Additive Chow groups of schemes

TL;DR

This paper extends the theory of additive Chow groups to a broader class of schemes, establishing key properties and a surjective regulator map, thus advancing algebraic cycle and K-theory research.

Contribution

It develops a graded module structure for additive Chow groups on smooth projective varieties and generalizes the theory to arbitrary finite-type schemes with key properties.

Findings

Established projective bundle and blow-up formulas for additive Chow groups.

Defined additive Chow groups with log poles at infinity for arbitrary schemes.

Proved the surjectivity of the regulator map to absolute Kähler differentials.

Abstract

We show how to make the additive Chow groups of Bloch-Esnault, Ruelling and Park into a graded module for Bloch's higher Chow groups, in the case of a smooth projective variety over a field. This yields a a projective bundle formula as well as a blow-up formula for the additive Chow groups of a smooth projective variety. In case the base-field admits resolution of singularieties, these properties allow us to apply the technique of Guille'n and Navarro Aznar to define the additive Chow groups "with log poles at infinity" for an arbitrary finite-type k-scheme X. This theory has the usual properties of a Borel-Moore theory on finite type k-schemes: it is covariantly functorial for projective morphisms, contravariantly functorial for morphisms of smooth schemes, and has a projective bundle formula, homotopy property, Mayer-Vietoris and localization sequences. Finally, we show that the regulator map defined by Park from the additive Chow groups of 1-cycles to the modules of absolute Kaehler differentials of an algebraically closed field of characteristic zero is surjective, giving an evidence of conjectured isomorphism between these two groups.