Studies on singular Hermitian metrics via L2 theoretic methods and their applications to algebraic geometry

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K20336
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Tokyo University of Science
• Principal Investigator: Inayama Takahiro 東京理科大学, 創域理工学部数理科学科, 助教 (00907404)
• Project Period (FY): 2021-08-30 – 2024-03-31
• Keywords: 特異エルミート計量 / L2評価法 / L2拡張定理 / 中野正値性 / Griffiths正値性 / 連接層

Outline of Final Research Achievements

Curvature is a crucial concept in geometry and has been widely studied from various perspectives. Roughly speaking, curvature corresponds to the second derivative of a metric and is usually defined only for smooth metrics. In this study, we have mainly researched how to define curvature and its positivity for metrics, specifically those called singular Hermitian metrics, which are not necessarily smooth. We have achieved certain results regarding problems such as the coherence of multiplier submodule sheaves associated with singular Hermitian metrics on vector bundles and the relationship between the positivity of metrics and L2 extension indices.

Free Research Field

複素解析幾何学

Academic Significance and Societal Importance of the Research Achievements

特異エルミート計量に付随する乗数部分加群層がいつ連接層になるかという問いは，自然でかつ重要な問題である．実際，直線束の場合は，Nadelによって乗数イデアル層の連接性が解明されて以降，複素解析学や代数幾何学において数々の応用をもたらしてきた．特に，特異エルミート計量が単にGriffiths正値である場合は，Nadelの証明方法が直接適用できないため，本質的に新しいアプローチが必要となる．実際私が提唱した予想及びそれに対する部分的な解明は，その後数々の研究者によって研究，改良されている．