Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups

S M Gusein-Zade,A Melle-Hernández,I Luengo

doi:10.3390/sym11070902

Abstract

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. 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 fGr ( Var C ) 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 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) 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.

Highlights

The notion of the orbifold Euler characteristic χorb came from physics at the end of the 1980s [1](see [2,3])
Coincidence of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric
Coincidence of the orbifold Euler characteristics is a necessary condition for orbifolds to be mirror symmetric

Summary

Introduction

The notion of the orbifold Euler characteristic χorb came from physics at the end of the 1980s [1]. On which these Euler characteristics are ring homomorphisms was defined in [9] It was called the Grothendieck ring of complex quasi-projective varieties with actions of finite groups. For a topological space X (say, a complex quasi-projective variety) with an action of a finite group G, one has the notions of the orbifold Euler characteristic χorb ( X, G ) and of the (orbifold) Euler characteristics χ(k) ( X, G ) of higher orders (see, e.g., [2,3,4]) They can be defined, in particular, in the following way. The orbifold Euler characteristic and the Euler characteristics of higher orders can be considered as functions on the Grothendieck ring K0G (VarC ) of quasi-projective G-varieties These functions are group homomorphisms, but not ring ones. Αk can be considered as a sort of a generalized version of the Euler characteristic of order k with values in K0fGr (VarC ). (In [13], the homomorphism α is called the inertia homomorphism.)

The Universal Euler Characteristic

Macdonald-Type Equations and λ-Structure Homomorphisms

A Substitute of a Macdonald-Type Equation for the Homomorphism α

Conclusions