The AbelJacobi map from the algebraic curve $C$ to its Jacobian $J(C)$ is given analitically 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 it is 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. 


This map is written thinking of J(C) as C^g/L where g = genus(C) and L is the lattice of periods. C^g/L can be made algebraic by embedding into a projective space via theta functions. These are section of a line bundle associated to some multiple of a theta divisor $\Theta$ on J(C) http://en.wikipedia.org/wiki/Theta_divisor Then it suffices to show that the restriction to $\Theta$ to C is an ample divisor; but J(C) is a smooth manifold and dim $\Theta$ + dim C = dim J(C) and so $C \cap \Theta$ is a finite number of points, hence ample. 

