2024/04/25 by Kerner, Dmitry, Mendes, Rodrigo
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2404.16968
Let X be a reduced complex-analytic germ of pure dimension n≥2, with arbitrary singularities (not necessarily normal or complete intersection). Various homology cycles on Link_\ep[X] vanish at different speeds when \ep→0. 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→ (Cn,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.