On the smoothings of non-normal isolated surface singularities

Abstract

We show that isolated surface singularities which are non-normal may have Milnor fibers which are non-diffeomorphic to those of their normalizations. Therefore, non-normal isolated singularities enrich the collection of Stein fillings of links of normal isolated singularities. We conclude with a list of open questions related to this theme.Comment: 14 pages, 1 figure. Compared to the first version on ArXiv, I added Remark 5.10, containing information communicated to me by J\'anos Koll\'ar. Proceedings of Singularities in Geometry and Applications III, Edinburgh, Scotland, 2013. J.-P. Brasselet, P. Giblin, V. Goryunov ed

Complex singularities and contact topology

“…It is probably the easiest way to construct smoothings, which explains why a drawing similar to Figure 4.4 was represented on the cover of Stevens' book [182]. My explanation follows the one I gave in [165,Section 4].…”

“…Therefore it is not a priori clear that even the topological types of taut singularities (see Definition 3.47) produce a finite number of Milnor fibers. Nevertheless, one may show that this is the case for the topological types of taut and rational singularities, as a consequence of results of Kollár (see [165,Remark 5.10] [97,Section A.4], this simple homotopy type is independent of the chosen resolution. The fact that its homotopy type has this property is a consequence of the previous work of Danilov [36].…”

“…Using this observation and the method of sweeping out the cone, I proved in [165] the following proposition which has to be contrasted with the fact that simple elliptic singularities are smoothable only for a finite number of topological types (see Example 4.19):…”

Unexpected Stein fillings, rational surface singularities and plane curve arrangements

“…It should also be noted that in the nonrational case, one should in principle consider nonnormal singularities as well, as these might generate additional Stein fillings; see [55] for a detailed discussion of this issue (which doesn't arise in the rational case).…”