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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] and let $r \\ge 0$ be an integer.\nA *holomorphic vector bundle* of rank $r$ over $X$ is a complex manifold $E$ with a [[holomorphic map|3MQTWKBBBT]] $\\pi \\colon E \\to X$ such that:\n\n1. Each fibre $E_x = \\pi^{-1}(x)$ has the structure of a complex vector space of dimension $r$.\n2. Each point of $X$ has an open neighbourhood $U$ and a biholomorphic map $\\varphi \\colon \\pi^{-1}(U) \\to U \\times \\C^r$ with $\\mathrm{pr}_1 \\circ \\varphi = \\pi$.\n3. For each $x \\in U$, the restriction $\\varphi \\colon E_x \\to \\set{x} \\times \\C^r$ is $\\C$-linear.\n\n## Remarks\n\nGiven two such trivializations $\\varphi_i$ over $U_i$ and $\\varphi_j$ over $U_j$, one has $\\varphi_i \\circ \\varphi_j^{-1}(x, v) = (x, g_{ij}(x) v)$ with $g_{ij} \\colon U_i \\cap U_j \\to GL_r(\\C)$ holomorphic.\nConversely, holomorphic maps $g_{ij}$ with $g_{ij} g_{jk} = g_{ik}$ on triple overlaps glue the products $U_i \\times \\C^r$ to a holomorphic vector bundle.\nA holomorphic vector bundle is in particular a smooth complex [[vector bundle|P4DZM975BV]].\n\nExample: the holomorphic tangent bundle $T X$ of a complex manifold of dimension $n$ has rank $n$.\nA holomorphic chart $z = (z_1, \\dots, z_n)$ trivializes it by the frame $\\partial/\\partial z_1, \\dots, \\partial/\\partial z_n$.\nThe transition functions are the complex Jacobian matrices of the coordinate changes, which are holomorphic.\n\nNon-example: cover the Riemann sphere by $U_0 = \\C$ and $U_1 = \\mathbb{P}^1 \\setminus \\set{0}$, and glue $U_0 \\times \\C$ and $U_1 \\times \\C$ over $\\C^*$ with $g_{01}(z) = \\bar z$.\nThis gives a smooth complex line bundle, but $g_{01}$ is not holomorphic.\nSo the trivializations of this atlas are not biholomorphic for any complex structure on the glued space, and clause 2 fails for them.\n\nThe sheaf of holomorphic sections of $E$ is a locally free sheaf of $\\mathcal{O}_X$-modules of rank $r$, and every such sheaf arises in this way.\n",
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