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):

C. Banica. Topologisch triviale Vektorbündel auf $\mathbb{P}^n(\mathbb{C})$. J. Reine Angew. Math. 344 (1983) 102-119.

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...

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