Poincar\'e series associated with surface singularities

TL;DR

This paper unifies and extends formulas related to Poincaré series for surface singularities, establishing positive results for certain classes and exploring their limits and connections to other conjectures.

Contribution

It generalizes existing formulas for Poincaré series and investigates their applicability and limitations across different surface singularity types.

Findings

Positive results for rational and minimally elliptic singularities

Examples and counterexamples illustrating the limits of the formulas

Discussion of connections with Seiberg-Witten and Semigroup Density Conjectures

Abstract

We unify and generalize formulas obtained by Campillo, Delgado and Gusein-Zade in their series of articles. Positive results are established for rational and minimally elliptic singularities. By examples and counterexamples we also try to find the `limits' of these identities. Connections with the Seiberg-Witten Invariant Conjecture and Semigroup Density Conjecture are discussed.