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2020 Fiscal Year Research-status Report

Localization formulae in Donaldson-Thomas theory of Calabi-Yau 4-folds

Research Project

Project/Area Number 19K23397
Research InstitutionThe University of Tokyo

Principal Investigator

Cao Yalong  東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (80791459)

Project Period (FY) 2019-08-30 – 2022-03-31
Keywordslocalization formulae / Donaldson-Thomas theory / Calabi-Yau 4-folds
Outline of Annual Research Achievements

During the support of the current JSPS Kakenhi funding, we have made several progress in computations of Donaldson-Thomas type invariants for Calabi-Yau 4-folds. For examples: (1) We used localization formulae to compute DT4 invariants for moduli spaces of one dimensional stable sheaves with descendent insertions and verified the conjectural relation with genus one GW invariants in several examples. (2) We used localization formulae to compute primary Pandharipande-Thomas stable pair invariants on local Calabi-Yau 4-folds and verified the conjectural formulae due to Cao-Maulik-Toda, (3) We used localization formulae to compute tautological invariants for moduli spaces of Le Potier stable pairs on local curves and verified our conjectural formula in many examples. (4) We found a K-theoretic localization formula and used it to compute K-theoretic Pandharipande-Thomas stable pair invariants and Hilbert scheme invariants on Calabi-Yau 4-folds. As a result of this computation, we found a K-theoretic DT/PT correspondence on toric Calabi-Yau 4-folds which generalises previous known DT/PT correspondence on toric Calabi-Yau manifolds of dimension three and four. The researches under the current JSPS support lead to the writing of six papers.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

We have made progress according to our original plan. The results we obtained had been written into 6 research papers (three are published in journals and three are under referee process).

Strategy for Future Research Activity

We will keep on applying the powerful localization formulae to other moduli spaces in different examples. This will provide more verifications of our previous conjectures and possibly shed new light to directions we have not touched before.

Causes of Carryover

Due to the outbreak of COVID-19, I am not able to use my travel funding to attend international conferences and visit my collaborators outside Japan.

  • Research Products

    (12 results)

All 2021 2020 Other

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

  • [Int'l Joint Research] Mathematical Institute/Utrecht university(オランダ)

    • Country Name
      NETHERLANDS
    • Counterpart Institution
      Mathematical Institute/Utrecht university
  • [Journal Article] Stable pair invariants of local Calabi-Yau 4-folds2021

    • Author(s)
      Cao Yalong, Kool Martijn, Monavari Sergej
    • Journal Title

      International Mathematics Research Notices

      Volume: in print Pages: in print

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Curve counting and DT/PT correspondence for Calabi-Yau 4-folds2020

    • Author(s)
      Cao Yalong、Kool Martijn
    • Journal Title

      Advances in Mathematics

      Volume: 375 Pages: 107371~107371

    • DOI

      10.1016/j.aim.2020.107371

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Gopakumar-Vafa type Invariants on Calabi-Yau 4-folds via descendent insertionsInsertions2020

    • Author(s)
      Cao Yalong、Toda Yukinobu
    • Journal Title

      Communications in Mathematical Physics

      Volume: 383 Pages: 281~310

    • DOI

      10.1007/s00220-020-03897-9

    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2021

    • Author(s)
      Yalong Cao
    • Organizer
      UCSD Algebraic Geometry Seminar
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2021

    • Author(s)
      Yalong Cao
    • Organizer
      Edinburgh Hodge Seminar, University of Edinburgh
    • Invited
  • [Presentation] Curve counting via stable objects in derived categories of Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      ETH Algebraic Geometry and Moduli Zoominar
    • Invited
  • [Presentation] Curve counting via stable objects in derived categories of Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      Algebraic Geometry Zoominar, Shanghai-Tech. University
    • Invited
  • [Presentation] Introduction to Donaldson-Thomas invariants on Calabi-Yau 4 folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      Online lecture series for Northeast Normal University of China
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      GTM seminar, Kavli IPMU, Japan
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      Online seminar, KIAS, South Korea
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      International Consortium of Chinese Mathematicians annual meeting
    • Invited

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Published: 2021-12-27  

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