Is there a nice way to characterise the derived equivalence induced by a flop? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:19:38Z http://mathoverflow.net/feeds/question/21998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21998/is-there-a-nice-way-to-characterise-the-derived-equivalence-induced-by-a-flop Is there a nice way to characterise the derived equivalence induced by a flop? babubba 2010-04-20T22:02:12Z 2010-04-22T19:47:43Z <p>Hello, I'm interested in the case when all varieties are projective threefolds over the complex numbers.</p> <p>Start with a flopping contraction $f:Y \rightarrow X$, with corresponding flop $f^+: Y^+ \rightarrow X$. It has been <a href="http://www.tombridgeland.staff.shef.ac.uk/papers/flop.pdf" rel="nofollow">proved</a> that there then exists an equivalence $\Phi : D^b(Y^+) \rightarrow D^b(Y)$.</p> <p>Is there a way to understand this (or any other) equivalence explicitly? </p> <p>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?</p> <p>I am mostly interested in what happens to sheaves on $Y^+$ supported on the exceptional locus.</p> <p>Thanks.</p> http://mathoverflow.net/questions/21998/is-there-a-nice-way-to-characterise-the-derived-equivalence-induced-by-a-flop/22000#22000 Answer by Graham Leuschke for Is there a nice way to characterise the derived equivalence induced by a flop? Graham Leuschke 2010-04-20T23:47:48Z 2010-04-20T23:47:48Z <p>As always, it depends on what you think "explicitly" means. It's a Fourier-Mukai transform; see, for example, <a href="http://alpha.uhasselt.be/Research/Algebra/Publications/hille.pdf" rel="nofollow">Van den Bergh and Hille's expository article</a>. It can also be explained in terms of so-called non-commutative crepant resolutions, <a href="http://alpha.uhasselt.be/Research/Algebra/Publications/flops.ps" rel="nofollow">see Van den Bergh</a>.</p> http://mathoverflow.net/questions/21998/is-there-a-nice-way-to-characterise-the-derived-equivalence-induced-by-a-flop/22061#22061 Answer by David Ben-Zvi for Is there a nice way to characterise the derived equivalence induced by a flop? David Ben-Zvi 2010-04-21T14:29:09Z 2010-04-21T14:56:52Z <p>There's a very explicit characterization of the derived equivalence -- in fact this is how Bridgeland <em>constructs</em> 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 <em>which</em> family of objects (and check some conditions).</p> http://mathoverflow.net/questions/21998/is-there-a-nice-way-to-characterise-the-derived-equivalence-induced-by-a-flop/22242#22242 Answer by ploughshare for Is there a nice way to characterise the derived equivalence induced by a flop? ploughshare 2010-04-22T19:47:43Z 2010-04-22T19:47:43Z <p>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.</p>