Singular McKay correspondence for normal surfaces

TL;DR

This paper extends the McKay correspondence to normal surfaces with certain singularities by defining new invariants like the singular orbifold elliptic genus and E-function, revealing deep links to rigidity phenomena.

Contribution

It introduces the singular orbifold elliptic genus and E-function for normal surfaces without strictly log-canonical singularities, generalizing previous stringy invariants.

Findings

Established the McKay correspondence analogue for these surfaces.

Connected the invariants' definability to elliptic genus rigidity.

Extended the framework of stringy invariants to broader singularity classes.

Abstract

We define the singular orbifold elliptic genus and $E$-function for all normal surfaces without strictly log-canonical singularities, and prove the analogue of the McKay correspondence in this setting. Our invariants generalize the stringy invariants defined by Willem Veys for this class of singularities. We show that the ability to define these invariants is closely linked to rigidity phenomena associated to the elliptic genus.