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Hartshorne's famous conjecture on vector bundles say that any rank $2$ vector bundle over a projective space $\mathbb{P}^n$ with $n\geq 7$ splits into the direct sum of two line bundles.

So my questions are the following:

1) what is an evidence for this conjecture?

2)why is the condition on $n\geq 7$, but not other numbers?

3)any recent survey or reference on this conjecture?

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Regarding 2): Page 4 of this file edocs.uis.edu/mmaji2/www/Research/Research%20statement.pdf contains a table of some small examples. –  Steven Sam Feb 3 '10 at 16:39
    
Thanks Steven, although I knew that that up to $\mathbb{P}^5$ non-spliting vector bundles exist, but this article is really helpful to get a quick survey on this topic. –  Fei YE Feb 3 '10 at 16:55

3 Answers 3

up vote 2 down vote accepted

This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).

As for 3), you can also look at Zolbani's thesis, which has a lot more details then his research statements mentioned by Steven.

(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).

EDIT: Today while answering another question I was reminded of a line of research which can be viewed as evidence for Hartshorne's conjecture: smooth subvarieties of small codimension behave cohomologically like complete intersections (this was discussed in Section 2 of Hartshorne original paper). A paper by Lyubeznik, especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references.

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I checked that question. Although I don't know precisely how $S_2$ condition plays its rule in Hartshorne conjecture, I knew that the vanishing of certain local cohomology groups with $i>0$ implies the splitting condition. By the way, thanks for the reference. –  Fei YE Feb 3 '10 at 21:55
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In codimension 2 situation, one only need to check that the local ring $R$ at the origin of the cone has depth 2 (since you already assume X is smooth, so $R$ is already an isolated singularity). Thus it is basically $H_m^1(R)=0$. –  Hailong Dao Feb 3 '10 at 22:08
    
The link to Zolbani's thesis doesn't seem to work... –  name Aug 14 '12 at 10:39

I think the original statement of Hartshorne was that for smooth $ X \subset \mathbb{P}^N$ with $ 3/2 \dim(X) > N$, then $X$ is a complete intersection (you recover your statement from this conjecture with the famous construction of Serre). Being a complete intersection implies that $ H^0(\mathbb{P}^N, O_{\mathbb{P}^N}(k)) \rightarrow H^0(X,O_X(k))$ is surjective for all $k$. A variety with such a surjective map for a fixed $k$ is called $k$-normal.

With surprisng new "topologico-geometric" methods, Fyodor Zak proved in the beginning of the 80's that if $3/2\dim(X) + 1 > N$ then $X$ is $1$-normal. He proved morever that for $ 3/2\dim(X) + 1 = N$ there exist only $4$ varieties which are not $1$-normal namely : the 2nd Veronese embedding of $\mathbb{P}^2$ projected down into $\mathbb{P}^4$, the Segre embedding of $\mathbb{P}^2 \times \mathbb{P}^2$ projected down into $\mathbb{P}^7$, the Plucker embedding of the grassmannian of $\mathbb{C}^2 \subset \mathbb{C}^6$ projected down into $ \mathbb{P}^{13}$ and the closed orbit of $E_6$ in $\mathbb{P}^{26}$ projected down into $\mathbb{P}^{25}$. You can read his wonderful book "Tangents and Secants of Algebraic Varieties" to get complete proofs of these facts.

If you restrict to the case of varieties defined by quadratic equations (which is a bit limitated but still interesting as a step toward a more complete answer), then Hartshorne conjecure has been proved in that setting by Ionescu and Russo (see http://arxiv.org/abs/0909.2763). Note also that Hartshorne conjecture completely fails for other toric projective Fano varieties, see http://arxiv.org/abs/1005.5546).

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Concerning the last reference: there is a complete description of toric vector bundles due to Klyachko 1989. In his long paper, among other things, he gives an iff condition for all rank 2 toric vector bundles to be split. –  Piotr Achinger Aug 13 '12 at 22:35
    
That's interesting, can you give the title of the paper? Nevertheless I think the result of the last reference I gave is a bit more subtle than the result by Klyachko you metionned. Indeed the bundles constructed by these authors are (if I remember correctly) never toric. –  Franz Aug 13 '12 at 22:44
    
It is strange then that they neither check if these bundles are toric nor make reference to Klyachko's article. –  Piotr Achinger Aug 13 '12 at 22:49
    
A. Klyachko, Equivariant vector bundles on toral varieties, Math. USSR Izv. 35 (1990), 337–375. –  Piotr Achinger Aug 13 '12 at 22:49

A remark similar to Hailong Dao's comment under his answer:

Let $E$ be a vector bundle on $\mathbb{P}^n$. A cohomological criterion (Horrocks' criterion) states that $E$ splits if and only if $H^i(\mathbb{P}^n, E(t))=0$ for $0 < i < n$ and all $t$.

There is a little less well known criterion, due to Evans and Griffiths, which says that we only need to check the vanishing of $H^i(\mathbb{P}^n, E(t))$ for $0 < i < \min(n, rank(E))$ and all $t$.

In particular, in the rank two case, the whole conjecture boils down to the simple claim that $H^1(\mathbb{P}^n, E) = 0$. Since $E$ is trivial on each "standard open" $U_i$, we can describe cohomology classes in this $H^1$ group using explicit Cech cocycles in this covering.

In summary, it is surprising how little we know about such a simple situation!

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I guess that you mean that $H^1(\mathbb{P}^n, E(t))=0$ for any $t\in \mathbb{Z}$. –  Fei YE May 23 '13 at 15:42
    
I mean $E$, not $E(t)$, because if we prove vanishing of $H^1$ for all rank two vector bundles, then in particular we have it for $E(t)$. –  Piotr Achinger May 23 '13 at 22:18

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