# Surgery in complex geometry

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

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