$\mu$-constant monodromy groups and Torelli results for marked singularities, for the unimodal and some bimodal singularities

TL;DR

This paper proves conjectures about the structure and monodromy groups of marked singularities, extending results to all remaining unimodal and certain bimodal singularities, and establishing Torelli-type results.

Contribution

It completes the proof of conjectures on marked singularities' monodromy groups and Torelli results for all unimodal and some bimodal singularities.

Findings

Connectedness of the moduli space $M_^{mar}$ established for remaining singularities.

Equality of monodromy group $G^{mar}$ and automorphism group $G_Z$ proved.

Torelli-type conjectures confirmed for all considered singularities.

Abstract

This paper is a sequel to [He7]. There a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_\mu^{mar}$ for marked singularities in one $\mu$-homotopy class of isolated hypersurface singularities was established. One can consider it as a global $\mu$-constant stratum or as a Teichm\"uller space for singularities. It comes together with a $\mu$-constant monodromy group $G^{mar}\subset G_Z$. Here $G_Z$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_\mu^{mar}$ is connected. This is equivalent to $G^{mar}= G_Z$. Also Torelli type conjectures were formulated. All conjectures were proved for the simple singularities and 22 of the exceptional unimodal and bimodal singularities. In this paper the conjectures are proved for the remaining unimodal singularities and the remaining exceptional bimodal singularities.