Semi-stable extensions over 1-dimensional bases

Given a family of Calabi.Yau varieties over the punctured disc or over the field of Laurent series, we show that, after a finite base change, the family can be extended across the origin while keeping the canonical class trivial. More generally, we prove similar extension results for families whose log-canonical class is semi-ample. We use these to show that the Berkovich and essential skeleta agree for smooth varieties over .((t)) with semi-ample canonical class.