Abel-Jacobi map (not) holomorphic

Question

It is well known (Miranda's Algebraic Curves and Riemann surfaces, p. 266) that if $X$ is a Riemann surface of genus one, then the Abel-Jacobi map, that is defined (locally) as $$J(p)=\int_{p_0}^p \omega,$$ where $p_0$ is a fixed basepoint on $X$, and $\omega$ is a holomorphic 1-form on $X$, is a holomorphic function (of $p$). This fact is used to show that Riemann surfaces of genus 1 and complex 1-tori are the same thing. If the genus of $X$ is $g\geq 2$, then we fix a basis $\omega_1, \ldots. \omega_g$ of the holomorphic 1-forms on $X$, and we define the Abel-Jacobi map as $$J(p)=\Big(\int_{p_0}^p \omega_1, \ldots, \int_{p_0}^p \omega_g\Big).$$ Now, this map cannot be holomorphic, because it would induce a biolomorphism between a Riemann surface of genus $g$ and a complex $g$-torus, right? But what prevents this map to be holomorphic?