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  "body": "\n## Example\n\nLet $n \\ge 1$.\n*Complex projective space* $\\mathbb{CP}^n$ is the [[quotient|0BR3FY5Q8M]] of $\\C^{n+1} \\setminus \\set{0}$ by the relation $z \\sim \\lambda z$ for $\\lambda \\in \\C \\setminus \\set{0}$; the class of $z = (z_0, \\ldots, z_n)$ is written $[z_0 : \\cdots : z_n]$.\nSo $\\mathbb{CP}^n$ is the set of complex lines through $0$ in $\\C^{n+1}$.\n\nIt is a [[complex manifold|JSGWAN2RJV]] of dimension $n$.\nFor $i = 0, \\ldots, n$, let $U_i = \\set{[z] : z_i \\ne 0}$ and $\\varphi_i[z] = (z_0/z_i, \\ldots, \\widehat{z_i/z_i}, \\ldots, z_n/z_i) \\in \\C^n$, where the hat means that the entry is left out.\nEach $\\varphi_i$ is a well-defined homeomorphism of $U_i$ onto $\\C^n$, and the $U_i$ cover $\\mathbb{CP}^n$.\nOn $\\varphi_i(U_i \\cap U_j)$ the coordinate $w$ that stands for $z_j/z_i$ is nonzero.\nThe components of the transition map $\\varphi_j \\circ \\varphi_i^{-1}$ are $1/w$ and the other coordinates divided by $w$, so it is holomorphic.\nThe space is Hausdorff and [[compact|0TVJZYB6SH]]: it is the continuous image of the unit sphere $S^{2n+1} \\subset \\C^{n+1}$, since every line meets the sphere.\n\n## Remarks\n\nFor $n = 1$ one gets the [[Riemann sphere|0J78KBPFZB]], and for $n = 2$ the [[complex projective plane|58WSWYBCWA]].\nThe map $S^{2n+1} \\to \\mathbb{CP}^n$ is a circle bundle, the [[Hopf fibration|M5AKNYC4QQ]] when $n = 1$.\nThe Fubini–Study metric makes $\\mathbb{CP}^n$ a [[Kähler manifold|X60JG9AEQJ]].\nBy [[Chow's theorem|6EFEG5VJEF]], every closed complex submanifold of $\\mathbb{CP}^n$ is a [[projective variety|A4H4CZ2V35]].\nBy the [[Kodaira embedding theorem|7G7CSM2NZ9]], a compact complex manifold embeds in some $\\mathbb{CP}^N$ if and only if it carries a positive line bundle.\nSince $\\mathbb{CP}^n$ is compact and connected, its only holomorphic functions are the [[constants|2SM64J81YC]].\n\n## Explanations\n\n### L1 human\n\nComplex projective space collects all complex lines through the origin of $\\C^{n+1}$ into one space, each line becoming a single point.\nNear any point it looks like $\\C^n$: one coordinate is nonzero, and dividing by it gives $n$ complex numbers.\nThe points where that coordinate vanishes are the \"points at infinity\", and they make the space compact.\nFor $n = 1$ this is the Riemann sphere: the complex plane plus one point at infinity.\n",
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