Complex projective space

Example

Let $n \ge 1$. Complex projective space $\mathbb{CP}^n$ is the quotient of $\C^{n+1} \setminus \set{0}$ by the relation $z \sim \lambda z$ for $\lambda \in \C \setminus \set{0}$; the class of $z = (z_0, \ldots, z_n)$ is written $[z_0 : \cdots : z_n]$. So $\mathbb{CP}^n$ is the set of complex lines through $0$ in $\C^{n+1}$.

It is a complex manifold of dimension $n$. For $i = 0, \ldots, n$, let $U_i = \set{[z] : z_i \ne 0}$ and $\varphi_i[z] = (z_0/z_i, \ldots, \widehat{z_i/z_i}, \ldots, z_n/z_i) \in \C^n$, where the hat means that the entry is left out. Each $\varphi_i$ is a well-defined homeomorphism of $U_i$ onto $\C^n$, and the $U_i$ cover $\mathbb{CP}^n$. On $\varphi_i(U_i \cap U_j)$ the coordinate $w$ that stands for $z_j/z_i$ is nonzero. The components of the transition map $\varphi_j \circ \varphi_i^{-1}$ are $1/w$ and the other coordinates divided by $w$, so it is holomorphic. The space is Hausdorff and compact: it is the continuous image of the unit sphere $S^{2n+1} \subset \C^{n+1}$, since every line meets the sphere.