Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology? - MathOverflow

Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?

2009-11-04T15:47:21Z

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.

I hope the answer is not "trivially yes" for some stupid reason, but I cannot guarantee that.

Answer by A-C for Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?

2009-11-04T16:14:43Z

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.

Answer by userN for Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?

2009-11-04T16:17:28Z

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.

Answer by Ilya Nikokoshev for Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?

2009-11-04T17:39:22Z

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?