Asian Journal of Mathematics

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

Robin De Jong

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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.

Article information

Source
Asian J. Math., Volume 18, Number 3 (2014), 507-524.

Dates
First available in Project Euclid: 8 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1410186669

Mathematical Reviews number (MathSciNet)
MR3257838

Zentralblatt MATH identifier
1360.14083

Subjects
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]

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

Citation

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


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