Algebra Seminar 

11/06/2007, 11:00 — 12:00 — Room P3.10, Mathematics Building
Paulo Lima-Filho, Texas A&M University

Integral Deligne Cohomology for Real Varieties

Given a real variety $X$, we use methods of equivariant topology to introduce integral Deligne cohomology groups $H_{𝒟}^{n,p}X$. These groups are related to the bigraded Bredon cohomology $H^{n,p}X(ℂ)$ of the $\mathrm{Gal}(ℂ/ℝ)$-space $X(ℂ{)}^{\mathrm{an}}$ in the same manner that the Deligne cohomology $H_{𝒟}^{n,p}{X}_{ℂ}$ of the complex variety $X_{ℂ}$ is related to its singular cohomology. These groups are natural recipients of cycle maps from motivic cohomology and introduce variants of intermediate Jacobians and refined Abel-Jacobi maps. Amongst the examples discussed will be the case where $X=\mathrm{Spec}(K\otimes ℝ),$ where $K$ is a number field, and a geometric interpretation of $H_{𝒟}^{\mathrm{2,2}}X$. In the case of number fields, we provide a homomorphism from the Milnor $K$-theory of $K$ to the “diagonal part” of Deligne cohomology which is an isomorphism away from dimension $2$ and relate change of coefficients homomorphism with classical regulators in algebraic number theory. This is joint work with Pedro F. dos Santos.