Equivariant Poincar\'e series of filtrations and topology

TL;DR

This paper explores how the equivariant Poincaré series associated with filtrations on analytic varieties encodes the topological information of curves and divisors under finite group actions.

Contribution

It investigates the extent to which the equivariant Poincaré series determines the topology of curves and divisors in the presence of group symmetries.

Findings

The equivariant Poincaré series captures significant topological data.

It clarifies the relationship between algebraic invariants and topological types.

Results apply to filtrations defined by valuations on analytic functions.

Abstract

Earlier, for an action of a finite group $G$ on a germ of an analytic variety, an equivariant $G$-Poincar\'e series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of $G$-sets with an additional structure. We discuss to which extend the $G$-Poincar\'e series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.