Slice filtration on motives and the Hodge conjecture

The slice filtration on Voevodsky’s triangulated category of motives is defined by effectivity conditions. It is constructed and studied in [HK]. An analogous filtration on the homotopy category was introduced by Voevodsky. In this note we try to get a conjectural picture of the properties of the slice filtration by systematic use of the realization functor to the derived category of Hodge structures. A key ingredient is Grothendieck’s Generalized Hodge Conjecture about the analogous filtration on pure Grothendieck motives. This approach is successful, even if the answers are not what we had hoped for originally. An old example of Griffith’s allows to deduce using deep but standard conjectures the following: