2022 Fiscal Year Final Research Report
Study on D-modules with irregular singular points and geometric monodromies
| Project/Area Number |
17H02848
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tohoku University (2020-2022)
University of Tsukuba (2017-2019) |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | D加群 / 不確定特異点 / 特異点理論 / 有理型関数 / フーリエ変換 / Bernstein-佐藤多項式 |
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| Outline of Final Research Achievements |
We clarified the basic properties of the Fourier transforms of irregular holonomic D-modules. In particular, we described explicitly their singular sets and showed that the irregularities (exponential factors) along them can be described by the characteristic varieties and the exponential factors of the original holonomic D-modules. These results largely extend those obtained by Brylinski about 30 years ago. In the course of our research, we studied also the singularities of meromorphic functions at infinity and at their points of indeterminacy. As a byproduct, we found also Bernstein-Sato polynomials for meromorphic functions and studied their basic properties.
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| Free Research Field |
代数解析学とその特異点理論への応用
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| Academic Significance and Societal Importance of the Research Achievements |
望月とKedlayaの理論により、近年不確定特異点を持つホロノミーD加群の理論は劇的な
進展を遂げた。我々の研究は、それをホロノミーD加群のフーリエ変換の基本的な性質の解明という、代数解析学における長年の未解決問題に応用したものである。またこの研究の過程において、有理型関数の無限遠点や不確定点における特異性について基礎的な研究を行った。これらは非常に自然な数学的対象であるが、なぜかこれまで研究されなかったものであり、今後自然科学の多方面に応用が期待できる。
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