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 $\mathbb{C}{\mathbb{P}}^{\mathrm{\infty }}$, 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 ($\mathrm{\infty }$)	countable-dimensional complex projective space

Algebraic topology

Homology

Further information: homology of complex projective space

Cohomology

Further information: cohomology of complex projective space

Homotopy

Further information: homotopy of complex projective space