Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral mixed Hodge structures.


Is $\Phi$ expected to be faithful? If not, what kind of information does one expect to lose?



Motives over the complex numbers versus mixed Hodge structures
Yes, this functor is expected to be faithful, but this has very little to do with Hodge theory. The reason for this is that, if $\Psi$ denotes the forgetful functor from integral mixed Hodge structures to abelian groups, then the composed functor $\Psi\circ\Phi$ simply is the Betti realization functor. As $\Psi$ itself is faithful, it is thus sufficient to prove that the Betti realization functor is faithful. The reason why the Betti realization functor is expected to be faithful is the following. The abelian category $\mathsf{MM}(\mathbf{C})$ is expected to be the heart of the motivic $t$-structure on the triangulated category $\mathsf{DM}\_{et}(\mathbf{C})$ of Voevodsky's constructible (or geometric) *étale motives* (the one obtained from étale sheaves with transfers; the triangulated category of motives obtained from Nisnevich sheaves with transfers (which is known to coincide with $\mathsf{DM}\_{et}(\mathbf{C})$ with rational coefficients) is known not to have a motivic $t$-structure with integral coefficients anyway). The faithfulness of the Betti realization functor as above boils down to the following two properties of the triangulated version of the Betti realization functor
$$R:\mathsf{DM}\_{et}(\mathbf{C})\to\mathsf{D}^b(\mathit{Ab})$$
1) it must be conservative;


2) it must be $t$-exact.


Note that the functor $R$ is actually well defined and that property 1) is equivalent to its version with rational coefficients (because, for torsion coefficients, the functor $R$ is known to be an equivalence, by the rigidity theorem of Suslin and Voevodsky). Property 2) is required in the very definition of a motivic $t$-structure. This essentially means that, for this problem, the main difficulty is about rational coefficients. This is then very related with the standard conjectures, as explained in this paper of Beilinson: arXiv:1006.1116.


Finally, it might be worth to mention that there is an actual candidate for $\mathsf{MM}(\mathbf{C})$, constructed by Madhav Nori, which, by definition, comes with an exact and faithful functor to $\mathsf{MHS}$.


