Equivariant characteristic classes of singular complex algebraic varieties

TL;DR

This paper introduces equivariant homology Hirzebruch characteristic classes for singular algebraic varieties with group actions, enabling calculations for quotient varieties and applications in monodromy and orbifold theories.

Contribution

It defines new equivariant characteristic classes for singular varieties and demonstrates their use in computing classes of quotient varieties and in monodromy problems.

Findings

Defined equivariant homology Hirzebruch classes for singular varieties.

Calculated classes for global quotient varieties.

Applied classes to monodromy and orbifold problems.

Abstract

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds.