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The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the Jacobian definition. Let $X$ be a compact connected Riemann surface, let $p_{0} \in X$ and let $u = u_{p_{0}} : X \to Jac (X)$ be the Abel-Jacobi map of The Abel-Jacobi map. Then:

Path independence. For any two paths $γ, γ^{'}$ from $p_{0}$ to $p$ the functionals $ω \mapsto \int_{γ} ω$ and $ω \mapsto \int_{γ^{'}} ω$ differ by an element of the period lattice $Λ$ , namely by $P ([γ * γ^{' - 1}], \cdot) = e ([γ * γ^{' - 1}])$ , with $*$ concatenation and $γ^{' - 1}$ the reversal (The period pairing and the period subgroup, The period pairing is well defined and computed by integration). Hence $u$ is well defined.
Derivative. $u$ is holomorphic in the atlas of the Jacobian definition; in a holomorphic chart $z$ at $p$ and a holomorphic lift $ξ$ of $u$ near $p$ , the derivative of $ξ$ at $z (p)$ is the linear map $C ⟶ C^{g} ≅ Ω (X)^{*}, 1 ⟼ (ω_{1} (\partial_{z}), \dots, ω_{g} (\partial_{z})),$ where $ω_{1}, \dots, ω_{g}$ is the basis of $Ω (X)$ used by the Jacobian definition and $ω_{i} (\partial_{z})$ is the local coefficient of $ω_{i}$ in the chart $z$ . This derivative is nonzero (equivalently, injective) if and only if some holomorphic differential does not vanish at $p$ (Every complex analytic function has a primitive on a neighbourhood of each point).
Base-point independence and additivity on degree zero. For degree-zero divisors $D = \sum_{p} n_{p} [p]$ the class $\sum_{p} n_{p} u_{p_{0}} (p)$ is independent of $p_{0}$ , and $u : {Div}^{0} (X) \to Jac (X)$ is a group homomorphism. Moreover $u ((q) - (p)) + u ((r) - (q)) = u ((r) - (p))$ for all $p, q, r \in X$ , and $u (D_{1} + D_{2}) = u (D_{1}) + u (D_{2})$ for degree-zero divisors $D_{1}, D_{2}$ (Divisors, principal divisors and canonical divisors on a Riemann surface).
Cocycle form. For each $ω \in Ω (X)$ the function $x \mapsto u (x) (ω)$ is locally a primitive of $ω$ , in the sense that near each point it coincides with a local primitive of $ω$ up to an additive constant modulo $Λ_{ω}$ , where $Λ_{ω} \subseteq C$ is the image of $Λ$ under evaluation at $ω$ . Consequently, along any path $γ$ from $p$ to $q$ , $u (q) (ω) - u (p) (ω) = \int_{γ} ω in C / Λ_{ω} .$

Facts & Assumptions

Given: Full AC, a compact connected Riemann surface $X$ of genus $g$ , the Jacobian $Jac (X) = Ω (X)^{*} / Λ$ , a base point $p_{0} \in X$ , and the map $u = u_{p_{0}}$ with its divisor extension.

[F1]

For a path $γ$ from $p_{0}$ to $p$ the point class is $u_{p_{0}} (p) = [ω \mapsto \int_{γ} ω] \in Jac (X)$ , this class is independent of $γ$ , and $u$ is holomorphic in the explicit chart sense of the definition (The Abel-Jacobi map, The Jacobian of a compact Riemann surface).

[F2]

$u$ satisfies the addition rule $u_{p_{0}} (q) - u_{p_{0}} (p) = [ω \mapsto \int_{p}^{q} ω]$ ; its linear extension to divisors is base-point independent on ${Div}^{0} (X)$ , is a group homomorphism there, and satisfies $u ((q) - (p)) = [ω \mapsto \int_{p}^{q} ω]$ (The Abel-Jacobi map, Divisors, principal divisors and canonical divisors on a Riemann surface).

