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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 bundleIn mathematics, the Hopf bundle (or Hopf fibration is a particular fiber bundle S''1 → S''3 → S''2 with base space S''2, total space S''3, and fiber S''1. It was discovered by Heinz Hopf in 1931. The Hopf bundle can actually be considered as a p. From this construction it is not hard to prove that CPⁿ is both compactSeveral specialized usages of the terms compact and compactness exist. Multiple definitions of the term "compact" are found in mathematics: The most common usage relates to topology, where one considers compact spaces . This article also includes the clos and simply connected.

In general, the algebraic topologyTopology Algebraic topology Abstract algebra Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants The goal is to take topological spaces, and further ca of CPⁿ is based on the rank of the homology groups being zero in odd dimensions; also H_2i(CPⁿ, Z) is infinite cyclic for i = 0 to n. Therefore the Betti numberIn algebraic topology, the Betti numbers of a topological space X are a sequence b b . of topological invariants. Each Betti number is a natural number, or infinity. For the most reasonable spaces (such as compact manifolds, finite simplicial complexes ors run

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

The Euler characteristicIn algebraic topology, the Euler characteristic is a topological invariant (infact homotopy invariant) defined for broad class of topological spaces. It is usually denoted by. Sometimes in topology also called as Euler number''. In case of 2-dimensional t 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 h^i,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.