Fast vanishing cycles on perturbations of complex weighted-homogeneous complete intersection germs

Abstract

Take a complex-analytic germ X in (C^N,o) with arbitrary singularity. In many cases Link[X] contains cycles that vanish faster than linearly, when Link[X] shrinks to the origin. These ``fast cycles" capture the crucial metric/Lipschitz properties of X. We consider germs (of arbitrary dimensions and codimensions) that are perturbations of weighted-homogeneous complete intersections. For such germs we determine the fast cycles (i.e. their homotopy type, tangent cone, vanishing rates) via the weights. This gives a vast zoo of fast cycles with prescribed properties. As an immediate application we get countable families of (distinct) exotic Lipschitz structures on germs of topological manifolds, for each fixed dimension, codimension, and multiplicity. Another application is the obstruction for germs to be inner metrically conical, e.g.: * (with certain assumptions) If X is IMC then the n lowest weights coincide. * Let the surface germ X in (C^3,o) be Newton-non-degenerate and IMC. Then for some of the faces of the Newton diagram the two lowest weights coincide.