Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $a$).

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).

share|improve this question

1 Answer 1

up vote 6 down vote accepted

The only holomorphic subbudles of $T\mathbb P^3$ are the null-correlation bundles coming from symplectic forms in 4 variables (see for example http://www.math.ubc.ca/~reichst/nesting.pdf, Corollary 1.6). The first Chern class of a null-correlation bundle is non-zero, so the answer is negative.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.