On equivalences of derived and singular categories

TL;DR

This paper proves that under certain Fourier-Mukai equivalences between derived categories of smooth stacks, the singular derived categories of their fibers are also equivalent, with applications to McKay correspondence and Calabi-Yau hypersurfaces.

Contribution

It establishes a new link between derived and singular categories for fibers of functions on stacks, extending known results and applying to Calabi-Yau and McKay contexts.

Findings

Singular derived categories of fibers are equivalent under Fourier-Mukai transforms.

Generalization of Orlov's result to products of Calabi-Yau hypersurfaces.

Application to McKay correspondence and toric varieties.

Abstract

Let X and Y be two smooth Deligne-Mumford stacks and consider a function f, resp. g, on X, resp. Y. Assume that there exists a complex F of sheaves on the fiber product of X and Y over A^1 (induced by f and g), such that the Fourier-Mukai transform with the kernel F gives an equivalence between the bounded derived categories of coherent sheaves on X and Y. If X_0 Y_0 are the fibers of f and g over zero, respectively, we show that the singular derived categories of X_0 and Y_0 are also equivalent. We apply this statement in the setting of McKay correspondence, and generalize a result 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.