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 letX be a smooth and projective 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.
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