This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'.

Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional curve iff $E^2=-1$. The proof in Beauville's book is to find a very ample divisor $H$ satisfying $H^1(S,\mathcal{O}_S(H))=0$ first, and then set $H'=H+kE$ where $k=H\cdot E$. The linear system of $H'$ gives a projective morphism from $S$ to $\mathbb{P}^n$ which contracts $E$, and then some topological arguments implies that the image of $S$ is actually smooth.

Although this proof is not difficult to understand, I still want a proof based on complex manifolds but not algebraic geometry.

**Question**: Is there any holomorphic version of the tubular neighborhood theorem?

I have several reasons to raise this question:

If we have some holomorphic tubular neighborhood theorem, we can identify some neighborhood $U$ of $E$ in $S$ with neighborhood $V$ of the zero section in $N_E$. Here $N_E$ is the holomorphic normal bundle of $E$. Then $E^2=-1$ easily implies $N_E\cong\mathcal{O}_{E}(-1)$, so $E$ can be contracted in $U$ directly. Thus we not only prove Castelnuovo's theorem but also generalize it to non-algebraic surfaces.

There exists a symplectic version of the tubular neighborhood theorem, so I guess the holomorphic case is also true.

Any answers or comments are welcome. I'll really appreciate your help.

On the implicit function theorem in algebraic geometry). He proves that a modification contracting a given closed subvariety exists (as an algebraic space) provided that it exists in the formal n.h. of the subvariety. This gives an approach to Castelnuovo's theorem (and generalizations thereof) which is similar in spirit to the one you are asking about (but with complex geometry replaced by formal geometry). Regards, $\endgroup$