Weights, t-structures, and mixed motivic sheaves

Deligne's weights are very important for various cohomology theories. It is conjectured that weights for cohomology could be lifted to weights for the abelian categories of mixed motives (and mixed motivic sheaves, i.e. mixed motives over a base scheme S). Since the existence of mixed motives is very much conjectural, the attempts to define weights for them previously didn't yield any interesting results. In his previous work the speaker defined the notion of a weight structure (for general triangulated categories); the Chow weight structure for Voevodsky's motives yields Deligne's weights for cohomology (and several other results). Recently the notion of transversal weight and t-structures was introduced; a weight structure and a t-structure are transversal if both of them could be described in terms of certain 'generators'. This notion axiomatizes the (conjectural) relations between the Chow weight structure and the motivic t-structure (for Voevodsky's motives). This picture becomes non-conjectural when restricted to the derived categories of Deligne's 1-motives (over a smooth base) and of Artin-Tate motives over number fields; weight structures transversal to the canonical t-structures also exist for the Beilinson's graded polarizable mixed Hodge complexes and for the derived category of (Saito's) mixed Hodge modules. This notion also allows to prove that the category of mixed motivic sheaves endowed with a weight filtration (with semi-simple factors) exists over a 'reasonable' base S if certain 'standard' conjectures hold for motives over algebraically closed fields.