[proofplan] Fix a base point $P_0\in X$ and construct the Abel-Jacobi homomorphism from degree-zero divisors to the quotient of $H^0(X,\Omega_X^1)^*$ by periods. The only ambiguity in the construction comes from changing integration paths, and that ambiguity is exactly a period. [Abel's theorem](/theorems/834) identifies precisely which degree-zero divisors map to zero, namely the principal divisors, so the map descends injectively to $\operatorname{Pic}^0(X)$. Jacobi inversion gives surjectivity, and the standard holomorphic dependence of Abelian integrals identifies the resulting bijective homomorphism as an isomorphism of complex tori. [/proofplan] [step:Define the period lattice and the Abel-Jacobi functional on degree-zero divisors] Let \begin{align*} V := H^0(X,\Omega_X^1) \end{align*} be the complex [vector space](/page/Vector%20Space) of holomorphic $1$-forms on $X$. For a piecewise smooth singular $1$-chain $c=\sum_{k=1}^s m_k\beta_k$, where $m_k\in\mathbb{Z}$ and $\beta_k:[0,1]\to X$ is piecewise smooth, define the integration functional \begin{align*} I_c: V &\longrightarrow \mathbb{C} \\ \omega &\longmapsto \sum_{k=1}^s m_k\int_{[0,1]} b_{k,\omega}(t)\,d\mathcal{L}^1(t), \end{align*} where $\beta_k^*\omega=b_{k,\omega}(t)\,dt$ on each smooth subinterval of $[0,1]$. Define the period homomorphism \begin{align*} \iota: H_1(X,\mathbb{Z}) &\longrightarrow V^* \\ [\gamma] &\longmapsto I_\gamma. \end{align*} By the classical period nondegeneracy theorem for compact Riemann surfaces, $\iota$ is injective and its image \begin{align*} \Lambda := \iota(H_1(X,\mathbb{Z})) \end{align*} is a lattice in the complex vector space $V^*$; that is, $\Lambda$ is a discrete subgroup of real rank $2g$. This is the period lattice of $X$. Fix a point $P_0\in X$. Let $\operatorname{Div}^0(X)$ denote the abelian group of degree-zero divisors on $X$. For a divisor \begin{align*} D=\sum_{i=1}^r n_i[P_i]\in \operatorname{Div}^0(X), \end{align*} where $n_i\in\mathbb{Z}$, $P_i\in X$, and $\sum_{i=1}^r n_i=0$, choose for each $i$ a piecewise smooth path \begin{align*} \alpha_i:[0,1]\to X \end{align*} with $\alpha_i(0)=P_0$ and $\alpha_i(1)=P_i$. Define a complex-linear functional \begin{align*} A_D: V &\longrightarrow \mathbb{C} \\ \omega &\longmapsto \sum_{i=1}^r n_i I_{\alpha_i}(\omega). \end{align*} Then define \begin{align*} \operatorname{AJ}_{P_0}: \operatorname{Div}^0(X) &\longrightarrow V^*/\Lambda \\ D &\longmapsto A_D+\Lambda. \end{align*} [guided] The target torus is built from periods of holomorphic differentials. We first set \begin{align*} V := H^0(X,\Omega_X^1), \end{align*} so $V$ is the $g$-dimensional complex vector space of holomorphic $1$-forms on $X$. If $c=\sum_{k=1}^s m_k\beta_k$ is a piecewise smooth singular $1$-chain, where $m_k\in\mathbb{Z}$ and $\beta_k:[0,1]\to X$ is piecewise smooth, define \begin{align*} I_c: V &\longrightarrow \mathbb{C} \\ \omega &\longmapsto \sum_{k=1}^s m_k\int_{[0,1]} b_{k,\omega}(t)\,d\mathcal{L}^1(t), \end{align*} where $\beta_k^*\omega=b_{k,\omega}(t)\,dt$ on each smooth subinterval. A homology class $[\gamma]\in H_1(X,\mathbb{Z})$ determines a functional on $V$ by \begin{align*} \iota([\gamma])=I_\gamma. \end{align*} The classical period nondegeneracy theorem for compact Riemann surfaces states that this map is injective and has image a lattice of real rank $2g$ in $V^*$. We denote this image by \begin{align*} \Lambda := \iota(H_1(X,\mathbb{Z})). \end{align*} This is the subgroup by which we quotient, because changing paths in Abelian integrals changes the answer by exactly an integral over a closed cycle. Now fix a base point $P_0\in X$. Let $\operatorname{Div}^0(X)$ be the group of degree-zero divisors. If \begin{align*} D=\sum_{i=1}^r n_i[P_i]\in \operatorname{Div}^0(X), \end{align*} then $n_i\in\mathbb{Z}$, $P_i\in X$, and $\sum_{i=1}^r n_i=0$. For each point $P_i$, choose a piecewise smooth path \begin{align*} \alpha_i:[0,1]\to X \end{align*} from $P_0$ to $P_i$. We define \begin{align*} A_D: V &\longrightarrow \mathbb{C} \\ \omega &\longmapsto \sum_{i=1}^r n_i I_{\alpha_i}(\omega). \end{align*} This is complex-linear because integration of $1$-forms is complex-linear in the form. The Abel-Jacobi value of $D$ is the class \begin{align*} \operatorname{AJ}_{P_0}(D):=A_D+\Lambda\in V^*/\Lambda. \end{align*} [/guided] [/step] [step:Show that changing the integration paths changes the functional by a period] Let $\alpha_i'$ be another piecewise smooth path from $P_0$ to $P_i$ for each $i$, and let $A_D'$ be the functional obtained from the paths $\alpha_i'$. For