Indices of collections of equivariant 1-forms and characteristic numbers

TL;DR

This paper introduces new indices for collections of equivariant 1-forms on singular complex G-varieties, generalizing classical characteristic numbers and invariants like the GSV-index and Euler obstruction.

Contribution

It constructs two novel analogues of characteristic numbers for singular G-varieties using indices of equivariant 1-forms, extending classical invariants to singular settings.

Findings

Indices coincide for G-cobordant manifolds.

New indices generalize the GSV-index and Euler obstruction.

Provides tools for studying singular G-varieties.

Abstract

If two G-manifolds are G-cobordant then characteristic numbers corresponding to the fixed point sets (submanifolds) of subgroups of G and to normal bundles to these sets coincide. We construct two analogues of these characteristic numbers for singular complex G-varieties where G is a finite group. They are defined as sums of certain indices of collections of 1-forms (with values in the spaces of the irreducible representations of subgroups). These indices are generalizations of the GSV-index (for isolated complete intersection singularities) and the Euler obstruction respectively.