Point-like limit of the hyperelliptic Zhang–Kawazumi invariant

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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Published 15 February 2017