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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 3, Pages 203–224 (Mi izv8234)  

This article is cited in 1 scientific paper (total in 1 paper)

On the Brauer group of an arithmetic model of a hyperkähler variety over a number field

S. G. Tankeev

Vladimir State University

Abstract: We prove Artin's conjecture on the finiteness of the Brauer group for an arithmetic model of a hyperkähler variety $V$ over a number field $k\hookrightarrow\mathbb C$ provided that $b_2(V\otimes_k\mathbb C)>3$. We show that the Brauer group of an arithmetic model of a simply connected Calabi–Yau variety over a number field is finite. We also prove that if the $l$-adic Tate conjecture on divisors holds for a certain smooth projective variety $V$ over a field $k$ of arbitrary characteristic $\operatorname{char}(k)\ne l$, then the group $\operatorname{Br}'(V\otimes_k k^{\mathrm{s}})^{\operatorname{Gal}(k^{\mathrm{s}}/k)}(l)$ is finite independently of the semisimplicity of the continuous $l$-adic representation of the Galois group $\operatorname{Gal}(k^{\mathrm{s}}/k)$ on the space $H^2_{ét}(V\otimes_kk^{\mathrm{s}},\mathbb Q_l(1))$.

Keywords: hyperkähler variety, Calabi–Yau variety, arithmetic model, Brauer group, Artin's conjecture, K3-surface, Abelian surface, Hilbert scheme of points, generalized Kummer variety, Hilbert modular surface.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00097
This paper was written with the financial support of the Russian Foundation for Basic Research (grant no. 12-01-00097).


DOI: https://doi.org/10.4213/im8234

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English version:
Izvestiya: Mathematics, 2015, 79:3, 623–644

Bibliographic databases:

UDC: 512.7
MSC: 14F22, 14K05
Received: 14.03.2014
Revised: 24.11.2014

Citation: S. G. Tankeev, “On the Brauer group of an arithmetic model of a hyperkähler variety over a number field”, Izv. RAN. Ser. Mat., 79:3 (2015), 203–224; Izv. Math., 79:3 (2015), 623–644

Citation in format AMSBIB
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\paper On the Brauer group of an arithmetic model of a~hyperk\"ahler variety over a~number field
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\pages 623--644
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    This publication is cited in the following articles:
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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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