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Stability conditions on derived categories and Donaldson-Thomas invariants

Research Project

Project/Area Number 26287002
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

Toda Yukinobu  東京大学, カブリ数物連携宇宙研究機構, 教授 (20503882)

Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥15,990,000 (Direct Cost: ¥12,300,000、Indirect Cost: ¥3,690,000)
Fiscal Year 2018: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2017: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2016: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2015: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2014: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Keywords連接層の導来圏 / 安定性条件 / Donaldson-Thomas不変量 / Gopakumar-Vafa不変量 / 3次元カラビヤウ多様体 / Bridgeland安定性条件 / d-臨界的スキーム / 導来圏の安定性条件 / 非可換変形
Outline of Final Research Achievements

We studied stability conditions on derived categories of coherent sheaves on Calabi-Yau 3-folds, and constructed moduli spaces of semistable objects. We applied this result to the study of Donaldson-Thomas invariants which count stable sheaves or curves on Calabi-Yau 3-folds. Moreover we gave mathematical definitions of Gopakumar-Vafa invariants on Calabi-Yau 3-folds and 4-folds, described conjectural relationships with Donaldson-Thomas invariants explicitly, and proved them in several cases.

Academic Significance and Societal Importance of the Research Achievements

3次元カラビヤウ多様体は物理学の超弦理論においても考察される数学的対象であり、特にこの上のGopakumar-Vafa不変量は物理学者のGopakumarとVafaによって提唱された重要な不変量である。よってその不変量の数学的に厳密な定義を与える事は数学・物理双方にとって意義深いものである。Maulik氏との共同研究でGopakumar-Vafa不変量の数学的定義を与えることに成功し、この不変量の更なる理解を深めることができた。

Report

(6 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
  • Research Products

    (60 results)

All 2019 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (3 results) Journal Article (24 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 24 results,  Acknowledgement Compliant: 14 results) Presentation (31 results) (of which Int'l Joint Research: 24 results,  Invited: 29 results) Book (1 results) Remarks (1 results)

  • [Int'l Joint Research] MIT(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] マサチューセッツ工科大学(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] マサチューセッツ工科(米国)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas invariants2019

    • Author(s)
      Dulip Piyaratne and Yukinobu Toda
    • Journal Title

      J. Reine Angew. Math.

      Volume: 747 Issue: 747 Pages: 175-219

    • DOI

      10.1515/crelle-2016-0006

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Stable pairs on nodal K3 fibrations2018

    • Author(s)
      Amin Gholampour, Artan Sheshmani and Yukinobu Toda
    • Journal Title

      International Mathematics Research Notices. IMRN

      Volume: 17 Issue: 17 Pages: 5297-5346

    • DOI

      10.1093/imrn/rnx035

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2018

    • Author(s)
      Yalong Cao, Davesh Maulik and Yukinobu Toda
    • Journal Title

      Advances in Mathematics

      Volume: 338 Pages: 41-92

    • DOI

      10.1016/j.aim.2018.08.013

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Gopakumar-Vafa invariants via vanishing cycles2018

    • Author(s)
      Davesh Maulik and Yukinobu Toda
    • Journal Title

      Invent. Math.

      Volume: 213 Issue: 3 Pages: 1017-1097

    • DOI

      10.1007/s00222-018-0800-6

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Moduli stacks of semistable sheaves and representations of Ext-quivers2018

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Geometry and Topology

      Volume: 22 Issue: 5 Pages: 3083-3144

    • DOI

      10.2140/gt.2018.22.3083

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Contraction algebra and invariants of singularities2018

    • Author(s)
      Zheng Hua and Yukinobu Toda
    • Journal Title

      International Mathematics Research Notices. IMRN

      Volume: 10 Issue: 10 Pages: 3173-3198

    • DOI

      10.1093/imrn/rnw333

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Gopakumar-Vafa invariants via vanishing cycles2018

    • Author(s)
      Davesh Maulik and Yukinobu Toda
    • Journal Title

      Invent. Math.

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Moduli stacks of semistable sheaves and representations of Ext-quivers2018

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Geometry & Topology

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-commutative thickening of moduli spaces of stable sheaves2017

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Compos. Math.

