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We introduce the categories of geometric mixed Hodge modules on algebraicvarieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, thecategories of geometric $p$-adic mixed Hodge modules on algebraic varietiesover a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We thengive an Hodge realization functor on the de derived category of relativemotives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodgerealization functor on the derived category of relative motives over $k\subsetK\subset\mathbb C_p$.

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