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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
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.
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
Linking options:
https://www.mathnet.ru/eng/im8234https://doi.org/10.1070/IM2015v079n03ABEH002755 https://www.mathnet.ru/eng/im/v79/i3/p203
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Abstract page: | 423 | Russian version PDF: | 136 | English version PDF: | 10 | References: | 45 | First page: | 14 |
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