Complex projective space and its holomorphic charts

For an integer $n\ge 1$, complex projective space is the set of complex lines in ${\mathbb{C}}^{n+1}$ with the quotient topology ${\mathbb{P}}^n(\mathbb{C}):=({\mathbb{C}}^{n+1}\setminus \{0\})\mathrm{/}{\mathbb{C}}^{\times },[z_0:\cdots :z_n]:={\mathbb{C}}^{\times }(z_0,\dots ,z_n)\mathrm{.}$ The quotient projection is continuous and surjective. Its restriction to the unit sphere $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ is still surjective, so ${\mathbb{P}}^n(\mathbb{C})$ is compact. It is Hausdorff: the map sending a nonzero vector $z$ to the orthogonal projection $z{z}^{\ast }\mathrm{/}\parallel z{\parallel }^2$ is continuous and constant on each complex line, hence descends by the quotient topology to a continuous injection from ${\mathbb{P}}^n(\mathbb{C})$ into the Hausdorff space of complex matrices. A continuous injection from a compact space to a Hausdorff space is a homeomorphism onto its image.

For open $V\subseteq {\mathbb{C}}^r$, a map $H:V\to {\mathbb{C}}^s$ is holomorphic if at every $a\in V$ there is a complex-linear map $A_a:{\mathbb{C}}^r\to {\mathbb{C}}^s$ such that $H(a+h)=H(a)+A_{a}h+R_a(h)$ with $\parallel R_a(h)\parallel \mathrm{/}\parallel h\parallel \to 0$ as $h\to 0$, $h\mathrm{\ne }0$. When $r=1$, this is equivalent to each component being holomorphic in the sense of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions, since a finite vector of scalar difference quotients converges exactly when each component does. A complex atlas consists of charts into ${\mathbb{C}}^n$ with holomorphic transitions in both directions.

For $j\in \{0,\dots ,n\}$ let $U_j:=\{[z_0:\cdots :z_n]:z_j\mathrm{\ne }0\}$. These open sets cover projective space: their inverse images under the quotient projection are the open saturated sets where $z_j\mathrm{\ne }0$. The standard chart is ${\phi }_j:U_j⟶{\mathbb{C}}^n,[z_0:\cdots :z_n]⟼(z_0\mathrm{/}{z}_j,\dots ,\widehat{z_j\mathrm{/}{z}_j},\dots ,z_n\mathrm{/}{z}_j),$ where the $j$th coordinate is omitted. It is well defined under rescaling. The coordinate-ratio map on the inverse image of $U_j$ is continuous and constant on each quotient fibre, so the quotient topology makes ${\phi }_j$ continuous; its inverse inserts $1$ in the $j$th position and is continuous by composition with the quotient projection. Hence it is a homeomorphism. On $U_j\cap U_k$ the transition from the $j$th chart to the $k$th chart sends each coordinate $w_{\mathrm{\ell }}=z_{\mathrm{\ell }}\mathrm{/}{z}_j$ ($\mathrm{\ell }\mathrm{\ne }j$) to $w_{\mathrm{\ell }}\mathrm{/}{w}_k$ for $\mathrm{\ell }\mathrm{\ne }k$, with the omitted $j$th coordinate equal to $1\mathrm{/}{w}_k$. To check holomorphy at $a$ with $a_k\mathrm{\ne }0$, set $a_j=1$ and $h_j=0$ for the omitted source coordinate. Each target coordinate has the expansion $(a_{\mathrm{\ell }}+h_{\mathrm{\ell }})\mathrm{/}(a_k+h_k)=a_{\mathrm{\ell }}\mathrm{/}{a}_k+h_{\mathrm{\ell }}\mathrm{/}{a}_k-a_{\mathrm{\ell }}{h}_k\mathrm{/}{a}_k^2+O(\parallel h{\parallel }^2)$ for $\mathrm{\ell }\mathrm{\ne }k$. The linear term is complex-linear, and the remainder estimate follows by expanding the reciprocal at the nonzero $a_k$. The inverse transition has the same form, so both are holomorphic. Thus these charts make ${\mathbb{P}}^n(\mathbb{C})$ a complex manifold of complex dimension $n$ and a topological manifold of real dimension $2n$ in the sense of Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces. The finite chart cover by second-countable copies of ${\mathbb{C}}^n$ gives a countable base. Its rational transition functions have smooth real coordinate expressions on their domains, so the atlas also defines the smooth structure in the sense of Smooth manifolds and their smooth charts. The space is path-connected: distinct lines represented by $u,v$ give the path $t\mapsto [(1-t)u+tv]$, whose vector never vanishes because $u,v$ are linearly independent; equal lines give a constant path. In particular, ${\mathbb{P}}^1(\mathbb{C})$ is a Riemann surface in the sense of Riemann surfaces and holomorphic atlases.

A map $F:X\to {\mathbb{P}}^n(\mathbb{C})$ from a Riemann surface is holomorphic when it is continuous and every component of each chart expression ${\phi }_j\circ F$, written in a local coordinate of $X$, is holomorphic on the open set $F^{-1}(U_j)$. On an overlap, the new components are $f_{\mathrm{\ell }}\mathrm{/}{f}_k$ and $1\mathrm{/}{f}_k$, where the $f_{\mathrm{\ell }}$ are the old components and $f_k\mathrm{\ne }0$. The scalar quotient rule gives their holomorphy (Linearity, product, reciprocal, and quotient rules for complex derivatives), so it suffices to check one target chart locally. Changing the source coordinate composes each scalar component with a one-variable holomorphic chart transition, where The chain rule for complex derivatives applies (Holomorphic maps and meromorphic functions on Riemann surfaces). A projective line is the image in ${\mathbb{P}}^n(\mathbb{C})$ of a two-dimensional complex subspace of ${\mathbb{C}}^{n+1}$. Every invertible linear map of ${\mathbb{C}}^{n+1}$ induces a holomorphic projective linear transformation: in source and target standard charts its coordinates are ratios of affine-linear functions with nonzero denominator, and the same reciprocal expansion gives a complex-linear derivative; these transformations act transitively on ${\mathbb{P}}^n(\mathbb{C})$.