Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$.
Is there any relation between $H^i(\hat Z,\mathcal{O})$ and $H^i(U, \mathcal{O})$ when $U$ is small enough? (Here $\mathcal{O}$ denotes the corresponding structure sheaf of holomorphic functions.) More precisely I would like to deduce from vanishing of one of the groups the vanishing of the other one.
I apologize that the question is vague. Thus I know that there is the notion of formal neighborhood of a submanifold in context of algebraic geometry. Here I assume that it also exists in the context of complex analytic geometry.
Remark. In my case $Z$ is isomorphic to $\mathbb{C}\mathbb{P}^1$ and its normal bundle is isomorphic to the direct sum of several copies of $\mathcal{O}(1)$.
