TITLE: $(-2)$-curves in complex $3$-folds
QUESTION [6 upvotes]: Let $X$ be a smooth complex $3$-fold, 
and let $C \subset X$ be an embedded smooth rational curve whose 
normal bundle $N_{C/X}$ is isomorphic to $\mathscr{O}(-1) \oplus \mathscr{O}(-1)$. 
Is it true that a neighborhood of $C$ in $X$ is biholomorphic 
to some neighborhood of $C$ in $N_{C/X}$? Could you please give a reference?

REPLY [2 votes]: Yes, this result appeared in "Uber Modifikationen und exzeptionelle analytische Mengen" (Mathematische Annalen, vol. 146, n.4) by H. Grauert. See the Corollary on p. 363, which also treats a much more general case.