Derived category of equivariant coherent sheaves on a smooth toric variety and Koszul duality

TL;DR

This paper explores the derived category of equivariant coherent sheaves on smooth toric varieties, establishing a combinatorial Koszul duality equivalence through sheaves of modules over a graded algebra, linking algebraic and geometric perspectives.

Contribution

It introduces a new combinatorial Koszul duality framework for equivariant coherent sheaves on smooth toric varieties, connecting algebraic modules with geometric categories.

Findings

Establishes an equivalence between module categories over a sheaf of graded algebras and equivariant coherent sheaves.

Describes the equivariant category and a related category O in terms of sheaves over the algebra.

Interprets Koszul duality via the Serre functor on the derived category of equivariant sheaves.

Abstract

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 A_{\Sigma} on {\Sigma} . The category of modules over 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 coh_{X,T} and a related (slightly bigger) equivariant category O_{X,T}-mod in terms of sheaves of modules over the sheaf of algebras A_{\Sigma} . Eventually (for a complete X ) the combinatorial Koszul duality is interpreted in terms of the Serre functor on D^b(coh_{X,T})