Let $Y$ be a complex manifold, $X\subset Y$ a compact submanifold, and $E\to X$ a holomorphic vector bundle. Can $E$ be extended to a bundle over an open neighborhood of $X$ in $Y$? (Four years ago I have asked this question on MO Extending the tangent bundle of a submanifold for the case $E=T_X$.)

After tinkering with this problem for a while I found a necessary condition, there is an invariant in $H^2(X, \mathcal{N}_{X/Y}^*\otimes End(E))$ which must be zero for an extension to be possible. So far, so good. Now I think about writing it down and submitting somewhere (assuming it is a new result). But, in a decent paper there are supposed to be references to known results in the same direction, right? And this is what the real problem is: damned if I have a clue where to look! It is all miles away from areas I am familiar with (mostly differential geometry), and this far I could not find anything remotely relevant. So, it would be nice if someone helps me with this.