Integral Deligne Cohomology for Real Varieties

TL;DR

This paper introduces an integral version of Deligne cohomology tailored for smooth proper real varieties, replacing singular cohomology with G-equivariant cohomology and providing new geometric insights.

Contribution

It develops a novel integral Deligne cohomology theory for real varieties using G-equivariant cohomology, extending the complex case framework.

Findings

Establishes basic properties of the new cohomology theory.

Provides geometric interpretation for groups in dimension 2, weights 1 and 2.

Abstract

We develop an integral version of Deligne cohomology for smooth proper real varieties. For this purpose the role played by singular cohomology in the complex case has to be replaced by ordinary bigraded G-equivariant cohomology, where G=Gal(C/R). This is the G-equivariant counterpart of singular cohomology. We establish the basic properties of the theory and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2.