The multiplicative group action on singular varieties and Chow varieties

TL;DR

This paper investigates the effects of multiplicative group actions on singular complex projective varieties, applies findings to Chow varieties to compute specific algebraic groups, and discusses their structure and rationality properties.

Contribution

It provides new insights into the structure of Chow varieties under group actions, including calculations of Chow groups and Lawson homology, and addresses rationality questions with counterexamples.

Findings

Chow groups of 0-cycles are computed for Chow varieties.

Lawson homology groups of 1-cycles are determined.

Counterexamples to Shafarevich's rationality question are provided.

Abstract

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A short survey on the structure of the Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's question on the rationality of the irreducible components of Chow varieties.