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Further to my question,

A Naive Question on Mixed Motives and Mixed Hodge Structures

that has received very good replies and suggestions, and I really appreciate it. I am going to ask a question on generalised Hodge conjecture which is closely related to last one. From conjecture 3.22 of Marc Levine's Mixed Motives in K-theory hand book

https://www.uni-due.de/~bm0032/publ/MixMotKHB.pdf

it conjectures a functor from Nori's mixed motives to rational MHS, \begin{equation} H:\mathcal{MM}_{\text{Nori}}(k,\mathbb{Q})\rightarrow \text{MHS}_{\mathbb{Q}} \end{equation} is fully faithful, which could be seen as the generalisation of Hodge conjecture. My first question is why it is Nori's mixed motives that fits into this conjecture?

If $H$ is conjectured to be fully-faithful, then (I guess) the derived functor of $H$ (by abuse of notation) \begin{equation} DH:D^b(\mathcal{MM}_{\text{Nori}}(k,\mathbb{Q}))\rightarrow D^b(\text{MHS}_{\mathbb{Q}}) \end{equation} is also fully faithful. In D. Harrer's PhD thesis, Comparison of the Categories of Motives defined by Voevodsky and Nori

https://arxiv.org/ftp/arxiv/papers/1609/1609.05516.pdf

From main theorem, 7.4.17, there exists a realization functor \begin{equation} R_{\text{Nori}}: DM_{gm}(k,\mathbb{Q})\rightarrow D^b(\mathcal{MM}_{\text{Nori}}(k,\mathbb{Q})) \end{equation} I guess the composition of $R_{\text{Nori}}$ and $DH$ \begin{equation} DH \circ R_{\text{Nori}}:DM_{gm}(k,\mathbb{Q}) \rightarrow D^b(\text{MHS}_{\mathbb{Q}}) \end{equation} is the usual Hodge realisation functor of Voevodsky's motive. My second question is, is there a generalised Hodge conjecture stated using Voevodsky's category instead of Nori's mixed motive? e.g. like $DH \circ R_{\text{Nori}}$ is fully faithful?

Any references, comments and answers will be fully faithfully appreciated!

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    $\begingroup$ I think $R_{\textrm{Nori}}$ is itself conjectured to be fully faithful, which would imply that the composition is fully faithful as well. $\endgroup$ – Will Sawin Jun 1 '17 at 15:58
  • $\begingroup$ Do you know any references for this? $\endgroup$ – Wenzhe Jun 1 '17 at 15:59
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    $\begingroup$ As I have already told you, $MHS$ and its derived category is definitely not the "optimal" category here. The reason is that there exist non-trivial extensions of pure Hodge structures of the same weight, whereas pure motives should form a semi-simple category. You should look at graded polarizable Hodge categories after all. $\endgroup$ – Mikhail Bondarko Jun 1 '17 at 17:05
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Here are a series of comments which might help.

  1. "Why it is Nori's mixed motives that fits into this conjecture?". The usual Hodge conjecture is known to be equivalent to the full-faithfulness of the realization from pure homological motives to Hodge structures. Suppose that one sought an extension to the mixed setting, where the target is $MHS_\mathbb{Q}$. Then one would like a source which is (expected to be) abelian together with an (exact) realization functor to $MHS$. Nori's category is abelian etc. and it may be the "right" category of mixed motives, so it seems natural to use (at least to me). A side remark: Nori's Hodge conjecture is really more of an analogue rather than a strict generalization of Hodge (it doesn't imply it with unless one also assume's Grothendieck's standard conjectures or something like it).
  2. "..then (I guess) the derived functor of H ... is also fully faithful. " I don't think this follows or is even reasonable to expect. There are no higher $Ext$'s beyond $Ext^1$ in $MHS$, whereas there should be on the other side. Similar objections would apply to conjecturing your $DH\circ R_{Nori}$ is fully-faithful.
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    $\begingroup$ Another reason to use Nori's motives here is that constructing realizations for them is easy (as far as I remember).:) $\endgroup$ – Mikhail Bondarko Jun 1 '17 at 16:48

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