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  "body": "\n## Statement\n\nLet $X$ be a [[Stein manifold|HMCD0JBFST]].\nThen two [[holomorphic line bundles|QAC7BYK9KX]] on $X$ are holomorphically isomorphic if and only if they are isomorphic as topological complex line bundles.\nMoreover, every topological complex line bundle on $X$ is isomorphic to a holomorphic one.\nEquivalently, the first Chern class map $c_1 \\colon H^1(X, \\mathcal{O}^*) \\to H^2(X, \\Z)$ is a group isomorphism.\n\n## Remarks\n\nThe proof uses the exponential sequence $0 \\to \\Z \\to \\mathcal{O} \\to \\mathcal{O}^* \\to 0$, where the second map is $f \\mapsto e^{2\\pi i f}$.\nIts long exact [[cohomology|PQ6Q143ARJ]] sequence contains $H^1(X, \\mathcal{O}) \\to H^1(X, \\mathcal{O}^*) \\to H^2(X, \\Z) \\to H^2(X, \\mathcal{O})$.\nBy [[Cartan's theorem B|R7VKRV8AGQ]], the two outer groups are zero, so the middle map, which is $c_1$, is an isomorphism.\nTopological complex line bundles are classified by $H^2(X, \\Z)$ through their Chern class, which gives the first two sentences.\nOka proved this result in 1939, in the language of the second Cousin problem.\nIn that form it says that on a domain of holomorphy, a second Cousin problem is solvable holomorphically if and only if it is solvable continuously.\nThe name Oka principle is also used for a general heuristic.\nOn Stein spaces, holomorphic problems with only topological obstructions should be solvable holomorphically as soon as they are solvable continuously.\nThe Stein hypothesis is needed.\nOn a compact complex torus of dimension $1$, the holomorphic line bundles of degree $0$ are all topologically trivial, but they form a family parametrized by the torus itself.\n\n## Explanations\n\n### L1 human\n\nOn a Stein manifold, a holomorphic line bundle is determined, up to isomorphism, by its topology alone.\nSo a question about holomorphic objects, which looks hard, is the same as a question about continuous objects, which is a question of algebraic topology.\nThis surprising rigidity is the original Oka principle.\n\n### L2 human\n\nThe result solves the second Cousin problem on Stein manifolds: it is solvable exactly when the associated line bundle has Chern class zero, in particular always when $H^2(X, \\Z) = 0$.\nGrauert extended the principle from line bundles to vector bundles and principal bundles with a complex Lie group as structure group.\nGromov and Forstnerič extended it further to sections of holomorphic submersions and to maps into Oka manifolds.\n",
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