6
$\begingroup$

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?

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ I'm pretty sure there is a theorem of Grauert for this. $\endgroup$
    – user691704
    Commented Jul 10, 2019 at 21:18

1 Answer 1

Reset to default
2
$\begingroup$

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.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .