Castelnuovo contraction for analytic surfaces

Question

Let $X$ be a smooth complex manifold of complex dimension $2$, and let $C$ be a ($-1$)-curve on $X$. Can $C$ be blown down?

Answer 1

Yes, the analytic counterpart to Castelnuovo's theorem is sometimes called the Grauert's criterion. See chapter 3 of Compact Complex Surfaces. This comes from a more general theorem by Grauert, se page 14 here. Hartshorne also compares the analytic vs algebraic aspects in chapter 5 of Algebraic Geometry.