 Complex projective plane

In mathematics, the complex projective plane, usually denoted CP^{2}, is the twodimensional complex projective space. It is a complex manifold described by three complex coordinates
where, however, the triples differing by an overall rescaling are identified:
That is, these are homogeneous coordinates in the traditional sense of projective geometry.
Contents
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 . The fundamental group is trivial and all other higher homotopy groups are those of the 5sphere, 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 nonsingular 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 nonsingular complex quadric in P^{3} is obtained from the plane by blowing up two points to curves, and then blowing down the line through these two points; the inverse of this transformation can be seen by taking a point P on the quadric Q, blowing it up, and projecting onto a general plane in P^{3} by drawing lines through P.
The group of birational automorphisms of the complex projective plane is the Cremona group.
Differential geometry
As a Riemannian manifold, the complex projective plane is a 4dimensional manifold whose sectional curvature is quarterpinched. The rival normalisations are for the curvature to be pinched between 1/4 and 1; alternatively, between 1 and 4. With respect to the former normalisation, the imbedded surface defined by the complex projective line has Gaussian curvature 1. With respect to the latter normalisation, the imbedded real projective plane has Gaussian curvature 1.
References
Weisstein, Eric W., "Complex Projective Plane" from MathWorld.
See also
Categories: Algebraic surfaces
 Complex surfaces
 Projective geometry
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