Lawson homology groups of Chow varieties

Let $C_{p,d}({\mathbb{P}}^n)$ denote the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective space ${\mathbb{P}}^n$. In this paper, we compute the rational Lawson homology groups $L_q{H}_k(C_{p,d}({\mathbb{P}}^n){)}_{\mathbb{Q}}$ for $0\le 2q\le k\le 2d$. Additionally, we prove that the rational Lawson homology groups of a natural completion of the Chow monoid of algebraic $p$-cycles in projective spaces are isomorphic to the corresponding rational singular homology groups. We also establish the stability of Lawson homology groups of Chow varieties under natural embeddings and algebraic suspension maps within a specified range.