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

Abstract

Let k k be a field finitely generated over the field of rational numbers, and Br ⁡ ( k ) \operatorname {Br}(k) the Brauer group of k k . For an algebraic variety X X over k k we consider the cohomological Brauer–Grothendieck group Br ⁡ ( X ) \operatorname {Br}(X) . We prove that the quotient of Br ⁡ ( X ) \operatorname {Br}(X) by the image of Br ⁡ ( k ) \operatorname {Br}(k) is finite if X X is a K 3 K3 surface. When X X is an abelian variety over k k , and X ¯ \overline {X} is the variety over an algebraic closure k ¯ \overline {k} of k k obtained from X X by the extension of the ground field, we prove that the image of Br ⁡ ( X ) \operatorname {Br}(X) in Br ⁡ ( X ¯ ) \operatorname {Br}(\overline {X}) is finite.