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Let $E$ be a vector bundle on projective space ${\bf P}^n$ whose Hilbert polynomial is the same as $\mathcal{O}^{{\rm rank}(E)}$.

Does there exist a vector bundle over ${\bf P}^n \times {\rm Spec}(R)$ where $R$ is a DVR so that the general fiber is $\mathcal{O}^{{\rm rank}(E)}$ and the special fiber is $E$?

The intuition is that the cohomology of twists of $E$ must be at least as large as the cohomology of the corresponding twists of the trivial bundle, and I'd like to see that realized as a flat degeneration.

The real goal is to use this to prove that $h^i(E \otimes F) \ge {\rm rank}(E) \cdot h^i(F)$ for all $i$ and all vector bundles $F$.

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  • $\begingroup$ This is probably not a very useful observation: if $E$ is stable, then a deformation of $E$ should be stable as well, hence cannot be trivial. Maybe for some obvious reasons I don't see, a v.b. with Hilbert polynomial the same as the trivial bundle cannot be stable... $\endgroup$
    – Piotr Achinger
    Commented Apr 22, 2015 at 1:18
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    $\begingroup$ This paper may answer your question - math.wustl.edu/~kumar/papers/kpr2.pdf $\endgroup$
    – meh
    Commented Apr 22, 2015 at 13:42

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