A categorification of Morelli’s theorem

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 _ℝ 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 _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.