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

TL;DR

This paper establishes conditions for detecting fast vanishing loops in complex-analytic germs, linking singularity data to the inner metric conical property without resolution of singularities, especially for surface germs.

Contribution

It introduces a new criterion based on discriminant and covering data for identifying fast loops in complex germs, applicable without resolution and providing necessary and sufficient conditions for surface germs.

Findings

Fast loops obstruct inner metric conicalness.

Conditions are necessary and sufficient for surface germs.

Numerous classes of IMC and non-IMC germs are characterized.

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.