On symplectic fillings of small Seifert $3$-manifolds

TL;DR

This paper classifies all minimal symplectic fillings of small Seifert 3-manifolds with canonical contact structures, showing they can be obtained via rational blowdowns from surface singularity resolutions.

Contribution

It provides a complete classification of minimal symplectic fillings for small Seifert 3-manifolds under certain conditions, linking them to surface singularity resolutions.

Findings

All minimal symplectic fillings are obtained by rational blowdowns.

Classification applies to small Seifert 3-manifolds with canonical contact structures.

Connections established between fillings and complex surface singularities.

Abstract

In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous complex surface singularity.