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: 10.5802/crmath.178
Scavia, Federico 1

1 Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
@article{CRMATH_2021__359_3_305_0,
     author = {Scavia, Federico},
     title = {Motivic classes and the integral {Hodge} {Question}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {305--311},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {3},
     year = {2021},
     doi = {10.5802/crmath.178},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.178/}
}
TY  - JOUR
AU  - Scavia, Federico
TI  - Motivic classes and the integral Hodge Question
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 305
EP  - 311
VL  - 359
IS  - 3
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.178/
DO  - 10.5802/crmath.178
LA  - en
ID  - CRMATH_2021__359_3_305_0
ER  - 
%0 Journal Article
%A Scavia, Federico
%T Motivic classes and the integral Hodge Question
%J Comptes Rendus. Mathématique
%D 2021
%P 305-311
%V 359
%N 3
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.178/
%R 10.5802/crmath.178
%G en
%F CRMATH_2021__359_3_305_0
Scavia, Federico. Motivic classes and the integral Hodge Question. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 305-311. doi : 10.5802/crmath.178. http://www.numdam.org/articles/10.5802/crmath.178/

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

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

[3] Colliot-Thélène, Jean-Louis 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 | DOI | Zbl

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

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

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

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

[8] Merkurjev, Alexander S. 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 | Zbl

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

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

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

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

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

Cited by Sources: