On equivalences of derived and singular categories

Vladimir Baranovsky; Jeremy Pecharich

Open Mathematics (2010)

  • Volume: 8, Issue: 1, page 1-14
  • ISSN: 2391-5455

Abstract

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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 space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.

How to cite

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Vladimir Baranovsky, and Jeremy Pecharich. "On equivalences of derived and singular categories." Open Mathematics 8.1 (2010): 1-14. <http://eudml.org/doc/269404>.

@article{VladimirBaranovsky2010,
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.},
author = {Vladimir Baranovsky, Jeremy Pecharich},
journal = {Open Mathematics},
keywords = {Derived category; Singular category; Landau-Ginzburg model; McKay correspondence; derived category; singular category},
language = {eng},
number = {1},
pages = {1-14},
title = {On equivalences of derived and singular categories},
url = {http://eudml.org/doc/269404},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Vladimir Baranovsky
AU - Jeremy Pecharich
TI - On equivalences of derived and singular categories
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 1
EP - 14
AB - 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.
LA - eng
KW - Derived category; Singular category; Landau-Ginzburg model; McKay correspondence; derived category; singular category
UR - http://eudml.org/doc/269404
ER -

References

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