Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → , g:Y → . Assuming that there exists a complex of sheaves on X × 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.
@article{bwmeta1.element.doi-10_2478_s11533-009-0063-y,
author = {Vladimir Baranovsky and Jeremy Pecharich},
title = {On equivalences of derived and singular categories},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {1-14},
zbl = {1191.14004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0063-y}
}
Vladimir Baranovsky; Jeremy Pecharich. On equivalences of derived and singular categories. Open Mathematics, Tome 8 (2010) pp. 1-14. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0063-y/
[1] Ballard M., Equivalences of derived categories of sheaves on quasi-projective schemes, preprint available at http://arxiv.org/abs/0905.3148
[2] Bezrukavnikov R., Kaledin D., McKay equivalence for symplectic resolutions of singularities, preprint available at http://arxiv.org/abs/math/0401002 | Zbl 1137.14301
[3] Bridgeland T., King A., Reid M., The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc., 2001, 14(3), 535–554 http://dx.doi.org/10.1090/S0894-0347-01-00368-X | Zbl 0966.14028
[4] Chen J.-C., Flop sand equivalences of derived categories for three folds with only terminal Gorenstein singularities, J. Differential Geom., 2002, 61(2), 227–261
[5] Cox D., Little J., Schenck H., Toric varieties, book in preparation, available at http://www.cs.amherst.edu/_dac/toric.html | Zbl 1223.14001
[6] Edidin D., Notes on the construction of the moduli space of curves, preprint available at http://arxiv.org/abs/math/9805101
[7] Fulton W., Introduction to toric varieties. Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993 | Zbl 0813.14039
[8] Fujino O., Sato H., Introduction to the toric Mori theory, Michigan Math. J., 2004, 52(3), 649–665 http://dx.doi.org/10.1307/mmj/1100623418 | Zbl 1078.14019
[9] Hernández Ruiprez D., López Martín A.C., de Salas F.S., Fourier-Mukai transforms for Gorenstein schemes, Adv. Math., 2007, 211(2), 594–620 http://dx.doi.org/10.1016/j.aim.2006.09.006 | Zbl 1118.14022
[10] Huybrechts D., Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2006 http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001
[11] Isik M.U., private communication
[12] Kawamata Y., Log crepant birational maps and derived categories, J. Math. Sci. Univ. Tokyo, 2005, 12(2), 211–231 | Zbl 1095.14014
[13] Kawamata Y., Derived Categories and Birational Geometry, preprint available at http://arxiv.org/abs/0804.3150 | Zbl 1175.14009
[14] Kresch A., On the geometry of Deligne-Mumford stacks, Algebraic geometry-Seattle 2005. Part 1, 259–271, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009 | Zbl 1169.14001
[15] Kontsevich M., Homological algebra of mirror symmetry, In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 120–139 | Zbl 0846.53021
[16] Kuznetsov A., Hyperplane sections and derived categories, Izv. Ross. Akad. Nauk Ser. Mat., 2006, 70(3), 23–128 (in Russian); English translation: Izv. Math., 2006, 70(3), 447–547 | Zbl 1133.14016
[17] Laumon G., Moret-Bailly L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 39, Springer-Verlag, Berlin, 2000
[18] Matsuki K., Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002 | Zbl 0988.14007
[19] Mehrotra S., Triangulated categories of singularities, matrix factorizations and LG models, PhD Thesis, University of Pennsylvania, Philadelphia, USA, 2005
[20] Neeman A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., 1996, 9(1), 205–236 http://dx.doi.org/10.1090/S0894-0347-96-00174-9 | Zbl 0864.14008
[21] Nironi F., Grothendieck Duality for Projective Deligne-Mumford Stacks, preprint available at http://arxiv.org/abs/0811.1955
[22] Orlov D., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, preprint available at http://arxiv.org/abs/math/0302304 | Zbl 1101.81093
[23] Orlov D., Derived categories of coherent sheaves and triangulated categories of singularities, preprint available at http://arxiv.org/abs/math/0503632 | Zbl 1200.18007
[24] Quintero-Vélez A., McKay Correspondence for Landau-Ginzburg models, Commun. Number Theory Phys., 2009, 3(1), 173–208 | Zbl 1169.14011
[25] Thomason R.W., Trobaugh T., Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, 247–435, Progr. Math., 88, Birkhäuser Boston, Boston, MA, 1990 http://dx.doi.org/10.1007/978-0-8176-4576-2_10
[26] Vistoli A., Intersection theory on algebraic tacks and on their moduli spaces, Invent. Math., 1989, 97(3), 613–670 http://dx.doi.org/10.1007/BF01388892 | Zbl 0694.14001