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.


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 $Y$.

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    $\begingroup$ Thank you, this is exactly what I missed. You are right, the invariant I have found is the first obstruction. The construction I used is more explicit then Griffiths', but it is probably still unpublishable. $\endgroup$ – Alex Gavrilov Jul 16 at 9:16

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