AI 中文总结
本文研究双共形网中由两个亚纯投影诱导的相对微分态射，证明其对数联络可由两网的曲率数据对称实现，留数给出相对分歧与度数差，并讨论双态射及超椭圆例子。
AI 中文摘要
设 $X$ 为紧致连通黎曼曲面，配备两个非常值亚纯投影 $q,z:X\to\mathbb{P}^1$。它们的微分确定一个典范的亚纯相对微分态射 $Z:T_q=q^*T\mathbb{P}^1\to T_z=z^*T\mathbb{P}^1$，其在坐标标架下的局部系数为 $Z=z_q\\,\partial_z\otimes dq$。其除子为 $\operatorname{Div}(Z)=R_z-R_q$，其局部对数导数定义了 $L=\operatorname{Hom}(T_q,T_z)$ 上的一个秩一对数联络。对数联络的一般理论是经典的。本文的主要观点是，在双共形网背景下，两个网的 Levi--Civita 曲率数据通过对称公式 $H^{-1}\widetilde{\omega}\\,dz-H\omega\\,dq =2i\\,d\log z_q =2i\\,Z^{-1}dZ$ 实现该对数联络，其中最后一个表达式通过 $L$ 的局部平凡化来理解。因此，两个双共形联络微分以相反的符号出现，它们的差是相对微分态射的对数 Maurer--Cartan 形式。其留数恢复相对分歧重数，$\operatorname{Res}_p\Theta=2i(r_z(p)-r_q(p))$，且全局地 $\sum_{p\in X}\operatorname{Res}_p\Theta=4i(\operatorname{deg}z-\operatorname{deg}q)$。因此，局部共形联络数据同时确定相对分歧和两个投影的度数之差。我们还讨论了双态射、平方根问题以及一个超椭圆例子。
英文摘要
Let $X$ be a compact connected Riemann surface equipped with two nonconstant meromorphic projections $q,z:X\to\mathbb{P}^1$. Their differentials determine a canonical meromorphic relative differential morphism $Z:T_q=q^*T\mathbb{P}^1\to T_z=z^*T\mathbb{P}^1$, whose local coefficient in coordinate frames is $Z=z_q\,\partial_z\otimes dq$. Its divisor is $\operatorname{Div}(Z)=R_z-R_q$, and its local logarithmic derivative defines a rank-one logarithmic connection on $L=\operatorname{Hom}(T_q,T_z)$. The general theory of logarithmic connections is classical. The main point of the present paper is that, in the dual conformal-net setting, the Levi--Civita curvature data of the two nets realize this logarithmic connection by the symmetric formula $H^{-1}\widetildeω\,dz-Hω\,dq =2i\,d\log z_q =2i\,Z^{-1}dZ$, where the last expression is understood through local trivializations of $L$. Thus the two dual conformal connection differentials occur with opposite signs, and their difference is the logarithmic Maurer--Cartan form of the relative differential morphism. Its residues recover the relative ramification multiplicities, $\operatorname{Res}_pΘ=2i(r_z(p)-r_q(p))$, and globally $\sum_{p\in X}\operatorname{Res}_pΘ=4i(\operatorname{deg}z-\operatorname{deg}q)$. Thus local conformal connection data determine both relative ramification and the difference of the degrees of the two projections. We also discuss the dual morphism, the square-root issue, and a hyperelliptic example.
Comments19 pages
