Equivalences of equivariant derived categories

TL;DR

This paper explores when Fourier-Mukai transforms between derived categories of sheaves on stacks with finite group actions can be lifted to equivariant categories, with applications to quotient stacks and Kummer stacks.

Contribution

It provides new criteria for lifting Fourier-Mukai transforms to equivariant derived categories and applies these to analyze derived equivalences of quotient stacks.

Findings

A condition under which a global quotient stack cannot be derived equivalent to a variety.

Application of techniques to generalized Kummer stacks.

Insights into derived equivalences involving symmetric quotients.

Abstract

We investigate conditions for a Fourier-Mukai transform between derived categories of coherent sheaves on smooth projective stacks endowed with actions by finite groups to lift to the associated equivariant derived categories. As an application we give a condition under which a global quotient stack cannot be derived equivalent to a variety. We also apply our techniques to generalised Kummer stacks and symmetric quotients.