# On rigid varieties with projective reduction

## TL;DR

This paper investigates smooth proper rigid varieties with projective reduction, proving their Picard varieties are proper and showing p-adic Hopf varieties cannot have projective reductions, using moduli of semistable sheaves.

## Contribution

It establishes the properness of Picard varieties for certain rigid varieties and rules out projective reduction for p-adic Hopf varieties, advancing understanding of their structure.

## Key findings

- Picard varieties of these rigid varieties are proper.
- p-adic Hopf varieties do not admit projective reduction.
- Main proof uses moduli of semistable coherent sheaves.

## Abstract

In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that p-adic Hopf varieties will never have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves.