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?



Tensoring by Line Bundles to Produce Holomorphic Sections
I am not sure what you mean by equivariant. What you want is certainly true, provided the line bundle is ample (positive), over any projective variety. This follows e.g. from asymptotic Riemann-Roch (which gives control of the Euler characteristic), together with the fact that tensoring with an ample line bundle kills higher cohomology. I do not know the best reference for this (it is stated in e.g. Larasfeld's Positivity in Algebraic Geometry Chapter 1). 


A more analytic approach which produces a "peaked section" is due to Xiaowei Wang, see Proposition 5.1 of "Canonical metrics on stable vector bundles". It is more direct than the method described above. You do not need any more than a first course in complex geometry to understand his approach, I think. One should probably first read the proof that a positive line bundle has a peaked section, which is due to Tian, and was explained in an elementary way by Donaldson in Appendix 2 here: <http://wwwf.imperial.ac.uk/~skdona/KENOTES.PDF> 


