Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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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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
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
Email: zarhin@math.psu.edu

DOI: https://doi.org/10.1090/S1056-3911-07-00471-7
Received by editor(s): May 13, 2006
Received by editor(s) in revised form: October 12, 2006
Published electronically: December 10, 2007

American Mathematical Society