Complex Projective Plane

# Complex projective plane

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Description:
In mathematics, the complex projective plane, usually denoted CP<sup>2</sup>, is the two-dimensional complex projective space. It is a complex manifold described by three complex coordinates

where, however, the triples differing by an overall rescaling are identified:

[itex](z_1,z_2,z_3) equiv (lambda z_1,lambda z_2, lambda z_3);quad lambdain mathbb,qquad lambda neq 0.[/itex]

That is, these are homogeneous coordinates in the traditional sense of projective geometry.

## Topology

The Betti numbers of the complex projective plane are

1, 0, 1, 0, 1, 0, 0, ....

The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. The nontrivial homotopy groups of the complex projective plane are [itex]pi_2=pi_5=mathbb[/itex]. The fundamental group is trivial and all other higher homotopy groups are those of the 5-sphere, i.e. torsion.

## Algebraic geometry

In birational geometry, a complex rational surface is any algebraic surface birationally equivalent to the complex projective plane. It is known that any non-singular rational variety is obtained from the plane by a sequence of blowing up transformations and their inverses ('blowing down') of curves, which must be of a very particular type. As a special case, a non-singular complex quadric in P<sup>3</sup> is obtained...

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