The Betti numbers of the complex projective plane are 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
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.
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.
An explicit demonstration of the Riemann and Ricci tensors is given in the n=2 subsection of the article on the Fubini-Study metric.