Complex projective space

From Academic Kids

In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. The case n = 1 gives the Riemann sphere (also called the complex projective line), and the case n = 2 the complex projective plane.

Complex projective space is a complex manifold that may be described by n+1 complex coordinates as

<math>(z_1,z_2,\ldots,z_{n+1}) \in \mathbb{C}^{n+1},

\qquad (z_1,z_2,\ldots,z_{n+1})\neq (0,0,\ldots,0)<math>

where the tuples differing by an overall rescaling are identified:

<math>(z_1,z_2,\ldots,z_{n+1}) \equiv

(\lambda z_1,\lambda z_2, \ldots,\lambda z_{n+1}); \quad \lambda\in \mathbb{C},\qquad \lambda \neq 0.<math>

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

CPn is a complex manifold of complex dimension n, so is has real dimension 2n. It is a special case of a Grassmannian, and is a homogeneous space for various Lie groups. It is a Kähler manifold carrying the Fubini-Study metric, which is essentially determined by symmetry properties.

One may also regard CPn as a quotient of the unit 2n+1 sphere in Cn+1 under the action of U(1):

CPn = S2n+1/U(1)

This is because every line in Cn+1 intersects the unit sphere in a circle. By first projecting to the unit sphere and then identifying under the natural action of U(1) one obtains CPn. For n=1 this construction yields the classical Hopf bundle. From this construction it is not hard to prove that CPn is both compact and simply connected.

In general, the algebraic topology of CPn is based on the rank of the homology groups being zero in odd dimensions; also H2i(CPn, Z) is infinite cyclic for i = 0 to n. Therefore the Betti numbers run

1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ...

The Euler characteristic of CPn is therefore n+1. By Poincaré duality the same is true for the ranks of the cohomology groups. In the case of cohomology, one can go further, and identify the graded ring structure, for cup product; the generator of H2(CPn, Z) is the class associated to a hyperplane, and this is a ring generator, so that the ring is isomorphic with

Z[T]/(Tn+1),

with T a degree two generator. This implies also that the Hodge number hi,i = 1, and all the others are zero.

There is a space CP which, in a sense, is the limit of CPn as n → ∞. It is the classifying space of U(1), in the sense of homotopy theory, and so classifies complex line bundles; equivalently it accounts for the first Chern class.

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