Given an isolated singular point in a complex surface, its link is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that reflects the topology of the singularity. The link has a natural contact structure, and one can ask questions about its contact and symplectic topology, for example, try to describe symplectic 4-manifolds that the link can bound. A family of symplectic (even Stein) fillings is provided by possible smoothings of the singularity; it is then interesting to compare algebro-geometric smoothings and general Stein fillings. We will discuss this question for sandwiched singularities, a subclass of rational singularities where smoothings are well understood due to de Jong-van Straten work from 1990s: a dimensional reduction allows to reconstruct all Milnor fibers from deformations of associated singular plane curve germs. We build a symplectic analog of this theory to describe Stein fillings in terms of certain immersed disk arrangements in a very similar way. However, we also show that this method allows to build Stein fillings whose topology is different from that of any Milnor fibers.
The talk is based on joint work with Starkston and Beke-Starkston.
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