Holomorphic line bundle

Remarks

Equivalently, $L$ is given by an open cover $(U_i)$ of $X$ and nowhere vanishing holomorphic functions $g_{ij} \colon U_i \cap U_j \to \C^*$ with $g_{ij} g_{jk} = g_{ik}$ on $U_i \cap U_j \cap U_k$.

Example: the tautological line bundle $\mathcal{O}(-1)$ on $\mathbb{P}^n$ is $\set{(\ell, v) \in \mathbb{P}^n \times \C^{n+1} : v \in \ell}$ with the projection to $\mathbb{P}^n$. On $U_i = \set{z_i \ne 0}$ the point $(\ell, v)$ goes to $(\ell, v_i)$. The fibre coordinates satisfy $v_i = (z_i / z_j) v_j$, and $z_i / z_j$ is holomorphic and nonzero on $U_i \cap U_j$. Its dual is $\mathcal{O}(1)$, and $\mathcal{O}(k)$ is the $k$-th tensor power of $\mathcal{O}(1)$.

Non-example: the holomorphic tangent bundle of $\mathbb{P}^2$ is a holomorphic vector bundle of rank $2$, so it is not a line bundle. Its top exterior power is a line bundle, the anticanonical bundle $\mathcal{O}(3)$.

The tensor product of two line bundles with transition functions $g_{ij}$ and $h_{ij}$ has transition functions $g_{ij} h_{ij}$, and the dual of the first has $g_{ij}^{-1}$.