Acta Mathematica Sinica, English Series

, Volume 27, Issue 2, pp 275–308 | Cite as

The coherent-constructible correspondence and Fourier-Mukai transforms

  • Bohan FangEmail author
  • Chiu-Chu Melissa Liu
  • David Treumann
  • Eric Zaslow


As evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In a previous paper, the authors showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on ℝ n . In this paper we investigate and reprove some of Kawamata’s results from this perspective.


Toric orbifolds coherent sheaves constructible sheaves Fourier-Mukai transforms 

MR(2000) Subject Classification

14M25 18E30 


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  1. [1]
    Fang, B., Liu, C.-C. M., Treumann, D., Zaslow, E.: A categorification of Morelli’s theorem. arXiv: 1007.0053Google Scholar
  2. [2]
    Treumann, D.: Remarks on the nonequivariant coherent-constructible correspondence. arXiv:1006.5756Google Scholar
  3. [3]
    Bondal, A: Derived categories of toric varieties. Convex and Algebraic geometry, Oberwolfach conference reports, EMS Publishing House 3, 2006, 284–286Google Scholar
  4. [4]
    Morelli R.: The K theory of a toric variety. Adv. Math., 100(2), 154–182 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Fang, B., Liu, C.-C. M., Treumann, D., Zaslow, E.: The coherent-constructible correspondence for toric Deligne-Mumford stacks. arXiv:0911.4711Google Scholar
  6. [6]
    Borisov L., Chen, L., Smith, G.: The orbifold chow ring of a toric Deligne-Mumford stack. J. Amer. Math. Soc., 18(1), 193–215 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Kawamata, Y.: Log crepant birational maps and derived categories. J. Math. Sci. Univ. Tokyo, 12(2), 211–231 (2005)MathSciNetzbMATHGoogle Scholar
  8. [8]
    Wang, C.-L.: On the topology of birational minimal models. J. Diff. Geom., 50, (1998)Google Scholar
  9. [9]
    Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Algebraic Geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987Google Scholar
  10. [10]
    Fantechi, B., Mann, E., Nironi, F.: Smooth toric DM stacks. J. Reine Angew. Math., to appear, arXiv:0708.1254Google Scholar
  11. [11]
    Borisov, L., Horja, R.: On the K-theory of smooth toric DM stacks. Snowbird lectures on string geometry, Contemp. Math. 401, Amer. Math. Soc., Providence, RI, 2006, 21–42MathSciNetGoogle Scholar
  12. [12]
    Fulton, W.: Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, 1993Google Scholar
  13. [13]
    Vistoli, A.: Intersection theory on algebraic stacks and their moduli spaces. Invent. Math., 97, 613–670 (1987)MathSciNetCrossRefGoogle Scholar
  14. [14]
    Kashiwara, M. Schapira, P.: Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschafte, 292, Springer-Verlag, 1994Google Scholar
  15. [15]
    Drinfeld, V.: DG quotients of DG categories. J. Algebra, 272(2), 643–691 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bohan Fang
    • 1
    Email author
  • Chiu-Chu Melissa Liu
    • 1
  • David Treumann
    • 2
  • Eric Zaslow
    • 2
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonUSA

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