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

The study of the existence problem in the Donaldson-Tian-Yau conjecture

Research Project

Project/Area Number 25287010
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field Geometry
Research InstitutionOsaka University

Principal Investigator

MABUCHI Toshiki  大阪大学, その他部局等, 名誉教授 (80116102)

Co-Investigator(Kenkyū-buntansha) 後藤 竜司  大阪大学, 理学研究科, 教授 (30252571)
中川 泰宏  佐賀大学, 工学(系)研究科(研究院), 教授 (90250662)
新田 泰文  東京工業大学, 理工学研究科, 助教 (90581596)
Project Period (FY) 2013-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥12,870,000 (Direct Cost: ¥9,900,000、Indirect Cost: ¥2,970,000)
Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2016: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2015: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2014: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2013: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
KeywordsDonaldson-Tian-Yau予想 / K-安定性 / テスト配位 / Donaldson-二木不変量 / 偏極代数多様体 / 定スカラー曲率Kaehler計量 / extremal Kaehler計量 / extremal Kaehler 計量 / 定スカラー曲率ケーラー計量カラー曲率ケーラー計量 / 定スカラー曲率ケーラー計量 / 強K-安定性
Outline of Final Research Achievements

The Donaldson-Tian-Yau conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal Kaehler cases. We mainly worked in this unsolved cases of the conjecture.
For a polarized algebraic manifold, by introducing the concept of strong relative K-stability, we showed that strong relative K-stability implies asymptotic relative Chow stability. In particular, a strong relative K-stable polarized algebraic manifold always admits a sequence of polybalanced metric that are expected to converge to an extremal Kaehler metric in the polarization class. This then shows that the sequence of polybalanced metrics admits a suitable a priori bound. Moreover, we obtained a couple of results on the related problems.

Academic Significance and Societal Importance of the Research Achievements

特殊計量の存在に関する Donaldson-Tian-Yau 予想は,複素幾何における中心問題のひとつとして知られている.たとえば,この予想の Kaehler-Einstein 計量の場合は,最近 Chen-Donaldson-Sun や Tian によって肯定的に解決されたが,これはケーラー幾何において、フィールズ賞受賞者である Yau がカラビ予想を解決したとき以来の,初めての本格的な結果として知られている.さらに予想を,より一般の extremal Kaehler 計量の場合に解決することも非常に大きな学術的意義があり,その意味でも我々の研究には少なからぬ価値があると考えられる.

Report

(7 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • Research Products

    (18 results)

All 2019 2018 2016 2015 2014 2013

All Journal Article (10 results) (of which Peer Reviewed: 10 results,  Open Access: 1 results) Presentation (8 results) (of which Int'l Joint Research: 3 results,  Invited: 8 results)

  • [Journal Article] Asymptotic Polybalanced kernels on extremal Kaehler manifolds2019

    • Author(s)
      T. Mabuchi
    • Journal Title

      Asian J. Math.

      Volume: Mok教授還暦記念号

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic polybalanced kernels on extremal Kaehler manifolds2018

    • Author(s)
      Toshiki Mabuchi
    • Journal Title

      Asian J. Math.

      Volume: 22 Issue: 4 Pages: 647-664

    • DOI

      10.4310/ajm.2018.v22.n4.a2

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Yau-Tian-Donaldson conjecture for general polarizations, I2016

    • Author(s)
      T. Mabuchi
    • Journal Title

      Springer Proc. Math. Stat.

      Volume: 154 Pages: 235-245

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An l-th root of a test configuration of exponent l2016

    • Author(s)
      T. Mabuchi
    • Journal Title

      Complex Manifolds

      Volume: 3 Issue: 1 Pages: 169-185

    • DOI

      10.1515/coma-2016-0005

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] The Yau-Tian-Donaldson conjecture for general polarizations, I2016

    • Author(s)
      T. Mabuchi
    • Journal Title

      Geometry and Topology of Manifolds, Springer Japan

      Volume: 1

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Donaldson-Futaki invariant for sequences of test configurations2015

    • Author(s)
      T. Mabuchi
    • Journal Title

      Progress in Math., Birkh\"auser

      Volume: 308 Pages: 395-403

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Strong K-stability and asymptotic Chow-stability2015

    • Author(s)
      T. Mabuchi and Y. Nitta
    • Journal Title

      Progress in Math., Birkhaeuser

      Volume: 308 Pages: 405-411

    • Related Report
      2015 Annual Research Report 2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Donaldson-Futaki invariant for sequences of test configurations2015

    • Author(s)
      T. Mabuchi
    • Journal Title

      Progress in Math., Birkhaeuser

      Volume: 308 Pages: 395-403

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Relative stability and extremal metrics2014

    • Author(s)
      T. Mabuchi
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 66 Issue: 2 Pages: 535-563

    • DOI

      10.2969/jmsj/06620535

    • NAID

      130003394369

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2014 Annual Research Report 2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] NEW EXAMPLES OF SASAKI-EINSTEIN MANIFOLDS2013

    • Author(s)
      T.Mabuchi and Y.Nakagawa
    • Journal Title

      Tohoku Mathematical Journal, Second Series

      Volume: 65 Issue: 2 Pages: 243-252

    • DOI

      10.2748/tmj/1372182724

    • NAID

      130005562143

    • ISSN
      0040-8735, 2186-585X
    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Presentation] Stabilities in the existence problem of extremal metrics2019

    • Author(s)
      Toshiki Mabuchi
    • Organizer
      Global Aspects of Projective and Kaehler Geometry, Tsinghua Sanya International Mathematics Forum
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Extremal Kaehler versions of the Yau-Tian-Donaldson Conjecture2018

    • Author(s)
      T. Mabuchi
    • Organizer
      Modern Mathematics Lecture Series, YMSC, Tsinghua University
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Polybalanced metrics associated to extremal Kaehler metrics2016

    • Author(s)
      T. Mabuchi
    • Organizer
      Complex Geometry Conference (in honor of Prof. Mok's 60th birthday)
    • Place of Presentation
      KIAS, Seoul
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Strong K-stability for general polarizations2015

    • Author(s)
      T. Mabuchi
    • Organizer
      Recent Advances in K\"ahler Geometry
    • Place of Presentation
      Vanderbilt University
    • Year and Date
      2015-05-19
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] K-stability and strong K-stability2014

    • Author(s)
      T. Mabuchi
    • Organizer
      Complex Geometry and Algebraic Geometry
    • Place of Presentation
      Beijing International Center for Mathematical Research
    • Year and Date
      2014-07-14
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Donaldson-Tian-Yau’s conjecture for general polarization2013

    • Author(s)
      T. Mabuchi
    • Organizer
      Ricci curvature: limit spaces and Kaehler geometry
    • Place of Presentation
      ICMS, Edinburgh
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The Donaldson-Tian-Yau conjecture for extremal metrics2013

    • Author(s)
      T. Mabuchi
    • Organizer
      The Asian Mathematical Conference 2013
    • Place of Presentation
      BEXCO, Busan, Korea
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] A new version of the Donaldson-Tian-Yau conjecture for extremal metrics2013

    • Author(s)
      T. Mabuchi
    • Organizer
      Workshop "Extremal Kaehler Metrics"
    • Place of Presentation
      CRM, Montral
    • Related Report
      2013 Annual Research Report
    • Invited

URL: 

Published: 2013-05-21   Modified: 2020-03-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi