Editing Complex projective plane

{{short description|2-dimensional complex projective space}}
{{refimprove|date=May 2010}}

In [[mathematics]], the '''complex projective plane''', usually denoted {{tmath|\mathbb{P}^2(\C)}} or {{tmath|\mathbb{CP}^2,}} is the two-dimensional [[complex projective space]]. It is a [[complex manifold]] of complex dimension 2, described by three complex coordinates

:<math>(Z_1,Z_2,Z_3) \in \C^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 \lambda \in \C, \qquad \lambda \neq 0.</math>

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

==Topology==
The [[Betti number]]s 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{Z}</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 {{tmath|\mathbb 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 {{mvar|P}} on the quadric {{mvar|Q}}, blowing it up,  and projecting onto a general plane in {{tmath|\mathbb P^3}} by drawing lines through {{mvar|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 4-dimensional manifold whose sectional curvature is quarter-pinched, but not strictly so. That is, it attains ''both'' bounds and thus evades being a sphere, as the [[sphere theorem]] would otherwise require.  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.

An explicit demonstration of the Riemann and Ricci tensors is given in the ''n''=2 subsection of the article on the [[Fubini-Study metric]].

==See also==
*[[Circular points at infinity]]
*[[del Pezzo surface]]
*[[Toric geometry]]
*[[Fake projective plane]]

==References==
* C. E. Springer (1964) ''Geometry and Analysis of Projective Spaces'', pages 140–3, [[W. H. Freeman and Company]].

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[[Category:Algebraic surfaces]]
[[Category:Complex surfaces]]
[[Category:Projective geometry]]