Convergence of Bergman measures towards the Zhang measure

TL;DR

This paper proves that Bergman measures along holomorphic curve families converge to the Zhang measure on Berkovich spaces, advancing understanding of measure convergence in non-Archimedean geometry.

Contribution

It establishes the convergence of Bergman measures to the Zhang measure on Berkovich spaces, confirming a longstanding folklore conjecture.

Findings

Bergman measures converge to Zhang measure on Berkovich space

Convergence occurs on a Berkovich hybrid space

Results extend to measures on metrized curve complexes

Abstract

We prove a folklore conjecture that the Bergman measure along a holomorphic family of curves parametrized by the punctured unit disk converges to the Zhang measure on the associated Berkovich space. The convergence takes place on a Berkovich hybrid space. We also study the convergence of the Bergman measure to a measure on a metrized curve complex in the sense of Amini and Baker.