Inventiones mathematicae

, Volume 186, Issue 1, pp 79–114 | Cite as

A categorification of Morelli’s theorem

  • Bohan Fang
  • Chiu-Chu Melissa Liu
  • David Treumann
  • Eric Zaslow


We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli’s description of the K-theory of a smooth projective toric variety (Morelli in Adv. Math. 100(2):154–182, 1993). Specifically, let X be a proper toric variety of dimension n and let \(M_{\mathbb{R}} = \mathrm{Lie}(T_{\mathbb{R}}^{\vee})\cong\mathbb {R}^{n}\) be the Lie algebra of the compact dual (real) torus \(T_{\mathbb{R}}^{\vee}\cong U(1)^{n}\). Then there is a corresponding conical Lagrangian Λ⊂T M and an equivalence of triangulated dg categories \(\mathcal{P}\mathrm{erf}_{T}(X) \cong\mathit{Sh}_{cc}(M_{\mathbb{R}};\Lambda)\), where \(\mathcal{P}\mathrm{erf}_{T}(X)\) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Sh cc (M ;Λ) is the triangulated dg category of complex of sheaves on M with compactly supported, constructible cohomology whose singular support lies in Λ. This equivalence is monoidal—it intertwines the tensor product of coherent sheaves on X with the convolution product of constructible sheaves on M .


Line Bundle Toric Variety Coherent Sheave Fukaya Category Constant Sheaf 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abouzaid, M.: Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Sel. Math. New Ser. 15, 189–270 (2009) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arinkin, D.: Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on ℙ1 minus 4 points. Sel. Math. New Ser. 7, 213–239 (2001) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Zvi, D., Francis, J., Nadler, D.: Integral transforms and Drinfeld centers in derived algebraic geometry. J. Am. Math. Soc. 23(4), 909–966 (2010) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bondal, A.: Derived categories of toric varieties. In: Convex and Algebraic Geometry. Oberwolfach conference reports, vol. 3, pp. 284–286. EMS Publishing House, Zürich (2006) Google Scholar
  5. 5.
    Costa, L., Miró-Roig, R.M.: Tilting sheaves on toric varieties. Math. Z. 248, 849–865 (2004) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Fang, B.: Homological mirror symmetry is T-duality for ℙn. Commun. Number Theory Phys. 2, 719–742 (2008) MathSciNetMATHGoogle Scholar
  8. 8.
    Fang, B., Treumann, D., Liu, C.-C., Zaslow, E.: T-duality and homological mirror symmetry of toric varieties. arXiv:0811.1228v4
  9. 9.
    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993) MATHGoogle Scholar
  10. 10.
    Hartshorne, R.: Residue and Duality. Lecture Notes in Math., vol. 20. Springer, Heidelberg (1966) Google Scholar
  11. 11.
    Karshon, Y., Tolman, S.: The moment map and line bundles over pre-symplectic toric manifolds. J. Differ. Geom. 38(3), 465–484 (1993) MathSciNetMATHGoogle Scholar
  12. 12.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Grundlehren der Mathematischen Wissenschafte, vol. 292. Springer, Berlin (1994) Google Scholar
  13. 13.
    Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, Zürich, 1994, pp. 120–139 (1995) Google Scholar
  14. 14.
    Morelli, R.: The K theory of a toric variety. Adv. Math. 100(2), 154–182 (1993) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Nadler, D.: Microlocal Branes are Constructible Sheaves. Sel. Math. New Ser. 15, 563–619 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Nadler, D.: Springer theory via the Hitchin fibration. Compos. Math., to appear. arXiv:0806.4566
  17. 17.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22, 233–286 (2009) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243–259 (1996) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Toën, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167, 615–667 (2007) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Treumann, D.: Remarks on the nonequivariant coherent-constructible correspondence. arXiv:1006.5756

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

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

Personalised recommendations