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A degree-zero complex analytic divisor $E$ on a compact Riemann surface is principal exactly when its Abel-Jacobi map of a compact Riemann surface is zero. Normalize a differential of the third kind with residues equal to the coefficients of $E$. The Riemann bilinear relations for a compact surface express its $b$-periods as $2\pi i$ times the Abel integrals. If those integrals are in the period lattice, subtract a suitable integral multiple of the normalized holomorphic differential forms; all periods then lie in $2\pi i\mathbb{Z}$, so exponentiation gives the required meromorphic function. Conversely, the logarithmic differential of a meromorphic function has integer residues and periods in $2\pi i\mathbb{Z}$, giving a zero Abel class by the same calculation.

A Riemann surface which is compact as a topological space. For a connected one of genus $g$, the Riemann-Roch theorem gives $g$ independent holomorphic differential forms. The Riemann bilinear relations for a compact surface make their normalized periods a full lattice in ${\mathbb{C}}^g$, and the Abel-Jacobi map of a compact Riemann surface turns this lattice into a complex torus encoding divisor classes.

Let $V=H^0(C,{\Omega }_C^1)$, of complex dimension $g$, and choose a symplectic basis $a_1,\dots ,a_g,b_1,\dots ,b_g$ of $H_1(C,\mathbb{Z})$, with $a_i\cdot b_j={\delta }_{ij}$. Integration gives a homomorphismWe first prove both injectivity and discreteness, rather than treating periods as a lattice by definition.

Cut the compact Riemann surface along this basis to a polygon. For closed one-forms $\omega ,\eta$, choose a primitive $F$ of $\omega$ on the cut polygon and apply Stokes theorem to $d(F\eta )=\omega \wedge \eta$. Pairing its boundary edges records the jumps of $F$ across the cuts and gives the Riemann bilinear relations for a compact surface:If a holomorphic differential form $\omega$ has all $a$-periods zero, take $\eta =\overline{\omega }$. The right side is zero, while in a holomorphic coordinate, for $\omega =f(z)dz$,Its integral is strictly positive unless $\omega =0$. Thus the $a$-period map $V\to {\mathbb{C}}^g$ is injective, hence an isomorphism. Choose a basis ${\omega }_1,\dots ,{\omega }_g$ with ${\int }_{a_j}{\omega }_i={\delta }_{ij}$, and put ${\tau }_{ij}={\int }_{b_j}{\omega }_i$. Applying the same boundary identity to ${\omega }_i,{\omega }_j$, whose wedge product is zero, gives ${\tau }_{ij}={\tau }_{ji}$. Applying it to $\omega =\sum c_i{\omega }_i$ and its conjugate givesHence $\mathrm{Im}\tau$ is positive definite.

In the dual coordinates, $I(\sum m_i{a}_i+n_i{b}_i)=m+\tau n$. The real-linear map $(x,y)\mapsto x+\tau y$ from ${\mathbb{R}}^{2g}$ to ${\mathbb{C}}^g$ is an isomorphism: taking imaginary parts of a zero gives $(\mathrm{Im}\tau )y=0$, then $y=0$ and $x=0$. Its image of ${\mathbb{Z}}^{2g}$ is therefore discrete and cocompact, and no nonzero integral cycle has zero image. Thus

The Abel-Jacobi map of a compact Riemann surface sends an effective complex analytic divisor $D=\sum Q_i$ to ${\sum }_i{\int }_{P_0}^{Q_i}({\omega }_1,\dots ,{\omega }_g)$ modulo $\Lambda$. Changing a path adds an integral period, so the map is well-defined. For two degree-$d$ complex analytic divisors, let $E=D-D^{\prime }={\sum }_P{n}_{P}P$, with $\sum n_P=0$, and putWe prove that $E$ is principal exactly when $v\in \Lambda$, including the construction of a meromorphic function.

First construct a differential of the third kind $\eta$ with residues $n_P$ at the support of $E$ and no other poles. For the reduced support $S$, the residue exact sequence isOn sheaf cohomology, the obstruction to prescribing residues belongs to $H^1(C,K_C)\cong \mathbb{C}$ by Serre duality; its pairing with the constant section of ${\mathcal{O}}_C$ is the sum of residues. This follows directly by integrating the local simple-pole terms around the boundary circles. Thus the obstruction is zero because $\sum n_P=0$, and the required $\eta$ exists. Subtract a linear combination of the normalized holomorphic differential forms so that all its $a$-periods are zero.

Take its poles and the integration paths inside the same cut polygon. Apply the boundary calculation above to a primitive $F_i(P)={\int }_{P_0}^P{\omega }_i$, removing small circles around the poles of $\eta$. Since $d(F_i\eta )=0$ away from the poles, the paired outer edges contribute $B_i(\eta )$, while the small circles contribute $2\pi i{\sum }_P{n}_P{F}_i(P)$ by the residue theorem. ThereforeThe same argument without normalizing $\eta$ gives the useful reciprocity identityThese identities use compatible cuts and paths; the final conclusions are unchanged by integral periods.

Suppose $v=m+\tau n$ with $m,n\in {\mathbb{Z}}^g$. DefineIts $a$-periods are $-2\pi i{n}_i$, and its $b$-periods are $2\pi i{m}_i$. Around a pole its integral is $2\pi i{n}_P$. These cycles generate the first homology group of the punctured surface, so every period of ${\eta }^{\prime }$ belongs to $2\pi i\mathbb{Z}$. Choosing an auxiliary point $Q$ away from the poles, setIt is single-valued away from the poles. Locally ${\eta }^{\prime }=n_{P}dt/t+\text{a holomorphic one-form}$, so $f=t^{n_P}$ times a nonvanishing holomorphic function. It extends meromorphically with $\mathrm{div}(f)=E$. This proves sufficiency and explicitly produces the required principal complex analytic divisor.

Conversely, if $E=\mathrm{div}(f)$, the meromorphic differential $df/f$ has residues $n_P$ and every period in $2\pi i\mathbb{Z}$, since each integral counts the winding number of $f$ about zero. Write its $a$-period vector as $2\pi ia$ and its $b$-period vector as $2\pi ib$, with $a,b\in {\mathbb{Z}}^g$. The reciprocity identity gives $v=b-\tau a\in \Lambda$. Since $E$ has degree zero, changing the base point cancels in its Abel sum; a base point at the support of $E$ causes no problem, because the Abel integrals use holomorphic forms. We have provedThis establishes the Abel theorem for divisors directly from the period and residue calculations.