具有任意分歧的 Hurwitz 空间的谱几何

中文摘要 AI 辅助

我们研究了紧黎曼曲面 X 上非常值亚纯函数 φ:X→P^1 拉回 P^1 上的圆度量所关联的 Friedrichs 拉普拉斯算子。对于任意分歧型（包括在同一分支值上有多个分歧点的情况），我们证明了局部公式 Det_ζ(Δ_[φ],F)=C det Im B |τ_B|^2 ∏_{k=1}^N ρ(z_k, z̄_k)^{c_k}。零特征值被省略。这里 B 是周期矩阵，τ_B 是局部 Bergman tau 函数，z_k 是分支值坐标，ρ(z,z̄)=4(1+|z|^2)^{-2}，且 c_k=1/12 Σ_j(n_{kj}-n_{kj}^{-1})，其中 n_{kj} 是第 k 个分支值上的分歧指数。常数 C>0 与 Hurwitz 坐标无关，在亏格为零时 det Im B 取为 1。证明结合了光滑平凡化和预解幂的迹范数变分，并与球面锥模型进行矩阵比较。Davies--Gaffney 估计提供了所需的均匀高能控制，而零能项通过 Schiffer 双微分和 Rauch 变分公式识别。

英文摘要

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.

发表机构

• National Cheng Kung University（国立成功大学）

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