Principal divisors have vanishing Abel-Jacobi class

A curve with the stated properties exists: choose a radial ray whose direction is different from the arguments of the finitely many nonzero finite branch values, and connect $\mathrm{\infty }$ to $0$ along this ray, parametrized piecewise smoothly in the sphere charts. Its interior avoids $R$. For the rest of the proof use the given curve $\gamma$; the computation applies to every curve whose interior avoids $R$, including curves with repeated image points.

Fix an interior parameter $t_0\in (0,1)$. By [F2] the map off the branch values is an $n$-sheeted covering. Starting at each of the $n$ points over $\gamma (t_0)$, lift the parametrized path on $(0,1)$ in both parameter directions by [F5], obtaining $c_1,\dots ,c_n$ with $f\circ c_j=\gamma$. At each fixed parameter their values are distinct and exhaust the fibre, since reverse lifting is inverse to forward lifting. They are piecewise smooth on the interior by the local holomorphic inverse branches, but their image subsets need not be disjoint. Near either endpoint, choose pairwise disjoint power-coordinate neighborhoods about the finite endpoint fibre; compactness and properness allow a target disk whose full preimage is contained in their union. A lifted tail is connected and thus remains in one of these neighborhoods, and the local equation $t=z^e$ forces its coordinate to tend to zero as the target approaches the endpoint. Hence every $c_j$ extends continuously to $[0,1]$, giving the chain counted with multiplicities.

Restrict the paths to a compact subinterval $[ϵ,1-ϵ]\subset (0,1)$ and subdivide it into finitely many intervals whose target images lie in evenly covered disks. On each interval the $n$ lifts use every inverse branch once by step 1.2. By [F7], summing their $\omega$ integrals equals the integral of the sum of inverse-branch pullbacks, namely $f_{\ast }\omega$ by [F3]. Summing the intervals gives ${\sum }_j{\int }_{c_j{\mathrm{\mid }}_{[ϵ,1-ϵ]}}\omega ={\int }_{\gamma {\mathrm{\mid }}_{[ϵ,1-ϵ]}}{f}_{\ast }\omega =0$ by [F4]. Near each endpoint, the lifts converge into a coordinate disk with a holomorphic primitive; its endpoint differences show that the omitted tail integrals tend to zero. Thus the continuous-path integrals converge to the full chain integral and ${\int }_c\omega =0$. This calculation retains multiplicity and uses no global inverse branch along a self-intersecting curve.

For a zero $q$ of order $e_q$, the local power model over a sufficiently small target disk has exactly $e_q$ points over each nearby regular value. At an interior parameter sufficiently close to the endpoint, the lifted points exhaust that fibre by step 1.2. Exactly $e_q$ of them lie in the neighborhood of $q$, their connected tails remain there, and they all converge to $q$. Thus exactly $e_q$ lifted paths end at $q$, without an embedded-arc assumption. The same argument in the infinity chart counts $e_p$ paths starting at each pole $p$ of order $e_p$. Summing the endpoint boundaries gives $\mathrm{\partial }c={\sum }_q{e}_q[q]-{\sum }_p{e}_p[p]=f^{\ast }(0)-f^{\ast }(\mathrm{\infty })=(f)$ by [F6].

By steps 2.2 and 2.1 the chain $c$ has $\mathrm{\partial }c=(f)$ and ${\int }_c\omega =0$ for all $\omega \in \mathrm{\Omega }(X)$; hence by [F8] the class $u((f))$ is represented by the zero functional modulo the period lattice, that is, $u((f))=0$. If $f$ is a nonzero constant then $(f)=0$ and $u(0)=0$ as well.

The two cases together prove that every nonzero meromorphic function has $u((f))=0$: both cases by the chain, endpoint, and class computations above, both under the inherited full AC of [F10].