Absolute Hodge cohomology, Hodge and Tate conjectures and Abel-Jacobi maps

Abstract

Absolute Hodge cohomology is presented as a Poincare duality theory that generalizes Deligne-Beilinson cohomology in the sense that it includes the weight filtration. In this way it applies to general schemes over the complex numbers. The relation with motivic cohomology is again given by a regulator map that is conjectured to have dense image, at least for smooth schemes that can be defined over a number field. This conjectured property induces the classical Hodge Conjecture for smooth, projective varieties.