On the connectedness of $p$-adic period domains

TL;DR

This paper proves that all $p$-adic period domains are geometrically connected, resolving a question by Hartl and impacting the understanding of Shimura varieties' geometry.

Contribution

It establishes the geometric connectedness of all $p$-adic period domains and their non-minuscule analogues, answering a longstanding open question.

Findings

All $p$-adic period domains are geometrically connected.

Implications for the geometry of Shimura and local Shimura varieties.

Resolution of Hartl's question from 2013.

Abstract

We prove that all $p$-adic period domains (and their non-minuscule analogues) are geometrically connected. This answers a question of Hartl [Har13] and has interesting consequences to the geometry of Shimura and local Shimura varieties.