Milnor fibers and symplectic fillings of quotient surface singularities

TL;DR

This paper establishes a detailed correspondence between Milnor fibers and minimal symplectic fillings of quotient surface singularities, using advanced techniques from algebraic geometry and symplectic topology.

Contribution

It provides an explicit algorithm to compare Milnor fibers and symplectic fillings, classifies minimal symplectic fillings up to diffeomorphism, and shows all fillings derive from a maximal resolution via rational blow-downs.

Findings

Milnor fibers associated to different components are mostly non-diffeomorphic.

Classified minimal symplectic fillings up to diffeomorphism.

All symplectic fillings can be obtained from a maximal resolution by rational blow-downs.

Abstract

We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that: -- Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism. -- Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called $P$-resolutions of a quotient surface singularity.