Derived categories of coherent sheaves and motives

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We discuss relations between the motives of two varieties with equivalent derived categories of coherent sheaves.

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The derived category of a GIT quotient (2012, background): “10 correspond, via Orlov’s analogy between derived categories and motives [20], to the claim that the motive X is a summand of X.”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, result): “fourfolds described by Kuznetsov in [Kuz09b]. Furthermore, in addition to predictions in mirror symmetry, our results conjecturally have nontrivial implications towards decompositions of motives, see [Orl05]. For example, although the group action is slightly diﬀerent, we get a similar motivic decom”
An update on semisimple quantum cohomology and F-manifolds (2008, background): “7 produces a weak version of the Landau–Ginzburg model for such manifolds and thus gives a partial solution of the mirror problem for them: compare with a much more detailed pictures in [1, 18].”