QC24 Projective Chart Dimension

If $[z]\in U_j$, multiplying $z$ by any nonzero complex scalar leaves every ratio $z_i\mathrm{/}{z}_j$ unchanged, so ${\varphi }_j$ is well-defined on rays. The representative $z\mathrm{/}{z}_j$ has $j$th coordinate $1$ and all other coordinates equal to those ratios, so the stated inverse sends ${\varphi }_j([z])$ back to $[z]$.

Conversely, applying ${\varphi }_j$ to the inverse representative with $j$th coordinate $1$ returns each entry of $w$; hence the two maps are inverse bijections. Let $\pi$ be the quotient map and $O_j=\{z:z_j\mathrm{\ne }0\}={\pi }^{-1}(U_j)$. Since $O_j$ is open, $U_j$ is open. The restriction ${\pi }_j:O_j\to U_j$ is a quotient map: for $B\subseteq U_j$, if ${\pi }_j^{-1}(B)$ is open in $O_j$, it is open in ${\mathbb{C}}^D\setminus \{0\}$, so $B$ is open in the quotient and hence in $U_j$; the reverse implication follows from continuity of $\pi$. The composite ${\varphi }_j\circ {\pi }_j$ consists of continuous coordinate ratios on $O_j$, so ${\varphi }_j$ is continuous. The inverse is the quotient map composed with the continuous representative map $w\mapsto e_j+{\sum }_{i\mathrm{\ne }j}{w}_i{e}_i$, hence is continuous.

Let $\pi :{\mathbb{C}}^D\setminus \{0\}\to {\mathbb{C}\mathbb{P}}^{D-1}$ be the quotient map. Then ${\pi }^{-1}(U_j)=\{z:z_j\mathrm{\ne }0\}$ is Euclidean open, so $U_j$ is open by the quotient topology. On $U_j\cap U_k$ with $j\mathrm{\ne }k$, write $w={\varphi }_j([z])$. Then $w_k\mathrm{\ne }0$, the transition coordinates satisfy $w_j^{\mathrm{\prime }}=1\mathrm{/}{w}_k$ and $w_i^{\mathrm{\prime }}=w_i\mathrm{/}{w}_k$ for $i\mathrm{\ne }j,k$. Conversely, $w_k=1\mathrm{/}{w}_j^{\mathrm{\prime }}$ and $w_i=w_i^{\mathrm{\prime }}\mathrm{/}{w}_j^{\mathrm{\prime }}$ for $i\mathrm{\ne }j,k$. These rational maps are smooth as real maps on their open coordinate domains, so the charts are compatible and form a smooth atlas.

The patches cover every ray because a nonzero vector has at least one nonzero coordinate. If $D\ge 2$, choose $k\mathrm{\ne }j$; the ray of the basis vector $e_k$ has $j$th coordinate zero and is not in $U_j$, so each fixed standard patch is proper. Finally, ${\mathbb{C}}^{D-1}$ has $D-1$ complex, or $2D-2$ real, coordinates.