Divergent Diagrams of Folds Associated with Reflections

Abstract

Abstract We analyze divergent diagrams of k -fold map-germs at $$(\mathbb {C}^n,0)$$ ( C n , 0 ) , for $$k, n \ge 2,$$ k , n ≥ 2 , associated with reflections, adapting to the complex setting the theory of folds associated with involutions at $$(\mathbb {R}^n,0)$$ ( 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.