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According to this post. In the affine case the formal completion of $A$ along an ideal $I$ coincides with the formal is isomorphic to the completion of the normal bundle along its zero section. I was wondering can something like this happen in non-affine case (projective varieties), where let's say the closed subvariety is some ample divisor? (I'd appreciate any interesting examples if it is possible) List of questions:

  1. Does any smooth proejctive variety $X$ admit a smooth ample divisor $Y$ such that completion of $X$ along $Y$ coincides with the completion of normal bundle of $Y$ along its zero section?
  2. If not what interesting examples are there that are not trivial (for example when both $X$ and $Y$ are projective spaces.)

As mentioned in the comment section here a necessary and sufficient.

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  • $\begingroup$ Does this answer your question? Are formal completions along a subvariety only dependent on the normal bundle? $\endgroup$
    – user20948
    Commented Feb 11, 2021 at 20:33
  • $\begingroup$ It is very close but it doesn't provide any examples for the situation I'm asking. Jason Starr also provides an if and only if condition which I don't quite understand it, if someone can expand it in more details, that'd be also helpful. $\endgroup$
    – user127776
    Commented Feb 11, 2021 at 20:57

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