Motivic description conjecture for the slice filtration

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$, and suppose that

$$M(X)=M_0(X)\oplus\cdots\oplus M_{2n}(X)$$

is a conjectural Chow–Künneth decomposition. Let ${\bf L}$ denote the Lefschetz motive, and use $CH_{hom,\mathbf Q}^d(X)$, $CH_{Jac,\mathbf Q}^d(X)$, $CH_{alg,\mathbf Q}^d(X)$, $J_{a,\mathbf Q}^d(X)$, and ${\rm NS}_{hom,\mathbf Q}^d(X)$ as the corresponding homologically trivial, Abel–Jacobi-kernel, algebraically trivial, algebraic intermediate-Jacobian, and homological Néron–Severi pieces. Motivic description conjecture. For any integer $d\in[1,d]$,

$$\underline{\operatorname{Hom}}({\bf L}^{n-d},M_{2n-2d}(X))={\rm NS}_{hom,\mathbf Q}^d(X),$$

and

$$\underline{\operatorname{Hom}}({\bf L}^{n-d},M_{2n-2d+1}(X))=CH_{hom,\mathbf Q}^d(X)/(CH_{Jac,\mathbf Q}^d(X)+CH_{alg,\mathbf Q}^d(X)))\oplus J_{a,\mathbf Q}^d(X).$$

This conjecturally identifies successive motivic pieces with cycle-theoretic and intermediate-Jacobian data; the displayed range $d\in[1,d]$ is reproduced exactly from the source and may contain a typographical issue.