Weighted homogeneous singularities and rational homology disk smoothings

TL;DR

This paper classifies weighted homogeneous surface singularities that can be smoothed to rational homology disks, using symplectic geometry and smoothing techniques, linking topological and geometric conditions for such smoothings.

Contribution

It provides a complete classification of resolution graphs for these singularities and establishes a criterion based on contact and symplectic geometry for their smoothability.

Findings

Nonexistence of smoothings shown via symplectic methods

Existence verified through smoothing of negative weights

Starshaped plumbing trees correspond to smoothable singularities

Abstract

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown that a (negative definite) starshaped plumbing tree gives rise to a weighted homogeneous singularity admitting a rational homology disk smoothing if and only if the Milnor fillable contact structure of the link admits a weak symplectic rational homology disk filling.