A topological characterization of symplectic fillings of Seifert 3-manifolds

TL;DR

This paper characterizes when minimal symplectic fillings of Seifert 3-manifolds can be obtained via rational blowdowns, linking topological surgeries to complex surface singularities and their Milnor fibers.

Contribution

It provides a necessary and sufficient condition for minimal symplectic fillings to be derived from rational blowdowns of surface singularities, connecting topology and complex geometry.

Findings

Characterizes when fillings are obtained by rational blowdowns.

Shows all fillings in a large family are Milnor fibers.

Establishes a topological criterion for symplectic fillings.

Abstract

In this paper, we investigate a relation between rational blowdown surgery and minimal symplectic fillings of a given Seifert 3-manifold with a canonical contact structure. Consequently, we determine a necessary and sufficient condition for a minimal symplectic filling of a Seifert 3-manifold satisfying certain conditions to be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity. Furthermore, as an application of the main results, we prove that every minimal symplectic filling of a large family of Seifert 3-manifolds with a canonical contact structure is in fact realized as a Milnor fiber of the corresponding weighted homogeneous surface singularity in the Appendix.