Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 4

Special Issue: In Honor of Eduard Looijenga, Part 1 of 3

Guest Editor: Gerard van der Geer

Point-like limit of the hyperelliptic Zhang–Kawazumi invariant

Pages: 633 – 653

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n4.a4

Author

Robin de Jong (Mathematical Institute, Leiden University, Leiden, The Netherlands)

Abstract

The behavior near the boundary in the Deligne–Mumford compactification of many functions on $\mathcal{M}_{h,n}$ can be conveniently expressed using the notion of “point-like limit” that we adopt from the string theory literature. In this note we study a function on $\mathcal{M}_h$ that has been introduced by N. Kawazumi and S. Zhang, independently. We show that the point-like limit of the Zhang–Kawazumi invariant in a family of hyperelliptic Riemann surfaces in the direction of any hyperelliptic stable curve exists, and is given by evaluating a combinatorial analogue of the Zhang–Kawazumi invariant, also introduced by Zhang, on the dual graph of that stable curve.

Keywords

Arakelov–Green’s function, hyperelliptic curve, point-like limit, stable curve, Zhang–Kawazumi invariant

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