoshi Takahashi
    • Journal Title

      Kyushu Journal of Mathematics

      Volume: 73 Issue: 1 Pages: 145-164

    • DOI

      10.2206/kyushujm.73.145

    • NAID

      130007728835

    • ISSN
      1340-6116, 1883-2032
    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Extended McKay correspondence for quotient surface singularities2018

    • Author(s)
      Akira Ishii and Iku Nakamura
    • Journal Title

      The Quarterly Journal of Mathematics

      Volume: - Issue: 2 Pages: 00-00

    • DOI

      10.1093/qmath/hay047

    • NAID

      120006767378

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Numerical boundedness on rational equivalences of zero cycles on algebraic varieties with trivial CH_02018

    • Author(s)
      Shun-ichi Kimura, Mao Sheng, Mingwei Zhang
    • Journal Title

      Asian J. Math.

      Volume: 22 Pages: 569-576

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Numerical Boundedness on Rational Equivalences of zero cycles on algebraic varieties with trivial Ch_02018

    • Author(s)
      Kimura, Sheng, Zhang
    • Journal Title

      Asian Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Holes of the Leech lattice and the projective models of K3 surfaces2017

    • Author(s)
      Shimada, Ichiro
    • Journal Title

      Math. Proc. Cambridge Philos. Soc.

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] The automorphism groups of certain singular K3 surfaces and an Enriques surface2016

    • Author(s)
      Ichiro Shimada
    • Journal Title

      Progr. Math.

      Volume: 315 Pages: 297-343

    • DOI

      10.1007/978-3-319-29959-4_12

    • ISBN
      9783319299587, 9783319299594
    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the topology of projective subspaces in complex Fermat varieties2016

    • Author(s)
      Degtyarev, Alex; Shimada, Ichiro
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 68 Issue: 3 Pages: 975-996

    • DOI

      10.2969/jmsj/06830975

    • NAID

      130005166133

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Automorphisms of supersingular K3 surfaces and Salem polynomials2016

    • Author(s)
      Ichiro Shimada
    • Journal Title

      Experimental Mathematics

      Volume: 25 Issue: 4 Pages: 389-398

    • DOI

      10.1080/10586458.2015.1073641

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic2016

    • Author(s)
      Shimada, Ichiro; Shioda, Tetsuji
    • Journal Title

      manuscripta mathematica

      Volume: - Issue: 1-2 Pages: 279-297

    • DOI

      10.1007/s00229-016-0886-3

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Quandle varieties, generalized symmetric spaces and \varphi-spaces2016

    • Author(s)
      Takahashi, Nobuyoshi
    • Journal Title

      Transformation Groups

      Volume: 21 Issue: 2 Pages: 555-576

    • DOI

      10.1007/s00031-015-9351-8

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] On an Enriques surface associated with a quartic Hessian surface2017

    • Author(s)
      Ichiro Shimada
    • Organizer
      Algebraic Geometry Conference
    • Place of Presentation
      Baeundae, Busan (韓国)
    • Year and Date
      2017-03-29
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On an Enriques surface associated with a quartic Hessian surface2017

    • Author(s)
      Ichiro Shimada
    • Organizer
      Topology of Singularities
    • Place of Presentation
      Vietnam Institute for Advanced Study in Mathematics (ベトナム)
    • Year and Date
      2017-02-21
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] G-constellations and the maximal resolution of a quotient surface singularity2017

    • Author(s)
      Akira Ishii
    • Organizer
      Moduli spaces of sheaves and related topics
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2017-02-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] G-constellations and the maximal resolution of a quotient surface singularity2016

    • Author(s)
      石井亮
    • Organizer
      非可換代数幾何学の大域的問題とその周辺
    • Place of Presentation
      高知大学理学部数学大セミナー室
    • Year and Date
      2016-12-18
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Connected components of the moduli of elliptic K3 surfaces2016

    • Author(s)
      Ichiro Shimada
    • Organizer
      The 10th Arithmetic and Algebraic Geometry
    • Place of Presentation
      東京大学
    • Year and Date
      2016-12-14
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Connected components of the moduli of elliptic K3 surfaces2016

    • Author(s)
      Ichiro Shimada
    • Organizer
      The 4th Franco-Japanese-Vietnamese Singularities
    • Place of Presentation
      University Savoie Mont Blanc (フランス)
    • Year and Date
      2016-11-09
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Rationality of Motivic Chow Series modulo infinitesimal ${\mathbb A} ^1$-homotopy2016

    • Author(s)
      Shun-ichi Kimura
    • Organizer
      The first congress of Algebra Geometry of Mexico
    • Place of Presentation
      Casa Mathematica Oaxaca (メキシコ)
    • Year and Date
      2016-10-05
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Dimer models with group actions2016

    • Author(s)
      Akira Ishii
    • Organizer
      Categorical and analytic invariants in Algebraic geometry 3
    • Place of Presentation
      Higher School of Economics, Moscow (ロシア)
    • Year and Date
      2016-09-12
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Introduction to the McKay correspondence and Artin-Verdier theory2016

    • Author(s)
      Akira Ishii
    • Organizer
      Non-commutative crepant resolutions, Ulrich modules and generalizations of the McKay correspondence
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-06-13
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A smooth surface containing 56 lines2016

    • Author(s)
      Ichiro Shimada
    • Organizer
      Algebra, geometry and topology of singularities
    • Place of Presentation
      Galatasaray University (トルコ)
    • Year and Date
      2016-05-13
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] 基幹講座 数学 線型代数 (基幹講座数学)2018

    • Author(s)
      木村俊一
    • Total Pages
      322
    • Publisher
      東京図書
    • ISBN
      9784489022487
    • Related Report
      2018 Research-status Report
  • [Funded Workshop] 熊本岡山北京広島代数解析幾何シンポジウム2017

    • Place of Presentation
      広島大学
    • Year and Date
      2017-03-20
    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2021-02-19  

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