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. For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, a projective space is a fundamental construction from any vector space. ... A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. ...

Complex projective space is a complex manifold that may be described by n+1 complex coordinates as In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...

where the tuples differing by an overall rescaling are identified:

That is, these are homogeneous coordinates in the traditional sense of projective geometry. In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ...

CPⁿ 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 Fubini-Study metric, which is essentially determined by symmetry properties. In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ...

One may also regard CPⁿ as a quotient of the unit 2n+1 sphere in Cⁿ⁺¹ under the action of U(1): In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. ...

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 Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... In mathematics, the Hopf bundle (or Hopf fibration) is a particular fiber bundle with base space S2, total space S3, and fiber S1: S1 &#8594; S3 &#8594; S2 It was discovered by Heinz Hopf in 1931. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

In general, the algebraic topology 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 numbers run Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ...

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 It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ... In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. ... A hyperplane is a concept in geometry. ...

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. In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M. It was developed by W. V. D. Hodge in the 1930s as an extension...

There is a space CP^∞ which, in a sense, is the limit of CPⁿ 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 infinite-dimensional projective unitary group; see that article for additional properties and discussion. In mathematics, the classifying space for U(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space. ... In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, the projective unitary group PU(N) is the quotient of the unitary group U(N) by the right multiplication of its center, U(1). ...