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 (X) over bar and (Y) over bar 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((X) over bar) circle plus Br((Y) over bar) has finite index in the Galois invariant subgroup of Br((X) over bar x (Y) over bar). This implies that the cokernel of the natural map Br(X) circle plus Br(Y) -> Br(X x 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.