Skip to main content Accessibility help
×
Home

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

Published online by Cambridge University Press:  27 October 2016

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
Corresponding

Abstract

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.

Type
Research Article
Creative Commons
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

References

Beauville, A., ‘Sur l’anneau de Chow d’une variété abélienne’, Math. Ann. 273 (1986), 647651.CrossRefGoogle Scholar
Beauville, A., ‘On the splitting of the Bloch–Beilinson filtration’, inAlgebraic Cycles and Motives, Vol. 2, London Mathematical Society Lecture Notes, 344 (Cambridge University Press, 2007), 3853.CrossRefGoogle Scholar
Beauville, A. and Voisin, C., ‘On the Chow ring of a K3 surface’, J. Algebraic Geom. 13 (2004), 417426.CrossRefGoogle Scholar
Cheah, J., ‘Cellular decompositions for nested Hilbert schemes of points’, Pacific. J. Math. 183(1) (1998), 3990.CrossRefGoogle Scholar
Fu, L., Tian, Zh. and Vial, Ch., ‘Motivic hyperKähler resolution conjecture for generalized Kummer varieties’, Preprint, 2016, arXiv:1608.04968.Google Scholar
Fulton, W., Intersection Theory, 2nd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , (Springer, 1998).CrossRefGoogle Scholar
Jannsen, U., ‘Motivic sheaves and filtrations on Chow groups’, inMotives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics, 55 (American Mathematical Society, Providence, RI, 1994), 245302.Google Scholar
Manin, Y., ‘Correspondences, motifs and monoidal transformations’, Mat. Sb. (N.S.) 77 (1968), 475507.Google Scholar
Murre, J., ‘On the motive of an algebraic surface’, J. Reine Angew. Math. 409 (1990), 190204.Google Scholar
Murre, J., ‘On a conjectural filtration on the Chow groups of an algebraic variety. I. The general conjectures and some examples’, Indag. Math. (N.S.) 4(2) (1993), 177188.CrossRefGoogle Scholar
Riess, U., ‘On the Chow ring of birational irreducible symplectic varieties’, Manuscripta Math. 145(3–4) (2014), 473501.CrossRefGoogle Scholar
Shen, M. and Vial, Ch., ‘The Fourier transform for certain hyperKähler fourfolds’, Mem. Amer. Math. Soc. 240(1139) (2016), vii+163 pp.Google Scholar
Vial, Ch., ‘Algebraic cycles and fibrations’, Doc. Math. 18 (2013), 15211553.Google Scholar
Vial, Ch., ‘On the motive of some hyperKähler varieties’, J. Reine Angew. Math. to appear, doi:10.1515/crelle-2015-0008.Google Scholar
Voisin, C., ‘Chow rings and decomposition theorems for families of K3 surfaces and Calabi–Yau hypersurfaces’, Geom. Topol. 16 (2012), 433473.CrossRefGoogle Scholar
Voisin, C., ‘The generalized Hodge and Bloch conjectures are equivalent for general complete intersections’, Ann. Sci. Éc. Norm. Supér. (3) 46 (2013), 449475.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 309 *
View data table for this chart

* Views captured on Cambridge Core between 27th October 2016 - 18th January 2021. This data will be updated every 24 hours.

Access
Open access
Hostname: page-component-77fc7d77f9-wd6lz Total loading time: 0.683 Render date: 2021-01-18T12:20:10.399Z Query parameters: { "hasAccess": "1", "openAccess": "1", "isLogged": "0", "lang": "en" } Feature Flags last update: Mon Jan 18 2021 11:54:11 GMT+0000 (Coordinated Universal Time) Feature Flags: { "metrics": true, "metricsAbstractViews": false, "peerReview": true, "crossMark": true, "comments": true, "relatedCommentaries": true, "subject": true, "clr": true, "languageSwitch": true, "figures": false, "newCiteModal": false, "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *