Chow groups of Chow varieties

TL;DR

This paper computes the rational Chow groups of Chow varieties of algebraic cycles in projective space, showing their isomorphism with rational singular homology groups and establishing stability and partial computations for these groups.

Contribution

It provides explicit calculations of Chow groups for Chow varieties, proves their isomorphism with singular homology, and demonstrates stability under natural maps.

Findings

Chow groups are isomorphic to singular homology groups in the studied range.

Chow groups are stable under natural embeddings and suspension maps.

Partial computations of Chow groups for fixed degree algebraic cycles.

Abstract

Let $C_{p,d}(\mathbb{P}^n)$ be the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective $n$-space $\mathbb{P}^n$. In this paper, we compute the rational Chow groups $\mathrm{Ch}_q(C_{p,d}(\mathbb{P}^n))_\mathbb{Q}$ for $0 \le q \le d$. We show that these Chow groups are isomorphic to the corresponding rational singular homology groups $H_{2q}(C_{p,d}(\mathbb{P}^n), \mathbb{Q})$ in this range, a result that was previously known. Furthermore, we prove that the rational Chow groups of a natural completion of the Chow monoid of algebraic $p$-cycles on projective spaces coincide with the corresponding rational singular homology groups. We also establish the stability of Chow groups of Chow varieties under natural embeddings and algebraic suspension maps within a certain range. Finally, we determine the Chow groups, up to a certain level, for the space of algebraic cycles of fixed degree.