Elliptic classes, McKay correspondence and theta identities

TL;DR

This paper explores elliptic classes in singular algebraic varieties, focusing on theta function identities linked to McKay correspondence, especially for quotient and symplectic singularities, revealing new geometric formulas.

Contribution

It adapts elliptic class construction to equivariant local settings and uncovers theta identities arising from McKay correspondence for quotient singularities.

Findings

Theta identities originate from McKay correspondence in quotient singularities

A remarkable formula is derived for Du Val surface singularity $A_n$

Elliptic classes are extended to equivariant local situations

Abstract

We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to equivariant local situation. We study theta function identities having geometric origin. In the case of quotient singularities $\mathbb C^n/G$, where $G$ is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity $A_n$ leads to a remarkable formula.