Spectral Geometry of Hurwitz Spaces with Arbitrary Ramification

Abstract

We study the Friedrichs Laplacian associated with the pullback of the round metric on \(\mb P^1\) by a nonconstant meromorphic function \(φ:X\to\mb P^1\) on a compact Riemann surface. For arbitrary ramification profiles, including several ramification points over the same branch value, we prove the local formula \[ \operatorname{Det}_ζ(Δ_{[φ],\mc F}) =C\,\det\operatorname{Im}B\,|τ_B|^2 \prod_{k=1}^Nρ(z_k,\overline{z_k})^{c_k}. \] The zero eigenvalue is omitted. Here \(B\) is the period matrix, \(τ_B\) is the local Bergman tau-function, \(z_k\) are the branch-value coordinates, \(ρ(z,\overline z)=4(1+|z|^2)^{-2}\), and \(c_k=\frac1{12}\sum_j(n_{kj}-n_{kj}^{-1})\), where \(n_{kj}\) are the ramification indices over the \(k\)-th branch value. The constant \(C>0\) is independent of the Hurwitz coordinates, and \(\det\operatorname{Im}B\) is taken to be \(1\) in genus zero. The proof combines smooth trivializations and trace-norm variation of resolvent powers with matrix comparison to spherical conic models. The Davies--Gaffney estimate provides the required uniform high-energy control, while the zero-energy terms are identified through the Schiffer bidifferential and Rauch variational formulas.