Comprehensive studies on derived algebraic geometry, bivariant theory, topology of spaces of morphisms and related topics

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03468
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kagoshima University
• Principal Investigator: Yokura Shoji 鹿児島大学, 理工学域理学系, 名誉教授 (60182680)
• Co-Investigator(Kenkyū-buntansha): 中岡 宏行 鹿児島大学, 理工学域理学系, 准教授 (90568677) ; 石田 裕昭 鹿児島大学, 理工学域理学系, 助教 (00722422)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: derived schemes / bivariant theory / cobordism / algebraic cobordism / rational homotopy theory

Outline of Final Research Achievements

As a joint work with Toni Annala we constructed a bivariant theoretic version of Lee-Pandharipande's algebraic cobordism with bundles. We constructed a bivariant theory Ω(X,Y) which is a mixture of bivariant theory B(X→Y) and a bivariant theory KK(X,Y). We obtained results on the relationship between poset-stratified spaces and decomposition spaces. We obtained some results related to Hilali conjecture which we have learned while investigating on stratified spaces. We constructed co-operational bivariant theory which is a dual version of operational bivariant theory.

Free Research Field

トポロジー、代数幾何学

Academic Significance and Societal Importance of the Research Achievements

代表者の導入したuniversal bivariant theory（以下UBT)を用いてLee-Pandharipandeの理論の双変理論版を構成できたことにより、UBTの有効性を示せた。operational bivariant theoryの双対版として導入したco-operational bivariant theoryがalgebraic topologyでよく知られたコホモロジー作用素を研究する新しい枠組みであると捉えることができることは興味深いと言える。射に対するHilali予想の導入により、有理ホモトピー論でよく知られたHilali予想の研究の幅を広げたと言って良い。