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  "body": "\n## Summary\n\nA complex analytic variety, or complex analytic set, in a [[complex manifold|JSGWAN2RJV]] $X$ is a closed subset with a local description.\nNear each point of $X$ it is the common zero set of finitely many [[holomorphic functions|507C34W0ZT]].\nIt is the analytic counterpart of an algebraic variety, with polynomials replaced by holomorphic functions.\nAnalytic varieties may have singular points, and their local structure is studied through [[germs of analytic sets|Q9B74BYZ8C]] and the ring of convergent power series.\nAbstract varieties of this kind, defined by gluing local models, are the [[complex analytic spaces|HXSZA44J5J]].\n\n## Explanations\n\n### L1 human\n\nTake a few holomorphic functions on $\\C^n$ and look at the points where all of them vanish.\nSuch a set may be smooth, like the line $\\set{z_2 = 0}$ in $\\C^2$, or have singular points, like the curve $\\set{z_1 z_2 = 0}$, which is two lines crossing at the origin.\nAn analytic variety is a set that looks like this near every point, even if no single finite family of functions defines it everywhere.\nFor example, the zero set of $\\sin z$ in $\\C$, the set of integer multiples of $\\pi$, is an analytic variety but not the zero set of a polynomial.\n",
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