Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that this isomorphism can be easily achieved by the Hodge-theoretical methods, but they do not give any reference.

Maybe someone can give any reference or explanation about it. Sorry if it is a very basic question, I do not a lot about Hodge theory.

Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that this isomorphism can be easily achieved by the Hodge-theoretical methods, but they do not give any reference.

Maybe someone can give any reference or explanation about it. Sorry if it is a very basic question, I do not a lot about Hodge theory. So a detailed explanation will be very useful for me.