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  "body": "\n## Example\n\nThe *complex projective plane* $\\mathbb{CP}^2$ is the set of 1-dimensional complex linear subspaces of $\\C^3$, the [[projective space|ZT9A6JK8QF]] of dimension $2$ over $\\C$.\nA point is written in [[homogeneous coordinates|FS9FRJ17D2]] $[z_0 : z_1 : z_2]$ with $(z_0, z_1, z_2) \\ne 0$, up to multiplication by a nonzero complex number.\nA line is the set $\\set{[z_0 : z_1 : z_2] : a_0 z_0 + a_1 z_1 + a_2 z_2 = 0}$ for some $(a_0, a_1, a_2) \\in \\C^3 \\setminus \\set{0}$.\n\nCheck that it is a [[projective plane|DANFNN8C4Y]]: the arguments for the real case work over any field.\nTwo distinct points span a unique 2-dimensional subspace of $\\C^3$, so they lie on a unique line.\nTwo distinct 2-dimensional subspaces of $\\C^3$ meet in a 1-dimensional subspace, so two distinct lines meet in a unique point.\nThe points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line.\n\n## Remarks\n\nEach line is a copy of $\\mathbb{CP}^1$, which is the [[Riemann sphere|0J78KBPFZB]].\nSo, as a real manifold, $\\mathbb{CP}^2$ is a compact simply connected manifold of real dimension $4$ that contains many 2-spheres meeting in single points.\nIt is a compact complex manifold of complex dimension $2$, and with the Fubini–Study metric it is a Kähler manifold.\nIts integral cohomology is $\\Z$ in degrees $0$, $2$, $4$ and zero otherwise, so its [[Euler characteristic|RNDDWDAFJ5]] is $3$.\nPlane algebraic curves over $\\C$ live naturally in $\\mathbb{CP}^2$, and there Bézout's theorem holds without exceptions.\nTwo curves of degrees $d$ and $e$ with no common component meet in exactly $de$ points, counted with multiplicity.\n\n## Explanations\n\n### L1 human\n\nThe complex projective plane is built like the real one, but with complex numbers as coordinates.\nEach point is a line through the origin in $\\C^3$, and parallel lines of $\\C^2$ are made to meet at infinity.\nEvery line in it is a sphere, so the plane is a four-dimensional space woven out of spheres, any two of which cross at exactly one point.\nIt is the natural home of plane algebraic curves: there, a line and a circle always meet in exactly two points, counted correctly.\n",
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