The Hodge realization functor on the derived category of relative motives

Abstract

We give, for a complex algebraic variety $S$, a Hodge realization functor ${\mathcal{F}}_S^{Hdg}$ from the derived category of constructible motives $D{A}_c(S)$ to the derived category $D(MHM(S))$ of algebraic mixed Hodge modules over $S$. Moreover, for $f:T\to S$ a morphism of complex quasi-projective algebraic varieties, ${\mathcal{F}}_{-}^{Hdg}$ commutes with the four operation $f^{\ast }$,$f_{\ast }$,$f_{!}$,$f^{!}$ on $D{A}_c(-)$ and $D(MHM(-))$, making the Hodge realization functor a morphism of 2-functor wich for a given $S$ sends $D{A}_c(S)$ to $D(MHM(S)$, moreover ${\mathcal{F}}_S^{Hdg}$ commutes with tensor product. We also give an algebraic and analytic Gauss-Manin realization functor from which we obtain a base change theorem for algebraic De Rham cohomology and for all smooth morphisms a realtive version of the comparaison theorem of Grothendieck between the algebaric De Rham cohomology and the analytic De Rham cohomology.