Brad Drew: Mixed motives and their underlying mixed Hodge modules

Deligne showed that the Betti cohomology of a complex variety carries two natural filtrations satisfying the axioms for a mixed Hodge structure. This mixed Hodge structure is a much finer invariant than the Betti cohomology itself. Saito's theory of mixed Hodge modules generalizes Deligne's constructions to the relative setting of families of complex varieties, where cohomology groups are replaced by perverse sheaves.

Using techniques from higher algebra, we construct a realization functor from Voevodsky's triangulated category of mixed motives over a complex variety S to Saito's derived category of mixed Hodge modules over S compatible with Grothendieck's six operations. By way of motivation, we will discuss why such a realization functor is relevant to the conservativity conjecture.