Let $k$ be a field of characteristic zero and $DM_k$ be the derived category of rational Voevodsky motives. As I understand, there are conjectures which state that there is a $t$ structure on $DM_k$ such that the betti realization functor $DM_k \to Ch_{\mathbb Q}$ is exact and conservative (see Beilinson's Remarks on Grothendieck’s standard conjectures).

I am confused because (as Beilinson argues) this implies that betti realization yields a faithful functor from the heart of $DM_k$ to abelian groups. In particular, singular cohomology restricts to a fully faithful functor on the subcategory of pure chow motives under rational equivalence. This seems wrong to me, because homological equivalence of cycles is much stronger than rational equivalence of cycles.

For example over $\mathbb C$, the Hodge conjecture should only imply that the realization of the category of Chow motives is full, not faithful--since there are many cycles which are homologically equivalent but not rationally equivalent.

So from my perspective, it seems that these conjectures are wrong. Certainly, I am misunderstanding something-- what is it that makes the conjectures plausible?

Edit: After Hoyois's helpful comment, and finding his answer: motivic t-structure and realisations, it seems that if $X$ is a smooth projective variety, we expect that $Mot(X) = \bigoplus_{i} h^i(Mot(X))[-i]$ where $h^i(Mot(X))$ is in the heart and is a numerical motive. Then we expect that $$Hom(Mot(X),Mot(Y)) = \bigoplus_{i} Hom(h^i(Mot(X)),h^i(Mot(Y))) \oplus \bigoplus_{j > 0} \bigoplus_i Ext^j(h^i(Mot(X)), h^{i-j} (Mot(Y))).$$

Now the first term of the homomorphism will be given by numerical equivalence classes of algebraic cycles. So somehow the difference between rational and numerical equivalence is contained in the extension groups between motives?

Edit 2: Yes, apparently this a key point of Beilinson's conjectural category of mixed motives (apologies for my ignorance). In particular, Beilinson conjectured that the Chow group $Ch^{p}(X)$ carries a canonical filtration whose associated graded is $Ext^i(\mathbb Q, h^{2p-i}(Mot(X))).$ So even though the morphisms in the category of mixed motives only have to do with homological/numerical equivalence, the extensions should encode rational equivalence.

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    $\begingroup$ The heart of DM does not contain Chow motives, only numerical/homological motives. Chow motives also embed in DM, but in a different way (and not in the heart of the t-structure). $\endgroup$ Mar 14, 2022 at 17:33
  • $\begingroup$ @MarcHoyois Thanks for the quick reply. Is it the case that DM will not be equivalent to the derived category of its heart? Or does the conservativity conjecture imply some sort of relationship between rational and numerical/homological equivalence? $\endgroup$ Mar 14, 2022 at 18:29


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