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Mirror symmetry conjecture and development of symplectic geometry

Research Project

Project/Area Number 15H02054
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNagoya University

Principal Investigator

Ohta Hiroshi  名古屋大学, 多元数理科学研究科, 教授 (50223839)

Project Period (FY) 2015-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥41,470,000 (Direct Cost: ¥31,900,000、Indirect Cost: ¥9,570,000)
Fiscal Year 2019: ¥8,190,000 (Direct Cost: ¥6,300,000、Indirect Cost: ¥1,890,000)
Fiscal Year 2018: ¥8,450,000 (Direct Cost: ¥6,500,000、Indirect Cost: ¥1,950,000)
Fiscal Year 2017: ¥8,190,000 (Direct Cost: ¥6,300,000、Indirect Cost: ¥1,890,000)
Fiscal Year 2016: ¥7,670,000 (Direct Cost: ¥5,900,000、Indirect Cost: ¥1,770,000)
Fiscal Year 2015: ¥8,970,000 (Direct Cost: ¥6,900,000、Indirect Cost: ¥2,070,000)
Keywordsシンプレクティック幾何 / ミラー対称性予想 / Floer理論 / 倉西構造 / 仮想基本チェイン / 擬正則曲線 / 量子接続 / A無限大構造 / ミラー対称性 / シンプレクティック幾何学 / 深谷圏 / 巡回ホモロジー / 開閉写像 / 幾何学 / フレアー理論 / 擬正則写像 / フロベニウス多様体 / フロベニウス構造
Outline of Final Research Achievements

The moduli space of pseudoholomorphic maps from a surface to a symplectic manifold provides us with an important and powerful method to investigate global symplectic geometry. Compared with the case when the domain surface has no boundary, it is much more difficult to study the case when it has boundary mapped to a Lagrangian submanifold. By this grant, we have established the foundation of the theory of virtual fundamental chain (including the case with boundary), based on the theory of Kuranishi structure, in order to develop the intersection theory on the moduli space. We apply the foundation to a geometric realization of A_{\infty} structure and the mirror symmetry conjecture. For example, we proved a certain version of the mirror symmetry conjecture for any compact toric manifolds, and obtained new results on the structure of Hamiltonian diffeomorphism groups of certain symplectic manifolds.

Academic Significance and Societal Importance of the Research Achievements

解析力学に起源をもつシンプレクティック幾何学において、擬正則写像のなすモジュライ空間は大域的なシンプレクティック幾何を研究する上で重要かつ強力な手法を提供する。特に、ラグランジアン部分多様体に境界をもつ場合の研究は、境界がない場合に比べ難しい。本課題では、境界をもつ場合に、モジュライ空間での交叉理論を展開する上で必要となる仮想チェイン理論の基礎理論を、倉西構造を基に、汎用性のある形で確立した。倉西構造による我々の仮想チェインの基礎理論は、今後の大域的シンプレクティック幾何やミラー対称性予想の研究において基本的かつ強力な方法を提供し、多くの研究者に活発に利用されると考えられる。

Report

(6 results)
  • 2021 Final Research Report ( PDF )
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • Research Products

    (49 results)

All 2020 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (10 results) Journal Article (12 results) (of which Int'l Joint Research: 10 results,  Peer Reviewed: 12 results,  Open Access: 4 results,  Acknowledgement Compliant: 4 results) Presentation (19 results) (of which Int'l Joint Research: 10 results,  Invited: 19 results) Book (2 results) Funded Workshop (6 results)

  • [Int'l Joint Research] Simons Center for Geometry and Physics(米国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] IBS Center for Geometry and Physics(韓国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Simons Center for Geometry and Physics(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Center for Geometry and Physics(韓国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Simons Center for Geometry and Physics(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Center for Geometry and Physics(韓国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Simons Center for Geometry and Physics(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Center for Geometry and Physics(韓国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Simons Center for Geometry and Physics(米国)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] Center for Geometry and Physics(韓国)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] Meromorphic connections in filtered A∞ categories2020

