I got a question in Complex Geometry-- An Introduction by Dianel Huybrechts.:
Suppose $X$ is a compact complex curve, then $\operatorname{Abl}(X)\cong \operatorname{Pic}^0(X)$, the Jacobian of $X$. Since by Serre duality we have $H^0(X,\Omega_X)^*\cong H^1(X,\mathscr{O}_X)$ and by Poincare duality we get $H_1(X,\mathbb{Z})\cong H^1(X,\mathbb{Z})$. Thus $\operatorname{Abl}(X)\cong H^0(X,\Omega_X)^*/H_1(X,\mathbb{Z})\cong H^1(X,\mathscr{O}_X)/H^1(X,\mathbb{Z})\cong \operatorname{Pic}^0(X)$.
Then the author claimed that through this isomorphism, we have the Albanese map $X\to \operatorname{Abl}(X)\cong \operatorname{Pic}^0(X)\hookrightarrow \operatorname{Pic}(X)$ coincides with the Abel-Jacobi map $X\to \operatorname{Pic}(X)$ defined by $x\mapsto \mathscr{O}(x-x_0)$, where $x_0$ is a base point for both maps. But I couldn't verify this directly, can anyone help me? Thanks in advance.