Editing Complex projective space (section)

{{Short description|Mathematical concept}}
[[File:Stereographic projection in 3D.svg|thumb|right|{{center|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 number]]s.  By analogy, whereas the points of a [[real projective space]] label the lines through the origin of a real [[Euclidean space]], the points of a complex projective space label the ''[[complex plane|complex]]'' lines through the origin of a complex Euclidean space (see [[#Introduction|below]] for an intuitive account).  Formally, a complex projective space is the space of complex lines through the origin of an (''n''+1)-dimensional complex [[vector space]].  The space is denoted variously as '''P'''('''C'''<sup>''n''+1</sup>), '''P'''<sub>''n''</sub>('''C''') or '''CP'''<sup>''n''</sup>.  When {{nowrap|''n'' {{=}} 1}}, the complex projective space '''CP'''<sup>1</sup> is the [[Riemann sphere]], and when {{nowrap|''n'' {{=}} 2}}, '''CP'''<sup>2</sup> is the [[complex projective plane]] (see there for a more elementary discussion).

Complex projective space was first introduced by {{harvtxt|von Staudt|1860}} as an instance of what was then known as the "geometry of position", a notion originally due to [[Lazare Carnot]], a kind of [[synthetic geometry]] that included other projective geometries as well.  Subsequently, near the turn of the 20th century it became clear to the [[Italian school of algebraic geometry]] that the complex projective spaces were the most natural domains in which to consider the solutions of [[polynomial]] equations – [[algebraic variety|algebraic varieties]] {{harv|Grattan-Guinness|2005|pp=445&ndash;446}}.  In modern times, both the [[topology]] and geometry of complex projective space are well understood and closely related to that of the [[N-sphere|sphere]].  Indeed, in a certain sense the (2''n''+1)-sphere can be regarded as a family of circles parametrized by '''CP'''<sup>''n''</sup>: this is the [[Hopf fibration]].  Complex projective space carries a ([[Kähler metric|Kähler]]) [[metric tensor|metric]], called the [[Fubini&ndash;Study metric]], in terms of which it is a [[Hermitian symmetric space]] of rank 1.

Complex projective space has many applications in both mathematics and [[quantum physics]].  In [[algebraic geometry]], complex projective space is the home of [[projective variety|projective varieties]], a well-behaved class of [[algebraic variety|algebraic varieties]].  In topology, the complex projective space plays an important role as a [[classifying space]] for complex [[line bundle]]s: families of complex lines parametrized by another space.  In this context, the infinite union of projective spaces ([[direct limit]]), denoted '''CP'''<sup>∞</sup>, is the classifying space [[K(Z,2)]].  In quantum physics, the [[wave function]] associated to a [[pure state]] of a quantum mechanical system is a [[probability amplitude]], meaning that it has unit norm, and has an inessential overall phase: that is, the wave function of a pure state is naturally a point in the [[projective Hilbert space]] of the state space.
Complex projective manifold is 2n dimensional space or it is n dimensional complex space.