Projective twisting line bundles by coordinate transitions

For $r\ge 1$, let ${\mathbb{P}}^r(\mathbb{C})$ be the set of complex lines in ${\mathbb{C}}^{r+1}$, with its quotient complex-manifold structure, and write $[Z_0:\cdots :Z_r]$ for the line through a nonzero vector $(Z_0,\dots ,Z_r)$. For $i\in \{0,\dots ,r\}$ set $U_i=\{[Z_0:\cdots :Z_r]:Z_i\mathrm{\ne }0\}$; its standard coordinates are the ratios $Z_j\mathrm{/}{Z}_i$ for $j\mathrm{\ne }i$. For each integer $k$, the projective twisting line bundle ${\mathcal{O}}_{{\mathbb{P}}^r}(k)$ is obtained by gluing the trivial holomorphic line bundles $U_i\times \mathbb{C}$ with local frames $e_i^{(k)}$ and transition rule $e_j^{(k)}={\left(\frac{Z_j}{Z_i}\right)}^k{e}_i^{(k)}\text{on }{U}_i\cap U_j\mathrm{.}$ The transition factors are holomorphic and nowhere zero on the overlaps, with $g_{ii}=1$, $g_{ij}=g_{ji}^{-1}$, and $g_{ij}{g}_{j\mathrm{\ell }}=g_{i\mathrm{\ell }}$ on every triple overlap; hence they define a holomorphic line bundle. The sheaf ${\mathcal{O}}_{{\mathbb{P}}^r}(k)$ assigns to an open set the holomorphic sections of this bundle, and ${\mathcal{O}}_{{\mathbb{P}}^r}:={\mathcal{O}}_{{\mathbb{P}}^r}(0)$ is the structure sheaf.

Complex projective space and its holomorphic charts

Complex manifolds, holomorphic vector bundles and Hermitian metrics

MIT OpenCourseWare, 18.117 lecture 7

Dervan–Kuang, Complex Manifolds, §5