Evidences on Hartshorne's conjecture? References? - MathOverflow

Evidences on Hartshorne's conjecture? References?

2010-02-03T16:07:56Z

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?

Answer by Hailong Dao for Evidences on Hartshorne's conjecture? References?

2010-02-03T17:21:11Z

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.

Answer by Franz for Evidences on Hartshorne's conjecture? References?

2012-08-13T15:08:48Z

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).

Answer by Piotr Achinger for Evidences on Hartshorne's conjecture? References?

2012-08-13T22:45:48Z

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!