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.
Keywords
Cite
@article{arxiv.2311.13423,
title = {Fast vanishing cycles on perturbations of (complex) weighted-homogeneous germs},
author = {Dmitry Kerner and Rodrigo Mendes},
journal= {arXiv preprint arXiv:2311.13423},
year = {2023}
}