Unramified Brauer groups of symmetric products and the Brauer-Manin obstructions

This article focuses on smooth, projective, and geometrically integral varieties $X$ defined over a number field $k$ with torsion-free geometric Picard groups. We establish an isomorphism between the Brauer groups of $X$ and its symmetric products. As applications, we deduce the relationship between the Brauer--Manin obstruction to the Hasse principle and to weak approximation for $0$-cycles of degree $n$ on $X$ and the corresponding obstruction for rational points on smooth projective models of its $n$-fold symmetric product.