The essential skeleton of a degeneration of algebraic varieties

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 projective $k(\negthinspace(t)\negthinspace)$-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 $k(\negthinspace(t)\negthinspace)$ is a pseudo-manifold.