Spectral Geometry of Hurwitz Spaces with Arbitrary Ramification
Abstract
We study the Friedrichs Laplacian associated with the pullback of the round metric on mb P1 by a nonconstant meromorphic function φ:X→mb P1 on a compact Riemann surface. For arbitrary ramification profiles, including several ramification points over the same branch value, we prove the local formula Detζ(Δ[φ],mc F) =C detImB |τB|2 ∏k=1Nρ(zk,zk̅)ck. The zero eigenvalue is omitted. Here B is the period matrix, τB is the local Bergman tau-function, zk are the branch-value coordinates, ρ(z,z̅)=4(1+|z|2)−2, and ck=1/12∑j(nkj−nkj−1), where nkj are the ramification indices over the k-th branch value. The constant C>0 is independent of the Hurwitz coordinates, and detImB 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.
Related
- Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies
- Topological Recursion on Spectral Curves with Infinite Ramification Loci
- The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces
- The $2$-co-lacunary subsequence principle for arbitrary noncommutative L1-spaces
- Artin-Schreier geproci configurations in projective spaces of arbitrary dimension
BibTeX
@misc{jiaming2026spectral,
title = {{Spectral Geometry of Hurwitz Spaces with Arbitrary Ramification}},
author = {Jia-Ming and Liou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19604},
eprint = {2609.19604},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19604},
}
RIS
TY - UNPB
TI - Spectral Geometry of Hurwitz Spaces with Arbitrary Ramification
AU - Jia-Ming
AU - Liou
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19604
UR - https://arxiv.org/abs/2609.19604
AB - We study the Friedrichs Laplacian associated with the pullback of the round metric on mb P^1 by a nonconstant meromorphic function φ:X→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 Det_ζ(Δ_[φ],mc F) =C detImB |τ_B|^2 ∏_k=1^Nρ(z_k,z_k̅)^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,z̅)=4(1+|z|^2)^−2, and c_k=1/12∑_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 detImB 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.
ER -