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  "body": "\n## Summary\n\nAn Abelian differential is a meromorphic $1$-form on a [[Riemann surface|5K0N8QMV6Q]], written locally as $f(z)\\,dz$ with $f$ meromorphic.\nThe holomorphic ones are the differentials of the first kind.\nTheir integrals along paths give the abelian integrals, the periods, and the [[Abel–Jacobi map|HE8F7KBF42]].\nOn a compact Riemann surface of genus $g$, the holomorphic Abelian differentials form a complex vector space of dimension $g$.\n\n## Explanations\n\n### L1 human\n\nA function on a Riemann surface has a value at each point, but a differential is the thing you integrate along a path.\nIn a local coordinate $z$ it looks like $f(z)\\,dz$, and changing the coordinate changes $f$ by the derivative of the change.\nOn the torus $\\C / \\Lambda$, the form $dz$ is a holomorphic differential that never vanishes.\nIntegrating it along the two basic loops of the torus gives the two generators of the lattice $\\Lambda$.\nOn the Riemann sphere there is no nonzero holomorphic differential, which matches genus $0$.\n",
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