Point-like limit of the hyperelliptic Zhang–Kawazumi invariant

Abstract

The behavior near the boundary in the Deligne-Mumford compactification of many functions on 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 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.