Smoothness of equivariant derived categories

TL;DR

This paper introduces a new concept of G-smoothness for complex G-varieties based on equivariant derived categories, linking geometric orbit conditions to algebraic smoothness in dg categories.

Contribution

It defines G-smoothness for G-varieties using dg algebra enhancements and characterizes it via orbit cohomology conditions, advancing the understanding of equivariant derived categories.

Findings

G-smoothness characterized by orbit cohomology conditions

Established criteria for G-smoothness when finitely many orbits exist

Proved results on smoothness of dg categories over graded rings

Abstract

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant derived category of X. If there are only finitely many G-orbits and all stabilizers are connected, we show that X is G-smooth if and only if all orbits O satisfy H^*(O; R)=R. On the way we prove several results concerning smoothness of dg categories over a graded commutative dg ring.