An exposition on finite dimensionality of Chow groups

TL;DR

This paper investigates conditions under which the natural map from Chow varieties to Chow groups is surjective for smooth projective varieties, exploring implications of this surjectivity.

Contribution

It provides criteria and consequences for the surjectivity of the map from Chow varieties to Chow groups, clarifying when all cycles are represented.

Findings

Identifies conditions for surjectivity of the Chow map.

Derives consequences of surjectivity for cycle representation.

Clarifies the relationship between Chow varieties and Chow groups.

Abstract

In this exposition we understand when the natural map from the Chow variety parametrizing codimension $p$ cycles on a smooth projective variety $X$ to the Chow group $\CH^p(X)$ is surjective. We derive some consequences when the map is surjective.