Complex projective space
Contents
Definition
Short definition
Complex projective space is defined as projective space over a field of complex numbers , 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 , denoted or , is defined as the quotient space under the group action where 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 . Using a Hermitian inner product on , it can also be identified with the set of hyperplanes of codimension 1 (i.e., -dimensional complex linear subspaces) in .
Countable-dimensional
This space, called countable-dimensional complex projective space and denoted , is defined as the quotient space of the nonzero elements of a countable-dimensional complex vector space (with the standard topology) over by the action of 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 , the projective space is defined as follows:we take with the subspace topology, and then put the quotient topology on its quotient under the action of .
Particular cases
Complex projective space | |
---|---|
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 () | 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