The Hodge realization functor on the derived category of relative motives

TL;DR

This paper constructs a Hodge realization functor linking the derived category of constructible motives to mixed Hodge modules for complex algebraic varieties, preserving key operations and tensor structures, and establishes related base change and comparison theorems.

Contribution

It introduces a functorial Hodge realization on the derived category of motives that respects operations and tensor products, extending the understanding of motives and Hodge modules.

Findings

Hodge realization functor is compatible with four operations and tensor product.

Established a base change theorem for algebraic De Rham cohomology.

Provided a relative comparison theorem between algebraic and analytic De Rham cohomology.

Abstract

We give, for a complex algebraic variety $S$, a Hodge realization functor $\mathcal F_S^{Hdg}$ from the derived category of constructible motives $DA_c(S)$ 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 $f^*$,$f_*$,$f_!$,$f^!$ on $DA_c(-)$ and $D(MHM(-))$, making the Hodge realization functor a morphism of 2-functor wich for a given $S$ sends $DA_c(S)$ 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.