The Brauer group and the Brauer–Manin set of products of varieties

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\Q$, and let $\ov X$ and $\ov Y$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of $\Br(\ov X)\oplus \Br(\ov Y)$ has finite index in the Galois invariant subgroup of $\Br(\ov X\times\ov Y)$. This implies that the cokernel of the natural map $\Br (X)\oplus\Br (Y)\to\Br(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the product of varieties is the product of their Brauer–Manin sets.