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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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  • $\begingroup$ Since a family of vector bundles on $\mathbb{P}^3$ with one member trivial has general member trivial, do you mean that any topologically trivial bundle is the limit of trivial bundles? There is at least a question by Peskine and Lazarsfeld asking that such limits are in fact trivial over complex numbers. $\endgroup$ – Mohan Aug 21 '14 at 14:52
  • $\begingroup$ @Mohan: I do not think that all limits are trivial - there are non-trivial limits of trivial bundles over $\mathbb{P}^1$ and $\mathbb{P}^2$. Moreover, once you have a non-trivial limit, it may be possible to further deform the bundle to something that is no longer a limit of trivial bundles. This happens for $\mathbb{P}^2$, and this is the reason for stating explicitly that the parameter space does not need to be irreducible. $\endgroup$ – Matthias Wendt Aug 21 '14 at 15:23
  • $\begingroup$ I did not mean on projective line or plane. Question of Peskine is just for 3-space and rank 2 bundles(and hence for all larger dimensions) $\endgroup$ – Mohan Aug 21 '14 at 15:29
  • $\begingroup$ Could you give me a reference? This seems to say that the elements in $M(2)$ of Banica are not deformable to the trivial bundle. Has this something to do with properties of local vs. global complete intersection of codimension 2 in $\mathbb{P}^3$? $\endgroup$ – Matthias Wendt Aug 21 '14 at 15:34
  • $\begingroup$ This question due to Peskine can be found in Springer Lecture Notes 1389 (1988), where he asks, given a family of smooth curves in 3-space with general member a complete intersection, is the special member a complete intersection. To the best of my knowledge, the answer is not known, but I answered it in the negative for positive characteristics. $\endgroup$ – Mohan Aug 21 '14 at 17:03
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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.

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  • $\begingroup$ Thanks very much, also for the discussion. My confusion was that bundles of the form $0\to\mathcal{O}(n)\to E\to\mathcal{O}(-n)\otimes\mathcal{I}_Y\to 0$ with $Y$ complete intersection are nonsplit but deformable to the split bundle for $\mathbb{P}^2$, but they are split for $\mathbb{P}^3$. Your answer seems to say that if Peskine's question has a positive answer, then there are at least two classes of rank two bundles up to deformation - the trivial bundle and the rest. $\endgroup$ – Matthias Wendt Aug 22 '14 at 13:01
  • $\begingroup$ You said that you answered Peskine's question negatively in positive characteristic. Does that give rise to interesting deformations from split bundles in characteristic $p$? $\endgroup$ – Matthias Wendt Aug 22 '14 at 13:02
  • $\begingroup$ They are `interesting', but not very explicit. It would be nice to know which bundles appear this way as a limit of decomposable bundles. $\endgroup$ – Mohan Aug 22 '14 at 22:18

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