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)? 



Equivalence of derived categories which is not Fourier-Mukai
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](https://arxiv.org/abs/0908.4187) 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.


