A finiteness theorem for the Brauer group of abelian varieties and 𝐾3 surfaces

Skorobogatov, Alexei; Zarhin, Yuri

doi:10.1090/S1056-3911-07-00471-7

A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces

Authors: Alexei N. Skorobogatov and Yuri G. Zarhin
Journal: J. Algebraic Geom. 17 (2008), 481-502
DOI: https://doi.org/10.1090/S1056-3911-07-00471-7
Published electronically: December 10, 2007
MathSciNet review: 2395136
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Abstract | References | Additional Information

Abstract: Let $k$ be a field finitely generated over the field of rational numbers, and $\operatorname {Br}(k)$ the Brauer group of $k$. For an algebraic variety $X$ over $k$ we consider the cohomological Brauer–Grothendieck group $\operatorname {Br}(X)$. We prove that the quotient of $\operatorname {Br}(X)$ by the image of $\operatorname {Br}(k)$ is finite if $X$ is a $K3$ surface. When $X$ is an abelian variety over $k$, and $\overline {X}$ is the variety over an algebraic closure $\overline {k}$ of $k$ obtained from $X$ by the extension of the ground field, we prove that the image of $\operatorname {Br}(X)$ in $\operatorname {Br}(\overline {X})$ is finite.

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Additional Information

Alexei N. Skorobogatov
Affiliation: Department of Mathematics, South Kensington Campus, Imperial College, London, SW7 2BZ England, United Kingdom; Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow, 127994 Russia
MR Author ID: 218233
Email: a.skorobogatov@imperial.ac.uk

Yuri G. Zarhin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802; Institute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow Region, Russia
MR Author ID: 200326
Email: zarhin@math.psu.edu

Received by editor(s): May 13, 2006
Received by editor(s) in revised form: October 12, 2006
Published electronically: December 10, 2007