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?

## 1 Answer

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.