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Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool people "don't care" so much about plain old derived categories anymore, preferring instead to look at souped up things like dg-categories/A-infinity categories/... (for example, triangulated vs. dg/A-infinity). At this level, FM partners are no longer equivalent.

From the point of view of dg-categories/infinity categories, what does it mean for two varieties to be Fourier-Mukai partners? Why keep studying these things?

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    $\begingroup$ By results of D. Orlov and B. To\"en, there is no difference between equivalence at the level of derived categories or at the level of dg- or infinity-categories, for smooth projective varieties. $\endgroup$ – AAK Oct 29 '14 at 19:37
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Basically Adeel has already answered this question.

First you know that the triangulated category D(X) has an essentially unique enhancement (Lunts-Orlov). This means that you can freely interpret D(X) as a dg(or $A_\infty$ or $\infty$ or)-category. Second, we know that any functor D(X) --> D(Y) (say with X and Y smooth and projective) is given by a Fourier-Mukai transform - and hence it's also a dg-functor. So, two varieties are partners in the triangulated sense if and only if they are partners in the dg-sense.

The point of the dg-world is that it makes constructions behave as you'd want them to behave (like taking cones and gluing). This of course comes at the cost of having two keep track of more data (the dg structure).

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