Complex Projective Space

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

:

(z_1,z_2,\ldots,z_{n+1}) \in \mathbb{C}^{n+1},
\qquad (z_1,z_2,\ldots,z_{n+1})
eq (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 
eq 0.

That is, these are Homogeneous Coordinates in the traditional sense of Projective Geometry .

CP^''n'' is a Complex Manifold of complex dimension ''n'', so is has real dimension 2''n''. It is a special case of a Grassmannian , and is a Homogeneous Space for various Lie Group s. It is a Kähler Manifold carrying the Fubini-Study Metric , which is essentially determined by symmetry properties.

One may also regard CP^''n'' as a Quotient of the unit 2''n''+1 Sphere in '''C'''^''n''+1 under the action of U(1) :
:CP^''n'' = ''S''^2''n''+1/U(1)
This is because every line in C^''n''+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 '''CP'''^''n''. For ''n''=1 this construction yields the classical Hopf Bundle . From this construction it is not hard to prove that '''CP'''^''n'' is both Compact and Simply Connected .

In general, the Algebraic Topology of CP^''n'' is based on the rank of the Homology Group s being zero in odd dimensions; also ''H''_2''i''(CP^''n'', '''Z''') is Infinite Cyclic for ''i'' = 0 to ''n''. Therefore the Betti Number s run
:1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ...
The Euler Characteristic of CP^''n'' is therefore ''n''+1. By Poincaré Duality the same is true for the ranks of the Cohomology Group s. In the case of cohomology, one can go further, and identify the Graded Ring structure, for Cup Product ; the generator of ''H''²(CPⁿ, '''Z''') is the class associated to a Hyperplane , and this is a ring generator, so that the ring is isomorphic with

Z

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 CP^''n'' as ''n'' → ∞. It is BU(1) , the Classifying Space of U(1) , in the sense of Homotopy Theory , and so classifies complex Line Bundle s; 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.

	CATEGORIES ABOUT COMPLEX PROJECTIVE SPACE

	algebraic varieties

	complex manifolds

	projective geometry

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Information About

Complex Projective Space