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. Specifically, let $X$ be a proper toric variety of dimension $n$ and let M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n be the Lie algebra of the compact dual (real) torus T_\bR^\vee\cong U(1)^n. Then there is a corresponding conical Lagrangian \Lambda \subset T^*M_\bR and an equivalence of triangulated dg categories \Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda), where \Perf_T(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and \Sh_{cc}(M_\bR;\Lambda) is the triangulated dg category of complex of sheaves on M_\bR with compactly supported, constructible cohomology whose singular support lies in $\Lambda$. This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on M_\bR.Comment: 20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v