Distinguished bases and Stokes regions for the simple and the simple elliptic singularities

Abstract

Isolated hypersurface singularities come equipped with a Milnor lattice, a ${\mathbb Z}$-lattice of finite rank, and a set of $distinguished$ ${\mathbb Z}$-bases of this lattice. Usually these bases are constructed from $one$ morsification and $all\ possible$ choices of distinguished systems of paths. But what does one obtain if one considers $all\ possible$ morsifications and $one$ fixed distinguished system of paths? Looijenga asked this question 1974 for the simple singularities. He and Deligne found that one obtains a bijection between Stokes regions in a universal unfolding and the set of distinguished bases modulo signs. This allows to see the base space of the universal unfolding as an atlas of Stokes data. Here we reprove their result and extend it to the simple elliptic singularities. We use more conceptual arguments, moduli spaces of marked singularities (i.e. Teichm\"uller spaces for singularities), extensions of them to F-manifolds, and the actions of symmetries of singularities on the Milnor lattices and these moduli spaces. We use and extend results of Jaworski on the Lyashko-Looijenga maps for the simple elliptic singularities. The sections 2 and 3 give a survey on singularities and the associated objects which allows to read the paper independently of other sources.