Speaker: Tohru Kohrita
Date: November 23, 2017
Affiliation: TIFR, Mumbai

Abstract: Any algebraic variety $X$ over an algebraically closed field $k$ is associated with its Albanese variety $\operatorname{Alb}_X.$ According to Rojtman, for smooth proper $X,$ the torsion part of the group of rational points $\operatorname{Alb}_X(k)$ is canonically isomorphic to $\operatorname{CH}_0(X)_{\operatorname{tor}}^0,$ 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 $CH^1(X)_{alg}\longrightarrow \operatorname{Pic}_X(k),$ where $CH^1(X)_{\operatorname{alg}}$ is the subgroup of $CH^1(X)$ consisting of algebraically trivial cycles and $\operatorname{Pic}_X$ is the Picard variety. <p> To extend this picture to other Chow groups, Samuel introduced the concept of regular homomorphisms. For divisors and zero cycles, the map $\operatorname{alb}_X$ and the Abel-Jacobi isomorphism are universal with respect to regular homomorphisms. The case of codimension $2$ cycles was also treated by Murre. <p> 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.