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](https://mathoverflow.net/questions/249050/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.



Extending a holomorphic vector bundle: a reference request
It seems to me that your result follows by Proposition 1.1 of the paper


P. A. Griffiths: [**The extension problem in complex analysis. II: Embeddings with positive normal bundle**](http://dx.doi.org/10.2307/2373200), Am. J. Math. **88**, 366-446 (1966). [ZBL0147.07502](https://zbmath.org/?q=an:0147.07502),


that can be freely downloaded [here](https://publications.ias.edu/sites/default/files/extensionproblemcomplexanalysisII.pdf). The statement is the following:



> 
> **Proposition 1.1 (Griffiths 1966).** If $\alpha$ is a holomorphic vector bundle $\mathbf{E} \to X$, then $$\omega(\alpha\_{\mu-1}) \in H^2(X,\,\Omega(\mathrm{Hom}(\mathbf{E}, \, \mathbf{E})(\mu))).$$
> 
> 
> 


Here $\omega(\alpha\_{\mu-1})$ is the obstruction to extending $\mathbf{E}$ to the $\mu$th infinitesimal neighborhood $X\_{\mu}$ of $X$ in $Y$, provided that you already have an extension $\alpha\_{\mu-1}$ to $X\_{\mu-1}$, and $$H^2(X,\,\Omega(\mathrm{Hom}(\mathbf{E}, \, \mathbf{E})(\mu))= H^2(X, \mathrm{End}(\mathbf{E}) \otimes \mathrm{Sym}^{\mu}(N\_{X/Y}^\*)).$$


In order to have an extension of $\mathbf{E}$ to a genuine analytic neighborhood of $X$ in $Y$, all these obstruction classes must vanish. In fact, if I understand correctly, you only provided the obstruction class for the extension of $\mathbf{E}$ to the first infinitesimal neighborhood $X \subset X\_1$ of $X$ in $Y$.


