Study of noncommutative Hodge structures in mirror symmetry

• Project/Area Number: 26610008
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Takahashi Atsushi 大阪大学, 理学研究科, 教授 (50314290)
• Co-Investigator(Kenkyū-buntansha): 入谷 寛 京都大学, 理学研究科, 准教授 (20448400) ; 小西 由紀子 京都大学, 理学研究科, 准教授 (30505649)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 代数学 / 幾何学 / 数理物理学 / ミラー対称性

Outline of Final Research Achievements

Mirror symmetry is a symmetry which interchanges the role of the symplectic geometry and complex algebraic geometry. The purpose of this reserch is a trial for a deep understanding of the non-commutative Hodge theory in the theorey of the mirror symmetry and for a solution to important problems in related areas.
The following are examples of our research results. Motivated by the non-commutative Hodge theory for categories of matrix factorizations, we proposed a set of axioms for orbifold Jacobian algebras, and we proved the existence and the uniqueness of them. We have also been studying the entropy of endo-functors on derived categories of coherent sheaves on smooth projective varieties and show the Gromov-Yomdin type theorem for some varieties.

Research Products

• [Int'l Joint Research] Universitaet Hannover/Universitaet Mannheim(Germany)
• [Int'l Joint Research] Higher School of Economics(ロシア連邦)
• [Int'l Joint Research] Imperial College London(United Kingdom)
• [Int'l Joint Research] Institute for Advanced Study(米国)
• [Journal Article] Strange duality between hypersurface and complete intersection singularities (2016)
  • Author(s): Wolfgang Ebeling, Atsushi Takahashi
  • Journal Title: Arnold Journal of Mathematics Volume: 印刷中 Issue: 3 Pages: 277-298
  • DOI: 10.1007/s40598-016-0044-8
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Extremal transition and quantum cohomology: Examples of toric degeneration (2016)
  • Author(s): Hiroshi Iritani, Jifu Xiao
  • Journal Title: Kyoto Math. J. Volume: 56-4 Issue: 4 Pages: 873-905
  • DOI: 10.1215/21562261-3664959
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Gamma classes and quantum cohomology of Fano manifolds: Gamma Conjectures (2016)
  • Author(s): Sergey Galkin, Vasily Golyshev, Hiroshi Iritani
  • Journal Title: Duke Math. J. Volume: 165 Issue: 11 Pages: 2005-2077
  • DOI: 10.1215/00127094-3476593
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On the existence of a global neighbourhood (2016)
  • Author(s): Tom Coates, Hiroshi Iritani
  • Journal Title: Glasgow Math. J. Volume: 58 Issue: 3 Pages: 717-726
  • DOI: 10.1017/s0017089515000427
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] On entropy of autoequivalences of smooth projective varieties (2016)
  • Author(s): Atsushi Takahashi
  • Organizer: Algebraic Geometry and Integrable Systems, Kobe 2016
  • Place of Presentation: 神戸大学
  • Year and Date: 2016-12-09
  • Int'l Joint Research / Invited
• [Presentation] Orbifold Jacobian algebras for invertible polynomials (2016)
  • Author(s): Atsushi Takahashi
  • Organizer: Categorical and analytic invariants in Algebraic geometry 4
  • Place of Presentation: 東京大学国際高等研究所数物連携宇宙研究機構
  • Year and Date: 2016-11-17
  • Int'l Joint Research / Invited
• [Presentation] On the Gamma conjecture associated with toric flips (2016)
  • Author(s): Hiroshi Iritani
  • Organizer: Workshop on Homological Mirror Symmetry: Methods and Structures
  • Place of Presentation: Institute for Advanced Study (USA)
  • Year and Date: 2016-11-17
  • Int'l Joint Research / Invited
• [Presentation] 齋藤構造の双対性と複素鏡映群への応用 (2016)
  • Author(s): 小西 由紀子
  • Organizer: 第４回 K3曲面・エンリケス曲面ワークショップ
  • Place of Presentation: 北海道教育大学札幌駅前サテライト
  • Year and Date: 2016-10-10
  • Invited
• [Presentation] On entropy of autoequivalences of projective varieties (2016)
  • Author(s): Atsushi Takahashi
  • Organizer: Categorical and analytic invariants in Algebraic geometry 3
  • Place of Presentation: Steklov Mathematical Institute (RUSSIA)
  • Year and Date: 2016-09-17
  • Int'l Joint Research / Invited
• [Presentation] Toric mirror symmetry via Seidel representation (2016)
  • Author(s): Hiroshi Iritani
  • Organizer: Workshop on geometric correspondence of gauge theories
  • Place of Presentation: ICTP Trieste (ITALY)
  • Year and Date: 2016-09-15
  • Int'l Joint Research / Invited
• [Presentation] Landau-Ginzburg mirror symmetry for toric stacks (2016)
  • Author(s): Hiroshi Iritani
  • Organizer: Number Theory and Quantum Field Theory "Mathematics of Quantum Field Theory"
  • Place of Presentation: IBS Center for Geometry and Physics (KOREA)
  • Year and Date: 2016-08-25
  • Int'l Joint Research / Invited
• [Presentation] On orbifold Jacobian algebras (2016)
  • Author(s): Atsushi Takahashi
  • Organizer: The 7th Pacific RIM Conference on Mathematics 2016
  • Place of Presentation: Seoul National University (KOREA)
  • Year and Date: 2016-06-27
  • Int'l Joint Research / Invited
• [Presentation] On entropies of autoequivalences on smooth projective varieties (2016)
  • Author(s): Atsushi Takahashi
  • Organizer: Intensive Lecture Series
  • Place of Presentation: IBS Center for Geometry and Physics (KOREA)
  • Year and Date: 2016-05-11
  • Invited
• [Presentation] On the categorical entropy and the topological entropy (2016)
  • Author(s): Atsushi Takahashi
  • Organizer: Oberseminar algebraic geometry
  • Place of Presentation: Leibniz University Hannover (GERMANY),
  • Year and Date: 2016-01-27
  • Invited
• [Presentation] From Calabi--Yau dg categories to Frobenius manifolds via primitive forms (2016)
  • Author(s): Atsushi Takahashi
  • Organizer: String field theory of the B-model
  • Place of Presentation: Center for Geometry and Physics (Korea)
  • Year and Date: 2016-01-09
  • Int'l Joint Research / Invited
• [Presentation] From Calabi--Yau dg categories to Frobenius manifolds via primitive forms (2015)
  • Author(s): Atsushi Takahashi
  • Organizer: Categorical and analytic invariants in Algebraic geometry 1
  • Place of Presentation: Steklov Mathematical Institute (Russia)
  • Year and Date: 2015-09-17
  • Int'l Joint Research / Invited
• [Funded Workshop] Mirror Symmetry and Related Topics, Kyoto 2016 (2016)
  • Place of Presentation: 京都大学数学教室
  • Year and Date: 2016-12-12
• [Funded Workshop] Mirror Symmetry and Algebraic Geometry 2015 (2015)
  • Place of Presentation: 京都大学数学教室
  • Year and Date: 2015-12-07