Semiorthogonal decompositions of derived categories of equivariant coherent sheaves

TL;DR

This paper constructs semiorthogonal decompositions of derived categories of G-equivariant coherent sheaves on algebraic varieties with full exceptional collections, linking them to twisted group representations and applying to classical varieties.

Contribution

It introduces a method to decompose derived categories of equivariant sheaves into components related to twisted representations, extending known results to new classes of varieties.

Findings

Decomposition of derived categories into twisted group representation components

Full exceptional collections obtained for finite or reductive groups

Applications to projective spaces, quadrics, Grassmannians, and Del Pezzo surfaces

Abstract

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of bounded derived category of G-equivariant coherent sheaves on X into components, equivalent to derived categories of twisted representations of the group. If the group is finite or reductive over the algebraically closed field of zero characteristic, this gives a full exceptional collection in the derived equivariant category. We apply our results to particular varieties such as projective spaces, quadrics, Grassmanians and Del Pezzo surfaces.