      Volume: 153 Issue: 6 Pages: 1153-1195

    • DOI

      10.1112/s0010437x17007047

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Generalized Donaldson-Thomas invariants on the local projective plane2017

    • Author(s)
      Yukinobu Toda
    • Journal Title

      J. Differential Geom.

      Volume: 106 Issue: 2 Pages: 341-369

    • DOI

      10.4310/jdg/1497405629

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Contraction algebra and invariants of singularities2017

    • Author(s)
      Zheng Hua and Yukinobu Toda
    • Journal Title

      IMRN

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Non-commutative thickening of moduli spaces of stable sheaves2017

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Compositio Math

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Generalized Donaldson-Thomas invariants on the local projective plane2017

    • Author(s)
      Yukinobu Toda
    • Journal Title

      JDG

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Gepner type stability condition via Orlov/Kuznetsov equivalence2016

    • Author(s)
      Yukinobu Toda
    • Journal Title

      IMRN

      Volume: 2016 Pages: 24-82

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Stable pair invariants on Calabi-Yau 3-folds containing P22016

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Geometry and Topology

      Volume: 20 Pages: 555-611

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Stable pair invariants on Calabi-Yau 3-folds containing P^22016

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Geometry and Topology

      Volume: 20 Pages: 555-611

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] S-duality for surfaces with A_n singularities2015

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Math Ann

      Volume: 363 Pages: 679-699

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Non-commutative width and Gopakumar-Vafa invariants2015

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Manuscripta Math

      Volume: 148 Pages: 521-533

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Gepner type stability conditions on graded matrix factorizations2014

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Algebr. Geom.

      Volume: 1 Pages: 613-665

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Stability conditions and birational geometry of projective surfaces2014

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Compos. Math.

      Volume: 150 Pages: 1755-1788

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Bridgeland stability conditions of threefolds II: An application to Fujita's conjecture2014

    • Author(s)
      Arend Bayer and Emanuele Macri and Yukinobu Toda
    • Journal Title

      J. Algebraic Geom.

      Volume: 23 Pages: 693-710

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A note on Bogomolov-Gieseker type inequality for Calabi-Yau 3-folds2014

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 142 Pages: 3387-3394

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Multiple cover formula of generalized DT invariants I: Parabolic stable pairs2014

    • Author(s)
      Yukinobu Toda
    • Journal Title

      Adv. Math. 257

      Volume: 257 Pages: 476-526

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities2014

    • Author(s)
      Arend Bayer and Emanuele Macri and Yukinobu Toda
    • Journal Title

      J. Algebraic Geom.

      Volume: 23 Pages: 117-163

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      String-Math 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Geometry and Topology inspired by Physics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gopakumar-Vafa invariants for Calabi-Yau 3-folds and 4-folds2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Geometric correspondences of gauge theories
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Yamabe Memorial Symposium: Moduli Spaces in Algebraic Geometry
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Moduli spaces and varieties in Shanghai
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Non-commutative deformatiosn and moduli spaces
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Semiorthogonal decompositions under d-critical flips2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Categorical and Analytic Invariants in Algebraic Geometry Ⅵ
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      D-modules, Quantum Geometry and related topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability conditions and wall-crossing in derived categories2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Introductory Workshop: Enumerative Geometry Beyond Numbers
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Higher dimensional algebraic geometry
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds2018

