---- > [!definition] Definition. ([[complex projective space]]) > The **complex projective space** $\mathbb{C}P^{n}$ is the [[quotient space|quotient]] of $\mathbb{C}^{n+1} - \{ 0 \}$ by the relation $z \sim \alpha z$ for all $\alpha \in \mathbb{C}_{ \neq 0}$ and $z \in \mathbb{C}^{n+1}_{\neq 0}$, i.e., the [[quotient space|quotient]] $\frac{\mathbb{C}^{n+1} \setminus \{ 0 \}}{\mathbb{C}^{\times}}$. > > The [[equivalence class|equivalence classes]]/[[orbit]] of $(z_{0},\dots,z_{n}) \in \mathbb{C}^{n+1} \setminus \{ 0 \}$ is denoted $[z_{0} : \cdots : z_{n}]$ (called **homogeneous coordinates**, though take care that such 'coordinates' only determine a point up to scaling). > >Note that we can identify $\mathbb{C}P^{n-1}$ with the set of points in $\mathbb{C}P^{n}$ whose last homogeneous coordinate is zero. ^definition > [!generalization] > [[projective space]] ^generalization > [!proposition] Cell decomposition of $\mathbb{C}P^{n}$. > Similar to how $\mathbb{R}P^{n}$ can be obtained as a [[cell complex]] by [[adjunction space|attaching]] an $n$-cell to $\mathbb{R}P^{n-1}$, $\mathbb{C}P^{n}$ can be obtained by attaching a $2n$-cell to $\mathbb{C}P^{n-1}$. Indeed, a given [[hyperplane]] $H \subset \mathbb{C}^{n+1}$ through the origin projects under the quotient map to a [[subspace topology|subspace]] of $\mathbb{C}P^{n}$ [[homeomorphism|homeomorphic]] to $\mathbb{C}P^{n-1}$. The complement of its image[^1] is [[homeomorphism|homeomorphic]] to $\mathbb{C}^{n} \cong \mathbb{B}^{2n}$. Thus $\mathbb{C}P^{n}=\mathbb{C}P^{n-1} \cup_{f}\underbrace{ \mathbb{C}^{n} }_{ =\mathbb{B}^{2n} }$ "Attaching map $f$ is [[Hopf fibration]]." Note that only even-dimensional cells exist, which makes the [[cellular homology]] immediate to compute (all the differentials are zero). [[cohomology groups of complex projective space|Can also compute cohomology groups]] using [[Mayer-Vietoris theorem|Mayer-Vietoris]]. To get the [[singular cohomology|cohomology ring]] structure, though, need more equipment. This is carried out both in [[Gysin sequence]] and in [[the perfect Poincare pairing]]. ^proposition > [!basicexample] > In the case $n=1$, one has the [[Riemann sphere]]. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```