A vector bundle with a given jumping line - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:44:46Z http://mathoverflow.net/feeds/question/79891 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79891/a-vector-bundle-with-a-given-jumping-line A vector bundle with a given jumping line Vamsi 2011-11-03T03:15:19Z 2011-11-04T00:50:24Z <p>I'd like to know if there exists a holomorphic rank 2 sub-bundle of $T\mathbb{P}^3$ which, when restricted to a given line is $\mathcal{O}(-a)\oplus \mathcal{O}(a)$, but is trivial when restricted to all other lines lying in a plane containing this line (i.e. this line is a jumping line of order $a$).</p> <p>EDIT: From Angelo's answer, we see that there are no subbundles of the tangent bundle satisfying this property. A related question: Is there a vector bundle that has a given jumping line? (which is not a subbundle of the tangent bundle of course).</p> http://mathoverflow.net/questions/79891/a-vector-bundle-with-a-given-jumping-line/79896#79896 Answer by Angelo for A vector bundle with a given jumping line Angelo 2011-11-03T05:37:18Z 2011-11-03T05:37:18Z <p>The only holomorphic subbudles of $T\mathbb P^3$ are the null-correlation bundles coming from symplectic forms in 4 variables (see for example <a href="http://www.math.ubc.ca/~reichst/nesting.pdf" rel="nofollow">http://www.math.ubc.ca/~reichst/nesting.pdf</a>, Corollary 1.6). The first Chern class of a null-correlation bundle is non-zero, so the answer is negative.</p>