Editing Complex projective space (section)

==Construction==
Complex projective space is a [[complex manifold]] that may be described by ''n''&nbsp;+&nbsp;1 complex coordinates as

:<math>Z=(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]].  The point set '''CP'''<sup>''n''</sup> is covered by the patches <math>U_i=\{ Z \mid Z_i\ne0\}</math>.  In ''U''<sub>''i''</sub>, one can define a coordinate system by
:<math>z_1 = Z_1/Z_i, \quad z_2=Z_2/Z_i, \quad \dots, \quad z_{i-1}=Z_{i-1}/Z_i, \quad z_i = Z_{i+1}/Z_i, \quad \dots, \quad z_n=Z_{n+1}/Z_i.</math>
The coordinate transitions between two different such charts ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> are [[holomorphic function]]s (in fact they are [[fractional linear transformation]]s).  Thus '''CP'''<sup>''n''</sup> carries the structure of a [[complex manifold]] of complex dimension ''n'', and ''[[a fortiori]]'' the structure of a real [[differentiable manifold]] of real dimension 2''n''.

One may also regard '''CP'''<sup>''n''</sup> as a [[Quotient space (topology)|quotient]] of the unit 2''n''&nbsp;+&nbsp;1 [[sphere]] in '''C'''<sup>''n''+1</sup> under the action of [[Unitary group|U(1)]]:
:'''CP'''<sup>''n''</sup> = ''S''<sup>2''n''+1</sup>/U(1).
This is because every line in '''C'''<sup>''n''+1</sup> 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'''<sup>''n''</sup>. For ''n''&nbsp;=&nbsp;1 this construction yields the classical [[Hopf bundle]] <math>S^3\to S^2</math>.  From this perspective, the differentiable structure on '''CP'''<sup>''n''</sup> is induced from that of ''S''<sup>2''n''+1</sup>, being the quotient of the latter by a [[compact group]] that acts properly.