Symplectically oriented cohomology theories of algebraic varieties

代数簇的面向辛的上同调理论

• 批准号: EP/H021566/1
• 负责人: Grigory Garkusha
• 金额: $ 1.93万
• 依托单位: Swansea University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH021566%2F1
• 关键词: Symplectically; oriented; cohomology; theories; algebraic

The versatility of A^1-homotopy theory and its associated range of cohomological techniques has made it an important branch of mathematics. Recently there have been several fundamental developments which have been used to solve a number of longstanding problems. The new strategically important developments are related to the names of V. Voevodsky (Fields Medal 2002), M. Rost, A. Suslin, I. Panin, M.Levine, F. Morel.The principal aim of this proposal is the study of symplectically oriented cohomology theories on algebraic varieties introduced recently by Nenashev-Panin-Walter. We plan to construct bijective correspondences between symplectic orientations, Pontriagin structures, Pontriagin-Thom structures and symplectic trace structures (symplectic integrations) respectively on a given ring cohomology theory. Another aim of the project is to give an explicitdescription for symplectic trace structures on a symplectically oriented cohomology theory. The theory is illustrated by the symplectic algebraic cobordism of Voevodsky, the hermitian algebraic K-theory and by other examples.The research will be undertaken in the Department of Mathematics, Swansea University.

A^1-同伦理论及其相关的上同调技术的多功能性使其成为数学的一个重要分支。最近出现了一些基础性的发展，这些发展已被用来解决许多长期存在的问题。新的具有重要战略意义的进展与 V. Voevodsky（2002 年菲尔兹奖）、M. Rost、A. Suslin、I. Panin、M.Levine、F. Morel 的名字有关。本提案的主要目的是研究Nenashev-Panin-Walter 最近提出了关于代数簇的面向辛的上同调理论。我们计划在给定的环上同调理论上分别构建辛取向、Pontriagin结构、Pontriagin-Thom结构和辛迹结构（辛积分）之间的双射对应。该项目的另一个目标是在辛导向的上同调理论上给出辛迹结构的明确描述。并通过Voevodsky的辛代数协边主义、厄密代数K理论等实例来说明该理论。该研究将在斯旺西大学数学系进行。

项目成果

K-motives of algebraic varieties

代数簇的 K-动机

• DOI: 10.48550/arxiv.1108.0375
• 发表时间: 2011
• 作者: Garkusha G

$K$-motives of algebraic varieties

$K$-代数簇的动机

• DOI: 10.4310/hha.2012.v14.n2.a13
• 发表时间: 2012
• 期刊: Homology, Homotopy and Applications
• 作者: Garkusha G