1
$\begingroup$

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?

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ If the connection is not holomorphic, how to give a counter example? $\endgroup$
    – Fei YE
    Commented Jan 6, 2010 at 6:08
  • 1
    $\begingroup$ 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 $(1:0:\dots:0)$. $\endgroup$
    – t3suji
    Commented Jan 6, 2010 at 15:02
  • $\begingroup$ 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? $\endgroup$
    – Fei YE
    Commented Jan 6, 2010 at 16:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.