二次型与Grothendieck-黎曼-罗赫公式的推广

近年来代数几何中新涌现的一些上同调理论，如埃尔米特K理论和Milnor-Witt上同调，把经典的上同调理论与二次型之间建立了联系。如何对从埃尔米特K理论到Milnor-Witt上同调的Borel特征标建立类似于Grothendieck-黎曼-罗赫类型的公式是该领域的重要问题之一，将有助于把向量丛和代数闭链之间的内在联系提升到二次型的层面。现有的研究方法主要运用上同调中的定向理论，以及其对应的陈类的性质，而无法应用于埃尔米特K理论这样不可定向的上同调理论。本项目拟运用广义定向和Borel类等工具，通过研究motivic同伦论中的基本类和Borel特征标之间的相互关系来尝试建立Grothendieck-黎曼-罗赫定理的一个对于二次型的推广，在此基础上深入研究埃尔米特K理论上的过滤和Brown-Gersten类型的谱序列具有的性质，以此来深入理解代数几何中与二次型有关不变量蕴含的算术-几何本质。

In the past years some new cohomology theories in algebraic geometry emerged, such as Hermitian K-theory and Milnor-Witt cohomology, which relate classical cohomology theories to quadratic forms. To establish a Grothendieck-Riemann-Roch type formula for the Borel character from Hermitian K-theory to Milnor-Witt cohomology is an important question in this domain, as such a formula would bring a quadratic refinement of the intrinsic relations between vector bundles and algebraic cycles. The current research mainly uses orientations in cohomology theories and the properties of the corresponding Chern classes, which cannot be applied to non-orientable cohomology theories such as Hermitian K-theory. In this project we plan to use tools such as generalized orientations and Borel classes, and attempt to establish a quadratic refinement of the Grothendieck-Riemann-Roch formula by considering the interaction between fundamental classes in motivic homotopy theory and the Borel character, which will further allow us to study the properties of the filtrations on Hermitian K-theory and the Brown-Gersten type spectral sequence, and therefore better understand the arithmetic-geometric nature of invariants related to quadratic forms in algebraic geometry.