The topology of the non-singular level set and the variation operator of a singularity

Abstract

Let f:$(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$ be a singularity, that is the germ ofa holomorphic function, with an isolated critical point at the origin. It follows from implicit function theorem that in a neighbourhood of the origin in the space $\mathbb{C}^{n}$the level setf-l ${f}^{-1}(\varepsilon)$for for ε≠0 is a non-singular analytic manifold and the level set ${f}^{-1}(0)$ is a nonsingular manifold away from the origin. At the point $0 \,\epsilon \,\mathbb{C}^{n}$ the level set has a singular point.

V.I. Arnold (deceased)