Deformation theory of submanifolds characterized by differential forms

• Project/Area Number: 17K14187
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Mie University
• Principal Investigator: Moriyama Takayuki 三重大学, 教養教育院, 准教授 (60532554)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 四元数構造 / ポワソン構造 / 複素接触多様体 / ツイスター空間 / 複素接触構造 / 幾何学 / 変形理論

Outline of Final Research Achievements

The space of certain Poisson structures is decided. We provide a splitting theorem of k-vector fields and vanishing of the cohomology in complex contact manifolds. We introduce quaternionic k-vector fields in quatenionic kahler manifolds and prove that such a k-vector field corresponds to a holomorphic k-vector field on the twistor space. Moreover, a quaternionic k-vector field which is a real vector field corresponds to a holomorphic k-vector field which is real with respect to the real structure on the twistor space.

Academic Significance and Societal Importance of the Research Achievements

４次元四元数ケーラー多様体（自己双対アインシュタイン多様体）は数学のみならず物理学的にも非常に重要な研究対象であると捉えられており、特に４次元球面はその最も基本的かつ重要な例である。本研究の対象であるポワソン構造は量子化の問題と深くかかわっている。また、本研究で用いられたツイスター空間による手法は高次元においても重要な研究手法であり、様々な分野への応用、関係性の発見が期待できる。

Research Products

• [Journal Article] Some examples of global Poisson structures on S4 (2019)
  • Author(s): Moriyama, Takayuki; Nitta, Takashi
  • Journal Title: Kodai Math. J Volume: 42 Pages: 223-246
  • NAID: 130007674787
  • Peer Reviewed
• [Journal Article] Splitting theorem for sheaves of holomorphic k -vectors on complex contact manifolds (2018)
  • Author(s): Takayuki Moriyama and Takashi Nitta
  • Journal Title: International Journal of Mathematics Volume: 29 Pages: 1-21
  • Peer Reviewed
• [Presentation] Some examples of global Poisson structures on $S^4$ (2018)
  • Author(s): 森山貴之、新田貴士
  • Organizer: 第６５回幾何学シンポジウム
• [Presentation] Some examples of global Poisson structures on $S^4$ (2018)
  • Author(s): 森山貴之、新田貴士
  • Organizer: ２０１８年度日本数学会幾何学分科会
• [Presentation] Splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds (2018)
  • Author(s): 森山貴之、新田貴士
  • Organizer: ２０１８年度日本数学会幾何学分科会