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The definition of mixed Hodge complexes by Saito is a very interesting one, since it's more a cohomology theoretic than geometric generalization of Hodge structures. Since Saito's motivation for mixed Hodge complexes comes from simplicial schemes, as standard mixed Hodge modules become useless for them. This brings us to the question of cohomology, since the definition of mixed Hodge complexes is uses a very category theoretic language.

Question 1: Could mixed Hodge complexes be used to generalize current cohomology theories? (especially de Rham and Deligne cohomologies)

Question 2: Could mixed Hodge complexes be used to construct new cohomology theories?

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  • $\begingroup$ Could you give hints / possibly incorrect rough ideas of what you are imagining? $\endgroup$
    – Artur Jackson
    Commented Apr 14, 2017 at 8:33
  • $\begingroup$ @ArturJackson I'd like to know applications for Deligne and de Rham cohomology especially, since both are used in Hodge theory. $\endgroup$
    – Tatu
    Commented Apr 14, 2017 at 9:19
  • $\begingroup$ I'm not a motives expert, but I think in Voevodsky's formalism, all of these cohomology theories will satisfy Brown representability, and thus be representable by motivic spectra. You are interested to produce new (concrete examples) of motivic spectra from mixed Hodge complexes? I'm trying to understand what you mean by "generalize current cohomology theories." I'm guilty of not understanding mixed Hodge complexes yet, but this sounds very interesting. Thank you for your clarification(s). $\endgroup$
    – Artur Jackson
    Commented Apr 14, 2017 at 9:28
  • $\begingroup$ @ArturJackson By "generalizing current cohomology theories" I mean to represent these cohomology theories in the language of mixed Hodge complexes. There's also the fact that mixed Hodge complexes are defined in a triangulated category, so they naturally induce cohomology. $\endgroup$
    – Tatu
    Commented Apr 14, 2017 at 10:01
  • $\begingroup$ I understand. Cool. I'll take a look. $\endgroup$
    – Artur Jackson
    Commented Apr 14, 2017 at 10:02

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