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On equivalences of derived and singular categories

Vladimir Baranovsky; Jeremy Pecharich — 2010

Open Mathematics

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $𝔸^{1}$ , g:Y → $𝔸^{1}$ . Assuming that there exists a complex of sheaves on X × $𝔸^{1}$ 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...

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