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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[Riemann surfaces|5K0N8QMV6Q]].\nA map $f \\colon X \\to Y$ is *holomorphic* if:\n\n1. $f$ is [[continuous|0P27W9RW0G]], and\n2. for every chart $\\varphi \\colon U \\to \\C$ of $X$ and every chart $\\psi \\colon V \\to \\C$ of $Y$, the map\n$$\\psi \\circ f \\circ \\varphi^{-1} \\colon \\varphi(U \\cap f^{-1}(V)) \\to \\C$$\nis [[holomorphic|0DG0Z72CQD]].\n\nA holomorphic map is a *biholomorphism* if it is bijective; its inverse is then also holomorphic.\n\n## Remarks\n\nContinuity makes $U \\cap f^{-1}(V)$ open, so clause 2 makes sense.\nIt suffices to check clause 2 for charts from one atlas of $X$ and one atlas of $Y$.\nWhen $X$ and $Y$ are open subsets of $\\C$ with the identity chart, this is the usual notion of a holomorphic function.\n\nExample: $f \\colon \\hat\\C \\to \\hat\\C$ with $f(z) = z^2$ for $z \\in \\C$ and $f(\\infty) = \\infty$.\nIn the chart $z$ near finite points it is $z \\mapsto z^2$.\nNear $\\infty$, in the chart $w = 1/z$ on both sides, it reads $w \\mapsto 1/(1/w)^2 = w^2$, which is holomorphic.\n\nNon-example: complex conjugation $z \\mapsto \\bar z$ on $\\C$ is continuous, but in the identity chart it is [[not holomorphic|0Z8ATBPM5W]], so clause 2 fails.\n",
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