Realisations de Hodge des motifs de Voevodsky

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”.

💡 Research Summary

The paper develops a comprehensive Hodge realization functor for geometric motives over a subfield (k\subset\mathbb{C}) in the sense of Voevodsky’s triangulated category (DM_{gm}(k)). The author’s main achievement is to represent the realization not merely as a cohomology group but as the cohomology of a mixed Hodge differential graded (DG) complex in the style of Deligne’s mixed Hodge theory. Two filtrations—weight and Hodge—are built explicitly by means of truncation functors on two well‑known complexes: the Bondarko weight complex for the weight filtration and the De Rham motivic complex for the Hodge filtration.

Weight filtration.
For each motive (M) the Bondarko weight complex (t_w(M)) lives in the bounded homotopy category of Chow motives (K^{b}(\mathrm{Chow}(k))). By applying the standard truncation functors (\tau_{\leq n}) and (\tau_{\geq n}) to this complex, a weight tower (W_{\bullet}) is obtained. The author proves that this tower coincides with Deligne’s weight filtration on the associated mixed Hodge structure, thereby giving a concrete DG‑model for the weight side of the realization.

Hodge filtration.
On the De Rham side, the motivic De Rham complex (\Omega^{\bullet}{\mathrm{mot}}(M)) is equipped with a decreasing Hodge filtration defined by the truncations (\tau^{\ge p}). The resulting filtered complex ((\Omega^{\bullet}{\mathrm{mot}}(M),F^{\bullet})) is shown to be quasi‑isomorphic to Deligne’s filtered complex ((K_{\mathbb{C}},F^{\bullet})). Consequently the Hodge filtration on cohomology agrees with the classical Hodge filtration on de Rham cohomology.

Mixed Hodge DG‑complex.
By putting the two filtrations together the author defines a mixed Hodge DG‑complex ((\Omega^{\bullet}{\mathrm{mot}}(M),W{\bullet},F^{\bullet})). This object satisfies the three Deligne axioms for a mixed Hodge structure: (i) the underlying cohomology groups are the usual Betti (or de Rham) cohomology of the motive, (ii) the weight and Hodge filtrations are opposite in the sense required for mixed Hodge structures, and (iii) the DG‑structure provides functorial morphisms compatible with both filtrations. In particular, the realization functor \