Invariants of ramification and characteristic cycles of sheaves

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K21020
• Project/Area Number (Other): 18H05828 (2018)
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund (2019); Single-year Grants (2018)
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Tokyo Institute of Technology (2020-2022); Saitama University (2018-2019)
• Principal Investigator: Yatagawa Yuri 東京工業大学, 理学院, 准教授 (90819343)
• Project Period (FY): 2018-08-24 – 2023-03-31
• Keywords: 分岐 / エタール層 / 分岐 / 導手 / 特性サイクル

Outline of Final Research Achievements

The purpose of this research is to compute the characteristic cycles of a smooth rank 1 sheaf in terms of ramification theory of the sheaf on a smooth variety, which is a generalization of ramification theory of an extension of a local field in equal characteristic. We especially considered the case where the logarithmic characteristic cycle is defined in terms of logarithmic ramification theory of the sheaf. By using a conductor measuring the logarithmic and non-logarithmic ramification of the sheaf efficiently, we obtained a candidate of the support of the characteristic cycle with admitting blow-ups of the variety. Further, in the case where the candidate has the base of codimension less than 3 in the variety, we obtained computations of the characteristic cycle and its support in terms of ramification theory of the sheaf.

Free Research Field

数論幾何学

Academic Significance and Societal Importance of the Research Achievements

構成可能層のオイラー数はWeil予想などとも関係の深い歴史的にも重要な大域的な不変量である。オイラー数と局所的な現象である層の分岐との関係はこれまでも盛んに研究されており、オイラー数の分岐の不変量による計算は大域的な不変量を局所的な不変量から得るという幾何学における中心的な問題の一つとも見ることができる。本研究はオイラー数の計算を与える特性サイクルに分岐の不変量による計算を与えることでオイラー数と分岐の不変量を明らかにしようとするものであり、得られた結果はこれまでに知られていた特性サイクルの代数多様体が曲線または曲面の場合の計算に続くものである。