I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something I'm missing.
In "Complex manifolds and deformations complex structures" Kodaira defines a surgery operation which takes two complex manifolds, makes certain choices, and returns a third complex manifold obtained by joining the two along a submanifold. The details are on page 52 of Kodaira's book, which may be found here.
After a quick look it seems to me that Kodaira uses this surgery to construct the Hirzebruch surfaces, the Hopf surface, and the blow-up of a point in $\mathbb C^n$. Elsewhere, the only similar examples I can find are the ones of a blow-up of a point or subvariety on a complex manifold (see, for example, page 93 of Zheng's "Complex differential geometry").
So my question is, why don't we hear more about surgery of complex manifolds? Heuristically I'd expect that it doesn't work all that well, because else I'd already seen it used to construct (counter-)examples, but I also can't figure out why it doesn't work.