Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:35:35Z http://mathoverflow.net/feeds/question/4104 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4104/is-a-holomorphic-vector-bundle-on-a-projective-variety-locally-trivial-in-the-zar Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology? Andrea Ferretti 2009-11-04T15:47:21Z 2009-11-04T17:39:22Z <p>By the GAGA principle we know that a holomorphic vector bundle E->X is analitically isomorphic to an algebraic one, say F->X, and by definition F is locally trivial in the Zariski topology. But since the isomorphism between E and F is analytic, I fail to see if this implies that E is Zariski locally trivial too.</p> <p>I hope the answer is not "trivially yes" for some stupid reason, but I cannot guarantee that.</p> http://mathoverflow.net/questions/4104/is-a-holomorphic-vector-bundle-on-a-projective-variety-locally-trivial-in-the-zar/4110#4110 Answer by A-C for Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology? A-C 2009-11-04T16:14:43Z 2009-11-04T16:14:43Z <p>You get (analytic) trivializations of E over Zariski-open sets just by composing a trivialization of E with the isomorphism between E and F. Of course, you do not get algebraic trivializations, but for this you would need an algebraic structure on E in the first place.</p> http://mathoverflow.net/questions/4104/is-a-holomorphic-vector-bundle-on-a-projective-variety-locally-trivial-in-the-zar/4112#4112 Answer by userN for Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology? userN 2009-11-04T16:17:28Z 2009-11-04T16:17:28Z <p>It's probably worth noting that etale-locally trivial principal GL(n)-bundles are automatically Zariski-locally trivial. This isn't necessarily true for general G.</p> http://mathoverflow.net/questions/4104/is-a-holomorphic-vector-bundle-on-a-projective-variety-locally-trivial-in-the-zar/4123#4123 Answer by Ilya Nikokoshev for Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology? Ilya Nikokoshev 2009-11-04T17:39:22Z 2009-11-04T17:39:22Z <p>My memory is fuzzy, but when you compute algebraic bundle cohomology, and it happens to be 0 here because of GAGA, I think, on all affine subschemes, wouldn't it automatically mean the bundle is locally trivial?</p>