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

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