I studied the Abel Jacobi Theorem defining the Picard group as
$$Pic(X) = \frac{ \{ D: X \rightarrow \mathbb{Z} \; \text{ with support free from accumulation point} \}}{ \{ \text{divisors of non constantly zero meromorphic functions} \}}$$
Now I am studying the Picard group as $$Pic(X) = \frac{\{ \text{Holomorphic line bundles on $X$}\}}{\sim}$$
Where $\sim$ means the isomorphism equivalence relation.
Considering the exponential sequence for a connected, compact holomorphic manifold we get:
$$Pic^{0}(X) = Ker(c_1 : H^{1}(X, \mathcal{O}^{\ast}_X) \rightarrow H^2(X, \mathbb{Z})) \cong \frac{H^{1}(X, \mathcal{O}_X)}{H^{1}(X, \mathbb{Z})}$$
I think this isomorphism is the Abel Jacobi one if $X$ is a Riemann Surfaces. Am I correct?
I would like to explicitly write down an isomorphism (it's a guess of mine these two spaces are isomorphic, so perhaps my efforts are meaningless, in that case please tell me)
$$\frac{H^{1}(X, \mathcal{O}_X)}{H^{1}(X, \mathbb{Z})} \rightarrow \frac{\Omega^{1}(X)^{\vee}}{\{ \text{subgroup of periods}\}}$$
I was thinking about something that uses the existence of a global non zero $(1,1)$ form: $dz \wedge d \, \overline{z}$, but I am not finding any ideas.
Thank you!