A Lefschetz fibration on minimal symplectic fillings of a quotient surface singularity

TL;DR

This paper constructs specific Lefschetz fibrations on minimal symplectic fillings of quotient surface singularities and shows these fillings can be derived from minimal resolutions via rational blowdowns.

Contribution

It provides explicit Lefschetz fibrations for all minimal symplectic fillings of non-cyclic quotient surface singularities and links these fillings to rational blowdowns from minimal resolutions.

Findings

Constructed genus-0 or genus-1 Lefschetz fibrations for all such fillings.

Proved all minimal symplectic fillings can be obtained via rational blowdowns.

Connected symplectic fillings to minimal resolutions through explicit geometric operations.

Abstract

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution.