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Hello, I'm interested in the case when all varieties are projective threefolds over the complex numbers.

Start with a flopping contraction $f:Y \rightarrow X$, with corresponding flop $f^+: Y^+ \rightarrow X$. It has been proved that there then exists an equivalence $\Phi : D^b(Y^+) \rightarrow D^b(Y)$.

Is there a way to understand this (or any other) equivalence explicitly?

I've heard there is a way to find an equivalence by considering a common resolution of $Y$ and $Y^+$ and then using derived pullback and pushforward, is it true?

I am mostly interested in what happens to sheaves on $Y^+$ supported on the exceptional locus.

Thanks.

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    $\begingroup$ "I've heard there is a way to find an equivalence by considering a common resolution of Y and Y+ and then using derived pullback and pushforward, is it true?" True. That's all it is. $\endgroup$
    – VA.
    Commented Apr 21, 2010 at 3:46

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As always, it depends on what you think "explicitly" means. It's a Fourier-Mukai transform; see, for example, Van den Bergh and Hille's expository article. It can also be explained in terms of so-called non-commutative crepant resolutions, see Van den Bergh.

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There's a very explicit characterization of the derived equivalence -- in fact this is how Bridgeland constructs the flop (a really gorgeous idea IMHO). Namely you can build a very simple t-structure on D(Y) by a "tilting" procedure, and then the moduli of point objects is the flop Y^+. I forget the exact details but you do a tilt along the curve you want to contract, so that "perverse point sheaves" are just points away from this curve and are perverse coherent sheaves (in this case rank two complexes with H^0 being a line bundle and H^1 being torsion I think? the paper is great, so easy to find the precise statements). The basic idea being that any derived equivalence (appropriately construed) can be characterized by a universal sheaf on the product, which you can interpret as saying the Y^+ will be a moduli of a particular family of objects in the derived category of Y -- so to build Y^+ you just need to say which family of objects (and check some conditions).

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  • $\begingroup$ What's the name of this paper? (or enough of the name to find it) $\endgroup$
    – Peter Samuelson
    Commented Apr 21, 2010 at 14:53
  • $\begingroup$ David, nice answer. I edited the capitals, I hope you don't mind. $\endgroup$
    – Hailong Dao
    Commented Apr 21, 2010 at 14:57
  • $\begingroup$ I think David is referring to the paper I linked in the question. @David: thanks for clearing some things up! $\endgroup$
    – babubba
    Commented Apr 22, 2010 at 22:48
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For threefolds an equivalence can be constructed as pull back and push forward through any common resolution. It is true. On the other hand, it is not true for general flops in higher dimensions.

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