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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.

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One reason might be that surgery theory fails miserably in any category less flabby than diff (I don't claim that the surgery fails, just the theory). The h-cobordism theorem is false for analytic manifolds, and with that goes every nontrivial result I know (admittedly not that many) in surgery theory. – Paul Siegel Dec 21 '10 at 23:54
Note that we have surgery in Ricci flow pioneered by Perelman "Ricci flow with surgery on three-manifolds".So when we want to know more about Kahler-Ricci flow,it's unavoidable to consider some surgeries on complex manifolds. – Unknown Jan 2 '11 at 14:36
Perelman did surgery on smooth manifolds, not complex manifolds. From what I understand (which isn't very much) surgery was needed to show existence of the Ricci flow in three real dimensions. I believe we have existence for the Kahler-Ricci flow on compact manifolds, but I'd be surprised if the same methods were used in that proof. – Gunnar Þór Magnússon Jan 2 '11 at 17:42
up vote 7 down vote accepted

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 (unfortunately that exact page is not available on google books). See also this and this 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.)

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Thank you for that Sándor. I do think there should be a completely complex analytic notion of surgery (1: like Kodaira's, 2: maybe it would coincide with flips in the projective case), because a surgery is a mostly local operation, and locally speaking we have more functions to glue with in the analytic category than in the algebraic one. Also I feel that surgery should make sense for non-algebraic manifolds, so I'd expect an operation which makes explicit use of algebraic properties to be a special case of something more general. – Gunnar Þór Magnússon Dec 21 '10 at 20:41
Also: do we know the topology of a manifold obtained as a flip? I ask because we know the Betti numbers of a blow-up of $X$ at a point: they're the Betti numbers of $X$, plus the Betti numbers of $\mathbb P^n$. If a blow-up is a special case of surgery, then I'd expect (or at least very much like) to see some sort of relation between the topology of the manifold obtained by surgery and the original manifolds. – Gunnar Þór Magnússon Dec 21 '10 at 20:55
@Gunnar: what about applying Mayer-Vietoris and/or Van Kampen? – Qfwfq Dec 22 '10 at 1:49
Gunnar: I totally agree that there should be a more general, complex analytic notion of which flips would be special cases. In fact, I wonder if that would perhaps lead to a new way to construct flips. In fact, I would expect that a reasonable definition of surgery would be more general than flips even in the algebraic case. As you mention blow-ups, those are not flips, so that's already one. – Sándor Kovács Dec 22 '10 at 4:54
Gunnar #2: regarding the topology of flips: I am not sure how much research has been done in this direction. Certain things are easy: anything that only depends on codimension 1 data remains the same as flips are isomorphisms in codimension 1. On the other hand, in general the topology changes and as far as I understand it is hard to control how (at least for an algebraic geometer). An extremely interesting relevant work is a series of papers of Kollár on the minimal model program over the reals. In particular, he is working very hard to not change the topology in an uncontrolled way (cont) – Sándor Kovács Dec 22 '10 at 5:00

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