The De Rham, complex Hodge and p-adic Hodge realization functor on the derived category of relative motives over a field of characteristic zero

Abstract : We introduce the categories of geometric complex mixed Hodge modules on algebraic varieties over a subfield k ⊂ C, and for a prime number p, the categories of p-adic mixed Hodge modules on algebraic varieties over a subfield k ⊂ Cp. We then give a complex Hodge realization functor on the derived category of relative motives over k ⊂ C and a p-adic Hodge realization functor on the derived category of relative motives over k ⊂ Cp.