The Abel-Jacobi map embeds a positive-genus surface

Suppose $p\mathrm{\ne }q$ and $u(p)=u(q)$. By [F3], $u((q)-(p))=u(q)-u(p)=0$, so Abel's theorem makes $(q)-(p)$ a principal divisor. By [F4] there is a meromorphic $f$ with $(f)=(q)-(p)$, whose associated map $X\to \widehat{\mathbb{C}}$ has degree $1$ and is therefore a biholomorphism; hence $X\cong \widehat{\mathbb{C}}$ has genus $0$, contradicting $g\ge 1$ by [F5]. Thus $u$ is injective.

By [F1] the derivative of a holomorphic lift $\xi$ at any $p$ is the evaluation map $\omega \mapsto \omega ({\mathrm{\partial }}_z)$ on the chart tangent direction; by [F2] some holomorphic differential is nonzero at $p$, so this linear map $\mathbb{C}\to {\mathbb{C}}^g$ is nonzero, hence injective. Thus $u$ is an immersion at every point.

For the generation statement, every difference $u(p)-u(q)$ lies in the subgroup generated by $u(X)-u(p_0)=\{u(x)-u(p_0):x\in X\}$, and conversely every element of that subgroup is a finite combination of such differences. Every degree-zero divisor is a finite $\mathbb{Z}$-combination of point differences, so its image under $u$ lies in the subgroup generated by $u(X)-u(p_0)$; by [F7] The degree-zero divisor extension $u:{\mathrm{Div}}^0(X)\to \mathrm{Jac}(X)$ is surjective, so that subgroup is all of $\mathrm{Jac}(X)$.

Fix $p$ and a holomorphic lift $\xi$ on a coordinate disk centered at $p$, translating the target so $\xi (p)=0$. By step 1.2 its derivative is nonzero, so an invertible complex-linear change of target coordinates makes the first component ${\xi }_1$ have nonzero derivative. The inverse function theorem [F6] in complex dimension one gives a smaller disk on which $s={\xi }_1(z)$ is a holomorphic coordinate with holomorphic inverse $z=h(s)$. In these coordinates the lifted image is exactly $(s,{\xi }_2(h(s)),\dots ,{\xi }_g(h(s)))$, a holomorphic graph; for $g=1$ there are no remaining components. The quotient atlas transfers this graph description to a neighborhood in the Jacobian.

Since $X$ is compact and the Jacobian is Hausdorff, the continuous injection $u$ is a homeomorphism onto its compact image by [F5]. Fix a graph disk $U$ about $p$ as in step 2.2 and a smaller neighborhood $U^{\mathrm{\prime }}$ whose closure lies in $U$. The compact set $X\setminus U^{\mathrm{\prime }}$ has compact, hence closed image disjoint from $u(p)$. Shrink the target graph chart about $u(p)$ to miss this image. In that chart the entire image $u(X)$ is the graph portion from $U$, so no other branches occur. This proves the complex-submanifold chart assertion as well as the closed topological embedding.

If $g=1$, then $u$ is injective by step 1.1 and an immersion by step 1.2, so in the local model of step 2.2 it is locally biholomorphic onto its image; the image is open (by local biholomorphy) and compact (by step 3.1), hence closed, and the connected torus $\mathrm{Jac}(X)$ is therefore entirely covered by the nonempty image: $u$ is a biholomorphism onto $\mathrm{Jac}(X)$.

Steps 1.1, 1.2, 2.2 and 3.1 establish the four statements, and step 4.1 the genus-one clause, all under the inherited AC of [F8].