Let $n \geq 0$ be an integer and let $K \in \{\mathbb{R}, \mathbb{C}\}$. Define

\begin{align*} \mathbb{P}^n_K := (K^{n+1} \setminus \{0\})/K^\times \end{align*}

under the [equivalence relation](/page/Equivalence%20Relation) identifying $x,y \in K^{n+1} \setminus \{0\}$ when $y = \lambda x$ for some $\lambda \in K^\times$. For each $0 \leq i \leq n$, let

\begin{align*} U_i := \{[x_0:\cdots:x_n] \in \mathbb{P}^n_K : x_i \neq 0\}. \end{align*}

Let $\varphi_i: U_i \to K^n$ be the standard affine coordinate map obtained from

\begin{align*} \left(\frac{x_0}{x_i},\ldots,\frac{x_n}{x_i}\right) \end{align*}

by deleting the $i$-th coordinate, which is equal to $1$.

If $K=\mathbb{R}$, then the collection $\{(U_i,\varphi_i)\}_{i=0}^n$ is a compatible smooth atlas on $\mathbb{P}^n_{\mathbb{R}}$ and defines a smooth manifold structure of real dimension $n$. If $K=\mathbb{C}$, then the collection $\{(U_i,\varphi_i)\}_{i=0}^n$ is a compatible holomorphic atlas on $\mathbb{P}^n_{\mathbb{C}}$ and defines a complex manifold structure of complex dimension $n$.