Theory of residues associated with localization of characteristic classes and its applications

2022 Fiscal Year Final Research Report

• Project/Area Number: 16K05116
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Hokkaido University
• Principal Investigator: SUWA Tatsuo 北海道大学, 理学研究院, 名誉教授 (40109418)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Keywords: 幾何学 / 複素解析幾何学 / 特性類の局所化 / 相対コホモロジー / 局所双対性 / 留数 / 佐藤超関数 / 特異多様体

Outline of Final Research Achievements

The purposes of this research project were to investigate the localization of characteristic classes using relative de Rham, Dolbeault and Bott-Chern cohomologies, to find explicitly the residues obtained via the local duality and to apply these to various problems.
The following are the research achievements: 1. Generalization of the Lefschetz coincidence formula (with J.-P. Brasselet), 2. Theory of relative Bott-Chern cohomollogies and applications (with M. Correa), 3. Behavior of the Hodge structure of a complex manifold under blowing-up (with D. Angella, N. Tardini and A. Tomassini), 4. Simple explicit representation of Sato hyperfunctions and their operations (with N. Honda and T. Izawa).

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

数学の研究により得られた知見により文化の発展に貢献する. 本研究代表者が推し進める局所化理論が発展し, さまざまな方面での応用が見出されている. 研究成果の概要欄で述べた成果の内, 特に当初予期されなかったこととして, 佐藤超関数およびそれに関連した演算, 局所双対性等が相対 Dolbeault コホモロジー論を用いると簡明かつ明示的に表せることが分かり, 超関数論の新たな展開を見た. これは複素解析幾何学, 解析学に新たな境地を拓くことが期待される．