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  "body": "\n## Summary\n\nThe Jacobian variety of a compact [[Riemann surface|5K0N8QMV6Q]] of genus $g$ is a [[complex torus|Y7MA6KBE9M]] of dimension $g$.\nIt is built from the periods of the holomorphic differentials of the surface.\nIt is a principally polarized [[abelian variety|R7W03TXRS9]].\nIts points correspond to divisor classes of degree $0$ on the curve, so it turns questions about divisors into questions about a group.\nThe [[Abel–Jacobi map|HE8F7KBF42]] maps the curve into its Jacobian.\n\n## Explanations\n\n### L1 human\n\nIntegrating the holomorphic differentials of a surface along loops gives a lattice of periods in $\\C^g$.\nDividing $\\C^g$ by this lattice gives a doughnut-shaped space of dimension $g$, which is the Jacobian.\nIt is a group, so points of the curve, and formal sums of them, can be added inside it.\nFor a torus $\\C / \\Lambda$ of genus $1$, the Jacobian is the torus itself.\n",
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