Motivic invariants of symmetric powers of curves

TL;DR

This paper explores how various motivic invariants of symmetric powers of smooth projective curves relate to the Jacobian, extending classical results to new invariants like higher Chow groups and K-theory.

Contribution

It generalizes Macdonald and Collino's results to include a range of motivic invariants such as Weil-cohomology, higher Chow groups, and rational K-groups.

Findings

Invariants of symmetric powers are expressed in terms of the Jacobian.

Extension of classical results to higher Chow groups and K-theory.

Unified framework for various motivic invariants of curves.

Abstract

We study the structure of various invariants of the symmetric powers of a smooth projective curve in terms of that of the Jacobian of the curve. We generalise the results of Macdonald and Collino to various invariants including the Weil-cohomology theory, the higher Chow groups, the additive higher Chow groups and the rational $K$-groups.