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# Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the litterature quite hard to follow -"from experts, for experts".

Voevodsky in "Triangulated categories of motives over a field" [4.3.8] shows that there is no reasonable t-structure on his category $DM^{eff}_{gm}(k)$ (for $k=\mathbb Q$ or most fields, characterized it seems by a certain cohomological dimension condition which flies above my head) but I do not quite understand the meaning of his proof, in particular how he came up with it, though I more or less follow individual steps. Can anyone help me?

Related to this are Bruno Kahn's remarks in his review article in the "Handbook of K-theory" that the etale version of $DM_{gm,et}^{eff}(k)$, with Homsets isomorphic to $DM^{eff}_{gm}(k)$'s after tensoring by $\mathbb Q$ (using some motivic scissoring magic I presume), should have a t-structure whose heart should be (equivalent to) Nori's category. Kahn says this is related to the Hodge conjecture (not "Hodge-type standard"), can anyone flesh this relation out here?

So I wonder what happens when we pass from $gm$ to $gm,et$. Does anybody know? Is it related to (Serre-type) supersingularity-based counterexamples to the existence of a Weil cohomology theory with $\mathbb Q$ or $\mathbb Q_p$ coefficients?

Also, what has been done to relate the cohomological t-structure on the bounded derived category of Nori mixed motives to $DM_{gm,et}^{eff}(k)$? ($DM_{gm}^{eff}(k)$ and $DM_{gm,et}^{eff}(k)$ have a canonical functor to $D^b(NMM(k))$, which should be an equivalence after tensoring with $\mathbb Q$ -Beilinson's "mixed motives" conjecture.)

Are these questions related to CM lifting results of Chai-Conrad-Oort? Do their constructions explain why there is no t-structure for Nisnevich triangulated motives but there is for etale triangulated motives? Should Voevodsky's example be interpreted in that context?

As you see I am starting to divagate, so I would greatly appreciate some enlightenment.

I would also appreciate any comment on the feeling (probably quite ill-informed) that the t-structure on $DM_{gm,et}^{eff}(k)$ should not be too hard to construct but rather a matter of technical mastery of the algebra/arithmetic involved. And along this line, would the existence of that t-structure yield insight on the Tate and Hodge conjectures, or other conjectures on algebraic cycles? Those seem harder but to confirm (and following Kahn's remark mentioned above): does the Hodge conjecture imply the existence of the motivic t-structure on $DM_{gm,et}^{eff}(k)$?

Finally, I hesitate asking more but... Has anything been tried regarding "bootstrapping" the thorough understanding we have of t-structures on Tate motives to construct t-structures on larger triangulated categories of motives? I think I remember something from Déglise, I have to check... And have those constructions on Tate motives been related to the Nori-Kontsevich tannakian philosophy -e.g. to justify/formalize a hypothetical bootstrapping?