Skeletons and tropicalizations

Abstract

Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^{\circ }$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^{\circ }$-model $\mathcal{X}$ of $X$ plus a suitable horizontal divisor $H$ a skeleton $S(\mathcal{X},H)$ 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 $S(\mathcal{X},H)$. 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.

44 pages, 2 figures. Version 3: minor errors corrected; Remark 3.14 expanded. Final version, to appear in Advances in Mathematics