Lawson homology groups of Chow varieties

TL;DR

This paper computes the rational Lawson homology groups of Chow varieties of effective algebraic cycles in projective spaces, establishing their stability and relation to singular homology.

Contribution

It provides explicit calculations of Lawson homology groups for Chow varieties and proves their stability and isomorphism to singular homology in certain cases.

Findings

Rational Lawson homology groups are computed for Chow varieties in specified degrees.

Lawson homology groups of a natural completion are isomorphic to singular homology.

Stability of Lawson homology groups under embeddings and suspension maps is established.

Abstract

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 $L_qH_k(C_{p,d}(\mathbb{P}^n))_\mathbb{Q}$ for $0 \leq 2q\leq k \leq 2d$. 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.