Algebraic Cycles of a Fixed Degree

TL;DR

This paper studies the stability of homotopy groups of Chow varieties of effective p-cycles of fixed degree, providing new insights and answering a question about degree two cycles.

Contribution

It proves the stability of homotopy groups for Chow varieties as dimensions grow and addresses a specific open question for degree two cycles.

Findings

Homotopy groups of Chow varieties are stable with increasing p or n.

Negative answer to Lawson and Michelsohn's question for degree two cycles.

Provides new understanding of the topology of cycle spaces.

Abstract

In this paper, the homotopy groups of Chow variety $C_{p,d}(P^n)$ of effective $p$-cycles of degree $d$ is proved to be stable in the sense that $p$ or $n$ increases. We also obtain a negative answer to a question by Lawson and Michelsohn on homotopy groups for the space of degree two cycles.