Abstract
Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → \( \mathbb{A}^1 \), g:Y → \( \mathbb{A}^1 \). Assuming that there exists a complex of sheaves on X × \( \mathbb{A}^1 \) Y which induces an equivalence of D b(X) and D b(Y), we show that there is also an equivalence of the singular derived categories of the fibers f −1(0) and g −1(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.
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Baranovsky, V., Pecharich, J. On equivalences of derived and singular categories. centr.eur.j.math. 8, 1–14 (2010). https://doi.org/10.2478/s11533-009-0063-y
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DOI: https://doi.org/10.2478/s11533-009-0063-y