[Paper Review] Equivariant characteristic classes of complex algebraic varieties

Sylvain E. Cappell, Laurenţiu Maxim+2 more|arXiv (Cornell University)|Apr 11, 2010

Homotopy and Cohomology in Algebraic Topology23 references11 citations

TL;DR

This paper introduces equivariant Hirzebruch characteristic classes for singular quasi-projective varieties with finite group actions, extending Brasselet-Schuermann-Yokura's homology Hirzebruch classes to the equivariant setting. It establishes a framework to compute these classes for global quotient varieties and applies them to monodromy problems and Atiyah-Meyer type formulae for twisted classes on orbifolds.

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.

Motivation & Objective

• To extend the homology Hirzebruch characteristic classes of singular varieties to the equivariant setting under finite group actions.
• To provide a systematic method for computing these classes on global quotient varieties arising from group actions.
• To apply the equivariant classes to monodromy problems involving equivariant fibrations over connected algebraic manifolds.
• To derive Atiyah-Meyer type formulae for twisted Hirzebruch classes on global orbifolds.
• To unify and generalize existing characteristic class theories (e.g., Chern, Todd, L-classes) in the context of group-equivariant singular algebraic varieties.

Proposed method

• Define equivariant characteristic classes using the equivariant Grothendieck group of coherent sheaves on singular quasi-projective varieties.
• Construct the classes via equivariant Riemann-Roch-type theorems relating K-theory and homology.
• Utilize the global quotient structure to express the equivariant classes in terms of fixed-point data and group representation theory.
• Apply the classes to fibrations with complex analytic topology to analyze monodromy behavior.
• Derive twisted Hirzebruch class formulae by incorporating equivariant Euler classes and characteristic classes of vector bundles.
• Leverage localization techniques in equivariant cohomology to compute the classes explicitly in specific geometric settings.

Experimental results

Research questions

• RQ1 How can homology Hirzebruch characteristic classes be generalized to singular varieties with finite group actions?
• RQ2 What is the structure of the equivariant characteristic classes for global quotient varieties?
• RQ3 How do these classes behave under equivariant fibrations and monodromy in the complex topology?
• RQ4 Can Atiyah-Meyer type formulae be extended to twisted Hirzebruch classes on global orbifolds?
• RQ5 What is the relationship between the new equivariant classes and classical characteristic classes such as Chern, Todd, and L-classes?

Key findings

• The paper successfully defines equivariant Hirzebruch characteristic classes for singular quasi-projective varieties with finite group actions, generalizing the non-equivariant theory.
• The classes are shown to be computable for global quotient varieties through fixed-point localization and representation-theoretic data.
• The framework enables explicit computation of homology Hirzebruch classes in the presence of group actions, particularly in monodromy contexts.
• Atiyah-Meyer type formulae are established for twisted Hirzebruch classes on global orbifolds, extending known results to the equivariant setting.
• The equivariant classes recover classical invariants such as MacPherson Chern classes and Baum-Fulton-MacPherson Todd classes in appropriate limits.
• The theory provides a unifying framework that connects singular characteristic classes with group-equivariant geometry and topology.