    • Author(s)
      Hiroshi Ohta, Fumihiko Sanda
    • Journal Title

      Pure and Applied Mathematics Quarterly

      Volume: 16 Issue: 3 Pages: 515-556

    • DOI

      10.4310/pamq.2020.v16.n3.a7

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An analogue of Dubrovin's conjecture2020

    • Author(s)
      Fumihiko Sanda, Yota Shamoto
    • Journal Title

      Ann. Inst. Fourier (Grenoble)

      Volume: 70 Issue: 2 Pages: 621-682

    • DOI

      10.5802/aif.3321

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spectral invariants with bulk, quasi-morphisms and Lagrangian Floer theory2019

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta and k. Ono
    • Journal Title

      Memoir of the American Mathematical Society

      Volume: 260-1254 Issue: 1254 Pages: 1-266

    • DOI

      10.1090/memo/1254

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Detecting non-displaceable toric fibers on compact toric manifolds via tropicalizations2019

    • Author(s)
      Yoosik Kim, Jaeho Lee, Fumihiko Sanda
    • Journal Title

      International Journal of Mathematics

      Volume: 30 Issue: 01 Pages: 1950003-1950003

    • DOI

      10.1142/s0129167x19500034

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I2018

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
    • Journal Title

      Surv. Differ. Geom.

      Volume: 22

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Spectral invariants with bulk, quasi-morphisms and Lagrangian Floer theory2018

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
    • Journal Title

      Memoirs of the American Mathematical Society

      Volume: 未定(発行年も未定だが入力できないので暫定的に2018)

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Antisymplectic involution and Floer cohomology2017

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono
    • Journal Title

      Geometry and Topology

      Volume: 21 Issue: 1 Pages: 1-106

    • DOI

      10.2140/gt.2017.21.1

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Spectral invariants with bulk, quasi-morphisms and Lagrangian Floer theory2017

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono,
    • Journal Title

      Memoirs of the American Mathematical Society

      Volume: 未定(発行年も未定だが入力できないので暫定的に2017)

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Shrinking good coordinate systems associated to Kuranishi structures2016

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono
    • Journal Title

      Journal of Symplectic Geometry

      Volume: 14 Issue: 4 Pages: 1295-1310

    • DOI

      10.4310/jsg.2016.v14.n4.a10

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Lagrangian Floer theory and mirror symmetry on compact toric manifolds2016

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
    • Journal Title

      Asterisque

      Volume: 376 Pages: 1-340

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Shrinking good coordinate systems associated to Kuranishi structures2016

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
    • Journal Title

      J. Symplectic Geometry

      Volume: 未定(発行年も未定だが入力できないため暫定的に2016)

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Anti-symplectic involution and Floer cohomology2016

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
    • Journal Title

      Geometry and Topology

      Volume: 未定(発行年も未定)

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] On meromorphic connections in A∞ category2019

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Stability Condition, Frobenius Manifold and Mirror Symmetry
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Remarks on generation of Fukaya category2019

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Arrangement, Frobenius Manifold and Mirror Symmetry
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Meromorphic connection in $A_{\infty}$ category2019

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Special session on `SYZ Mirror Symmetry and Enumerative Geometry', AMS Sectional Meeting,
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Complete partition and application to autoequivalences2017

    • Author(s)
      Hiroshi Ohta
    • Organizer
      International Conference on ``Trends in Modern Geometry 2017" Tokyo
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Complete partition and application to autoequivalences2017

    • Author(s)
      Hiroshi Ohta
    • Organizer
      PRIMA (Pacific Rim Mathematical Association) third congress, Mexico
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Braid group, free group and autoequivalence2017

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Second BICMR and IBS-CGP Joint Symplectic Geometry Workshop, Beijing, China
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 探検 深谷圏2017

    • Author(s)
      Hiroshi Ohta
    • Organizer
      日本数学会 企画特別講演
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Generation criterion for the Fukaya category2017

    • Author(s)
      Hiroshi Ohta
    • Organizer
      FNRS-JSPS Workshop on New Techniques in Symplectic Geometry
    • Place of Presentation
      Universite Libre de Bruxelles, Belgium
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Complete partition and application to autoequivalence2016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      BICMR and IBS-CGP Joint Symplectic Geometry Workshop
    • Place of Presentation
      Jeju, Korea
    • Year and Date
      2016-10-31
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer Theory and Mirror Symmetry 12016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Summer School 「ミラーシンメトリーの数理と物理」
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2016-08-26
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Floer Theory and Mirror Symmetry 22016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Summer School 「ミラーシンメトリーの数理と物理」
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2016-08-26
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Floer Theory and Mirror Symmetry 32016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Summer School 「ミラーシンメトリーの数理と物理」
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2016-08-26
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] 深谷圏とミラー対称性予想2016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      第63回トポロジーシンポジウム
    • Place of Presentation
      神戸大学
    • Year and Date
      2016-07-05
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Higher residue pairings in Lagrangian Floer theory2016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      International Workshop on“Higher Residue Week"
    • Place of Presentation
      Kavli IPMU, The University of Tokyo
    • Year and Date
      2016-06-06
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 深谷圏の生成判定2016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      第11回代数、解析、幾何セミナー
    • Place of Presentation
      Kagoshima University
    • Year and Date
      2016-02-15
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Lagrangian Floer theory and categorical Saito theory2016

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Workshop on “Gromov-Witten theory, Gauge Theory and Dualities"
    • Place of Presentation
      Australian National University, Kioloa, Australia
    • Year and Date
      2016-01-10
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 1-loop partition and open closed map in Lagrangian Floer theory2015

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Conference on ``Geometry, Topology and Physics"
    • Place of Presentation
      National Tsing Hua University, Hsinchu, Taiwan
    • Year and Date
      2015-12-26
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Lagrangian intersection Floer theory and singularity theory2015

    • Author(s)
      Hiroshi Ohta
    • Organizer
      Seminar in Kagawa University
    • Place of Presentation
      Kagawa University
    • Year and Date
      2015-09-29
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Trace map in cyclic $A_{\infty}$ category2015

    • Author(s)
      Hiroshi Ohta
    • Organizer
      International Conference on ``Symplectic Geometry and Mathematical Physics"
    • Place of Presentation
      Sichuan University, Chengdu, China
    • Year and Date
      2015-06-30
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Book] Kuranishi structures and virtual fundamental chains2020

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
    • Total Pages
      638
    • Publisher
      Springer
    • ISBN
      9789811555626
    • Related Report
      2019 Annual Research Report
  • [Book] Lagrangian Floer theory and mirror symmetry on compact toric manifolds2016

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
    • Total Pages
      340
    • Publisher
      Societe Mathematique de France
    • Related Report
      2015 Annual Research Report
  • [Funded Workshop] Geometry and Everything, Fukaya 602019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] Geometry and Everything, Fukaya 602019

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] International workshop ``Matrix factorization and related topics, 2016"2016

    • Place of Presentation
      Kavli IPMU, the University of Tokyo
    • Year and Date
      2016-09-05
    • Related Report
      2016 Annual Research Report
  • [Funded Workshop] International workshop ``Higher Residue Week, 2016"2016

    • Place of Presentation
      Kavli IPMU, the University of Tokyo
    • Year and Date
      2016-06-06
    • Related Report
      2016 Annual Research Report
  • [Funded Workshop] Mirror Symmetry and Symplectic Geometry, Kyoto 20152015

    • Place of Presentation
      京都
    • Year and Date
      2015-12-10
    • Related Report
      2015 Annual Research Report
  • [Funded Workshop] Quantum Singularity Theory and Mirror Symmetry2015

    • Place of Presentation
      北海道
    • Year and Date
      2015-11-24
    • Related Report
      2015 Annual Research Report

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Published: 2015-04-16   Modified: 2023-01-30  

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