  We study the Lipschitz Geometry of Complex Algebraic Singularities. For this
purpose we introduce the notion of choking horns. A Choking horn is a family of
cycles on the family of the sections of an algebraic variety by very small
spheres centered at a singular point, such that the cycles cannot be boundaries
of nearby chains. The presence of choking horns is an obstruction to metric
conicalness as we can see with some classical isolated hypersurfaces
singularities which we prove are not metrically conic. We also show that there
exist infinitely countably many singular varieties, which are locally
homeomorphic, but not locally bi-Lipschitz equivalent with respect to the inner
metric.
