QC24 Complex Projective Space and Its Standard Patches

For an integer $D\ge 1$, let ${\mathbb{C}\mathbb{P}}^{D-1}$ be the quotient of ${\mathbb{C}}^D\setminus \{0\}$ by $z\sim w$ when $w=\lambda z$ for a nonzero complex scalar $\lambda$. Give ${\mathbb{C}}^D\setminus \{0\}$ its usual Euclidean topology and the quotient the topology in which a subset is open exactly when its inverse image is open. Every nonzero ray has a unit representative, and two unit representatives of one ray differ by a unit scalar, so this is the pure-state ray space of Pure state and ray.

For $j\in \{0,\dots ,D-1\}$ define the standard patch $U_j=\{[z]\in {\mathbb{C}\mathbb{P}}^{D-1}:z_j\mathrm{\ne }0\}$ and the candidate coordinate map ${\varphi }_j([z])=(z_i\mathrm{/}{z}_j{)}_{i\mathrm{\ne }j}$, with the coordinates kept in increasing original index order. Its candidate inverse sends $w\in {\mathbb{C}}^{D-1}$ to the ray of the vector with $j$th coordinate $1$ and other coordinates given by $w$ in that same order. QC24 Projective Chart Dimension ↗ proves these are smooth charts, computes their real dimension, and notes that a single standard patch omits part of projective space when $D\ge 2$.