I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would like to relate these motivic weights with certain weights for pervese sheaves (a-la the ones defined in [BBD]). Yet I wonder which of these "perverse weights" are "more explicit" than the motivic ones.
I am aware of three types of "perverse weights" related to the ones in [BBD]. Those are defined on the following categories:
1) $\mathbb{Q}_l$-adic perverse sheaves on varieties over finite fields.
2) Huber's perverse sheaves on varieties over number fields.
3) Saito's weights for Hodge modules.
If $f:Y\to X$ is a finite type morphism of schemes corresponding to one of these categories, one can take the constant object $C_Y$ over $Y$ and consider the weights on the perverse (co)homology of the "mixed complex" $Rf_*(C_Y)$.
So, my question is: did anyone compute the weights of this sort for any of the three "types of weights" mentioned (the case when $X$ is the spectrum of a field is of certain interest to me; yet I would also like to consider other cases)? For example, are there any cases when one can make the corresponding computation for Huber's weights? Maybe, there exist some more constructions of "weights" and perverse sheaves of this type?
The corresponding references are:
[BBD]= Beilinson, A. A., Bernstein, J., Deligne, P.: ‘Faisceaux pervers’, in B. Teissier, J. L.Verdier, ‘Analyse et Topologie sur les Espaces singuliers’ (I), Astérisque 100, Soc. Math. France 1982.
Huber, Annette, Mixed perverse sheaves for schemes over number fields. Compositio Math. 108 (1997), no. 1, 107–121.
