1. There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is isomorphic to the derived category of the corresponding abelian category. Are there any analogues of this construction known for mixed Galois modules? It seems that Beilinson's method could work in the Galois case also; yet I am far from being sure here.

  2. Is it true that the functor from the derived category of graded polarizable mixed Hodge complexes to the 'ordinary DHS' is injective on morphisms?

  3. A related question: in section 21 of her book (Mixed Motives And Their Realization In Derived Categories) Huber considers the derived category of mixed polarizable realizations. In the Remark preceding Section 21.2 she says: "Fixing a P-structure reduces the number of extensions of pure structures of fixed weight .... However, it does not have influence on extensions of pure polarizable structures of different weight." Can anyone give a hint how can the second statement be proved?


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