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

S. M. Gusein-Zade ; A.-M. Ya. Rauch

Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 1, pp. 56-64

Résumé

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 ${\mathbb Z}_3$.

Détail

Mots-clés : Group action, invariant germ
Keywords: simple singularity.