# Extension of a holomorphic vector bundle

Let $E$ be a holomorphic vector bundle over $\mathbb{P}^n\setminus\begin{Bmatrix}[1,0,0,\cdots,0]\end{Bmatrix}$. Let $D$ be a connection on $E$. Let $\widetilde{E}$ be an extension of $E$. Since $\widetilde{E}$ is reflexive, i.e. double dual of $E$ is isomorphic to itself, then up to isomorphism, $\widetilde{E}$ is unique. My questions are

1) Is it possible that $\widetilde{E}$ is a vector bundle?

2) If $\widetilde{E}$ is a vector bundle, does it admit a connection $\widetilde{D}$ which is naturally induced by $D$?

Edit:For the first question, I just proved that $\widetilde{E}$ is a vector bundle if and only if $\widetilde{E}$ is splits. I am wondering if this result was already known. If so, does any one know any reference on this result?

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## 1 Answer

Presumably, you assume $n\ge 2$.

1) Is it possible that $\tilde E$ is a vector bundle? Yes. Is it always a vector bundle for any $E$? No. Unless, of course, you assume that the connection is flat and holomorphic, then it extends essentially for topological reasons.

2) Does the connection extends to $\tilde E$ if it is a vector bundle? Assuming the connection is holomorphic, the answer is yes: locally, the connection is given by a bunch of holomorphic functions that extend across codimension two.

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If the connection is not holomorphic, how to give a counter example? –  Fei YE Jan 6 '10 at 6:08
If the connection is not holomorphic, the question is not particularly interesting, but here it goes: Take $E$ to be trivial rank one bundle. It extends to trivial rank one bundle. A connection on such an object is a differential form. Let coefficients of this differential form be smooth functions with a singularity at $(1:0:\dots:0)$. –  t3suji Jan 6 '10 at 15:02
Thanks! I think however, in general, it is very hard to find a holomorphic connection on a vector bundle over projective space. It is well-known that if a holomorphic connection exists, then the vector bundle splits into line bundles. As for the first question, is there any criterion on that the extension is a vector bundle? I mean necessary and sufficient conditions. Any reference on this subject? –  Fei YE Jan 6 '10 at 16:16
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