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The Abel image in its Jacobian

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a compact connected Riemann surface of genus $g \geq 1$ , let $p_{0} \in X$ and let $u = u_{p_{0}} : X \to Jac (X)$ be the Abel-Jacobi map (The Abel-Jacobi map). Then:

The image $u (X)$ is a compact subset of the complex torus $Jac (X)$ with a one-dimensional complex-graph chart at each of its points, and $u$ is a homeomorphism of $X$ onto $u (X)$ ; for $g \geq 2$ the image has complex dimension one inside the $g$ -dimensional torus, so it is neither open nor a complex subtorus (The Abel-Jacobi map embeds a positive-genus surface).
The subgroup of $Jac (X)$ generated by $u (X) - u (p_{0}) = {u (p) - u (p_{0}) : p \in X}$ is all of $Jac (X)$ : every class $u ((q) - (p))$ is a difference of two image points, every degree-zero divisor is a finite sum of such classes, and Jacobi inversion gives the whole group (Jacobi inversion).
For $g = 1$ the image is the whole torus: $u : X \to Jac (X)$ is a biholomorphism (Periods of a complex torus). For the explicitly constructed genus-two pentagon curve $X_{0}$ of Period matrix and Jacobian of the pentagon curve, $X_{0}$ embeds by its Abel-Jacobi map as a compact curve generating the two-dimensional torus $Jac (X_{0}) = C^{2} / Λ$ of Period matrix and Jacobian of the pentagon curve.
The map is a bijection onto its image with holomorphic inverse: $u (p) = u (q)$ implies $p = q$ by the embedding theorem, and the inverse is holomorphic because $u$ is locally biholomorphic onto its image.

Facts & Assumptions

Given: Full AC, a compact connected Riemann surface $X$ of genus $g \geq 1$ , a base point $p_{0}$ , and the Abel-Jacobi map $u$ .

[F1]

$u$ is injective, its derivative is nonzero at every point, it is a homeomorphism onto its compact image, and every point of the image has a chart of $Jac (X)$ in which the image is the graph of a holomorphic map (The Abel-Jacobi map embeds a positive-genus surface, Biholomorphic maps between complex domains).

[F2]

On ${Div}^{0} (X)$ the map $u$ is a group homomorphism with $u ((q) - (p)) = u (q) - u (p)$ , and the kernel is the principal divisors (The Abel-Jacobi map, The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent, Abel's theorem for divisors).

[F3]

The degree-zero divisor extension of $u$ is surjective, so every class is $u (D)$ for a degree-zero divisor $D$ , and every degree-zero divisor is a finite $Z$ -combination of point differences (Jacobi inversion, Abel's theorem for divisors).

[F4]

The Jacobian is the complex torus $Ω (X)^{*} / Λ$ of complex dimension $g$ ; for $g = 1$ the Abel-Jacobi map is a biholomorphism onto the whole torus, and ${Pic}^{0} (X) ≅ Jac (X)$ (The Jacobian of a compact Riemann surface, The Abel-Jacobi map embeds a positive-genus surface, Picard zero is the Jacobian, Periods of a complex torus).

[F5]

The pentagon supplier constructs the smooth projective curve $y^{2} = x^{5} - 1$ , identifies it analytically with the translation double-pentagon, and proves the integral orbit basis and nonzero normalized periods. Its genus-two model therefore has Jacobian $C^{2} / Λ$ with a full period lattice $Λ$ , and every point of it is represented by a degree-zero divisor (Period matrix and Jacobian of the pentagon curve, Jacobi inversion).

[F6]

Full AC is inherited from the Jacobian construction (The Axiom of Choice).

Verification

Given: The objects and conventions in the Statement.

1.1F1F4

By [F1], $u$ is injective with nonzero derivative, is a homeomorphism onto its compact image, and is locally biholomorphic onto the graph charts of that image. Thus $u (X)$ is a one-dimensional complex submanifold in the stated chart sense. For $g \geq 2$ , the ambient dimension is $g > 1$ by [F4]; in every graph chart a small change in a transverse complex coordinate leaves the graph, so the image has empty interior and is not open.

1.2F1F4F5

For $g = 1$ claim 3 is [F4]: $u$ is a biholomorphism onto the whole torus. For the pentagon clause, specialize to $X = X_{0}$ and take any base point on $X_{0}$ . By [F5] this curve has genus two and $Jac (X_{0}) = C^{2} / Λ$ with full period lattice. Claims 1 and 2 therefore apply to its Abel-Jacobi map: its image is a compact embedded curve generating that torus.

2.1F2F3F4step 1.1algebra

Every difference $u (q) - u (p)$ is the class $u ((q) - (p))$ by [F2]. Finite sums of such differences are images of degree-zero divisors, and conversely every degree-zero divisor is a finite integer sum of point differences. By [F3] the divisor extension is surjective, so the subgroup generated by $u (X) - u (p_{0})$ is the whole Jacobian. Since $u (p_{0}) = 0$ , if $u (X)$ were a complex subtorus, it would already be a subgroup and would equal the subgroup it generates, hence would equal the Jacobian. This is impossible for $g \geq 2$ by the dimension and empty-interior calculation in step 1.1. This completes claims 1 and 2.

2.2F1step 1.1

Claim 4 is contained in claim 1: $u$ is injective, so it is a bijection onto its image, and the local biholomorphy of step 1.1 gives a holomorphic inverse on the image.

3.1F6step 1.1step 2.1step 1.2step 2.2∎

Claims 1-4 are steps 1.1, 2.1, 1.2 and 2.2, under the inherited AC of [F6].

Source notes

McMullen's Theorem 15.7 and the surrounding discussion (Riemann Surfaces, printed p. 130) exhibit the curve in its Jacobian as a smooth embedded curve generating the torus; Looijenga's Ch. 7 §2 (printed pp. 60-63) shows the point map, its injectivity for positive genus and the genus-one biholomorphism. The example packages the general embedding theorem with the two explicit models of claims 3.

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Sources

Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)
Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)