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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]].\nA *holomorphic line bundle* on $X$ is a [[holomorphic vector bundle|TKA958Z4KZ]] $\\pi \\colon L \\to X$ of rank $1$.\n\n## Remarks\n\nEquivalently, $L$ 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 $U_i \\cap U_j \\cap U_k$.\n\nExample: the [[tautological line bundle|BDB06N91FN]] $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$ is $\\set{(\\ell, v) \\in \\mathbb{P}^n \\times \\C^{n+1} : v \\in \\ell}$ with the projection to $\\mathbb{P}^n$.\nOn $U_i = \\set{z_i \\ne 0}$ the point $(\\ell, v)$ goes to $(\\ell, v_i)$.\nThe fibre coordinates satisfy $v_i = (z_i / z_j) v_j$, and $z_i / z_j$ is holomorphic and nonzero on $U_i \\cap U_j$.\nIts dual is $\\mathcal{O}(1)$, and $\\mathcal{O}(k)$ is the $k$-th tensor power of $\\mathcal{O}(1)$.\n\nNon-example: the holomorphic tangent bundle of $\\mathbb{P}^2$ is a holomorphic vector bundle of rank $2$, so it is not a line bundle.\nIts top exterior power is a line bundle, the anticanonical bundle $\\mathcal{O}(3)$.\n\nThe 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 $g_{ij}^{-1}$.\n",
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