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Bi-Lipschitz geometry of weighted homogeneous surface singularities. (English) Zbl 1153.14003

Any complex singularity can be quipped with a natural metric space structure, the so-called inner metric structure, well defined up to bi-Lipschitz equivalence. A natural question of metric theory of complex analytic singularities is the existence of a metrically conical structure near the singular point; this means that the inner metric space is bi-Lipschitz equivalent with the cone metric. For example, singularities of complex algebraic curves have this property. On the other hand, it was discovered by L. Birbrair and A. Fernandes [Commun. Pure Appl. Math. 61, No. 11, 1483–1494 (2008; Zbl 1156.32019)] that this is not always the case for surface singularities, even if we consider only the weighted homogeneous germs. This result is greatly extended in the present article, showing that these germs are rarely metrical: a weighted homogeneous surface singularity is metrically conical only if its two lowest weights are equal. The article also provides an example of a pair of germs which are topologically equivalent but not bi-Lipschitz equivalent.

MSC:

14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
32S25 Complex surface and hypersurface singularities

Citations:

Zbl 1156.32019
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References:

[1] Birbrair, L., Fernandes, A.: Metric geometry of complex algebraic surfaces with isolated singularities. Comm. Pure Appl. Math. (to appear) · Zbl 1156.32019
[2] Birbrair, L., Fernandes, A., Neumann, W.: Bi-Lipschitz geometry of complex surface singularities. (preprint arXiv:0804.0194) · Zbl 1164.32005
[3] Brasselet, J.P., Goresky, M., MacPherson, R.: Simplicial differential forms with poles. Am. J. Math. 113, 1019–1052 (1991) · Zbl 0748.55002 · doi:10.2307/2374899
[4] Neumann, W., Jankins, M.: Lectures on Seifert manifolds. Brandeis Lecture Notes, vol. 2. Brandeis University, Waltham (1983)
[5] Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983) · Zbl 0561.57001 · doi:10.1112/blms/15.5.401
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