Divergent diagrams of folds associated with reflections

TL;DR

This paper studies divergent diagrams of k-fold map-germs in complex spaces related to reflections, classifying singularities and pairs of linear reflections under certain conditions.

Contribution

It adapts the theory of folds associated with involutions to the complex setting and classifies pairs of transversal linear reflections and their divergent diagrams.

Findings

Complete classification of pairs of transversal linear reflections.

Identification of invariants related to nontrivial eigenvalues.

Analysis of divergent diagrams associated with reflections in complex spaces.

Abstract

We analyse divergent diagrams of \(k\)-fold map-germs on \((\mathbb{C}^n,0)\), for $k, n \geq 2$, associated with reflections, adapting to the complex setting the theory of folds associated with involutions on \((\mathbb{R}^n,0)\). In the complex case, a \(k\)-fold is naturally related to a cyclic group generated by a reflection, which guides the analytic classification of singularities. Under the conditions of transversality and linearity of the associated reflections, certain conditions related to the nontrivial eigenvalues appear as invariants by simultaneous conjugacy. We also provide a complete classification of pairs of transversal linear reflections and the corresponding divergent diagrams.