The motive of the Hilbert cube

The Hilbert scheme $X^{[3]}$ of length-$3$ subschemes of a smooth projective variety $X$ 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 $X^3 \dashrightarrow X^{[3]}$. 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 $X$ has a multiplicative Chow-Kuenneth decomposition, then the Chow rings of its powers $X^n$ have a filtration, which is the expected Bloch-Beilinson filtration, that is split.