# Extending a holomorphic vector bundle: a reference request

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.

## 1 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 $$Y$$.

• 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. – Alex Gavrilov Jul 16 at 9:16