Coverings in p-adic analytic geometry and log coverings II: Cospecialization of the p'-tempered fundamental group in higher dimensions

TL;DR

This paper develops a framework for cospecialization homomorphisms between p'-tempered fundamental groups in higher-dimensional p-adic geometry, extending previous work on families of curves and utilizing log structures for polystable varieties.

Contribution

It introduces a method to construct cospecialization homomorphisms for p'-tempered fundamental groups in higher dimensions using log structures.

Findings

Constructed cospecialization homomorphisms for p'-tempered fundamental groups.

Described pro-(p') tempered fundamental groups via log structures.

Extended the theory from curves to higher-dimensional varieties.

Abstract

This paper constructs cospecialization homomorphisms between the (p') versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction (the tempered fundamental group is a sort of analog of the topological fundamental group of complex algebraic varieties in the p-adic world). We studied the question for families of curves in another paper. To construct them, we will start by describing the pro-(p') tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tempered fundamental groups for log polystable varieties.