Convergence of Measures on Non-Archimedean Hybrid Spaces

Abstract

We study the convergence of certain classes of complex geometric measures to certain non-Archimedean measures. This convergence takes place on the non-Archimedean hybrid space introduced by Boucksom and Jonsson.

Given a family X of complex analytic spaces parametrized by the punctured unit complex disk, the hybrid space associated to this family is a partial compactification of this family obtained by filling in the puncture with the Berkovich analytification of X. The topology of the hybrid space is given by local logarithmic convergence. Furthermore, if each of the complex analytic spaces in the family carry a natural measure, we can think of these measures as being supported on the hybrid space, then their weak limit is a measure supported on the Berkovich space.

First, we study the convergence of volume forms on a degenerating holomorphic family of log Calabi–Yau varieties, 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 form on X with poles along B such that the restriction of eta is a top-dimensional form on each of the fibers. We show that the (possibly infinite) measures induced by the restriction of eta to a fiber weakly converge to a measure on the Berkovich analytification of X - B as we approach the puncture. The limit measure is a sum of suitably normalized Lebesgue measures supported on certain skeletal subsets of the Berkovich space.

Secondly, we prove a folklore conjecture that the Bergman measures along a holomorphic family of curves parametrized by the punctured unit disk weakly converge to the Zhang measure on the associated Berkovich space. We also study the convergence of the Bergman measures to a measure on a metrized curve complex in the sense of Amini and Baker.