On Simple ${\mathbb Z}_3$-Invariant Function Germs

Abstract

V. I. Arnold classified simple (i.e., having no moduli for classification) singularities (function germs) and also simple boundary singularities, that is, function germs invariant with respect to the action $\sigma(x_1; y_1,\dots, y_n)=(-x_1; y_1,\dots, y_n)$ of the group ${\mathbb Z}_2$. In particular, he showed that a function germ (a germ of a boundary singularity) is simple if and only if the intersection form (respectively, the restriction of the intersection form to the subspace of anti-invariant cycles) of a germ in $3+4s$ variables stably equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding space is finite. In a previous paper the authors obtained analogues of the latter statements for function germs invariant with respect to an arbitrary action of the group ${\mathbb Z}_2$ and also for corner singularities. This paper presents an analogue of the simplicity criterion in terms of the intersection form for functions invariant with respect to a number of actions (representations) of the group ${\mathbb Z}_3$.