  We study a conjecture, due to Voisin, on 0-cycles on varieties with $p_g=1$.
Using Kimura's finite dimensional motives and recent results of Vial's on the
refined (Chow-)K\"unneth decomposition, we provide a general criterion for
Calabi-Yau manifolds of dimension at most $5$ to verify Voisin's conjecture. We
then check, using in most cases some cohomological computations on the mirror
partners, that the criterion can be successfully applied to various examples in
each dimension up to $5$.
