Stability conditions on derived categories and Donaldson-Thomas invariants

• Project/Area Number: 26287002
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: The 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: ¥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。

Research Products

• [Int'l Joint Research] MIT(米国)
• [Int'l Joint Research] マサチューセッツ工科大学(米国)
• [Int'l Joint Research] マサチューセッツ工科(米国)
• [Journal Article] Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas invariants (2019)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Stable pairs on nodal K3 fibrations (2018)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds (2018)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Gopakumar-Vafa invariants via vanishing cycles (2018)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Moduli stacks of semistable sheaves and representations of Ext-quivers (2018)
  • Author(s): Yukinobu Toda
  • Journal Title: Geometry and Topology Volume: 22 Issue: 5 Pages: 3083-3144
  • DOI: 10.2140/gt.2018.22.3083
  • Peer Reviewed
• [Journal Article] Contraction algebra and invariants of singularities (2018)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Gopakumar-Vafa invariants via vanishing cycles (2018)
  • Author(s): Davesh Maulik and Yukinobu Toda
  • Journal Title: Invent. Math. Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Moduli stacks of semistable sheaves and representations of Ext-quivers (2018)
  • Author(s): Yukinobu Toda
  • Journal Title: Geometry & Topology Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Non-commutative thickening of moduli spaces of stable sheaves (2017)
  • Author(s): Yukinobu Toda
  • Journal Title: Compos. Math. Volume: 153 Issue: 6 Pages: 1153-1195
  • DOI: 10.1112/s0010437x17007047
  • Peer Reviewed
• [Journal Article] Generalized Donaldson-Thomas invariants on the local projective plane (2017)
  • Author(s): Yukinobu Toda
  • Journal Title: J. Differential Geom. Volume: 106 Issue: 2 Pages: 341-369
  • DOI: 10.4310/jdg/1497405629
  • Peer Reviewed
• [Journal Article] Contraction algebra and invariants of singularities (2017)
  • Author(s): Zheng Hua and Yukinobu Toda
  • Journal Title: IMRN Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Non-commutative thickening of moduli spaces of stable sheaves (2017)
  • Author(s): Yukinobu Toda
  • Journal Title: Compositio Math Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Generalized Donaldson-Thomas invariants on the local projective plane (2017)
  • Author(s): Yukinobu Toda
  • Journal Title: JDG Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Gepner type stability condition via Orlov/Kuznetsov equivalence (2016)
  • Author(s): Yukinobu Toda
  • Journal Title: IMRN Volume: 2016 Pages: 24-82
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Stable pair invariants on Calabi-Yau 3-folds containing P2 (2016)
  • Author(s): Yukinobu Toda
  • Journal Title: Geometry and Topology Volume: 20 Pages: 555-611
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Stable pair invariants on Calabi-Yau 3-folds containing P^2 (2016)
  • Author(s): Yukinobu Toda
  • Journal Title: Geometry and Topology Volume: 20 Pages: 555-611
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] S-duality for surfaces with A_n singularities (2015)
  • Author(s): Yukinobu Toda
  • Journal Title: Math Ann Volume: 363 Pages: 679-699
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Non-commutative width and Gopakumar-Vafa invariants (2015)
  • Author(s): Yukinobu Toda
  • Journal Title: Manuscripta Math Volume: 148 Pages: 521-533
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Gepner type stability conditions on graded matrix factorizations (2014)
  • Author(s): Yukinobu Toda
  • Journal Title: Algebr. Geom. Volume: 1 Pages: 613-665
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Stability conditions and birational geometry of projective surfaces (2014)
  • Author(s): Yukinobu Toda
  • Journal Title: Compos. Math. Volume: 150 Pages: 1755-1788
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bridgeland stability conditions of threefolds II: An application to Fujita's conjecture (2014)
  • Author(s): Arend Bayer and Emanuele Macri and Yukinobu Toda
  • Journal Title: J. Algebraic Geom. Volume: 23 Pages: 693-710
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A note on Bogomolov-Gieseker type inequality for Calabi-Yau 3-folds (2014)
  • Author(s): Yukinobu Toda
  • Journal Title: Proc. Amer. Math. Soc. Volume: 142 Pages: 3387-3394
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Multiple cover formula of generalized DT invariants I: Parabolic stable pairs (2014)
  • Author(s): Yukinobu Toda
  • Journal Title: Adv. Math. 257 Volume: 257 Pages: 476-526
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities (2014)
  • Author(s): Arend Bayer and Emanuele Macri and Yukinobu Toda
  • Journal Title: J. Algebraic Geom. Volume: 23 Pages: 117-163
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories (2018)
  • Author(s): Yukinobu Toda
  • Organizer: String-Math 2018
  • Int'l Joint Research / Invited
• [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Geometry and Topology inspired by Physics
  • Int'l Joint Research / Invited
• [Presentation] Gopakumar-Vafa invariants for Calabi-Yau 3-folds and 4-folds (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Geometric correspondences of gauge theories
  • Int'l Joint Research / Invited
