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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.

Fix $p_0\in X$ and use the normalized holomorphic one-forms and period lattice of Solution 1. The Abel-Jacobi map of a compact Riemann surface isChanging paths changes the vector by a period, so this is well defined and holomorphic. Adding these vectors gives a symmetric map on $X^d$ and therefore the Abelian sum map $u_d:X^{(d)}\to \mathrm{Jac}(X)$. Locally, even where points coincide, symmetric holomorphic functions descend to the elementary symmetric coordinates of the symmetric product of a curve. For any divisor $D={\sum }_p{n}_{p}p$ of degree zero, define $u(D)={\sum }_p{n}_{p}u(p)$. Abel's theorem says that $D$ is a principal divisor if and only if $u(D)=0$. Equivalently, two effective divisors of the same degree have the same Abel sum exactly when they are linearly equivalent.

We first establish the residue version of the bilinear identity. Let $\eta$ be a meromorphic differential on a Riemann surface with only simple poles and total residue zero, and let $F={\int }_{p_0}^p\omega$ on the cut polygon, for a holomorphic one-form $\omega$. Excise small disks about the poles. Since $d(F\eta )=\omega \wedge \eta =0$ off the poles, the outer boundary integral is the sum of the positive small-circle integrals. The latter are $2\pi i{\mathrm{res}}_p(\eta )F(p)$. Pairing the outer edges as in Solution 1 therefore givesThe paths and the cut polygon are fixed together in this equality. Changing them changes the Abel vector by its period lattice, which is precisely the ambiguity we need.

For the sufficiency direction, we need a differential of the third kind with prescribed residues $n_p$. Let $S$ be the reduced support of $D$, consisting of $s$ points. When $s>0$, the Riemann-Roch theorem gives $h^0(K_X(S))=g-1+s$. The residue map from this space has kernel $H^0(K_X)$, of dimension $g$: a differential with at most a simple pole and residue zero has no pole. Its image therefore has dimension $s-1$. The residue theorem puts the image inside the $s-1$ dimensional space of tuples with sum zero, so it is all that space. In particular there is a meromorphic differential on a Riemann surface $\eta$ with residues $n_p$. Subtracting a combination of the normalized holomorphic one-forms makes all its $a$-periods zero. The bilinear identity then sayswhere on the right we temporarily choose the representative determined by the paths. If $D=0$, we simply take $\eta =0$.

Suppose now that $u(D)=0$ in the Jacobian variety, so the chosen vector is $m+Bn$, with $m,n\in {\mathbb{Z}}^g$. Replace $\eta$ byIts $a$-periods are $-2\pi i{n}_j$ and its $b$-periods are $2\pi i{m}_i$. The periods around the punctures are $2\pi i{n}_p$. These loops together generate the homology of the punctured Riemann surface, so every period of ${\eta }^{\prime }$ lies in $2\pi i\mathbb{Z}$. Consequently $f(p)=\exp ({\int }^p{\eta }^{\prime })$ is a single-valued nonzero holomorphic function off the support of $D$. At a support point, ${\eta }^{\prime }=n_{p}dt/t+$ a holomorphic differential, so $f=t^{n_p}$ times a nonvanishing holomorphic function. It extends meromorphically and has divisor exactly $D$. This proves sufficiency, including negative coefficients.

Conversely, if $D=\mathrm{div}(f)$, take $\eta =df/f$. Its residues are the coefficients of $D$, while its $a$- and $b$-periods are integer multiples of $2\pi i$, by the winding numbers of $f$ along these loops. Writing them as $2\pi ia$ and $2\pi ib$, the residue bilinear identity gives $u(D)=b-Ba\in \Lambda$. Thus $u(D)=0$ in the Jacobian variety, proving necessity and henceHere the displayed necessity of degree zero is also the residue theorem applied to $df/f$.

There is a useful geometric version of the conclusion. For an effective divisor $D$, the fiber of $u_d$ through $D$ consists precisely of the effective divisors linearly equivalent to $D$. A nonzero section of ${\mathcal{O}}_X(D)$ is a meromorphic function $f$ with $\mathrm{div}(f)+D\ge 0$, and its zero divisor as a section is $\mathrm{div}(f)+D$. Two such sections give the same effective divisor exactly when their ratio is a nonzero constant. Thus the fiber is the complete linear system of a divisor $|D|=\mathbb{P}{H}^0(X,{\mathcal{O}}_X(D))$.

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.