Complex projective plane

Complex Projective Plane

Complex projective plane

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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

<math>(z_1,z_2,z_3) in mathbb^3,qquad (z_1,z_2,z_3)neq (0,0,0)</math>


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

<math>(z_1,z_2,z_3) equiv (lambda z_1,lambda z_2, lambda z_3);quad lambdain mathbb,qquad lambda neq 0.</math>


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 <math>pi_2=pi_5=mathbb</math>. 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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