Complex projective space: Difference between revisions

Latest revision as of 14:17, 2 April 2011

Definition

Short definition

Complex projective space is defined as projective space over a field of complex numbers $C$ , where the topological structure is induced from the structure of the field of complex numbers as a topological field.

Finite-dimensional

Complex projective space of dimension $n$ , denoted $C P^{n}$ or $P^{n} (C)$ , is defined as the quotient space under the group action $C^{n + 1} ∖ {0} / C^{*}$ where $C^{*}$ acts by scalar multiplication. It is equipped with the quotient topology.

As a set, we can think of it as the set of complex lines (which are planes in the real vector space sense) through the origin in $C^{n + 1}$ . Using a Hermitian inner product on $C^{n + 1}$ , it can also be identified with the set of hyperplanes of codimension 1 (i.e., $n$ -dimensional complex linear subspaces) in $C^{n + 1}$ .

Countable-dimensional

This space, called countable-dimensional complex projective space and denoted $C P^{\infty}$ , is defined as the quotient space of the nonzero elements of a countable-dimensional complex vector space (with the standard topology) over $C$ by the action of $C^{*}$ by scalar multiplication.

Others

We can also consider the real projective space corresponding to any topological complex vector space, possibly infinite-dimensional, which is a real vector space equipped with a compatible topology. If the vector space is $V$ , the projective space is defined as follows:we take $V ∖ {0}$ with the subspace topology, and then put the quotient topology on its quotient under the action of $C^{*}$ .

Particular cases

$n$	Complex projective space $C P^{n}$
0	one-point space
1	complex projective line, which turns out to be homeomorphic to the 2-sphere
2	complex projective plane
3	link: Fill this in later
countable ( $\infty$ )	countable-dimensional complex projective space

Algebraic topology

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 ===Finite-dimensional===
-'''Complex projective space''' of dimension <math>n</math>, denoted <math>\matbb{C}\mathbb{P}^n</math> or <math>\mathbb{P}^n(\mathbb{C})</math>, is defined as the quotient space under the group action <math>\mathbb{C}^{n+1} \setminus \{ 0 \}/\mathbb{C}^*</math> where <math>\mathbb{C}^*</math> acts by scalar multiplication. It is equipped with the [[quotient topology]].
+'''Complex projective space''' of dimension <math>n</math>, denoted <math>\mathbb{C}\mathbb{P}^n</math> or <math>\mathbb{P}^n(\mathbb{C})</math>, is defined as the quotient space under the group action <math>\mathbb{C}^{n+1} \setminus \{ 0 \}/\mathbb{C}^*</math> where <math>\mathbb{C}^*</math> acts by scalar multiplication. It is equipped with the [[quotient topology]].
 As a set, we can think of it as the set of complex lines (which are planes in the real vector space sense) through the origin in <math>\mathbb{C}^{n+1}</math>. Using a Hermitian inner product on <math>\mathbb{C}^{n+1}</math>, it can also be identified with the set of hyperplanes of codimension 1 (i.e., <math>n</math>-dimensional complex linear subspaces) in <math>\mathbb{C}^{n+1}</math>.