Intersection theory and cobordism with a quadratic twist

相交理论和二次扭曲的协边

• 批准号: 437860477
• 负责人: Privatdozent Dr. Olivier Haution
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/437860477?language=en
• 关键词: Intersection; theory; cobordism; quadratic; twist

The aim of this project is to obtain new results in algebraic geometry via methods stemming from A1-homotopy theory. Using this theory, classical invariants with integral values can be refined into quadratic forms-valued ones, which carry new arithmetic information. The proposed research will concentrate in the following directions:(1) Refined enumerative geometry, a programme started by Levine and Kass-Wickelgren. In particular, the case of del Pezzo surfaces will be considered.(2) Use of refined invariants to provide new restrictions on fixed points of finite group actions on smooth projective varieties.(3) Properties of cohomology theories oriented by one of the groups SL, SLc, Sp, SO, O, Spin. The corresponding cobordism theories will be studied from a geometric point of view.(4) Applications of the stable Adams operations in hermitian K-theory.(5) Study of isotropic motivic categories.(6) Possible generalisations of Smirnov-Vishik "subtle Stiefel-Whitney classes" of quadratic forms to the context of other linear algebraic groups.

这个项目的目的是通过源于A1-同伦理论的方法在代数几何中获得新的结果。利用这一理论，经典的整数值不变量可以精化为二次型值不变量，从而携带新的算术信息。建议的研究将集中在以下几个方向：（1）精化枚举几何，一个计划开始的莱文和Kass-Wickelgren。特别是，德尔佩佐表面的情况下，将被考虑。(2)利用精化不变量对光滑射影簇上有限群作用的不动点提供新的限制。(3)由SL，SLc，Sp，SO，O，Spin群之一定向的上同调理论的性质。相应的配边理论将从几何的角度进行研究。(4)稳定的亚当斯运算在厄米特K理论中的应用。(5)各向同性动机范畴的研究。(6)二次型的Smirnov-Vishik“微妙的Stiefel-Whitney类”的可能推广到其他线性代数群的背景下。