Symplectic monodromy, quasi-homogeneous polynomials and spectral flow

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.