A categorification of Morelli's theorem

TL;DR

This paper establishes a categorification of Morelli's theorem by relating torus-equivariant coherent sheaves on toric varieties to constructible sheaves on a vector space, enriching the understanding of their K-theoretic and categorical structures.

Contribution

It introduces an equivalence of dg categories between perfect complexes of equivariant sheaves on toric varieties and constructible sheaves with specific singular support, extending Morelli's K-theory description.

Findings

Proves an equivalence of triangulated dg categories.

Shows the monoidal structure is preserved under the equivalence.

Recovers Morelli's description of K-theory for smooth projective toric varieties.

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 $M_\bR$.