Grothendieck ring of varieties with finite groups actions

TL;DR

This paper introduces a new Grothendieck ring framework for varieties with finite group actions, enabling the definition of orbifold Euler characteristics and higher order invariants as ring homomorphisms, with applications to wreath product series.

Contribution

It defines a Grothendieck ring for varieties with group actions, introduces natural λ-structures and power structures, and extends to equivariant vector bundles with motivic Euler characteristics.

Findings

Defined a Grothendieck ring with finite group actions.

Established homomorphisms for orbifold and higher order Euler characteristics.

Derived a Macdonald type formula for wreath product series.

Abstract

We define a Grothendieck ring of varieties with finite groups actions and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural $\lambda$-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized ("motivic") Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher order Euler characteristics of wreath products.