• [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Yamabe Memorial Symposium: Moduli Spaces in Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Moduli spaces and varieties in Shanghai
  • Int'l Joint Research / Invited
• [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Non-commutative deformatiosn and moduli spaces
  • Int'l Joint Research / Invited
• [Presentation] Semiorthogonal decompositions under d-critical flips (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Categorical and Analytic Invariants in Algebraic Geometry Ⅵ
  • Int'l Joint Research / Invited
• [Presentation] Birational geometry for d-critical loci and wall-crossing in derived categories (2018)
  • Author(s): Yukinobu Toda
  • Organizer: D-modules, Quantum Geometry and related topics
  • Int'l Joint Research / Invited
• [Presentation] Stability conditions and wall-crossing in derived categories (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Introductory Workshop: Enumerative Geometry Beyond Numbers
  • Int'l Joint Research / Invited
• [Presentation] Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Higher dimensional algebraic geometry
  • Int'l Joint Research / Invited
• [Presentation] Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds (2018)
  • Author(s): Yukinobu Toda
  • Organizer: Structures in Enumerative Geometry
  • Int'l Joint Research / Invited
• [Presentation] DT invariants and NC deformations (2017)
  • Author(s): Yukinobu Toda
  • Organizer: Geometry, Derived categories and birational geometry
  • Place of Presentation: 大阪大学
  • Year and Date: 2017-02-20
  • Int'l Joint Research / Invited
• [Presentation] Gopakumar-Vafa invariants via vanishing cycles (2017)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Donaldson-Thomas invariants on abelian 3-folds and Fourier-Mukai transforms (2017)
  • Author(s): Yukinobu Toda
  • Organizer: Developments of mathematics at IPMU: in honor of Kyoji Saito
  • Int'l Joint Research / Invited
• [Presentation] Donaldson-Thomas invariants on abelian 3-folds and Fourier-Mukai transforms (2017)
  • Author(s): Yukinobu Toda
  • Organizer: Inaugural conference for Mirror Symmetry Laboratory of HSE
  • Int'l Joint Research / Invited
• [Presentation] Gopakumar-Vafa invariants and wall-crossing (2017)
  • Author(s): Yukinobu Toda
  • Organizer: "Modern Moduli Theory" Clay workshop
  • Int'l Joint Research / Invited
• [Presentation] Gopakumar-Vafa invariants and wall-crossing (2017)
  • Author(s): Yukinobu Toda
  • Organizer: Categorical and Analytic Invariants in Algebraic Geometry V
• [Presentation] Gopakumar-Vafa invariants and wall-crossing (2017)
  • Author(s): Yukinobu Toda
  • Organizer: 2017 Taiwan Mathematical Society Annual Meeting
• [Presentation] Gopakumar-Vafa invariants via vanishing cycles (2017)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Gopakumar-Vafa invariants via vanishing cycles (2016)
  • Author(s): Yukinobu Toda
  • Organizer: Algebraic geometry and Integrable systems
  • Place of Presentation: 神戸大学
  • Year and Date: 2016-12-05
  • Int'l Joint Research / Invited
• [Presentation] Gopakumar-Vafa invariants via vanishing cycles (2016)
  • Author(s): Yukinobu Toda
  • Organizer: KMS-MSJ joint meeting
  • Place of Presentation: ソウル大学
  • Year and Date: 2016-10-20
  • Int'l Joint Research / Invited
• [Presentation] S-duality conjecture and Donaldson-Thomas invariants (2015)
  • Author(s): Yukinobu Toda
  • Organizer: String-Math 2015
  • Place of Presentation: Sanya International Mathematical Forum, Sanya, China
  • Year and Date: 2015-12-31
  • Int'l Joint Research / Invited
• [Presentation] Non-commutative virtual structure sheaves (2015)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Non-commutative thickening of moduli spaces of stable sheaves (2015)
  • Author(s): Yukinobu Toda
  • Organizer: Algebraic geometry 2015
  • Place of Presentation: University of Utah, Saltlake city, USA
  • Year and Date: 2015-07-19
  • Int'l Joint Research / Invited
• [Presentation] Non-commutative thickening of moduli spaces of stable sheaves (2015)
  • Author(s): Yukinobu Toda
  • Organizer: Derived categories and birational geometry
  • Place of Presentation: Warwick University, Covently, UK
  • Year and Date: 2015-05-31
  • Int'l Joint Research / Invited
• [Presentation] Bogomolov-Gieseker type inequality and Donaldson-Thomas invariants (2015)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Moduli of Bridgeland semistable objects on 3-folds and DT invariants (2015)
  • Author(s): Yukinobu Toda
  • Organizer: Algebraic Geometry
  • Place of Presentation: Oberwolfach, ドイツ
  • Year and Date: 2015-03-16
  • Invited
• [Presentation] Bogomolov-Gieseker type inequality and Donaldson-Thomas invariants (2015)
  • 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
  • Invited
• [Presentation] Bogomolov-Gieseker type inequality and Donaldson-Thomas invariants (2015)
  • 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
  • Invited
• [Presentation] Stable pair invariants on Calabi-Yau 3-folds containing P^2 (2014)
  • Author(s): Yukinobu Toda
  • Organizer: Current developments of mirror symmetry
  • Place of Presentation: Sanya, 中国
  • Year and Date: 2014-12-17
  • Invited
• [Presentation] Derived category of coherent sheaves and counting invariants (2014)
  • 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
  • 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
• [Remarks] 研究機関が作成したwebページ
  • URL: http://db.ipmu.jp/member/personal/755ja.html