The Eta invariant on the Milnor fibration of a quasihomogeneous polynomial

TL;DR

This paper computes the eta-invariant for the odd signature operator on Milnor fibrations of quasihomogeneous hypersurface singularities, linking it to topological and algebraic data, and provides explicit formulas for Brieskorn polynomials.

Contribution

It introduces a method to calculate the eta-invariant using boundary conditions and fiberwise isometries, connecting analytical, topological, and algebraic aspects of singularities.

Findings

Eta-invariant expressed as a Maslov-type number

Explicit formula for Brieskorn polynomials

Topological invariant of isolated singularity

Abstract

We calculate the eta-invariant for the odd signature operator relative to a specific submersion metric on the Milnor fibration of a quasihomogeneous hypersurface singularity using certain global boundary conditions in terms of the data of its fibre intersection form, monodromy and variation mapping resp. the monomial data of its Milnor algebra. This is done by representing this eta-invariant as the eta-invariant of the odd signature operator on a certain closed fibrewise double of the original bundle and expressing the latter as the mapping cylinder of a specific fibrewise isometry. In this situation, well-known cutting and pasting-laws for the Eta-invariant apply and give equality (modulo the integers) to a certain real-valued Maslov-type number, first introduced by Lesch and Wojciechowski, whose value in this case is a topological invariant of the isolated singularity. We finally give an explicit formula for the eta-invariant in the case of Brieskorn polynomials in terms of combinatorial data.