Geometry Non-Euclidean geometries Example

Complex projective plane

Builds on Homogeneous coordinates Projective space Riemann sphere Euler characteristic

Example

The complex projective plane $\mathbb{CP}^2$ is the set of 1-dimensional complex linear subspaces of $\C^3$ , the projective space of dimension $2$ over $\C$ . A point is written in homogeneous coordinates $[z_0 : z_1 : z_2]$ with $(z_0, z_1, z_2) \ne 0$ , up to multiplication by a nonzero complex number. A 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 $(a_0, a_1, a_2) \in \C^3 \setminus \set{0}$ .

Check that it is a projective plane: the arguments for the real case work over any field. Two distinct points span a unique 2-dimensional subspace of $\C^3$ , so they lie on a unique line. Two distinct 2-dimensional subspaces of $\C^3$ meet in a 1-dimensional subspace, so two distinct lines meet in a unique point. The points $[1 : 0 : 0]$ , $[0 : 1 : 0]$ , $[0 : 0 : 1]$ , $[1 : 1 : 1]$ have no three on a line.

Remarks

Each line is a copy of $\mathbb{CP}^1$ , which is the Riemann sphere. So, 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. It is a compact complex manifold of complex dimension $2$ , and with the Fubini–Study metric it is a Kähler manifold. Its integral cohomology is $\Z$ in degrees $0$ , $2$ , $4$ and zero otherwise, so its Euler characteristic is $3$ . Plane algebraic curves over $\C$ live naturally in $\mathbb{CP}^2$ , and there Bézout's theorem holds without exceptions. Two curves of degrees $d$ and $e$ with no common component meet in exactly $de$ points, counted with multiplicity.

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Tag	`58WSWYBCWA`
URI	`m2://58WSWYBCWA`
Kind	example
Writing	W0 stub
MSC	51N15, 14N05, 32Q15
Instance of	Projective plane
Uses	Homogeneous coordinates, Projective space, Riemann sphere, Euler characteristic
Same as	Wikidata Q4426755 Wikipedia Complex projective plane

Complex projective plane

Example

Remarks

Reading path

Further reading

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