2021 Fiscal Year Final Research Report
Interdisciplinary research of arithmetic geometry and quantum field theory related to the moduli space of hyperbolic curves
| Project/Area Number |
18K13385
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Keywords | モジュライ空間 / 微分方程式 / oper / p曲率 / 接続 / 代数曲線 / 正標数 |
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| Outline of Final Research Achievements |
In this research, we established the theory of opers (generalizations of ordinary differential operators) defined on pointed stable curves in arbitrary characteristic and their moduli. In particular, it includes the development of the study of dormant opers, which are opers of a certain sort in positive characteristic. As a main result of this research, we given an explicit formula, conjectured by Joshi, for the generic number of dormant opers. We do so by obtaining a detailed understanding of the moduli space of dormant opers and computing the Gromov-Witten invariant for relative Grassmaniann varieties. This result reveals an interaction between studies in p-adic Teichmuller theory and certain areas of enumerative geometry, including Gromov-WItten theory.
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| Free Research Field |
代数幾何学，数論幾何学，モジュライ理論
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| Academic Significance and Societal Importance of the Research Achievements |
微分方程式あるいはその解の数論的性質に関する研究は数学において重要なテーマの一つである．本研究では，とくに正標数の場合における微分方程式およびその一般化に対する基礎理論を拡張・構築した．その結果，p進Teichmuller理論において展開されるモジュライ理論と組み合わせ論やGromov-Witten理論などの数え上げ幾何との間にある顕著な繋がりを明らかにした．その応用として，微分方程式の数え上げに関する未解決問題を証明した．このように本研究の成果は，代数的微分方程式論に対する新たな手法と観点を導入し，様々な分野の相互的発展を可能にさせるものであり，多大な波及効果が今後期待できる．
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