Skeletons and tropicalizations

TL;DR

This paper develops a generalized theory of skeletons and tropicalizations for algebraic varieties over non-archimedean fields, extending previous constructions to higher dimensions and unbounded faces, and establishing foundational properties like piecewise linearity and balancing conditions.

Contribution

It introduces a new class of skeletons with unbounded faces, generalizes tropicalization constructions to higher dimensions, and proves key properties like piecewise linearity, slope formulas, and faithful tropicalizations.

Findings

Skeletons have integral polyhedral structures.

Valuations of rational functions are piecewise linear on skeletons.

Every skeleton can be faithfully tropicalized.

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 $\mathscr X$ of $X$ plus a suitable horizontal divisor $H$ a skeleton $S(\mathscr X,H)$ 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 $S(\mathscr X, H)$. 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.