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A conjecture of Hartshorne states that any vector bundle of a small rank on a projective space of large dimension is split (i.e. isomorphic to a direct sum of line bundles). I would like to know the current state of the art concerning this conjecture for vector bundles of rank $3$.

I work over a field $k$ (of characteristic $0$ if you prefer, but not necessarily algebraically closed).

What is the smallest value of $n_0$ known for which every vector bundle of rank $3$ on $\mathbb{P}_k^n$ is split for all $n > n_0$?

My knowledge on the subject is pretty poor. For example, I don't even know if $n_0$ is known to exist.

Much of the literature seems to be concerned with just rank $2$ vector bundles, and I was struggling to find anything about vector bundles of higher rank.

Daniel Loughran
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    First of all, I am surprised that you assign this conjecture to Hartshorne -- do you have a reference? I am not an expert, but my impression is that there has been no significant progress on these questions since the 80's. At that time, the best result was Horrocks' construction of an indecomposable rank 3 bundle on $\mathbb{P}^5$. I would guess that there are still no example known on $\mathbb{P}^n$ for $n\geq 6$. On the other hand, proving that they do not exist (i.e. the existence of the integer $n_0$ in your question) is wide open. – abx Jan 17 '17 at 13:34
  • @abx: Thanks for the comment, it is very useful. As for references there is for example http://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references. Admittedly this is only stated for rank 2 bundles, but I had assumed that an analogous conjecture exists for arbitrary rank vector bundles. – Daniel Loughran Jan 17 '17 at 14:17
  • This was stated by Hartshorne as an equivalent conjecture regarding the fact that varieties of low codimension in projective spaces should be complete intersections. See [R. Hartshorne, *Algebraic vector bundles on projective spaces: a problem list], in particular the discussion following Problem 3. http://www.sciencedirect.com/science/article/pii/0040938379900302 – Francesco Polizzi Jan 17 '17 at 14:20
  • The problem seems to be closely related to Hartshorne's conjecture that every smooth subvariety of projective space of small codimension is a complete intersection. There are particular results towards this conjecture (e.g. Corollary 3 of http://www.ams.org/journals/jams/1991-04-03/S0894-0347-1991-1092845-5/S0894-0347-1991-1092845-5.pdf), so I was hoping there would also be partial results towards the closely related problem on vector bundles. – Daniel Loughran Jan 17 '17 at 14:20
  • @Francesco Polizzi: Yes, but this is only stated in the reference for rank $2$ vector bundles. Are the two conjectures equivalent in general? – Daniel Loughran Jan 17 '17 at 14:22
  • @abx: This is definitively known as Hartshorne's conjecture, but maybe should be called Hartshorne's question. See http://www.ams.org/journals/bull/1974-80-06/S0002-9904-1974-13612-8/S0002-9904-1974-13612-8.pdf especially the discussion following Conjecture 6.3. – Count Dracula Jan 17 '17 at 14:25
  • @DanielLoughran: I do not know. Hartshorne says that the equivalence of the two conjectures for rank $2$ vector bundles in $\mathbb{P}^6$ follows from a result of Barth, one should read that paper. – Francesco Polizzi Jan 17 '17 at 14:28
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    @Count Dracula: I read in the paper you mention: "I do not feel that I have sufficient evidence to formulate a conjecture about bundles of rank >2" (says Hartshorne). That doesn't seem to me sufficient ground to talk about the "Hartshorne conjecture" for rank >2 bundles. – abx Jan 17 '17 at 14:30
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    I do not think whether the two questions mentioned above are known to be equivalent, except in the codimension 2 case and rank 2 vector bundles. Even in this case, one of the important ingredients is Barth's theorem, which is valid for all low codimensional varieties. For codimension 2, of course Serre's construction plays the deciding tool in one direction. There is also the published (Inv. Math.) incorrect proof that unstable bundles of rank 2 are split if $n\geq 4$ over complex numbers, due to Remmert and Schneider (I think). – Mohan Jan 17 '17 at 14:36
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    @abx: Thanks for pointing this out and apologies for the misattribution to Hartshorne for the case of rank bigger than $2$. If anyone knows of results towards my question I would still be interested, but it's looking increasingly doubtful something useful is known... – Daniel Loughran Jan 17 '17 at 14:41
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    @abx It is a time honored tradition in Algebraic Geometry to misattribute conjectures and I for one will continue to do so... – Count Dracula Jan 17 '17 at 16:13
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    @CountDracula. In that case, you should offer us some questions that we can misattribute to you as conjectures :) – Jason Starr Jan 17 '17 at 16:54
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    "On a conjecture of Count Dracula" - I'd like to see that on the arxiv. – byu Jan 19 '17 at 17:37

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