Motivic and derived motivic Hirzebruch classes

TL;DR

This paper provides formulas for the Hirzebruch $ ext{chi}_y$-genus and motivic Hirzebruch class for singular varieties, introducing a new derived class inspired by higher Euler characteristics.

Contribution

It introduces a formula for motivic Hirzebruch classes of singular varieties and defines a novel derived motivic Hirzebruch class based on higher Euler characteristics.

Findings

Formulas for $ ext{chi}_y$-genus and motivic Hirzebruch class for singular varieties

Introduction of derived motivic Hirzebruch class

Connection to secondary and higher Euler characteristics

Abstract

In this paper we give a formula for the Hirzebruch $\chi_y$-genus $\chi_y(X)$ and similarly for the motivic Hirzebruch class $T_{y*}(X)$ for possibly singular varieties $X$, using the Vandermonde matrix. Motivated by the notion of secondary Euler characteristic and higher Euler characteristic, we consider a similar notion for the motivic Hirzebruch class, which we call a \emph{derived motivic Hirzebruch class}