Differentiable Structure on the Complex Projective Plane

Define the complex projective plane $\mathbb{C}P^n=(\mathbb{C}^{n+1}\setminus\{0\})/\sim$ where $x\sim y$ if and only if $x=\lambda y$ for some $\lambda\in\mathbb{C}\setminus\{0\}$ . Which of the following is a differentiable atlas in the differentiable structure of $\mathbb{C}P^n$ ?

Options:

• A. \(\{(\mathbb{C}P^n,𝟙_{\mathbb{C}\to\mathbb{R}}^{n\to2n}\circ([x]\mapsto\frac{x}{|x|}))\}\)
• B. \(\{(\mathbb{C}P^n,𝟙_{\mathbb{C}\to\mathbb{R}}^{n\to2n}\circ([\langle x_1,\dots,x_{n+1}\rangle]\mapsto\langle\frac{x_2}{x_1},\dots,\frac{x_{n+1}}{x_1}\rangle))\}\)
• C. \(\{(\{[x]\in\mathbb{C}^{n+1}\,\big|\,x_i\neq0\},𝟙_{\mathbb{C}\to\mathbb{R}}^{n\to2n}\circ([\langle x_1,\dots,x_{n+1}\rangle]\mapsto\langle\frac{x_1}{x_i},\dots,\widehat{1},\dots,\frac{x_{n+1}}{x_i}\rangle))\}_{i=1,\dots,n+1}\)

In the options, \(𝟙_{\mathbb{C}\to\mathbb{R}}^{n\to2n}\) represents the natural map from $\mathbb{C}^n$ to $\mathbb{R}^{2n}$ and $\widehat{1}$ indicates that the component there is to be omitted.