624 research outputs found
Lipschitz normal embedding among superisolated singularities
Any germ of a complex analytic space is equipped with two natural metrics:
the outer metric induced by the hermitian metric of the ambient space and the
inner metric, which is the associated riemannian metric on the germ. A complex
analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner
metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface
singularities; the only known LNE surface germs outside the trivial case
(straight cones) are the minimal singularities. In this paper, we show that a
superisolated hypersurface singularity is LNE if and only if its projectivized
tangent cone has only ordinary singularities. This provides an infinite family
of LNE singularities which is radically different from the class of minimal
singularities.Comment: 17 pages, 8 figures. Minor errors and misprints corrected. Comments
are welcome
Lipschitz geometry of complex surfaces: analytic invariants and equisingularity
We prove that the outer Lipschitz geometry of a germ of a normal
complex surface singularity determines a large amount of its analytic
structure. In particular, it follows that any analytic family of normal surface
singularities with constant Lipschitz geometry is Zariski equisingular. We also
prove a strong converse for families of normal complex hypersurface
singularities in : Zariski equisingularity implies Lipschitz
triviality. So for such a family Lipschitz triviality, constant Lipschitz
geometry and Zariski equisingularity are equivalent to each other.Comment: Added a new section 10 to correct a minor gap and simplify some
argument
Lipschitz geometry does not determine embedded topological type
We investigate the relationships between the Lipschitz outer geometry and the
embedded topological type of a hypersurface germ in . It is
well known that the Lipschitz outer geometry of a complex plane curve germ
determines and is determined by its embedded topological type. We prove that
this does not remain true in higher dimensions. Namely, we give two normal
hypersurface germs and in having the same
outer Lipschitz geometry and different embedded topological types. Our pair
consist of two superisolated singularities whose tangent cones form an
Alexander-Zariski pair having only cusp-singularities. Our result is based on a
description of the Lipschitz outer geometry of a superisolated singularity. We
also prove that the Lipschitz inner geometry of a superisolated singularity is
completely determined by its (non embedded) topological type, or equivalently
by the combinatorial type of its tangent cone.Comment: A missing argument was added in the proof of Proposition 2.3 (final 4
paragraphs are new
Minimal surface singularities are Lipschitz normally embedded
Any germ of a complex analytic space is equipped with two natural metrics:
the {\it outer metric} induced by the hermitian metric of the ambient space and
the {\it inner metric}, which is the associated riemannian metric on the germ.
We show that minimal surface singularities are Lipschitz normally embedded
(LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and
inner metrics, and that they are the only rational surface singularities with
this property.Comment: This paper is a major revision of the 2015 version. It now builds on
the paper arXiv:1806.11240 by the same authors which gives a general
characterization of Lipschitz normally embedded surface singularitie
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