Tropical and non-archimedean limits of degenerating families of volume forms

Abstract

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich–Soibelman and Gross–Wilson, bearing on maximal degenerations of Calabi–Yau manifolds.