实代数簇双分次Cech上同调与KH理论

There is a natural Galois group Gal(C/R)=Z/2Z-action on real algebraic varieties. Hence the equivariant cohomology theory and equivariant K-theory are indispensable tools for the study of real algebraic varieties. This project, on one hand, will establish the sheaf theoretic bigraded Cech cohomology theory, which combines the equivariant cohomology theory and Cech type cohomology. While the equivariant cohomology theory has excellent algebraic and topological properties, the Cech cohomology is strongly cohesive to the geometric structure of real varieties. Meanwhile, this cohomology theory helps us construct some kind of Deligne type cohomology theory on the category of real algebraic varieties. On the other hand, this project investigates the KH-theory and the above bigraded Cech cohomology theory on real algebraic varieties from homotopy viewpoint. KH-theory is a kind of topological equivariant K-theory on real varieties which exhibits the anti-symmetric structure of complex vector bundles over real varieties. This project builds the morphisms from KH-groups of real varieties to bigraded Cech cohomology groups via constructing the proper classifying spaces for KH-theory and bigraded Cech cohomology theory, respectively. Furthermore, we link this to the Deligne cohomology theory, and by which we introduce K-theory to the study of Deligne cohomology theory over real algebraic varieties.

实代数簇具有Galois群Gal(C/R)=Z/2Z作用，这使得等变上同调和等变K理论成为研究实代数簇的有力工具。本项目一方面将在一般实代数簇范畴上建立基于层论的"双分次Cech上同调理论"，把具有良好代数和拓扑性质的等变上同调理论与反映了研究对象几何性质的Cech上同调结合起来，同时这个上同调理论也是有助于我们在一般实代数簇范畴上建立某种Deligne型上同调理论。 另一方面，本项目从同伦观点深入研究实代数簇上的KH理论和双分次Cech上同调理论，KH理论是实代数簇上一种等变拓扑K理论，反映了实代数簇上复向量丛的反对称结构。本项目构造关于KH理论及双分次Cech上同调理论的分类空间，建立实代数簇KH群到双分次Cech上同调群的态射，进而连接到实代数簇Deligne上同调理论，将K理论引入到实代数簇Deligne上同调理论研究的框架中。

项目组对有限群G，建立了由正交表示环RO(G)分次的层论等变Bredon上同调理论，建立了RO(G)分次等变Bredon上同调与等变Cech 超上同调的同构关系。.本项目研究实代数簇上的Cech上同调和KH理论，因实代数簇具有Galois群Gal(C/R)=Z/2Z作用，实代数簇上的RO(G)分次等变Bredon上同调可具体表示为由RO(Z/2Z)=Z+Z分次，即由整数p, q双分次。本项目在一般实代数簇范畴上建立了基于层论的“双分次Cech上同调理论”，把具有良好代数和拓扑性质的等变上同调理论与反映了研究对象几何性质的Cech上同调结合起来。.另一方面，本项目研究了实代数簇上的KH理论和双分次Cech上同调理论间的联系，KH理论是实代数簇上一种等变拓扑K理论，反映了实代数簇上复向量丛的反对称结构。本项目构造了关于KH理论及双分次Cech上同调理论的分类空间，建立实代数簇KH群到双分次Cech上同调群的态射。