 We can define

and therefore speak of the value of the Abel–Jacobi map on divisors.The theorem is then that if "D" and "E" are two "effective" divisors, meaning that the formula_62 are all positive integers, then

Jacobi proved that this map is also surjective, so the two groups are naturally isomorphic.The Abel–Jacobi theorem implies that the Albanese variety of a compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its Jacobian variety (divisors of degree 0 modulo equivalence).