Boundary asymptotics of the relative Bergman kernel metric for curves

TL;DR

This paper investigates the asymptotic behavior of the relative Bergman kernel metrics on degenerating hyperelliptic Riemann surfaces, revealing explicit formulas near singularities and differences in convergence behavior.

Contribution

It provides precise asymptotic formulas for Bergman kernels near nodes and cusps, highlighting the influence of singularities and complex structures on their behavior.

Findings

Explicit asymptotic formulas near nodes and cusps

Bergman kernels do not always converge on degenerating families

Both singularity and complex structure affect asymptotics

Abstract

We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical Taylor expansion for Abelian differentials and period matrices.