Abel–Jacobi map

Summary

The Abel–Jacobi map sends a point of a compact Riemann surface of genus $g \ge 1$ to its Jacobian variety, a complex torus of dimension $g$. It is defined by integrating a basis of holomorphic Abelian differentials from a fixed base point. Abel's theorem says that it detects which divisors of degree $0$ are divisors of meromorphic functions. It is the basic bridge between a curve and its Jacobian.