Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a **t**-structure? If a **t**-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?


My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that


(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a **t**-structure, or


(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a **t**-structure is unknown, or


(3) no such conditions are known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in that generality.


I believe from my reading that option (1) is not true, but I've included it just to make sure. Thanks!



When does a triangulated category have a heart?
A silly remark is that "trivial" $t$-structures always exist. You should probably say that you want a bounded or a non-degenerate $t$-structure. As far as I remember, non-zero negative $K$-groups of $T$ should give an obstruction for the former condition if you believe that the heart is noetherian or something like this. Note also that
these groups for $DM\_{gm}$ are isomorphic to that of Chow motives; see Sosnilo, Vladimir, Theorem of the heart in negative K-theory for weight structures. Doc. Math. 24 (2019), 2137–2158.


As for comparing $DM\_{gm}$ with $D^b(MM)$: try to read (the introduction to?) Positselski, Leonid, Mixed Artin-Tate motives with finite coefficients. Mosc. Math. J. 11 (2011), no. 2, 317–402.


