On the Milnor fibers of cyclic quotient singularities

TL;DR

This paper proves Lisca's conjecture that Milnor fibers of cyclic quotient singularities correspond bijectively to Stein fillings of lens spaces, and establishes their unique diffeomorphism types using smoothing equations and additional structures.

Contribution

It confirms Lisca's conjecture by explicitly relating Milnor fibers to Stein fillings and introduces the concept of order to distinguish non-diffeomorphic fibers.

Findings

Proves the bijective correspondence between Milnor fibers and Stein fillings.

Identifies Milnor fibers with Lisca's fillings using smoothing equations.

Establishes the uniqueness of Milnor fibers up to order-preserving diffeomorphisms.

Abstract

The oriented link of the cyclic quotient singularity $\mathcal{X}_{p,q}$ is orientation-preserving diffeomorphic to the lens space $L(p,q)$ and carries the standard contact structure $\xi_{st}$. Lisca classified the Stein fillings of $(L(p,q), \xi_{st})$ up to diffeomorphisms and conjectured that they correspond bijectively through an {\it explicit} map to the Milnor fibers associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of $\mathcal{X}_{p,q}$. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibers given by de Jong and van Straten, we also canonically identify these fibers with Lisca's fillings. Using these and a newly introduced additional structure - the order - associated with lens spaces, we prove that the above Milnor fibers are pairwise non-diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.