The essential skeleton of a degeneration of algebraic varieties

TL;DR

This paper investigates the relationship between the minimal model program and Berkovich spaces, demonstrating that the essential skeleton of certain algebraic varieties aligns with minimal models and has topological properties like being a pseudo-manifold.

Contribution

It establishes the equivalence of the essential skeleton with the skeleton of minimal dlt-models and shows it is a strong deformation retract of the Berkovich analytification.

Findings

Essential skeleton coincides with minimal dlt-model skeleton.

The essential skeleton is a strong deformation retract of the Berkovich space.

The essential skeleton of a Calabi-Yau variety is a pseudo-manifold.

Abstract

In this paper, we explore the connections between the Minimal Model Program and the theory of Berkovich spaces. Let $k$ be a field of characteristic zero and let $X$ be a smooth and proper $k((t))$-variety with semi-ample canonical divisor. We prove that the essential skeleton of $X$ coincides with the skeleton of any minimal $dlt$-model and that it is a strong deformation retract of the Berkovich analytification of $X$. As an application, we show that the essential skeleton of a Calabi-Yau variety over $k((t))$ is a pseudo-manifold.