[F3]

The period pairing and homomorphism are $e (γ) (ω) = P (γ, ω)$ and $Λ = e (H_{1} (X; Z))$ ; for $g \geq 1$ a basis $ω_{1}, \dots, ω_{g}$ of $Ω (X)$ is fixed (The period pairing and the period subgroup, The Jacobian of a compact Riemann surface).

[F4]

For every $γ \in H_{1} (X; Z)$ and every continuous singular cycle $c$ representing $γ$ , $P (γ, ω) = \int_{c} ω$ for all $ω \in Ω (X)$ (The period pairing is well defined and computed by integration).

[F5]

The path integral of a holomorphic differential is additive under concatenation, changes sign under reversal, and is $C$ -linear in the differential; on a simply connected coordinate disk a holomorphic differential $ω = h (z) d z$ has a holomorphic primitive $H$ with $H^{'} = h$ (Path integral of a holomorphic differential on a Riemann surface, Meromorphic differentials, orders and residues).

[F6]

A map into $C^{g}$ is holomorphic exactly when its components are holomorphic, and holomorphic functions are differentiable with the derivative computed in coordinates (Holomorphic maps $C^{m} \to C^{n}$ and the complex Jacobian matrix, A map into $C^{n}$ is holomorphic exactly when each of its components is, Every complex analytic function has a primitive on a neighbourhood of each point).

[F7]

Divisors on compact $X$ have finite support and their degree is the sum of the coefficients, $\deg D : = \sum_{p} D (p)$ ; hence degrees add and the divisors of degree zero form a subgroup (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F8]

Full AC is inherited from the Jacobian definition, which uses it to select the bases of [F3]; no additional arbitrary selection is made here (The Axiom of Choice, The Jacobian of a compact Riemann surface).

Proof

technique · direct

1.1F1F3F4F5

Let $p \in X$ and let $γ, γ^{'}$ be paths from $p_{0}$ to $p$ . By [F5], $\int_{γ} ω - \int_{γ^{'}} ω = \int_{γ * γ^{' - 1}} ω$ for every $ω \in Ω (X)$ , and the closed curve $γ * γ^{' - 1}$ is a continuous singular cycle with class $δ \in H_{1} (X; Z)$ . By [F4] and [F3], the right-hand side equals $P (δ, ω) = e (δ) (ω)$ , an element of $Λ$ as a functional. Hence the two functionals differ by $e (δ) = P (δ, \cdot) \in Λ$ , and by [F1] they define the same class $u (p)$ ; this is clause 1.

1.2F1F3F5F6

Fix $p \in X$ and a holomorphic chart $z : U \to D$ at $p$ with $z (p) = 0$ . Write $ω_{i} = h_{i} (z) d z$ on $U$ and let $H_{i}$ be the holomorphic primitive of $h_{i}$ on $D$ with $H_{i} (0) = 0$ , which exists by [F5]. Fixing a path from $p_{0}$ to $p$ , the same concatenation argument as [F1] shows that the lift $ξ (x) = ξ_{0} + (H_{1} (z (x)), \dots, H_{g} (z (x)))$ , where $ξ_{0}$ is the functional of the fixed path from $p_{0}$ to $p$ , satisfies $π \circ ξ = u$ on $U$ . Each component is holomorphic, so $ξ$ is holomorphic by [F6], and its derivative at $0$ is $D H_{i} (0) = h_{i} (0) = ω_{i} (\partial_{z})$ , giving the displayed linear map $C \to C^{g}$ . This map is zero exactly when $h_{i} (0) = 0$ for all $i$ , i.e. when every holomorphic differential vanishes at $p$ , since the $ω_{i}$ are a basis. This is clause 2.

