The Hilbert scheme of length- subschemes of a smooth projective
variety is known to be smooth and projective. We investigate whether the
property of having a multiplicative Chow-Kuenneth decomposition is stable under
taking the Hilbert cube. This is achieved by considering an explicit resolution
of the map . The case of the Hilbert square was
taken care of in previous work of ours. The archetypical examples of varieties
endowed with a multiplicative Chow-Kuenneth decomposition is given by abelian
varieties. Recent work seems to suggest that hyperKaehler varieties share the
same property. Roughly, if a smooth projective variety has a multiplicative
Chow-Kuenneth decomposition, then the Chow rings of its powers have a
filtration, which is the expected Bloch-Beilinson filtration, that is split.