mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the complex projective plane, usually denoted P²(C), is the two-dimensional complex projective space. It is a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

of complex dimension 2, described by three complex coordinates :

(Z_1,Z_2,Z_3) \in \mathbf^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 \mathbf,\qquad \lambda \neq 0.

That is, these are

homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...

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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...

, 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, a rational surface is a surface birational geometry, birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 1 ...

is any

algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

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 P³ 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 P³ by drawing lines through ''P''. The group of birational automorphisms of the complex projective plane is the

Cremona group In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the n-dimensional projective space over a field It is denoted by Cr(\mathbb^n(k)) or Bir(\mathbb^n(k)) or Cr_n(k). The Cremona group is natura ...

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 In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. ...

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.

References

* C. E. Springer (1964) ''Geometry and Analysis of Projective Spaces'', pages 140–3,

W. H. Freeman and Company W. H. Freeman and Company is an imprint of Macmillan Higher Education, a division of Macmillan Publishers. Macmillan publishes monographs and textbooks for the sciences under the imprint. History The company was founded in 1946 by William H. ...

. {{DEFAULTSORT:Complex Projective Plane Algebraic surfaces Complex surfaces Projective geometry

Topology

Algebraic geometry

Differential geometry

See also

References