Symplectic fillings of lens spaces as Lefschetz fibrations

TL;DR

This paper constructs Lefschetz fibrations on symplectic fillings of lens spaces and shows they are related to rational blowdowns of resolutions of cyclic quotient singularities.

Contribution

It provides a new Lefschetz fibration construction for minimal symplectic fillings of lens spaces and characterizes these fillings via rational blowdowns.

Findings

Lefschetz fibrations are constructed on all minimal weak symplectic fillings.

All such fillings are obtained from minimal resolutions through rational blowdowns.

The construction links symplectic topology with complex surface singularities.

Abstract

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.