A Tower of Riemann Surfaces whose Bergman Kernels Jump at the Roof

Abstract

It is shown that, for any Fuchsian group $\Gamma$ acting on the complex upper half plane $\mathcal{H}$ such that $\mathcal{H}/\Gamma$ is a compact hyperelliptic Riemann surface, there exists a sequence of subgroups ${\Gamma }_n\subset \Gamma (n=1,2,\dots )$ satisfying ${\Gamma }_1=\Gamma$ and ${\bigcap }_{n=1}^{\infty }{\Gamma }_n=\{\mathrm{id}\}$ such that the associated sequence of the Bergman kernels of $\mathcal{H}/{\Gamma }_n$, pulled back to $\mathcal{H}$, does not converge to the Bergman kernel of $\mathcal{H}$.