How do you identify the tangent space of the Jacobian variety with differential forms of the first kind?

Question

I know that the Jacobian variety of a Riemann surface $X$ can be defined as $$Jac(X) = (\Omega_{hol}^{1}(X))^{*}/\text{H}_1(X, \mathbb{Z}).$$ My question is how do you canonically identify its tangent space with the space $\Omega_{hol}^1(X)$, because as far as I understand, its tangent space is canonically isomorphic to the dual space $(\Omega_{hol}^1(X))^*$. I've searched for a while and I could only find a proof of this that uses algebraic geometry machinery, is there any way to see this at the level of manifolds?