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.
On the Chow groups of the variety of lines of a cubic fourfold
Finite-dimensionality and cycles on powers of K3 surfaces
About Chow groups of certain hyperkähler varieties with non-symplectic automorphisms
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