THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$
Forum of Mathematics, Sigma, Tome 4 (2016)
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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 $X^{3}{\dashrightarrow}X^{[3]}$ . 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 $X^{n}$ have a filtration, which is the expected Bloch–Beilinson filtration, that is split.
@article{10_1017_fms_2016_25,
author = {MINGMIN SHEN and CHARLES VIAL},
title = {THE {MOTIVE} {OF} {THE} {HILBERT} {CUBE} $X^{[3]}$},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {4},
year = {2016},
doi = {10.1017/fms.2016.25},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2016.25/}
}
MINGMIN SHEN; CHARLES VIAL. THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$. Forum of Mathematics, Sigma, Tome 4 (2016). doi: 10.1017/fms.2016.25
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