On the structure of the complement of skeleton

TL;DR

This paper investigates the geometric structure of Berkovich spaces, proposing a conjecture relating the skeleton to virtual open disks and proving it under specific conditions involving semistable models.

Contribution

It introduces a conjecture connecting the Berkovich skeleton with virtual open disks and proves it for certain models with semiample canonical class.

Findings

Conjecture that the skeleton is the complement of virtual open disks.

Proof of the conjecture for strictly semistable models with semiample canonical class.

Advances understanding of Berkovich space geometry.

Abstract

We study the higher dimensional geometry of Berkovich spaces using virtual open disks, which are given by fibration of relative dimension $1$. Inspired by birational geometry, we conjecture that the Berkovich skeleton is the complement of the union of all virtual open disks, and prove this conjecture for $\mathcal{X}$ admitting a strictly semistable model with semiample canonical class.