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Izvestiya: Mathematics, 2015, Volume 79, Issue 3, Pages 623–644
DOI: https://doi.org/10.1070/IM2015v079n03ABEH002755
(Mi im8234)
 

This article is cited in 2 scientific papers (total in 2 papers)

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

S. G. Tankeev

Vladimir State University
References:
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 $H^2_{\text{\'et}}(V\otimes_kk^{\mathrm{s}},\mathbb Q_l(1))$.
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).
Received: 14.03.2014
Revised: 24.11.2014
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2015, Volume 79, Issue 3, Pages 203–224
DOI: https://doi.org/10.4213/im8234
Bibliographic databases:
Document Type: Article
UDC: 512.7
MSC: 14F22, 14K05
Language: English
Original paper language: Russian
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
\jour Izv. RAN. Ser. Mat.
\yr 2015
\vol 79
\issue 3
\pages 203--224
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\jour Izv. Math.
\yr 2015
\vol 79
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\pages 623--644
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  • https://www.mathnet.ru/eng/im8234
  • https://doi.org/10.1070/IM2015v079n03ABEH002755
  • https://www.mathnet.ru/eng/im/v79/i3/p203
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:423
    Russian version PDF:136
    English version PDF:10
    References:45
    First page:14
     
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