I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial bundles, i.e., where the Chern classes as well as Atiyah-Rees-$\alpha$-invariant are trivial. 


A precise question would be: 
**Is it true that every topologically trivial rank two vector bundle on $\mathbb{P}^3$ can be deformed to the trivial bundle? If not, how could one define a reasonable invariant to distinguish non-deformable bundles?** (The parameter space of the deformation could be anything, not necessarily irreducible or smooth.) 


I guess this question could possibly be phrased asking if the moduli of vector bundles of topologically trivial rank two bundles is connected but this is a bit delicate because we are dealing with unstable bundles and so there is no coarse moduli space as such. 


I only know a single paper discussing questions like this for rank two bundles on $\mathbb{P}^3$ (apart from the huge body of literature which focuses only on stable bundles): 



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> C. Banica. Topologisch triviale Vektorbündel auf $\mathbb{P}^n(\mathbb{C})$. J. Reine Angew. Math. 344 (1983) 102-119. 
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The paper shows that the moduli space $M(2)$ of topologically trivial bundles with splitting type $d=2$ has two connected components, but the methods suggest that it would be fairly difficult to study $M(d)$ in general. Certainly the results in Banica's paper do not decide the deformability question. **Has there been any further work on such questions?**


I would greatly appreciate references to further literature dealing with unstable bundles on projective spaces (beyond what is in the book of Okonek-Schneider-Spindler) as well as hints on possible approaches to try on this problem...



Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$
As I mentioned earlier, Peskine (and possibly Kollar too) asked whether given a family of smooth curves in 3-space with general member a complete intersection, is the special member also a complete intersection. To the best of my knowledge, the answer is not known (over complex numbers). Under the above hypothesis, it is immediate that $\omega\_{C\_t}=\mathcal{O}\_{C\_t}(d)$ for some $d$ for all $t$ in the family where $\omega\_C=K\_C$, the canonical bundle. Thus by Serre construction, we get a family of rank two vector bundles $E\_t$ such that $E\_t$ maps onto the ideal sheaf of $C\_t$. Again by Serre, $E\_t$ is a direct sum of line bundles if and only if $C\_t$ is a complete intersection. Thus, Peskine's question is equivalent to, given a family of rank 2 vector bundles with general member a direct sum of lines, is it true also for the special member. Of course, this does not answer your question, since Peskine's question is unresolved.


