2023 Fiscal Year Final Research Report
Fusion of nonarchimedean geometry and Arakelov geometry
| Project/Area Number |
18K03211
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | University of Tsukuba (2021-2023)
Kyoto University (2018-2020) |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | 非アルキメデス的幾何 / ベルコビッチ解析空間 / トロピカル幾何 / 骨格 / 忠実トロピカル化 |
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| Outline of Final Research Achievements |
In Arakelov geometry, we study an arithmetic variety, i.e., something defined as the common zero of a polynomial system of integer coefficients. If we fix a prime p and regard the coefficients of the polynomials definining the variety as p-adic numbers, it can be regarded as a nonarchimedean analytic geometric object over the p-adic number Thus, nonarchimedean geometric objects appear naturally in Arakelov geometry.
Among the nonarchimedean geometric objects, this study focuses mainly on Berkovich analytic spaces and tropical varieties. In particular, we obtained results on the relation between the “skeleton,” which is an important closed subset of the Berkovich analytic space, and its tropicalization; in particular, the existence of an identitification, called a “faithful tropicalization".
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| Free Research Field |
代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
研究成果は、ベルコビッチ解析空間上における様々な解析的対象を調べる上で大きな意義がある。忠実トロピカル化を通じて、解析空間上の対象をトロピカル幾何的枠組みにおいて組み合わせ論的に考察することが可能となる点が大きい。また、忠実トロピカル化問題はトロピカル化という操作における基本的問題であり、この分野の基礎理論構築という視点からも意義が深い。社会的意義は、現時点では特に見当たらないが、トロピカル幾何は他の様々な分野で応用されていることから、将来的に意味のある応用が得られることは期待できる。
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