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As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any similar results for etale cohomology since it is not known whether the corresponding bilinear forms are positive definite? Or maybe for varieties in characteristic 0 one still can prove that these form are always positive definite using the comparison of etale cohomology with singular one?

Is anything known here in the case of 'mixed' realizations (as defined by Jannsen and Huber)?

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    $\begingroup$ Your first question sounds a lot like Grothendieck's standard conjecture "Hdg", which would still be open in general. Regarding your last question, it seems that such statements are proved in Jannsen's book in chapter 1, sections 1 and 4. $\endgroup$
    – Donu Arapura
    Commented Nov 5, 2010 at 15:15

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