On the structure of equivariant derived categories

TL;DR

This paper explores the structure of equivariant derived categories, emphasizing restricted local cohomology, and discusses applications like birational modifications, categorical completions, and braid group actions on GIT quotients.

Contribution

It introduces a new perspective on equivariant derived categories using restricted local cohomology and presents applications including categorical completions and braid group actions.

Findings

Analysis of birational modifications on derived categories

Construction of categorical completions of equivariant derived categories

Implementation of braid group actions on GIT quotients

Abstract

In this expository note, we discuss some results of the author on the structure of derived categories of equivariant coherent sheaves and the derived categories of geometric invariant theory quotients. We take a recent perspective, emphasizing the theory of restricted local cohomology. We also discuss several applications and concrete examples: studying the effects of birational modification on derived categories, constructing categorical completions of equivariant derived categories, and constructing actions of generalized braid groups on derived categories of GIT quotients. This is a contribution to the proceedings of the International Congress of Basic Science, held in July 2024.