Abel's theorem for divisors

Given: Full AC, a compact connected Riemann surface $X$, and a degree-zero divisor $D\in {\mathrm{Div}}^0(X)$.

For any chain $c$ of continuous curves with $\mathrm{\partial }c=D$ the class $u(D)$ is represented by the functional $\omega \mapsto {\int }_c\omega$, and $u$ is a base-point-free group homomorphism on ${\mathrm{Div}}^0(X)$ (The Abel-Jacobi map, The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent).

$\mathrm{\Lambda }=e(H_1(X;\mathbb{Z}))\subseteq \mathrm{\Omega }(X{)}^{\ast }$ with $e(\gamma )(\omega )=P(\gamma ,\omega )$, and for every continuous singular cycle $c_{\alpha }$ representing a class $\alpha \in H_1(X;\mathbb{Z})$, $P(\alpha ,\omega )={\int }_{c_{\alpha }}\omega$ for all holomorphic $\omega$ (The Jacobian of a compact Riemann surface, The period pairing and the period subgroup, The period pairing is well defined and computed by integration).

Weak-solution lemma: for a degree-zero divisor and a chain with boundary it, there is a weak solution $f$ of $D$ with $\frac{1}{2\pi i}{\int }_X\frac{\bar{\mathrm{\partial }}f}{f}\wedge \omega ={\int }_c\omega$ for every $\omega \in \mathrm{\Omega }(X)$, and $f$ is unique up to a smooth nowhere-vanishing factor (Weak solutions of a degree-zero divisor and the logarithmic-derivative identity).

$\bar{\mathrm{\partial }}$-solvability criterion: a smooth $(0,1)$-form $\theta$ on $X$ equals $\bar{\mathrm{\partial }}g$ for a smooth $g$ if and only if ${\int }_X\theta \wedge \omega =0$ for every $\omega \in \mathrm{\Omega }(X)$ (The dbar-solvability criterion and the holomorphic-orthogonality pairing).

On a Riemann surface a smooth function is holomorphic exactly where $\bar{\mathrm{\partial }}h=0$; moreover $\bar{\mathrm{\partial }}(f{e}^{-g})=e^{-g}(\bar{\mathrm{\partial }}f-f\bar{\mathrm{\partial }}g)$ and the product of a weak solution with a smooth nowhere-vanishing factor is again a weak solution of the same divisor, with local powers $z^{n_p}$ (The d, partial and dbar identities, The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions, Bigraded complex forms and the Dolbeault operators, Holomorphic maps and meromorphic functions on Riemann surfaces).

The forward direction: if $D=(f)$ is a principal divisor then $u(D)=0$ (Principal divisors have vanishing Abel-Jacobi class).

On a connected Riemann surface any two points are joined by a continuous path; hence every divisor of degree zero is the boundary of a finite chain of paths (A connected, locally path-connected space is path-connected, because its path components are open, Riemann surfaces and holomorphic atlases).

Divisor orders, principal divisors and degrees are those of Divisors, principal divisors and canonical divisors on a Riemann surface; a weak solution of $D$ whose local factors are holomorphic is a meromorphic function with divisor $D$.

Full AC is inherited from the Hodge and Riemann-Roch interfaces used by the weak-solution and solvability suppliers (The Axiom of Choice).

The path integral is additive under concatenation and reverses sign under reversal (Path integral of a holomorphic differential on a Riemann surface). Repeating paths realizes positive integer weights, and reversing paths realizes negative weights with the same boundary and integrals.