Abel-Jacobi theorem

In mathematics, 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 Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

In complex algebraic geometry, the Jacobian of a curve C is constructed using path integration. Namely, suppose C has genus g, which means topologically that

Geometrically, this homology group consists of (homology classes of) cycles in C, or in other words, closed loops. Therefore, we can choose 2g loops ${\displaystyle \gamma _{1},\dots ,\gamma _{2g}}$ generating it. On the other hand, another more algebro-geometric way of saying that the genus of C is g is that

By definition, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms ${\displaystyle \omega _{1},\dots ,\omega _{g}}$. Given forms and closed loops we can integrate, and we define 2g vectors

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