Symplectic monodromy, quasi-homogeneous polynomials and spectral flow

TL;DR

This paper links the variation of quasihomogeneous polynomials with isolated singularities to spectral flows of signature operators, revealing conditions under which symplectic monodromy has infinite order in the symplectic isotopy group.

Contribution

It introduces a novel method of encoding the variation structure via spectral flows and interprets inequalities as conditions for infinite order symplectic monodromy.

Findings

Spectral flow variations encode the polynomial's singularity structure.

An inequality condition implies the symplectic monodromy has infinite order.

Potential for extending results to broader classes of algebraic singularities.

Abstract

We encode the variation structure of a quasihomogeneous polynomial with an isolated singularity as introduced by Nemethi in a set of spectral flows of the signature operator on the Milnor bundle by varying global elliptic boundary conditions in a specific way using the quasihomogeneous circle action on the Brieskorn lattice. For this, we use adiabatic techniques and well-known results on spectral flow and Maslov index. Furthermore we interpret the inequality of a certain member of this family of spectral flows with a spectral flow induced by a Reeb flow on the boundary of the Milnor fibre as giving a sufficient condition for the 'symplectic monodromy' of the fibration to define an element of infinite order in the relative symplectic isotopy group of the Milnor fibre, this uses previous results of P. Seidel resp. of the author. We expect generalizations of the results to wider classes of (algebraic) singularities.