Determine complex analytic germ along a smooth compact curve via normal bundle?

Question

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?

Answer 1

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).