Line bundles on rigid varieties and Hodge symmetry

TL;DR

This paper investigates the properties of line bundles and Hodge numbers on smooth proper rigid analytic varieties over non-archimedean fields, introducing a new rigid analytic Albanese and leveraging p-adic Hodge theory.

Contribution

It establishes results on low-degree Hodge numbers, utilizes structure theorems for Picard varieties, and defines a new rigid analytic Albanese.

Findings

Results on low-degree Hodge numbers of rigid varieties

Definition of a rigid analytic Albanese variety

Application of p-adic Hodge theory advances

Abstract

We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory. We also define a rigid analytic Albanese naturally associated with any smooth proper rigid space.