Fast vanishing cycles on perturbations of (complex) weighted-homogeneous germs

Abstract: Let X_o be a complex weighted-homogeneous complete intersection germ, (possibly non-reduced). Let X be a perturbation of X_o by ``higher-order-terms". We give sufficient criteria to detect fast cycles on X, via the weights of X_o. This is an easy obstruction to be non-metrically conical. A simple application of our results gives, e.g. * Suppose the germs X_o,X,X\cap V(x_1) are ICIS. If X is IMC then the n lowest weights of X_o coincide. * Let the surface germ X=V(f)\subset (C^3,o) be Newton-non-degenerate and IMC. Then for each of the faces of the Newton diagram the two lowest weights coincide. As an auxiliary result we prove (under certain assumptions, for \k=\R,\C): the weighted-homogeneous foliation of the pair X_o \sset (\k^N,o) deforms to a foliation of the pair X \sset (\k^N,o). In particular, the deformation by higher order terms is ambient-trivializable by a semialgebraic Lipschitz homeomorphism.