Equivalence of derived categories which is not Fourier-Mukai 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)?  
 A: 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.
A: 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.
