On the Brauer group of an arithmetic model of a~hyperk\"ahler variety over a~number field

S. G. Tankeev

Izvestiya. Mathematics , Tome 79 (2015) no. 3, pp. 623-644

Résumé

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 $H^2_{\text{\'et}}(V\otimes_kk^{\mathrm{s}},\mathbb Q_l(1))$.

Détail

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.