Complex projective space

Definition

Short definition

Complex projective space is defined as projective space over a field of complex numbers $\mathbb {C}$ , 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 $\mathbb {C} \mathbb {P} ^{n}$ or $\mathbb {P} ^{n}(\mathbb {C} )$ , is defined as the quotient space under the group action $\mathbb {C} ^{n+1}\setminus \{0\}/\mathbb {C} ^{*}$ where $\mathbb {C} ^{*}$ 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 $\mathbb {C} ^{n+1}$ . Using a Hermitian inner product on $\mathbb {C} ^{n+1}$ , it can also be identified with the set of hyperplanes of codimension 1 (i.e., $n$ -dimensional complex linear subspaces) in $\mathbb {C} ^{n+1}$ .

Countable-dimensional

This space, called countable-dimensional complex projective space and denoted $\mathbb {C} \mathbb {P} ^{\infty }$ , is defined as the quotient space of the nonzero elements of a countable-dimensional complex vector space (with the standard topology) over $\mathbb {C}$ by the action of $\mathbb {C} ^{*}$ 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 $V\setminus \{0\}$ with the subspace topology, and then put the quotient topology on its quotient under the action of $\mathbb {C} ^{*}$ .

Particular cases

$n$	Complex projective space $\mathbb {C} \mathbb {P} ^{n}$
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