Extended Abel-Jacobi theorem

Question

Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole or zeros on $\{P_1, ..., P_n\}$. Let $d\log(u)$ be the logarithmic differential form of $u$. Do we have the following equality in $\mathbf{C}^{\times}$: $$\exp(\int_{div(f)} d\log(u)) = \prod_{i=1}^n f(P_i)^{\text{ord}_{P_i}(u)}\text{ ?}$$ Here, $\text{div}(f)$ is the divisor of $f$, and the integral is well-defined (independant of the choice of the path) up to $2\pi i \mathbf{Z}$.

This identity is true if for all $i$ we have $f(P_i)=1$, as it relates to the analytic Abel-Jacobi description of the generalized Jacobian of $X$ with respect to the divisor $(P_1)+...+(P_n)$.

Answer 1

This is true. Given two points $a,b\in X$ with $u(a)$ and $u(b)\neq 0$, one can choose a path from $a$ to $b$ and a determination of $\log u$ along that path, so that $\exp\int^b_a d\log u=u(b)/u(a)$. Thus, if $ \operatorname{div}(f)=\sum n_jQ_j$, the left-hand side is $\prod u(Q_j)^{n_j}$.

Recall that if $f,g$ are two meromorphic functions on $X$, of order $n$ and $m$ at a point $P$, the tame symbol $(f,g)_{P}$ is $(-1)^{nm}[g^{n}/f^m](P)$. Thus, if $\operatorname{div}(u)=\sum m_iP_i$, we have $(f,u)_{P_i}= f(P_i)^{-m_i}$, $(f,u)_{Q_j}= u(Q_j)^{n_j}$, and $(f,u)_{P}=1$ at the other points. Now the product formula $\prod (f,u)_P=1$ gives your equality.