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.

arXiv commentMinor changes. To appear in Journal de l'Ecole polytechnique