Motivic Milnor fiber of a quasi-ordinary hypersurface

TL;DR

This paper expresses the motivic Igusa zeta function, Milnor fibre, and Hodge-Steenbrink spectrum of a quasi-ordinary hypersurface in terms of its topological invariants, linking analytic and topological data.

Contribution

It provides a new description of motivic invariants of quasi-ordinary hypersurfaces using their topological invariants, advancing the understanding of their singularities.

Findings

Explicit formulas for motivic Igusa zeta function

Description of motivic Milnor fibre in topological terms

Connection between Hodge-Steenbrink spectrum and topological invariants

Abstract

Let $f$ be a germ of complex analytic function at $({\mathbf{C}}^{d+1}, 0)$ such that its zero level defines an irreducible germ of quasi-ordinary hypersurface $(S,0)$. We describe the motivic Igusa zeta function, the motivic Milnor fibre and the Hodge-Steenbrink spectrum of $f$ at 0 in terms of topological invariants of the quasi-ordinary hypersurface $(S,0)$.