Asian Journal of Mathematics
- Asian J. Math.
- Volume 18, Number 3 (2014), 507-524.
Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces
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.
Asian J. Math., Volume 18, Number 3 (2014), 507-524.
First available in Project Euclid: 8 September 2014
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14H15: Families, moduli (analytic) [See also 30F10, 32G15]
Secondary: 14D06: Fibrations, degenerations 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]
De Jong, Robin. Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. Asian J. Math. 18 (2014), no. 3, 507--524. https://projecteuclid.org/euclid.ajm/1410186669