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  "body": "\n## Summary\n\nA holomorphic line bundle is a [[holomorphic vector bundle|TKA958Z4KZ]] of rank one.\nIts transition functions are nowhere vanishing holomorphic functions.\nSo the isomorphism classes of holomorphic line bundles on a complex manifold $X$ form the [[Picard group|3Z9DCWYXQZ]] $H^1(X, \\mathcal{O}_X^*)$ under tensor product.\nLine bundles encode divisors, maps to projective space and positivity, as with the [[canonical bundle|W05VYY4HBR]] and the bundles $\\mathcal{O}(k)$ on $\\mathbb{P}^n$.\n\n## Explanations\n\n### L1 human\n\nA holomorphic line bundle attaches a complex line to each point of a complex manifold, in a holomorphic way.\nLocally it is just $U \\times \\C$, and two local pictures are related by multiplying by a nowhere zero holomorphic function.\nA section is then like a holomorphic function, except that its values are only defined up to these gluing factors.\nFor example, the homogeneous polynomials of degree $k$ in $z_0, z_1$ are not functions on the Riemann sphere, but they are exactly the holomorphic sections of the line bundle $\\mathcal{O}(k)$.\n",
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