Unramified cohomology and Brauer--Manin pairing

We show that the Tate module of the $\mathrm{\ell }$-adic unramified cohomology $H_{\text{nr}}^3(X,{\text{Q}}_{\mathrm{\ell }}\mathrm{/}{\mathbb{Z}}_{\mathrm{\ell }})$ of a smooth projective surface $X$ over a local or strictly local field $k$ of residue characteristic prime to $\mathrm{\ell }$ vanishes under suitable conditions. We apply this to answer some questions concerning the left and right kernels of the Brauer–Manin pairing for open subvarieties of $X$. To prove the vanishing theorem, we establish Bloch’s formula for the Chow group of codimension-two cycles on a semistable model of $X$, and derive applications to the restriction map for such cycles to the closed fiber of the model.