This research is supported by the Narodowe Centrum Nauki under the OPUS grant number 2023/49/B/ST1/00848.
Title of the project - On Zariski pairs of surface singularities
Team - Christophe Eyral, Masaharu Ishikawa, Mutsuo Oka
The project deals with the topology of complex singularities.
It is well known that if two isolated surface singularities in the complex 3-space have the same embedded topology, then they have the same Milnor number. On the other hand, it is quite possible for two isolated surface singularities to have the same Milnor number (or even the same Teissier μ*-invariant) and distinct embedded topologies. Disproving a conjecture of Yau, Artal Bartolo further showed that the embedded topology of a surface singularity is not determined by the piece of data consisting of the abstract topology of its link and the characteristic polynomial of its monodromy.
However, in practice, given two isolated surface singularities with the same characteristic polynomial (equivalently, with the same monodromy zeta-function), the same Teissier μ*-invariant and with homeomorphic abstract links, it is extremely difficult to determine whether the two singularities have the same embedded topology or not. In this project, we are interested in certain types of surface singularities which are "likely to systematically produce" pairs of surface-germs sharing all these invariants but having different embedded topologies.
Publications
C. Eyral and M. Oka, On Milnor-Orlik's theorem and admissible simultaneous good resolutions, to appear in Ann. Polon. Math.