The Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of 
Niels Abel and Carl Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

Let $C$ be a smooth projective curve of genus g. Let $ω_1, ω_2, …, ω_g$ be a basis for $Ω_ℂ(ℂ(C))$ over $ℂ$. Let $P_0$ be any arbitrary point on $C$. We define the Abel-Jacobi map as a map from the curve $C$ to its Jacobian $J\left(C\right)$ given by