Skeletons and tropicalizations

Abstract

ABSTRACT. Let K be a complete, algebraically closed non-archimedean field with ring of integers K • and let X be a K-variety. We associate to the data of a strictly semistable K • -model X of X plus a suitable horizontal divisor H a skeleton 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). 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 wellknown 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.

Faithful tropicalisation and torus actions

“…In particular, we will give another, more geometric proof of the main result of [7] that Grassmannians of planes admit such a section. In the recent paper [13] (written concurrently with our paper) it is proved that, if X is a subvariety of G n m , then a section exists on the locus in Trop(X ) where the tropical multiplicity equals one [13]. This beautiful general theorem implies parts of our results, e.g.…”

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

“…As in [9,47,43,28], one has a canonical embedding S (X ,H) → X an and a canonical strong deformation retraction from X an to S (X ,H) . In this paper, we will only be concerned with the retraction map at time one, which we denote by τ : X an → S (X ,H) .…”

“…First, we describe a general setup of tropicalization using snc pairs (see also [9,36,47,10,28,52]). Then we recall the relation between toroidal blowups of formal models and polyhedral subdivisions of the tropicalization following [30].…”

A tropical approach to nonarchimedean Arakelov geometry

“…Let H : N U,R → R 2 be the canonical affine map with N U the dual of M U . Moreover, we have a canonical surjective map G from the minimal skeleton S(W ) onto the canonical tropicalization Trop(U ) which is affine on every segment and every leave of the minimal skeleton and such that trop U • ϕ an = G • τ for the canonical retraction τ onto the skeleton S(W ) (see [GRW16,Section 5]). Using our description of O(U ) × , we deduce that…”