A $p$-adic local invariant cycle theorem with applications to Brauer groups

In this article, we prove a $p$-adic analogue of the local invariant cycle theorem for $H^2$ in mixed characteristics. As a result, for a smooth projective variety $X$ over a $p$-adic local field $K$ with a proper flat regular model $\mathcal{X}$ over $O_K$, we show that the natural map $Br(\mathcal{X})\to Br(X_{\bar{K}}{)}^{G_K}$ has a finite kernel and a finite cokernel. And we prove that the natural map $Hom(Br(X)\mathrm{/}Br(K)+Br(\mathcal{X}),\mathbb{Q}\mathrm{/}\mathbb{Z})\to Al{b}_X(K)$ has a finite kernel and a finite cokernel, generalizing Lichtenbaum’s duality between Brauer groups and Jacobians for curves to arbitrary dimensions.