R\'ealisations de Hodge des motifs de Voevodsky

TL;DR

This paper constructs Hodge realizations of Voevodsky motives over subfields of complex numbers, using mixed Hodge complexes and filtrations, and introduces a Deligne-Beilinson realization, advancing the understanding of motivic cohomology.

Contribution

It provides a new construction of Hodge and Deligne-Beilinson realizations for Voevodsky motives using filtrations and complexes, improving previous frameworks.

Findings

Hodge realization as cohomology of mixed Hodge DG-complex

Representation of filtrations via truncation functors

Construction of Deligne-Beilinson realization

Abstract

Over a subfield of the field of complex numbers, the Hodge realization of a geometrical motive is defined and represented as the cohomology of a mixed Hodge DG-complex in the sense of Deligne. Both filtrations are represented by truncation functors, on a Bondarko weight complex for the weight filtration and on the De Rham motivic complex for the Hodge one. The Deligne-Beilinson realization is also constructed. This preprint partially replaces the former "R\'ealisation des complexes motiviques de Voevodsky".