Beilinson's conjectural filtration on Chow groups

Let $k$ be a field, let $\mathcal V_k$ be the category of smooth projective varieties over $k$, and let $A^j(X)$ denote the codimension-$j$ Chow group of $X$. Write $(A^j(X))_{hom}$ for the subgroup of homologically trivial cycles. For a decreasing filtration $F^\bullet$ on $A^j(X)$, set

$$Gr^\nu_F A^j(X)=F^\nu A^j(X)/F^{\nu+1}A^j(X).$$

Beilinson's conjecture. For every $X\in\mathcal V_k$ there exists a decreasing filtration $F^\bullet$ on $A^i(X)$ such that: (a) $F^0A^j(X)=A^j(X)$ and $F^1A^j(X)=(A^j(X))_{hom}$; (b) $F^\bullet$ is compatible with intersection products; (c) it is compatible with $f^*$ and $f_*$ for morphisms $f:X\to Y$; (d), if the Künneth components of $\Delta_X$ are algebraic, $Gr^\nu_F A^j(X)$ depends only on the motive $h^{2j-\nu}(X)$ in $\mathcal M_{hom}$; and (e) $F^{j+1}A^j(X)=0$ for all $j$.

The conjecture predicts a canonical filtration on Chow groups governed by homological motives and functorial under correspondences. Its existence would organize the relationship between rational and homological equivalence; the source also notes that, if it exists, the filtration is unique.