1.3F2F7

Let $q_{0} \in X$ be a second base point and $D = \sum_{p} n_{p} [p]$ . By the addition rule of [F2], $u_{p_{0}} (p) - u_{p_{0}} (q_{0}) = [ω \mapsto \int_{q_{0}}^{p} ω] = u_{q_{0}} (p)$ for every $p$ , so $u_{q_{0}} (p) = u_{p_{0}} (p) - u_{p_{0}} (q_{0})$ ; summing with coefficients $n_{p}$ shows that the two linear extensions differ by $(\deg D) u_{p_{0}} (q_{0})$ , which is zero when $\deg D = 0$ . For $D_{1}, D_{2} \in {Div}^{0} (X)$ the linear extension satisfies $u (D_{1} + D_{2}) = u (D_{1}) + u (D_{2})$ because finite sums in an abelian group add, and $u ((q) - (p)) = u (q) - u (p)$ for all $p, q \in X$ . This is clause 3.

1.4F1F2F5

Fix $ω \in Ω (X)$ and a chart $z : U \to D$ at a point $p$ with local primitive $H$ of the coefficient of $ω$ , so $H^{'} = h$ and $ω = h d z$ on $U$ . For $x \in U$ the class $u (x)$ is represented by the functional $ω \mapsto \int_{p_{0}}^{x} ω$ , which by [F5] differs from $H (z (x))$ by an additive constant. Hence $x \mapsto u (x) (ω)$ is, modulo the constant and modulo $Λ_{ω}$ , the primitive $H$ ; and along a path $γ$ from $p$ to $q$ contained in $U$ the increment $u (q) (ω) - u (p) (ω)$ equals $H (z (q)) - H (z (p)) = \int_{γ} ω$ in $C / Λ_{ω}$ . For a general path subdivide it into finitely many chart pieces, on each of which the increment equals the corresponding path integral; the increments telescope and give the same identity. This is clause 4.

2.1F2F7F8step 1.1step 1.2step 1.3step 1.4∎

By steps 1.1, 1.3 and 1.4, respectively, the map is well defined, its divisor extension on ${Div}^{0} (X)$ is base-point free and additive, and each evaluation $x \mapsto u (x) (ω)$ is locally a primitive; together with the derivative computation of step 1.2 this proves all four clauses under the inherited full AC of [F8]. In particular $u ((q) - (p)) + u ((r) - (q)) = u ((r) - (p))$ follows by applying the homomorphism property of step 1.3 to $(q) - (p) + (r) - (q) = (r) - (p)$ , and $u (D_{1} + D_{2}) = u (D_{1}) + u (D_{2})$ is the additivity just recalled.

Source notes

Clause 1 is Looijenga's computation ${\tilde{I}}_{γ^{'}} - {\tilde{I}}_{γ} = e ([γ * γ^{'}])$ in Riemann Surfaces, Ch. 7 §2 (printed p. 60); clause 2 is the holomorphy of the point map in Lemma 7.2 there and the derivative formula $D φ_{P} (Q) = (ω_{1} (Q), \dots, ω_{g} (Q))$ in McMullen, Riemann Surfaces, Ch. 15 (printed pp. 129-130); clause 3 is Forster, Lectures on Riemann Surfaces, §21.6 (printed pp. 170-171), where $Φ$ is determined by $D$ up to a period and is a homomorphism; clause 4 is the local read-off of the path integral from a chart primitive. The item proves the four clauses from the definition's explicit chart construction and the local-primitive interface, so the topological side-loop representatives may be integrated without smoothness assumptions.

The scaffold's direct edge to prop-reversal-and-concatenation-of-complex-line-integrals was removed: the reversal and concatenation identities used here are those of the Riemann-surface path integral in def-path-integral-of-a-holomorphic-differential-on-a-riemann-surface, and the plane contour identities are not invoked.

Depends on

Used by

Picard zero is the Jacobian Corollary
Base-point cancellation for degree-zero divisors Example
Periods of a complex torus Example
Principal divisor tests via the Abel-Jacobi map Example
The Abel image in its Jacobian Example
Principal divisors have vanishing Abel-Jacobi class Lemma
Abel's theorem for divisors Theorem
Jacobi inversion Theorem
The Abel-Jacobi map embeds a positive-genus surface Theorem

Dependency tree · two levels

107 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)
Karl Otto Forster, Lectures on Riemann Surfaces, GTM 81, 4th corrected printing (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)