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2010 | 8 | 1 | 1-14
Tytuł artykułu

On equivalences of derived and singular categories

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
8
Numer
1
Strony
1-14
Opis fizyczny
Daty
wydano
2010-02-01
online
2010-02-02
Twórcy
  • UC-Irvine
  • UC-Irvine
Bibliografia
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [13] Kawamata Y., Derived Categories and Birational Geometry, preprint available at http://arxiv.org/abs/0804.3150
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [23] Orlov D., Derived categories of coherent sheaves and triangulated categories of singularities, preprint available at http://arxiv.org/abs/math/0503632
  • [24] Quintero-Vélez A., McKay Correspondence for Landau-Ginzburg models, Commun. Number Theory Phys., 2009, 3(1), 173–208
  • [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
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-009-0063-y
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