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  "body": "\n## Definition\n\nLet $X$ be a [[Riemann surface|5K0N8QMV6Q]].\nAn *Abelian differential* on $X$ is a meromorphic $1$-form on $X$.\nThat is, to each chart $z$ of $X$ it assigns a [[meromorphic function|0ZZ7ZYDR8W]] $f$ on the chart domain, written $f(z)\\,dz$.\nOn the overlap of two charts $z$ and $w$ with forms $f(z)\\,dz$ and $g(w)\\,dw$, these must satisfy\n$$f(z) = g(w(z))\\, \\frac{dw}{dz}.$$\nThe differential is classified as follows.\n\n1. It is of the *first kind* if every $f$ is holomorphic.\n2. It is of the *second kind* if every [[residue|0PHN07KVQT]] of every $f$ is zero.\n3. It is of the *third kind* in general, that is, with arbitrary poles and residues.\n\n## Remarks\n\nExample: on the torus $X = \\C / \\Lambda$ for a lattice $\\Lambda$, the form $dz$ is well defined, since the coordinate changes are translations $w = z + \\lambda$ with $dw/dz = 1$.\nIt is of the first kind and has no zeros.\n\nNon-example: on the Riemann sphere, $e^{1/z}\\,dz$ is not an Abelian differential, because $e^{1/z}$ has an essential singularity at $0$ and so is not meromorphic there.\nAlso $dz$ itself on the sphere has a pole of order $2$ at $\\infty$: in the chart $w = 1/z$ it becomes $-w^{-2}\\,dw$.\nSo it is an Abelian differential of the second kind but not of the first kind.\n\nSome authors, notably in the study of translation surfaces, use \"Abelian differential\" for a holomorphic $1$-form only.\n",
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