The thick-thin decomposition and the bilipschitz classification of normal surface singularities

• 成果类型: Article
• 署名作者: Birbrair, Lev; Neumann, Walter D.; Pichon, Anne
• 署名单位: Universidade Federal do Ceara; Columbia University; Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-014-0111-8
• 发表日期: 2014
• 页码: 199-256
• 关键词: algebraic-surfaces; complex-surfaces; reductive groups; polar varieties; geometry; CURVES; SPACES; LIMITS; sets; l2-cohomology

摘要

We describe a natural decomposition of a normal complex surface singularity (X, 0) into its thick and thin parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of (X, 0) in terms of its topology and a finite list of numerical bilipschitz invariants.