Derived Category of Equivariant Coherent Sheaves on a Smooth Toric Variety and Koszul Duality

Let \(X\) be a smooth toric variety defined by the fan \(\Sigma\). We consider \(\Sigma\) as a finite set with topology and define a natural sheaf of graded algebras \(\mathcal{A}_\Sigma\) on \(\Sigma\). The category of modules over \(\mathcal{A}_\Sigma\) is studied (together with other related categories). This leads to a certain combinatorial Koszul duality equivalence.

We describe the equivariant category of coherent sheaves \(\mathrm{coh}_{X,T}\) and a related (slightly bigger) equivariant category \(\mathcal{O}_{X,T}\text{-}\mathrm{mod}\) in terms of sheaves of modules over the sheaf of algebras \(\mathcal{A}_\Sigma\). Eventually (for a complete \(X\)), the combinatorial Koszul duality is interpreted in terms of the Serre functor on \(D^b(\mathrm{coh}_{X,T})\).