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  "body": "\n## Summary\n\nA Riemann surface is a connected space that looks locally like an open piece of the complex plane, with changes of coordinates that are holomorphic.\nIt is a complex manifold of complex dimension one.\nOn a Riemann surface one can speak of [[holomorphic|0DG0Z72CQD]] and [[meromorphic|0NSD1TPHR0]] functions.\nThis makes it the natural home of multivalued functions such as $\\sqrt{z}$ and $\\log z$.\nCompact Riemann surfaces are the same objects as smooth projective complex algebraic curves.\n\n## Explanations\n\n### L1 human\n\nImagine a surface built by gluing pieces of the complex plane, where the gluing maps are holomorphic.\nBecause the gluing respects holomorphy, you can tell whether a function on the surface is holomorphic by looking at it in any piece.\nThe plane itself, the [[Riemann sphere|0CDMXGPSES]] and the torus $\\C/(\\Z + i\\Z)$ are examples.\nThe graph of $w^2 = z$ is a Riemann surface on which $\\sqrt{z}$ becomes a single-valued function, namely $w$.\n",
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