# Generalized orbifold Euler characteristics on the Grothendieck ring of varieties with actions of finite groups

## TL;DR

This paper extends the concept of orbifold Euler characteristics to a motivic setting, defining generalized versions as ring homomorphisms from a Grothendieck ring of varieties with finite group actions to the classical Grothendieck ring, and explores their properties.

## Contribution

It introduces generalized orbifold Euler characteristics as ring homomorphisms from the Grothendieck ring of varieties with finite group actions to the classical Grothendieck ring, expanding their theoretical framework.

## Key findings

- Defined generalized orbifold Euler characteristics as ring homomorphisms
- Established analogues of Macdonald equations for symmetric products
- Connected orbifold Euler characteristics with motivic invariants

## Abstract

The notion of the orbifold Euler characteristic came from physics at the end of 80's. There were defined higher order versions of the orbifold Euler characteristic and generalized ("motivic") versions of them. In a previous paper the authors defined a notion of the Grothendieck ring $K_0^{\rm fGr}(Var)$ of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from $K_0^{\rm fGr}(Var)$ to the Grothendieck ring $K_0(Var)$ of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.