Example 10.6.601B4AD44BAC3

Example 10.6.6.01B4AD44BAC3 Let $N\geq 0$. The projective space $\mathbb P^ N$ is the quotient of $\mathbb C^{N+1}\setminus \{ 0\} $ by the action of $\mathbb C^*$ by multiplication, with the quotient topology; we write $q$ for the quotient map and $[Z]=[Z_0:\cdots :Z_ N]$ for the class of $Z$. For $0\leq i\leq N$ the sets $U_ i=\{ [Z]:Z_ i\neq 0\} $ are open and cover $\mathbb P^ N$, and the map $\phi _ i\colon U_ i\to \mathbb C^ N$, $[Z]\mapsto (Z_ k/Z_ i)_{k\neq i}$, is a homeomorphism: it is continuous because $\phi _ i\circ q$ is, and its inverse is $q$ composed with the map inserting the coordinate $1$ in the $i$th place. On $\phi _ i(U_ i\cap U_ j)=\{ w_ j\neq 0\} $, where we put $w_ i=1$, the transition map $\phi _ j\circ \phi _ i^{-1}$ is $w\mapsto (w_ k/w_ j)_{k\neq j}$, which is holomorphic, and so is its inverse $\phi _ i\circ \phi _ j^{-1}$. The space $\mathbb P^ N$ is compact and connected, as the image under $q$ of the unit sphere of $\mathbb C^{N+1}$. It is Hausdorff, because the continuous map $[Z]\mapsto Z\overline Z^{\mathsf T}/|Z|^2$ to the space of complex matrices is injective, the matrix being the orthogonal projection onto the line $\mathbb CZ$; and being compact, it is paracompact. Thus $\mathbb P^ N$ is a connected compact complex manifold of dimension $N$ (definition 5.2.3).

The tautological line bundle $\mathcal O_{\mathbb P^ N}(-1)$ has the fiber $\mathbb CZ$ over $[Z]$. In the sense of definition 6.14.9 it is given by the frames $e_ i([Z])=Z/Z_ i$ over $U_ i$, the points of the line with $i$th coordinate one. Since $e_ i=(Z_ j/Z_ i)e_ j$ on $U_ i\cap U_ j$, its transition functions are $g_{ij}=Z_ j/Z_ i$, which are holomorphic and nowhere zero on $U_ i\cap U_ j$ and satisfy $g_{ij}g_{jk}=g_{ik}$. The Euclidean norm of $\mathbb C^{N+1}$ gives the smooth metric $|e_ i|^2=|Z|^2/|Z_ i|^2$, which satisfies $|e_ i|^2=|g_{ij}|^2|e_ j|^2$. The dual line bundle $\mathcal O_{\mathbb P^ N}(1)$ has the dual frames $e_ i^*$, with $e_ i^*(e_ i)=1$ and transition functions $g_{ij}^{-1}$, and the dual metric $|e_ i^*|^2=|e_ i|^{-2}=|Z_ i|^2/|Z|^2$, the Fubini–Study metric $h_{\mathrm{FS}}$. A linear form $\ell $ on $\mathbb C^{N+1}$ defines a holomorphic section $s_\ell $ of $\mathcal O_{\mathbb P^ N}(1)$, the restriction of $\ell $ to the fibers of $\mathcal O_{\mathbb P^ N}(-1)$: on $U_ i$ it is $s_\ell =\ell (e_ i)e_ i^*=(\ell (Z)/Z_ i)\, e_ i^*$, and these expressions agree on overlaps, because $\ell (e_ i)e_ i^*=\ell (g_{ij}e_ j)\, g_{ij}^{-1}e_ j^*$. For the coordinate forms we write $X_ k=s_{Z_ k}$. Thus $X_ i=e_ i^*$ on $U_ i$, and

The tensor power $\mathcal O_{\mathbb P^ N}(k)=\mathcal O_{\mathbb P^ N}(1)^{\otimes k}$, $k\geq 1$, has the frames $(e_ i^*)^{\otimes k}$, the transition functions $g_{ij}^{-k}$ and the metric $h_{\mathrm{FS}}^ k$; products of $k$ sections of $\mathcal O_{\mathbb P^ N}(1)$, such as the monomials $X_0^{a_0}\cdots X_ N^{a_ N}$ with $a_0+\cdots +a_ N=k$, are holomorphic sections of $\mathcal O_{\mathbb P^ N}(k)$.