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Comptes Rendus. Mathématique
Algebraic geometry
Motivic classes and the integral Hodge Question
Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 305-311.

We prove that the obstruction to the integral Hodge Question factors through the completion of the Grothendieck ring of varieties for the dimension filtration. As an application, combining work of Peyre, Colliot-Thélène and Voisin, we give the first example of a finite group G such that the motivic class of its classifying stack BG in Ekedahl’s Grothendieck ring of stacks over is non-trivial and BG has trivial unramified Brauer group.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.178
Federico Scavia 1

1. Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
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     title = {Motivic classes and the integral {Hodge} {Question}},
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     language = {en},
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Federico Scavia. Motivic classes and the integral Hodge Question. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 305-311. doi : 10.5802/crmath.178. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.178/

[1] Michael F. Atiyah; Friedrich Hirzebruch Analytic cycles on complex manifolds, Topology, Volume 1 (1962), pp. 25-45 | Article | MR 145560 | Zbl 0108.36401

[2] Franziska Bittner The universal Euler characteristic for varieties of characteristic zero, Compos. Math., Volume 140 (2004) no. 4, pp. 1011-1032 | Article | MR 2059227 | Zbl 1086.14016

[3] Jean-Louis Colliot-Thélène Un théorème de finitude pour le groupe de Chow des zéro-cycles d’un groupe algébrique linéaire sur un corps p-adique, Invent. Math., Volume 159 (2005) no. 3, pp. 589-606 | Article | Zbl 1080.14012

[4] Jean-Louis Colliot-Thélène; Claire Voisin Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J., Volume 161 (2012) no. 5, pp. 735-801 | Article | Zbl 1244.14010

[5] Torsten Ekedahl A geometric invariant of a finite group (2009) (https://arxiv.org/abs/0903.3148)

[6] Torsten Ekedahl The Grothendieck group of algebraic stacks (2009) (https://arxiv.org/abs/0903.3143)

[7] Ivan Martino The Ekedahl invariants for finite groups, J. Pure Appl. Algebra, Volume 220 (2016) no. 4, pp. 1294-1309 | Article | MR 3423448 | Zbl 1332.13007

[8] Alexander S. Merkurjev Invariants of algebraic groups and retract rationality of classifying spaces, Algebraic Groups: Structure and Actions (Proceedings of Symposia in Pure Mathematics), Volume 94, American Mathematical Society, 2017, pp. 277-294 | MR 3645070 | Zbl 1397.14024

[9] Emmanuel Peyre Unramified cohomology of degree 3 and Noether’s problem, Invent. Math., Volume 171 (2008) no. 1, pp. 191-225 | Article | MR 2358059 | Zbl 1155.12003

[10] David J. Saltman Noether’s problem over an algebraically closed field, Invent. Math., Volume 77 (1984) no. 1, pp. 71-84 | Article | MR 751131 | Zbl 0546.14014

[11] Burt Totaro The motive of a classifying space, Geom. Topol., Volume 20 (2016) no. 4, pp. 2079-2133 | Article | MR 3548464 | Zbl 1375.14027

[12] Claire Voisin Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, 2007 | MR 2451566 | Zbl 1129.14019

[13] Claire Voisin Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, 77, Cambridge University Press, 2007 | MR 2449178 | Zbl 1129.14020

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