Detecting fast vanishing loops in complex-analytic germs (and detecting germs that are inner metrically conical)

Abstract

Let X be a reduced complex-analytic germ of pure dimension n\ge2, with arbitrary singularities (not necessarily normal or complete intersection). Various homology cycles on Link_\ep[X] vanish at different speeds when \ep\to0. We give a condition ensuring fast vanishing loops on X. The condition is in terms of the discriminant and the covering data for "convenient" coverings X\to (C^n,o). No resolution of singularities is involved. For surface germs (n=2) this condition becomes necessary and sufficient. A corollary for surface germs that are strictly complete intersections detects fast loops via singularities of the projectivized tangent cone of X. Fast loops are the simplest obstructions for X to be inner metrically conical. Hence we get simple necessary conditions to the IMC property. For normal surface germs these conditions are also sufficient. We give numerous classes of non-IMC germs and IMC germs.