In BBD mixed sheaves and weights for them were only defined for ($\overline{\mathbb{Q}_l}$-)sheaves over a variety $X_0$ defined over a finite field $F$. Weights start to behave better when one extends coefficients from $F$ to its algebraic closure i.e. passes from $X_0$ to $X$.

Now, BBD was published in 1982. Are any significant improvements and/or generalizations known in this field now? There is a paper by Huber and a book by Jannsen where mixed sheaves and weights for them are mentioned. Yet these authors were not able to generalize the results of BBD (in fact, an example of Jannsen shows that this is probably impossible). They also didn't extend scalars. So, are there any other papers on this subject?

Upd. There seems to be two basic ways to define weights (for sheaves) explicitly. The first method uses weights of Hodge structures. It seems that this method can work only for something like the category of mixed Hodge modules. Possibly, I will study these categories in the future. Yet at the moment I study motives, and it seems that 'motivic' people usually do not understand mixed Hodge modules (and so did not relate them with motives).

So, I am currently interested in the second method. It uses the eigenvalues of the Frobenius action. So, was anything interesting done using THIS approach after 1982?

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