Asian Journal of Mathematics

Volume 18 (2014)

Number 3

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

Pages: 507 – 524



Robin Dejong (Mathematical Institute, University of Leiden, The Netherlands)


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.


Arakelov metric, Ceresa cycle, Green’s functions, Kawazumi-Zhang invariant, stable curves

2010 Mathematics Subject Classification

Primary 14H15. Secondary 14D06, 32G20.

Full Text (PDF format)