Skeletons and tropicalizations

Let K   be a complete, algebraically closed non-archimedean field with ring of integers K∘K∘ and let X be a K  -variety. We associate to the data of a strictly semistable K∘K∘-model XX of X plus a suitable horizontal divisor H   a skeleton S(X,H)S(X,H) in the analytification of X. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H  . It also generalizes constructions by Tyomkin and Baker–Payne–Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on S(X,H)S(X,H). For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well-known result by Sturmfels–Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.