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 Mℝ = Lie(T∨ℝ ≅ ℝn be the Lie algebra of the compact dual (real) torus T∨ℝ ≅ U(1)n. Then there is a corresponding conical Lagrangian Λ⊂T*Mℝ and an equivalence of triangulated dg categories PerfT(X) ≅ Shcc(Mℝ; ∨), where PerfT(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Shcc(Mℝ;Λ) is the triangulated dg category of complex of sheaves on Mℝ 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 Mℝ.
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