Symplectic monodromy, Leray residues and quasi-homogeneous polynomials

TL;DR

This paper establishes conditions under which the symplectic monodromy of certain singularities has infinite order, using topological and algebraic methods without Floer theory, and discusses potential generalizations.

Contribution

It provides new criteria for infinite order symplectic monodromy of quasihomogeneous singularities, avoiding Floer theory and introducing alternative approaches.

Findings

Conditions for infinite order monodromy established

Alternative proof methods using Maslov classes and Morse theory

Discussion on extending results beyond quasihomogeneous cases

Abstract

We formulate certain sufficient conditions for the symplectic monodromy of an isolated quasihomogeneous singularity to be of infinite order in the relative symplectic mapping class group of the Milnor fibre and give a proof using Maslov classes, stability theory for Lagrangian folds resp. stable Morse theory for generating families as well as algebraic results about relative cohomology of smoothings of isolated singularities. Our conditions being slightly more restrictive than Seidel's, in contrary to Seidel's proof, we do not use Floer theory to derive this result. An alternative approach using bounding disks in fibred Lagrangian families is given and its possible application to generalizations to the non-quasihomogeneous case is discussed.