each $i$, define the closed singular $1$-cycle \begin{align*} \gamma_i := \alpha_i-\alpha_i'. \end{align*} Then \begin{align*} (A_D-A_D')(\omega) &= \sum_{i=1}^r n_i\left(I_{\alpha_i}(\omega)-I_{\alpha_i'}(\omega)\right) \\ &= \sum_{i=1}^r n_i I_{\gamma_i}(\omega) \\ &= I_{\sum_{i=1}^r n_i\gamma_i}(\omega). \end{align*} The chain $\sum_{i=1}^r n_i\gamma_i$ is a closed integral $1$-cycle, so it defines a class in $H_1(X,\mathbb{Z})$. Hence \begin{align*} A_D-A_D'=\iota\left(\left[\sum_{i=1}^r n_i\gamma_i\right]\right)\in \Lambda. \end{align*} Therefore $A_D+\Lambda$ is independent of all path choices. [guided] We must check that the definition does not depend on the auxiliary paths. Suppose $\alpha_i'$ is a second path from $P_0$ to $P_i$. The formal difference \begin{align*} \gamma_i:=\alpha_i-\alpha_i' \end{align*} is a closed singular $1$-cycle: both paths have the same initial point and the same terminal point, so their boundaries cancel. Let $A_D'$ be the functional obtained using the paths $\alpha_i'$. For every $\omega\in V$, linearity of path integration gives \begin{align*} (A_D-A_D')(\omega) &= \sum_{i=1}^r n_i\left(I_{\alpha_i}(\omega)-I_{\alpha_i'}(\omega)\right) \\ &= \sum_{i=1}^r n_i I_{\gamma_i}(\omega) \\ &= I_{\sum_{i=1}^r n_i\gamma_i}(\omega). \end{align*} The cycle $\sum_{i=1}^r n_i\gamma_i$ has integral coefficients, so it represents a homology class \begin{align*} \left[\sum_{i=1}^r n_i\gamma_i\right]\in H_1(X,\mathbb{Z}). \end{align*} By the definition of the period homomorphism $\iota$, the functional $I_{\sum_i n_i\gamma_i}$ lies in $\Lambda$. Hence the two choices of paths give the same element of $V^*/\Lambda$. [/guided] [/step] [step:Descend the Abel-Jacobi map to divisor classes by Abel's theorem] The assignment $D\mapsto \operatorname{AJ}_{P_0}(D)$ is a group homomorphism because both divisor addition and integration are additive. Let \begin{align*} \operatorname{Prin}(X)\subset \operatorname{Div}^0(X) \end{align*} denote the subgroup of principal divisors. By Abel's theorem for compact Riemann surfaces, a divisor $D\in\operatorname{Div}^0(X)$ is principal if and only if $\operatorname{AJ}_{P_0}(D)=0$ in $V^*/\Lambda$ (citing a result not yet in the wiki: Abel's Theorem for compact Riemann surfaces). Therefore \begin{align*} \ker(\operatorname{AJ}_{P_0})=\operatorname{Prin}(X). \end{align*} Since \begin{align*} \operatorname{Pic}^0(X)=\operatorname{Div}^0(X)/\operatorname{Prin}(X), \end{align*} the homomorphism $\operatorname{AJ}_{P_0}$ descends to an injective group homomorphism \begin{align*} \overline{\operatorname{AJ}}_{P_0}: \operatorname{Pic}^0(X)&\longrightarrow V^*/\Lambda \\ [D]&\longmapsto \operatorname{AJ}_{P_0}(D). \end{align*} [/step] [step:Use Jacobi inversion to prove surjectivity] Let $\xi\in V^*/\Lambda$. By the classical Jacobi inversion theorem, there exist points $Q_1,\dots,Q_g\in X$ such that \begin{align*} \xi=\operatorname{AJ}_{P_0}\left(\sum_{j=1}^g[Q_j]-g[P_0]\right) \end{align*} in $V^*/\Lambda$ (citing a result not yet in the wiki: Jacobi Inversion Theorem). The divisor \begin{align*} E:=\sum_{j=1}^g[Q_j]-g[P_0] \end{align*} has degree $g-g=0$, so it defines a class $[E]\in\operatorname{Pic}^0(X)$. Hence \begin{align*} \overline{\operatorname{AJ}}_{P_0}([E])=\xi. \end{align*} Thus $\overline{\operatorname{AJ}}_{P_0}$ is surjective. [/step] [step:Identify the bijective homomorphism as an isomorphism of complex tori] The quotient $V^*/\Lambda$ is a complex torus because $V^*$ is a complex vector space of complex dimension $g$ and $\Lambda$ is a lattice of real rank $2g$. The standard analytic structure on $\operatorname{Pic}^0(X)=J(X)(\mathbb{C})$ is the one for which the Abel-Jacobi map from symmetric products of $X$ is holomorphic. Since the local Abelian integration functionals \begin{align*} \omega\longmapsto I_{\alpha_P}(\omega) \end{align*} depend holomorphically on $P\in X$ locally, where $\alpha_P:[0,1]\to X$ is a locally chosen piecewise smooth path from $P_0$ to $P$, the induced map \begin{align*} \overline{\operatorname{AJ}}_{P_0}: J(X)(\mathbb{C})\longrightarrow V^*/\Lambda \end{align*} is holomorphic. Its inverse is holomorphic by the analytic form of Jacobi inversion, which constructs local inverse charts by effective divisors of degree $g$ away from the usual special-divisor locus. Therefore $\overline{\operatorname{AJ}}_{P_0}$ is an isomorphism of complex tori: \begin{align*} J(X)(\mathbb{C})\cong H^0(X,\Omega_X^1)^*/\iota(H_1(X,\mathbb{Z})). \end{align*} This is the desired Abel-Jacobi description of the Jacobian. [/step]