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THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$

Part of: Cycles and subschemes

Published online by Cambridge University Press:  27 October 2016

MINGMIN SHEN  and
CHARLES VIAL
MINGMIN SHEN
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands; m.shen@uva.nl
CHARLES VIAL
Affiliation:
DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK; c.vial@dpmms.cam.ac.uk

Abstract

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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.

MSC classification

Primary: 14C05: Parametrization (Chow and Hilbert schemes)
Secondary: 14C25: Algebraic cycles 14C15: (Equivariant) Chow groups and rings; motives
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e30
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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