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

TL;DR

This paper establishes a $p$-adic analogue of the local invariant cycle theorem for $H^2$, leading to new finiteness results for Brauer groups of varieties over $p$-adic fields and generalizing classical duality results.

Contribution

It proves a $p$-adic local invariant cycle theorem for $H^2$ and extends Lichtenbaum's duality to higher-dimensional varieties, with applications to Brauer groups.

Findings

Finite kernel and cokernel for the Brauer group map from model to generic fiber.

Finite kernel and cokernel for the duality map involving Brauer groups and Albanese varieties.

Generalization of Lichtenbaum's duality from curves to arbitrary dimensions.

Abstract

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 $Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}$ has a finite kernel and a finite cokernel. And we prove that the natural map $Hom(Br(X)/Br(K)+Br(\mathcal{X}), \mathbb{Q}/\mathbb{Z}) \rightarrow Alb_X(K)$ has a finite kernel and a finite cokernel, generalizing Lichtenbaum's duality between Brauer groups and Jacobians for curves to arbitrary dimensions.