Atiyah-Segal completion theorem for singular varieties

Algebraic Geometry and Number Theory Seminar

We prove the AS completion theorem for almost toric actions on singular varieties in characteristic 0.
The strategy is based on the two ingredients:
1. the A1-invariant AS completion theorem proved by Amalendu Krishna;
2. a completion theorem for Hochschild homology proved by Harrison Chen.
To get the result the two ingredients are combined using the cyclotomic trace techniques.
As an application, we construct a complete motivic filtration on the AS completion of the equivariant K-theory, extending Elmanto-Morrow filtration from the case of schemes.
This is based on joint work in progress with Elden Elmanto and Dmitry Kubrak.