Complex Projective Space - Construction

Construction

Complex projective space is a complex manifold that may be described by n + 1 complex coordinates as

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)

where the tuples differing by an overall rescaling are identified:

(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.

That is, these are homogeneous coordinates in the traditional sense of projective geometry. The point set CPn is covered by the patches . In Ui, one can define a coordinate system by

The coordinate transitions between two different such charts Ui and Uj are holomorphic functions (in fact they are fractional linear transformations). Thus CPn carries the structure of a complex manifold of complex dimension n, and a fortiori the structure of a real differentiable manifold of real dimension 2n.

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 bundle . From this perspective, the differentiable structure on CPn is induced from that of S2n+1, being the quotient of the latter by a compact group that acts properly.

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