paper

A categorification of Morelli's theorem

arXiv:1007.0053 · doi:10.1007/s00222-011-0315-x

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 be a proper toric variety of dimension 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 $Λ\subset T^*M_\bR$ and an equivalence of triangulated dg categories $\Perf_T(X) \cong \Sh_{cc}(M_\bR;Λ),$ where $\Perf_T(X)$ is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on and $\Sh_{cc}(M_\bR;Λ)$ is the triangulated dg category of complex of sheaves on $M_\bR$ with compactly supported, constructible cohomology whose singular support lies in . This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on with the convolution product of constructible sheaves on $M_\bR$.

20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v4

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