Convergence of volume forms on a family of log-Calabi-Yau varieties to a non-Archimedean measure

Abstract

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured unit disk. Let $\eta$ be a meromorphic volume form on $X$ with poles along $B$. We show that the (possibly infinite) measures induced by the restriction of the $\eta$ to a fiber converge to a measure on the Berkovich analytification as we approach the puncture. The convergence takes place on a hybrid space, which is obtained by filling in the space $X\setminus B$ with the aforementioned Berkovich space over the puncture.