The Abel-Jacobi map from an algebraic curve $C$ to its Jacobian $J(C)$ is given analytically by $$p\to \left( \ldots, \int^{p}\_{p\_0} \omega\_i,\ldots\right),$$
where $p\_0$ is some point on $C$ and $\omega\_i$ form a basis of $H^0(C,K)$. Why is it an algebraic morphism?



Why is the Abel-Jacobi map an algebraic morphism?
Maybe there are easier ways to see it, but Chow's theorem/GAGA certainly gives you the result, since you have an analytic morphism of projective analytic varieties.


