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  "body": "\n## Definition\n\nLet $E$, $B$ and $F$ be [[complex manifolds|JSGWAN2RJV]].\nA *holomorphic fiber bundle* over $B$ with fibre $F$ and total space $E$ is a surjective [[holomorphic map|3MQTWKBBBT]] $\\pi \\colon E \\to B$ with the following property.\nEach point of $B$ has an open neighbourhood $U$ and a [[biholomorphism|FX92PP3HZR]] $\\varphi \\colon \\pi^{-1}(U) \\to U \\times F$ with $\\mathrm{pr}_1 \\circ \\varphi = \\pi$ on $\\pi^{-1}(U)$.\n\n## Remarks\n\nThe underlying smooth map is a [[fiber bundle|CX4M8K61AB]] in the usual sense.\nOn overlaps the trivializations differ by maps $(b, f) \\mapsto (b, g_{ij}(b)(f))$, where each $g_{ij}(b)$ is a biholomorphism of $F$ that depends holomorphically on $b$.\nWhen $F = \\C^r$ and each $g_{ij}(b)$ is linear, the bundle is a holomorphic vector bundle.\n\nExample: the map $\\pi \\colon \\C^2 \\setminus \\set{0} \\to \\mathbb{P}^1$, $(z_0, z_1) \\mapsto [z_0 : z_1]$, with fibre $F = \\C^*$.\nOver $U_0 = \\set{z_0 \\ne 0}$ the map $(z_0, z_1) \\mapsto ([z_0 : z_1], z_0)$ is a biholomorphism $\\pi^{-1}(U_0) \\to U_0 \\times \\C^*$, with inverse $([1 : w], c) \\mapsto (c, c w)$, and similarly over $U_1$.\n\nNon-example: $\\pi \\colon \\C^2 \\to \\C$, $(z, w) \\mapsto zw$.\nEach fibre over $t \\ne 0$ is biholomorphic to $\\C^*$, but the fibre over $0$ is the union of the two axes, which is not a manifold.\nSo no neighbourhood of $0$ has a trivialization.\n\nBy a theorem of Fischer and Grauert, a proper holomorphic submersion between connected complex manifolds whose fibres are all biholomorphic to one another is a holomorphic fiber bundle.\n",
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