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  "body": "\n## Summary\n\nA holomorphic vector bundle is a complex [[vector bundle|P4DZM975BV]] over a [[complex manifold|JSGWAN2RJV]].\nIts total space is a complex manifold, and its local trivializations are holomorphic.\nEquivalently, its transition functions are holomorphic maps into $GL_r(\\C)$.\nHolomorphic vector bundles carry sheaves of holomorphic sections, and their cohomology is a main tool of complex geometry.\nExamples are the holomorphic tangent bundle, the bundles of holomorphic forms, and the line bundles $\\mathcal{O}(k)$ on projective space.\n\n## Explanations\n\n### L1 human\n\nOver each point of a complex manifold sits a complex vector space, and these spaces vary holomorphically from point to point.\nLocally the bundle looks like a product $U \\times \\C^r$, and two such local pictures are glued by matrices whose entries are holomorphic functions.\nThe holomorphic sections are then the \"vector-valued holomorphic functions\" of the bundle.\nOn the Riemann sphere, the tautological line bundle, whose fibre over a line in $\\C^2$ is that line, is the simplest nontrivial example.\n",
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