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Twisted polytope sheaves and coherent–constructible correspondence for toric varieties

  • Peng ZhouEmail author


Given a smooth projective toric variety \(X_\Sigma \) of complex dimension n, Fang–Liu–Treumann–Zaslow (Invent Math 186(1):79–114, 2011) showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves \(Coh(X_\Sigma )\) into the dg derived category of constructible sheaves on a torus \(Sh(T^n, \Lambda _\Sigma )\). Recently, Kuwagaki (The nonequivariant coherent-constructible correspondence for toric stacks, 2016. arXiv:1610.03214) proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors.

Mathematics Subject Classification

53D37 (Homological Mirror Symmetry) 



It is a pleasure to thank Xin Jin, Linhui Shen, Lei Wu, Elden Elmanto, Dima Tamarkin for many helpful discussions, and David Treumann and David Nadler for their interests in this work. I am grateful for Elden for carefully reading the draft and giving many useful comments. I am greatly indebted to my advisor Eric Zaslow for inspirations and encouragements (and patience!). The discussion with David Treumann at IAS inspired the current approach, which uses the twisted polytope sheaves to corepresent the stalk functor. I am thankful for the referee for many useful suggestions.


  1. 1.
    Bondal, A.: Derived categories of toric varieties. In: Convex and Algebraic Geometry, Oberwolfach Conference Reports. EMS Publishing House, vol. 3, pp. 284–286 (2006)Google Scholar
  2. 2.
    Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and mutations. Math USSR IZV 35(3), 519–541 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borisov, L., Chen, L., Smith, G.: The orbifold Chow ring of a toric Deligne-Mumford stack. J. Am. Math. Soc. 18(1), 193–215 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. American Mathematical Society, Providence (2011)CrossRefGoogle Scholar
  5. 5.
    Craw, A., Smith, G.G.: Projective toric varieties as fine moduli spaces of quiver representations. Am. J. Math. 130, 1509–1534 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Efimov, A.: Maximal lengths of exceptional collections of line bundles. J. Lond. Math. Soc. 90(2), 350–372 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: A categorification of Morelli’s theorem. Invent. Math. 186(1), 79–114 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: The coherent constructible correspondence for toric Deligne-Mumford stacks. Int. Math. Res. Not. 4, 914–954 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fantechi, B., Mann, E., Nironi, F.: Smooth toric Deligne Mumford stacks. J. Reine Angew. Math. 648, 201–44 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hille, L., Perling, M.: A counterexample to Kings conjecture. Compos. Math. 142(6), 1507–1521 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kawamata, Y.: Derived categories of toric varieties. Mich. Math. J. 54(3), 517–536 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Keller, B.: On differential graded categories. arXiv:math/0601185
  14. 14.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin, Heidelberg (1990).
  15. 15.
    Karshon, Y., Tolman, S.: The moment map and line bundles over presymplectic toric manifolds. J. Differ. Geom. 38(3), 465–484 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kuwagaki, T.: The nonequivariant coherent-constructible correspondence for toric surfaces. J. Differential Geom. 107(2), 373–393 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kuwagaki, T.: The nonequivariant coherent-constructible correspondence for toric stacks (2016). arXiv:1610.03214
  18. 18.
    Nadler, D.: Microlocal branes are constructible sheaves. Sel. Math. 15(4), 563–619 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nadler, D.: Wrapped microlocal sheaves on pairs of pants. arXiv:1604.00114 [math.SG]
  20. 20.
    Nadler, D., Zaslow, E.: Constructible sheaves and the fukaya category. J. Am. Math. Soc. 22, 233–286 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Scherotzke, S., Sibilla, N.: The non-equivariant coherent-constructible correspondence and a conjecture of King. Sel. Math. (N.S.) 22(1), 389–416 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schapira, P.: A short review on microlocal sheaf theory.
  23. 23.
    Shende, V., Treumann, D., Williams, H.: On the combinatorics of exact Lagrangian surfaces. arXiv:1603.07449
  24. 24.
    Treumann, D.: Remarks on the nonequivariant coherent-constructible correspondence for toric varieties (2010). arXiv:1006.5756
  25. 25.
    Vaintrob, D.: Microlocal mirror symmetry on the torus, available at the authors homepageGoogle Scholar
  26. 26.
    Zhou, P.: Variation of GIT quotients and constructible sheaves. In preparationGoogle Scholar
  27. 27.
    Zhou, P.: Sheaf Quantization of Legendrian Isotopy. arxiv:1804.08928

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

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