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](http://dl.dropbox.com/u/1963234/Kodaira%20-%20Complex%20manifolds%20and%20deformation%20of%20complex%20structures.djvu).


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.



Surgery in complex geometry
There is an operation in algebraic geometry, called a *flip*, which is a (kind of a special) surgery, so one could say that you hear about it, but under a different name. You can see the definition of a flip on page 41 of [Birational geometry of algebraic varieties by János Kollár and Shigefumi Mori](http://books.google.com/books?id=Lrsvfxuaw7QC&printsec=frontcover&dq=birational+geometry&hl=en&ei=UfYQTcL7KZP2tgP0ncmQCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCYQ6AEwAA#v=onepage&q=flip&f=false) (unfortunately that exact page is not available on google books). See also [this](https://mathoverflow.net/questions/5321/flips-in-the-minimal-model-program/40983#40983) and [this](https://mathoverflow.net/questions/5321/flips-in-the-minimal-model-program/42320#42320) MO answers.


One possible reason for the general lack of mentioning surgery in general may be that it is extremely hard to prove that it exists (at least for flips, but I suppose if one came up with a more general definition it would be also hard to prove existence). Shigefumi Mori was awarded the Fields Medal for proving the existence of flips in dimension 3 and it was only proved very recently by Hacon and McKernan that flips exist in any dimension. (Of course, here one would have to mention the works of Shokurov and Siu that influenced them and give references and try to give credit to everyone, but I don't want to write a book here, so let me just leave researching this in detail for the reader for now.)


