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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asked Oct 27, 2009 at 20:07
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$\begingroup$ 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? $\endgroup$– Ben Webster ♦Commented Oct 27, 2009 at 20:11
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1$\begingroup$ 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. $\endgroup$– Ilya NikokoshevCommented Oct 27, 2009 at 20:16
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$\begingroup$ Exactly, I am asking for an example of such an equivalence which is not of Fourier-Mukai type. $\endgroup$– Andrei HalanayCommented Oct 27, 2009 at 20:24
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$\begingroup$ Just a sec, as stated, this is nonsense. You're asking for something that contradicts Orlov's result. Maybe you phrased the question inaccurately? $\endgroup$– Kim MorrisonCommented Nov 3, 2009 at 3:26
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1$\begingroup$ @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. $\endgroup$– Kevin H. LinCommented Nov 3, 2009 at 15:00
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2 Answers
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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 (in Equivalences of derived categories of sheaves on quasi-projective schemes) 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 The homotopy theory of dg-categories and derived Morita theory which treats DG enhancements but shows that the philosophy of integral transforms and "bimodules" is a very general one.
answered Oct 27, 2009 at 20:25
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1$\begingroup$ 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). $\endgroup$ Commented Oct 27, 2009 at 20:57
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1$\begingroup$ Also, Ben-Zvi, Francis, and Nadler proved an extension of Toen's result to hold for arbitrary perfect stacks. $\endgroup$– S. Carnahan ♦Commented Oct 28, 2009 at 2:00