2009/07/21 by Vladimir Baranovsky, Baranovsky, Vladimir, Jeremy Pecharich +1
Mathematics · #14D21 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.0907.3717
openalex publication_date 2009/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A1 (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 X0 Y0 are the fibers of f and g over zero, respectively, we show that the singular derived categories of X0 and Y0 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.