In mathematics, complex projective space, P(C^{n+1}), P_{n}(C) or CP^{n}, in fact preferably
is the projective space of (complex) lines in C^{n+1}. The case n = 1 gives the Riemann sphere (also called the complex projective line), and the case n = 2 the complex projective plane. The infinite direct union (direct limit), denoted is of particular importance as a universal object, see K(Z,2).
Contents 
CP^{n} is a complex manifold of complex dimension n, so it 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 FubiniStudy metric, which is essentially determined by symmetry properties. It also plays a central role in algebraic geometry; by Chow's theorem, any compact complex submanifold of CP^{n} is the zero locus of a finite number of polynomials, and is thus a projective algebraic variety.
Complex projective space is a complex manifold that may be described by n + 1 complex coordinates as
where the tuples differing by an overall rescaling are identified:
That is, these are homogeneous coordinates in the traditional sense of projective geometry.
One may also regard CP^{n} as a quotient of the unit 2n + 1 sphere in C^{n+1} under the action of U(1):
This is because every line in C^{n+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 CP^{n}. For n = 1 this construction yields the classical Hopf bundle.
Projective space is compact and connected, being a quotient of a compact, connected space.
From the fiber bundle
or more suggestively
is simply connected, has , and higher homotopy agrees with that of S^{2n + 1}
In general, the algebraic topology of CP^{n} is based on the rank of the homology groups being zero in odd dimensions; also H_{2i}(CP^{n}, Z) is infinite cyclic for i = 0 to n. Therefore the Betti numbers run
That is, 0 in odd dimensions, 1 in even dimensions up to 2n. The Euler characteristic of CP^{n} 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 H^{2}(CP^{n}, Z) is the class associated to a hyperplane, and this is a ring generator, so that the ring is isomorphic with
with T a degree two generator. This implies also that the Hodge number h^{i,i} = 1, and all the others are zero.
It follows from induction and Bott periodicity that
The tangent bundle satisfies
where denotes the trivial line bundle. From this, the Chern classes and characteristic numbers can be calculated.
There is a space CP^{∞} which, in a sense, is the limit of CP^{n} as n → ∞. It is BU(1), 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. CP^{∞} is also the same as the infinitedimensional projective unitary group; see that article for additional properties and discussion.
The natural metric on is the FubiniStudy metric, and its isometry group is the projective unitary group PU(n + 1), where the stabilizer of a point is
It is a Hermitian symmetric space, represented as a coset space
It has sectional curvature ranging from 1/4 to 1, and is the roundest manifold that isn't a sphere (or covered by a sphere): by the 1/4pinched sphere theorem, any complete, simply connected Riemannian manifold with curvature strictly between 1/4 and 1 is homeomorphic to the sphere. Complex projective space shows that 1/4 is sharp.
