mu-constant monodromy groups and marked singularities

TL;DR

This paper investigates the structure of mu-constant monodromy groups and introduces global moduli spaces for marked singularities, proposing conjectures that connect these groups with the connectedness of moduli spaces and Torelli problems, with proofs for simple and most exceptional singularities.

Contribution

It defines and studies a monodromy group for mu-constant families and establishes global moduli spaces for marked singularities, formulating and proving related conjectures.

Findings

Conjecture that the monodromy group contains all automorphisms respecting certain forms, modulo {id,-id}.

Connectedness of moduli spaces is linked to the monodromy group conjecture.

Proved conjectures for simple and 22 of 28 exceptional singularities.

Abstract

mu-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo {id,-id}. Second, marked singularities are defined and global moduli spaces for right equivalence classes of them are established. The conjecture on the group would imply that these moduli spaces are connected. The relation with Torelli type problems is discussed and a new global Torelli type conjecture for marked singularities is formulated. All conjectures are proved for the simple and 22 of the 28 exceptional singularities.