Jacobian variety

Summary

The Jacobian variety of a compact Riemann surface of genus $g$ is a complex torus of dimension $g$. It is built from the periods of the holomorphic differentials of the surface. It is a principally polarized abelian variety. Its points correspond to divisor classes of degree $0$ on the curve, so it turns questions about divisors into questions about a group. The Abel–Jacobi map maps the curve into its Jacobian.