Extending a holomorphic vector bundle: a reference request

Question

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.

Answer 1

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, Am. J. Math. 88, 366-446 (1966). ZBL0147.07502,

that can be freely downloaded here. 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 $Y$.