Poincar\'e series and monodromy of the simple and unimodal boundary singularities

TL;DR

This paper explores the relationship between Poincaré series and monodromy in simple and unimodal boundary singularities, extending classical correspondences to boundary cases and analyzing their algebraic properties.

Contribution

It generalizes the McKay correspondence to simple boundary singularities and investigates the monodromy's characteristic polynomial in relation to Poincaré series.

Findings

Monodromy characteristic polynomial relates to Poincaré series

Generalization of McKay correspondence to boundary singularities

Analysis of monodromy in various boundary singularity types

Abstract

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V. I. Arnold and V. I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincar\'e series of the coordinate algebra of the ambient singularity.