On equivariant derived categories

TL;DR

This paper investigates the structure of equivariant derived categories under finite group actions on smooth projective varieties, establishing decomposition, Serre functor existence, Calabi-Yau criteria, and auto-equivalence obstructions, with applications to elliptic curves.

Contribution

It provides new insights into the structure and properties of equivariant derived categories, including criteria for Calabi-Yau conditions and auto-equivalence actions.

Findings

Equivariant categories admit decompositions and have a Serre functor.

Criteria for when an equivariant category is Calabi-Yau are established.

The equivariant category of a symplectic action on an elliptic curve is equivalent to the derived category of an elliptic curve.

Abstract

We study the equivariant category associated to a finite group action on the derived category of coherent sheaves of a smooth projective variety. We discuss decompositions of the equivariant category and faithful actions, prove the existence of a Serre functor, give a criterion for the equivariant category to be Calabi--Yau, and describe an obstruction for a subgroup of the group of auto-equivalences to act. As application we show that the equivariant category of any symplectic action on the derived category of an elliptic curve is equivalent to the derived category of an elliptic curve.