The derived category of a GIT quotient

TL;DR

This paper explores the relationship between equivariant derived categories and GIT quotients, providing criteria for derived equivalences and applying them to Calabi-Yau manifolds with torus actions.

Contribution

It generalizes classical results on coherent sheaves and offers new criteria for derived equivalences between different GIT quotients.

Findings

Established a criterion for derived equivalence of GIT quotients.

Proved that generic GIT quotients of certain Calabi-Yau manifolds are derived equivalent.

Connected derived categories of quotients to classical geometric invariant theory results.

Abstract

Given a quasiprojective algebraic variety with a reductive group action, we describe a relationship between its equivariant derived category and the derived category of its geometric invariant theory quotient. This generalizes classical descriptions of the category of coherent sheaves on projective space and categorifies several results in the theory of Hamiltonian group actions on projective manifolds. This perspective generalizes and provides new insight into examples of derived equivalences between birational varieties. We provide a criterion under which two different GIT quotients are derived equivalent, and apply it to prove that any two generic GIT quotients of an equivariantly Calabi-Yau projective-over-affine manifold by a torus are derived equivalent.