  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.
