The Abel-Jacobi map

Assume the Axiom of Choice (The Axiom of Choice), inherited from the Jacobian definition. Let $X$ be a compact connected Riemann surface of genus $g$ with Jacobian $\mathrm{Jac}(X)=\mathrm{\Omega }(X{)}^{\ast }\mathrm{/}\mathrm{\Lambda }$ and quotient projection $\pi :\mathrm{\Omega }(X{)}^{\ast }\to \mathrm{Jac}(X)$ (The Jacobian of a compact Riemann surface), and let $p_0\in X$ be a base point.

Point map. Coordinate disks make $X$ locally path connected; its connectedness therefore makes it path connected (Riemann surfaces and holomorphic atlases, A connected, locally path-connected space is path-connected, because its path components are open). For $p\in X$ choose a path $\gamma$ in $X$ from $p_0$ to $p$ and set $u_{p_0}(p):=[\omega \mapsto {\int }_{\gamma }\omega ]\in \mathrm{Jac}(X),$ where $[\xi ]=\xi +\mathrm{\Lambda }$ denotes the class modulo the period lattice and ${\int }_{\gamma }\omega$ is the path integral of the holomorphic differential (Path integral of a holomorphic differential on a Riemann surface). The value is independent of the path: if $\gamma ,{\gamma }^{\mathrm{\prime }}$ are two paths from $p_0$ to $p$, then the closed curve $\gamma \ast {\gamma }^{\mathrm{\prime }-1}$ is a continuous singular cycle and ${\int }_{\gamma }\omega -{\int }_{{\gamma }^{\mathrm{\prime }}}\omega ={\int }_{\gamma \ast {\gamma }^{\mathrm{\prime }-1}}\omega =P([\gamma \ast {\gamma }^{\mathrm{\prime }-1}],\omega )=e([\gamma \ast {\gamma }^{\mathrm{\prime }-1}])(\omega )$ for every $\omega \in \mathrm{\Omega }(X)$ (The period pairing and the period subgroup, The period pairing is well defined and computed by integration), so the two functionals differ by the element $e([\gamma \ast {\gamma }^{\mathrm{\prime }-1}])\in \mathrm{\Lambda }$ of the period lattice. The resulting map $u_{p_0}:X\to \mathrm{Jac}(X)$ is the Abel-Jacobi map of $X$ with base point $p_0$. It is holomorphic in the atlas of the Jacobian definition, in the following explicit sense: for every $q\in X$ there are an open neighbourhood $U$ of $q$ and a holomorphic map $\xi :U\to \mathrm{\Omega }(X{)}^{\ast }$ with $\pi \circ \xi =u_{p_0}{\mathrm{\mid }}_U$. It satisfies the addition rule $u_{p_0}(q)-u_{p_0}(p)=[\omega \mapsto {\int }_p^q\omega ]$ for all $p,q\in X$, where the integral is taken along any path from $p$ to $q$.

Divisors. Writing points as degree-one divisors, extend $u_{p_0}$ by linearity: for $D={\sum }_p{n}_p[p]\in \mathrm{Div}(X)$ set $u_{p_0}(D):={\sum }_p{n}_p{u}_{p_0}(p)\in \mathrm{Jac}(X);$ the sum is finite because divisors on a compact Riemann surface have finite support (Divisors, principal divisors and canonical divisors on a Riemann surface). For $\mathrm{deg}D=0$ the class $u(D)$ is independent of the base point $p_0$: the addition rule gives $u_{q_0}(p)=u_{p_0}(p)-u_{p_0}(q_0)$ for every $p$, so replacing $p_0$ by $q_0$ adds $-\left({\sum }_p{n}_p\right)u_{p_0}(q_0)=0$ whenever ${\sum }_p{n}_p=\mathrm{deg}D=0$. Hence on the subgroup ${\mathrm{Div}}^0(X)$ of degree-zero divisors the notation $u(D)$ is base-point free, $u:{\mathrm{Div}}^0(X)\to \mathrm{Jac}(X)$ is a group homomorphism, and $u((q)-(p))=[\omega \mapsto {\int }_p^q\omega ]\mathrm{.}$ Base-point dependence for divisors of nonzero degree is recorded explicitly: for $\mathrm{deg}D\mathrm{\ne }0$ the two base points give different classes exactly when $(\mathrm{deg}D)u_{p_0}(q_0)\mathrm{\ne }0$. A nonzero torsion shift can therefore cancel in nonzero degree.

The construction is the standard integration of holomorphic differentials along paths, modulo the period lattice. Looijenga, Riemann Surfaces, Ch. 7 §2, Lemma 7.2 and Corollary 7.3 (printed pp. 60-61), computes ${\tilde{I}}_{{\gamma }^{\mathrm{\prime }}}-{\tilde{I}}_{\gamma }=e([\gamma \ast {\gamma }^{\mathrm{\prime }}])$ and extends the point map to ${\mathrm{Div}}^0(S)$; McMullen, Riemann Surfaces, Ch. 15 (printed p. 129), defines $\phi$ and the point map $f(Q)=\phi (Q-P)$; Forster, Lectures on Riemann Surfaces, §21.6 (printed pp. 170-171), defines the same map through chains and states that it is determined by $D$ up to the period lattice. The item proves the well-definedness, holomorphy and the addition rule from the local path-integral interface, which accepts the continuous paths used by the polygon side-loop model.

The original scaffold cited def-complex-line-integral-over-a-rectifiable-path and def-complex-contours-reversal-concatenation-and-closedness for ${\int }_{\gamma }\omega$; those interfaces concern plane contour integrals, while the integral here is taken on a Riemann surface along continuous paths. The direct dependency is now the local-primitive path integral def-path-integral-of-a-holomorphic-differential-on-a-riemann-surface, which supplies additivity, reversal and the holomorphy computations used above. The unused scaffold edge to thm-riemann-bilinear-relations was removed: only the quotient structure of the Jacobian, not the full-lattice property, enters the definition.