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D. Orlov proved that any equivalence of bounded derived categories F:Db(X) -> Db(Y) is a Fourier-Mukai transform, when X and Y are smooth projective varieties. Is there any example of such equivalence, which is not a Fourier-Mukai transform (it is not an integral transform)?

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Perhaps you could be a little clearer: do you mean an equivalence of such categories when X and Y are varieties that are not projective? –  Ben Webster Oct 27 '09 at 20:11
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He means an example of the equivalence between two derived categories together with the proof that this equivalence doesn't come from a Fourier-Mukai transform, imho. –  Ilya Nikokoshev Oct 27 '09 at 20:16
    
Exactly, I am asking for an example of such an equivalence which is not of Fourier-Mukai type. –  Andrei Halanay Oct 27 '09 at 20:24
    
Just a sec, as stated, this is nonsense. You're asking for something that contradicts Orlov's result. Maybe you phrased the question inaccurately? –  Scott Morrison Nov 3 '09 at 3:26
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@Scott: I think he's asking for an equivalence of such categories when X and Y are not smooth projective varieties, with a proof that the equivalence does not come from a Fourier-Mukai. –  Kevin H. Lin Nov 3 '09 at 15:00

2 Answers 2

up vote 6 down vote accepted

Schlichting gave an example of two categories of singularities which are derived equivalent but whose K-groups are not isomorphic. Dugger and Shipley (arXiv:0710.3070) expanded on this example and noted that it gives two dga's which are derived equivalent but not by an integral transform.

Otherwise, Lunts and Orlov's results on uniqueness of enhancements give a large class of triangulated categories for which one might lift exact functors to dg-functors and apply Toen's result.

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I don't know of a counterexample but I can tell you some more situations in which it is true. Ballard has extended Orlov's result as well as getting a result in this direction for the case of quasi-projective varieties. There is also section 8.3 of Toën's paper which treats DG enhancements but shows that the philosophy of integral transforms and "bimodules" is a very general one.

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I am interested more of the case of compact complex manifolds or analytic spaces. Ballard also poses a similar question in the Introduction of his paper (after theorem 1.4). –  Andrei Halanay Oct 27 '09 at 20:57
    
Also, Ben-Zvi, Francis, and Nadler proved an extension of Toen's result to hold for arbitrary perfect stacks. –  S. Carnahan Oct 28 '09 at 2:00

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