nLab
Jacobian variety

Jacobian varieties are the most important class of abelian varieties.

To every nonsingular algebraic curve $C$ (over the complex numbers) of genus $g$ one associates the Jacobian variety or simply Jacobian $J(C)$ either via differential 1-forms or equivalently via line bundles: Jacobian is the the moduli space of degree $0$ line bundles over $C$, i.e. the connected component of the identity of the Picard group of $C$.

The Abel-Jacobi map $C\to J(C)$ is defined with help of periods.

(one should also cover Abel-Jacobi theorem)

• wikipedia Jacobian variety, Abel-Jacobi map
• P. Griffiths, J. Harris, Principles of algebraic geometry
• A. Beauville, Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables, Acta Math. 164 (1990), 211-235.