Inspired by the line bundle case, I have the following question:

Given an equivariant holomorphic vector bundle over complex projective space, is it true that tensoring it by line bundles often enough can change it from a bundle with no holomorphic sections to a bundle with holomorphic sections. Where could one find a proof of this - a more global, differential geometric approach would be preferable. Can it be deduced from any of the big basic results of complex geometry, i.e. the one you learn about in a first course on complex manifolds?

How about more general spaces like the Grassmannians?