Convergence of Narasimhan–Simha measures on degenerating families of Riemann surfaces

Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan–Simha measure. For $m = 1$, both these measures coincide with the Bergman measure on $Y$ . We also extend the definition of both of these measures to the singular curves on the boundary of $\overline{\mathcal{M}_g}$ in such a way that they form a continuous family of measures on the universal curve over $\overline{\mathcal{M}_g}$.

Keywords

Narasimhan–Simha measures, hybrid spaces, Berkovich spaces, moduli of curves

2010 Mathematics Subject Classification

14H15

Full Text (PDF format)

Received 5 April 2022

Accepted 20 July 2022

Published 13 April 2023