Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces

TL;DR

This paper derives precise asymptotic formulas for the Kawazumi-Zhang invariant as Riemann surfaces degenerate, refining previous results and providing explicit calculations for genus two surfaces.

Contribution

It provides exact asymptotic formulas, including constants, for the Kawazumi-Zhang invariant during surface degeneration, and refines earlier asymptotic results for the beta-invariant.

Findings

Asymptotic formulas for the Kawazumi-Zhang invariant are established.

Refined asymptotics for the beta-invariant are derived.

Explicit calculations are performed for genus two degenerating surfaces.

Abstract

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces.