The Motive of the Hilbert Cube X^[3]

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–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map X3⇢X[3]. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety X has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers Xn have a filtration, which is the expected Bloch–Beilinson filtration, that is split.