Let $X\_1, X\_2$ be two smooth complex manifold and $C\_1 \subset X\_1, C\_2 \subset X\_2$ be two smooth projective curves. Assume that $C\_1 \simeq C\_2$ as complex curves and their normal bundles are isomorphic. 


Q. Is it possible to take two analytic open neighbourhoods $U\_1 \subset X\_1$ and $U\_2 \subset X\_2$ of $C\_1$ and $C\_2$ such that there is an isomorphism of complex manifolds $U\_1 \simeq U\_2$ which sends $C\_1$ to $C\_2$? 
That is, are two complex germs $(U\_1, C\_1)$ and $(U\_2, C\_2)$ same? 


If not, is there some useful criterion?



Determine complex analytic germ along a smooth compact curve via normal bundle?
This is equivalent to asking whether for any smooth curve $C$ on a complex manifold $X$, there is an analytic neighborhood of $C$ in $X$ that is equivalent to a neighborhood of $C$ in the normal bundle $N$. However, this implies that the exact sequence
$$
0 \longrightarrow T\_C \longrightarrow T\_X\mid\_{C} \longrightarrow N \longrightarrow 0
$$
splits, and it is easy to give examples in which this does not happen (for example, plane curves).


