8
$\begingroup$

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } i>0. $$

Is it true that for any open neighborhood $U$ of $Z$ there exists an open neighborhood $V\subset U$ of $Z$ such that $$H^i(V, A)=0 \mbox{ for any } i>0.$$

A reference would be helpful.

Remark. In my case $Z$ is isomorphic to $\mathbb{C}\mathbb{P}^1$ and $A$ is the trivial line bundle. Moreover the normal bundle of $Z$ is isomorphic to direct sum of several copies of $\mathcal{O}(1)$ (I am not sure if it is relevant).

$\endgroup$
2
  • $\begingroup$ Ah, ok, $Z =\Bbb{CP}^1$ is a quite different situation! I should have assumed that's what you meant - my apologies. $\endgroup$
    – mme
    Oct 14, 2018 at 18:36
  • $\begingroup$ @MikeMiller: Sorry, there was a misprint. I fixed it. Thank you. $\endgroup$
    – asv
    Oct 14, 2018 at 18:37

0

Your Answer

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