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July 2014 Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces
Robin De Jong
Asian J. Math. 18(3): 507-524 (July 2014).

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.

Citation

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Robin De Jong. "Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces." Asian J. Math. 18 (3) 507 - 524, July 2014.

Information

Published: July 2014
First available in Project Euclid: 8 September 2014

zbMATH: 1360.14083
MathSciNet: MR3257838

Subjects:
Primary: 14H15
Secondary: 14D06 , 32G20

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

Rights: Copyright © 2014 International Press of Boston

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Vol.18 • No. 3 • July 2014
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