Advertisement

Inventiones mathematicae

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

A categorification of Morelli’s theorem

  • Bohan Fang
  • Chiu-Chu Melissa LiuEmail author
  • David Treumann
  • Eric Zaslow
Article

Abstract

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 .

Keywords

Line Bundle Toric Variety Coherent Sheave Fukaya Category Constant Sheaf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abouzaid, M.: Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Sel. Math. New Ser. 15, 189–270 (2009) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fang, B.: Homological mirror symmetry is T-duality for ℙn. Commun. Number Theory Phys. 2, 719–742 (2008) MathSciNetzbMATHGoogle 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) zbMATHGoogle 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) MathSciNetzbMATHGoogle 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) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Nadler, D.: Microlocal Branes are Constructible Sheaves. Sel. Math. New Ser. 15, 563–619 (2009) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243–259 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Toën, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167, 615–667 (2007) MathSciNetzbMATHCrossRefGoogle 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
    Email author
  • David Treumann
    • 2
  • Eric Zaslow
    • 2
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonUSA

Personalised recommendations