TITLE: Biholomorphism between Neighborhood of a complex submanifold and a Neighborhood of zero section of its normal bundle
QUESTION [5 upvotes]: Let X be a compact kahler manifold, $M\subseteq X$ be a complex submanifold, is there a biholomorphism between a neighborhood of the zero section of the normal bundle $NM$ of $M$ and a neighborhood $U(M)\subseteq X$?

REPLY [7 votes]: The answer is "no" because in general there does not even exist a neighbourhood $U$ of your submanifold that holomorphically retracts to the submanifold.
In fact Finnur Lárusson has proved  the following   (warning: I have kept his notations, so as to faithfully quote him. In particular $[X]$ is $\mathcal O(X)$, the line bundle on $M$ associated to the divisor $X$)

Theorem Let $X$ be a connected hypersurface in a compact complex manifold
  M of dimension at least $2$. If
  (1) $H^0 (M, TM) = 0$
  (2) $dim H^0(M; [X]) \geq 2$, and
  (3) $H^1(M;  [X]^{-1}  \otimes TM) = 0$,
  then no neighbourhood of $X$ retracts holomorphically onto $X$.

Note that the hypotheses are very easy to satisfy: (1) holds for $K3$ surfaces for example  and (2), (3) will follow from $X$ being sufficiently ample.
Here is a freely downloadable version of Lárusson's   article, from his homepage.