Elliptic classes, McKay correspondence and theta identities

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 the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities $${\mathbb {C}}^n/G$$ 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 $$A_n$$ A n leads to a remarkable formula.