In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety $X$ over a field $k$:
The regulator which Beilinson defines using Chern classes, should be replaced by an Abel-Jacobi map $\text{CH}^r(X,n)\to J$, where $J$ is the intermediate Jacobian associated to the Hodge structure $H^{2r-1}(X\times\Delta^n, X\times S^{n-1})$ ($S^{n-1}$ being the union of the $n-1$ faces in $\Delta^n$).
Questions:
(1) Has this actually been done in some later paper by Bloch or anybody else?
(2) What does he mean by "Hodge structure associated to $H^{2r-1}(X\times\Delta^n, X\times S^{n-1})$"?
What even is $H^{2r-1}(X\times\Delta^n, X\times S^{n-1})$?
I can imagine a few possibilities, but I'd like to get some insight from someone who has been acquainted with this for longer than I have.
