Semi-stable extensions over 1-dimensional bases

TL;DR

This paper proves that families of Calabi-Yau varieties over punctured bases can be extended across singular points after base change, and shows the equivalence of Berkovich and essential skeleta for certain smooth varieties.

Contribution

It establishes semi-stable extension results for Calabi-Yau families over 1-dimensional bases and compares Berkovich and essential skeleta in this context.

Findings

Families can be extended after finite base change

Berkovich and essential skeleta coincide for certain varieties

Extension results apply to semi-ample log-canonical classes

Abstract

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 exended 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 ${\mathbb C}((t))$ with semi-ample canonical class.