Lipschitz geometry does not determine embedded topological type

Walter D. Neumann (1) , Anne Pichon (2)

1 Columbia University [New York]
2 I2M - Institut de Mathématiques de Marseille

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in $(\mathbb C^n,0)$. 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 $(X_1,0)$ and $(X_2,0)$ in $(\mathbb C^3,0)$ 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.

Domaines: Géométrie algébrique [math.AG]; Variables complexes [math.CV]; Géométrie métrique [math.MG]

Walter D. Neumann, Anne Pichon. Lipschitz geometry does not determine embedded topological type. José Luis Cisneros-Molina, Dũng Tráng Lê, Mutsuo Oka, Jawad Snoussi. Singularities in Geometry, Topology, Foliations and Dynamics A Celebration of the 60th Birthday of José Seade, Merida, Mexico, December 2014, Birkhäuser, pp.183--195, 2017, Trends in Mathematics, Birkhäuser. ⟨hal-01175532⟩