Let $X$ be a compact, connected Riemann surface and $\text{Jac}(X)$ be the associated Jacobian. Let $$ J: X \to \text{Jac}(X)$$
be the holomorphic embedding known as the Abel-Jacobi map.
Since this is a non-constant holomorphic map from a compact Riemann surface to a connected Riemann surface I believe this map should be surjective.
I know this map factors through the degree zero Picard group of $X$ and that the intermediate map $\text{Pic}^0(X) \to \text{Jac}(X)$ is an isomorphism. It follows that the embedding from $X \to \text{Pic}^0(X)$ must be surjecitve, however I believe this is often not the case, i.e. when $X$ has genus higher than 1.
My question: Why is the Abel-Jacobi map not surjective. Which of the following fail:
1) J is holomorphic. 2) Jac(X) is connected.