The representability of motivic cohomology

Abstract:

Any algebraic variety $X$ over an algebraically closed field $k$ is
associated with its Albanese variety $Alb_X.$ According to Rojtman, for
smooth proper $X,$ the torsion part of the group of rational points
$Alb_X(k)$ is canonically isomorphic to $CH_0(X)_{tor}^0,$ the torsion
part of the degree zero part of the Chow group of zero cycles. For a curve
$X,$ this isomorphism agrees with the Abel-Jacobi isomorphism
$CH^1(X)_{alg}\longrightarrow Pic_X(k),$ where $CH^1(X)_{alg}$ is the
subgroup of $CH^1(X)$ consisting of algebraically trivial cycles and
$Pic_X$ is the Picard variety.

To extend this picture to other Chow groups, Samuel introduced the concept
of regular homomorphisms. For divisors and zero cycles, the map $alb_X$
and the Abel-Jacobi isomorphism are universal with respect to regular
homomorphisms. The case of codimension $2$ cycles was also treated by
Murre.

In this talk, we explain how to extend this picture to other motivic
invariants. If time permits, we explain the relation with Griffiths's
intermediate Jacobians.