# $(-2)$-curves in complex $3$-folds

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?

• I'm pretty sure there is a theorem of Grauert for this. Jul 10, 2019 at 21:18