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  "body": "\n## Summary\n\nThe Abel–Jacobi map sends a point of a compact [[Riemann surface|5K0N8QMV6Q]] of genus $g \\ge 1$ to its Jacobian variety, a complex torus of dimension $g$.\nIt is defined by integrating a basis of holomorphic [[Abelian differentials|P014C13K5H]] from a fixed base point.\nAbel's theorem says that it detects which divisors of degree $0$ are divisors of meromorphic functions.\nIt is the basic bridge between a curve and its Jacobian.\n\n## Explanations\n\n### L1 human\n\nPick a base point on the surface and a point $P$, and integrate the $g$ holomorphic differentials along a path from the base point to $P$.\nThis gives $g$ complex numbers, but they depend on the path.\nTwo paths differ by a loop, and the integrals over loops form a lattice, so the answer is well defined modulo that lattice.\nFor a torus $\\C / \\Lambda$ with the differential $dz$, the map is just $P \\mapsto P - P_0$, so the torus is its own Jacobian.\n",
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