  Let $X$ be a complete intersection inside a variety $M$ with finite
dimensional motive and for which the Lefschetz-type conjecture $B(M)$ holds. We
show how conditions on the niveau filtration on the homology of $X$ influence
directly the niveau on the level of Chow groups. This leads to a generalization
of Voisin's result. The latter states that if $M$ has trivial Chow groups and
if $X$ has non-trivial variable cohomology parametrized by $c$-dimensional
algebraic cycles, then the cycle class maps $A_k(X) \to H_{2k}(X)$ are
injective for $k<c$. We give variants involving group actions which lead to
several new examples with finite dimensional Chow motives.
