isogenic homotopy theory and its applications to geometry and derived algebraic geometry

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04872
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya Institute of Technology
• Principal Investigator: Minami Norihiko 名古屋工業大学, 工学(系)研究科(研究院), 教授 (80166090)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 代数幾何 / モチビックホモトピー論 / ホモトピー論 / 高次単線織性 / 高次ファノ多様体

Outline of Final Research Achievements

Classical (stable) homotopy theory is now known to be subsumed by the Morel-Voevodsky A1-(stable)homotopy theory, which originates in algebraic geometr, which deals with common zeros of polynomials and polynomial maps between them. In search of some hierachy structure in the algebr-geometrically origined A1-(stable)homotopy theory, such as the Hopkins-Smith hierachy structure in the classical (stable) homotopy theory, I obtanied some unexpected very interesting results concerning some hierachy structures about higher uniruledness structure. Here, higher uniruledness structure is, roughly speaking ,
represents how many independent lines are passing through each point of a given algebro-geometric object. Surfaces whose arbitrary point is contained in at least one line are called ruled surfaces, and are sometimes found in architecture also.

Free Research Field

(モチビック)ホモトピー論

Academic Significance and Societal Importance of the Research Achievements

考察の対象が複雑な場合，適当な同値関係を導入して考察の対象を簡単なものにすることは，日常生活でも良く行います．このような操作を抽象的に研究するのがホモトピー論ですが，より直観的な幾何的に展開しても納得できる意味で同値であることがGrothendieck,Kanらにより知られています．そして，Hopkins-Smithらにより，幾何的対象の客観的データを与えるコホモロジー論というものを用いて，更なる同一視を要請した安定ホモトピー論において，有る階層構造が得られました．本研究は，このような古典的ホモトピー論を含むA1(安定)ホモトピー論への大域的性質への応用と，代数幾何自身への応用が期待されます．