Complex projective plane


Complex projective plane

In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. It is a complex manifold described by three complex coordinates

(z_1,z_2,z_3) \in \mathbb{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 \mathbb{C},\qquad \lambda \neq 0.

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

Contents

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 \pi_2=\pi_5=\mathbb{Z}. 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 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.

Differential geometry

As a Riemannian manifold, the complex projective plane is a 4-dimensional manifold whose sectional curvature is quarter-pinched. 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.

References

Weisstein, Eric W., "Complex Projective Plane" from MathWorld.

See also


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Complex projective space — The Riemann sphere, the one dimensional complex projective space, i.e. the complex projective line. In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a …   Wikipedia

  • Projective plane — See real projective plane and complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is… …   Wikipedia

  • Fake projective plane — For Freedman s example of a non smoothable manifold with the same homotopy type as the complex projective plane, see 4 manifold. In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have… …   Wikipedia

  • Gromov's inequality for complex projective space — In Riemannian geometry, Gromov s optimal stable 2 systolic inequality is the inequality: mathrm{stsys} 2{}^n leq n!;mathrm{vol} {2n}(mathbb{CP}^n),valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound… …   Wikipedia

  • Complex space — In mathematics, n dimensional complex space is a multi dimensional generalisation of the complex numbers, which have both real and imaginary parts or dimensions. The n dimensional complex space can be seen as n cartesian products of the complex… …   Wikipedia

  • Plane geometry — In mathematics, plane geometry may mean:*geometry of a plane, *geometry of the Euclidean plane,or sometimes a plane is any flat surface that extends without end in all directions.*geometry of a projective plane, most commonly the real projective… …   Wikipedia

  • Projective geometry — is a non metrical form of geometry, notable for its principle of duality. Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th …   Wikipedia

  • Projective space — In mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non zero vectors which are equal up to a multiplication by a non zero scalar. A formal… …   Wikipedia

  • Plane (geometry) — Two intersecting planes in three dimensional space In mathematics, a plane is a flat, two dimensional surface. A plane is the two dimensional analogue of a point (zero dimensions), a line (one dimension) and a space (three dimensions). Planes can …   Wikipedia

  • Projective line — In mathematics, a projective line is a one dimensional projective space. The projective line over a field K , denoted P1( K ), may be defined as the set of one dimensional subspaces of the two dimensional vector space K 2 (it does carry other… …   Wikipedia