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?

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    $\begingroup$ I'm pretty sure there is a theorem of Grauert for this. $\endgroup$
    – user691704
    Jul 10, 2019 at 21:18

1 Answer 1


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.


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