Seminar in Real and Complex Geometry

Thursday, June 1, 2023, 16:15-17:45, Schreiber 309 and online via zoom

Dmitry Kerner
(Ben-Gurion University)

Which ICIS are IMC's ?


Let (X,o) be a complex analytic germ. How to visualize it? The conic structure theorem reads: (X,o) is homeomorphic to the cone over Link[X].
In ``most cases" this homeomorphism cannot be chosen differentiable (in whichever sense). The natural weaker question is: whether (X,o) is ``inner metrically conical" (IMC), i.e. whether (X,o) is bi-Lipschitz homeomorphic to the cone over its link.
Any curve-germ is inner metrically conical. In higher dimensions the (non-)IMC verification is more complicated.
We study this question for complex-analytic ICIS, giving necessary/sufficient criteria to be IMC. For surface germs this becomes an ``if and only if'' condition. So we get (explicitly) a lot of ICIS that are IMC's, and the other lot of ICIS that are not IMC's.
Our criteria are of two types: via the polar locus/discriminant (in the general case) and via weights (for semi-weighted homogeneous ICIS).
Joint work with L. Birbrair and R. Mendes Pereira