## 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_{ℝ} = Lie(T^{∨}_{ℝ} ≅ ℝ^{n} be the Lie algebra of the compact dual (real) torus T^{∨}_{ℝ} ≅ U(1)^{n}. Then there is a corresponding conical Lagrangian Λ⊂T^{*}M_{ℝ} and an equivalence of triangulated dg categories Perf_{T}(X) ≅ Sh_{cc}(M_{ℝ}; ∨), where Perf_{T}(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Sh_{cc}(M_{ℝ};Λ) is the triangulated dg category of complex of sheaves on M_{ℝ} 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_{ℝ}.

Original language | English (US) |
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Pages (from-to) | 79-114 |

Number of pages | 36 |

Journal | Inventiones Mathematicae |

Volume | 186 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2011 |

## ASJC Scopus subject areas

- Mathematics(all)