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Transactions of the American Mathematical Society

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Twisting derived equivalences


Author: Oren Ben-Bassat
Journal: Trans. Amer. Math. Soc. 361 (2009), 5469-5504
MSC (2000): Primary 14D22, 14K99, 18E30
DOI: https://doi.org/10.1090/S0002-9947-09-04832-6
Published electronically: April 21, 2009
MathSciNet review: 2515820
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Abstract: We introduce a new method for ``twisting'' relative equivalences of derived categories of sheaves on two spaces over the same base. The first aspect of this is that the derived categories of sheaves on the spaces are twisted. They become derived categories of sheaves on gerbes living over spaces that are locally (on the base) isomorphic to the original spaces. Secondly, this is done in a compatible way so that the equivalence is maintained. We apply this method by proving the conjectures of Donagi and Pantev on dualities between gerbes on genus-one fibrations and comment on other applications to families of higher genus curves. We also include a related conjecture in Mirror Symmetry.


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  • 1. M. Aldi and E. Zaslow, Seidel's mirror map for abelian varieties, 2005, arXiv:math.SG/ 0512229.
  • 2. B. Andreas and D. Hernández Ruipérez, Fourier Mukai transforms and applications to string theory, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 99 (2005), no. 1, 29-77. MR 2174259
  • 3. D. Arinkin, Fourier transform for quantized completely integrable systems, Ph.D. thesis, Harvard University, 2002.
  • 4. O. Ben-Bassat, J. Block, and T. Pantev, Non-commutative tori and Fourier-Mukai duality, 2005, math.AG/0509161, to appear in Compositio Mathematica.
  • 5. J. Block, Duality and equivalence of module categories in noncommutative geometry I, 2005, math.QA/0509284.
  • 6. -, Duality and equivalence of module categories in noncommutative geometry II: Mukai duality for holomorphic noncommutative tori, 2006, math.QA/0604296.
  • 7. A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, 1995, alg-geom/9506012.
  • 8. T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25-34. MR 1651025 (99k:18014)
  • 9. T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535-554 (electronic). MR 1824990 (2002f:14023)
  • 10. T. Bridgeland and A. Maciocia, Fourier-Mukai transforms for $ K3$ and elliptic fibrations, J. Algebraic Geom. 11 (2002), no. 4, 629-657. MR 1910263 (2004e:14019)
  • 11. V. Brinzanescu and R. Moraru, Twisted Fourier-Mukai transforms and bundles on non-Kahler elliptic surfaces, Math. Res. Lett. 13 (2006), no. 4, 501-514. MR 2250486 (2007e:32018)
  • 12. U. Bunke, P. Rumpf, and T. Schick, The topology of T-duality for Tn-bundles, 2005, math.GT/0501487.
  • 13. I. Burban and B. Kreussler, On a relative Fourier-Mukai transform on genus one fibrations, 2004, math.AG/0410349.
  • 14. A. Căldăraru, Derived categories of twisted sheaves on calabi-yau manifolds., Ph.D. thesis, Cornell University, 2000.
  • 15. -, Derived categories of twisted sheaves on elliptic threefolds, J. Reine Angew. Math. 544 (2002), 161-179. MR 1887894 (2003a:14022)
  • 16. D. Hernández Ruipérez, A.C. López Martın, and F.S. de Salas, Fourier-Mukai transforms for Gorenstein schemes, Adv. Math. 211 2007, 594-620. MR 2323539 (2008e:14019)
  • 17. U. V. Desale and S. Ramanan, Classification of vector bundles of rank $ 2$ on hyperelliptic curves, Invent. Math. 38 (1976/77), no. 2, 161-185. MR 0429897 (55:2906)
  • 18. Igor Dolgachev and Mark Gross, Elliptic threefolds. I. Ogg-Shafarevich theory, J. Algebraic Geom. 3 (1994), no. 1, 39-80. MR 1242006 (95d:14037)
  • 19. R. Donagi and T. Pantev, Torus fibrations, gerbes, and duality, Mem. Amer. Math. Soc. 193 (2008), vi + 90 pp. MR 2399730
  • 20. -, Langlands duality for Hitchin systems, 2006, arXiv:math.AG/0604617.
  • 21. P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448. MR 0232821 (38:1144)
  • 22. Nigel Hitchin, Lectures on special Lagrangian submanifolds, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math., vol. 23, Amer. Math. Soc., Providence, RI, 2001, pp. 151-182. MR 1876068 (2003f:53086)
  • 23. D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, IJM Paris, Oxford University Press, 2006. MR 2244106 (2007f:14013)
  • 24. Anton Kapustin and Dmitri Orlov, Remarks on A-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys. 48 (2003), no. 1, 84-99. MR 2006226 (2004f:14053)
  • 25. M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990, With a chapter in French by Christian Houzel. MR 1074006 (92a:58132)
  • 26. M. Lieblich, Moduli of Twisted Sheaves, 2004, arXiv:math.AG/0411337.
  • 27. V. Mathai and J. Rosenberg, $ T$-duality for torus bundles with $ H$-fluxes via noncommutative topology, Comm. Math. Phys. 253 (2005), no. 3, 705-721. MR 2116734
  • 28. -, T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group, 2005, hep-th/0508084.
  • 29. S. Mukai, Duality between $ D(X)$ and $ D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153-175. MR 82f:14036
  • 30. A. Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Mathematics, vol. 153, Cambridge University Press, Cambridge, 2003. MR 1987784
  • 31. J. Sawon, Twisted Fourier-Mukai transforms for holomorphic symplectic fourfolds, 2005, arXiv:math.AG/0509222.
  • 32. Y. Toda, Deformations and Fourier-Mukai transforms, 2005, math.AG/0502571.

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Additional Information

Oren Ben-Bassat
Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
Email: oren.benbassat@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-09-04832-6
Received by editor(s): December 14, 2007
Published electronically: April 21, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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