Obviously, if the first Chern classes for $X$ and $Y$ are surjective, then the Brauer groups are isomorphic (even $H^{3}_{sing}(X)_{tors} \cong H^{3}_{sing}(Y)_{tors}$ is sufficient).

Obviously, if the first Chern classes for $X$ and $Y$ are surjective, then the Brauer groups are isomorphic (even $H^{3}_\text{sing}(X)_\text{tors} \cong H^{3}_\text{sing}(Y)_\text{tors}$ is sufficient).

Obviously, if the first Chern classes for $X$ and $Y$ are surjective, then the Brauer groups are isomorphic.

Obviously, if the first Chern classes for $X$ and $Y$ are surjective, then the Brauer groups are isomorphic (even $H^{3}_{sing}(X)_{tors} \cong H^{3}_{sing}(Y)_{tors}$ is sufficient).