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

Research Project

Project/Area Number 19K23397
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeMulti-year Fund
Review Section 0201:Algebra, geometry, analysis, applied mathematics,and related fields
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
Project Status Granted (Fiscal Year 2020)
Budget Amount *help
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordslocalization formulae / Donaldson-Thomas theory / Calabi-Yau 4-folds
Outline of Research at the Start

In 2013-2015, the applicant with Leung and Borisov with Joyce made progress on the theory of Donaldson-Thomas invariants on Calabi-Yau 4-folds. Computations of such invariants are in general very difficult. The proposal aims to develop localization formulae to effectively compute such invariants when there are torus actions on the corresponding moduli spaces. In particular, we aim to compute DT4 invariants for: (1) moduli spaces of stable pairs, (2) Hilbert schemes.

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.

Report

(2 results)
  • 2020 Research-status Report
  • 2019 Research-status Report

Research Products

(22 results)

All 2021 2020 2019 Other

All Int'l Joint Research (1 results) Journal Article (8 results) (of which Int'l Joint Research: 8 results,  Peer Reviewed: 8 results) Presentation (12 results) (of which Int'l Joint Research: 1 results,  Invited: 12 results) Book (1 results)

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

    • Related Report
      2020 Research-status Report
  • [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

    • Related Report
      2020 Research-status Report
    • 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

    • Related Report
      2020 Research-status Report
    • 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

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao, Jacob Gross and Dominic Joyce
    • Journal Title

      Advances in Mathematics

      Volume: -

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2020

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

      Journal of the European Mathematical Society

      Volume: -

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Remarks on Kawamata’s effective non-vanishing conjecture for manifolds with trivial first Chern classes2020

    • Author(s)
      Yalong Cao, Chen Jiang
    • Journal Title

      Mathematische Zeitschrift

      Volume: -

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds II: Fano 3-folds2019

    • Author(s)
      Cao Yalong
    • Journal Title

      Communications in Contemporary Mathematics

      Volume: - Pages: 1950060-1950060

    • DOI

      10.1142/s0219199719500603

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Counting conics on sextic 4-folds2019

    • Author(s)
      Cao Yalong
    • Journal Title

      Mathematical Research Letters

      Volume: 26 Pages: 1343-1357

    • DOI

      10.4310/mrl.2019.v26.n5.a5

    • Related Report
      2019 Research-status Report
    • 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
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2021

    • Author(s)
      Yalong Cao
    • Organizer
      Edinburgh Hodge Seminar, University of Edinburgh
    • Related Report
      2020 Research-status Report
    • 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
    • Related Report
      2020 Research-status Report
    • 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
    • Related Report
      2020 Research-status Report
    • 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
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      GTM seminar, Kavli IPMU, Japan
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      Online seminar, KIAS, South Korea
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Gopakumar-Vafa type invariants for Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      International Consortium of Chinese Mathematicians annual meeting
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Curve counting via stable objects in derived categories of Calabi-Yau 4-folds2020

    • Author(s)
      Yalong Cao
    • Organizer
      Algebraic Geometric Seminar, Mathematical Institute, University of Oxford
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Introduction to Donaldson-Thomas theory on Calabi-Yau 4-folds I, II, III2019

    • Author(s)
      Yalong Cao
    • Organizer
      Hua Luo-Keng Youth Forum for Algebraic Geometry, The Chinese Academy of Sciences
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Introduction to Donaldson-Thomas theory on Calabi-Yau 4-folds2019

    • Author(s)
      Yalong Cao
    • Organizer
      Department's Geometry seminar, Boston University
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Curve counting via stable objects in derived categories of Calabi-Yau 4-folds2019

    • Author(s)
      Yalong Cao
    • Organizer
      UTokyo-NTU Joint Symposium in Mathematics
    • Related Report
      2019 Research-status Report
    • Invited
  • [Book] Tsinghua Lectures in Mathematics2019

    • Author(s)
      Yalong Cao, Naichung Conan Leung
    • Publisher
      International Press of Boston, Inc.
    • ISBN
      9781571463722
    • Related Report
      2019 Research-status Report

URL: 

Published: 2019-09-03   Modified: 2021-12-27  

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