The multiplicative group action on singular varieties and Chow varieties

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 brief survey on the structure of Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

In this paper, we address two questions raised by Carrell about a singular complex projective variety with a multiplicative group action, providing both positive and negative results. These results are then utilized in the context of Chow varieties, yielding Chow groups of 0-cycles and Lawson homology groups of 1-cycles. Additionally, a brief survey on the structure of Chow varieties is conducted for comparison and completeness, along with the presentation of counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties.