Extending a holomorphic vector bundle: a reference request

Question

Let $Y$  be a complex manifold,   $Xsubset Y$ a compact submanifold, and $Eto X$ 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.

Francesco Polizzi · Answer

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 $Y$.

Extending a holomorphic vector bundle: a reference request

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