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  "body": "\n## Definition\n\nA *translation surface* is a pair $(X, \\omega)$ with the following data.\n$X$ is a compact connected [[Riemann surface|5K0N8QMV6Q]].\n$\\omega$ is a holomorphic 1-form on $X$, an [[abelian differential|P014C13K5H]], that is not identically zero.\nTwo translation surfaces $(X, \\omega)$ and $(X', \\omega')$ are the same if there is a biholomorphic map $\\phi \\colon X \\to X'$ with $\\phi^* \\omega' = \\omega$.\n\n## Remarks\n\nLet $\\Sigma$ be the finite set of zeros of $\\omega$.\nNear a point $p \\notin \\Sigma$, the function $z(q) = \\int_p^q \\omega$ is a local coordinate in which $\\omega = dz$, and two such coordinates differ by a translation $z \\mapsto z + c$.\nSo $X \\setminus \\Sigma$ gets a flat metric $\\abs{\\omega}$, and a zero of order $k$ is a cone point of angle $2\\pi(k + 1)$.\nIf $X$ has [[genus|NJ787GZS1N]] $g$ and the zeros have orders $k_1, \\dots, k_n$, then $k_1 + \\cdots + k_n = 2g - 2$.\n\nEquivalently, a translation surface is given by finitely many polygons in $\\C$ whose sides are glued in pairs by translations.\nEach side is glued to a parallel side of the same length with opposite outward normal.\nThe form $\\omega$ is $dz$ on each polygon.\n\nExample: the unit square with opposite sides glued is the [[torus|03NM7DZXXC]] $X = \\C / (\\Z + i\\Z)$ with $\\omega = dz$.\nThe form $dz$ is invariant under the translations by $\\Z + i \\Z$, so it descends to $X$, and it has no zeros, in line with $2g - 2 = 0$.\n\nExample: the regular octagon with opposite sides glued has all eight vertices glued to one point, with total angle $8 \\cdot \\frac{3\\pi}{4} = 6\\pi$.\nSo $\\omega$ has one zero of order $2$, and $2 = 2g - 2$ gives genus $g = 2$.\n\nNon-example: $(X, 0)$ is not a translation surface, since the zero form gives no charts.\n",
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