    • Author(s)
      Yukinobu Toda
    • Organizer
      Structures in Enumerative Geometry
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] DT invariants and NC deformations2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      Geometry, Derived categories and birational geometry
    • Place of Presentation
      大阪大学
    • Year and Date
      2017-02-20
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gopakumar-Vafa invariants via vanishing cycles2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      Geometry, Analysis and Mathematical Physics
    • Place of Presentation
      京都大学数学教室
    • Year and Date
      2017-02-13
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Donaldson-Thomas invariants on abelian 3-folds and Fourier-Mukai transforms2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      Developments of mathematics at IPMU: in honor of Kyoji Saito
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Donaldson-Thomas invariants on abelian 3-folds and Fourier-Mukai transforms2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      Inaugural conference for Mirror Symmetry Laboratory of HSE
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gopakumar-Vafa invariants and wall-crossing2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      "Modern Moduli Theory" Clay workshop
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gopakumar-Vafa invariants and wall-crossing2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      Categorical and Analytic Invariants in Algebraic Geometry V
    • Related Report
      2017 Annual Research Report
  • [Presentation] Gopakumar-Vafa invariants and wall-crossing2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      2017 Taiwan Mathematical Society Annual Meeting
    • Related Report
      2017 Annual Research Report
  • [Presentation] Gopakumar-Vafa invariants via vanishing cycles2017

    • Author(s)
      Yukinobu Toda
    • Organizer
      Geometry of Modui spaces of sheaves and related topics
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gopakumar-Vafa invariants via vanishing cycles2016

    • Author(s)
      Yukinobu Toda
    • Organizer
      Algebraic geometry and Integrable systems
    • Place of Presentation
      神戸大学
    • Year and Date
      2016-12-05
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gopakumar-Vafa invariants via vanishing cycles2016

    • Author(s)
      Yukinobu Toda
    • Organizer
      KMS-MSJ joint meeting
    • Place of Presentation
      ソウル大学
    • Year and Date
      2016-10-20
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] S-duality conjecture and Donaldson-Thomas invariants2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      String-Math 2015
    • Place of Presentation
      Sanya International Mathematical Forum, Sanya, China
    • Year and Date
      2015-12-31
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-commutative virtual structure sheaves2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      Categorical and Analytic invariants in Algebraic Geometry
    • Place of Presentation
      Stelkov Institute, Moscow, Russia
    • Year and Date
      2015-09-13
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-commutative thickening of moduli spaces of stable sheaves2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      Algebraic geometry 2015
    • Place of Presentation
      University of Utah, Saltlake city, USA
    • Year and Date
      2015-07-19
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-commutative thickening of moduli spaces of stable sheaves2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      Derived categories and birational geometry
    • Place of Presentation
      Warwick University, Covently, UK
    • Year and Date
      2015-05-31
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Bogomolov-Gieseker type inequality and Donaldson-Thomas invariants2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      Algebraic Geometry and applications to physics and dynamics
    • Place of Presentation
      Chebyshev Laboratory, St Petursburg, Russia
    • Year and Date
      2015-05-26
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Moduli of Bridgeland semistable objects on 3-folds and DT invariants2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      Algebraic Geometry
    • Place of Presentation
      Oberwolfach, ドイツ
    • Year and Date
      2015-03-16
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Bogomolov-Gieseker type inequality and Donaldson-Thomas invariants2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      Geometry from Stability Conditions
    • Place of Presentation
      Warwick大学、イギリス
    • Year and Date
      2015-02-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Bogomolov-Gieseker type inequality and Donaldson-Thomas invariants2015

    • Author(s)
      Yukinobu Toda
    • Organizer
      Homological mirror symmetry and geometry
    • Place of Presentation
      Miami大学, アメリカ
    • Year and Date
      2015-01-28
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Stable pair invariants on Calabi-Yau 3-folds containing P^22014

    • Author(s)
      Yukinobu Toda
    • Organizer
      Current developments of mirror symmetry
    • Place of Presentation
      Sanya, 中国
    • Year and Date
      2014-12-17
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Derived category of coherent sheaves and counting invariants2014

    • Author(s)
      Yukinobu Toda
    • Organizer
      国際数学者会議
    • Place of Presentation
      ソウル、韓国
    • Year and Date
      2014-08-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Book] 連接層の導来圏に関わる諸問題2016

    • Author(s)
      戸田 幸伸
    • Total Pages
      276
    • Publisher
      数学書房
    • Related Report
      2015 Annual Research Report
  • [Remarks] 研究機関が作成したwebページ

    • URL

      http://db.ipmu.jp/member/personal/755ja.html

    • Related Report
      2015 Annual Research Report

URL: 

Published: 2014-04-04   Modified: 2022-02-21  

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