Study of algebraic cycles in algebraic and arithmetic geometry

2019 Fiscal Year Final Research Report

• Project/Area Number: 15H03606
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokyo (2018-2019); Tokyo Institute of Technology (2015-2017)
• Principal Investigator: Saito Shuji 東京大学, 大学院数理科学研究科, 教授 (50153804)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: algebraic cycles / motives

Outline of Final Research Achievements

The research consists of two parts: (I) Generalization of theory of motives. (II) Construction of K-theory for rigid analytic spaces.
(1) We extended Voevodksy's theory of motives. Voevodksy’s theory is based on homotopy invariant sheaves. In order to extend Voevodsky's theory, we introduced reciprocity sheaves as generalization of homotopy invariant sheaves. We proved several basic properties on reciprocity sheaves, which gives a motivic interpretation of some classical theorems on cohomology such as the projective bundle formula and Grothendieck duality. Moreover, we applied theory of reciprocity sheaves to ramification theory.
(2) Motivated by works of Bloch-Esnault-Kerz and Morrow on Grothendieck's variational Hodge conjecture, we constructed a new theory of analytic K-theory of rigid spaces.It sheds new light on Grothendieck's variational Hodge conjecture from the new perspective of rigid analytic geometry.

Free Research Field

代数幾何学　数論幾何学

Academic Significance and Societal Importance of the Research Achievements

モチーフ理論は代数幾何学や数論幾何学の指導原理である．その存在を最初に予見したのはGrothendieckで1970年代のことである．仮想的なモチーフの理論は代数幾何学や数論幾何学の様々な分野に多大な影響を与えてきた．今世紀に入りVoevodskyが、滑らかな多様体にたいしてはうまく機能するモチーフ理論を構成することに成功した．当該研究では、Voevodskyの理論を拡張して一般の多様体に対しても機能するモチーフ理論を構成しつつある．この拡張により、Voevodskyの理論では不可能であったガロア表現の暴分岐や微分方程式の不確定特異点での挙動をモチーフ理論の枠組みで捉えることが可能になった．