Question

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).

Answer 1

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.