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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?

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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

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  • $\begingroup$ Thanks Ruadhaí for your answer. Could check the link to the Donaldson notes please, it doesn't seem to be working. $\endgroup$ – Tomasz Köner May 14 '15 at 17:21
  • $\begingroup$ Sometimes Imperial's website is a bit buggy, the link I gave works for me. The link is at the bottom of Donaldson's homepage: wwwf.imperial.ac.uk/~skdona Hope that works! If it does not, this technique is also described here, in notes by Ross: dpmms.cam.ac.uk/~jar62/kodairaembedding.pdf The main difference compared to Donaldson's notes is that Donaldson describes how to prove the Hörmander estimate needed. $\endgroup$ – Ruadhaí Dervan May 14 '15 at 17:29
  • $\begingroup$ The links are working now. Another question though: How does one show that tensoring with an ample line bundle kills higher cohomology? Is this in Larasfeld as well? $\endgroup$ – Tomasz Köner May 14 '15 at 19:28
  • $\begingroup$ Also, does asymptotic Riemann Roch follow from the ordinary Riemann Roch (in the case of CPN at least)? $\endgroup$ – Tomasz Köner May 14 '15 at 19:29
  • $\begingroup$ From a complex geometric point of view, that tensoring with a positive line bundle kills cohomology is Proposition 5.2.7 in Huybrecht's book "Complex Geometry". I am not sure which version of Riemann-Roch you mean, but it does follow from Hirzebruch-Riemann-Roch. Huybrechts discusses that result a bit at the start of Chapter 5. Asymptotic Riemann-Roch also follows from the peaked section techniques I mentioned before, too. I should also mention that there is an excellent exposition of peaked section techniques in Section 6 of these notes: www3.nd.edu/~gszekely/notes.pdf $\endgroup$ – Ruadhaí Dervan May 14 '15 at 20:25

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