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


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 
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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

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