Intersection theory and cobordism with a quadratic twist

本课题的目的是利用源于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，可以将经典的整型不变量细化为二次型不变量，从而携带新的算术信息。建议的研究将集中在以下方向：(1)由Levine和Kass-Wickelgren发起的精细枚举几何项目。特别地，我们将考虑del Pezzo曲面的情况。(2)利用改进不变量，给出了光滑射影变上有限群作用不动点的新约束条件。(3)由SL， SLc, Sp， SO， O， Spin群之一取向的上同论的性质。相应的共乘理论将从几何的角度进行研究。(4)稳定Adams运算在厄米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Stiefel-Whitney类”在其他线性代数群中的可能推广。

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.