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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the complex projective plane, usually denoted or is the two-dimensional
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
. It is a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2,Z_3)\neq (0,0,0) where, however, the triples differing by an overall rescaling are identified: :(Z_1,Z_2,Z_3) \equiv (\lambda Z_1,\lambda Z_2, \lambda Z_3); \quad \lambda \in \C, \qquad \lambda \neq 0. That is, these are homogeneous coordinates in the traditional sense of
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
.


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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
, lying in the plane. The nontrivial homotopy groups of the complex projective plane are \pi_2=\pi_5=\mathbb. 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 In algebraic geometry, a branch of mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sc ...
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 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 on the quadric , blowing it up, and projecting onto a general plane in by drawing lines through . The group of birational automorphisms of the complex projective plane is the Cremona group.


Differential geometry

As a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, 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 In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
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. {{DEFAULTSORT:Complex Projective Plane Algebraic surfaces Complex surfaces Projective geometry