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  "body": "\n## Summary\n\nA complex manifold is a space that looks locally like an open subset of $\\C^n$, glued together by holomorphic maps.\nIt is the setting for complex analysis beyond domains in $\\C^n$, just as a [[smooth manifold|F45VQ47GPV]] is the setting for calculus.\n[[Riemann surfaces|5K0N8QMV6Q]] are the complex manifolds of dimension $1$; complex projective space and complex tori are basic compact examples.\nCompact complex manifolds behave very differently from smooth ones: for instance, on a compact connected complex manifold the only global holomorphic functions are the constants.\n\n## Explanations\n\n### L1 human\n\nA complex manifold is built from pieces of $\\C^n$ that are glued along overlaps.\nThe gluing rules must be holomorphic, so that the notion of a holomorphic function makes sense on the whole space.\nThe Riemann sphere is an example: it is made of two copies of the complex plane, glued by $z \\mapsto 1/z$ away from $0$.\nEvery complex manifold of dimension $n$ is also a smooth manifold of real dimension $2n$, but a smooth manifold can carry many complex structures, or none.\n",
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