Evidences on Hartshorne's conjecture? References? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:44:56Z http://mathoverflow.net/feeds/question/13990 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references Evidences on Hartshorne's conjecture? References? Fei YE 2010-02-03T16:07:56Z 2012-08-14T10:52:42Z <p>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.</p> <p>So my questions are the following:</p> <p>1) what is an evidence for this conjecture?</p> <p>2)why is the condition on $n\geq 7$, but not other numbers?</p> <p>3)any recent survey or reference on this conjecture? </p> http://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references/14005#14005 Answer by Hailong Dao for Evidences on Hartshorne's conjecture? References? Hailong Dao 2010-02-03T17:21:11Z 2010-02-05T21:03:59Z <p>This <a href="http://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689" rel="nofollow">answer</a> 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).</p> <p>As for 3), you can also look at Zolbani's <a href="https://edocs.uis.edu/mmaji2/www/Research/Dissertation.pdf" rel="nofollow">thesis</a>, which has a lot more details then his research statements mentioned by Steven. </p> <p>(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).</p> <p><strong>EDIT</strong>: 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 <a href="http://www.ams.org/bull/1974-80-06/S0002-9904-1974-13612-8/" rel="nofollow">paper</a>). A paper by <a href="http://www.jstor.org/pss/2946619" rel="nofollow">Lyubeznik</a>, especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references. </p> http://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references/104620#104620 Answer by Franz for Evidences on Hartshorne's conjecture? References? Franz 2012-08-13T15:08:48Z 2012-08-14T10:52:42Z <p>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.</p> <p>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 "<strong>Tangents and Secants of Algebraic Varieties</strong>" to get complete proofs of these facts.</p> <p>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 <a href="http://arxiv.org/abs/0909.2763" rel="nofollow">http://arxiv.org/abs/0909.2763</a>). Note also that Hartshorne conjecture completely fails for other toric projective Fano varieties, see <a href="http://arxiv.org/abs/1005.5546" rel="nofollow">http://arxiv.org/abs/1005.5546</a>).</p> http://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references/104652#104652 Answer by Piotr Achinger for Evidences on Hartshorne's conjecture? References? Piotr Achinger 2012-08-13T22:45:48Z 2012-08-13T22:45:48Z <p>A remark similar to Hailong Dao's comment under his answer:</p> <p>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 $0 &lt; i &lt; n$ and all $t$. </p> <p>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 $0 &lt; i &lt; \min(n, rank(E))$ and all $t$.</p> <p>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.</p> <p>In summary, it is surprising how little we know about such a simple situation! </p>