Higher order generalized Euler characteristics and generating series

TL;DR

This paper introduces higher order generalized Euler characteristics for complex quasi-projective manifolds with group actions, extending classical invariants into the Grothendieck ring and deriving their generating series for Cartesian products under wreath product actions.

Contribution

It defines new higher order invariants in the Grothendieck ring and computes their generating series for Cartesian products with wreath product symmetries.

Findings

Defined higher order generalized Euler characteristics in the Grothendieck ring.

Computed generating series for these invariants for Cartesian products with wreath actions.

Extended classical Euler characteristics to a more general algebraic setting.

Abstract

For a complex quasi-projective manifold with a finite group action, we define higher order generalized Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the class of the affine line. We compute the generating series of generalized Euler characteristics of a fixed order of the Cartesian products of the manifold with the wreath product actions on them.