The Hodge spectrum of analytic germs on isolated surface singularities

TL;DR

This paper introduces a topological approach to study the Hodge spectrum of analytic germs on isolated surface singularities, using a new matrix analogue and establishing semicontinuity and inequality properties.

Contribution

It introduces the fractured Seifert matrix and demonstrates its role in determining the Hodge spectrum and related semicontinuity inequalities.

Findings

The fractured Seifert matrix determines the Hodge spectrum.

Semicontinuity properties of the Hodge spectrum are established.

Murasugi type inequalities relate to spectrum semicontinuity.

Abstract

We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine--Tristram signatures associated with it, defined for null-homologous links in arbitrary three dimensional manifolds. Moreover, we establish Murasugi type inequalities in the presence of cobordisms of links. It turns out that the fractured Seifert matrix determines the Hodge spectrum and the Murasugi type inequalities can be read as spectrum semicontinuity inequalities.